Breakdown Phenomena in Semiconductors and Semiconductor Devices 9812563954

BREAKDOWN PHENOMENA IN SEMICONDUCTORS AND SEMICONDUCTOR DEVICES

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BREAKDOWN PHENOMENA IN SEMICONDUCTORS AND SEMI CON DUCTOR D EVI CES

Michael Levinshtein Russian Academy of Sciences, Russia

Juha Kostamovaara Sergey Vainsh tein

University of Oulu, Finland

vp World Scientific N E W J E R S E Y - L O N D O N * S I N G A P O R E * BEIJING - S H A N G H A I * H O N G K O N G * T A I P E I * C H E N N A I

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BREAKDOWN PHENOMENA IN SEMICONDUCTORS AND SEMICONDUCTOR DEVICES

Copyright Q 2005 by World Scientific Publishing Co. Pte. Ltd.

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ISBN 981 -256-395-4

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To the memory of Julia Titova M. L. To my family J . K.

s. v. To my parents Serafima and Naum Vainshtein

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Preface

One form of avalanche breakdown has been known to mankind from ancient times: lightning, the terrifying gas discharge, the fear of which is inscribed in the tales and myths of all primitive tribes.

The first known practical application of the avalanche breakdown principle goes back to the first century of our era. There is a fish in the Mediterranean, the electric ray, or skate, which was called LLnurcuell by the ancient Greeks, a word which means “paralyzing”. It is known nowadays that the voltage generated by this fish can reach 200 Volts. The Roman physician Scribonius, in his famous writing “De Compositiones Medicamentorum” , published in AD 40, described the using of this narcue for the treatment of headaches, gout and some other diseases. The treatment was rather painful. This may be the reason why the term “breakdown” is associated very often with such unpleasant concepts as “failure” and “destruction”.

Electrical breakdown itself is not connected with any form of destriiction, however. One widely used microwave device, the IMPATT diode, for example, has a characteristic operation frequency of about 100 GHz ( lo l l Hz), which means that it goes into a mature avalanche breakdown regime 10l1 times a second. Since the guaranteed lifetime of a commercial IMPATT diode is a t least 5000 hours, each diode will go into this regime safely no less than N 3 x 1 O I 8 times. Moreover, impact ionization, avalanche and breakdown phenomena form the basis of many very interesting and very important semiconductor devices, such as avalanche photodiodes, avalanche transistors, suppressors, sharpening diodes (diodes with delayed breakdown), and IMPATT and TRAPATT diodes.

We should note at the same time that avalanche phenomena are always associated with high electric fields F, and that the optimal regimes of many devices can be realised only at high current densities j. Thus the power density Po = j x F can be extremely large. The value of the characteristic breakdown field Fi for a silicon IMPATT diode with an operation frequency of about 100 GHz, for example, is about 5 x lo5 V/cm, its characteristic current density j is approximately lo5 A/cm2, and Po is about 5 x lolo W/cm3. As a result, the breakdown phenomena are often accompanied by a high temperature. It is probable, of course, that if the temperature is too high, the device may be destroyed due to melting or decomposition of

vii

viii Breakdown Phenomena in Semiconductors and Semiconductor Devices

the material of which it is constructed. This is not a n electric breakdown as such, but only “overheating”, ( (‘heat breakdown”) causes the device destruction.

It worth noting that operation in high electric fields is the backbone of modern semiconductor electronics. Indeed, the mainstream of the modern electronics lies in increasing the operation frequency and velocity of semiconductor device “switching”. Both the operation frequency and the velocity of switching are inversely proportional to the length of the ”active region” of the device, L. For the most important devices used in semiconductor electronics, Field Effect Transistors (FETs) and Bipolar Transistors (BJTs), the characteristic length of the active region (gate or base) is about 0.1 pm. With a standard operation bias Vo of about 1 V, the average value of the electric field Fo across the active region of the device is approximately lo5 V/cm, which means that the maximal value of the electric field in the active region can be as large as (2-3)x105 V/cm, i.e. practically equal to the characteristic breakdown field Fi. Generally speaking, in order to provide maximal speed and maximal power, many semiconductor devices must operate either under breakdown conditions or very close to these. Consequently, an acquaintance with breakdown phenomena is very important and useful for any scientist or engineer dealing with semiconductor devices.

Many books contain chapters or sections devoted to the principal features of the avalanche and breakdown phenomena, and there are many good books and out- standing reviews concerning certain special aspects of these phenomena. The aim of this book is to summarize the main experimental results on avalanche and breakdown phenomena in semiconductors and semiconductor devices and to analyse them from a unified point of view. This book has been written by experimentalists for experimentalists. We will scarcely deal at all with fundamental theoretical aspects such as the distribution function of hot electrons, nuances of the band structure at high energy, etc., but instead we will focus our attention on the phenomenology of avalanche multiplication and the various kinds of breakdown phenomena and their qualitative analysis.

The book is organised as follows. In the introductory chapter (Chapter 1) we will briefly discuss the main definitions and establish the main approaches to describing breakdown phenomena.

Chapter 2 will be devoted to avalanche multiplication phenomena, and the main parameters of avalanche photodiodes will be discussed and analysed on this basis.

In Chapter 3 we will consider the reverse current-voltage characteristic of semiconductor diodes over an extremely wide range of current densities, including pre- breakdown leakage current, microplasma breakdown, mature (homogeneous) breakdown, the part of the current-voltage characteristic with negative differential resistance at very high current densities, and the second part with positive differential resistance. The operation regimes and main characteristics of two important devices: suppressor diodes and IMPATT diodes, will be also observed in this chapter.

The phenomenon of avalanche injection will be discussed in Chapter 4 for sam-

Preface ix

ples of the n+ - n - nf and p+ - p - p f types and for bipolar transistors. The operation of Si avalanche transistors will be analysed for both a conventional regime and a very effective, fast operation regime realised at extremely high current densities (Section 4.4). In Section 4.5 we will discuss the recently discovered effect of extremely fast switching of GaAs avalanche transistors a t high current densities.

The phenomena of so called “dynamic breakdown” will be analysed in Chapter 5 . This regime is realized under conditions in which the avalanche ionization front moves along the samples with a velocity which is higher than the saturated velocity of free carriers (the TRAPATT zone or streamer). The operation regimes of Silicon Avalanche Sharpers (SAS) and Diodes with Delayed Breakdown (DDB) will be considered in this chapter.

The main ideas of the book will be summarised in the Conclusion. We are deeply indebted to Dr. Pave1 Rodin (The Ioffe Institute) for valuable

discussions. We would like to thank our wives and children for their understanding and patience.

We will greatly appreciate any comments and suggestions which can be e-mailed to

M . E. Levinshtein (melev@nimis. ioffe .rssi. ru) , Juha Kostamovaara ([email protected]),

and Sergey Vainshtein ([email protected]).

The Authors


Contents

Preface vii

1 . Introductory Chapter 1

1.1 Elementary act of impact ionization . . . . . . . . . . . . . . . . . . 1 1.2 Auger recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Energy of electrons and holes as a function of electric field . . . . . . 1.4 Main approaches for describing ionization phenomena . . . . . . . . 10

1.4.1 Approximation of the characteristic breakdown field Fi . . . 10 1.4.2 Monte-Carlo simulation . . . . . . . . . . . . . . . . . . . . . 13 1.4.3 Approximation of ionization rates . . . . . . . . . . . . . . . . 14

8

2 . Avalanche Multiplication 21

2.1 Fundamentals of avalanche multiplication . . . . . . . . . . . . . . . 21 2.2 Avalanche photodiodes . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.1 Spectral sensitivity . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2.2 Dark current . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2.3 Quantum efficiency . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2.4 Time response . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2.5 Multiplication factor . . . . . . . . . . . . . . . . . . . . . . . 32 2.2.6 Avalanche excess noise . . . . . . . . . . . . . . . . . . . . . . 36

3 . Static Avalanche Breakdown 39

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2 General form of the static “breakdown” current-voltage characteristic 39

3.2.1 Microplasma breakdown . . . . . . . . . . . . . . . . . . . . . 41 3.2.2 Homogeneous (“mature”) breakdown . . . . . . . . . . . . . 44

3.2.2.1 Contact resistivity . . . . . . . . . . . . . . . . . . . 44 3.2.2.2 Thermal resistance . . . . . . . . . . . . . . . . . . . 45 3.2.2.3 Space-charge resistance . . . . . . . . . . . . . . . . . 47

xi

xii Breakdown Phenomena i n Semiconductors and Semiconductor Devices

3.2.3 Negative differential resistance . . . . . . . . . . . . . . . . . . 3.2.3.1 Qualitative consideration . . . . . . . . . . . . . . . . 3.2.3.2 The zero doping ( p - i - n) structure . . . . . . . . . 3.2.3.3 Computer simulation . . . . . . . . . . . . . . . . . .

differential resistance at very high current densities 3.3 Avalanche suppressor diodes . . . . . . . . . . . . . . . . . . . . . . .

3.3.1 Principle of operation . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Main parameters . . . . . . . . . . . . . . . . . . . . . . . . .

3.4 IMPATT diodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Principle of operation . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Some physical problems that arise at very high frequencies . .

3.2.4 Second part of the current-voltage characteristic, with positive . . . . . .

50 50 50 55

58 60 60 62 65 66 75

4 . Avalanche Injection 81

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.2 Avalanche injection in n+ - n - n+ ( p f . p . p+) structures 81 4.3 Avalanche injection in bipolar transistors . . . . . . . . . . . . . . . 91

4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.3.2 Avalanche transistor: conventional regime of operation . . . . 92

4.3.2.1 Difference in breakdown voltages of a BJT between the common-base and common-emitter configurations 92

4.3.2.2 Dependence of the bipolar transistor gain coefficient 98

4.3.2.3 Main features of ABT operation in a conventional regime . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.4 Operation regime of a Si avalanche transistor at very high current densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.4.2 Steady-state collector field distribution . Residual collector

voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.4.3 Transient properties of Si avalanche transistor at extreme cur-

rent densities . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.5 Operation regime of GaAs avalanche transistor at very high current

densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.5.1 Experimental results . . . . . . . . . . . . . . . . . . . . . . . 122 4.5.2 Breakdown in moving Gunn domain in GaAs: qualitative

analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.5.3 Computer simulations of superfast switching in GaAs

avalanche transistor . . . . . . . . . . . . . . . . . . . . . . . . 133

. . . .

QO on current density . . . . . . . . . . . . . . . . . .

5 . Dynamic Breakdown 137

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Contents xiii

5.2 Impact ionization front (TRAPATT zone) . . . . . . . . . . . . . . 140 5.3 Silicon Avalanche Sharpers (SAS) . . . . . . . . . . . . . . . . . . . . 142

5.3.1 Computer simulations and comparison with experimental results144 5.3.2 Stability of the plane ionization front . . . . . . . . . . . . . . 148

5.3.2.1 Short-wavelength instability of the plane ionization front . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5.3.2.2 Long-wave length instability of the plane ionization front . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

5.3.3 The problem of the initial carriers . . . . . . . . . . . . . . . . 154 5.4 GaAs diodes with delayed breakdown . . . . . . . . . . . . . . . . . 157 5.5 Superfast switching of GaAs thyristors . . . . . . . . . . . . . . . . . 162 5.6 Main features of streamer breakdown . . . . . . . . . . . . . . . . . 168

5.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 5.6.2 Analytical theory of a streamer discharge . . . . . . . . . . . 169 5.6.3 Computer simulation . . . . . . . . . . . . . . . . . . . . . . . 176

Conclusion 179

List of Symbols 181

Bibliography 185

Index 195


International Journal of High Speed Electronics and Systems

@ World Scientific Publishing Company Val. 14, NO. 4 (2004) 921-939 World Scientific

w w w wor ldsc lent l f lc corn

Chapter 1

Introductory Chapter

1.1 Elementary act of impact ionization

The basis of all ionization, breakdown and avalanche effects, without exception, is the elementary act of ionization (Fig. 1.1).

electron atom electron

electron

4 - 0 = @ hole collision 3 particles (2 new)

Fig. 1.1 an atom produces two new free carriers: an electron and a hole.

The elementary act of impact ionization. Collision of an energetic electron (or hole) with

A free carrier (electron or hole) “impact” on the atom of a semiconductor. If the energy of the carrier is large enough, this carrier will “knock out” the electron from the valence shell of the atom. As a result, two new free carriers, an electron and a hole, appear. In other words, if an initial carrier has enough energy, it can initiate the transition of an electron from a valence band to a conduction band. The minimal energy necessary to carry out the act of impact ionization is called the threshold energy Eth. It is clear from the law of energy conservation that the threshold energy cannot be less than the energy gap of the semiconductor E,.

The laws of energy conservation and momentum conservation must nevertheless be satisfied simultaneously in the process of an elementary act of ionization. As a result, Eth > E,. In the case of the simplest dispersion law for electrons and holes (Fig. 1.2), the relation between the energy of the particles E and their wave vector

k is defined as E = - (an approximation for the isotropic effective mass m*).

tion threshold for the electrons, &he is

h2k2 2m*

In this case the threshold energy can be calculated fairly simply [l]. The ioniza-

1

922 Breakdown Phenomena an Semiconductors and Semiconductor Devices

Fig. 1.2 acterized by isotropic effective mass m: and m:, respectively.

The simplest “parabolic” dispersion law. At any energy, electron and hole can be char-

Analogously, for ionization initiated by holes, we obtain for the hole ionization threshold Ethh

It is worth noting that when the effective masses of the electron and hole are equal (ma = mi) , &he = Ethh = 3 / 2 E g . On the other hand, &he + Ethh = 3E, at any effective mass ratio mz/m;l .

The band structure of real semiconductors a t high electron or hole energies can never be described by this simple parabolic law, however. A schematic of a GaAs band structure [2] is shown in Fig. 1.3.

At low energy, practically all the electrons are located at the bottom of the central r-valley, and can be characterized by an isotropic effective mass m: = 0.063mo (mo = 0.911 x kg is the mass of a free electron at rest). At high electron energy, however, the electron effective mass even in the r-valley depends to an appreciable extent on the electron energy (nonparabolicity). In addition, as seen in Fig. 1.3, there are two side valleys in the conduction band: the L-valley, in the < 111 > direction of the Brillouin Zone (with energy separation between the bottoms of the L - and r -valleys ErL = 0.29 eV), and the X- valley, in the < 100 > direction of the Brillouin Zone (with energy separation between the bottoms of the X - and r- valleys Erx = 0.48 eV). In both the L - and X - valleys the surfaces of equal energy are ellipsoids with a high ratio of longitudinal to transverse effective mass. As for valence band, three bands, those of heavy holes and light holes together with the split-off band, must be taken into account in order to calculate the threshold energy Eth (for details, see Review [3] ) .

Introductory Chapter 923

Energy 300K Eg=1.42 eV

E~=1.71 eV Ex=1.90 eV

t X-valley

._ E,,= 0.34eV

L-valley

<Ill>

Wave vector w- Heavy holes Light holes

\Split-off band

Fig. 1.3 band and maxima of the valence band are indicated.

Schematic of a GaAs band structure. The most important minima of the conduction

As a consequence, Eth may be substantially larger than the energy gap, and the threshold energy will demonstrate considerable variation with crystallographic orientation. For GaAs, for example, &he = 2.01 eV for impact ionization by electrons propagating in the < 110 > direction and &he = 2.05 eV for electrons propagating in the < 100 > direction, while electrons moving in the < 111 > direction do not cause impact ionization at all [4].

A schematic Si band structure is shown in Fig. 1.4 [2]. As seen in the figure, the existence of several valleys in the conduction and

valence bands must be taken into account in order to calculate the threshold energy Eth. Nevertheless, the estimates show that the magnitude of Eth in the Si is close to the energy gap E, (1.1 eV at room temperature) [5].

If the energy of an electron (or hole) is exactly equal to the threshold energy Eth, the cross-section of the impact ionization is zero. As the carrier energy increases, the probability of ionization j? increases approximately in the manner [6]:

P 0~ ( E - Eth)2 (1.3)

However, the number of very “energetic” carriers, with an energy E exceeding Eth, decreases exponentially as E increases. Thus the effective ionization energy lies very close to the threshold value.

In zero or very low electric fields (close to equilibrium), the role of impact ionization depends to a critical extent on the energy gap E,. In relatively wide-gap

3

924 Breakdown Phenomena in Semiconductors and Semiconductor Devices

Energy 300K Eg=1.12 eV E~=2.0 eV Ex=1.2 eV

/ Eso= 0.044 eV Erl=3.4 E n = 4.2

c

<I oo> <Ill> Wave vector

&Of

-7)

' I Heavy holes

Light holes

Split-off band

eV eV

Fig. 1.4 and maxima of the valence band are indicated.

Schematic of the Si band structure. The most important minima of the conduction band

semiconductors such as Si and GaAs the role of impact ionization is negligible. of finding an electron that is able to cause an

act of impact ionization in Si at room temperature. The equilibrium thermal energy

of a free electron at 300 K F = -kT M 0.039 eV (here k = 8.617 x lop5 eV K-' is the Boltzmann constant). With a threshold energy Eth M E, M 1.1 eV, this probability is equal to

Let us estimate the probability

3 2

In a sample with an equilibrium concentration of, say, no = 10l8 omp3, the concentration of electrons which are able to cause impact ionization is about 6 x lo5 ~ m - ~ . Given a modern field-effect transistor with characteristic dimensions of O.lpm x0.05pm x50pm = 2.5 x cm-3 and an equilibrium concentration in the channel - 10l8 cmp3, the probability of finding just a single electron which can cause the ionization act is about In the largest device known in semiconductor electronics, a silicon power rectifier diode, with a characteristic operation area of approximately 10 cm2, a base of thickness about 500 pm and a characteristic electron equilibrium concentration in the base of about 1013 ~ m - ~ , the total number of electrons in the base is 5 x lo1'. Even in this huge device there are on average only 2-3 electrons with this threshold energy.

However, as seen from Eq. (1.4), the probability p increases exponentially with the mean energy of the carrier E, so that in a strong electric field, when the mean energy is large enough, the effects of impact ionization becomes very important in semiconductors at any value of E,.

4


In narrow-gap semiconductors with a small E,, the probability p can be large enough even at equilibrium, in the absence of an electric field. For example, in InSb (Eg = 0.17 eV) at 300 K, taking Eth = E,, we have p x lo-’. It is ap- parent that impact ionization processes are very important for such narrow-band semiconductors, even when at equilibrium.

1.2 Auger recombination

According to the principle of detailed balance, a state of equilibrium implies that the number of carriers that appear per unit of time due to a distinct generation process must be equal to the number that disappear due to the inverse recombination process. For band-to-band generation, for example, the inverse process is band-to-band recombination, in which the electron and hole recombine and the energy E, is transformed into the energy of a photon or a number of phonons.

The inverse of impact ionization is Auger recombination (Fig. 1.5), which is similarly a “three-particle” process, in that an electron and hole recombine and the energy E that is released ( E 2 Eth) is transferred to a third particle, which can be either an electron or a hole. Two electrons and a hole are involved in the Auger recombination process in an n- type material (e-e-h process) and two holes and an electron in a p-type material (h-h-e process).

,w electron

Fig. 1.5 Schematic representation of a three-particle electron-electron-hole (e-e-h) Auger recombination process. An electron and a hole recombine, and the energy released E 2 Eth is transferred to the electron.

Decrease in the number of excess carriers (e.g. electrons) due to Auger recombination can be described by the expression 171:

d n - = - ( C n n + C p p ) . n . p , d t

where C, and C, are the Auger coefficients for e-e-h and h-h-e processes, respec-

5


tively. If n = p (intrinsic semiconductor or the case of a high injection level):

For a low injection level in an n- type semiconductor ( p << no M N d , where Nd is the donor doping level), we have

For a low injection level in a p- type semiconductor (n << PO M Na, where Na is the acceptor doping level), we obtain

d n / d t = -C,npi = -C,nN; (1.8)

Comparing the expressions for Auger recombination and conventional (linear) Shockley-Read recombination via traps, d p / d t M -PIT, for an n- type semiconductor and d n / d t M -n/rn for a p- type semiconductor, the recombination lifetimes associated with Auger recombination can be written in the form:

In narrow band semiconductors it is Auger recombination that determines the maximum achievable lifetime in a pure material. In InSb, for example, the intrinsic concentration, i.e. the minimum possible concentration at room temperature, is ni = p i M 2 x 1016 cm-3 [ 2 ] . With C = 5 x cm6sP1 [ 2 ] , T,A = T,A = l/Cnp M 5 x s. Hence, the lifetime in InSb at 300 K cannot exceed this value even at a zero concentration of recombination traps (Fig. 1.6).

Fig. 1.6 in InSb (Curve 1) and InAs (Curve 2) at 300K.

Dependencies of lifetimes associated with Auger recombination on carrier concentration

6


In InAs (E, = 0.354 eV) at 300 K, ni = p i M 1015 cmP3 [a], and with c = 2.2 x s. This value is rather large, and at room temperature the lifetime, even in pure material, is limited by either Shockley-Read or band-to-band recombination . The rate of Auger recombination nevertheless increases with temperature or doping in a manner that is directly proportional to the square of the doping (Fig. 1.6, Curve 2), and it is Auger recombination that limits the maximum achievable lifetime a t elevated temperatures (large intrinsic concentration) and / or moderate doping levels.

In relatively widegap semiconductors such as Si (E , = 1.12 eV, ni = pi M

lo1' a t 300K [2] ) , the part played by Auger recombination is important either a t a high doping level or in the event of a high carrier concentration caused by optical pumping or injection.

The dependence of the hole lifetime rp in pure silicon on the concentration of shallow donors ( n o = Nd) is demonstrated in Fig. 1.7. At a low doping level, T~

falls into a very broad range from several milliseconds to nanoseconds, depending on the concentration of deep traps, and the lifetime is determined by the linear Shockley-Read recombination lifetime TSR. Auger recombination comes into play

cm6 sK1 [ a ] , T,A = 7 p A = 1/Cn: z 4.5 x

when TSR N TA = 1/CNj or Nd 2

Donor density Nd (cm-3)

Fig. 1.7 Hole lifetime T~ as a function of shallow donor density in pure n- type Si. The fact that, at N d 2 5 x l O I 7 ~ r n - ~ , T~ - 1/Nj indicates the decisive importance of Auger recombination at a high doping level.

At C = 1.1 x cm6 s-l and TSR M s (Si of very high purity), the role of Auger recombination will be decisive at Nd >> 3 x 10l6 cmP3, while a t a fairly

7


typical TSR of s Auger recombination predominates at Nd >>

1.3 Energy of electrons and holes as a function of electric field

In an electric field F the carriers take their energy from the field:

- = ev. F , dE d t

(1.10)

where e is the electron charge and v is the electron velocity. On the other hand, they give up their excess energy to the crystal lattice due to

various types of scattering (acoustic tering, impurity scattering, etc.):

dE dt

Here Eo = -kT is the equilibrium

-

3 2

phonons, optical phonons, piezoelectric scat-

(1.11)

energy of the carrier and re is the effective electron energy relaxation time (re is usually of the order of steady state,

- s). In a

E - Eo r e

evF = ~ (1.12)

and

E = Eo + evF . re (1.13)

In a low electric field, Ohm’s law is satisfied, so that u = p F , where p is a low field mobility. Then

E = Eo + epF2re (1.14)

and the excess energy A E = E - EO is proportional to F2. In strong electric fields the carrier drift velocity saturates in the majority of

important cases and becomes almost independent of the electric field (see Fig. 1.8). In this case, u = us, and

A E = E - Eo = ev, Fre (1.15)

The excess energy in this case is proportional to F : A E Although re depends on the electric field in strong electric fields, Eqs. (1.14)

and (1.15) give a qualitatively correct idea of the dependencies of carrier energy on the electric field. The field dependencies of the mean electron energy En (a) and mean hole energy Ep (b) in Si, calculated by the Monte-Carlo technique, are shown in Fig. 1.9. As seen, En and Ep are at their equilibrium value En = Ep M 0.039 eV in low electric fields. The mean energy of the carriers then increases with F , so that in relatively low fields the dependencies E ( F ) come close to the law AE - F 2

F .

8


Field F (Vcm-1) Field F (Vcm-1)

Fig. 1.8 field at different temperatures [8]. (With kind permission from Elsevier)

Dependencies of (a) electron drift velocity and (b) hole drift velocity in Si on the electric

(see Eq. (1.14)) and in relatively strong fields (v = u s ) , AE F (Eq. (1.15)). The higher the doping level, the smaller is the low field mobility p and the higher value of F required to reach a given magnitude of E.

0.02 1 o3 1 o4 10'

Field F (Vcm-1) Field F (Vcm-1)

Fig. 1.9 Field dependencies of (a) mean electron energy and (b) mean hole energy in Si at 300 K, calculated by the Monte-Carlo technique. The dependencies for electrons are calculated for three doping levels N d ( ~ r n - ~ ) : 1 - 0 (high purity Si), 2 - 4 x 10l8 and 3 - 4 x 1019. The dependence E ( F ) in Fig. 1.9 b is calculated for pure p-Si [8]. (With kind permission from Elsevier)

As mentioned earlier, the probability p of finding an electron that is able to cause an act of impact ionization increases vastly with an increase in the strength of the field. For Si, for example, the characteristic field for impact ionization Fi is about 3 x lo5 V/cm. As seen in Fig. 1.9a, at a field F of 2 x lo4 V/cm, which is an order of magnitude less than Fi, the mean energy of electrons in pure Si (curve 1) is approximately 0.1 eV. Using Eq. 1.4 it is easy to estimate p at Eth = 1.1 eV and F = 0.1 eV: p M 1.7 x and by comparing this value with the result for

9


equilibrium conditions at room temperature (Eq. (1.4)) one can see that a 2.5-fold increase in E will cause i? to increase by seven orders of magnitude.

1.4 Main approaches €or describing ionization phenomena

There are 3 main approaches to the study of ionization phenomena:

0 Approximation of the characteristic breakdown field Fi, 0 Monte-Carlo simulation,

Approximation of ionization rates.

1.4.1 Approximation of the characteristic breakdown field F;

Approximation of the characteristic breakdown field Fi is the simplest way of describing impact ionization and breakdown phenomena. In the framework of this approach one assumes that avalanche breakdown occurs if the maximum field F, at any point in the structure exceeds a value known as the characteristic breakdown electric field Fi. If Fm is less than Fi, there will be no impact ionization phenomena at all.

This approach is illustrated with an abrupt reverse biased p - n junction in Fig. 1.10. The field F reaches its maximum at the boundary of the p and n regions and (in the simplest case of homogeneous doping) decreases linearly with distance from the junction. The slope F ( s ) is determined by the one-dimensional

U log I

e

x L

t

Fig. 1.10 Approximation of characteristic breakdown field F,. If Fm < F, and, accordingly, the bias applied to the structure, V , is Iess than breakdown voltage V,, there are no ionization effects. The current flowing through reverse biased p - n junction is the leakage current. At F, = F, (and, respectively V = Vz), the “mature” breakdown occurs, and current is controlled by external load resistance Ri.

10


Poisson equation dF/dx = eN/&&o, where N = Na in the p region and N = Nd in the n region, E is the dielectric constant of the semiconductor, and € 0 is the permittivity of the vacuum. The width of the space-charge region W in each region is connected with the maximum field F, and the doping level N by the obvious relation W = &&oF/eN. At F, = Fi, Wi = EEOFi/eN, and the breakdown volt-

age drop V, across each ( p and n) region is V, = - - - - The breakdown 2 2eN '

voltage of this structure, calculated in the framework of this approximation, is

FiWi EEOF~

V . - * &&OF2 ($ + A). 2 -

If the applied bias V < V,, and accordingly F, < Fj, the leakage current will flow through the reverse biased junction. Mature breakdown occurs at V =

V,, and (F, = Fi), and (in the framework of this approximation) the current flowing through the structure is controlled only by the external load resistance R1 (Fig. 1.10b).

This is obviously a very rough approximation indeed, since the transient problems cannot be even formulated in the framework of this approach, for example. Moreover, it is clear that at V > V,, the maximum field F, exceeds Fi value and ionization effects occur not a t a single point but across the whole region L , where F, > Fj (Fig. 1.10a). In reality, the current-voltage characteristic at breakdown does not take the form of a vertical straight line with zero differential resistance dV/dI = 0 (Fig. 1.10b), etc.

Nevertheless, this approximation is not infrequently used, e.g. to estimate the magnitude of the breakdown voltage at p - n junctions and in Schottky diodes with arbitrary doping distribution across the base, in field-effect and bipolar transistors (FETs and BJTs), in thyristors and in many other semiconductor devices. On the other hand, it can be used for qualitative analysis in very complicated situations connected with breakdown phenomena (see Chapters 4 and 5).

The characteristic breakdown electric field Fi in all semiconductor materials depends on the doping level and the temperature. Its dependence on the doping level N for Si is shown in Figure 1.11, where Fj is seen to increase monotonically from 2 x lo5 V/cm to 6 x lo5 V/cm with a rise in doping from - 1014 cm-3 to N 1017 ~ m - ~ . Analogous dependencies are common in other semiconductors. In GaAs Fi increases from - 3 x lo5 to 8 x lo5 V/cm in the same doping concentration range, while in Sic, Fi falls over a range between 2 x lo6 and 6 x lo6 V/cm.

Such type of Fi (N) and V , ( N ) dependencies are also characteristic of other semiconductors (Fig. 1.12).

As a rule, both the breakdown voltage V, and the breakdown field Fi increase with temperature (Fig. 1.13), so that the dependencies shown in the figure are typical of most semiconductors and semiconductor structures. Vj increases monotonically with temperature, so that the smaller the doping level, the stronger is the temperature dependence of V, .

The characteristics of the dependencies can be explained by simple qualitative

11


Concentration N (cm-3)

Fig. 1.11 (A', >> Nd) Si p - n junction as a function of doping level.

Breakdown field Fi and breakdown voltage Vi of an abrupt and highly asymmetrical

Impurity concentration (cm-3)

Fig. 1.12 on the basis of the Si, GaAs and Sic .

Breakdown voltage as a function of doping level for abrupt pf-n junctions fabricated

considerations. As the impact ionization process is defined by the energy of the carrier, gaining from the electric field between scattering collisions, the probability of impact ionization decreases as scattering events become more frequent. Thus, since the frequency of phonon scattering increases with temperature, it becomes more difficult for an electron (hole) to take a large amount of energy from the electric field. (This can be described formally as a decrease in re in Eqs. (1.13-1.15) as temperature increases.) As a result, the breakdown field and breakdown voltage increase with temperature.

The lower the doping level, the larger is the relative contribution of phonon scattering to the total scattering processes. That is why the temperature dependence

12


1.5 - 1016,,-3

Fig. 1.13 different doping levels [g].

Normalized breakdown voltage versus temperature for an abrupt Si p n junction at

of V, becomes greater as the doping level decreases. There are exceptions to this rule, however, as voltage breakdown Vi can decrease

with increasing temperature in semiconductors with a high concentration of deep levels, due to thermal ionization of the traps. As we will see later, such a situation is very dangerous from the point of view of possible thermal instabilities.

1.4.2 Monte- Carlo simulation

The Monte-Carlo technique is a very powerful numerical method that allows us to simulate any transport phenomena in semiconductors, including ionization and breakdown effects. It is based on the approach suggested in Ref. [lo]. The idea is to simulate carrier motion in a k-space (and generally speaking x-space) under the action of an electric field and scattering processes. By observing the motion of a single electron (hole) in a k-space for a sufficiently long time, we obtain a distribution function f(k). All the transport parameters, such as the drift velocity v ( F ) , diffusion coefficient D ( F ) , etc., can then easily be calculated.

Between the scattering events electron (hole) moves in the electric field, and the change in the carrier wave vector k is determined by electric field F :

(1.16)

A scattering event is defined as occurring at an instant tl determined by a computer- generated random number r1. Another random number 7-2 defines which scattering process occurs: acoustic scattering, polar optical scattering, impurity scattering,

13


etc. The next random number (or numbers) will be taken to define the parameters of the electron state after scattering, and so on . . .The probabilities of the scattering events should be known from microscopic theory or from experimental data. When considering ionization phenomena one must take into account the probabilities attached to the elementary acts of impact ionization 11; 11; 12; 131. Details of the Monte-Carlo algorithm can be found in many books and handbooks (see, for example [14]).

Monte-Carlo technique has become a standard numerical method nowadays, and is a conventional attribute of many commercial simulators (ATLAS, DESSIS, MEDIC1 etc.). The accuracy of its calculations in the present instance is limited by only the accuracy of our knowledge of the band structure and scattering rates.

This technique is rarely used to calculate the operating regimes of devices, however, because it usually takes up too much computer time. It is used as a rule to check the principal problems and to calculate ionization rates, and it has also been successfully used to simulate extremely small semiconductor devices when all other techniques have failed due to the large space inhomogeneities and very high space derivatives that are characteristic of small devices.

1.4.3 Approximation of ionization rates

The approximation of ionization rates is the LLworkhorsel’ of the theory of ionization phenomena. It is a very productive and effective compromise between the “oversimplified” approach of the effective breakdown field Fi and the rigorous but rather complicate Monte-Carlo simulation procedure.

In the framework of this approach one assumes that impact ionization is characterized by ionization rates of C Y ~ for electrons and ,& for holes, which are defined as probabilities of impact ionization per unit length.

For example, if in a given electric field F an electron moves an average distance of li = lop3 cm between two acts of impact ionization, then C Y ~ is equal to lo3 cm-’. If li is equal to lop5 cm, then ai = lo5 cm-l, and so on.. .Ionization rates are assumed to be instant functions of the electric field F : a i ( F ) and ,&(F) (the local model ) . This assumption has obvious limitations, however.

Let us assume that a t t = 0 the field F increases instantly from F = 0 to a high value Fa (Fig. 1.14). It takes some time for an electron (or hole) to acquire the threshold energy & which is necessary to produce an elementary act of impact ionization. Roughly speaking, this time will be equal to the energy relaxation time T,(- 10-12-10p13 s in high electric fields). Hence, when considering processes with characteristic times of some picoseconds or less (i.e. frequencies of some hundreds of Gigahertz and higher), we must remember that the local model may not be valid.

A similar situation emerges if an electric field F changes very sharply in a space (Fig. 1.15). It is clear that if a field changes notably along the mean free path l o , it will be impossible to say which value of F should we use to calculate a i ( F ) or Pi(F) .

14

Introductory chapter 935

Fig. 1.14 (10-l' - represents the time dependence of F and the dashed line the qualitative time dependence of E .

If field F increases instantly from F = 0 to F = Fo, it will take the time t N N

s) for an electron or hole to acquire the appropriate energy E. The solid line

Fig. 1.15 cluding in part the ionization rates approach) will not be valid.

If an electric field F changes very sharply (dF/dx 2 1011 V/cm2), local models (in-

Taking a characteristic mean free path of l o - cm and a characteristic Fi of lo5 V/cm, one can estimate a characteristic magnitude for dF/dx of about 1011 V/cm2. Such large values of dF/dx are realized either in extremely small semiconductor structures with characteristic sizes of about 100-1000 A and less or in the case of very high doping levels N 2 1018 cmP3. In these cases Monte-Carlo simulation should be used to describe the ionization processes correctly.

The approximation of ionization rates is nevertheless the most popular and most efficient tool for studying ionization and breakdown phenomena in its region of applicability: f 5 400 - 500 GHz, L 2 0.1 pm, and N 5 10'' cmP3.

Even in very strong electric fields it is the case as a rule that only a small portion

15


of the electrons (or holes) have an energy which exceeds the characteristic critical energy Eo(E0 N &h). On average the carrier energy is much smaller, and it is limited by optical phonon scattering with an energy of Eph = LO << Eo. The values concerned are LO = 0.063 eV for Si, for example, tUJ0 = 0.035 eV for GaAs and L o = 0.1 eV for Sic.

In order to achieve an energy of Eo, an electron has to move without collision for a distance

The probability of such an event is

(where lo is the mean free path). Hence, the expressions for cri(F) and Pi(F) take the forms

ai = aoezp[-F,o/F]

and

(1.17)

(1.18)

(1.19)

Pi = Poezp[-F,o/F] (1.20)

where Fo = Eo/elo. To calculate a0 and PO and define the exponential parts of these expressions more

precisely, it is necessary to decide on the form of the energy distribution function for electrons (holes) [ll; 12; 131 (see Review [IS])

The experimental a ( F ) and P ( F ) dependencies for the most important semiconductor materials are usually described by the following empirical equations:

(1.21)

In Si, for example, m, = mp = 1, and

ai = 3.318. 105ezp [-1.174. 106/F] (cm-l)

To describe the P ( F ) dependence correctly over a wide range of F values, it is necessary to use a “two-piece” approximation:

If 2 . 105V/cm < F < 5.3. lo5 V/cm,

Pz = 2 . 106ezp [-1.97. 106/F](cm-1),

while for F > 5.3. lo5 V/cm,

,l?i = 5.6. 106ercp [-1.32. 106/F](cm-1),

16


Fig. 1.16 300K [16; 171. (With kind permission from Elsevier)

Dependencies of the ionization rates of electrons, ai, and holes, &, in Si on 1/F at

(see Figure 1.16). The a i ( F ) and Pi(F) dependencies for many semiconductor materials can be

found in Refs. [2; 18; 191. It is worth noting that if the electric field F is relatively small ( F << Fo), ai

and Pi will be very strongly dependent on the field strength, while if F - Fo, a( and ,& will show a fairly weak dependence on F .

In very strong fields, F >> Fo, ai and pi tend towards their limiting values of a0 and Po, respectively, which fall within the range - lo4 to - lo6 cm-' for different semiconductors. These limiting values correspond to a situation in which the distance between two elementary acts of impact ionization 1 = 1/ao is equal to the mean free path l o , i.e. in which electrons (holes) ionize at every scattering act.

Let us consider the fluxes of electrons and holes passing through a region of a semiconductor (Figure 1.17). While travelling a distance d x , each electron will create an average of (ai d x ) electron-hole pairs. The increase in the electron current due to electron multiplication will thus be

(1.22)

In addition, the electron current density will increases due t o hole multiplication:

(1.23)

Hence

djn _ - - aijn + Pijp d x (1.24)

17

938 Breakdown Phenomena in Semaconductors and Semiconductor Devices

, du, I I =

l l 1

Fig. 1.17 ionization by electrons (a) and holes (b).

Augmentation of the electron and hole current densities j , and j , caused by impact

Analogously,

d3P - = -p& - a& dx

and

The total current density

(1.25)

(1.26)

(1.27)

d n dP dx dx

where j n = envn(F) + eDn--; j p = epwp(F) - eDp- . As we will see, by solving simultaneously the set of equations (1.25)-(1.27) and

the Poisson equation

dF e - = - ( N d - N A + p - n ) , dx EEO

(1.28)

with appropriate boundary conditions allow us to describe (in a one-dimensional approximation) the steady state electron and hole distributions under conditions of avalanche multiplication and breakdown.

The transient characteristics can be described by a set of partial differential

18


equations:

d n 1 dj, 1 _ - + -(aij, + Pijp) d p 1aj, 1 + - (ai jn + Pijp)

- -- d t e dx e

d t e dx e _ - - _-I

dn jn = enun (F) + eD, - dx

dx jp = epup(F) - eD,- dP

dF e _ - - - ( N d - N A + p - n ) dx EEO

with appropriate boundary and initial conditions,

(1.29)

19


World Scientific www.worldscientific.com

International Journal of High Speed Electronics and Systems Vol. 14, No. 4 (2004) 941-958 (Q World Scientific Publishing Company

Chapter 2

Avalanche Multiplication

2.1 Fundamentals of avalanche multiplication

Let us consider the following simple situation. An electron is injected into a sample of length L with a homogeneous field F (Fig. 2.1). Let us suppose that a t this given field F the electron ionization rate a; is much larger than the hole ionization rate

I= L a

+

Fig. 2.1 An electron is injected into a sample from the left. In travelling a distance li = l /a , (on average) it will create a new electron and a hole. The hole will move to the left (without ionization), while emergent electrons will move to the right and create further electron-hole pairs, and so on.

21


(ai >> Pi), so that aiL >> 1 but PiL << 1 This means that the electron performs numerous elementary acts of impact ionization as it passes through the sample. On the other hand, the probability of a hole making even one act of impact ionization is practically zero.

In this simplest case we have

- aijn; j n ( z ) = jn(0)eaia: djn - _ dx

and a multiplication factor Mn for electrons injected at the cathode M, = i n (L ) / jn (0 ) is equal to

For a non-homogeneous field distribution across the sample:

It is worth noting that in this simplest case (PiL << 1) the multiplication factor M is always a finite value. It may be very large (aiL >> l), however, if there is no electron injection at x = 0 ( j n ( 0 ) = 0), there will be no carriers at z = L. To cause the “avalanche”, an extrinsic carrier should be injected in the avalanche region.

The qualitative time response to a “bunch” of electrons injected at the cathode of the sample at t = 0 is shown in Fig. 2.2.

Fig. 2.2 Electrons injected into a sample at t = 0 are multiplied during the time t , = L/wsn (where vsn is the electron saturation velocity). During time t , x L / v s p (where wsp is the hole saturation velocity) holes move from the right boundary of the sample to the cathode (Fig. 2.1). The total time response is t , x t , + t,.

22

Avalanche Multiplication 943

Now let us consider the case in which both types of carrier are able to create electron-hole pairs (Fig. 2.3).

0

-

t

L x

+

Fig. 2.3 As before (cf. Fig. 2.1, an electron is injected into the sample from the left. In travelling the distance li, Y l/ai it creates an electron-hole pair. The hole will then move to the left and, in travelling the distance l i p % l/Pi(oi N P i ) , will also create an electron-hole pair. The new electron created by the hole will now move to the right and create a new electron-hole pair, and so on.

For this general case in which both electron and hole impact ionization must be taken into account, the multiplication factor M may be obtained using Eqs. (1.24- Eq. (1.27), from which we obtain:

With the boundary condition j n ( 0 ) = j (electron injection) the solution of (2.4) takes the form [13; 201:

X (2.5) 1

1 - s ai exp[- s(ai - ,&)dx’]drc L

0 0

Mn =

Similarly, for hole injection ( j p ( L ) = j ) , the hole multiplication factor M p is:

23


1

1 - S a d x L

0

Mn = M p = M =

L

0 It is seen from (2.7) that a t Saidrc -+ 1, Mn + 00. The condition Adn + m

corresponds to avalanche breakdown. For homogeneous field distribution along the avalanche region, the condition of

breakdown has the simplest form: a L = 1. This means that avalanche breakdown occurs (at a( = pi) when the electron (and hole) creates just one electron-hole pair on average while travelling through the avalanche region L. The physical reason for such a situation is made clear in Fig. 2.3. An electron creates a hole, and the newly emergent hole in turn creates an electron, and so on. Just this positive feedback provides the appearance of an avalanche breakdown.

Three important points are worth noting:

With Mn + 00, it is not necessary to have any external carrier to support the avalanche breakdown process. Breakdown is a self-supporting process. The condition

L X

0 0 S a i e x p [ - S(ai - & ) d x ' ] d x = 1

or L

J a d x = 1 (ai = pi) (2.8) 0

m u s t be satisfied at any bias voltage VO (for VO > K h ) and at any current density j. With a change in current density, the field distribution along the avalanche region is reconstructed to support the main condition of a breakdown (2.8).

The larger the ratio ai/pi (or Pi/ai) is, the larger the multiplication factor M that can be reached before breakdown occurs. The most favourable condition for reaching maximum values of M is ai >> pi (or ai << pi) , and the most unfavourable condition is ai = pi (Fig. 2.4).

In the framework of the approximation of ionization rates one can easily under- stand the reason for the dependence of the characteristic field Fi on the doping level (Fig. 1.11).

The field distribution along the reverse-biased p f - n junction at two doping

levels is demonstrated qualitatively in Figure 2.5. As seen, the integral S a ( F ) d x

(see Eq. 2.8) is obviously less than the integral S a ( F ) d x at the same maximum

L1

0 LZ

0

24


M 120

100

80

60

40

20

1 0

a. 5 p. 1 - 1 -

.Breakdown

i/ : v, +

vth2 Vth 1 VO

Fig. 2.4 Qualitative dependence of the multiplication factor M on the bias VO for different ratios ail,&. At pi = 0, M is the finite value at an arbitrary bias Vo. The minimum achievable value of M is realized at ai = pi.

Fig. 2.5 Field distribution along a reverse-biased p + -n junction. The dashed line represents the field versus x dependence for a relatively highly doped base, and the dotted line shows the F ( z ) dependence for a relatively low-doped base. The lower the doping level is, the wider is the “zone of multiplication” at the same value of Fm.

field F,. That is why the higher the doping level is, the larger the “characteristic” field F, = Fi must be to provide the same value for the integral (2.8). It is not enough to satisfy the condition F = Fi just at one point. It is necessary t o have an avalanche zone of length L (determined by the expression (2.8)).

25


2.2 Avalanche photodiodes

The phenomenon of avalanche multiplication is used directly in one of the most important types of photodetector: the avalanche photodiode.

There are many types of photodetectors: photoconductors, reverse-biased p - i - n diodes, Schottky diodes, phototransistors etc. [20; 211. Avalanche photodiodes, which combine fairly high internal gain (up to several hundred times) with a high limiting frequency of operation (up to several gigahertz), are widely used in fibre- optic communication systems.

From a physical point of view, an avalanche photodiode (APD) is simply a conventional photodiode (reverse-biased p - n or p - i - n structure, or a reverse-biased Schottky diode) that is under a bias which provides a substantial multiplication factor M in the reverse-biased barrier. In such a regime, the carriers generated by light create other carriers that lead to internal gain in the APD. Special precautions must be taken, however, to guard against surface breakdown in APDs (by means of guard rings, junction termination extensions formed by implantation, etc. [22]-1241. There is a rich variety of APDs constructions and modifications, and we will discuss here only the principal features of APDs.

2.2.1 Spectral sensit ivity

This parameter depends mainly on the semiconductor material. The region of maximum spectral sensitivity, as in conventional photodiodes (PDs) , is usually close to X x 1.239/Eg, where X is the wavelength of maximal sensitivity (measured in pm) and Eg is the band gap of the semiconductor (in eV). Hence X M 0.88pm for GaAs, 1.12pm for Si, and 1.7pm for Ge. Values of E, for many semiconductors and their temperature dependencies can be found in Refs. [2; 18; 191.

As a rule, E, decreases with temperature growth and Am increases accordingly. There are several AIVBV' compounds, e.g. PbS, however, in which the opposite is the case, i.e. Eg increases with temperature growth (Fig. 2.6, Curves 2,3).

The long-wave boundary of spectral sensitivity is determined by the magnitude of A, (the photoelectric threshold). At X > A, the absorption coefficient is too small to provide any appreciable photogeneration. The short-wave boundary of spectral sensitivity is caused by the effective surface recombination. At X << A, the absorption coefficient is usually very large (- lo6 cm-' for Si, Gel GaAs and InSb), and photons are absorbed very close to the surface of the photodiode, where recombination lifetimes are small due to the high concentration of defects.

The semiconductors that are most widely used for the manufacture of photodiodes are Si, Gel and A"'BV compounds, including binary compounds (GaAs, Gap) , ternary compounds (InGaAs, GaAlSb) and quaternary compounds (In- GaAsP, GaInAsSb). Visible blind S ic and GaN-based APDs have been successfully demonstrated recently [25; 261.

26


'rn.

F: 1

c.l .r.

d 3 0.8 W

1-

.M 2

+% 0.4 2 % 2 " 1 2 3 4 5 6 7 8 i% CA Wavelength h (pm)

-

Fig. 2.6 1 - CdS (300 K), 2,3 - PbS. 2 - 300 K, 3 - 77 K. 4 - PbSe (300 K) , 5 - InSb (77 K).

Qualitative dependencies of the spectral sensitivity of several semiconductor materials:

2.2.2 Dark current

Dark current in APDs is the leakage current of a reverse-biased p - n junction multiplied M times. There are usually two components that make the main contribution to the leakage current: the minority carrier current through the p - n junction (known as the Shockley component [27]) and the generation current in the space-charge region of the p - n junction (the Sah-Noyce-Shockley component [28]).

The Shockley component does not depend on the reverse bias VO (at VO 2 (2 - 3)kT) and is determined by the charge of the minority carriers generated per unit time within the diffusion length Ld close to the p - n junction. The electron component of this current (generated in the p-region of the p - n junction) can be written in the form "7; 20; 211:

where np = n5/Na is the equilibrium electron concentration in the p-layer and Ln =

Analogously, the hole component of this current (generated in the n-region of the p - n junction) is:

is the electron diffusion length.

(2.10)

where p , = n:/Nd is the equilibrium hole concentration in the n-layer and L, = is the hole diffusion length. The total magnitude of the Shockley component

of the leakage current density, j , , is equal to the sum of the appropriate electron and hole components: j, = j,, + j,,. In the most widely used asymmetrical p - n junction, the current of the minority carriers from the weakly doped layer (base)

27


makes the main contribution to the total current j,. For a p+ - n junction ( N , >> Nd), for example,

(2.11)

The current density of the Sah-Noyce-Shockley component "281 can be written as foHows [7; 20; 211:

. n eni W (2.12)

where W is the space charge width (at a given bias Vo), Ei is the Fermi level position in an intrinsic semiconductor, and ~~0 and r,o are electron and hole lifetimes in a heavily doped semiconductors of n- and p- type, respectively.

Expression (2.12) describes the total current generated in the space charge region of the pf - n junction via Ic traps with energies Etk. It is seen that the main contribution to this current comes from the trap (or traps) located close to the intrinsic level Ei, i.e. close to the middle of the energy gap. That is why, the following simplified expression is widely used to obtain a rough estimate of j,:

(2.13)

where r is the effective carrier lifetime, which characterizes the generation- recombination processes in the space-charge region.

The j g / j , ratio can be estimated as:

(2.14)

The values of Lp and W are usually of the order of several microns, and T~ and r can differ in magnitude by several orders. However, the main contribution to the j g / j s ratio in fact comes from the term N/ni.

The magnitude of the doping level N typically falls between 1015 and 10'' ~ m - ~ . As for nt, this depends exponentially on temperature and on the band gap E,. At room temperature, ni is equal to N 1.3 x 10l6 cm-3 for InSb (E , x 0.17 eV), N 2.4 x 1013 cm-3 for Ge (E , x 0.66 eV), N 1.1 x lo1' cm-3 for Si (E , M 1.1 eV), N 1.4 x lo7 cm-3 for GaAs (E , x 1.4 eV), and N lo-' ~ r n - ~ for S ic (4H-Sic polytype, E, x 3.2 eV).

In diodes based on relatively narrow-band semiconductors (InSb, InAs, Ge), it is the bias-independent Shockley component that predominates in the leakage current. In Si-based p+ -n junctions, however, the Shockley component predominates rather infrequently, only in p + - n junctions with an extremely low concentration of deep levels in the space-charge region (large values of r ) . In diodes based on relatively

28


wide-band semiconductors such as GaAs, Gap, Sic, GaN, etc., the contribution of the Shockley component is negligible.

As seen from expressions (2.9), (2.10), and (2.13), j , is proportional to ni squared ( j , 0: n!), and j , 0; ni. At low temperatures, when ni becomes very small (even in relatively narrow-band semiconductors) , the diode reverse current can be determined by the surface leakage (even in structures with guard rings), or by volume leakage through bulk structure defects.

2.2.3 Q u a n t u m e f i c i ency

The quantum efficiency v p h is defined as a ratio of the number of carriers flowing in the external circuit of the photodiode, Npc = Ipc /e , to the number of photons incident on the window of the photodiode per second, Nph = P/hu, where Ipc is the photocurrent and P is the incident light power.

If one assumes that only the photons, which are absorbed in the space-charge region give a contribution to the photocurrent, it is easy to estimate the quantum efficiency as

- e - f f p h w (2.15)

where R is the reflectance, a p h is the absorption coefficient and W is the space- charge width.

Not all carriers generated by light contribute to the photocurrent, as some of them recombine on the surface of the photodiode and in the quasi-neutral regions adjoining the space-charge region. Eq. (2.15) nevertheless gives a correct estimate for the quantum efficiency of an optimally designed photodiode.

As seen from (2.15), the quantum efficiency is high enough if aphW 2 1. Any increase in W , however, will result in an increase in carrier transit time through the space-charge region, and accordingly in a decrease in the speed of response of the APD (see Section 2.2.4). On the other hand, W should not be too small. Any decrease in W will result in an increase in the barrier capacity C, which in turn will increase the time constant R1C. A rough estimate for the optimum value of W for conventional PDs can be obtained from the condition W zz vs/2f0, where fo is the maximum operation frequency. With fo = 20 GHz and w, - lo’ cm/s, for example, the optimum value of W is close to 2.5 p m . For APDs, the multiplication factor M must also be taken into account (see Section 2.2.5).

2.2.4 T i m e response

Generally speaking, four time constants determine the time response of an APD, and consequently the limiting frequency of its operation:

1. Carrier transit time through the space-charge region t,,

29


2. Diffusion time through the quasi-neutral regions adjoining the space-charge re-

3. The time constant RlC, where C(V0) is the capacitance of the barrier ( p - n

4. The multiplication time tM.

gion t o ,

junction) and Rl is the load resistance,

First three of these time constants, ts , tD, and Rc, also determine the time response of conventional PDs

The carrier transit time t , can be estimated as t , M W/v,, where the value of W lies in the range from 0.5 pm to 40+50 pm. Accordingly, t , lies in the range

from 5 . lo-'' s (characteristic limiting frequency of operation fo M - M 50 GHz) 1

4ts to 5 . lo-'' s (fo M 500 MHz).

The diffusion time through quasi-neutral regions adjoining the space-charge re-

The size of the quasi-neutral region L,, is much less than the diffusion length gion t o can be estimated for two limiting cases:

Ld (Lqn << Ld). In this case, the magnitude of can be estimated as

tdl M - L& (2.16) 2 0

where D is the diffusion coefficient of the minority carriers in the quasi-neutral region.

The size of the quasi-neutral region L,, is much larger than the diffusion length Ld (Lqn >> Ld). Then

(2.17)

where r is the lifetime of the minority carriers. tD is determined in this case only by the diffusion length, because the carriers generated by illumination over the distance L > Ld will recombine before they can reach the space-charge region and contribute to the photocurrent.

The R1C time constant can easily be estimated provided the doping level N in the base is known. The capacitance C is

(2.18)

where S is the operation area of p - n junction, Vbi is a built-in voltage, and W(V0) is the width of the space-charge region at a given bias.

Taking R1 = 2 Ohm, N = l O l 5 cmP3, E = 11.7 (Si), Ki = 1 V, VO = 200 V, we obtain W x 16 pm, and C x 1.6 pF with S = (500 x 500) pm'. An appropriate RlC time constant would be 3.2 ps.

The multzplication time (or "effective transit time") tM is defined as follows. Let a photocurrent j (0) be excited at a point z = 0 at instant t = 0 (Fig. 2.1). The

30


steady state output signal M x j ( 0 ) will appear at x = L at time t M .

It is easy to see that the magnitude oft^ depends critically on the ratio a&. For the limiting case Pi = 0 (a i /p i + oo), the total multiplication factor Mn is

reached for the transit time t “N L/v, (Figures 2.1 and 2.2). In spite of the fact that with a large multiplication factor M = eaL = eL/‘, an electron will ionize many times along the sample length L, (the number of ionization acts being L / L ) , all these ionization acts occur for one effective electron transit through the avalanche region (Figures 2.1 and 2.2). In this case t M = t,.

In the opposite limiting case, ai = pi, the number of ionization acts per transit for both electrons and holes, should be less than unity to avoid the avalanche breakdown (see Figure 2.3 and Eq. (2.7)). It is clear that to provide a large multiplication factor M , the electrons (and holes) must transit the avalanche region L many times. Hence, in this case t M >> t,.

The dependence of t M on the ratio ailpi and on the value of M is calculated in Ref. [as]. As has been shown in this paper, the multiplication time t M can be approximated by the expression:

t M = MtsN(az /P i ) (2.19)

where M is the multiplication factor t , = L/v , is the transit time through the avalanche region and N(ai/Pi) is a coefficient determined by the ratio ai/Pi. The value of N lies in the range from 1/3 (a i lp i = lo3) to 2 (ai = pi). The larger the ratio c.i/Pi is, the smaller is the multiplication time t M at the same magnitude of the multiplication factor M . At large magnitudes of A4, the multiplication time t~ is usually the largest time constant which determines the time response (or limiting operation frequency) of an APD.

It is possible by measuring t M to determine the coefficient N , provided the length L of the avalanche region is known (Fig. 2.7).

Multiplication factor M 2

Fig. 2.7 Experimental dependence of multiplication time t~ on the multiplication factor A4 for a Ge pf - i - n APD (301. The experimental points correspond to the different wavelengths of the illumination and different frequencies of analysis. t M = Mt,N = ( 5 x 10-12)M sec.

31


2.2.5 Multiplication factor

The main parameter which determines the maximum practically achievable multiplication factor M is the ratio ail,& or ( P i l a i ) . The larger this ratio is, the larger the magnitude of M that can be obtained.

The field dependencies of oi and ,@ for several semiconductor materials are shown in Figure 2.8.

The ionization rates of the electrons ai in Si are seen to be much larger than those of the holes pi, especially a t F 5 3 . lo5 V/cm (Fig. 2.8a). From this one can

n

n - 106 v 5 .- a

105

104

m b)

rd c.

.- c) R .- H ' I o 3 , 2 3 4 5 6

Io3 ' ' '5;O' ' ' ' 5.5 ' ' ' ' 6.C '

d n 1 /F (1 0-6 cmV- 1)

2 3 4 u

11F ( 10-7cmV-1) l/F (10-7 cmV-1)

Fig. 2.8 171, Ge (b) [31], 4H-SIC (c) [32], and GaN (d) [33]. 300 K.

Dependencies of ionization rates C Y ~ and pi for electrons and holes on 1/F for Si (a) [16;

conclude that fairly large values of M can be achieved with Si-based APDs, especially with a relatively large space charge width W (W 210-20 pm). Indeed, values of about 200-500 for the multiplication factor M were observed experimentally in Si APDs with W - 20-30 pm.

The ionization rates for electrons and holes in Ge are shown for two crystallographic directions in Fig. 2.8b. As one can see, the ailpi ratio is much smaller in

32


Ge than in Si, and thus the practically achievable multiplication factor M for Ge APDs is usually M ~ 2 0 - 8 0 .

The ai and ,& field dependencies for one of the most practically important S ic polytypes, 4H-SiC1 are shown in Figure 2 . 8 ~ . It is worth noting first of all that the characteristic values of ai and ,& of about lo5 cm-I are achieved in S ic (and GaN, see Fig. 2.8d) in fields F that are approximately an order magnitude larger than those for Si and Ge.

Due to the large forbidden gap in S ic and GaN (3.23 eV for 4H-SiC, 3.39 eV for GaN), the elementary act of impact ionization requires a higher threshold energy Eth and greater magnitudes of F . Note that the characteristic magnitude of the breakdown field Fi in Sic is also an order magnitude larger than that in Si: (2 - 6) x lo5 V/cm in Si, and (2 - 6) x lo6 V/cm in Sic, and also that in 4H-Sic pi >> ai, i.e. the situation is quite the opposite to that in Si (Fig. 2.8a). The authors of Ref. [32] associate this situation with the discontinuity of the electron spectrum in the conduction band of hexagonal Sic polytypes [MI. The problem of the ai and pi field dependencies in Sic cannot yet be considered to be finally settled, and alternative data have been reported in Ref. [35].

As a rule, the multiplication factor M decreases with temperature growth (Fig.2.9).

The reason for the temperature dependence of M is just the same as for the breakdown voltage increasing with temperature growth. The higher the temperature is, the more intensive is the phonon scattering, and the more difficult it is for electrons or holes to achieve the threshold energy needed to provide an elementary

1 50 100 150 200

Voltage ( y)

Fig. 2.9 Si-based APD [36]. (With kind permission from Elsevier)

Dependence of the multiplication factor A4 on bias voltage at different temperatures.

33

954 Breakdown Phenomena in Semaconductors and Semiconductor Devices

act of impact ionization. The temperature dependence of M in S ic may be more complicated than this, however, [37]-[39].

A number of methods have been suggested for increasing the ai//?i ratio “arti- ficially” ; the effectiveness of some of which has been demonstrated experimentally. Let us consider, for example, the use of a graded gap structure as proposed in Ref. [40].

It is well known that band-offsets (band discontinuities) appear a t the boundary of two semiconductors with different band gaps (heterojunction). The conductive band discontinuity AE, for the well-studied GaAs/Gal-,Al,As heterojunction, for example, is AEJx) = AE,(x) - AE,(x) (Figure 2.10), where AE,(x) is the dif-

t r

Fig. 2.10 Schematic diagram of an abrupt GaAs/Gal-,Al,As heterojunction.

ference between the band gaps of GaAs and Gal-,Al,As, AE, is the conduction band discontinuity, and AE, is the valence band discontinuity:

AE, x 1.247x(eV) for 0 < x < 0.45 AE, x 0.476 + 0 . 1 2 5 ~ + 0.1432’ (eV) for 0.45 < x < 1,

AE, x 0.792 (eV) for 0 < x < 0.41 AE, M 0.475 - 0 . 3 3 5 ~ + 0.143~’ (eV) for 0.45 < x < 1 ,

(2.20)

AE, x -0.46x(eV) for 0 < x < l

By changing the concentration of Al, “smoothly” along the y-coordinate, one can achieve a “graded gap” structure (Figure 2.11). It is easy to see that in such a structure the electric field acts on electrons in the conduction band:

(2.21)

34


Fig. 2.11 tion of A1 decreases gradually along the structure (cf. Fig. 2.10).

Schematic diagram of graded-gap GaAs/Gal-,Al,As heterostructure. The concentra-

and on holes in the valence band:

F,, = e . d(AE,)/dy , (2.22)

where AE,(y) and AEv(y) are the energies of the bottom of the conductive band and top of the valence band, respectively.

These fields push the electrons and holes in the same direction (from left to right). When an external (reverse) bias is applied to the graded gap structure, however, the external electric field F in the space-charge region pushes the electrons and holes in different directions (Figure 2.12). As a result, the internal field of the graded gap semiconductor is added to the external field for the electrons (F, = F + F,,), however, it is deducted from external field for holes (Fp = F - Fgv). Knowing the band discontinuity and the thickness of the device W , it is easy to estimate the strength of the related electric field. For example, taking AE, = 0.3 eV and W = lo-* cm, we have F, = 3 . lo3 V/cm. This field is rather small by comparison with the characteristic breakdown field Fi, although due to the very strong (exponential) field dependences of ai and pi (see Eqs. (1.19)-(1.21)), the Lyi/pi ratio can be increased appreciably.

An even more important contribution to the ai and pi ratio increase in such a structure can give a different effect. In Fig. 2.12, the initial electron-hole pair is denoted by 1-1’. Note that the next act of impact ionization will be made by the electron in the region with the smaller gap E,, and correspondingly with a smaller magnitude of the ionization threshold field E t h (the pair 2-2’). Conversely, the hole will create its next ionization act in the region with the larger energy gap, and consequently with the larger value of Eth (pair 3-3’). Hence the probability of impact ionization increases for electrons and decreases for holes.

A number of other approaches to the question of increasing the ratio ai lp i (or

956 Breakdown Phenomena i n Semiconductors and Semiconductor Devices

- F Bf Ec-,+-- 1 GaAlAs F, =F+Fg,

+

Y

Fig. 2.12 A graded-gap GaAs/Gal-,AI,As heterostructure under reverse bias. The internal field of the graded-gap semiconductor is added to the ext,ernal field for the electrons ( F , = F+ Fgc) but deducted from the external field for the holes (Fp = F - F g U ) . Besides, an electron performs every subsequent act of impact ionization in the region with the smaller magnitude of Eth, whereas a hole will perform every subsequent act of impact ionization in the region with the larger magnitude of Eth. Both these effects increase the effective cri/,Bi ratio.

pi/ai) are discussed in Ref. [Is].

2.2.6 Avalanche excess noise

The main source of excess noise in the conventional PDs is a shot noise [41]-[43] arising from the statistical nature of photon flux:

< i2 >= 2eIph (2.23)

where < i2 > is the current noise per unit bandwidth (in a bandwidth of 1Hz). Shot noise is frequency independent over an extremely wide frequency region ( “white noise”).

The main source of noise in APDs is, as a rule, Avalanche Excess Noise (AEN), arising from the statistical nature of the ionization process. The noise in APDs can usually be expressed as:

< i2 >= 2eIphM2F, (2.24)

where Fa is the excess noise factor. The nature of the factor M 2 is clear: the current noise is multiplied together

with the signal I p h . The factor Fa appears as a result of the contribution of the ionization processes to the noise of APDs.

Taking the variance of the multiplication distribution to be 0

a2 =< M 2 > - < M > 2 or a2 =< M2 > -M2 (2.25)

36


we have

or

< M 2 > 0 2 = 1 + -

M2 M2 Fa = (2.26)

Just like all the other parameters of APDs, Fa depends fundamentally on the ratio ailpi. The larger this ratio ai lp i (or ,&/ai), the smaller the noise factor Fa , providing the avalanche is initiated by carriers having a larger ionization rate. Thus, if ai/pi >> 1, a low noise level will be achieved in a situation where the avalanche is initiated by electrons, whereas if &/ai >> 1, a low magnitude of Fa will be achieved when the avalanche process is initiated by holes.

Analytical expressions for Fa can be obtained in the case of a field-independent ratio ailpi [44]:

F a p = M p [1+ (7) (?)'I if avalanche multiplication is initiated by holes, and

(2.27)

(2.28)

if the only carriers injected into the avalanche zone are electrons. Here k = ,&/ai. The noise factor Fan is shown as a function of the multiplication factor M , at

different values of k = ,&/ai in Figure 2.13. One can see that if the multiplication process is initiated by electrons and ai/pi >> 1 (k << I ) , the noise factor will be very small ( F M 2 over a wide range of Mn from Mn M 3 to Mn M 100). As k increases, the noise factor increases, and at k = 1 (a , = pi) , F = M . In this case the noise power < i2 > is proportional to M 3 :

< i2 >= 2eIPhM2F = 2eIp,hM3 (2.29)

As the noise increases very fast with growth in the multiplication factor, it is necessary to reach a compromise between high gain (large multiplication factor) and high detectivity (low noise, large signal-to-noise ratio).

The experimental dependencies for Si APDs agree reasonably well with the theoretical estimates (Fig. 2.14).

A well-designed APD is one in which the optimal compromise is achieved between acceptable quantum efficiency (which requires a relatively large value for the space-charge width W ) , a small time response (which requires a relatively small value for W and not too great magnitudes of the multiplication factor M ) , high gain and low noise [46].

37


Fig. 2.13 The excess noise factor F,, as a function of the multiplication factor M , at different values of k = &/ai for the case of electron injection. Just the same dependences are valid for the case of hole injection, with replacement of F,, by FRp, Mn by M p and k by l / k [15].(With kind permission from Elsevier)

0 Multiplication factor A4

Fig. 2.14 Experimental dependencies of noise power versus multiplication factor for two Si APDs. f = 30 MHz, bandwidth B = 1 MHz [45]. Excess noise factor Fa was found to be 4-5 at a multiplication factor M = 100.

38

World Scientific w w w . w o r l d s c i e M i c . c o m


@ World Scientific Publishing Company Vol. 14, NO. 4 (2004) 959-999

Chapter 3

Static Avalanche Breakdown

3.1 Introduction

Systematic studies of breakdown phenomena in solids began more than 80 years ago, in the early 1920s. Approximately at the same time it was found that the scenario for breakdown depends critically on the magnitude of the ratio d V o / d t , where VO is the bias applied to the structure and t is time. Over a very wide range of magnitudes of dVo/d t from very small (quasi-static) to fairly large, just the same LLconventional’l scenario is followed, that usually known simply as “breakdown” I but if dVo/d t becomes extremely large the picture changes dramatically to that termed “dynamic breakdown” .

The critical value of dVo/d t depends on the parameters of the object in which the breakdown is observed. We will establish the related criteria in Chapter 5, where the phenomenon of dynamic breakdown will be considered. For the moment we will remark only that LLconventional” (quasi-static or static) breakdown takes place, roughly speaking, if the bias is applied to the sample in a time t which exceeds the carrier transit time through the sample at saturated velocity: t 2 t , = L/v,. With 0.1 pm< L < 100 pm and us M lo7 cm/s, we have t 2 10-12-10-9 s. Taking the characteristic breakdown field Fi N lo5 V/cm, we find that the breakdown voltage V, = FiL falls into a range from 1 V to lo3 V. Hence we can conclude that conventional, quasi-static, or static breakdown occurs over a very wide range of dVo/d t ramps 0 5 dVo/d t 5 10l2 V/s.

3.2 General form of the static “breakdown” current-voltage characteristic

The qualitative current-voltage ( I - V ) characteristic of a reverse biased p n junction (or Schottky diode) over a very wide range of current densities j is presented in Figure 3.1. Seven characteristic parts can be distinguished in this curve.

Part 1 is associated with conventional leakage current (see Section 2.2.2). Part 2 is associated with the avalanche multiplication phenomenon considered

39


Bias Vo (arb. units)

Fig. 3.1 diode) in a condition of static breakdown.

Qualitative current-voltage characteristic of a reverse-biased p - n junction (or Schottky

in the previous chapter. Part 3, an area of “microplasma breakdown” occurs between Parts 2 and 4 in

any semiconductor diode structures of large area (and even in diodes of small area if fabricated on the basis of relatively new semiconductor materials). In this part of the curve, avalanche breakdown occurs only at local points in the reverse-biased junction (the microplasma channels).

Part 4, representing homogeneous (“mature”) breakdown, is the most important and best-studied part of the current-voltage characteristic. Sometimes this part is regarded as a “breakdown” current-voltage characteristic itself, as it is this part of the curve that is used in most applications. Part 4 is characterized by a very sharp increase in j with growth in Vo and a small positive differential resistance

R d = -l which is the main parameter of this part of the I - V Characteristic. dV0 dj

Part 5 is characterized by a very sharp increase in Rd with further growth in VO. It is very often difficult to observe this part of the curve, because the appropriate range of Vo can be rather narrow.

Part 6 is the section of Negative Differential Resistance (NDR). To observe this part experimentally “point by point” it is necessary to use a circuit with a large load resistance Rl. In the circuit with a low load resistance the current density will “jump” from the point 5 to point 7 (Fig. 3.1). As the amplitude of such a jump can be very large (several orders of magnitude) “overheating” of the device is possible, leading to its destruction. The characteristic switching time from the initial state (point 5) to the “final” one (point 7) is approximately (2 + 3) t , = (2 + 3)L/v,. 1.e.

40

Static Avalanche Breakdown 961

for a device with a base width L = 5 p m , the switching time is about 5 x s. As in other systems with an NDR of the S-type (current increases as the bias decreases), the current filaments in devices can appear in this NDR part of the I - V characteristic. This filamentation increases the local current density, and consequently the “overheating” of the “hot points” in the structure.

In Part 7, with extremely high current densities, the differential resistance of the structure becomes positive again. This effect appears mainly as a result of saturation of the ionization rates ai and ,& in very high electric fields. Electron- hole scattering and the recombination of carriers can also contribute to this effect.

Below we will discuss the physical nature of each part of the I - V characteristic shown in Fig. 3.1 and the most critical parameters, which are important in various applications.

3.2.1 Microplasma breakdown

The phenomenon of microplasma breakdown was observed and studied for the first time in Si p - n junctions (for references, see Review [47]). Later, however, it was demonstrated that it is observed in reverse-biased junctions in any semiconductor materials: Ge [48], GaAs, Gap, ternary A3B5 compounds, SIC, GaN, etc.

This type of breakdown reveals usually the presence of any imperfections in the space-charge region of a reverse-biased junction, as an imperfection, especially when located close to a p - n junction, will causes a local increase in the field a t some point in the junction (Figure 3.2).

a b

Fig. 3.2 of conductivity u o and permittivity E [47]. a) particle of conductivity u >> un (metallic inclusion model). F,,, = 3Fo. b) dielectric particle (u = 0, E << E O ) (second phase model) F,,, cz 1.5Fo.

Qualitative electric field distribution for two small spherical particles located in a material

As seen, both, quasi-metal and dielectric particles cause an increase in the elec-

41


tric field. As a result, with the reverse bias increasing, the breakdown occurs not across the whole area of the junction but only at one local point where the field is at its maximum and the breakdown condition is satisfied at the smallest magnitude of the reverse bias Vo (known as the “first microplasma”). The presence of dislo- cations, micro-occurrences of %econd phase”, metallic or dielectric particles, etc., will cause breakdowns at local points.

As a rule, the beginning of a microplasma breakdown manifests itself in a series of the current pulses (Figure 3.3).

a

I c 0 t

b

I 0 t

C

0 f

Fig. 3.3 Qualitative picture of microplasma current pulses at different values of the bias Vo: v03 > V02 > Val. The “unstable region” for each microplasma does not as a rule exceed 0.2 - 2 V. The amplitude of the microplasma pulses depends very weakly on the bias Vo, but as the bias increases, the duration of the “switched-on” state increases monotonically and that of the “switched-off’’ state decreases.

At a given bias V O ~ < VO < VO, (see Fig. 3.4), one can observe current pulses of practically constant amplitude which lie in the range - 10 to - 200 pA depending on the junction. As a rule, the pulse amplitude is higher for high-voltage junctions.

The existence of a current pulse shows that the microplasma is “switched on”, i.e local breakdown appears at one point. Although the pulse amplitude remains practically constant as the bias increases, the duration of the “switched-on” state increases and that of the “switched-off” state decreases. At bias Vo4 = VOl + AV, the current is practically constant. The AV value is usually 0.2 + 2 V.

The probability of a microplasma being switched on is determined by the random arrival of free carriers (originating from the leakage current) in the region of the

42

Stat ic Avalanche Breakdown 963

n

d 1

v1 c .r(

d 8 W

Homogenius (mature) n

d 1

v1 c .r(

W Ls

Homogenius (mature) I

First microplasma

vO v04 Vi Vo (arb. units)

Fig. 3.4 Qualitative current-voltage characteristic for part of a microplasma breakdown. Vol is the bias at which the first microplasma appears, and it is switched on permanently at Vo = V04. The current-voltage characteristic of a switched-on microplasma is linear.

high field. The turn-off probability is merely the probability of such random current fluctuation] which reduces the number of carriers entering the microplasma channel to a value at which stable breakdown is no longer possible [49; 501.

At Vo > VO, the first microplasma has been switched on permanently, whereupon its current-voltage characteristic is usually linear (Figure 3.4). The characteristic value of the switched-on microplasma resistance R,, lies in a range from several dozen Ohms for relatively low voltage junctions to several kilo-Ohms for p n junctions with a high breakdown voltage.

With a further increase in the bias, the second microplasma can appear (Fig. 3.4), and so on . . . . In the part of the I - V characteristic between the first and second microplasmas, the differential resistance Rd = dV/dI is equal to Rmp, while between the second and third microplasmas dV/dI M Rm,/2 (the two microplasmas are “connected in parallel”) , etc. The transition to homogeneous ( “mature11) breakdown is usually characterized by a small, bias-independent Rd.

The regions of instability of the first and second and/or second and third microplasmas, etc., can very often “overlap” l i.e. the second microplasma appears a t a bias at which the first microplasma is still unstable. In this case the pulses from the microplasmas are overlaid and the current-time dependence can assume a fairly complicated form.

In the case of large imperfections] microplasma breakdown can begin at a much

43


lower bias than calculated magnitude of the homogeneous breakdown voltage Vi. In this case the device can be destroyed at a relatively low current, because the density of the current flowing through the first microplasma can reach the critical value even at small average current magnitudes.

On the other hand, in materials of good, modern quality (Si, Ge), the magnitude of the microplasma breakdown bias can be only a few percent smaller than Vi even in devices with a large operation area. In this case the microplasma processes are important only at relatively low current densities (“at the beginning” of breakdown), while at high current densities homogeneous (mature) breakdown is dominant.

It is worth noting that only microplasma-free reverse bias structures can be used with devices operating at very high current densities (IMPATT and TRAPATT diodes).

Detailed analyses of microplasma breakdown can be found in reviews [51; 521.

3.2.2 Homogeneous ( ccmature77) breakdown

As far as homogeneous breakdown (part 4 in Fig. 3.1) is concerned, the differential resistance R d = dVo/dj is determined mainly by three components:

where R, is the contact resistivity, Rth is the thermal component, R,, is the resistivity of the space-charge region.

3.2.2.1 Contact resistivity

Until now, the fabrication of ohmic contacts with low resistivity (low specific resistance rc) has been an art rather than a science. The most common used way of obtaining a low resistance ohmic contact is to create a layer of very high doping between the semiconductor and the metal (the nf layer for an n-type semiconductor and the pf layer for a p-type semiconductor). In this case the depletion layer of the appropriate Schottky barrier is very thin, and current transport through it is determined by tunnelling (field emission regime) [14; 53; 54; 551.

Generally speaking, T, decreases with the doping level N in the manner T , - 1/N (see Figure 3.5 where the data for GaAs are represented). Similar dependences are observed also for other semiconductors. Usually, the larger the band gap E,, the more difficult is to make a low-resistance contact. For n-Si with Nd M 10” cmP3, T, is about Ohm.cm2 (and increases linearly as N decreases).

For GaAs with just the same magnitude Nd, the value of T, is equal to T , M

lov5 Ohm.cm2, while for Sic the value of T, for middle-quality contacts ranges from to Ohm.cm2.

44


-2 I .

-3 n

r 4

Ej 6 -4 W

-6

-7

15 16 17 18 19 20 log Nd (cm-3)

Fig. 3.5 Specific contact resistance r c as a function of the doping level N of an active layer for n-type GaAs. The points represent experimental data from different papers, and the dotted line shows the predicted minimum specific contact resistance for n-GaAs [14].

3.2.2.2 Thermal resistance

As mentioned in Chapter 1 (see Fig. 1.13), the breakdown voltage generally increases with temperature, an effect which contributes to the increase in the differential resistance. In fact, the greater current density is, the higher is the temperature of the device. Hence it is necessary to apply a higher bias to support the same current density. As a result, the differential resistance r d = dVo/dj increases (Figure 3.6). The dashed lines in Fig. 3.6 represent isothermal current-voltage characteristics measured at different ambient temperatures. (These characteristics can be measured using short pulses).

Under direct current (d.c.) measurements, every point on the current-voltage characteristic corresponds to an appropriate temperature of the device, so that the higher the current density is, the greater will be the temperature of the device due to self-heating. It is seen that self-heating increases the differential resistance r d .

The temperature dependence of the breakdown voltage is defined by the temperature breakdown coeficient /3:

1 dV, p=- - V, dT

The dependence of /3 on the breakdown voltage for abrupt Si p+ - n junctions is shown in Figure 3.7. As seen, /3 increases in magnitude monotonically with Vi (cf. Fig. 1.13).

45


Voltage (arb. units) VO

Fig. 3.6 Qualitative breakdown current-voltage characteristics of a reversebiased barrier ( p n junction, Schottky diode, heterojunction, etc.). The dashed lines represent isothermal (pulse) current-voltage characteristics measured at different temperatures (TI < Tz < T3 < T4), and the solid line shows the d.c. I - V characteristic. The higher the current density is, the greater will be the temperature of the device due to self-heating.

X

t I l l I l l I t

102 1 o3 lo4 "0 '

0

E Breakdown voltage Vi ( V ) c-" Fig. 3.7 Dependence of p on the breakdown voltage Vi for abrupt Si p + - n junctions. The solid line represents the result of the calculation [56]. The experimental points are taken from papers [57]-[59].

46


It is clear that, besides the current density and breakdown voltage (power density), the level of self-heating is also conditioned by the heat sink. The operating temperature of the device at a given power density depends strictly on the thermal resistance between the device and the heat sink and on the material and size of the heat sink, etc. Hence, to calculate the thermal component of Rd, Rth, it is necessary as a rule to solve simultaneously the set of the equations describing the device and the set of thermal conductivity equations for the device and the heat sink.

Analytical solutions for many important thermal problems can be found in the encyclopaedic monograph [60].

In some cases the breakdown voltage Vi decreases with increasing temperature (negative temperature breakdown coeficient). As a rule, such a situation occurs as a result of the presence of deep levels in the forbidden gap of a semiconductor. When the temperature becomes high enough to cause the thermal ionization of these deep traps, the concentration of carriers in the base of the structure increases, and the breakdown voltage V, decreases accordingly.

Generally speaking, a negative temperature breakdown coefficient can appear in some semiconductors with a ‘hatural superlattice” in the conduction (or valence) band. One assumes that such a situation can occur in particular in the hexagonal polytypes of silicon carbide: 4H-SiC, 6H-Sic, and so on [61]-[63]. Regardless of the reasons for it, however, a negative temperature coefficient of breakdown can determine the appearance of a negative differential resistance part in the current-voltage breakdown characteristic (Figure 3.8). If a small load resistance Rl is connected in series with the sample, the current will increase unlimitedly at VO = V, . As a consequence, thermal instability and destruction of the device is not improbable. Many countries have regulations that prohibit the use of materials with a negative temperature breakdown coefficient for the fabrication of devices based on the effects of avalanche ionization and breakdown, especially for military applications.

3.2.2.3 Space-charge resistance

In a well-designed device with low resistive contacts and an effective heat sink, the main contribution to Rd comes as a rule from the intrinsic resistance of the space- charge region Rsc. This resistance appears as a result of space-charge alterations in response to free carriers generated by impact ionization. The idea of this effect can be clarified in the framework of the approximation known as the “infinite narrow breakdown region’’ [64].

Let us consider a reverse-biased junction with a homogeneous doping level across the base. Suppose that impact ionization occurs only in an extremely narrow region close to the junction (Figure 3.9). If the breakdown current is very small, the slope of the field in the base will be determined by the Poisson equation

47


Fig. 3.8 Qualitative breakdown current-voltage characteristics of a reverse-biased barrier for the case of a negative temperature coefficient of breakdown. The dashed lines represent isothermal (pulse) current-voltage characteristics (TI < TZ < T3 < T4), and the solid line shows the d.c. 1 - V characteristic. The negative differential resistance (NDR) appears as a result of self-heating of the sample.

where Nd = no is the concentration of shallow donors. With F,,, = Fi (Fig. 3.9), the width of a space-charge region is

P+ “ I n

- 0 wo WO’) X

(3.4)

Fig. 3.9 Field distributions across a reversebiased p f - n junction. The solid line corresponds to the field distribution at current density j = 0, and the dashed line represents the dependence F ( z ) at a relatively small current density j.

48


and the breakdown voltage V, is

Let us now assume that an appreciable breakdown current with density j flows through the diode but the breakdown still takes place only in an extremely narrow region (an approximation to an infinite narrow breakdown region). The holes created by the impact ionization move away towards a high-doped p f - emitter and have practically no effect on the field distribution in the emitter. Electrons travelling across the base can nevertheless essentially change the field distribution in the base.

At current density j , the concentration of free electrons travelling across the base n is n = j/ev,. These negatively charged electrons partly compensate for the positive charge of the ionized shallow donors, and the Poisson equation should be written as:

dF e j dx EEO ev,

Let us introduce the notation j~ = ev,Nd. Then - = -(1 - -), and

- = -(Nd - -)

dF eNd j dx EEO 3 N

J 1-- 3 N

j For j << j ~ , V ( j ) M K ( 1 + -) and the specific space-charge resistance T,, is 3 N

Since v, is inversely proportional to Nd (K - 1/Nd), and J” is directly proportional to Nd ( j ~ - Nd), r,, is inversely proportional to 1/Nj (rSc - l /Nj).

Taking (for si) E = 11.7, Nd = 1014 ~ m - ~ , and Fi = lo5 V/cm, we have V, 1300V, j~ = 160 A/cm2 and T,, = 8 Ohm.cm2. With Nd = l O I 7 cmP3 we have V, F lOV, J” = 1.6. lo5 A/cm2 and r,, E 6.2 . l o p 5 Ohm.cm2.

It is seen that the main contribution to Rd in high-voltage diodes can come from the resistance of the space-charge region. For low-voltage structures, special attention should be paid to the contact resistance R, and the thermal component Rth in order to reach the minimum value of R d .

At j - j ~ , we obtain from (3.7)

49


It is seen that in this case r,, increases monotonically with increasing current density (Part 5 of the current-voltage characteristic, see Fig. 3.1).

3.2.3 Negative di’erential resistance

3.2.3.1 Qualitative consideration

Formally speaking, it follows from Eq. (3.9) that r,, tends to infinity as j tends to j ~ . It is obvious, however, that the “narrow breakdown region” approximation fails when j + j ~ . Indeed, at j = j ~ , it follows from Eq. (3.6) that d F / d x = 0, and consequently the field F is distributed homogeneously along the base. This means in turn that avalanche ionization takes place not only in the narrow layer close to the p+ - n junction but in the whole base.

1.

2.

Two important conclusions can be reached from this simple consideration:

In the situation where j - j ~ , it is necessary to take into account the second boundary of the sample. At d F / d x + 0, W ( j ) --+ 00 Hence, for any length of sample, the region of the strong field reaches the second contact (Figure 3.10a). When avalanche ionization takes place in the whole base, the holes inevitably contribute to the field distribution along the base. The Poisson equation must now be written as follows:

(3.10)

One can show that the contribution of the holes to the field distribution along the base causes the emergence of a negative differential resistance (NDR) in the current-voltage characteristic (Part 6 in Fig. 3.1).

The simplest way of considering the physical nature of the NDR is to discuss the case of a very low-doped base where Nd = 0 (known as the p - i - n structure). In such a structure, J N = eNdus z 0, and NDR begins just at the beginning of the avalanche breakdown.

3.2.3.2

The Poisson equation for the p - j - n structure has an especially simple form:

The zero doping ( p - i - n) structure

j n (3.11)

Equation (3.11) takes into account the fact that electrons and holes move in the fields corresponding to the breakdown with the saturated velocities us, and u , ~ , respectively (see Chapter 1).

The concentration of electrons increases exponentially from left to right, and that of holes from right to left (see Fig. 3.10). At the left boundary of the sample ( x = 0, Figure 3.11), the total current is practically equal to the hole component:

d F e e -) _ - - - (P-n )=- (2L

da: EEO EEO evsp ev,,

50


- I I

X

X

Fig. 3.10 a) Field distribution along the base of the reversebiased p+ - n junction at j N j~ in the approximation of electron ionization only. The breakdown region ( F > Fi) extends across the whole base, reaching the opposite contact (n+). b) If both electrons and holes ionize, the electron concentration will increases from left to right and the hole concentration from right to left, on account of impact ionization. c) Field distribution along the base in the case where F > Fi. NDR appears on account of partial compensation of the electron space charge by holes.

Field and carrier distributions under conditions of breakdown at j N j,.

51


j E j , = epv,,. And at x = 0 we have:

(3.12)

can see that at x = 0 the derivative dF/dx increases in a manner directly proportional to the current density j . Note now that due to the very powerful exponential dependences of ail and pi on F , a very small increase in F, provides very large increase in j,, and accordingly, a very sharp increase in dFldx at x close to the boundary x = 0 (compare curves 1 and 2 in Figure 3.11). Roughly speaking, with approximately the same value of F,, we have a marked increase in the slope of the dependence F ( x ) .

Ep t i - base

Fig. 3.11 left boundary of a p - i - n structure for two values of the current density j (jl < j z ) .

Qualitative field distributions (solid lines) and hole distributions (dashed lines) at the

Just the same situation evidently prevails at the right boundary of the sample (Figure 3.12):

(3.13)

It is seen that the fields Fm close to the boundaries of the sample increase only slightly with increasing current, whereas the field in the middle part of the sample decreases markedly. As a result, the NDR appears.

Let us illustrate this conclusion by means of a simple analytical consideration [64]. Suppose that the bias VO 2 FiL is applied to a p - i - n structure. The electric field F is equal to Fi across the whole base, and a small breakdown current j o flows across the diode.

52


Let us now assume that a slightly larger current j flows across the diode: j = j o + Sj. The new value of the bias, Vo + SVo, corresponds to this new current j. To calculate the value and sign of 6V0, note that for every point on a sample one can write:

Fig. 3.12 Qualitative field distributions across the base of a p - i - n structure for two values

of the current density j 1 and j z (jl < j 2 ) . For j , , the bias V1 = J Fldx, and for j z , the bias

V2 = J Fzdx. It is seen that Vz < V1. The NDR therefore occurs (cf. Fig. 3.10).

L

0 L

0

L

0 Let us now recall that the main condition for breakdown, Jcui(F)da: = 1, must

be satisfied at a n y current densi ty j (see Eq. 2.8). Hence

and L / cui(Fi + 6 F ) d z = 1

0

Expanding cu in a Taylor series we have:

cui(F,+bF)"cu,(F,)+-bF+--(bF)2+.. da. 1 d2ai dF 2 dF2

(3.15)

(3.16)

(3.17)

53


Then

dai 1 d2ai [ai(Fi) + -6F + - - ( (sF)2]d~ = 1 s dF 2 dF2

0

(3.18)

L

0 However, J a i ( F i ) d x = 1, hence

GFdX = bVo 0

(3.19)

(3.20)

Hence

(3.21)

and

(3.22)

L The value of J ( 6 F ) 2 d x is always positive.

Taking the dependence ai (F) in the form ai = croexp(-Fo/F) (see expression ( 1 ~ 1 9 ) , it is easy to ascertain that a; = dai/dF is also positive for any F . It is also easy to see that the second derivative a: is positive if F < Fo/2. At F > F0/2, a: < 0. Hence one must conclude that the differential resistance of a p - i - n structure is negative (NDR) as long as Fm < F0/2.

Since the magnitude of Fm increases with current density (see Figs. 3.11, 3.12), the differential resistance becomes positive again when the current density is large enough and F, > Fo/2 (Part 7 in Fig. 3.1).

Such a result has a clear physical interpretation. Indeed, when F << Fo, the ionization rates cq and pi increase very sharply with increasing F . The very small increase in F,,, at the boundaries of the base causes a drastic increase in j, as a result of which the field in the middle of the base decreases appreciably (NDR) . Conversely, a t F, N FO the ionization rates cq and pi are rather weakly dependent on F . Hence, to achieve any noticeable increase in the current density j , it is necessary to increase Fma, very markedly at the boundaries of the base. In this

0

54


case, the decrease in the field in the middle of the base cannot compensate for the increase in F,,, , and NDR disappears (Figure 3.13).

’ I

Fig. 3.13 Qualitative breakdown current-voltage characteristic for p - i - T I structure. V, N F,L.

At p - n junctions with a doped base, just the same scenario of NDR appearance prevails at j 2 j ~ , when the concentration of free carriers is comparable to the doping level Nd and the electrons and holes are able to bring about an appreciable change in the field distribution (see expression (3.10)). It is easy to estimate that the threshold value for NDR, j N j~ = eNdv,, is N 160 A/cm2 for Nd = l O I 4 cmP3, and j~ N 1.6 x lo5 A/cm2 for Nd = loi7 ~ m - ~ .

3.2.3.3 Computer simulation

Steady state field distributions and current-voltage characteristics for a p f - n - n f diode structure under conditions of breakdown were calculated in Ref. [65] using the conventional set of equations (see Eqs. (1.25)-(1.28)) with appropriate donor and acceptor distributions Nd(5) and N a ( s ) in the n f and p+ regions of the structures. Realistic dependences of vn(F), v p ( F ) , a i ( F ) and ,&(F) were used for simulation purposes. It was mentioned that the results of the simulation were not sensitive to the form of the approximations vn(F) and v p ( F ) .

The results are presented in Figures 3.14-3.17. Figure 3.14 shows field distributions for a diode with a base length L M 7 ,urn and a base doping level Nd = 5 x cmP3 for three magnitudes of the breakdown current density j . At a relatively small current density, j = 80 A/cm2 (curve l), the space-charge region occupies only part of the base (the punch-through voltage V,, is larger than the breakdown voltage K). For curve 1, the field distribution is determined mainly by the density of ionized donors Nd (n, p << Nd in Eq. (3.10), compare with the solid line in Fig. 3.9), and as j increases ( j = 790 A/cm2, curve 2), the width of the space-charge region increases (compare with the dashed line in Fig. 3.9). Curve 3

55


4

0

Distance (pm)

Fig, 3.14 Field distributions across a reverse-biased pf - n - nf structure for three magnitudes of the breakdown current density j : 1- j = 80 A/cm2, 2- j = 790 A/cm2 and 3 - j = 7900 A/cm2. The dashed lines show the boundaries of the pf - n and n - n+ junctions. Nd = 5 x l O I 5 cmP3.

104

100 120 140 Bias (V)

Fig. 3.15 Current-voltage characteristics of a reverse biased p + - n - n+ structure with L = 7 pm (see Fig. 3.14) for two magnitudes of the doping level Nd: 1 - Nd = 5 x 1015 cmP3 (see associated field distributions in Fig.3.14) and 2 - Nd = 3 x l O I 5 ~ m - ~ .

corresponds to j = 7900 A/cm2. It is seen that multiplication is negligible in the right-hand part of the sample, where the negative charge of the free electrons compensates almost entirely the positive charge of the ionized donors. (As expected,

56


such a situation corresponds practically to the critical current density j~ = ev, Nd z 8000 A/cm2 for N d = 5 x l O I 5 ~ m - ~ . )

4

3 n

E Y % L - k

1

0

3

Distance (pm) 1 2 3 4 5 6 7

Fig. 3.16 Field distributions across a reverse-biased p + -n-n f structure for three magnitudes of the breakdown current density j: 1- j = 80 A/cm2, 2 - j = 7900 A/cm2 and 3 - j = 32000 A/cm2. N,j = 1 x 1015 crnp3.

140 148 156 Bias (V)

Fig. 3.17 Current-voltage characteristic of a reverse-biased p f - n - n+ structure with Nd = 1 x 1015 cm-3 (see associated field distributions in Fig. 3.16)

It is seen in Fig. 3.14 that an increase of two orders of magnitude in j is achieved

57


with a very small increase in the maximal electric field in the base. Curve 1 in Figure 3.15 shows the current-voltage characteristic calculated for

the structure in question. One can see that the differential resistance R d of the structure is relatively small at small values of j , but that it increases as j increases (part 5 in Fig. 3.1). As j increases further, NDR occurs.

Curve 2 in Fig. 3.15 is calculated for the structure shown in Fig. 3.14 but with a smaller donor concentration Nd = 3 x loi5 crnp3. As Nd decreases, the breakdown voltage V, increases and the threshold current density for NDR decreases, as expected. It is important to note that the voltage drop across the structures remains fairly large even at a very high current density j - lo5 A/cm2. This means that the NDR mechanism in question is not able to provide a deep modulation of the base resistance and small residual voltage across the structures.

The field distributions across the base for the same diode with L = 7 pm and a doping level Nd = 1 x 1015 cm-3 are shown in Figure 3.16. At such a small donor concentration the punch-through voltage V,O is much smaller than the breakdown voltage V,, and the situation is rather similar to that in the p - i - n structure (see Figure 3.12). The current-voltage characteristic of this diode is shown in Figure 3.17.

Since V,O << x, the space charge occupies the whole base at a bias that is much smaller than V,. As a result, the space charge cannot spread in the depth of the base as j increases, and the part of the current-voltage characteristic with a large differential resistance (part 5 in Fig. 3.1) is absent. One can see that the differential resistance of this structure is slightly negative even at a very small breakdown current density j. The transition to the part with a pronounced NDR occurs for this structure at j “N 2 x lo3 A/cm2. A simple estimate of the critical current density j~ for Nd = 1 x 1015 cmP3 (w 1600 A/cm2) agrees well with this calculated result.

The calculated field distributions are fairly similar to those predicted on the base of qualitative considerations. It is seen that the field at the boundaries of the structure increases as the current increases, conversely, the field in the depth of base decreases with increasing current, thus leading to NDR.

Note again that even at very high current densities, j 2 lo5 A/cm2, the residual voltage drop is fairly large (VO = 140 V). The difference between the maximal voltage drop (Vo = 158 V) and the voltage at j = lo5 A/cm2 is only N 10%.

3.2.4 Second part of the current-voltage characteristic, with positive diflerential resistance at very high current densities

Part 7 of the I-VO characteristic, involving positive differential resistance (PDR) at extremely high current densities (see Fig. 3.1), was calculated for the first time in Ref. [66].

It is worth noting that earlier calculations [67]-[69] did not recognise any part with positive differential resistance at high currents, but assumed NDR even at

58


extremely high current densities. As was shown in (661, the reason for this incor- rect result was the use of an erroneous approximation for the ionization rate field dependences, in the form ai = ezp(XF), which gave finite magnitudes for ai and pi at F = 0 and did not provide for saturation of ail pi in high electric fields. In practice, as has been demonstrated previously in the framework of the qualitative consideration (see Section 3.2.3.2), it is precisely the saturation of ai and pi in high electric fields that is mainly responsible for the appearance of PDR at high current densities.

In the analytical model developed in Ref. [66] only the saturation of ai and pi in high electric fields was taken into account as a reason for PDR. The numerical estimates obtained in the framework of this model show that the characteristic current density at which NDR shifts to PDR lies in the range lo7 to lo8 A/cm2.

Several other factors such as the recombination of carriers, electron-hole scattering, the doping profile, etc., can make a considerable contribution to the formation of the current-voltage characteristic at high current densities [70; 711.

The possible role of different recombination processes can be elucidated as follows. At every point on the steady-state current-voltage characteristic there is a balance between the generation of free carriers in the base of the structure and their disappearance due to various processes including removal by the electric field and different forms of recombination.

The characteristic time of carrier sweeping is equal to the transit time t , = L/v,. For L = 10 pm, t , M 10-l' s, while at L = 1 pm, t , M

Semiconductor materials with such small Shockley-Read (linear) recombination lifetime T values, and even much shorter ones (- 10-12-10-13 s), can be obtained using various technologies (low temperature growth of A3B5 compounds, irradia- tion by high energy particles, etc.). The use of such short-lifetime semiconductor materials is not unusual in modern fast photodiodes. If 7 5 t,, the contribution of linear recombination must be taken into account when calculating the current- voltage characteristics under breakdown conditions.

The contribution of Auger recombination is appreciable if the associated recombination lifetimes T,A and T ~ A (see Eq. 1.9) are comparable to t,: TA = 1/Cn2 N

L/v,. Or:

s.

(3.23)

With L = 100pm and C = 1.1 x cm6 s-l (Si), we have n 2 3 ~ l ' c m - ~ , and j = enu, - 5 x lo7 A/cm2. Such large magnitude of current density is very characteristic for PDR part [66; 70; 711.

The current-voltage characteristic of a reverse-biased p - i - n- diode in the region of large breakdown current densities is calculated taking into account linear and Auger recombination in Ref. [70] (Figure 3.18). As expected, the inclusion of

59


the recombination effects reduces the characteristic current density at which NDR becomes PDR . This density was found to be 1.3 x lo6 A/cm2 for the structure with L = 100pm (Fig. 3.18) in comparison with lo8 A/cm2 found in Ref. [66] for the structure with the same value of L but without taking into account the contribution of recombination processes.

Fig. 3.18 Current-voltage characteristic of a p - i - n diode at high current densities calculated by taking into account the saturation of oli and pi, linear recombination, and Auger recombination [70]. L = 100pm, C = cm6 s-', concentration of the recombination traps Nt = l O I 3 ~ r n - ~ , capture cross-section of the recombination trap 00 = cm-2.

As it was mentioned for the first time in Ref. [65], doping gradients close to p + and n f regions of the diode can remarkably affect current-voltage characteristics in the condition of the breakdown at high current densities. Detailed analytical and numerical studies of this effect were made in Ref. [71].

3.3 Avalanche suppressor diodes

Below we will discuss the main characteristics of two devices based on the properties of reverse biased p - n junctions in the breakdown regimes: Avalanche Suppressor Diodes (ASD) and IMPATT (Impact and Transient Time) diodes. The purpose of suppressor diode is to protect electrical circuits against overload.

3.3.1 Principle of operation

The principle of ASD operation can be illustrated easily using the simplest protection circuit (Figure 3.19) and idealized current-voltage characteristic of suppressor

60


diode.

Fig. 3.19 Principle of operation of a suppressor diode.

An “ideal” suppressor diode (SD) is characterized only by a single parameter Vi. If V < V,, the current flowing through the ideal SD will be zero, while at V > V, it will be Is0 = (V - V,)/Ro (assuming that the differential resistance of the ideal SD is zero). In the simplest case, the circuit to be protected by the SD may be thought of as an active resistor Rl placed in parallel with SD (Fig. 3.19). A resistor &(R << Rl) is connected in series with the parallel SD and R1. The bias for the normal regime of operation (quiescent voltage) Vn is selected to be somewhat less than V,, so that the ideal SD will not expend any power ( I s 0 = 0). The magnitude of V, is

(3.24)

where VO is an external bias. Redundant power consumption is equal to I 2 x Ro << 1’ x Ri.

If the external voltage Vo occasionally increases to a marked extent (overload regimes), then: (i) - without protection, this voltage will be applied to the load R1. (ii) - with protection from the ideal SD, the voltage applied to the load will

be V,, and the excess current Iez = Ro

As is clear from its principle of operation, the SD must provide extremely sharp current-bias dependence. There are three main physical mechanisms which make this possible: diffusion forward current in the p - n junction, tunneling, and avalanche breakdown.

will flow through the SD. Voverload - V,

61


1.

2.

3.

Accordingly, three main types of SD can be recognised:

Forward-biased p - n junctions (or Schottky diodes) with a high-doped base [72]. Such SDs provide a V, of about 0.5 + 1.2 V. A set of forward-biased Schottky diodes ( p - n junctions) connected in series can give the necessary protection voltage in a range N 0.5 to N 4 t 6 V. SDs based on the ”tunnel breakdown” principle (see [20]). The operation of these SDs is based on tunnelling in very high-doped reverse-biased p - n junctions. The operation voltage of a single element of this type is several volts (2 - 6 V) depending on the type of the semiconductor. A set of these elements can provide SDs with an operation bias of 30 - 60 V. Reverse-biased avalanche diodes can serve as SDs in the range 50 + 80 V to 5 + 6 kV. Modern technology allows Si p+ - n diodes to be designed and manufactured with operation reverse voltages of more than 15 kV, but such high-voltage structures are never used as ASDs, as it is much more convenient to connect several comparatively low-voltage structures in series to protect high-voltage circuits. The typical operation voltage of a single ASD lies in the range 30 V to 300 V.

3.3.2 Main parameters

Unlike an “ideal” SD, a real ASD can be characterized by several parameters: not only the most important parameter, V, , but also pre-breakdown current, differential resistance dV/dj in the “breakdown” part of the current-voltage characteristic, turn- on time and overload capability.

The breakdown voltage V,, for an abrupt p - n junction can be estimated as

v. - - (- + -) (see Section 1.3.1) and calculated for any doping profile by

means of Eqs. 1.25-1.28 with appropriate boundary conditions (see [65]). Pre-breakdown current. The current flowing through an “ideal” ASD at V < Vi

is zero, but in real reverse-biased ASDs the leakage current of the reverse-biased p - n junction constitutes a pre-breakdown current (see Section 2.2.2). This current is as a rule fairly small a t room temperature (typically ranging from to 1 0 - ~ A/cm2 for silicon reverse biased p - n junctions, for example), but a t elevated temperatures the leakage current can be an important source of energy losses in ASDs and must be taken into account to avoid self-heating of the device, especially in high-voltage operation.

The dzflerential resistance R d is one of the most important parameters of an ASD. As mentioned above (see Section 3.2.2), Rd is the sum of three components: contact, thermal, and space-charge resistance. The space-charge component r,, may be expressed by the formula (3.8). Taking Vi = ~~oF:/2eNd and j~ = ev,Nd,

&&& 1 1 2e Na Nd 2 -

62


we have

(3.25)

Comparing the magnitudes of rSc calculated according to (3.25) with the characteristic contact resistance values (see Fig. 3.5) one can conclude that contact resistance does not make any distinct contribution to the total differential resistance of a well-designed ASD.

The contribution of the thermal component r t h depends notably on the duration of the overload pulse t,. Depending on the application, the characteristic duration of an overload pulse will usually lie in the range from a fraction of a nanosecond (t, N 5 x lo-” s) to a few dozen milliseconds (t, N 10W2 s).

Device overheating can easily be estimated for a very short t , (t, 5 1ps) by assuming that all the energy in the overload pulse goes into device heating:

(3.26)

where j , is the maximal current density, V, is the operation (breakdown) voltage, CT is the specific heat, p is the density, WO is the width of space-charge region at given bias voltage V, (see formula (3.4)), d = ,& is the characteristic length of the heat diffusion in the pulse time t,, x = K / ~ C T is a thermal diffusivity, and K is the thermal conductivity. In the frame of this estimation one assumes that all the heat energy (per unit area) j , . V, . t, is released in the space charge region and extends for a length of d.

For relatively long pulses, the properties of the heat sink must be taken into account [GO].

Figure 3.20 illustrates the effect of the self-heating on the differential resistance of Si ASD. As seen in Figure 3.20, self-heating has a considerable effect on the differential resistance R d of a Si ASD even at t, = G p . Estimation according to (3.26) shows that at V, = 48.5 V, j , = 300 A/cm2, and t, = 6ps, the overheating is AT M 20 K. For these magnitudes of V, and j,, the overheating is really small (AT < 1 K) only for t , 5 500 ns.

Note that for any pulse duration, R d at first decreases with j growth, and then, with further increase in j (at j 2 GO + 80 A/cm2) Rd remains practically constant. This means that it is at j 2 60 -+ 80 A/cm2 that the whole diode area is included in the breakdown process (homogeneous breakdown). It is worth noting that even at modern levels of technology it is impossible to achieve homogeneous breakdown at a very small breakdown current density in diodes with a large operation area. On the other hand, as mentioned in Section 3.2.2, the “old” materials of semiconductor electronics (Si, Ge), can achieve a first microplasma appearance that is only a few percent smaller than the bias of homogeneous breakdown & even in devices with a very large operation area (2 10 cm2).

63


"

48 50 52 54 Bias (V)

Fig. 3.20 Experimental current-voltage characteristics of a Si ASD with V, = 48.5 V (5' 0.55 cm2) at different pulse duration t p . t , (ps): 1 - 6, 2 - 10, 3 - 15, 4 - 20, 5 - 60. Repetition frequency f = 100 Hz. [73]

=

Turn-on t ime . When an overload pulse is applied to the circuit, a stabilization time t,t elapses before the ASD is turned on and the load is protected. Generally speaking, tSt depends on the overload pulse amplitude and on the stabilizing resistance & (see Fig. 3.19). Hence, the calculation of t,t is a non-linear problem that can be solved precisely in the framework of ionization rate approximation, taking into account the real geometry and doping distribution along the ASD base. As a crude estimate, it is occasionally assumed that t,t - t , - W/v,, where the space- charge width W at the breakdown voltage V, is calculated from expression (3.4). With v, M lo7 cm/s, W M 2.6pm (for V, = 30 V) and W M 20pm (for V, = 300 V), we have t,t - t , - 2.6.10-l' s for an ASD with V, = 30 V, and t,t - 2 . 10-l' s for an ASD with V, = 300 V.

Very often, however, the real delay time required for protection of the circuit is limited by the RC time constant. For an ASD with an operation area S of about 1 cm2, the capacitance at V, = 30 V is about C - EEOSIW - 3.8. lo3 pF, and that a t V , = 300 V will be about 500 pF. The related RC time constants will then be - 2 x s (200 ns) and 2.5xlO-'s ( 2 5 ns) for ASDs with V, = 30 V and V, = 300 V, respectively (with Ro = 50 Ohm).

Overload capability. The ability of an ASD to protect the load is determined by the maximum admissible current and maximum overload power. The maximum current density j c , is limited by a sharp increase in R d with current density (part 5 in the I - V breakdown characteristic, Fig. 3.1). It can be estimated i ~ s j,, N (0.2 - 0 . 5 ) j ~ w (0.2 - 0.5)evSNd. Hence, for an ASD with V, M 30 V ( N d M ~ r n - ~ ) ; j c , is about (3 - 8) x lo3 A/cm2 and for V, M 300 V ( N d M ~ m - ~ ) , j,, is about

64


(300 - 800) A/cm2. The maximum overload power depends strictly on the duration t , of the over-

load pulse. For ASDs with homogeneous (or quasi-homogeneous) breakdown, the overload power can reach N 20 kW/cm2 at t, N s.

3.4 IMPATT diodes

IMPATT diodes are very important microwave devices that are widely used as generators in the frequency range N 30 to N 400 GHz. In fact, practically any diode in the homogeneous avalanche breakdown regime can operate as an IMPATT diode if the parasitic resistances are not too large. To achieve maximum efficiency and the highest operation frequency, however, the parameters of the diode must be optimized. The operating point of a diode in the IMPATT regime is located in part 4 of the current-voltage characteristic (Fig. 3.1), i.e. corresponding to homogeneous breakdown with a small positive differential resistance.

When the operating point is located in the part of the I - V characteristic with a negative differential resistance (Figure 3.21), the phase deviation between the a.c. voltage and the a.c. current, 6 at low frequencies is 7r. The device consumes a power of IoVo from the bias source and discharges an usable power of 1 /2 . IlVl into the external circuit. Such devices are sometimes used for amplification and generation (see [74]).

I0

vO 5 vb Bias V

I c Time t

I - - Time t

Fig. 3.21 phase shift between the a.c. voltage and ax . current is 7r.

When the operating point is located in the NDR part of the I - V characteristic, the

When the operating point is located in the part of the I - V characteristic with a positive differential resistance, the phase deviation between the a.c. voltage and a.c. current, 6 at low frequencies is zero. Then the diode functions as a passive

65


resistance and consumes power from the external a.c. circuit. However, as we will see, the real part of the total impedance R e ( 2 ) can become negative at an adequate high frequency w .

The first IMPATT structure was suggested by W. Read in 1958 [XI, and generation in an IMPATT regime was achieved in practice for the first time in 1959 [76]. In Read’s structure (Figure 3.22) breakdown occurs only in a narrow layer close to the reverse-biased p - n junction, but it has been shown in Ref. [77] that an IMPATT regime can also be achieved in other limiting cases of homogeneous breakdown (a p - i - n diode or Misawa’s diode). Since then it has been demonstrated that diodes with a more or less arbitrary doping profile can operate in an IMPATT regime. Read’s structures provide higher efficiency and larger negative resistance, but in a relatively narrow frequency range, whereas the negative resistance value of Misawa’s diode is relatively small, but it can be achieved in a wider frequency region.

Detailed analysis of the design and operation regimes of IMPATT diodes can be found in many books, selected chapters of books, and reviews (see, for example, [20; 64; 78; 791). Thus we restrict ourselves here to an analysis of the physical principle of operation and a discussion of some physical problems which appear when operating at a very high frequency.

3.4.1 Principle of operation

In any type of IMPATT diode two mechanisms contribute to the phase shift between the a.c. voltage and a.c. current: delay in avalanche multiplication and transit-time delay. Both are represented in the name of the diode: IMP - “IMPact” reminds us of the delay in avalanche multiplication and ATT - “And Transient Time” reminds us of the second delay mechanism. As an example, we will discuss here the contribution of these mechanisms to the operation of the Read diode (Fig. 3.22).

As seen in Fig. 3.22 a, the n+-n base of the Read structure, of length L, consists of two parts: the narrow avalanche region (path) close to the p + - n+ junction, of length W,, and the drift region, of length ( L - Wa). The electric field F in this path is too small to cause impact ionization, but it is larger than the characteristic field F,, which is the field of drift velocity saturation. It is worth noting that F, for electrons in Si, for example, can be estimated to be 5 - 10 kV/cm (see Fig.l.8), while the characteristic field of impact ionization Fi is about 200-500 kV/cm (see Fig. 1.11).

Holes created in the narrow avalanche region by impact ionization progress to the p + region very rapidly and do not affect the operation of the device. Electrons generated in the avalanche path go to the drift region.

Let us consider the diode when operating in a microwave circuit (resonator). Be- sides the direct bias Vo from an external source, an alternating voltage of amplitude VI is also applied to the device: V = VO + VI sin w (Figure 3.23 a).

66

i I I I I I I I I I I

I

I

L

I

0


/ n+colltact a

I ’ w X

- X

b

Fig. 3.22 structure. b) Field distribution F ( z ) at operating point I0 - VO.

Schematic diagram of the Read diode. a) Doping distribution along the p f - n f - n -

Let us neglect the hole multiplication in the avalanche region. The generation d n at of electrons in the avalanche region can then be described (see Eq. (1.29)) as: - N -

aiv,;n = gnn . The electron generat ion rate gn = aivsn instantly “tracks” the bias oscillation, but the avalanche current I,, which is proportional to n, (I , 0: n), does n o t follow the instant value of the voltage V . When the a.c. voltage reaches a maximum and begin to decrease, the avalanche current continues to rise because gn is still larger than at V = VO. The avalanche current reaches its peak when the ax . voltage is (approximately) equal to VO (Fig. 3.23). Hence the avalanche region acts as an effective inductance providing effective phase deviation B between the a.c. voltage and an avalanche current of approximately 7r/2 (Fig. 3.23).

This qualitative consideration can be illustrated by the following simplified example. Let us assume that electrons and holes differ only in their sign (wsn = vSp =

67


I I I I I I

- 2n 3,7c 4? at

I I I I b I I I

I I I I

Fig. 3.23 circuit:V = Vi f V1 (a), and the avalanche current through the diode (b).

Qualitative time diagrams for the voltage applied to an IMPATT diode in a radio

us, and ai = pi = a). Then, neglecting the diffusion terms in the equations for the electron and hole currents (Eq. (1.29)) in the avalanche region, we can write:

jn = enu,

j , = epu, (3.27)

(3.28)

(3.29)

We will also assume that even at a.c.

j = j , + j p (3.30)

Strictly speaking, at a.c. the displacement current must be taken into account in the equalities (3.28) and (3.29). It can be shown, however, that this simplification does not appreciably affect the final result [79].

Summarizing (3.28) and (3.29); integrating over the avalanche region from 2 = 0 to x = W,, and taking into account (3.30), we obtain:

(3.31)

68


where t , = Wa/v, is the carrier transit time through the avalanche region at saturation velocity.

At x = 0, the boundary condition can be taken in the form j M j , M 0 due to the very large multiplication factor in the avalanche region. Analogously, at z = W,, j M j , M 0. With these boundary conditions, we obtain from (3.31):

(3.32)

Note that with the cessation of multiplication (a = 0), the current falls with a time constant t , due to removal of the carrier from the avalanche path at a saturation velocity 21,.

Denoting & as the average magnitude of a over the avalanche region:

h = 1 Wa [ pd.1

we obtain:

dj 2 j d t t , - = -(&Wa - 1)

In a small signal approximation

(3.33)

where a‘ = d a / d F ;

SWa = 1 + W,~!$’~; le~~~; j = j o + j l e iwt ; F = Fo + FIeiwt (3.34)

Substituting (3.34) in (3.33), we obtain in a standard small signal approximation:

. 2cY’jOVl 31 = ~ i w t ,

(3.35)

where Vl = W,j, is the amplitude of the a.c. voltage applied to the avalanche region. It is obvious from (3.35) that the avalanche path acts as an inductance La(La 1/ j0 ) :

t , La = - 2joff’

(3.36)

A further shift between the voltage applied to the Read diode and the current comes from the transit time path. As seen in Fig. 3.23, the avalanche current I , at 6 > 7r/2 decreases very sharply to a negligible magnitude, but the current through the diode does n o t decrease (Figure 3.24).

The narrow electron “packet” generated in the avalanche path (0 - Wa), is “thrown” in the drift region (W, - L ) at 19 M 7r (see Fig. 3.23). If the electric field in the drift region F exceeds F, (even at a minimum magnitude VO - Vl), the

69


l a

I I I I

n; 2n; 3n; 4n; at I

b I

I I

c, I

0

cb 4

+ G

I L

n; 27c 37L 47L at

Fig. 3.24 Qualitative time diagrams for the voltage applied to the IMPATTdiode: I/ = VotV1 (a) and total current through the diode (b) (compare with Fig. 3.23). If the electric field F in the drift region exceeds F,, the total current 1, remains constant until the “electron packet” has moved along the drift region (Ramo-Shockley theorem). To provide this situation, the condition F > F, must be satisfied even at 19 = 3n/2 , when the voltage applied V reaches a minimum (V = Vo - V1).

total current through the diode will remain constant until the “electron packet” has moved along the drift region (Ramo-Shockley theorem [80; 811).

The statement which is known in the theory of electron tubes as the Ram0 theorem [8O] and in semiconductor device theory as the Shockley theorem [81] can in the simplest case be formulated as follows. Let a charged plane with the density of charge p of width d move between the plates of the plane capacitor with the velocity u. The distance between the plates is L , and they are connected through an ammeter. What is the ammeter reading I when the charged plane moves in the space between the plates of the capacitor? According to the Ramo-Shockley theorem, I = evN/L, where N = pSd is the total number of charged carriers (electrons) in the plane and S is the plate area [79]-[81].

Consequently, until the electron packet has moved along the drift region with constant velocity v,, the current through the diode will remain constant. If the IMPATT diode is designed to operate at a frequency f = 1/T, the optimal length of the drift region ( L - Wa) M L (W, << L ) can be found using the expression t M L/v, M T/2 (see Fig. 3.24), or

L x v,/2 f (3.37)

70


For a diode with such a length, the total phase shift between the voltage and the current will be close to 7r (Fig. 3.24). This means that a t a given frequency f, the real part of the impedance Re(2) will be negative (Figure 3.25). In the vicinity of this frequency (and its harmonics) an IMPATT diode can be used as a generator (and amplifier).

Fig. 3.25 Qualitative frequency dependence of the real part of the impedance Re(2) for an IMPATT diode. Re(2) is positive for d.c. (w = 0). The diode can be used as an active element in an electrical circuit in the vicinity of w1 = 2 s f l N TW,/L and (sometimes) close to the harmonics of w1.

Computer simulations confirm these qualitative considerations. The results of a simulation for a silicon Read IMPATT p + - n - n+ diode of length L M 6.5pm [82] are presented in Figure 3.26. The field (solid lines) and electron (dashed lines) distributions at different values of the phase angle 6 are shown in Figures 3.26 a- 3.26 d (see Fig. 3.24).

Although the amplitude of the voltage applied to the diode in Fig. 3.26 a is close to the maximum (6 M 7r/2), electrons are practically absent (and the current is close to its minimum value, Fig. 3.24). The avalanche current I , reaches its maximum at 6 x 7r (Fig. 3.26 b), whereupon the electron “packet” generated in the vicinity of the p + - n junction is “injected” into the drift region. The amplitude of the voltage applied in Fig. 3 . 2 6 ~ is close to the minimum (6 x 3 ~ / 2 ) , and the electron packet is positioned approximately in the middle of the drift region. It is also much smaller than in Fig. 3.26 b, because the packet is “spread” as it moves along the drift region due to the diffusion process. The situation in Figure 3.26 d corresponds approximately to a voltage amplitude 6 M 27r (see Fig. 3.24), whereupon the electron packet “disappears” in the n f region and the current decreases to its initial value.

It is clear from the principle of IMPATT diode operation that soft excitation is characteristic of this device. It means that the NDR is realized in this device even at very small amplitude of a.c. voltage, which means in turn that in an appropriate microwave circuit even the inevitable fluctuations are able to excite oscillations. The oscillations arising with a small a.c. amplitude bV1 will grow until their amplitude is limited by non-linear processes.

71


2

1

W 3 E 0 0 i vl 0 5 k

4

3

2

1

0

M

C -

-

-

-

1 2 3 4 5 6 7

Distance (pm)

Fig. 3.26 Electron (dashed lines) and field (solid lines) distributions along the n base of a Si p + - n - n+ Read diode at different magnitudes of the phase angle 19 = wt [82]. a) 19 % 7r/2, b) 19 = 7r, c) 19 = 37r/2, and d) 19 M 27r (cf. Fig. 3.24).

Some non-linear processes which are characteristic of IMPATT diodes can be discussed on the basis of a simple qualitative consideration. Let us suppose that the a.c. amplitude V1 is so large that at d FZ 3 ~ / 2 when the voltage applied V FZ Vo + Vlsinwt is close to its minimum value V M VO - Vl, the magnitude of the field in the drift region becomes less that the characteristic field of drift velocity saturation F, (Figure 3.27 a). According to the Ramo-Shockley theorem this means that the current flowing through the diode is not constant and coincides in phase with the voltage applied (Fig. 3.2713). It is clear that during this part of the period the diode acts as a non-linear (due to the non-linear v ( F ) dependence) but passive resistance. Hence this effect reduces NDR and detracts from the efficiency of the

72

Static Avalanche

V l a

Breakdown 993

b

I I I

I I

- wt ;7c ;27c ;37c 147c

I I

I

wt ‘0 r 7c 2.n 7c 4n

Fig. 3.27 Qualitative field distribution across IMPATT diode (a) and voltage and current waveforms (b) when field in the drift region at I9 x 3 n / 2 is less than F,. (a) Solid line represents F ( z ) distribution at V = Vo. Dashed line represents F ( z ) distribution at I9 z 3 n / 2 when V x Vo - Vl. (b) When F < F,, the drift velocity of electrons and, consequently, the total current oscillates in phase with voltage applied. As a result, the active losses increase in this part of the period.

IMPATT regime. If this complementary mechanism for the losses cannot stop further Vl growth,

then the field at the end of the drift region can become zero at 6 M 3n/2 (incomplete punch-through, Figure 3.28 a). As a result, three effects can affect the IMPATT operation. First of all, the effective length of the drift region evidently decreases. The transit time of the electron packet is equal now to ( L - S)/w, (instead of L/w, in the punch-through situation). At given circuit frequency fo, optimal phase relations between bias and current are deteriorated. Secondly, the part of the drift region of the length of S is now a parasitic “ohmic” resistor. Additional active resistor causes further increase of the losses. Third, it is worth noting that at complete punch- through, the total capacitance of the diode does not depend on bias. In incomplete punch-through (the situation in question), the parametric modulation of the capacitance appears at operation frequency. This results the parasitic modulation of oscillations [78].

On the other hand, increasing V1 causes some non-linear effects at 6 M ./a, where the bias voltage V is close to its maximum magnitude V + V1. The amplitude of the avalanche current 1, increases sharply with V, (see Fig. 3.23), and as a result, the average over a period current 10 increases monotonically with V1 (see Fig. 3.24), which in turn results in an autoshift of the operation point away from its optimal value.

73


F

1 - 0

Fig. 3.28 .9 zz 3 ~ 1 2 the field at the end of the drift region can become zero (incomplete punch-through).

Qualitative field distribution across an IMPATT diode at high a.c. amplitude V1. At

According to the Gauss theorem, an electron packet of charge Q travelling along a sample will reduce the field F2 behind itself and increases the field Fl ahead of itself Fz - FI = Q / E (Fig. 3.29). Thus, when the electron packet Q is generated

F

Fo

n-no I-; t

x

c 0 L x

Fig. 3.29 Illustration of the effect of the electron packet on the field distribution in the simplest case, that of an electron packet of charge Q travelling along the sample at a constant bias V . Fo = V / L , F2 - Fi = Q / E .

in the avalanche region, the field behind the packet will decreases, attenuating the avalanche process. At large magnitudes of Vl , the amplitude of the generated charge

74


Q can be large, and the decrease in the field behind the electron packet can be very considerable.

All the non-linear processes mentioned above can be taken into account and analysed in the framework of a conventional approach based on the set of equations (1.25)-(1.28) and the Poisson equation.

It is worth noting, however, that GaAs IMPATT diodes operate up to frequencies f > 120 GHz, while the high frequency limit of Si IMPATT diodes is about 400 GHz. As it follows from the expression L M vs /2 f established above that the length L of a diode with operation frequency f N lo1’ Hz is several tenths of a micron. In such short samples there are several important physical processes which are not covered by the conventional transport equations and should be discussed separately.

3.4.2 Some physical problems that arise at very high frequencies

Let us note first of all that the shorter L is, the larger must be the doping level Nd of the base of the IMPATT structures. For Read’s structure, for example, such a conclusion follows directly from the obvious condition that over a length W, << L the electric field F must fall from a magnitude of Fi to F, (F, << Fi, see Figs. 3.22 and 3.27). This statement must be easily generalized to any IMPATT structure, however [ 781.

cm, and E = 10 it is easy to see that the doping level Nd in the highly doped part of the base Wa (see Fig. 3.22) should be Nd x 3 x 10’’ cmP3. The breakdown voltage V, for such a high-doped p + - nt junction will be several volts (see Fig. 1.12). In the meantime, it is well-known that it is not avalanche breakdown that predominates in thin, very highly doped p + - n+ junctions but direct tunnelling from the valence band of the p + layer to the conduction band of the n+ layer (tunnelling breakdown). It is usually believed that tunnelling breakdown predominates a t V, 5 4E,/e, i.e. a t V, 2 4.4 V for Si and 5 5.6 V for GaAs, while in the range 4Eg/e 5 V, 5 6Eg/e both mechanisms, avalanche and tunnelling, contribute to the breakdown current [20]. In the case of tunnelling breakdown, phase shift between the voltage and tunnelling current is practically zero, and consequently the transition to tunnelling breakdown in very short (and therefore very highly doped) IMPATT structures will disrupt the generation.

The other restriction on the high frequency limit of IMPATT diodes can be derived from the evident condition that “broadening” of the electron packet during its transit along a drift region of length ( L - Wa) M L should be much less than the length L. Given a transit time t , M Llu , and a characteristic diffusion broadening L o M ( D . t , ) 1 / 2 , we have L >> L o , or L >> Dlu,. Taking into account expression (3.37) we obtain

For Read’s structure, with Fi M 5 x lo5 V/cm, Wa M

V2 f<<“ 2 0

(3.38)

75


A more precise estimate gives f i v;/167rD [83]. It should be emphasized in particular that neither v, nor D in expression (3.37)

can be considered conventional values for long samples. Indeed, to establish a balance between the energy taken from the field (Eq. (1.10)) and the energy released to the crystal lattice (Eq. ( l . l l ) ) , the electron (or hole) must experience not one act of scattering but several. Estimates show that several dozen collisions are required to reach a steady state. Meanwhile, the electron transit time in a high electric field in a very short sample may become so short that the electrons (or holes) will not even have time to experience any collisions during transit (ballistic transport [84]). In this limiting case, the drift velocity of the carrier will be determined merely by the difference in potentials (energy of the carrier) with regard to the band structure of the semiconductor. The calculated dependence of the drift velocity of an electron in GaAs on its energy in a “pure” ballistic regime is shown in Figure 3.30. As seen, the achievable maximum drift velocity in this idealized case can be as high as N 1.1 x lo8 cm/s, i.e. more than ten times larger than the conventional saturation velocity v, in GaAs in high electric fields (w lo7 cm/s).

0 0.1 0.2 0.3 0.4 0.5 0.6 Energy E (eV)

Fig. 3.30 ballistic regime. 300 K. Diffusion and reflection from the anode are neglected.

Calculated dependence of the drift velocity of an electron in GaAs on its energy in a

Another case emerges much more often in practice, however, that in which the carrier experiences just a few collisions during transit (overshoot). The distance dependences of the electron drift velocity Vd calculated for electrons injected at x = 0 to the corresponding electric field F in GaAs are shown in Figure 3.31.

At relatively large distances 0.5 < x < 1.5pm (depending on the magnitude of F ) , v d tends to its steady state magnitude v, after several dozen collisions. (It should be remembered that the steady-state drift velocity in GaAs decreases with a n increase in the field at F 2 Fth % 3 kV/cm, where Fth is the threshold field of the Gunn effect).

At small distances x 5 0.5 - 1 pm, however, the maximum achievable drift veloc-

76

6.0

4.0


a

2 “ 0 1 .o 2.0 2,

I I I I

1 .o 2.0 Distance x (pm)

Fig. 3.31 Dependences of the electron drift velocity vd on distance for electrons injected at 5 = 0 in a sample with a homogeneous field F distribution. GaAs. Monte-Carlo calculations [85]. a) T = 300 K, doping level Nd = l O I 7 cmP3, b) T = 77 K, Nd = 0 (pure sample with low doping level).

ity even at room temperature and with a relatively high doping level (Fig. 3.31 a) can be very large (overshoot). At F = 25 kV/cm, u d is approximately 5.5 x lo’ cm/s, i.e. more than 5 times higher than us under steady-state conditions. In pure samples of GaAs at 77 K (Fig. 3.31 b), the maximum value of U d is as large as 10 x lo7 cm/s (compare with the ballistic regime, Fig. 3.30).

The overshoot phenomenon provides a good illustration of the obvious fact discussed in Chapter 1 (see Fig. 1.14) that for a given electric field F , the drift velocity in extremely short diodes depends on the coordinate, because it takes a time of about 7, and a distance of about 7,ud to reach the steady-state drift velocity. This must be taken into account in precise calculations for high frequency devices, because the

77


total length of the active region L of such a device will be only a few tenths of a micron.

The diffusion coefficient D in expression (3.38) under non-stationary conditions is also dependent on time (and distance). Moreover, even in a steady state, the diffusion coefficient will depends on the magnitude of the electric field F. Under equilibrium ( F = 0) or quasi-equilibrium (low electric field) conditions, the diffusion coefficient of the electrons D, will be connected with the low-field mobility p, by the Einstein relation

Under non-equilibrium conditions, when the mean energy of the electrons < En > markedly exceeds the equilibrium energy Eo = 3/2 . lcT0 (TO is the lattice temperature, see Eqs. (l.ll), (1.12)), the Einstein relation will not valid, strictly speaking, but in spite of this, equation (3.39) is sometimes used to estimate the D, value if < En > and EO do not differ very greatly in magnitudes. In doing so, the “electron temperature” T, is determined from the calculated or measured mean energy of the electrons < En >= 312 ’ ICT,.

In strong electric fields, however, it is necessary to take into account the fact that D is a tensor and the magnitudes of its longitudinal component Dll and its transverse component D l depend on the electric field and, generally speaking, on the crystallographic direction. By way of an example, the dependences of the longitudinal electron (Fig. 3.32 a) and hole (Fig. 3.32 b) diffusion coefficients D in Si on the electric field are shown in Figure 3.32 for the case F//(lll) at T = 300 K and T = 77 K. As seen, the magnitudes of D, and D, decrease considerably as the field increases.

For very short time intervals t 5 1 - 2ps, the values of D can be even smaller than those observed in a steady state. The non-stationary diffusion coefficients Dll and D l can be determined as follows (see [87; 881):

I d 2 Dll = -- (< z > -(?J#) 2 dt (3.40)

D l = - - < y I d 2 > (3.41) 2 dt

At t 4 00, the magnitudes of D I I and DI determined by Eqs. (3.40) and (3.41) tend towards the steady state values of Dll and DI (at appropriate magnitudes of the field F ) .

Expressions (3.40) and (3.41) describe the space broadening (%pread”) of the electron packet along the field (Dll) and in the perpendicular direction (01). It is clear that in a ballistic regime, when all the carriers are moving along the field without any collisions, both Dll and D l are very small (Dill DI -+ 0 at t -+ 0), but with further movement the diffusion coefficients will increase due to scattering


' 101 102 103 104 lo5 a" Field F (V cm-1)

U Field F (V em-1)

Fig. 3.32 Field dependences of the longitudinal electron (a) and hole (b) diffusion coefficients in Si for P//(lll), Solid lines represent Monte-Carlo calculations. The experimental points were obtained using the time-of flight technique 1861.

acts, which randomize the carrier motion (overshoot regime, cf. Fig. 3.31). After the time interval each carrier (on average) experiences several dozen collisions, and the diffusion coefficients reach their steady-state magnitudes.

The time dependences of Vd and D must be taken into account in precise calculations for high-frequency devices [89].

79


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Chapter 4

Avalanche Injection

4.1 Introduction

The term L‘avalanche injection” was introduced by John Gunn in 1957 [go] to refer to a situation in which “the avalanche heh,aues as a copious source of electrons (holes) which are injected into the material. . . ’’ The term “copious” implies that the concentration of the free carriers created by avalanche breakdown exceeds the doping level in the semiconductor sample. In this case, as we have seen in Chapter 3, the space charge and electric field distribution are determined not by the fixed charge of the impurities, but by the charge of the free carriers (electrons and holes) (see Sections 3.2.3 and 3.2.4).

In reverse-biased diodes, free carriers can appear at a notable concentration only due to avalanche breakdown (Chapter 3). In this chapter we will consider the avalanche injection phenomenon in “semiconductor resistors” (i.e. slabs of the semiconductor with two ohmic contacts (n+ - n - n+ or p+ - p - p + ) ) and semiconductor transistors and thyristors, i.e. in the structures in which an appreciable concentration of the free carrier can be caused by injection. It can be injection from the contacts in n+ - n - nf or p+ - p - p+ structures, injection from the emitter in transistor structures, and injection from the gate or one of the emitters in thyristors. Free carriers (electrons or holes) injected from the outside into a high field region provide a reconstruction of the field domain such that breakdown occurs a t relatively small applied voltages, notably smaller than those at which the breakdown is possible without injection. Hence, in the cases under discussion, breakdown appears under the influence of free carriers.

4.2 Avalanche injection in n f - n - n+ ( p f - p - p + ) structures

Let us consider for the sake of definiteness a n+ - n - n+ structure. If the bias VO is applied to a dielectric bar of length L , the field F across the sample will simply be equal to Vo/L. For a semiconductor sample, however, the situation can be more complicated. To explain the point, let us start from the most illustrative

81


case, that in which the carrier drift velocity in the semiconductor decreases with field growth at high fields (negative differential conductivity; NDC). As mentioned in Chapter 3, such a situation occurs in GaAs, for example. In fact there are many semiconductors, such as InP, InSb, InAs [a], many A3B5 ternary and quaternary compounds [18], GaN, and (presumably) Sic [19], in which the electron drift velocity in a certain range of electric fields decreases with an increase in the electric field (Figure 4.1).

3

vth h - c?

E 7- - 2

3 .C s

8 1

6

Vth

0 0 c

G .-

I I I Fth I I c

.th F th 5 10 15 20

Field F (kVcm-1)

Fig. 4.1 Field dependencies of the electron drift velocity in GaAs, InP, and Ino.5Gw.5Sb. The electron drift velocity in a certain range of electric fields decreases with an increase in the electric field in many other semiconductors, such as InSb, InAs, GaN, S i c [2; 18; 191.

The current density

j = en(x)v [F(x ) ] = const (4.1)

does not depend on the coordinate x . (Here we neglect the diffusion current, for the sake of simplicity). The carrier distribution is determined by the Poisson equation

dF - e ( n - no) dx EEO --

and by the boundary condition. We consider the simplest and very frequent case F = 0 at x = 0 (injecting cathode contact) (Figure 4.2a).

82

Avalanche Inject ion 1003

a

0 L Cathode Anode

h

Fig. 4.2 Field (b) and carrier (c) profiles along a n+ - n - n f semiconductor slab (a) with an injecting cathode contact ( F = 0 at 5 = 0) at three magnitudes of bias Vo. 1 - Vo = VO, < &h = FthL; 2 - vo = vo2 > Vth; 3 - VO = v03 > &2.

Then the field distribution F ( z ) is defined by the integral

dF e (4.3)

83


which can easily be calculated numerically for a given v ( F ) dependence. Let us consider, however, the qualitative character of possible solutions at dif-

ferent magnitudes of the applied bias VO. If VO < &h = FthL, the field distribution is given by the “conventional” curve 1 in Fig. 4.2 b. The field is small near the cathode contact x = 0 and then increases to FO M Vol/L. Indeed, since the field is small a t x = 0, the velocity v ( x ) at this point is also very small. Hence, to support the given current density j , the magnitude of n must be very large. Accordingly, dF/dx in the Poisson equation is very high. At the next point to the right, the value of F is somewhat higher, the magnitude of v ( x ) is some higher as well, and n is smaller. Accordingly, dF/dx in the Poisson equation is also smaller, and so on. The F ( x ) curve becomes progressively flatter with the advance from cathode to anode.

In this case, moving to the right from the cathode, we inevitably reach a point where F = Fth. Beyond this point any increase in the electric field leads to a decrease in the electron velocity w (see Fig. 4.1). Hence, the electron concentration n has to increase to provide a constant current density along the sample. Now the higher the magnitude of F , the smaller is w, the higher is n and the larger is dF/dx. Accordingly, the F ( x ) curve becomes steeper with any advance from the cathode to the anode (curve 2 in Fig. 4.2 b).

With v& > v02 > Kh, the point of intersection F = Fth is positioned closer to the cathode. Accordingly, the value of n is higher a t every point on the n ( x ) profile (compare curves 2 and 3 in Fig. 4.2 c) and the dependence F ( x ) becomes steeper.

It is worth noting certain important points. First, a t VO > &h, there is no point in the structure where n = no (n > no everywhere), i.e. this is a “space-charged limited (SCL)“ regime of operation [91].

Secondly, as the applied voltage increases (at VO > &h), the field F reaches Fth closer to the cathode, where n is larger (compare curves 2 and 3 in Fig. 4.2 c). The drift velocity a t this point is equal to the Wth (see Fig. 4.1). Hence one must conclude that the current density increases with voltage increase (in spite of NDC). The decrease in drift velocity in the part of the sample where F 2 Fth is compensated for by the growth in electron concentration. Note that the electrons are injected in the sample from the cathode.

Third, it is obviously that the field at the anode in a SCL regime reaches the breakdown value Fi at a smaller magnitude of VO than in the case where the field is homogeneously distributed along the no layer (base).

In the case when the drift velocity is saturated at high fields (Si, Ge), the SCL regime inevitably occurs at an applied bias VO > V,, where V, = F,L (Figure 4.3). In such a case, moving from the cathode (where F = 0) to the right, we encounter a point at which F = F,.

Beyond this point the field increases linearly towards the anode. Indeed, since v = v, a t F 2 F,, we can combine the equation for the current density j = en(x)w,

The other situation applies to the case in which v, = Vo2 > Kh.

84

Avalanche Injection 1005

‘ S

Electric field F

Fig. 4.3 (compare with Fig. 1.8).

Qualitative field dependence of the carrier (electron and hole) drift velocity in Si and Ge

and the Poisson equation to obtain:

Equation (4.4) determines the linear growth in the electric field from left to right, provided that the current density j exceeds the critical value j~ = enov,. A more detailed analysis of this case can be found in Ref. [92], for example.

The magnitude of F, can be easily estimated in the frame of a Ywo-piece approximation”, in which it is assumed that the v ( F ) dependence is linear (v = p F ) at v 5 v, (i. e. at F 5 F,), and v = v, at F 2 F,. Hence, F, M v,/p. In low-doped silicon, the low field mobilities p n and p p are 1400 cm2/Vs and 450 cm2/Vs for electrons and holes, respectively, while in low-doped germanium pn = 3900 cm2/Vs, and p p = 1900 cm2/Vs. The v, in all these cases can be estimated to be v, M lo7 cm/s. Hence F, M 7 kV/cm for n-Si, F, M 22 kV/cm for p-Si, F, M 2.5 kV/cm for n-Ge and F, M 5 kV/cm for p-Ge. Mobility decreases with doping level growth, and F, increases accordingly. In all situations that are of practical importance, however, F, is much smaller than Fi.

Finally, a SCL regime is realized even in the case of linear dependence of drift velocity on the field (see detailed analyses in Ref. [91]). Let us suppose that Ohm’s low is valid, i.e. the expression (4.1) has the simplest form: j = e p n ( z ) F ( z ) = const. It is seen from Fig. 4.4 that a t a given current density j , the concentration n reaches its equilibrium value of no at a characteristic distance 1. It is easy to show that the value of 1 increases monotonically with the growth in j (compare curves 1 and 2 in Fig. 4.4). Hence, if j (or the bias VO) is large enough, 1 becomes larger than the length of the sample L. In this case, even at x = L, i.e. a t the anode of the sample, dF/dx > 0, and n > no (curve 4 in Fig. 4.4). Just as in the case shown in Figs. 4.2a, b (curves 2 and 3), the whole base is negatively charged and the field F increases monotonically towards the anode (SCL regime). (It is worth

85


noting that, on the whole, the sample is neutral: the negative charge accumulated in the base is compensated for by an equal positive charge in the right n+ region, where the field F decreases sharply, as shown in Fig. 4.4).

Fig. 4.4 Field (a) and carrier (b) distributions along a n+ - 7~ - n f semiconductor resistance with injecting cathode contact at different current densities (different bias VO). Drift velocity v is proportional to the field: w = p F ; j , > j , > j z > j ~ .

In the case of the linear dependence u versus F , the SCL regime occurs when the transit time tt = L / p F becomes less than the differential Maxwellian time of dielectric relaxation 7, = &&,/a = &Eo/enop [91]:

(4.5)

where V ~ C , is the bias at which the SCL regime occurs:

86


Comparing Eq. (4.6) with the condition of SCL due to velocity saturation V, > F,L, one can see that the SCL regime occurs under conditions of a linear w(F) dependence if Vsct < V,, or:

(4.7)

For silicon of the n type, for example (E = 11.7, F, = 7 kV/cm), the condition (4.7) is satisfied for noL < 4.5 x lo1’ cm-’.

Independently of the type of SCL regime (NDC, velocity saturation, SCL at linear w(F) dependence), it is easy to see that impact ionization at the anode contact can occur at a relatively low bias Vo.

The main characteristics of breakdown process in silicon n f - n - n+ structures are investigated theoretically and experimentally in Ref. [%I. The current-voltage characteristics calculated for samples with a length L of 10 pm, 20 pm and 30 pm and a doping level Nd of 1014 cm-3 are presented in Figure 4.5.

Bias (V)

Fig. 4.5 nesses and doping concentrations. 300K [93].

Calculated current-voltage characteristics of silicon nf - n - nf slabs of different thick-

For a sample with L = 10 pm, the current-voltage characteristics have also been calculated for Nd = 1015 ~ r n - ~ and Nd = 1OI6 ~ r n - ~ . The appropriate “three- piece approximations” for the electron and hole velocity-field characteristics w, ( F ) and wp(F) have been used. The boundary conditions corresponded to the injecting

87


contact a t the cathode ( F = 0 at x = 0) and the n+ - n junction at the anode ( p = 0 at x = L ) [MI.

For silicon with a Nd of 1014 ~ m - ~ , the characteristic breakdown field Fi = 2 x lo5 V/cm (see Fig. 1.11). Hence, one can expect the breakdown voltage V, for dielectric slabs with L of 10 pm, 20 pm and 30 pm and a doping level Nd of 1014 cmP3 to be 200 V, 300 V and 600 V, respectively. Meanwhile, as seen in Fig. 4.5, not only the beginning of the breakdown, but even the threshold of the NDR parts of the I - V characteristics are approximately 115 V, 205 V, and 275 V for the 10 pm, 20 pm, and 30 pm slabs, respectively,

The calculated field distributions across the slab with L = 10pm and no = 1015 cm-3 [93] are shown in Figure 4.6.

Fig. 4.6 Field distributions along the sample at different current densities. L = 10pm, n o = 1015 cm-3; Injecting ' . cathode contact. j (A/cm2): 1 - 2 x lo3; 2 - 3 x lo3; 3 - 10 X lo3; 4 - 1 x 105; 5 - 5 x 105 [931.

Let us note, first of all, that for no = 1015 crnp3, the characteristic current density j~ = eNdv, = 1600 A/cm2. (As seen in Fig. 4.5, the form of the current- voltage characteristic changes markedly at j M j , M 1600 A/cm2). In Fig. 4.6, the current density even for the curve 1 (j = 2000 A/cm2) exceeds the j~ magnitude (SCL regime) and the field increases linearly towards the anode in accordance with Eq. (4.4). When no breakdown exists (curves 1 and 2), dF/dx is proportional to the current density. Curve 3 (j = lo4 A/cm2) corresponds to the beginning of the

88


breakdown: in the major part of the sample the field increases linearly toward the anode, although dF/dx increases in the region near to the anode due to electron multiplication at the anode contact. With a further current increase (curves 4 and 5), the field increases sharply towards the anode and decreases in the middle part of the sample (NDR regime, compare with Fig. 4.5).

The calculated current-voltage dependencies of two Si samples (no = 1.65 x ~ m - ~ , L =20 pm and 30 pm) are compared with related experimental data in

Figure 4.7.

0.8

0.6 h

Q

= 0.4 2

Ilh

0.2

v 4 *

L 3

I I I I 7

0 50 100 150 200 250 300

Bias (V)

Fig. 4.7 of n+ - n - n+ Si samples of no = 1.65 x 1015 cm-3 and L = 20 pm and 30 pm [93].

Experimental (solid lines) and calculated (dashed lines) current-voltage characteristics

As seen, the experimental I-V characteristics agree very well with the calculated dependencies at current values I smaller than the threshold current of NDR Ith. At I > Ith, a qualitative distinction is observed. The calculated current-voltage characteristics demonstrate a very slow decrease in the residual voltage with current increase. In the experimental dependencies a voltage “jump” is observed to rather low magnitudes of the residual bias (S-type switching).

The authors of Ref. [93] suppose that the current filament with very small area and accordingly an extremely high current density j of about lo6 - lo7 A/cm2 appears in the sample at I > Ith. As we will see below, rather low magnitudes for the residual voltage drop can be observed in structures with a high field domain at the anode under conditions of intensive impact ionization at j - l o6 - lo7 A/cm2.

An alternative treatment of the experimental results has been suggested in

There are two possible explanations for this disagreement.

89


Refs. [66; 92; 95; 961, in the framework of which it has been shown that the current distribution at 1 > 1th can acquire the conical form (Figure 4.8). Given a conical

Fig. 4.8 Conical current distribution after S-type switching under breakdown conditions [GG].

n d

Bias (V) Bias (V)

Fig. 4.9 with one dot contact (conical current distribution) [97].

Current-voltage characteristics of germanium (a) and silicon (b) n f - n - n+ structures

90


distribution of the current lines, the residual voltage after switching can be much less than in the case of parallel current lines. Where one-dimensional numerical calculations for a Si p - i - n structure of L = lop2 cm, for example, give a threshold voltage for S-type switching of &h = 1300 V and a minimum residual voltage drop V, = 540 V (at j - 10' A/cm2), the estimates for a conical current distribution give V, N 45 V (with the same I& magnitude). Experimental results demonstrate directly the possibility of a conical current distribution after S-type switching [95; 961. Analytical estimates for the case of a conical current line distribution can be found in Ref. [92].

A very low residual voltage drop after S-type switching was also observed experimentally with a conical current distribution in n - i - n (nf - n - n+) and p - i - n configurations with one dot contact in Si, Ge, and GaAs structures (Figure 4.9) [97; 981.

4.3 Avalanche injection in bipolar transistors

4.3.1 Introduction

The phenomenon of avalanche injection in bipolar junction transistors (BJTs) is extremely important in practice and very interesting from a physical point of view. It is this that is responsible for the destructive second (thermal) breakdown in all types of BJT, and it has also been shown to be responsible for very effective and very fast switching of avalanche bipolar transistors (ABTs).

Although avalanche injection can be highly essential in any type of BJT, this phenomenon is especially important in high voltage ones. Indeed, to block the bias of several hundreds (or even a couple of thousand) volts, it is necessary to have a depleted space-charge region WO of several dozens microns in the structure (see Fig. 3.9 and Eqs. (3.4) and (3.5)). The base layer, which is the lowest-doped layer in a conventional BJT, should be thin enough to maintain the current gain p at an acceptable magnitude. The maximum possible width of the base is usually close to l p m . On the other hand, to block the bias V = 1000 V, the width of the appropriate depleted layer W in a Si structure must be W > WO = 2V/Fi M 70pm (with Fi z 3 x lo5 V/cm). It is for this reason that the collector layer is the lowest-doped region in a high-voltage BJT (Figure 4.10).

As we know (see Section 3.2.3), appreciable reconstruction of the space-charge region by free carriers occurs if the current density j is equal to or larger than the critical current value j~ = eNv,. As the current density (in a one-dimensional approximation) is the same in all parts of the structure, and the collector is the lowest-doped region ( N , << Nb << N e ) , reconstruction of the field in the collector space-charge region of a high voltage BJT by free carriers begins at current densities at which the field distributions in the base and the emitter remain practically unchanged. (Here N,, Nb and N, are the doping levels in the collector, base and

91


n P

i c emitter base collector Wo x

Fig. 4.10 region is the collector.

Qualitative field distribution along a high-voltage BJT structure, The lowest-doped

emitter, respectively). If the density of the current injected from the emitter to the collector essen-

tially exceeds the critical current density for the collector j ~ , = eNcv,, breakdown and avalanche injection can occur even in a case where the initial collector bias VO = F m W 0 / 2 (see Fig. 4.10) is considerably smaller than the breakdown voltage of the collector V,. The higher VO is, however, the easier it is to achieve the conditions required for avalanche injection. Therefore this phenomenon manifests itself especially obviously in ABTs, because the magnitude of VO in these is always very close to the V, value.

As a first step, we consider in this section the functioning of ABTs in the well- studied conventional regime of operation (see, for example, [21; 991). In this regime the current density j < j ~ ~ , and avalanche injection does not apply.

4.3.2

The operation of an ABT in the conventional regime is based on two phenomena:

the difference in the breakdown voltages of a BJT between the common-base and

the dependence of the BJT gain coefficient 010 on current density

Avalanche transistor: conventional regime of operation

common-emitter configurations, and

4.3.2.1 Difference in breakdown voltages of a BJT between the common- base and common-emitter configurations

A schematic diagram of a common-base circuit configuration for a n - p - n bipolar junction transistor (BJT) is presented in Figure 4.11. The reverse collector bias V, is applied to the collector p - n junction. Even though V, is much smaller than the breakdown voltage V,, the leakage current I,o flows through the collector junction (see Eqs. (2 .9)-(2.14)) . Neglecting recombination in a reverse-bias collector junction, one can say that with given leakage current Ice, every second Nh holes

92


( N h = Ico/e) come from the base-collector junction to the base of BJT

emitter n+ n- collector

Fig. 4.11 junction transistor (BJT).

Schematic representation of a common-base circuit configuration for a n - p - n bipolar

As a first step, let us discuss the “fate” of these holes in the case of an open emitter. It is obvious that in this case the collector-base junction operates as a conventional reverse-biased p - n junction: all the holes generated at the collector- base junction leave the base and go to the base-collector bias source.

We should note here that if there is an emitter-base source the situation will remain unchanged. Indeed, to leave the base through the emitter-base source, the holes must overcome the emitter-base potential barrier, whereas there is no potential barrier for holes leaving the base through the base-collector bias source. Hence, practically all the holes will leave the base by this latter route.

Thus the collector-base junction always operates as a conventional reverse-biased p - n junction in the common-base circuit configuration. Hence, the breakdown voltage yb = V, for this circuit configuration is consequently reached when multiplication factor for this junction, Mcb, tends to infinity (Mcb 4 03, compare with

A schematic diagram of a common-emitter circuit configuration for the same Eqs. (2.7)-( 2.8)).

n - p - n type of BJT is shown in Figure 4.12.

collector + n

Fig. 4.12 Schematic presentation of a common-emitter circuit configuration for a n - p - n BJT.

93


The classical collector-emitter breakdown is achieved in the common-emitter configuration with an open base (Fig. 4.12). As one can see, holes which come from a reverse biased collector-base junction to the base cannot leave the base in this case. The base contact is open and the emitter and collector junctions constitute very high potential barriers for the holes. As a consequence, all the holes have to recombine with electrons entering the base from the forward-biased emitter-base junction.

However, according to the fundamentals of BJT operation, to “kill” one hole,

electrons to the base, because for every 1

the emitter should send not one but - 1 - Qo

N electrons injected from the emitter to the base, QON electrons will pass through the base without recombination and will be captured by the reverse-biased collector- base junction and just (1 - ao)N electrons entering the base recombine with holes there.

If there is no multiplication at the collector-base junction and a conventional leakage current I,o flows through the collector-base junction, the well-known “balance” equation for BJTs with an open base will assume the form:

I , = I , (4.8)

or

Then

L o I , = - 1 - Qo

(4.10)

As one can see, the “initial” leakage current of the collector-base junction is ampli-

fied by a factor of - due to the inner positive feedback of the BJT.

Just the same situation takes place when holes are generated at the collector p-n. junction not only due to the thermal generation but also on account of avalanche multiplication. In the presence of avalanche multiplication at the collector p - n. junction] the “balance” equation for a BJT with an open base obviously takes the form:

1 - a0

I e = Ic = Mca(Ico + QoI,) (4.11)

or

(4.12)

L o where Ieo = - 1 - Q o

94


and

(4.13)

is the multiplication factor Mc, of a common-emitter configuration.

configuration (Mce 4 co) occurs at One can see from Eq. (4.13) that the breakdown in the common-emitter circuit

(4.14)

Since the characteristic value of a0 in a BJTs lies in the range 0.9-0.99, the condition for breakdown in a common-emitter configuration corresponds to characteristic Mcb values of 1.1-1.01. It is evident that breakdown in a common-emitter configuration is achieved at a collector bias V,.., which is appreciably smaller than the breakdown voltage in a common-base configuration vb = V,. (As mentioned above, the breakdown voltage in a common-base configuration corresponds to the condition Mcb + co and coincides entirely with the breakdown voltage of a reverse- biased collector-base p - n junction).

The multiplication factor Mcb is very often approximated using the following empirical expression:

1 Mcb =

(4.15)

The magnitude of m depends on the doping profile in the collector region and on the temperature. For Si BJTs m usually lies in the range between 2 and 5.

For the breakdown voltage in a common-emitter configuration we obtain:

(4.16)

Or

vy E V,(1 - a o ) l / m (4.17)

As the value (YO is close to unity, we have: (1 - 0 0 ) << 1 and V,.. << V,. The multiplication factors Mcb and Mc, calculated for a BJT with a0 = 0.99

are shown in Figure 4.13 for common-base and common-emitter configurations [21]. As one can see, the V,/v,"" ratio can be really large.

Let us now consider a situation in which the emitter and base terminals are connected by an external resistor Rb (Figure 4.14). It is clear that in the limiting case Rb 4 co (i.e. Rb >> Rb,), where Rbe is the resistance of base-emitter junction, such a situation corresponds to a common-emitter configuration with an open base. The breakdown voltage in this case is equal to V y .

95


50

40 53 0 0 c

30 6 2 -& 20

.- c

.-

.- e c

10 5

0

~ = 0 . 9 9

I Mce

0

Reverse voltage (V)

Fig. 4.13 figurations calculated for a BJT with cyo = 0.99 [21].

Multiplication factors h f c b and M,, for common-base and common-emitter circuit con-

Fig. 4.14 and emitter terminals.

Common-emitter circuit configuration with an external resistor Rb between the base

In the opposite limiting case, Rb << Rbe, the situation is quite similar to that of a common-base circuit configuration, for as in the latter, all the holes generated in the collector-base junction have the opportunity to leave the base and go to the source, bypassing the base-emitter junction. Accordingly, the breakdown voltage in this case is equal to V,.

The solid lines in Figure 4.15 show the current-voltage characteristics of the

96


Fig. 4.15 Breakdown voltages of an avalanche transistor (FMMT-417, ZETEX Semiconductors) [loo]. Solid lines show experimental current-voltage characteristics measured with a curve tracer and dashed line represents schematically the current voltage characteristic of an ABT with a variable resistor Rb placed between the emitter and base terminals.

commercial avalanche transistor FMMT-417 (ZETEX Semiconductors) in breakdown regime for two circuit configurations, as measured with a curve tracer [loo]. As one can see, the breakdown voltage in common-emitter configuration with an open base is 170 V, while that in common-base configuration (or common-emitter configuration with Rb = 0) is twice as large, 340 v.

Let us now imagine that the emitter and base terminals are connected by a variable resistor Rb. If R b is small ( R b << Rbe), the breakdown voltage will be equal to vb = V,.

Assuming that V,, is practically equal to V,, let us progressively increase Rb.

It is easy to see that the breakdown voltage will decrease monotonically with the growth in Rb. Indeed, the larger the resistor Rb is, the smaller will be the proportion of the holes injected from the collector to the base that can leave the base through the resistor Rb, and the larger the proportion that recombine into the base. Thus an increase in Rb is equivalent to an increase in QO, and in accordance with expression (4.17), Ye decreases. The collector current-voltage characteristic will thus follow the dashed line in Fig. 4.15, tending towards the breakdown characteristic in the common-emitter configuration y.

It is also possible to interpret the dependence represented by the dashed line in Fig. 4.15 in a different way. Let the emitter and base terminals be connected by a fixed resistor R b . The differential resistance of the emitter-base p - n junction

97

1018 Breakdown Phenomena an Semiconductors and Semiconductor Deuice9

R b e = dvbe/dIbe will then decrease with growth in the forward emitter current:

Ibe Io[eXP(qVbe/kT) - 11 10 exp(qVbe/kT) (4.18)

and

R b e = l /(drbe/dvbe) o( exP(-qvbe/kT) 0: 1 / I b e (4.19)

The Rbe is large at small current values, the condition Rb << Rbe is satisfied. At V,, = vb = V,, breakdown occurs, and the current increases very sharply with further growth in Vce. As the current increases, the R b e decreases, according to Eq. (4.19). The &,/Rbe ratio increases monotonically, and the collector current- voltage characteristic again follows the dashed line 2 in Fig. 4.15.

Last but not the least, there is an even more fundamental reason for the appearance of an S-shaped fo rm for the collector current-voltage characteristic under conditions of multiplication in the collector of a BJT: the dependence of the bipolar transistor gain coefficient (YO on the current.

4.3.2.2 Dependence of the bipolar transistor gain coefficient QO o n current densi ty

The qualitative dependence of QO on the current density j passing through a BJT is shown in Figure 4.16.

0.0 I , I I I I

10-2 10-1 100 101 102 103 104 105 logj (arb. units)

Fig. 4.16 Qualitative dependence of the gain coefficient ( Y O in a BJT on the current density j .

The ao(j) dependencies of this kind are characteristic of any type of BJTs. As seen, there is fairly wide range of moderate magnitudes of j for which 010 is at its maximum and is practically independent of j . This value of (YO is usually referred

98


to in the handbooks as the "BJT gain coefficient". The a0 tends towards a very small quantity at small magnitudes of j ( j tends to zero), however, and decreases appreciably at high magnitudes of j.

The common-base current gain CYO can be represented as a product of two factors [14; 201:

Qo = aT ' 7, (4.20)

where CYT = 81nc/81ne is the base transport factor and y = 81,",/81e the emitter injection efficiency. Here In, is the electron current of the collector and I,, the electron diffusion component of the emitter current In, .

The value of (YT shows what proportion of the electrons injected from the emitter to the base pass through the collector. Evidently the rest of the electrons, (1 - a ~ ) , will recombine into the base. It is clear that the ratio of the quantity of electrons, which recombine into the base, to the total quantity of the electrons injected from the emitter is approximately equal to the ratio T d / r . The time Td W w,"/Dn is the time required for the diffusion of electrons through the base (here Wb is the base width) and T = L i / D n is the electron lifetime. Such an estimate gives 1 - CYT M W,"/Li, or CUT z 1 - W,"/Li. A more precise calculation [14; 201 gives:

W2 C q W l - -

2Lg (4.21)

In the majority of cases CUT does not in practice depend on j , and the relationship between (YO and j (Fig. 4.16) is determined by the dependence y(j) .

The value of y shows what proportion of the total emitter current I , comes from the electron diffusion current of the emitter I$. It should be emphasized that only the electron diffusion current I:, provides the electrons which diffuse through the base and pass through the collector. At I:, = I,, y would be equal to unity, although such a situation never occurs.

Two main components contribute to the emitter current I , at small forward current densities: the diffusion electron component I:, and the recombination current in the space-charge region of the emitter I , [14; 201. (The diffusion current of the holes I t , is negligible):

I,, M I& + I , = I,[exp(eV/kT) - 11 + 1 ~ [ c x p ( e ~ / 2 ~ ) - 11 (4.22)

where I , = j,S and I , = j gS, and j,, j , are determined by the expressions (2.9)- (2.13).

As deduced in Section 2.2.2 (see comment on the expression (2.14)), j g / j , >> 1 in the majority cases of interest. The qualitative current-voltage characteristic of a forward-biased (emitter) junction, which is valid in this case, is shown in Figure 4.17.

As seen, the recombination current in the space-charge region I , is much larger than the diffusion current ( I , >> I:,) at small forward currents. Hence, at small

99


-2 -

-4 -

I I I

0 1 2 3 4 Forward bias V

(linear scale)

Fig. 4.17 Qualitative current-voltage characteristic of a forward-biased n+ - p junction. In the part of the curve where j N exp(eV/2kT) there is no injection of electrons into the base, while in the part where j - exp(eV/kT) practically the whole electron current In, is a diffusion current

d Ine.

magnitudes of the forward bias V (small emitter current) we have:

d d d I n e = - < < < I y7Z-7Z- I n e Ine

Ine I ie +IT IT (4.23)

The diffusion current I i e increases much more sharply with V than the recombination current I,, and at relatively large emitter current the opposite situation prevails, I& >> I,. When this takes place, y increases and reaches its maximum value (the ‘Lplateau” in Fig. 4.16). It can be shown that if the emitter doping level Ne is much higher than the base doping level Nb, y can come very close to unity in this region.

Let us consider the passage of a current through the emitter-base n+ - p junction when I:e > > I, and recombination into the emitter-base space-charge region can be neglected (Fig. 4.18).

Neglecting processes in the emitter-base space-charge region, we can represent the emitter n f - p junction as an abrupt energy barrier between the base and the emitter (Fig. 4.18). As seen, the injection of electrons into the base is inevitably accompanied by an injection of holes into the emitter. Hence the current at the emitter-base boundary consists of two components:

where j i e is the diffusion current of holes injected from the base into the emitter.

100


n+-emitter n =N,

P(O), __I

Fig. 4.18 emitter layers of a BJT at a low injection level (n(0) << Nb).

Distribution of the electrons (solid lines) and holes (dashed lines) along the base and

The emitter injection coefficient y is

(4.25)

The concentrations of the carriers, electrons and holes on the two sides of the emitter-base energy barrier obey the Boltzmann ratio:

(4.26)

where E = (p - V ) is the height of the barrier. Here p M E, is the height of the energy barrier of the n+ - p junction in the absence of bias. Then

(4.27)

It is worth noting that the ratio Ne/Nb is usually very large and falls within the range between lo2 and lo5.

Since the diffusion length of the electrons in the base L , is usually much greater than the base width W , the electron concentration in the base varies linearly with distance (Fig. 4.18):

"1 W

The diffusion current of electrons in the base at x = 0 is then

(4.28)

(4.29)

On the other hand, the hole diffusion length in the emitter L, is as a rule small by comparison with the emitter thickness, so that the hole diffusion current in the

101


emitter at IC = 0 is

(4.30)

And the injection coefficient y is

Expression (4.31) determines the maximum magnitude of y which corresponds to the plateau in Fig. 4.16. Due to the very large ratio Ne/Nb, this value of y can be very close to unity. The maximum y value in a conventional homojunction BJT can be as high as 0.99, for example, and even 0.995.

The decrease in y with further growth in the current density (Fig. 4.16) is caused by a transition to a high injection level in the base (Fig. 4.19).

n-collector

Nb - X

Fig. 4.19 Distribution of electrons (solid lines) and holes (dashed lines) along the base and emitter layers of a BJT at a high injection level in the base (n(0) >> Nb). The dotted line shows the doping level in the p-base Nb.

At a high injection level the boundary electron concentration n ( 0 ) exceeds the doping level in the base Nb by an appreciable amount. Due to the neutrality (or, more strictly speaking, quasi-neutrality) of the base, the boundary concentration of the holes p1 in such a situation is practically equal to n(0): p 1 M n(0) + Nb M n(0).

The boundary electron concentration n(0) is connected with the emitter doping level Ne by the ratio (4.26): n(0) = N,exp(-E/kT). On the other hand, the boundary hole concentration is now p ( 0 ) = p l exp ( -E/kT) M n(0) exp ( -E/kT)

102


and the expression for y at a high injection level takes the form:

1 .d

jge +jpd, 3ne - -

PI w DP 1 + - . - . - Ne Lp Dn

7’

From Eq. (4.29), we obtain:

.i$W 3~ n(0) “ p 1 x - - eD, eD,

and 1

‘+’(eiVeLPD:)

N N

1 y = Pl w D, W2D, I + - . - . -

Ne Lp Dn

(4.32)

(4.33)

(4.34)

As seen, y decreases monotonically with increasing j at a high injection level in the base.

With a further increase in j , other non-linear processes contribute essentially to the decrease in y at high emitter current densities, namely electron-hole scattering (for references, see [102; 103]), narrowing of the band-gap [104; 1051 and Auger recombination see [104; 1061.

4.3.2.3

Now let us assume that the bias Vo has been applied to the collector of an ABT connected in series with a load resistor Rl (Fig. 4.20). ABTs operate in a common- emitter configuration. When no input signal exists, the current passing through the ABT is very small, so that the magnitude of a0 will also be very small (cf. Fig. 4.16) and the breakdown voltage in the common-emitter configuration V . will in practice be equal to V,.’ (see expressions (4.13) and (4.17)).

When an input (base) signal is applied, the current passing through the ABT increases, and (YO increases correspondingly (see Fig. 4.16) and tends to its maximum magnitude a O m a r = C Y T Y ~ ~ ~ M IYT. In this case the breakdown voltage in common-emitter configuration, V,.., is described by expressions (4.13) and (4.17). A new stable state can be reached now only on the branch of the current-voltage characteristic corresponding to ye at aornax = cq- (point B in Fig. 4.20). Hence, when the input base signal is applied, the collector current “jumps” from point A to point B and the voltage AV = V,.. - Vb forms the legitimate signal across the load.

The qualitative field distribution along the collector of the ABT in such a regime is shown Figure 4.20 b. As seen in Fig. 4.20, the regime under discussion corresponds to the current density j << J” = ev,N, (see Figs. 3.9 and 3.14 and Eq. (3.7)), with the slopes of the F ( z ) dependences in the collector region equal (or close) to one another at the operation points A and B. Note that with a small load resistor, and

Main features of ABT operation in a conventional regime

103


accordingly a high current value on branch 2, the voltage across the transistor could be expected to increase beyond the magnitude of (see dashed line at high currents in Fig. 4.15). This is because a0 decreases at a high current density (Fig. 4.16) and the breakdown voltage should increase accordingly (expression (4.17)).

As pointed out as far back as 1966, however, AS/ can be appreciably larger at high current densities than that in the “classical” mode described above. The residual voltage after the current “jump” is much smaller than that in the conventional mode, and the current-voltage characteristic in the low-voltage high-current state does not follow Eqs. (4.13) and (4.17) [107; 1081

A qualitative interpretation of these effects has been put forward in Ref. [92] on the base of the avalanche injection phenomenon; although the highly non-linear processes caused by avalanche injection in the collector of the ABT mean that an adequate description can be achieved only in the framework of a computer simulation.

a h collector

Collector voltage V,,

Fig. 4.20 Collector current-voltage characteristics of an ABT (a) and field distribution along its collector region (b) in the conventional regime of operation. a. Bias Vo is applied to the collector of the ABT and the load resistance R1 connected in series. R1 = l / t a n a . When no input (base) signal exists, the point A is stable, while in the presence of a base input signal, the stable condition corresponds to the point B. b. Field distributions along the collector region corresponding to the points A and B, respectively. The area of the hatched part corresponds to the voltage AV.

104


4.4 Operation regime of a Si avalanche transistor at very high current densities

4.4.1 Introduction

In early simulations of Si avalanche transistors with avalanche injection involved, the collector of the ABT was considered simply to be a reverse-biasedp+-n-n+ (or n f - p - p f ) diode. The true electron injection from the emitter is simulated in this approach by the electron (or hole) flow at 5 = 0 (cf. Fig. 4.20 b). This oversimplified approach ignores the intrinsic positive feedback that in fact determines the operation of any BJTs. Nevertheless, even this approach still allows us to study important peculiarities of ABT operation in the avalanche injection mode [log; 1101.

The steady-state electric field profiles calculated in Ref. [log] are shown in Fig. 4.21 for a reverse-biased p f - n - n f structure with an n-region W of 50 pm thickness and a doping concentration Nd = 1015 cm-3 at various densities of the injected electron current.

tB h

40 A 2

Lc" 0 20

5" 10

d

2 30 22

b

x

.- 0

0 10 20 30 40 50 Distance (pm)

Fig. 4.21 Distributions of the electric field along a reverse-biased p f - n - n + structure at different current densities j . j (A/cm2): 1 - 0; 2 - lo2; 3 - lo3; 4 - 1 . 5 ~ 1 0 ~ ; 5 - 1 . 7 5 ~ 1 0 ~ ; 6 - 2 . 5 ~ 1 0 ~ ; 7 - lo4; 8 - lo5. Points A and B show the maximum electric fields for curves 7 and 8. A: F,,, = 3.6 x lo5 V/cm (Curve 7); B: F,,, = 4.9 x lo5 V/cm (Curve 8).

One can see that at an initial bias Vo = 320 V (curve 1 in Fig. 4.21), the maximum electric field has a value of 3 x lo5 V/cm (at 5 = 0). According to the approximations for ai and ,& adopted in Ref. [log]:

(4.35) ai = 3.8 . 106exp(-1.75. 106/F) cm-l pi = 2.25. 107exp(-3.26. 10°F) cm-'

105


at x = 0 the ionization rates are equal to ai M 1.1 x lo4 cm-' for electrons and pi M 4.3 x 102 cm-l for holes.

As seen, the entire n-region is fully punched-through at j = lo3 A/cm2 (Curve 3). At j = 1 . 7 5 ~ lo3 A/cm2 (Curve 4), the field is distributed almost homogeneously across the whole n-layer. (Note that with a doping level of Nd = 1015 ~ m - ~ , the characteristic critical current density j~ = ev,Nd M 1.6 x lo3 A/cm2). At j = 2.5 x lo3 A/cm2 (Curve 5 ) , the sign of the dF/dx is reversed due to the space charge of the free electrons (see Figs. 3.10 and 3.12) and the field increases monotonically towards the n+ layer.

With a further increase in j, avalanche breakdown occurs near the n+ contact of the p f - n - n+ structure. The holes generated in the breakdown region compensate for the space charge of the free electrons. As a result of this avalanche injection effect, the field in the major part of the n-region is very small, and the residual voltage drop across the structure is also small, being mainly determined by a sharp peak in the electric field near the n+ collector contact.

These calculations illustrate the experimental results regarding the low residual voltage drop [107; 1081. The full set of boundary conditions was not specified in the paper [log], however, which makes it difficult to evaluate whether the analysis presented for a diode structure is related to processes in the collector of the avalanche transistor.

As seen in Fig. 4.21, an appreciable current (j > j ~ ) introduced at the reverse- biased p - n junction is able to totally rebuild the field distribution in the low- doped n-region. A reverse bias applied to the collector junction of an ABT provides a notable multiplication of M , and under such conditions rebuilding of the field distribution and avalanche breakdown can occur even when the input injection current is much smaller than the critical current density j~ [110]. The initial phase of this process is considered in Ref. [110].

The transient characteristics of a reverse-biased p f - n - n+ diode with an n- layer of width W = 10pm and doping level Nd = 1015 has been calculated. With a reverse bias of VO = 222 V, the n-layer of the structure is totally punched- through and the maximum field (at x = 0) is 3 x lo5 V/cm (compare with curve 1 in Fig. 4.21). The simulation is made for a case in which electrons with a concentration nin = j in/ev , are injected into the n-layer a t x = 0 as a step function of time at the instant t = 0. As mentioned above, the field in the n-layer is strong enough to provide considerable avalanche multiplication of the carriers.

The increase in the current during the transient process for a diode with an operation area S of 1000 p m 2 at different magnitudes of nin is shown in Fig- ure 4.22a. With nin = lolo cmP3 (curve l), the input current density jin = ev,nin is M 1.7 x cm2, the input current Iin is 1.7 x lop7 A. As seen in Fig. 4.22 a, the multiplication coefficient M at the bias applied is A M 59. The current increases from input to the steady state value in a time of about 1000 ps. As seen in Fig. 4.22 a, the

A/cm2 (with v, = 1.07 x lop7 cm/s [ l lo]) , and with S =

A/1.7 x

106


characteristics of the transient processes and the value of the multiplication factor do not depend on the input concentration nin at ni, < 1013 ~ m - ~ .

The travelling and multiplication of carriers is illustrated in Figure 4.22 b. Elec- trons (solid lines) with a concentration nin = 10l2 cm-3 (Curve 3 in Fig. 4.22a) are injected into the n-layer (z = 0) at time t = 0. The dotted lines represent the hole distribution. It is seen that the electrons reach the “anode” (z = 10,um) in time t M 100 ps, which corresponds approximately to the transit time through the space-charge region at a saturation velocity t , M W/us M 10-l’ s. The total duration of the multiplication process is nevertheless about 500 + 1000 ps.

As it is clear from Fig. 4.22, the multiplication regime is achieved at n,, < 1013 cm-3 (ji, = ew,nin) M 17 A/cm2), since for a given Vo magnitude the current saturates at t -+ 0;) and the “output current” is proportional to the input current (see Section 2.1).

At nin 2 1014 ~ m - ~ , however, the injected carriers initiate avalanche breakdown and unlimited current growth (Figure 4.23 a).

It should be noted that since the concentration ni, = 1014 cmP3 is still much lower than the doping level Nd M lo1’ ~ r n - ~ , the injected electrons themselves cannot distort the field distribution to any marked extent. The electrons (and holes) .are multiplied while travelling through the n-layer, however, and at some instant t (t M 160 ps in Fig. 4.23b) their concentrations will exceed the doping level. The reconstructed field provides stronger carrier multiplication, and as a result, a further increase in current.

Unlimited current growth in the model developed in Ref. [110] is caused by the absence of any load resistance R1 connected in series with the diode. Rl is always present in a real circuit, and the voltage drop across it increases with current growth, causing an appropriate decrease in voltage drop across the reverse-biased diode. Consequently, the multiplication intensity decreases and the system reaches a new steady state,

The model under discussion does not claim to describe the transition of a system to a high current-low voltage steady state even in the diode structure. Moreover, the set of boundary conditions used in Ref. [110] can hardly pretend to describe the processes that take place in the collector of an avalanche transistor.

The most comprehensive analysis of steady-state and transient processes in a Si ABTs in the avalanche injection regime is provided in Refs. [loo; 111; 1121.

107


Time (ps)

a

10

h

Distance (Fm)

Fig. 4.22 Transient processes in a reverse-biased p + - n - n+ diode at different input electron concentrations nin(nin = j in /ev , ) injected into the n-layer at x = 0 as a step function. The width of the n-layer is 10 pm. Operation area S = l op5 cm2. Bias voltage VO = 222 V. a) Current growth during the transient process at different values of nin ( ~ m - ~ ) : 1 - 1O1O; 2 -

b) Travelling and multiplication of carriers. Electrons (solid lines) are injected into the n-layer (z = 0) at time t = 0. Dotted lines represent the hole distribution. The input electron concentration nin = 10l2 cm-3 (Curve 3 in Fig. 4.22 a)

10'1. 3 - 1012. 4 - 1013.

108


Time (ps)

Distance (pm)

Fig. 4.23 Transient processes in a reverse-biased p+ - n - n+ diode at nin = 1014 ~ m - ~ . All data apart from nin are the same as in Fig. 4.22. a) Current versus time dependence. The exponential current increase begins at the moment when n = Nd (compare with Fig. 4.23 b). b) Temporal profiles of carrier density determined by carrier transport and multiplication. The carrier concentration reaches the doping level N d = 1015 cmP3 at instant t N 160 ps. After that rebuilding of the space charge occurs and the current increases exponentially with time at a given vo .

4.4.2 Steady-state collector field distribution. Residual collector voltage

Switching processes in transistors with an n-collector (n+ - p - no - n f structures) and a p-collector (p+ - n - po - p f structures) were investigated experimentally in Ref. [loo] in a low-inductance circuit (Lee N 3.5 nH) at a low load resistance

Several commercial types of pf - n - po - p+ transistor with a breakdown voltage in the common-base configuration yb ranging from 200 to 350 V were investigated, and in all cases the switching corresponded to the conventional regime of ABT operation (Section 4.3.2).

On the other hand, very effective high current switching was observed in the n-collector transistors (n f - p - no - n+ structure) (Figure 4.24). The current- voltage breakdown characteristics of the avalanche transistor FMMT-417 (ZETEX Semiconductors) employed in Ref. [lo01 were measured with a curve tracer a t low

(RL N 10).

109


currents for two circuit configurations (Fig. 4.15), the breakdown voltage in the common-base configuration yb being 340 V, while the minimum possible breakdown voltage, in the common-emitter configuration ye7 was 170 V. At the same time, as can be seen in Fig. 4.24, the “quasi-static” residual voltage drop across the transistor is as low as V, - 70 V at a current amplitude I , M 70 A (curves 3 and 3’) , and V, M 95 V at I , x 120 A (curves 1 and 1’).

0 5 10 15 20 Time (ns)

Fig. 4.24 Current (curves 1-3) and voltage (curves 1’-3’) across the avalanche transistor with no-collector (FMMT-417, ZETEX SEMICONDUCTORS) as a function of time for various initial biasing values Vo: 1 , 1’ - 290 V; 2, 2’ - 240 V; 3, 3’ - 180 V. The current waveforms I ( t ) are derived taking into account the parasitic load inductances. The instant t = 0 corresponds to the input of the base pulse.

Several important points should be noted here. First, the collector voltage is supplied by a storage capacitor of a relative small capacitance (Cce = 6.6 nF), and the decrease in the collector current after it has reached the peak magnitude I , (and also the slow reduction in the residual collector voltage) is associated simply with discharge of the capacitor.

Secondly, the whole transient process, including fast turn-on and a further reduction in the collector current, is significantly shorter than the deduced carrier lifetime in the no region. Thus the achieving of a quasi steady state is determined by the balance between carrier generation by impact ionization and the sweeping out of the carriers by the electric field (not by the carrier recombination).

Third, the operational transistor area was estimated to be about cm2, which means that the current densities corresponding to different values of I , in

110


Fig. 4.24 ranged from 7 x lo4 to 1.2 x lo5 A/cm2. These current densities exceed by a factor of - 50 - 100 the critical current density j~ = eNdv, which is 1600 A/cm2 for a doping level of - l O I 5 cmP3 in the collector layer.

The conventional set of equations was solved (neglecting recombination and diffusion) to simulate the quasi-steady state at the reverse-biased collector junction after high current-low voltage switching (Fig. 4.20 b). The field dependences of the ionization rates for electrons and holes were taken to be of the form defined in expression (4.34). The electron and hole velocities were determined using conventional expressions (see, for example, [8]):

where the coefficients a t 300 K are us, F, x 6.98 kV/cm; Fp x 18 kV/cm; check that the electron and hole drift

u p = vsp FIFP (4.36) (1 + (F/Fp)Pp)l’Pp

x 1.07. lo’ cm/s; uSp x 0.834. lo7 cm/s; pn = 1.1087; p p = 1.213. It is easy to velocities follow Ohm’s law, u = /IF, a t

F << Fn(Fp), while a t F >> F,(Fp) they tend towards their saturated values us, and usp1 respectively.

With appropriate boundary conditions, the set of equations was solved for both the no and po types of collector. As in Refs. [log; 1101, electron injection from the emitter was simulated by the injection electron current jn0 at 2 = 0 (for a n o collector).

In full agreement with the experimental data, the simulations show just a conventional operation mode for transistors with po collectors and a very effective high current-low voltage switching mode for transistors with no collectors. The results for the latter (n+ - p - n o - nS structure) are shown in Figures 4.25 - 4.27.

The dashed line in Fig. 4.25 corresponds to jn0 = 0. Hence it represents the conventional current-voltage characteristic of a reverse-biased p + - n junction under breakdown conditions. As seen, the passing of a high current ( j N lo5 A/cm2) requires that the residual collector voltage should be very high (> 500 V) in this case (cf. Fig. 3.15), which makes it impossible to attain a highly conductive state. An appropriate injection current would substantially reduce the residual voltage.

Let us consider as an example the collector voltage magnitudes at a total current density j = 7000 A/cm2 and different values for the injection current jn0 (straight line AA in Fig. 4.25). At jn0 = 0, the voltage drop V, is as high as 685 V (point 1’). On the other hand, with the same j value and jn0 = 5000 A/cm2, V, = 296 V (point 4’). An increase in jn0 (at the same total current j ) causes a marked decrease in the collector voltage (see also Fig. 4.26).

As seen in Fig. 4.26, to provide an appreciable decrease in V,, the injection current jn, must account for a notable proportion of the total current j ( j n o / j 2 0.6). The decrease in V, is attributed to two effects.

The first is associated with the fact that the higher the injection current is,

111


0 100 300 500 700 Collector voltage (V)

Fig. 4.25 Current-voltage characteristics for various electron injection current densities j,o (A/cm2): 1 - 1750, 2- 3000, 3 - 5000,4 - 12000, 5 - 26000, 6 - 40000, 7 - 70000. The dashed line corresponds to j,o = 0. The critical current density J,, = 1712 A/cm2 [loo].

Fig. 4.26 total current j n o / j . j (kA/cm2): 1 - 40 , 2 - 70 and 3 - 100 [loo].

Dependence of the collector voltage on the ratio of the electron injection current to the

the weaker is the electric field on the left side of the collector, where the field is constant and very close to the value F(,,o) = FO (see Curves 4-7 in Fig. 4.27). The second is the formation of a narrow peak F near the right boundary of the structure (Fig. 4.27). The higher the injection current, the narrower this peak is, and the

112


higher is the maximum field F, (Fig. 4.27). Assuming dF/dx = 0 in the quasi-neutral region in the left part of the collector,

and neglecting the donor concentration with respect to the electron (hole) concentration at the high current density in question, an analytical expression connecting Fo and j n o / j was derived in Ref. [loo]:

Fig. 4.27 Electric field distributions along the no-collector for various combinations of the injection current j , ~ and total current j . j,o (kA/crn2)/J (kA/cm2): 1 - 010, 2 - 0.85/0.856, 3 - 5/7.15, 4 - 12118.6, 5 - 26/41.3, 6 - 40164, 7 - 70/110, 8 - 40/81.1 (1001.

% p F n

V d p It is seen that this relation tends towards the limit 1 + - G 1.3 as Fo 4 0,

which is determined by the ratio of the effective mobilities of electrons and holes - 1530 cm2/Vs and p p e f f = zl,p = 463 cm2/Vs. Correspondingly, referring to Fig. 4.26, we can conclude that the minimum collector voltage is reached when j n o / j is equal to l/(b,pf + 1) M 0.77 at a fixed value of j .

At high current densities it is also possible to obtain an approximate analytical expression for the amplitude of the high field domain F, at the right boundary of

2),n- FP

b e f f = P n e f f / p p e f f , where p n e f f = Fn

113


the collector (Fig. 4.27). Neglecting the hole ionization (ai >> pi) and taking into account the fact that F o is small with respect to F in the avalanche region, one can obtain the following expression for F,:

w

where F2(s) = J exp(-zt)t-2dt is the tabulated function. At a given magnitude

of j , the maximum value of F, is again reached in the limiting case j n o / j M 0.77. For example, at j = 100 kA/cm2, F, = 3.65. lo5 V/cm. Formula (4.38) and the curves in Fig. 4.27 show that the distribution of the electric field near the right-hand boundary at high currents ceases to depend on the collector thickness starting from a no region a few pm in thickness.

Curve 8 in Fig. 4.27 corresponds to the situation in which the ratio j n o / j is somewhat smaller than 0.5. As seen, the quasi-neutral region of the weak field is not formed in this case, and the sustaining of a sufficient electron flux at a high total current density requires impact ionization at the left boundary of the no collector. The appearance of a field peak at the left boundary (curve 8 in Fig. 4.27) is analogous to that in the diode in the absence of any electron injection (cf. Fig. 3.16).

Comparing the results presented in Fig. 4.27 with those of Ref. [log] (see Fig. 4.21), one can conclude that the appearance of a high-field domain at the right boundary is similar to that obtained in [log]. The role of the injection current cannot be understood from Ref. [log], however, since it does not specify the full set of boundary conditions.

In full agreement with the experimental data, the simulation performed in Ref. [loo] for a po collector shows that effective switching to a high current - low voltage state is impossible for a transistor of this type. At first glance one could assume that the reason for this qualitative difference in behaviour lies in the difference in ionisation rates between electrons and holes. This has been a common opinion for many years [113]. The simulations performed for a po-collector with the ionisation coeficients for the electrons and holes interchanged did not show any appreciable reduction in the residual collector voltage, however, whereas the simulations for an no-collector with interchanged ionisation coeficients provided only a slightly higher residual voltage than that shown in Fig. 4.27.

The analysis shows that the diflerence in velocity between the electrons and holes plays a key role in this effect. In transistors with an no-collector, the carriers with a relatively LLsmall” velocity (the holes) are generated in the high field domain at the right boundary of the collector, while the “fast” electrons injected from the left can compensate fully for the space charge of these holes. As a result, formation of a quasi-neutral region with a weak field is possible in the left part of the collector. In the structures with a po-collector, in which “fast” electrons are generated in the high field domain, such compensation is impossible.

1

114


It is fairly obvious in principle that inducing a high electron current (in n f - p - no - n+ structures) should create a steep slope in the high field region close to the collector boundary. This circumstance was mentioned in early papers (see, for example, [log; 1131). The possible existence of a low-field quasi-neutral region, however, can be assumed rather than demonstrated in models that adopt simplification approaches. In part, the model presented in [I131 fails to recognize any difference between n f - p - n o -nf and p+ - n - p o -p+ structures. Meanwhile, as was demonstrated in Ref. [loo], the low-field quasi-neutral region in the collector cannot exist in Si p+ - n - po - p+ structures.

4.4.3 Transient properties of Si avalanche transistor at extreme current densities

Transient processes in an Si avalanche transistor at extreme current densities were studied in Ref. [lll] for the same commercial ABT (type FMMT417, ZETEX SEMI- CONDUCTORS) with the characteristics shown in Figs. 4.15 and 4.24. Detailed comparison of the simulation results with the experiment requires information concerning both the transistor structure and the external circuit. It is particularly essential to have precise information on the parasitic inductances, since these drastically affect the transient, which has a characteristic time of N 1 ns in a circuit with an impedance of N 1 Ohm.

The doping profile used in Ref. [ill] for simulating the transient processes and parasitic inductances measured in a set of special test experiments are shown in Figure 4.28.

The current and electric field distributions are always multidimensional in any bipolar transistor, but multidimensional (especially 3D) simulations are usually rather complicated. On the other hand, a relatively simple one-dimensional approach often provides an adequate and reasonable result which can be clearly interpreted. One essential problem is how to induce an external triggering (base) current in a one-dimensional model.

The approach used in Ref. [lll] implies a source for the majority carriers that is uniformly distributed across the base. The areas of the collector, base and emitter are considered to be equal (each N cm2 for the selected transistor type), and the base current is treated as the generation of the majority carriers in the base region. The base current value in the external circuit is comprehended as the generation rate integrated over the base volume.

The experimental and simulated base ( I b ) and collector ( I c ) currents and the collector-emitter voltage (Vce) during switching are shown in Fig. 4.29. As seen, the data agreed fairly well. The most pronounced difference concerns the switching delay, which is N 2 ns larger in the experiment than in the simulations. An obvious source of this discrepancy is the limitation to a one-dimensional approach, which implies hole generation in the volume of the base.

115


'E 0 l O ' * j . ' ' I ' ' ' ' ' ' ' ' . ' . ' ' ' ' ' ' ' . ' ' ' ' E . EMITTEF 0

I

I :

0 2 4 6 8 10 12 14 16 4- L . , . , . I . , , , . I . , 'I,

Fig. 4.28 and experimentally evaluated parasitic inductances [lll].

Doping profile used for simulation of transient processes in an FMMT417 transistor

In a real transistor the characteristic time required for electron diffusion from the emitter to the collector across the base, to - W,"/De x 2.7 ns, should be taken into account. This value is comparable with the difference between the experimental and simulated delay times.

The electric field profiles in the base and collector regions during switching are shown in Fig. 4.30.

The process can be conditionally separated into three stages. The first stage (0 - 1.5 ns) corresponds to an increase in the collector current to a critical value j~ = evsNd M 4.8 x lo2 A/cm2, where Nd = 3 x 1014 cm-3 (see Fig. 4.28). There is no appreciable rebuilding in the collector field distribution until the electron and hole densities remain below the donor concentration. At the second stage (t M 1.5 - 3 ns), the electron density exceeds that of the donors, and the derivation dF/dx changes its sign. The third stage (see Fig. 4.30 b) is responsible for the high d I / d t rate of the collector current, which is caused by a rapid reduction in the emitter-collector voltage due to shrinkage of the collector field domain. The latter is determined by the spread of a quasi-neutral region from the base towards the collector contact.

The quasi-neutral region is formed by an accumulation of the electrons injected from the emitter, and by compensation of their charge by the holes generated due to impact ionization near the no - n f boundary.

A non-trivial result was obtained in Ref. [lll] from an investigation of the effect of collector thickness on the switching parameters. It was found that the thickness

116


Time (ns)

Fig. 4.29 two values of the bias voltage Vo(V): (a) - 200, (b) - 290.

Measured (solid lines) and simulated (dashed lines) current and voltage waveforms for

of a no collector affects the switching delay much more significantly than the collector current rise time or the maximum d I / d t rate within the third switching stage (Fig. 4.31)

An increase in the collector thickness from 15 to 35 pm causes a rise in the switching delay (second stage) to N 20 ns. A qualitative difference was observed within the delay stage relative to the case of a “thin” collector. Hole accumulation in a “thick” collector occurs at moderate currents when the impact ionization effect is negligible. This accumulation (see Fig. 4.31 b, t = 2 - 20 ns) is determined by hole diffusion from the base and can be attributed to the Kirk effect [114].

The third switching stage begins after the peak in the electric field near the collector contact has provided a high rate of hole generation. The thickness of the high-field domain at this instant is comparable to that of a “thin” collector (compare Fig. 4.31 b, t = 20 ns, and Fig. 4.30 b, t = 4 ns), resulting in a comparable duration of the rising current in both cases. The somewhat higher voltage drop across the LLquasi-neutral” region in a “thick” collector correspondingly leads to a lower amplitude for the collector current.

117


,g- 2.0

2

0 -.. ' 1.5

x 1.0' k

- 0.5

0.0:

2.5

: i

. . , ' r ' 3 u :

I : ' I

- Y , , , , , i' 5 : I

0 5 10 15 Distance

Fig. 4.30 Electric field distributions across the p-base (z = -4 t Opm) and in the n-collector (z = 0 t 16 pm) regions at various instants (t = 0 + 9 ns). (a) t = 0 t 3 ns; (b) t = 4 + 9 ns. The profiles correspond to the simulated current and voltage waveforms shown in Fig. 4.29a.

One can conclude that one-dimensional simulation provides a fairly good description of the switching transient in an Si avalanche transistor. There is one very important problem, however, that cannot be solved in the framework of a 1D simulation.

As seen, a very high current density j N lo5 A/cm2 (a collector current I , N 100 A across a device of area N cm2) is achieved simultaneously with a high electric field ( F N 2.5 x lo5 V/cm, see Figs. 4.30, 4.31). Accordingly, the local high power density j x F can generally cause severe local heating and destroy the device within a single pulse. Consequently, time-dependent local heating is an extremely important problem for ABTs.

It is also obvious that an estimate of the actual peak current density j,,, requires a knowledge of the current distribution across the structure, whereas the 1D model implies a homogeneous current distribution across the whole switching area. This assumption cannot in principle be valid for any kind of BJT, on account of the very well-known emitter crowding effect [115]-[117].

Two-dimensional simulations of the switching transient for the same type of avalanche transistor (FMMT417) were performed in Ref. [112] using the ATLAS device simulator (Silvaco Inc.) that provided perfect agreement between the simulated and measured current and voltage waveforms, as shown by solid lines in Fig. 4.29(b), including the switching delay. The current reached its maximal value I,,, M 90.2 A at the instant t M 8.4 ns.

The cross-section of the device in the lateral plane is shown in Fig. 4.32a1 the doping profiles are shown in Fig. 4.32 b.

The distribution of the current density along the z-axis with different values of

118


1 "0 2 4 6 8 10 12 14 16 18 20 22 24

Time (ns)

Distance x (pm)

Fig. 4.31 (a) - Simulated collector current (solid lines) and emitter-collector voltage (dashed lines) for a n+ - p - n o - n+ transistor with various n o collector thicknesses W,. Vo = 250 V, the parasitic inductances (Fig. 4.28) were excluded, and the load resistor RL = 1 R. (b) - Electric field profiles across the collector for the transistor structure with W, = 35 pm at various instants.

y at the peak magnitude of the collector current I,,,, M 90.2 A (t x 8.4 ns, cf. Fig. 4.29 b) is shown in Fig. 4.32 c. As seen, a very appreciable current crowding is observed in 2D simulations. Even at y = 2.6 pm, where the current density j is at a minimum, it is still five times larger than in the 1D simulation, and at y = 19 pm, near the peak of the electric field in the collector domain, it is nearly 20 times larger ( j N 1.7 x lo6 A/cm2). It worth noting that the residual voltage across the structure is fairly similar in the 1D and 2D simulations and the electric field domain in the channel is reconstructed in the same manner in both cases (cf. Fig. 4.30 b).

The "nucleus" of localization at the very beginning of the switching transient originates from the emitter crowding effect [115]-[117], but much stronger lateral current crowding is observed near the collector later on, and finally a very narrow conducting channel is formed across the collector region. It was shown that the electric field reconstruction in the direction y at x M 6 0 p m and the non-rebuilt field domains at z < 30pm and z > 80pm cause a lateral electric field which extrudes electrons from the channel (injected from the emitter) and retracts the

119


b a

, 1.0

2 0.5 3 - 0.0 5 0 10 20 30 40 50 60 70 80 90

.- I-

Fig. 4.32 (a) - Cross section of the simulated device. (b) - Doping profile of the device. (c) - Distribution of the current density at different values of y at peak current (90.2 A). y (pm): 1 - 2.6; 2 - 10; 3 - 19.

holes (generated near the n f collector) into it. Thus strong current localization causes well-localized heat generation in the

channel near the n f collector, where both the electric field and the current density reach their peak values (Fig. 4.33).

As seen in Fig.4.33, the peak value for the local temperature can be as high as N 750°K at the instant t = 13.5 ns (at the end of the current pulse). (Note that the estimates for the maximum possible increase in the lattice temperature A T in a 1D approximation give AT M 30°K for almost the same switching parameters [ l l l ] ) .

Although the local temperature at the end of the pulse is rather high, it does not cause any harm to the device in single pulse operation. Indeed, the intrinsic carrier concentration in Si at a lattice temperature of - 750°K (ni - 7 . 10l6 ~ m - ~ )

120


n E A-

A W

0

4

8

12

16

20

& 400 300

0 10 20 30 Lu) 50 60 70 80 90

Fig. 4.33 Temperature map of the structure at the end of the current pulse (t = 13.5 ns, compare with solid lines in Fig. 4.29 b) and distribution of the temperature at y = 19 pm at various instants [112].

is still less than the doping in the n+ collector and comparable to that in the quasi- neutral region. This should mean that the pronounced local heating observed in the simulations does not cause destruction of the device in the single pulse mode, which is in full agreement with experimental results.

121


4.5 Operation regime of GaAs avalanche transistor at very high current densities

4.5.1 Experimental results

Very fast switching in GaAs BJTs was observed in Refs. [118; 1191. The GaAs n+ - p - no - n+ structures with parameters which were close to those of the Si ABTs discussed in Section 4.4.3 demonstrated high current-low voltage switching within a time of about 200 ps. The switching was characterized by the authors as “superfast” for two reasons. First, the switching time was approximately 15 times shorter than that in analogous Si BJTs. Second, it was shorter than the transit time required for the carriers to cross the base regions of the transistor, provided the carrier velocity was saturated.

Doping profiles across the structure, established by means of various electro- physical test measurements, are shown in Figure 4.34a (cf. Fig. 4.32b).

u

0 5 10 15 20 25 30 35 40 45 b

Y (Pm) - so0 300 - a 260 260

L > W

s

0 2 > 150 150

0

a

- 2 200 200 g

100 5 b 3 100 - a 0 - -

60

o a _---- n i , -0 1 2 3 4 6 6 7 8 -

Time (ns)

Fig. 4.34 Doping profile of a GaAs transistor (a) and temporal dependences of the collector voltage and current across GaAs (solid lines) and Si (dashed lines) transistors in the high-current avalanche mode (b). The time dependence of the collector voltage characterizes the speed of the switch, while the d I / d t rates of the transistors are limited by the circuit parameter L p / R ~ (cf. Figs. 4.28, 4.29)

The transient characteristics of a GaAs transistor are compared with the appro-

122


priate characteristics of an Si transistor in Fig. 4.34b (cf. Figs. 4.24, 4.29). One can see that extremely fast switching is observed in a GaAs avalanche

transistor. The time required for a reduction in the collector voltage from the initial value of 300 V to N 110 V is about 200 ps, i.e. shorter than the time required for the carriers to traverse the no collector region structure at the maximum possible (saturation) velocity. It should be noted that the observed low residual voltage manifests relatively deep modulation of the conductivity of the collector region. Meanwhile, the essential modulation of the conductivity in this depleted layer by the carrier drift or diffusion from the emitter should take much longer than the minimum time for the passage of a carrier across the region.

A spatial picture of the switching in a GaAs transistor is given in Figure 4.35. A number of light-emitting channels are observed along the perimeter of the emitter- base interface in single transistor switching.

h

c

d

Fig. 4.35 (a) - Schematic diagram of the top view of a GaAs transistor. The collector is located beneath it (cf.Fig. 4.32 a). (b)-(d) - Emission patterns corresponding to a single switching of a GaAs avalanche transistor: (b) - upper view of the transistor, with external lighting, (c) and (d) - light emission from the switching channels at different magnifications. The light is emitted from the gap between the emitter and the base ohmic contacts.

These channels have a quasi-periodic spatial distribution, their location varying from one current pulse to another. The characteristic size of a single channel (at half-maximum optical power) typically varied from N 4 to N 8 pm, and the typical number of channels was N 10 - 12. Hence one must conclude that the characteristic

123


area of the device participating in high-current switching is much smaller than its total area and is about - 3 x lop6 cm2. With a current amplitude of 100 A (Fig. 4.34b), this means that the current density in the channels must be as large as N 3 x lo7 A/cm2.

In an attempt to explain the phenomenon of superfast switching, simulations were performed with the ATLAS 2D device simulator (Silvaco Inc.) [119], aiming at a step by step evaluation of the following factors: (a) the difference in electron and hole mobilities, which should be larger than in Si, (b) the effect of the small operating area, and (c) the effect of negative electron mobility.

At the first stage, the simulations were performed with an Si-like dependence of the electron velocities on the electric field (see Fig. 4.3) but with all the material parameters typical of GaAs, including the electron and hole mobilities. The simulations were performed for both a device with a large-perimeter emitter-base interface ( 2 mm, according to the geometry of the experimental chip), and a small-perimeter interface (10 pm), aiming at a characterization of the switching by narrow channels observed in the experiment (the operating area in these simulations was about - 3 . 1 0 - ~ cm’).

The simulation for the device of perimeter 2 mm showed that switching was twice as fast as for a Si transistor, but still much slower than in the experiment. A certain acceleration of the transient in these simulations of a GaAs transistor relative to a Si one is attributable to faster electron penetration into the avalanching region due to higher electron mobility and slower sweeping of holes out of the plasma region, which maintains a high density of the electron-hole concentration in the quasi-neutral region.

A reduction in the perimeter of the device and corresponding increase in the current density causes only a slowing down in the switching process.

Very good agreement between the simulations and the experiment was achieved when the simulations were performed with a real dependence of the electron drift velocity on the electric field, with a negative drift conductivity part included in the v ( F ) dependence (see Fig. 4.1). It has been shown that superfast switching occurs due to the appearance of a number of Gunn domains (up to N 20 domains across a collector region -30 microns in thickness). These domains of huge amplitude (up to - 600 kV/cm) are moving towards the anode and provide a very effective ionization across the volume of the channel in the no collector region.

4.5.2 Breakdown in moving Gunn domain in GaAs: qualitative analysis

J. Gunn discovered in the early 1960s that when the bias V applied to an n- GaAs or n-InP “semiconductor resistor” (i.e. to a semiconductor bar with two ohmic contacts) exceeds the threshold magnitude V t h = F t h L , spontaneous current oscillations appear (Fig. 4.36a). Here F t h is the threshold field for the Negative

124


Differential Conductivity (NDC) part of the field dependence of the electron drift velocity w(F) (Fig.4.1) and L is the sample length [120; 1211. At V > I&, the travelling high field domain nucleates in a sample near the cathode.

30 50 70 90 t (ns)

F I b

Fig. 4.36 domain propagates towards the anode.

(a) Current oscillations across a Gunn diode of length 2 mm [122]. (b) A high field

It propagates towards the anode with the velocity of the order of lo7 cm/s and disappears into the anode. As the domain formation leads to a current drop, the domain disappearance results in an increase in current. The current remains approximately constant while the domain is travelling across the sample (Fig. 4.36 b). A qualitative explanation for the Gunn effect was put forward by H. Kroemer [123].

In a homogeneous sample of the n-type, the electric current is a flux of electrons moving from the cathode to the anode at a velocity w(F). Let us assume that a small fluctuation of field bF appears in the sample and that the field in this small region is slightly higher than the average field F M V/L . Such would be the situation if a small positive charge appeared from the right-hand side of this region and an equal negative charge from the left-hand side. Combining with the external electric field F M V / L , the field of these local charges would create a fluctuation in the electric

125


field bF. Let us suppose that the applied voltage V is equal to the threshold voltage

Vth. In this case the increasing field inside the small region will cause a decrease in electron velocity, due to NDC (Fig. 4.1). This means that the electrons situated inside this region will move more slowly than those located ahead of the front of this region or those moving behind it. As a result, electron depletion within the leading edge and electron accumulation within the trailing edge will both increase with time. Hence, the charge of the “dipole layers” a t the boundaries of the fluctuation will increase, the field in this region will grow, the velocity of the region will decrease, etc. In this manner a small fluctuation can intensify and transform into a high field domain in the NDCpart of the w(F) dependence.

Since the voltage drop across the growing domain increases in time, the electric field outside the domain at given applied bias V decreases. Accordingly, the electron velocity outside the domain decreases (since the electric field outside the domain remains below the threshold Fth), and the current across the sample j = e . n o . w, will also be reduced.

The domain will cease growing when the electron velocity outside it w, becomes equal to the domain velocity u, and then a stable high field domain will move across the sample (Fig. 4.36 b). After the domain reaches the anode, its annihilation causes an increase in the field outside the domain. When the electric field outside the domain approaches the value Fch, and the current density grows to the threshold value jth = e . no . vth accordingly, a new domain nucleates near the cathode and the process repeats itself.

The characteristic time for the growth of a small fluctuation of this kind is determined by the differential dielectric relaxation (Maxwellian relaxation) time 7, :

(4.39)

where p d = dw/dF is the characteristic negative magnitude of the differential mobility in the NDC part of the w(F) dependence.

It is obviously that such a domain can only be formed if 7, is markedly smaller than the transit time tt = L/w. This condition can be written in the form:

or

(4.40)

(4.41)

Equation (4.41) is known as the Kroemer criterion.

126


Taking for a GaAs E = 12.9, v M lo7 cm/s, lpdl M 400 cm2/Vs, we can see from (4.41) that the travelling domains in the GaAs appear a t noL _> (noL)I M

5 x cm-2. Several thousands of original papers, dozens of reviews and many books devoted

to various aspects of this phenomenon have been published since discovery of the Gunn effect (see, for example, [14; 79; 124; 125; 1261 and references therein). Thus we will only summarize here some of the major observations which are important for a qualitative understanding of the results described in Section 4.5.1.

The parameters of the high field domain are determined by the doping level no, the sample length L , and the parameters of the dependence of the electron drift velocity on the electric field v ( F ) . The v(F) dependence in a GaAs over a wide range of fields and temperatures is shown in Figure 4.37.

As one can see from Fig. 4.37, the drift velocity at a given temperature decreases monotonically with growth in the electric field up to extremely high field magnitudes.

The simplified v(F) dependence which is often used to obtain simple analytical estimates for the domain parameters is shown in Figure 4.38.

The following analytical approximation is not infrequently used for the v ( F ) dependence shown in Fig. 4.38:

Field F (kV/cm)

Fig. 4.37 Dependences of the electron drift velocity in a GaAs at different temperatures T (K): 1 - 130; 2, 2' - 158; 3 - 210; 4, 4' - 300; 5, 5' - 340 1127; 1281

(4.42)

127


Fig. 4.38 Simplified v ( F ) dependence. The electron drift velocity decreases with growth in the field at F t h < F < F,, but is constant at F > F,. The shaded areas illustrate the “equal area rule” (Eq. 4.43).

where us = 1.13 x lo7 - 1.2 x 104T (cm/s) 11291. As can be seen, the drift velocity w tends towards us at F >> Fth, and w(F) FZ poF at F << Fth.

For any v ( F ) dependence, the “equal area rule” is valid for a stable travelling domain [130]:

F,

/ [ w ( F ) - w(Fr)]dF = 0

F,

(4.43)

where F, and F, are the maximum domain field and the field outside the domain, respectively (see Fig.4.36 b). The expression (4.43) has a very simple geometrical interpretation, in that the shaded areas in Fig. 4.38 are equal. As seen in Fig. 4.38, the smaller the field outside the domain Fr, the higher is the maximum domain field F,. At Fr -+ Fr min, F, 4 00.

For a domain of high amplitude Fm(Fm >> F,) the equal area rule can be written in a simpler form:

where

(4.44)

(4.45)

128


(the notations are evident from Fig. 4.38). Hence,

F m ( E - Fr min) M F,“ (4.46)

An estimate for GaAs provides F, M 2.8 kV/cm. Analytical expressions describing the high field domain can be obtained for two

limiting cases of a small (no << n,,) and a large (no >> n,,) electron density [130; 13 11 , where

(4.47)

For GaAs ( P O M 7000 cm2/Vs, Fth 3 kV/cm, Fr min M 1.5 kV/cm, F, M

12 kV/cm, and D x 180 cm2/s), n,, M 2 x 1015 cmP3. For Gunn domains in GaAs avalanche transistors (Section 4.5.1), very high

values of the carrier density n o are typical: no = j / e v N lo1’ cm-3 >> n,,, providing the current density N lo7 A/cm2 and Thus we will consider below the limiting case no >> nc,.

In a sample of finite length, a stable high field domain must simultaneously satisfy two conditions: the equal area rule and the voltage balance across the sample:

M lo7 cm/s.

V = F(x )dx = F,L + VdO , i 0 (4.48)

where Vdo is the domain voltage. The necessity for satisfying the conditions (4.43) and (4.48) simultaneously is

responsible for the appearance of a domain of finite (and generally speaking rather large) amplitude F, even at the threshold voltage Vth = FthL (“hard excitation”). If (noL) >> (noL)I (see formula (4.41)) and no >> n,,, the maximum domain field F, >> F,, and the field outside the domain is close to F,,in [131]. This is just one case related to switching in a GaAs avalanche transistor] since given a carrier density no N lo1’ cm-3 and a characteristic length L N cm of the region in which Gunn instability takes place, the noL product is (noL) M 10l6 cm2 >> (noL)1.

At no >> nc, and (noL) >> (noL)1, the expressions for the domain voltage V ~ O , the maximum domain field F,, and the effective domain width d can be written as follows [14; 1311:

(4.50)

129


and

(4.51)

It is important to note that at a given domain voltage V d o the maximum domain field F, increases and the domain width decreases with growth in the concentration

It is obvious that the estimate (4.39) for the characteristic time required for domain formation is a rather rough approximation, as it is impossible to expect the formation of the domain of high amplitude Fm to be adequately described by a small signal Maxwellian relaxation time T ~ .

The domain dynamics are discussed in detail in Refs. [14; 1251. For the case in question, i.e. a large domain amplitude F, >> F,, the dimensionless equation governing domain formation takes the form:

no.

(4.52)

where u = V d ( t ) / V d O is the dimensionless domain voltage and i = t / T f is the normalized characteristic time constant of domain formation. The time constant ~f can be expressed as

where RO = L/ (eponoS) is a low resistance of the diode and C d = EEOS/d is the domain capacitance. With the initial condition ~ l ~ , ~ = 0, the solution of Eq. (4.52) is

(4.54)

Estimates (4.50)-(4.53) are valid, strictly speaking, only if ~f is much larger than the energy relaxation time: ~f >> 7, - lo-’’ s, and if the domain width d exceeds markedly the electron mean free path l o : d >> lo = 0.1pm (see Figs. 1.14, 1.15, 3.31). It is easy to check that these conditions are not satisfied at no 2 1017 ~ m - ~ . For accurate calculations at such high concentrations the Monte-Carlo technique or “temperature model” [132] may be used.

Let us now consider the current-voltage characteristic of a Gunn diode (Fig- ure 4.39). This characteristic follows the v ( F ) dependence j = enov (F) at a small voltage v < Kh, but at V = &h domain instability appears and the current oscillates between the threshold magnitude j t h = enovth and the value of j , = enou,

130


A

vth

Fig. 4.39 ionization in the high field domain. The characteristic is timeaveraged at V > Vth.

Qualitative current-voltage characteristic of a Gunn diode under conditions of avalanche

(ur is the electron field outside the domain, see Fig. 4.38). Hence at V > Kh, the current-voltage characteristic is time-averaged.

At (noL) >> (noL)1, the field outside the domain is close to the F,,i, (see the “equal area rule” and Fig. 4.38), and 21,. M 21, M poFTmin. Taking into account that at L >> d the domain exists in the sample over the major part of its transit time, it is clear that the value j averaged over time is close to the j r m i n = enov, =

enopoFr ,,,in. Since the ratio J&/ j r ,in in a GaAs is about 2.4, an appreciable current “jump” from j = j t h to j M j r m i n is observed at v = &, (left part of Figure 4.39).

With further V increase, the average current passing across the sample remains practically constant and equal to until the maximum field in the domain F, reaches the characteristic field of the impact ionization Fi, and this inevitably causes impact ionization within the high field domain at a sufficiently high bias V . The obvious consequence of impact ionization is band-to-band light emission [133; 1341.

Let us assume first that F, is not too high and that the rate of impact ionization is not very large, so that the characteristic electron-hole pair generation time is larger than the domain transit time [135]. Under such conditions the extra concentration of electrons (and holes) generated during one domain transit is small by comparison with the initial concentration no.

Every subsequent domain, however, will now appear in the sample with a somewhat increased electron concentration (no + Anl). Meanwhile, as seen from the expression (4.50), the maximum electric field F, grows as the electron concentra- t ion n increases. Hence the next domain creates a larger excess carrier density An, > An1, etc. This positive feedback provides incremental growth in both the maximum domain field F, and the excess carrier density n. Consequently, the

131


current may grow to a very high value, provided that a voltage generator is used in the external circuit (line A-A in Fig. 4.39).

If the sample is fed by a current generator, the current (i.e. the rate of electron- hole pair generation) is determined by the external circuit. In this case the intrinsic sample parameters determine the voltage drop across the device. It is evident that a t a high ionization rate the voltage applied to the sample will decrease with any increase in current. This corresponds to the S-type of current-voltage characteristic (averaged over a time interval which is larger than the domain transit time (right- hand part of Fig. 4.39)).

Indeed, the field outside the domain does not depend on the electron concentration and is equal to Frmjn. The voltage drop across the domain V& evidently decreases with n growth, because due to the very strong dependences of the electron and hole ionization rates a, and pa on the electric filed (see Section 1.4.3), an increase of several orders of magnitude in n is achieved with a fairly small enhance- ment of the domain amplitude F,. As the domain width d decreases monotonically with the increase in n, the voltage drop in the sample, V M Fr,i,L + 1/2Fmd, decreases with increasing n (i.e. with increasing in current density).

A current-voltage characteristic of the S-type has been observed experimentally in Gunn diodes in numerous works (see, for example, 1136; 1371. The parameters of this characteristic were calculated in Refs. [138; 1391.

5’-type negative resistance leads as a rule to current filamentation. The current density in the filaments may be very high and stimulated band-to-band light emission from the filaments may occur [140; 141; 1421. The threshold density of electron-hole pairs in the filaments a t which stimulated emission occurs was estimated in Ref. [143], giving reasonable agreement with the experimental data [140; 141; 142; 1441. The stimulated light is emitted from thin filament,s, whereas the spontaneous recombination radiation observed at relatively small current densities is distributed uniformly over the whole cross-section of the sample.

It should be noted that the maximum domain field F, at high current densities observed in the filaments must exceed substantially the characteristic breakdown field F, in GaAs ( F , - 300 - 400 kV/cm 1201).

Gunn domains with a very high maximum field F, were observed in the channel of high voltage discharge (V = 55 kV) excited between two point contacts in a GaAs bulk crystal in Ref. [145]. A large number of the domains (up to a few dozen) were observed in the discharge channel. It should be emphasized, however, that according to the authors’ estimates, the voltage drop across each domain was extremely high (N 1 kV).

Simulation of Gunn domain generation in an n-collector of an n - p - n GaAs/GaAlAs Heterojunction Bipolar Transistor (HBT), performed in Ref. 11461, showed that the Kroemer criterion (4.41) can be satisfied at a high collector current and that Gunn oscillations occur. The conditions chosen for simulation in Ref. [146]: a small collector-emitter bias V,, = 4 V and moderate doping level in the collector

132


layer Nd = ~ m - ~ , nevertheless excluded impact ionization in the domain.

4.5.3 Computer simulations of superfast switching in GaAs avalanche transistor

A comparison of the voltage waveform measured across a GaAs transistor during the switching transient (curve 1) with the result of 1D simulations [119] (curve 2) is presented in Fig. 4.40. The switching device area in the simulations (3 x lop6 cm’) was equal to the total area of the switching channels observed in the experiment [118], and the electron drift velocity was assumed to be monotonically decreasing with increasing electric field up to extremely high field magnitudes (Fig. 4.37).

300

- 250 5 3 200 4

0

c 6)

u

Y * 150

g 100

0 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8

Time (ns)

Fig. 4.40 ing process. The “small” switching area of the transistor structure (3 . simulation corresponds t o the total area of the switching channels observed in the experiment.

Measured (solid line) and simulated (dashed line) voltage waveforms during the switch- cm2) used in the

Despite marked high-frequency oscillations in the simulated collector voltage, the agreement between the simulations with the 1D model and the experiment is very good, and even the switching delay is comparable.

The temporal evolution of the electric field profiles is shown in Fig. 4.41. At relatively small current density j < 1 . 4 j ~ (Fig. 4.41 a), the filed distribution is qualitatively analogous to that in long-collector Si avalanche transistor (cf. Fig. 4.31 b). However, with further increase in current density a qualitative difference is observed in the field profile evolution in Si and GaAs ABTs.

In Si ABTs, a stationary “anode” high field domain appeared at n-n+ boundary, and just the avalanche ionization within this anode domain provided further current growth and low-voltage high-current switching (Figs. 4.30 b, 4.31 b). As seen in

133


Fig. 4.41, in GaAs ABT the “anode” (collector) domain does not appear at this switching stage. Several relatively broad high-field domains appear instead across no collector region, with the amplitude growing in time (cf. Figs. 4.41 b and 4.41 c). It is worth noting that appearance of these domains with relatively small amplitudes does not yet cause any appreciable reduction in the collector voltage (see Fig. 4.40). Actual switching starts at t M 2.98 ns (Fig. 4.41d) with the steepest reduction in the collector voltage corresponding to t M 3.05 ns (Fig. 4.41 f and Fig. 4.41 g). The voltage then tends to saturation at t >= 3.2 ns (Fig. 4.41i), but a still further reduction in voltage is observed within the time range 3.20-3.25 ns (curve 2 in Fig. 4.40, Figs. 4.41 h - 4.41 k).

The temporal evolution of the field domains clearly demonstrates an effective positive feedback which is characteristic of impact ionization in high-field domains (See Section 4.5.2), namely that growth in the domain amplitude causes an increase in ionisation rates, which in turn leads to an appreciable increment in the carrier density. An increase in the carrier concentration causes growth in the domain amplitude Fm, followed by a further increase in ionisation intensity, etc. One can see from Fig. 4.41 that the domain amplitude increases and the domain width decreases with growth in the current density (carrier concentration) (cf. Figs. 4.41 a-h).

A very fast reduction in the collector voltage is obviously caused by drastic narrowing in the width of the travelling Gunn domains, from - 1.5 pm (Fig. 4.41 d) to - 0.1 pm (Fig. 4.41 k). Despite a simultaneous increase in the number of domains from - 5 to - 20, the average voltage per domain is reduced from - 60 V (Fig. 4.41d) to - 5 V (Fig. 4.41 k), thus reducing the total collector voltage from - 300 V to - 100 V (Fig. 4.40).

Carrier generation by high field domains is distributed practically homogeneously across the collector, since the characteristic distance between the domains (ranging from - 6 to - 0.5 pm) can be covered by a domain travelling at a velocity of - lo7 cm/s within a time of 60 - 5 ps, which is less than the total switching time of the transistor by a factor of - 3 to - 40.

The simulations show that after nucleation every domain travels towards the anode with a velocity of about lo7 cm/s. A domain disappears after it travels a distance of LO, a value which ranges from several microns to 20-30 pm for different domains. Hence the travelling domains can be considered “quasi-stable” , because their “lifetime” is much larger than the characteristic time constant of domain formation ~f defined by formula (4.52).

Estimates for ~f in the frame of the local field model, when the transport coefficients D, p and V d are regarded as instant functions of the electric field, give values of 7f at n - 10” - lo1’ cm-3 which are much smaller than the energy relaxation time T, - 1 ps. Strictly speaking, this means that the local field model employed in the simulations cannot be used to describe correctly the processes that occur at the end of switching, when n 2 10” (t 2 3.1 ns, Figs. 4.41, f - k).

Moreover, the situation looks very unusual from the point of view of the “clas-

134


X t=3.077 ns. P=..O7OLi

I 0’’

I Ola

10’’

1 o’6

Fig. 4.41 Electric field (u - k) and carrier density ( I ) profiles across the structure simulated for a transistor of area 3 x lop6 cm2. The time instants corresponding to each profile and the current densities in the units of critical value J” x 1.1. lo3 A/crn2 are shown in the figure. The solid lines in ( 1 ) represent the electron density and the dotted lines correspond to the hole concentration.

sical” Gunn effect. First of all, just one stable travelling domain can usually exist in a sample. Indeed, at a constant bias V applied to the sample, domain nucleation leads to a decrease in the electric filed outside the growing domain to a value below the threshold field Fth. A “multidomain regime” [147]-[149] can be achieved when the bias increases with time very fast. In this case the field outside the domain can even rise despite domain formation. The conventional condition for a multidomain regime [125] is:

135


(4.55)

but in the case studied here the bias applied to the collector decreases in time. Second, Gunn domains of a very high amplitude (F, - 4 - 6 x lo5 V/cm)

provide very large values for the ionization coefficients cri, pi for both electrons and holes (w 1.6 x lo4 - lo5 cm-I). These values are quite comparable to the reciprocal domain width, thus providing a considerable probability of an ionization event occurring within a single domain. This situation differs significantly from the approach adopted in the “classical” theory of Gunn effect, where F, is assumed to be not too high, so that the characteristic time required for electron-hole pair generation is larger than the domain transit time [135].

Third, the theory of the Gunn effect in the presence of free holes developed in Refs. [150; 1511 assumes “ohmid’ hole behaviour, i.e. that Ohm’s law is valid for holes in any electric fields: wp = p,F. It is obvious that this assumption is not valid in the very strong fields under consideration.

Hence one can conclude that further efforts are needed in order to obtain a detailed description of the physical nature of the domains responsible for superfast switching in GaAs avalanche transistors. There is no doubt, however, that the simulations performed in Ref. [119] give a qualitatively correct description of the effect. It should be noted that various examples are known in which the local field model gives a fairly correct description of the effects beyond the strict boundary of its applicability (see, for example, the theory of the “anode domain” in the Gunn effect [152; 1531).

136



Chapter 5

Dynamic Breakdown


5.1 Introduction

As mentioned in Section 2.1, the breakdown scenario depends critically on the ramp d&/dt . Over a very wide range of magnitudes of dVo/dt the conventional (static or quasi-static) breakdown scenario discussed in Chapter 3 is followed, but if dVo/dt ramp becomes extremely large the picture changes dramatically.

To establish the criteria for static and dynamic breakdown, let us consider the situation in which a relatively small voltage Vo << V, (where V, is the static breakdown voltage) is applied to a reverse-biased p - n junction (Figure 5.1, curve 1).

P+

Fig. 5.1 Qualitative field distribution across the base of a reverse-biased p + - R. junction when a large reverse current 10 is flowing through the diode. At every point in the space-charge region (SCR) the field increases in time according to the law: d F / d t = IO/EEOS = ~ O / E E O . Point 20 moves

to the right with a velocity wo = ws-, where J” = eNdWs. j0

3 N

137


The left-hand part of the base region of W length is completely depleted (space- charge (SCR) or depleted region). The electric field in this region decreases linearly with x according to the Poisson equation dFldx = e N d / E E O . The right-hand side of the base, of length ( L - W ) , is neutral (no = Nd).

Let us assume that at a certain point in time a bias with a large ramp d V / d t

is applied to the diode, so that an appreciable current I0 = C- flows across it (where C is the capacitance of the diode). The total current at every point in the SCR will be the displacement current j d :

d V d t

Hence, the field will increases with time at every point in the SCR according to the law

(curves 2 and 3 in Fig. 5.1). The boundary between the SCR and the neutral region moves to the right with

a velocity bounded by the saturation carrier velocity us, while the field in neutral region increases somewhat more slowly than that in the SCR, because the density of the displacement current in the neutral region is equal to the difference between j o and the conduction current density j c = enov(Fn), where Fn(t) is the field in the neutral region:

(5 .3)

Let us denote the instant when the maximum field (the field at the pf-n boundary) reaches the threshold value for avalanche ionization, Fi, as t = 0 (curve 3 in Fig. 5.1). From this instant onwards, avalanche multiplication of the carriers (electrons and holes) begins in the left-hand part of the sample.

The initial (background) concentrations of the carriers nb, p b are typically 8-10 orders of magnitude lower than the base doping level Nd. Meanwhile, to affect the electric field profile across the SCR to a significant extent, the free carrier density must be comparable to Nd (see Eqs. (1.28), (3.10) and (3.11)). Therefore, the shape of the field profile and the rate dF/d t in the SCR will not change within a certain time despite the fact that avalanche multiplication has already started at the emitter-base junction (curves 4 and 5 in Fig. 5.1).

Let us now find the velocity wo of the point xo where the field F = Fi (Fig. 5.1). At instant t = 0, the field distribution in the base is defined by the obvious equation

138

So that at an arbitrary instant t

For F ( z ) = Fi (point 2 0 ) we have:

Dynamic Breakdown 1059

(5.4)

As can be seen, the velocity of the “avalanche point”, i.e. the point which corresponds to the boundary of the “ionization zone”, is directly proportional to the current density j o and inversely proportional to the base doping level.

If vo is less than the saturation carrier velocity w,, the electrons created in the avalanche region (in which F > Fi) will move faster than the point ZO. The carriers generated in the avalanche region will modify the field distribution along the base, and we will in effect be dealing with a conventional static (or ‘Lquasi-static”) avalanche breakdown as considered in Chapter 3.

A qualitatively different situation arises if vo > w,. In this case the LLavalanche point” moves faster than the maximum possible carrier velocity ws, which means that the region to the right of the point zo has no chance ‘<to know” anything about the processes occurring to the left of this point. Until a constant current 10 flows through the diode, the current density in the SCR to the right of the point xo is merely equal to the displacement current, and the field at every point of this region increases with the same ramp dF/dt = ~ O / E E ~ . Unlike the situation in static breakdown, when the main breakdown processes are concentrated at the boundaries of the diode (see Figs. 3.10-3.12, 3.14 and 3.16), in the case under consideration (dynamic breakdown, vo > v, ), the whole base is included in the breakdown process.

Note that the condition of dynamic breakdown wo > w, is fully equivalent to the condition

We encountered Eq. (5.7) earlier while discussing a conventional static breakdown (Chapter 3) as this condition corresponded to an appreciable reconstruction of the field by the free carriers (Sections 3.2.2, 3.2.3). However, in the case of static breakdown the current j , is the conduction current, whereas in the case in question here (dynamic breakdown), the current j o is the displacement current.

139


5.2 Impact ionization front (TRAPATT zone)

Now let us consider the processes taking place in the region to the left of the point x = $0 (Figure 5.2). As mentioned above, the initial density of free carriers, electrons and holes, is 8-10 orders of magnitude lower than the doping in the base Nd. Until the condition n(t) M p ( t ) << N d is satisfied, the maximum field F,,, at x = 0 increases with the same ramp - = - as the field at any other point

in the SCR (curves 1 and 2 in Fig. 5.2). Due to the delay between the instant when the field F, achieves Fi and the instant when the condition n(t) M p ( t ) x Nd is accomplished, the maximum electric field achievable in dynamic breakdown is appreciably larger than that achievable in static breakdown.

d F j o d t EEO

L n I n+

Fig. 5.2 ev, Nd.

Formation and motion of the impact ionization front (TRAPATT zone) at j o > j~ =

As soon as the condition n(t) M p ( t ) M Nd is satisfied at the point of maximum

f ield (x = 0), the - ramp at x = 0 becomes smaller that that at other points in the SCR (curve 3).

Indeed, when the free carrier concentration n, p becomes comparable to Nd, the conduction current j , becomes comparable to the displacement current j d , and the d F l d t ramp at x = 0 is equal to

dF d t

140


where the conduction current density j , = e [n(t)- tp( t )] v, M 2en( t )v , . Nevertheless, the field F ( t ) at z = 0 will still exceed Fi, and the carrier concentration will increase. When j , becomes equal to j o , d F / d t is zero. The carrier concentration, and consequently the conduction current j,, will nevertheless continue to grow.

When the conduction current j , becomes larger than j o , the displacement current j , becomes negative, to support a constant total current j o . The field F at z = 0 then decreases with time (curve 4 in Fig. 5.2) , although remaining larger than Fi, and the multiplication process will continue. Finally, when the electron and hole concentrations become high enough, the electric field is expelled from the region of high free carrier concentration, and a dipole layer of electrons on the right and holes on the left is formed, screening the external field. The field behind the impact ionization front falls to a very small value, FO (curve 5), and a region with the high electron and hole concentrations n T I p~ and a low field FO is created behind the “avalanche zone”. The current density in this region is again equal to j o , although it is not a displacement current but the conduction current.

The resulting impact ionization front moves to the anode with a velocity determined by Eq. (5.6). As this happens the voltage drop across the base decreases (Curves 6-8). The residual voltage drop at the end of the process (curve 9) can fall to a low value of FoL.

Three important circumstances should be noted. First, the maximum voltage drop across the base just before formation of the impact ionization front (curve 3 in Fig. 5.2) can significantly exceed the static breakdown voltage V , GZ (Fi x W i ) / 2 , where Wi is the space-charge width at static breakdown when F,,, = Fi (Fig. 5.2). Second, the voltage drop across the diode falls from a very high value (curve 3) to a rather low one (curve 9) within the transit time of the ionization front across the

base tt N L/uo N - . -. And third, the carrier concentrations n T , p~ behind the

avalanche zone can drastically exceed the doping of the base Nd.

The concept of the formation and spread of a plane impact ionization front in semiconductor devices was formulated for the first time in Ref. [154] on the basis of a computer simulation of the “anomalous mode” of IMPATT diode operation (TRAPATT mode).

A high efficiency mode of IMPATT diode operation was first reported in Ref. [155], being referred to as anomalous because the oscillation frequency was appreciably lower than the conventional frequency of IMPATT operation f = v , / 2 L (see Eq. (3.37)) while the efficiency 77 (> 60%) was much higher than the maximum possible magnitude of 77 for any IMPATT structure (-J 37%). Operation in a multiresonant circuit was a necessary condition for this “anomalous” mode of oscillation. Shortly afterwards this oscillation mode was observed with germanium

L j N

us j o

141


IMPATT diodes 11561. Simulations showed that in multiresonant circuit the “anomalous” mode can

occur when the a.c. voltage of the first harmonic combines with a voltage wave of a higher harmonic (2nd, 3rd or even 12th) to give a sharp L L ~ ~ e r ~ h o ~ t ’ ’ of the bias with a large ramp dV/d t . As a result, an avalanche zone appears in the diode, leading to switching of the diode from the high-resistance to the low-resistance state (Fig. 5.2).

The full cycle of the TRAPATT mode involves one or more conventional IM- PATT periods, formation of the ionization front and its passage across the base, and a recovery period during which the electrons and holes of high concentration generated in the base by the ionization front are removed (swept out) of the base. The magnitude of the field FO in the base (curve 9 in Fig. 5.2) is fairly small during the sweeping out process, and the drift velocity of the carriers is usually notably smaller than the saturation velocity w,. As a result, the carriers are removed relatively slowly, and the term trapped plasma is usually used to describe this part of the TRAPATT cycle. An analytical theory describing the TRAPATT regime was put forward in Refs. 1157; 1581.

It was assumed that, due to their extremely high efficiency, TRAPATT oscillators will be widely used in microwave applications. Their relatively low limiting frequency of operation and extremely high level of high-frequency and low-frequency noise nevertheless prevented any practical utilization of these devices, although the idea of an impact ionization front was successfully used to analyse some other types of superfast switching in semiconductor devices.

5.3 Silicon Avalanche Sharpers (SAS)

Although the analytical theory and numerical simulations of the TRAPATT regime give a fairly good description of its main properties, it is necessary to note that only indirect comparison is possible between the theory and experimental data for these structures. Indeed, TRAPATT structures are “discs” of thickness L N 3 - 5 pm and a diameter d of about 50-200 pm with a characteristic transit time of the avalanche zone of approximately

Besides, as mentioned above, implementation of the TRAPATT mode is possible only in a complicated multiresonant circuit. Consequently, direct measurements of either the transit time of the avalanche front or current-time dependences during the switching-on process are impossible, and the correctness of the theoretical approaches and calculations can only be evaluated by reference to fairly indirect data, comparing the experimental and simulated values for efficiency, the oscillation pe-

142


riod, output power, etc. An experiment described in Ref. [159] appears a t first glance to have reproduced

all the main features of excitation of the impact ionization front in TRAPATT structures, but the diode structure used in the experiments [159] was of much larger size ( L = 280pm, d = 0.2 cm) than that typical of TRAPATT diodes and it was operated in a very simple circuit that consisted, in fact, of the diode and a load resistor Rl connected in series. The expected transit time of the avalanche front transient across the structure, estimated using formula (5.9), was about 1 . 4 ~ s, and this could also be reliably measured.

The samples studied in [159] were p+ - n - nf diodes with base doping Nd =

1014 ~ r n - ~ . Their static breakdown voltage V, was found to be 1600-1700 V. A reverse d.c. bias VO M 400 V (VO << V,) was initially applied to the diode, and a voltage pulse with a high dVo/dt ramp (dVo/dt N (1-2) x 10l2 V/s) was additionally applied in the same reverse direction at a certain instant in time. Under such conditions the maximum voltage Vo,,, - 2000 V can be applied to the diode without any appreciable current in the circuit and the voltage then dropped to a fairly low value within a short time of - 0.5 ns. (The current across the load resistor Rl = 50 Ohm grows during the same time to a value practically equal to Vomax/Rl M 40 A). It was mentioned that the measured switching time of 0.5 ns corresponded to the time resolution of the circuit rather than to the real duration of the process.

The recorded time of 0.5 ns for the voltage drop across the diode corresponded to the avalanche front velocity vo M L/tt M 5.6 x lo7 cm/s, which is larger than the characteristic ionization front velocities observed in the TRAPATT regime but still comparable to them. An improvement in the circuit time resolution has nevertheless allowed a switching time of 5 100 ps [I601 to be recorded later. This means that an estimate for the velocity of the ionization front would be vo 2 25 x lo7 cm/s. As will be shown later, such a high value for the plane ionization front velocity 2r0

cannot be explained in the framework of a TRAPATT-like model even with much higher dV/dt ramps than those used in the experiments.

The devices in which this phenomenon was observed were called Silicon Avalanche Sharpers (SAS), since at an input dV/dt ramp of about (1 - 2) x 10l2 V/s the output dV/dt ramp exceeds - 2 x 1013 V/s. They are also sometimes termed “Diodes with Delayed Breakdown” (DDB) , since well developed impact ionization does not take place at the instant when the bias reaches the static breakdown value of V, but rather the breakdown is delayed by the time td which is necessary to multiply the carriers from small densities 726, p b to concentrations of n, p - Nd.

143


5.3.1 Computer simulations and comparison with experimental results

The first computer simulation of SAS operation, as reported in Ref. [161], was performed for a p+ - n diode with a base length L = 200pm and a doping level N d = 1014 cm-3 (Fig. 5 . 3 ) .

Fig. 5.3 Field and carrier density profiles across the base of a p+ - n diode during the superfast switch-on process. 1 , l‘, 1” - t = 0; 1 - F ( z ) , 1’ - n ( z ) , 1” - p ( z ) ; 2-5 - field profiles at different instants: 2 - t = 1.23 ns, 3 - t = 1.47 ns, 4 - t = 1.6 ns, 5 - t = 1.82 ns (compare with the temporal dependences of the current and voltage (Fig. 5.4)). L = 200pm, s = 2 x cm2, Nd = 1014 cmP3, v = [400$5000sin(4 x lost)] V, Rl = 100 Ohm.

In accordance with the experimental conditions, a reverse bias of 400 V was applied to the diode and a load resistance Rl = 100 Ohm connected in series with it (curve 1). At this bias the SCR width W is N 70pm and the length of the neutral region ( L - W ) is N 130pm. The concentration of free carriers n b , p b

in the SCR is determined by the leakage current j co : j c o x e w , ( n b f p b ) . The magnitude of jco at room temperature in the structures studied in [159] was about

A/cm2, which corresponds to n b , p b N“ lo5 - 10‘ crnp3. The electron concentration in the neutral region no = N d = 1014 and the hole density p = n:/nO M 2 x 10‘ cmp3.

At t = 0, the bias V = Vo + V(t) was applied to the diode and series resistance, where V(t) = VO + VI sinwt at t < 7r/2w and V = VO + V1 at t > 7r/2w. This waveform corresponds well to the experimental conditions (cf. Fig. 5.5).

As seen in Fig. 5.3, the voltage across the diode reaches its peak value V,,,, - 2300 V at t = 1.23 ns (curve 2 in Fig. 5.3, cf. curve 3 in Fig. 5.4). After that instant

-

144


of time, the plane impact ionization front is formed, and its propagation across the base causes the appearance behind the front of a quasi-neutral region with high carrier (electron and hole) density (curves 3-5). The voltage across the diode decreases and the current increases (cf. curves 2 and 3 in Fig. 5.5). It is interesting to note that the “recovery” process, i.e. removal of the electrons and holes from the base, starts before the ionization front reaches the anode and disappears (see left-hand parts of curves 4 and 5 in Fig. 5.3). The slope of the field profiles at the left boundary of the base dF/dx is determined by the electron density nT in the quasi-neutral region. Comparison of these d F / d x slopes with the slope of the F ( s ) dependence at the front of the avalanche zone (which is determined by the doping level Nd) shows clearly that the carrier density behind the ionization front exceeds Nd by a significant margin ( n ~ > > Nd) .

The transit time of the plane impact ionization front across the base is approximately (1.82 - 1.23) = 0.59 ns, which corresponds to the average velocity of the impact ionization front wo N“ 2 x lo-’ (cm)/0.59 (ns) N“ 3.4 x lo7 cm/s, i.e. vo M 3.4 21,.

Comparing the simulation results with the experimental data presented in

0 2 4 Time f (ns)

Fig. 5.4 (curve l), current (curve 2) and voltage across the diode (curve 3).

Calculated time dependences of the bias applied to the diode and the series load resistance

145


2400

A 1600

h z 0.2 2 - 800 9 d

0 1 2 3 4 5 6 7 8 9 Time t (ns)

Fig. 5.5 (curve l ) , current (curve 2), and voltage across the diode (curve 3) [162].

Experimental time dependences of the bias applied to the diode and series load resistance

Fig. 5.5 (see also [159; 160; 162; 163]), one can conclude that in terms of the plane impact ionization front approach, the switching process can be divided into two stages. Fairly good agreement between the simulation and the experiment is observed at the first stage, which starts from the application of the dV/d t ramp and ends when the voltage across the diode reaches its peak value Vo,,,. The second stage (which is the most important one), the very fast voltage drop across the diode, cannot be adequately described in terms of the plane impact ionization front approach.

Let us first consider the main properties of the first stage. A reduction in the d V / d t ramp causes a reduction in the current amplitude Imax, and the switching effect disappears a t d V / d t 5 10l2 V/s in both the simulations [161] and the experiment [162; 1631. The calculated characteristic values for the delay time t d and the voltage peak VO max also agree fairly well with the experimental data. Intensive illumination of the diode (which causes a marked increase in the free carrier concentration, and correspondingly in the leakage current) , suppresses the switching in the experiment 1162; 1631. Accordingly, an increase in the carrier densities n b , Pb to N lo1' cm-3 in the simulation causes pronounced suppression of switching, while the switching effect disappears completely at 726, pb - 10l2 ~ m - ~ .

A qualitative difference between the simulated and experimental results exists at the most important stage of fast growth in the current with a reduction in the voltage, however. This primarily concerns the speed of switching. As was observed experimentally, the switching time does not exceed 100 ps even at relatively low magnitudes of d V / d t 2 10l2 V/s [160]. This transit time tt corresponds to the velocity of the ionization front vo 2 25 x lo7 cm/s. The simulated velocity of the front grows as dV/dt increases; however, the simulated velocities remain significantly lower than the experimental value even at much higher values for the

146


d V / d t ramps. For example, the switching process was calculated in Ref. [164] with d V / d t values of up to 4 x 1012 V/s. However, even at so high d V / d t ramp, the calculated velocity of the front remains still much lower than that experimentally observed value (Fig. 5.6).

0 1

Fig. 5.6 Calculated time dependencies of the voltage at the diode during the switching process at different dV/dt ramps (V/s): 1 - 1.6 x 10l2 V/s; 2 - 2 x 10l2 V/s; 3 - 2.6 x 10l2 V/s; 4 - 4 x v/s. L = 150pm, Nd = loi4 ~ r n - ~ [164].

Besides, the residual voltage drop at the end of the switching process increases in the simulations as the d V / d t ramp increases. At d V / d t = 4 x 1OI2 V/s (curve 4 in Fig. 5.6), the calculated value of the residual voltage drop reaches - 800 V, although such a high value for the voltage drop has newer been observed experimentally, even with a significantly higher switching speed (cf. curve 3 in Fig. 5.5).

The temporal dependence of the voltage across the diode as calculated in Ref. [161] is non-monotonic, i.e. there is a pronounced “burst” (see also curve 3 in Fig. 5.4). As has been shown in Ref. [164], the existence of this “burst” corresponds to the formation of the initial region of electron-hole plasma behind the rear edge of the ionization front. It is worth noting, however, that no non-monotonic behaviour of the voltage during the switching process has been observed experimentally.

The details of the V(t) characteristic depend considerably on the base length L (Fig. 5.7). In short diodes all the majority carriers have been removed from the base of the diode before the ionization front reaches the n+ contact. In this case the front continues to propagate up to the moment when the whole base is filled with electrons and holes of concentration n ~ , p ~ . In long samples, however, the ionization front must inevitably run down the neutral region, because the velocity of the left boundary of the neutral region is bounded by the saturation carrier velocity 21,.

The fairly good agreement between the main experimental results and the simulation for the first stage of the switching process attests that the ionization front forms and starts its propagation in plane form. The difference between the the-

147

1068 Breakdown P h e n o m e n a in Semiconduc tors and S e m i c o n d u c t o r Devices

4

3 h

& 2 b

1

0

Fig. 5.7 Calculated time dependences of the voltage across the diode during the switching process for different base lengths L (pm): 1 - 50; 2 - 100; 3 - 150; 4 - 200; 5 - 250. Nd = 1014 ~ r n - ~ , d V / d t = 2 x 1OI2 V/S [164].

ory and the experiment for the second stage of the process demonstrates, however, that the plane ionization front is subsequently transformed into some other kind of breakdown phenomenon.

5.3.2

Let us consider the propagation of the plane ionization front in a sample of length L and transverse size d ( L << d, inset in Fig. 5.8a). A schematic representation of the field profile in the moving plane ionization front is shown in Fig. 5.8. One should bear in mind that the impact ionization takes place only in the ionization region of length l f , where F > Fi (If << L << d). Let us assume that Q = ai = pi, and define a mean ionization rate 6 (across the ionization region):

Stability of the plane ionization front

(5.10)

The velocity of propagation of the front can then be evaluated qualitatively as follows [165; 166; 168; 169; 1701. The carrier density at the rear edge of the plane ionization wave nT can be expressed as

nT N n b exp(c?iv,.rf) (5.11)

where n b is the background electron concentration ahead of the impact ionization front and rf = I f /vo is the time during which the front travels a distance I f . We then have

148

F+ I I I < . I I I I,

Fi I

I I

FO 4 ; L I

I I


(I

A

h Avalanche “plane”

L-XO

0 L x

Fig. 5.8 of the plane ionization front (cf. Fig. 5.3).

Schematic diagram of the field distribution across the base of length L during propagation

(5.12)

On the other hand, an estimate for the electron concentration behind the impact ionization wave nT can be obtained if one takes into account that its propagation is a self-congruent process consisting of carrier generation in the ionization region of length 1, with a characteristic frequency of about &I, and of the extrusion of the electric field out of the region situated behind the ionization front during the differential Maxwellian time 7,. This means that the value for 7, should be of the order of

(5.13)

And

149


(5.14)

The expression (5.14) is valid for the case in which the drift velocity in the channel behind the ionization front obeys Ohm's law.

For the case in which the carrier velocity reaches its saturated magnitude v,, the value of nT can be found from the condition that the total charge flowing through the ionization region during the time interval ~f must be equal to the charge at the rear edge of the ionization region. As we have already seen in Section 5.2, the field behind the ionization front drops when the charge of the electrons and holes p N J i d t that has accumulated in the ionization region separates out spatially to form the rear wall of the electric field domain (Fig. 5.3). The surface charge p, is obviously equal to EEOF. Thus

(nT >> n b ) and

E E O & F nT N -

e

Here E is the mean field across the ionization region:

(5.15)

(5.16)

F = - Fdx > Fi (5.17) - ' S

lf lf

As one can see from the expression (5.12), the ionization front velocity wo grows as the ionization rate 6 increases, i.e. when the electric field in the ionization region grows. The velocity vo also increases with increasing length l f .

5.3.2.1

Curve 1 in Fig. 5.9 corresponds to the beginning of the propagation of the ionization front (cf. Fig. 5.8 b). Let us assume that the front has been distorted by a small fluctuation, as shown in Fig. 5.9, curve 2. Within a small part of the front with a characteristic size of A/2 (A << L << d , short wavelength fluctuation), the velocity of the ionization region occasionally becomes larger than that across the other parts of the plane ionization front. Now let us see what happens to this fluctuation.

It is clear from the simple electrostatic consideration that the electric field F near the fluctuation is somewhat larger that in the other parts of the ionization front (curve 2 in Fig. 5.9).

Sh.ort-wavelength instability of the plane ionization front

150


I *’

1 1 t2 t3

0 Distance x

Fig. 5.9 Schematic diagram of the field distributions under conditions of short-wavelength instability (cf. Fig. 5.8 b). The positive feedback accelerates the propagation of the fluctuation and re- tards the propagation of other parts of the front. As a result, “protuberance-like’’ short-wavelength instability can provide a transformation of the plane ionization front in a “streamer-like” discharge.

As a result, the ionization rate sic increases (see (5.10))] and in accordance with Eq. (5.12), the velocity of the fluctuation ZI increases as well. In turn, the velocity of the other parts of the ionization front decreases. This is because with an increase in F the concentration of the electrons and holes nT = p ~ , and consequently the current density behind the fluctuation] will increase (see (5.16), (5.17)). Hence, if the current generator is used in an external circuit, an increase in the current in one part of the structure will inevitably cause a reduction in the current in other parts of the sample. A small fluctuation will thereby grow and transform into a “protuberance-like” form [160) (curve 3 in Fig, 5.9).

A transition from a plane ionization front to a streamer discharge [165; 166; 1671 is apparently possible, provided that the rise time of the short-wavelength fluctuation is sufficiently small. It is well-known that the spread velocity of streamers can be as high as N lo9 cm/s and the electron-hole density in the channel behind the streamer head can reach 2 10l8 ~ m - ~ . Transition from the plane ionization front to streamer discharge could thus explain both the short switching time (- 100 ps) in long diodes ( L - 300 pm) and the relatively small values for the residual voltage drop after switching.

Unfortunately, no analytical or numerical calculations of the short-wavelength instability of the plane ionization front have so far been produced.

151


5.3.2.2

The theory for the long-wavelength instability of the plane ionization front is presented in the papers by Minarskii and Rodin [168; 169; 1701.

In Ref. [168] the long-wavelength instability of the plane ionization front is considered for TRAPATT-like diodes in which the base region has been depleted completely, even at a very low initial constant bias Vo before the d V / d t pulse is applied (Fig. 5.10).

Following Ref. [168], let us consider a transverse fluctuation of the position of the front xf with a characteristic length X that is much larger than the length of the sample L. (It should be noted that the width of the samples under discussion, d N S1/' N 10-1 cm, is usually much greater than L. Hence, under conditions of long-wavelength instability, the condition L << X < d can be satisfied).

Such fluctuation, unlike short-wavelength fluctuation does n o t cause a n y distor- t ion of the field ahead of the front. Interaction between the fluctuation and the remaining part of the structure occurs only through the external circuit.

Let us assume that in the fore-part of the structure shown in Fig. 5.10 (0 < y < d/2), the ionization front passes ahead of the front in the rear part of the device (d/2 < y < d ) by a value of Z = zf1 - ~ f 2 .

The same voltage is applied to both the leading and lagging parts of the structure, but as follows from the sample geometry, the field profiles F~(z) and F ~ ( z ) must be different. The maximum field F,,, and, consequently, the length of the ionization region I f in the lagging part of the structure, should be smaller than those in the leading part: (Fmz < Fml) , ( l f z < lf l) .

To make this important statement even clearer it is worth comparing curves 3 and 4 in Fig. 5.3. It is easy to check that the voltage applied to the diode at t = 1.47 ns (curve 3) is larger than that applied at t = 1.6 ns (curve 4). Nevertheless, the maximum field F,,,, and consequently the length of the ionization region l f for curve 3, are smaller than those for curve 4. The difference in F,,, and l f values would obviously be even larger if the voltages across the structure corresponding to the two instants (curves 3 and 4 in Fig. 5.3) were equal.

Long-wave length instability of the plane ionizat ion f ront

The difference I f 1 - l f z can be written as:

As follows from expression (5.12), the smaller the length of the ionization region l f , the lower is the velocity of the front 210. This means that the lag 2 increases with time:

dZ Z d t 7( t )

- (5.19)

where

152


Fig. 5.10 Schematic diagram of the field distribution in the base of the diode under conditions of long-wavelength instability. It is assumed that the ionization front in the forepart of the diode (0 < y < d / 2 ) passes ahead of the front in its rear part ( d / 2 < y < d) .

(5.20)

The logarithmic component in Eq. (5.12) is disregarded in (5.20). If the lag 5 becomes larger than Zf, the field in the lagging part of the sample

falls below the characteristic field of ionization Fi, and the lagging part stops. After that, all the current passes through the moving part of the front.

At F ( L ) x Fi/3, the time constant ~ ( t ) is approximately equal to the time required for the plane ionization front to move a distance equal to its own length 1,. At F ( L ) x Fi, (which corresponds to a low doping level in the base, and consequently high velocities of the front WO), the value of T is very short and is determined by the Maxwellian time T, (Eq. (5.13)). At F ( L ) << Fi, the value of T is large, and the plane front must be classified as stable. The corresponding distribution F ( z ) does not satisfy the approximations of the model in question, however.

The authors of Ref. [168] believe that the stratification of the plane ionization front considered in the framework of the model can explain the peculiari-

153

1074 Breakdown Phenomena in. Semiconductors and Semiconductor Devices

ties of superfast switching in GaAs diodes [171]. The punch-through criterion, L < ( 2 ~ ~ o V / e N d ) ~ / ~ , was not satisfied in the samples investigated in Ref. [171], however.

The long-wavelength instability of the ionization front for diodes with a non- depleted base when the punch-through criterion L < (2€€0V/eNd)~/’ is not satisfied has been studied in Refs. 1169; 1701. Such “long-base” structures exhibit several peculiarities which have no analogues in TRAPATT structures (or in the “short-base” structures considered in Ref. [168]). The main effect observed in these structures is “stabilization” of the plane ionization front with respect to long-wave instability, owing to presence of the neutral region. Two new phenomena can be also distinguished: slowing-down of the ionization front due to voltage redistribution between the neutral region and the space charge region (see Fig. 5.3), and the avalanche breakdown in neutral region.

Analytical estimates for the criteria of stopping the ionization front, breakdown in the neutral region and the instability increments at different values of the ratio L / s f (see Fig. 5.10) have been obtained in [169; 1701. One can conclude, however, that long-wavelength perturbations cannot explain the very short switching times corresponding to the extremely high effective velocity of the ionization front ‘UO 2 25 x lo7 cm/s.

5.3.3

Considering the operation of Silicon Avalanche Sharpers (SAS) one faces the fundamental problem of “initial carriers”. Indeed, to initiate the formation and further propagation of the ionization front, it is necessary to have a certain number of free carriers (electrons and/or holes) in the region of a strong electric field F > Fi.

It should be noted first of all that it is totally insufficient to have just one free carrier in an ionization region of length If and volume Vf - If x S. In such a situation breakdown will occur only in a very narrow channel, and as a rule irreversible failure of the structure must be expected. This was demonstrated experimentally with the very ”short-base” p - i - n silicon diode [160]. With a length of the i-layer L = 120pm and a specific resistance of the base material p of 270 Ohm.cm, the punch-through of the base region occurred at a bias Vo, M 200 V. The voltage of the stationary breakdown was about 2800 V. A short pulse of 3 ns duration was applied to the diode in addition to the constant bias VO = 1000 V. When the total voltage across the diode reached approximately 5000 V, only a displacement current was passing through the diode. Such a “currentless” state existed for 3 ns. With a further increase of - 10% in the voltage, an irreversible breakdown occurred.

Thus, the condition for a reversible and reproducible breakdown should be written in the form pVf >> 1. With Vf - I, x S - loe3 cmx10-2 cm2 N cm3, the required electron (hole) concentration is p >> lo5 ~ m - ~ .

As mentioned above, the leakage current density j C o in an SAS a t room tem-

The problem of the initial carriers

154


perature is approximately - A/cm2 [159], which corresponds to a hole (electron) density of about lo5 - lo6 ~ m - ~ . As seen, the free carrier concentration provided by the leakage current even at room temperature is close to the lowest limit of the required value of p .

The equilibrium minori ty carriers (holes) located in the neutral region ( L - W ) (see Fig. 5.3) form the second evident source of carriers which are able to initiate the breakdown. When the d V / d t ramp is applied to the diode, the electrons move to the right whereas the holes move to the left, towards the high-field domain (Fig. 5.3). The density of these holes at room temperature is p = $/Nd M 2 x lo6 cmP3 for the base doping level N d = lOI4 ~ m - ~ .

An experiment has shown [163], however, that reversible and reproducible superfast switching is also successfully observed even at T = 78 K, when the concentration of free carriers provided by both sources: leakage current and minority carriers in the neutral region, is reduced by many orders. As mentioned in Ref. [163], variation in the sample temperature from 78 to 350 K does not entail any qualitative change in the characteristics of the switching. The decrease in temperature causes just a minor reduction in the delay time t d (not more than 30 %) and a decrease in the maximum achievable voltage V,,,. This result demonstrates quite clearly that neither the leakage current nor minori ty carviers in the neutral region can play a dominant role in the initiation of superfast switching or in the formation and spread of the ionization front.

It should also be noted that at a low free carrier concentration ( p V f << 1) the free carriers are simply absent from the ionization region for fairly long intervals in time. At a given leakage current j C o , the number of carriers passing through the cross-section of the device S per second is equal to N - I / e - ( j C o x S) /e . The value of jc , typically does not exceed lo-'' - A/cm2 at T = 78 K. Hence, at this temperature N N lo5 - lo6. The carrier transit time through the space-charge region at saturation velocity, t,, is about lo-' s , so that the probability that just one carrier will be presented in the high-field region during the switching delay time (N lo-' s) can be estimated as being N x t , N lov4 - loP3.

In addition, the instant at which an initiating carrier appears in the ionization region is in no way synchronized with the instant at which the d V / d t pulse is applied. Hence one could expect a huge dispersion between the beginning of the d V / d t pulse and the beginning of the switching (jitter). Meanwhile, the experimentally measured jitter does not exceed N 50 ps over the whole temperature range from 78 to 350 K [160].

One more possible source of initiating carriers is considered in Ref. [164]. As seen in Figs. 5.1 - 5.3, the electric field in the neutral region increases when the d V / d t ramp is applied (although this increase takes place more slowly than that in the space-charge region). Thus one can assume that the holes generated due t o impact ionization in the neutral region can ensure the presence of initiating carriers, provided the electric field is sufficiently high. Indeed, the calculations performed in

155


Ref. [I641 show that this mechanism can in principle be brought into effect. When an initial reverse bias VO N 400 V is applied to the structure, however, impact generation by the holes in the neutral region becomes essential only at very high magnitudes of the d V / d t ramp, exceeding - 2.5 x lo1’ V/s. Hence, in real SAS operation modes y o N 400 V, d V / d t M 1.0 x 10l2 V/s) this mechanism i s of practically n o importance.

It should be noted that the simulations performed in the framework of the conventional, so called “continuum approximation” approach (see Equations (1.29)) may give qualitatively erroneous results. Let us assume, for example, that a small number of holes is generated in the system by means of a certain physical mechanism (leakage current, minority carriers, etc.). These holes move towards the region of the strong field F, where their concentration increases exponentially (Figs. 5.1 - 5.3), reaches the physically relevant level p V f >> 1 and initiates switching. Con- sequently, a simulation predicts that the ionization front is excited by any concentration of initial free carriers, even if it corresponds to just a fraction of a hole (electron).

As mentioned in Ref. [164], such a result might be either quantitatively or qualitatively erroneous. First, the simulation can predict an excessively early start for the ionization front, which will consequently propagate at a lower bias voltage. Sec- ond, the continuum approximation can predict the formation of the ionization front under conditions when, in reality, there are no carriers in the system to initiate its formation and propagation. An approach which allows this problem to be partially overcome is considered in Ref. [164].

The radiation recombination coefficient in Si is extremely small, so that pho- toionization i s negligible in an SAS.

Band-to-band tunnelling can generally speaking provide the appearance of carriers which are able to initiate switching at very high peak magnitudes of the electric field F,,,, and calculations show [I721 that this mechanism for the generation of initiating carrier can make an appreciable contribution to switching, provided the d V / d t ramp is of the order of 10 kV/ns and F,,, - lo6 Vjcm, i.e. at the magnitude of maximum voltage across a diode V,,, of about 8 - 10 kV. These values of F,,, and V,,, exceed several times corresponding experimental magnitudes in conventional SAS modes. Therefore, band-to-band tunnelling cannot play a n important role in a conventional SAS.

However, extremely fast switching (- 200 ps) of 20 reverse-biased Si diodes connected in series in a tunnel-assisted delayed breakdown regime has been reported in Ref. [173]. The d V / d t ramp was about 10 kV/ns per structure, and the switching occurred with a delay of N 1 ns when the maximum electric field F,,, in each diode reached approximately lo6 V/cm. The amplitude of the output pulse was N 150 kV across a 50 Ohm load resistance, and the maximum current density across the diode structures was estimated to be

Impur i ty tunnelling could be one possible mechanism for initiating carrier gener- 13 kA/cm2.

156


ation in a conventional SAS, but a rigorous consideration of this mechanism requires detailed information on the type and concentration of the unintentionally introduced impurities. The possible contribution of this mechanism to the switching of an SAS has not yet been investigated.

To summarize, one may conclude that the problem of initiating carriers remains an open question, and additional experiments should be performed to establish the nature of these in an SAS.

In spite of many unsolved problems, the SAS has found practical applications in sub-nanosecond high voltage generators (see, for example [174]). A series of fast power generators with pulse amplitudes from 1 kV to 200 kV, pulse rise times from 100 ps to 1 ns and a pulse width from 0.1 to dozens of nanoseconds has been developed.

5.4 GaAs diodes with delayed breakdown

Superfast switching in reverse-biased GaAs diodes was reported for the first time in Ref. [175]. GaAs p + - po - no - n f diodes with a thickness of the n o region L = 170pm, operation area S of 5 x cm2 and static breakdown voltage V, =

1000 - 1200 V were tested in the experiments. The reverse voltage pulses with a rise time of about 2 ns generated by a Si SAS were applied to a GaAs diode.

The conditions under which the effect was observed differed somewhat from the typical conditions for a standard Si SAS. (Figure 5.11). As seen in Fig. 5.11, the switching occurs at a constant applied voltage (curves 1,l' and 2,2' in Fig. 5.11, cf. Fig. 5.5), although the switching characteristics are qualitatively similar to those in a Si SAS. There is a time interval of about 2 ns in which the current across the diode is very small (the delay time), and after that it increases sharply while the voltage decreases from - 2000 V to 100 - 200 V.

The switching time tt of about 200 ps corresponds to an avalanche front velocity wo M L/tt M 8.5 x lo7 cm/s. The maximum voltage achieved, V,,,, was 3400 V (curve 3 ) , and the switching time tt in this case was - 70 ps (WO M 25 x lo7 cm/s).

The maximum current density j,,, achieved in the process of superfast switching in Ref. [I751 was - lo4 A/cm2 ( I M 50 A, S = 5 x lop3 cm2). As the voltage across the sample after switching, Vmin, was - 100 - 200 V, the average electric field across the base after the switching, Fo, can be estimated to be V&/L M lo4 V/cm. The drift of the electron and hole velocities 'ud in this field is approximately lo7 cm/s, and the carrier concentration behind the impact ionization front n T , p~ is about nT M p~ M jmax/ewd - 6 x 1015 cmp3.

As mentioned in Ref. [176], population inversion can be achieved in a GaAs sample after switching at a current density as high as j,,, - lo6 A/cm2 and at carrier concentration behind the impact ionization front n T , p T 2 lo1' cm-3 at room temperature. Under these conditions the stimulated band-to-band light emission can occur.

157


n

9 , 8 30

u 0 2 4 6 8

Time t (ns)

Fig. 5.11 resistance Rl = 50 Ohm (curves 1-3) and current through the diode (curves 1’-3‘) (cf. Fig. 5.5).

Experimental time dependences of a reverse bias applied to the diode and serial load

Superfast switching with a current density j,,, of N 2.8 x lo5 A/cm2 was observed in GaAs diodes with an operation area S = 6 x cm2 ( L = 200pm) in Ref. [176], where voltage pulses with a d V / d t ramp of 3 x 1013 V/s were used. The peak voltage before switching V,,, was N 5 kV at room temperature and N 4 kV at T = 77 K, the residual voltage drop after switching, Vmin, was 300 V and the switching time tt was N 70 ps (VO M L/tt M 30 x lo7 cm/s). The carrier concentration behind the impact ionization front n T , p~ was estimated to be 2 x 1017 ~ m - ~ . Such values of TIT, PT are too low to provide population inversion at room temperature, but the conditions for population inversion are satisfied at T = 77 K.

Stimulated band-to-band light emission from a GaAs diode under conditions of superfast switching was observed at room temperature in Ref. [177]. The samples were chipped off the wafer of a large area, with no special protection against surface breakdown.

Pulses with a d V / d t ramp of 5 x 1013 V/s and amplitude of up to 7 kV were used to switch the samples on. The maximum voltage prior to switching was about 5.5 kV. (This value exceeded the static breakdown voltage V , by a factor of - 3). The duration of the fast part of the V ( t ) dependence did not exceed - 100 ps, which corresponded to the average velocity of the ionization front vo M 20 x lo7 cm/s. The maximum current density j,,, was about N 2 x lo6 A/cm2 at room temperature.

The pattern of band-to-band recombination emission recorded from the screen of an electro-optical Image Convector (IC) at a single pulse of 1 ns duration and a current density after switching j,,, N 1.1 x lo6 A/cm2 is shown in Fig. 5.12 a. The IC operates in the static mode without any time sweep. As can be seen, the picture

158


demonstrates fairly homogeneous light emission from the side of the 150 pmx 100 pm sample.

a

b c

1 1 1 " ' " " ' I

0.4 0.3 02 0.1 0 Time t (ns)

Fig. 5.12 a) Photography of band-to-band recombination emission (GaAs sample, single voltage pulse of 1 ns duration, current density after switching j,,, N 1.1 x lo6 A/cm2 [177]). An image converter was operated in the static mode. b) Chronogram of the emission at current density after switching j,,, N 1.1 x lo6 A/cm2. c) Chronogram of the emission at current density j,,, N 2.0 x lo6 A/cm2 (Note the change in sweep speed).

Chronograms of the emission from the same region of the samples obtained with the same duration of pulse applied (N 1 ns) at two sweep speeds and at two magnitudes of the current density after switching j,,, are shown in Figures 5.12, b, c.

The pattern shown in Fig. 5.12 b corresponds to j,, N 1.1 x lo6 A/cm2, and it is seen that practically homogeneous emission intensity is set up within a time of - 1 ns. The characteristic fall time of the emission intensity is about 10 ns. As noted in Ref. [177], the emission from the other sides of the sample has virtually the same character. A reduction in the current density to j,,, N 4 x lo5 A/cm2 does not essentially alter the character of the emission.

The chronogram shown in Fig. 5.12 c was recorded at the same duration of the applied voltage pulse but with a higher dV/d t ramp and accordingly a somewhat larger current density j,,, N 2.0 x lo6 A/cm2. As seen from Fig. 5 . 1 2 ~ ~ the

159


duration of the intensive light pulse in this case is two orders of magnitude shorter than that in Fig. 5.12 b, and a very sharp rise in the emission brightness takes place. This brightness enlargement and the reduction in the duration of the light pulse (at the same duration of the applied voltage pulse) are interpreted in Ref. [177] as manifestations of a transition to a regime of stimulated emission.

The data reported in Ref. [I771 are considered by the authors to provide proof of the statement that the switching in GaAs diodes occurs homogeneously across the entire diode area. One should take into account, however, several circumstances which do not allow this statement to be regarded as entirely proven.

First of all, as mentioned by the authors of Ref. [177], one could expect surface breakdown along the non-protect,ed surface of the reverse-biased diode in the samples in question. It is assumed in [I771 that surface breakdown has no time to develop within the extremely short time that corresponded to the experimental conditions. There are, however, no arguments to support the statement that the development of surface breakdown takes a longer time than that of volume breakdown.

Then, due to the re-absorption of radiative recombination (see Ref. [178] for references) the light emission from the different cross-sections of the diode should be observed from the front surface with fairly insignificant attenuation. This means that the experimentally observed “homogeneous” emission pattern may appear as a result of the superposition of the radiation emitted by several streamer or quasi- streamer channels. Channels of this kind were experimentally observed in Ref. [171] in superfast switching in GaAs diodes.

cm2 ( L = 45 - 50 pm, donor concentration in the base Nd - (5-8) x 1014 ~ m - ~ ) were studied in Ref. [171]. The static avalanche breakdown voltage Vi was - 350 - 400 V. The shape of the samples (a), the time dependences of the current with different d V / d t ramps (b) (cf. Figs. 5.5, and 5.11 b), and the distribution of light-emitting regions over the surface of the sample on the face of a mesa structure (c) are shown in Figure 5.13.

The diodes investigated in [I711 have a thinner base region than the diode structures studied in Ref. [176] and are partly protected against surface breakdown by their edge profile (mesa structure). Nevertheless, the experiment has shown that both static and dynamic breakdown occur on the surface of the mesa structure.

The d V / d t ramp, even for curve 1 in Fig. 5.13 b, is as large as - 3 x lo1’ V/s, but the switching time is somewhat larger than that typical of a standard SAS. This result can be associated with a higher base doping level N d - (5-8) x 1014 cm-3 and a lower maximum voltage V,,, across the diode prior to switching (cf. Fig. 5.11). The current rise time for curve 1 in Fig. 5.13 b is about 200 ps, which corresponds to an average velocity of the ionization front wo w 2.5 x lo7 cm/s. This value only slightly exceeds the maximum possible electron velocity in GaAs (see Fig. 4.37). Assuming homogeneous switching across the whole diode area, the current density after switching can be estimated to be j - I / S N 5 x lo3 A/cm2. Direct observation of the light emission with an infrared (IR) image converter shows, however, that

GaAs diodes with an operation area S of about 7 x

160


cathode - T a

a300 p a p+ I +

anode

C I*

1 2 3 4

Fig. 5.13 b) Time dependences of the current through the diode with different d V / d t ramps (cf. Figs. 5.5 and 5.11 b). The rise time of the voltage pulse (applied to the diode and a 50 R load resistor connected in series) is 200 ps for all the curves. Amplitudes of the reverse voltage pulse V (V): 1

c) Distribution of the light-emitting regions over the surface of a mesa structure. The numbers (1-5) of the images (c) correspond to the numbers of the curves (1-5) in (b).

a) The shape of the mesa diodes studied in Ref. [171];

- 600, 2 - 800, 3 - 1200, 4 - 2000, 5- 2500;

the surface breakdown takes place in a number of the switching filaments of small diameter (Fig. 5 . 1 3 ~ )

The pictures 1-5 in Fig. 5 . 1 3 ~ correspond to the images (viewed from above) on the screen of an IR image converter from the side of the cathode contact with different d V / d t ramps. (Note that the left side of the sample cannot be observed due to the particular experimental conditions). The white spots on the surface of the mesa structure (the regions of light emission) correspond to the regions of current flow.

Picture 1 is related to curve 1 in Fig. 5.13 b. As seen, the current passes through two channels of - 30-50 pm in diameter. Assuming the existence of two conducting channels in the invisible left-hand side of the structure, one can evaluate the current density in the channels as j & - I /Sc - 41/N7rd2 - lo5 A/cm2 (here N is the

161


number of channels). The current rise time for curve 2 iii Fig. 5.13 b is practically equal to that for

curve 1, but the number of channels on the surface of the mesa-structure is increased (picture 2 in Fig. 5 .13~) .

Both the switching delay and the current rise time decrease with a further increase in the magnitude of the reverse voltage pulse V. At large values of V (curves 4 and 5), the current rise time corresponds to the circuit time resolution, and the duration of the steep part of the current front does not exceed about 100 ps. The switching related to curves 4 and 5 in Fig. 5.13 is fairly stable, so that the jitter does not exceed 30-50 ps.

At a maximum value of V = 2500 V (dV/dt - 1.2 x 1013 V/s, curve 5), the number of breakdown channels on the visible part of the sample surface is 12, which means that more than 20 channels appear in the sample at a current value of - 18 A. Note that the density of the current passing through every channel j,, of about (0.5 - 1) x lo5 A/cm2 is approximately constant at a particular current value. At the saturated carrier velocity 21, - lo7 cm/s, this current density corresponds to a carrier concentration of n N p - 1017 cm-3 in the channels.

The very interesting feature of the switching process is that while there is absolutely no instability of the front at an accuracy of better than 50 ps, switching occurs through several dozen apparently independent channels.

Note that due to re-absorption of the radiative recombination [178], the superposition of light emission from separate channels can cause an illusion of “homogeneous” bulk breakdown. This may happen if one observes the radiation in a direction perpendicular to the direction of current flux (as was done in Ref. [177]), i.e. in the direction A-A’ shown in picture 5 of Fig. 5 . 1 3 ~ ) .

To summarize, one can conclude that further studies should be performed to clarify the nature of superfast switching in GaAs reverse-biased diodes. Besides the somewhat contradictory data and statements discussed above, one should also bear in mind that the investigation of superfast switching in GaAs avalanche transistors (see Section 4.5) has shown the important role of Gunn instability, which might also take place in the breakdown channels of GaAs diodes during switching.

5.5 Superfast switching of GaAs thyristors

Reducing of the switch-on time of thyristors is one of the most important problems in power semiconductor electronics. There are hundreds of publications devoted to the investigation of turn-on processes in Si thyristors (see, for example [179; 180; 1811 and references therein). The shortest values for the characteristic switching-on time 70 achieved in Si thyristors are 70 = 5 ns for an operation bias VO = 800 V [I821 and 70 = 8 ns for VO = 600 V [183]. It is important to note that in all cases the TO observed in a Si thyristor was appreciably larger than the carrier transit time through the base of the thyristor at a saturated carrier velocity, t,.

162


As mentioned for the first time in Ref. [184], the 70 in GaAs thyristors decreases very sharp with increasing VO, and even at relatively low magnitudes of VO the switching time may become much shorter than in Si thyristors (Fig. 5.14).

2OL I I 1 1 1 1 1 1 I 1 I I l l

2 3 4 5 7 10 15 20 30 40 5060

b

Fig. 5.14 Dependences of the switch-on time constant TO on the operation bias Vo for a GaAs thyristor (different symbols correspond to different samples, curve 1) and a rated Si thyristor (curve 2). a) 2 V < Vo < 50 V; b) 60 V< Vo < 200 V.

The GaAs thyristors studied in Ref. [I841 were grown by vapour-phase epitaxy. The widths of the base layers W, and W, were 50- 55 pm and 7-8 pm, respectively,

163


the doping level of the blocking n-base was Nd N 5 x 1014cm-3. The magnitude of the static breakdown voltage V, was 500 - 600 V. The silicon analogue of these GaAs thyristor structures is a commercial Si thyristor KU-103 (manufactured in Russia), in which the width of the blocking n-base, W,, varies between 40 and 100 pm. The doping level Nd is about - 3 x 1014cm-3. Depending on the width of the n-base, the blocking voltage V, varies from 200 V to 600 V. KU-103 thyristors with V, x 500 - 600 V were chosen for the experiments in order to compare the switch-on properties of Si and GaAs thyristor structures.

As seen in Fig. 5.14a1 the dependences TO(VO) are practically identical at low voltages Vo. For both Si and GaAs thyristors these dependences approximate to T 0: Vo-1’2. Just the same character of ~o(V0) dependences is mentioned for different types of Si thyristor in Ref. [185], but the TO in a GaAs thyristor decreases much more sharply with a further increase in VO than does that in a rated Si thyristor (Fig. 5.14 b). As seen in Fig. 5.14 b, the value of TO in a GaAs thyristor at Vo w 200 V is about 1 ns, i.e. smaller than in a rated Si thyristor by a factor of 7.

As is demonstrated in Refs. [186; 1871, 70 in a GaAs thyristor can become as small as 10-l’ s with a further increase in VO, i.e. approximately 6 times shorter than the carrier transit time through the bases of the thyristor at the saturation velocity, t , (superfast switching).

The physical nature of this switching was investigated in Ref. [187]. The GaAs thyristors were switched-on by laser light pulses from a semiconductor GaAs laser structure. The transient characteristics of the switching process were found to depend qualitatively on the intensity of the light pulses (Fig. 5.15).

The time dependences of the current at two intensities of the light controlling pulse are shown in Figure 5.15, where the lower intensity (curve 1) corresponds to a longer delay (- 44 ns) and subnanosecond switching, while at a sufficiently high intensity of the triggering light pulse the current rise time increases markedly (curve 2 ) and the characteristic current rise time 70 becomes larger than the carrier transit time through the bases of the thyristor at the saturation velocity, t , (conventional switching).

The spatial distribution of the recombination radiation from a sample during the switching process, observed using a three-stage infrared image converter with high time resolution (IC in Fig. 5.15), was practically homogeneous across the “window” in the upper part of the structure with conventional switching. It means that the region of light emission (the region of initial thyristor switching) was just slightly smaller than the size of the “window” (Fig. 5.15).

With subnanosecond superfast switching, the characteristic size of the initially switched region, as observed on the IC screen during an interval of about 5 ns, did not exceed - 20 pm (the bright “spot” in Fig. 5.16), and the brightness of the light emitted by this region was appreciably higher than that in case of conventional switching.

It should be noted that the recorded diameter of - 20pm was governed by

164


recombination radiation

*O

2 3 0- - - - - - 4t 1

I I 4 I I I

1 I

48

Fig. 5.15 Time dependences of the current rise at two intensities of controlling light laser pulses. 1 - low intensity. 2 - high intensity. The decay in the photocurrent excited by the laser pulse can be seen in the left-hand part of the time diagram. The inset shows the shape of the mesa thyristor structure (cf. Fig. 5.13a) and a schematic of observation of radiation recombination.

the resolution of the optical system. Hence, this value represents merely an upper estimate for the size of the initially switched region.

At a diameter d N 20pm and a current amplitude I - 4 A (see Fig. 5.15), the current density j in the initially switched ‘(channel” is about lo6 A/cm2. In the case of conventional switching the current density in the initially switched region is two orders of magnitude smaller. If the nature of subnanosecond and conventional switching were caused by qualitatively the same physical mechanism, one would expect the lower current density to correspond to a lower residual voltage drop across the thyristor. The experimental results clearly indicate, however, that precisely the opposite situation occurs.

The voltage drop across the thyristor V ( t ) can be found at any instant of time in terms of the difference between the initial operating voltage Vo and the product of I x R1: [V( t ) = VO - ( I x Rl)]. As can be seen in Fig. 5.15, in the case of conventional slow switching (curve 2 in Fig. 5.15), the voltage drop across the thyristor at. the end of the current rise is relatively high and continues to decrease for several tens of nanoseconds. In the case of superfast switching (curve 1 in Fig. 5.15), on the other hand, this drop (when the current density j exceeds 106 A/cm2) is less

165


Fig. 5.16 In the case of conventional switching the recombination radiation is distributed practically homogeneously across the “window” in the upper part of the structure (cf. Fig. 5.15). In the case of superfast switching the characteristic diameter d of the initially switched region does not exceed N 20pm (a bright “spot”).

than at the end of the conventional switching process (when j 5 lo4 A/cm2), obviously indicating that the concentration of non-equilibrium carriers in the channel is appreciably larger with superfast switching.

As mentioned in Ref. [187], heating of the thyristor up to a temperature T M 440 K does not qualitatively alter the characteristics of superfast switching. A very intensive additional illumination of the thyristor during the delay time by a GaAs laser diode of - 200 W maximum output power focused on the “window” in the cathode contact also fails to produce any qualitative alteration in the characteristics of subnanosecond superfast switching up to a fairly high intensity of the laser beam. Suppression of the superfast switching and a change from subnanosecond to conventional nanosecond switching was observed only when the photocurrent excited by the additional GaAs laser exceeded N 100 A/cm2.

These experimental data provide convincing evidence that the superfast switching in GaAs thyristors is not caused by spread of the ionization front across the base regions of the structure. Indeed, as mentioned in Sections 5.3 and 5.4, intensive illumination and/or heating of the structures (which will cause a pronounced increase in free carrier concentration and in the leakage current), will suppress the formation of an ionization front. To suppress superfast switching it is sufficient to increase the leakage current across a SAS by 3 to 4 orders of magnitude, from lov6 - l ov7 A/cm2 to lov2 - lov3 A/cm2. The data presented above show that superfast switching in a GaAs thyristor still takes place at a “leakage current” as high as N 10 A/cm2 or even higher.

166


One can assume that the mechanism of superfast switching in GaAs thyristors is similar to that in GaAs avalanche transistors (see Section 4.5), and that it is caused by the formation of Gunn domains of high amplitude in the breakdown channel. Such an assumption allows us to explain all the main properties of this superfast switching.

Let us note first of all that the mechanism of superfast switching in GaAs structures, being associated with the formation of Gunn domains, is not sensitive to either heating or illumination of the structure. Secondly, the standard mechanism of thyristor positive feedback, which is associated with exchange by minority carriers between the bases of the thyristor, cannot operate within the characteristic time scale of superfast switching. Superfast switching in both transistor and thyristor structures can be caused only by impact ionization.

It is worth noting that the reconstruction of the field in the blocking base of the thyristor during the delay stage at high current densities is virtually the same as that in the collector of an avalanche transistor. The peak of the electric field shifts from the blocking junction of the thyristor towards the injecting junction (cf. Figs. 4.30, 4.31), and the maximum field F, can reach fairly high magnitudes [188].

At a high amplitude of the triggering signal (curve 2 in Fig. 5.15) the charge of non-equilibrium carriers introduced into the base of the thyristor significantly exceeds the minimum critical charge required to switch the thyristor on, as a result of which switching occurs practically homogeneously across whole area of the structure and the current density is relatively low. Conversely, given the minimum triggering signal which will still cause switching, the necessary density of the critical charge is achieved only at the L L ~ e a k e ~ t 7 1 point in the structure and switching occurs in a narrow channel (filament) in the vicinity of this “weakest” point. The current density is much larger in this case than with homogeneous switching, and the conditions for the reconstruction of the field domain and for the beginning of impact ionization are satisfied in such a filament (cf. Figs. 4.21, 4.27, 4.31). This reasoning explains the effect of the intensity of the triggering signal on the character of the switching (Fig. 5.15).

Last but not least, an indirect but fairly convincing argument in favour of the Gunn domain nature of superfast switching in GaAs thyristors is the fact that superfast switching has never been observed in Si thyristors, even though practically all possible designs and configurations of Si p - n - p - n structures and all switching regimes have been tested over more than half a century of investigations. The absence of superfast switching in Si thyristors leads us to the conclusion that the reason for such a qualitative difference in behaviour between GaAs and Si thyristors should be associated with some fundamental difference in the physical properties of these semiconductor materials. One obvious feature is the presence of the NDC region in the field dependence of electron drift velocity in GaAs in contrast to its absence in Si.

167


An opportunity for using the superfast switching in GaAs thyristors to form power (100 A, 600 V) and short (600 ns) pulses has been demonstrated in Ref. [189].

5.6 Main features of streamer breakdown

5.6.1 Introduction

The experimental data on superfast switching in SAS and GaAs diodes in the regime of delayed breakdown discussed in previous sections show that the observed switching time should correspond to an effective velocity of the impact ionization front wo of about 10' cm/s or even higher. Computer simulations employing a one- dimensional approximation show clearly that the propagation of a plane ionization front cannot provide such a high speed of switching.

The only known type of breakdown in semiconductors which is able to provide such a high velocity in the spread of ionization is the streamer. A streamer is a thin, highly conductive filament that elongates a t a very high velocity (N 10' - lo9 cm/s) due to impact ionization in the vicinity of its head (Figure 5.17). Streamer discharge in liquids, solids and gases (including lightning) has been investigated in innumerable papers (see, for example, [165; 1671, [190]-[194] and references therein).

channel head r,

t F

Fig. 5.17 Schematic diagram of the forefront and longitudinal field distribution of a streamer.

The physical processes which determine the propagation of a streamer are very similar to those for a plane ionization front. A very high electric field near the head

168


provides extremely intensive avalanche ionization. Carrier generation ahead of the streamer front continues until the field is expelled as a result of the increased conductivity, after which the field of the head moves in the direction of streamer spread and the resulting electron-hole plasma is redistributed via a Maxwell relaxation process. The qualitative field distribution under conditions of streamer discharge, as shown in Fig. 5.17, is similar to that observed in the case of a plane ionization front (cf. Fig. 5.3) . As will be seen later, the much higher velocity of streamer discharge than of a plane ionization front is caused by the substantially higher field F, at the streamer head relative to the characteristic maximum field F, in case of a plane front.

The main properties of streamer discharge had already been described in the 1950s, but a quantitative theory has been lacking until now. A qualitative analytical theory of streamers, taking into account the specific properties of the semiconductors, is presented in Refs. [166; 195; 1961. It is assumed in this theory that the electrons and holes differ only in the sign of their electric charge, i.e. that their mobilities, saturation carrier velocities, ionization rates, etc., are equal. In particular, it is assumed (see Eqs. (1.19) and (1.20)) that

ai = pi = a0 exp(-Fo/F) (5.21)

5.6.2

Following [166], let us show that stable propagation of a streamer is possible only if the maximum field F, at its head (Fig. 5.17) is of the order of the field Fo at which carrier ionization rates reach their saturation (see (5.21)).

Let us suppose, for example, that F, >> Fo. This means that the ionization rate a is at its maximum possible value a N a0 in a region which is much larger in size than the radius of the streamer head T O (Fig. 5.18a).

Since the field near the streamer head decreases with distance as 1/r, it is clear that the characteristic radius of the region where a ( F ) N 010 (dotted line in Fig. 5.18a) is of the order of

Analytical theory of a streamer discharge

(5.22)

As a result, the radius of the head increases (and the field at the head decreases). Conversely, if F, << Fo, effective avalanche ionization takes place only in a

small region close to the forefront of the head (Fig. 5.18 b). Owing to pronounced a ( F ) dependence at F << Fo, the characteristic radius of the region where noticeable generation takes place is of the order of

169


' * ' .r /'

Fig. 5.18 Schematic diagram of the streamer dynamics at different values of the ratio Fm/F0. Avalanche ionization at a maximum rate of cy - cyo exists in the region bounded by the dotted line. a) F, >> Fo. The radius of the head increases; b) F,,, << Fo. The radius of the head decreases; c) F, N Fo. A stable streamer is propagated with an invariable radius T O .

(5.23) FO Frn

To- << To

The radius of the head decreases (so that the field a t the head increases). Hence, given a steady state of streamer propagation, the radius of the head has

to be adjusted to a value which provides a field a t the head F, that is of the same order as FO (I?, - PO).

In Si the characteristic values of FO are 1.8 x lo6 V/cm and 1.3 x lo6 V/cm for electrons and holes, respectively, while in GaAs Fo is of the order of 6 x lo6 V/cm for both electrons and holes. In S ic FO is of the order of 5 x lo7 V/cm for electrons and 1.6 x lo7 V/cm for holes [2; 18; 191.

An estimate for the streamer velocity vo can be obtained by means of specula- tions analogous to those used for deriving the expressions (5.11) and (5.12). Taking into account that the effective value of the ionization rate 6 is now at its maximum magnitude ( Y O (01 N (YO), the expression for vo can be written as

(5.24)

170


where A1 = ln(nT/nb) (cf. formula (5.12). Expression (5.24) for the streamer velocity was first obtained in Ref. [193].

If the conductivity in the channel behind the head obeys Ohm’s law (the drift carrier velocity w is smaller centration behind the head

than the saturation carrier velocity w,) the carrier con- nT will be defined by

(5.25)

(cf. expression (5.14)).

the carrier density in the channel takes a form [166]: When the carrier velocity in the channel is saturated (w = u s ) , the estimate for

(5.26)

(cf. expression (5.16)). A streamer discharge in semiconductors is usually investigated in two configu-

rations. In high-resistivity semiconductor crystals (and dielectrics) streamers are studied as a rule in “dielectric configuration” (Figure 5.19), in which a metal tip to which a voltage is applied is pressed onto the surface of the semiconductor (dielectric). The streamer nucleates near the tip and spreads into the volume of the material, provided that the voltage V(t) applied to the tip grows sufficiently fast. It can be shown [166] that the length 1 of the streamer is limited at a constant applied voltage VO.

Adopting the simplest assumption of negligible voltage drop across the channel length, one can assume that the potential of the streamer head is equal to the potential of the metal tip VO. The charge distribution along the streamer channel is known for this case from the classical electrostatic problem of a thin metal filament with a given potential Vo, i.e. the charge is distributed along the filament with a linear density pz.

(5.27)

The electric field ahead of the front of the streamer F, is of an order of magnitude

where A2 = ln(lFo/Vo). Taking into account

210 :

(5.28)

(5.24), one can write for the velocity of the streamer head

171


met

plane I 4ectrode

Fig. 5.19 Excitation of streamer discharge in a “dielectric configuration”.

(5.29)

Expressions (5.28) and (5.29) describe the parameters of the streamer at the initial stage of its spread, when the voltage drop along the channel can be neglected. In this case both the streamer radius TO and its velocity vo are proportional to the potential of the tip VO.

When the streamer elongates with a velocity ‘UO the newly formed parts of the channel acquire the linear charge pi N EEOFO x T O . Hence the current I , N pluo must flow along the channepduring the streamer spread, and a longitudinal electric field F, is necessary to provide this current:

(5.30)

Employing (5.25) we obtain F, N Fo/Al. Since A1 >> 1, one may conclude that F, << Fo, and furthermore, one can conclude that the field F, is much smaller than the characteristic field of drift velocity saturation F, = v s / p . Indeed, if we

172


assume that the carrier velocity in the channel is saturated, then the relation (5.26) instead of (5.25), should be used for the carrier density n T , and the expression for Fx will take the form:

F. Fx - - A1

(5.31)

Thus the validity of expressions (5.28) and (5.29), which requires a small potential drop across a channel of the length 1, has the form:

(5.32)

With an increase in the channel length I, the potential a t the head is reduced, which causes a reduction in both the velocity and the radius of the streamer head (expressions (5.28), (5.29)). The streamer will cease to spread when its velocity is reduced to the value 210 N us (see expressions (5.6) and (5.7) and the general definition of dynamic breakdown in Section 5.1). The estimates show [166] that the maximum possible length of the streamer channel is close to the value 10 defined by formula (5.32).

It is well known that the potential Vo(t) of the tip should increase sufficiently fast to provide for formation of the streamer in the “dielectric configuration”. On the other hand, if a rectangular voltage pulse of amplitude Vo is applied to the tip, then the streamer can be formed only at a sufficiently large value of VO, exceeding the threshold value V&. An approach developed in [166] allows us to estimate both the threshold values, V t h ~ and (dV~/dt)~ho:

and

(5.33)

(5.34)

where R is the radius of the tip. For FO - lo6 V/cm, us - lo7 cm/s, QO - lo6 cm-l and R - 10-1 cm, we have

I& - lo4 V and ( d V ~ / d t ) ~ h o N 10l2 V/s. These values agree reasonably well with the experimental data.

At u, - lo7 cm/s and p N lo3 cm2/Vs, the characteristic field of drift velocity saturation F, = u,/p - lo4 V/cm. The estimate for the ratio nT/n,, can be taken to be n T / n b N lo9 (see comment to Fig. 5.3). Then, at VO - lo4, we have l o N 10 cm (formula (5.32)), A2 = ln(lFo/Vo) - 7, TO N VO/FOAZ N cm, vo N lo9 cm/s (see expression (5.28), (5.29)). All these estimates agree reasonably well with the

173


experimental data obtained for a streamer in the “dielectric configuration” (see, for example, [145]).

Neither high voltages (of an order of several dozens of kilovolts) nor large lengths of the “discharge gap” (of several centimetres) are typical of semiconductor devices, however. The main characteristics of the streamer which are typical of semiconductor devices (the “device configuration” in Figure 5.20) are considered in Ref. [195].

Fig. 5.20 figuration”).

Schematic diagram of streamer propagation in a given external field Fez (“device con-

Comparing Figs. 5.19 and 5.20, it is possible to see that where the streamer in the configuration shown in Fig. 5.19 moves away from the metal tip in a medium in which there is practically no external electric field (the second electrode is ac- tually assumed to be infinitely distant), the streamer in the configuration shown in Fig. 5.20 moves in a homogeneous external electric field Fez. This causes fairly substantial differences in the character of streamer propagation between the two configurations.

Expressions (5.24) - (5.25) still remain valid, since they are independent of the configuration of the discharge gap. The value of the field Fm at the streamer head, however, is now determined by the magnitude of the external field Fez (and by the streamer length 1 ) rather than by the potential VO at the metal tip.

Indeed, in the electrostatic approximation the distribution of the linear charge density pl along a conducting filament of radius TO situated in an external field parallel to its axis takes the form [197]:

(5.35)

where A, = ln(l/ro) (see expression (5.28)). The field at the end of the filament F, is then proportional to the length of the filament I :

174


(5.36)

When the end of the conducting filament moves, the expressions (5.35) and (5.36) are valid if the length 1 is fairly small and the charge manages to spread along the filament, screening the field inside it. One can show [195] that the electrostatic approximation is valid if 1 5 l o , where

(5.37)

(cf. Eq. (5.32)). The main attention in Ref. [195] is concentrated on the case 1 2 l o , and it is

shown that only one magnitude of the electric field F,, = F, exists at which a steady-state streamer is propagated with a constant radius ro and velocity WO. This field value F, is defined by the expression

n

(5.38)

At F,, < F, the radius of the streamer TO and its velocity vo decrease, so that the length of the streamer propagation becomes finite. At F,, > F,, ro increases and the streamer propagates unrestrictedly. The estimates €or the maximum radius value TO and the maximum streamer velocity vo take the form

c

(5.39)

(5.40)

The case 1 < l o , is the most important from the point of view of streamer propagation in the devices, however, since the value of 10 defined by expression (5.37) ( lo N 1 cm) significantly exceeds the thickness of the base region in any semiconductor device (see Sections 5.2 - 5.5). At 1 < 10 the length 1 grows during streamer propagation and the field at its head increases (see expression (5.36)). Hence the radius and velocity of the streamer increase as well. The contacts of the sample become electrically shortened at the instant when the streamer approaches the opposite electrode (Fig. 5.20).

It is necessary to mention that in a real situation involving SAS or GaAs diodes with delayed breakdown we are faced with fairly complicated experimental conditions. The external electric field Fez cannot be regarded as constant in either time or in space even as a first approach (see, for example, Fig. 5.3), and the conditions

175


for the conversion of a plane ionization front to streamer discharge are unclear, etc. In this situation the most adequate description of the processes can be obtained using only computer simulations.

5.6.3 Computer simulation

Unfortunately, no adequate computer simulations of streamer formation and propagation in real semiconductor devices have been available until now, chiefly because of the complexity of the system of non-linear two (or rather three)-dimensional partial differential equations required for describing streamer discharge.

As mentioned in Refs. [195; 1981, there are a number of papers devoted to the analysis of streamer properties, but the set of equations used in most of them includes many secondary processes that do not play a substantial role in streamer propagation. On the other hand, the simplifying assumptions that are employed in these papers are not justified even qualitatively.

A more appropriate approach is that chosen in Ref. 11991, which uses a set of simple equations that take into account only the basic processes. The initiation and propagation of a streamer in a plane-parallel discharge gap (Fig. 5.20) of length L = 0.5 cm in nitrogen (Nz ) at atmospheric pressure is considered, with an applied voltage of 26 kV, giving an external field Fez of 52 kV/cm. The primer (high- conducting nucleus of the streamer) was of a spherical or hemispherical form, with a characteristic radius of about 250 pm and an initial plasma concentration of -

1015 ~ m - ~ . The initial concentration of free carriers homogeneously distributed throughout the gap n b was lo5 - 10’ ~ m - ~ .

It is obvious that the results of the simulation obtained in [199] cannot be compared with experimental data for real semiconductor devices. It is sufficient to note that the size of the primer in Ref. [199] is practically equal to the length of the “plane-parallel discharge gap” in SAS and GaAs diodes with delayed breakdown. The data obtained in [199] can be used, however, to evaluate the qualitative streamer theory proposed by Dyakonov and Kachorovskii [166; 195; 1961. The results obtained in [199] are compared with qualitative theoretical estimates in Ref. 11951.

The ratio of the maximum field F, at the head of the streamer to the field Fo was foand to be 0.4-0.8 in computer calculations (for an external field Fez ranging from 0.18 to 0.3 Po). This result is in agreement with the analytical estimate F, - Fo.

The carrier concentration behind the streamer front, n ~ , was found in simulations to be smaller than that derived from the analytical estimates by a factor of 5-10, but the relation between the streamer velocity propagation 210 and the head radius TO was found to be in good agreement with the estimate (5.23).

The simulation described in Ref. [I981 had a specific purpose, to calculate numerical coefficients for the expressions obtained in the analytical theory [166; 195; 1961. Since the simplified set of equations used in [198] did not contain any pa-

176


rameters with a length dimension, the dimensionless parameter xo could be used arbitrarily. The characteristic streamer dimensions were determined in terms of the streamer radius ro, while the ro value at the initial instant t o was in turn defined by the parameters of the primer, on the assumption that the primer, being situated near one of the electrodes, is characterized by the initial carrier density distribution

(5.41)

where x is the direction of streamer propagation, r l x is the radial coordinate, and n 1 >> n b is the initial characteristic concentration in the primer. The length L between the electrodes was chosen to be 20x0.

The results of a calculation for the case TO = XO, n 1 = nT ( n ~ is defined by expression (5.25)), F,, = 0.25F0, and n b = 10W'n~ are shown in Figure 5.21. The dimensionless time t is measured in units t o = ( Q ~ V , ) - ~ .

As can be seen in Fig. 5.21, the streamer becomes quasi-stationary fairly soon, and its parameters are practically constant in the process of further propagation. (A similar result is obtained in Ref. [199]). It is evident that the maximum field at the streamer head F, is very close to FO in this quasi-stationary regime, and it should be noted that similar results were also obtained with external parameters F,, and F0 differing from those in Fig. 5.21. According to the calculations performed in [198], the relation between F, and FO can be expressed as F, = CoFO, where

c o = 0.9 k 0.1 (5.42)

It is seen that the actual magnitude of the conductivity ahead of the streamer front is not constant (Fig. 5.21 a) but increases slowly with time due to impact ionization in the external field Fex. Taking this effect into account, the authors of Ref. [198] were able to show that the equality

(5.43)

is valid with

C1 = 0.18 f 0.2 (5.44)

In summary, we can state that there is a graceful analytical theory of streamer discharge and some examples of simulations which confirm the main results of this theory, but there is no adequate simulation of streamer discharge in the most important and interesting semiconductor devices (SAS, GaAs diodes with delayed breakdown), in which there are good grounds to expect a nucleation and spread of the streamer. The primer (high-conducting nucleus of the streamer) should be

177

109% Breakdown Phenomena in Semiconductors and Semiconductor Devices

100

10-2

\ &- 10-4

10-6

10-8

E:

n

0 5 10 15 20

1 .o 0.8

0.6

0.4

0.2

4

0 5 10 15 20 x/xo

Fig. 5.21 Time dependences of the carrier density (a) and field (b) distributions along the longitudinal axis (T = 0) in the process of streamer propagation [198]. t / t o : 1 - 40; 2 - 120; 3 - 200; 4 - 280; 5 - 360; 6 - 440; 7 - 520.

absent from simulations of a real experimental situation, and the streamer should possibly appear as a result of filamentation of the plane ionization front.

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International Journal of High Speed Electronics and Systems Vol. 14, No. 4 (2004) 1099-1100 @ World Scientific Publishing Company

Conclusion

The problems discussed in this little book could perhaps be described more exactly by a title such as “Main approaches to electrical breakdown phenomena in “good” single-crystal semiconductors and devices based on them”, where the term “good semiconductors” is understood as referring to “semiconductors with relatively high carrier mobility”. Roughly speaking, the phenomena considered in this book apply to semiconductors with a low-field mobility p of a magnitude exceeding N (10- 100) cm2/Vs. The term “main approaches” means that the breakdown phenomena discussed in this book are considered, as a rule, in the context of fairly simple spatial configurations.

In semiconductors with relatively small values for the low-field mobility p i.e. in polycrystalline, amorphous, and most polymer semiconductors, etc., thermal breakdown plays a very important role. The most detailed review of phenomena associated with thermal breakdown can be found in Ref. [200], by this book is unfortunately available only in Russian. The fundamentals of thermal breakdown theory are expounded in Refs. [201]-[205].

It is very difficult as a rule to separate the contributions of electrical and thermal breakdown in the case of semiconductors of low mobility. The history of breakdown switching in chalcogenide glassy semiconductors research appears to provide one of the most instructive examples of this problem. The question of the breakdown mechanism in this materials was already raised in the pioneer works [206]-[208], but the problem still cannot be considered to have been completely solved [209;

The prevention of surface breakdown (the edge termination problem) in reverse- biased devices is one of the most valuable examples of the importance of breakdown studies under conditions involving complicated spatial configurations. The characteristic surface breakdown field Fi, is usually appreciably lower than that in the “volume”, Fi, due to the high density of defects on the surface. Meanwhile, for majority of semiconductor devices (avalanche photodiodes, suppressor diodes, IM- PATT diodes, silicon avalanche sharpers, avalanche bipolar transistors, high-voltage rectifier diodes and thyristors, and many other devices), it is very important, and sometimes absolutely essential for the breakdown in response to an increase in the

2101.

179

1100 Conclusion

reverse bias to appear in the volume (but not a t the surface). Many techniques have been developed for protecting the devices from surface breakdown (guard rings, junction termination extensions etc.) , but calculations of the field distribution at the surface and in its vicinity nevertheless require the solution of two-dimensional and sometimes even three-dimensional problems [22]-[24], [211]-[213].

A necessity for 2D and 3D simulations frequently arises when the breakdown in a high-field domain between the drain and the gate is analysed in different types of field effect transistors in a deep saturation regime, when breakdown in bipolar transistors is considered under conditions of a pronounced crowding effect, or when devices of fairly complicated geometry are being investigated] etc. It should be noted, however, that all the basic principles and approaches considered in this book are applicable to such calculations in a practically unchanged form.

Although investigations of breakdown phenomena in solid states started about a hundred years ago, this is still a lively and powerful branch of the mighty tree of semiconductor physics. One can easily experience this by typing into any browser the combination ”Breakdown Semiconductors”. Several thousands of papers devoted to breakdown studies involving semiconductors and semiconductor devices are published every year. This situation will continue to prevail all the time that new materials are becoming involved in semiconductor electronics, all the time that still new devices are being proposed, and all the time that modifications and im- provements are being developed for known devices within semiconductor electronics.

To anybody who doubts in the usefulness of the efforts spent to get to know such an interesting and important subject as ”Breakdown Phenomena in Semiconductors and Semiconductor Devices” we would like to answer by the words of William Shakespeare’s 76-th sonnet:

Why is my verse so barren of new pride, So far from variation or quick change? Why with the time do I not glance aside To new-found methods and compounds strange? ....

For as the sun is daily new and old, So is my love still telling what is told.

180



List of Symbols

Ei

Auger coefficient for e - e - h process Auger coefficient f for e - e - h process h - h - e specific heat electron diffusion coefficient hole diffusion coefficient electron charge energy energy gap Fermi level in intrinsic semiconductor energy of the trap threshold energy of impact ionization frequency operation frequency electric field excess noise factor breakdown field characteristic field of drift velocity saturation threshold field of Gunn effect field on the back side of the impact ionization front electron generation rate hole generation rate Planck constant optical phonon energy current avalanche current leakage current of reverse-biased p - n junction photocurrent current density conduction current density displacement current density Boltzmann constant

181

1102 List of Symbols

wave vector thermal conductivity mean free path ionization region length distance between two acts of impact ionization length of the device active region avalanche inductance diffusion length electron diffusion length hole diffusion length free electron mass effective mass electron effective mass hole effective mass multiplication factor of B JT in common-base circuit configuration multiplication factor of B JT in common-emitter configuration electron multiplication factor hole multiplication factor electron concentration equilibrium electron concentration in n-type semiconductor background electron concentration before impact ionization front intrinsic electron concentration electron concentration behind of impact ionization front acceptor concentration donor concentration concentration of the traps hole concentration equilibrium hole concentration in p-type semiconductor background hole concentration before impact ionization front intrinsic hole concentration hole concentration behind of impact ionization front power power density radius of the streamer specific contact resistance specific space-charge resistance reflectance differential resistance load resistance resistance of a single microplasma resistance of space charge region device operation area

182

Last of Symbols 1103

time delay time in superfast switching diffusion time multiplication time duration of overload pulse carrier transit time through the space-charge region with saturation velocity transit time temperature carrier velocity velocity of impact ionization front saturation carrier velocity electron saturation velocity hole saturation velocity operation bias breakdown voltage punch-t hrough voltage space charge width avalanche region length the width of BJT base limiting electron ionization rate at very high electric field electron ionization rate absorption coefficient temperature breakdown coefficient limiting hole ionization rate at very high electric field hole ionization rate emitter injection efficiency thermal diffusivity dielectric constant of the semiconductor permittivity of vacuum quantum efficiency phase deviation wavelength wavelength of maximum spectral sensitivity electron low-field mobility hole low-field mobility density capture cress section of the recombination trap carrier lifetime energy relaxation time differential Maxwellian time of dielectric relaxation electron lifetime

183

1104 List of Symbols

T P hole lifetime TnA

T ~ A

TSR

70

w = 27rf circular frequency

electron lifetime associated with Auger recombination hole lifetime associated with Auger recombination carrier lifetime associated with linear Shockley-Read recombination characteristic time of switch-on process

184




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165. Loeb, L. B. (1939). Fundamental Processes of Electrical Discharge in Gases, J. Wiley & Sons, Chapman & Hall.

166. Dyakonov, M. I., Kachorovskii, V. Yu. (1988). Theory of streamer discharge in semiconductors Sou. Phys. JETP, 67, 5, pp. 1049-1054.

167. Bazelyan, E. M., Raizer, Yu P. (2001). Lightning Physics and Lightning Protection, Institute of Physics Publishing.

168. Minarskii, A. M., and Rodin, P. B. (1994). Long-wavelength transverse instability of shock-ionization waves in diode structures, Techn. Phys. Lett., 20, 6, pp. 49&491.

169. Minarskii, A. M., and Rodin, P. B. (1997). Transverse stability of an impact-ionization front in a Si p+-n-n+ structure, Semiconductors, 31, 4, pp. 366-370.

170. Minarskii, A. M., and Rodin, P. B. (1997). Transverse instability and inhomogeneous dynamics of superfast impact ionization waves in diode structures, Solid State. Elec- tron., 41, 5, pp. 813-823.

pp. 1611-1613.

51, 8, pp. 984-985.

Php , 92, 4, pp. 1971-1980.

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171. Vainshtein, S. N., Zhilyaev, Yu. V., Levinshtein, M. E. (1989). Vizualization of subnanosecond switching of gallium arsenide diode structures, Sov. Tech. Phys. Lett.,

172. Rodin, P., Ebert, U., Hundsdorfer, W., Grekhov, I. V. (2002). Tunneling-assisted impact ionization front in semiconductors, Journ. Appl Phys., 92, 2, pp. 958-964.

173. Lyubutin, S., Mesyats, G., Rukin, S., Slovikovsky, B., Tsyranov, S. (2004). Ultra-fast solid state switching based on tunneling-assisted delayed breakdown device, Ab- stracts of 26th Power Modulator Conference, May 23-26, San-Francisco, CA, p. 147.

174. http://www.fidtechnology.com/Products/fastpowergens.htm 175. Alferov, Zh. I., Grekhov, I. V., Efanov, V. M., Kardo-Sysoev, A. F., and Stepanova, M.

N. (1988). Formation of high picosecond-range volage drops across gallium arsenide diodes, Sov. Tech. Phys. Lett., 13, 9, pp. 454-455.

176. Grekhov, I. V., Efanov, V. M. (1988). Possibility of a rapid production of a dense large-volume electron-hole plasma in gallium arsenide, Sov. Tech. Phys. Lett., 14,

177. Grekhov, I. V., Efanov, V. M. (1990). Possible generation of simulated emission using collisional ionization waves in semicondutors, Sov. Tech. Phys. Lett., 16, 9, pp. 645- 647.

178. Delimova, L. A., Zhilyaev, Yu. V., Kachorovsky, V. Yu., Levinshtein, M. E., and Rossin, V. V. (1988). Forward current-voltage characteristics of gallium arsenide power diodes at high current densities, Solid State. Electron., 31, 6, pp. 1101-1104.

179. Bergman, G. D. (1965). The gate-triggered turn-on process in thyristors, Solid State Electronics, 8, 9, pp. 757-765.

180. Blicher, A. (1976). Thyristor physics, Springer Verlag, New York-Heidelberg-Berlin. 181. Geriach, W. (1981). Thyristoren, Springer-Verlag, Heidelberg. 182. Kuzmin, V. A., Pavlik, V. Ya., Shuman, V. B. (1980). On the maximum turn-on

speed of p-n-p-n structures, Radiotechnika i Electronika, 26, p. 1270 (in Russian). 183. Brylevskii, V. I., Levinshtein, M. E., Chashnikov, I. G. (1984). Dynamic localization

of the current during turn-on transient in thyristors, Sou. Phys. Tech. Phys., 29,

184. Vainshtein, S. N., Zhilyaev, Yu. V., and Levinshtein, M. E. (1986). Comparative study of the turn-on of gallium arsenide and silicon thyristors, Sov. Phys. Techn. Phys.,

185. Levinshtein, M. E., Shenderei, S.V. (1979). Rate of propagation of the “on” state in a thyristor at high current density, Sov. Phys. Semicond, 13, pp. 593-595.

186. Belyaeva, 0. A., Vainshtein, S. N., Zhilyaev, Yu. V., Levinshtein, M. E., Chelnokov, V. E. (1987). Subnanosecod turn-on of gallium arsenide thyristors, Sov. Tech. Phys. Lett., 12, 8, pp. 383-384.

187. Vainshtein, S. N., Zhilyaev, Yu. V., Levinshtein, M. E. (1988). Investigation of subnanosecod switching of gallium arsenide thyristor structures, Sov. Phys. Semicond,

188. Grekhov, I. V., Kardo-Sysoev, A. F., Levinshtein, M. E., Sergeev, V. G. (1971). Turn-on process in a thyristor, Sov. Phys. Semicond, 5 , 1, pp. 157-159.

189. Vainshtein, S. N., Kilpela, A. J., Kostamovaara, J. T., Myllyla, R. A., Starobinets, S. V., Zhilyaev, Yu. V. (1994). Multistreamer regime of GaAs thyristor switching, IEEE Trans. Electron Devices, ED-41, 8, pp. 1444-1449.

14, 8, pp. 664-665.

12, pp. 920-921.

pp. 69-70.

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22, 6, pp. 717-718.

190. Meek, J. M., and Grass, J. D. (1953). Electrical breakdown in gases, Oxford. 191. Skanavi, G. I. (1958). Physics of dielectrics (strong electric field), Fizmatgiz, Moscow

192. Raether, H. (1964). Electron avalanches and breakdown in gases, Butterworths, Lon- (in Russian).

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1426. 194. O'Dwyer, J. J. (1973). The theory of electrical conduction and breakdown in solid

dielectrics, Clarendon Press. 195. Dyakonov, M. I., Kachorovskii, V. Yu. (1989). Streamer discharge in a homogeneous

electric field, Sov. Phys. JETP, 68, 5 , pp. 1070-1074. 196. Dyakonov, M. I., Kachorovskii, V. Yu. (1990). Velocity of streamer propagating from

a point during linear voltage increase, Sou. Techn. Phys. Lett., 16, 1, pp. 32-33. 197. Landau, L. D., and Lifshits, E. M. (1984). Electrodynamics of Continuous Media,

Pergamon. 198. Evlakhov, N. V., Kachorovskii, V. Yu., Chistyakov, V. M. (1992). Numerical simula-

tion of a streamer discharge in a uniform field, Sou. Phys. JETP, 75 , 1, pp. 31-36. 199. Dhali, S. K., Williams, P. F. (1987). Two-dimensional studies of streamers in gases,

JOU~Z. Appl Phys., 62, 12, pp. 4696-4707. 200. Electron phenomena in chalcogenide glassy semiconductors, (1996). Ed. by K. D.

Tsendin, Nauka, St. Petersburg (Chapter 6: E. A. Lebedev and K. D. Tsendin, Switching effect in chalcogenide glassy semiconductors) (in Russian).

201. Fock, V. A.(1927). Zur Warmetheorie des elektrischen Durchschlages, Archiv fur Elek- trotechnik, 19, 1, pp. 71-81.

202. F'ranz, W. (1956). Dielektrischer Durchschag, Springer Verlag, Berlin. 203. F'ritzsche, H. (1974). Electronic Properties of Amorphous Semiconductors, Plenum

204. Adler, D., Henisch, H. K., Mott, N. (1978). The mechanism of threshold switching in

205. Maden, A., Shaw, M. P. (1978). The Physics and Application of Amorphous Semi-

206. Kolomiets, B. T., Lebedev, E. A. (1963). Radiotekhnika i Elektronika, 8, p. 2097. 207. Ovshinsky, S. R. (1966). US Patent, No 3.721.591, September 06; Ovshinsky, S. R.

(1968). Phys. Rev. Lett., 21, p. 1450. 208. Kolomiets, B. T., Lebedev, E. A., Taksami, I. A. (1969). On the mechanism

of the breakdown in chalcogenide glassy semiconductor films, Phizika i Tekhnika Poluprovodn., 3, 2, pp. 312-314.

209. Voronkov, E. N. (2002). Pulsed breakdown of chalcogenide glassy semiconductor films, Journ. Optoelectron. and Adv. Materials, 4, 3, pp. 793-798.

210. Senkader, S., Wright, C. D. (2004). Models for phase-change of GezSbzTes in optical and electrical memory devices, Journ. Appl. Phys., 95, 2, pp. 504-511.

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Press, London-NY.

amorphous alloys, Rev. Mod. Phys., 50, 2, pp. 209-220.

conductors, Academic Press.

194

World Scientific www.worldscienl i f ic .com



Index

Absorption coefficient, 26, 29 Approximation of ionization rates, 14, 15,

Auger coefficients, 5 Auger recombination, 5-8, 59 Avalanche bipolar transistor (ABT), xii,

Avalanche breakdown, vii, 24 Avalanche current, 67, 69 Avalanche diodes, 62 Avalanche excess noise, 36 Avalanche injection, 81, 91, 92, 106 Avalanche multiplication, viii, 26, 37, 39 Avalanche photodiode, viii, 26 Avalanche region, 31, 66, 69 Avalanche suppressor diodes (ASD), xii,

Avalanche zone. 141

21, 22

91, 92, 103, 105, 115, 122, 133

60, 62

Background carrier concentration, 138, 146, 148, 176

Ballistic regime, 77 Ballistic transport, 76 Band discontinuity, 34 Band-to-band generation, 5 Band-to-band recombination, 5, 7 Band-offsets, 34 Bipolar junction transistor (BJT), 91,

Breakdown field, vii, 10-12, 33 Breakdown in the common-emitter circuit

Breakdown voltage, 11, 12, 49 Brillouin Zone, 2 Brillouin zone, 2 Built-in voltage, 30

105, 122

configuration, 95, 97, 103, 110

Carrier concentration behind the impact ionization front, 141, 149, 151, 157, 158, 171, 176

Carrier transit time, 30 Collector field domain, 116, 133 Common-base circuit configuration, 92,

Common-base current gain coefficient, xii,

Common-emitter circuit configuration,

Conical current distribution, 90, 91 Contact resistivity, 44, 45, 49 Critical current density, 85, 91, 92, 106,

Crowding effect, 119 Crystallographic orientation, 3 Current filaments, 41, 89, 132

93, 95, 96

92, 98, 99

93-96

111

Dark current, 27 Device configuration, 174 Dielectric configuration, 171 Differential Maxwellian time, 86, 126, 130,

Differential resistance, 40, 45 Diffusion broadening, 75 Diffusion coefficient, 13 Diodes with delayed breakdown (DDB),

xiii, 143, 157, 160, 162, 175 Displacement current, 138-140 Drift region, 66, 73, 75 Drift velocity, 13 Dynamic breakdown, ix, xii, 39, 137

149, 153

Effective ionization energy, 3 Einstein relation, 78

195

1116 Index

Elementary act of impact ionization, 1,

Emitter injection efficiency, 99 Energy gap, 1, 3 Energy relaxation time, 8, 14, 130, 134 Equilibrium electron concentration, 27 Equilibrium energy, 8 Equilibrium hole concentration, 27 Excess noise, 36 Excess noise factor, 36

14, 33, 35

Fermi level position, 28 First microplasma, 44, 63

GaAs band, 2 Graded gap, 35 Graded gap structure, 34 Gunn domains, xii, 124, 128, 129, 131,

Gunn effect, 125, 127, 135, 136 132, 134, 136, 167

High injection level, 6 Homogeneous breakdown, viii, xi, 11, 40,

44, 63, 65

Impact ionization, 3-5, 10 Impact ionization front, xiii, 140, 141,

IMPATT, vii, viii, xii, 60, 65, 66, 70-73,

Infinite narrow breakdown region, 47, 49,

Initial carriers, xiii, 154-156 Injecting cathode contact, 82, 83 Injection current, 111-114 Intrinsic concentration, 6 Intrinsic level, 28 Intrinsic semiconductor, 6, 28 Ionization rates, 10, 14, 32 Ionization region, 148, 150

143, 147, 156

75, 141, 142

50

Leakage current, viii, 11, 27, 39 Lifetime, 6, 7 Long-wave boundary of spectral

Low injection level, 6 sensitivity, 26

Mean energy, 8, 9, 78 Mean free path, 15-17, 130 Microplasma, 42

Microplasma breakdown, viii, xi, 40, 41,

Microplasma current pulses, 42 Monte-Carlo technique, 10, 13-15 Multiplication factor, 37 Multiplication factor, 22-24, 26, 29,

Multiplication time, 30, 31

43, 44

31-33, 37

Narrow-gap semiconductors, 5 Negative differential conductivity (NDC),

82, 89, 125 Negative differential resistance (NDR),

xii, 40, 41, 48, 50, 52-55, 58-60, 71 Negative temperature breakdown

coefficient, 47 Noise factor, 37

Optical phonon scattering, 16 Overheating, 63 Overload, 63-65 Overshoot, 76, 77, 79

p-i-n diode, xii, 50, 52, 53, 58, 59, 66 Photocurrent, 29 Photodiodes, 59 Photoelectric threshold, 26 Plane ionization front, xiii, 141, 145, 146,

Poisson equation, 11, 18 Positive differential resistance (PDR),

Probability of ionization, 3 Punch-through voltage, 55, 58, 106

148-153

58-60

Quantum efficiency, 29, 37

Ramo-Shockley theorem, 70, 72 RC time, 64 Read’s diode, 66, 69, 72, 75 Residual voltage, 58, 104, 106, 141, 147,

Resistivity of the space-charge region, xi, 158

44, 47

S-type of current-voltage characteristic,

S-type switching, 89-91 Sah-Noyce-Shockley component, 27, 28 Scattering processes, 13

132

196

Index 1117

Schottky diodes, 11 Second microplasma, 43 Self-heating, 63, 120 Semiconductor resistor, 81, 124 Shallow donors, 7 Shockley component, 27-29 Shockley-Read recombination, 6, 7 Short-wave boundary of spectral

Shot noise, 36 Si band, 3 Silicon Avalanche Sharpers (SAS), xiii,

Space-charge region, 11, 28, 49, 55, 137 Space-charged limited (SCL) regime,

Spectral sensitivity, 26 Speed of response, 29 Static avalanche breakdown, xi, 39 Streamer, xiii, 151, 168, 169, 171, 176 Streamer head, 171 Suppressor diode (SD), viii, 61, 62

sensitivity, 26

142-144, 154, 156, 157, 175

84-87, 91

Temperature breakdown coefficient, 45 Thermal resistance, xi, 44, 45, 49 Threshold energy, 1, 3, 4, 33 Time response, 29, 37 Transit time, 31 TRAPATT, vii, xiii, 140-142 Tunnelling breakdown, 75

Valence band discontinuity, 34

Wide-gap semiconductors, 4, 7

197


World Scientific www.worldscientilic.com


AUTHOR INDEX Volume 14 (2004)

Adam, T. N., see Troeger, R. T. Agarwal, A,, see Ryu, S. H. Aggarwal, S. K., see Gupta, R. S. Aleksiejunas, R., see Mickevicius, J. Alexander, D., see Nowlin, N. Anwar, A. F. M., see Faraclas, E. Anwar, A. F. M., see Islam, S. S. Asbeck, P. M., see Keogh, D. M. Asbeck, P. M., seeLi, J. C. Asif Khan, M., see Rumyantsev, S. L. Asif Khan, M., see Simin, G. Bailey, J., see Nowlin, N. Barnaby, H. J., Total Dose Effects in Linear Bipolar Integrated Circuits Bates, G. M., see De Salvo, G. C. Baumann, R. C., Soft Errors in Commercial Integrated Circuits Belyaev, A. E., see Vitusevich, S. A. Ben-Yaacov, I., see Gao, Y. Bilenko, Y., see Sawyer, S. Blking, J., see Neuburger, M. Brandes, G. R., see Lee, J. Brandes, G., see Faraclas, E. Bu, G., Ciplys, D., Shur, M. S., Schowalter, L. J., Schujman, S. B. and

Gaska, R., Leaky Surface Acoustic Waves in SingleCrystal A1N Substrate

Buchner, S., see Fouillat, P. Bude, J., see Ye, P. D. Butler, J. E., see Huang, W. Buttari, D., Chini, A., Chakraborty, A,, McCarthy, L., Xing, H., Palacios,

T., Shen, L., Keller, S. and Mishra, U. K., Selective Dry Etching of GaN over AlGaN in BC13/SF6 Mixtures

Oxide MOS Structures

Study on Breakdown Behavior of Field-Plate S i c MESFETs

Cester, A. and Paccagnella, A,, Ionizing Radiation Effects on Ultra-Thin

Cha, H.-Y., Choi, Y. C., Eastman, L. F. and Spencer, M. G., Simulation

Cha, H.-Y., see Choi, Y. C. Chakraborty, A., see Buttari, D.

670-675 879-883 897-905 696-701 367-378 750-755 853-859 831-836 825-830 175-195 197-224 367-378 519-541 906-908 299-309 762-768 245-264 702-707 7 8 5 - 7 9 0 805-809 750-755

837-846 327-339 791-796 872-878

756-761

563-574

884-889 909-914 756-761

199

1120 Author Index

Chakraborty, A., see Rajan, S. Chao, P. C., see Chu, K. K. Chavarkar, P. M., see Wu, Y.-F. Chen, C., Lu, Z., Shi, S. and Prather, D. W., Self-Guiding in

Chen, Y. K., see Veksler, D. Chini, A., see Buttari, D. Chini, A., see Xie, S. Chini, A,, see Xu, H. Cho, W., see Xi, J.-Q. Choi, Y. C., Cha, H.-Y., Eastman, L. F. and Spencer, M. G., Influence of

Low-Index-Contrast Planar Photonic Crystals

the N-diffusion Layer on the Channel Current and the Breakdown Voltage in 4H-Sic SIT

Choi, Y. C., see Cha, H.-Y. Chow, T. P., see Huang, W. Chow, T. P., see Ruan, J. Chow, T. P., see Zhu, L. Chu, K. K., Chao, P. C. and Windyka, J. A., Stable High Power

GaN-on-GaN HEMT Ciplys, D., see Bu, G. Clarke, D. R., see Etzkorn, E. V. Clarke, D. R., see Tavernier, P. R. Clarke, R. C., see De Salvo, G. C. Collazo, R., Schlesser, R. and Sitar, Z., High Field Transport in A1N Conway, A. M., see Keogh, D. M. Conway, A., see Li, J. C. Cook, Jr., T. E., Fulton, C. C., Mecouch, W. J., Davis, R. F., Lucovsky,

G. and Nemanich, R. J., Electronic Properties of GaN(0001) - Dielectric Interfaces

Heterojunction Bipolar Transistors

Context, Radiation Effects, and Future Trends

Cressler, J. D., Total-Dose and Single-Event Effects in Silicon-Germanium

Cristoloveanu, S. and Ferlet-Cavrois, V., Introduction to SO1 MOSFETs:

Dadgar, A., see Neuburger, M. Danylyuk, S. V., see Vitusevich, S. A. Das, M. K., Sumakeris, J. J., Hull, B. A., Richmond, J., Krishnaswami, S.

Das, M., see Ryu, S. H. Daumiller, I., see Neuburger, M. Davis, R. F., see Cook, Jr., T. E. Davis, R. F., see Einfeldt, S. Davis, R. F., see Lee, J. Davis, R. F., see Roskowski, A. M.

and Powell, A. R., High Power, Drift-Free 4H-Sic PIN Diodes

732-737 738-744 816-818

720-725 632-639 756-761 847-852 810-815 726-731

909-914 884-889 872-878 797-804 865-871

738-744 837-846 63-81 51-62 906-908 155-174 831-836 825-830

107-125

489-501

465-487 785-790 762-768

860-864 879-883 785-790 107-125 39-50 83-105 21-37

200

Author Index 1121

De Salvo, G. C., Esker, P. M., Flint, T. A., Ostop, J. A., Stewart, E. J., Knight, T . J., Petrosky, K. J., Van Campen, S. D., Clarke, R. C. and Bates, G. M., Ion Implanted S i c Static Induction Transistor with 107 W Output Power and 59% Power Added Efficiency Under CW Operation at 750 MHz

DenBaars, S., see McCarthy, L. S. Dupuis, R. D., see Keogh, D. M. Dusseau, L. and Gasiot, J., Online and Realtime Dosimetry Using

Dutta, P. S., see Kumar, A. Dutta, P. S., see Pino, R. Eastman, L. F., see Cha, H.-Y. Eastman, L. F., see Choi, Y . C. Eastman, L. F., see Vitusevich, S. A. Einfeldt, S., Reitmeier, Z. J. and Davis, R. F., Strain of GaN Layers

Einfeldt, S., see Roskowski, A. M. Esker, P. M., see De Salvo, G. C. Estrada, S., Hu, E. and Mishra, U., n-AlGaAs/p-GaAs/n-GaN

Optically Stimulated Luminescence

Grown Using 6s-SiC(OOO1) Substrates with Different Buffer Layers

Heterojunction Bipolar Transistor: The First Transistor Formed Via Wafer h s i o n

Etzkorn, E. V. and Clarke, D. R., Cracking of GaN Films Etzkorn, E. V., see Tavernier, P. R. Faccio, F., Radiation Issues in the New Generation of High Energy Physics

Faraclas, E., Webster, R. T., Brandes, G. and Anwar, A. F. M., Experiments

Dependence of R F Performance of GaN/AlGaN HEMTs upon AlGaN Barrier Layer Variation

Fareed, Q., see Mickevicius, J. Feng, M., see Keogh, D. M. Feng, M., see Lai, J. W. Ferlet-Cavrois, V., see Cristoloveanu, S. Fleetwood, D. M., see Rashkeev, S. N. Flint, T. A., see De Salvo, G. C. Flynn, J. S., see Lee, J. Fonstad Jr, C. G.., see Giziewicz, W. P. Fouillat, P., Pouget, V., Lewis, D., Buchner, S. and McMorrow, D.,

Investigation of Single-Event Transients in Fast Integrated Circuits with a Pulsed Laser

Fulton, C. C., see Cook, Jr., T. E. Galloway, K. F., see Shenai, K. Gao, Y., Ben-Yaacov, I., Mishra, U. and Hu, E., Etched Aperture GaN

Gasiot, J., see Dusseau, L. Gaska, R., see Bu, G. Gaska, R., see Mickevicius, J. Gaska, R., see Rumyantsev, S. L.

Cavet Through Photoelectrochemical Wet Etching

906-908 225-243 831-836

605423 652457 658-663 884-889 909-914 762-768

3%50 21-37 906-908

265-284 63-81 51-62

379-399

750-755 696-701 831-836 625431 465-487 575-580 906-908 805-809 7 14-7 19

327-339 107-1 2 5 445-463

245-264 6055623 837-846 696-701 175-195

201

1122 Author Index

Gaska, R., see Sawyer, S. Gaska, R., see Simin, G. Ghori, A. and Ghosh, P., Analysis of Operational Transconductance

Ghosh, P., see Ghori, A. Gill, W. N., see Xi, J.-Q. Giziewicz, W. P., Fonstad Jr, C. G.. and Prasad, S., High Speed 0.9 prn

Lateral P-I-N Photodetectors Fabricated in a Standard Commercial GaAs VLSI Process

Goel, K., see Gupta, R. S. Gonye, G., see Xuan, G. Gorev, N. B., Kodzhespirova, I. F., Privatov, E. N., Khuchua, N.,

Amplifier for Application in GHz Frequency Range

Khvedelidze, L. and Shur, M. S., Photocapacitance of GaAs Thin-film Structures Fabricated on a Semi-Insulating Compensated Substrate

Grober, R., see Roskowski, A. M. Grossman, E. N., see Luukanen, A. Gupta, M., see Gupta, R. S. Gupta, M., see Gupta, R. S. Gupta, R. S., Aggarwal, S. K., Gupta, R., Haldar, S. and Gupta, M.,

Analytical Model for Non-Self Aligned Buried P-Layer S ic MESFET Gupta, R. S., Goel, K., Saxena, M. and Gupta, M., Two-Dimensional

Analytical Modeling and Simulation of Retrograde Doped HMG MOSFET

Gupta, R., see Gupta, R. S. Gunther, M., see Neuburger, M. Hafez, W., see Lai, J. W. Haldar, S., see Gupta, R. S. Halder, S., see Ye, P. D. Heidergott, W. F., System Level Single Event Upset Mitigation Strategies Heikman, S., see Xie, S. Heikman, S., see Xu, H. Holman, W. T., Radiation-Tolerant Design for High Performance

Mixed-Signal Circuits Hopkinson, G. R. and Mohammadzadeh, A., Radiation Effects in

Charge-coupled Device (CCD) Imagers and CMOS Active Pixel Sensors Houtsma, V. E., see Veksler, D. Hu, E., see Estrada, S. Hu, E., see Gao, Y. Huang, W., Chow, T. P., Yang, J. and Butler, J. E., High-Voltage

Huang, W., see Ruan, J. Hull, B. A., see Das, M. K. Hwang, J. C. M., see Ye, P. D. Islam, S. S. and Anwar, A. F. M., Spice Model of AlGaN/GaN HEMTs

Islam, S. S., see Mukherjee, S. S. Kapoor, V., see Xuan, G.

Diamond Schottky Rectifiers

and Simulation of VCO and Power Amplifier

702-707 197-224

690-695 690-695 726-731

714-719 676-683 684489

775-784 21-37 664469 676-683 897-905

897-905

676-683 897-905 785-790 625-631 897-905 791-796 341-352 847-852 810-815

353-366

419443 632439 265-284 245-264

872-878 797-804 860-864 791-796

853-859 890-896 684489

202

Author Index 1123

Keller, S., see Buttari, D. Keller, S., see Xu, H. Keogh, D. M., Li, J. C., Conway, A. M., Qiao, D., Raychaudhuri, S.,

Asbeck, P. M., Dupuis, R. D. and Feng, M., Analysis of GaN HBT Structures for High Power, High Efficiency Microwave Amplifiers

Keogh, D. M., see Li, J. C. Khuchua, N., see Gorev, N. B. Khvedelidze, L., see Gorev, N. B. Kim, H., Lee, J. and Lu, W., Trap Behavior in AlGaN/GaN HEMTs by

Kim, H., see Lee, J. Kim, K. W., Kochelap, V. A., Sokolov, V. N. and Komirenko, S. M.,

Kim, S., see Troeger, R. T. Klein, N., see Vitusevich, S. A. Knight, T. J., see De Salvo, G. C. Knudson, A. R., see McMorrow, D. KO, Y., see Pino, R. Kochelap, V. A., seeKim, K. W. Kodzhespirova, I. F., see Gorev, N. B. Kohn, E., see Neuburger, M. Kolodzey, J., see Troeger, R. T. Kolodzey, J., see Xuan, G. Komirenko, S. M., see Kim, K. W. Konkapaka, P., see Wu, H. Kostamovaara, J., Breakdown Phenomena in Semiconductors and

Semiconductor Devices Kosterin, P. V., see Sawyer, S. Kotecki, D. E., see Turner, S. E. Krishnaswami, S., see Das, M. K. Krishnaswami, S., see Ryu, S. H. Krost, A,, see Neuburger, M. Krtschil, A., see Neuburger, M. Kumar, A,, Sridaran, S. and Dutta, P. S., Atomically Flat 111-Antimonide

Kunze, M., see Neuburger, M. Label, K. A., see Reed, R. A. Lai, J. W., Hafez, W. and Feng, M., Vertical Scaling of Type I InP HBT

Lee, J., Davis, R. F. and Nemanich, R. J., Direct Bonding of GaN and

Lee, J., Liu, D., Kim, H., Schuette, M. L., Lu, W., Flynn, J. S. and

Post-Gate- Annealing

Quasi-Ballistic and Overshoot Transport in Group 111-Nitrides

Epilayers Grown Using Liquid Phase Epitaxy

With FT > 500 GHz

Sic; A Novel Technique for Electronic Device Fabrication

Brandes, G. R., Fabrication of Self-Aligned T-Gate AlGaN/GaN High Electron Mobility Transistors

Lee, J., see Kim, H. Levinshtein, M. E., see Rumyantsev, S. L. Levinshtein, M. E., see Rumyantsev, S. L.

756-761 810-815

831-836 825-830 775-784 775-784

769-774 805-809

127-154 670-675 762-768 906-908 311-325 658463 127-154 775-784 785-790 670475 684489 127-154 745-749

921-1 118 702-707 646451 860-864 879-883 785-790 785-790

652-657 785-790 401-417

625-631

83-105

805-809 76+774 1-19 175-195

203

1124 Author Index

Levinshtein, M. E., Breakdown Phenomena in Semiconductors and Semiconductor Devices

Lewis, D., see Fouillat, P. Li, J. C., Keogh, D. M., Raychaudhuri, S., Conway, A., Qiao, D. and

Asbeck, P. M., Analysis of High DC Current Gain Structures for GaN/InGaN/GaN HBTs

Li, J. C., see Keogh, D. M. Liu, D., see Lee, J. Liu, Q., Sutar, S. and Seabaugh, A,, Tunnel Diode/Transistor Differential

Long, S. I., see Xie, S. Losee, P., see Zhu, L. Lu, W., see Kim, H. Lu, W., see Lee, J. Lu, Z., see Chen, C. Lucovsky, G., see Cook, Jr., T. E. Luukanen, A., Grossman, E. N., Moyer, H. P. and Schulman, J. N., Noise

and THz Rectification Characteristics of Zero-Bias Quantum Tunneling Sb-Heterostructure Diodes

Lv, P.-C., see 'lkoeger, R. T. Makarov, Y., see Wu, H. Marshall, P. W., see Reed, R. A. Matocha, K., see Ruan, J. McCarthy, L. S., Zhang, N-Q., Xing, H., Moran, B., DenBaars, S. and

Mishra, U. K., High Voltage AlGaN/GaN Heterojunction Transistors McCarthy, L., see Buttari, D. McMorrow, D., Melinger, J. S. and Knudson, A. R., Single-Event Effects

in 111-V Semiconductor Electronics McMorrow, D., see Fouillat, P. Mecouch, W. J., see Cook, Jr., T. E. Melinger, J. S., see McMorrow, D. Mickevicius, J., Aleksiejunas, R., Shur, M. S., Zhang, J. P., Fareed, Q.,

G a s h , R. and Tamulaitis, G., Lifetime of Nonequilibrium Carriers in AlGaN Epilayers with High A1 Molar Fraction

Comparator

Miraglia, P. M., see Roskowski, A. M. Mishra, U. K., see Buttari, D. Mishra, U. K., see McCarthy, L. S. Mishra, U. K., see Rajan, S. Mishra, U. K., see Xie, S. Mishra, U. K., see Xing, H. G. Mishra, U. K., see Xu, H. Mishra, U., see Estrada, S. Mishra, U., see Gao, Y. Mohammadzadeh, A., see Hopkinson, G. R. Moore, M., see Wu, Y.-F. Moran, B., see McCarthy, L. S. Moyer, H. P., see Luukanen, A.

92 1-1 118 327-339

825-830 831-836 805-809

640-645 847-852 865-871 76S774 805-809 720-725 107-125

664469 670475 745-749 401417 797-804

225-243 756-761

311-325 327-339 107-125 3 11-325

696-701 21-37 756-761 225-243 732-737 847-852 819-824 810-815 265-284 245-264 41+443 816-818 225-243 664469

204

Author Index 1125

Mukherjee, S. S. and Islam, S. S., Effects of Buffer Layer Thickness and

Nemanich, R. J., see Cook, Jr., T. E. Nemanich, R. J., see Lee, J. Neuburger, M., Zimmermann, T., Kohn, E., Dadgar, A., Schulze, F.,

Doping Concentration on S i c MESFET Characteristics

Krtschil, A., Gunther, M., Witte, H., Blasing, J., Krost, A,, Daumiller, I. and Kunze, M., Unstrained InAlN/GaN HEMT Structure

Ng, K. K., see Ye, P. D. Normand, E., Single Event Effects in Avionics and on the Ground Nowlin, N., Bailey, J., Turfler, B. and Alexander, D., A Total-Dose

Hardening-By-Design Approach for High-speed Mixed-Signal CMOS Integrated Circuits

Ojha, M., see Xi, J.-Q. Oldham, T . R., Switching Oxide Traps Ostop, J. A., see De Salvo, G. C. Paccagnella, A,, see Cester, A. Paidi, V., see Xie, S. Pala, N., see Rumyantsev, S. L. Pala, N., see Sawyer, S. Palacios, T., see Buttari, D. Palmour, J., see Ryu, S. H. Pantelides, S. T., see Rashkeev, S. N. Parikh, P., see Wu, Y.-F. Pease, R. L., Hardness Assurance for Commercial Microelectronics Petrosky, K. J., see De Salvo, G. C. Petrychuk, M. V., see Vitusevich, S. A. Pino, R., KO, Y. and Dutta, P. S., Native Defect Compensation in

Plawsky, J. L., see Xi, J.-Q. Poblenz, C., see Rajan, S. Pouget, V., see Fouillat, P. Powell, A. R., see Das, M. K. Prasad, S., see Giziewicz, W. P. Prather, D. W., see Chen, C. Preble, E. A., see Roskowski, A. M. Privalov, E. N., see Gorev, N. B. Qiao, D., see Keogh, D. M. Qiao, D., see Li, J. C. Rajan, S., Chakraborty, A., Mishra, U. K., Poblenz, C., Waltereit, P. and

Speck, J. S., MBEGrown AlGaN/GaN HEMTs on S i c Rashkeev, S. N., Fleetwood, D. M., Schrimpf, R. D. and Pantelides, S. T.,

Hydrogen at the Si/SiOz Interface: From Atomic-Scale Calculations to Engineering Models

111-Antimonide Bulk Substrates

Ray, S. K., see Troeger, R. T. Raychaudhuri, S., see Keogh, D. M. Raychaudhuri, S., see Li, J. C.

890-896 107-125 83-105

785-790 791-796 285-298

367-378 726-731 581403 906-908 563-574 847-852 175- 195 702-707 756-761 879-883 575-580 816-818 543-561 906-908 762-768

658463 726-731 732-737 327-339 860-864 714-719 720-725 2 1-37 775-784 831-836 825-830

732-737

575-580 670475 831-836 825-830

205

1126 Author Index

Reed, R. A., Marshall, P. W. and Label, K. A,, Space Radiation Effects in

Reitmeier, Z. J., see Einfeldt, S. Richmond, J., see Das, M. K. Richmond, J., see Ryu, S. H. Rodwell, M. J. W., see Xie, S. Roskowski, A. M., Preble, E. A., Einfeldt, S., Miraglia, P. M., Schuck, J.,

Optocouplers

Grober, R. and Davis, R. F., Kinetics, Microstructure and Strain in GaN Thin Films Grown Via Pendeo-Epitaxy

Electron Mobility on EPI Channel Doping in GaN MOSFETs

Asif Khan, M. and Simin, G., Generation-Recombination Noise in GaN-Based Devices

Properties of Nitrides. Summary

Ruan, J., Matocha, K., Huang, W. and Chow, T. P., Dependence of

Rumyantsev, S. L., Pala, N., Shur, M. S., Levinshtein, M. E., Gaska, R.,

Rumyantsev, S. L., Shur, M. S. and Levinshtein, M. E., Materials

Rumyantsev, S. L., see Sawyer, S. Ryu, S. H., Krishnaswami, S., Das, M., Richmond, J., Agarwal, A.,

Palmour, J. and Scofield, J., 2 kV 4H-Sic DMOSFETs for Low Loss High Frequency Switching Applications

Salzberg, B. M., see Sawyer, S. Sanabria, C., see Xu, H. Sawyer, S., Rumyantsev, S. L., Pala, N., Shur, M. S., Bilenko, Y., Gaska,

R., Kosterin, P. V. and Salzberg, B. M., Noise Characteristics of 340 nm and 280 nm GaN-Based Light Emitting Diodes

Saxena, M., see Gupta, R. S. Saxler, A., see Wu, Y.-F. Schlesser, R., see Collazo, R. Schowalter, L. J., see Bu, G. Schrimpf, R. D., see Rashkeev, S. N. Schrimpf, R. D., see Shenai, K. Schrimpf, R. D., Gain Degradation and Enhanced Low-Dose-Rate

Schubert, E. F., see Xi, J.-Q. Schubert, E. F., see Xi, Y . Schuck, J., see Roskowski, A. M. Schuette, M. L., see Lee, J. Schujman, S. B., see Bu, G. Schulman, J. N., see Luukanen, A. Schulze, F., see Neuburger, M. Scofield, J., see Ryu, S. H. Seabaugh, A., see Liu, Q. Shen, L., see Buttari, D. Shen, L., see Xie, S. Shenai, K., Galloway, K. F. and Schrimpf, R. D., The Effects of Space

Shi, S., see Chen, C.

Sensitivity in Bipolar Junction Ransistors

Radiation Exposure On Power MOSFETs: A Review

401-417 39-50 860-864 879-883 847-852

21-37

797-804

175- 195

1-19 702-707

879-883 702-707 810-815

702-707 676483 816818 155-174 837-846 575-580 445-463

503-517 726-731 708-713 2 1-37 805-809 837-846 664469 785-790 879-883 640-645 756-761 847-852

445-463 720-725

206

Author Index 1127

Shur, M. S., see Bu, G. Shur, Mi S., see Gorev, N. B. Shur, M. S., see Mickevicius, J. Shur, M. S., see Rumyantsev, S. L. Shur, M. S., see Rumyantsev, S. L. Shur, M. S., see Sawyer, S. Shur, M. S., see Simin, G. Shur, M. S., see Veksler, D. Simin, G., Asif Khan, M., Shur, M. S. and Gaska, R., Insulated Gate 111-N

Simin, G., see Rumyantsev, S. L. Sitar, Z., see Collazo, R. Sokolov, V. N., see Kim, K. W. Speck, J. S., see Rajan, S. Spencer, M. G., see Cha, H.-Y. Spencer, M. G., see Choi, Y. C. Spencer, M. G., see Wu, H. Sridaran, S., see Kumar, A, Stewart, E. J., see De Salvo, G. C. Sumakeris, J. J., see Das, M. K. Sutar, S., see Liu, Q. Tamulaitis, G., see Mickevicius, J. Tavernier, P. R., Etzkorn, E. V. and Clarke, D. R., Growth of Thick GaN

Th. Gessmann, see Xi, J.-Q. Troeger, R. T., Adam, T. N., Ray, S. K., Lv, P.-C., Kim, S. and Kolodzey,

J., Temperature Dependence of Terahertz Emission from Silicon Devices Doped with Boron

Heterostructure Field-Effect Transistors

Films and Seeds for Bulk Crystal Growth

Turfler, B., see Nowlin, N. Turner, S. E. and Kotecki, D. E., Benchmark Results for High-speed 4-Bit

Vainshtein, S., Breakdown Phenomena in Semiconductors and

Van Campen, S. D., see De Salvo, G. C. Veksler, D., Shur, M. S., Houtsma, V. E., Weimann, N. G. and Chen, Y.

Accumulators Implemented in Indium Phosphide DHBT Technology

Semiconductor Devices

K., Numerical Investigation of the Effect of Doping Profiles on the High Frequency Performance of InP/InGaAs Super Scaled HBTs

Vertiatchikh, A., see Vitusevich, S. A. Vitusevich, S. A., Danylyuk, S. V., Klein, N., Petrychuk, M. V., Belyaev,

A. E., Vertiatchikh, A. and Eastman, L. F., Low Frequency Noise Parameters in an AlGaN/GaN Heterostructure with 33% and 75% A1 Mole Fraction

Waltereit, P., see Rajan, S. Webster, R. T., see Faraclas, E. Wei, Y., see Xu, H. Weimann, N. G., see Veksler, D. Wilk, G. D., see Ye, P. D.

837-846 775-784 696-701 1-19 175-195 702-707 197-224 632439

197-224 175-195 155-174 127-1 54 732-737 884-889 909-914 745-749 652457 906-908 860-864 640445 696-701

5 1 4 2 726-731

670-675 367-378

646451

921-1118 906-908

632439 762-768

762-768 732-737 750-755 81G815 632439 791-796

207

1128 Author Index

Windyka, J. A., see Chu, K. K. Wisleder, T., see Wu, Y.-F. Witte, H., see Neuburger, M. Wu, H., Konkapaka, P., Makarov, Y. and Spencer, M. G., Thick GaN

Layer Grown by Ga Vapor Transport Technique Wu, Y.-F., Moore, M., Wisleder, T., Chavarkar, P. M., Parikh, P. and

Saxler, A,, Noise Characteristics of Field-Plated GaN HEMTs Xi, J.-Q., Ojha, M., Cho, W., Gessmann, Th., Schubert, E. F., Plawsky, J.

L. and Gill, W. N., Omni-Directional Reflector using a Low Refractive Index Material

Xi, Y. and Schubert, E. F., Junction-Temperature Measurements in GaN UV Light-Emitting Diodes using the Diode Forward Voltage

Xie, S., Paidi, V., Heikman, S., Shen, L., Chini, A,, Mishra, U. K., Rodwell, M. J. W. and Long, S. I., High Linearity GaN HEMT Power Amplifier with Pre-Linearization Gate Diode

Characteristics of AlGaN/GaN HBTs and GaN BJTs Xing, H. G. and Mishra, U. K., Temperature Dependent I-V

Xing, H., see Buttari, D. Xing, H., see McCarthy, L. S. Xu, H., Sanabria, C., Chini, A., Wei, Y., Heikman, S., Keller, S., Mishra,

U. K. and York, R. A., A New Field-Plated GaN HEMT Structure with Improved Power and Noise Performance

DNA Molecules on Silicon Field Effect Transistor Xuan, G., Kolodzey, J., Kapoor, V. and Gonye, G., Electrical Effects of

Yang, B., see Ye, P. D. Yang, J., see Huang, W. Ye, P. D., Yang, B., Ng, K. K., Bude, J., Wilk, G. D., Halder, S. and

Hwang, J. C. M., GaN MOS-HEMT Using Atomic Layer Deposition A1203 as Gate Dielectric and Surface Passivation

York, R. A,, see Xu, H. Zhang, J. P., see Mickevicius, J. Zhang, N-Q., see McCarthy, L. S. Zhu, L., Losee, P. and Chow, T. P., Design of High Voltage 4H-Sic

Superjunction Schottky Rectifiers Zimmermann, T., see Neuburger. M

738-744 816-818 785-790

745-749

8 16-8 18

726-731

708-713

847-852

81S-824 75G761 225-243

810-815

684489 791-796 872-878

791-796 810-815 696-701 225-243

865-871 785-790

208

Documents

Breakdown Phenomena in Semiconductors and Semiconductor Devices 9812563954