Chapter 3

IMPATT (Impact Avalanche Transit Time) DEVICES

INTRODUCTION

IMPATT (Impact Avalanche Transit Time) devices are solid state oscilla­tors or amplifiers for microwave and millimeter wave frequencies up to above 200 GH., generally with high power output (for a solid state device). Efficien­cies as high as 25% have recently been obtained with GaA. devices, but the traditional IMPATT is a silicon device with up to 15% efficiency. The basic principle involves a 1800 phase-delay of the current with respect to the volt­age, which clearly will make the device exhibit a negative resistance. Shockley (1954) had introduced this general idea, not yet involving impact avalanching to produce the phase-shift. A number of different p-n-junction type structures are used today. During operation, the diode is undergoing (controlled) reverse break-down.

The history of the development of the IMP ATT device has some similari­ties to that of the Gunn-device. Both were being developed in parallel in the early 1960's, for example one Bell Laboratories group worked simultaneously on both devices (DeLoach, 1976). In the case of the IMPATT, there was also a theoretical prediction of one version of the IMP ATT operation, published in 1958 by W.T. Read (Read, 1958). Work was ongoing to translate Read's ideas into practice, but this work had not yet been successful when, as described by DeLoach (@1976 IEEE):

"In order to check the electrical continuity of one of these cir­cuits, Ralph [Johnston) borrowed a silicon diode (from Ray Rulison's junk box) that had been manufactured years before and inserted it into one of the cavities. Upon observing the pulse current-voltage characteristics of this diode in the cavity with a sampling oscillo­scope, Ralph observed oscillations and later informed Barry [Cohen] and me that he had a "box full" of high-frequency oscillators. Would we like to see them?"

The event described above, resulted in the first publication regarding an IMPATT type oscillator (Johnston, DeLoach and Cohen, 1965). Only a couple of months later, Read-type IMPATT operation was also observed (Lee et al., 1965).

S. Yngvesson, Microwave Semiconductor Devices© Kluwer Academic Publishers 1991

60 Microwave Semiconductor Devices

+ E-field P ! Em n

~--~----~~----------~X

x=o x=W

Figure 3.1. Electric field versus position in a p+ on-diode.

ENERGY BANDS (Reverse Bias)

Electrons

Holes, accelerated

Figure 3.2. Energy band" of a p+ on-junction in reverse bias.

v

In this chapter, we will discuss reverse break-down due to impact ioniza­tion avalanching as well as phase-delay of the current due to drifting carriers. The combination of these two effects yields the required 180· of phase-shift.

Chapter 3 61

OPERATION OF IMPATT DEVICES-PHYSICAL DISCUSSION

The Impact Ionization Process

Reverse break-down is most conveniently first discussed for a p+-n diode, usually implemented with a p+-n-n+ structure, see Figure 3.1. The corre­sponding energy band diagram for large reverse bias is given in Figure 3.2. The reverse current consists primarily of holes which diffuse up to the poten­tial barrier from the n-side, and then quickly traverse the junction, accelerated by the very strong electric field there. A smaller number of electrons cross the junction in the opposite direction, smaller because the p-side is more highly doped, and because it is the minority carriers which constitute the reverse di­rection current. The holes will become energetic enough to be able to ionize an electron pair, somewhere in the depletion layer. This process requires about 1.5 times the bandgap energy, i.e. about 1.6 eV for silicon. This is considerably more energy than was required for electron transfer in GaAs used in Gunn­devices, as described in the previous chapter, and consequently much higher fields must be used to accelerate the electrons (200 kV fcm or higher). Such fields do exist in p-n-junctions. The impact ionization will be able to sustain an avalanche if the probability that a carrier will cause an ionization event on its transit of the depletion region is equal to 1.0 or higher, and under these circumstances, the process will start to grow rapidly (in time) in the manner of an avalanche. The avalanche condition can be expressed:

l W o:(E)dz = 1 (3.1)

Here, 0: is the probability of impact ionization occurring in unit length of travel, or the "ionization rate". The units are usually cm-1 • To be more realistic, we must recognize that the ionization rates for electrons and holes are not the same. Also, the ionization rates depend very strongly on the electric field. A typical dependence is:

o:(E) = 0:0 • E6 (3.2a)

For GaAs, a better fit is obtained by using for electrons (see Chapter 1):

(3.2b)

and for holes:

f3(E) = cexp [ - (i) n] (3.2c)

Many sets of experimental data have been published for the parameters in (3.2b) and (3.2c) (for a review, see (Bulman et al., 1985)). In particular, it has not been clear if 0: and f3 differ significantly. Recent results due to Ito et al. (1978), Bulman et al. (1985), and Masse' et al. (1985) are quite consistent, however, especially in the region of very high fields, which is the important

62 Microwave Semiconductor Devicell

region for IMPATT devices. Masse' et al. (1985) use m = 2, while Bulman et al. (1985) arrive at a best-fit to their extensive data for m = 1.82 and n = 1.75. It should be noted that the parameters a through d are temperature-dependent. Based on measurements by Holway et aI. (1979), Masse' et aI. (1985) quote the following values for a and b:

a = 1.61 x 105 [1 + 0.0007{T - 25)]

b = 5.41 x 105 [1 + 0.00097{T - 25)]

(3.3a)

(3.3b)

where a is in cm-1 , b in V /cm, and T in °C. The different values for a and f3 can be taken care of by defining an effective a, as discussed below.

Figure 3.3 shows a versus electric field for a number of semiconductors. Note that larger electric fields are required for larger bandgap semiconductors, as expected. The most recent data for GaAa due to Bulman et al. (1985) are given in Figure 3.3c. Note that a and f3 are very close at high electric fields.

A generalized avalanche condition can be derived for the case of unequal rates for electrons and holes (Sze, 1981):

l W f3exp[-1-(f3-a)dz']dZ=1 (3.4)

Define the average ionization rate for electrons and holes:

<a >=f3exp [- [(f3-a)dZ']

which leads to: l W < a > dz = 1

By using (3.2b) and (3.2c), the effective ct can be shown to be: a

a-II = (k - 1) ink

Here,

k{E) = ~~!~.

(3.4a)

(3.4b)

(3.5)

(3.5a)

This value for < a > can be used with very accurate results for GaAa (Masse' et al., 1985).

Due to the strong dependence of a on electric field, we find that the avalanching occurs almost excl usi vely very close to the peak of the electric field (in Figure 3.1). This region of the diode is termed the "avalanche region". In a device with field distribution as in Figure 3.1, the holes generated will drift to the left into the contact and will be out ofthe picture as far as device operation is concerned, whereas the electrons will drift to the right through the so called "drift region" of the device, see Figure 3.4.

Chapter 3 63

e {lO!> Vlcml

5){1O~ 108 6 5 • 3

-51

10·

~ !" 10' ;i

:\ ~ >= ~ Z

'" 10'

a.

102 0 • 6

1/C (10-6 em/v)

(a)

I.'!I 20 2'.!o l.O '!I 111.0 III,!! 5..0 liE (0Tl1V! _10"

(b)

Figure 3.S. (a) Impact ionization rate, at 300 K for Si and GaA" ver,,,,, reciprocal electric field. Reprinted from SZE, S.M. (1985). "Semiconduc­tor Device,: Phy,ic6 and Technology," John Wiley fJ Son6, New York, with permiuion. (b) Recent measured impact ionization coeiJicient6 for GaA6, versus reciprocal electric field, reprinted from BULMAN, G.E., ROBBINS, V.M., and STILLMAN, G.E. (1985). "The Determination of Impact Ion­ization CoeiJicient6 in (100) GaA, U,ing Avalanche Noi,e and Photocurrent Multiplication Mea,urements," IEEE Tran8. Electron Device" ED-32, 2-15-1, @1985 IEEE. The top curve applies to electrons, and the lower one to holes. The ,olid curve, drawn are the re6ult of fib to Equation' (3.2b) and (9.2c), re'pectively.

64 Microwave Semiconductor Devices

E ,---- Avalanche Region -----

Drift Region

Figure 3.4. Definition of avalanche and drift regions in a p+ -n IMPATT diode.

Reverse Break-Down Voltage

It is convenient to observe that the maximum field of the triangular field­distribution in a p-n-junction diode at breakdown is fairly independent of the doping and the type of junction (see Figure 3.5). The explanation for this fact is again the very high E-field dependence of the ionization-coefficient, cr. Thus, when the field has reached a certain magnitude, then it becomes very easy to satisfy the avalanche condition when the integral is taken over typical junction widths, and the field for which this happens is basically a universal constant for the material. The reverse break-down voltage can now be derived in terms ofthe maximum E-field at break-down, Em (see Chapter 1, p. 10). The results are for the standard cases of a one-sided (p+ - n) diode, and two-sided diode, respecti vely,

Vi - 1 E W _ E,E~ B-- m. ---

2 2elVB (3.5)

Vi -!E W- E,E~ B- 2 m - elVB (3.6)

Note that the break-down voltage is inversely proportional to the doping in both cases. This can be seen from Figure 3.6. Typical break-down voltages in the 10 to 100 volt range are obtained if the doping is of the order of 1016

to 1017cm-3 (on the low side in one-sided junctions!). This corresponds to avalanche region widths of 1 micrometer or less, as seen from Figure 3.7. These widths are 10-20% of the width of the entire depletion region, justifying the model, which assumes that the drift region occupies most of the depletion region.

Chapter 3

10XI0~r--------'------r------T----7)1

9

8.~--------+---------~---------r--~¥-~~

l-e7

""" 0"" {GOAS-<IOO> jj ~ 6f-----ONE SIDED

u:: Z Si ~~5 ,,0 ~~ x"" 4 ~-------I---:;::_-:::::;;""''''''''=--+-'''S:-a.:::.-=::------'' ""w ~:li3

2~~------~--------~-------i---------;

~OLI~4-L-~~~~IOUI~5-J--LJ-LLLUI0~1~6-L-~~~UI0~1~7-~~~LU~10~18

NB(cm-3 )

65

Figure 3.S. Mazimum electric field at breakdown, ver.u. doping, for Si and GaA. one-aided and two-.ided abruptjunctiona. Reprinted from SCHROEDER, W_E_, and HADDAD, G_I. (1971). "Avalanche Region Width in VarioU6 Structure. of IMPATT Diode.," Proc. IEEE, 59, 1145, @1971 IEEE.

1000 ONE-SIDED ~

GaAs (100) ABRUPT JUNCTION S :j f': ll-:i " +"""1 f'.,.SII" ~ap "I

1'-.. ., Ge-- i' 10-f'

I·' 1 10'4 1015 10'6 '0'7

IMPURITY CONCENTRATION NB(Cm-3)

Figure 3.6. A valanche breakdown voltage verau. impurity concentration for one-.ided abrupt junction. in Ge, Si, (100) GaA., and GaP_ Be1l0nd the dotted line, tunneling mechanuma dominate the breakdown. Reprinted from SZE, S.M. (1981). "Phy,ie, of Semiconductor Device,," Second Edition, John Wile1l tJ Son., New York, with permi""ion.

66 Microwave Semiconductor Device"

SI (SYMMETRICAL)

T"300 I(

GoA, (ONE SIDED)

Si (n+p)

Figure 3.7. Avalanche region width for abrupt Si and GaAs junctions, and depletion region widths for lIymmetrical Si and GaAII junctions. Reprinted from SCHROEDER, W.E., and HADDAD, G.I. (1971). "Avalanche Region Width in Varia1" Structures of IMPATT Diodell," Proc. IEEE, 59, 12.45, @1971 IEEE.

Chapter 3 67

Saturated Drift Velocity

From Figure 3.8a we can see that the electron and hole drift velocities in silicon reach a saturation value of about 107 em/sec at 3000 K. The sat­uration velocity decreases as the temperature increases. It is important that the velocity is almost independent of the electric field for the high fields which exist in IMPATT devices - this means that we can regard the carrier velocity as essentially constant and equal to the saturation velocity anywhere in the drift-region.

The saturation velocity of electrons and holes in GaAB is somewhat lower. The saturation velocity is essentially the same for electrons and holes. Its temperature-dependence is given by Masse' et al. (1985):

6 [ 262 1.21 x 104 ] vs(T) = 6.12 x 10 0.268 + 273 + T - (273 + T)2 (3.7)

where v is the em/sec and T is in GC.

DisplaceDlent Current

Due to the large amounts of rapidly moving electric charge in an IMP ATT device, it is necessary to take the displacement current into account. The avalanche gives rise to the "injected" current which is localized to the avalanche region. Due to the nature of the avalanche process, the injected current is concentrated toward the end of the first half period of the AC-voltage across the device (to be discussed in more detail below). The time dependence of the injected charge is therefore similar to a short pulse. As this pulse of charge drifts through the drift region, we can model it with a constant charge which moves through a capacitor. In general, the total current is continuous, and is represented by a displacement current in any plane of the capacitor which does not contain the charge, see Figure 3.9. IT the charge is moving with constant velocity, then the total current is also constant in time until the charge has traversed the capacitor. We can calculate the average current when the charge, q, moves across a capacitor of width, W, in a time at, as

q q·v 1=-=-at W

(3.8)

If the velocity of the charge is equal to the saturated velocity, v., then the current becomes

q. v. 1=-­

W (3.9)

We should note that this current appears almost instantaneously in the contacts of the device, once the charge starts moving.

68 Microwave Semiconductor DevicelJ

~ GoAs (ELECTRONS) E

i 'O'~~~!I~~~~~€lJlII"-~_~_-~·~~-!l~~~~~ w Ge -->

t;:: / K -~ 106~~~~'~~~~~'~~II~~~T~=~3~0~0~K~~::;~~ - ~ Si ELECTRONS = ~ , ----- HOLES -

~ " , IIIIIII IIIIIII

105 ,--V ~/'-J..J.' ""-I..I..I..I-~..LU..I.L.-.-~II ............ III-,----,-I,-,-,JI~lll d d d d d

"iii ..... E u

to

g '" >

>-I-0 0 ...J W > z 0

~ II: ::;)

!:t en

ELECTRIC FIELD (V/cm)

(a) 14....-----------'---'----,--------,

12

10

O.B

0.6

04

0.2 10

2.4xl07 Si v~-------------

'+OB exp (T/SOOK)

(b) 100

T(K)

Vs- liT

1000

Figure 3.S. (a) Mea.ured carrier velocity verlJUIJ electric field for high purity Ge, Si, and GaA., (b) Saturated electron velocity verlJulJ temperature in Si and GaAIJ. Reprinted from SZE, S.M.(1981). "PhYlJic. of Semiconductor Device.," Second Edition, John Wiley £j SonlJ, New York, with permil.ion.

Chapter 3

Total Current is -1 I v r-- Continuous [I -e I

I (Displacement + ~ Conduction Current)

w

Figure 3.9. An electron moving between two metal plates.

Diagram of p+-n Diode IMPATT Device Operation

69

We are now ready to put the above facts together in a diagram (Figure 3.10). We assume that the E-field consists of a DC component, adjusted so that reverse break-down just occurs, and an AC component, which is added uniformly across the device (compare the Gunn-device case, where the electric field could not be uniform). The structure of the device is shown in Figure 3.11. We use the same sign conventions as in Chapter 2, i.e. the E-field and voltage are positive when they accelerate carriers in the positive z-direction, and the current is positive in the same direction as the electric field (see Fig. 3.11). Referring to Figure 3.10, the AC electric field is shown in the lowest set of curves, as a function of time. In the top of the diagram, we see in the first "frame" the DC electric field distribution. As the AC field component grows positive, the avalanche is initiated. The growth of the charge produced by the avalanche is pictured in the second set of frames from the top. In the second frame (t = T/4), the AC field is at its maximum, but the maximum avalanche charge injection does not occur until the third frame (t = T/2), just before the AC voltage goes negative and drives the total voltage below the break­down voltage. In other words, the avalanche continues to grow as long as the electric field is larger than E.... As the AC voltage becomes more negative, the avalanche injected charge drifts through the drift region, and the device current, shown in the next set offrames, is constant with time until the charge reaches the right contact. If we compare the two lowest sets of frames, we notice that the current has an approximately square wave-form, being oft' for the first half period, and constant and positive during the second half period, which represents the negative resistance which we sought.

70

C'un.:nt

.l:d e.~C(t)

~. ElccU'h;:flcld

L Ol .... t:c Density

l::T/"

~

Eu. IS I::T/oI :\

-AvaianchcGro .... 'i -

Microwave Semiconductor Devices

t .. T/2

~ I =3'f/4

~time ,.T

~'ime lime:

Ub ~ Ime ~.

~ ~x L

Figure 3.10. Operation of an IMPATT device, illustrating the electric field, charge density, current and AC voltage for the device.

Chapter 3 71

The negative resistance arises as a result of three effects:

1) The delay of the maximum of the avalanche until it just turns off, i.e. by T/2 with respect to t = O.

2) The transit time delay of T /2 for the drifting charges in the drift region.

3) The effect of the displacement current, which produces a square current pulse from a short pulse of injected charge. This factor effectively speeds up the onset of the current in the contact by T /2.

Taking into account all of the three above effects, we find a total delay of T/2 of the current with respect to the voltage. Specifically, we see that the correct value of the transit time delay in the drift-region should be about T/2. We can also calculate the estimated frequency of operation as follows (note that there is a factor of two ratio of the transit times, in periods, for the IMPATT device compared with a transit-time mode Gunn device.):

1 WD V. TC = T/2 == 21 = -:v,- => 1 = 2WD

(3.10)

As an example, a 10 GHz IMPATT device using silicon should have a drift region length of about 5 micron.

We also find it useful to introduce the concept of transit angle, defined as:

(3.11)

SMALL-SIGNAL THEORY OF IMPATT DEVICE IMPEDANCE

In discussing the microwave properties of devices, we will often use the concepts of 6mall-6ignal and large-6ignal models. In a small-signal model, all RF quantities are very small compared with the DC (bias) quantities. Such a model is linear in the AC quantities and can be used, for example, to find the gain of a low-noise (i.e. low-power) amplifier. It will also be able to describe an oscillator during .tart-up, i.e. a short time after the bias supply has been turned on. For either a high-power amplifier or an oscillator in steady-state, the small-signal condition is not satisfied. For the latter devices we need to employ a (non-linear) large-signal model. The transition of an oscillator from start-up toward the steady-state condition will be discussed in some detail in Chapter 6. We may also note that the diagram of Figure 3.10 implicitly assumed a large-.ignal RF electric field and voltage. In this chapter we will now develop analytical models of IMPATT devices for both the small-signal and the large-signal cases.

72 Microwave Semiconductor Devices

Avalanche Region (p+)

Drift Region + 8-Electrons

~ Pos.E ~ X=O - X=WD

Injection Pos. I Plane -

Figure 3.11. Definition of positive directions for electron velocity, electric field, and current in an IMPATT device.

Optimum Phase Delays

In order to begin to explore the different phase delays analytically, we use a model of the device as shown in Figure 3.11. Lower-case symbols indicate small-signal quantities. We assume that the avalanche and drift regions can be regarded separately. The current has two components, as discussed before, i.e. the conduction current and the displacement current:

(3.12)

conduction displacement

The injected current at z = 0 is clearly a conduction current and is as­sumed to have a phase delay tP with respect to the total current. Although it would be more straight-forward conceptually to count the phase-delay with respect to the total voltage, it is easier to do the calculations at this stage with this definition. Thus,

(3.13)

At an arbitrary z, the conduction current will be further phase-shifted according to the time it takes for the pulse of injected charge to reach that point, moving with the saturated velocity. Adding the displacement current, we thus have,

iTOT(Z) = inez = 0) . e-i..,m/w, + iwee(z) (3.14)

Chapter 3 73

We can solve for the electric field:

jTOT [1 - 'Y • e-; .. ,"/e.-Nj (3.15) e(z) = .

JW€

If we now integrate the electric field over the drift region, and divide by the total current, we find the impedance of the drift region*

_ fowD e(z)dz __ 1_ [ _ 'Y' e-N (l- e-j'D)] ZD -. - . C 1 .()

JTOT 1w D J D (3.16)

We have defined the following quantities:

CD == capacitance/unit area = ;D ()D == transit angle = ":'0 (the transit angle is 211' when the transit time

is one RF period)

Before we consider a more complete model we will neglect the details of the avalanche process, and simply set 'Y = 1.

Taking real and imaginary parts of Equation (3.16), we finally have the following expressions for the drift region equivalent resistance and reactance:

RD = costfJ - cOS(tfJ+()D)jXD = __ 1_ + sin(()D + tfJ) - sintfJ (3.17) WCD()D WCD WCD()D

These expressions are useful for a parametric study of the effect of chang­ing the two phase-delays, tfJ and ()D' The resistance is shown versus ()D for three different values of tfJ in Figure 3.12. For tfJ = 0, there is no negative resis­tance for any value of ()D, while for non-zero tfJ a region for ()D develops which shows a negative resistance. The optimum value for (JD decreases, such that, very approximately, the sum of ()D and tfJ is constant and equal to 211'. The largest negative resistance occurs when tfJ = 11', with an optimum for ()D of just under 11'. These are the same phase delays which we found in the discussion of Figure 3.10, based on physical considerations. Note that in this case, we are predicting a negative resistance for the total current, i.e. our phase delay is with respect to the positive going part of the total voltage at t = 0, and it is seen that tfJ is indeed 11'. The operation of the BARITT device corresponds closest to tfJ = j while "regular" IMP ATT operation corresponds to tfJ ~ 11'. BARITTs will be discussed further in Chapter 4.

Device Impedance

We next consider the avalanche region. We assume small-signal, AC, values for all parameters: The equations are simplified by assuming an average ionization probability Ii for the avalanche region.

(3.18)

* The sign convention for the RF voltage is to take VRF positive at 011 = W D

in Fig. 3.11.

74 Microwave Semiconductor Devicell

i

0.8 <--PHI-PI/2

0.6

0.4 <--PH 1-0

u ~ CD 0.2 IlJ % 0 0 ...... c a=

-0.2

-0.4 <--PHI-PI -0.6

-0.8 0 2 3

THETA. UNITS of PI

Figure 3.12. Plot of the resi,ltive part of the IMPATT diode impedance, all a function of the angle 9, all calculated from the ,mall-lignal theory. Three different valuell of cp have been allsumed, all marked.

The avalanche condition, (3.4b), is assumed to be satisfied by the DC (bias) component of the electric field. The total product of ciWA is made somewhat greater than 1 by the additional AC electric field:

(3.19)

Also,

J J . dJA •• E E A= O+1Ai--a:;:=1WJ1Ai A= o+eAi (3.20)

Chapter 3 75

We will also use the so-called "Read's Equation" (derived in Sze, (1981), p.581-2) which describes the time development of the avalanche:*

dJ rWA TA d: = 3JA()0 Q(E)dz -1); (3.21)

Here, TA = .!f1- is the avalanche region transit time; If we use aWA =

IowA a(E)dz, (3.21) simplifies to:

dJA ~ 3JA(aWA -1) (3.22) dt TA

We can now substitute (3.19) into (3.22), and use the fact that aA(Eo)WA = 1:

dJA ., 3Jo I W -- = ]IN]A ~ - X QA AeA dt TA

(3.23)

In terms of small-signal quantities, the relation (3.23) tells us that the voltage and current in the avalanche region are related as if the avalanche were a circuit inductance, LA:

ilNLAiA = VA(= WAe .. )

The value for LA is (see (3.23)):

LA=~ 3JOQ A

The region also has a capacitance, of course, i.e.

(per unit area)

(3.24)

(3.25)

(3.26)

This results in a standard parallel resonance equivalent circuit which is shown below in Figure 3.13.

The resonant frequency of this circuit is called the "avalanche resonance frequency", and is given by (introduce the area of the device, A, such that lolA is the current density):

1 2 3aAW A l o INA = v'LACA j INA = EAT A (3.27)

Note that INA is proportional to the square root of the current. An alternative expression uses TA = WAlv, :

2 3aAv,lo (3.28) INA=~j

* Different authors disagree about the numerical factor in the Read equa­tion. We are using a factor of '3' which is nowadays generally accepted (see Masse' et al. (1985), Holway et al. (1979))

76 Microwave Semiconductor Device!

Figure 3.13. Equivalent circuit for the avalanche region of an IMPATT device.

which is now independent of the width of the avalanche region. From the equivalent circuit, we find the total impedance of the avalanche region by standard circuit theory manipulations:

Thus, ZA is a capacitive reactance for W > WA'

Another quantity which is easy to derive from the equivalent circuit in Figure 3.13 is the injection phase delay, t/J, which we introduced earlier (Eq. (3.13)):

(3.30)

An IMPATT device is used at a frequency which is typically somewhat higher than the avalanche resonance frequency, and this case corresponds to t/J = 71', consistent with our earlier findings. We will see below that this gives us a negative resistance for the entire device.

Now that we have an expression for the injection phase delay, (3.30), we can substitute this into (3.16) - the drift region impedance. The total length of the device is Wand the length of the drift region is thus W - WA. The voltage across the drift region becomes:

The drift region impedance then is found to be:

ZD = .VD STOT

(3.31 )

Chapter 3 77

Figure 3.14. Equivalent circuit of an IMPATT diode with a load, at reso­nance.

1 [ 1 (1-COS8D)] j [1 1 (sin8D )] = WCD 1 - (W/WA)2 8D + WCD - + 1 - (W/WA)2 9;;-(3.32)

Adding the impedances for the drift and avalanche regions, and also a series resistance to represent losses in the contact regions, we finally get:

+_3_ SInI7D -1 _ 8D W-WA n . [(." ) (!!!!.!D. + ~)] WCD 8D 1 - (WA/w)2

(3.33)

As anticipated, we see that the real part becomes negative for frequencies above the avalanche resonance frequency. The reactive part then becomes capacitive, so that the equivalent circuit for the entire device for this ca,e i, a negative re,i,tance in ,erie, with a capacitance. The value of the capacitance for W » WA and ()D < i is simply the parallel plate capacitance of the entire device. In most devices, W is only somewhat> WA and ()D ~ 11"; then the value of the device capacitance is modified from CD and depends on WA, ()D and ~ W-WA·

ESTIMATE OF THE POWER CONVERSION EFFICIENCY OF IMPATT DEVICES - A SIMPLE LARGE SIGNAL MODEL

We want to calculate the efficiency of the IMP ATT diode for power gen­eration, when it is placed in a resonant circuit such as shown in Figure 3.14. At resonance, the reactance cancels, and the diode can be represented by a

78 Microwave Semiconductor Devices

negative conductance, -G D. The power is absorbed in the load conductance, G L. As we will discuss in detail in Chapter 6, the oscillation condition is:

IRF IGDI = GL : -GD =-V RF

The power delivered to the load is:

and the efficiency becomes:

(3.34)

(3.35)

(3.36)

We assume simplified voltage and current waveforms as in Figure 3.15 in order to estimate the optimum efficiency for RF power generation. In this figure, the injected current is the current injected by the avalanche into the drift region, and the induced (displacement) current is the current in the anode due to the displacement effect. It is intuitively obvious that the pulse of the injected current should be as narrow as possible in order to result in optimum efficiency, and we thus assume that the angular width of the injected current pulse 9w is «1['. Also note that the phase angle at which the injection takes place is 9" while the transit angle for the drift region is 9 D. Since the induced current results from the injected charge at z = 0, we find by using the condition for conservation of the total charge:

(3.37)

Further,

(3.38)

which is useful for specifying the amplitude of the induced current, and

(3.39)

The RF power generated can be obtained from the integral (note that this is the negative of the expression commonly used to calculate dis'ipatedpower):

Chapter 3 79

VT

0 VRF btl S "0 :> ;:; c

Vdc .~

~ a

Imj E ~

.--- -::: U

~ - f-aw 0

S I I I I a

'2'

Imd E ~

aD ::: U '3 u ::: "0 .s

7r/2 a

0 7t 37r/2 2x

Figure 3.15. Simplified voltage and current waveform6, u6ed to calculate the o6cillator efficiency.

and we can thus express the power conversion efficiency, 1], as follows:

VRF cos fI; - cos(fI; + flD) 11 = - VB X flD (3.41)

The optimum value for this expression occurs for flD = 0.7411', and fI; = 11', and the efficiency is then:

_ 2.27 (VRF) lImax - -- --

11' VB (3.42)

80 Microwave Semiconductor Device8

IT the width of the injected charge can not be neglected, a further factor should be included:

sin (Ow /2)

Ow/2 (3.43)

It remains to estimate how large the RF voltage can be compared with the DC voltage, and what fraction ofthe voltage that appears across the avalanche­and drift-regions, respectively. The latter ratio has optima at 10 GHz of 0.65 and 1.1 for GaAs and Si, respectively (Seidel et al., 1974), see Figure 3.16. (These early results have been modified in more recent work, see the section further on which deals with an analytical large-signal model.) The magni­tude of the AC field is limited because the field in the drift region must stay above the field for saturated drift velocity. Also the field will redistribute itself (with constant total voltage) due to the traveling charge. The field becomes depressed to the left of the moving charge and increased to the right of it, as shown in Figure 3.17. If the magnitude of the charge "bunch" is QA(Z) , then the induced current is found by first integrating Poisson's law:

j dE dz = JQ(z)dz =?AE= JQ(z)dz dz E E

(3.44)

Here, AE is the total change in E-field due to the passing charge, as shown in Figure 3.17. The induced current density is obtained from the transit time (Eq. (3.10)) and the integrated charge,

J = JtD Q(z)dz = E' AE. 2/ Tj

(3.45)

and

JDC=E·AE·/ (3.46)

since the average (DC) current is about half of the maximum current in the simplified model of the IMPATT device (see Figure 3.15).

The change in field, AE, must not exceed the avalanche critical field, Em, or avalanche will occur in the drift region also, diminishing the efficiency. We therefore limit AE to equal Em. The maximum permissible average current density thus is:

JDC,max = t/AEmax ~ E/Em (3.47)

JDC,max is predicted to be about 4 x 103A/cm2 for / = 10 GHz. In practise, limiting current densities for 10 GHz IMPATT devices are from 1,000 A/cm' to a few times this value (somewhat higher for millimeter wave devices, in agreement with (3.47)). If we somewhat arbitrarily limit the amplitude of the RF field to be equal to 1/2 x (E-field in the drift region) == !ED, then break­down will be avoided in the drift-region. The corresponding peak voltage is

(3.48)

Chapter 3 81

I I

32 / I

I

~ / f:' /

/ ",'"

16 / ",,,,,,,: .... ~e.".."'" ",,,,, .... '..,

~.".. GoAs{opll

~ 00 0.2 0.4 1.2 1.4 1.6 1.8 2.0

Va/Vd

Figure 3.16. IMPATT o.cillator efficiency for Si and GaA. diode., 1Ier.u. VA/VD. Reprinted from SEIDEL, T.E., NIEHAUS, w.e., and IGLESIAS, D.E. (1974), "Double-Drift Silicon IMPATT. at X-Band," IEEE 7ran •. Elec­tron De1lice., ED-~5, 660, @1974 IEEE.

A typical unperturbed field-distribution is as in Figure 3.18, with the drift field ED = 1/2 of Em, and EA = Em. We now can estimate both AC and DC voltages, and substitute these in (3.42):

1 I+WA /WD 1/ ~ ;: x 1 + 2WA/WD (3.49)

The value of WA/WD can be made quite small, and optimum efficiencies can be expected to be around 33%. This conclusion agrees with empirical simulations for GaA •. Silicon devices, however, have higher threshold fields for velocity saturation (about 10 kV /cm), not allowing quite as large voltage swings. Theoretical efficiencies for Si diodes are at most about 20%, and for GaA. about 30%. We will return to a comparison of Si and GaA. IMPATT device efficiencies in the next section, which reviews the large-signal analysis.

DOPING PROFILES FOR IMPATT DIODES

A great variety of doping profiles have been used for IMPATT diodes and a few are illustrated in Figures 3.19 through 3.21. The recent trend has been toward structures with very narrow (higher efficiency) avalanche regions, such as the profile originally proposed by Read (1958). The "lo-hi-Io" structure has a uniform field in the avalanche region, while the narrow clump of (donor) charge allows the field to decrease to values well below Em immediately as the carriers enter the drift region, preventing any avalanche there. Another popular structure is the double-drift one, which allows both electrons and holes to drift

82

..... ...../BREAKDOWN FIELD

/. " "

Microwave Semiconductor Devicell

" -- ELECTRIC FIELD

+ + + + + + + __ lis

+ + + .

+

~ ......... + HOLES - - - - ELECTRONS

DISTANCE (01

/ ... - ..... v'BREAKDDWN F(ELD

I "-I "

I " I \ I ;t I I I

Figure 3.17. Field and charge di6tributionll in an IMPATT diode at two different time. during the cycle of o.ci"ation. (a) Immediately after the avalanching hall terminated (b) At the time when the charge concentration ill about to complete it. tranI/it acrou the diode. Reprinted from SZE, S.M. (1981). "PhYllicll of Semiconductor Devicell," Second Edition, John Wiley fJ Son., New York, with permillilion.

through separate drift regions. This structure is more efficient and produces more power, as we shall see in the next chapter, and Problem 3.

Even a diode which has an avalanche region which stretches through the entire device will produce a negative conductance, as shown by Misawa (1967). This case is represented by a PIN-diode, which has a constant electric field in the intrinsic region. Figure 3.22. shows Misawa's small-signal calculations for the conductance of devices with varying relative avalanche widths, WA/W, as a function of the transit angle. As expected, the narrowest avalanche region devices yield the largest negative conductance, while the smaller negative con­ductance for especially the PIN diode (WA/W = 1) are much less sensitive to the transit angle. The latter translates to wider bandwidth for the negative conductance.

Chapter 9 83

a

r~A b

b ~A L

c

Figure 3.1S. Simplified field distribution in an IMPATT diode at three dif­ferent times a, b, and c, as marked on the voltage waveform.

AN ANALYTICAL LARGE-SIGNAL MODEL OF IMPATT DEVICES

In performing the large-signal analysis, we will use the article by Masse' et al. (1985). This article specifically deals with a double-drift diode, and devices for millimeter wave frequencies, but the general principles are well illustrated. The structure of the device is shown in Figure 3.23. Both sides of the diode are versions of the "lo-hi-lo" structure which we described earlier, with the avalanche "10" regions coincident in the center of the device. The combination of the "lo-hi-lo" structure and the double-drift feature is expected to make the device very efficient.

The current wave-form (which is not expected to be sinusoidal), is ex­panded in a Fourier series as follows:

00

i(z, t) = L: im(z)eim,.t (3.50)

84 Microwave Semiconductor Devices

Read Diode

p' I n I I(OR 1) n'

10 20 '1 Nl 15X10 16

'" I 10 13

N2

0 2 3 x(~m)

5 .. 4 ~

3 .. ~ 2 C!. 1 w

0 x

_ 10 .. ~ B

1<11> dx.l .. ':'1

6

4

" 2 tS v

0 0 2 3

x

Figure 3.19. Distribution of doping, electric field and ionization probability in a Read type IMPATT diode.

As before (see (3.12)) the total current density consists of a conduction and a displacement component

. . ( ) BE(z, t) 1TOT = 1 .. z, t + E--a-t - (3.51)

It may be useful to quote a generalization of the expression (3.41) for the efficiency to the double-drift device case (using the same assumptions otherwise) :

-1 VRF 1/= x--

",(Wn + WI') VB

X {v .. [-cos(6, +6 .. ) + cos 6,) +vp [-cos(6, +61') + cos6,)} (3.52)

To find a more realistic form for the conduction current pulse, we start from the Read equation (3.22):

TA 8:: = 3jA(t)(aWA -1) (3.53)

Chapter 3 85

'liE 10 20 2' .2-.,. p+ n+ 'li p+ n+

c 8 ! 10 '6 P n

0 2 3 x (11m) x

.. 5 Em Em 5! 4 x 3 IEdx-vB-60V UI

~ 2 IEdx -100V

ie 1 UI 0 X (11m) x

..

~L 0 ... x

A

l tj r I<cx>dx&1 y

A tj Y o 0 1 2 3 x (11m) x

(a) (b)

Figure 3.20. Di6tribution of doping, electric field and ionization probability in (a) a p+-n (6ingle drift) diode (b) a p+-p-n-n+ 611mmetric double-drift diode.

As noted above, we use a factor of'S' in the Read equation. Masse' et al. (1985) also define the avalanche region transit time (TA) somewhat differently than Sze (1981). They introduce:

WA or. = Sv,

The Read equation in their notation becomes:

or. 8:: = (iiWA - 1)iA(t)

The electric field is assumed to be of the form:

E(t) = EDC + ERF sin(wt + 1/1);

(S.54)

(S.55)

(S.56)

Note that 1/1 is referenced w.r.t. vet). This AC component orthe field modulates the ionization coefficient a in the Read equation, and we use a Taylor expansion of ii, retaining only the first term, to obtain:

(S.57)

86 Microwave Semiconductor Devices

p+ n+ p+ '1 E co & Nl C

co 8-c 'ii c 8

N, N2

X 0

X w b w

E

Em Em

E2 (xl E2 (xl

x x w 0 bzxA W

XA ea .. ea ..

L-O.c.lo..Lb------i~ X L-~O~b--~w--~~X

(al (b)

Figure 3.21. Distribution of doping, electric field and ionization probability in (a) an IMPATT device with Hi-lo structure (b) an IMPATT device with Lo-hi-lo structure.

As noted elUlier, ao WA = 1. Substituting in (3.55), we have

WA 8iA . 'w E . ( "") ---8 = 3A'0< A RFsm wt + 'I' 3v. t

The solution to this differential equation can be found to be :

iA = imaxexp{ -b [1 + cos(wt +,p)]}

(3.58)

(3.59)

Here, the parameter 'b' is proportional to the AC field amplitude, and equal to

b _ 30<'v.ERF ~ 30<'v. VRF - W - w(W" + Wp )

(3.60)

The last part of this expression follows if we assume the approximation that the AC electric field is uniform in the drift region. It can be shown that this is a good approximation ifb » 1. The parameter 'b' determines the half-

Chapter 3 87

3 .3

J·0.0704(J,) SCALE

.2

... u z .. I-U :> a z a

0 0 u a ,

1 1 ... ;'

!::! \ 1 / ;'

-' 1/ " .. \ \ ,,, .- '" 20 -.I a: -I \ \ '/ '" 0 --~t:-.2..- _ ........... /

Z -- , , ..... _- , \ 5

, '" \

, -.2 -2 \ READ , ,

\ 4'---

-3 -.3 0 2 3 6 7 8 9 10

TRANSIT ANGLE IN RADIANS

Figure 3.22. Normalized conductance versus transit angle (8) for .iz 1M­PATT diode. with different avalanche region widths. Case 1 WA/W = 0.1 and case 6 has WA/W = 1. Reprinted from M1SAWA, T. (1967). "Multi­ple Uniform Layer Approzimation in Analysis of Negative Resistance in p-n Junctions in Breakdown," IEEE Trans.Electron Devices, ED-l-4, 795, @1967 IEEE.

width of the current pulse by the relation (the shape of the pulse is illustrated in Figure 3.24)

8w = arccos [1 - (In2)/b] (3.61)

A typical value of "b' for an X-band device is about 8 (at about the max­imum possible VRF ), which makes the "ON/OFF ratio" of the current very large. Since ERF is limited by the break-down field, as discussed in the pre­vious section, then 'b' is essentially inversely proportional to the frequency. It is therefore fundamentally much more difficult to achieve a well confined charge pulse in millimeter wave devices. At 40 GHz, using 0;' = 0.22V-l, and VRF = 10V, we find 2 8w =120°. The efficiency should be multiplied by the sinc factor from (3.43), which for this angle is 0.83, i.e. not a disastrous decrease in efficiency. At even higher frequencies, though, confinement of the carriers becomes a major problem. Fourier-expanding the avalanche-current

88 Microwave Semiconductor Devicell

'x'D' 3 7.0

1 J 6.0 1

5 1x10,J B ". 5 5.0 ~ O. On U>

" '" ~ i5

> .2 4.0 '" i:

0 . ~ "-:, :' ~

'? -E r: 3.0 ,; ... N • I!i 0;

<» 1 i! r: 1

r: .!.! Q. ~ = 0 1 ::J 2.0 jl 0 1 5.,0'5 1 U 1x10-1 W

1 1 "'--Wp -I" 1 • '_Wn---l 1.0

1 w. 1 1 1 ill

1x10-2 w 0.0

Distance, 101'11 0.0 0.4 0.8 1.2 1.6

a Distance. I'm

b

Figure 3.23. Doping profile (a) and DC 1I0lution (b) for a typical double­drift IMPATT diode. Adapted from MAINS, R.K., EL-GABALY, M.A., HADDAD, G.!" and SUN, J.P. (1984). "Comparison of Theoretical and Ezperimental Re.ultl for Millimeter- Wave GaA. IMPATT., II IEEE Electron Devicell, ED-31, 1273, @1984 IEEE.

given by (3.59), we obtain coefficients which contain modified Bessel functions, 1m, using:

00

exp {-bcos ,",t} = L (_1)mlm(b)eimw1 (3.62) m=-oo

The Fourier expansion is given in (3.50). At z = 0, the m'th Fourier component is:

. (0) = (_l)m . IDC . lm(b) im¢ 3m A loeb) e

(3.63)

The avalanche resonance frequency also becomes dependent on the AC voltage amplitude in the large signal analysis, and is given by (compare (3.28» (A is the cross-sectional area):

h(b) ,....-....,

2 _ h(b) 2 _ 2Il(b) 3o:'v, I /A '"'R - '"'A - --_._-. DC b loeb) e

(3.64)

Chapter 3 89

o TT

~~::=--t----4-t----~~~======t-_wt (0)

1-----8;

lav ~----+L--~-+--~--t-----t--.wt (b)

Iterm f-~"'---+-"'==----+-+-----+-----t--wt (c)

\4---- 8 ---1--+----8 ---~ 'I

o ...

Figure 3.24. (a) Large-6ignal avalanche current waveform (b) .implified avalanche current waveform (c) the re6ulting terminal current waveform. Reprinted from MASSE', D., ADLERSTEIN, M.G., and HOLWAY, Jr., L.H. (1985). "Millimeter Wave GaA. IMPATT diode.," in Infrared and Millimeter Wave", K.J. Button, Ed., Academic Pre"" Orlando, FL., Vol. 1., Ch. 5, p. £91, with permi""ion.

We would like to find the AC impedance at the fundamental frequency, CAl. The current density component at this frequency, and at z = 0, can be expressed as follows, using CAlR :

(3.65)

After some manipulations, the total device admittance is found to be:

Y = jWleA[l - (Wik/CAl 2)] W" + Wp - (Wlk/",2)[W,,(1 + F,,) + Wp(l + Fp)]

(3.66)

where F" = exp(i~:n)-l and Fp is an analogous function ofB,.

If we take the extreme large-signal limit (b very large), then "'k « ",2,

and ifB" = Bp = BD, then we again find an expression for the negative conduc-

90 Microwave Semiconductor Devices

tance which is consistent with our calculation of the efficiency in the previous section (see (3.41)):

(3.67)

It is also useful to quote the conductance in the small-signal limit:

3 I 1 - cos 8D [ 9 (cr') 2 2] G = --0< • . IDc 1 - - - VRF 2 81 32 8D

(3.68)

In this limit, G depends on the actual current, not the current density. This equation predicts how the device behaves close to start-up.

As was the case in the previous section, the efficiency increases with in­creasing AC voltage amplitude, as illustrated in Figure 3.25. The conductance and susceptance of the device also change sign at the avalanche resonance fre­quency, as in the small-signal theory. A typical curve of G and B versus '-i~/'-i2 is shown in Figure 3.26. The measured variation of the negative conductance with AC voltage is given in Figure 3.27. It follows the theory well up to a certain voltage, and above this drops off much faster than predicted by the present theory.

The faster drop-off of the negative conductance occurs close to the point where the device has its maximum measured power output of 10 watts (Figure 3.27 includes constant power lines - note that the analytical model predicts no saturation of the output power). In order to explain the more rapid change in -G, we need to include some further phenomena in the model, and numerical simulation becomes necessary. The performance of several GaAB IMPATT devices of the type discussed above was simulated by Mains et al. (1984). The simulations were carried out by using either (1) the drift-diffusion equations, or by using (2) a transport model with the average electron energy and momentum as variables. Collision processes were included in the latter approach through the energy and momentum relaxation times of the carriers. Approach (2) was shown to agree best with the experimental data, especially at the highest frequency (44 GHz in this case). Specifically, it was found that above a certain critical RF voltage, the field in the drift region went to zero, and carriers even reversed their direction of travel. At this RF voltage, the negative conductance began to decrease rapidly. It appears that a collapse of the electric field in the drift region is the correct general cause of the sudden drop in negative conductance. Several competing effects (to be discussed in the next section) occur in millimeter wave IMPATTs, however, and further refinements in the model are likely to be required if one wants to be able to correctly predict the point at which the field collapses.

The analytical model is obviously much more useful in practise, compared with numerical simulation, since it can be used as a design tool. We may note that the maximum value for VRF /VDC is close to 0.6 in the above GaA8 devices, and an approximate modeling approach could thus consist in using

Chapter 3 91

20

f. >-u z '" <:; it '" 10

V,f (V)

Figure 3.25. Efficiency a6 a function of the voltage amplitude, VRF, for a millimeter wave diode with fized VDC. Parameter, are f = 40 GHz, IDC = 480 mA, VDC = 26 V, and A = 6 X 10- 5 cm'. Reprinted from MASSE', D., ADLERSTEIN, M.G., and HOLWAY, Jr., L.H. (1985). "Millimeter Wave GaA" IMPATT diode8," in Infrared and Millimeter Wave8, K.J. Button, Ed., Academic Preu, Orlando, FL., Vol. 14, Ch. 5, p. 191, with permi68ion.

the analytical model of this section, while restricting the RF voltage to 0.6 of the DC voltage. Also, an approximate analytical fit to the rapid decrease of -G versus VRF might be useful. This is likely to give a reasonable first order estimate for the maximum RF voltage and the output power.

NON-STEADY STATE LARGE SIGNAL MODELS FOR IMPATT DEVICES

In Chapter 2, we found that recently investigated transient phenomena, such as energy relaxation, and the finite time required for inter-valley transfer, limit the ultimate speed of Gunn devices. A similar development has occurred during the last few years in our understanding of IMPATT device operation, especially at millimeter wave frequencies. We will briefly review below some of the new concepts which have been introduced.

The most complete studies of such non-steady state phenomena have made use of Monte Carlo simulation (Lippens et al., 1985, 1986), in which individual particles are being followed through the device, while scattering probabilities

92 Microwave Semiconductor Devices

(,l 1

< 0.9 CD w % O.B c c 0.7 l-

e 0.6 w N 1-1

0.5 ..J < % a: 0.4 c z

0.3 iii e 0.2 z < 0.1 CD I

(OMEGA-R/OMEGA)SaUARED

Figure 3.26. Normalized value, of G and B a, a function of "'1/",2, for 9n = 9, = '11". An iterative solution to (3.66) wa, used for thi, plot.

have been assigned to the relevant processes, which are allowed to occur at random times. After a sufficient number of particles have been followed in this manner, the average behavior is calculated. In non-steady state modeling of IMPATT devices it is important to apply the ionization condition not as an integral of the ionization rate, versus 21, as in (3.1), which was used for the steady-state analysis. Instead, we find the equivalent probability per unit time of an ionization event, as a function of the carrier energy. As mentioned in Chapter 1, an energy of about 1.5 times the bandgap energy is required for ionization. In devices for millimeter wave frequencies, the lengths of the avalanche and drift regions are only a few tenths of microns. If the electric field drops very fast as a function of 21, the carriers will continue into the lower field region with essentially no change in energy. They may therefore still give rise to an ionization event, in a region in which the electric field is well below the "critical" value. This explains why it is necessary to give the ionization probability as a function of the carrier energy, not the electric field, in modeling millimeter wave devices.

Retarded Injection

In their Monte Carlo simulations, Lippens et al. (1985, 1986) found that the maximum earrier generation occurred with a delay of more than one half

Chapter 3 93

100r---------------~------~--~~~------------__,

U)

E 10 C) ,

Read theory

To • 2.73 psec

10 100 Vrf (V)

Figure 3.27. Mea6ured and predicted negative conductance ver'U6 VRF for an X-band diode. Constant power contour, are also 6hown. The parameter marked on the theoretical curve. (dashed line6) i6 the avalanche re'pon.e time, Ti. ReprintedfromADLERSTEIN, M.G., McCLYMONDS, J.W., and STATZ, H. (1981). "Avalanche Re.pon.e Time in GaA, a. Determined from Microwave Admittance Mea.urement6," IEEE TranI. Electron Devices, ED-28, 808, @1981 IEEE.

period (referenced to the RF voltage phase, as before). This can be under­stood in terms of the above discussion, if we note that the avalanche process is still occurring when the RF voltage goes negative. The carriers have substan­tial average energy at this point, and will continue to give rise to ionization as they drift, as long as their energy is still above the one needed for ioniza­tion. This explains the extra delay in the maximum of the injected current pulse, the so-called "retarded injection". This phenomenon will contribute to an increased negative resistance, and improved efficiency. Figure 3.28 shows

94 Microwave Semiconductor Devices

simulated waveforms ofthe voltage and current for a GaAs double-drift device at 204 GHz. The carrier density from the same simulation is given in a three­dimensional plot versus distance and time in Figure 3.29. This figure shows how the carrier "pulse" is widely spread in the spatial dimension.

The "Dark Space"

An effect which is related to the retarded injection is the "dark space". The carriers which are being accelerated in the avalanche region require a finite time before they reach a sufficient energy to be able to ionize. Thus, there will be a short space at the edge of the avalanche region in which no avalanching occurs, as illustrated in Figure 3.30. Another effect to take into account is the fact that after an ionizing event a carrier has essentially zero energy, and thus zero probability of giving rise to an ionization event.

Modulation of the Depletion Region Length

As the total voltage decreases, the drift region may become less depleted, and will act as a series resistance. This will seriously degrade the efficiency. Generally, it is found that high efficiency devices are designed to be well "punched through" in their drift region(s).

Over- and Under-shoot Velocities

The simulations show that the effective transit time of electrons in a very short device is less than what one would find by using (3.8). The reason is that a transient velocity "over-shoot" occurs during the initial stage of the acceleration process (before the scattering processes have impressed a steady­state value on the electron drift velocity). We will discuss this phenomenon in more detail in Chapter 10. Under-shoot velocities can also occur, if the carriers lose enough energy in the middle of the drift region, and they begin to be accelerated again as the RF voltage rises toward the end of the period. This effect can also be seen in the simulated data in Figure 3.28.

Secondary Avalanching in the Drift Region

There is some evidence (Thoren, 1983, Masse' et al., 1985), that secondary avalanching occurs in the drift region in some millimeter wave devices. This effect may either result in an increase or a decrease in the efficiency, depending on the phase of the injected current it gives rise to.

Chapter 3 95

Electric field -N

E u C 1.5

1ft 0 :::::. C Time-. GI t: 1.0 :l U GI c:; E C'G a.

T 12 T

Figure 3.28. Simulated particle current and terminal voltage as a function of time, for a GaAs double-drift diode, oscillating at £08 GHz. The inset shows the delay of the particle mean energy increa8e, and the avalanche generation, with re8pect to the terminal voltage. Based on LIPPENS, D., NIERUCHAL­SKI, J.L., DALLE, C., and ROLLAND, P.A.(1986}. "Comparative Studie, of Si, GaAs, and InP Millimeter- Wave IMPATT Diode" /I Intern. J. Infrared Millimeter Wave" 7, 771, with permi88ion.

p n n+ F.208GHx

Distance pm Distance pm

Figure 3.29. Evolution of the 8patial di8tribution of the hole and electron demities, respectively, through two RF cycles, in a double-drift GaAs diode operating at £08 GHz. Total active length of the device is 0.£8 p.m. Reprinted /rom LIPPENS, DALLE, C., and ROLLAND, P.A.(1986}. "Comparative Studies of Si, GaAs, and InP Millimeter- Wave IMPATT Diodes," Intern. J. Infrared Millimeter Waves, 7, 771, with permis8ion.

96

9 w

v iii I-

~ w

w ..... ... o g:

5 ;::: « N

Z Q

Microwave Semiconductor Devices

DISTANCE

Figure 3.30. Rlu6tration of the effect! of energy relazation on the location and width of the avalanche region in a single drift IMP ATT device. Reprinted from LIPPENS, D., NIERUCHALSKI, J.L., and CONSTANT, E. (1985). "Simplified Particle Simulation of Millimeter Wave IMPATT Devices, " IEEE Trans. Electron Devices, ED-SIl, 1l1l69, @1985 IEEE.

Potential of Different Materials for Millimeter Wave Operation

In discussing the potential of different semiconductor materials for use in millimeter wave IMPATT devices, it had previously been thought that GaAs was a poor choice. The present (1990) experimental evidence shows that com­parable power can be produced by silicon and GaAs IMPATTs up to 60 GHz. The efficiency and power of GaAs devices drops much more quickly than that of silicon devices as the frequency is increased to 94 GHz, however. No such dramatic difference is seen in the simulations discussed in this section. The ultimate frequency limit of IMP ATT devices based on different semiconductor materials is therefore still not clear, and new developments may occur which substantially increase the power of for example GaAs IMPATTs at the higher millimeter wave frequencies.

Problems, Chap 3

1. You have available a silicon p+-n diode, and you want to determine which frequency it will oscillate at as an IMPATT device. You measure a break­down voltage of 20 volts. Assume that you can treat the diode as a "one­sided" device (i.e. one side of the depletion region is much thicker - which one in this case?). First determine the doping on the side with lowest doping, and then the maximum electric field in the junction at break­down, by using curves given in Chapter 3. Then calculate the approximate width of the drift region and finally the expected frequency of oscillation.

Chapter 3 97

Assume reasonable values for any other materials parameters which you may need to use.

2. "Design" a silicon IMPATT diode with the "lo-hi-lo" doping profile, with n-doping in the avalanche-region and the drift-region. Fig. 3.21.b. shows the doping profile, and defines band w.

a) Assume b = 0.5~ meter. Calculate the value of Em required for break­down. Assume that the ionization rate versus E is given by the curve for electrons in Sf. in Fig. 3.3.

b) Find the width of the drift-region for optimum operation at 15 GHz. Assume the optimum transit angle given on p. 79 and v, from Figure 3.8.

c) The cross-sectional area is A = lO-4cm2. The doping in the drift­region and in the "10" portion of the avalanche region are both 5 x lO15cm-3. £. = 11.9 for Sf.. For a lo-hi-lo profile, (£, = £.£0)

N2 and Q are defined in Fig. 3.21. Find the total charge Q per cm2

in the doping "spike" (i.e. the "hi" part) and its doping density ifthe width of the spike is 0.05~.

d) Find the bias voltage (VB)' While doing this, calculate the electric field in the drift-region by integrating Poisson's equation. What is the ratio of the voltages in the avalanche and drift regions?

e) Assume that the current density is 500A/cm2 • Find the avalanche resonance frequency. Estimate a ' from Fig. 3.3.

f) Find the small-signal device impedance, ZIMP (Eq. (3.33)). Use reasonable values for (JD and R,.

3. This problem explores the saturation of a double-drift GaAs IMPATT oscillator. Assume the same frequency and area as in problem 2. Find the saturation velocity from Figure 3.8. The transit angle is 11' for both electrons and holes. Find the device admittance from the large-signal theory, as a function of RF voltage. The value of (a /) is 0.22 V-l. The (bias) current density is 1,000 A/cm2 • For small values of b, h(b) is given by

A plot of modified Bessel functions is also enclosed (see Figure 3.31), so that you can find h(b) for larger values of b, when necessary.

Plot the real part of the device admittance versus RF voltage. Compare it with the experimentally measured curve in the text.

98 Microwave Semiconductor Devices

3000

2500

2000 ~

B ~

1500 •

1000

500

0 0

Fig. 3.31.a.

Fig. 3.31.b.

Chapter 3

o H

~

I ----.----,------.----,------,

a.s .

u.B .

0.7 .

oL---~---~--~---~--~ o 6 10

Fig. 3.31.c.

99

Figure 3.31. Plob of Modified Beuel Function, (a) 11 and 10 for 0 < z < 10,0 (b) 11 and 10 for 0 < z < 4,0 (c) The ratio 11/10 for 0 < z < 10 (refer to Problem 3).

Also plot the output power from the IMPATT oscillator, if for each value of the RF voltage, the circuit conductance is adjusted to be equal to the magnitude of the negative conductance of the device (and the device capacitance is resonated with the circuit reactance). Does the output power reach a maximum for these values?

REFERENCES

BULMAN, G.E., ROBBINS, V.M., and STILLMAN, G.E. (1985). "The De­termination of Impact Ionization Coefficients in (100) Gallium Arsenide Us­ing Avalanche Noise and Photocurrent Multiplication Measurements," IEEE Trans.Electron Devices, ED-32, 2454.

DeLOACH, B.C., Jr. (1976). "The IMPATT Story," IEEE Trans. Electron Devices, ED-23, 657.

HOLWAY, L.H., Jr. (1979). "Electron-Hole Avalanches with Constant Ioniza­tion Coefficients," IEEE Trans. Electron Devices, ED-26, 991.

__ , ADLERSTEIN, M.G., and STEELE, S.R. (1979). Proc. 7th Biennial Cornell Electric. Eng. Conf., 199.

ITO, T., KAGAWA, S., KANEDRA, T., and YAMAOKA, T(1978). "Ioniza­tion Rates for Electrons and Holes in GaAII." J. Appl. Phys., 49, 4607.

100 Microwave Semiconductor Devices

JOHNSTON, R.L., DeLOACH, B.C., Jr. and COHEN, B.G. (1965). "A Silicon Diode Oscillator," Bell Syst. Tech. J., 44, 369.

LEE, C.A., BATDORF, R.L., WIEGMAN, W. and KAMINSKY, G. (1965). "The Read Diode and Avalanche, Transit-Time Negative-Resistance Oscilla­tor," Appl. Phys. Lett., 6, 89.

LIPPENS, D., NIERUCHALSKI, J.-L. and CONSTANT, E. (1985). "Simpli­fied Particle Simulation of Millimeter Wave IMPATT Devices," IEEE Trans. Electron Devices, ED-32, 2269.

__ , NIERUCHALSKI, J.-L., DALLE, C. and ROLLAND, P.A. (1986). "Comparative Studies of Si, GaAs, and InP Millimeter Wave IMPATT Diodes," Intern. J. Infrared and Millimeter Waves, 7, 77l.

MAINS, R.K., EL-GABALY, M.A., HADDAD, G.!. and SUN, J.P. (1984). "Comparison of Theoretical and Experimental Results for Millimeter-Wave GaAs IMPATTs," IEEE Trans. Electron Devices, ED-31, 1273.

MASSE', D., ADLERSTEIN, M.G., and HOLWAY, Jr., L.H. (1985). "Milli­meter-Wave GaAs IMPATT Diodes" , in Infrared and Millimeter Waves, K.J. Button, Ed., Vol. 14, Ch. 5, 29l.

MISAWA, T. (1967). "Multiple Uniform Layer Approximation in Analysis of Negative Resistance in rrn Junctions in Breakdown," IEEE Trans. Electron Devices, ED-14, 795.

READ, W.T. (1958). "A Proposed High-Frequency Negative Resistance Diode," Bell Syst. Tech. J., 37, 40l.

SEIDEL, T.E., NIEHAUS, W.C. and IGLESIAS, D.E. (1974). "Double-Drift Silicon IMPATTs at X-Band," IEEE Trans. Electron Devices, ED-21, 523.

SHOCKLEY, W. (1954). "Negative Resistance Arising from Transit-Time in Semiconductor Diodes," Bell Syst. Tech. J., 33, 799.

SZE, S.M. (1981). "Physics of Semiconductor Devices," 2nd Edition, John Wiley & Sons, New York.

THOREN, G.R. (1983). "New GaAs IMPATT Theory Explains Millimeter Wave Operation," Microwaves and RF, February 1983, p. 74.

ADDITIONAL READING

CARROL, J .E. (1970). "Hot Electron Microwave Generators," American El­sevier Publishing Company, New York.

HADDAD, G.I.(1973). "Avalanche Transit-Time Devices," Artech House, Dedham, MA.

Chapter 3 101

HOWES, M.J. and MORGAN, D.V., Eds. (1976). "Microwave Devices," Wiley, New York.

SZE, S.M., and RYDER, R.M.(1971). "Microwave Avalanche Diodes," Proc. IEEE, 59, 1140.

___ , (1981). "Physics of Semiconductor Devices," 2nd Edition, John Wiley & Sons, New York.

___ , (1985). "Semiconductor Devices: Physics and Technology," John Wi­ley, New York.

WATSON, H.A. (1969). "Microwave Semiconductor Devices and Their Circuit Applications," McGraw-Hill, New York.