Chapter 2

TRANSFERRED ELECTRON DEVICES (TED)­GUNN DEVICES

INTRODUCTION

It is possible, in principle at least, to have structures which are much smaller in one dimension than the other two, and which exhibit negative resistance, and thus give AC power at frequencies compa­rable to the reciprocal of the small dimension. W. Shockley (1954), with permission, Bell System Techn. Journal.

Transferred electron devices (TED) or GUNN-devices are moderate power devices, with low oscillator phase noise, which are in use at all microwave fre­quencies up to just above 100 GHz. They are "bulk" devices, i.e. they contain no junctions, etc., which all other microwave semiconductor devices employ in one form or another. Although not necessarily simpler to understand, the bulk nature makes the GUNN devices a natural starting point for our discus­sion in this book. Present devices use the III-V compounds GaA. and I nP exclusively, but in the future some other compound semiconductors are almost certain to be added to the list.

Many active devices to be discussed in this book can be described in terms of a negative resistance. If the device represents a microwave load with a negative real part, then it is easy to see that the magnitude of the reflection coefficient of such a load will be greater than one, i.e. we have a device with gain.

(2.1)

The property of a negative real part for the device impedance, a "negative re6i.Jtance" will thus be a recurring theme in the book.

GUNN-devices rely for their operation on transfer of electrons between two different valIeys in the conduction band ofthe semiconductor, and this effect is reflected in the al ternati ve name "transferred electron devices". The idea that such electron transfer effects might lead to negative resistance and be useful for devices originated in papers by Hilsum (1962) and Ridley and Watkins (1961), (Ridley, 1963). The experimental discovery was made by James Gunn (Gunn, 1963). In Figure 2.1 we reproduce an excerpt from his original paper, which describes the discovery of oscillating microwave currents in bulk GaAs

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

24 Microwave Semiconductor Devices

Figure 2.1. Sampling oscilloscope recording of "GUNN oscillation" current waveform. A voltage pulse of 16 volt amplitude and 10 nanosec. duration was applied to a specimen ofn-type GaAB 2.5 x 10-3 cm in length. The frequency of the oscillating component is 4.5 GHz. The scales for the lower trace are 2 nanosec.jdiv. horizontally, and 0.23 A/div. vertically. The upper trace is an ezpanded version 0/ the lower trace. Reproduced with permission /rom G UNN, J.B. (1963), "Microwave Oscillation 0/ Current in 111- V Semiconductors," Solid State Commun., 1, 88. Copyright 1966, Pergamon Press.

samples with ohmic contacts, which were being subjected (pulsed) to high electric fields.

Clearly, already in this rudimentary form of a GUNN device, sinusoidal oscillations resulted. GUNN was interested in what happened to the mobility of electrons in GaAs at very high fields, and does not arrive at a satisfactory explanation of his observations in the original paper, although the previously mentioned papers were already published. Instead, Kroemer (1964) explained the oscillations observed by Gunn by using the results from Hilsum (1962) and Ridley and Watkins (1961). In following this explanation, we shall first discuss how electron transfer may lead to negative differential mobility (NDM), or equivalently negative differential resistivity (NDR). Later, we will discuss the different types of instabilities, which arise in a medium with NDM, and their role in the device operation. Several other device characteristics will finally be treated.

Chapter .2 25

ELECTRON TRANSFER AND NEGATIVE DIFFERENTIAL MOBILITY

In order to discuss electron transfer effects, we must look at the details of the energy bands, in particular the conduction band, of GaA8, see Figure 2.2. GaA8 is a "direct bandgap" semiconductor, i.e. the top of the valence band and the bottom of the conduction band both occur at k = O. The bottom of the conduction band, the "lower valley", is of course where most electrons reside at room temperature. The conduction band has two other types of minima, or "valleys" ,however. One of these occurs in the < 111 > crystallographic direction, at the Brillouin zone boundary (the dotted vertical line in the figure), and has an energy which is 0.32 eV higher than the "lower valley". We will refer to this minimum as the "upper valley". Note that there really are 8 equivalent directions in k-space which define the position of the upper valley, see Figure 2.3*. In addition to this valley, there is a somewhat higher energy valley in the < 100 > direction, see Figure 2.2. This valley does not tend to become populated, assuming typical conditions for GUNN devices, and it will be ignored in what follows. Also note that the size of the bandgap, 1.4 eV, is considerably larger than the 0.32 eV energy difference between the two valleys.

The essence of the model we shall use to explain the GUNN-effect, is that as electrons are accelerated by a high electric field, they may gain sufficient energy in order to be able to transfer to the upper valley, and we will assume that electrons reside in either of these two valleys, characterized by their usual parameters for that valley. The parameters we need are:

LOWER VALLEY: Effective mass: Mobility: Electron concentration:

UPPER VALLEY: Effective mass: Mobility: Electron concentration: Total electron cone. :

mt = 0. 067mo 1'1 = 8,500 cm2y-1 sec-1

n1 m; = 0.55mo 1'2 = 150 cm2y-1sec-1

n2

no = n1 + n2

Thus the effective mass is larger in the upper valley by a factor of about 8.2, while the mobility is smaller by an even larger ratio of about 53. Note from the expression for the mobility (I' = .;[., see (1.20» that the carrier collision times must be shorter in the upper valley in order to explain that the mobility ratio is larger than the effective mass ratio! Another important factor is the respective densities of states, which are proportional to the effective mass to the 3/2 power. The density of states in the upper valley must therefore be higher by about 23.5, times the number of valleys (eight as explained above, but only half of each valley "counts", since the other half is in a different

* Until 1976, the lowest satellite valley was believed to be the one in the < 100 > (X) direction. It was then discovered that the < 111 > valley was lower (Sze, (1981), p. 645)

26

4

3

> 2

.!.. >-(9 a: w z w

0

-1

GaAs

T-300K

r <111>- -<100>

Microwave Semiconductor Devices

x l

InP

T-300K

r <111>- -<100>

WAVE VECTOR

x

Figure 2.2. Energy bands for GaA8 and InP. Note that the positive and negative k-azes represent the < 100 > and < 111 > directions, respective/yo

Brillouin zone, see Fig. 2.3), and we end up with a density of states ratio, R, for the two valleys of 94, with the upper valley having the largest density of states.

In the two-valley model we simply add the contributions to the current by the electrons in each valley, with their respective concentrations and mobilities:

(2.2)

We will assume a sign convention for the electric field such that it is positive in the negative z-direction in this and several subsequent discussions of the GUNN-effect. See Figure 2.4 below:

It is now easy to estimate the current density and the mobility. We intro­duce an average mobility for all electrons, ji,. For very low and very high fields we thus get straight lines if we plot J versus electric field, see Figure 2.5. For intermediate fields,

IT = e(nllLl + n21L2)

= enoji, (2.3a)

Chapter 2 27

Figure 2.3. Constant energy surfaces for the < 111 > "valleys" in the con­duction band of GaAB and ,imilar compounds (these are the "upper valley,," discu."ed in the tezt). The ellipsoidal Bur/ace, become truncated at the Bril­louin zone boundarie" which occur at the point along the k-azi" marked 'L' in the energy band diagram in Figure 2.2. Constant energy ,urface" in the central "lower" valley would inBtead be represented by spheres at the center of the Brillouin zone (the 'T" point, or k == 0).

Device

- cathOdelL. _-=-.... -_-:=_-==_-==_-=:_ .. ::=_~_ .. __ x...J1 A~de +

(Fig. 2.4)

Figure 2.4. Definition of positive direction, for vector quantities which are used to de,cribe a GaAB bulk device. Note that the positive electric field i, in the negative z-direction.

28 Microwave Semiconductor Devices

J

Ev enoJ.i.2 (exaggerated!)

Figure 2.5. Sketch of J ver6U6 E for a bulk GaAs device, using a two-valley model, and assuming a uniform electric field.

from this,

and i = IT E = enoji.E =

= -enoii

(2.3b)

(2.4)

(In these equations, e is a positive number.) Note that i and ii are propor­tional, i.e. the plots of either one versus electric field will look similar.

Guessing the shape of the connecting part of the full curve, we arrive at something like Figure 2.5. If the transition from ILl to IL2 occurs reasonably quickly, there will be a portion of the curve with negative slope - that is the part which we shall focus our interest on. But first we take a look at the actual velocity/field relationship for GaAB at room temperature in Figure 2.6, which shows data both from measurements and a more sophisticated theory. We quote two measured curves: (1) From the by now classical papers by Ruch and Kino (1967, 1968) and (2) From a much more recent measurement by Masselink et al. (Masselink, 1989). The latter reference used a microwave technique, which appears to eliminate some difficulties in obtaining the cor­rect average drift velocity, caused by charge instabilities (to be discussed in detail later on in this chapter). Despite the general acceptance of Ruch-Kino's curve for many years, it seems that Masselink et al.'s data may be more nearly correct. It appears that both the differential mobility (i.e. the slope of the ve­locity /field curve) and the differential conductivity (slope of the current density versus field curve) are negative. Does this mean that we have accomplished a

Chapter 2 29

negative resistance device? As it turns out, yes, but in a more roundabout way than it first appears, as we shall see later on. In the connecting part of the velocity /field curve, there are substantial concentrations of electrons in both valleys. The curve turns down rather sharply at a critical electric field Ec>it

(about 3.2 kV /cm) once electrons in the central valley begin to have enough energy to be able to transfer to the upper valley. The probability of transfer is proportional to the density of states in the upper valley, which as we re­marked is about 94 times that in the lower valley. This helps speed up the transfer of electrons to the upper valley, and thereby the down- turn of the velocity, once electrons gain enough energy to transfer. The momentum of the transfer-transition is supplied by an optical phonon.

Calculation of the Velocity versus Field Curve.

Electrons lose energy primarily by emitting optical phonons as they are accelerated through the crystal lattice. The average time between collisions which change the momentum (the "momentum relaxation time") is of the order of 0.4 psec (derived from the low-field conductivity) while the average time be­tween collisions in which energy is lost (the "energy relaxation time") is about 1.0 psec, in GaAa. The effective time for transfer between valleys is estimated to be 1.5 psec (Kroemer, 1978). Most of this time is actually expended for acceleration of the electron from the bottom of the band to E - Ec ~ 0.31 eV, while the actual inter band transition is very fast once .the electron has the re­quired energy. A similar argument shows that the effective transfer time from the upper valley back to central valley is about the same (Kroemer, 1978). As a curve such as those in Figures 2.5 and 2.6 is measured, the electrons come to equilibrium in each point, i.e. they arrive at an equilibrium distribution between the two valleys (because the establishment of a steady state condi­tion requires of the order of 1.5 picoseconds, the velocity/field curve is valid for fields which do not vary substantially during that time - we will return to the implication of this for millimeter wave GUNN devices). Since there are frequent transitions between the two valleys in order to maintain the equilib­rium, early theories of the GUNN-effect assumed that the electrons in the two valleys are in thermal equilibrium at some higher temperature, the electron temperature, T., corresponding to their increased energy. We are also assum­ing that the type of statistical distribution function (Maxwell-Boltzmann) of the electrons is un-changed, something which may not be true for all cases. The average thermal energy of the accelerated electrons is then

3/2kB T. (2.5)

(kB is Boltzmann's constant)

and the excess energy, beyond the thermal energy at the ambient temperature, T, must be supplied by the electric field on the average as fast as it is being

30 Microwave Semiconductor Devices

25 '-------,--,----,

'" 300 K

§ 20

Wo 15

>-." ~ 10 > Bulk GaAs c 2 2DEG x=0.3 ~ 2DEG x=O_5 UJ

(Ioo;tri<;f .. kl. E(kV/cm) 23456

Electric Field (k V fem)

b

Figure 2.6. Theoretical and ezperimental velocity-field characteri,tic, of GaAa. (a) Reproduced with permil6ion from RUCH, J.G. and KINO, G.S. (1968), "Tran'port Propertie, of GaAa," Phy,. Rev., 174, 921. (b) Repro­duced with permiB6ion from MASSELINK, W.T. (1989), "Electron Velocity in GaAa :Bulk and Selectively Doped Hetero,tructure" "Semicond. Sci. Tech­nol., 4, 503. 2DEG curve, are ezplained in Ch. 11.

lost by relaxation (characterized by the energy relaxation time T.), and we can write:

(2.6)

Velocity/field curves based on a unique electron temperature, and the above simplified two-valley model, can now be calculated. One of the problems at the end of this chapter illustrates this calculation. In order to find T., we

. t n1P.1+ n 2P.2 ~ ~ . W al apprOXlma e Po = n1 + n2 - ni--F n2' SInce /J.1 > > /J.2. e so use

Chapter 11 31

~ = Rxe -e;; = (density ofstates ratio) x (Boltzmann factor). Combining these results with (2.4) and (2.6), we arrive at:

T. = T 2er.J.'1 E2[1 R (_ tJ.E )]-1 • + 3kB + exp kBT.

(2.7a)

and

(2.7b)

These equations thus enable us to estimate the electron temperature for electrons in GaA, at a given electric field and lattice temperature. They are plotted in Figure 2.7. One feature to note is that the electron temperature initially increases with the square of the electric field in this model. Also note that we regard the temperature of the crystal lattice (T) as a constant. We have just seen an example of the phenomenon called "hot electrons", which is a concept to which we shall return in many other connections in this book.

A better model uses one electron temperature for the lower valley, and another one for the upper valley. The electrons in the upper valley tend to be close to the lattice temperature, because the mobility is so low, and because the electrons arriving from the lower valley have little kinetic energy left after transfer. Results from calculations with a two-temperature model are quoted from (Fawcett and Bott, 1968) in Figure 2.8. This curve agrees somewhat better with the measured one (Figure 2.6) than the single-temperature model curve. This calculation shows how T1 rises much faster than Tz, as the field is increased, as expected from the above discussion.

The good agreement between measured and calculated velocity/field char­acteristics for GaA, is evidence that the explanation presented is correct. An­other convincing piece of evidence was provided by experiments in which the energy-separation between the two valleys was changed, either by using hydro­static pressure, or by changing the composition. As expected, when the energy separation decreased to zero, no GUNN-type effect was observed (Hutson et al., 1965; Allen et aI., 1965).

The main results from this section are (1) the v versus E curve, and (2) the "hot electron" concept (T.).

Is there a negative resistance for a uniform GUNN device in steady &tate?

We return to the question alluded to above: Is there a static negative re­sistance across a GUNN-device with a uniform electric field distribution? This

32 Microwave Semiconductor Device.

Figure 2.1. Re.ult. from calculation. based on the 'ingle electron tem­perature model. (a) Average drift velocit" ver.u. electric field; (b) Electron temperature ver.u. electric field; (c) Normalized population, in the two valle". ver.w electric field.

question was answered by McCumber and Chynoweth (1966). First, we use Poisson's equation:

BE -e -eN" ( n ) - = -(n - N,,) = -- - - 1 Bz E E N"

(2.8)

With the electric field assumed to be in the negative z-direction as in Figure 2.4, (2.8) will have a positive sign on the right side of the equation. Here, N" = doping == equilibrium concentration.

Chapter 2 33

3 X 103

1.0

r£' +

f ~

0.5

T'2 .,-""" ---------------~-~~------o 10 20 30 40

Electric field. E (kV!cm)

Figure 2.8. Result& from calculations based on a two-temperature model. The graph shows the average drift velocity (v), and the electron temperature in the upper (Tt) and lower (T3) valleys, respectively. After BOTT, I.B. and FAWCETT, W. (1968), "The GUNN Effect in GaAB," in Advances in Mi­crowaves, L. Young, Ed., Vol. 3, 233, Academic Prells, New York, with per­millsion.

We are seeking a static solution, and thus the current density is constant versus z and equal to (using the same sign conventions):

enii(E) = J (2.9)

While J does not depend on z, nand E (and thus v = v(E)), may do so. Combining these two equations, we have (J is directed in the negative z-direction!)

{)E = eNo ( J _ 1) {)z E eNov(E)

(2.10)

We assume that we initially apply a voltage V across the sample of length L such that V / L is greater than the critical field. The only quantity in the equation above which depends on z is veE). For z less than zero (the contact region), we assume that the doping is very large, and the electric field must be

34

c ,g ~

1 B § ~ ca

.J:. 0

I 1 w

, I I I \

0

Ev

0

o 0

Cathode

'" ;-

Microwave Semiconductor Devices

Donors No

Carriers n I

-t , I

...... ""----,

L Anode

Distance x ~

Figure 2.9. Profile of donor and free carrier concentrations, electric field and carrier velocity for the time independent .fOlution of a bulk device with NDM. After BULMAN, P.J., HOBSON, G.S. and TAYLOR, B.C. (1972), "Transferred Electron Devices, " Academic Press, London and New York.

small. For continuity reasons, E thus must start out low for small z. Since v is then also low, the equation above shows that dE/dz must be fairly large and positive. Thus, the magnitude of the electric field must increase with z. As this happens, v increases also, and dE/dz must decrease from the high initial value. Specifically, it is not possible to have a uniform electric field.

The electric field must at some point z reach the critical field. If, beyond this point, the electric field did not increase further, then we would not reach the correct integral of the electric field = the applied voltage, thus E must continue to increase. We get a curve of E versus position qualitatively as in Figure 2.9. If we replot the curve of E versus z for a higher current, then for all points z we find that the new curve is above the previous one, and thus the integral of the electric field, i.e. the voltage over the device, has increased. In other words, increased voltage means increased current. * This conclusion means that the device DC resistance is positive, and not negative, despite the fact that we biased the sample into the N DM region ofthe velocity/field curve.

* The polarity of the voltage across the device is defined such that it drives a positive current in agreement with our definition of the sign of the electric field (Fig. 2.4).

Chapter 2 35

The fact that a uniform field-distribution is impossible is crucial for in­terpreting the behavior of an N DM device. In reality, it is also difficult to achieve the above described steady state distribution due to instabilities which we shall now describe.

Shockley's Positive Conductance Theorem

In an article entitled "Negative Resistance Arising from Transit Time in Semiconductor Diodes," William Shockley (Shockley, 1954) discussed devices in which carriers have a velocity lfield relation with an NDM region. He showed that the DC resistance of the device must be positive. Later work by Kroemer (1970) demonstrated that this theorem holds regardless of the geometry ofthe device and the doping distribution, provided that the current can be written as in (2.9). The conclusions in the previous section clearly are in agreement with this theorem.

HIGH-FIELD DIPOLE DOMAINS IN GUNN DEVICES

It had been predicted by Ridley and Watkins (1961) that high-field dipole domains would form and propagate in biased materials with N DM. In a clever experiment, Gunn was able to position capacitive probes along the sample, and detect the propagation of such high-field domains from the cathode to the anode, see Figure 2.10. Under typical conditions, a domain would "nucleate" at the cathode, travel through the sample, and another domain would appear when it had reached the anode, etc ..

Dielectric Relaxation Time Argument

The dielectric relaxation time is defined as: E e

'7',,= -=--(T enp,

Typical values for GaA. of the parameters in this expression are:

n =:: 1015cm- 3 ; p, =:: 8, 500cm2 IV .ec; E = 12.8 x Eo

(2.11)

A typical value for the relaxation time in GaA. thus is found to be about 6 x 10-11 sec by plugging in the above data. This is the characteristic time for any excess charge to decay. If, however, we bias the GaA. to the N DM region, then the differential mobility will be negative, and the expression for the dielectric relaxation time predicts that excess charge will grow instead of decay. We thus have a fundamentally unstable situation whenever an N DM material is created - any small charge fluctuation in the sample will tend to grow indefinitely. The initial growth rate ofthe excess charge can be predicted from the relaxation time, but one should realize that the relaxation time con­cept only applies to small signal conditions. As the excess charge continues to

36 Microwave Semiconductor Devices

CATHODE ANODE

5

10

15

20

25

30

35

Figure 2.10. Direct detection of the motion of a high-field domain in a bulk GaA8 device. The recordings show the derivative of the potential with respect to time, taken at 8ucce8ive time intervals, versus position between cathode and anode. After GUNN, J.B. (1964), "Instabilitie8 of Current and of Potential Distribution in GaA8 and InP," 7th Intern. Conf. Ph1ls. Semicond., Vol. R, Plasma Effects in Solid., Academic Pres8, N. Y., p. 199, with permis8ion.

Chapter 2 37

-idTHODE ANODE ~ 2.l

- (\. (\. I V I • I

A !B -. r---' r I I I I I I

4 0 40 l 200 ~ 5 ~ ---' r-IDGH-FIELD "DOMAJN"

'-

I I 'I x

Figure 2.11. Computer simulation of a bulk GaAs device, showing the devel­opment of a high-field domain. The frames are numbered in a time Bequence, encompassing a total time of about one transit time. After McCUMBER, D.E. and CHYNOWETH, A.G. (1966), "Theory of Negative Conductance Application and GUNN Instabilities in 'Two- Valley' Semiconductors," IEEE Tran". Electron Device", ED-13, -4, @1966 IEEE.

grow, a large-signal calculation must be performed, as will be discussed in the next section.

A COInputer SiInulation

Results from a computer simulation of a typical GUNN device sample are shown in Figure 2.11 (McCumber, Chynoweth, 1966). The sample is first bi­ased above the critical field and a small "notch" is introduced in the electric field distribution at the cathode, in accordance with the observation that do­mains tend to nucleate there. As the calculation proceeded to later times, the notch was found to deepen, and finally develop into a fairly thin region with very high field, the "high-field domain". Qualitatively, we can explain what happens as follows:

Refer to the velocity/field curve during this discussion and remember that the initial field is in the N DM region. The electrons in the region marked A are in a lower field, and thus have a higher velocity, while the electrons in the region marked B are moving more slowly. Thus electrons will tend to

38 Microwave Semiconductor Device"

accumulate in the boundary area between these two regions, i.e. the notch. The accumulating charge will cause a larger discontinuity in electric field, which in turn will make the velocities in the regions A and B even more different, etc .. Thus the charge accumulation will continue, just in the manner which we expect from the discussion about the dielectric relaxation time. Figure 2.12 shows that in the case illustrated by the simulation, we actually get a double ("dipole") layer, with accumulation of charge to the left in the Figure, and depletion of charge to the right. The depleted layer has the positive charge of the donors in the n-type material. The electric field distribution is very similar to that of a pn-junction. Outside the stable domain, the electric field is below the critical field, and as we will explore in the problem sets and the example at the end of the chapter, a stable domain is obtained when the two areas in Figure 2.13. are equal (this is known as the "equal-areas rule"). The entire stable domain as well as the electrons on either side of it will move with the velocity given by v(Eo). Further, the voltage across the device consists oftwo terms:

(2.12)

Here, V.a is the excess voltage due to the moving high-field domain. V. a and Eo adjust themselves (in steady state) so that (2.12) and the equal areas rule are satisfied.

Domain Transit Time

Now that we can estimate the domain propagation velocity, we can also find the domain transit time

L Tt = v(Eo) (2.13)

As will be discussed in more detail later, many GUNN devices oscillate at a frequency which is the inverse of the transit time, i.e., with ii about = 107

em/sec, f = ~ = ii(Eo) _ 107cm/sec _ 100(GHz)

Tt L - L - L(~m) (2.14)

A sample with a length of 10 micrometers will oscillate at about 10 GHz. Gunn's original sample was about 25 micrometers long, and we predict oscil­lations at 4 GHz, whereas Gunn measured 4.5 GHz.

We should note that the stable high-field domain is not the only possible solution, i.e. for different conditions other types of instabilities may occur. We will come back to some of these later in this chapter.

Estimate of the Condition For Instability in an NDM Material

We can estimate the condition for stable formation of a high-field dipole domain through some fairly simple calculations. The first fact to observe is

Chapter 2

n ACCUMULATION c:; LAYER

OVESAS ONE UNIT

• x

I -v

-

39

E , . +

- +

+ +V -+ - +

Figure 2.12. Sketch of the carrier density and electric field distribution, in a bulk GaAe device in which a 'teady ,tate high field domain propagate •.

40 Microwave Semiconductor Devicell

x107 2.5 .

u 2 /Ecrit -CD

10 ...... e u

1.5

1\

->-~ 4"4 U a ''-... 2 I r-t 1 -"'" CD EO >

Ed ~ 01-4"4

0.5 -t. C

0 • • 0 2 4 6 B 10

El. Field. V/cm x104

Figure 2.13. fllulltration to the "Equal Area" Rule".

that the depletion region tends to be so much wider than the accumulation region, so that we can approximate the width of the domain with that of the former. The limiting case for formation of a stable domain would clearly be when the entire domain just fits the length of the sample, as shown in Figure 2.14. Also, the voltage must be equal to the critical field times the length, L. Since the field in the depletion layer varies linearly with "', it becomes clear that the field outside the domain is roughly 1/2E •• it, see Figure 2.14. Integrating the field distributions in Figure 2.14, we find that half of the applied voltage is accounted for as the excess voltage of the domain, V.". With pn-junction theory, we can now find V ... in terms of other parameters of the device as follows:

i.e.

(Ed - Eo) = c:..No . L (Poisson's Equation; see (2.8» £

L V." = (Ed. - Eo) . -

2

(2.15)

(2.16)

(2.17)

Chapter 2

E

x=o CATHODE

x=L ANODE

x

41

Figure 2.14. Sketch of the electric field for a high field domain which just fits into the length of a bulk GaA8 device. Approzimate values for Ed and Eo are indicated.

Using this result, we can formulate the "instability condition"

(2.18)

The above estimate is clearly on the low side, since the domain must be narrower than the device length in order to be able to propagate, and would typically have a higher peak field. We may guess that a better value may be about an order of magnitude higher. In fact, a more careful calculation shows that the critical product of doping and device length is

(2.19)

Sze (1981) derives the instability condition based on the relaxation time con­cept (see pp. 651-652).

To summarize, above the critical value, high-field dipole domains will de­velop and propagate, while below the critical value, accumulation layers and other less drastic instabilities occur. The No x L product is significant because (1) larger No makes the domain grow faster, and (2) a longer device gives the domain more time to grow.

42 Microwave Semiconductor Devices

I

~ Current wave-form

-------------------------[1 fl

I I

~

_______ J U I

Voltage wave-form

Figure 2.15. Current and electric field waveforms in an ideal GUNN device with uniform electric field, such that the liE curve follows the viE curve.

MODES OF OPERATION OF GUNN DEVICES

Efficiency of a Solid State Device

The purpose of a typical solid state oscillator device is to convert DC power to microwave power. The efficiency will therefore be defined as the ratio of the output microwave power to the DC power supplied. Before we describe the different modes of operation of GUNN devices, and compare their efficiencies, we take a look at what efficiencies that might be expected.

A hypothetical highly efficient GUNN device might be constructed such that it takes maximum advantage of the NDM region by assuming that the electric field is uniform (although we know from the above discussion that this is impossible in practise), and by allowing the voltage to switch the current from the peak value to the minimum obtained for a higher electric field (if the field is uniform, then the current/voltage curve will have the same shape as the velocity /field curve). This is illustrated in Figure 2.15. Note the definitions of

(2.20a)

and (2.20b)

where Vo is the DC bias voltage, and all quantities are marked in Figure 2.15. The microwave power is obtained by finding the power for the fundamental

Chapter .2 43

c

(Typical Circuit)

Figure 2.16. A GUNN device in a typical circuit.

Fourier component of the square waveforms for voltage and current, i.e.,

(2.21)

The DC power is v;. x {3(lp + 1.)/2 and the efficiency then becomes

11 = (1 - ct)({3 - 1) X (8/11'2) (1 + ct){3

(2.22)

The maximum efficiency is obtained if the DC voltage is very high (large (3) and if the peak-to-valley ratio (1/ct) in the current (i.e. velocity) curve is a maximum. In GaA6, ct = 0.43, and the hypothetical maximum efficiency is 32 % . Due to the fact that a uniform field-distribution through the device is impossible, and that waveforms of voltage and current are non-ideal, practical efficiencies are much lower. One useful feature of (2.22) is that it predicts the efficiency as a function of the valley-to-peak velocity ratio, ct. This may be useful for a rough comparison of different materials for use in devices.

Dipole DolWliD Modes

Now that we have investigated the physics of the transferred electron ef­fect in GaA8, we will look at how the effect is being put to use in a device. The device is typically packaged and inserted in a microwave resonant circuit. The schematic circuit will be as shown in Figure 2.16. The bias supply biases the device to the N DM region, and is filtered with the inductances indicated which prevent microwave leakage. A by-pass capacitor is used to couple the

44 Microwave Semiconductor Device,

120

CD 100 Q CD ..., .... 0 80 >

60 ..., c: CD

40 '-'-:;, u

20

0 0 200 400 600 800

Phase Angle. Degrees Figure 2.17. Voltage and current waveform, for a G UNN device in the tran,it

time mode.

microwave resonant circuit, represented by L, C, and GL. The resonant fre­quency of the microwave circuit (without the GUNN device) is fR, and the corresponding period of oscillation is TR = 1/IR. The active device will oscil­late at a frequency which we will call lou.The transit time of electrons across the full length of the device is '7"&. There are three types of dipole domain modes which can typically occur:

The Transit TUne Mode. In this mode, TR = '7"&. This mode occurs in many cases, as is evidenced by Gunn's initial experiment, for example. It is often also used in simple practical devices. The voltage and current wave-forms are shown in Figure 2.17. A domain starts at the cathode ofthe device when the voltage increases above the critical or "threshold" voltage, VT. A roughly constant current flows while the domain travels through the device. No major displacement current occurs in the contacts since the domain contains equal amounts of positive and negative charge. There is a short pulse of current when the domain arrives at the anode. The efficiency is predicted to have a maximum of about 10 % for No x L = 1012cm- Z (Thim, 1980). It is fairly low because the current pulses are typically quite short.

Chapter f

DOMAIN CHARGE

v

CURRENT

45

NO DOMAIN

Figure 2.1S. Voltage waveform, high field domain propagation, and current waveform for a GUNN device in the delayed domain mode.

The Delayed Domain Mode. In this mode, 2Tt > TR > Tt. Figure 2.18 shows the voltage waveform, the moving domain charge, and the current wave­form for this mode. Note that once the domain has been developed, a somewhat smaller voltage than VT (designated Vs) is sufficient to sustain the domain. The domain nucleates when the total circuit voltage passes the threshold voltage and then grows as it passes through the device. The domain decreases in size somewhat as the voltage again goes negative, but is sustained until it reaches the anode. The device current is given by the velocity of the carriers outside the domain (see Figure 2.11, note that the current is independent of z). This current decreases as the domain develops, as shown in Figure 2.18. After the domain has reached the anode and disappeared, the average velocity will again increase, and the current will go up. The next domain nucleates when the voltage reaches VT and the current stays high until this happens. The current pulse therefore is somewhat broader than in case 1) above, resulting in higher efficiency, which could reach 20% (Thim, 1980). The frequency of

46 Microwave Semiconductor Device"

oscillation becomes lower than 1/ Tt because of this delay. The delayed domain mode requires more careful tuning of the device than the transit-time mode.

The Quenched Domain/Limited Space-Charge Accumulation (LSA) Mode. The wave forms for the Quenched Domain Mode are shown in Figure 2.19. The condition for the occurrence of this mode is that

and

/0'. > l/Tt

Here, TS is the time which it takes for a domain to be quenched. The voltage must swing below Vs in order for the domain to be quenched. The time TS

is roughly equal to the dielectric relaxation time (with positive differential mobility) which we mentioned earlier. The devices can be made longer since the transit time condition no longer applies. If they are very long, more than one domain may tend to form, although this may be handled if all domains can be quenched. The condition for the full LSA mode (domain never develops) requires that (1) the excess-charge must not grow to a full domain in one period of the oscillation and (2) the quenching (dielectric relaxation) time must be less than the period. Using (2.11), we then obtain:

(2.23)

The range of acceptable values of No / /0'. is from about 104 to 105 for GaA •. Theoretically, efficiencies up to 25 % are predicted, but in practise only about 15 % has been obtained.

The LSA mode is the most demanding in terms of the conditions on the circuit used for the oscillator. It is not likely to occur in devices for frequencies higher than 20 GHz (Kroemer, 1978).

Other Modes

Accumulation Layer Modes. If the doping x length product is at the most about 1012, then accumulation layers will be formed instead of dipole domains. These modes can also be quenched or delayed, as for the dipole domain modes.

Relaxation Mode. The circuit may be designed so that the voltage waveform is nearly a half sinusoid. This mode is termed a relaxation oscillation, in analogy with other types of such oscillators. The efficiencies are somewhat higher. This mode is mostly used in conjunction with the LSA mode.

Chapter 2 47

QUENCHED MODE

V Voc

VT Vs

TIME i , , I ,

DOMAIN I

DOMAIN QUENCHED!

BY : CHARGE VOLTAGE! , , , ,

TIME NO DOMAIN

I , , , CURRENT

, I ,

~ ,

Iy

Figure 2.19. Voltage waveform, high field domain propagation, and current waveform for a GUNN device in the quenched domain mode.

Amplifier Mode. A device with sub-critical No x L can be used as a stable negative resistance amplifier. We will return to this ease in Chapter 6, when we look at circuit properties of two-terminal devices.

Summary of Different Modes

We summarize the different modes in a diagram due to Copeland (1967), see Figure 2.20. Millimeter wave operation of GUNN devices depends on some phenomena which we have neglected in the more elementary treatment pre­sented so far. The next section is devoted to these phenomena and the recent development of more efficient millimeter wave GUNN devices.

48 Microwave Semiconductor Devices

10 8 J!! E S. Quenched :; domain 0> c: ~ )(

(;' ~ GUNN ::> 0-e

LL

Delayed domain

10 6

10" 10'2 10'3 10'4

Doping x length (cm·2)

Figure 2.20. Mode diagram for transferred electron devices. After COPELAND, l.A. (1967), "LSA Oscillator-Diode Theory", l. Appl. Phys., 38, 3096, with permis8ion.

INDIUM PHOSPHIDE TRANSFERRED ELECTRON DEVICES/ MILLIMETER WAVE OPERATION OF TED's

Indium phosphide was shown to exhibit the GUNN-eft'ect in Gunn's orig­inal experiments. The energy bands are fairly similar, as seen from Figure 2.2. The energy gap between the two valleys is somewhat larger, and consequently the electrons in the lower valley must be accelerated more strongly in order to transfer to the upper valley, which increases the critical field to 10.0 kV /cm, compared with 3.2 kV /cm for GaAs. The velocity/field characteristics for the two materials are compared in Figure 2.21. As can be seen, the peak velocity is also somewhat higher for InP. The valley/peak velocity ratio f3 is lower, which leads to the potential for higher efficiency, as can be seen from (2.22) which predicts lImax = 45%. InP GUNN devices have been successfully fabri­cated in the last few years, and demonstrated the validity of these theoretical predictions (Eddison, 1984).

Frequency Lhnitations of GUNN·devices

As the frequency is increased, the electrons will not be able to transfer completely back and forth between the two valleys in response to the rapidly changing microwave field. Instead of following the steady-state curve, the average velocity will follow a curve like curve b in Figure 2.22. As a result, the efficiency will gradually decrease as the frequency goes up. The maximum frequency for GaAs is dose to 50 GHz.

Chapter 2 49 3 u .,

~ E u

'0 !:

?: 'u 0 Q; >

0

0 10 20 30 40 60

Electric field I kV/cm I

Figure 2.21. Electron drift velocity ver.us electric field in GaAB and InP. After EDDISON, I.G. (1984). "Indium Phosphide and Gallium Ar6enide Tran.ferred-Electron Device.," in Infrared and Millimeter Wave", K.J. But­ton, Ed., Academic Pre66, Orlando, FL, Vol. 11, Ch. 1, p. 1, with permis­sion.

Applied field E

Figure 2.22. Dynamic electron drift velocity versus electric field, showing energy relazation effect •. (a) 10 GHz, (b) 50 GHz. After EDDISON, I.G. (1984). "Indium Pho.phide and Gallium Ar8enide Tran6ferred-Electron De­vice8," in Infrared and Millimeter Waves, K.J. Button, Ed., Academic Pre"., Orlando, FL, Vol. 11, Ch. 1, p. 1, with permission.

80 laO 120 140 150 180 200

Frequencv IGHzl

Figure 2.23. Predicted output power and efficiency ofn+-n-n+ InP TEOs. (a) fundamental mode (b) .econd harmonic mode. AfterFRISCOURT, M.R., ROLLAND, P.A., CAPPY, A., CONSTANT, E., and SALMER, G. (1983). IEEE Tran8. Electron Devices, ED-30, 223, @1983 IEEE.

50 Microwave Semiconductor Device6

Another parameter which depends on frequency is the quenching time for excess charge. Because of this effect, quenched mode GaAs oscillators are not possible above about 25 GHz.

One can extend the frequency range of GaAB TED oscillators to somewhat above 100 GHz by providing an extra circuit at the second harmonic frequency. The effective transfer time for InP is about 0.75 psec, i.e. considerably shorter than for GaAs. The bulk of the transfer time consists of the time required to accelerate the electrons in the lower valley - the actual transition time is fairly short once the energy is in the correct range required for transfer. While InP requires a higher critical field, the higher field will also accelerate the electrons faster, leading to the shorter transfer time. Computer simulations predict good output power from fundamental frequency InP oscillators above 160 GHz, and the potential for second harmonic operation above 200 GHz, see Figure 2.23. InP oscillators with 50-100 mW CW output power at 94 GHz are now available. Power limitations will be discussed further in Chapter 5.

"Dead-Zone Effects" /Injection-Controlled Devices

Computer simulations such as those illustrated in Figure 2.24 have re­vealed that the electrons do not immediately acquire enough energy to be able to transfer to the upper valley, as they are injected from the cathode. This leads to a "dead zone" with a length of about a micron near the cathode. The resistance of this zone remains positive, and thus causes considerable losses. The situation becomes particularly critical if one attempts to design an InP millimeter wave device, which has to be very short. The device simulated in Figure 2.24 is an InP device at 100 GHz. A solution to this problem is to create a high-field current-limiting cathode contact which injects hot electrons into the active n-layer. This will allow the elimination of the dead zone and also allow a much more uniform electric field distribution, which will lead to higher efficiency. Efficiencies above 20 % have been obtained at 12 GHz for an InP device with a nonohmic contact by Gray et al (1975). Much more work needs to be done in this area.

EXAMPLE: GROWTH RATE OF A HIGH·FIELD DIPOLE DOMAIN - THE "EQUAL AREAS" RULE

In the small-signal approximation we can use the dielectric relaxation time to estimate the growth rate of a domain. As the domain grows, this method becomes inaccurate. In particular, we know that the size of the domain will eventually come to a steady state condition. We derive some useful expressions in this context below. First, we use Poisson's equation, and the same sign conventions as earlier.

8E _ e(n- N.,) 8z - E

(2.24)

Chapter 2 51

_ 1.6 2 n+ n'

~ R

f12 0 0' c ~ 0.8

W 0.4

2 - 1

a: -2

- 3

-4 Active laver (urn)

1.1 - 5

Ibl

Figure 2.24 (a) Evolution of the electron energy, and (b) diode resistance, plotted versus position in the active layer of a 100 GHz InP TED oscil­lator device. The graphs in part (a) show the electron energy at varying times during one RF period. The bold dotted curve represents the beginning of the period. From FRISCOURT, M.R., ROLLAND, P.A., CAPPY, A., CONSTANT, E., and SALMER, G. (1983). IEEE Trans. Electron Device" ED-30, 223, @1983 IEEE.

The total current density for this case includes the displacement current, and is constant throughout the device (diffusion is neglected, which is not completely accurate, but this assumption simplifies matters a lot)

BE J = env(E) + EEit (2.25)

In particular, outside the domain the current density is given by (n = No and E = Eo)

(2.26)

We now substitute n = No + ~ x * from (2.24) into (2.25), and subtract (2.26), noting that J is independent of z, to obtain

B eNo _ _ _ BE Bt(E - Eo) = -E-(v(Eo ) - v(E» - v(E). Bz (2.27)

We integrate this equation between two points, Zl and zz, on either side of the domain, as shown in Figure 2.25. Because the integration limits are outside

52 Microwave Semiconductor Device8

40

~5

30 Ed

" .... !! 25 ... u ~O ... ~ .. u 15 ~ IU

10

5

0 0 20 40 60 BO 100

x-coordinate

Figure 2.25 Electric field of a traveling domain in a GUNN device, with definition8 of quantitie6 wed in the derivation of the "Equal Area." Rule.

the domain, where the electric field and the velocity are independent of :1:, the second term will cancel, and

(2.28)

The left side of this equation yields the time derivative of the excess voltage of the domain, V. a , and is a useful quantity to represent the growth rate of the domain. The right side involves the drift velocity, and can be manipulated further if we assume a typical shape of the domain electric field versus :1:, in which the accumulation layer is much thinner than the depletion layer. Thus, most of the excess voltage originates in the depletion layer, in which n = O. For this layer, (2.24) yields

tJE eND -=---tJz E

(2.29)

This enables us to re-write the right side integral in (2.28) to an integral in terms of E instead of :1::

dV: I.E. I.E. d;Z ~ - (V(ED) - v(E))dE == (v(Eo) - v(E))dE E., Eo

(2.30)

The useful feature of this equation is that we can predict the growth rate of the domain based on the well-known v / E curve. A special case is the steady state case, for which the right side of (2.30) is zero. Comparing with Figure 2.13, we see that this gives us the earlier quoted "equal areas" rule.*

* The equal areas rule is still obtained for the more general case, which incorporates a diffusion current term, provided that the diffusion constant is independent of electric field. If the latter is not the case a more complicated rule must be used (Butcher et al., 1967, and Fawcett and Bott, 1968).

Chapter 2 53

As an exercise, you may use (2.30) to predict how the domain grows, starting from a sample which is biased above the critical field. A problem at the end of this chapter also explores this relation further. As a further exercise, you might assume a sinusoidal, small signal voltage, added to the bias voltage which is "propelling" a domain in steady state through a GaAs sample, and derive the resulting sinusoidal current from (2.30). The result should show a negative resistance, in contrast to the DC steady state case which we discussed previously.

STATIONARY DOMAIN AT THE ANODE

As discussed by Thim (1971), it is not obvious that the domain will dis­appear into the anode when it arrives there. A different possibility is that it becomes stationary at the anode, and the field will remain low in the rest of the device. The first case occurs ifthe domain (assumed to be an accumulation layer) moves into the anode fast compared with the speed at which the accu­mulation layer can re-adjust. Then the field in the rest of the device rises until it reaches the critical field, and a new domain is formed at the cathode. Upon this, the field and electron velocity to the left of the first accumulation layer will decrease, and not enough electrons will arrive from the left of that layer to sustain it. In the second case, the accumulation layer adjusts fast enough to prevent other layers from forming, and a new steady state situation occurs with a stationary accumulation layer formed at the anode. Thim derives the condition for a stable anode domain as:

-2 1 N f1l0 0> x-eJ- JLJD 4

(2.31)

where D is the diffusion coefficient. The critical density is a few x1014cm-3

for GaAs.

Problems, Chapter 2

1. Find the thermal 1Ielocity of an electron in GaAs at room temperature (290K). To do this problem, use the fact that the average thermal energy is 3/2kBT. What is the group velocity of an electron in a parabolic band with effective mass m'? Values for constants are kB = 1.38 x 1O-23J /0 K, mo = 0.91095 X 10-30kgj take m' from the text. Also find the drift velocity of an electron in GaAs in a sample 10 micrometers long, with 2 volts applied. Compare the two velocities!

2. Look up data for a minimum of 2 and a maximum of 5 different semicon­ductors other than GaAs, and explain why they mayor may not exhibit the GUNN effect. lllustrate with band diagrams, and use qualitative dis­cussions based on the band diagrams if you can't find quantitative data for some case.

54 Microwave Semiconductor Devices

3. Assume that a conduction band electron in GaAs is accelerated from the bottom of this band until it acquires an energy of 0.31 eV, making an (oversimplified) assumption that no collisions occur. The electric field is a constant 3.2 kV /cm. How long does it take before the electron reaches an energy of 0.31 eV? Assume m" to be constant. What may this calculation be useful for? Note that 1 eV = 1.60218 x 10-19 Joule, assume the value for the effective mass as given in text.

4. Use Eq. (2.7) to estimate the electron temperature, T., for an electron accelerated by a field of a) 3.2 kV /cm b) 10 kV /cm. Assume parameter values from the text. Observe that the equation is transcendental, iter­ation should be suitable for solving it. Next, also find the electron drift velocities for these two cases. If you like computer programming, you could go on and plot the entire curve of velocity versus electric field based on the model used for Eq. (2.7). How do your results agree with data given in the figures in the notes?

5. For problem 5, assume standard parameters for GaA .. where necessary.

a) A GUNN-device using GaA .. operates at 20 GHz in the transit-time mode. Find the length of the device.

b) The bias voltage is 8 volts. Find the initial electric field when the bias supply is just being turned on, and plot it on curve "I" of v/ E given below (Figure 2.26a).

· · , ~

" a

· ,. " a

2."~<1'~O'~~r-__ ~r-__ ~r-__ ~r-__ ~r-__ -,

1.5

0.::;

°OL-----O.~5----~----~1.~.----~----2~:~.----~

Elu:trlc. FlIiiI1Cl. VIc.

Figure 2.26a. Rlwtration for Problem 5.

Chapter 2 55

V (em/sec)

t Curve "2"

--f---....l------L...-----I~E(kV/cm)

3.2 10

Figure 2.26b. fllustration for Problem 5.

c) For this problem, assume curve "2" of 'VIE above (Figure 2.26b), which approximates the real curve with straight-line sections. First discuss, using Eq. (2.30) in the text, how the excess voltage of the domain grows with time. Mark qualitatively in the 'VIE diagram how Ed and Eo change as this happens. Use a geometric argument to show for which Edl Eo combination the growth rate is a maximum. Mark this in the diagram. Estimate Ed and Eo at this point and find the maximum value for the growth rate of V ••. The product No x L is 2 x 1012cm- 2 • Does it seem likely that the domain will develop fully with this growth rate? (The frequency of oscillation is still assumed to be 20 GRz.)

d) Indicate in the 'VIE curve the values of Ed and Eo for which the domain has reached the steady state condition. Make a guess for the value of Eo from the diagram, and go on to calculate Ed. Then use the "equal areas rule" to find a better estimate for 'V(Eo) at the steady state condition.

e) Estimate the current through the device (1) when the domain is trav­eling through it, in steady state, and (2) after the domain disappears, using your data from the problems above. The area of the device is 10-4 cm- 2 • Plot qualitatively how the current varies with time, com­pared with how the voltage varies with time.

6. (This one is more difficult.) The I-V-characteristic of a GUNN device, which is not in a cavity, is measured by pulsing the voltage on and off. The peak current which results is plotted versus the peak voltage of the

56 Microwave Semiconductor Devices

applied pulse. Different types of curves are obtained, depending on the pulse frequency (Fig. 2.27):

-t-----:-",-- V peak

Low and high pulse frequencies (low is 0.1 Hz, high is MHz)

negative resistance??? +--""'t,...---- V peak

Pulse frequencies -100 Hz - 1 kHz - 10 kHz

Figure 2.27. fllustration for Problem 6.

Explain! Hints: Which phenomenon has a time-constant in the msec range? You may also check Fig. 7, p. 647 in Sze (1971).

REFERENCES

ALLEN, J.W., SHYAM, M., CHEN, Y.S., and PEARSON, G.L. (1965). "Mi­crowave Oscillations in GaA8t_.P. Alloys," Appl. Phys. Lett., 7, 78.

BOTT, I.B. and FAWCETT, W. (1968). "The GUNN Effect in GaAs" , in Microwaves, L. Young, Ed., Vol. 3,233, Academic Press, New York.

BUTCHER, P.N., FAWCETT, W. and OGG, N.R. (1967). "Effect of Field­Dependent Diffusion on Stable Domain Propagation in the GUNN Effect," Brit. J. Appl. Phys., 18, 755.

COPELAND, J .A. (1967). "LSA Oscillator-Diode Theory," J. Appl. Phys., 38,3096.

EDDISON, I.G. (1984). "Indium Phosphide and Gallium Arsenide Transfer­red-Electron Devices", in Infrared and Millimeter Wave", K.J. Button, Ed., Academic Press, Orlando, FL, Vol. 11, Ch. 1, p. 1.

FRISCOURT, M.R., ROLLAND, P.A., CAPPY, A., CONSTANT, E., and SAL MER, G. (1983). IEEE Trans. Electron Devices, ED-30, 223.

GRAY, K.W., PATTISON, J.E., REES, J.E., PREW, B.A., CLARKE, R.C. and IRVING, L.D. (1975). "InP Microwave Oscillator with 2-Zone Cath­odes," Electron. Lett., 11, 402.

Chapter f! 57

GUNN, J.B. (1963). "Microwave Oscillation of Current in IIl-V Semiconduc­tors," Solid State Commun., I, 88.

___ , (1967). "On the Shape of Traveling Domains in GaAs," IEEE Tran6. Electron Device6, ED-14, 720.

HILSUM, C. (1962). "Transferred Electron Amplifiers and Oscillators," Proc. IRE, 50, 185.

HUTSON, A.R., JAYARAMAN, A., CHYNOWETH, A.G., CORIELL, A.S. and FELDMANN, W.L., "Mechanism of the GUNN Effect from a Pressure Experiment," Phys. Rev. Lett., 14, 639. (1965)

KROEMER, H. (1964). "Theory of the GUNN Effect," Proc.IEEE, 52, 1736.

___ , (1970). "Generalized Proof of Shockley's Positive Conductance Theo­rem," Proc. IEEE, 58, 1844.

___ , (1978). "Hot Electron Relaxation Effects in Devices," IEEE TraM. Electron Devices, ED-15, 819.

MASSELINK, W.T. (1989). "Electron Velocity in GaAs:Bulk and Selectively Doped Heterostructures," Semicond. Sci. Technol., 4, 503.

McCUMBER, D.E. and CHYNOWETH, A.G. (1966). "Theory of Negative Conductance Application and GUNN Instabilities in 'Two-Valley' Semicon­ductors," IEEE Tran6. Electron Device6, ED-13, 4.

RIDLEY, B.K. and WATKINS, T.B. (1961). "The Possibility of Negative Resistance Effects in Semiconductors," Proc. PhYIl. Soc. Lond., 78, 293.

RIDLEY, B.K. (1963). "Specific Negative Resistance in Solids," Proc. Phys. Soc. Lond., 82, 954.

RUCH, J.G. and KINO, G.S. (1967). "Measurement of the Velocity-Field Characteristics of GaAs," Appl. Phys. Lett., 10, 40.

RUCH, J .G. and KINO, G.S. (1968). "Transport Properties of GaAs," Phys. Rev., 174, 921.

SHOCKLEY, W. (1954). "Negative Resistance Arising from Transit Time in Semiconductor Diodes," Bell Sy6t. Tech. J., 33, 799.

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

THIM, H. (1971). "Stability and Switching in Over critically Doped GUNN Diodes," Proc. IEEE, 59, 1285.

---, (1980). "Solid State Microwave Sources," in C. Hilsum, Ed., Hand­book on Semiconductors, Vol. 4, Device Physics, North-Holland, Ams­terdam.

58 Microwave Semiconductor Devicell

FURTHER READING

BULMAN, P.J., HOBSON, G.S. and TAYLOR, B.C. (1972). "'Transferred Electron Devices," Academic Press, London and New York.

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

EASTMAN, L. (1972). "GaAs Microwave Bulk and Transit-Time Devices," Artech House,Dedham,MA.

GAYLORD, T.K., SHAH, P.L. and ROBSON, T.A. (1968). "GUNN Effect Bibliography," IEEE TranI. Electron Devicell, ED-15, 777.

HILSUM, C. (1978). "Historical Background of Hot Electron Physics (A Look over the Shoulder)," Solid State Electronic" 21, 5.

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

JONES, D. and REES, H.D. (1973). «A Reappraisal of Instabilities due to the Transferred Electron Effect," J. PhYII. C: Solid State Phy"iclI, 6, 1781.

SHAW, M.P., GRUBIN, H.L. and SOLOMON, P.R. (1979). "The Gunn­Hilsum Effect," Academic Press, New York.

STERZER, F. (1971). "Transferred Electron (GUNN) Amplifiers and Oscilla­tors for Microwave Applications," Proc. IEEE, 59, 1155.

SZE, S.M. (1985). "Semiconductor Devices: Physics and Technology," John Wiley, N ew York.

__ , (1981) - listed above.

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