IEEE TRANSACTIOKS ON ELECTRON DEVICES VOL. ED-13, KO. 1 J4XUARY, 1966

High-Field Distribution Function in GaAs E. 111. CONWELL AND M. 0. VASSELL

Absfract-Use of the drifted Maxwellian distribution is shown to be unjustified for GaAs samples in which the Gunn effect is usually observed, because the carrier concentration is much too low for electron-electron collisions to pre- dominate. It is pointed out that solution of the Boltzmann equation is considerably simpler at fields high enough so that the average electron energy exceeds several times the optical phonon energy. The simplification occurs because the polar optical scattering may then be considered elastic. Relaxation times and rates of energy loss in acoustic, opti- cal, and intervalley scattering are examined in order to determine which scattering processes must be included in the Boltzmann equation. It is found that intervalley scatter- ing is very likely to be more important than polar optical scattering for high-energy electrons in either the central or the outer valleys. Approximate solutions of the Boltzmann equation for electrons in the lower valley are given for a wide range of fields. The number of electrons per unit energy range, calculated from the solution of the Boltzmann equation, is given for electrons in both valleys for a field of 2.4 X lo3 volts/cm, where the approximation mentioned should be reasonably good.

CCURATE calculations of transport prop- erties in high electric fields require a know- ledge of the distribution function of the

carriers. This is particularly so for the Gunn effect [l], [ 2 ] . The detailed distribution of the carriers among the energy levels, particularly in the high-energy region where the carriers are di- vided between the central and outer valleys, is all-important in this case. The most straightfor- ward way of determining the distribution function is to solve the Boltzmann equation. This is fre- quently difficult, particularly for high fields. To circumvent this it has been usual to assume that the distribution has the drifted Maxwellian form, l.e.,

f - exp { -m(v - ~ , ) ~ / 2 k , T , ] (1)

where m is the effective mass, v and vd the velocity and drift velocity, respectively, k , the Boltzmann constant, and T , the electron temperature. The quantities vd and T , are treated as parameters, to be determined as a function of electric-field intensity by using the conditions of conservation of energy and momentum. Calculations using this approach have been carried out recently for GaAs [3j, [4]. In order for the introduction of an electron temperature to be valid, the rate of energy loss of a fast electron to the other electrons must be greater than the rate of loss of an electron to other

Manusrript received September 20, 1965. The authors are with General Telephone and Electronics

Laboratories, Inc., Bayside, N. Y.

scattering mechanisms. We can easily show that this condition will not be satisfied in GaAs at the con- centrations for which the Gunn effect is usually studied.

At not too high fields, at any rate, the pre- dominant scattering mechanism for electrons in GaAs is the polar optical one [5]. The rate at which an electron of energy & loses energy in polar optical scattering is given by [6]

($j = - 2eEoRwz [ (N*i + 1) sinh-' (xy/a E - b l (2m E ) l''

For the first term in the angular bracket, which represents the emission rate, & must be greater than Awl, the optical phonon energy. N a l is the steady-state number of optical phonons, e the electronic charge, and E, an effective field defined by

x,, and X, being the dielectric constants for zero and infinite frequencies, respectively. The rate of loss of a fast electron to other electrons is given by 171

where n is the electron density and e" an effective charge, usually approximated by e divided by the square root of the average dielectric constant. A logarithmic factor of the order of unity has been neglected in (3). The electron concentration re- quired to make the two loss rates equal is obtained by equating (2) and (3). To evaluat'e it numer- ically, the following values were used: kwi = IC, X 418" [5] , m = 0.072 m, (m, is the free electron mass), X, = 11.6, and X, = 13.5, leading to E, = 5950 volts/cm. The concentration required for the rate of loss to other electrons to match that to the polar optical modes is found to be 3 X 10"/cm3 for an electron with energy 3 R w i , and is larger for larger electron energies. At high fields other types of loss will become important, as will be seen, and the concentration required for a Maxwellian dis- tribution would be higher still. It is clear that the more reasonable assumption for the GaAs samples in which the Gunn effect is usually observed is that electron-electron collisions can be neglected.

22

1966 CONWELL AND VASSELL: HIGH-FIELD DISTRIBUTION FUNCTION 23

The Boltzmann equation, for the case of present interest, is the statement that in the steady state the rate of change of the distribution due to the acceleration of the carriers by the field must be balanced by the rate of change due to scattering or collisions, i.e.,

where P is the crystal momentum of an electron. The collision term may be written as the difference between the rate at which electrons are scattered into the state with momentum P and the rate at which they are scattered out, i.e.:

- 1 f(P)W(P 4 P’) dP’, (5)

W(P’ -+ P) being the probability per unit time of an electron being scattered from P’ into P. This form of the collision term makes the Boltzmann equation in general a complicated integro-dif- ferential equation. A situation in which i t can be simplified occurs when the energy exchange in a collision is relatively small. For elastic collisions it is well known that even in high fields the dis- tribution function may be written

f(P) = f o ( €1 + €1, P,g( E ) << f o (6)

where P E is the component of momentum, meas- ured from the valley center, along the field direc- tion. When (6) is substituted into the collision operator (5), f(P’) is replaced by terms involving /,(E’) and g(&’). Since for elastic collisions &’ E E , fo(&’) may, by a Taylor series expansion, be ex- pressed in terms of fo(&), dfo /d& and dzfo/dE2. The function g(&’), since it is small compared t,o fo(&’), may be expressed in terms of g(&) and dg/dC. The result is that the Boltamann equation is converted into a differential equation, or more exact,ly, a pair Qf coupled differential equations for f o and g, which greatly simplifies solution.

For low fields in GaAs, where the average energy ‘of the carriers is of the order of hul, the optical phonon scattering is not at all elastic. When the field is increased sufficiently, however, so that the average energy of the carriers has gone up by perhaps a factor 3 or 4, the type of approximation discussed in the foregoing becomes quite reason- able. We have made this approximation in order to solve the Boltzmann equation for GaAs. As a result, our solutions will not be accurate until fairly high fields, perhaps a couple of thousand volts/cm, but even a t lower fields they should give at least a fair approximation to the distribution function.

To set up the Boltzmann equation properly, it is necessary to include scattering processes other than the polar optical one. The other processes that might be expected to contribute are acoustic mode (deformation potential) intravalley scatter- ing, scattering from the central (000) valley to the outer (100) valleys, and scattering among the (100) valleys. To describe the latter processes accurately it is necessary to know the location in the Brillouin zone of the (100) valley minima. The best informa- tion a t present points to their being at the zone edge or just inside it [SI. In the former case they would be three in number and have X , symmetry; in the latter they would be six in number and have A, symmetry. The constant energy surfaces are expected to be ellipsoidal in either case. In the absence of knowledge about the principal masses it is not possible to determine the various combina- tions, such as geometric mean mass 6 and inertial mass, that enter into the transport properties. The only mass that is known for this case [5] is the density-of-states mass m‘N’, equal to 6 ( N , ) ” 3 , where N , is the number of valleys. We shall in the following approximate all masses other than the density-of-states mass by 1 n ( ~ ) / 6 ~ / ~ , since they are generally equal to the geometric mean mass or smaller. This mass is of the correct order of mag- nitude for either N , = 3 or N , = 6. It will be seen that the other features of the calculation are also reasonably applicable to either case, although conforming better to the three-valley case.

As a rough rule for setting up the Boltzmann equation, the processes that should be included in (dg/at),,ll are those that are important for mo- mentum loss, while (afo/at),,l, should include those that are important for energy loss. To determine which processes are mainly responsible for momen- tum loss we have plotted in Figs. 1 and 2 the momentum relaxation times for electrons in the central and outer valleys, respectively. The relax,a- tion time ra for intravalley acoustic mode scattering has been calculat’ed from

where E, is the deformation potential, p the density of the crystal, T the lattice temperature, ul the velocity of longitudinal acoustic waves, and Eo the energy of the valley minimum. For the central valley we shall take Eo = 0, which makes Eo = 0.36 eV for the outer valleys [Fi]. E , = 6 eV for the central valley [9], and, arbitrarily, the same value was used for the outer valleys. The quant.ities p and uI were taken as 5.31 gm/cm3 and 5.22 >(: lo5 cm/sec, respectively. For the central valley m = 0.72 m,, while for the outer va,lleys, as dis- cussed earlier, m was taken as (m(N)/6P’3) =Q.364mO.

For polar optical scattering, i t is well known

- 1,V vsTIME

l - 7 E vs DISTANCE

\i 4 I vs TIME

n

j 1

I i u E vs DISTANCE

24 IEEE TRANSACTIONS OK ELECTROX DEVICES JANCARY

I , 0.18

0 0.36

5 0.54 0.72 E in eV

IO 15 20 & / h w ,

Fig. 1. Relaxation times for scattering of electrons in the (000) valley as a function of energy.

t

1 E i n eV

2 0 'i y P ? " , .dd

Fig. 2. Relaxation times for scattering of electrons in a

taken equal to 5 X IO8 eV/cm. (100) vdley 58 a funct ion o f elrergy. For T Z - ~ , Dl2 was

that a relaxation time does not exist [SI unless 8 >> hal. In that limit, the reiaxatiou time is given by

This T has been plotted in Figs. 1 and 2 down to & - &, = 4 h E , wllich is sorncwhat below its limit of validity. For the outer valleys the same

E,, value was used as given earlier for the lower valley.

Scattering from the central valley to one of the outer ones involves a large change in crystal mo- mentum, and therefore a phonon of large crystal momentum, e.g., one a t or near the zone edge, depending on whether the valley minimum is a t or near the zone edge. In the former case, a group- theoretical study of the selection rules [lo] shows that the phonon must belong to the longitudinal optical (LO) branch, while in the latter it may belong to either LO or the longitudinal acoustic (LA) branch. The possible frequencies for the latter case, if the valley minimum is actually close to the zone edge, are quite close to the LO frequency at the band edge. We have threrefore chosen the angular frequency of the phonon required for the transition to be the experimentally determined [ll] LO frequency a t the band edge, 0.8 w l , correspond- ing to a characteristic temperature of 330".

The relaxat'ion time for intervalley scattering of a carrier initially in a state with energy & in the i th valley is given by [I21

*[ (N , i i + 1)( E - f iw i i - + N,ij( E + hwaj - E o j ) 1 ' 2 ] (9)

where the subscript j refers to the valley into which the electron is scattered, the summation over j is to be taken over all possible final valleys, and D i , refers to the coupling constant or deforma- tion potential for the process. The first term in the angular bracket represents transitions involving emission of a phonon (with angular frequency mi,), while the second represents transitions involving absorption. In the case of scattering from the central valley, to be denoted by the subscript 1, to the set of (100) valleys, to be denoted by the subscript 2, ail the quantities in (9) are the same for each of the (100) valleys, and the summation over j gives h7, times the quantity inside the summation sign. As noted earlier, N,.(m,)"" = ( m : N ' ) 8 / P = 1.2172, [ 5 ] . Of the other quantities in (9), Gni --= E,, = 0.36 eT', and uti = w , ~ = 0.8 w z , leaving only D l , unknown. We have chosen to consider D 1 2 a parameter in our calrulations, to he determined by conlparirlg with experiment the I-V curves obtained eventually for different values of this parameter. In I'ig. 1 we havc plotted the int'ervalley seatterirlg relaxation t,ime for D , 2 = I X lox e V ~ c m , arid -5 X 10' eV/cm, the latter being t8he d u e fotlrld for nonequivalerlt inte~valley scat,tering in ( ;e 1131.

I t is seen f l o n l I*.ig. 1 that the acollstic mode scattering is always less effective than the opticai,

1966 CONWELL AND VASSELL: HIGH-FIELD DISTRIBUTION FUNCTIOK 25

and it was therefore omitted from (dg /d t ) , , l l for electrons in the central valley. Because T,, a E”’, polar optical scattering gets less effective as the electron energy increases, and is likely to be over- shadowed by the intervalley scattering when & > E o z . If this is the case, then an electron with 8 > E O 2 will make much less contribution Do the cur- rent than an electron with & < EO2, and could, as will be seen, make a cont,ribution comparable to or smaller than an electron in a (100) valley. In that case, the simple model that has been used to treat the Gunn effect, in which all electrons in the central valley are assigned a high mobility while those in the outer valleys are assigned a low mobility, [14], [3] would be quite a poor approximation.

For electrons in the outer valleys, we consider separately scattering to the central valley and scattering t’o the other (100) valleys. The relaxation time T ~ + ~ for scattering to the central valley differs from T ~ + ~ only in having Eoi = Eol = 0, and xi fii:” = m;”. This time has been plotted in Fig. 2 for D l , = 5 x 10’ eV/cm. It is seen that this process is relatively ineffective, a t least for this value of D12. This is expected because of the small density of states in the central valley.

The details of scattering among the (100) valleys differ considerably for the two possible locations of the minima. For the minima at the zone edges electrons may scatter to the two other (100) valleys with absorption or emission of the same LO phonon as required for the (000) to (100) transitions [lo]. For the minima inside the zone edges transitions between valleys on opposite sides of the zone re- quire LO or LA phonons of small g, while transi- tions between adjacent valleys may take place through interaction with a large g phonon from any of the four branches of the spectrum [lo]. Thus in this case a variety of small and high-energy phonons may be involved in the intervalley transi- tions. The situation is quite similar to that in the conduction band of silicon, where the minima lie inside the zone edges along the (100) directions. Scattering in silicon has been analyxed in detail by Long [15]. To make the analysis feasible, he grouped the possible phonons into a low-energy and a high-energy group, and then represented each group by a single frequency and a single coupling constant. Comparing his theoretical re- sults with the experimentally observed lattice mobility, Long concluded that the coupling to the low-energy phonons was considerably smaller than that to the high-energy phonons. In our analysis we shall assume that scattering among the (100) valleys requires a phonon with angular frequency w,;, = 0.8 wJ. This is correct for the minima at the zone edges and, in the light of the foregoing discussion, may not be a bad approximation for

stant for this process unknown, and we have arbitrarily taken it to be 1 X 10’ eV/cm, the value obtained [15] for scattering between the (100) minima in silicon. (This quantity may also be considered a parameter to be determined by com- parison of the Calculated I-V curve with experi- ment.) The resulting relaxation time, calculated from (9), is plotted in Fig. 2 as T+,. It is selen that acoustic, polar optical, and equivalent inter- valley scattering must all be included in ( d g / d t ) , , l l

for the outer valleys. Actually, the scattering to the central valley was also included, since it would be significant for a larger value of D,2 .

The rates of energy loss for electrons in the central valley are plotted in Fig. 3. The rate of loss to polar modes is calculated from ( 2 ) . The rate of loss in intervalley scattering is hwI2 times the difference between the emission and absorption rates. This difference is obtained from (9) by re- placing the plus sign between the two terms in the angular bracket with a minus. The loss to polar modes is seen to rise to a maximum at 3hcoJ, and then decrease slowly. For energies above Goz the rate of loss in intervalley scattering is likely to overshadow that to optical modes. The rake of loss to acoustic modes is too small to show on the scale used in Fig. 3, and acoustic modes were omit- ted from ( d f o / d t ) for the central valley electrons.

We consider now the solution of the Boltzmann equation. For energies less than &,,, electrons can occupy only the central valley and, if we neglect the very small region just below Goz where thley can make a transition to one of the outer valleys by absorbing a phonon, or~ly optical modes nesd be considered. Making the approximations dis- cussed earlier, we have converted the Boltzmann equation into a second order differential equation. This equation has been integrated numerically for various fields. It is of interest, however, that for energies greater than several hw, the differential equation is well approximated by a simpler equation that can be integrated once by inspection. The first-order equation that results from this integra- tion is

where x = €/hut . In the limit of large fields, i.e., E >> E,, a good approximate solution for (10) is

f o = eY’z(4x)Y’y/z (1 11 where

the other ‘case: This still leaves the coupling con- -

I

I,V vsTitYE

1

J u E vs DISTANCE

2 I VSTIME

26 IEEE TRANSACTIONS ON ELECTRON DEVICES JANUARY

0 5 IO 15 20 & / ) l W )

Fig. 3. Rates of energy loss of electrons in the (000) valley in polar optical and intervalley scattering. For the latter plot, D12 was taken equal to 5 X 108 eV/cm.

In Fig. 4 solutions of (10) have been plotted as a function of x for various values of EIE,,. These solutions have been plotted down to x = 0.3, although, as indicated earlier, t,hey are not expected to be valid below about 3 or 4 h w , . For all fields f o has been set equal to unity at x = 0.3. Zero and very low fields have been included although, in principle, the solutions should not be good there. The solution for E = 0 turns out to be a Maxwellian at 340°, whereas all the calculations mere done for a lattice temperature of 300". The exact solution for E = 0, a Maxwellian a t 300°, is shown as a dashed line. It is seen that the error is not too large even at zero field. As expected, when the field increases the number of electrons a t high energies increases. The non-Maxwellian character, particularly at the intermediate fields, is easily seen. As compared with a Maxwellian of the same average energy, which would be a straight line intersecting the curve for the given field value, there are relatively fewer low-energy electrons, and relatively more high-energy electrons. This results from the decreasing effectiveness of the polar optical scattering with increasing energy.

For 8 > Eoz the Boltzmann equation takes the form of coupled differential equations for the dis- tributions in the (000) and (100) valleys. By means of the approximations discussed earlier, these have been put into a form in which numerical integration is straightforward. Complete distribution functions have been obtained at a few fields for an intervalley coupling constant of 5 x lo8 eV/cm. More in- formative than the distribution function is the number of electrons per unit energy range, which is the product of the distribution function and the number of states per unit energy range, p(8 ) . Apart from constant factors, p ( 8 ) is proportional to (m(N))3'Z(& - I n Fig. 5 we show the number of electrons per unit energy range, in arbitrary units, for the central and outer valleys, calculated

Fig. 4. Approximate solutions of the Boltzmann equation for the electrons in the central valley.

0.5

0.4

0.3

0.2

0.1

0 0 4 8 12 16

6 in units of %tu,

Fig. 5 . Number of electrons per unit energy range in the central and outer valleys calculated from the solution of the Boltzmann equation for E = 0.4E0 = 2.4 X lo3 eV/cm.

for E = 0.4 E, or 2.4 X lo3 V/cm. This is a high enough field so that the approximations we have been discussing should be good. It is seen that the population of the upper valleys is substantial a t this field.

L

16

IEEE TRANSACTIONS ON ELECTRON DEVICES VOL. ED-13, NO. 1 JANUARY, 1966

REFERENCES 7th Internat’l Conf., on Physics of Semiconductors, [9] E. Haga and H. Kimura, “Free-carrier infrared ab-

[l] J. B. Gunn, “Instabilities of current in 111-V semi- sorption in 111-V semiconductors, 111. GaAs, InP, GaP conductors,” IBBM J. Res. Dev., vol. 8, pp. 141-159, and GaSb,” J . Phys . Soe. ( J a p a n ) , vol. 19, pp. 658-669, April 1964. May 1964.

[2] H. Kroemer, “Theory of t.he Gunn effect,” Proc. IEEE [lo] M. Lax and R. Loudon, “Intervalley scattering selec-

paper cites other relevant references. (Correspondence), vol. 52, p. 1736, December 1964. This tion rules in Gallium arsenide,” submitted to Phys . .Rev.

We are indebted to Dr. Lax for making the ms. avail- [3] C. Hilsum, “Transferred electron amplifiers and oscil- able before publication. E vs DISTANCE [4] P. N. Butcher and W. Fawcett, “Int’ervalley transfer

lators,” Proc. IRE, vol. 50, pp. 185- 189, February 1962. [Ill G. Dolling and J. L. T. Waugh, “Normal vibrations in

of hot electrons in gallium arsenide,” Phys. Letters, gallium arsenide,” Proc. 1963 Internat’l. Conf. on Lattice Dynamics, Copenhagen, Denmark, pp. 19-32.

to Mr. Butcher for making the ms. available before vol. 17, pp. 216-217, July 15, 1965. We are indebted [12] C. Herring, “Transport properties of a many-valley

semiconductor,” Bell Sys. Tech. J . , vol. 34, pp. 237-290, publication. March 1955.

[5] H. Ehrenreich, “Band structure and electron transport [13] S. M. De Veer and H. J. G. Meyer, “Infra-red absorp- of GaAs,” Phys. Rev., vol. 120, pp. 1951-1963, Decem- tion by free electrons in germanium,” Proc. 1962 ber 1960. Internat’l. Conf. on Physics of Semiconductors, Exeter,

[6] E. M. Conwell, “Rate of energy loss to polar modes,” England, pp. 358-366. submit,ted for publication.

[7] H. Frohlich and B. V. Paranjape, “Dielectric break- [14] B. K. Ridley and T. B. Watkins, “The possiPjlity of

negative resistance effects in semiconductors, Proc. down in solids,’’ Proc. Phys. SOC., vol. 69B, pp. 21-32, 1956.

[SI C. Hilsum, “Band structure, effective c h y e and [15] D. Long, “Rcattermg of conduction electrons by lattice

scattering mechanisms in 111-V compounds, Proc. vibrat~ons in silicon,” Phys. Rev., vol. 120, pp. 2024- 2032, December 1960.

pp. 1127-1139, 1964.

”

I ,V vsTIME

Phys. SOC., V O ~ . LXXVIII, pp. 293-304, 1961.

Nonlinear Space = Charge Domain Dynamics in a Semiconductor with Negative

Differential Mobility -

Abstract-On a semiphenomenological level, the explana- tion of the Gunn effect is one in terms of a time-independent, negative differential, bulk conductivity. This mechanism is based on the conduction-band structure of GaAs, which provides for two kinds of electrons, light and heavy ones. Light electrons dominate at low fields, heavy ones at high fields. Since the mobility of the heavy electrons is much lower than that of the light ones, there is a range of current decrease with increasing field, Le., a negative conductivity.

This negative conductivity leads to an electrical breakup of the crystal into alternating traveling domains of high and low fields, accompanied by alternating current. In a “mathematically perfect” crystal this instability would take the form of traveling negatively charged electron accumula- tion layers, separating the domains of high and low fields. In real crystals the inevitable spatial fluctuations in the impurity distribution lead to the experimentally observed dipole mode, wherein both negatively charged electron accumulation layers and positively charged electron deple- tion layers occur.

was supported by the Research and Technology Division Manuscript received September 13, 1965. This work

of the U. S. Air Force, under Contract AF 33(657)-11015. The author is with Varian Associates, Palo Alto, Calif.

I. INTRODUCTION H E WORK described in this paper was motivated by the discovery, by J. B. Gunn, of the effect that is now commonly called

after him [1]-[4]. The objective of this work was a semiphenomenological rather than a microscopic one, namely, to gain a fairly detailed understanding of the space-charge domain dynamics in a medium of field-controlled negative differential bulk con- ductivity, without the all too common restrictions of mathematical linearity. The work is based on the Ridley-Watkins-Hilsum (RWH) model of the nega- tive mobility of the Gunn effect [5]-[8] , that is, on the assumption that there are two classes of electrons, of densities n, and n,, with different mobilities p, and pzl such that

B = S < I . !J

Pl (1.)

Furthermore, the local distribution ratio n,/n, of the electrons over the two classes is assumed to be

1 I I vsTIME

27