asymptotic solution of the boltzmann equation for hot carriers

M. MOCKER: Solution of the Boltzmann Equation for Hot Carriers 621

phys. stat. sol. (b) 46, 621 (1971)

Subject classification: 13 and 14.3; 6; 22.1.1

Hektion Phyaik der Humboldt- Universitat zu Berlin, Bereich Theoretische Halbleiterphysik

Asymptotic Solution of the Boltzmann Equation for Hot Carriers

BY M. MOCKER

The steady-state Boltzmann equation for hot carriers is solved asymptotically for the case of predominate inelastic scattering by nonpolar optical phonons in the limit of high carrier energies. This: solution has a simple analytic shape and permits one to define a temperature T and an anisotropy parameter q for the hot carriers. The analytic relation between T , q, the lattice temperature TL, and the applied field strength E is demonstrated. In the high-energy limit the applicability of conventional approaches, which are based on a Legendre polynomial expansion, is discussed by comparison with the asymptotic solution.

Die stationare Boltzmann-Gleichung fiir heiBe LadungstrPger wird bei Vorherrschen der inelastischen Streuung an nichtpolaren optischen Phononen fur den Grenzfall hoher La- dungstragerenergien asymptotisch gelost. Diese Losung besitzt eine einfache analytische Gestalt und gestattet die Definition einer Temperatur T und eines Anisotropieparameters Q fur die heiBen Ladungstrager. Der analytische Zusammenhang yon T, q, der Gittertem- peratur T, und der angelegten Feldstiirke E wird dargestellt. Die Anwendbarkeit iiblicher Naherungsverfahren, die auf einer Entwicklung nach Legendreschen Polynomen beruhen, wird durch Vergleich mit der asymptotischen Losung diskutiert.

1. Introduction

The solution of the Boltzmann equation for hot carriers is affected by consid- erable analytical difficulties (see, e.g., [l] and [2]). I n particular the existence of inelastic scattering mechanisms for cooling of hot carriers complicates the mathematical situation. With the predominance of inelastic scattering processes an external electric field causes a strongly anisotropic distribution of carriers in the momentum space. This is because the momentum gained in the electric field on the free path is not distributed equally over all directions in the momentum space owing to the strong energy exchange caused by inelastic collisions. Thus the mean drift velocity of the carriers is no longer small compared with the root mean square velocity and the higher coefficients of an expansion of the distribution function in a series of Legendre polynomials P,(,u) are no longer negligible. Thus the diffusion approximation, which works well for weakly inelastic scattering [ I , 21, does not appear t o be an adequate approach for hot- carrier transport processes with strongly inelast.ic scattering (see, e.g., [3]). Baraff's maximum anisotropy approximation [4] can describe carrier distribution with maximum anisotropy ; its applicability however is not clear in cases where the diffusion approximation has already failed, but there is no maximum anisotropy.l)

~ _ _ _ l) Maximum anisotropy means all the carriers travelling in the field direction.

40'

622 M. MOCKER

In addition to the difficulties connected with the angular dependence of the distribution function in the momentum space further analytical complications exist. Thus the collision integral in the Boltzmann equation couples the energy distribution function f0 ( W ) for energies which differ by the amount of the collision energy W,. For inelastic scattering processes, approximating these functions fo ( W f We) by the first terms of a Taylor expansion in powers of Wc is in general no longer a good approximation.

In recent years these difficulties in the transport theory of hot carriers have been overcome by numerical techniques such as the Monte Carlo and the path- variable methods [5 t o 81. Of course the necessity for analytic investigation remains. For instance, the study of limiting cases which are analytically solvable can throw light on general relation like the relation between the inelasticity of the scattering, the electric field strength, and the anisotropy of the carrier distribution.

In this work we solve analytically the transport equation of hot carriers with mainly inelastic scattering on nonpolar optical phonons and with elastic scattering on acoustic phonons in the limit of high carrier energies. The asymptotic solution represents a new approximation for energies sufficiently high above the emission threshold of optical phonons. In order to make comparisons with earlier numerical or experimental work [5,6,9], the calculation is adapted to the situation of p-Ge.

-

2. Boltzmann Equation in t,ho Limit of High Carrier Energies

The steady-state Boltzmann equation in our case has the form

-(:) ficltl = C"'.f + C O P . / ,

where CaC. and Cop. denote the collision operators for acoustic and nonpolar optical phonon scattering, respect,ively. We assume spheric parabolic energy surfaces. The distribution function f ( k ) then only depends on the energy W and the angle 0 between the electric field E and the momentum lc. We introduce the dimensionless variables

h2 k2 w=-- - and p = cos 0 , 2 m Wop.

where Wop. is the energy of optical phonons. The following notations will be used : E amount of electric field strength, .lac. mean free path of acoustic phonon scattering, lop. mean free path of the optical phonon scattering in the limit W + 00,

T, lattice temperature, k Boltzmann constant, rn effective mass of carriers. Further we introduce the quantities 6, l,, BL, and v by the relations

Asymptotic Solution of the Boltzmann Equation for Hot Carriers 623

where v is the ratio of the optical phonon energy and the energy increase of carriers induced by the electric field in field direction. The analytic form of the collision operators is [l] .

Here the collisions with acoustic phonons are supposed to be ideally elastic and the scattering is therefore spherically symmetrical in the both cases (4a) and (4b) (see, e.g., [S]). With (4) the Boltzmann equation (1) in the variables W and p reads

I n the following we are dealing with the distribution of carriers the energies of which lie far above the threshold. I n this energy range, terms in equation ( 5 ) whose ratio to other t,erms is of the order 1 / W can be neglected. There are two such terms in equation ( 5 ) :

(i) the deviation of the square roots I/( W f 1)/ W from 1. We therefore set

(6) V'(W f l ) / W = 1 ; (ii) the second part of the. field term on the left-hand side in equation (5). With (6) the ratio of the last term and, for instance, the out-term Cout f in

the collision integral given by

(7)

624 M. MOCKER

is of the order 1 / W if aflap vanishes asymptotically with f . Under this assumption we neglect the second field term part too, and we shall prove the validity of this assumption when solving the asymptotic equation, which now has the final form

af P a w =

This equation for f( W, p) is an ordinary difference-differential equation in the variable W and an integral equation of Fredholm type in the variable p.

3. Asymptotic Solution of the Boltzmann Equation

Physically significant solutions of equation (8) satisfy the conditions

f ( W , p ) > 0 and f ( W , p ) = o ( W - ~ / ~ ) 2, for W -+ 0 0 . (9)

The last relation guarantees that the normalizing integral for the distribution converges. As the coefficients in the asymptotic equation do not depend upon t,he energy, the two variables can be separated by putting

f ( w, p) = c exp [-I WI w-4 (10)

with the function Y(p) describing the angular dependence of the distribution. The energy distribution exp (-B W) demonstrates the existence of a separate “temperature” T of hot carriers in the range of validity of the asymptotic approximation (in correspondence with [S]). For Y(p) we obtain the following integral equation:

+I

(1 - r p ) W p ) = K ( 7 , v, B L , 6) 1 Wp’) f (11) -1

Here the parameter 17 and the function K were introduced by

B 17= v ( 1 + 6 )

and

The integral equation (11) has a singular solution for 171 > 1 that does not fulfil the condition (9). For 171 < 1 the equation (11) is solved uniquely except

2, We use the definitions lim O(zn)/xn = const and lim o(xn)/zn = 0 for the expressions O ( 2 ) and o ( 2 ) respectively. S * O z+o


for a constant factors) in the range of definition of Y(p) by the regular solution

with

and q satisfying a condition of solubility. It takes the form

For the given parameters v, ,!IL, and 6 there is exactly one q in the range 1q I < 1 that solves (16). This value is always positive or zero.

The distribution Y(p) is illustrated for several q in Fig. 1. It is seen that 7 determines the anisotropy of the distribution. We therefore call 7 the anisotropy parameter. If 7 equals zero, the distribution will be isotropic. There is a maximum anisotropy in the limit q + 1. In this limit the solution (14) takes the singular form4)

lim Y(p) = 4 6 (1 - p) (17) q + l - O

with the condition of solubility K(l, v, a, 6) = 0. This condition can only be fulfilled for the parameter values PL = 00, v = 00, and 6 = 0, i.e. for vanishing lattice temperature, for vanishing field strength, and in the absence of elastic scattering. We see that there is no maximum anisotropy of the distribution function for real physical parameters.

With 7 given by (16) and with (12) the solution parameter is also given as a function of physical parameters BL, v, and 6. Therefore the asymptotic distribution (10) is known, except for the constant factor c, which cannot be determined in this work because it can only be calculated by normalizing the distribution function in the entire energy range.

We obtain the result : The Boltzmann equation (5 ) is asymptotically solved by (lo), (14), and (16). In this particular case it was possible to treat the problem of nonlocal energy dependence (the terms f (W f 1, p) in (8)) without employing a Taylor series expansion, and to describe the momentum distribution without using a Legendre polynomial expansion.

Fig. 1. Angular distribution Y ( p ) = B(q) / ( l - q p ) versus p for wrious values of the anisotropy parameter q , qo = 0, q, = 0.4,

qr = 0.8, = 0.95, qr sz 1 (B(q,) = 0.1) P-

s, In (14) the constant factor is chosen to satisfy

4) The 6-distribution character of this limit can be demonstrated by using the complete-

!!& y(,u') = I . -1 1' 2

ness of the set of Legendre polynomials 501).

626 M. MOCKER

4. Discussion of the Asymptotic Distribution The condition of solubility (16) yields a relation between theanisotropy param-

eter q, the carrier temperature T , and the electric field strength E and the parameters T L , Wop., lop., and 6 of the system. The transcendent equation (16) cannot be solved in a closed form for q ; however, the elimination of v is possible :

In the following we discuss the cases of strong or weak anisotropy. can be achieved if

the argument of the inverse cosh function in (18) is greater than 1 for this q, i.e:if Strong anisotropy. A state with the anisotropy parameter

By means of equation (19) an anisotropy parameter qs(BL, 6) is defined which is maximal for the given system (BL, 6). From (19) with (18) and (12) the rela- t,ions

BL

v(%) = 2 qa ( 1 + 6) and

follow. The state of highest anisotropy even exists a t a carrier temperature T double the lattice temperature and a field strength E ,

E, = rls kTL (1 + 6) . e lop.

The anisotropy parameter q decreases as a function of E to values of E smaller as well as greater than E,. In Fig. 2 v and ,!? are shown as functions of q for a given system. At lattice temperatures for which the relation BL 2 3 holds (for p-Ge this means TL below some hundred deg Kelvin) and a t a weak interaction level of carriers with acoustic phonons (6 < 0.2) the value of q, is near 1. I n this case a large field strength range exists for which the anisotropy parameter of the distribution is near qs. This range can be estimated under the condition 0.95 < q .< qs by

Yig. 2. The quantity u, the reciprocal carrier temperature 8 , and the ratio q'/fl versus anisotropy parameter q for an assumptive physical situation

with @L = 3.5 (i.e. TL r 123 OK) and 8 = 0.3

$symptotic Solution of the Boltzmann Equation for Hot Carriers 627

For p-Ge at TL = 77 O K we get E, = 1870 V cm-l. Here the inequality (21) gives the interval

1.1 x lo3 V cm-’ < E < 6.4 x lo3 V cm-l (21’)

for the range of nearly maximum anisotropy. With (12) we obtain an increase in carrier temperature T proportional to the field strength E in this range:

Such a behaviour of the carrier temperature of a strongly anisotropic hot-carrier distribution corresponds to the “Shockley spike”, e.g. discussed by Baraff [4].6)

Weak anisotropy. We consider equation (18) for the case of weak anisotropy (q < 1). Neglecting terms of the order o(q2) we can write

B(q) = 1 v2 3 ’

In this approach we obtain

where the positive sign is correct €or small fields (E < E,) and the negative sign for strong fields (E EJ. At small fields the carrier temperature varies only weakly according to

and the anisotropy parameter increases proportional to the field E :

For strong fields the carrier temperature increases with the square of the field strength :

Such carriers are so “hot” that the inelastic character of nonpolar optical phonon collisions loses its effect on the anisotropy of the distribution.s)

5, Namely with (22), for the energy distribution follows exp (- %‘/kT) = exp (- @/Q) where the notation of Baraff Q = E e l h is used and where % again denotes the energy in original energy units.

6, It is only in this case that a treatment using the first terms of a Taylor expansion around W of the.expressions f,, ( W f l), which have the form c exp -B ( W f 1) here, yields the same result as ours, because the series converges rapidly for < 1.

628 M. MOCKER

For the anisotropy parameter q we get the small value

and the distribution also becomes nearly isotropic.

(8). With (7), (12), and (14) we get Now we can prove the assumption used in the derivation of asymptotic equation

The ratio q2/B is of order zero in 1/ W . This shows that the assumption of same order vanishing for f and af/ap is consistent with the asymptotic solution (14).

Summarizing, the relations (6) and (29) yield for the applicability of the asymptotic approximation a t carrier energies W the conditions

and (ii) w > wop. r" B '

respectively, where the term q2/B has the upper bound 3 tanh (BL/2)/(1 + 6); however, for field strengths below E,, q2/B is smaller than 2 q:/BL (see Fig. 2). One expects that for energies satisfying both conditions of (30), the asymptotic solution (lo), (14) is also a good approximation to the true distribution. Of course, the applicability of the asymptotic approximation is restricted by the validity of the basic equation ( 5 ) in the limit of high carrier energies.

6. Comparison with Diffusion Approximation and Maximum Anisotropy Approximation

The convential approaches in solving the Boltzmann equation of the kind treated in this paper are based on the expansion of the distribution function f ( W , p) in a series of Legendre polynomials Pl(p) and on a defined truncation of the infinite set of equations for the Legendre coefficients of the distribution function. The truncation methods can only be justified by assumptions restrict- ing the anisotropy of the distribution [l t o 41. Because of the exact solubility of the asymptotic Boltzmann equation (8) one can discuss the applicability of the approximation methods in the high-energy limit. The results will also give an idea about the situation in the other energy range. As the approximation methods only differ in the treatment of angular dependence, we can begin with equation ( 1 1). Using a Legendre expansion equation (1 1) yields the following set of equations :

l = 0 , 1 , 2 ) . . . ) where

+ 1

are the Legendre coefficients of Y(p). The parameter q again fulfils the relation (12).


The infinite set (31) is solved by the Legendre coefficients of the exact solution (14), which have the form

The Qz are the Legendre functions of second kind [lo]:

We are considering the diffusion approximation of n-th order (Pn-approximation) and Baraff's maximum anisotropy approximation [a] of n-th order (MA,-approximation) both defined by the truncation at the (n + 1)-th equation in (31) with the relations

u/,+, = 0 (P,-approximation) , (35a)

Yn (MA,-approximation) . (35 b) 2 n + 3

Y n + 1 = 2 + 1

The now finite set of equations (31) has the solution

where the solution parameter T,I = qlfn(v, &, 6), which here obviously acts as an anisotropy parameter, follows from a condition of solubility7)

B*"(T,I) = K(q, 8, a, 6) ' (37)

and where the indices +n and -n indicate the P,-approximation and the MA,,- approximation, respectively. The exact asymptotic solution is denoted by the index 00 in this section. B * ~ ( T , I ) is a finite continued fraction and reads

1 - . . 1

I -

In solution (36) we have taken the free coefficient !Po equal to 1 owing to the same normalization of Y(p) as in (14) in all cases.

The approximate solution (36) and the corresponding condition of solubility (37) only differ from the exact solution (33) and the condition of solubility (16)

') It is interesting that Reik and Risken [ll] using the PI-approximation also obtain the equation B+1 = K in the energy range W > 2 Wop. for scattering processes which correspond to ours.

- -

630 M. MOCKER

by the fact that here B(q) is replaced by B'n(q).*) The value of the expressions B f n ( q ) decreases monotonously, and the value of the expressions B-,(T,I) increases monotonously with increasing n. It follows from the solution (36), the equation (12), and the properties of the expressions B**(q) that for equivalent physical situations both the values of calculated Legendre coefficients !Pz (1 > 1 ) and the calculated values for the carrier temperature T lie under the exact values for the P,-approximation and abovee) the exact values for the MA,- approximation. Moreover the approximated values approach the exact ones for increasing n or m. We have in fact the relation

lim B*n(.);) = Bw(q) == B(q) . n + w

(39)

We obtain the important result that both the diffusion approximation and the maximum anisotropy approximation appear as convergent approximations t o the solution of the asymptotic Boltzmann equation.

We again consider the cases of weak or strong anisotropy. Weak anisotropy. It is only in physical situations with vanishing qm that the

q*" and the !P?"coincide with the true result (qw = 0 , !Pr = a!,,) for all approximation orders. This coincidence is also approximately true in the case of weak anisotropy discussed in the last section of this paper, and follows because the left-hand sides of equations (37) and (16) for q < 1 only differ by quantities o(qzn) where n again is the order of approximation. I n this case we obtain for all approximationmethods (n 2 1) theresults (23) to (28) ofthe last section. Neglect- ing the differences in the .);+", the solution (36) has the form

where the expansion of !Py(q) by powers of reads

The order of the error in the approximated function !P?"(q) is higher than the order of the exact function !Pr if 1 5 n. Then equation (40) in all the cases treated here reduces t o

(42)

i.e., within the range of weak anisotropy the Legendre coefficients describing the anisotropy of the distribution increase proportionally with the powers of the anisotropy parameter q.

With (42) it becomes clear that the maximum anisotropy approximation also yields satisfactory results in the case of weak anisotropy. By the truncation

l ! !Pz(q) = (2 - I ) ! ! q1 + o w ) 9

6) Moreover it is clear that the n-th order of approximation does not say anything about Yn+2 and higher Legendre coefficients, whereas the exact solution includes all the Legendre coefficients.

s, For single values of 7 equality between the exact and the approximated results exists independent of n or m.


(35b) a more or less arbitrary approaching assumption is made, but that has a minor effect on the solution because the approximated term vanishes for r --f 0 one order higher than the remaining terms in the corresponding equation of set (31). However, we realize that in the range of weak anisotropy the diffusion approximation of the same order leads to smaller deviations from the exact result than the maximum anisotropy approximation does, because the el-ror of Yl in the diffusion approximation with (40) is an order higher in 11 than the error in the maximum anisotropy approximation.

Strong anisotrogy. I n the case of maximum anisotropy (11 = 1) the MA,- approximation of all orders n yields the exact result. On account of

B-,(l) = B"(1) = B(1) = 0 (43)

this follows from (36) and (37), because the truncation condition (35b) gives the exact ratio of the Legendre coefficients Yr+l and YE which have the form

Yl = 2 I + 1 , (44)

in conformity with the singular solution (17). In the range of nearly maximum anisotropy (defined by equation (21)) the

MA,-approximation already turns out to be a possible approach towards the physical distribution for n = 1, while the diffusion approximation of the same order leads to Legendre coefficients that here are essentially too small. However, the quality of the maximum anisotropy approximation decreases fairly quickly with the distance of the nonphysical limit of maximum anisotropy, and for values of q of about 0.8 the MA,-approximation is already less exact than the Pn-approximation.

Now that the quality of approximation procedures with respect to the indi- vidual Legendre coefficients is known, the question remains as to how the total angular distribution is approximated by the first n terms of a Legendre polynomial expansion. The partial sum

n

will only describe the angular distribution adequately if the higher coefficients ( I > n) yield a negligible contribution to Y ( p ) . That is the case for 17 < 1, because then

Y(p ) = Y*, (p ) + o(q") Y (46)

coinciding with (40) and (42). Thus the error in the description of the total angular distribution Y(p) by !P*"(p) always increases (which is also the case for the MA-approximation) with the real anisotropy of the distribution in this range.

In the case of maximum anistropy the truncation condition (35b) is correct and !PI = 2 1 + 1 also follows for the higher coefficients (1 > n). But for nearly maximum anisotropic behaviour of the distribution the last relation already contains a strong deviation from the true Y, for 1 > n, and therefore the MA- approximation does not yield any better information on these higher coefficients.

632 M. MOCKER

However, this situation need not disturb us, because only the first Legendre coefficients are relevant in describing most of the interesting physical quantities. Consequently one has a t least to use the n-th order approximation if Yn is still of physical significance.

Finally one can say that for small n the MAn-approximation can only be applied for very weak or very strong anisotropy and the P,-approximation only for weak anisotropy as the quality of these approximation methods sinks in the region of medium anisotropy for the Pn-approximation as well as for the MA,- approximation (contrary t o Baraff's suggestion [4]). If these methods are to be used in this region, higher orders of approximation must be employed (at least n = 2). The MA-approximation of the same order is to be given priority over the diffusion approximation only in the sense that the MA-approximation yields for any anisotropy of the angular distribution a fairly suitable approximation, while, in fact, the diffusion approximation gives a better description of the range of small q but fails, however, for values of q near to unity.

6. R.esults for p-Ge I n conformity with various other authors (cf., e.g., [5 ] and [6]) we apply t.he

model (4) for the collision mechanisms of the carriers to the heavy holes in p-Ge. Because of the strong coupling of the holes in p-Ge to the nonpolar optical phonons neglecting the contribution of acoustic phonons to the energy relaxation of the carriers is justified.

The physical parameters used to calculate the stationary distribution of heavy holes were taken from a paper by Brown and Bray 1121. In our notations that means: lop, = 7 . 7 8 ~ lo-* m, T, = 77 OK, m = 0.35 me, 6 = 0.0935, and Wop, = 0.0371 eV. The electric field strength varies in the range 0 < E < < lo6 V cm-l.l0) I n Fig. 3, 4, and 5 the anisotropy parameter q, the reciprocal carrier temperature b, and the Legendre coefficient !PI are shown as functions of the applied field strength E not only as exactly asymptotic but also in PI-,

a5

0 .

I I

?03 E, 704

Fig. 3. Calrulated anisotropy parameters versiis field strength in tlie case of u-Ge with TL = 77 "K. The indivi- dual curves correspond ti] the various appro~~~.Iies or to tlw exact asymptotic account, respectively. Curves ( +Z) aiid (il): P,- and P,-approximation: curvcs ( - 2 ) and ( - 1 ) : MA?- and MA,-approximation; i'urve (m): esavt asymp-

totic calculation ( E s = 1870 Vrm-I) E(Vcm-7- - lo) additional effects occurring at strong field strength as, for instance, ionization arc

not considered.


Fig. 4. Calculated values of the reciprocal carrier temperature p versus field strength under the same conditions and with the same

denotations as in Fig. 3

I I I

E (Vcm-’) --

7

A lo5

E (Vcm-’) - Fip. 6. Legendre coefficient V: versus field xtreiigth under the same conditions and with

thc same denotations as in Fig. 3

P,-, MA,-, MA,-approximation. The analytical results are seen to be valid; in part,icular the MA,-approximation turns out to be sufficient in the entirc range of field strength.

A comparison of the asymptotic solution with known results for p-Ge is restricted to the distribution function in the energy range where the asymptotic approximation is already applicable. Numerical results for the energy distribution of the heavy holes with a lattice temperature T, = 77 OK are given in [5] and [6] up to energies % 2 Wop,. For those energies which are immediately

N

634 M. MOCKER: Solution of the Boltzmann Equation for Hot Carriers

above the emission threshold, the asymptotic approximation would still be a poor approachll) ; however, the asymptotic values for the hole temperature are already quantitatively comparable with the values given by Kurosawa [5] and Budd [6] in the energy range immediately above the threshold. This is shown in Table 1. The values in the table are not given explicitly in [5] and [6], but are taken from the figures in [5] and [6].

field strength 1 (V cm-l)

Table 1

hole temperature ( f > Fop.) -

asymptotic values Budd [6] I Kurosawa [5] 1

WOK ~ 88OK 1 :ilg 128OK 1 - 113 OK

103OK 1 - ;:: 1 1370

The angular dependence of the distribution function in the momentum space a t E = 600 V cm-' is exhibited explicitly in [5] . The corresponding asymptotic distribution shows a qualitative agreement for the highest energies ($ x G o , ) which are taken into consideration there. The Legendre coefficients f z ( @) given by Budd [6] for E = 926 V cm-1 and E = 695 V om-l are about double the calculated asymptotic values. Othcr comparable results to the angular dependence &re not available to us.

Arkaou~ledgenient

The author is greatly indebted t o Prof. R. Enderlein for suggesting the problem and for continuous help in preparing this paper.

References [l] E. M. CONWELL, Solid State Phys., Suppl. 9 (1967). [2] M. ASCHE and 0. G. SARBEI, phys. stat. sol. 33, 9 (1969). [3] E. G. S. PAIGE, Proc. VIII. Internat. Conf. Semicond., Kyoto 1966 (p. 397). [4] G. A. BARAFF, Phys. Rev. 133, A26 (1964). [5] T. KIJROSAWA, Proc. VIII. Internat. Conf. Semicond., Kyoto 1966 (p. 424). [6] H. BUDD, Proc. VIII. Internat. Conf. Semicond., Kyoto 1966 (p. 451); Phys. Rev. 168,

171 W. FAWCETT, A. D. BOARDMAN, and S. SWAIN, J. Phys. Chem. Solids 31, 1963 (1970). [8] H. D. REES, J. Phys. Chem. Solids 30, 643 (1969). [9] W. E. PINSON and R. BRAY, Phys. Rev. 136, A1449 (1964).

798 (1967).

[lo] E. JAHNKE, F. EMDE, and F. LOSCH, Tafeln hoherer Funktionen, B. G. Teubner Ver-

[ll] H. G. REIK and H. RISKEN, Phys. Rev. 124, 777 (1961).

[12] D. M. BROWN and R. BRAY, Phys. Rev. 127, 1593 (1962).

lagsgesellschaft, Stuttgart 1960.

H. G. REIK, Festkorperprobleme 1, 89 (1962).

(Received April 20, 1971)

11) In particular the approximation of the expressions v (W f l ) / W by 1 fails here.

asymptotic solution of the boltzmann equation for hot carriers

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