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Effect of electron-electron collisions on properties of hot electrons
Solid State Communications, Vol. 65, No. 10, pp. 1221-1225, 1988. Printed in Great Britain.
0038-1098/88 $3.00 + .00 Pergamon Press plc
E F F E C T OF E L E C T R O N - E L E C T R O N COLLISIONS ON P R O P E R T I E S O F H O T E L E C T R O N S M. Combescot Groupe de Physique des Solides de l'Ecole Normale Suprrieure, 24 rue Lhomond, 75231 Paris Cedex 5, France
(Received 13 November 1987 by M. Balkanski) We calculate analytically the electron distribution, electron temperature and electric current of a non degenerate electron gas in a large electric field including finite electron-electron collisions. We demonstrate that these quantities vary very slightly with the electron-electron collision time: this establishes for the first time the validity of the results obtained from the famous but unjustified displaced Maxwellian distribution function.
P R O B L E M S R E L A T E D TO T H E behaviour of elec- studied how the electron distributuion f ( k ) is driven trons in presence of large electric fields are of great toward a displaced Maxwellian by a finite e-e collision technological interest for electronic devices: the minia- time; they solved analytically a simplified form of the turization, in very-large-scale integration (VLSI) tech- Boltzmann equation. But within their relaxation- time nology, leads to high-field strengths well outside the approximation for e-ph scattering [7], they cannot Ohmic response region. The first attempt to solve the address the nonlinear high-field behavior of the curBoltzmann equation in the limit of large electric field rent nor the effect of e-e scattering on the current and E, was to use a displaced Maxwelliam distribution [1]. the electron temperature. In this Communication, for the first time, we Although this procedure is justified only when electron-electron (e-e) collisions are extremely fast com- calculate analytically the electron drift velocity, tempared with the electron-phonon (e-ph) ones - - con- perature and distribution function of non degenerate dition which is usually not satisfied experimentally hot electrons in semiconductors, including a finite e-e - - the displaced Maxwellian had a great success as it collision time ze~ and we show h o w f ( k ) evolves towgives, after rather simple algebra, most of the trans- ards a displaced Maxwellian when re-~ decreases. We port properties of hot electron systems. An analytical pay particular attention to the treatment of the e ph solution has also been obtained neglecting e-e colli- collision integral (to remedy the defect of W.S.W. sions [2]: this case corresponds to an extremely large calculation) and we mainly focus on non polar matee~e collision time r ~ , while for the displaced Max- rials. We show that, while the isotropic part o f f ( k ) is wellian, ze~ ~ 0 is implied, ze-~depends on the carriers essentially unchanged by e-e collisions, the anisotropdensity n: the larger n, the smaller Te_~. For usual ic part o f f ( k ) varies from E . k to E'k/k when the doped materials, ~_~ is neither very large nor very effect of e-e collision decreases. But this modification small so that a solution of the Boltzmann equation for o f f ( k ) induces very tiny changes on physical quanfinite ze_~is needed. Although this is a quite old pro- tities such as the electron current and temperature in blem, up to now only numerical answers have been agreement with the results of the numerical methods; given using Monte-Carlo methods [3, 4]. The intro- this explains a posteriori the success of the displaced duction of e-e collisions is technically difficult and Maxwellian procedure which is, in usual experimental requires powerful computers and very long calcula- situations, never valid as e-ph collisions are always tions, due to non linearity problems [4] and long range faster than e-e ones. We also consider the case of polar potential associated to it: the numerical resolutions optical phonons and show where it differs from non polar materials. are not yet totally free from approximations [5]. An analytical solution of the Boltzmann equation Many analytical resolutions of the Boltzmann at intermediate Te~ is still today highly valuable as it equation use an expansion in powers of the electric would allow a physical understanding of the exact role field E. While it is justified in the Ohmic (low-field) played by carrier-carrier collisions in hot electrons limit, there is no ground for it in the nonlinear high systems. Recently Wingreen, Stanton and Wilkins [6] field regime. Here we demonstrate that, in most expe(W.S.W.) tried such an analytical approach and have rimental cases, there is a small parameter through the 1221
E F F E C T OF E L E C T R O N - E L E C T R O N COLLISIONS
1222
entire range of electric field, a fact which has been overlooked up to now. This parameter is the modulus of a dimensionless vector 8, parallel to E: e can be seen as the energy eE2 gained by an electron over a characteristic length 2 (depending on the e-ph scattering) divided by the electron thermal energy ks T*: for non polar materials in high-field, T* increases as E 2, so that e decreases as 1/E. Therefore, for large E, we can indeed proceed to an expansion in e: this is at the basis of our analytical solution; quite generally, this expansion preserves non linearity in E. In part 1, we rewrite the Boltzmann equation in a dimensionless form: all the physical quantities appear inside two parameters e and z* ( ,-, Te_e/Ze_ph). We then show that e < 1: this is the crucial point for the mathematical resolution of the reduced Boltzmann equation done in part 2. The results are given in part 3.
Vol. 65, No. 10
wellian distribution, the corresponding function F(K) is (1 -- 2K'2/3) -3/2
FDM(K) =
× e x p [ - I K - K'I:/(1 - 2K'2/3)], (7a) FoM(K) ~
e x p ( - K 2 ) [ 1 + 2K.K* + 2((K'K*) 2
- K2K*2/3)) + . . . ]
if K* ,~ 1. (7b)
Let us now consider the Boltzmann equation, and start by the e-ph collisions integral. We rewrite it in dimensionless form using our reduced quantities, Ie-ph (k) = Ie-ph (K)/Ze-ph where (2rt) -~ I d 3 K ' [ - F ( K ) M ( K , K')
Ie ph(K) =
+ F(K')M(K', K)], 1. R E D U C E D B O L T Z M A N N E Q U A T I O N
M(K,K')
The problem is to determine the electron distribution f ( k ) for large electric field E, including e-ph and e-e collisions. We introduce the average wavevector k* and the effective temperature T*, defined as nk* =
2(2a) -3 fd3kf(k)k,
(3/2)nks T* = 2(2zc)-3 yd3kf(k)htk2/2m,
(1) (2)
n is the e density. T*, which reduces to the e temperature in case of a Maxwellian distribution, can be defined even if f (k) is not related to any temperature. Although yet unknown, T* is used to define reduced momentum K and distribution function F(K) K2 = h2k2/2mksT * =
k222r.,
(413gI/2)(T.IT*)312F(K),
f(k)
=
ksT,
= fiz(31r2n)2/3/2m,
(3) (4) (5)
T, is a measure of the e density n in term of a temperature. Physically, it is the temperature below which the plasma becomes degenerate. The momentum K and the function F are now expected to be of the order of 1. More precisely F(K) satisfies, due to equations (1) and (2) and the fact that the e density is n
X3/2[1 or K* or 3/2] = fd3KF(K)[1 or K or K2].
(6) F(K) depends only on one parameter, the reduced momentum K* (from k*) which characterizes its anisotropic part, its isotropic part being always of the order of 1. In the particular case of a displaced Max-
=
(Sa)
IK - K'Is[nLf(K 2 + D . o - K a) + (nL + 1)6(K'2 + f l 0 -
K2)],
(Sb)
D.o = fie)o/ks T* is the reduced optical phonon energy. nL is the phonon occupation number, s = - 2 or 0 depending if the material is polar or non polar. All the physical parameters of the e-ph collision integral appear in 27e_ph 1/ze_ph = fi '[2/3n3/2][Ao/2~SksT*](T./T*) 3/2,
(9)
=_ h-lksT*[2/3rt3/2](T./T*)3/2 x (Te_ph/T*) °-')/2,
(10)
the thermal wavelength 2r, is defined from equation (3). The constant A0 comes from the usual e-ph coupling 2Ma-lA0; in equation (10) we re-express it in terms [8] of a temperature like parameter Te_p,. For Ge(s = 0) we find [9]: T~_ph = 0.56 K and for GaAs (s = - 2 ) : Te., = 370°K. Similarly we rewrite the LHS of the Boltzmann equation V j ( k ) . ( - e E / h ) = VKF(K)'n/r,~h, the reduced electric field being =
2E2r._ph ( T* ,~-~/2 <--phi '
(11)
2r~_p, is the thermal wavelength for T.,_ph. e can be interpreted as the ratio of the energy gained by an electron in the electric field E over the characteristic length 2r,_ph divided by k s T * for non polar optical phonons (s = 0) or by some e-ph characteristic energy ks T~_phfor polar optical phonons (s = - 2). It is crucial for the analytical resolution of the Boltzmann equation, to note that (fortunately) this reduced field [10] ~ is smaller than 1 in usual physical cases: for
Vol. 65, No. 10
E F F E C T OF E L E C T R O N - E L E C T R O N COLLISIONS
polar phonons s = - 2 and e increases linearly with E; but even for a field of E ,-~ 5 KV cm -~, e is not larger than 0.3 in GaAs. For non polar phonons, s = 0 and e ~ E/T*: for E_-__ I KV cm -~ and T* ~ 600 K, one find e ~ 0.5 in Ge. But anticipating the conclusion that T* is the electron temperature which increases for hot electrons as E 2, e is indeed small for large field: for non polar optical phonons, e first increases as E for cold electrons and then decreases as 1/E for hot electrons, being maximum in the warm region but still small. For the e-e collision integral I¢_~, we follow W.S.W. and write it IN(k) = [ - F ( K ) + FDM(K)]/ z ~ . This preserves the most important features of e-e scattering, namely that the number of particles, momentum and energy of the e system are conserved in each collision, property of the displaced Maxwellian. W.S.W. obtained z~_¢by calculating the quadrupole moment [11]. One can also get z ~ directly by extracting [12] the T, and T* dependence from the e-e collision integral; one finds fl/%_e ~- kaTRyd(T,/T*)3/ZLnQ;I
~
nLn(n),
(12)
in agreement with W.S.W. result, kn TRyd is the Rydberg energy me4/2h2e 2 and Q~ the reduced screening wavevector [Q2 = (8/3rOT3n Tr~2o/T.Z].
2. R E S O L U T I O N FOR A F I N I T E e-e COLLISION TIME We are finally left with the resolution for small of the following reduced Boltzmann equation VF(K).n =
I~_ph(K) + [--F(K) + FDM(K)]/Z*,
1223
+ I [F(K'2)JQ'(K" K ) - F(K2)aI(K, K')] d3K ' , h,~I(K, K') being the angular average of M(K, K'). The second term of equation (l 5) disappears for non polar phonons as M = M (s being 0) while it remains for polar phonons: I~-ph contains in that case an integral operator for g(K) (which is of the order of I whatever is fl0) and this leads to a more complex resolution of the Boltzmnn equation. In the remaining part of this Communication, we will restrict to non polar optical phonons. Using equations (8b) and (1 5), the e-ph integral reads in this case, with K+ = ( K 2 + ~.o) j/2 Ie-ph(K) =
--(nLK+ + (nL + 1)K )g(K) + J(KZ), (16)
J(K 2) = K+ F(K 2) - K _F(K2) + nL K+ [F(K2+) - F(K 2)1 + nLK [F'(K2 ) -- F(K2)].
(17)
As we have shown that e is small for hot electrons, the Boltzmann equation (13), valid for all K, implies Ie_~ and I¢ph small. I N small means either z* large or, if z* is small, F close to FDM as expected. Ie-ph small implies J small: this happens only either if the e distribution F is a Maxwellian at the lattice temperature or if fl0 is small [7]: the first case corresponds to cold electrons (i.e. the Ohmic low-field limit). So that, in the hot electrons case, ff~0has to be small. J ( K 2) then reduces [13] to D.0K - ~ d ( K Z F ) / d K 2 . Ie-ph small implies also g(K) small (as K ~ 1). As its angular average is zero, an expansion ofg(K) in Legendre polynomials of e-K is then justified g(K) = 8 . K g j ( K 2) + [(g.K) 2 - e2K2/3]g2(K2 )
(13) + .... where FDM, Ie-ph and e are given in equations (7), (8) and (1 1); the parameter z* is simply 1/z* = Z~_ph/Z~ ,~, (TRyd/T*)(T,/T*) 3/2 X (T*/Te_ph) 0 s)/2LnQj-l.
(14)
The displaced Maxwellian procedure is justified if 1/r* is large. This happens only for large n, and small T*: one has then to conclude that the displaced Maxwellian is unjustified at high temperature i.e. for hot electrons... To go further, it turns out to be useful to separate F(K) from its angular average _P(K2) and write F(K) = F(K 2) + g(K); the e-ph integral equation (8) then reads 2~zle_ph( K )
=
-
with gl, & ~ 1. We are now in position to solve easily the Boltzmann equation (13) for all z*. This is what we do in the rest of this paragraph. Conditions (6) for the e distribution function imply F ,-~ 1 and ~z3/2K* =
(1/3)~ ~ K2gl(K2)d3K + O(e2) -
~3/2~8" (19)
This defines a coefficient ~ of the order of 1. We then use this value of K* (,~ 1) in the expression (7b) of FDM(K). The identification of the coefficients of the two first Legendre Polynomials [14] in the Boltzmann equation gives immediately eZ[gI + z~.2,q3,61J , = D.oK I(KZF)" - (F -- e-K2)/z *,
g(K) ~d3K'21~r(K, K')
+ Ig(K')[M(K', K) - 3~r(K', K)]d3K '
(18)
(20a) (15)
2F =
--(2nL + 1)Kgl -- (gl -- 2ae-k2)/z *,
(20b)
E F F E C T OF E L E C T R O N - E L E C T R O N COLLISIONS
1224
e and ~ being small, equation (20a) implies [15] for a finite z* that P = exp ( - K 2) to lowest order in e: this means that the isotropic part of the e distribution is a Maxwellian at T*. Equation (20b) gives the anisotropic part gl =
2(z* + ~)[(2nL + 1)Kz* + 1] lexp(--K2).
The coefficient ~ is determined self consistently from equation (19). One finds, with X = (2nL + 1)z*.
1 =
XG(X)
A(X)
(2nL + 1) G(0) - G(X) -
2nL + 1'
(21)
G(X) = Id3KK2e-X2(XK + 1) -I. A(X) is an increasing function of X which varies from A(0) = 3 ~ ~ 0.66 to A ( ~ ) = 4/3~/-~ ,~ 0.75, so that A(X) is essentially independent of X, i.e. ~ is independent of r*. The simplest way to get the reduced phonon energy ~0 to lowest order in e, is to calculate the second moment of equation (20a) as the part coming from e-e collisions cancels exactly. It is then straightforward to find f~0 0~7~1/2e2~-- /rl/Z~-lK*2, using equation (19).
Vol. 65, No. 10
A(0)8. K to 8. K/K. This variation will give only small changes in the averages appearing in most physical quantities as A(0) ,,~ 0.66 and the leading terms in these averages will come from K's of the order of 1, due to exp ( - K 2) in FOK). In conclusion, I have shown the existence in the Boltzmann equation of a small parameter which allows to obtain, in the case of non polar materials, analytical expressions for the electron distribution, electron temperature and electric current of electrons in a large electric field including a finite electronelectron collision time. I have demonstrated that their variations with this collision time are very very small. This explains why, although a priori unjustified because electron-electron collisions are essentially never faster than electron-phonon ones, the Displaced Maxwellian procedure has a great success.
Acknowledgements - - I wish to thank R. Combescot, J. Bok and N. Pottier for stimulating discussions.
=
REFERENCES
3. RESULTS Translating the above results in unreduced units, one finds for the temperature and drift momentum
1.
h2k .2 2m
2.
-
ksT*
-
3 A(z*(2nL + 1)) hog0 , 8(2nL + 1) A(0)
(22)
3.
3rr3 (eE2"r~-Ph)2A(z*(2nL + 1)), 2(2nL + 1) rio90 A(0)
4. 5.
(23) where the function A(X) is defined in equation (21). One recognizes the well-known results of the displaced Maxwellian method with some correction A(X)/A(O) coming from a finite z*. These corrections are very small as the function A(X) is essentially constant: A(X)/A(O) increases from 1 to 32/9n = 1.13. This means that the drift velocity slightly increases when e-e collisions become less effective. One also notes that the other well-known result h2k*2/2m = (4/3.q/-~)h~o0/ (2nL + 1), obtained by neglecting e-e collisions [2], is simply the limit of equation (22) for z* infinite. The electron distribution reads to lowest order in e:
6. 7.
8. 9. 10.
F(K) = e-~[1 + (2nL + l ) 8 . k X + A(x) kx+ 1 + . . . ],
(27)
where X = (2nL + 1)z*: the main term e x p ( - K 2) is a Maxwellian with an electron temperature T*. The anisotropic part varies, with increasing z*, from
11.
M. Fr61ich, Proc. Roy. Soc. A188, 521 (1947); M. Fr61ich & B.V. Paranjape, Proc. Roy. Soc. B69, 21 (1956). E.M. Conwell, Solid State Physics 9 Academicques. G. Bauer, Solid State Physics. p. 1, Springer Tracts (1974). C. Jacoboni & L. Reggiani, Rev. Mod. Phys. 55, 645 (1983). As an example the final state of the collision is chosen randomly according to a Displaced Maxwellian (see [4]). N.S. Wingreen, C.J. Stanton & J.W. Wilkins, Phys. Rev. Lett. 57, 1084 (1986). For large E, e-ph collisions do not relax f ( k ) toward a Maxwellian at the lattice temperature but instead leads to f~0 small (see our equation (17)). Ao - 23+~(ksT*)2 (Te_ph/T*)~l-s)/2 is indeed a constant independent of T*. We used the values of the e-ph coupling constants given in K. Seeger, Semiconductors Physics, Springer. One may think that the numerical value obtained for e is somewhat arbitrary as more (2r0's can be included in the definition of I~-ph. Our choice of the prefactor of I~-phis in fact such that no numerical factor appears in its final expression equation (20) so that e contains indeed all the constants. See also M. Combescot & R. Combescot, Phys. Rev. B35, 7986 (1987) and M. Combescot, R. Combescot, Phys. Rev. Lett. 59, 375 (1987).
Vol. 65, No. 10 12. 13.
14.
E F F E C T OF E L E C T R O N - E L E C T R O N COLLISIONS
M. Combescot, (preprint). For polar optical phonons, the same argument would also apply; so that fl01nfl0 ~ e2, g(K) ,~ ~. KG(KQ and fl0 In D.0 ~ e2 as the term corresponding to the last bracket of equation (17) is of the order of f",.oLnO, o for f~0 small. This implies nL~o "~ 1. If not, one can keep the t e r m nL~'~o .
15. 16.
1225
The term (2/3) e)K2g't in equation (20a) comes from restoring the P2 symetry in the equation for g2(K2). For infinite T*, equation (20a) and (20b) leads to a second order differential e~uation, the solution of which is F = e x p ( - K ) with f~0 = 452/3 (nL + 1), due to equation (6): this is precisely our limiting value for r* ~ oo.
0038-1098/88 $3.00 + .00 Pergamon Press plc
E F F E C T OF E L E C T R O N - E L E C T R O N COLLISIONS ON P R O P E R T I E S O F H O T E L E C T R O N S M. Combescot Groupe de Physique des Solides de l'Ecole Normale Suprrieure, 24 rue Lhomond, 75231 Paris Cedex 5, France
(Received 13 November 1987 by M. Balkanski) We calculate analytically the electron distribution, electron temperature and electric current of a non degenerate electron gas in a large electric field including finite electron-electron collisions. We demonstrate that these quantities vary very slightly with the electron-electron collision time: this establishes for the first time the validity of the results obtained from the famous but unjustified displaced Maxwellian distribution function.
P R O B L E M S R E L A T E D TO T H E behaviour of elec- studied how the electron distributuion f ( k ) is driven trons in presence of large electric fields are of great toward a displaced Maxwellian by a finite e-e collision technological interest for electronic devices: the minia- time; they solved analytically a simplified form of the turization, in very-large-scale integration (VLSI) tech- Boltzmann equation. But within their relaxation- time nology, leads to high-field strengths well outside the approximation for e-ph scattering [7], they cannot Ohmic response region. The first attempt to solve the address the nonlinear high-field behavior of the curBoltzmann equation in the limit of large electric field rent nor the effect of e-e scattering on the current and E, was to use a displaced Maxwelliam distribution [1]. the electron temperature. In this Communication, for the first time, we Although this procedure is justified only when electron-electron (e-e) collisions are extremely fast com- calculate analytically the electron drift velocity, tempared with the electron-phonon (e-ph) ones - - con- perature and distribution function of non degenerate dition which is usually not satisfied experimentally hot electrons in semiconductors, including a finite e-e - - the displaced Maxwellian had a great success as it collision time ze~ and we show h o w f ( k ) evolves towgives, after rather simple algebra, most of the trans- ards a displaced Maxwellian when re-~ decreases. We port properties of hot electron systems. An analytical pay particular attention to the treatment of the e ph solution has also been obtained neglecting e-e colli- collision integral (to remedy the defect of W.S.W. sions [2]: this case corresponds to an extremely large calculation) and we mainly focus on non polar matee~e collision time r ~ , while for the displaced Max- rials. We show that, while the isotropic part o f f ( k ) is wellian, ze~ ~ 0 is implied, ze-~depends on the carriers essentially unchanged by e-e collisions, the anisotropdensity n: the larger n, the smaller Te_~. For usual ic part o f f ( k ) varies from E . k to E'k/k when the doped materials, ~_~ is neither very large nor very effect of e-e collision decreases. But this modification small so that a solution of the Boltzmann equation for o f f ( k ) induces very tiny changes on physical quanfinite ze_~is needed. Although this is a quite old pro- tities such as the electron current and temperature in blem, up to now only numerical answers have been agreement with the results of the numerical methods; given using Monte-Carlo methods [3, 4]. The intro- this explains a posteriori the success of the displaced duction of e-e collisions is technically difficult and Maxwellian procedure which is, in usual experimental requires powerful computers and very long calcula- situations, never valid as e-ph collisions are always tions, due to non linearity problems [4] and long range faster than e-e ones. We also consider the case of polar potential associated to it: the numerical resolutions optical phonons and show where it differs from non polar materials. are not yet totally free from approximations [5]. An analytical solution of the Boltzmann equation Many analytical resolutions of the Boltzmann at intermediate Te~ is still today highly valuable as it equation use an expansion in powers of the electric would allow a physical understanding of the exact role field E. While it is justified in the Ohmic (low-field) played by carrier-carrier collisions in hot electrons limit, there is no ground for it in the nonlinear high systems. Recently Wingreen, Stanton and Wilkins [6] field regime. Here we demonstrate that, in most expe(W.S.W.) tried such an analytical approach and have rimental cases, there is a small parameter through the 1221
E F F E C T OF E L E C T R O N - E L E C T R O N COLLISIONS
1222
entire range of electric field, a fact which has been overlooked up to now. This parameter is the modulus of a dimensionless vector 8, parallel to E: e can be seen as the energy eE2 gained by an electron over a characteristic length 2 (depending on the e-ph scattering) divided by the electron thermal energy ks T*: for non polar materials in high-field, T* increases as E 2, so that e decreases as 1/E. Therefore, for large E, we can indeed proceed to an expansion in e: this is at the basis of our analytical solution; quite generally, this expansion preserves non linearity in E. In part 1, we rewrite the Boltzmann equation in a dimensionless form: all the physical quantities appear inside two parameters e and z* ( ,-, Te_e/Ze_ph). We then show that e < 1: this is the crucial point for the mathematical resolution of the reduced Boltzmann equation done in part 2. The results are given in part 3.
Vol. 65, No. 10
wellian distribution, the corresponding function F(K) is (1 -- 2K'2/3) -3/2
FDM(K) =
× e x p [ - I K - K'I:/(1 - 2K'2/3)], (7a) FoM(K) ~
e x p ( - K 2 ) [ 1 + 2K.K* + 2((K'K*) 2
- K2K*2/3)) + . . . ]
if K* ,~ 1. (7b)
Let us now consider the Boltzmann equation, and start by the e-ph collisions integral. We rewrite it in dimensionless form using our reduced quantities, Ie-ph (k) = Ie-ph (K)/Ze-ph where (2rt) -~ I d 3 K ' [ - F ( K ) M ( K , K')
Ie ph(K) =
+ F(K')M(K', K)], 1. R E D U C E D B O L T Z M A N N E Q U A T I O N
M(K,K')
The problem is to determine the electron distribution f ( k ) for large electric field E, including e-ph and e-e collisions. We introduce the average wavevector k* and the effective temperature T*, defined as nk* =
2(2a) -3 fd3kf(k)k,
(3/2)nks T* = 2(2zc)-3 yd3kf(k)htk2/2m,
(1) (2)
n is the e density. T*, which reduces to the e temperature in case of a Maxwellian distribution, can be defined even if f (k) is not related to any temperature. Although yet unknown, T* is used to define reduced momentum K and distribution function F(K) K2 = h2k2/2mksT * =
k222r.,
(413gI/2)(T.IT*)312F(K),
f(k)
=
ksT,
= fiz(31r2n)2/3/2m,
(3) (4) (5)
T, is a measure of the e density n in term of a temperature. Physically, it is the temperature below which the plasma becomes degenerate. The momentum K and the function F are now expected to be of the order of 1. More precisely F(K) satisfies, due to equations (1) and (2) and the fact that the e density is n
X3/2[1 or K* or 3/2] = fd3KF(K)[1 or K or K2].
(6) F(K) depends only on one parameter, the reduced momentum K* (from k*) which characterizes its anisotropic part, its isotropic part being always of the order of 1. In the particular case of a displaced Max-
=
(Sa)
IK - K'Is[nLf(K 2 + D . o - K a) + (nL + 1)6(K'2 + f l 0 -
K2)],
(Sb)
D.o = fie)o/ks T* is the reduced optical phonon energy. nL is the phonon occupation number, s = - 2 or 0 depending if the material is polar or non polar. All the physical parameters of the e-ph collision integral appear in 27e_ph 1/ze_ph = fi '[2/3n3/2][Ao/2~SksT*](T./T*) 3/2,
(9)
=_ h-lksT*[2/3rt3/2](T./T*)3/2 x (Te_ph/T*) °-')/2,
(10)
the thermal wavelength 2r, is defined from equation (3). The constant A0 comes from the usual e-ph coupling 2Ma-lA0; in equation (10) we re-express it in terms [8] of a temperature like parameter Te_p,. For Ge(s = 0) we find [9]: T~_ph = 0.56 K and for GaAs (s = - 2 ) : Te., = 370°K. Similarly we rewrite the LHS of the Boltzmann equation V j ( k ) . ( - e E / h ) = VKF(K)'n/r,~h, the reduced electric field being =
2E2r._ph ( T* ,~-~/2 <--phi '
(11)
2r~_p, is the thermal wavelength for T.,_ph. e can be interpreted as the ratio of the energy gained by an electron in the electric field E over the characteristic length 2r,_ph divided by k s T * for non polar optical phonons (s = 0) or by some e-ph characteristic energy ks T~_phfor polar optical phonons (s = - 2). It is crucial for the analytical resolution of the Boltzmann equation, to note that (fortunately) this reduced field [10] ~ is smaller than 1 in usual physical cases: for
Vol. 65, No. 10
E F F E C T OF E L E C T R O N - E L E C T R O N COLLISIONS
polar phonons s = - 2 and e increases linearly with E; but even for a field of E ,-~ 5 KV cm -~, e is not larger than 0.3 in GaAs. For non polar phonons, s = 0 and e ~ E/T*: for E_-__ I KV cm -~ and T* ~ 600 K, one find e ~ 0.5 in Ge. But anticipating the conclusion that T* is the electron temperature which increases for hot electrons as E 2, e is indeed small for large field: for non polar optical phonons, e first increases as E for cold electrons and then decreases as 1/E for hot electrons, being maximum in the warm region but still small. For the e-e collision integral I¢_~, we follow W.S.W. and write it IN(k) = [ - F ( K ) + FDM(K)]/ z ~ . This preserves the most important features of e-e scattering, namely that the number of particles, momentum and energy of the e system are conserved in each collision, property of the displaced Maxwellian. W.S.W. obtained z~_¢by calculating the quadrupole moment [11]. One can also get z ~ directly by extracting [12] the T, and T* dependence from the e-e collision integral; one finds fl/%_e ~- kaTRyd(T,/T*)3/ZLnQ;I
~
nLn(n),
(12)
in agreement with W.S.W. result, kn TRyd is the Rydberg energy me4/2h2e 2 and Q~ the reduced screening wavevector [Q2 = (8/3rOT3n Tr~2o/T.Z].
2. R E S O L U T I O N FOR A F I N I T E e-e COLLISION TIME We are finally left with the resolution for small of the following reduced Boltzmann equation VF(K).n =
I~_ph(K) + [--F(K) + FDM(K)]/Z*,
1223
+ I [F(K'2)JQ'(K" K ) - F(K2)aI(K, K')] d3K ' , h,~I(K, K') being the angular average of M(K, K'). The second term of equation (l 5) disappears for non polar phonons as M = M (s being 0) while it remains for polar phonons: I~-ph contains in that case an integral operator for g(K) (which is of the order of I whatever is fl0) and this leads to a more complex resolution of the Boltzmnn equation. In the remaining part of this Communication, we will restrict to non polar optical phonons. Using equations (8b) and (1 5), the e-ph integral reads in this case, with K+ = ( K 2 + ~.o) j/2 Ie-ph(K) =
--(nLK+ + (nL + 1)K )g(K) + J(KZ), (16)
J(K 2) = K+ F(K 2) - K _F(K2) + nL K+ [F(K2+) - F(K 2)1 + nLK [F'(K2 ) -- F(K2)].
(17)
As we have shown that e is small for hot electrons, the Boltzmann equation (13), valid for all K, implies Ie_~ and I¢ph small. I N small means either z* large or, if z* is small, F close to FDM as expected. Ie-ph small implies J small: this happens only either if the e distribution F is a Maxwellian at the lattice temperature or if fl0 is small [7]: the first case corresponds to cold electrons (i.e. the Ohmic low-field limit). So that, in the hot electrons case, ff~0has to be small. J ( K 2) then reduces [13] to D.0K - ~ d ( K Z F ) / d K 2 . Ie-ph small implies also g(K) small (as K ~ 1). As its angular average is zero, an expansion ofg(K) in Legendre polynomials of e-K is then justified g(K) = 8 . K g j ( K 2) + [(g.K) 2 - e2K2/3]g2(K2 )
(13) + .... where FDM, Ie-ph and e are given in equations (7), (8) and (1 1); the parameter z* is simply 1/z* = Z~_ph/Z~ ,~, (TRyd/T*)(T,/T*) 3/2 X (T*/Te_ph) 0 s)/2LnQj-l.
(14)
The displaced Maxwellian procedure is justified if 1/r* is large. This happens only for large n, and small T*: one has then to conclude that the displaced Maxwellian is unjustified at high temperature i.e. for hot electrons... To go further, it turns out to be useful to separate F(K) from its angular average _P(K2) and write F(K) = F(K 2) + g(K); the e-ph integral equation (8) then reads 2~zle_ph( K )
=
-
with gl, & ~ 1. We are now in position to solve easily the Boltzmann equation (13) for all z*. This is what we do in the rest of this paragraph. Conditions (6) for the e distribution function imply F ,-~ 1 and ~z3/2K* =
(1/3)~ ~ K2gl(K2)d3K + O(e2) -
~3/2~8" (19)
This defines a coefficient ~ of the order of 1. We then use this value of K* (,~ 1) in the expression (7b) of FDM(K). The identification of the coefficients of the two first Legendre Polynomials [14] in the Boltzmann equation gives immediately eZ[gI + z~.2,q3,61J , = D.oK I(KZF)" - (F -- e-K2)/z *,
g(K) ~d3K'21~r(K, K')
+ Ig(K')[M(K', K) - 3~r(K', K)]d3K '
(18)
(20a) (15)
2F =
--(2nL + 1)Kgl -- (gl -- 2ae-k2)/z *,
(20b)
E F F E C T OF E L E C T R O N - E L E C T R O N COLLISIONS
1224
e and ~ being small, equation (20a) implies [15] for a finite z* that P = exp ( - K 2) to lowest order in e: this means that the isotropic part of the e distribution is a Maxwellian at T*. Equation (20b) gives the anisotropic part gl =
2(z* + ~)[(2nL + 1)Kz* + 1] lexp(--K2).
The coefficient ~ is determined self consistently from equation (19). One finds, with X = (2nL + 1)z*.
1 =
XG(X)
A(X)
(2nL + 1) G(0) - G(X) -
2nL + 1'
(21)
G(X) = Id3KK2e-X2(XK + 1) -I. A(X) is an increasing function of X which varies from A(0) = 3 ~ ~ 0.66 to A ( ~ ) = 4/3~/-~ ,~ 0.75, so that A(X) is essentially independent of X, i.e. ~ is independent of r*. The simplest way to get the reduced phonon energy ~0 to lowest order in e, is to calculate the second moment of equation (20a) as the part coming from e-e collisions cancels exactly. It is then straightforward to find f~0 0~7~1/2e2~-- /rl/Z~-lK*2, using equation (19).
Vol. 65, No. 10
A(0)8. K to 8. K/K. This variation will give only small changes in the averages appearing in most physical quantities as A(0) ,,~ 0.66 and the leading terms in these averages will come from K's of the order of 1, due to exp ( - K 2) in FOK). In conclusion, I have shown the existence in the Boltzmann equation of a small parameter which allows to obtain, in the case of non polar materials, analytical expressions for the electron distribution, electron temperature and electric current of electrons in a large electric field including a finite electronelectron collision time. I have demonstrated that their variations with this collision time are very very small. This explains why, although a priori unjustified because electron-electron collisions are essentially never faster than electron-phonon ones, the Displaced Maxwellian procedure has a great success.
Acknowledgements - - I wish to thank R. Combescot, J. Bok and N. Pottier for stimulating discussions.
=
REFERENCES
3. RESULTS Translating the above results in unreduced units, one finds for the temperature and drift momentum
1.
h2k .2 2m
2.
-
ksT*
-
3 A(z*(2nL + 1)) hog0 , 8(2nL + 1) A(0)
(22)
3.
3rr3 (eE2"r~-Ph)2A(z*(2nL + 1)), 2(2nL + 1) rio90 A(0)
4. 5.
(23) where the function A(X) is defined in equation (21). One recognizes the well-known results of the displaced Maxwellian method with some correction A(X)/A(O) coming from a finite z*. These corrections are very small as the function A(X) is essentially constant: A(X)/A(O) increases from 1 to 32/9n = 1.13. This means that the drift velocity slightly increases when e-e collisions become less effective. One also notes that the other well-known result h2k*2/2m = (4/3.q/-~)h~o0/ (2nL + 1), obtained by neglecting e-e collisions [2], is simply the limit of equation (22) for z* infinite. The electron distribution reads to lowest order in e:
6. 7.
8. 9. 10.
F(K) = e-~[1 + (2nL + l ) 8 . k X + A(x) kx+ 1 + . . . ],
(27)
where X = (2nL + 1)z*: the main term e x p ( - K 2) is a Maxwellian with an electron temperature T*. The anisotropic part varies, with increasing z*, from
11.
M. Fr61ich, Proc. Roy. Soc. A188, 521 (1947); M. Fr61ich & B.V. Paranjape, Proc. Roy. Soc. B69, 21 (1956). E.M. Conwell, Solid State Physics 9 Academicques. G. Bauer, Solid State Physics. p. 1, Springer Tracts (1974). C. Jacoboni & L. Reggiani, Rev. Mod. Phys. 55, 645 (1983). As an example the final state of the collision is chosen randomly according to a Displaced Maxwellian (see [4]). N.S. Wingreen, C.J. Stanton & J.W. Wilkins, Phys. Rev. Lett. 57, 1084 (1986). For large E, e-ph collisions do not relax f ( k ) toward a Maxwellian at the lattice temperature but instead leads to f~0 small (see our equation (17)). Ao - 23+~(ksT*)2 (Te_ph/T*)~l-s)/2 is indeed a constant independent of T*. We used the values of the e-ph coupling constants given in K. Seeger, Semiconductors Physics, Springer. One may think that the numerical value obtained for e is somewhat arbitrary as more (2r0's can be included in the definition of I~-ph. Our choice of the prefactor of I~-phis in fact such that no numerical factor appears in its final expression equation (20) so that e contains indeed all the constants. See also M. Combescot & R. Combescot, Phys. Rev. B35, 7986 (1987) and M. Combescot, R. Combescot, Phys. Rev. Lett. 59, 375 (1987).
Vol. 65, No. 10 12. 13.
14.
E F F E C T OF E L E C T R O N - E L E C T R O N COLLISIONS
M. Combescot, (preprint). For polar optical phonons, the same argument would also apply; so that fl01nfl0 ~ e2, g(K) ,~ ~. KG(KQ and fl0 In D.0 ~ e2 as the term corresponding to the last bracket of equation (17) is of the order of f",.oLnO, o for f~0 small. This implies nL~o "~ 1. If not, one can keep the t e r m nL~'~o .
15. 16.
1225
The term (2/3) e)K2g't in equation (20a) comes from restoring the P2 symetry in the equation for g2(K2). For infinite T*, equation (20a) and (20b) leads to a second order differential e~uation, the solution of which is F = e x p ( - K ) with f~0 = 452/3 (nL + 1), due to equation (6): this is precisely our limiting value for r* ~ oo.