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Theoretical analysis of the surface-polar optical phonon limited distribution function of hot electrons in GaAs-quantum wells
Solid State Communications, Vol. 65, No. 10, pp. 1241-1246, 1988. Printed in Great Britain.
0038-1098/88 $3.00 + .00 Pergamon Press plc
THEORETICAL ANALYSIS OF THE SURFACE-POLAR OPTICAL PHONON LIMITED DISTRIBUTION FUNCTION OF HOT ELECTRONS IN GaAs-QUANTUM WELLS C. Kiener, G. Zandler and E. Vass Institut fiJr Experimentalphysik, University of Innsbruck, A-6020 Innsbruck, Austria
(Received 30 October 1987 by B. Miihlschlegel) The steady state nonequilibrium distribution function of quasi two-dimensional (2D) electrons confined to move in a square n-GaAs quantum well is determined from the Boltzmann integral-equation, which is solved iteratively for the interaction of the charge carriers with polar optical phonons taking into account electron-electron collisions. The carrier concentration, field and temperature dependence of the distribution function is investigated systematically for the first time.
1. INTRODUCTION THE DISTRIBUTION function of hot carriers in the bulk of semiconductors has been investigated extensively in terms of the Boltzmann transport equation. To solve this equation different analytical and numerical techniques have been used [1-3]. In [1, 4] the distribution function was expanded in Legendre polynomials, in order to transform the integro-differential equation into an infinite system of differential equations for the expansion coefficients. In this approach it is usually assumed, that only the first two terms of this expansion are essential, implying the drift velocity of the electrons in the external electric field to be much smaller than the thermal velocity of the charge carriers. A lot of theoretical investigations concerning hot carrier distributions are bas.ed on the electrontemperature model [5, 6], in which the distribution function is approximated by a hot drifted Maxwell Boltzmann or Fermi-Dirac function. In this model it is assumed that the electron-electron collisions take place more frequently than those between electrons and phonons. This approximation is therefore only valid if the carrier concentration is sufficiently high. The most frequently used numerical technique to calculate the exact distribution function is the Monte Carlo method [7]. In this method the motion of the charge carriers is simulated by random numbers. In order to achieve sufficiently accurate results for the transport quantities a large number of simulations is necessary, which makes this method very time consuming. Numerous theoretical works have been presented in the past in which the electron--electron scattering was included phenomenologically into the Monte Carlo analysis [8-10], whereby a hot drifted I
Maxwell-Boltzmann distribution was used to calculate the electron-electron scattering rate. This approach is however unsatisfactory, since the existence of an electron-temperature parameter is postulated "a
priori". The above described methods to determine the hot carrier distribution function have been applied also to 2D electronic systems such as n-Si inversion layers [11, 12] as well as GaAs-heterostructures [1315]. In [16] the 2D-Boltzmann integro-differential equation was transformed to an integral equation similar to the 3D case [17, 18], and was applied for the first time to calculate the energy relaxation time in GaAs-heterostructures. However no detailed analysis of the distribution function has been presented. A hybrid technique based on the 2D integralequation method combined with a Monte Carlo simulation was used to calculate the hot electron distribution function in GaAs [19]. In order to simplify the calculation, the method of selfscattering [20] was applied. Furthermore the 2DEG was considered to be nondegenerate. In this Communication the influence of the external electric field, the carrier concentration and the lattice temperature onto the distribution function of hot carriers in a n-GaAs square quantum well is investigated systematically for the first time, when the charge carriers interact with polar optical phonons. In the calculation the electron-electron interaction is taken into account without any unphysical simplification of the corresponding collision term. 2. THEORY In our calculation we start from the Boltzmann integral equation [16], which can be written as
1241
HOT ELECTRONS IN GaAs-QUANTUM WELLS
1242 f(k)
;(e)[; ( e)] dtK
=
k -~Et
exp
-
0
x).
k - ~Et'
dt' 0
(1)
,
whereby the exponential factor describes the probability that the electron is not scattered during the time interval [0, t]. E is the electric field applied parallel to the 2DEG. K and 2 are given by
Vol. 65, No. 10
where .4 is the area of the 2D system and a;q is the phonon angular frequency. The upper (lower) sign is for emission (absorption) of a polar optical phonon. In order to evaluate the matrixelements, the 2D electron-wavefunction is taken to be [21]
(tkr)'~(z).
~(k) = A J/2 exp
(8a)
The envelope function ~(z) is defined by
K = Kpo + K,,~, where d is the width of the quantum well. With (8) the interaction matrixelements for polar optical phonons and electrons are calculated to be [16, 22, 23]
with
K~o(k)
=
/
d ~ 'k W0o(k' , k)f(k'),
1 e2ksTD.(l
(2)
~) (
1
~)
BZ
and K~ (k)
X
= fffd2k'd2k2d2k'EW~e(k',k'2;k, k2)
IF ~ P ° k / ]
(9)
" ~k,k' ~ q '
BZ
× f ( k ' ) f ( k ~ ) I1 - f(k2)],
being the kernels for the polar optical phonon and electron-electron scattering. The scattering rates for both mechanism are defined by )~po(k)
=
~ d 2k, {Wpo(k, k)f(k')
d BZ
+ Wpo(k, k')[l - f(k')]}, 2~(k)
= fff d2k'd2k'2{W~(k ', k~; k,
Fee
(3)
(4) k2)
BZ
X
(10)
6k+lt2,k,+k ~ .
Nq is the phonon occupation number which is replaced in the following by the Planck-function, e0 is the permittivity of the free space, e~ and e~ are the static and optical dielectric constants. To denotes the Debye temperature and fl is the screening length in the random phase approximation [21]. The formfactors F~(i = po, ee) are defined by
× f(k')f(k~)[1 - f(k2)] gpo =
+ W~(k, k2; k', k~)f(k2)
i dzl((z)[ 2' e -q'z 2,
(1 la)
0
× [1 - f(k~)][1 - f(k')]}.
(5)
In (2) and (4) k and k' are the wavevectors of an electron before and after the collision with a polar optical phonon. In (3) and (5) the initial and final wavevectors are denoted by k, k2, and k', k2. The (1 - f ) - f a c t o r s are due to the Pauli-principle. All integrations are carried out over the 2D-Brillouin zone (BZ). The transition probabilities per unit time are defined by A 27r Wpo(k, k') = (2~z)2 h IMPS'J2
x 6(ek, - ek +__hwq), W~(k, k2; k', k~) =
(6)
(~2_~12 --fl--}M~k2g%l 2~ ~ 2 \-
- /
x ~(e~, + ek~ - ek - %), (7)
Fee =
tis
dz dz2l~(z)[2.l~(ze)12.e qqz-z21 ,
(lib)
0
where q = k - k' is the scattering vector. To reduce the number of integrations in equations (2)-(5), we introduce polar coordinates and integrate over the delta-functions with respect to the absolute value of the electron wavevectors. In the case of electron-electron scattering all integrations are carried out in the center of mass coordinate system [23] defined by k k2
= =
G + ½g
G-
½g
k'
=
k~ =
G + ½g'
G_
(12)
~gl',
where G is the center of mass wavevector, g and g' are the corresponding relative wavevectors.
having the energy of the optical phonon. ~0, ~0', q~, ~O are the polar angles between the vectors k, k', g, g' and the electric field E. The angle between the collisionfree trajectory k - e ~ Et and E is denoted by
As a result we obtain 2n
Kv°(k(t))
=
1243
HOT ELECTRONS IN GaAs-QUANTUM WELLS
Vol. 65, No. 10
I dqY[(N~q + 1)Spo(k~(t), qg') 0
+ N°Spo(kE(t), q~')f(ke(t), q¢)], Z(t) = K~(k(t)) =
dg'g'
dO
0
0
k(t)
dffSe~(g', O, ~b) 0
1
t
1
1 x [1 - f ( I G - ~gl, Ol)], 2n
f dtp'{Spo(ka(t), q~')[_f(kA(t), ¢p') 0
+ N~q] + Spo(kE(t), go') x [N o + 1 - f ( k E ( t ) , dg'g'
2~e(k(t)) = 0
dO 0
go')]},
d~bSCe(g',O, ~)
[24].
0
1 t x {f(lG + ~g I, ~kl)[f(lG - ~1g , I, ~ )
-f(lG
- ½gl, O1)] + f ( I G - ½gl, O,)
x [1 -- f ( I G
_
1
t
~g 1, ~2)]}
where Ol, ~l and Ip2 a r e the angles between E and k2, k' and k~ defined by (12). The functions Spo and S~ are defined by
Spo(k,(t), qY) =
ekT°8o (1
1)m* h
[kZ(t) + k~(t) - 2k(t)ki(t)cos(z(t ) - tp')] '/2 (i= A,E) See(g', O, ~) -
1 ( e2 )2m*o (4~) 3 \27teoes,] ~ {[lg,Z X
(l
--
C O S ((I) - -
@))]1/2
..[_ j~}
2.
The zeros of the delta-function in (6) are given by
kA(t) = [k2(t) + x2] 1/2
(for absorption)
kE(t) = [kZ(t) - x~]1/2
(for emission),
where k2(t) =
Ik(t)l z = Ik --
e
"
The solution of the integral equation (1) is performed iteratively. In the first step of the iteration procedure we choose an arbitrary test-function f0(k) which is substituted into the kernel K as well as the total scattering rate 2 on the right hand side of (1). In the next step we generate the functionf(k) by evaluating numerically all integrals of the right hand side of equation (1). After then f ( k ) is resubstituted on the right hand side of (1). The procedure is then repeated until convergence is achieved. The convergence of this iteration method has been proved analytically in ref.
r
x f ( I G + ~g l, ~b~)f(IG - ~g l, ~q)
2po(k(t)) =
arcsin(--k'sinq~.)
Et2,
3. N U M E R I C A L RESULTS AND DISCUSSION In this section the above described iterative procedure is applied to solve (1) for different values of the external controllable parameters E (electric field), Ns (carrier concentration) and T (lattice temperature). In the calculation the occupation of the lowest (ground-state) subband is assumed. Intervalley transitions are neglected in order to work out the essential novel features of the method. In the analysis the following numerical values of the material parameters were used: m* = 0.0665.m0, es = 12.9, e~ = 10.9, To = 420 K.
3.1 Convergence of the iteration procedure Figure 1(a) shows that any arbitrary trial function f0(k) leads to the same stable steady state solution provided the kernel of (1) remains positive definite in agreement with [24]. Figure l(b) demonstrates the quickness of the iterative procedure to solve (1) by means of a Gaussian trial function (represented by the dotted line). It can be seen that the distribution functions obtained after the third and sixth iteration nearly coincide showing that the exact distribution can be calculated already after a small number of iterations with high accuracy (If,+t - f,[ ~< 10-3, for n - 6), which cannot be achieved by the standard Monte Carlo methods [8-11] within the same computation time.
and
[2m* /~d
=
~'--~'-
"X112 h(/)q)
,
(13)
is the magnitude of the wavevector of an electron
3.2 Electric field, carrier density and lattice temperature dependence of the distribution function In Figs. 2(a)-(c) the distribution function is plotted vs the electron wavevector for different values of
1244
HOT ELECTRONS IN GaAs-QUANTUM WELLS .-,,
1.0
1 st
....
initial
iterated
....
DF DF
2 ndinltlal
--
iterated
DE
N s
=
2.5
x
T
=
100
K
E
=
500
V/cm
.
,,-<....~" .
.
-5
10L~cm-2
1.0
.
.
.
.
.
77"::.,.
---
E
=
1000
V/cm
.....
E
- 2000
V/cm
0 k [ 1 0 6 c m -I]
.....
initial
i st
O.
5
N s
= 2.5
iteration
T
=
100
K
- - - 3 rd
iteration
E
=
500
V/cm
--
iteration
6 th
DF
x
x
~ -5
1 0 ~ c m -2
...... 5
0 k [ i oBcITI -i ]
Fig. 2(a). Field dependence of the hot carrier distribu. tion function.
l.O
--Ns
=
--- N s
= 2.5x
l. O x l O 1 t c m -z 1011crn-2
5.Ox
T
=
100K
E
=
500
V/cm
1 0 " c m /'" -2 ........,, ..
l O " c m "2 :
\
: / ~
~H
= 2.5
= 100K
.,f+..I+-""
: .... N s =
....
N s
T
t',
....
"
Fig. 1(a). Surface polar optical phonon limited steady state distribution function of hot electrons in a GaAs square quantum well vs the magnitude of the electron wavevector parallel to the direction of the external electric field E calculated with different trial functions.
1.O
5oov/~
.s
N
.
=
--Z
DF
,'/"
Vol. 65, No. 10
.5
~ ' :/]"
Ii
,/ I I i :
.5
/ ." // ," /
O.
, .........
-5
0
"L !, I "'.
/
5
Fig. l(b). Initial, first-, third- and sixth-iteration of the distribution function. the external electric field, the carrier density as well as the lattice temperature. The presented curves are intersection lines of the two-dimensional distribution function f ( k ) with a plane parallel to the direction of E. By means of these figures we come to the following conclusions: (1) For given values of N+, T and E the polar optical phonon limited distribution function shows a maximum at a certain k-value in the direction of the external electric field. According to Fig. 2(a) the position of this maximum is shifted to higher k-values i f E increases. Simultaneously the shape of the distribution function becomes more and more asymmetric while the height decreases continuously due to the increase of the average carrier energy. (2) At k = where is defined by (13) the distribution functions show a kink (indicated by the arrow in Fig. 2(a)) which is due to the onset of the phonon emission process. The position of this kink
Xd
xa
r
,
....
£,...
~"
-5
> w ......
0 k [ 1 06cm -~]
.
,
0 k [ i 0 6 c m -~]
5
Fig. 2(b). Carrier concentration dependence of the distribution function.
i.@
--T
-
--- T
= 200
i00 K K
..... T
=
K
300
~q-a. 5
i/
N s
=
2.5
x
E
=
500
V/on
1 0 ~ o n -2
%
• .,.",/./l"
.
-5
0 k [ 106cm -: ]
5
Fig. 2(c). Lattice temperature dependence of the distribution function. remains fixed if Ns, T or E are changed since the activation energy of the polar optical phonons involved in the scattering process is constant. In the range the distribution function is mainly determined by the absorption process. (3) According to Fig. 2(b) the area below the
k < Xd
HOT ELECTRONS IN GaAs-QUANTUM WELLS
Nol. 65, No. 10
distribution function increases with increasing values of N, and approaches to a Fermi-Dirac like distribution as a consequence of the Pauli-principle. (4) The shape of the distribution function is strongly altered if the lattice temperature T increases. This behaviour can be attributed to the T-dependence of the phonon occupation number which exponentially increases with T.
3.3 Influence of the electron-electron scattering Figures 3(a,b) show the influence of the electronelectron scattering onto the distribution function for two different carrier concentrations. We observe that this additional interaction mechanism significantly changes the distribution function due to the enhanced randomization of the electron momenta. Three different effects can be seen: (i) The shape of the distribution function becomes more symetric. (ii) The polar optical phonon kink is flattened out. (iii) The drift of the distribution function caused by the external electric field is less pronounced than the drift of the distribution function without electron-electron scattering taken into account.
1.0
q-" .5
N s
-
5 x
1 0 * * c m "2
T
=
100
K
E
=
500
V/cm
(~) /
--
with
[. . . . .
:i~
e-e
without
According to (9) and (10) the po-phonon interaction matrixelement decreases proportional to Ik' - k[-WE. On the other hand the electron-electron interaction matrixelement is ~lk' - kl -~ • This implies that the electron-electron interaction is more effective in the wave-vector range k < Xd (indicated by the arrow in Fig. 3(a)), while for wave-vectors k > Xd the distribution functions calculated with and without electron-electron scattering nearly coincide. As the carrier concentration increases the influence of the electron-electron collisions onto the distribution function is compensated by the Pauli-principle. Therefore we observe that both distributions shown in Fig. 3(b) have a similar shape in the whole wavevector range. Moreover we can conclude that the distribution function remains asymetric even when electron-electron scattering is taken into account. In a forthcoming paper the influence of the formfactors as well as the screening will be studied in more detail by means of the integral equation method. It will be shown that both effects have to be taken into account absolutely in the calculation of the distribution function and the drift velocity in real 2D systems.
scarf. e-e
scatt.
Acknowledgements - - The authors are grateful to Prof. Dr. E. Gornik for helpful comments. This work was supported by the Fond zur F6rderung der wissenschaftlichen Forschung in Austria; Project No. P6128P.
"'"i, "~
REFERENCES 1.
.
-5
N~
1.0
0 k [ i 06cm -~ ] -
2.5 x
l O ~ c m -2
TE -= 500100V/craK
--
with
5
e-e scatt.
..... wl thout e-e scatt
2. 3. 4. 5. 6. 7. 8.
.5
9. .
-5
0 k [ i 0Scm -~ ]
1245
5
Fig. 3. Influence of the inter-carrier collisions onto the polar optical phonon limited hot carrier distribution function for carrier concentration Ns = 2.5.10" cm -2 (a) and for Ns = 5"10~cm -2 (b).
10. 11. 12.
E.M. Conwell, High Field Transport in Semiconductors (Academic Press, New York) (1967). P.J. Price, IBM J. Res. Dev. 14, 12 (1970). R. Kubo, Acta Physica Austriaca, Suppl. X, 301 (1973). G.A. Baraff, Phys. Rev. 133, A26 (1964). K. Seeger, Semicond. Phys. p. 734, Springer, Wien (1973) and references cited therein. G. Bauer, Solid State Physics, Springer Tracts in Modern Physics Vol 74, Springer, Berlin (1974). C. Jacoboni & L. Reggiani, Rev. Mod. Phys. 55, 645 (1983) and references cited therein. R. Brunetti, C. Jacoboni, A. Matulonis & Dienys, (Proc. Int. Conf. Hot Electrons in Semiconductors, Innsbruck, 1985) Physica 134 B + C, 396 (1985). N. Takenaka, M. Inoue & Y. Inuishi, J. Phys. Soc. Japan 47, 861 (1979). M. Inoue & J. Frey, J. Appl. Phys. 51, 4234 (1980). K. Hess & C.T. Sah, Phys. Rev. BI0, 3375 (1974). K. Hess, Physics of Nonlinear Transport in Semiconductors, Nato Advanced Study Institut Series B52, 1 (1980).
1246 13. 14. 15. 16. 17.
HOT ELECTRONS IN GaAs-QUANTUM WELLS P.K. Basu, K. Bhattacharyya & B.R. Nag, Solid State Commun. 48, 981 (1983). D. Chattopadhyay, Solid State Commun. 62, 395 (1987). P. Lugli & S.M. Goodnick, Phys. Rev. B59, 716 (1987). E. Vass, (Proc. Int. Conf. Hot Electrons in Semiconductors, Innsbruck, 1985) Physica 134 B+C, 337 (1985). R.G. Chambers, Proc. Phys. Soc. A65, 458 (1952).
18. 19. 20. 21. 22. 23. 24.
Vol. 65, No. 10
H. Budd, Phys. Rev. 158, 798 (1967). J,B. Socha & J.A. Krumhansl, (Proc. Int. Conf. Hot Electrons in Semiconductors, lnnsbruck, 1985) Physica 134 B+C, 142 (1985). H.D. Rees, J. Phys. Chem. Solids 30, 643 (1969). T. Ando, A.B. Fowler & F. Stern, Rev. Mod. Phys. 54, 437 (1982). Y. Takada & Y. Uemura, J. Phys. Soc. Japan 43, 139 (1977). S.K. Lyo, Phys. Rev. B34, 7129 (1986). M.O. Vassel, J. Math. Phys. 11, 408 (1970).
0038-1098/88 $3.00 + .00 Pergamon Press plc
THEORETICAL ANALYSIS OF THE SURFACE-POLAR OPTICAL PHONON LIMITED DISTRIBUTION FUNCTION OF HOT ELECTRONS IN GaAs-QUANTUM WELLS C. Kiener, G. Zandler and E. Vass Institut fiJr Experimentalphysik, University of Innsbruck, A-6020 Innsbruck, Austria
(Received 30 October 1987 by B. Miihlschlegel) The steady state nonequilibrium distribution function of quasi two-dimensional (2D) electrons confined to move in a square n-GaAs quantum well is determined from the Boltzmann integral-equation, which is solved iteratively for the interaction of the charge carriers with polar optical phonons taking into account electron-electron collisions. The carrier concentration, field and temperature dependence of the distribution function is investigated systematically for the first time.
1. INTRODUCTION THE DISTRIBUTION function of hot carriers in the bulk of semiconductors has been investigated extensively in terms of the Boltzmann transport equation. To solve this equation different analytical and numerical techniques have been used [1-3]. In [1, 4] the distribution function was expanded in Legendre polynomials, in order to transform the integro-differential equation into an infinite system of differential equations for the expansion coefficients. In this approach it is usually assumed, that only the first two terms of this expansion are essential, implying the drift velocity of the electrons in the external electric field to be much smaller than the thermal velocity of the charge carriers. A lot of theoretical investigations concerning hot carrier distributions are bas.ed on the electrontemperature model [5, 6], in which the distribution function is approximated by a hot drifted Maxwell Boltzmann or Fermi-Dirac function. In this model it is assumed that the electron-electron collisions take place more frequently than those between electrons and phonons. This approximation is therefore only valid if the carrier concentration is sufficiently high. The most frequently used numerical technique to calculate the exact distribution function is the Monte Carlo method [7]. In this method the motion of the charge carriers is simulated by random numbers. In order to achieve sufficiently accurate results for the transport quantities a large number of simulations is necessary, which makes this method very time consuming. Numerous theoretical works have been presented in the past in which the electron--electron scattering was included phenomenologically into the Monte Carlo analysis [8-10], whereby a hot drifted I
Maxwell-Boltzmann distribution was used to calculate the electron-electron scattering rate. This approach is however unsatisfactory, since the existence of an electron-temperature parameter is postulated "a
priori". The above described methods to determine the hot carrier distribution function have been applied also to 2D electronic systems such as n-Si inversion layers [11, 12] as well as GaAs-heterostructures [1315]. In [16] the 2D-Boltzmann integro-differential equation was transformed to an integral equation similar to the 3D case [17, 18], and was applied for the first time to calculate the energy relaxation time in GaAs-heterostructures. However no detailed analysis of the distribution function has been presented. A hybrid technique based on the 2D integralequation method combined with a Monte Carlo simulation was used to calculate the hot electron distribution function in GaAs [19]. In order to simplify the calculation, the method of selfscattering [20] was applied. Furthermore the 2DEG was considered to be nondegenerate. In this Communication the influence of the external electric field, the carrier concentration and the lattice temperature onto the distribution function of hot carriers in a n-GaAs square quantum well is investigated systematically for the first time, when the charge carriers interact with polar optical phonons. In the calculation the electron-electron interaction is taken into account without any unphysical simplification of the corresponding collision term. 2. THEORY In our calculation we start from the Boltzmann integral equation [16], which can be written as
1241
HOT ELECTRONS IN GaAs-QUANTUM WELLS
1242 f(k)
;(e)[; ( e)] dtK
=
k -~Et
exp
-
0
x).
k - ~Et'
dt' 0
(1)
,
whereby the exponential factor describes the probability that the electron is not scattered during the time interval [0, t]. E is the electric field applied parallel to the 2DEG. K and 2 are given by
Vol. 65, No. 10
where .4 is the area of the 2D system and a;q is the phonon angular frequency. The upper (lower) sign is for emission (absorption) of a polar optical phonon. In order to evaluate the matrixelements, the 2D electron-wavefunction is taken to be [21]
(tkr)'~(z).
~(k) = A J/2 exp
(8a)
The envelope function ~(z) is defined by
K = Kpo + K,,~, where d is the width of the quantum well. With (8) the interaction matrixelements for polar optical phonons and electrons are calculated to be [16, 22, 23]
with
K~o(k)
=
/
d ~ 'k W0o(k' , k)f(k'),
1 e2ksTD.(l
(2)
~) (
1
~)
BZ
and K~ (k)
X
= fffd2k'd2k2d2k'EW~e(k',k'2;k, k2)
IF ~ P ° k / ]
(9)
" ~k,k' ~ q '
BZ
× f ( k ' ) f ( k ~ ) I1 - f(k2)],
being the kernels for the polar optical phonon and electron-electron scattering. The scattering rates for both mechanism are defined by )~po(k)
=
~ d 2k, {Wpo(k, k)f(k')
d BZ
+ Wpo(k, k')[l - f(k')]}, 2~(k)
= fff d2k'd2k'2{W~(k ', k~; k,
Fee
(3)
(4) k2)
BZ
X
(10)
6k+lt2,k,+k ~ .
Nq is the phonon occupation number which is replaced in the following by the Planck-function, e0 is the permittivity of the free space, e~ and e~ are the static and optical dielectric constants. To denotes the Debye temperature and fl is the screening length in the random phase approximation [21]. The formfactors F~(i = po, ee) are defined by
× f(k')f(k~)[1 - f(k2)] gpo =
+ W~(k, k2; k', k~)f(k2)
i dzl((z)[ 2' e -q'z 2,
(1 la)
0
× [1 - f(k~)][1 - f(k')]}.
(5)
In (2) and (4) k and k' are the wavevectors of an electron before and after the collision with a polar optical phonon. In (3) and (5) the initial and final wavevectors are denoted by k, k2, and k', k2. The (1 - f ) - f a c t o r s are due to the Pauli-principle. All integrations are carried out over the 2D-Brillouin zone (BZ). The transition probabilities per unit time are defined by A 27r Wpo(k, k') = (2~z)2 h IMPS'J2
x 6(ek, - ek +__hwq), W~(k, k2; k', k~) =
(6)
(~2_~12 --fl--}M~k2g%l 2~ ~ 2 \-
- /
x ~(e~, + ek~ - ek - %), (7)
Fee =
tis
dz dz2l~(z)[2.l~(ze)12.e qqz-z21 ,
(lib)
0
where q = k - k' is the scattering vector. To reduce the number of integrations in equations (2)-(5), we introduce polar coordinates and integrate over the delta-functions with respect to the absolute value of the electron wavevectors. In the case of electron-electron scattering all integrations are carried out in the center of mass coordinate system [23] defined by k k2
= =
G + ½g
G-
½g
k'
=
k~ =
G + ½g'
G_
(12)
~gl',
where G is the center of mass wavevector, g and g' are the corresponding relative wavevectors.
having the energy of the optical phonon. ~0, ~0', q~, ~O are the polar angles between the vectors k, k', g, g' and the electric field E. The angle between the collisionfree trajectory k - e ~ Et and E is denoted by
As a result we obtain 2n
Kv°(k(t))
=
1243
HOT ELECTRONS IN GaAs-QUANTUM WELLS
Vol. 65, No. 10
I dqY[(N~q + 1)Spo(k~(t), qg') 0
+ N°Spo(kE(t), q~')f(ke(t), q¢)], Z(t) = K~(k(t)) =
dg'g'
dO
0
0
k(t)
dffSe~(g', O, ~b) 0
1
t
1
1 x [1 - f ( I G - ~gl, Ol)], 2n
f dtp'{Spo(ka(t), q~')[_f(kA(t), ¢p') 0
+ N~q] + Spo(kE(t), go') x [N o + 1 - f ( k E ( t ) , dg'g'
2~e(k(t)) = 0
dO 0
go')]},
d~bSCe(g',O, ~)
[24].
0
1 t x {f(lG + ~g I, ~kl)[f(lG - ~1g , I, ~ )
-f(lG
- ½gl, O1)] + f ( I G - ½gl, O,)
x [1 -- f ( I G
_
1
t
~g 1, ~2)]}
where Ol, ~l and Ip2 a r e the angles between E and k2, k' and k~ defined by (12). The functions Spo and S~ are defined by
Spo(k,(t), qY) =
ekT°8o (1
1)m* h
[kZ(t) + k~(t) - 2k(t)ki(t)cos(z(t ) - tp')] '/2 (i= A,E) See(g', O, ~) -
1 ( e2 )2m*o (4~) 3 \27teoes,] ~ {[lg,Z X
(l
--
C O S ((I) - -
@))]1/2
..[_ j~}
2.
The zeros of the delta-function in (6) are given by
kA(t) = [k2(t) + x2] 1/2
(for absorption)
kE(t) = [kZ(t) - x~]1/2
(for emission),
where k2(t) =
Ik(t)l z = Ik --
e
"
The solution of the integral equation (1) is performed iteratively. In the first step of the iteration procedure we choose an arbitrary test-function f0(k) which is substituted into the kernel K as well as the total scattering rate 2 on the right hand side of (1). In the next step we generate the functionf(k) by evaluating numerically all integrals of the right hand side of equation (1). After then f ( k ) is resubstituted on the right hand side of (1). The procedure is then repeated until convergence is achieved. The convergence of this iteration method has been proved analytically in ref.
r
x f ( I G + ~g l, ~b~)f(IG - ~g l, ~q)
2po(k(t)) =
arcsin(--k'sinq~.)
Et2,
3. N U M E R I C A L RESULTS AND DISCUSSION In this section the above described iterative procedure is applied to solve (1) for different values of the external controllable parameters E (electric field), Ns (carrier concentration) and T (lattice temperature). In the calculation the occupation of the lowest (ground-state) subband is assumed. Intervalley transitions are neglected in order to work out the essential novel features of the method. In the analysis the following numerical values of the material parameters were used: m* = 0.0665.m0, es = 12.9, e~ = 10.9, To = 420 K.
3.1 Convergence of the iteration procedure Figure 1(a) shows that any arbitrary trial function f0(k) leads to the same stable steady state solution provided the kernel of (1) remains positive definite in agreement with [24]. Figure l(b) demonstrates the quickness of the iterative procedure to solve (1) by means of a Gaussian trial function (represented by the dotted line). It can be seen that the distribution functions obtained after the third and sixth iteration nearly coincide showing that the exact distribution can be calculated already after a small number of iterations with high accuracy (If,+t - f,[ ~< 10-3, for n - 6), which cannot be achieved by the standard Monte Carlo methods [8-11] within the same computation time.
and
[2m* /~d
=
~'--~'-
"X112 h(/)q)
,
(13)
is the magnitude of the wavevector of an electron
3.2 Electric field, carrier density and lattice temperature dependence of the distribution function In Figs. 2(a)-(c) the distribution function is plotted vs the electron wavevector for different values of
1244
HOT ELECTRONS IN GaAs-QUANTUM WELLS .-,,
1.0
1 st
....
initial
iterated
....
DF DF
2 ndinltlal
--
iterated
DE
N s
=
2.5
x
T
=
100
K
E
=
500
V/cm
.
,,-<....~" .
.
-5
10L~cm-2
1.0
.
.
.
.
.
77"::.,.
---
E
=
1000
V/cm
.....
E
- 2000
V/cm
0 k [ 1 0 6 c m -I]
.....
initial
i st
O.
5
N s
= 2.5
iteration
T
=
100
K
- - - 3 rd
iteration
E
=
500
V/cm
--
iteration
6 th
DF
x
x
~ -5
1 0 ~ c m -2
...... 5
0 k [ i oBcITI -i ]
Fig. 2(a). Field dependence of the hot carrier distribu. tion function.
l.O
--Ns
=
--- N s
= 2.5x
l. O x l O 1 t c m -z 1011crn-2
5.Ox
T
=
100K
E
=
500
V/cm
1 0 " c m /'" -2 ........,, ..
l O " c m "2 :
\
: / ~
~H
= 2.5
= 100K
.,f+..I+-""
: .... N s =
....
N s
T
t',
....
"
Fig. 1(a). Surface polar optical phonon limited steady state distribution function of hot electrons in a GaAs square quantum well vs the magnitude of the electron wavevector parallel to the direction of the external electric field E calculated with different trial functions.
1.O
5oov/~
.s
N
.
=
--Z
DF
,'/"
Vol. 65, No. 10
.5
~ ' :/]"
Ii
,/ I I i :
.5
/ ." // ," /
O.
, .........
-5
0
"L !, I "'.
/
5
Fig. l(b). Initial, first-, third- and sixth-iteration of the distribution function. the external electric field, the carrier density as well as the lattice temperature. The presented curves are intersection lines of the two-dimensional distribution function f ( k ) with a plane parallel to the direction of E. By means of these figures we come to the following conclusions: (1) For given values of N+, T and E the polar optical phonon limited distribution function shows a maximum at a certain k-value in the direction of the external electric field. According to Fig. 2(a) the position of this maximum is shifted to higher k-values i f E increases. Simultaneously the shape of the distribution function becomes more and more asymmetric while the height decreases continuously due to the increase of the average carrier energy. (2) At k = where is defined by (13) the distribution functions show a kink (indicated by the arrow in Fig. 2(a)) which is due to the onset of the phonon emission process. The position of this kink
Xd
xa
r
,
....
£,...
~"
-5
> w ......
0 k [ 1 06cm -~]
.
,
0 k [ i 0 6 c m -~]
5
Fig. 2(b). Carrier concentration dependence of the distribution function.
i.@
--T
-
--- T
= 200
i00 K K
..... T
=
K
300
~q-a. 5
i/
N s
=
2.5
x
E
=
500
V/on
1 0 ~ o n -2
%
• .,.",/./l"
.
-5
0 k [ 106cm -: ]
5
Fig. 2(c). Lattice temperature dependence of the distribution function. remains fixed if Ns, T or E are changed since the activation energy of the polar optical phonons involved in the scattering process is constant. In the range the distribution function is mainly determined by the absorption process. (3) According to Fig. 2(b) the area below the
k < Xd
HOT ELECTRONS IN GaAs-QUANTUM WELLS
Nol. 65, No. 10
distribution function increases with increasing values of N, and approaches to a Fermi-Dirac like distribution as a consequence of the Pauli-principle. (4) The shape of the distribution function is strongly altered if the lattice temperature T increases. This behaviour can be attributed to the T-dependence of the phonon occupation number which exponentially increases with T.
3.3 Influence of the electron-electron scattering Figures 3(a,b) show the influence of the electronelectron scattering onto the distribution function for two different carrier concentrations. We observe that this additional interaction mechanism significantly changes the distribution function due to the enhanced randomization of the electron momenta. Three different effects can be seen: (i) The shape of the distribution function becomes more symetric. (ii) The polar optical phonon kink is flattened out. (iii) The drift of the distribution function caused by the external electric field is less pronounced than the drift of the distribution function without electron-electron scattering taken into account.
1.0
q-" .5
N s
-
5 x
1 0 * * c m "2
T
=
100
K
E
=
500
V/cm
(~) /
--
with
[. . . . .
:i~
e-e
without
According to (9) and (10) the po-phonon interaction matrixelement decreases proportional to Ik' - k[-WE. On the other hand the electron-electron interaction matrixelement is ~lk' - kl -~ • This implies that the electron-electron interaction is more effective in the wave-vector range k < Xd (indicated by the arrow in Fig. 3(a)), while for wave-vectors k > Xd the distribution functions calculated with and without electron-electron scattering nearly coincide. As the carrier concentration increases the influence of the electron-electron collisions onto the distribution function is compensated by the Pauli-principle. Therefore we observe that both distributions shown in Fig. 3(b) have a similar shape in the whole wavevector range. Moreover we can conclude that the distribution function remains asymetric even when electron-electron scattering is taken into account. In a forthcoming paper the influence of the formfactors as well as the screening will be studied in more detail by means of the integral equation method. It will be shown that both effects have to be taken into account absolutely in the calculation of the distribution function and the drift velocity in real 2D systems.
scarf. e-e
scatt.
Acknowledgements - - The authors are grateful to Prof. Dr. E. Gornik for helpful comments. This work was supported by the Fond zur F6rderung der wissenschaftlichen Forschung in Austria; Project No. P6128P.
"'"i, "~
REFERENCES 1.
.
-5
N~
1.0
0 k [ i 06cm -~ ] -
2.5 x
l O ~ c m -2
TE -= 500100V/craK
--
with
5
e-e scatt.
..... wl thout e-e scatt
2. 3. 4. 5. 6. 7. 8.
.5
9. .
-5
0 k [ i 0Scm -~ ]
1245
5
Fig. 3. Influence of the inter-carrier collisions onto the polar optical phonon limited hot carrier distribution function for carrier concentration Ns = 2.5.10" cm -2 (a) and for Ns = 5"10~cm -2 (b).
10. 11. 12.
E.M. Conwell, High Field Transport in Semiconductors (Academic Press, New York) (1967). P.J. Price, IBM J. Res. Dev. 14, 12 (1970). R. Kubo, Acta Physica Austriaca, Suppl. X, 301 (1973). G.A. Baraff, Phys. Rev. 133, A26 (1964). K. Seeger, Semicond. Phys. p. 734, Springer, Wien (1973) and references cited therein. G. Bauer, Solid State Physics, Springer Tracts in Modern Physics Vol 74, Springer, Berlin (1974). C. Jacoboni & L. Reggiani, Rev. Mod. Phys. 55, 645 (1983) and references cited therein. R. Brunetti, C. Jacoboni, A. Matulonis & Dienys, (Proc. Int. Conf. Hot Electrons in Semiconductors, Innsbruck, 1985) Physica 134 B + C, 396 (1985). N. Takenaka, M. Inoue & Y. Inuishi, J. Phys. Soc. Japan 47, 861 (1979). M. Inoue & J. Frey, J. Appl. Phys. 51, 4234 (1980). K. Hess & C.T. Sah, Phys. Rev. BI0, 3375 (1974). K. Hess, Physics of Nonlinear Transport in Semiconductors, Nato Advanced Study Institut Series B52, 1 (1980).
1246 13. 14. 15. 16. 17.
HOT ELECTRONS IN GaAs-QUANTUM WELLS P.K. Basu, K. Bhattacharyya & B.R. Nag, Solid State Commun. 48, 981 (1983). D. Chattopadhyay, Solid State Commun. 62, 395 (1987). P. Lugli & S.M. Goodnick, Phys. Rev. B59, 716 (1987). E. Vass, (Proc. Int. Conf. Hot Electrons in Semiconductors, Innsbruck, 1985) Physica 134 B+C, 337 (1985). R.G. Chambers, Proc. Phys. Soc. A65, 458 (1952).
18. 19. 20. 21. 22. 23. 24.
Vol. 65, No. 10
H. Budd, Phys. Rev. 158, 798 (1967). J,B. Socha & J.A. Krumhansl, (Proc. Int. Conf. Hot Electrons in Semiconductors, lnnsbruck, 1985) Physica 134 B+C, 142 (1985). H.D. Rees, J. Phys. Chem. Solids 30, 643 (1969). T. Ando, A.B. Fowler & F. Stern, Rev. Mod. Phys. 54, 437 (1982). Y. Takada & Y. Uemura, J. Phys. Soc. Japan 43, 139 (1977). S.K. Lyo, Phys. Rev. B34, 7129 (1986). M.O. Vassel, J. Math. Phys. 11, 408 (1970).