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Screened electron-polar optic phonon interaction and power loss of hot electrons in bulk GaAs and in quantum wells
526
Surface Science 170 (1986) 526--530 North-Holland, Amsterdam
SCREENED ELECTRON-POLAR OPTIC PHONON INTERACTION AND P O W E R L O S S OF H O T E L E C T R O N S IN BULK GaAs AND IN Q U A N T U M W E L L S P.K. BASU and Sudakshina K U N D U Centre of Advanced Study in Radio Physics and Electronics, 92 Aeharya Prafulla Chandra Road, Cak'utta 700009, India Received 13 July 1985; accepted for publication 13 September 1985
The power loss of hot two-dimensional electrons in MQWs calculated by taking the proper matrix element due to the bulk nature of the phonons and the screening agrees well with the experimental data for 2 K. It is also found that the electron temperature is higher in MQWs than in bulk for the same power input.
I. Introduction
The electron temperature and power loss of hot two-dimensional electrons in G a A s - A I G a A s multiple quantum wells (MQWs) have recently been measured by a few workers [1-3] from photoluminescence experiments. Shah et al. [1,2] applied a DC field to heat the majority electrons and photoexcited electron-hole pairs and found that at 2 K the power loss of electrons, predominantly due to emission of polar optic (LO) phonons for electron temperature exceeding 40 K, is substantially lower than the values calculated for electrons in bulk GaAs or from the available two-dimensional theory [4]. In the experiment of Xu and Tang [3], the optical energy in excess of the effective band gap energy heated the carriers. They found that for the same input energy and the same carrier density, the electron temperature is higher in MQWs than in bulk GaAs, The purpose of the present work is to point out that the results may be explained if proper matrix elements considering the bulk nature of the phonons and the screening of electron LO phonon interaction are taken into account [5]. The essential points of the theory and the results obtained are given below. 2. Theory We assume each quantum well to be isolated and the 2D wave function along the direction (z) of quantization to be sine functions. The power loss 0039-6028/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) and Yamada Science Foundation
P.K, Basu, S. Kundu / H o t electrons in bulk GaAs and in Q W
527
( d E / d t ) due to emission of LO phonons of energy h~o0 is given by [6] ( d E / d t ) = h~%~_,fo(k ) ~ , e e ( k , k')[1 - f o ( k ' ) ] , k
(1)
k'
where )Co is the distribution function characterized by the electron temperature T~. The matrix element appearing in the transition rate Pe is given by [7]
M( k, k') = ( C/q3d)Sk,,k_ q B(qz),
(2)
B( qz) = -~ 2 foLSinZ(~rz/L) exp(iqzZ ) dz
(3)
with
C = e( ho~o/2Vcp) ~/2 ,
~p_ 1 = , ~ 1 _ , ; 1 ,
q2d = q2+q2,
where L is the width of the well and the other symbols are defined in ref. [7]. The matrix element associated with the z-component of the phonon wave vector, B(qz ), may be evaluated analytically. Since there is no selection rule for qz, all values of qz should be considered while summing over all the final state wave vectors k' in eq. (1). The qz integration may be performed exactly [8]; we prefer, however, to use the simpler procedure due to Ridley [9], known as momentum conservation approximation (MCA). The power loss becomes then
(dE/dt)
e26°°
ln(1 + exp ~ ) - l f ~
8~pL
xo
1 e~-'-x° 1 + e x-n 1 + e x-"-~°
× 2+(a6yo~+8yo(2-X-Xo)+X~),/2
dx,
(4)
where X
~
E kBT~ '
h ~oo xo=kaTe,
Yo=
( ~r/L)2 kaTe ,
Ev *1=kaTe.
In order to include the effect of screening, the unscreened matrix element in eq. (2) is divided by a factor [8] e(q, ¢0o, T~) = [1 + (e]/2,sq) F(q) X(q, ~oo, T~)],
(5)
where F(q) is the form factor [10] and X is the two-dimensional polarizability calculated under RPA [10]. The energy loss rate of the bulk electrons may be calculated from eq. (1) and the expression for it for unscreened interaction is given in ref. [11]. We have included the effect of screening by including a factor (q3d/q3d 2 2 + q2), where qD is the inverse Debye length, in the matrix dement [11].
528
P.K. Basu, S. Kundu / Hot electrons in bulk GaAs and in Q W
3. Results and discussions The power loss of hot 2 D E G at 2 K has been evaluated by assuming the 2D concentration to be 7 × 101~ cm -2 as in ref. [1]. The inverse electron temperature (T~- ] ) is plotted against the power loss per electron (Pc) in fig. 1. It is found that the available 2D theory using B(q=)= 1, non-degeneracy and unscreened interaction ( X = 0 ) (curve 1) gives values of P~ about 12 times higher than the experimental values (curve E). When the values of B ( q : ) calculated under MCA are included (curve 2), the discrepancy is reduced. Curve 3 indicates the results considering degeneracy, but hardly alters the results. Curves 4 - 6 are obtained by including screening. The results given by curve 4 are obtained in the static 0 K limit but considering the q dependence of X. Curve 6 gives values obtained by taking exact X, while curve 5 is obtained by neglecting the damping terms in X [5]. The results given by curvers 5 and 6 do not differ much, however. It appears therefore that the wide discrepancy between the values of power loss calculated from the available 2D theory [4] and the experimental data at 2 K may be reduced significantly if the proper matrix elements due to the 3D nature of the phonon wave vector and the screening of the electron-pbonon interaction are considered. The possibility of higher subband occupation and intersubband scattering has not been considered in the present work. It has been found earlier [5] by considering the unscreened potential that a larger power loss results when two subbands are occupied. It is expected that such a " - .-~.
MQW \
2K
1
,% :,,,
.
2.0
1 lill
J
10-12
J
I
I I ill
I
]
I
I I ii
10-11 10-10 POWER LOSS {WATT) Fig. l. Power loss per electron versus reverse electron temperature. (E) experimentalcurve. Curves 1-3: no screening; (1) B ( q z ) = 1 , nondegenerate (N); (2) MCA, N; (3) MCA. degenerate; (4) X(q, O, 0); (5) X(q, o~o, Te), no damping; (6) X(q, ~oo, T~).
P.K. Basu, S. Kundu / Hot electrons in bulk GaAs and in Q W
529
77K
1 I--
t
LK
/
/ / t13 o. . j
/ / /
O a..
-
Id I
200
I
I
I
I
I
I
I
300 TEMPERATURE (K)
I
I
I
400
Fig. 2. Power loss per electron versus electron temperature in bulk GaAs and in GaAs MQWs at 77 K.
conclusion will be valid even when screening is considered. Thus there will be some more deviation between theory and experiment than indicated by curves 6 and E. We have not considered in this work any deviation of the phonon distribution function from equilibrium [12]. Such a hot phonon effect is expected to lower the values of power loss and the agreement between theory and experiment is likely to improve. A complete calculation including intersubband scattering, proper multisubband screening and hot phonon effects is, however, fairly complicated. The results for power loss in bulk GaAs and in GaAs based MQWs at 77 K as a function of electron temperature are given in fig. 2. The carrier densities in both cases were taken to be about 10 ]8 cm -3. It is found that for the same power loss (i.e., same power input in the steady state), the electron temperature is higher in MQWs than in bulk. In the present calculation the simple Debye screening model is employed for the calculation of electron-phonon interaction in bulk, which exaggerates somewhat the effect of screening. The power loss at the same temperature will increase if a proper q, o~0, T~-dependent screening parameter is included in the calculation. The difference between the electron temperatures for the same input will increase therefore. The above results are in conformity with the experimental data obtained by Xu and Tang [3]. The present results cannot be directly compared with their data, however, since in the experiments more than one subband is occupied. The present results only indicate that screening is important in 2D systems and the large difference in electron temperatures in the bulk and in M Q W s may partly be
530
P.K. Basu, S. Kundu / H o t electrons in bulk GaAs and in Q W
explained when the matrix element for and screening of electron-LO phonon interaction are properly taken into account. The work is supported in part by the Indian National Science Academy through the award of INSA Research Fellowship to P.K.B.S.K. is thankful to the University Grants Commission for a Fellowship.
References [1] J. Shah, A. Pinczuk, H.L. St~Srmer, A.C. Gossard and W. Wiegmann, Appl. Phys. Letters 42 (1983) 55. [2] J. Shah, A. Pinczuk, H.L. St~3rmer, A.C. Gossard and W. Wiegmann, Appl. Phys. Letters 44 (1984) 322. [3] Z.Y. Xu and C.L. Tang, Appl. Phys. Letters 44 (1984) 692. [4] K. Hess, N. Holonyak, Jr., W.D. Laidig, B.A. Vojak, J.J. Coleman and P.D. Dapkus, Solid State Commun. 34 (1980) 749. [5] A preliminary report of the theory has been given earlier, see P.K. Basu and S. Kundu, Appl. Phys. Letters 47 (1985) 264. [6] P.K. Basu, Phys. Letters 68A (1978) 459. [7] J.B. Roy, P.K. Basu and B.R. Nag, Solid State Commun. 40 (1981) 491. [81 P-J. Price, J. Vacuum Sci. Technol. 19 (1981) 599. [9] B.K. Ridley, J. Phys. C15 (1982) 5899. [10] T. Ando, A.B. Fowler and F. Stern, Rev. Mod. Phys. 54 (1982) 437. [11] B.R. Nag, Electron Transport in Compound Semiconductors (Springer, Berlin, 1980) p. 311. [12] P.J. Price, private communication; P.J. Price, Physica 134B (1985) 164.
Surface Science 170 (1986) 526--530 North-Holland, Amsterdam
SCREENED ELECTRON-POLAR OPTIC PHONON INTERACTION AND P O W E R L O S S OF H O T E L E C T R O N S IN BULK GaAs AND IN Q U A N T U M W E L L S P.K. BASU and Sudakshina K U N D U Centre of Advanced Study in Radio Physics and Electronics, 92 Aeharya Prafulla Chandra Road, Cak'utta 700009, India Received 13 July 1985; accepted for publication 13 September 1985
The power loss of hot two-dimensional electrons in MQWs calculated by taking the proper matrix element due to the bulk nature of the phonons and the screening agrees well with the experimental data for 2 K. It is also found that the electron temperature is higher in MQWs than in bulk for the same power input.
I. Introduction
The electron temperature and power loss of hot two-dimensional electrons in G a A s - A I G a A s multiple quantum wells (MQWs) have recently been measured by a few workers [1-3] from photoluminescence experiments. Shah et al. [1,2] applied a DC field to heat the majority electrons and photoexcited electron-hole pairs and found that at 2 K the power loss of electrons, predominantly due to emission of polar optic (LO) phonons for electron temperature exceeding 40 K, is substantially lower than the values calculated for electrons in bulk GaAs or from the available two-dimensional theory [4]. In the experiment of Xu and Tang [3], the optical energy in excess of the effective band gap energy heated the carriers. They found that for the same input energy and the same carrier density, the electron temperature is higher in MQWs than in bulk GaAs, The purpose of the present work is to point out that the results may be explained if proper matrix elements considering the bulk nature of the phonons and the screening of electron LO phonon interaction are taken into account [5]. The essential points of the theory and the results obtained are given below. 2. Theory We assume each quantum well to be isolated and the 2D wave function along the direction (z) of quantization to be sine functions. The power loss 0039-6028/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) and Yamada Science Foundation
P.K, Basu, S. Kundu / H o t electrons in bulk GaAs and in Q W
527
( d E / d t ) due to emission of LO phonons of energy h~o0 is given by [6] ( d E / d t ) = h~%~_,fo(k ) ~ , e e ( k , k')[1 - f o ( k ' ) ] , k
(1)
k'
where )Co is the distribution function characterized by the electron temperature T~. The matrix element appearing in the transition rate Pe is given by [7]
M( k, k') = ( C/q3d)Sk,,k_ q B(qz),
(2)
B( qz) = -~ 2 foLSinZ(~rz/L) exp(iqzZ ) dz
(3)
with
C = e( ho~o/2Vcp) ~/2 ,
~p_ 1 = , ~ 1 _ , ; 1 ,
q2d = q2+q2,
where L is the width of the well and the other symbols are defined in ref. [7]. The matrix element associated with the z-component of the phonon wave vector, B(qz ), may be evaluated analytically. Since there is no selection rule for qz, all values of qz should be considered while summing over all the final state wave vectors k' in eq. (1). The qz integration may be performed exactly [8]; we prefer, however, to use the simpler procedure due to Ridley [9], known as momentum conservation approximation (MCA). The power loss becomes then
(dE/dt)
e26°°
ln(1 + exp ~ ) - l f ~
8~pL
xo
1 e~-'-x° 1 + e x-n 1 + e x-"-~°
× 2+(a6yo~+8yo(2-X-Xo)+X~),/2
dx,
(4)
where X
~
E kBT~ '
h ~oo xo=kaTe,
Yo=
( ~r/L)2 kaTe ,
Ev *1=kaTe.
In order to include the effect of screening, the unscreened matrix element in eq. (2) is divided by a factor [8] e(q, ¢0o, T~) = [1 + (e]/2,sq) F(q) X(q, ~oo, T~)],
(5)
where F(q) is the form factor [10] and X is the two-dimensional polarizability calculated under RPA [10]. The energy loss rate of the bulk electrons may be calculated from eq. (1) and the expression for it for unscreened interaction is given in ref. [11]. We have included the effect of screening by including a factor (q3d/q3d 2 2 + q2), where qD is the inverse Debye length, in the matrix dement [11].
528
P.K. Basu, S. Kundu / Hot electrons in bulk GaAs and in Q W
3. Results and discussions The power loss of hot 2 D E G at 2 K has been evaluated by assuming the 2D concentration to be 7 × 101~ cm -2 as in ref. [1]. The inverse electron temperature (T~- ] ) is plotted against the power loss per electron (Pc) in fig. 1. It is found that the available 2D theory using B(q=)= 1, non-degeneracy and unscreened interaction ( X = 0 ) (curve 1) gives values of P~ about 12 times higher than the experimental values (curve E). When the values of B ( q : ) calculated under MCA are included (curve 2), the discrepancy is reduced. Curve 3 indicates the results considering degeneracy, but hardly alters the results. Curves 4 - 6 are obtained by including screening. The results given by curve 4 are obtained in the static 0 K limit but considering the q dependence of X. Curve 6 gives values obtained by taking exact X, while curve 5 is obtained by neglecting the damping terms in X [5]. The results given by curvers 5 and 6 do not differ much, however. It appears therefore that the wide discrepancy between the values of power loss calculated from the available 2D theory [4] and the experimental data at 2 K may be reduced significantly if the proper matrix elements due to the 3D nature of the phonon wave vector and the screening of the electron-pbonon interaction are considered. The possibility of higher subband occupation and intersubband scattering has not been considered in the present work. It has been found earlier [5] by considering the unscreened potential that a larger power loss results when two subbands are occupied. It is expected that such a " - .-~.
MQW \
2K
1
,% :,,,
.
2.0
1 lill
J
10-12
J
I
I I ill
I
]
I
I I ii
10-11 10-10 POWER LOSS {WATT) Fig. l. Power loss per electron versus reverse electron temperature. (E) experimentalcurve. Curves 1-3: no screening; (1) B ( q z ) = 1 , nondegenerate (N); (2) MCA, N; (3) MCA. degenerate; (4) X(q, O, 0); (5) X(q, o~o, Te), no damping; (6) X(q, ~oo, T~).
P.K. Basu, S. Kundu / Hot electrons in bulk GaAs and in Q W
529
77K
1 I--
t
LK
/
/ / t13 o. . j
/ / /
O a..
-
Id I
200
I
I
I
I
I
I
I
300 TEMPERATURE (K)
I
I
I
400
Fig. 2. Power loss per electron versus electron temperature in bulk GaAs and in GaAs MQWs at 77 K.
conclusion will be valid even when screening is considered. Thus there will be some more deviation between theory and experiment than indicated by curves 6 and E. We have not considered in this work any deviation of the phonon distribution function from equilibrium [12]. Such a hot phonon effect is expected to lower the values of power loss and the agreement between theory and experiment is likely to improve. A complete calculation including intersubband scattering, proper multisubband screening and hot phonon effects is, however, fairly complicated. The results for power loss in bulk GaAs and in GaAs based MQWs at 77 K as a function of electron temperature are given in fig. 2. The carrier densities in both cases were taken to be about 10 ]8 cm -3. It is found that for the same power loss (i.e., same power input in the steady state), the electron temperature is higher in MQWs than in bulk. In the present calculation the simple Debye screening model is employed for the calculation of electron-phonon interaction in bulk, which exaggerates somewhat the effect of screening. The power loss at the same temperature will increase if a proper q, o~0, T~-dependent screening parameter is included in the calculation. The difference between the electron temperatures for the same input will increase therefore. The above results are in conformity with the experimental data obtained by Xu and Tang [3]. The present results cannot be directly compared with their data, however, since in the experiments more than one subband is occupied. The present results only indicate that screening is important in 2D systems and the large difference in electron temperatures in the bulk and in M Q W s may partly be
530
P.K. Basu, S. Kundu / H o t electrons in bulk GaAs and in Q W
explained when the matrix element for and screening of electron-LO phonon interaction are properly taken into account. The work is supported in part by the Indian National Science Academy through the award of INSA Research Fellowship to P.K.B.S.K. is thankful to the University Grants Commission for a Fellowship.
References [1] J. Shah, A. Pinczuk, H.L. St~Srmer, A.C. Gossard and W. Wiegmann, Appl. Phys. Letters 42 (1983) 55. [2] J. Shah, A. Pinczuk, H.L. St~3rmer, A.C. Gossard and W. Wiegmann, Appl. Phys. Letters 44 (1984) 322. [3] Z.Y. Xu and C.L. Tang, Appl. Phys. Letters 44 (1984) 692. [4] K. Hess, N. Holonyak, Jr., W.D. Laidig, B.A. Vojak, J.J. Coleman and P.D. Dapkus, Solid State Commun. 34 (1980) 749. [5] A preliminary report of the theory has been given earlier, see P.K. Basu and S. Kundu, Appl. Phys. Letters 47 (1985) 264. [6] P.K. Basu, Phys. Letters 68A (1978) 459. [7] J.B. Roy, P.K. Basu and B.R. Nag, Solid State Commun. 40 (1981) 491. [81 P-J. Price, J. Vacuum Sci. Technol. 19 (1981) 599. [9] B.K. Ridley, J. Phys. C15 (1982) 5899. [10] T. Ando, A.B. Fowler and F. Stern, Rev. Mod. Phys. 54 (1982) 437. [11] B.R. Nag, Electron Transport in Compound Semiconductors (Springer, Berlin, 1980) p. 311. [12] P.J. Price, private communication; P.J. Price, Physica 134B (1985) 164.