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Static screening approximations for calculations of intersubband electron–electron scattering rates in semiconductor quantum wells
Physica B 272 (1999) 237}240
Static screening approximations for calculations of intersubband electron}electron scattering rates in semiconductor quantum wells S.-C. Lee*, I. Galbraith Department of Physics, Heriot-Watt University, Edinburgh EH14 4AS, UK
Abstract We compare intersubband electron}electron scattering rates calculated with both dynamic and static screening models. We "nd that the use of the intrasubband static dielectric function in the long-wavelength limit incorrectly models the screening of the interactions involving intersubband transitions. The use of this model results in a large underestimate of the intersubband scattering rates. We present here the correct analytical form of the static dielectric function for these intersubband interactions, which takes into account the multisubband nature of the interactions. ( 1999 Elsevier Science B.V. All rights reserved. Keywords: Intersubband; Electron}electron scattering rates; Static screening; Dynamic screening
Estimates of intersubband scattering rates are necessary for the design of optical devices based on intersubband transitions within a semiconductor quantum well; a well-known example being the quantum cascade laser. The two main non-radiative processes which determine these scattering rates, and which compete with the lasing process, are the electron}electron and electron}LO-phonon scattering processes. For far-infrared applications, the intersubband energy separation is less than the LO-phonon energy and the electron}electron scattering process becomes signi"cant. In this work we focus on the calculation of the intersubband electron}electron scattering rates. In particular, we study and compare di!erent
* Corresponding author. Fax: #44-131-451-3136. E-mail address: [email protected] (S.-C. Lee)
approximations that are used in modelling the screening of the interaction between the electrons. A realistic model of the screened electron}electron interaction should incorporate the dynamic response of the electron gas. One such model is the dynamic dielectric function derived in the random phase approximation (RPA). However, the evaluation of the full dynamic RPA dielectric function is numerically intensive, and a static approximation to this dielectric function, derived in the longwavelength limit, is often used. Details of the multisubband dynamic-screening model derived in the random phase approximation, and the multisubband Boltzmann collision rates for the electron}electron scattering processes are given in Ref. [1]. In the static, long-wavelength limit (u"0, qP0) the RPA polarization functions, [1] which encapsulate the screening behaviour of the electrons, can
0921-4526/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 2 7 7 - X
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S.-C. Lee, I. Galbraith / Physica B 272 (1999) 237}240
be written as AmH P (qP0, 0)"! n k ii p+2 i, /0
(1)
and A (N !N ), iOj, (2) P (qP0, 0)" i j ij *E ij where *E "E !E is the energy separation beij i j tween subbands i and j, and N is the carrier density i in subband i. A is the sample area parallel to the quantum well plane, n k is the occupation probi, /0 ability at the bottom of subband i, and mH is the electron e!ective mass. Using these expressions for the polarization functions, we can de"ne the two-subband dielectric function for screening interactions resulting in intersubband transitions [1]: e (qP0, 0)"1# 1212 i (q)/q where 12 e2(N !N ) 1 2 F i (q)" (q). (3) 12 1212 e *E s 21 The form factor F (q)":L8 dz:L8 [email protected](z)t (z) i{ i 0 0 i{ij{j [email protected]~z{@tH ([email protected])t ([email protected]), where ¸ is the well width, and j{ j 8 t (z) is the electron envelope function in subband i. i
A simplifying approximation that is sometimes made to model the screening of the interactions involving intersubband transitions is to use the single-subband dielectric function e (qP0, 0)" 1 1#i (q)/q, with the quasi-2D single-subband 1 screening wave number i (q)"mHe2F (q) 1 1111 n k /2pe +2. We show below that the screening 1, /0 s wave vector i (q) is only applicable to intrasub1 band interactions between electrons within the subband. In the literature it has sometimes been applied, incorrectly, to interactions involving intersubband transitions. In the following calculations of carrier}carrier scattering rates for equilibrium and nonequilibrium distributions we consider only conduction-band electrons in two subbands of a wide quantum well, with intersubband energy separation *E less 21 than the LO-phonon energy. We assume that the well has in"nite barriers and a parabolic electron energy dispersion. We use GaAs parameters. Fig. 1 shows electron}electron scattering rates for interactions involving intersubband transitions, calculated for equilibrium distributions at 10 and 300 K, using (i) full dynamic RPA screening, e (q, u), (ii) intersubband static screening in RPA the long-wavelength limit, e (qP0, 0), and 1212
Fig. 1. Intersubband electron}electron scattering rates in equilibrium distributions at 10 and 300 K. n "1011 cm~2 and ¸ "300 As . % 8
S.-C. Lee, I. Galbraith / Physica B 272 (1999) 237}240
239
Fig. 2. Intersubband electron}electron scattering rates in nonequilibrium distributions with 50% and 70% of the carriers in the upper subband. n "1.5]1011 cm~2 and ¸ "307 As . % 8
(iii) the single-subband static dielectric function in the long-wavelength limit, e (qP0, 0). There is 1 good agreement between the scattering rates calculated with the dynamic and intersubband static screening models. The single-subband static screening model e (qP0, 0) results in a large under1 estimation of the scattering rate. This discrepancy has its origin in the di!erent behaviour of F (q) 1111 and F (q) in the long-wavelength limit [1]. 1212 Fig. 2 shows intersubband electron}electron scattering rates in highly nonequilibrium distributions: (1) with 50% of the carriers in the upper subband, (2) a population inversion with 70% of the carriers in the upper subband. We again compare results using di!erent screening models: e (q, 0) and e (qP0, 0). In addition, since RPA 1212 there is now a large fraction of carriers in the upper subband, we calculate rates using the dielectric function e (qP0, 0)"1#[i (q)#i (q)]/q with 1,2 1 2 i (q)"(mHe2/2pe +2)F (q) n k , instead of i s iiii i, /0 e (qP0, 0). The rates calculated with the intersub1 band static screening model tend to be larger than the rates calculated with the dynamic screening model particularly at low energies. The use of the intrasubband screening wave vectors i and 1
i again overestimates the screening resulting in 2 a large underestimation of the intersubband scattering rates. In the case with a 70% population inversion, we also show intersubband scattering rates calculated without any screening. These rates are also fairly close to the rates calculated with the full dynamic RPA screening model. Unlike the interactions causing intrasubband transitions which diverge as qP0, the unscreened interaction < (q) for in1212 tersubband transitions tends to a "nite value even without screening, and thus the screening for these interactions has less importance than for intrasubband transitions. We point out here a notable property of the intersubband screening wave vector i [Eq. (3)]: 12 its dependence on the di!erence of the total carrier densities in the two subbands. A population inversion implies a negative i , giving rise to an en12 hanced interaction (i.e., a dielectric function (1) and leading eventually to a sign change of the interaction. In Fig. 3, we plot e (qP0, 0) against 1212 carrier fraction, x, in the upper subband. When x"0.5, e (qP0, 0)"1 implying that there is 1212 no screening when there are equal populations in
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S.-C. Lee, I. Galbraith / Physica B 272 (1999) 237}240
Fig. 3. e (qP0, 0) versus carrier fraction in upper subband. 1212
both subbands. When x'0, we "nd the dielectric function falls between 0 and 1 at 1011 cm~2, and for low population inversions at 1012 cm~2, giving rise to antiscreening or an enhanced interaction. At 1012 cm~2, for larger population inversions, the dielectric function is negative implying that at least the static, long-wavelength component of the interaction changes sign. In conclusion, we have compared intersubband scattering rates calculated using the full dynamic multisubband temperature-dependent RPA screening model, and static screening models in the
long-wavelength limit. We "nd that the single-subband static dielectric function gives a gross underestimation of the intersubband scattering rates. We give an analytical result which correctly models the behaviour of these interactions in the limit qP0. It is these interactions that determine intersubband population relaxation rates, hence it is particularly crucial to have a correct determination of the interaction strength. For equilibrium distributions, there is good agreement between the rates calculated with the dynamic model and the intersubband static screening model we have presented. In the nonequilibrium case, the agreement, particularly at low energy, is less satisfactory indicating the breakdown of the static qP0 approximation. This is not surprising since for intersubband transitions the transfer momentum q and transfer energy +u cannot simultaneously tend to zero.
Acknowledgements Financial support from The Royal Society and EPSRC is acknowledged.
References [1] S.-C. Lee, I. Galbraith, Phys. Rev. B 59 (1999) 15 796.
Static screening approximations for calculations of intersubband electron}electron scattering rates in semiconductor quantum wells S.-C. Lee*, I. Galbraith Department of Physics, Heriot-Watt University, Edinburgh EH14 4AS, UK
Abstract We compare intersubband electron}electron scattering rates calculated with both dynamic and static screening models. We "nd that the use of the intrasubband static dielectric function in the long-wavelength limit incorrectly models the screening of the interactions involving intersubband transitions. The use of this model results in a large underestimate of the intersubband scattering rates. We present here the correct analytical form of the static dielectric function for these intersubband interactions, which takes into account the multisubband nature of the interactions. ( 1999 Elsevier Science B.V. All rights reserved. Keywords: Intersubband; Electron}electron scattering rates; Static screening; Dynamic screening
Estimates of intersubband scattering rates are necessary for the design of optical devices based on intersubband transitions within a semiconductor quantum well; a well-known example being the quantum cascade laser. The two main non-radiative processes which determine these scattering rates, and which compete with the lasing process, are the electron}electron and electron}LO-phonon scattering processes. For far-infrared applications, the intersubband energy separation is less than the LO-phonon energy and the electron}electron scattering process becomes signi"cant. In this work we focus on the calculation of the intersubband electron}electron scattering rates. In particular, we study and compare di!erent
* Corresponding author. Fax: #44-131-451-3136. E-mail address: [email protected] (S.-C. Lee)
approximations that are used in modelling the screening of the interaction between the electrons. A realistic model of the screened electron}electron interaction should incorporate the dynamic response of the electron gas. One such model is the dynamic dielectric function derived in the random phase approximation (RPA). However, the evaluation of the full dynamic RPA dielectric function is numerically intensive, and a static approximation to this dielectric function, derived in the longwavelength limit, is often used. Details of the multisubband dynamic-screening model derived in the random phase approximation, and the multisubband Boltzmann collision rates for the electron}electron scattering processes are given in Ref. [1]. In the static, long-wavelength limit (u"0, qP0) the RPA polarization functions, [1] which encapsulate the screening behaviour of the electrons, can
0921-4526/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 2 7 7 - X
238
S.-C. Lee, I. Galbraith / Physica B 272 (1999) 237}240
be written as AmH P (qP0, 0)"! n k ii p+2 i, /0
(1)
and A (N !N ), iOj, (2) P (qP0, 0)" i j ij *E ij where *E "E !E is the energy separation beij i j tween subbands i and j, and N is the carrier density i in subband i. A is the sample area parallel to the quantum well plane, n k is the occupation probi, /0 ability at the bottom of subband i, and mH is the electron e!ective mass. Using these expressions for the polarization functions, we can de"ne the two-subband dielectric function for screening interactions resulting in intersubband transitions [1]: e (qP0, 0)"1# 1212 i (q)/q where 12 e2(N !N ) 1 2 F i (q)" (q). (3) 12 1212 e *E s 21 The form factor F (q)":L8 dz:L8 [email protected](z)t (z) i{ i 0 0 i{ij{j [email protected]~z{@tH ([email protected])t ([email protected]), where ¸ is the well width, and j{ j 8 t (z) is the electron envelope function in subband i. i
A simplifying approximation that is sometimes made to model the screening of the interactions involving intersubband transitions is to use the single-subband dielectric function e (qP0, 0)" 1 1#i (q)/q, with the quasi-2D single-subband 1 screening wave number i (q)"mHe2F (q) 1 1111 n k /2pe +2. We show below that the screening 1, /0 s wave vector i (q) is only applicable to intrasub1 band interactions between electrons within the subband. In the literature it has sometimes been applied, incorrectly, to interactions involving intersubband transitions. In the following calculations of carrier}carrier scattering rates for equilibrium and nonequilibrium distributions we consider only conduction-band electrons in two subbands of a wide quantum well, with intersubband energy separation *E less 21 than the LO-phonon energy. We assume that the well has in"nite barriers and a parabolic electron energy dispersion. We use GaAs parameters. Fig. 1 shows electron}electron scattering rates for interactions involving intersubband transitions, calculated for equilibrium distributions at 10 and 300 K, using (i) full dynamic RPA screening, e (q, u), (ii) intersubband static screening in RPA the long-wavelength limit, e (qP0, 0), and 1212
Fig. 1. Intersubband electron}electron scattering rates in equilibrium distributions at 10 and 300 K. n "1011 cm~2 and ¸ "300 As . % 8
S.-C. Lee, I. Galbraith / Physica B 272 (1999) 237}240
239
Fig. 2. Intersubband electron}electron scattering rates in nonequilibrium distributions with 50% and 70% of the carriers in the upper subband. n "1.5]1011 cm~2 and ¸ "307 As . % 8
(iii) the single-subband static dielectric function in the long-wavelength limit, e (qP0, 0). There is 1 good agreement between the scattering rates calculated with the dynamic and intersubband static screening models. The single-subband static screening model e (qP0, 0) results in a large under1 estimation of the scattering rate. This discrepancy has its origin in the di!erent behaviour of F (q) 1111 and F (q) in the long-wavelength limit [1]. 1212 Fig. 2 shows intersubband electron}electron scattering rates in highly nonequilibrium distributions: (1) with 50% of the carriers in the upper subband, (2) a population inversion with 70% of the carriers in the upper subband. We again compare results using di!erent screening models: e (q, 0) and e (qP0, 0). In addition, since RPA 1212 there is now a large fraction of carriers in the upper subband, we calculate rates using the dielectric function e (qP0, 0)"1#[i (q)#i (q)]/q with 1,2 1 2 i (q)"(mHe2/2pe +2)F (q) n k , instead of i s iiii i, /0 e (qP0, 0). The rates calculated with the intersub1 band static screening model tend to be larger than the rates calculated with the dynamic screening model particularly at low energies. The use of the intrasubband screening wave vectors i and 1
i again overestimates the screening resulting in 2 a large underestimation of the intersubband scattering rates. In the case with a 70% population inversion, we also show intersubband scattering rates calculated without any screening. These rates are also fairly close to the rates calculated with the full dynamic RPA screening model. Unlike the interactions causing intrasubband transitions which diverge as qP0, the unscreened interaction < (q) for in1212 tersubband transitions tends to a "nite value even without screening, and thus the screening for these interactions has less importance than for intrasubband transitions. We point out here a notable property of the intersubband screening wave vector i [Eq. (3)]: 12 its dependence on the di!erence of the total carrier densities in the two subbands. A population inversion implies a negative i , giving rise to an en12 hanced interaction (i.e., a dielectric function (1) and leading eventually to a sign change of the interaction. In Fig. 3, we plot e (qP0, 0) against 1212 carrier fraction, x, in the upper subband. When x"0.5, e (qP0, 0)"1 implying that there is 1212 no screening when there are equal populations in
240
S.-C. Lee, I. Galbraith / Physica B 272 (1999) 237}240
Fig. 3. e (qP0, 0) versus carrier fraction in upper subband. 1212
both subbands. When x'0, we "nd the dielectric function falls between 0 and 1 at 1011 cm~2, and for low population inversions at 1012 cm~2, giving rise to antiscreening or an enhanced interaction. At 1012 cm~2, for larger population inversions, the dielectric function is negative implying that at least the static, long-wavelength component of the interaction changes sign. In conclusion, we have compared intersubband scattering rates calculated using the full dynamic multisubband temperature-dependent RPA screening model, and static screening models in the
long-wavelength limit. We "nd that the single-subband static dielectric function gives a gross underestimation of the intersubband scattering rates. We give an analytical result which correctly models the behaviour of these interactions in the limit qP0. It is these interactions that determine intersubband population relaxation rates, hence it is particularly crucial to have a correct determination of the interaction strength. For equilibrium distributions, there is good agreement between the rates calculated with the dynamic model and the intersubband static screening model we have presented. In the nonequilibrium case, the agreement, particularly at low energy, is less satisfactory indicating the breakdown of the static qP0 approximation. This is not surprising since for intersubband transitions the transfer momentum q and transfer energy +u cannot simultaneously tend to zero.
Acknowledgements Financial support from The Royal Society and EPSRC is acknowledged.
References [1] S.-C. Lee, I. Galbraith, Phys. Rev. B 59 (1999) 15 796.