- Email: [email protected]
Exciton levels and optical absorption in coupled double quantum well structures
ARTICLE IN PRESS
Journal of Luminescence 112 (2005) 216–219 www.elsevier.com/locate/jlumin
Exciton levels and optical absorption in coupled double quantum well structures S.C. Arapana,b,, M.A. Libermana a Department of Physics, Uppsala University, Box 530, SE-751 21, Uppsala, Sweden LISES, Institute of Applied Physics, Academy of Sciences of Moldova, MD-2028, Academiei str.5, Chisinau, Moldova
b
Available online 26 November 2004
Abstract We study exciton states in a coupled double quantum well (CDQW) semiconductor structure. Exciton levels and binding energies of direct and indirect excitons are calculated for a symmetric CDQW system with an applied electric field. The exciton states are obtained by solving the exciton effective-mass equation in the momentum space using the modified Gaussian quadrature method. Within this approach we perform realistic calculations of the exciton states by taking into account the coupling between different subband pairs and calculate optical-absorption coefficients. The calculated values of the exciton binding energy are in a good agreement with the experiment and the calculated absorption spectra qualitatively agree with the measured photoluminescence excitation spectra. r 2004 Elsevier B.V. All rights reserved. PACS: 73.21.Fg; 71.35.Cc; 78.67.De Keywords: Coupled quantum wells; Absorption spectra
A coupled double quantum well (CDQW) system has attracted a considerable interest during the last decade [1–9]. Originally such interest was caused by a potential application of the CDQWs to optical devices due to their unique electronic properties. In a CDQW system electron and hole states, as well as exciton states are influenced by Corresponding author. Department of Physics, Uppsala University, Box 530, SE-751 21, Uppsala, Sweden. Tel.:+46 18 4713550; fax: +46 18 4713524. E-mail address: [email protected] (S.C. Arapan).
the barrier separating quantum wells. For a suitably thin width of that barrier the significant coupling of the two wells occurs leading to a considerable enhancement of the electro-optic effects, such as the quantum-confined Stark effect and the quadratic electro-optic effect. The physical interest in these systems, however, is determined by their excitonic properties, which have attracted attention both with respect to the single excitonproblem and the exciton–exciton interaction. In a CDQW the spatial separation of electrons and holes can be externally controlled by an electric
0022-2313/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jlumin.2004.09.059
ARTICLE IN PRESS S.C. Arapan, M.A. Liberman / Journal of Luminescence 112 (2005) 216–219
nmk
where n and m are the electron and hole subband indices, and k is the wave vector in the x–y plane. The amplitude fX nm ðkÞ satisfies the effective-mass equation for excitons in a momentum space E X ÞfX nm ðkÞ
ðE nm ðkÞ X 0 hnmjV kk0 jn0 m0 ifX n0 m0 ðk Þ ¼ 0:
ð2Þ
n 0 m 0 k0
In the following we will consider only the exciton states made up of single-particle electron and heavy-hole states, which are well described within the one-band approximation with anisotropiceffective mass. Then E nm ðkÞ ¼ E G þ E en ðkÞ þ E hm ðkÞ;
(3)
where E G is the energy gap between the conduction and valence bands in the well, and E eðhÞ nðmÞ ðkÞ is the parabolic dispersion of electron and heavy-
hole subbands. The electron–hole interaction with the quantum-well size effect is given by Z Z e2 dzh eqjze zh j dze hnmjV q jn m i ¼ 2q
f n ðze Þg m ðzh Þf n0 ðze Þgm0 ðzh Þ; 0
0
ð4Þ
where q ¼ k k0 ; and f n ðze Þ and gm ðzh Þ are electron and hole envelope functions, respectively. The exciton levels and corresponding eigenfunctions can be obtained with high accuracy by solving Eq. (2) using the modified Gaussian quadrature method [14]. First, we will consider GaAs=Al0:3 Ga0:7 As double quantum wells separated by thin AlAs barriers (see Ref. [8]). In our calculations we have used the following values for the GaAs=Alx Ga1x As band parameters: me ðxÞ ¼ 0:0665 þ 0:0835x; mhz ðxÞ ¼ 0:34 þ 0:42x; mh? ðxÞ ¼ 0:18 þ 0:12x; ¼ 13:2; E G ¼ 1:519 (eV), DE G ¼ 1:247x and a conduction band offset ratio of 0.65. In Fig. 1 are shown the numerical results for the normalized absorption coefficient for CDQW structures with different width of the separating barrier, considering only the contribution from the heavy-hole excitons. The two spectral lines of higher intensity in the absorption spectra can be attributed to transitions between the symmetric (the highest peak) and antisymmetric single-particle ground states. The
1.0
normalized absorption coefficient
field applied perpendicular to the semiconductor layers. As a result two types of excitons, which are either direct or indirect in real space, can be formed in a CDQW structure. For the spatially direct exciton the bound electron–hole pair is confined in the same quantum well, while the indirect exciton consists of an electron and a hole confined in separate quantum wells. Due to a spatial separation of the particles different physical properties are expected for the indirect excitons in contrast to the direct excitons. The most striking difference is the strongly increased recombination lifetime of the indirect excitons [10], which make them a promising candidate for the Bose–Einstein condensation of excitons. Recently novel effects involving the excitonic states in CDQW systems was reported [11–13], stimulating a more detailed theoretical study of the coupled quantum well structures. In this paper we investigate theoretically the exciton binding energy and the spatial character of excitons in GaAs=Alx Ga1x As CDQW structure. We perform realistic calculations for both ground and excited exciton states in two physical systems used in the experiments of Refs. [8,12]. The exciton state jX i can be expanded as a linear combination of the free electron and hole state jnmki X jX i ¼ fX (1) nm ðkÞjnmki;
217
0.8
0.6
0.4
0.2
0.0 1.55
1.56
1.57
1.58
1.59
1.60
photon energy (eV)
Fig. 1. Normalized absorption coefficient (the heavy-hole excitons contributions only) of the GaAs=Al0:3 Ga0:7 As CDQWs with AlAs barrier width of 1 ML (dotted line), 3 ML (dashed line), and 5 ML (solid line).
ARTICLE IN PRESS S.C. Arapan, M.A. Liberman / Journal of Luminescence 112 (2005) 216–219
218
calculated absorption spectra agree qualitatively with the photoluminescence excitation spectra from Ref. [8]. In Fig. 2 is shown the exciton binding energy of the ground state exciton as a function of the separating barrier width for a CDQW system with 7:5 nm quantum wells. The dependence E X B ðl B Þ shows a nonmonotonic behavior with a minimum for a barrier thickness of 1 monolayer (ML). This nontrivial dependence of the exciton binding energy versus the width of the inter-well barrier was theoretically predicted earlier by Kamizato and Matsuura [5]. The minimum in the exciton binding energy dependence arises from the redistribution of the free particle envelope function with the variation of the separating barrier and the corresponding reduction of the electron–hole Coulomb interaction. The calculated values of the exciton binding energy are in a good agreement with the experiment [8], except the case of 1 ML AlAs barrier. For small l B the electron and hole envelope functions have dips in the middle of the separating barrier, resulting, as can be seen from Eq. (4), in a smaller electron–hole interaction. However, the redistribution of the wave function due to the presence of the barrier does not lead, according to our calculations, to such a dramatic decrease of the binding energy. There is also a small discrepancy between the calculated and
experimentally obtained value of the binding energy for the structure without the middle barrier, which corresponds to a single quantum well (QW) with a width of 15 nm. In paper [8] the exciton binding energy was estimated by multiplying the 2s–1s splitting by a factor of 98 valid for an ideal two-dimensional exciton. A 15 nm QW exciton cannot be considered as a two-dimensional one, so the validity of such an estimation of the exciton binding energy for large QWs is questionable. The Coulomb interaction mixes the symmetric and antisymmetric single-particle states. This mixing becomes stronger with the increasing of the width of the separating barrier and the exciton ground states acquire predominantly direct or indirect spatial character. The numerical results show that the ground state exciton has a direct, intrawell character, while the exciton state corresponding to the second peak in the absorption spectra has an indirect, interwell character. By applying an external electric field E it is possible to change the spatial character of the exciton ground state from a direct to an indirect one. In the following we will consider a 8 nm GaAs CDQWs separated by a 4 nm Al0:33 Ga0:67 As barrier, which corresponds to the physical system used in the experiment of Ref. [12]. In Fig. 3 are shown the 1.568 8
1.566
10
EX (eV)
EB (meV)
7
8
1.562 1.560
EB (meV)
1.564
9
6 1.558
7
1.556
5
6 0.0
0.2
0.4
0.6
0.8
1.0
electric field strength (mV/nm) 0.0
0.5
1.0
1.5
2.0
2.5
lB (nm)
Fig. 2. Exciton binding energy for the ground state exciton versus the width of the separating AlAs barrier. The values (filed squares) for the binding energy obtained from the experiment are also shown [8].
Fig. 3. Exciton ground state and exciton excited states that correspond to the absorption line with maximum intensity (line+circles), as well as the binding energy for the ground state exciton (dash-dotted line) versus the strength of the applied electric field. Indirect exciton states are denoted by open circles and the direct exciton states by closed circles.
ARTICLE IN PRESS S.C. Arapan, M.A. Liberman / Journal of Luminescence 112 (2005) 216–219
ground state exciton binding energy and the energy levels corresponding to the exciton ground state and two excited states, which yield a spectral line with the highest peak in the absorption spectrum. For electric fields less than 0.5 mV/nm the ground state is a direct exciton and indirect exciton appears as an excited state. The binding energy of the ground state exciton decreases rapidly with electric field until the switch between the direct and indirect excitons occurs at E ’ 0.5 mV/nm. The energy of the indirect exciton decreases significantly with electric field, as well as the intensity of the corresponding line in the absorption spectra due to the reduced overlap of the electron and hole states localized in opposite wells. The energy of the direct exciton states remains approximately constant at larger values of the electric field and the spectral line due to these transitions becomes dominant. One can note that the indirect excitons correspond to symmetryallowed transition in the absence of the electric field, whereas the direct excitons correspond to symmetry-forbidden transitions. These results are in good agreement with the experimentally obtained dependence of the exciton levels with electric field [2] and coincide with the theoretical results of Ref. [9]. In conclusion, we have performed realistic calculations of the heavy-hole exciton states in a CDQW structure. The exciton levels and exciton binding energies are obtained by solving the exciton effective-mass equation in the momentum space using the Gaussian quadrature method and are in good agreement with the experiment. The
219
calculated absorption spectra qualitatively agree with the photoluminescence excitation spectra. We have also studied the dependence of the spatial character of the exciton states versus the separating barrier and the strength of the applied electric field.
References [1] M.N. Islam, R.L. Hillman, D.A.B. Miller, D.S. Chemla, A.C. Gossard, J.H. English, Appl. Phys. Lett. 50 (1987) 1098. [2] Y.J. Chen, E.S. Koteles, B.S. Elman, C.A. Armiento, Phys. Rev. B 36 (1987) 4562. [3] S. Charbonneau, M.L.W. Thewalt, E.S. Koteles, B. Elman, Phys. Rev. B 38 (1988) 6287. [4] C.C. Phillips, R. Eccleston, S.R. Andrews, Phys. Rev. B 40 (1989) 9760. [5] Tsuneo Kamizato, Mitsuru Matsuura, Phys. Rev. B 40 (1989) 8387. [6] M.M. Dignam, J.E. Sipe, Phys. Rev. B 43 (1991) 4084. [7] Yutaka Takahashi, Yoshimine Kato, S.S. Kano, Susumu Fukatsu, Yasuhiro Shiraki, Ryoichi Ito, J. Appl. Phys. 76 (1994) 2299. [8] M. Bayer, V.B. Timofeev, F. Faller, T. Gutbrod, A. Forchel, Phys. Rev. B 54 (1996) 8799. [9] M.H. Szymanska, P.B. Littlewood, Phys. Rev. B 67 (2003) 193305. [10] J.E. Golub, K. Kash, J.P. Harbison, L.T. Florez, Phys. Rev. B 41 (1990) 8564. [11] L.V. Butov, C.W. Lai, A.L. Ivanov, A.C. Gossard, D.S. Chemla, Nature 417 (2002) 47. [12] L.V. Butov, A.C. Gossard, D.S. Chemla, Nature 418 (2002) 751. [13] D. Snoke, S. Denev, Y. Liu, L. Pfeiffer, K. West, Nature 418 (2002) 754. [14] Shun-Lieng Chuang, Stefan Schmitt-Rink, A. David, B. Miller, D.S. Chemla, Phys. Rev. B 43 (1991) 1500.
Journal of Luminescence 112 (2005) 216–219 www.elsevier.com/locate/jlumin
Exciton levels and optical absorption in coupled double quantum well structures S.C. Arapana,b,, M.A. Libermana a Department of Physics, Uppsala University, Box 530, SE-751 21, Uppsala, Sweden LISES, Institute of Applied Physics, Academy of Sciences of Moldova, MD-2028, Academiei str.5, Chisinau, Moldova
b
Available online 26 November 2004
Abstract We study exciton states in a coupled double quantum well (CDQW) semiconductor structure. Exciton levels and binding energies of direct and indirect excitons are calculated for a symmetric CDQW system with an applied electric field. The exciton states are obtained by solving the exciton effective-mass equation in the momentum space using the modified Gaussian quadrature method. Within this approach we perform realistic calculations of the exciton states by taking into account the coupling between different subband pairs and calculate optical-absorption coefficients. The calculated values of the exciton binding energy are in a good agreement with the experiment and the calculated absorption spectra qualitatively agree with the measured photoluminescence excitation spectra. r 2004 Elsevier B.V. All rights reserved. PACS: 73.21.Fg; 71.35.Cc; 78.67.De Keywords: Coupled quantum wells; Absorption spectra
A coupled double quantum well (CDQW) system has attracted a considerable interest during the last decade [1–9]. Originally such interest was caused by a potential application of the CDQWs to optical devices due to their unique electronic properties. In a CDQW system electron and hole states, as well as exciton states are influenced by Corresponding author. Department of Physics, Uppsala University, Box 530, SE-751 21, Uppsala, Sweden. Tel.:+46 18 4713550; fax: +46 18 4713524. E-mail address: [email protected] (S.C. Arapan).
the barrier separating quantum wells. For a suitably thin width of that barrier the significant coupling of the two wells occurs leading to a considerable enhancement of the electro-optic effects, such as the quantum-confined Stark effect and the quadratic electro-optic effect. The physical interest in these systems, however, is determined by their excitonic properties, which have attracted attention both with respect to the single excitonproblem and the exciton–exciton interaction. In a CDQW the spatial separation of electrons and holes can be externally controlled by an electric
0022-2313/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jlumin.2004.09.059
ARTICLE IN PRESS S.C. Arapan, M.A. Liberman / Journal of Luminescence 112 (2005) 216–219
nmk
where n and m are the electron and hole subband indices, and k is the wave vector in the x–y plane. The amplitude fX nm ðkÞ satisfies the effective-mass equation for excitons in a momentum space E X ÞfX nm ðkÞ
ðE nm ðkÞ X 0 hnmjV kk0 jn0 m0 ifX n0 m0 ðk Þ ¼ 0:
ð2Þ
n 0 m 0 k0
In the following we will consider only the exciton states made up of single-particle electron and heavy-hole states, which are well described within the one-band approximation with anisotropiceffective mass. Then E nm ðkÞ ¼ E G þ E en ðkÞ þ E hm ðkÞ;
(3)
where E G is the energy gap between the conduction and valence bands in the well, and E eðhÞ nðmÞ ðkÞ is the parabolic dispersion of electron and heavy-
hole subbands. The electron–hole interaction with the quantum-well size effect is given by Z Z e2 dzh eqjze zh j dze hnmjV q jn m i ¼ 2q
f n ðze Þg m ðzh Þf n0 ðze Þgm0 ðzh Þ; 0
0
ð4Þ
where q ¼ k k0 ; and f n ðze Þ and gm ðzh Þ are electron and hole envelope functions, respectively. The exciton levels and corresponding eigenfunctions can be obtained with high accuracy by solving Eq. (2) using the modified Gaussian quadrature method [14]. First, we will consider GaAs=Al0:3 Ga0:7 As double quantum wells separated by thin AlAs barriers (see Ref. [8]). In our calculations we have used the following values for the GaAs=Alx Ga1x As band parameters: me ðxÞ ¼ 0:0665 þ 0:0835x; mhz ðxÞ ¼ 0:34 þ 0:42x; mh? ðxÞ ¼ 0:18 þ 0:12x; ¼ 13:2; E G ¼ 1:519 (eV), DE G ¼ 1:247x and a conduction band offset ratio of 0.65. In Fig. 1 are shown the numerical results for the normalized absorption coefficient for CDQW structures with different width of the separating barrier, considering only the contribution from the heavy-hole excitons. The two spectral lines of higher intensity in the absorption spectra can be attributed to transitions between the symmetric (the highest peak) and antisymmetric single-particle ground states. The
1.0
normalized absorption coefficient
field applied perpendicular to the semiconductor layers. As a result two types of excitons, which are either direct or indirect in real space, can be formed in a CDQW structure. For the spatially direct exciton the bound electron–hole pair is confined in the same quantum well, while the indirect exciton consists of an electron and a hole confined in separate quantum wells. Due to a spatial separation of the particles different physical properties are expected for the indirect excitons in contrast to the direct excitons. The most striking difference is the strongly increased recombination lifetime of the indirect excitons [10], which make them a promising candidate for the Bose–Einstein condensation of excitons. Recently novel effects involving the excitonic states in CDQW systems was reported [11–13], stimulating a more detailed theoretical study of the coupled quantum well structures. In this paper we investigate theoretically the exciton binding energy and the spatial character of excitons in GaAs=Alx Ga1x As CDQW structure. We perform realistic calculations for both ground and excited exciton states in two physical systems used in the experiments of Refs. [8,12]. The exciton state jX i can be expanded as a linear combination of the free electron and hole state jnmki X jX i ¼ fX (1) nm ðkÞjnmki;
217
0.8
0.6
0.4
0.2
0.0 1.55
1.56
1.57
1.58
1.59
1.60
photon energy (eV)
Fig. 1. Normalized absorption coefficient (the heavy-hole excitons contributions only) of the GaAs=Al0:3 Ga0:7 As CDQWs with AlAs barrier width of 1 ML (dotted line), 3 ML (dashed line), and 5 ML (solid line).
ARTICLE IN PRESS S.C. Arapan, M.A. Liberman / Journal of Luminescence 112 (2005) 216–219
218
calculated absorption spectra agree qualitatively with the photoluminescence excitation spectra from Ref. [8]. In Fig. 2 is shown the exciton binding energy of the ground state exciton as a function of the separating barrier width for a CDQW system with 7:5 nm quantum wells. The dependence E X B ðl B Þ shows a nonmonotonic behavior with a minimum for a barrier thickness of 1 monolayer (ML). This nontrivial dependence of the exciton binding energy versus the width of the inter-well barrier was theoretically predicted earlier by Kamizato and Matsuura [5]. The minimum in the exciton binding energy dependence arises from the redistribution of the free particle envelope function with the variation of the separating barrier and the corresponding reduction of the electron–hole Coulomb interaction. The calculated values of the exciton binding energy are in a good agreement with the experiment [8], except the case of 1 ML AlAs barrier. For small l B the electron and hole envelope functions have dips in the middle of the separating barrier, resulting, as can be seen from Eq. (4), in a smaller electron–hole interaction. However, the redistribution of the wave function due to the presence of the barrier does not lead, according to our calculations, to such a dramatic decrease of the binding energy. There is also a small discrepancy between the calculated and
experimentally obtained value of the binding energy for the structure without the middle barrier, which corresponds to a single quantum well (QW) with a width of 15 nm. In paper [8] the exciton binding energy was estimated by multiplying the 2s–1s splitting by a factor of 98 valid for an ideal two-dimensional exciton. A 15 nm QW exciton cannot be considered as a two-dimensional one, so the validity of such an estimation of the exciton binding energy for large QWs is questionable. The Coulomb interaction mixes the symmetric and antisymmetric single-particle states. This mixing becomes stronger with the increasing of the width of the separating barrier and the exciton ground states acquire predominantly direct or indirect spatial character. The numerical results show that the ground state exciton has a direct, intrawell character, while the exciton state corresponding to the second peak in the absorption spectra has an indirect, interwell character. By applying an external electric field E it is possible to change the spatial character of the exciton ground state from a direct to an indirect one. In the following we will consider a 8 nm GaAs CDQWs separated by a 4 nm Al0:33 Ga0:67 As barrier, which corresponds to the physical system used in the experiment of Ref. [12]. In Fig. 3 are shown the 1.568 8
1.566
10
EX (eV)
EB (meV)
7
8
1.562 1.560
EB (meV)
1.564
9
6 1.558
7
1.556
5
6 0.0
0.2
0.4
0.6
0.8
1.0
electric field strength (mV/nm) 0.0
0.5
1.0
1.5
2.0
2.5
lB (nm)
Fig. 2. Exciton binding energy for the ground state exciton versus the width of the separating AlAs barrier. The values (filed squares) for the binding energy obtained from the experiment are also shown [8].
Fig. 3. Exciton ground state and exciton excited states that correspond to the absorption line with maximum intensity (line+circles), as well as the binding energy for the ground state exciton (dash-dotted line) versus the strength of the applied electric field. Indirect exciton states are denoted by open circles and the direct exciton states by closed circles.
ARTICLE IN PRESS S.C. Arapan, M.A. Liberman / Journal of Luminescence 112 (2005) 216–219
ground state exciton binding energy and the energy levels corresponding to the exciton ground state and two excited states, which yield a spectral line with the highest peak in the absorption spectrum. For electric fields less than 0.5 mV/nm the ground state is a direct exciton and indirect exciton appears as an excited state. The binding energy of the ground state exciton decreases rapidly with electric field until the switch between the direct and indirect excitons occurs at E ’ 0.5 mV/nm. The energy of the indirect exciton decreases significantly with electric field, as well as the intensity of the corresponding line in the absorption spectra due to the reduced overlap of the electron and hole states localized in opposite wells. The energy of the direct exciton states remains approximately constant at larger values of the electric field and the spectral line due to these transitions becomes dominant. One can note that the indirect excitons correspond to symmetryallowed transition in the absence of the electric field, whereas the direct excitons correspond to symmetry-forbidden transitions. These results are in good agreement with the experimentally obtained dependence of the exciton levels with electric field [2] and coincide with the theoretical results of Ref. [9]. In conclusion, we have performed realistic calculations of the heavy-hole exciton states in a CDQW structure. The exciton levels and exciton binding energies are obtained by solving the exciton effective-mass equation in the momentum space using the Gaussian quadrature method and are in good agreement with the experiment. The
219
calculated absorption spectra qualitatively agree with the photoluminescence excitation spectra. We have also studied the dependence of the spatial character of the exciton states versus the separating barrier and the strength of the applied electric field.
References [1] M.N. Islam, R.L. Hillman, D.A.B. Miller, D.S. Chemla, A.C. Gossard, J.H. English, Appl. Phys. Lett. 50 (1987) 1098. [2] Y.J. Chen, E.S. Koteles, B.S. Elman, C.A. Armiento, Phys. Rev. B 36 (1987) 4562. [3] S. Charbonneau, M.L.W. Thewalt, E.S. Koteles, B. Elman, Phys. Rev. B 38 (1988) 6287. [4] C.C. Phillips, R. Eccleston, S.R. Andrews, Phys. Rev. B 40 (1989) 9760. [5] Tsuneo Kamizato, Mitsuru Matsuura, Phys. Rev. B 40 (1989) 8387. [6] M.M. Dignam, J.E. Sipe, Phys. Rev. B 43 (1991) 4084. [7] Yutaka Takahashi, Yoshimine Kato, S.S. Kano, Susumu Fukatsu, Yasuhiro Shiraki, Ryoichi Ito, J. Appl. Phys. 76 (1994) 2299. [8] M. Bayer, V.B. Timofeev, F. Faller, T. Gutbrod, A. Forchel, Phys. Rev. B 54 (1996) 8799. [9] M.H. Szymanska, P.B. Littlewood, Phys. Rev. B 67 (2003) 193305. [10] J.E. Golub, K. Kash, J.P. Harbison, L.T. Florez, Phys. Rev. B 41 (1990) 8564. [11] L.V. Butov, C.W. Lai, A.L. Ivanov, A.C. Gossard, D.S. Chemla, Nature 417 (2002) 47. [12] L.V. Butov, A.C. Gossard, D.S. Chemla, Nature 418 (2002) 751. [13] D. Snoke, S. Denev, Y. Liu, L. Pfeiffer, K. West, Nature 418 (2002) 754. [14] Shun-Lieng Chuang, Stefan Schmitt-Rink, A. David, B. Miller, D.S. Chemla, Phys. Rev. B 43 (1991) 1500.