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Scattering rate via electron-acoustic phonon interaction in quantum wire
Materials Science and Engineering B75 (2000) 130 – 133 www.elsevier.com/locate/mseb
Scattering rate via electron-acoustic phonon interaction in quantum wire Yunlong He a, Zhen Yin a, Ming-Sheng Zhang a,* , Tianquan Lu¨ b, Yisong Zheng b a
National Laboratory of Solid State Microstructures and Center for Materials Analysis, Nanjing Uni6ersity, Nanjing 210093, PR China b Physics Department, Jilin Uni6ersity, Changchun 130013, PR China
Abstract Using the continuum phonon model and effective-mass approximation, we have calculated the scattering rate via electron– acoustic phonon interaction in a quantum wire. Both taking and not taking into account electron effective-mass mismatch (EEMM), the scattering rates decrease with increasing electron energy. This is attributed to rapid decrease of the state density of the quantum wire. The rate for the case with EEMM is lower than that without EEMM, since EEMM can make the barrier drop along the growth direction. The scattering rate increases with increasing Al concentration, which is attributed to lower well height and smaller wave vector along the growth direction. The relative deviation of the scattering rates between the without and with EEMM cases decreases with increasing Al concentration because of the lowering of the effective-barrier height caused by EEMM. With increasing well width along the growth direction the scattering rate decreases, which is ascribed to the decrease of wave vector a along the z-axis, that is, weakening of the bounding for electron in the quantum well. © 2000 Published by Elsevier Science S.A. All rights reserved. Keywords: Scattering rate; Quantum wire; Electronic–acoustic phonon interaction; Electron effective-mass mismatch
1. Introduction Recently electron conductivity in two-dimensional semiconductors has attracted much interest because of its technical and fundamental importance. Many models such as infinite height quantum well (IHQW) and finite height quantum well (FHQW) have been used to study the electron properties in two-dimensional systems [1,2]. Among them the FHQW model is more appropriate for describing electron motion in a quantum well [3]. The scattering rate due to electron–optic and electron–acoustic phonon interaction in a finite quantum well was investigated where electron effectivemass mismatch (EEMM) was taken into account [4,5]. There are also other studies on electron – phonon interaction in quantum well [6,7]. Based on continuum phonon model under an effective-mass approximation the optical phonon scattering rate in quantum wires * Corresponding author: Tel.: +86-25-3593975; fax: + 86-253595535. E-mail address: [email protected] (M.-S. Zhang)
was rigorously calculated [8]. However, to our knowledge, none of research discussed the electron–acoustic phonon interaction in a quantum wire, while the research is closely related to electrical resistivity at low temperature. In this paper, we report results on the scattering rate in quantum wires via electron–acoustic phonon interaction. We discuss the influence of EEMM on the rate, the relative deviation of the rates versus Al concentration and dependence of the rate on both Al concentration and well width along the growth direction.
2. Theory A typical quantum wire structure is schematically drawn in Fig. 1 where d denotes the thickness of GaAs and b denotes the width of the quantum wire. Such a structure was experimentally prepared [9–11]. From Fig. 1, it can be seen that electrons are confined in semi-infinite quantum wire along the z direction, so d is the well width. Thus, the wave function has the following form
0921-5107/00/$ - see front matter © 2000 Published by Elsevier Science S.A. All rights reserved. PII: S 0 9 2 1 - 5 1 0 7 ( 0 0 ) 0 0 3 4 7 - 0
c(x,y,z)=e ikx x
'
Y. He et al. / Materials Science and Engineering B75 (2000) 130–133
2 np sin y Z(z) b b
(1)
Putting Eq. (1) into the Schro¨dinger equation, we obtain ' ( Z(z) = E1Z(z) for 0 Bz Bd 2m0 (z 2 2
−
where E1 = E−
n 2' 2p 2 ' 2k 2x − 2m0 2m0
(7)
2p D 2 % % q× { n e iqy y n0 2 × l e iqz z l0 2 ' Vrs kxnylz qx,qy,qz × d(kx− kx0 + qx )
(3)
× nq × d(E− E0 − 'v)+ n e − iqy y n0 2
where
× l e − iqz z l0 2 × d(kx− kx0 − qx )
n 2' 2p 2 ' 2k 2x E2 = E−V− − 2m 2m
×(nq + 1)× d(E− E0 + 'v))
Setting
−2mE2
2m0E1 , b= a= ' ' we have the following equations which determine the electron energy
2V ma 2 1 1 a2 + 2 = + − m0 m 0 tan2(ad) ' 2 m m0
n 2p 2 +k 2x b2
' 2a 2 n 2' 2p 2 ' 2k 2x + + 2m0 2m0 2m0
(4) (5)
Considering an electron – acoustic phonon interaction the Hamiltonian under the deformation has the following expression
− iq r a*) H. = % (Cqe iq·raq +C*e q q
(6)
q
n
where Cq =
2p % f H. i 2d(Ef − Ei) ' f
where i and f are the initial and final states of the electron, respectively. Substituting the wave function of the electron into Eq. (7), we have
=
'2 (2 Z(z)=E2Z(z) for z\d 2m (z 2
E=
W(E)=
W(E)
and −
following electron scattering rate from Fermi’s golden rule
2
(2)
131
'qD 2 2rds
1/2
where r, s, D and q denote the density, acoustic velocity, the deformation potential constant and the acoustic phonon wave vector, respectively. We can obtain the
(8)
where E0 and E are the energy of the initial and final states, respectively, 'v is the phonon energy, nq is the number of phonon with energy of 'v, qx, qy and qz are the wave vectors of phonon along the x, y and z axes, and both kx0, n0, l0 and kx, n, l are the quantum numbers of the initial and final states, respectively. With integration substituting for summation, Eq. (8) becomes
& & &
W(E) =
2p D 2 V % dqx dqy dqzq ' Vrs ' 3 kx,n,l ×{ n e − iqy y n0 2 × l e − iqz z l0 2
× d(kx− kx0 − qx )× nq × d(E− E0 − 'v)+ n e − iqy y n0 2 × l e − iqz z l0 2 × d(kx− kx0 +qx ) (nq + 1)× d(E −E0 + 'v)
(9)
Finally, we can simplify Eq. (9) as the following expression W(E) =
& &
(E−E0)2 D2 % dq dq x z 4p 2rs 4' 3 n,l qy 0 × { n e − iqy y n0 2 × l e − iqz z l0 2
× nq × H1 + n e − iqy y n0 2 × l e − iqz z l0 2 ×(nq +1) (10)
× H2 where nq = Fig. 1. Schematic diagram of a GaAs/GaAlAs quantum wire.
1 'v −1 exp kBT
132
Y. He et al. / Materials Science and Engineering B75 (2000) 130–133
H1 =
!
1······E \ E0 0······E B E0
H2 =
!
0······E \ E0 1······E B E0
(for absorption)
(for emission)
3. Results and discussion
Fig. 2. Dependence of intrasubband scattering rate on electron energy. Curves (a)and (b) correspond to the results without and with EEMM, respectively.
Fig. 3. Intrasubband scattering rates (E= 0.3 eV) as a function of Al concentration with and without EEMM. Curves (a) and (b) correspond to the results without and with EEMM, respectively.
Fig. 4. Dependence of the relative deviation of the scattering rates (r1 − r2)/r1 caused by EEMM on Al concentration with the electron energy of E = 0.3 eV. Parameters r1 and r2 stand for the scattering rates without and with EEMM, respectively.
We calculate the intrasubband rates in GaAs/ GaAlx As1 − x quantum wire. All the parameters in the calculation are taken as follows: D= 13.5 eV, rd =5.3 g cm − 1, s= 5370 m s − 1, T =77 K, mo = 0.0657me and m= mo (1.0+ 1.24x), where x is the Al concentration in GaAlx As1 − x. Dependence of intrasubband scattering rates on electron energy is plotted in Fig. 2 where curves (a) and (b) represent the scattering rates for both the without and with EEMM cases, respectively. Curves (a) and (b) are of similar electron energy behavior. When the energy is below the ground state, both of the scattering rates equal zero. With increasing electron energy, the scattering rates increase and reach a maximum at Ea and Eb, then drop quickly. The rapid drop of the rate in curve (a) without EEMM is opposite to the two-dimensional case where the rate shows a flat shape relationship with energy [5]. We attribute the difference to the following reasons. The state density in a two-dimensional system has nothing to do with the electron energy, whereas the state density is inversely proportional to the square root of the energy in a quantum wire, so the rate quickly decreases with increasing electron energy. Taking into account EEMM, the scattering rate becomes smaller with increasing energy, which is attributed to a remarkable drop of the barrier along the z direction because of ascending of wave vector kx along the x direction. With increasing electron energy the scattering rate tends to the two-dimensional case. The ground-state energy is decreased in the EEMM case. This is caused by the decrease of wave vector a along the growth direction (z-axis). Fig. 3 shows that the intrasubband scattering rates increase with increasing Al concentration. The rate under considering EEMM shown in curve (b) is lower than that under not considering EEMM shown in curve (a). This is ascribed to the fact that the electron barrier along the z direction makes a larger contribution to the rate for the latter than for the former, since EEMM can cause a decrease in the effective-barrier height. The relative deviation of the scattering rates r1 and r2 as a function of Al concentration is shown in Fig. 4 where r1 and r2 correspond to the rate without and with EEMM, respectively. With increasing Al concentration, the relative reduction of the rate becomes smaller since the relative change of the effective-barrier height is decreased caused by EEMM.
Y. He et al. / Materials Science and Engineering B75 (2000) 130–133
Fig. 5. Intrasubband scattering rate (E =0.3 eV) versus the well width d along the z direction. Curves (a) and (b) correspond to the results without and with EEMM, respectively.
Dependence of the intrasubband scattering rate on well width d (z direction) is shown in Fig. 5 where curve (a) stands for without EEMM and (b) for with EEMM. The scattering rates equal zero as the well is below the critical width. With increase of well width, the rate jumps to a maximum and then drops quickly. The reasons can be described as follows. Eq. (4) after several-steps deriving can obtain a relationship that wave vector a along the z-axis decreases with increasing well width d, thus leading to lowering of the scattering rate. As we know, the increase of electron state density caused by the increase of well width enhances the scattering rate. However, the former effect makes more of a contribution to the rate than the latter does, leading to a total decrease of the rate. It is obvious that the electron undergoes a weaker bonding due to increase of the well width, thus the rate decreases to tend to the two-dimensional case. Preliminary study on the intersubband scattering rate indicates that it has the same electron energy behavior as those of the intrasubband rate, but the critical energy is different from that of the intrasubband rate. Further work is under way. .
4. Conclusions We used the continuum phonon model and effective-
133
mass approximation to calculate the electron scattering rate in GaAs/GaAlx As1 − x quantum wire via electron– acoustic phonon interaction. Not taking into account electron effective-mass mismatch, the decrease of the scattering rate with increasing electron energy is attributed to the decrease of the energy state density. Taking into account electron effective-mass mismatch, reduction of the rate is due to a remarkable drop of the barrier along the growth direction. The scattering rate increases with increasing Al concentration. The ability of the electron barrier to diminish the rate is larger for the case without EEMM than for that with EEMM. The lowering of the effective-barrier height caused by EEMM makes the relative deviation of the scattering rates between the without and with EEMM cases decrease with increasing Al concentration. The decrease of the scattering rate with increasing well width is attributed to decrease of wave vector a along the growth direction, that is, weakening of the bonding for the electron in the quantum well.
Acknowledgements This work was supported by The National Climbing Project of Fundamental Research of China.
References [1] P.J. Price, Ann. Phys. 133 (1981) 217. [2] H. Fenniche, L. Mandhour, S. Jaziri, R. Bennaceur, Semicond. Sci. Technol. 12 (1997) 796. [3] T. Lu¨, Y. Zheng, Phys. Rev. B 53 (1996) 1438. [4] Y. Zheng, T. Lu, J. Liu, W. Su, Semicond. Sci. Technol. 12 (1997) 1235. [5] Y. Zheng, T. Lu¨, Y. Wang, X. Wu, C. Zhang, W. Su, Semicond. Sci. Technol. 12 (1997) 296. [6] J. Pozela, V. Juciene, K. Pozela, Semicond. Sci. Technol. 10 (1995) 1555. [7] J. Pozela, G. Butkus, V. Juciene, Semicond. Sci. Technol. 9 (1994) 1480. [8] P.A. Knipp, T.L. Reinecke, Phys. Rev. B 48 (1993) 5700. [9] A.A. Darhuber, G. Bauer, P.D. Wang, C.M. Sotomaryer Torres, J. Appl. Phys. 83 (1998) 126. [10] H. Sakaki, T. Someya, H. Akiyama, Y. Nakamura, N. Kondo, D. Kishimoto, Phys. Status Solidi A 184 (1998) 241. [11] A. Yacoby, H.L. Stormer, N.S. Wingreen, L.N. Pfeiffer, K.W. Baldwin, K.W. West, Phys. Rev. Lett. 77 (1996) 4612.
Scattering rate via electron-acoustic phonon interaction in quantum wire Yunlong He a, Zhen Yin a, Ming-Sheng Zhang a,* , Tianquan Lu¨ b, Yisong Zheng b a
National Laboratory of Solid State Microstructures and Center for Materials Analysis, Nanjing Uni6ersity, Nanjing 210093, PR China b Physics Department, Jilin Uni6ersity, Changchun 130013, PR China
Abstract Using the continuum phonon model and effective-mass approximation, we have calculated the scattering rate via electron– acoustic phonon interaction in a quantum wire. Both taking and not taking into account electron effective-mass mismatch (EEMM), the scattering rates decrease with increasing electron energy. This is attributed to rapid decrease of the state density of the quantum wire. The rate for the case with EEMM is lower than that without EEMM, since EEMM can make the barrier drop along the growth direction. The scattering rate increases with increasing Al concentration, which is attributed to lower well height and smaller wave vector along the growth direction. The relative deviation of the scattering rates between the without and with EEMM cases decreases with increasing Al concentration because of the lowering of the effective-barrier height caused by EEMM. With increasing well width along the growth direction the scattering rate decreases, which is ascribed to the decrease of wave vector a along the z-axis, that is, weakening of the bounding for electron in the quantum well. © 2000 Published by Elsevier Science S.A. All rights reserved. Keywords: Scattering rate; Quantum wire; Electronic–acoustic phonon interaction; Electron effective-mass mismatch
1. Introduction Recently electron conductivity in two-dimensional semiconductors has attracted much interest because of its technical and fundamental importance. Many models such as infinite height quantum well (IHQW) and finite height quantum well (FHQW) have been used to study the electron properties in two-dimensional systems [1,2]. Among them the FHQW model is more appropriate for describing electron motion in a quantum well [3]. The scattering rate due to electron–optic and electron–acoustic phonon interaction in a finite quantum well was investigated where electron effectivemass mismatch (EEMM) was taken into account [4,5]. There are also other studies on electron – phonon interaction in quantum well [6,7]. Based on continuum phonon model under an effective-mass approximation the optical phonon scattering rate in quantum wires * Corresponding author: Tel.: +86-25-3593975; fax: + 86-253595535. E-mail address: [email protected] (M.-S. Zhang)
was rigorously calculated [8]. However, to our knowledge, none of research discussed the electron–acoustic phonon interaction in a quantum wire, while the research is closely related to electrical resistivity at low temperature. In this paper, we report results on the scattering rate in quantum wires via electron–acoustic phonon interaction. We discuss the influence of EEMM on the rate, the relative deviation of the rates versus Al concentration and dependence of the rate on both Al concentration and well width along the growth direction.
2. Theory A typical quantum wire structure is schematically drawn in Fig. 1 where d denotes the thickness of GaAs and b denotes the width of the quantum wire. Such a structure was experimentally prepared [9–11]. From Fig. 1, it can be seen that electrons are confined in semi-infinite quantum wire along the z direction, so d is the well width. Thus, the wave function has the following form
0921-5107/00/$ - see front matter © 2000 Published by Elsevier Science S.A. All rights reserved. PII: S 0 9 2 1 - 5 1 0 7 ( 0 0 ) 0 0 3 4 7 - 0
c(x,y,z)=e ikx x
'
Y. He et al. / Materials Science and Engineering B75 (2000) 130–133
2 np sin y Z(z) b b
(1)
Putting Eq. (1) into the Schro¨dinger equation, we obtain ' ( Z(z) = E1Z(z) for 0 Bz Bd 2m0 (z 2 2
−
where E1 = E−
n 2' 2p 2 ' 2k 2x − 2m0 2m0
(7)
2p D 2 % % q× { n e iqy y n0 2 × l e iqz z l0 2 ' Vrs kxnylz qx,qy,qz × d(kx− kx0 + qx )
(3)
× nq × d(E− E0 − 'v)+ n e − iqy y n0 2
where
× l e − iqz z l0 2 × d(kx− kx0 − qx )
n 2' 2p 2 ' 2k 2x E2 = E−V− − 2m 2m
×(nq + 1)× d(E− E0 + 'v))
Setting
−2mE2
2m0E1 , b= a= ' ' we have the following equations which determine the electron energy
2V ma 2 1 1 a2 + 2 = + − m0 m 0 tan2(ad) ' 2 m m0
n 2p 2 +k 2x b2
' 2a 2 n 2' 2p 2 ' 2k 2x + + 2m0 2m0 2m0
(4) (5)
Considering an electron – acoustic phonon interaction the Hamiltonian under the deformation has the following expression
− iq r a*) H. = % (Cqe iq·raq +C*e q q
(6)
q
n
where Cq =
2p % f H. i 2d(Ef − Ei) ' f
where i and f are the initial and final states of the electron, respectively. Substituting the wave function of the electron into Eq. (7), we have
=
'2 (2 Z(z)=E2Z(z) for z\d 2m (z 2
E=
W(E)=
W(E)
and −
following electron scattering rate from Fermi’s golden rule
2
(2)
131
'qD 2 2rds
1/2
where r, s, D and q denote the density, acoustic velocity, the deformation potential constant and the acoustic phonon wave vector, respectively. We can obtain the
(8)
where E0 and E are the energy of the initial and final states, respectively, 'v is the phonon energy, nq is the number of phonon with energy of 'v, qx, qy and qz are the wave vectors of phonon along the x, y and z axes, and both kx0, n0, l0 and kx, n, l are the quantum numbers of the initial and final states, respectively. With integration substituting for summation, Eq. (8) becomes
& & &
W(E) =
2p D 2 V % dqx dqy dqzq ' Vrs ' 3 kx,n,l ×{ n e − iqy y n0 2 × l e − iqz z l0 2
× d(kx− kx0 − qx )× nq × d(E− E0 − 'v)+ n e − iqy y n0 2 × l e − iqz z l0 2 × d(kx− kx0 +qx ) (nq + 1)× d(E −E0 + 'v)
(9)
Finally, we can simplify Eq. (9) as the following expression W(E) =
& &
(E−E0)2 D2 % dq dq x z 4p 2rs 4' 3 n,l qy 0 × { n e − iqy y n0 2 × l e − iqz z l0 2
× nq × H1 + n e − iqy y n0 2 × l e − iqz z l0 2 ×(nq +1) (10)
× H2 where nq = Fig. 1. Schematic diagram of a GaAs/GaAlAs quantum wire.
1 'v −1 exp kBT
132
Y. He et al. / Materials Science and Engineering B75 (2000) 130–133
H1 =
!
1······E \ E0 0······E B E0
H2 =
!
0······E \ E0 1······E B E0
(for absorption)
(for emission)
3. Results and discussion
Fig. 2. Dependence of intrasubband scattering rate on electron energy. Curves (a)and (b) correspond to the results without and with EEMM, respectively.
Fig. 3. Intrasubband scattering rates (E= 0.3 eV) as a function of Al concentration with and without EEMM. Curves (a) and (b) correspond to the results without and with EEMM, respectively.
Fig. 4. Dependence of the relative deviation of the scattering rates (r1 − r2)/r1 caused by EEMM on Al concentration with the electron energy of E = 0.3 eV. Parameters r1 and r2 stand for the scattering rates without and with EEMM, respectively.
We calculate the intrasubband rates in GaAs/ GaAlx As1 − x quantum wire. All the parameters in the calculation are taken as follows: D= 13.5 eV, rd =5.3 g cm − 1, s= 5370 m s − 1, T =77 K, mo = 0.0657me and m= mo (1.0+ 1.24x), where x is the Al concentration in GaAlx As1 − x. Dependence of intrasubband scattering rates on electron energy is plotted in Fig. 2 where curves (a) and (b) represent the scattering rates for both the without and with EEMM cases, respectively. Curves (a) and (b) are of similar electron energy behavior. When the energy is below the ground state, both of the scattering rates equal zero. With increasing electron energy, the scattering rates increase and reach a maximum at Ea and Eb, then drop quickly. The rapid drop of the rate in curve (a) without EEMM is opposite to the two-dimensional case where the rate shows a flat shape relationship with energy [5]. We attribute the difference to the following reasons. The state density in a two-dimensional system has nothing to do with the electron energy, whereas the state density is inversely proportional to the square root of the energy in a quantum wire, so the rate quickly decreases with increasing electron energy. Taking into account EEMM, the scattering rate becomes smaller with increasing energy, which is attributed to a remarkable drop of the barrier along the z direction because of ascending of wave vector kx along the x direction. With increasing electron energy the scattering rate tends to the two-dimensional case. The ground-state energy is decreased in the EEMM case. This is caused by the decrease of wave vector a along the growth direction (z-axis). Fig. 3 shows that the intrasubband scattering rates increase with increasing Al concentration. The rate under considering EEMM shown in curve (b) is lower than that under not considering EEMM shown in curve (a). This is ascribed to the fact that the electron barrier along the z direction makes a larger contribution to the rate for the latter than for the former, since EEMM can cause a decrease in the effective-barrier height. The relative deviation of the scattering rates r1 and r2 as a function of Al concentration is shown in Fig. 4 where r1 and r2 correspond to the rate without and with EEMM, respectively. With increasing Al concentration, the relative reduction of the rate becomes smaller since the relative change of the effective-barrier height is decreased caused by EEMM.
Y. He et al. / Materials Science and Engineering B75 (2000) 130–133
Fig. 5. Intrasubband scattering rate (E =0.3 eV) versus the well width d along the z direction. Curves (a) and (b) correspond to the results without and with EEMM, respectively.
Dependence of the intrasubband scattering rate on well width d (z direction) is shown in Fig. 5 where curve (a) stands for without EEMM and (b) for with EEMM. The scattering rates equal zero as the well is below the critical width. With increase of well width, the rate jumps to a maximum and then drops quickly. The reasons can be described as follows. Eq. (4) after several-steps deriving can obtain a relationship that wave vector a along the z-axis decreases with increasing well width d, thus leading to lowering of the scattering rate. As we know, the increase of electron state density caused by the increase of well width enhances the scattering rate. However, the former effect makes more of a contribution to the rate than the latter does, leading to a total decrease of the rate. It is obvious that the electron undergoes a weaker bonding due to increase of the well width, thus the rate decreases to tend to the two-dimensional case. Preliminary study on the intersubband scattering rate indicates that it has the same electron energy behavior as those of the intrasubband rate, but the critical energy is different from that of the intrasubband rate. Further work is under way. .
4. Conclusions We used the continuum phonon model and effective-
133
mass approximation to calculate the electron scattering rate in GaAs/GaAlx As1 − x quantum wire via electron– acoustic phonon interaction. Not taking into account electron effective-mass mismatch, the decrease of the scattering rate with increasing electron energy is attributed to the decrease of the energy state density. Taking into account electron effective-mass mismatch, reduction of the rate is due to a remarkable drop of the barrier along the growth direction. The scattering rate increases with increasing Al concentration. The ability of the electron barrier to diminish the rate is larger for the case without EEMM than for that with EEMM. The lowering of the effective-barrier height caused by EEMM makes the relative deviation of the scattering rates between the without and with EEMM cases decrease with increasing Al concentration. The decrease of the scattering rate with increasing well width is attributed to decrease of wave vector a along the growth direction, that is, weakening of the bonding for the electron in the quantum well.
Acknowledgements This work was supported by The National Climbing Project of Fundamental Research of China.
References [1] P.J. Price, Ann. Phys. 133 (1981) 217. [2] H. Fenniche, L. Mandhour, S. Jaziri, R. Bennaceur, Semicond. Sci. Technol. 12 (1997) 796. [3] T. Lu¨, Y. Zheng, Phys. Rev. B 53 (1996) 1438. [4] Y. Zheng, T. Lu, J. Liu, W. Su, Semicond. Sci. Technol. 12 (1997) 1235. [5] Y. Zheng, T. Lu¨, Y. Wang, X. Wu, C. Zhang, W. Su, Semicond. Sci. Technol. 12 (1997) 296. [6] J. Pozela, V. Juciene, K. Pozela, Semicond. Sci. Technol. 10 (1995) 1555. [7] J. Pozela, G. Butkus, V. Juciene, Semicond. Sci. Technol. 9 (1994) 1480. [8] P.A. Knipp, T.L. Reinecke, Phys. Rev. B 48 (1993) 5700. [9] A.A. Darhuber, G. Bauer, P.D. Wang, C.M. Sotomaryer Torres, J. Appl. Phys. 83 (1998) 126. [10] H. Sakaki, T. Someya, H. Akiyama, Y. Nakamura, N. Kondo, D. Kishimoto, Phys. Status Solidi A 184 (1998) 241. [11] A. Yacoby, H.L. Stormer, N.S. Wingreen, L.N. Pfeiffer, K.W. Baldwin, K.W. West, Phys. Rev. Lett. 77 (1996) 4612.