- Email: [email protected]
In0.52Al0.48As heterostructure
Physica E 6 (2000) 767–770
www.elsevier.nl/locate/physe
Observation of the zero- eld spin splitting of the second subband in an inverted In0:53Ga0:47As= In0:52Al0:48As heterostructure C.M. Hua; ∗ , J. Nittaa , T. Akazakia , H. Takayanagia , J. Osakab , P. Pfe erc , W. Zawadzkic b NTT
a NTT Basic Research Labs, 3-1 Wakamiya, Morinosato, Atsugi-shi, Kanagawa 243-0198, Japan System Electronics Laboratories Labs, 3-1 Wakamiya, Morinosato, Atsugi-shi, Kanagawa 243-0198, Japan c Institute of Physics, Polish Academy of Sciences, Al. Lotnikow 32/46, 02-668 Warsaw, Poland
Abstract A gated inverted In0:52 Al0:48 As=In0:53 Ga0:47 As=In0:52 Al0:48 As quantum well is studied via magneto transport. By analysing the gate-voltage-dependent beating pattern observed in the Shubnikov-de Haas oscillation, we determine the gate voltage (or electron concentration) dependence of the spin-orbit coupling parameter . Our experimental data and their analysis show that the band nonparbolicity e ect cannot be neglected. For electron concentrations above 2 × 1012 cm−2 , it causes a reduction of up to 25%. We report for the rst time the value for the second subband. ? 2000 Elsevier Science B.V. All rights reserved. PACS: 73.20.Dx; 71.70.Ej; 75.25.+z Keywords: Two-dimensional electron gas; Shubnikov–de Haas oscillation; Spin–orbit coupling
Spin degeneracy in the single-electron energy spectra of solid is the combined e ect of inversion symmetry in space and time. In semiconductor quantum wells both the bulk inversion asymmetry (BIA) and the structure inversion asymmetry (SIA) break the spatial inversion symmetry and lift the spin degeneracy. The later (known as Bychkov–Rashba spin splitting) can be formulised as spin–orbit interaction of the electron (or hole) moving in the asymmetric con ning potential [1]. It causes both macroscopic e ects like ∗ Corresponding author. Tel.: +81-462-40-3490; fax: +81-462-40-4722. E-mail address: [email protected] (C.M. Hu)
a beating pattern in the Shubnikov–de Haas (SdH) oscillation [2–5] and mesoscopic e ects such as antilocalisation [6]and spin–orbit Berry phase [7,8]. Recently it was found that a surface gate could control the spin–orbit coupling parameter [9 –12]. This is the rst step to realise a spin-transistor proposed by Datta and Das [13]. However, the understanding of this subject is still controversial. For example, a contribution of the average electric eld to the spin splitting is estimated very di erently in di erent theoretical models [10,11, 14 –16]. Experiments on InGaAs=InAlAs [9,10] and InAs=AlSb [17] quantum wells, respectively, found rather di erent behaviour of the gate voltage (or
1386-9477/00/$ - see front matter ? 2000 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 9 9 ) 0 0 2 0 0 - 3
768
C.M. Hu et al. = Physica E 6 (2000) 767–770
electron concentration) dependence of . More precise and comprehensive experimental data are required. In addition, except for a general discussion given by Das et al. [4], the band nonparabolicity effect was often neglected both in the evaluation of the spin splitting from SdH data and in the self-consistent band structure calculation. While earlier work on Bychkov–Rashba spin splitting focused on mostly GaAs=AlGaAs heterostructures [1,14], recently there is growing interest of using InGaAs or InAs quantum wells [2–11,15 –17] where the band nonparabolicity e ect is not negligible due to their smaller energy gap. In this paper, we report on an investigation of the gate voltage Vg dependent SdH oscillations in an inverted In0:52 Al0:48 As=In0:53 Ga0:47 As=In0:52 Al0:48 As quantum well. Fourier analysing the SdH oscillations as functions of 1=B con rms for the rst time the existence of the zero- eld spin splitting of the second electron subband. The gate voltage dependence of the spin–orbit coupling parameter of both the rst and second subband is determined. We nd that taking into account the correction from band nonparbolicity leads to a reduction of up to 25% at high concentrations. This e ect was not reported in previous studies. Our device is a gated Hall bar structure fabricated on an inverted modulation doped In0:52 Al0:48 As= In0:53 Ga0:47 As=In0:52 Al0:48 As quantum well sample [9]. The SdH measurement is performed in lock-in technique with an excitation current of 73 nA at a temperature of 0.4 K. Fig. 1a shows the typical results obtained under di erent gate voltage bias. Beating patterns are observed at the low magnetic eld region, in accordance with a previous study on a di erent sample from the same wafer [9]. The origin of the beating pattern is known to be the zero- eld spin splitting, which results in two closely spaced SdH oscillation frequency components with similar amplitudes. At positive gate voltage, a di erent low SdH oscillation frequency component appears which becomes clearly visible for Vg ¿ 0:5 V. It indicates the occupation of the second subband with small carrier density. In Fig. 1b, we show the correspondent fast Fourier transform (FFT) results performed on the SdH oscillations as functions of 1/B. The (horizontal) frequency axis is normalised to give the unit in the spin-dependent carrier concentration. The FFT data shown here are analysed within the eld range of
Fig. 1. (a) Shubnikov–de Haas oscillations measured at T = 0:4 K with di erent gate voltage applied. (b) The correspondent Fourier power spectra of traces in (a). The horizontal axis is normalised to give the unit in the spin-dependent carrier concentration. Traces are shifted vertically for clarity.
0.7–9 T. To check the contribution from Zeeman term, the data are compared with FFT analysis within the eld range of 0.7–3 T and 0.7–5 T, respectively. We nd the resolution decreases by choosing smaller eld ranges, however the peak positions remain unchanged. With increasing gate voltage, carrier concentration of the rst subband is found to linearly increase till reaching a saturation value of about 2:8 × 1012 cm−2 for Vg ¿ 0:1 V. The clearly resolved double-peak structures allow us to determine the concentration of carriers on the same subband but of di erent spin orientations. Of particular interests is the double-peak structure of the second subband observed for high positive gate voltage. Although it is dicult to identify beating patterns related with the second subband, the FFT analysis shows that the second subband also splits into two spin resolved sublevels. By analysing the spin-dependent density of state taking into account both the spin–orbit interaction and band nonparabolicity e ect [18], i of the ith subband could be determined from the total (ni ) and the di erence (ni ) of the concentrations of the spin-resolved sublevels by the form r p ˜2 p i = ( ni + ni − ni − ni )(1 − i ); (1) ∗ 2 m0 where i = (2˜2 =m∗0 Eg )ni is the normalised modi cation factor arising from the band nonparabolicity. Here m∗0 ≈ 0:042 me and Eg = 0:81 eV are the bandedge
C.M. Hu et al. = Physica E 6 (2000) 767–770
e ective mass and the energy gap for In0:53 Ga0:47 As, respectively. For large Eg or small ni where i ≈ 0; i deduced from Eq. (1) is approximately equal to a form deduced recently from a parabolic band [10,11]. 1 The reason why band nonparabolicity should be considered is straightforward. In the In0:53 Ga0:47 As= In0:52 Al0:48 As quantum well structure studied here with a typical total electron concentration of about 2 × 1012 cm−2 , the nonparabolicity-induced energy correction at the Fermi level can be estimated to be EF2 =Eg ≈ 10 meV. Here EF is the Fermi energy measured from the subband edge. As found in this work this energy correction is larger than the Bychkov– Rashba spin splitting energy at the Fermi level (of about 5 meV). Therefore one has to take into account the modi cation of the density of states by the band nonparabolicity to get the correct value of from the measured electron concentrations. In Fig. 2a and b we plot the carrier concentration of the spin resolved subbands and the spin–orbit coupling parameter as a function of the gate voltage, respectively. In Fig. 1a up- and down-wards triangular marks represent the concentration of electrons on di erent spin-resolved sublevels. For the gate voltage below 1 V, the spin splitting for the second subband was not resolved. In Fig. 1b we comparatively plot the value of determined with (solid marks) and without (open marks) band nonparabolicity correction, respectively. Circles and squares represent the spin–orbit coupling constant of the rst and second subband, respectively. Clearly, the band nonparabolicity e ect is not negligible when the subband is highly populated. With the saturation density of about 2:8 × 1012 cm−2 for the rst subband, the modi cation of 1 due to the band nonparabolicity e ect reaches about 25%. We therefore con rm that in our structure 1 of the rst subband could be modulated from a value of 10 × 10−12 eV m at −1 V to about 5 × 10−12 eV m at +1:5 V. Such a 100% change of the value of could be applied to modulate the spin precession angle from to 2, which is required to get the maximum current modulation in the eld e ect spin transistor [13]. Another commonly used method to determine is to t the measured beating pattern in SdH oscillations with the use of Landau fan chat [2,3,9 –11,17]. Instead 1 The small di erence is caused by an approximation in the integration used in Ref. [8], which is not used in this work.
769
Fig. 2. (a) Up () and down (∇) spin electron concentrations obtained from experiment as a function of the gate voltage. (b) Spin–orbit coupling parameter of the rst (circle) and second (square) subband obtained with (solid) and without (open) band nonparabolicity correction as a function of the gate voltage.
of using the slight di erence of the density of states of the two spin-resolved sublevels, this method relies on the slight di erence in the spin-dependent modi cation of the otherwise equally spaced Landau levels. Similar to the above discussion, if the modi cation of the electron kinetic energy due to the band nonparabolicity is comparable or larger than that due to the spin–orbit coupling, both e ects have to be treated on the same level to get the correct value of . We emphasise that the correction to due to the band nonparabolicity depends on both the band gap energy and the electron concentration. The recent study of a gated InAs=AlSb quantum well [17], where the spin– orbit coupling constant was found to be gate voltage independent, need probably be revised since InAs has a small band gap energy.
770
C.M. Hu et al. = Physica E 6 (2000) 767–770
Finally, we would like to discuss the di erent value of found for the rst and second subband, respectively. Recent theories [15,19] showed that in a square quantum well should be determined by the penetration of the wave function into the barriers and its asymmetry at both interfaces. Since the wave function for the second (higher) subband penetrates more into the barriers than that of the rst subband, Andrada e Silva et al. [19] predict 2 ¿ 1 , if the asymmetry of both wave functions is similar. That is what we observe. An estimation based on the theory of Pfe er and Zawadzki [15] found that 1 can be either smaller or larger than 2 , depending on the detail of the potential shape of the quantum well. A quantitative evaluation of both values would require a detailed knowledge of electric eld distribution in the well, which is at present unknown. In conclusion, the Rashba spin–orbit coupling of the conduction band electrons in an inverted In0:53 Ga0:47 As=In0:52 Al0:48 As heterostructure with different surface gate voltage bias was investigated. Both the concentration and the subband dependence of the spin-orbit coupling parameter is determined. The band nonparabolicity e ect is quantitatively analysed which is found to be not negligible at high density or in heterostructures with small energy gap. Acknowledgements One of us (C.M.H.) would like to thank Y. S. Gui, Z. H. Chen, and P. D. Ye for many helpful discussions.
References [1] Y.A. Bychkov, E.I. Rashba, J. Phys. C 17 (1984) 6039. [2] J. Luo, H. Munekata, F.F. Fang, P.J. Stiles, Phys. Rev. B 38 (1988) 10 142. [3] J. Luo, H. Munekata, F.F. Fang, Phys. Rev. B 41 (1990) 7685. [4] B. Das, D.C. Miller, S. Datta, R. Reifenberger, W.P. Hong, P.K. Bhattacharya, J. Singh, M. Ja e, Phys. Rev. B 39 (1989) 1411. [5] B. Das, D.C. Miller, S. Datta, R. Reifenberger, W.P. Hong, P.K. Bhattacharya, J. Singh, M. Ja e, Phys. Rev. B 41 (1990) 8278. [6] W. Knap et al., Phys. Rev. B 53 (1996) 3912. [7] A.G. Aronov, Y.B. Lyanda-Geller, Phys. Rev. Lett. 70 (1993) 343. [8] A.F. Morpurgo, J.P. Heida, T.M. Klapwijk, B.J. van Wees, Phys. Rev. Lett. 80 (1998) 1050. [9] J. Nitta, T. Akazaki, H. Takayanagi, T. Enoki, Phys. Rev. Lett. 78 (1997) 1335. [10] G. Engels, J. Lange, Th. Schapers, H. Luth, Phys. Rev. B 55 (1997) 1958. [11] G. Engels, J. Lange, Th. Schapers, H. Luth, J. Appl. Phys. 83 (1998) 4324. [12] J.P. Lu, J.B. Yau, S.P. Shukla, M. Shayegan, L. Wissinger, U. Rossler, R. Winkler, Phys. Rev. Lett. 81 (1998) 1282. [13] S. Datta, B. Das, Appl. Phys. Lett. 56 (1990) 665. [14] F. Malcher, G. Lommer, U. Rossler, Superlattices and Microstruct. 2 (1986) 267. [15] P. Pfe er, W. Zawadzki, Phys. Rev. B 59 (1999) R5312. [16] P. Pfe er, W. Zawadzki, Phys. Rev. B 52 (1995) R14 332. [17] J.P. Heida, B.J. van Wees, J.J. Kuipers, T.M. Klapwijk, Phys. Rev. B 57 (1998) 11 911. [18] C.M. Hu, J. Nitta, T. Akazaki, H. Takayanagi, J. Osaka, P. Pfe er, and W. Zawadzki, Phys. Rev. B 15 September 1999, to be published. [19] E.A. de Andrada e Silva, G.C. La Rocca, F. Bassani, Phys. Rev. B 55 (1997) 16 293.
www.elsevier.nl/locate/physe
Observation of the zero- eld spin splitting of the second subband in an inverted In0:53Ga0:47As= In0:52Al0:48As heterostructure C.M. Hua; ∗ , J. Nittaa , T. Akazakia , H. Takayanagia , J. Osakab , P. Pfe erc , W. Zawadzkic b NTT
a NTT Basic Research Labs, 3-1 Wakamiya, Morinosato, Atsugi-shi, Kanagawa 243-0198, Japan System Electronics Laboratories Labs, 3-1 Wakamiya, Morinosato, Atsugi-shi, Kanagawa 243-0198, Japan c Institute of Physics, Polish Academy of Sciences, Al. Lotnikow 32/46, 02-668 Warsaw, Poland
Abstract A gated inverted In0:52 Al0:48 As=In0:53 Ga0:47 As=In0:52 Al0:48 As quantum well is studied via magneto transport. By analysing the gate-voltage-dependent beating pattern observed in the Shubnikov-de Haas oscillation, we determine the gate voltage (or electron concentration) dependence of the spin-orbit coupling parameter . Our experimental data and their analysis show that the band nonparbolicity e ect cannot be neglected. For electron concentrations above 2 × 1012 cm−2 , it causes a reduction of up to 25%. We report for the rst time the value for the second subband. ? 2000 Elsevier Science B.V. All rights reserved. PACS: 73.20.Dx; 71.70.Ej; 75.25.+z Keywords: Two-dimensional electron gas; Shubnikov–de Haas oscillation; Spin–orbit coupling
Spin degeneracy in the single-electron energy spectra of solid is the combined e ect of inversion symmetry in space and time. In semiconductor quantum wells both the bulk inversion asymmetry (BIA) and the structure inversion asymmetry (SIA) break the spatial inversion symmetry and lift the spin degeneracy. The later (known as Bychkov–Rashba spin splitting) can be formulised as spin–orbit interaction of the electron (or hole) moving in the asymmetric con ning potential [1]. It causes both macroscopic e ects like ∗ Corresponding author. Tel.: +81-462-40-3490; fax: +81-462-40-4722. E-mail address: [email protected] (C.M. Hu)
a beating pattern in the Shubnikov–de Haas (SdH) oscillation [2–5] and mesoscopic e ects such as antilocalisation [6]and spin–orbit Berry phase [7,8]. Recently it was found that a surface gate could control the spin–orbit coupling parameter [9 –12]. This is the rst step to realise a spin-transistor proposed by Datta and Das [13]. However, the understanding of this subject is still controversial. For example, a contribution of the average electric eld to the spin splitting is estimated very di erently in di erent theoretical models [10,11, 14 –16]. Experiments on InGaAs=InAlAs [9,10] and InAs=AlSb [17] quantum wells, respectively, found rather di erent behaviour of the gate voltage (or
1386-9477/00/$ - see front matter ? 2000 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 9 9 ) 0 0 2 0 0 - 3
768
C.M. Hu et al. = Physica E 6 (2000) 767–770
electron concentration) dependence of . More precise and comprehensive experimental data are required. In addition, except for a general discussion given by Das et al. [4], the band nonparabolicity effect was often neglected both in the evaluation of the spin splitting from SdH data and in the self-consistent band structure calculation. While earlier work on Bychkov–Rashba spin splitting focused on mostly GaAs=AlGaAs heterostructures [1,14], recently there is growing interest of using InGaAs or InAs quantum wells [2–11,15 –17] where the band nonparabolicity e ect is not negligible due to their smaller energy gap. In this paper, we report on an investigation of the gate voltage Vg dependent SdH oscillations in an inverted In0:52 Al0:48 As=In0:53 Ga0:47 As=In0:52 Al0:48 As quantum well. Fourier analysing the SdH oscillations as functions of 1=B con rms for the rst time the existence of the zero- eld spin splitting of the second electron subband. The gate voltage dependence of the spin–orbit coupling parameter of both the rst and second subband is determined. We nd that taking into account the correction from band nonparbolicity leads to a reduction of up to 25% at high concentrations. This e ect was not reported in previous studies. Our device is a gated Hall bar structure fabricated on an inverted modulation doped In0:52 Al0:48 As= In0:53 Ga0:47 As=In0:52 Al0:48 As quantum well sample [9]. The SdH measurement is performed in lock-in technique with an excitation current of 73 nA at a temperature of 0.4 K. Fig. 1a shows the typical results obtained under di erent gate voltage bias. Beating patterns are observed at the low magnetic eld region, in accordance with a previous study on a di erent sample from the same wafer [9]. The origin of the beating pattern is known to be the zero- eld spin splitting, which results in two closely spaced SdH oscillation frequency components with similar amplitudes. At positive gate voltage, a di erent low SdH oscillation frequency component appears which becomes clearly visible for Vg ¿ 0:5 V. It indicates the occupation of the second subband with small carrier density. In Fig. 1b, we show the correspondent fast Fourier transform (FFT) results performed on the SdH oscillations as functions of 1/B. The (horizontal) frequency axis is normalised to give the unit in the spin-dependent carrier concentration. The FFT data shown here are analysed within the eld range of
Fig. 1. (a) Shubnikov–de Haas oscillations measured at T = 0:4 K with di erent gate voltage applied. (b) The correspondent Fourier power spectra of traces in (a). The horizontal axis is normalised to give the unit in the spin-dependent carrier concentration. Traces are shifted vertically for clarity.
0.7–9 T. To check the contribution from Zeeman term, the data are compared with FFT analysis within the eld range of 0.7–3 T and 0.7–5 T, respectively. We nd the resolution decreases by choosing smaller eld ranges, however the peak positions remain unchanged. With increasing gate voltage, carrier concentration of the rst subband is found to linearly increase till reaching a saturation value of about 2:8 × 1012 cm−2 for Vg ¿ 0:1 V. The clearly resolved double-peak structures allow us to determine the concentration of carriers on the same subband but of di erent spin orientations. Of particular interests is the double-peak structure of the second subband observed for high positive gate voltage. Although it is dicult to identify beating patterns related with the second subband, the FFT analysis shows that the second subband also splits into two spin resolved sublevels. By analysing the spin-dependent density of state taking into account both the spin–orbit interaction and band nonparabolicity e ect [18], i of the ith subband could be determined from the total (ni ) and the di erence (ni ) of the concentrations of the spin-resolved sublevels by the form r p ˜2 p i = ( ni + ni − ni − ni )(1 − i ); (1) ∗ 2 m0 where i = (2˜2 =m∗0 Eg )ni is the normalised modi cation factor arising from the band nonparabolicity. Here m∗0 ≈ 0:042 me and Eg = 0:81 eV are the bandedge
C.M. Hu et al. = Physica E 6 (2000) 767–770
e ective mass and the energy gap for In0:53 Ga0:47 As, respectively. For large Eg or small ni where i ≈ 0; i deduced from Eq. (1) is approximately equal to a form deduced recently from a parabolic band [10,11]. 1 The reason why band nonparabolicity should be considered is straightforward. In the In0:53 Ga0:47 As= In0:52 Al0:48 As quantum well structure studied here with a typical total electron concentration of about 2 × 1012 cm−2 , the nonparabolicity-induced energy correction at the Fermi level can be estimated to be EF2 =Eg ≈ 10 meV. Here EF is the Fermi energy measured from the subband edge. As found in this work this energy correction is larger than the Bychkov– Rashba spin splitting energy at the Fermi level (of about 5 meV). Therefore one has to take into account the modi cation of the density of states by the band nonparabolicity to get the correct value of from the measured electron concentrations. In Fig. 2a and b we plot the carrier concentration of the spin resolved subbands and the spin–orbit coupling parameter as a function of the gate voltage, respectively. In Fig. 1a up- and down-wards triangular marks represent the concentration of electrons on di erent spin-resolved sublevels. For the gate voltage below 1 V, the spin splitting for the second subband was not resolved. In Fig. 1b we comparatively plot the value of determined with (solid marks) and without (open marks) band nonparabolicity correction, respectively. Circles and squares represent the spin–orbit coupling constant of the rst and second subband, respectively. Clearly, the band nonparabolicity e ect is not negligible when the subband is highly populated. With the saturation density of about 2:8 × 1012 cm−2 for the rst subband, the modi cation of 1 due to the band nonparabolicity e ect reaches about 25%. We therefore con rm that in our structure 1 of the rst subband could be modulated from a value of 10 × 10−12 eV m at −1 V to about 5 × 10−12 eV m at +1:5 V. Such a 100% change of the value of could be applied to modulate the spin precession angle from to 2, which is required to get the maximum current modulation in the eld e ect spin transistor [13]. Another commonly used method to determine is to t the measured beating pattern in SdH oscillations with the use of Landau fan chat [2,3,9 –11,17]. Instead 1 The small di erence is caused by an approximation in the integration used in Ref. [8], which is not used in this work.
769
Fig. 2. (a) Up () and down (∇) spin electron concentrations obtained from experiment as a function of the gate voltage. (b) Spin–orbit coupling parameter of the rst (circle) and second (square) subband obtained with (solid) and without (open) band nonparabolicity correction as a function of the gate voltage.
of using the slight di erence of the density of states of the two spin-resolved sublevels, this method relies on the slight di erence in the spin-dependent modi cation of the otherwise equally spaced Landau levels. Similar to the above discussion, if the modi cation of the electron kinetic energy due to the band nonparabolicity is comparable or larger than that due to the spin–orbit coupling, both e ects have to be treated on the same level to get the correct value of . We emphasise that the correction to due to the band nonparabolicity depends on both the band gap energy and the electron concentration. The recent study of a gated InAs=AlSb quantum well [17], where the spin– orbit coupling constant was found to be gate voltage independent, need probably be revised since InAs has a small band gap energy.
770
C.M. Hu et al. = Physica E 6 (2000) 767–770
Finally, we would like to discuss the di erent value of found for the rst and second subband, respectively. Recent theories [15,19] showed that in a square quantum well should be determined by the penetration of the wave function into the barriers and its asymmetry at both interfaces. Since the wave function for the second (higher) subband penetrates more into the barriers than that of the rst subband, Andrada e Silva et al. [19] predict 2 ¿ 1 , if the asymmetry of both wave functions is similar. That is what we observe. An estimation based on the theory of Pfe er and Zawadzki [15] found that 1 can be either smaller or larger than 2 , depending on the detail of the potential shape of the quantum well. A quantitative evaluation of both values would require a detailed knowledge of electric eld distribution in the well, which is at present unknown. In conclusion, the Rashba spin–orbit coupling of the conduction band electrons in an inverted In0:53 Ga0:47 As=In0:52 Al0:48 As heterostructure with different surface gate voltage bias was investigated. Both the concentration and the subband dependence of the spin-orbit coupling parameter is determined. The band nonparabolicity e ect is quantitatively analysed which is found to be not negligible at high density or in heterostructures with small energy gap. Acknowledgements One of us (C.M.H.) would like to thank Y. S. Gui, Z. H. Chen, and P. D. Ye for many helpful discussions.
References [1] Y.A. Bychkov, E.I. Rashba, J. Phys. C 17 (1984) 6039. [2] J. Luo, H. Munekata, F.F. Fang, P.J. Stiles, Phys. Rev. B 38 (1988) 10 142. [3] J. Luo, H. Munekata, F.F. Fang, Phys. Rev. B 41 (1990) 7685. [4] B. Das, D.C. Miller, S. Datta, R. Reifenberger, W.P. Hong, P.K. Bhattacharya, J. Singh, M. Ja e, Phys. Rev. B 39 (1989) 1411. [5] B. Das, D.C. Miller, S. Datta, R. Reifenberger, W.P. Hong, P.K. Bhattacharya, J. Singh, M. Ja e, Phys. Rev. B 41 (1990) 8278. [6] W. Knap et al., Phys. Rev. B 53 (1996) 3912. [7] A.G. Aronov, Y.B. Lyanda-Geller, Phys. Rev. Lett. 70 (1993) 343. [8] A.F. Morpurgo, J.P. Heida, T.M. Klapwijk, B.J. van Wees, Phys. Rev. Lett. 80 (1998) 1050. [9] J. Nitta, T. Akazaki, H. Takayanagi, T. Enoki, Phys. Rev. Lett. 78 (1997) 1335. [10] G. Engels, J. Lange, Th. Schapers, H. Luth, Phys. Rev. B 55 (1997) 1958. [11] G. Engels, J. Lange, Th. Schapers, H. Luth, J. Appl. Phys. 83 (1998) 4324. [12] J.P. Lu, J.B. Yau, S.P. Shukla, M. Shayegan, L. Wissinger, U. Rossler, R. Winkler, Phys. Rev. Lett. 81 (1998) 1282. [13] S. Datta, B. Das, Appl. Phys. Lett. 56 (1990) 665. [14] F. Malcher, G. Lommer, U. Rossler, Superlattices and Microstruct. 2 (1986) 267. [15] P. Pfe er, W. Zawadzki, Phys. Rev. B 59 (1999) R5312. [16] P. Pfe er, W. Zawadzki, Phys. Rev. B 52 (1995) R14 332. [17] J.P. Heida, B.J. van Wees, J.J. Kuipers, T.M. Klapwijk, Phys. Rev. B 57 (1998) 11 911. [18] C.M. Hu, J. Nitta, T. Akazaki, H. Takayanagi, J. Osaka, P. Pfe er, and W. Zawadzki, Phys. Rev. B 15 September 1999, to be published. [19] E.A. de Andrada e Silva, G.C. La Rocca, F. Bassani, Phys. Rev. B 55 (1997) 16 293.