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Magnetotransport in a wide parabolic quantum well
83
Surface Science 229 ( 1990) 83-87 North-Holland
~AGNETO~NSPORT
IN A WIDE PARABOLIC
QUA~UM
WELL
M. SHAYEGAN, T. SAJOTO, J. JO, M. SANTOS and L. ENGEL Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA
Received 1I July 1989; accepted for publication 14 September 1989
Measurements of the magnetoresistance of a high-quality electron system realized in a selectively doped, parabolic Al,Ga, _,As well are reported. Quantum oscillations in the magnetoresistance are analyzed to obtain the electron densities of the electric subbands. These densities are in agreement with the predictions of a self-consistent calculation of the subband structure for this system. With the magnetic field B slightly tilted away from the sample plane, we observe a dramatic m~ifes~tion of the subband-~dau-level coupling. At low B, the magnetic depopulation of the hybrid (elect~c-ma~etic) subbands occurs, while at higher tields, Shubnikov-de Haas-like oscillations (periodic in 1/S) are observed as the Fermi level crosses the quantized energy levels associated with the lowest hybrid subband.
Recently, the fabrication of selectively-doped semiconductor structures which may be considered as a first step towards the realization of a quasi threedimensional (3D) dilute electron system with low disorder was reported [ l-3 1. These electron systems are of considerable interest because a variety of collective phenomena such as charge-density waves, spin-density waves, and Wigner c~stallization have been theoretically proposed to occur in dilute 3Dfiee electron systems under appropriate conditions, e.g., at very low temperatures and in intense applied magnetic fields [ 41. In this paper we report on ( 1) the fabrication of a new structure with improved quality, (2) the measured and calculated subband structure of the electron system, (3 ) the electron states in this system in tilted B, and (4) the observation of the fractional quantum Hall effect in this structure. The structure was grown by molecular beam epitaxy and consists of a wide, undoped, parabolically graded AI,Ga,_,As well bounded by undoped (spacer) and doped layers of A&Gal _fis (y> X) on two sides. The width of the well is 1600 A, x is quadratically varied between zero at the well center to x=0.15 at the well edge, the spacer is 450 8, of A10.,Ga0.6As,and the doping consists of five &layers of Si (each with a density of 1.3 x lO”~rn-~ and separated by 30 A) on each side of the well. In the absence of any space charge, and assuming that the conduction band offset for GaAs/Al,Ga, +As is lin0039-6028/90/$03.50 ( Nosh-Holland )
0 Elsevier Science Publishers B.V.
early proportional to x, this compositional grading in the well is expected to result in a parabolic potential well as shown in fig. 1 (curve a). In order to determine the subband structure of the electron system, we performed self-consistent calculations by solving Schriidinger and Poisson equations [ 3 ] , and taking into account the exchange correlation via local density-functional approximation [ 51. The results of this calculation are shown in fig. 1 and indicate that for the measured area1 density in our structure (n,~l.7xlO” cm-2), two electric subbands are occupied. The experimental and quantitative evidence for the realization of the designed electron system is provided by our magnetotransport measurements. In fig. 2a we present the transverse (pXX)and Hall (p,) resistivities, measured in the conventional manner (B perpendicular to the sample plane ) . The data exhibit the integral and fractional quantum Hall effects, with the positions (in B) of pu minima and pX,,plateaus for B>, 1 T being consistent with an area1 density n,= 1.8~ IO" cm-‘. In the low-lieid range (Bs 1 T), many Landau levels of each subband are occupied. The oscillatory magnetoresistance data in this range are periodic in 1/B and contain more than one oscillation frequency (fig. 2b). We determined these frequencies from the fast Fourier transform (FFT) of the p,.. versus l/B data (fig. 2~). The positions of the peaks in the FFT power spectrum give
84
M. Shayegan et al./Magnetotransport in a parabolic well
Fig. 1. The conduction band edge of the structure in the absence of any space charge is shown by curve a. We assumed that the conduction band offset for GaAs/Al,Ga, _xAs heterostructure is equal to 750x (in units of meV) (ref. [ 61). Curve b shows the self-consistently calculated potential for n,= 1.7 x 10” cmm2. The total charge distribution, calculated by appropriately summing the wave functions of the two occupied subbands is shown by curve c.
the areas of the Fermi surface cross sections (circles in our case), and therefore the electron densities for the different occupied electric subbands (ni for i= 0,1,... ). These measured densities are no= 1.16, n, =0.54 in units of 10” cm-‘. The total density (n,,+n,=1.7~10”cm-~) isingoodagreementwith the high-field data (fig. 2a). The electron mobility deduced from the data of fig. 2 is P!Y 3.0~ 1O5 cm21 V-s, indicating the high quality of the structure. To compare the measured subband densities with the results of the self-consistent calculations, in fig. 2c we have indicated (solid arrows) the expected positions of the FFT peaks (based on the calculation whose results are shown in fig. 1). Considering the uncertainties in the growth parameters and in the parameters used in the calculation, the agreement between the measured and calculated subband densities is quite good. We conclude that our magnetotransport data and the self-consistent calculations provide clear evidence for the realization of a low disorder (high mobility), dilute electron system with a wave function about 500 A wide. We now describe our magnetotransport measurements in this system with B tilted with respect to the sample plane (the x-y plane) [ 7 1. In a narrow quan-
turn well subjected to a moderately strong B, tilted with respect to the x-y plane, the spatial quantization still dominates the shape of the electron wave function. This is true as long as I >> w, where I= (fi/ eB) ‘/* is the magnetic length and w is the well width. The motion in the x-y plane is quantized into Landau orbits with the cyclotron frequency determined by the component of B normal to the plane [S]. In a sufficiently strong B (so that 1CKw), however, the electron orbits are constrained to spiral along the oblique magnetic field lines. We report here the observation of such electron states, which we call oblique states, in our wide quantum well system. The Hamiltonian for the case of a magnetic field oriented at an angle 0 with respect to the x axis in the x-z plane is given by: H= [py+eB (zcos B-xsin +p,2/2m* +ps/2m*
0)]‘/2m*
+ V(z) ,
(1)
where V(z) is the self-consistent confining potential. The general solutions to H are stationary states in the x-z plane and plane waves in they direction. The spectrum is discrete with a degeneracy equal to (eB/ h) sin 0. For the case of a quadratic potential V(z),
M. Shayegan et al./Magnetotransport
85
in a parabolic well
Ml%Bb
B(T)
24
OoO 1
1
0
(cl
+
-A!-I
_I
0
FREQUENCY (T)
5
Fig. 2. Magnetotransport coeffkients pu- and pm are shown in (a). The vertical arrows indicate the Landau-level tilling factors (v) at which the integral or fractional quantum Hall effect is observed. The pu data is shown in detail at low magnetic fields in (b). The FFI power spectrum of the oscillating pu versus l/Bdatain (b) isshownin (c).
this problem can be solved analytically and describes coupled harmonic oscillator and Landau level orbits [ 9,10 1. The low lying states are then a set of equally spaced levels with a spacing of fiQ sin 0 where Q is the frequency of the harmonic oscillator states in zero magnetic field. For the sake of illustration, in fig. 3 we show the energy spectrum calculated for a parabolic well whose zero-field subband energies are the same order as those of our well (we used M2=2.6 meV). The calculation is for 0= 10”. Also shown in fig. 3 is the position of EF as a function of B, using the fact that the degeneracy of each of the quantized energy levels is equal to (2eBlh) sin 8 [ 9,101. In
2
R (T)
3
4
5
Fig. 3. Energy versus B fan diagram corresponding to & 10” otientation is shown. The position of Er as a function of B is also shown. Spin splitting is not included.
principle, if the magnetoresistance is measured at finite 8, it should show oscillations as EF crosses each of the energy levels. In a real system, however, the energy levels will have finite widths (because of disorder). Therefore, the density of states at the Fermi level at low B ( 6 1.8 Tin fig. 3), where a great number of closely-spaced energy levels (corresponding to several subbands) are occupied, will be a nearly smooth function of B, except when a subband is depopulated in which case it has a step-like discontinuity. For small 8, therefore, strong low-field oscillations in p should occur as the subbands are depopulated. At sufficiently high B ( >, 1.5 T in fig. 3), only energy levels corresponding to the lowest subband are occupied. New oscillations may then be expected provided that the width of the levels is small compared to the separation of the energy levels and that the temperature is sufficiently low. These oscillations should be periodic in 1/B since the degeneracy of each level is proportional to B. Our experimental observations (fig. 4) are in agreement with the above predictions. First, for 06 10” we note a strong low-field oscillation (with a minimum at around 2 T) which corresponds to the depopulation of the upper subband. This oscillation
M. Shayegan et al./Magnetotransport in a parabolic well
86
1
MlZS-Bb T=0.4K
0
3
6
9
12
B CT) Fig. 4. Magnetotransport data measured at different orientations of B with respect to the sample plane are shown.
is present even for 6= 0” (dashed arrow). Secondly, assuming that the level degeneracy is equal to (eB/ h ) sin 19,we have indicated in fig. 4 (vertical arrows) the field positions at which an integer number of (spin-split ) levels are expected to be occupied. The minima observed in the p(B) data in fig. 4 agree with the positions of these arrows in most cases. For the self-consistent potential in the wide well described here, I’(z) is expected to be better approximated by a square well. In this case, the Hamiltonian of eq. ( 1) can be solved in the high field limit by using the adiabatic approximation [ 111, and gives the spectrum of the oblique states. Within this approximation, the quantized energies scale as &tan2 0, where E, are the energies of the zero field electric subband states in the square well. As 8+O” and B-co the oblique states go over into plane waves along the field in the lowest bulk Landau level. Since the degener-
acy of each state is given by (eB/h)sin 19,the oblique states should nevertheless produce Shubnikov-de Haas oscillations (periodic in 1/B) as we have observed experimentally. However, since the energy levels of the oblique states for a square well are nonuniformly spaced in energy, accidental overlapping of the spin-split energy levels can occur. This accidental degeneracy can account for the missing minima at some tilling factors in the data of fig. 4. Further work aimed at a more quantitative understanding of the energy level structure of a square well in a tilted magnetic field is planned. Finally, we point out that the fractional quantum Hall effect (FQHE) observed at v = 3 and i is evidence for the high quality of the electron system (fig. 2a). We studied the temperature dependence of the v= 3 minimum in the range 25 mKI TI 1 K. Our preliminary results indicate a very small activation energy (E,S 100 mK), which may be attributed to the disorder in the structure as well as the large thickness of the electron system [ 12 1. We plan to carry on a more detailed study of the temperature dependence of the FQHE in our wide electron systems in the future. Part of this work was performed at the Francis Bitter National Magnet Laboratory, which is supported at MIT by the NSF. We thank H.D. Drew, D.C. Tsui, and K. Karrai for useful discussions and B. Brand& L. Rubin, and J. Moodera for technical assistance. Support of this work by NSF grant nos. ECS8553110 and DMR-8705002, AR0 Grant No. DAAL03-89-K-0036 and the New Jersey Commission on Science and Technology is acknowledged.
References [ 11 M Shayegan,
T. Sajoto, M. Santos and C. Silvestre, Appl. Phys. Lett. 53 (1988) 791; T. Sajoto, J. Jo, M. Santos and M. Shayegan, Appl. Phys. Lett. 55 (1989) 1430. [2] E.G. Gwinn, R.M. Westervelt, P.F. Hopkins, A.J. Rimberg, M. Sundaram and A.C. Gossard, Phys. Rev. B 39 (1989) 6260, and references therein. [ 3 ] T. Sajoto, J. Jo, L. Engel, M. Santos and M. Shayegan, Phys. Rev. B 39 (1989) 10464. [4] B.I. Halperin, Jpn. J. Appl. Phys. 26, Suppl. 26-3 (1987) 1913. [ 51 F. Stern and S. Das Sarma, Phys. Rev. B 30 ( 1984) 840.
h4. Shayegan et al./Magnetotransport [ 6 ] See, e.g., R.C. Miller, AC. Gossard, D.A. Kleinman and 0. Munteanu, Phys. Rev. B 29 (1984) 3740. [ 71 M. Shayegan, T. Sajoto, J. Jo, M. Santos and H.D. Drew, Phys. Rev. B 40 (1989) 3476. [8] T. Ando, A.B. Fowler and F. Stem, Rev. Mod. Phys. 54 (1982) 437.
in a parabolic well
[ 9 ] J.C. Maan, in: Two Dimensional
87
Systems, Heterostructures, and Superlattices, Eds. G. Bauer et al. (Springer, Berlin, 1984) p. 183. [lo] R. Merlin, Solid State Commun. 64 (1987) 99. [ 111 W. Trzeciakowski, M. Baj, S. Huant and L.C. Brunel, Phys. Rev. B 33 (1986) 6846. [ 121 F.C. Zhang and S. Das Sarma, Phys. Rev. B 33 (1986) 2903.
Surface Science 229 ( 1990) 83-87 North-Holland
~AGNETO~NSPORT
IN A WIDE PARABOLIC
QUA~UM
WELL
M. SHAYEGAN, T. SAJOTO, J. JO, M. SANTOS and L. ENGEL Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA
Received 1I July 1989; accepted for publication 14 September 1989
Measurements of the magnetoresistance of a high-quality electron system realized in a selectively doped, parabolic Al,Ga, _,As well are reported. Quantum oscillations in the magnetoresistance are analyzed to obtain the electron densities of the electric subbands. These densities are in agreement with the predictions of a self-consistent calculation of the subband structure for this system. With the magnetic field B slightly tilted away from the sample plane, we observe a dramatic m~ifes~tion of the subband-~dau-level coupling. At low B, the magnetic depopulation of the hybrid (elect~c-ma~etic) subbands occurs, while at higher tields, Shubnikov-de Haas-like oscillations (periodic in 1/S) are observed as the Fermi level crosses the quantized energy levels associated with the lowest hybrid subband.
Recently, the fabrication of selectively-doped semiconductor structures which may be considered as a first step towards the realization of a quasi threedimensional (3D) dilute electron system with low disorder was reported [ l-3 1. These electron systems are of considerable interest because a variety of collective phenomena such as charge-density waves, spin-density waves, and Wigner c~stallization have been theoretically proposed to occur in dilute 3Dfiee electron systems under appropriate conditions, e.g., at very low temperatures and in intense applied magnetic fields [ 41. In this paper we report on ( 1) the fabrication of a new structure with improved quality, (2) the measured and calculated subband structure of the electron system, (3 ) the electron states in this system in tilted B, and (4) the observation of the fractional quantum Hall effect in this structure. The structure was grown by molecular beam epitaxy and consists of a wide, undoped, parabolically graded AI,Ga,_,As well bounded by undoped (spacer) and doped layers of A&Gal _fis (y> X) on two sides. The width of the well is 1600 A, x is quadratically varied between zero at the well center to x=0.15 at the well edge, the spacer is 450 8, of A10.,Ga0.6As,and the doping consists of five &layers of Si (each with a density of 1.3 x lO”~rn-~ and separated by 30 A) on each side of the well. In the absence of any space charge, and assuming that the conduction band offset for GaAs/Al,Ga, +As is lin0039-6028/90/$03.50 ( Nosh-Holland )
0 Elsevier Science Publishers B.V.
early proportional to x, this compositional grading in the well is expected to result in a parabolic potential well as shown in fig. 1 (curve a). In order to determine the subband structure of the electron system, we performed self-consistent calculations by solving Schriidinger and Poisson equations [ 3 ] , and taking into account the exchange correlation via local density-functional approximation [ 51. The results of this calculation are shown in fig. 1 and indicate that for the measured area1 density in our structure (n,~l.7xlO” cm-2), two electric subbands are occupied. The experimental and quantitative evidence for the realization of the designed electron system is provided by our magnetotransport measurements. In fig. 2a we present the transverse (pXX)and Hall (p,) resistivities, measured in the conventional manner (B perpendicular to the sample plane ) . The data exhibit the integral and fractional quantum Hall effects, with the positions (in B) of pu minima and pX,,plateaus for B>, 1 T being consistent with an area1 density n,= 1.8~ IO" cm-‘. In the low-lieid range (Bs 1 T), many Landau levels of each subband are occupied. The oscillatory magnetoresistance data in this range are periodic in 1/B and contain more than one oscillation frequency (fig. 2b). We determined these frequencies from the fast Fourier transform (FFT) of the p,.. versus l/B data (fig. 2~). The positions of the peaks in the FFT power spectrum give
84
M. Shayegan et al./Magnetotransport in a parabolic well
Fig. 1. The conduction band edge of the structure in the absence of any space charge is shown by curve a. We assumed that the conduction band offset for GaAs/Al,Ga, _xAs heterostructure is equal to 750x (in units of meV) (ref. [ 61). Curve b shows the self-consistently calculated potential for n,= 1.7 x 10” cmm2. The total charge distribution, calculated by appropriately summing the wave functions of the two occupied subbands is shown by curve c.
the areas of the Fermi surface cross sections (circles in our case), and therefore the electron densities for the different occupied electric subbands (ni for i= 0,1,... ). These measured densities are no= 1.16, n, =0.54 in units of 10” cm-‘. The total density (n,,+n,=1.7~10”cm-~) isingoodagreementwith the high-field data (fig. 2a). The electron mobility deduced from the data of fig. 2 is P!Y 3.0~ 1O5 cm21 V-s, indicating the high quality of the structure. To compare the measured subband densities with the results of the self-consistent calculations, in fig. 2c we have indicated (solid arrows) the expected positions of the FFT peaks (based on the calculation whose results are shown in fig. 1). Considering the uncertainties in the growth parameters and in the parameters used in the calculation, the agreement between the measured and calculated subband densities is quite good. We conclude that our magnetotransport data and the self-consistent calculations provide clear evidence for the realization of a low disorder (high mobility), dilute electron system with a wave function about 500 A wide. We now describe our magnetotransport measurements in this system with B tilted with respect to the sample plane (the x-y plane) [ 7 1. In a narrow quan-
turn well subjected to a moderately strong B, tilted with respect to the x-y plane, the spatial quantization still dominates the shape of the electron wave function. This is true as long as I >> w, where I= (fi/ eB) ‘/* is the magnetic length and w is the well width. The motion in the x-y plane is quantized into Landau orbits with the cyclotron frequency determined by the component of B normal to the plane [S]. In a sufficiently strong B (so that 1CKw), however, the electron orbits are constrained to spiral along the oblique magnetic field lines. We report here the observation of such electron states, which we call oblique states, in our wide quantum well system. The Hamiltonian for the case of a magnetic field oriented at an angle 0 with respect to the x axis in the x-z plane is given by: H= [py+eB (zcos B-xsin +p,2/2m* +ps/2m*
0)]‘/2m*
+ V(z) ,
(1)
where V(z) is the self-consistent confining potential. The general solutions to H are stationary states in the x-z plane and plane waves in they direction. The spectrum is discrete with a degeneracy equal to (eB/ h) sin 0. For the case of a quadratic potential V(z),
M. Shayegan et al./Magnetotransport
85
in a parabolic well
Ml%Bb
B(T)
24
OoO 1
1
0
(cl
+
-A!-I
_I
0
FREQUENCY (T)
5
Fig. 2. Magnetotransport coeffkients pu- and pm are shown in (a). The vertical arrows indicate the Landau-level tilling factors (v) at which the integral or fractional quantum Hall effect is observed. The pu data is shown in detail at low magnetic fields in (b). The FFI power spectrum of the oscillating pu versus l/Bdatain (b) isshownin (c).
this problem can be solved analytically and describes coupled harmonic oscillator and Landau level orbits [ 9,10 1. The low lying states are then a set of equally spaced levels with a spacing of fiQ sin 0 where Q is the frequency of the harmonic oscillator states in zero magnetic field. For the sake of illustration, in fig. 3 we show the energy spectrum calculated for a parabolic well whose zero-field subband energies are the same order as those of our well (we used M2=2.6 meV). The calculation is for 0= 10”. Also shown in fig. 3 is the position of EF as a function of B, using the fact that the degeneracy of each of the quantized energy levels is equal to (2eBlh) sin 8 [ 9,101. In
2
R (T)
3
4
5
Fig. 3. Energy versus B fan diagram corresponding to & 10” otientation is shown. The position of Er as a function of B is also shown. Spin splitting is not included.
principle, if the magnetoresistance is measured at finite 8, it should show oscillations as EF crosses each of the energy levels. In a real system, however, the energy levels will have finite widths (because of disorder). Therefore, the density of states at the Fermi level at low B ( 6 1.8 Tin fig. 3), where a great number of closely-spaced energy levels (corresponding to several subbands) are occupied, will be a nearly smooth function of B, except when a subband is depopulated in which case it has a step-like discontinuity. For small 8, therefore, strong low-field oscillations in p should occur as the subbands are depopulated. At sufficiently high B ( >, 1.5 T in fig. 3), only energy levels corresponding to the lowest subband are occupied. New oscillations may then be expected provided that the width of the levels is small compared to the separation of the energy levels and that the temperature is sufficiently low. These oscillations should be periodic in 1/B since the degeneracy of each level is proportional to B. Our experimental observations (fig. 4) are in agreement with the above predictions. First, for 06 10” we note a strong low-field oscillation (with a minimum at around 2 T) which corresponds to the depopulation of the upper subband. This oscillation
M. Shayegan et al./Magnetotransport in a parabolic well
86
1
MlZS-Bb T=0.4K
0
3
6
9
12
B CT) Fig. 4. Magnetotransport data measured at different orientations of B with respect to the sample plane are shown.
is present even for 6= 0” (dashed arrow). Secondly, assuming that the level degeneracy is equal to (eB/ h ) sin 19,we have indicated in fig. 4 (vertical arrows) the field positions at which an integer number of (spin-split ) levels are expected to be occupied. The minima observed in the p(B) data in fig. 4 agree with the positions of these arrows in most cases. For the self-consistent potential in the wide well described here, I’(z) is expected to be better approximated by a square well. In this case, the Hamiltonian of eq. ( 1) can be solved in the high field limit by using the adiabatic approximation [ 111, and gives the spectrum of the oblique states. Within this approximation, the quantized energies scale as &tan2 0, where E, are the energies of the zero field electric subband states in the square well. As 8+O” and B-co the oblique states go over into plane waves along the field in the lowest bulk Landau level. Since the degener-
acy of each state is given by (eB/h)sin 19,the oblique states should nevertheless produce Shubnikov-de Haas oscillations (periodic in 1/B) as we have observed experimentally. However, since the energy levels of the oblique states for a square well are nonuniformly spaced in energy, accidental overlapping of the spin-split energy levels can occur. This accidental degeneracy can account for the missing minima at some tilling factors in the data of fig. 4. Further work aimed at a more quantitative understanding of the energy level structure of a square well in a tilted magnetic field is planned. Finally, we point out that the fractional quantum Hall effect (FQHE) observed at v = 3 and i is evidence for the high quality of the electron system (fig. 2a). We studied the temperature dependence of the v= 3 minimum in the range 25 mKI TI 1 K. Our preliminary results indicate a very small activation energy (E,S 100 mK), which may be attributed to the disorder in the structure as well as the large thickness of the electron system [ 12 1. We plan to carry on a more detailed study of the temperature dependence of the FQHE in our wide electron systems in the future. Part of this work was performed at the Francis Bitter National Magnet Laboratory, which is supported at MIT by the NSF. We thank H.D. Drew, D.C. Tsui, and K. Karrai for useful discussions and B. Brand& L. Rubin, and J. Moodera for technical assistance. Support of this work by NSF grant nos. ECS8553110 and DMR-8705002, AR0 Grant No. DAAL03-89-K-0036 and the New Jersey Commission on Science and Technology is acknowledged.
References [ 11 M Shayegan,
T. Sajoto, M. Santos and C. Silvestre, Appl. Phys. Lett. 53 (1988) 791; T. Sajoto, J. Jo, M. Santos and M. Shayegan, Appl. Phys. Lett. 55 (1989) 1430. [2] E.G. Gwinn, R.M. Westervelt, P.F. Hopkins, A.J. Rimberg, M. Sundaram and A.C. Gossard, Phys. Rev. B 39 (1989) 6260, and references therein. [ 3 ] T. Sajoto, J. Jo, L. Engel, M. Santos and M. Shayegan, Phys. Rev. B 39 (1989) 10464. [4] B.I. Halperin, Jpn. J. Appl. Phys. 26, Suppl. 26-3 (1987) 1913. [ 51 F. Stern and S. Das Sarma, Phys. Rev. B 30 ( 1984) 840.
h4. Shayegan et al./Magnetotransport [ 6 ] See, e.g., R.C. Miller, AC. Gossard, D.A. Kleinman and 0. Munteanu, Phys. Rev. B 29 (1984) 3740. [ 71 M. Shayegan, T. Sajoto, J. Jo, M. Santos and H.D. Drew, Phys. Rev. B 40 (1989) 3476. [8] T. Ando, A.B. Fowler and F. Stem, Rev. Mod. Phys. 54 (1982) 437.
in a parabolic well
[ 9 ] J.C. Maan, in: Two Dimensional
87
Systems, Heterostructures, and Superlattices, Eds. G. Bauer et al. (Springer, Berlin, 1984) p. 183. [lo] R. Merlin, Solid State Commun. 64 (1987) 99. [ 111 W. Trzeciakowski, M. Baj, S. Huant and L.C. Brunel, Phys. Rev. B 33 (1986) 6846. [ 121 F.C. Zhang and S. Das Sarma, Phys. Rev. B 33 (1986) 2903.