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Anomalous cyclotron resonance linewidth and splitting in heterojunctions displaying the fractional quantum Hall effect
160
Surface Science 170 (1986) 160 166 North-Holland, Amsterdam
ANOMALOUS CYCLOTRON RESONANCE LINEWIDTH AND SPLITrlNG IN HETEROJUNCTIONS DISPLAYING THE FRACTIONAL QUANTUM HALL EFFECT G.L.J.A. R I K K E N , H.W. M Y R O N , C.J.G.M. L A N G E R A K
a n d H. S I G G
High Field Magnet Laboratory and Research Institute for Materials, University of Nijrnegen. Toernooiveld, 6525 ED Nijmegen, The Netherlands
Received 16 July 1985; accepted for publication 13 September 1985
We report anomalous structure in the cyclotron resonance linewidth as a function of filling factor for very high mobility GaAs-AIGaAs heterojunctions. These structures are found at or near the filling factors where the fractional quantum Hall effect in these samples occurs. We have also observed a splitting of the CR for ~,< 1 in a fixed frequency-swept field configuration. We attribute this splitting to the crossing of Landau levels of the two lowest electric subbands. Calculations of this effect on a two-dimensional electron gas have been performed.
There is much interest in the p r o p e r t i e s of the high m o b i l i t y t w o - d i m e n sional electron gas ( T D E G ) since the discovery of the q u a n t u m Hall effect ( Q H E ) [1]. In G a A s - A 1 G a A s heterojunctions the fractional q u a n t u m Hall effect ( F Q H E ) was first discovered [2], which is a t t r i b u t e d to the increasing role of the e l e c t r o n - e l e c t r o n ( e - e ) interactions over that of e l e c t r o n - p h o n o n ( e - p ) interactions. A technique used to s t u d y T D E G s is c y c l o t r o n r e s o n a n c e (CR), which yields i n f o r m a t i o n on the L a n d a u level (LL) width. T h e o r y predicts [3] that this width increases as v ~ , where B is the m a g n e t i c field, a p p l i e d p e r p e n d i c u l a r to the interface. This has been e x p e r i m e n t a l l y verified [4]. M o r e recently, an oscillatory b e h a v i o r of the C R linewidth was o b s e r v e d in h e t e r o j u n c t i o n s [5] and explained in terms of filling factor d e p e n d e n t screening [5,6]. Here, we r e p o r t on the C R linewidth observed on h e t e r o j u n c t i o n s that show the F Q H E . These m e a s u r e m e n t s show structure in the C R linewidth at or n e a r the fractional tilling factors. The m e a s u r e m e n t s were p e r f o r m e d on very high mobility, M B E grown m o d u l a t e d d o p e d G a A s - A I G a A s heterojunction. Samples 1355, 1367 a n d 1368 have respectively carrier densities of 3.1, 2.5 a n d 2.0 x 10 ~ cm -2 a n d mobilities of 0.7, 1.0 a n d 1.0 x 106 c m 2 / V • s at 4.2 K in the dark. T h e s u b s t r a t e cleaning and growth p r o c e d u r e s were d e s c r i b e d in detail elsewhere [7]. The substrates were slightly wedged to eliminate interference effects, a n d the 2 D electron c o n c e n t r a t i o n could be varied b y m e a n s of a s e m i t r a n s p a r a n t 0 0 3 9 - 6 0 2 8 / 8 6 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division) a n d Y a m a d a Science F o u n d a t i o n
G.L.J.A. Rikken et aL / Anomalous CR linewidth in heterojunctions
161
& B\V-~ (10 .3 T 1/2 ) ',}
'/i
50
{- f{4+" l {,"
+,,
t
++'
20 I
I
I
I
/
]
i
/
f'
I
1
50
?,+ / 1 1 Y 2/3 i
I
4,/3 i
1
5/3 i
I
2
7/3 i
V
Fig. 1. Normalized cyclotron resonance linewidth versus filling factor J, at T = 1.4 K for sample 1367 (upper curve) and 1368 (lower curve).
backgate, consisting of 15 n m Cr evaporated on a 6 /~m Mylar foil. This e n a b l e d us to apply electric fields in excess of 106 V / m at the interface, a n d to vary the 2D electron c o n c e n t r a t i o n by 30%. The C R linewidth was d e t e r m i n e d b y recording the t r a n s m i s s i o n of the far infrared radiation ( F I R ) as a function of magnetic field B at T = 1.4 K. The linewidth A B was d e t e r m i n e d as the full width half m i n i m u m ; values as low as 50 m T at B = 12 T were observed, which are the narrowest C R m e a s u r e m e n t s ever reported o n a 2D electron gas. Fig. 1 shows versus filling factor for two different samples, where Be, is the cyclotron resonance field for a given incident energy. The curves show a strong m a x i m u m a r o u n d u = 2 a n d a m i n i m u m somewhat below u - 1, as was
AB/~
162
G.L.J.A. Rikken et al. / Anomalous CR linewidth in heterojunctions
previously reported by Englert et al. [5]. However, there are additional features in these curves, not reported earlier. The linewidth minima can be observed around v = 5 / 3 and structure around v = 4/3. For the 1367 sample a minimum also occurs at v - - 7 / 3 . Nothing can be said about a minimum at v = 2 / 3 and 4 / 5 because of the presence of the level crossing nearby. We find that the cyclotron effective mass, defined as m * = eBcr/Wfir, shows no anomalies at these filing factors, within the experimental accuracy of 0.3%, and seems only to be influenced by the GaAs conduction band nonparabolicity. As yet, no theory exists on the interaction of the highly correlated ground states, believed to be responsible for the FQHE, with electromagnetic radiation. Therefore our previous interpretation of the observed linewidth minima in terms of such ground states is somewhat speculative [8]. Our measurements indicate that at larger filling factors, the CR linewidth has a local minimum, which implies that the reduction of e - e scattering is the dominant effect, whereas at v = 4 / 3 only a shoulder can be seen, which might suggest that the screening reduction comes into play. However, it can also be that e - e scattering becomes relatively unimportant as the Landau orbit radius decreases. Due to the relatively high temperature at which the experiment was performed, only a small part of the electrons will be in the correlated ground state, and the ultimate effect on the CR linewidth can only be small. Therefore these anomalies should be more pronounced at lower temperatures. There seems to be a strong maximum around v = 0.75. Furthermore, there is a strong scattering of the data around that filling factor, which indicates that AB/v~ is here no longer solely a function of the filling factor, but also of the F I R energy. We attribute these effects to the coupling between the second LL of the ground subband and the lowest LL of the first excited electric subband by means of a small magnetic field component parallel to the interface [9,10]. This lifts the degeneracy that occurs at a certain magnetic field. Due to the narrow LLs in the samples used, even a very small parallel field component, which can hardly be avoided with wedged substrates, may result in a significant broadening of the CR. Our measurements show that this broadening is altered by rotating the sample with respect to the magnetic field direction and that the angular dependence of the CR magnetic field is of the same order as observed by Schlesinger et al. [9]. The level-anti-level crossing shown in fig. 2 can be altered by means of a backgate voltage, V~. By varying Vg, the resonance at 13.9 T disappears and the one at 14.2 T appears. Fig. 3 shows the angular dependence of such a splitting at a backgate voltage where both minima are approximately equally strong. The fact that the splitting does not drop to zero with tilt angle is most likely due to a small tilt over the axis perpendicular to both the magnetic field and the rotation axis. In order to explain the observed splitting, we have calculated the subband
G.L.J.A. Rikken et a L / Anomalous CR linewidth in heterojunctions
163
MAGNETIC FIELD ( T ) - - - 12.4 / 12.3
I //
TRANSMISSION (orb. u n i t s ) -
/// ' 1
12.2 a
~
f ~
/
12.1 , . . ~ / 12.0 11.9
11.8
/~ = 53.86 I J m 1~3.5
14.0
] 5'/, 14.5 B ( T )
Fig. 2. Transmission for sample 1367 at T = 1.4 K and 0 = 2.5° for various gate voltages: (a) 105 V/m, (b) 0 V/m; (c) - 105 V/m.
11.7
\ \\\ \\ 5 10 TILT ANGLE (degrees)
Fig. 3. Angular dependence of the split CR, with Vg so that minima are approximately equally strong.
LLs, which is straightforward [11]. We have performed our calculations using a simple triangular potential well with a conveniently chosen slope instead of using a selfconsistent potential approach. The observed splitting can be explained in the following way. Around the crossing region the higher states are linear combinations of the pure subband states, and the energy deviates somewhat from the unperturbed value. The mixing of the pure states makes transitions for an incident energy equal to E m possible at magnetic fields below and above the crossing as long as the hybrid states have not been shifted over much more than the level width by the coupling. At fields too close to the crossing field these shifts are larger, and transitions are no longer possible; at fields too far away the mixing is negligible and transitions are also not possible because of the AN = 1 selection rule. This results in two absorption maxima. The transition probability from the ground state to this mixed state at an energy is calculated using Fermi's Golden Rule, approximating the electron photon interaction with the electric dipole Hamiltonian, while using the as-calculated subband eigenfunctions.
164
G.L.J.A. Rikken et aL / Anomalous CR linewidth in heterojunctions
TRANSITION PROBABILITY (arb. units) 400 300 200 100 0 100
/ / /
80
f/ 60
e= 2.0" ,../
//~/",
40
i
20
/
'\ -.. e:1.5 o
\ ', i \i,,
\.
---_
e=10* I
h
12.5
130
Fig. 4. C a l c u l a t e d a b s o r p t i o n
I
I
i
13.5 140 14.5 MAGNETIC FIELD(T) for El0 = 21.85 meV and linewidth = 75 /~V.
In order to calculate the absorption, we have to assign each eigenstate a lineshape with a given linewidth. Using a homogeneously broadened Lorentzian lineshape gives a very good agreement of the calculated absorption with the C R away from the level crossing. In the region where the hybridization occurs, this seems no longer valid. Assuming a Gaussian lineshape we can reproduce a splitting. Fig. 4 shows the calculated absorption at a given incident energy of 21.85 meV. F r o m fig. 4 one can see that the two absorption maxima are approximately equally strong as the incident F I R energy equals Em. Therefore, by tuning El0 with the backgate voltage so that, at a given F I R wavelength, the transmissions minima are equally deep, one can find E m as a function of backgate voltage, as is shown in fig. 5. Both the value for E m and its tunability found this way are close to the values found by Schlesinger et al. [9]. In
G.LJ.A. Rikken et al. / Anomalous CR linewidth in heterojunctions
165
ENERGY (meV) 23
22
21
20
\ 19 I
L
-2
-1
I
I
I
I
0 1 2 3 xl03 ELECTRIC FIELD (V/cm)
Fig. 5. FIR energy versus Vg at which split resonances are equally strong.
addition, we have reproduced the splitting observed by them using a fixed field-swept frequency configuration. We gratefully acknowledge the assistance of the technical staff of the Nijmegen High Field Magnet Laboratory, K. van Hulst, H. Muileman and J. Rook. We kindly thank G. Weimann for providing heterojunctions for these experiments. Part of this work was supported by the Stichting voor Fundamenteel Onderzoek der Materie (FOM) with the financial support from the Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek (ZWO).
References [1] [2] [3] [4] [5]
K. von Klitzing, G. Dorda and M. Pepper, Phys. Rev. Letters 45 (1980) 494. D.C. Tsui, H . L St~Srmer and A.C. Gossard, Phys. Rev. Letters 48 (1982) 1559. T. Ando and Y. Uemura, J. Phys. Soc. Japan 36 (1974) 959. G. Abstreiter, J.P. Kotthaus, J.F. Koch and G. Dorda, Phys. Rev. B14 (1976) 2480. Th. Englert, J.C. Maan, Ch. Uihlein, D.C. Tsui and A.C. Gossard, Solid State C o m m u n . 46 (1983) 545. [6] R. Lassnig and E. Gornik, Solid State Commun. 47 (1983) 959.
166
G.L.J.A. Rikken et al. / Anomalous CR linewidth in heterojunctions
[7] G. Weimann and W. Schlapp, in: Two-Dimensional Systems, Heterostructures and Superlattices, Springer Series in Solid State Sciences, Vol. 53, Eds. G. Bauer, F. Kuchar and H. Heinrich (Springer, Berlin, 1984) p. 88. [8] G.L.J.A. Rikken, H.W. Myron, P. Wyder, G. Weimann, W. Schlapp, R.E. Horstman and J. Wotter, J. Phys. C (Solid State Phys.) 18 (1985) L175. [9] Z. Schlesinger, J.C.M. Hwang and S.J. Allen, Jr., Phys. Rev. Letters 50 (1983) 2098. [10] T. Ando, Phys. Rev. B19 (1979) 2106. [11] T. Ando, A.B. Fowler and F. Stern, Rev. Mod. Phys. 54 (1982) 437, and references therein.
Surface Science 170 (1986) 160 166 North-Holland, Amsterdam
ANOMALOUS CYCLOTRON RESONANCE LINEWIDTH AND SPLITrlNG IN HETEROJUNCTIONS DISPLAYING THE FRACTIONAL QUANTUM HALL EFFECT G.L.J.A. R I K K E N , H.W. M Y R O N , C.J.G.M. L A N G E R A K
a n d H. S I G G
High Field Magnet Laboratory and Research Institute for Materials, University of Nijrnegen. Toernooiveld, 6525 ED Nijmegen, The Netherlands
Received 16 July 1985; accepted for publication 13 September 1985
We report anomalous structure in the cyclotron resonance linewidth as a function of filling factor for very high mobility GaAs-AIGaAs heterojunctions. These structures are found at or near the filling factors where the fractional quantum Hall effect in these samples occurs. We have also observed a splitting of the CR for ~,< 1 in a fixed frequency-swept field configuration. We attribute this splitting to the crossing of Landau levels of the two lowest electric subbands. Calculations of this effect on a two-dimensional electron gas have been performed.
There is much interest in the p r o p e r t i e s of the high m o b i l i t y t w o - d i m e n sional electron gas ( T D E G ) since the discovery of the q u a n t u m Hall effect ( Q H E ) [1]. In G a A s - A 1 G a A s heterojunctions the fractional q u a n t u m Hall effect ( F Q H E ) was first discovered [2], which is a t t r i b u t e d to the increasing role of the e l e c t r o n - e l e c t r o n ( e - e ) interactions over that of e l e c t r o n - p h o n o n ( e - p ) interactions. A technique used to s t u d y T D E G s is c y c l o t r o n r e s o n a n c e (CR), which yields i n f o r m a t i o n on the L a n d a u level (LL) width. T h e o r y predicts [3] that this width increases as v ~ , where B is the m a g n e t i c field, a p p l i e d p e r p e n d i c u l a r to the interface. This has been e x p e r i m e n t a l l y verified [4]. M o r e recently, an oscillatory b e h a v i o r of the C R linewidth was o b s e r v e d in h e t e r o j u n c t i o n s [5] and explained in terms of filling factor d e p e n d e n t screening [5,6]. Here, we r e p o r t on the C R linewidth observed on h e t e r o j u n c t i o n s that show the F Q H E . These m e a s u r e m e n t s show structure in the C R linewidth at or n e a r the fractional tilling factors. The m e a s u r e m e n t s were p e r f o r m e d on very high mobility, M B E grown m o d u l a t e d d o p e d G a A s - A I G a A s heterojunction. Samples 1355, 1367 a n d 1368 have respectively carrier densities of 3.1, 2.5 a n d 2.0 x 10 ~ cm -2 a n d mobilities of 0.7, 1.0 a n d 1.0 x 106 c m 2 / V • s at 4.2 K in the dark. T h e s u b s t r a t e cleaning and growth p r o c e d u r e s were d e s c r i b e d in detail elsewhere [7]. The substrates were slightly wedged to eliminate interference effects, a n d the 2 D electron c o n c e n t r a t i o n could be varied b y m e a n s of a s e m i t r a n s p a r a n t 0 0 3 9 - 6 0 2 8 / 8 6 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division) a n d Y a m a d a Science F o u n d a t i o n
G.L.J.A. Rikken et aL / Anomalous CR linewidth in heterojunctions
161
& B\V-~ (10 .3 T 1/2 ) ',}
'/i
50
{- f{4+" l {,"
+,,
t
++'
20 I
I
I
I
/
]
i
/
f'
I
1
50
?,+ / 1 1 Y 2/3 i
I
4,/3 i
1
5/3 i
I
2
7/3 i
V
Fig. 1. Normalized cyclotron resonance linewidth versus filling factor J, at T = 1.4 K for sample 1367 (upper curve) and 1368 (lower curve).
backgate, consisting of 15 n m Cr evaporated on a 6 /~m Mylar foil. This e n a b l e d us to apply electric fields in excess of 106 V / m at the interface, a n d to vary the 2D electron c o n c e n t r a t i o n by 30%. The C R linewidth was d e t e r m i n e d b y recording the t r a n s m i s s i o n of the far infrared radiation ( F I R ) as a function of magnetic field B at T = 1.4 K. The linewidth A B was d e t e r m i n e d as the full width half m i n i m u m ; values as low as 50 m T at B = 12 T were observed, which are the narrowest C R m e a s u r e m e n t s ever reported o n a 2D electron gas. Fig. 1 shows versus filling factor for two different samples, where Be, is the cyclotron resonance field for a given incident energy. The curves show a strong m a x i m u m a r o u n d u = 2 a n d a m i n i m u m somewhat below u - 1, as was
AB/~
162
G.L.J.A. Rikken et al. / Anomalous CR linewidth in heterojunctions
previously reported by Englert et al. [5]. However, there are additional features in these curves, not reported earlier. The linewidth minima can be observed around v = 5 / 3 and structure around v = 4/3. For the 1367 sample a minimum also occurs at v - - 7 / 3 . Nothing can be said about a minimum at v = 2 / 3 and 4 / 5 because of the presence of the level crossing nearby. We find that the cyclotron effective mass, defined as m * = eBcr/Wfir, shows no anomalies at these filing factors, within the experimental accuracy of 0.3%, and seems only to be influenced by the GaAs conduction band nonparabolicity. As yet, no theory exists on the interaction of the highly correlated ground states, believed to be responsible for the FQHE, with electromagnetic radiation. Therefore our previous interpretation of the observed linewidth minima in terms of such ground states is somewhat speculative [8]. Our measurements indicate that at larger filling factors, the CR linewidth has a local minimum, which implies that the reduction of e - e scattering is the dominant effect, whereas at v = 4 / 3 only a shoulder can be seen, which might suggest that the screening reduction comes into play. However, it can also be that e - e scattering becomes relatively unimportant as the Landau orbit radius decreases. Due to the relatively high temperature at which the experiment was performed, only a small part of the electrons will be in the correlated ground state, and the ultimate effect on the CR linewidth can only be small. Therefore these anomalies should be more pronounced at lower temperatures. There seems to be a strong maximum around v = 0.75. Furthermore, there is a strong scattering of the data around that filling factor, which indicates that AB/v~ is here no longer solely a function of the filling factor, but also of the F I R energy. We attribute these effects to the coupling between the second LL of the ground subband and the lowest LL of the first excited electric subband by means of a small magnetic field component parallel to the interface [9,10]. This lifts the degeneracy that occurs at a certain magnetic field. Due to the narrow LLs in the samples used, even a very small parallel field component, which can hardly be avoided with wedged substrates, may result in a significant broadening of the CR. Our measurements show that this broadening is altered by rotating the sample with respect to the magnetic field direction and that the angular dependence of the CR magnetic field is of the same order as observed by Schlesinger et al. [9]. The level-anti-level crossing shown in fig. 2 can be altered by means of a backgate voltage, V~. By varying Vg, the resonance at 13.9 T disappears and the one at 14.2 T appears. Fig. 3 shows the angular dependence of such a splitting at a backgate voltage where both minima are approximately equally strong. The fact that the splitting does not drop to zero with tilt angle is most likely due to a small tilt over the axis perpendicular to both the magnetic field and the rotation axis. In order to explain the observed splitting, we have calculated the subband
G.L.J.A. Rikken et a L / Anomalous CR linewidth in heterojunctions
163
MAGNETIC FIELD ( T ) - - - 12.4 / 12.3
I //
TRANSMISSION (orb. u n i t s ) -
/// ' 1
12.2 a
~
f ~
/
12.1 , . . ~ / 12.0 11.9
11.8
/~ = 53.86 I J m 1~3.5
14.0
] 5'/, 14.5 B ( T )
Fig. 2. Transmission for sample 1367 at T = 1.4 K and 0 = 2.5° for various gate voltages: (a) 105 V/m, (b) 0 V/m; (c) - 105 V/m.
11.7
\ \\\ \\ 5 10 TILT ANGLE (degrees)
Fig. 3. Angular dependence of the split CR, with Vg so that minima are approximately equally strong.
LLs, which is straightforward [11]. We have performed our calculations using a simple triangular potential well with a conveniently chosen slope instead of using a selfconsistent potential approach. The observed splitting can be explained in the following way. Around the crossing region the higher states are linear combinations of the pure subband states, and the energy deviates somewhat from the unperturbed value. The mixing of the pure states makes transitions for an incident energy equal to E m possible at magnetic fields below and above the crossing as long as the hybrid states have not been shifted over much more than the level width by the coupling. At fields too close to the crossing field these shifts are larger, and transitions are no longer possible; at fields too far away the mixing is negligible and transitions are also not possible because of the AN = 1 selection rule. This results in two absorption maxima. The transition probability from the ground state to this mixed state at an energy is calculated using Fermi's Golden Rule, approximating the electron photon interaction with the electric dipole Hamiltonian, while using the as-calculated subband eigenfunctions.
164
G.L.J.A. Rikken et aL / Anomalous CR linewidth in heterojunctions
TRANSITION PROBABILITY (arb. units) 400 300 200 100 0 100
/ / /
80
f/ 60
e= 2.0" ,../
//~/",
40
i
20
/
'\ -.. e:1.5 o
\ ', i \i,,
\.
---_
e=10* I
h
12.5
130
Fig. 4. C a l c u l a t e d a b s o r p t i o n
I
I
i
13.5 140 14.5 MAGNETIC FIELD(T) for El0 = 21.85 meV and linewidth = 75 /~V.
In order to calculate the absorption, we have to assign each eigenstate a lineshape with a given linewidth. Using a homogeneously broadened Lorentzian lineshape gives a very good agreement of the calculated absorption with the C R away from the level crossing. In the region where the hybridization occurs, this seems no longer valid. Assuming a Gaussian lineshape we can reproduce a splitting. Fig. 4 shows the calculated absorption at a given incident energy of 21.85 meV. F r o m fig. 4 one can see that the two absorption maxima are approximately equally strong as the incident F I R energy equals Em. Therefore, by tuning El0 with the backgate voltage so that, at a given F I R wavelength, the transmissions minima are equally deep, one can find E m as a function of backgate voltage, as is shown in fig. 5. Both the value for E m and its tunability found this way are close to the values found by Schlesinger et al. [9]. In
G.LJ.A. Rikken et al. / Anomalous CR linewidth in heterojunctions
165
ENERGY (meV) 23
22
21
20
\ 19 I
L
-2
-1
I
I
I
I
0 1 2 3 xl03 ELECTRIC FIELD (V/cm)
Fig. 5. FIR energy versus Vg at which split resonances are equally strong.
addition, we have reproduced the splitting observed by them using a fixed field-swept frequency configuration. We gratefully acknowledge the assistance of the technical staff of the Nijmegen High Field Magnet Laboratory, K. van Hulst, H. Muileman and J. Rook. We kindly thank G. Weimann for providing heterojunctions for these experiments. Part of this work was supported by the Stichting voor Fundamenteel Onderzoek der Materie (FOM) with the financial support from the Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek (ZWO).
References [1] [2] [3] [4] [5]
K. von Klitzing, G. Dorda and M. Pepper, Phys. Rev. Letters 45 (1980) 494. D.C. Tsui, H . L St~Srmer and A.C. Gossard, Phys. Rev. Letters 48 (1982) 1559. T. Ando and Y. Uemura, J. Phys. Soc. Japan 36 (1974) 959. G. Abstreiter, J.P. Kotthaus, J.F. Koch and G. Dorda, Phys. Rev. B14 (1976) 2480. Th. Englert, J.C. Maan, Ch. Uihlein, D.C. Tsui and A.C. Gossard, Solid State C o m m u n . 46 (1983) 545. [6] R. Lassnig and E. Gornik, Solid State Commun. 47 (1983) 959.
166
G.L.J.A. Rikken et al. / Anomalous CR linewidth in heterojunctions
[7] G. Weimann and W. Schlapp, in: Two-Dimensional Systems, Heterostructures and Superlattices, Springer Series in Solid State Sciences, Vol. 53, Eds. G. Bauer, F. Kuchar and H. Heinrich (Springer, Berlin, 1984) p. 88. [8] G.L.J.A. Rikken, H.W. Myron, P. Wyder, G. Weimann, W. Schlapp, R.E. Horstman and J. Wotter, J. Phys. C (Solid State Phys.) 18 (1985) L175. [9] Z. Schlesinger, J.C.M. Hwang and S.J. Allen, Jr., Phys. Rev. Letters 50 (1983) 2098. [10] T. Ando, Phys. Rev. B19 (1979) 2106. [11] T. Ando, A.B. Fowler and F. Stern, Rev. Mod. Phys. 54 (1982) 437, and references therein.