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Screening of collective cyclotron resonance in inversion layers on InSb
0038-1098/84 $3.00 + .00 Pergamon Press Ltd.
Solid State Communications, Vol. 49, No. 7, pp. 707-709, 1984. Printed in Great Britain.
SCREENING OF COLLECTIVE CYCLOTRON RESONANCE IN INVERSION LAYERS ON InSb M. Horst, U. Merkt and J.P. Kotthaus Institut for Angewandte Physlk, Universit~it Hamburg, D-2000 Hamburg 36, Federal Republic of Germany (Received 21 October 1983 by B. Miihlschlegel)
Cyclotron Resonance (CR) of inversion electrons on InSb is studied in parallel magnetic fields with normally incident light. Whereas in this Voigt configuration collective CR is observed in thin bulk samples, we observe single particle CR in the case of inversion layers. The apparent screening of the collective mode is explained by the inhomogeneity of the inversion electron gas in crossed electric and magnetic fields. CYCLOTRON RESONANCE in thin bulk crystals is usually detected in the infrared transmission with normally incident light of wavevector k. Two principally different geometries are commonly employed, namely the Faraday and the Voigt configuration. When the static magnetic field B is applied in the Faraday configuration (k [I B) single particle CR is observed at the resonance frequency 60c = eB/m*. In the Voigt configuration (k ± B) the magnetic field lies in the film plane and the cyclotron motion of individual carriers is coupled by their macroscopic depolarization field [ 1]. With the plasma frequency 60p the resonance of this collective CR appears at the frequency (602 + 6o2)1/2. In polar semiconductors, the macroscopic depolarization field also provides strong coupling to LO phonons of long wavelength. Consequently, normal modes with frequencies co_+that result from the mixing of collective CR and LO phonons will be excited in Voigt configuration [1 ]. However, as long as the light frequency 60 strongly differs from the LO phonon frequency, the coupling to LO phonons can be ignored. Both Faraday and Voigt configuration have extensively been studied in bulk InSb and the collective modes were observed at the positions co_+that were calculated taking into account the macroscopic depolarization field of the cyclotron motion [1,2]. In inversion layers of MOS-structures the surface electric field F s and the steep potential barrier at the semiconductor-oxide interface quantize the electron motion perpendicular to the semiconductor surface, but still allow free motion parallel to it. This leads to electric subbands with two-dimensional continua of states [3]. The most simple description of these electric subbands is obtained in the triangular potential approximation [4]. In this model, the electric length L, that is essentially the inversion layer thickness, is given by L = (h2/2m*eFs) 1/3,
Fs ~-- ens/eoes,
n s being the density, m* the effective mass of the
carriers and e s the dielectric constant of the semiconductor. Cyclotron motion of inversion electrons has mainly been studied in Faraday configuration [5, 6], because of the restrictions imposed by the quasi 2-D nature of this electron gas. However, as a rule of thumb, cyclotron motion also becomes possible in Voigt configuration, if the cyclotron radius l is less than the electric length L [7]. We have recently studied this situation more thoroughly and have calculated the hybrid electric and magnetic subbands [8]. The subband structure is governed by the dimensionless parameter kDl, where kD = m*Fs/hB is the wavevector belonging to the classical drift velocity v D = Fs/B in crossed electric and magnetic fields. The physical meaning of the kDl parameter is further illustrated by its relation kDl = (l/L)3/2 to the electric and magnetic length. Figure 1 shows the first two hybrid subbands for a situation that applies to the present experiments at low carrier densities (n s = 2.5 x 1011 cm -2, h60e = 44.3 meV). Unlike the bulk case, the surface levels are not degenerate, but the energy and the subband splitting depend on the center coordinate zofl. For center coordinates zo/l ~ + 1 the motion of electrons can be visualized as skipping trajectories (see Fig. 1) that come from periodic specular reflection at the surface. Because of their similarity with those in purely electric subbands [7], we will call such states 2-D like. For center coordinates Zofl >~ + 1, the wavefunctions of the surface electrons coincide with those of bulk Landau electrons [9] and their classical trajectories can be visualized as Landau circles in a frame moving with the drift velocity v 9 (see Fig. 1). The energy spectrum of these 3-D like electrons is E ""
h 2k~2
2m*
+ h60c(i + 1/2) +
m*v~
2
+ eFsz o.
Note, that the dispersion h2k2x/2m * is the same for all electrons, also for center coordinates zofl ~ + 1. Because the polarization vector of the incident 707
(1)
(2)
708
SCREENING OF COLLECTIVE CYCLOTRON RESONANCE z=o
E
I p
hZkxz
-
I
I
I
lnSb(111)
huh: = 44.3meV T = 4.2K
--'£~m" \
I
Vol. 49, No. 7
n s (1012cm-2
8
ht°c
I,
0.04
10/
0.02 0.01
I
-2
I
-1
/
I
I
0
1
2
I
I, Z o / I
I
I
5
Fig. 1. Surface levels (i = 0, 1) of InSb inversion electrons in a parallel magnetic field (n~ = 2.5 × 1011 cm -2, B = 7.0 T). Occupied states are indicated by the hatched region. Classical trajectories for center coordinates Zofl "~ 0 and Zofl ~ + 1 are shown in a frame moving with the drift velocity. radiation lies in the plane of the inversion layer, only resonances o f 3-D-like electrons can be observed, if the small coupling of the parallel and perpendicular motion due to nonparabolicity is neglected [10]. Excitation of 2-D-like electrons would require a polarization vector perpendicular to the surface. The excitations o f the 3-D electrons are expected to be cyclotron resonances, because the corresponding wavefunctions are identical with Landau oscillations in bulk crystals. In addition, because of the Voigt configuration, coupling to the plasma, i.e. collective CR has been expected [11 ]. Here we experimentally demonstrate that screening effects completely suppress collective CR and that the resonance observed is the single particle resonance. This astonishing effect is explained by the influence of the 2-D-like electrons that only weakly contribute to the total absorption, but strongly influence the observed 3-D-like CR. The experiments are performed on Cd-doped (NA 3 x 1014 cm -3) InSb (1 11) platelets with silicon dioxide gate insulators and semitransparent NiCr gates [12]. The transmission spectra are recorded at fixed inversion electron densities n s and laser energy hco in a sweep of the magnetic field B at liquid helium temperatures. For the p_resent experiments we have chosen a line o f a discharge molecular laser (H20) with a relatively high
m F
. . . . . . . . . . .
I
I
I
I
0
2
B(TESLA)
6
8
10
Fig. 2. Absorption spectra of inversion electrons on InSb in parallel magnetic fields. The traces have been successively displaced upward for clarity. The small dip at B = 6.4 T results from a bound hole transition in the p-InSb substrate. photon energy (hco = 44.3 meV) to achieve high resonance magnetic fields. The laser radiation is incident perpendicular to the surface and is unpolarized. This can be tolerated because we know from previous experiments that only the component E ± B is responsible for the resonant structures [8]. Some spectra for various electron densities n8 are presented in Fig. 2. The change in transmission AT caused by the presence of inversion electrons is normalized to the transmission of the sample without inversion electrons in zero magnetic field. The small structure at B = 6.4 T is due to a bound hole transition in the p-InSb substrate [13]. The most striking feature of the present spectra is the sharp resonance whose position (m* = 0.0182) is totally independent of density within the precision of our measurements (~ 1%). The position coincides with single particle CR in n-type InSb [2]. There must be a very effective screening of collective CR as can be realized by a simple estimation of the plasma energy h ~ v. If the plasma energy is calculated with the density ns/L , hcov ~ 1 0 - 1 0 0 m e V is obtained in the n~ = 101110 '2 cm -2 range. These plasma energies have to be compared with the laser energy h w = 44.3 meV and accordingly should cause drastic plasma shift. However, no effect at all is experimentally seen.
Vol. 49,'No. 7
SCREENING OF COLLECTIVE CYCLOTRON RESONANCE
Before proposing a solution to this problem, we briefly discuss the observed line shapes, that in turn provide a hint to a solution. At low carrier densities the resonances are very similar to those expected for classical CR but with some weak field dependent background absorption. At high carrier densities (n 8 >~ 101: cm -2) the background absorption increases, but the sharp resonance does not. The absorption of the sharp resonance remains essentially constant (--AT/T ~ 3%) but the line shape broadens on the high field side. These effects can qualitiatively be understood in terms of the surface band structure in Fig. l, if the nonparabolicity of InSb is included. The broad background is presumably caused by Drude absorption of 2-D-like electrons and by transitions of 2-D-like electrons (Zofl ~ 1 in Fig. 1) from the ground to the first excited hybrid subband. Such transitions are weakly allowed by nonparabolicity, despite the chosen polarization of the incident light [ 1]. The asymmetry of the sharp resonance is expected to result from nonparabolicity. Depending on their center coordinate 3-D-like electrons that contribute to the sharp resonance occupy a relatively broad range ofkxvalues. The electrons, therefore, represent also heavier masses as compared to the subband edge mass at k x = O. These heavier masses contribute to the CR at higher magnetic fields. A similar effect is expected because of the spin splitting of hybrid subbands that has been ignored in the calculations underlying Fig. 1. From the discussion of the line shapes it is clear that only part of the induced electrons, namely the 3-Dlike, contribute to the sharp CR signal. This is confirmed by the strength of the signal. If one fits the n s = 2.5 × 1011 cm -2 curve with a classical CR one finds that only about 20% of the induced electrons participate in the sharp resonance. We think that all the other electrons are essentially 2-D-like and provide the very effective screening of the macroscopic depolarization field that the 3-Dlike electrons would like to establish. Thus, the inhomogeneity of the electron gas provides the basic reason for the missing plasma shifts. The mechanism of screening can be visualized by typical classical trajectories as those in Fig. 1. Most trajectories actually permeate each other and are not in phase, because the 2-D-trajectories have much higher frequencies ( ~ 2we) than the 3-D cyclotron motion. All electrons can only move in phase if there is a common subband splitting hwc, i.e. no 2-D-like states (Zofl ~ + I) are occupied. This will be the case if the Fermi energy EF is much smaller than the cyclotron energy hwc. From equation 2 we obtain (kDl ~ 1)
12.
EF/h6°c "~ 181/az'r2"tn21S/a*~2/3~s /, :
13.
,
where a* is the effective Bohr radius of the
(3)
709
semiconductor. Experimentally this situation can not be reached in InSb inversion layers even in very high magnetic fields, since at low densities n s ~ 1011 cm -2 surface mobilities decrease and the depletion potential [3] becomes important. In conclusion, we have experimentally demonstrated that there is no collective CR in Voigt configuration in inversion layers as long as the electric length L is still comparable to the cyclotron radius l. Instead of this, single particle CR is observed. This is explained by the inhomogeneity of the inversion electron gas in crossed electric and magnetic fields. The interesting question arises: Is collective CR likely to develop under conditions when L >> l in space charge layers? For an experimental test accumulation layers on a semiconductor that combines small effective mass and high dielectric constant with the smallest possible nonparabolicity seem to be the best candidates. Corresponding experiments are planned for the near future. Acknowledgement - We thank F. Koch for stimulating discussions on plasma effects in surface cyclotron resonance. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
R. Kaplan, E.D. Palik, R.F. Wallis, S. Iwasa, E. Burstein & Y. Sawada, Phys. Rev. Lett. 18, 159 (1967). C.R. Pidgeon, Handbook o f Semiconductors, Vol. 2, (Edited by M. Balkanski) pp. 223-328. NorthHolland, Amsterdam (1980). T. Ando, A.B. Fowler & F. Stern, Rev. Mod. Phys. 54,437 (1982). F. Stern, Phys. Rev. B5, 4891 (1972). A. Daerr, J.P. Kotthaus & J.F. Koch, Solid State Commun. 17,455 (1975). M. Horst, U. Merkt & J.P. Kotthaus,Phys. Rev. Lett. 50,754 (1983). F. Koch, Physics in High Magnetic Fields, (Edited by S. Chikazumi & N. Miura) pp. 262-273. Springer, Berlin, (1981). J.H. Crasemann, U. Merkt & J.P. Kotthaus, Phys. Rev. B28, 2271 (1983). W. Zawadzki, Physics o f Solids in lntense Magnetic Fields, (Edited by E.D. Haidemenakis) pp. 311328. Plenum, New York, (1969). Y. Takada, K. Arai, N. Uchimura & Y. Uemura, J. Phys. Soc. Japan 49, 1851 (1980); Y. Takada, J. Phys. Soc. Japan. 50, 1998 (1981). W. Zhao, C. Mazure, J.F. Koch, J. Ziegler & H. Maier, Proceedings o f the fifth International Conference on Electronic Properties o f TwoDimensional Systems, Oxford, England, (to be published) (1983). U. Mackens & U. Merkt, Thin Solid Films 97, 53 (1982). C.L. Litter, D.G. Seiler, R. Kaplan & R.J. Wagner, Phys. Rev. B27, 7473 (1983).
Solid State Communications, Vol. 49, No. 7, pp. 707-709, 1984. Printed in Great Britain.
SCREENING OF COLLECTIVE CYCLOTRON RESONANCE IN INVERSION LAYERS ON InSb M. Horst, U. Merkt and J.P. Kotthaus Institut for Angewandte Physlk, Universit~it Hamburg, D-2000 Hamburg 36, Federal Republic of Germany (Received 21 October 1983 by B. Miihlschlegel)
Cyclotron Resonance (CR) of inversion electrons on InSb is studied in parallel magnetic fields with normally incident light. Whereas in this Voigt configuration collective CR is observed in thin bulk samples, we observe single particle CR in the case of inversion layers. The apparent screening of the collective mode is explained by the inhomogeneity of the inversion electron gas in crossed electric and magnetic fields. CYCLOTRON RESONANCE in thin bulk crystals is usually detected in the infrared transmission with normally incident light of wavevector k. Two principally different geometries are commonly employed, namely the Faraday and the Voigt configuration. When the static magnetic field B is applied in the Faraday configuration (k [I B) single particle CR is observed at the resonance frequency 60c = eB/m*. In the Voigt configuration (k ± B) the magnetic field lies in the film plane and the cyclotron motion of individual carriers is coupled by their macroscopic depolarization field [ 1]. With the plasma frequency 60p the resonance of this collective CR appears at the frequency (602 + 6o2)1/2. In polar semiconductors, the macroscopic depolarization field also provides strong coupling to LO phonons of long wavelength. Consequently, normal modes with frequencies co_+that result from the mixing of collective CR and LO phonons will be excited in Voigt configuration [1 ]. However, as long as the light frequency 60 strongly differs from the LO phonon frequency, the coupling to LO phonons can be ignored. Both Faraday and Voigt configuration have extensively been studied in bulk InSb and the collective modes were observed at the positions co_+that were calculated taking into account the macroscopic depolarization field of the cyclotron motion [1,2]. In inversion layers of MOS-structures the surface electric field F s and the steep potential barrier at the semiconductor-oxide interface quantize the electron motion perpendicular to the semiconductor surface, but still allow free motion parallel to it. This leads to electric subbands with two-dimensional continua of states [3]. The most simple description of these electric subbands is obtained in the triangular potential approximation [4]. In this model, the electric length L, that is essentially the inversion layer thickness, is given by L = (h2/2m*eFs) 1/3,
Fs ~-- ens/eoes,
n s being the density, m* the effective mass of the
carriers and e s the dielectric constant of the semiconductor. Cyclotron motion of inversion electrons has mainly been studied in Faraday configuration [5, 6], because of the restrictions imposed by the quasi 2-D nature of this electron gas. However, as a rule of thumb, cyclotron motion also becomes possible in Voigt configuration, if the cyclotron radius l is less than the electric length L [7]. We have recently studied this situation more thoroughly and have calculated the hybrid electric and magnetic subbands [8]. The subband structure is governed by the dimensionless parameter kDl, where kD = m*Fs/hB is the wavevector belonging to the classical drift velocity v D = Fs/B in crossed electric and magnetic fields. The physical meaning of the kDl parameter is further illustrated by its relation kDl = (l/L)3/2 to the electric and magnetic length. Figure 1 shows the first two hybrid subbands for a situation that applies to the present experiments at low carrier densities (n s = 2.5 x 1011 cm -2, h60e = 44.3 meV). Unlike the bulk case, the surface levels are not degenerate, but the energy and the subband splitting depend on the center coordinate zofl. For center coordinates zo/l ~ + 1 the motion of electrons can be visualized as skipping trajectories (see Fig. 1) that come from periodic specular reflection at the surface. Because of their similarity with those in purely electric subbands [7], we will call such states 2-D like. For center coordinates Zofl >~ + 1, the wavefunctions of the surface electrons coincide with those of bulk Landau electrons [9] and their classical trajectories can be visualized as Landau circles in a frame moving with the drift velocity v 9 (see Fig. 1). The energy spectrum of these 3-D like electrons is E ""
h 2k~2
2m*
+ h60c(i + 1/2) +
m*v~
2
+ eFsz o.
Note, that the dispersion h2k2x/2m * is the same for all electrons, also for center coordinates zofl ~ + 1. Because the polarization vector of the incident 707
(1)
(2)
708
SCREENING OF COLLECTIVE CYCLOTRON RESONANCE z=o
E
I p
hZkxz
-
I
I
I
lnSb(111)
huh: = 44.3meV T = 4.2K
--'£~m" \
I
Vol. 49, No. 7
n s (1012cm-2
8
ht°c
I,
0.04
10/
0.02 0.01
I
-2
I
-1
/
I
I
0
1
2
I
I, Z o / I
I
I
5
Fig. 1. Surface levels (i = 0, 1) of InSb inversion electrons in a parallel magnetic field (n~ = 2.5 × 1011 cm -2, B = 7.0 T). Occupied states are indicated by the hatched region. Classical trajectories for center coordinates Zofl "~ 0 and Zofl ~ + 1 are shown in a frame moving with the drift velocity. radiation lies in the plane of the inversion layer, only resonances o f 3-D-like electrons can be observed, if the small coupling of the parallel and perpendicular motion due to nonparabolicity is neglected [10]. Excitation of 2-D-like electrons would require a polarization vector perpendicular to the surface. The excitations o f the 3-D electrons are expected to be cyclotron resonances, because the corresponding wavefunctions are identical with Landau oscillations in bulk crystals. In addition, because of the Voigt configuration, coupling to the plasma, i.e. collective CR has been expected [11 ]. Here we experimentally demonstrate that screening effects completely suppress collective CR and that the resonance observed is the single particle resonance. This astonishing effect is explained by the influence of the 2-D-like electrons that only weakly contribute to the total absorption, but strongly influence the observed 3-D-like CR. The experiments are performed on Cd-doped (NA 3 x 1014 cm -3) InSb (1 11) platelets with silicon dioxide gate insulators and semitransparent NiCr gates [12]. The transmission spectra are recorded at fixed inversion electron densities n s and laser energy hco in a sweep of the magnetic field B at liquid helium temperatures. For the p_resent experiments we have chosen a line o f a discharge molecular laser (H20) with a relatively high
m F
. . . . . . . . . . .
I
I
I
I
0
2
B(TESLA)
6
8
10
Fig. 2. Absorption spectra of inversion electrons on InSb in parallel magnetic fields. The traces have been successively displaced upward for clarity. The small dip at B = 6.4 T results from a bound hole transition in the p-InSb substrate. photon energy (hco = 44.3 meV) to achieve high resonance magnetic fields. The laser radiation is incident perpendicular to the surface and is unpolarized. This can be tolerated because we know from previous experiments that only the component E ± B is responsible for the resonant structures [8]. Some spectra for various electron densities n8 are presented in Fig. 2. The change in transmission AT caused by the presence of inversion electrons is normalized to the transmission of the sample without inversion electrons in zero magnetic field. The small structure at B = 6.4 T is due to a bound hole transition in the p-InSb substrate [13]. The most striking feature of the present spectra is the sharp resonance whose position (m* = 0.0182) is totally independent of density within the precision of our measurements (~ 1%). The position coincides with single particle CR in n-type InSb [2]. There must be a very effective screening of collective CR as can be realized by a simple estimation of the plasma energy h ~ v. If the plasma energy is calculated with the density ns/L , hcov ~ 1 0 - 1 0 0 m e V is obtained in the n~ = 101110 '2 cm -2 range. These plasma energies have to be compared with the laser energy h w = 44.3 meV and accordingly should cause drastic plasma shift. However, no effect at all is experimentally seen.
Vol. 49,'No. 7
SCREENING OF COLLECTIVE CYCLOTRON RESONANCE
Before proposing a solution to this problem, we briefly discuss the observed line shapes, that in turn provide a hint to a solution. At low carrier densities the resonances are very similar to those expected for classical CR but with some weak field dependent background absorption. At high carrier densities (n 8 >~ 101: cm -2) the background absorption increases, but the sharp resonance does not. The absorption of the sharp resonance remains essentially constant (--AT/T ~ 3%) but the line shape broadens on the high field side. These effects can qualitiatively be understood in terms of the surface band structure in Fig. l, if the nonparabolicity of InSb is included. The broad background is presumably caused by Drude absorption of 2-D-like electrons and by transitions of 2-D-like electrons (Zofl ~ 1 in Fig. 1) from the ground to the first excited hybrid subband. Such transitions are weakly allowed by nonparabolicity, despite the chosen polarization of the incident light [ 1]. The asymmetry of the sharp resonance is expected to result from nonparabolicity. Depending on their center coordinate 3-D-like electrons that contribute to the sharp resonance occupy a relatively broad range ofkxvalues. The electrons, therefore, represent also heavier masses as compared to the subband edge mass at k x = O. These heavier masses contribute to the CR at higher magnetic fields. A similar effect is expected because of the spin splitting of hybrid subbands that has been ignored in the calculations underlying Fig. 1. From the discussion of the line shapes it is clear that only part of the induced electrons, namely the 3-Dlike, contribute to the sharp CR signal. This is confirmed by the strength of the signal. If one fits the n s = 2.5 × 1011 cm -2 curve with a classical CR one finds that only about 20% of the induced electrons participate in the sharp resonance. We think that all the other electrons are essentially 2-D-like and provide the very effective screening of the macroscopic depolarization field that the 3-Dlike electrons would like to establish. Thus, the inhomogeneity of the electron gas provides the basic reason for the missing plasma shifts. The mechanism of screening can be visualized by typical classical trajectories as those in Fig. 1. Most trajectories actually permeate each other and are not in phase, because the 2-D-trajectories have much higher frequencies ( ~ 2we) than the 3-D cyclotron motion. All electrons can only move in phase if there is a common subband splitting hwc, i.e. no 2-D-like states (Zofl ~ + I) are occupied. This will be the case if the Fermi energy EF is much smaller than the cyclotron energy hwc. From equation 2 we obtain (kDl ~ 1)
12.
EF/h6°c "~ 181/az'r2"tn21S/a*~2/3~s /, :
13.
,
where a* is the effective Bohr radius of the
(3)
709
semiconductor. Experimentally this situation can not be reached in InSb inversion layers even in very high magnetic fields, since at low densities n s ~ 1011 cm -2 surface mobilities decrease and the depletion potential [3] becomes important. In conclusion, we have experimentally demonstrated that there is no collective CR in Voigt configuration in inversion layers as long as the electric length L is still comparable to the cyclotron radius l. Instead of this, single particle CR is observed. This is explained by the inhomogeneity of the inversion electron gas in crossed electric and magnetic fields. The interesting question arises: Is collective CR likely to develop under conditions when L >> l in space charge layers? For an experimental test accumulation layers on a semiconductor that combines small effective mass and high dielectric constant with the smallest possible nonparabolicity seem to be the best candidates. Corresponding experiments are planned for the near future. Acknowledgement - We thank F. Koch for stimulating discussions on plasma effects in surface cyclotron resonance. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
R. Kaplan, E.D. Palik, R.F. Wallis, S. Iwasa, E. Burstein & Y. Sawada, Phys. Rev. Lett. 18, 159 (1967). C.R. Pidgeon, Handbook o f Semiconductors, Vol. 2, (Edited by M. Balkanski) pp. 223-328. NorthHolland, Amsterdam (1980). T. Ando, A.B. Fowler & F. Stern, Rev. Mod. Phys. 54,437 (1982). F. Stern, Phys. Rev. B5, 4891 (1972). A. Daerr, J.P. Kotthaus & J.F. Koch, Solid State Commun. 17,455 (1975). M. Horst, U. Merkt & J.P. Kotthaus,Phys. Rev. Lett. 50,754 (1983). F. Koch, Physics in High Magnetic Fields, (Edited by S. Chikazumi & N. Miura) pp. 262-273. Springer, Berlin, (1981). J.H. Crasemann, U. Merkt & J.P. Kotthaus, Phys. Rev. B28, 2271 (1983). W. Zawadzki, Physics o f Solids in lntense Magnetic Fields, (Edited by E.D. Haidemenakis) pp. 311328. Plenum, New York, (1969). Y. Takada, K. Arai, N. Uchimura & Y. Uemura, J. Phys. Soc. Japan 49, 1851 (1980); Y. Takada, J. Phys. Soc. Japan. 50, 1998 (1981). W. Zhao, C. Mazure, J.F. Koch, J. Ziegler & H. Maier, Proceedings o f the fifth International Conference on Electronic Properties o f TwoDimensional Systems, Oxford, England, (to be published) (1983). U. Mackens & U. Merkt, Thin Solid Films 97, 53 (1982). C.L. Litter, D.G. Seiler, R. Kaplan & R.J. Wagner, Phys. Rev. B27, 7473 (1983).