- Email: [email protected]
Ground state properties of interacting electrons in semiconductor quantum dots: Exact and unrestricted hartree-fock results
Solid-State Electronics Vol. 37. Nos 4-6, pp. 1179-1182, 1994
Pergamon
Copyright ~ 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038o1101/94 $6.00 + 0.00
0038-1101(93)E0039-4
GROUND STATE PROPERTIES OF INTERACTING ELECTRONS IN SEMICONDUCTOR QUANTUM DOTS: EXACT AND UNRESTRICTED HARTREE-FOCK RESULTS L. MARTiN-MORENOj, J. J. PALACIOS2, C. TEJEDOR2, G. CHIAPPE 3 and E. Louis 3 qnstituto de Ciencia de Materiales (CSIC) and 2Departamento de Fisica de la Materia Condensada, Universidad Aut6noma de Madrid, Cantoblanco, 28049 Madrid and 3Departamento de Fisica Aplicada, Universidad de Alicante, Apartado 99, 03080 Alicante, Spain Abstract--The ground state properties of up to 8 interacting electrons confined to square quantum boxes are calculated by exact diagonalization of the full Hamiltonian. The behavior with the magnetic field is analyzed as well as its possible implications on the measurables properties of such systems. Finally, comparison with an Unrestricted Hartree--Fock approach reveals the latter one can be used to go beyond the few-electrons limit of exact calculations.
Properties of quantum dots (QD) have been extensively studied in the last couple of years, both experimental and theoretically[l]. Measurements have been performed mainly on the transport properties, but some work has appeared, as well, on the capacitance of such systems[2]. In both cases, it has been shown that the properties of QD are greatly altered by adding or removing a single electron. The calculations that have appeared in the literature on the properties of such systems can be classified in two groups: large number of electrons and small number of electrons in the QD. In the first case, the approximation of representing the electron-electron interaction by a classical capacitive term has proved to be qualitatively satisfactory[3,4]. On the other hand, when the number of electrons is small and correlations are strong, this model is expected to fail. This has been shown in exact calculations performed in this regime, considering the full electron-electron interaction, for small number of electrons N (N ~< 5), in one-dimensional QDs[5], and in two-dimensional (2D) QD with high symmetry in the confining potential[6-8]. Two quantities related to the isolated dot are of special interest, due to their relation to measurable properties (e.g. the conductance): the energies and the spectral weights[9,10]:
tL~(N) = <,t, ~,N - '~1 dil ,t, ~"> \.-~'~'~ ,,ja' -.'~('~- ,,>
(1)
(UHF) approach can be used as an accurate way to calculate properties of the dot, especially in the intermediate- and high-N regime. We present exact calculations for N up to 8, for different magnetic fields. We compare these exact results with those obtained with the U H F approach and, given the good agreement found, we extend the study to higher N. We consider electrons confined in a 2D square box, under the influence of a perpendicular magnetic field B (B was represented by a symmetric gauge vector potential), and interacting via the Coulomb interaction, which has been cut off to take into account the finite width of 2D gases ( ~ 200/~) in real semiconductor systems. In order to obtain eigenvalues and eigenfunctions, the Hamiltionian is first expressed in a second quantisation form in the basis set of oneelectron wave functions. The calculation of each matrix element of the interacting part of the Hamiltonian in this basis set reduces to the calculation of four-dimensional integrals involving singleparticle wave functions. The ground state wave function and those of the firsts few excited states for N fixed were calculated by using Lanczos method. The U H F approximation consist of finding the selfconsistent one-electron wave functions that are eigen-functions of the Hamiltonian:
where i,j index one-electron eigenstates, a, fl index many-body eigenstates at fixed N, and di(d~) is the anhilation (creation) operator for the one-electron eigenstate i. The purpose of this paper is double: (i) to show the changes that a perpendicular magnetic field induces on the electronic properties of QDs, i.e. on eigenenergies, eigenstates and spectral weights [in this paper we concentrate on A ( N ) = Z,A°;°(N)] and (ii) to show that the Unrestricted Hartree-Fock 1179
H U"F = y~ ~,.od~od,.° La
+
Z
v,,,,
i,j,k,l,a,°'
× (a~o dk.o -½) -
~ i,j,k,l,°,o"
vok,(4.oa~.o, - ½)
x (a~,.4.o)
(2)
1180
L. MARTiN-MORENO et al.
t6 15-
(a)
2 e/ectro'n,s]
32 //
/.'"
(b)
//" ~/
14/
3 electrons
30
"'14 "
,....,;"
/
////~/
;=28
~12/.//
~q26
.......
-
24
10-
I
I
I
]
I
I
l
2
3
4
5
6
22
I
I
I
I
I
I
I
!
2
3
4
5
6
7
¢/~,
~/~,
Fig. 1. First lowest-laying many-body energies for a square quantum dot as a function of the magnetic flux though the dot for (a) two electrons in the dot (S singlet and T triplet) and (b) three electrons in the dot (D doublet and C quadruplet). For the dot considered (see text), the unit of energy = h -'~z:i(2 mL: ) = 0.56 meV.
where a labels the electron spin, E,.~ are the eigenenergies for n o n i n t e r a c t i n g electrons, V,ik~ are matrix elements of the C o u l o m b interaction in the oneelectron basis set, a n d the expectation values, and therefore the H a m i l t o n i a n , have to be recalculated in each iteration until convergency is reached. All parameters, as dielectric c o n s t a n t and effective mass, are taken to have the values in the 2D .-
1.0
electron gas formed in A I G a A s - G a A s . The results presented in this paper are for a square dot of area L 2 = 0.01 Ixm". Figure 1 shows the calculated eigenvalues as a function of the magnetic flux q$ t h r o u g h the dot [for the dot considered, this corresponds to a perpendicular magnetic field B=O.51dk/cho(T), with 4)0 = h/e, h being the Plank c o n s t a n t a n d e the electron charge] for (a) N = 2, a n d (b) N = 3. 12
(b)
(a)
~Ni@: 0.6-
;
:
[
:
I
I .... I;
,iii !
i
0,6-
<~
o.4- ,
;
0.4-
o.20.0
0.2!
0
I
0I 2
~
2
3.5 .....
3 --
4 --
5
6
7
8
9
3
4
5
6
7
8
9
5
6
7
8
9
N
N
3.53.0-
2.52.0~I~1.0O.5
2.54
¢a2.o! 1.5d 1.oI . . . . . .
I
2
3
4
5
N
6
?
8
o.o-
9
0
1
2
3
4
N
Fig. 2. Spectral weight A(N) and total spin S(N) for (a) 0 = 3q~0. and (b) = 15~ 0, for a square quantum dot calculated from the exact many-body ground state wave functions.
1181
Ground state properties of interacting electrons Table I. Ground state energies in meV (exact and UHF) N electrons in a square box of 0.1 pm of side with a transversal magnetic flux ~ (in units of the flux quanta ~b0 =h/e). The overlap between the exact and U H F wavefunctions is also given. 12 monoelectronic wave functions are included in the calculations (24 including spin). The Zeeman contribution has been neglected d~'~b~,
2
3
4
5
6
7
8
Energy (UHF) Energy (exact) (~rt~t:Hi)
N
6.0021 5.8481 0.919
13.371 13.131 0.972
23.309 22.952 0.959
36.279 35.878 0.958
51.659 51.217 0.959
69.825 69.423 0.961
90.010 89.958 0.978
Energy (UHF) Energy (exact) (~EI~L,HF)
12.884 12.774 0.736
22.851 22.726 0.735
35.022 34.994 0.991
50.739 50.615 0.824
68.840 68.719 0.833
89.654 89.525 0.918
111.55 I I 1.32 0.934
one-electron wave functions (24 including spin) were used throughout the calculations. In both figures there are crossings and anticrossings between energy levels. Crossing occur between eigenvalues corresponding to levels with different total spin (as the Zeeman energy term has not been taken into account in the calculation, spin multiplets are degenerate), while anticrossings occur between levels with the same total spin, and are reminescent of the crossings between states with different angular momentum that have been found in parabolic 2D QD. Both after a crossing or anticrossing between the two lowest laying eigenstates, the character of the ground state wave function changes abruptly. As we show later on, this abrupt change is going to be reflected on the values of spectral weights. Figure 2 shows A(N) and the spin total S(N) up to N = 8 for two different magnetic fluxes: (a) ~b =3q~ 0, and (b) ~ = 15~0, (~0 = h/e, h being the Planck constant and e the electron charge) which are typical representatives of the low- and high-magnetic-field regimes, respectively. In the low B case, the ground state eigenfunctions have minimum total spin, that is, S ( N ) = 0 for N even and S ( N ) = I / 2 for N odd. In this case, A(N) is a very smooth, although not monotonous, function of N. For high B, the ground state eigenfunctions are spin polarised, for small N. Even when, in order to obtain the results shown in Fig. 2(b), the electron-electron interaction has been fully taken into account, this may be understood so as to minimize the exchange energy (the relevance of the kinetic energy contribution to the total energy decreases with B, as a consequence of the condensation of one-electron energy levels into Landau levels). Spectral weights change rapidly as function of N, for N small, in this regime. For example, while A(I) = I always, A(2) decreases as B increases, reflecting the fact that the occupancy of the first one-electron eigenstate decreases for ~b > 3.9 ~b0 (i.e. after the first crossing between the ground and the first excited state). The reason for this is that the charge is more spread out when occupying one-electron edge states than when occupying the first one-electron state, which is localised in the center of the QD. Eventually, for large enough N, all the firsts one-electron levels are occupied and, after that, A(N) seems to tend to 1 for large N. Notice that Kinaret et al.[9] predict a quenching of A(N) in a different situation: they calculated A(N) for states
such that the filling factor remains constant, and these are not necessarily the ground states of the many-body system. It is also worth mentioning the disappearance of A(8) due to the change of S for N = 8 electrons (in comparison with N = 7) by more than 1/2. Our U H F calculations produce an excellent approximation to the ground state energy as well to the wave function as shown in Table i. The U H F ground state energy approximates better than 2% the exact ground state energy, and it is always in between the energies of the ground state and the first excited state in the exact many-body calculation. The excellent agreement found for the U H F calculations leads us to extent the calculation of both the manybody energies and spectral weights to large N. The A(N) and S(N) calculated from the U H F wave functions are shown in Fig. 3 for g, = 15~b0. Hartree-Fock reflects qualitatively all the features found in the exact calculations and the agreement between them is qualitatively rather good for A(N), when N t> 3. U H F calculation supports the idea that A(N) tends to 1 for large N for the dot considered, 1.0 0.80.6-
i !iii ii
O.Pl 0
i!i!
i!i~ii~i! !!ii!i~iiiii]: 5
I
I
10
15
20
25
15
20
25
N
3.53.02.,5
2.0-
1.5t.00.5
0.00
5
10
N Fig. 3. As for Fig. 2(b), but using the ground state wave functions obtained from the Unrestricted Hartree-Fock method and extended to 25 electrons.
1182
L. MARTiN-MORENOet al.
while showing a clear paramagnetic trend of the ground state. An observation, that deserves a more detailed investigation in the future, is that the overlaps between the exact and the U H F ground state wave functions remain above 0.9 in all cases, except close to the fractional filling factors v considered (defining v = N / O ) , suggesting a very correlated groundstate for these cases. A more detailed description of these effects for different geometries and of the intermediate B regime, where A(N) presents abrupt changes as a function of B, will be published elsewhere. To conclude, we have calculated the many-body few lowest-lying eigenenergies and their correspondent eigenstates in a square 2D Q D for up to 8 electrons and different magnetic fields. While for small B the spectral weights are smooth function of N, for high B they are strongly modulated. A similar effect is expected to appear in the conductance through these systems at low temperatures. Finally, it has been used the Hartree-Fock approximation for
the calculation of properties on "typical" dots defined in the A I G a A s - G a A s electron gas.
REFERENCES
1. M. A. Kastner, Phys. Today, January, p. 24, and references therein (1993). 2. R. C. Ashoori et al., Phys. Ret'. Lett. 68, 3088 (1992). 3. C. W. J. Beenakker, Phys. Rev. B 44, 1646 (1991). 4. P. L. McEuen et al,, Phys. Rev. Lett. 66, 1926 (1991). 5. W. Hausler, B. Kramer and J. Masek, Z. Phys. B 85, 435 (1991), 6. P. A. Maksym and T. Chakraborty, Phys. Rev. Lett. 65, 108 (1990); ibid., Phys. Rev. B 45, 1947 (1992). 7. U. Merkt, J. Huser and M. Wagner, Phys. Rev. B 43, 7320 (1991); M. Wagner, U. Merkt and A. V. Chaplik, Phys. Rev. B 45, 1951 (1992). 8. D. Pfannkuche and R. R. Gerhardts, Phys. Rev. B 44, 13132 (1991). 9. J. M. Kinaret et al., Phys. Rev. B 45, 9489 (1992); ibid. 46, 4681 (1992). 10. J. J. Palacios, L. Martin-Moreno and C. Tejedor, Surf Sei. In press.
Pergamon
Copyright ~ 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038o1101/94 $6.00 + 0.00
0038-1101(93)E0039-4
GROUND STATE PROPERTIES OF INTERACTING ELECTRONS IN SEMICONDUCTOR QUANTUM DOTS: EXACT AND UNRESTRICTED HARTREE-FOCK RESULTS L. MARTiN-MORENOj, J. J. PALACIOS2, C. TEJEDOR2, G. CHIAPPE 3 and E. Louis 3 qnstituto de Ciencia de Materiales (CSIC) and 2Departamento de Fisica de la Materia Condensada, Universidad Aut6noma de Madrid, Cantoblanco, 28049 Madrid and 3Departamento de Fisica Aplicada, Universidad de Alicante, Apartado 99, 03080 Alicante, Spain Abstract--The ground state properties of up to 8 interacting electrons confined to square quantum boxes are calculated by exact diagonalization of the full Hamiltonian. The behavior with the magnetic field is analyzed as well as its possible implications on the measurables properties of such systems. Finally, comparison with an Unrestricted Hartree--Fock approach reveals the latter one can be used to go beyond the few-electrons limit of exact calculations.
Properties of quantum dots (QD) have been extensively studied in the last couple of years, both experimental and theoretically[l]. Measurements have been performed mainly on the transport properties, but some work has appeared, as well, on the capacitance of such systems[2]. In both cases, it has been shown that the properties of QD are greatly altered by adding or removing a single electron. The calculations that have appeared in the literature on the properties of such systems can be classified in two groups: large number of electrons and small number of electrons in the QD. In the first case, the approximation of representing the electron-electron interaction by a classical capacitive term has proved to be qualitatively satisfactory[3,4]. On the other hand, when the number of electrons is small and correlations are strong, this model is expected to fail. This has been shown in exact calculations performed in this regime, considering the full electron-electron interaction, for small number of electrons N (N ~< 5), in one-dimensional QDs[5], and in two-dimensional (2D) QD with high symmetry in the confining potential[6-8]. Two quantities related to the isolated dot are of special interest, due to their relation to measurable properties (e.g. the conductance): the energies and the spectral weights[9,10]:
tL~(N) = <,t, ~,N - '~1 dil ,t, ~"> \.-~'~'~ ,,ja' -.'~('~- ,,>
(1)
(UHF) approach can be used as an accurate way to calculate properties of the dot, especially in the intermediate- and high-N regime. We present exact calculations for N up to 8, for different magnetic fields. We compare these exact results with those obtained with the U H F approach and, given the good agreement found, we extend the study to higher N. We consider electrons confined in a 2D square box, under the influence of a perpendicular magnetic field B (B was represented by a symmetric gauge vector potential), and interacting via the Coulomb interaction, which has been cut off to take into account the finite width of 2D gases ( ~ 200/~) in real semiconductor systems. In order to obtain eigenvalues and eigenfunctions, the Hamiltionian is first expressed in a second quantisation form in the basis set of oneelectron wave functions. The calculation of each matrix element of the interacting part of the Hamiltonian in this basis set reduces to the calculation of four-dimensional integrals involving singleparticle wave functions. The ground state wave function and those of the firsts few excited states for N fixed were calculated by using Lanczos method. The U H F approximation consist of finding the selfconsistent one-electron wave functions that are eigen-functions of the Hamiltonian:
where i,j index one-electron eigenstates, a, fl index many-body eigenstates at fixed N, and di(d~) is the anhilation (creation) operator for the one-electron eigenstate i. The purpose of this paper is double: (i) to show the changes that a perpendicular magnetic field induces on the electronic properties of QDs, i.e. on eigenenergies, eigenstates and spectral weights [in this paper we concentrate on A ( N ) = Z,A°;°(N)] and (ii) to show that the Unrestricted Hartree-Fock 1179
H U"F = y~ ~,.od~od,.° La
+
Z
v,,,,
i,j,k,l,a,°'
× (a~o dk.o -½
~ i,j,k,l,°,o"
vok,(4.oa~.o, - ½
x (a~,.4.o)
(2)
1180
L. MARTiN-MORENO et al.
t6 15-
(a)
2 e/ectro'n,s]
32 //
/.'"
(b)
//" ~/
14/
3 electrons
30
"'14 "
,....,;"
/
////~/
;=28
~12/.//
~q26
.......
-
24
10-
I
I
I
]
I
I
l
2
3
4
5
6
22
I
I
I
I
I
I
I
!
2
3
4
5
6
7
¢/~,
~/~,
Fig. 1. First lowest-laying many-body energies for a square quantum dot as a function of the magnetic flux though the dot for (a) two electrons in the dot (S singlet and T triplet) and (b) three electrons in the dot (D doublet and C quadruplet). For the dot considered (see text), the unit of energy = h -'~z:i(2 mL: ) = 0.56 meV.
where a labels the electron spin, E,.~ are the eigenenergies for n o n i n t e r a c t i n g electrons, V,ik~ are matrix elements of the C o u l o m b interaction in the oneelectron basis set, a n d the expectation values, and therefore the H a m i l t o n i a n , have to be recalculated in each iteration until convergency is reached. All parameters, as dielectric c o n s t a n t and effective mass, are taken to have the values in the 2D .-
1.0
electron gas formed in A I G a A s - G a A s . The results presented in this paper are for a square dot of area L 2 = 0.01 Ixm". Figure 1 shows the calculated eigenvalues as a function of the magnetic flux q$ t h r o u g h the dot [for the dot considered, this corresponds to a perpendicular magnetic field B=O.51dk/cho(T), with 4)0 = h/e, h being the Plank c o n s t a n t a n d e the electron charge] for (a) N = 2, a n d (b) N = 3. 12
(b)
(a)
~Ni@: 0.6-
;
:
[
:
I
I .... I;
,iii !
i
0,6-
<~
o.4- ,
;
0.4-
o.20.0
0.2!
0
I
0I 2
~
2
3.5 .....
3 --
4 --
5
6
7
8
9
3
4
5
6
7
8
9
5
6
7
8
9
N
N
3.53.0-
2.52.0~I~1.0O.5
2.54
¢a2.o! 1.5d 1.oI . . . . . .
I
2
3
4
5
N
6
?
8
o.o-
9
0
1
2
3
4
N
Fig. 2. Spectral weight A(N) and total spin S(N) for (a) 0 = 3q~0. and (b) = 15~ 0, for a square quantum dot calculated from the exact many-body ground state wave functions.
1181
Ground state properties of interacting electrons Table I. Ground state energies in meV (exact and UHF) N electrons in a square box of 0.1 pm of side with a transversal magnetic flux ~ (in units of the flux quanta ~b0 =h/e). The overlap between the exact and U H F wavefunctions is also given. 12 monoelectronic wave functions are included in the calculations (24 including spin). The Zeeman contribution has been neglected d~'~b~,
2
3
4
5
6
7
8
Energy (UHF) Energy (exact) (~rt~t:Hi)
N
6.0021 5.8481 0.919
13.371 13.131 0.972
23.309 22.952 0.959
36.279 35.878 0.958
51.659 51.217 0.959
69.825 69.423 0.961
90.010 89.958 0.978
Energy (UHF) Energy (exact) (~EI~L,HF)
12.884 12.774 0.736
22.851 22.726 0.735
35.022 34.994 0.991
50.739 50.615 0.824
68.840 68.719 0.833
89.654 89.525 0.918
111.55 I I 1.32 0.934
one-electron wave functions (24 including spin) were used throughout the calculations. In both figures there are crossings and anticrossings between energy levels. Crossing occur between eigenvalues corresponding to levels with different total spin (as the Zeeman energy term has not been taken into account in the calculation, spin multiplets are degenerate), while anticrossings occur between levels with the same total spin, and are reminescent of the crossings between states with different angular momentum that have been found in parabolic 2D QD. Both after a crossing or anticrossing between the two lowest laying eigenstates, the character of the ground state wave function changes abruptly. As we show later on, this abrupt change is going to be reflected on the values of spectral weights. Figure 2 shows A(N) and the spin total S(N) up to N = 8 for two different magnetic fluxes: (a) ~b =3q~ 0, and (b) ~ = 15~0, (~0 = h/e, h being the Planck constant and e the electron charge) which are typical representatives of the low- and high-magnetic-field regimes, respectively. In the low B case, the ground state eigenfunctions have minimum total spin, that is, S ( N ) = 0 for N even and S ( N ) = I / 2 for N odd. In this case, A(N) is a very smooth, although not monotonous, function of N. For high B, the ground state eigenfunctions are spin polarised, for small N. Even when, in order to obtain the results shown in Fig. 2(b), the electron-electron interaction has been fully taken into account, this may be understood so as to minimize the exchange energy (the relevance of the kinetic energy contribution to the total energy decreases with B, as a consequence of the condensation of one-electron energy levels into Landau levels). Spectral weights change rapidly as function of N, for N small, in this regime. For example, while A(I) = I always, A(2) decreases as B increases, reflecting the fact that the occupancy of the first one-electron eigenstate decreases for ~b > 3.9 ~b0 (i.e. after the first crossing between the ground and the first excited state). The reason for this is that the charge is more spread out when occupying one-electron edge states than when occupying the first one-electron state, which is localised in the center of the QD. Eventually, for large enough N, all the firsts one-electron levels are occupied and, after that, A(N) seems to tend to 1 for large N. Notice that Kinaret et al.[9] predict a quenching of A(N) in a different situation: they calculated A(N) for states
such that the filling factor remains constant, and these are not necessarily the ground states of the many-body system. It is also worth mentioning the disappearance of A(8) due to the change of S for N = 8 electrons (in comparison with N = 7) by more than 1/2. Our U H F calculations produce an excellent approximation to the ground state energy as well to the wave function as shown in Table i. The U H F ground state energy approximates better than 2% the exact ground state energy, and it is always in between the energies of the ground state and the first excited state in the exact many-body calculation. The excellent agreement found for the U H F calculations leads us to extent the calculation of both the manybody energies and spectral weights to large N. The A(N) and S(N) calculated from the U H F wave functions are shown in Fig. 3 for g, = 15~b0. Hartree-Fock reflects qualitatively all the features found in the exact calculations and the agreement between them is qualitatively rather good for A(N), when N t> 3. U H F calculation supports the idea that A(N) tends to 1 for large N for the dot considered, 1.0 0.80.6-
i !iii ii
O.Pl 0
i!i!
i!i~ii~i! !!ii!i~iiiii]: 5
I
I
10
15
20
25
15
20
25
N
3.53.02.,5
2.0-
1.5t.00.5
0.00
5
10
N Fig. 3. As for Fig. 2(b), but using the ground state wave functions obtained from the Unrestricted Hartree-Fock method and extended to 25 electrons.
1182
L. MARTiN-MORENOet al.
while showing a clear paramagnetic trend of the ground state. An observation, that deserves a more detailed investigation in the future, is that the overlaps between the exact and the U H F ground state wave functions remain above 0.9 in all cases, except close to the fractional filling factors v considered (defining v = N / O ) , suggesting a very correlated groundstate for these cases. A more detailed description of these effects for different geometries and of the intermediate B regime, where A(N) presents abrupt changes as a function of B, will be published elsewhere. To conclude, we have calculated the many-body few lowest-lying eigenenergies and their correspondent eigenstates in a square 2D Q D for up to 8 electrons and different magnetic fields. While for small B the spectral weights are smooth function of N, for high B they are strongly modulated. A similar effect is expected to appear in the conductance through these systems at low temperatures. Finally, it has been used the Hartree-Fock approximation for
the calculation of properties on "typical" dots defined in the A I G a A s - G a A s electron gas.
REFERENCES
1. M. A. Kastner, Phys. Today, January, p. 24, and references therein (1993). 2. R. C. Ashoori et al., Phys. Ret'. Lett. 68, 3088 (1992). 3. C. W. J. Beenakker, Phys. Rev. B 44, 1646 (1991). 4. P. L. McEuen et al,, Phys. Rev. Lett. 66, 1926 (1991). 5. W. Hausler, B. Kramer and J. Masek, Z. Phys. B 85, 435 (1991), 6. P. A. Maksym and T. Chakraborty, Phys. Rev. Lett. 65, 108 (1990); ibid., Phys. Rev. B 45, 1947 (1992). 7. U. Merkt, J. Huser and M. Wagner, Phys. Rev. B 43, 7320 (1991); M. Wagner, U. Merkt and A. V. Chaplik, Phys. Rev. B 45, 1951 (1992). 8. D. Pfannkuche and R. R. Gerhardts, Phys. Rev. B 44, 13132 (1991). 9. J. M. Kinaret et al., Phys. Rev. B 45, 9489 (1992); ibid. 46, 4681 (1992). 10. J. J. Palacios, L. Martin-Moreno and C. Tejedor, Surf Sei. In press.