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Quantum states of interacting electrons in a 2D elliptical quantum dot
Physica B 249—251 (1998) 233—237
Quantum states of interacting electrons in a 2D elliptical quantum dot P.A. Maksym* Department of Physics and Astronomy, University of Leicester, University Road, Leicester LE1 7RH, UK
Abstract The quantum states of interacting electrons in a 2D elliptical quantum dot in a magnetic field are investigated. A Fock—Darwin basis is used to calculate energies of the ground and low-lying states of 2 and 3 electrons accurately while keeping the dimension of the Hamiltonian matrix manageable. Ground state transitions as a function magnetic field are studied and it is shown that only some of the transitions found in a circular dot survive in the lower symmetry of the elliptical dot. The surviving transitions are accompanied by changes in the total spin and parity of the ground state. Rules for predicting which transitions survive are given and some implications for experimental addition spectra are discussed. At zero field there is the possibility of observing different forms of addition spectra, depending on whether the ratio of the x and y confinement energies is rational or irrational. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Quantum dots; Addition energy; Correlated electrons; Magic numbers
1. Introduction The quantum states of 2D circular quantum dots have been the subject of intensive theoretical and experimental research [1—4]. One of the interesting features of interacting electrons in circular dots is that the ground state of the dot in a magnetic field goes through a series of transitions in which the orbital (and in some cases spin) angular momentum changes when the field in increased. There is evidence that some of the transitions have been observed [5,6]. Generally, the total angular momentum increases with field but only certain values * Tel.: #44 116 252 3579; fax: #44 116 252 2770; e-mail: [email protected].
of the total angular momentum quantum number, J, are selected. These values, dependent on both the electron number, N, and total spin, S, are known as the “magic” angular momenta. The selection rules that lead to the magic numbers can be understood in terms of symmetry [7,8] so it is natural to ask what happens to the transitions when the symmetry of the system is reduced. In the present work the consequences of reducing the symmetry from circular to elliptical are examined. It is shown that some of the transitions survive and rules for predicting those that do are given. The work begins with a summary of the properties of non-interacting electrons in an elliptical dot (Section 2) and this is followed by results for the interacting system (Section 3).
0921-4526/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 1 0 5 - 7
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2. Non-interacting electrons The confining potential of a 2D elliptic dot is taken to have the form »(x, y)" m*(u2x2#u2y2)/2) y x where u Ou . The quantum states of non-interacx y ting electrons in this potential and a magnetic field perpendicular to the plane of the dot can be found analytically, using, for example, the methods of Schuh [11] or Dippel et al. [12]. Alternatively, the energies and states may be obtained from a general result due to Maksym [7] for the eigenstates of a quadratic Hamiltonian in which co-ordinates and momenta are coupled. The latter approach relates the quantum states and energies to classical vibrational modes and is used here. The energy eigenvalues, (identical to those given in Refs. [11,12]), have the form E "(n #1/2)+u #(n #1/ n`n~ ` ` ~ 2)+u , where n and n are quantum numbers and ~ ` ~ the vibrational frequencies are given by 2u2 "(u2#u2# u2)$[(u2# u2#u2)2! c y x c y x B 4u2u2]1@2, where u is the cyclotron frequency. # x y The corresponding states have the form (a`)n`(a`)n~exp( f (x, y)) where a` and a` are rais` ~ ~ ` ing operators, linear in x and y and derivatives with respect to x and y and f(x,y) is a quadratic form in x and y. Both the quadratic form and the operators can be obtained from general formulae given by Maksym [7]. Because the raising operators are linear in x,y and their derivatives, the states can be classified according to whether they change sign under the inversion, x,yP!x,!y. In other words they have a definite parity, (!1)n``n~. Another important property of the states is that their spatial extent becomes larger for larger values of n and n . ` ~ The magnetic field dependence of the eigenvalue spectrum is illustrated in Fig. 1. This shows the low lying part of the spectrum for a GaAs dot in the case when u /u is irrational (+u "J17 meV, x y x +u "J10 meV, upper frame) and rational y +u "(J17/2) meV, lower (+u "J17 meV, y x frame). Even and odd parity levels are distinguished by solid and broken lines respectively. The confinement energies are chosen to be typical of real dots. In the case when u /u is irrational the spectrum at x y B"0 is non-degenerate. Some level crossings occur as B increases and at large B the levels form broadened Landau levels. Within each Landau
Fig. 1. Energy levels as a function of magnetic field (excluding Zeeman energy) for one electron in an elliptic GaAs dot with irrational (upper frame) and rational (lower frame) confinement energy ratio as detailed in the text.
level the sub-levels alternate in parity. Similar behaviour is found in the case when u /u is rational, x y except that there are some degenerate levels at B"0. For the case shown u /u "2, and the levels x y above the first excited state are degenerate. Generally, if u /u is a rational number, p/q, the number x y of non-degenerate levels becomes larger as p and q increase and non-degenerate levels appear between the degenerate ones.
3. Interacting electrons In a circular dot, the transitions in the ground state are driven by magnetic field induced compression of the wavefunction. This increases the Coulomb energy and makes it favourable for the
P.A. Maksym / Physica B 249—251 (1998) 233—237
system to adopt a new and larger ground state at certain critical values of the magnetic field. The states of interacting electrons in the circular dot are eigenstates of total orbital angular momentum and spin and at the critical field values two levels, corresponding to different orbital angular momentum (and possibly spin) values, cross leading to a transition to a ground state with a higher angular momentum. Because the spatial extent of the states at fixed B increases with angular momentum the transition is accompanied by an abrupt expansion [9]. In an elliptical dot the states within a broadened Landau level alternate in parity and increase in size. Therefore one might expect that similar transitions could occur and would be accompanied by changes in the ground state parity. However, it is not clear that all the transitions of the circular dot can survive in the reduced symmetry of the elliptical dot. This question is investigated here with the aid of numerical calculations. The numerical procedure used to find the eigenstates is a standard exact diagonalization in a Fock—Darwin basis. These basis states are the eigenstates of a single electron in a circular dot and are labelled by radial and angular momentum quantum numbers n and l [10]. In principle it would be possible to use the one electron states discussed in Section 2 as a basis however this is inconvenient because the raising operators generate high order polynomials of both x and y which have a large number of terms. In contrast the polynomials in the Fock—Darwin basis are only functions of the radial co-ordinate. The numerical accuracy of calculations in the Fock—Darwin basis has been tested both by comparison with the analytic results of Section 2 and by changing the size of the basis for calculations for interacting systems. Typically the ground state and first few excited states for up to 3 electrons are obtained to about 1% accuracy with 3 n values and 17 l values for 0.5) u /u )2. After making use of parity the maxx y imum dimension of the Hamiltonian matrix is 1617. The numerical results for the energy levels of 2 and 3 interacting electrons in a GaAs dot are shown in Fig. 2 and Fig. 3 (upper frame). For clarity, only the lowest 10 levels for each parity are
235
Fig. 2. Energy levels as a function of magnetic field (including Zeeman energy) for two electrons in an elliptic GaAs dot.
Fig. 3. Energy levels as a function of magnetic field (including Zeeman energy) for three electrons in an elliptic GaAs dot (upper frame) together with ground state spin and parity (lower frame). Odd parity is denoted by !1 and even parity by 1.
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shown. The results include all possible spin states but the spin states are not distinguished in the figures. The confinement energies are +u " x J17 meV and +u "J10 meV. In the two electron y case the lowest even parity level crosses the lowest odd parity level at BK2 T but there are no other ground state transitions. The corresponding states have S"0 below the transition and S"1 above it and the crossing corresponds to the singlet-triplet transition of the circular dot. In the 3-electron case there are many more ground state transitions with accompanying changes in the spin and parity of the ground state (Fig. 3, lower frame). These transitions are very similar to the angular momentum transitions of the 3-electron circular dot. The different behaviours of the 2- and 3-electron systems can be understood in terms of the magic angular momentum values for the circular dot and the effect of the coupling between them in an elliptical dot. For u 'u the perturbation to the cirx y cular confinement is K(u2!u2)x2 and in polar y x co-ordinates this is (u2!u2)r2(1#cos2h)/2. Therey x fore the perturbation couples states whose angular momenta differ by 0 or $2. The J value of a spin polarised or unpolarised 2-electron circular dot respectively must be an odd or even integer. When two levels corresponding to the same spin cross their J values differ by 2. These states can be coupled by the perturbation so the degeneracy at the crossing point is lifted and the transition disappears. The spin transitions survive but only one ground state transition appears in Fig. 2 because spin-polarised states are favoured in higher magnetic fields. In the case of 3 electrons the magic J values are 3k for the spin polarised states and 3k#1 or 3k#2 for spin unpolarised states. Now when two levels corresponding to the same spin cross, the J values differ by either 1 or 3. In each case the states cannot be coupled by the perturbation so the crossings survive. The rules that predict whether the level crossings of a circular dot survive in the elliptic case can be generalised to more electrons. Once the magic angular momenta for the circular case are known it is simple to determine whether the states at the crossing are coupled by the perturbation. A new feature that occurs for 4 or more electrons is that the coupling can vanish to first order but higher
order couplings remain possible. For example, the magic angular momenta of a 4-electron spin-polarised circular dot are of the form 4k#2 so the J values at a crossing differ by 4. In this case first order coupling is impossible but second order coupling can occur. Numerical results confirm that the splitting is weak in this case. For more electrons the splitting would become very weak if the difference in J values at the crossing point was even and large. A possible way of probing the crossings is to look for structure in the addition spectrum. In circular dots, structure in the form of “bumps” is seen in the energy required to add an electron to a dot as a function of magnetic field [5,6,12]. If the shape of a dot was changed from circular to elliptic some of the structure should disappear in an electron number dependent manner. Another distinguishing feature of an elliptical dot is that systems with irrational and rational values of u /u should behave in x y a different way at zero field. For example, Tarucha et al. [6] have recently reported observation of structure in the second difference of the ground state energy, E !2E #E , which they exN`1 N N~1 plained as a shell filling effect, related to the degeneracy of the Fock—Darwin levels at zero field. In an elliptic dot there is the possibility of tuning the degeneracy by varying u /u and this should affect x y the structure in the second difference of the ground state energy.
4. Conclusion The ground state of interacting electrons in a 2D elliptic dot can undergo transitions similar to those of 2D circular dot but some of the transitions in the circular dot do not survive the lowering of the symmetry. The transitions that survive correspond to cases where the angular momenta on either side of the transition in the circular dot differ by an odd integer. Transitions between states of the same spin whose orbital angular momenta differ by 2 are forbidden in the elliptic case and are forbidden to first order if the orbital angular momentum difference is a higher multiple of 2. The surviving transitions are accompanied by changes in the ground state parity or spin or both. These effects could possibly be observed in the addition spec-
P.A. Maksym / Physica B 249—251 (1998) 233—237
trum. The success of numerical calculations in the Fock—Darwin basis for the elliptical dot opens the possibility of calculations for dots with other types of point symmetry. Generally, lowering the circular symmetry would affect the transitions in a way that depends on both the point symmetry and the electron number. Acknowledgements I thank Dr. H. Akera for drawing my attention to reference [11]. This work was supported by the UK Engineering and Physical Sciences Research Council. References [1] T. Chakraborty, Comm. Condens. Matter Phys. 16 (1992) 35.
237
[2] D. Heitmann, J.P. Kotthaus, Physics Today 46 (1993) 56. [3] R.C. Ashoori, Nature 379 (1996) 413. [4] P.A. Maksym, Springer Lecture Notes in Physics, in press. [5] R.C. Ashoori, H.L. Sto¨rmer, J.S. Weiner, L.N. Pfeiffer, K.W. Baldwin, K.W. West, Phys. Rev. Lett 71 (1993) 613. [6] S. Tarucha, D.G. Austing, T. Honda, R.J. Vanderhage, L.P. Kouvenhoven, Phys. Rev. Lett 77 (1996) 3613. [7] P.A. Maksym, Phys. Rev. B 53 (1996) 10871. [8] H. Imamura, P.A. Maksym, H. Aoki, Physica B 249—251 (1998), these proceedings. [9] P.A. Maksym, L.D. Hallam, J. Weis, Physica B 212 (1995) 213. [10] P.A. Maksym, T. Chakraborty, Phys. Rev. Lett 65 (1990) 108. [11] B. Schuh, J. Phys. A 18 (1985) 803. [12] O. Dippel, P. Schmelcher, L.S. Cedarbaum, Phys. Rev. A 49 (1994) 4415. [13] L.D. Hallam, J. Weis, P.A. Maksym, Phys. Rev. B 53 (1996) 1452.
Quantum states of interacting electrons in a 2D elliptical quantum dot P.A. Maksym* Department of Physics and Astronomy, University of Leicester, University Road, Leicester LE1 7RH, UK
Abstract The quantum states of interacting electrons in a 2D elliptical quantum dot in a magnetic field are investigated. A Fock—Darwin basis is used to calculate energies of the ground and low-lying states of 2 and 3 electrons accurately while keeping the dimension of the Hamiltonian matrix manageable. Ground state transitions as a function magnetic field are studied and it is shown that only some of the transitions found in a circular dot survive in the lower symmetry of the elliptical dot. The surviving transitions are accompanied by changes in the total spin and parity of the ground state. Rules for predicting which transitions survive are given and some implications for experimental addition spectra are discussed. At zero field there is the possibility of observing different forms of addition spectra, depending on whether the ratio of the x and y confinement energies is rational or irrational. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Quantum dots; Addition energy; Correlated electrons; Magic numbers
1. Introduction The quantum states of 2D circular quantum dots have been the subject of intensive theoretical and experimental research [1—4]. One of the interesting features of interacting electrons in circular dots is that the ground state of the dot in a magnetic field goes through a series of transitions in which the orbital (and in some cases spin) angular momentum changes when the field in increased. There is evidence that some of the transitions have been observed [5,6]. Generally, the total angular momentum increases with field but only certain values * Tel.: #44 116 252 3579; fax: #44 116 252 2770; e-mail: [email protected].
of the total angular momentum quantum number, J, are selected. These values, dependent on both the electron number, N, and total spin, S, are known as the “magic” angular momenta. The selection rules that lead to the magic numbers can be understood in terms of symmetry [7,8] so it is natural to ask what happens to the transitions when the symmetry of the system is reduced. In the present work the consequences of reducing the symmetry from circular to elliptical are examined. It is shown that some of the transitions survive and rules for predicting those that do are given. The work begins with a summary of the properties of non-interacting electrons in an elliptical dot (Section 2) and this is followed by results for the interacting system (Section 3).
0921-4526/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 1 0 5 - 7
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P.A. Maksym / Physica B 249—251 (1998) 233—237
2. Non-interacting electrons The confining potential of a 2D elliptic dot is taken to have the form »(x, y)" m*(u2x2#u2y2)/2) y x where u Ou . The quantum states of non-interacx y ting electrons in this potential and a magnetic field perpendicular to the plane of the dot can be found analytically, using, for example, the methods of Schuh [11] or Dippel et al. [12]. Alternatively, the energies and states may be obtained from a general result due to Maksym [7] for the eigenstates of a quadratic Hamiltonian in which co-ordinates and momenta are coupled. The latter approach relates the quantum states and energies to classical vibrational modes and is used here. The energy eigenvalues, (identical to those given in Refs. [11,12]), have the form E "(n #1/2)+u #(n #1/ n`n~ ` ` ~ 2)+u , where n and n are quantum numbers and ~ ` ~ the vibrational frequencies are given by 2u2 "(u2#u2# u2)$[(u2# u2#u2)2! c y x c y x B 4u2u2]1@2, where u is the cyclotron frequency. # x y The corresponding states have the form (a`)n`(a`)n~exp( f (x, y)) where a` and a` are rais` ~ ~ ` ing operators, linear in x and y and derivatives with respect to x and y and f(x,y) is a quadratic form in x and y. Both the quadratic form and the operators can be obtained from general formulae given by Maksym [7]. Because the raising operators are linear in x,y and their derivatives, the states can be classified according to whether they change sign under the inversion, x,yP!x,!y. In other words they have a definite parity, (!1)n``n~. Another important property of the states is that their spatial extent becomes larger for larger values of n and n . ` ~ The magnetic field dependence of the eigenvalue spectrum is illustrated in Fig. 1. This shows the low lying part of the spectrum for a GaAs dot in the case when u /u is irrational (+u "J17 meV, x y x +u "J10 meV, upper frame) and rational y +u "(J17/2) meV, lower (+u "J17 meV, y x frame). Even and odd parity levels are distinguished by solid and broken lines respectively. The confinement energies are chosen to be typical of real dots. In the case when u /u is irrational the spectrum at x y B"0 is non-degenerate. Some level crossings occur as B increases and at large B the levels form broadened Landau levels. Within each Landau
Fig. 1. Energy levels as a function of magnetic field (excluding Zeeman energy) for one electron in an elliptic GaAs dot with irrational (upper frame) and rational (lower frame) confinement energy ratio as detailed in the text.
level the sub-levels alternate in parity. Similar behaviour is found in the case when u /u is rational, x y except that there are some degenerate levels at B"0. For the case shown u /u "2, and the levels x y above the first excited state are degenerate. Generally, if u /u is a rational number, p/q, the number x y of non-degenerate levels becomes larger as p and q increase and non-degenerate levels appear between the degenerate ones.
3. Interacting electrons In a circular dot, the transitions in the ground state are driven by magnetic field induced compression of the wavefunction. This increases the Coulomb energy and makes it favourable for the
P.A. Maksym / Physica B 249—251 (1998) 233—237
system to adopt a new and larger ground state at certain critical values of the magnetic field. The states of interacting electrons in the circular dot are eigenstates of total orbital angular momentum and spin and at the critical field values two levels, corresponding to different orbital angular momentum (and possibly spin) values, cross leading to a transition to a ground state with a higher angular momentum. Because the spatial extent of the states at fixed B increases with angular momentum the transition is accompanied by an abrupt expansion [9]. In an elliptical dot the states within a broadened Landau level alternate in parity and increase in size. Therefore one might expect that similar transitions could occur and would be accompanied by changes in the ground state parity. However, it is not clear that all the transitions of the circular dot can survive in the reduced symmetry of the elliptical dot. This question is investigated here with the aid of numerical calculations. The numerical procedure used to find the eigenstates is a standard exact diagonalization in a Fock—Darwin basis. These basis states are the eigenstates of a single electron in a circular dot and are labelled by radial and angular momentum quantum numbers n and l [10]. In principle it would be possible to use the one electron states discussed in Section 2 as a basis however this is inconvenient because the raising operators generate high order polynomials of both x and y which have a large number of terms. In contrast the polynomials in the Fock—Darwin basis are only functions of the radial co-ordinate. The numerical accuracy of calculations in the Fock—Darwin basis has been tested both by comparison with the analytic results of Section 2 and by changing the size of the basis for calculations for interacting systems. Typically the ground state and first few excited states for up to 3 electrons are obtained to about 1% accuracy with 3 n values and 17 l values for 0.5) u /u )2. After making use of parity the maxx y imum dimension of the Hamiltonian matrix is 1617. The numerical results for the energy levels of 2 and 3 interacting electrons in a GaAs dot are shown in Fig. 2 and Fig. 3 (upper frame). For clarity, only the lowest 10 levels for each parity are
235
Fig. 2. Energy levels as a function of magnetic field (including Zeeman energy) for two electrons in an elliptic GaAs dot.
Fig. 3. Energy levels as a function of magnetic field (including Zeeman energy) for three electrons in an elliptic GaAs dot (upper frame) together with ground state spin and parity (lower frame). Odd parity is denoted by !1 and even parity by 1.
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shown. The results include all possible spin states but the spin states are not distinguished in the figures. The confinement energies are +u " x J17 meV and +u "J10 meV. In the two electron y case the lowest even parity level crosses the lowest odd parity level at BK2 T but there are no other ground state transitions. The corresponding states have S"0 below the transition and S"1 above it and the crossing corresponds to the singlet-triplet transition of the circular dot. In the 3-electron case there are many more ground state transitions with accompanying changes in the spin and parity of the ground state (Fig. 3, lower frame). These transitions are very similar to the angular momentum transitions of the 3-electron circular dot. The different behaviours of the 2- and 3-electron systems can be understood in terms of the magic angular momentum values for the circular dot and the effect of the coupling between them in an elliptical dot. For u 'u the perturbation to the cirx y cular confinement is K(u2!u2)x2 and in polar y x co-ordinates this is (u2!u2)r2(1#cos2h)/2. Therey x fore the perturbation couples states whose angular momenta differ by 0 or $2. The J value of a spin polarised or unpolarised 2-electron circular dot respectively must be an odd or even integer. When two levels corresponding to the same spin cross their J values differ by 2. These states can be coupled by the perturbation so the degeneracy at the crossing point is lifted and the transition disappears. The spin transitions survive but only one ground state transition appears in Fig. 2 because spin-polarised states are favoured in higher magnetic fields. In the case of 3 electrons the magic J values are 3k for the spin polarised states and 3k#1 or 3k#2 for spin unpolarised states. Now when two levels corresponding to the same spin cross, the J values differ by either 1 or 3. In each case the states cannot be coupled by the perturbation so the crossings survive. The rules that predict whether the level crossings of a circular dot survive in the elliptic case can be generalised to more electrons. Once the magic angular momenta for the circular case are known it is simple to determine whether the states at the crossing are coupled by the perturbation. A new feature that occurs for 4 or more electrons is that the coupling can vanish to first order but higher
order couplings remain possible. For example, the magic angular momenta of a 4-electron spin-polarised circular dot are of the form 4k#2 so the J values at a crossing differ by 4. In this case first order coupling is impossible but second order coupling can occur. Numerical results confirm that the splitting is weak in this case. For more electrons the splitting would become very weak if the difference in J values at the crossing point was even and large. A possible way of probing the crossings is to look for structure in the addition spectrum. In circular dots, structure in the form of “bumps” is seen in the energy required to add an electron to a dot as a function of magnetic field [5,6,12]. If the shape of a dot was changed from circular to elliptic some of the structure should disappear in an electron number dependent manner. Another distinguishing feature of an elliptical dot is that systems with irrational and rational values of u /u should behave in x y a different way at zero field. For example, Tarucha et al. [6] have recently reported observation of structure in the second difference of the ground state energy, E !2E #E , which they exN`1 N N~1 plained as a shell filling effect, related to the degeneracy of the Fock—Darwin levels at zero field. In an elliptic dot there is the possibility of tuning the degeneracy by varying u /u and this should affect x y the structure in the second difference of the ground state energy.
4. Conclusion The ground state of interacting electrons in a 2D elliptic dot can undergo transitions similar to those of 2D circular dot but some of the transitions in the circular dot do not survive the lowering of the symmetry. The transitions that survive correspond to cases where the angular momenta on either side of the transition in the circular dot differ by an odd integer. Transitions between states of the same spin whose orbital angular momenta differ by 2 are forbidden in the elliptic case and are forbidden to first order if the orbital angular momentum difference is a higher multiple of 2. The surviving transitions are accompanied by changes in the ground state parity or spin or both. These effects could possibly be observed in the addition spec-
P.A. Maksym / Physica B 249—251 (1998) 233—237
trum. The success of numerical calculations in the Fock—Darwin basis for the elliptical dot opens the possibility of calculations for dots with other types of point symmetry. Generally, lowering the circular symmetry would affect the transitions in a way that depends on both the point symmetry and the electron number. Acknowledgements I thank Dr. H. Akera for drawing my attention to reference [11]. This work was supported by the UK Engineering and Physical Sciences Research Council. References [1] T. Chakraborty, Comm. Condens. Matter Phys. 16 (1992) 35.
237
[2] D. Heitmann, J.P. Kotthaus, Physics Today 46 (1993) 56. [3] R.C. Ashoori, Nature 379 (1996) 413. [4] P.A. Maksym, Springer Lecture Notes in Physics, in press. [5] R.C. Ashoori, H.L. Sto¨rmer, J.S. Weiner, L.N. Pfeiffer, K.W. Baldwin, K.W. West, Phys. Rev. Lett 71 (1993) 613. [6] S. Tarucha, D.G. Austing, T. Honda, R.J. Vanderhage, L.P. Kouvenhoven, Phys. Rev. Lett 77 (1996) 3613. [7] P.A. Maksym, Phys. Rev. B 53 (1996) 10871. [8] H. Imamura, P.A. Maksym, H. Aoki, Physica B 249—251 (1998), these proceedings. [9] P.A. Maksym, L.D. Hallam, J. Weis, Physica B 212 (1995) 213. [10] P.A. Maksym, T. Chakraborty, Phys. Rev. Lett 65 (1990) 108. [11] B. Schuh, J. Phys. A 18 (1985) 803. [12] O. Dippel, P. Schmelcher, L.S. Cedarbaum, Phys. Rev. A 49 (1994) 4415. [13] L.D. Hallam, J. Weis, P.A. Maksym, Phys. Rev. B 53 (1996) 1452.