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Current distributions and the de Haas-van Alphen oscillation in a planar system of Hall ele


CURRENT DISTRIBUTIONS AND THE DE HAAS–VAN ALPHEN OSCILLATION

arXiv:cond-mat/9607148v2 12 Dec 1997

IN A PLANAR SYSTEM OF HALL ELECTRONS

K. SHIZUYA
Yukawa Institute for Theoretical Physics Kyoto University, Kyoto 606-01, Japan

Abstract
The current distribution is studied for a ?nite-width two-dimensional system of Hall electrons, with a clear distinction drawn between the equilibrium edge current and the Hall current. It is pointed out that both the distribution and direction of the equilibrium edge current change dramatically as the number of electron edge states increases, and that this alternating edge current is another manifestation of the de Haas–van Alphen e?ect. The Hall-current distribution is substantially di?erent from the edge current distribution, and it is shown numerically that the fast-traveling electrons along the sample edge are not the main carriers of the Hall current.

1

1. Introduction One basic question inherent to the quantum Hall e?ect1 (QHE) is where in the sample the current ?ows. The traditional explanations2–5 for the QHE are based on a picture that the Hall current is carried by electrons in the sample bulk and regard local disorder as crucial for the formation of visible conductance plateaus. On the other hand, considerable attention has recently been directed to another picture,6–8 the edge-state picture of the QHE, where the current is taken to ?ow in channels along the edges of a sample. Experiments9 appear to favor interpretation of observed results in terms of the edge channels but little is known about the current distribution in real samples so far, except that some information is obtained from observed potential distributions.10,11 Theoretically the current distribution as well as the potential distribution has been studied in some models,12–19 and possible connections between the bulk-state and edge-state pictures of the QHE have been discussed.16,17 The purpose of the present paper is to study the current-carrying properties of the electrons in the sample bulk and near the edges in detail. In equilibrium a prominent current ?ow arises along the edges of a Hall sample. It is pointed out that both the distribution and direction of the edge current change dramatically as electron edge states increase in number and that this alternating edge current is another manifestation of the de Haas – van Alphen (dHvA) e?ect. The Hall-current distribution turns out substantially di?erent from the equilibrium current distribution, and it is shown numerically that the fast-traveling electrons along the sample edges, though carrying a large amount of current per electron, are not the main carriers of the Hall current. In Sec. 2 we study the equilibrium current distribution in a disorder-free Hall sample of ?nite width. In Sec. 3 we examine the Hall current distribution and discuss the transport properties of the edge current. Sec. 4 is devoted to concluding remarks where the e?ect of disorder is discussed.

2

2. Current distributions in equilibrium Consider electrons con?ned to an in?nitely long strip of width W (or formally, a strip bent into a loop of circumference Lx ? W ) in the presence of a uniform magnetic ?eld B normal to the plane, described by the Hamiltonian: 1 H0 = ω ?2 p2 + (1/?2 )(y ? y0 )2 , y 2 (2.1)

√ written in terms of ω ≡ eB/m, the magnetic length ? ≡ 1/ eB and y0 ≡ px /(eB); the Landau-gauge vector potential (?By, 0) has been used to supply a uniform magnetic ?eld. We take explicit account of the two edges y = ±W/2, where the wave function is bound to vanish. The eigenstates of H0 in the sample bulk are Landau levels with discrete spectrum
1 ω(n + 2 ) labeled by integers n = 0, 1, 2, · · ·, and degenerate in y0 = ?2 px ; the y0 labels the

center-of-mass position of each orbiting electron. The eigenfunctions in the presence of sharp edges are still labeled by N = (n, y0 ): ψN (x, y) = (Lx )?1/2 eixpx φN (y), (2.2)

√ with φN (y) given6 by the parabolic cylinder functions20 Dν (± 2(y ? y0 )/?) for electrons residing near the edges y = ?W/2. The energy spectrum of each Landau level n ?n (y0 ) = ω νn (y0 ) + 1 2 (2.3)

is ?xed from the boundary condition φN = 0 at y = ±W/2. The spectra νn (y0 ) are deterW mined numerically and are functions of a dimensionless combination y0 /?: W νn (y0 ) = νn [y0 /?],

(2.4)

W where y0 ≡ y0 + W/2 refers to the value of y0 measured relative to y0 = ?W/2 near the W “lower” edge; y0 ≡ y0 ? W/2 near the “upper” edge y0 ? W/2. As shown in Fig. 1,

νn (y0 ) = νn (?y0 ) rise sharply only in the vicinity of the edges y0 ? ±W/2, and recover the bulk values n as y0 moves to the interior a few magnetic lengths away from the edges;6 for
W the lowest n = 0 level, 0 ≤ ν0 < 0.003 for y0 > 2.5?. Applying the Feynman-Hellman

3

theorem to ?H0 /?y0 reveals that a single electron state N = (n, y0 ) has its center-of-mass at position
cm yn ≡ N|y|N = y0 ? ?2 νn ′ (y0 )

(2.5)

cm on the y axis. This yn stays within the interval [?W/2, W/2] while y0 = ?2 px can take

values outside it for electrons residing near the edges.
W The normalized wave functions φN (y), taken to be real, are functions of y ? y0 and y0 :

√ W φN (y) = (1/ ?)φn (y ? y0 )/?; y0 /? ,

(2.6)

where we have taken φn dimensionless so that its dependence on dimensional quantities ?, y0, etc. is made explicit. Let us denote the spatial distribution of a single electron state N as ρN (y) ≡ ρn (y; y0) = 1 W φn (y ? y0 )/?; y0 /? ?
2

,

(2.7)

which is uniform in x and which is normalized so that
W/2 ?W/2

dy ρn (y; y0) = 1.

(2.8)

The associated current density per electron is written as a product of charge (?e), velocity (px ? eBy)/m and density ρn /Lx : jx (y|N) = eω (y ? y0 ) ρn (y; y0). Lx (2.9)

The distributions ρn (y; y0) are highly localized and are nonzero only for |y ? y0 | ≤ O(?). In the sample bulk [i.e., |y0 | < W/2 ? O(?)], in particular, ρn are even functions of y ? y0 : √ 2 ρbulk (y ? y0 ) = Hn (Y )2 e?Y /(?2n n! π), n (2.10)

with Y ≡ (y ? y0 )/?. The pro?le of ρn (y; y0) gets gradually deformed as y0 lies closer to
W the edge, as shown in Fig. 2 (a) for the n = 0 level. There the y0 = 3.5? electron state,

depicted with a solid curve, has ν0 ≈ 10?5 and thus exhibits essential characteristics of an electron in the bulk; clearly the associated current distribution, depicted with a solid curve in Fig. 2 (b), is characteristic of an orbiting electron and carries no net current. In contrast, 4

for an electron lying closer to the edge the cyclotron motion gets deformed, yielding a net current,5,6 which, in view of Eq. (2.5), is proportional to the gradient of the spectra νn :
W/2 ?W/2

dy jx (y|N) = ?

eω 2 ′ ? νn (y0 ). Lx

(2.11)

The current-carrying property of each electron state N is made manifest by decomposing
cm the current jx (y|N) into the circulating component ∝ (y ? yn ) ρn (y; y0) (that carries no

net current) and the traveling component ∝ ??2 νn ′ (y0 ) ρn (y; y0); see Fig. 2 (c) and (d). It is clearly seen from these ?gures that the electrons residing near the edge (the edge states) carry a large amount of current. Classically these edge states are visualized as electrons hopping along the sample edges.21 They travel fast (with velocity vx = ω?2 νn ′ ? ±ω?) in opposite directions at opposite sample edges. Let us next examine the current distributions ?lled Landau levels support. We shall focus on the case of well-?lled levels close to integer ?lling. It is convenient to use the Fermi potential EF ≡ ω(nF + 1/2) or nF to specify the ?lling of each level: For a given nF Landau
? levels n with n ≤ nF are populated densely with electron states over the range (y0 )n < y0 < + ± + ? (y0 )n so that νn ((y0 )n ) = nF ; the ?lling fraction of each level is fn = [(y0 )n ?(y0 )n ]/W with

the number of electrons given by (Ne )n = (eB/2π)Lx W fn . The nF stays constant nF = n while the nth level is gradually ?lled with the electron states residing in the sample bulk (the bulk states), and it increases as soon as the edge states start to be ?lled. Such ?lled states (n, y0 ) give rise to the current distribution eω jx (y) = 2π?2
+ (y0 )n

n

? (y0 )n

dy0 (y ? y0 ) ρn (y; y0).

(2.12)

In view of Eq. (2.10), it is clear that jx (y) = 0 in the sample bulk. Figure 3(a) shows the current distribution near the lower edge y = ?W/2 for various values of nF . For 0 < nF < 1 only the n = 0 level carries current, for 1 < nF < 2 two levels (n = 0, 1) contribute, and so on. There are two notable features read from the ?gure: (1) The current ?ows along the edge in a channel whose width ? O(?) increases with the number of current-carrying levels. This feature has been well known5,6 and is naturally expected from the current fraction 5

per electron in Eq. (2.11). Somewhat unexpected is the following: (2) Both the spatial distribution and direction of the current change dramatically with nF . For small nF ? 0.1 the edge current ?ows predominantly in the negative x direction while a large fraction of current ?ows in the opposite direction for nF ? 0.9; the integrated amount of current changes its sign at nF = 0.5. Analogous patterns of current ?ow are also seen for each integer interval n < nF < n+1. The amount of current integrated over one edge oscillates with nF , as shown in Fig. 3(b). These changing and oscillating current distributions may appear puzzling if one notes Eq. (2.11) alone. There is, however, a simple resolution21 : Associated with each orbiting
cm electron is a circulating current ∝ (y ? yn ) ρn (y; y0) that produces magnetic moment point-

ing opposite to the applied magnetic ?eld B. For a dense collection of electrons in the sample interior this circulating diamagnetic current is averaged out to vanish locally, but leaves a current circulating along its periphery. Indeed, collecting the circulating current components
circ alone yields a current distribution jx (y) localized near the edge, as shown in Fig. 4 (a) for tr the n = 0 level with di?erent nF . Also shown in Fig. 4 (b) is the current distribution jx (y)

formed by the traveling current components alone for the same n = 0 level. They combine to build up the current distributions of Fig. 3(a). It is now clear that the current distribution for nF = 0.1 in Fig. 3 (a) primarily derives from the orbital diamagnetic current whereas the edge-driven traveling current dominates in the nF = 0.9 case. Note that the edge-driven current works to cancel the orbital diamagnetic current. The cancellation is partial in quantum theory,21 leading to the Landau diamagnetism of electrons for three-dimensional samples. In the present two-dimensional case it is possible to calculate both the diamagnetic and edge-driven components of the edge current explicitly. As a preliminary step, let us ?rst consider a collection of electron bulk states (of the nth level) that ?ll up a half-in?nite interval y0 ≥ 0. They lead to the current distribution
bulk jx (y) = ?

eω 2π?2

∞ ?y

dz zρbulk (z) n

(2.13)

√ 2 2 localized around y = 0 with spread △y = O(?); for n = 0 this is given by ?(2 π)?1 e?y /? times (eω/2π?). The integrated amount of this diamagnetic current increases with n: 6

y?0

bulk dy jx (y) = ? n +

1 eω . 2 2π

(2.14)

? Consider next a collection of electron states that ?ll up the interval (y0 )n ≤ y0 ≤ η with η

lying somewhere far in the sample interior. The circulating current components associated with these states, though carrying no net current, combine to build up two prominent current distributions localized near the edge y = ?W/2 and around y = η, which are equal in net amount of current and opposite in sign. This shows that the diamagnetic component of the current localized near the edge y = ?W/2 supports a ?xed amount22 equal to ?(eω/2π)(n + 1/2), irrespective of its distribution as well as the shape of edge
circ potentials. In particular, the four di?erent distributions of jx (y) in Fig. 4(a) support the

same amount of current equal to ?(eω/4π). On the other hand, the edge states of the nth level arise for nF > n and, as seen from Eq. (2.11), carry the amount of current equal to (eω/2π)(nF ? n) at the lower edge y ? ?W/2. With the two current components put together the nth level alone supports the amount of edge current
edge Jn =

eω (nF ? 2n ? 1/2) 2π

for nF ≥ n.

(2.15)

For the integer interval n ≤ nF < n + 1 the lower (n + 1) ?lled levels combine to support the amount of current J edge = eω 1 (n + 1) nF ? (n + ) , 2π 2 (2.16)

localized near the edge y = ?W/2. This current-nF relation accounts for the alternating edge current of Fig. 3(b), and shows that the (integrated) edge current changes its direction precisely at nF = n + 1/2. The alternating edge current of Fig. 3 is intimately connected to an oscillatory variation of magnetic susceptibility of an electron gas with varying magnetic ?eld, known as the de Haas – van Alphen (dHvA) e?ect.21,23 Indeed, the current J edge , circulating along the sample edge, gives rise to uniform magnetization normal to the sample plane and of magnitude (per unit area) 7

Mz = ?B (1/2π?2 )(n + 1)[2(nF ? n) ? 1]

(2.17)

for n ≤ nF < n + 1, apart from corrections of O(?/W ) that vanish as W → ∞, where ?B = eω?2 /2 = e/(2m) is the Bohr magneton (with e?ective mass m). This result is readily veri?ed by thermodynamic methods as well. The simplest way is to consider how the energy of the present ?nite-width system, E = (Lx /2π?2 )
n W dy0 ω νn [y0 /?] + 1/2 ,

(2.18)

responds to an in?nitesimal variation of the magnetic ?eld B. Keep px = y0 /?2 ?xed and calculate the magnetization Mz = ?(1/W Lx )δE/δB: Mz = ? ?B 2π?2
n

dy0 W

W ′ 2νn (y0 ) + 1 ? (2y0 ? y0 )νn (y0 ) .

(2.19)

′ Note that νn = n in the sample bulk and that νn = 0 only in the edge regions. The y0 integral

therefore is equal to [2n + 1 ? 2(nF ? n)] for each ?lled level n, apart from corrections of O(?/W ); this result is consistent with Eq. (2.15) and thus recovers Eq. (2.17). It is also enlightening to con?rm Eq. (2.19) by a direct calculation of the thermodynamic potential ?(?, T ; B) = ?T
i

ln(1 + e(???i )/kT ) with T → 0. fn ∝ Ne /B. As seen from

With the spectra νn (y0 ) determined numerically, it is a simple task to express the magnetization (2.19) as a function of B or the ?lling factor f =
n

Fig. 5, the magnetization per electron, mz ≡ Mz /(Ne /W Lx ), plotted as a function of f shows an oscillatory variation characteristic of the dHvA e?ect. As usual, the gradual variation of mz in each integer interval n < f < n + 1 ? O(?/W ) is ascribed to the diamagnetism of orbiting electrons in the bulk. In each tiny interval n + 1 ? O(?/W ) < f < n + 1, mz makes a drastic change from diamagnetism to paramagnetism that is caused by the edge states of the nth level. Note here that the electron edge states, because of their tiny ?lling fraction? O(?/W ), scarcely contribute to the internal energy E. Thus, even without explicit account of the edge states, one can still arrive at the dHvA e?ect by thermodynamic methods (i.e., through E or ?), except that Mz now shows discontinuous jumps at integer ?lling f = n. The edge states are invisible in such derivations but are certainly present physically: Note that magnetization 8

by nature is continuous as a function of B/Ne , as seen clearly from the W = 30? case of Fig. 5. In view of this continuity, the prominent sign change of mz near integer ?lling f = n (which can be derived by thermodynamics without the edge states) does imply the presence of the edge states and, in particular, the alternation in direction of the edge current. In this sense, the alternating edge current in Fig. 3 is another aspect of the dHvA e?ect. The alternation of the edge current is caused by competition between the circulating and edge-driven components of the current near the edge. The edge-driven component is naturally very sensitive to the shape of the edge potential while the circulating component is not; see Fig. 2 (c) and (d). Correspondingly the edge current in general varies in distribution according to the shape of edge potentials. Still, in net amount per edge, both current components remain unaltered so that Eqs. (2.15) and (2.16) hold, irrespective of the details of edge potentials.

3. Hall-current distributions The alternation of the edge current, which we have just seen, does not necessarily imply that the information the edge current carries also changes the direction of propagation. In this section we clarify this point by studying how the edge current responds to a Hall ?eld. For simplicity let us consider a uniform ?eld Ey = ??y A0 (y). Its e?ect is readily taken care of by making the shift y0 → y0 ≡ y0 ? (e?2 /ω)Ey ? (3.1)

in H0 of Eq. (2.1), and the full Hamiltonian H0 ? eA0 (y) leads to the new spectrum
2 ?n (?0 ) + eEy y0 + O(Ey ). y ?

The normalized wave function φN (y) is given by Eq. (2.6)

with the replacement y0 → y0 , and the current per electron jx (y|N) by Eq. (2.9) with ? ρn (y; y0) → ρn (y; y0). The current distribution jx (y|N) itself barely changes thereby be? cause, under conditions of practical interest, the deviation y0 ? y0 is negligibly small. ? Actually the Hall current we are interested in is the small deviation, △jx (y|N) = eω (y ? y0 ) {ρn (y; y0) ? ρn (y; y0)} , ? Lx 9 (3.2)

representing the response to an applied ?eld. As seen from Fig. 6(a), unlike jx (y|N), the Hall current per electron △jx (y|N) is primarily composed of traveling components. Here we see that, while the edge states and bulk states are drastically di?erent in the amount of current they carry, they are essentially the same in the Hall-current transport. Numerically an edge state supports even a smaller amount of Hall current than a bulk state, as seen from Fig. 6(b), where the numerically-integrated amount of Hall current per electron, △Jx ≡ dy △jx (y|N), is compared with the net current per electron, Jx ≡ dy jx (y|N) ∝ ?νn ′ (y0 ). It is possible to understand the current-carrying properties of each electron state in a
cm more general way: Of the current jx (y|N), the circulating component ∝ (y ? yn ) ρn carries

no net current. It is the traveling component, associated with the drift of a Hall electron with velocity ω?2 νn ′ (?0 ) + Ey /B, y (3.3)

that carries a net current (and hence information with it). Accordingly disturbances caused upon a sample, e.g., by varying a magnetic ?eld or electron population will propagate in a direction ?xed by the edge with velocity? ω?. In contrast, the e?ect of a Hall ?eld (i.e., the Hall current) propagates in a direction ?xed by the polarity of Ey with velocity ? Ey /B. [Numerically ω? ? 107 cm/s for typical values ω ? 10 meV and ? ? 100 ? while A Ey /B ? 103 cm/s for Ey = 1 V/cm and B = 5 T.] It follows from Eq. (3.3) that the net amount of Hall current per electron is proportional to 1 ? ?2 ν0 ′′ (y0 ), which reproduces the numerical result in Fig. 6(b) very accurately. Filled Landau levels support the prominent edge-current distributions of Fig. 3(a), which remain essentially unchanged in the presence of a Hall ?eld as well. These edge currents ?ow in opposite directions at the two opposite edges of a sample, and in equilibrium with Ey = 0 they combine to vanish.5,6 When a Hall ?eld Ey is turned on, the Hall current emerges as a small deviation from the equilibrium distribution, as shown in Fig. 6(c) for the present impurity-free and uniform-Ey case. It is clear from Fig. 6, contrary to some tempting idea8 , that the electron edge states, because of their tiny ?lling fraction ? O(?/W ), share only a tiny portion of the Hall current.17 10

4. Concluding remarks In this paper we have examined current distributions in a Hall bar in the regime of the integer QHE, and shown, in particular, that the edge current changes its distribution and direction as the number of electron edge states increases. This dramatic change is a consequence of competition between the circulating diamagnetic component and edgedriven traveling component of the current carried by electrons near the sample edge, and is closely related to the dHvA e?ect. It should be emphasized that the dHvA oscillation of magnetization is indirect evidence for the edge states; it is clear physically that without them no oscillation would arise, since orbital magnetization alone leads to diamagnetism.21 In this connection we have seen explicitly that magnetization is continuous as a function of B/Ne when the edge states are properly taken into account. We have also seen that the Hall current ?ows in a manner quite independent of the equilibrium edge-current distribution. Finally we would like to comment on the in?uence of disorder. Rapid motions of Hall electrons like cyclotron motion and acceleration by the edge are potentially not very sensitive to disorder. The edge current of Fig. 3, resulting from such rapid motions, will therefore remain prominent in the presence of weak disorder as well, and continue to alternate in direction as the number of edge states increases. In contrast, disorder will substantially modify the current carried by slowly-drifting electrons in the sample bulk: In the presence of disorder a large fraction of electron bulk states get localized and cease to carry current; at the same time, the surviving extended states carry more current and achieve exact compensation.2,3 This exact current compensation is a consequence of electromagnetic gauge invariance and takes place under general circumstances involving both bulk and edge states.17 Note now that, of the electron bulk states, those residing near the edge of the sample bulk (“bulk edge”), though in?uenced by disorder, would have a better chance of staying extended than those far in the bulk. In view of current compensation, a considerable portion of the Hall current, redistributed via disorder, would therefore ?ow along the sample edges. In this way it has been pointed out17 that there are two kinds, fast and slow, of edge current in Hall samples in the regime of the integer QHE.

11

These two kinds of edge current di?er in channel width and in direction of ?ow. The fast component is nothing but the alternating edge current discussed so far, consisting of the two (circulating and traveling) subcomponents; it has a channel width of O(?) and ?ows in opposite directions at opposite sample edges. In contrast, the slow component, the “bulk-edge” Hall current, will have a channel width, related to some localization length characteristic to the bulk edge, which could well be larger than O(?). This slow edge current will ?ow in the same direction at opposite edges and reverse direction when the polarity of the Hall ?eld is ?ipped; an observed Hall-potential distribution11 appears to be in support of this feature. A numerical experiment is now under way to study the current distribution for small samples with randomly distributed impurities, and is yielding results that appear to con?rm the e?ect of disorder on the current distribution described above; details will be reported elsewhere.

Acknowledgments The author wishes to thank B. Sakita and Y. Nagaoka for useful discussions. This work is supported in part by a Grant-in-Aid for Scienti?c Research from the Ministry of Education of Japan, Science and Culture (No. 07640398).

12

REFERENCES
1

K. von Klitzing, G. Dorda and M. Pepper, Phys. Rev. Lett. 45, 494 (1980). For a review see, The Quantum Hall E?ect, edited by R.E. Prange and S.M. Girvin (Springer Verlag, Berlin, 1987).

2

H. Aoki and T. Ando, Solid State Commun. 38, 1079 (1981). R. E. Prange, Phys. Rev. B23, 4802 (1981). R. B. Laughlin, Phys. Rev. B23, 5632 (1981). B. I. Halperin, Phys. Rev. B25, 2185 (1982). A. H. MacDonald, and P. Streda, Phys. Rev. B29, 1616 (1984). P. Streda, J. Kucera and A. H. MacDonald, Phys. Rev. Lett. 59, 1973 (1987); J. K. Jain and S. A. Kivelson, Phys. Rev. B37, 4276 (1988).

3

4

5

6

7

8

M. B¨ ttiker, Phys. Rev. B38, 9375 (1988). u S. Washburn, A. B. Fowler, H. Schmidt, and D. Kern, Phys. Rev. Lett. 61, 2801 (1988); B. J. van Wees et al., Phys. Rev. Lett. 62, 1181 (1989); B. W. Alphenaar, P. L. McEuen, R. G. Wheeler and R. N. Sacks, Phys. Rev. Lett. 64, 677 (1990).

9

10

F. Fang and S. Stiles, Phys. Rev. B29, 3747 (1984); H. Z. Zheng, D. C. Tsui, A. M. Chang, Phys. Rev. B32, 5506 (1985); E. K. Sichel, H. H. Sample and J. P. Salerno, Phys. Rev. B32, 6975 (1985).

11

P. F. Fontein, et al., Phys. Rev. B43, 12090 (1991). A. H. MacDonald, T. M. Rice, and W. F. Brinkman, Phys. Rev. B28, 3648 (1983). O. Heinonen and P. L. Taylor, Phys. Rev. B32, 633 (1985). T. Otsuki and Y. Ono, J. Phys. Soc. Jpn. 58, 2482 (1989). D. B. Chklovskii, B. I. Shklovskii, and L. I. Glazman, Phys. Rev. B46, 4026 (1992).

12

13

14

15

13

16

D. J. Thouless, Phys. Rev. Lett. 71, 1879 (1993); C. Wexler and D. J. Thouless, Phys. Rev. B49, 4815 (1994).

17

K. Shizuya, Phys. Rev. Lett. 73, 2907 (1994); Phys. Rev. B45, 11143 (1992). Y. Avishai, Y. Hatsugai, and M. Kohmoto, Phys. Rev. B47, 9501 (1993); Y. Avishai and M. Kohmoto, Int. J. Mod. Phys. B9, 2949 (1995).

18

19

M. R. Geller and G. Vignale, Phys. Rev. B52, 14137 (1995). E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, (Cambridge University Press, London, 1927).

20

21

L. Landau, Z. Phys. 64, 629 (1930); E. Teller, Z. Phys. 67, 311 (1931); R. E. Peierls, Surprises in Theoretical Physics, (Princeton Univ. Press, Princeton, 1979). The author wishes to thank M. Stone for informing him of Teller’s paper.

22

This value (eω/2π)(n + 1/2) for the edge current was derived earlier in Ref. 19 within a derivative expansion for a slowly-varying con?ning potential.

23

For the dHvA e?ect in two dimensions, see A. Ishihara, Solid State Phys. 42, 271 (1989); D. Yoshioka and H. Fukuyama, J. Phys. Soc. Jpn. 61, 2368 (1992).

14

FIGURES

3 2.5 2 1.5 1 0.5 0 1

ν2 (y0) ν1 (y0) ν0 (y0)
2 3 4 5 6

y0W

W FIG. 1. Spectra νn (y0 ) plotted as a function of y0 ≡ y0 + W/2 (measured in units of the W magnetic length ?) near the lower edge y0 ≈ 0.

15

ρ (y ; y0 )
0

ν0 =0.5 ν0 =0.1 ν0

(a)
0 ( y0W=3.5 )

0.6 0.4 0.2

yW
1 2 3 4 5 6

jx(y | N)
0.2 0 -0.2 -0.4 -0.6 1 2 3

ν0

0

(b)

yW
4 5 6

ν0 =0.1 jx
circ

ν0 =0.5
(c)

(y| N)

0.2 0 -0.2 0
-0.2 -0.4 -0.6 jxtr(y| N)

yW
1
1

2
2

3
3

4
4

5
5

6
6

yW

(d)

FIG. 2.

(a) Spatial distributions ρn (y; y0 ) of single electron states with di?erent ν0 (y0 ) in

W the n = 0 level. The solid curve represents the ν0 ≈ 10?5 electron state with y0 = 3.5?, the W long-dashed curve the ν0 = 0.1 state with y0 ≈ 1.40?, and the dashed curve the ν0 = 0.5 state W with y0 ≈ 0.541?. The coordinate y W = y + W/2, de?ned relative to the edge y = ?W/2,

is measured in units of ?. (b) Distributions of the associated current per electron. Here, for convenience, the current jx (y|N ) is plotted in units of ?eω/Lx so that its sign refers to that of the velocity vx along the x axis. (c) Decomposition of jx (y|N ) into the circulating component
circ tr jx (y|N ) and the traveling component jx (y|N ).

16

0.4 0.2 0 -0.2 -0.4

jx (y)
1

(a)
n F=0.1 n F=0.5
2 3 4 5

yW
6

n F=0.9 n F=1.1 n F=1.5
1 2 3 4 5

0.4 0.2 0 -0.2 -0.4

yW
6

n F=1.9 n F=2.1 n F=2.5
1 2 3 4 5

0.4 0.2 0 -0.2 -0.4

yW
6

n F=2.9

2 1.5 1 0.5 0 -0.5 -1 -1.5

J edge

(b)

nF
0.5 1 1.5 2 2.5 3 3.5

FIG. 3. (a) Current distributions ?lled Landau levels support near the sample edge y = ?W/2; nF = EF /ω ? 1/2 refers to the Fermi potential EF . The current jx (y) is measured in units of ?eω/(2π) and the coordinate y W = y + W/2 in units of ?. (b) The integrated amount of the current per edge [in units of ?eω/(2π)] oscillates with nF .

17

circ jx (y)

0.5 0.4 0.3 0.2 0.1 1

nF=0.9 nF=0.5 nF=0.1 nF 0
W (y0 ) =3.5

(a)

2

3

4

5

6

yW yW

1 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6

2

3

4

5

6

nF=0.1

(b)

nF=0.5

jxtr(y)

nF=0.9

FIG. 4. (a) Current distributions near the edge y = ?W/2 built up by the circulating current components alone for the n = 0 level with di?erent values of nF . The nF ≈ 10?5 case refers to the
W n = 0 level ?lled with electron states y0 ≥ 3.5?. The current is measured in units of ?eω/(2π),

and the coordinate y W in units of ?. (b) Current distributions formed by the traveling current components alone for the same n = 0 level.

18

1 0.5

mz

0.5 -0.5

1

1.5

2

2.5

3

f

W=30 l
-1

W=50 l W=100 l

FIG. 5. Magnetization per electron [in units of ?B /2π?2 ] v.s. the ?lling factor f for a Hall bar of ?nite width W = 30?, 50?, and 100?.

19

0.4 0.3 0.2 0.1

? jx(y|N)

ν0 =0.1

ν0

(a)
W

0 y0 =3.5

1 -0.1

2

3

4

5

6

7

yw

ν0=0.5 ν0=0.9

1 0.8 0.6 0.4 0.2 0

Jx

? Jx

(b)

1

2

3

4

5

6

w y0
(c)

1 0.8 0.6 0.4 0.2

? jx(y) nF =0.9
0.5 nF =0.1

nF 0
(y0W ) =3.5

yw
1 2 3 4 5 6 7

FIG. 6.

(a) Distributions of the Hall current per electron, △jx (y|N ), for electron states

with di?erent ν0 (y0 ) in the n = 0 level. The current is plotted in units of 10?3 × (?eω/Lx ) for e?Ey /ω = 10?3 . (b) The Hall current per electron △Jx = dy △jx (y|N ) (in units of

?e2 ?2 Ey /Lx ) for electron states in the n = 0 level, in comparoson with the net current per electron Jx = dy jx (y|N ) (in units of eω?/Lx ). (c) Hall-current distributions near the edge for the n = 0

level. The edge states increase in number with nF .

20



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