Interaction effects on persistent currents in magnetic fields

22 April 1996

PHYSICS LETTERS A

Physics Letters A 213 (1996) 197-202

ELSEWIER

Interaction effects on persistent currents in magnetic fields J. Yi, M.Y. Choi’ Dep~rt~nt

of Physics and Centerfor

Theoretical Physics. Seoul ~ationui ~~iversi~, Seoul 151-742, Souih Korea

Received 7 August 1995; revised manuscript received 31 January 1996; accepted for publication 5 February 1996 Communicated by J. Flouquet

Abstract We investigate the effects of the Coulomb interaction on the persistent current displayed by electrons on a two-dimensionaf annulus, in the presence of a perpendicular magnetic field as well as a magnetic flux threading its center. In the appropriate regime, the electrons form an incompressible quantum fluid, which is described by the Laughlin wave function. It is shown that the Coulomb interaction here tends to enhance the persistent current. On the other hand, in the case of a Wigner solid, which may be induced by strong magnetic fields, the persistent current is suppressed by the Coulomb interaction. PACS:

73.SO.Jt; 72.10.88; 73.2O.D~

One of the remarkable manifestations of the Ah~onov-Bohm effect [ 1 ] is the persistent currents in mesoscopic systems with appropriate geometries [2]. The existence of such persistent currents was indeed confirmed in experiments, stimulating much interest in the problem. In particular, while the experiment performed in the ballistic regime yielded the current in agreement with the theoretical value 131, the magnitude of the current observed in the diffusive regime was much larger than the theoretical prediction [4] : This apparent quantitative discrepancy has caused controversy as to its origin, and yielded a number of studies which attempt to attribute the origin to the Coulomb inte~ction; the latter was neglected in the original theory. However, in a one-dimensional ring, which has been studied mostly, strong Coulomb repulsion between electrons in general leads to suppression of the persistent current [5], and fails to account for the discrepancy. This motivates one to t E-mai1: [email protected]. Elsevier Science B.V. PII SO375-9601(96)00127-2

consider persistent currents in systems with two- or t~eedimensional geometries; recent investigation of multich~nel rings indeed seems to favor the enhancement of the persistent currents due to the Coulomb interaction [ 61. In such higher dimensions, some tiny flux can smear into the sample, which is unavoidable in experiment. It is thus of relevance to study the persistent current in the presence of the magnetic field penetrating through the sample. This has been considered for noninteracting electrons in a two-dimensional (2D) annulus geometry by numerically integrating the Schriidinger equation [ 71. On the other hand, the persistent current of interacting electrons in an annulus subject to a magnetic field has not been investigated, even in the ballistic regime without disorder. Such 2D interacting electrons in magnetic fields are known to form an incompressible quantum fluid or a Wigner solid. Unfo~unately, the many-body nature of the problem makes it very difficult to solve even numeric~ly the

198

J. Yi. M.Y. Choi/ Physics Letters A 213 (1996) 197-202

corresponding Schrijdinger equation, let alone analytical treatment. This paper investigates in an approximate way the persistent current displayed by electrons on a 2D annulus, which is subject to a perpendicular magnetic held and pierced by a magnetic flux at its center, with emphasis paid to the role of the Coulomb interaction. For simplicity, we consider only the ballistic regime and concentrate on the interplay between the magnetic field and the Coulomb interaction. The 2D electrons in appropriate magnetic fields are expected to form an incompressible quantum fluid, which is described by the Laughlin wavefunction [ 81, We thus begin with the Laughlin wavefunction on the annulus [ 91 instead of solving directly the Schrodinger equation, circumventing the difficulty associated with the many-body formulation. First, the kinetic energy is shown not to contribute to the persistent current, which corresponds to the absence of a current-carrying ground state in the bulk. We then take into account the confining potential [ 91, which yields the persistent current linear in the threading flux. We finally examine the effects of the Coulomb interaction on the persistent current carried by the Laughlin state. It is found that the persistent current is in general enhanced by the Coulomb interaction. The enhancement depends on the strength of the magnetic field: Strong magnetic fields, which correspond to low filling factors, tend to suppress the enhancing effects of the Coulomb interaction. Here, strong magnetic fields tend to localize the electrons, which eventually form a Wigner solid as the ground state. This observation suggests that the Coulomb interaction might suppress the persistent current in a 2D Wigner solid. To confirm this expectation, we also investigate the persistent current in a Wigner solid, and indeed find the suppressing effects of the Coulomb interaction, which are in contrast to those for the Laughlin state. We consider a system of N interacting spinless electrons confined to a 2D annulus, the inner radius and the outer radius of which are RI and R2, respectively. The annulus, located on the x-y-plane, is threaded by a magnetic flux @ = c&u (@a = he/e) at its center, and a magnetic field B is applied on the surface of the annulus along the direction perpendicular to the surface. For convenience, we assume that the magnetic field is applied along the negative z-direction, while the threading flux can have either direction depending

on the sign of LY.The Hamiltonian

of the system reads

H=~[~(~r+~A(ri))l+o(r; e2

+C-= i_,j Iri -

Ho+ZJ+Y

(1)

rjl

where m and -e are the mass and the charge of an electron, U( ri) is the confining edge potential, and Vb(ri) represents the interaction between electrons and the neutralizing positive background. The vector potential A in the kinetic energy term Ho has contributions from both the threading flux @ and the perpendicular magnetic field B:

while the Coulomb interaction term V includes the electron-electron interactions described by the last term as well as vb. The persistent current is given by the derivative of the energy with respect to the threading flux: I =

-&-,

where E is the total energy of the system. It is given by the expectation value of the Hamiltonian in Eq. ( 1) :

where ]W) is the ground state of the system. Accordingly, three contributions to the current may be considered: I = iK + Ip + I& where IK = -(e/2di)(d(Ho)/tla), etc. To compute the current precisely, it is desirable to find the eigenvalues or eigenfunctions of the full Hamiltonian; this requires impractical numerical works for the interacting system, making it inevitable to rely upon approximate treatment. Here we expect that such a 2D interacting system forms an incompressible quantum liquid, which is rather well described by the Laughlin wavefunction in the appropriate regime [ 81. In the absence of the confining potential (U = 0), the widely accepted Laughlin wavefunction describing the system of filling factor v = l/q is given by

J. Yi, M.Y. Choi/Physics

!P,(z1,22,...,z~)

=fi(Zj

j-ck

-

199

Letters A 213 (1996) 197-202

zk)4~e+‘12~4e2~ i

where z; is the complex coordinate of the ith electron and ! E dm denotes the magnetic length. For 9 = 1, the Laughlin wavefunction describes the exact ground state of the noninteracting electrons filling completely the lowest Landau levels. In the presence of the confining potential, which introduces an annulus geometry, the appropriate Laughlin wavefunction has been proposed [ 91: ~(Zl,Z2....,

ZN) = ~lzil"z:ur,~z1,z2,...,ZN~,

(3) where lyq denotes the Laughlin wavefunction on a disc, given above. In Q. (3) the prefactor in front of Py accounts for the existence of the hole in the annulus, i.e. the absence of electrons for ]z 1 < RI, by generating quasiholes which cancel out the electrons in the hole region. For precise cancelation, the number of quasiholes should be q times that of the electrons in the region Iz 1 < RI; this condition gives the relation between the integer n and the flux LY(= @/Qio): ffi-n=s+S,

-0.5

1

R:

where 6 is a constant in the range -q/2 < 6 < q/2. We first consider the kinetic energy Ha in the Hamiltonian ( 1). It is straightforward to obtain the corresponding energy eigenvalue: (4) where wc is the cyclotron frequency. This corresponds to the ground state of the noninteracting system in which all the electrons occupy the lowest Landau level. Thus the kinetic energy does not depend on the threading flux, and has no contribution to the persistent current, 1~ = 0. It should be noted here that the Laughlin wavefunction given by Eq. (2) does not vanish on the edges of the annulus, and does not satisfy the hard-wall boundary conditions. Here we follow Ref. [ 91, and consider the effects of the confining potential in an approximate way, which should be sensible in a thick annulus with a small hole. Suppose that the threading flux is

0 ck

0.5

1

Fig. 1. Noninteracting contributions to the persistent current in units of In.

increased. This results in the redistribution of the electrons described by the Laughlin wavefunction, and in turn leads to the energy change since the distribution changes with respect to the confining potential. Such a change of the total energy due to redistribution is described by [ 9 1 --where U(r) is the confining potential, and CJ’ E XJ/&. This yields a parabolic energy spectrum, and accordingly, generates the (piecewise) linear persistent current via E$. (2), Ip = -10%

(6)

for -l/2 < (Y < l/2, where la G ( 2ee2/qh) x [ U’( R2) / RZ - U’( R1 )/RI ] and the confining potential has been assumed to give the minimum contribution to the energy at (Y= 0. In general U’( Rl)/Rl is larger in magnitude, and is of the order of -Fio,/C2 [7]. Fig. 1 shows the schematic behavior of the resulting persistent current, contributed by the confining potential. The noninteracting contribution to the persistent current is displayed in units of la, which is of the order of ew,/27rq. It is pleasing that this simple estimate gives the envelope of the behavior, without such details as plateaus and fluctuations, of the current in the noninteracting system, which has been obtained by extensive numerical integrations [ 71. We now consider effects of the Coulomb interaction, which are determined by its contribution to the total

J. Yi, M.Y. Chni/Physics

200

Letters A 213 (1996) 197-202

energy or its ground-state expectation value. Although the exact ground-state wavefunction is unknown, the Laughlin wavefunction still gives a rather accurate description of the ground state of the interacting electrons [ 101. In particular, the estimation by taking the expectation value of the noninteracting part of the Hamiltonian yields qualitatively correct overall behavior of the noninteracting contributions ( ZK and 1~). It is thus expected that the interacting contribution to the persistent current is reasonably estimated from the expectation value of the Coulomb interaction taken with respect to the Laughlin wavefunction. The expectation value of the Coulomb interaction is given by the 2Ndimensional integral:

0

f

0.04

i

. H

(V) :

Fig. 2. Expectation values of the Coulomb interaction taken with respect to the Laughlin fluid states with 9 = 1 (diamonds) and 9 = 3 (squares). For convenience, the energy is set equal to zero in the absence of the threading flux, and presented in units of e*/e. (pl”l!f’)

3

n J

X”(ZI

&idz:

p*(Zl,

...,ZN)

i

, . ..ZN)ly(ZI.

. . . . ZN). 0.1

We scale the length of the system in units of the magnetic length, and perform the integral by the Monte Carlo method to obtain the Coulomb energy (V) as a function of the flux LYin a system of N = 10 to 50 electrons. In particular, we investigate the dependence on the strength of the magnetic field and consider several values of the filling factor q = 1, 3, 5, 7, and 9. Note that the strength of the magnetic field determines the magnetic length, which is the natural length scale of the system. Thus we can change the effective size of the system by adjusting the magnetic field: The magnetic length e is inversely proportional to the square root of the magnetic field, the dimensionless radii 81 = RI/C and & E Rz/C in a given sample of fixed RI, R2, and N increase proportionally to the square root of the magnetic field or to the square root of q. Here we have used several values of the inner radius, Rt ranging from 0.4& to 4fi. Fig. 2 displays the typical behavior of the Coulomb energy of N = 10 electrons in the annulus of i?t = J;s and I?2 = m, for q = 1 and q = 3. In both cases as well as other cases not displayed (q 3 5), the Coulomb energy exhibits approximately linear increase with respect to the flux; in particular, the tendency to the linear behavior becomes more apparent within error bars as N is increased. Since the contribution of the Coulomb interaction to the persistent cur-

1

d

0.08 -

F

0.06

0.04 i t 0.02 c

i

I

5

7

?D f

I

3

9

9

Fig. 3. Contributions of the Coulomb interaction to the persistent current in the Laughlin states with various values of the filling factor.

rent is given by Ic = -(e/2di)~?(V)/&x, the positive slope in Fig. 2 indicates enhancement of the persistent current due to the Coulomb interaction. The slope (in units of e’/e), however, tends to decrease with q, as shown in Fig. 3: It appears to behave as l/q, approaching zero eventually. The corresponding contribution to the persistent current may be estimated: M Ae3/2rqfi!, where A is a constant of the order k 10-l depending on the geometry of the annulus. This interacting contribution can be compared with the noninteracting contribution 10 x ew,/2rq, which yields

J. Yi, M.Y. Choi/Physics

k -AL 10

a0 ’

Letters A 213 (1996) 197-202

201

(7)

with the effective Bohr radius ao. In typical samples, we have C/a0 N 1 to 10, which leads to Ic/le N 10-t to I. This indicates that the Coulomb interaction can give considerable contributions to the persistent current. In particular the interacting contribution can be comparable to the noninteracting contribution in the presence of a relatively weak magnetic field. As the magnetic field is increased, however, the interacting contribution decreases while the noninteracting contribution does not change. It is thus concluded that the magnetic field in the fluid state tends to suppress the enhancing effect of the interaction relative to the noninteracting contribution. Here it is of interest to note that very strong magnetic fields in general tend to favor the Wigner solid over the quantum fluid as the ground state of the interacting electron system. Thus the competition between the two states in the system has recently attracted much interest [ 111. The persistent current carried by such a Wigner solid has also been addressed in a onedimensional ring [ 121. In a two-dimensional annulus, our results on the effect of the Coulomb interaction imply that the persistent current might be suppressed in the Wigner solid state which exists in very strong magnetic fields. To confirm this, we consider the wave function for a Wigner solid on the annulus, again in the form of Eq. (2)) but with Pq replaced by lyw,-, the wave function for a Wigner solid on a disc with electrons localized on sites Ri [ 111. The localization sites on the annulus are chosen to form the analogue of a square lattice, i.e. equally spaced along the radial direction and equally separated along the azimuthal direction: In the complex coordinate, we thus have Zi = reiH, with r = n( R2 - R,)/N, + RI and t9 = 2vm/N2, where n and m are integers running from 0 to Nt and N2, respectively, and the total number of lattice sites is Nt N2 = N. Specifically, we have investigated the system of N = 40 electrons in the annulus of Rt = 0.4 and )?2 = &%, and show the resulting Coulomb energy in the Wigner solid state in Fig. 4, where the Coulomb energy in the fluid state of the same system is also displayed for comparison. The Coulomb energy in the Wigner solid indeed exhibits a negative slope, and suppresses the persistent current, which is in sharp contrast to that in the fluid.

P -i

-0.1 +

(V)

4

1 iI

/

-i

-0.2

.o,3i ,,Qii 0

0.1

0.2

0.3

0.4

cy

Fig.4. Expectation

values of the Coulomb interaction taken with respect to the Wigner solid state (diamonds). For comparison, the corresponding values in the fluid state are also displayed (squares). Note the difference in scale from Fig. 2.

In summary, we have investigated the persistent current displayed by electrons on an annulus, with emphasis on the roles of the magnetic field and the Coulomb interaction. Based on the Laughlin wave function, we have found the null contribution of the kinetic energy, which reflects the absence of a currentcarrying ground state. Thus the noninteracting contribution comes solely from the confining potential, which is shown to yield the piecewise linear persistent current. We have then examined the effects of the Coulomb interaction on the persistent currents, and found that the effects depend on the nature of the states in a crucial way: In the Laughlin liquid state the interaction in general gives an enhancing contribution, which can be comparable to the noninteracting contribution in the presence of a relatively weak magnetic field. This appears suggestive to the discrepancy between the theoretical and experimental results, if it is confirmed by further study that the presence of disorder does not alter qualitatively such enhancing effects of the interactions. Here the magnetic field tends to suppress the enhancing effects of the interaction relative to the noninteracting contribution. In particular strong magnetic fields stabilize the Wigner solid state, and yield suppressing effects of the Coulomb interaction on the persistent current. This work was supported in part by the SNUDaewoo Research Fund, by the Basic Science Re-

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J. Yi, M.Y. Choi/Physics

search Institute Program, Ministry of ~ucation, and by the Korea Science and Engineering Foundation through the SRC Program. References III Y. Aharonov and D. Bohm, Phys. Rev. I 15 (19.59) 485. 121 M. Biittiker, Y. lmry and R. Iandauer, Phys. Lett. A 96 ( 1983) 365. 131 D. Mailly, C. Chapelier and A. Benoit, Phys. Rev. I&t. 70 ( 1993) 2020. 141 L.I? Levy, Cl. Dolan, J. Dunsmuir and H. Bouchiat, Phys. Rev. Lett. 64 (1990) 2074; V. Chandrasekhar, R.A. Webb, M.J. Brady, M.B. Ketchen, W.J. Gallabher and A. Kleinsasser, Phys. Rev. Len. 67 (1991) 3578. 151D. Loss, Phys. Rev. Len. 69 (1992) 343;

Letters A 213 (1996) 197-202

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