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arXiv:cond-mat/9710052v1 [cond-mat.mes-hall] 6 Oct 1997

Density modulation and electrostatic self-consistency in a two-dimensional electron

gas subject to a periodic quantizing magnetic field

Ulrich J. Gossmann, Andrei Manolescu∗and Rolf R. Gerhardts

Max-Planck-Institut f¨ ur Festk¨ orperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Federal Republic of Germany

∗Institutul de Fizica ¸ si Tehnologia Materialelor, C.P. MG-7 Bucure¸ sti-M˘ agurele, Romˆ ania

We calculate the single-particle states of a two-dimensional

electron gas (2DEG) in a perpendicular quantizing mag-

netic field, which is periodic in one direction of the elec-

tron layer. We discuss the modulation of the electron den-

sity in this system and compare it with that of a 2DEG in

a periodic electrostatic potential.

induced potential within the Hartree approximation, and cal-

culate self-consistently the density fluctuations and effective

energy bands. The electrostatic effects on the spectrum de-

pend strongly on the temperature and on the ratio between

the cyclotron radius Rc and the length scale aδ ρ of the density

variations. We find that aδ ρ can be equal to the modulation

period a, but also much smaller. For Rc ∼ aδ ρ the spectrum

in the vicinity of the chemical potential remains essentially

the same as in the noninteracting system, while for Rc ≪ aδ ρ

it may be drastically changed by the Hartree potential: For

noninteger filling factors the energy dispersion is reduced, like

in the case of an electrostatic modulation, whereas for even-

integer filling factors, on the contrary, the dispersion may be

amplified.

We take account of the

I. INTRODUCTION

The interest in nonuniform magnetic fields, with spa-

tial variations on a nanometer scale, has been stimu-

lated by several recent experimental realizations, like

magnetic quantum wells1or magnetic superlattices.2–5

In the quasi-classical regime of low magnetic fields, the

theoretical investigations have concentrated on the com-

mensurability oscillations of the resistivity,6–9which are

equivalent to the Weiss oscillations10,11that occur in the

presence of a periodic electrostatic potential.

The quantum regime of nonuniform magnetic fields

with a strong variation of the order of 1 T within a dis-

tance of a few hundred nanometers is now experimentally

accessible.12For this regime single-particle quantum me-

chanical calculations, concerning the tunneling through

magnetic barriers or the bound states in magnetic wells,

have recently been performed by Peeters, Matulis, and

Vasilopoulos.13,14Coulomb interaction effects have been

discussed by Wu and Ulloa15who studied the electron

density distribution and the collective excitations in a

magnetic superlattice with a short period, comparable

to the average magnetic length. They found that the

periodic magnetic field gives rise to an electron-density

modulation, which is reduced due to the counteracting

induced electric field.

In the present paper we consider a two-dimensional

electron gas (2DEG) in a perpendicular magnetic field of

the form B = B0+ Bmod where B0 is a homogeneous

part and Bmodis, in the plane of the 2DEG, periodic in

one direction with zero average. We describe in detail

the charge-density response to the periodic part and the

effects of the associated electrostatic potential. It will be

very instructive to compare this situation with the modu-

lation by a unidirectional periodic electrostatic potential

Vext(x) (we include the charge −e in the definition of the

electrostatic potentials, which are therefore rather po-

tential energies). The homogeneous part of the magnetic

field is assumed to be strong enough so that a descrip-

tion in terms of Landau levels is adequate; we denote by

l0=

?¯ h/eB0and by ω0= eB0/m the magnetic length

field B0. For a fixed mean electron density ρ0, the num-

ber of relevant Landau levels is of the order of the filling

factor ν = (2πl2

0)ρ0 and thus inversely proportional to

the average magnetic field.

We first note that classically a magnetic field - modu-

lated or not - has no influence in thermodynamic equilib-

rium because it drops out of the integral over momenta

in the partition function (Bohr-van Leeuwen theorem16).

Especially a magnetic modulation does not lead to a

position dependence of the electron energy, which re-

mains mv2/2, and does not affect the equilibrium elec-

tron density. In contrast one expects (e. g. from Thomas-

Fermi-theory) that modulation by an electrostatic po-

tential should lead also to a modulation of the density

δρ(x) = −(Vext(x)/µ)ρ0, where µ is the chemical poten-

tial. In the classical limit, i. e. for low average magnetic

field with ¯ hω0≪ kBT, the density response of the 2DEG

to electrostatic and to magnetic modulations is thus very

different. A pure magnetic modulation does classically

not give rise to electrostatic effects, whereas an external

electrostatic potential is screened by the induced Hartree

potential.

However, in the quantum regime of low filling factors

the two types of modulation affect the density in a very

similar manner: Both lift the degeneracy of the Landau

levels and lead to dispersive bands. The homogeneous

part B0 of the magnetic field restricts the spatial ex-

tent of the relevant wavefunctions to the order of the

cyclotron radius Rc = l0√2nF+ 1 where nF is the in-

dex of the Landau level at the chemical potential (that

is nF is the largest integer smaller than ν/gs; we assume

and the cyclotron frequency associated with the uniform

1

Page 2

spin degeneracy gs = 2 in this work). If nF is small

the modulation is typically slowly varying on the scale

Rcand we can represent the Landau levels as functions

ǫn(x), varying on the same length scale as the modula-

tion. The width ∆ǫn of these bands is of the order of

¯ he∆B/m or ∆V , where ∆B and ∆V stand for the am-

plitude of the modulation of magnetic field and electro-

static potential, respectively. In this situation we expect

that the density is determined by a “local” filling fac-

tor ν(x) = gs

?

function. For temperatures satisfying kBT ≪ ∆ǫn, this

must lead to a density modulation of order (gs/2πl2

both electric and magnetic modulations.

For an electric modulation it is known that, due to this

strong effect of the Landau level dispersion on the density,

the inclusion of the Coulomb interaction in the Hartree-

approximation (which we will refer to as the electrostati-

cally self-consistent system) changes drastically the spec-

trum of the system for low filling factors. Wulf et. al.17

found that, for filling factors not too close to an even-

integer value, the self-consistent result corresponds to a

nearly perfect screening of the modulating external po-

tential and the Landau bands are flat within the order of

kBT. If for even-integer filling factor the chemical poten-

tial lies in a gap, the screening is much weaker although

still considerable. For potentials strong enough to yield

overlapping bands, |∆ǫn| > ¯ hω0, the screening around

even-integer filling factors becomes nonlinear, featuring

two bands touching the Fermi level, and the width of the

band nF therefore locked to ¯ hω0.

The width of the Landau band at the Fermi level, how-

ever, is the basic element for understanding transport

measurements6,3,18and a major aim of this work is to in-

vestigate its behaviour for a magnetic modulation when

electrostatic self-consistency is properly accounted for.

The paper is organized as follows. In section II we

describe our model and the self-consistency problem in

detail. In section III we treat the case of low filling fac-

tors, corresponding to a magnetic modulation varying

slowly on the length scale Rc. In section IV we discuss

the regime of lower average magnetic fields, where the

cyclotron radius is not small against the period of the

modulation. The numerical results we present are ob-

tained using the material parameters of GaAs, namely

the effective mass m = 0.067meand the dielectric con-

stant κ = 12.4. The average electron density is fixed to

ρ0= 2.4 · 1011cm−2, chosen such that νB0= 10 T, the

period of the modulation is a = 800 nm and we consider

values of B0 between 10 T and 0.1 T corresponding to

cyclotron radii between 7 nm and 740 nm.

n

f(ǫn(x)), where f(ǫ) denotes the Fermi

0) for

II. DESCRIPTION OF THE MODEL

We consider an idealized 2DEG confined to the plane

{r=(x,y)} and subject to a magnetic field B(r) =

(0,0,B(x)) which,7–9,15within the plane, is directed in

z-direction, does not depend19on y, and has a simple

periodic dependence on x:

B(x) = B0+ B1cosKx,(1)

where K = 2π/a is the wave vector of the modula-

tion. We start with the noninteracting 2DEG described

by the standard single-electron Hamiltonian H0= (p +

eA)2/2m in which we use the Landau gauge for the vec-

tor potential, A(x) = (0,B0x + (B1/K)sinKx,0). The

eigenfunctions of H0depend on y only through a plane-

wave prefactor,

ψnX0(x,y) = L−1/2

y

e−iX0y/l2

0φnX0(x), (2)

with Lybeing a normalization length and X0the center

coordinate. The functions φnX0(x) are the eigenvectors

of the one-dimensional Hamiltonian

H0(X0) = ¯ hω0

?

−l2

0

2

d2

dx2+

1

2l2

0

?

x − X0+s

KsinKx

?2?

(3)

,

where s = B1/B0 will be refered to as the modulation

strength.

For the homogeneous system, s = 0, the functions

φnX0(x) are oscillator wave functions centered on X0,

also known as Landau wave functions ϕL

with the degenerate Landau levels εL

The degeneracy is lifted for s ?= 0, and the resulting en-

ergy bands ǫn(X0), together with the corresponding wave

functions, can be obtained by diagonalizing the reduced

Hamiltonian (3) in the basis of the Landau wave func-

tions. The matrix elements can be written as

nX0, associated

n= (n + 1/2)¯ hω0.

?ϕL

nX0| H0(X0) | ϕL

??

?

+s2

8z

n′X0? =

¯ hω0

n +1

2

?

δnn′ +

s

2z

?

Enn′(z) +

√nn′En−1,n′−1(z)

−

(n+1)(n′+1)En+1,n′+1(z)

?

cos

?

KX0+ (n−n′)π

???

2

?

(4)

?

δnn′ − Enn′(4z)cos

?

2KX0+ (n−n′)π

2

,

where z = (Kl0)2/2. We have used the notation:

Enn′(z) =

?n′!

n!

?1/2

e−z/2z(n−n′)/2Ln−n′

n′

(z)

= (−1)n−n′En′n(z)

n(z) being a Laguerre polynomial. Applying first-

order perturbation theory we get from (4) the energy

levels as simple cosine-shaped bands

, (5)

with Lm

ǫPT1

n

(X0) = ¯ hω0

?

(n + 1/2) + sGn(z) cos(KX0)

?

(6)

where the factor 2s¯ hω0Gn(z) = s¯ hω0e−z/2(2L1

L0

n(z)) has an oscillatory dependence on the ratio

n(z) −

2

Page 3

l0

rability oscillations seen in transport experiments.3,8The

limits of validity of Eq. (6) will be discussed below.

The single-particle density is given by the formula

√2n + 1/a which is the basic reason for the commensu-

ρ(x) =

gs

2πl2

0

∞

?

n=0

+∞

?

−∞

dX0f(ǫn(X0)) | φnX0(x) |2

(7)

where f(ǫ) denotes the Fermi function and gs = 2 ac-

counts for spin degeneracy.

The density determines the electrostatic (Hartree) po-

tential, which we treat by Fourier expansion VH(x) =

?

VH

η =

4πǫ0κη

η≥1

VH

η cos(ηKx). Here

e2

a

ηρη =

1

a

2π aB

(2πl2

0ρη) ¯ hω0, (8)

with ρ(x) =

?

η≥0

ρηcos(ηKx) and aB the effective Bohr

radius. For GaAs, 2π aB ≈ 63 nm.

the system is electrically neutral such that the average

density ρ0does not contribute to VHbut only determines

the chemical potential contained in the Fermi function.

The Hartree potential has to be added to the Hamiltonian

(3) and gives a contribution

We assume that

?ϕL

=

nX0| VH(x) | ϕL

?

to the matrices (4). The strongest influence on the in-

duced potential originates from the low Fourier com-

ponents of the density, which are related to the long-

range charge fluctuations. We diagonalize the Hamilto-

nian H0(X0) + VHself-consistently with Eq. (7) by a

numerical iterative scheme.

To understand the way the system achieves self-

consistency, we occasionally consider also a 2DEG sub-

ject to a homogeneous magnetic field B0 and a cosine

electrostatic potential

n′X0?

η

VH

ηEnn′(η2z)cos

?

ηKX0+ (n − n′)π

2

?

(9)

V1(x) = V1cos(Kx) (10)

with modulation strength v1 = V1/¯ hω0 instead of the

magnetically modulated system (1). First-order pertur-

bation theory yields for the electric modulation (10) the

spectrum

ǫPT1

n

(X0) = ¯ hω0

?

(n + 1/2) + v1Fn(z) cos(KX0)

?

(11)

with Fn(z) = e−z/2Ln(z).

III. THE LIMIT OF LONG PERIOD

In this section we deal with a long-period magnetic

modulation, with a strong average magnetic field, such

that Kl0 ≪ 1 and Rc ∼ l0 (but not necessarily with

B1≪ B0). Approximate analytical results for both elec-

tric and magnetic modulations will be developed for a

better understanding of the energy spectra and electron

density. We first describe the properties of the noninter-

acting system.

A. Noninteracting electrons

Treating the magnetic modulation as a perturbation

presents some difficulties in this limit, since for nonvan-

ishing s and z → 0 the matrix elements ?ϕL

ϕL

n+mX0? given by Eq. (4) diverge for m = 0,±1, while

those with m = ±2 are finite and those with | m | > 2

vanish. Thus the Hamiltonian matrix becomes band-

diagonal, and the divergent elements cancel in a compli-

cated way in order to yield finite eigenvalues. Therefore,

for Kl0 ≪ 1 an accurate numerical diagonalization re-

quires a large matrix (4) and the Landau level mixing

is strong, except if s → 0. This complication does not

occur for the electric modulation (10) for which the Lan-

dau wave functions diagonalize the matrix in the long-

period limit for any v1, and first-order perturbation the-

ory gives the exact energy spectrum for z → 0, namely18

ǫnX0= (n + 1/2)¯ hω0+ V cos(KX0).

Instead of using standard perturbation theory with re-

spect to the modulation stregth s, we can handle the

Hamiltonian (3) by performing a Taylor expansion of the

potential term

?

its minimum X1given by

n X0| H0(X0) |

¯ hω0/2l2

0

??x − X0+

s

KsinKx?2around

X1= X0−s

KsinKX1.(12)

For fixed KX0 and |s| < 1 this has a unique solution

KX1 with X1 = X0 for KX0 = 0,π. The parabolic

approximation reads

H0(X0) ≈ ¯ hω0

?

−l2

0

2

d2

dx2+(1 + scosKX1)2

2l2

0

(x − X1)2

?

,

(13)

with an error term of order s¯ hω0Kl0((x−X1)/l0)3from

which we can show that (13) yields the eigenvalues and

the density for low filling factors correct to leading or-

der in Kl0. The Hamiltonian (13) is equivalent to the

unperturbed one, Eq.(3), but with modified center coor-

dinate X1, cyclotron frequency ˜ ω0= ω0(1+scos(KX1))

and magnetic length˜l0 = l0/?1 + scos(KX1).

tions is the shift (12) of their center of weight. We see

that, since K(X1− X0) is independent of K, the abso-

lute shift X1− X0increases with increasing modulation

period (K → 0) at equivalent positions within the pe-

riod (i. e. for fixed KX0) except for KX0= 0,π. This

explains the difficulties with the standard perturbation

The

main effect of the magnetic modulation on the wave func-

3

Page 4

theory which expands the shifted Landau functions with

center X1(X0) in the basis of Landau functions centered

around X0.

The Landau bands resulting from Eq.(13) are

ǫn(X0) = ¯ hω0

?

1 + scos?KX1(X0)???

n +1

2

?

. (14)

The appearance of X1instead of X0in Eq. (14) leads to a

substantial deviation of the simple cosine band shape pre-

dicted by first-order perturbation theory; the bandwidth

is, however, given correctly by Eq. (6). In calculating

the density, the indicated replacement of l0by˜l0leads to

corrections of higher order in Kl0 and, since this order

is not included correctly, is not to be used. We there-

fore insert just shifted Landau-functions into Eq. (7) and

obtain

ρ(x) =

gs

2πl2

0

?

?

n

dX1dX0

?

dX0f(ǫn(X0)) | ϕL

n,X1(X0)(x) |2

=

gs

2πl2

0

?

n

dX1f(ǫn(X0(X1)) | ϕL

n,X1(x) |2. (15)

From (12) we have dX0/dX1= 1+scos(KX1) and with

(14) for the energy spectrum we finally get

ρ(x) =

gs

2πl2

0

?

n

?

dX1

?

1 + scos(KX1)

?

n,X1(x) |2. (16)

×f

?

¯ hω0

?1 + scos(KX1)??n + 1/2??

Both results (14) and (16) turn out to be reliable within

a relative accuracy of (sKRc) . They can also be derived

by a simple variational approach, using a set of translated

oscillator states ϕL

n,X0+u(x) as trial wave functions and

taking the limit z → 0 after minimizing the expectation

value of the energy.20The numerical results which we

shall present are obtained from a diagonalization of (4),

however.

We note that for the electric modulation (10) the re-

sults corresponding to Eqs. (12), (14) and (16) read

X1= X0 + v1Kl2

| ϕL

0sinKX1,

ǫn(X0) = ¯ hω0

?

n + 1/2

+ v1

?

1 − (1/2)?n + 1/2?(Kl0)2?

cos(KX1)

?

(17)

and

ρ(x) =

gs

2πl2

0

?

n

?

dX1

?

1 − v1(Kl0)2cos(KX1)

?

×f

?

¯ hω0

?n + 1/2 + v1cos(KX1)??

The error term in the Taylor expansion around X1is here

of order v1¯ hω0(Kl0)3((x−X1)/l0)3and permits inclusion

of the (Kl0)2terms. The total width of the bands from

| ϕL

n,X1(x) |2. (18)

(17) is also obtained from the result (11) of perturbation

theory by expansion around Kl0= 0 up to order (Kl0)2.

We see from Eqs. (17,18) that for a long-period cosine

electric modulation the bands follow the potential with

constant width and the states are not changed by the

modulation; consequently the density is only affected by

the dispersion of the levels via the argument of the Fermi

function. In contrast, for a magnetic modulation accord-

ing to (14) the widths of the Landau bands increase lin-

early with n and the X1-dependent prefactor of the Fermi

function in (16) does not decrease with increasing period.

In Fig. 1 the dashed lines show the modulation of the

density of the noninteracting system with a magnetic

modulation of amplitude B1= 0.1 T for different values

of the filling factor between 4 and 6 obtained by sweeping

B0; the temperature is 1 K, so that kBT is much smaller

than s¯ hω0. The density is given in units of 1/(2πl2

that the mean value of each line equals ν. The lines for

the even-integer values of the filling factor, ν = 4,6, are

marked with circles. They show a cosine form, where

the amplitude is larger for ν = 6 than for ν = 4. This

behaviour is easily derived from Eq. (16); since here the

chemical potential lies in a gap, the Fermi function is

either 0 or 1, and the integral gives to leading order in

Kl0

δρ(x) |ν small and even= (sν/2πl2

In the corresponding result for the electric modulation,

the factor s is replaced by −v1(Kl0)2which has a differ-

ent sign and vanishes for Kl0 → 0. The persistence of

an finite density modulation at even-integer filling factors

for a period much longer than the magnetic length consti-

tutes a major difference between the two types of modula-

tion for strong average magnetic fields. Note that the re-

sult (19) can also be written in the form ρ(x) = ν/2πl2(x)

(ν small and even) where l(x) =

netic length corresponding to the local field B(x). This

means that we can in the long-period limit think of the

magnetic modulation as changing the local degeneracy of

the Landau levels, thus leading to a modulated density

even for spatially constant filling factor (in this work we

use the notion of an x-dependent filling factor ν(x) as just

counting the number of locally occupied bands, which

makes sense of course only in the long-period limit).

The dashed lines between the ones with circles in Fig. 1

show the behaviour of the density while the n = 2 level is

successively filled. Due to the energy dispersion (14) and

the low temperature, the n = 2 states around KX0= π

are occupied first, forming a region with local filling fac-

tor ν(x) = 6 while around KX0 = 0,2π we still have

ν(x) = 4 until the total filling closely approaches 6. Since

the spatial extent of the wavefunctions is small compared

to the period a, the difference in density between these

two regions is of order gs/(2πl2

that within a region of constant local filling factor the

density is not constant but follows the cosine shape im-

posed by Eq.(19) with ν replaced by the appropriate local

filling factor ν(x).

0) so

0)cosKx. (19)

?¯ h/eB(x) is the mag-

0). We observe, however,

4

Page 5

B. Self-Consistent System

Since the density profiles of the noninteracting system

correspond, according to Eq. (8), to electrostatic poten-

tials with amplitudes larger than ¯ hω0, we expect substan-

tial changes in the spectrum when we take electrostatic

self-consistency properly into account, as we do now. The

resulting densities are plotted as solid lines in Fig. 1 and

show much smaller fluctuations. In Fig. 2 we show results

for a magnetic modulation of B1= 0.1 T at filling factors

(a) ν = 5, (b) ν = 4 and (c) ν = 14.3 corresponding to

average fields B0= 2.0 T,2.5 T and 0.7 T, respectively.

The upper panel displays the self-consistent spectra for

temperatures T = 1 K and T = 0.1 K together with the

noninteracting spectra, the lower panel shows the cor-

responding self-consistent densities. More data for the

self-consistent bandwidths and the density amplitudes in

this regime are also displayed in Fig. 5 and 6 which are

discussed in section IVB.

When the total filling factor is small and not too close

to an even-integer value, the regions of increased den-

sity correspond to the minima of the Landau bands (cf.

Fig. 1). Therefore the generated Hartree potential, which

is maximum at maximum electron density, will act to re-

duce the dispersion of the not fully occupied band with

index nF. The self-consistent solution yields then a very

flat (“pinned”) band with deviations of only the order of

kBT from the chemical potential, and the local filling fac-

tor is fractional over the whole period. This situation is

shown in Fig. 2 (a) for an odd-integer average filling fac-

tor ν = 5. The self-consistent potential here has to cancel

the dispersion of the not fully occupied level nF, which

is larger than the dispersion of the levels with n < nF.

Thus the potential is VH(x) ≈ −s¯ hω0(nF+1/2)cosKx

and the dispersion of the levels with n < nF is reversed

in sign.

For even-integer filling factor, however, according to

Eq. (19) the regions of increased density correspond to

maxima of the Landau bands, since both ρ(x) and ǫn(X0)

follow the shape of the magnetic modulation with posi-

tive sign. Consequently, the potential generated by the

density modulation (19) increases the dispersion of the

highest occupied band nF instead of acting against the

modulation broadening. If the magnetic modulation is

sufficiently weak, the resulting self-consistent potential

can be calculated by combining Eqs. (16), (18) and (8)

as VH(x) =˜VHcosKx (ν small and even) where

˜VH=

sw¯ hω0ν

1 + w(Kl0)2ν

(20)

and w = a/2πaB ≫ 1. This linear behaviour breaks

down, however, if the resulting bandwidth |∆ǫnF| =

2s¯ hω0(nF + 1/2) + 2˜VH exceeds ¯ hω0. In this case the

next-higher band reaches the chemical potential around

KX0= π and the self-consistent solution (shown in Fig.2

(b) for ν = 4) features a region around KX0 = 0,2π

where the band nF is pinned to µ, a region around

KX0 = π where the band nF + 1 is pinned to µ and

a region in between where the chemical potential lies in

a gap and the density still follows the cosine shape (19).

As described in section I, similar effects of electrostatic

self-consistency are obtained for an electrically modu-

lated system,17but there the bandwidth of the highest

occupied band is always reduced compared to the non-

interacting results. Then, a modulation strength v1> 1

is needed to produce the formation of pinned regions at

even-integer filling factors, whereas in the magnetic case

only sw ∼ 1 must be satisfied.

For the parameters of Fig. 2 (c) the bands of the non-

interacting system do overlap at the Fermi level due to

the linear increase of their width with n. In this situa-

tion the density fluctuations and the induced potentials

consist mainly of higher Fourier components. The cor-

responding wavelengths 2π/ηK, with η > 1, are compa-

rable to Rc, even though Rc ≪ a is still satisfied. We

therefore cannot discuss the effects of the Hartree poten-

tial here within the limit of a long period but instead we

have to consider the density response for electric modu-

lations with a ∼ Rc. This is done in the next section. We

observe, however, that in Fig. 2 (c) the spectrum around

the Fermi level remains essentially unchanged although

we can tell from the behaviour of the lowest level that a

considerable electrostatic potential does exist.

For the lower temperature T = 0.1 K the density traces

in Fig. 2 (b) and very pronounced in (c) show also su-

perimposed short-period oscillations. These have their

origin in the nodes and maxima of the wavefunctions

and can also be reproduced by Eq. (16) with the Lan-

dau functions ϕL

n(x).

IV. RC COMPARABLE WITH PERIOD

In this section we discuss properties of the modulated

system obtained when for fixed modulation amplitude

B1the average field B0is lowered so that we enter the

regime where Rcis no longer small compared to the pe-

riod a. In this case the approximation of the modulation

by the first terms of a Taylor series breaks down and its

actual functional form becomes important. However, the

numerical method outlined in section II is still valid pro-

vided that s < 1, i. e. the total magnetic field B(x) is

nowhere vanishing. We consider first the noninteracting

system.

A. Noninteracting Electrons

The quantity we are most interested in is the ampli-

tude ∆ǫnFof the Landau level nF at the Fermi energy.

In Fig.3 (a) (solid line) this bandwidth is shown for a

weak magnetic modulation B1 = 0.01 T and average

fields B0 in the range 10 T > B0 > 0.125 T. Starting

from high fields at B0 = 10 T we have first nF = 0

5

Page 6

and the bandwidth is s¯ hω0. When the field is lowered,

|∆ǫnF| increases in steps of 2s¯ hω0at even-integer filling

factors, that is when nF jumps by one, as follows from

(14). When 2Rc/a becomes larger than about 1/4 the in-

crease of |∆ǫnF| becomes visibly slower and goes over into

an oscillatory behaviour with the first maximum at about

2Rc= 0.6a. This can be understood with the result (6)

of first-order perturbation theory. Using the asymptotic

relation between Laguerre polynomials and Bessel func-

tions we obtain for the bandwidth at the Fermi level from

(6) the formula9

|∆ǫnF| ≈

????2s¯ hω0AmJ1(KRc)

????

(21)

where J1 is the Bessel function of order 1, Rc =

l0

describes well the bandwidths (also for small filling fac-

tors) as long as the average field is strong enough to en-

sure s ≪ 1. It has zeros at approximately

KRc= (λ + 1/4)π , λ = 1,2,...

√2nF+ 1, and Am= (Rc/Kl2

0). The expression (21)

(22)

corresponding to a flat-band with vanishing dispersion

at the Fermi level.6For our parameters we encounter

only the first (λ = 1) of these magnetic flat-band situ-

ations around B−1

0

= 6.2 T−1. From Eq.(21) we infer

further that the maximum values of the bandwidth at

the Fermi level are of order 2s¯ hω0Amrather than 2s¯ hω0.

If we replace in Rcthe discrete 2nF+ 1 by ν, the factor

Am becomes

?a2ρ0/πgs and depends thus only on the

e. g. for our parameters Am = 15.6. Consequently a

seemingly weak modulation strength s ∼ (1/Am) ≪ 1 is

sufficient to yield for KRc around a maximum of the

Bessel function J1 a bandwidth |∆ǫnF| > ¯ hω0 which

means that the bands around the Fermi level do over-

lap. In Fig. 4 (a) the bandwidth at the Fermi level for a

modulation of B1= 0.1 T is plotted; it is larger than ¯ hω0

for B0< 1 T. For this stronger modulation the spectra

show, at fields B0< 0.25 T, also substantial deviations

from the first-order perturbation expression (6), because

the modulation strength s then becomes too large. The

bands around the Fermi level are not cosine-shaped in

this regime but have extrema away from KX0 = 0,π.

As a consequence, the bandwidth does not go through

a zero at the flat-band condition (22) although its be-

haviour still resembles the oscillations described by (21).

For an electric modulation the result corresponding

to (21) is |∆ǫnF| =

KRc = (λ − 1/4)π , λ = 1,2,... (these are the elec-

tric flat-band situations) and the bandwidth is always

smaller than 2v1¯ hω0. In Fig.(3) (a) the bandwidth at

the Fermi level for an electric modulation of amplitude

V1= 0.27 meV = Am(¯ he/m)·0.01 T is shown as dashed

line; the modulation amplitude is chosen such that the

bandwidths are comparable to the ones induced by the

magnetic modulation also depicted in this figure.

period and mean density; typically we have Am > 10,

???2v1¯ hω0J0(KRc)

??? with zeros at

In describing the induced density fluctuations δρ(x) =

ρ(x)−ρ0we face the difficulty that these have in general

no simple shape (see Fig. 1). As a measure of their mag-

nitude we therefore concentrate on their amplitude |∆ρ|.

This quantity is displayed in units of (1/2πl2

interacting case at temperatures T = 1 K and T = 0.1 K

in Fig. 3 (b) for a weak magnetic modulation not leading

to band overlap and in Fig. 4 (b) for a stronger modu-

lation with overlapping bands. The lower temperature

T = 0.1 K corresponds to kBT = 8.6 · 10−3meV which

in the displayed range of B0can be considered as small

compared to ¯ hω0 whereas for the higher temperature

T = 1 K the finite size of kBT becomes important for

about B−1

0

> 3 T−1. The results for the density can be

summarized as follows: If the bands around the Fermi

level do not overlap and kBT is small against ¯ hω0 and

also against the gap between the bands nF and nF+ 1,

then for even-integer filling factor the density is cosine-

shaped δρ(x)|ν

the distortion of the occupied wavefunctions by the mod-

ulated magnetic field. The amplitude rmis for KRc→ 0

equal to the total density ρ0(see Eq. (19)) and shows for

lower fields oscillations in KRcwith zeros at both elec-

tric and magnetic flat-band situations. During the filling

of each band, i. e. when the filling factor is not an even

integer, an additional density fluctuation of the order of

(1/2πl2

0) is produced due to the dispersion of the bands,

like in Fig. 1. Both effects have a tendency to cancel

each other. If kBT is not small compared to ¯ hω0or to

the gap between the bands nF and nF+ 1, this cancel-

lation becomes almost perfect and the resulting density

modulation is minute. If the bands around the Fermi

level do overlap (as in Fig. 4 (b) for B0< 1 T), the den-

sity has a complicated shape with several extrema and

its amplitude shows an irregular dependence on the fill-

ing factor. For kBT ≪ ¯ hω0the amplitude of the density

fluctuations is still of the order of 1/2πl2

whereas for a higher temperature again no appreciable

modulation of the density is produced.

We see thus that the modulated magnetic field affects

the density only if Rcis small compared to the modula-

tion period or if the temperature is very low; in any case

the resulting density modulation is limited in amplitude

by gs/2πl2

Fermi-prediction, namely no modulation of the density,

holds to good accuracy as soon as Rc> a/4.

The density amplitudes resulting from an electric mod-

ulation are shown in Fig. 3 (c). As discussed in section

IIIA, we find for Rc< a/8 density fluctutations of order

gs/2πl2

tially no modulation of the density at even-integer filling

factors. For lower fields B0, however, the density modu-

lation becomes dominated by a cosine contribution whose

amplitude is not related to gs/2πl2

Thomas-Fermi value

0) for the non-

even= srm(KRc)cos(Kx). This is due to

0(but not larger)

0. For not too low temperatures the Thomas-

0if the level nF is partially occupied, and essen-

0but rather equals the

δρTF(x) = −(V (x)/µ)ρ0

.(23)

Since δρTFis independent of B0it appears in the plotted

6

Page 7

quantity 2πl2

For the higher temperature we find that for Rc> a/4 the

density is described accurately by Eq. (23), i. e. δρ(x) =

−(v1¯ hω0/µ)ρ0cos(Kx). For the lower temperature devi-

ations from this result appear which are of similar mag-

nitude as the corresponding deviations from zero for the

magnetic modulation (except that at even-integer filling

factors they do not vanish at the magnetic flat-band con-

dition but only at the electric one). The density mod-

ulation for the low temperature and even-integer filling

factor is entirely due to the distortion of the occupied

wavefunctions and follows very well the formula derived

by Aleiner and Glazman21from first-order perturbation

theory. Most important for our purposes is the fact that

in any case for Rc> a/4 there exists an appreciable mod-

ulation of the density, whose main part is a wave-function

effect and follows the electrostatic potential linearly, in-

dependent of the spectrum at the Fermi level. Due to

the linearity in V , it is clear that this applies also to the

density response to those higher Fourier components of a

non-cosine electrostatic potential whose wavevectors ηK

satisfy ηKRc> 1.

0|∆ρ| as a linearly increasing background.

B. Self-Consistent System

Having discussed the density response induced by mod-

ulated magnetic and electric fields, we now proceed

with the investigation of the effects of electrostatic self-

consistency for the magnetically modulated system. In

Fig. 5 and 6 (a) and (b) the bandwidth at the Fermi level

of the self-consistent system and the amplitude of the

Hartree potential |∆VH| are shown for the same param-

eters as used in Fig. 3 and 4, respectively. The bandwidth

of the noninteracting system and ¯ hω0are also shown for

comparison. We can clearly distinguish the high-field

regime (limited by Rc < a/4) discussed in section III

where the spectrum around the Fermi level is dominated

by the electrostatic effects: Instead of the monotonous in-

crease in the noninteracting system the bandwidth at the

Fermi level is here of order kBT when the filling factor is

not close to an even-integer value and has at even-integer

filling factors sharp maxima with a height of the order

¯ hω0(this value is reached only for the stronger modula-

tion in Fig. 6 while in Fig. 5 Eq. (20) remains valid). The

amplitude of the potential equals the difference of the

self-consistent and noninteracting bandwidths, reflecting

the fact that electrostatic self-consistency is achieved by

adjusting the spectrum.

For lower fields with Rc> a/4 the electrostatic correc-

tions to the bandwidth at the Fermi level become much

less pronounced and the validity of (21) is eventually

restored well before the first magnetic flat-band situa-

tion. The main reason for this is that now the Hartree

potential is able to affect the density independently of

the dispersion at the Fermi level.

consistency can be achieved without changing |∆ǫnF|.

Consequently, self-

We first discuss the weaker modulation without band

overlaps (Fig. 5). Here, for the higher temperature, the

density modulation produced by the periodic magnetic

field is already small without inclusion of the Hartree po-

tential so that the self-consistent potential is also minute.

But also for the lower termperature in (b) the situation

changes around Rc = a/4 and the bandwidths become

close to the noninteracting values, although the ampli-

tude of the self-consistent potential remains appreciable.

Around Rc = (3/8)a an electric flat-band condition is

satisfied and the first Fourier component of the Hartree-

potential has no effect on |∆ǫnF|. Therefore here the

change of the bandwidth at the Fermi level due to the

Hartree-potential must be small but this is not reflected

in the amplitudes of the self-consistent densites and po-

tentials. For still lower fields, away from the electric flat-

band situation, the Hartree potential yields again notice-

able corrections to the bandwidths but the noninteract-

ing curve remains essentially valid.

For the stronger modulation displayed in Fig. (6), the

noninteracting bandwidth at the Fermi level is recovered

as soon as it gets larger than ¯ hω0. Since the bands at the

Fermi level then overlap, the density modulation consists

mainly of higher Fourier components with wave vectors

ηK, η > 1. The induced potential therefore belongs al-

ready to the regime of validity of the linear relation (23)

although we still have Rc< a/8. Therefore the Hartree

potential here reduces the density modulation in ampli-

tude but does not much alter its shape, while the bands

around the Fermi level remain dominated by the cosine

form imposed by the periodic magnetic field.

V. CONCLUSION

We have calculated the density response of a 2DEG in a

quantizing magnetic field B0to a magnetic cosine mod-

ulation B1cos(Kx) and compared it with the response

to an electric cosine modulation. We also included self-

consistently the induced electrostatic potentials which re-

duce the density fluctuations. We investigated in detail

the changes in the energy spectrum brought about by the

requirement of electrostatic self-consistency. In contrast

to the case of an electric modulation, where the Hartree

potential always tends to decrease the width of the Lan-

dau bands, for a magnetic modulation the Hartree poten-

tial may either decrease or increase the band dispersion,

depending on strength and period of the modulation and

on the filling factor.

In any case the produced density modulation depends

crucially on the temperature. If kBT is not small com-

pared to ¯ hω0 the behaviour of the density is quasi-

classical, i. e. the periodic magnetic field does not lead

to an appreciable density modulation, while the density

modulation induced by an electrostatic potential essen-

tially follows the Thomas-Fermi formula. For temper-

atures satisfying kBT ≪ ¯ hω0 (which we assume in the

7

Page 8

remainder of this section) both types of modulation lead

to an appreciable non-classical inhomogeneity of the den-

sity and thus to a non-trivial electrostatic self-consistency

problem.

We were mainly interested in the effect of the Hartree-

potential on the spectrum around the chemical potential.

We found that an important parameter is the ratio of

the cyclotron radius to the length scale aδ ρ of the den-

sity variation. If the bands in the vicinity of the Fermi

level do not overlap we have aδ ρ∼ a, whereas for over-

lapping bands the density fluctuations consists of higher

Fourier components and aδ ρis significantly smaller than

the period a of the modulation. Concerning electrostatic

effects we can clearly distinguish two regimes by the con-

ditions Rc≪ aδ ρ/4 and Rc>

total density a distinction between high and lower aver-

age magnetic fields. The value of B0around which the

regime changes depends for a weak modulation only on

the period while for a sufficiently strong modulation the

transition takes place when the bands start to overlap.

For aδ ρmuch larger than the cyclotron radius, i. e for

high enough B0, the dispersion of the energy levels is

changed drastically by the inclusion of electrostatic self-

consistency. As in the case of a purely electrostatic mod-

ulation, for which similar screening effects are known,17

the effective Landau bands may be pinned to the Fermi

level over regions comparable to the period. The nonuni-

form magnetic field affects the density also in regions

where the chemical potential lies in a gap between two

bands. In those regions the density is not constant but

reproduces the profile of the magnetic field, since the lat-

ter alters the number of states in the vicinity of each

center coordinate.

If the cyclotron radius is not small enough against aδ ρ,

Rc>

∼aδ ρ/4, the inclusion of electrostatic self-cosistency

does not lead to an appreciable change of the dispersion

of the bands around the Fermi level. The behaviour of

the latter in the regime where Rc∼ a can therefore safely

be calculated from the noninteracting system.

∼aδ ρ/4. This defines for fixed

VI. ACKNOWLEDGEMENTS

We thank Daniela Pfannkuche for fruitful discussions.

One of us (A. M.) is grateful to the Max-Planck-Institut

f¨ ur Festk¨ orperforschung, Stuttgart, for hospitality and

support. This work was supported by the German Bun-

desministerium f¨ ur Bildung und Forschung (BMBF).

1M. A. McCord and D. D. Awschalom, Appl. Phy. Lett. 57,

2153 (1990).

2H. A. Carmona et al., Phys. Rev. Lett. 74, 3009 (1995).

3P. D. Ye, et al., Phys. Rev. Lett. 74, 3013 (1995).

4S.

J. Phys. Soc. Japan 64, 706 (1995)

5P. D. Ye, et al., Appl. Phys. Lett. 67, 1441 (1995).

6P. Vasilopoulos and F. M. Peeters, Superlatt. Microstruct.

7, 393 (1990).

7D. P. Xue and G. Xiao, Phys. Rev. B 45, 5986 (1992).

8F. M. Peeters and P. Vasilopoulos, Phys. Rev. B 47, 1466

(1993).

9R. R. Gerhardts, Phys. Rev. B 53, 11 064 (1996).

10D. Weiss, K. v. Klitzing, K. Ploog, and G. Weimann in

Proc. of Int. Conf. Application of High Magnetic Fields in

Semi-conductor Physics, W¨ urzburg, 1988, G. Landwehr,

ed., Vol. 87 of Springer Series in Solid State Sciences,

Springer Verlag 1989; Europhys. Lett. 8, 179 (1989)

11R.R.Gerhardts,D. Weiss,

Phys. Rev. Lett. 62, 1173 (1989); R. W. Winkler, J. P. Kot-

thaus, K. Ploog ibid. 62, 1177 (1989).

12D. Weiss and P. D. Ye, private communication.

13F. M. Peeters and A. Matulis, Phys. Rev. B 48, 15 166

(1993).

14A. Matulis, F. M. Peeters,

Phys. Rev. Lett. 72, 1518 (1994).

15X. Wu and S. E. Ulloa, Phys. Rev. B 47, 7182 (1993).

16see e. g. J. H. van Vleck, The theory of Electric and Mag-

netic Susceptibilities, Oxford University Press, London,

1932.

17U. Wulf, V. Gudmundsson,

Phys. Rev. B 38, 4218 (1988).

18C. Zhang and R. R. Gerhardts, Phys. Rev. B 41, 12 850

(1990).

19Energy spectra for a 2D magnetic superlattice were cal-

culated by R. R. Gerhardts, D. Pfannkuche, and V. Gud-

mundsson, Phys. Rev. B 53, 9591 (1996).

20A. Manolescu, Phys. Rev. B 45, 11 829 (1992).

21I.L. Aleiner and L.I. Glazman Phys. Rev. B 52, 11296

(1995).

Izawa,S.Katsumoto, A. Endo,and Y. Iye,

andK.v.Klitzing,

and P. Vasilopoulos,

and R. R. Gerhardts,

FIG. 1.

netic modulation of amplitude B1 = 0.1 T in average fields

B0 = 2.5 − 1.7 T corresponding to filling factors ν = 4 − 6.

The temperature is T = 1 K. The dashed lines are for the

noninteracting system; the results for ν = 4 and ν = 6 are

marked with circles. The lines between these two show the

successive filling of the n = 2 level when ν is increased in

steps of 1/3. The solid lines display the results for the same

filling factors after establishing electrostatic self-consistency.

Density ρ(x) in units of 1/2πl2

0 for a mag-

FIG. 2.

tude B1 = 0.1 T at (a) ν = 5 (B0 = 2.0 T), (b) ν = 4

(B0 = 2.5 T) and (c) ν = 14.3 (B0 = 0.7 T). The upper

panel displays the spectra (dashed lines: noninteracting, solid

lines: self-consistent solution at T = 1 K, dash-dotted lines:

self-consistent solution at T = 0.1 K; the horizontal straight

line with dots indicates the position of the chemical potential

which, to the accuracy of the figure, is the same in all three

cases). The lower panel shows the density fluctuation in units

of 1/2πl2

T = 1 K and the dash-dotted line for T = 0.1 K).

Results for a magnetic modulation of ampli-

0for the self-consistent situations (the solid line is for

8

Page 9

FIG. 3.

the induced density fluctuations for the noninteracting sys-

tems under a weak magnetic modulation B1 = 0.01 T and an

electric modulation V1 = 0.27 meV (leading to comparable

maximal bandwidth at the Fermi level, i. e. v1 = Ams) for

average fields 10 T > B0 > 0.125 T. The data are plotted

versus inverse average field indicated on the top x-axis. The

bottom x-axis displays the ratio 2Rc/a ∝ B−1

width of the band at the Fermi level (solid line for the mag-

netic, dashed line for the electric modulation; the dash-dotted

line is ¯ hω0). The amplitude of the density fluctuation in units

of 1/2πl2

0is shown in (b) for the magnetic and in (c) for the

electric modulation. The solid lines are for T = 1 K and the

dashed lines for T = 0.1 K and the values for even-integer

filling factors are marked with circles or diamonds, respec-

tively. In (c) the dash-dotted line displays the prediction of

Thomas-Fermi-theory |∆ρ| = 2(V1/µ)ρ0.

Bandwidth at the Fermi level and amplitude of

0. (a) shows the

FIG. 4.

density modulation for the noninteracting system as in Fig.

3 but under a stronger magnetic modulation of amplitude

B1 = 0.1 T. In (a) the solid line is the width of the band at

the Fermi level and the dash-dotted line is ¯ hω0. (b) displays

the amplitude of the density; the solid line is for T = 1 K and

the dashed line for T = 0.1 K; the circles and diamonds mark

the values for even-integer filling factors.

Bandwidth at the Fermi level and amplitude of

FIG. 5.

tion of amplitude B1 = 0.01 T for 10 T > B0 > 0.18 T

plotted against 2Rc/a. The upper two panels show for (a)

T = 1 K and (b) for T = 0.1 K the bandwidth |∆ǫnF| at the

Fermi level (dashed lines with diamonds) and the amplitude

of the self-consistent potential |∆VH| (dashed lines with cir-

cles). The filling factor was increased in steps of 1/3 so that

each symbol corresponds to a calculated value and the lines

are only guides to the eye. The dash-dotted line in (a) and

(b) is ¯ hω0 and the solid line displays the noninteracting band-

width at the Fermi level for comparison. In (c) the amplitude

of the self-consistent densities are shown; here the solid line

with circles is for T = 1 K and the dashed line with diamonds

for T = 0.1 K.

Self-consistent results for a magnetic modula-

FIG. 6.

ulation of amplitude B1

10 T > B0 > 0.3 T. The other parameters and the mean-

ing of the lines is the same as in Fig. 5.

Self-consistent results for a magnetic mod-

=0.1 T and average fields

9

Page 10

0

π/2

Kx

π

3.5

4.0

4.5

5.0

5.5

6.0

6.5

(2πl0

2) ρ(x)

prb gossmann fig 1

Page 11

0

π/2

π

−0.04

−0.02

0.00

0.02

0.04

2πl0

2(ρ(x)−ρ0 )

0

π/2

π

0.0

5.0

10.0

15.0

εn(X0) [meV]

(a)

0

π/2

Kx

π

0

π/2

π

KX0

(b)

0

π/2

π

00

π/2

π/2

ππ

(c) (c)

prb gossmann fig 2

Page 12

0 1/41/23/41 5/43/2

2Rc /a

0.0

1.0

2.0

3.0

4.0

5.0

(2πl0

2) |∆ρ|

0

π/4

π/23π/4

π

5π/43π/2

0.0

1.0

2.0

(2πl0

2) |∆ρ|

0.01.0 2.0 3.04.05.0 6.07.0 8.0

B0

−1 [

T

−1 ]

0.0

0.2

0.4

|∆εnF| [meV]

prb gossmann fig 3

(a)

(b)

(c)

Page 13

0 1/41/23/415/43/2

2Rc /a

0.0

1.0

2.0

(2πl0

2) |∆ρ|

0.01.02.03.04.05.0 6.07.08.0

B0

−1 [

T

−1 ]

0.0

1.0

2.0

3.0

4.0

|∆εnF| [meV]

prb gossmann fig 4

(a)

(b)

Page 14

0 1/4 1/23/41

2Rc/a

0.00

0.02

0.04

0.06

2πl0

2|∆ρ|

0.0

0.2

0.4

0.6

0.8

1.0

E [meV]

0.01.0 2.03.04.0 5.0

B0

−1 [

T

−1 ]

0.0

0.2

0.4

0.6

0.8

1.0

E [meV]

prb gossmann fig 5

(a)

(b)

(c)

T = 1 K

T = 0.1 K

Page 15

01/41/2

2Rc/a

0.00

0.02

0.04

0.06

2πl0

2|∆ρ|

0.0

2.0

4.0

6.0

E [meV]

0.01.02.03.0

B0

−1 [

T

−1 ]

0.0

2.0

4.0

6.0

8.0

E [meV]

prb gossmann fig 6

(a)

(b)

(c)

T = 1 K

T = 0.1 K