Electron in a magnetic quantum dot

Solid State Communications, Vol. 96, No. 7, pp. 471475, 1995 Elsevier Science Ltd

Pergamon

Printed in Great Britain 0038-1098/!Z

$9.50+.00

003%1098(95)00439-4

ELECTRON

IN A MAGNETIC L. Solimany

I. Institut

fiir Theoretische

Physik,

Universitgt

QUANTUM

DOT

and B. Kramer Hamburg,

JungiusstraBe

9, 20355 Hamburg,

Germany

(Received and accepted 15 June 1995 by A.H. MacDonaM)

We have considered

decreased

there

where

electronic

has been an increasing

a two dimensional

It became

possible

to fabricate

a scanning

netic flux tubes numerically3

ing in a spatially pendent

ered6.

random

equation

of a magnetic

vector

quantum

shifts for an electron

analytically. dot considered

in this case.

numerical

tum mechanical In the present the (~,‘p)-plain magnetic

The

the system

quantum

momentum. bution.

momentum,

surprising

for the motion

than

with

in a homogeneous

to that

the region

of

it needs

negative

angular

magnetic

term that originates

We calculated

namely

free region),

field,

is

in the angular

also the level spacing

It shows an exponential

for an integrable

result, within

potential

with

as compared

behavior,

distri-

as expected

system.

number The Hamiltonian

to

step,

of the system

is

(1)

(r
H=&(P:+$+P:)

barriers

investigated4.

A

in a quan-

m,

moving

in excan

Aharonov-Bohm

and can be characterized

mass.

The

homogeneous

magnetic

fills the whole space except of a region

of radius ro, which we describe

in cylindrical

coordinates

by B(r) =

arrange-

leads to the result

is the electron

field in z-direction

of a perpendicular

which is non-zero

is integrable

energy

The elec-

energy for the motion

is localized

effective

due to an additional

of the radius TO. The system treatment

the electron

the

poten-

This

from the one

gauge.

to the mag-

an electron

under the influence

classical

levels.

momentum,

dot was done5.

as an inverse

angular

de-

The phase

of the wave functions

paper we consider

a cylinder

be considered ment.

In contrast

A magnetic

magnetic

which is different

model in symmetrical

the

with an effec-

below there are no bound

field in the z-direction,

cept within

a higher

spatial

from a single flux vortex

well were analytically

calculation

momentum

states

Also the inverse

dot was considered6.

scattered

netic quantum states

although

We solve the SchrGdinger of the dot and calculate

We find bound

(negative)

the dot (i.e.

mov-

40 = h/e), it is possible potential.

were calculated

and a magnetic

the Landau

field was consid-

to a rational

space.

tron needs a lower (higher)

have studied The

spectrum.

tive angular

if and only if the flux through

(in units of the flux quantum a periodic

levels.

localization

inside and outside

using

which complicates

of the magnetic

It was found that,

angular

for a classically

in the phase

positive

particle

field.

quantum

mag-

field causes a velocity

the unit cell of the field is equal construct

realized

(localized),

for the Landau

from 30

more than an electrostatic

variation

and negative

as expected

of

Schrijdinger

to the Landau

energy

microscope

authors

magnetic

term in the Hamiltonian,

Periodic

The

as compared

a torus

mag-

with random

a charged

of the magnetic

Schradinger tial.

Several

and analytically5

inhomogenity

tunneling

has been experimentally

type II superconductors’.

magnetic

equations

are for positive

equation

‘magnetic

to 100 nm by using

A 2D system

states

inter-

perpendicular

10 to 30 nm and heights

method.

orbits

is Possonian,

(2D) electron

dots’ with diameters lithographic

respectively,

level spacings

nanostructures,

to an inhomogeneous

field’.

the eigenenergies

and increased,

of energy

for periodic

classical

system.

Keywords:

years,

and the condit.ion

is solved analytically,

integrable

netic

in an inhomogeneous derived.

are solved,

The distribution

gas is exposed

confined

motion momenta

In recent

ma.gnetically

The

equation

est in systems

an electron

field which is B = 0 for r < ro and B # 0 elsewhere.

that

We have calculated

by 471

B for r > rg 0 for r
the corresponding

vector

potential

412

ELJXI’RGN

IN A MAGNETIC

QUANTUM

DOT

Vol. %, No. 7

of two successive crossing points of the trajectory with the circumference of the magnetic dot has to be equal to distance between two sucessive reflections in the conventional circle billiard for a closed orbit. First we express the cyclotron radius rC in terms of the angle cp between the trajectory and the tangent to the circumference at the crossing point as shown in Fig. 1 (corresponds to the reflecting angle in the conventional circle billiard) and then set it equal to the classical cyclotron radius r, = z, where u is the velocity of the electron. This yields meu eB

-=

rosin(v)

I.0

cos(2cp - 5) =pcos(cp);

~p=$; l,jEN

.

3

FIG. 1. Classical orbit in coordinate space, dashed line indicates r-0. Inset: derivation of condition for closed orbits.

tic eB0 the magnetic length, the coordinate r --+ lnr, r and the energy E + liw,e with w, = eBo/m,c, the cyclotron frequency. Both, the wave functions inside and outside of the dot are separable

from the flux @ =

@=

J

The vector potential

f

ids = 2mA(r)

i?df=

nB(r2 - r;f)

in q-direction

Following Hannay and Berry” the quantisation in phase space in the form of a torus leads to eigenstates, which correspond to periodic orbits in the classical limit. Only closed classical orbits correspond to quantum mechanical eigenstates. We solve the Schrodinger equation in two regions, outside and inside of the dot and calculate the energies by continuously matching the wave function at ro. We express all quantities in dimensionless units, with 1~ =

.

Q(r, ‘p, 2) = $,(r)eimlPe-ik~Z .

is then given by

(6) One of the classical trajectories, calculated numerically from the Poisson brackets9 is shown in Fig. 1. We considered here only trajectories, which cross the discontinuity in the magnetic field. The system is integrable, energy and angular momentum are conserved. The trajectories correspond to a torus in phase space. In the limit of an infinitely large magnetic field, the system can be considered as a conventional circle billiards. Similar as for the circle billiard we can set up a condition for periodic orbits in our magnetic billiard. Periodic orbits occur in the conventional circle billiard only if the reflection angle cp (Fig. 1) is a rational multiple’ of R. For the determination of the condition for closed classical orbits in magnetic dot, we require that the distance

(8)

Here m E 2 is the angular momentum and k, is the wave vector in z-direction. It corresponds to an additive term in the energy. Therefore we can set Ic, = 0 and deal with a lD-Schriidinger equation. For r > ro the Schrijdinger equation takes a similar form as for an electron in a homogeneous magnetic field in cylindrical coordinates

4f (&;;-,(

- X2r2 + 2e,fj

)

1c,= 0,

(9)

with X = l/2, rn,lf = m + X9$ an effective angular momentum, and ecfl = Am $ E + X2$ an effective energy. In contrast to a homogeneous magnetic field, r. = 0, the eigenvalues are not degenerate with respect to the angular momentum. The ansatz &(r) = rlm=~fle-‘1/4u(r) , and a substitution t = r2/2 transforms this differential equation into Kummers differential equation. The general solution is a combination of two independent confluent hypergeometric functions”.

Vol. 96, No. 7

ELECTRON IN A MAGNETIC QUANTUM DOT

The confluent hypergeometric function I Fl [a, c, t] is not defined for c = -n,n E N, we have to take in this case, namely for ri/2 E N, the logarithmic solution. On the other hand, if a = -n, n E N than iFr[a,c,z] becomes a polynomial of de ree n and the wave function 8 vanishes for r + cc as e-’ j2. This determines the Landau levels in a homogeneous magnetic field. The wave function in the magnetic quantum dot has to satisfy two conditions. First, it has to vanish for r + 00. Furthermore the energy has to be a continuous variable. In order to fulfill both conditions we can usei an asymptotic form of the confluent hypergeometric function for large r and set this equal zero. From this is it possible to calculate the ratio of the wave function coefficients.

where a1 = -n + -n-

b.ff

_-m,ff

2

2

, ~2 =

=

> Cl

Ihtt +m.fL

5

1 -m,ff,

=

l+m,ff,

a2 =

r2/2.

For r < rs we can describe the electron by the free particle wave function within a circle, which fulfills the the Bessel differential equation. It is a combination of Bessel and Neumann functions. Because of the nonregularity of the Neumann function at r = 0, d&(r)

The that

=

&(dZr)

.

(11)

of the effective potential for m = 0 suggests, wave functions

their

derivatives

473

leads to the energy eigenvalues E,. The dependence of the energy eigenvalues on m are shown in Fig. 2 for two values of radius ro. The comparison with B = const. shows, that for m > 0 the Landau levels are decreased, while for m < 0 they are increased. The lowest eigenvalue corresponds to rn,/f = $2, i.e. m = 0. For r < r. the effective angular momentum is m$, = m, outside it is different, ma; = m + ri/2 . This difference leads to an increase or decrease of the eigenvalues depending on the sign of m. The electron needs a higher energy for the motion with an negative angular momentum than in a system with a homogeneous magnetic field, but a lower one for the motion with positive angular momentum in the neighborhood of the dot. For large values of m the wave function is localized at a lar e distance from the dot, namely at < r >X &x5 and the wave function is no longer influenced by the presence of the dot. The dependence of the ground state energy from the radius of the dot rc in the asymptotic case of large magnetic field or large r. is ~0 N l/r:, which corresponds to the same result as in a system with a potential well. We introduce now a statistical quantity that characterizes the system. Although the solutions to the Schriidinger equation are not at all random, we can, of course, use a probabilistic language for their description. Such a language has been used previously, in order to distinguish between integrable, and nonintegrable systems12. A characteristic feature of integrable quantum systems is that the distribution of the spacings AE of their eigenvalues is Poissonian. They behave like completely random entities. On the other hand, non-integrable systems show level repulsion. The

0.8

0.6 P(a) 0.4

E

1

2. Dependence of the energy eigenvalues effective angular

momentum

m,/,

= m + $2

ru = 3. Dashed lines show the Landau

2

3

8

5

4

6

7

E on the r. = 1 and

levels (rn = 0).

FIG.

3.

Distribution

(histogram);em2

of energy

level

spacings

P(s)

dotted line. Inset: density of states p(E).

474

ELECTRON

IN A MAGNETIC

level spacings distributions are given approximately by the ‘Wigner surmise’14. We have calculated the eigenvalues for our system in the regime 0 < E < 50 for -20 < m < 20 and for different ro. Fig. 3 shows the distribution of the level spacings and the density of states (DOS), calculated from approximately 10000 eigenvaiues. We have ignored the degeneracies in the Landau levels. They lead to delta peaks at E = n + 5, n f N. Because the DOS is not constant, we unfolded the spectrum?. The DOS is given by p(& - pi) = xi S(E - E,), while the integrated density of states becomes N(e) = S’p(c)dc = CiO(E - &;). ei is the i-th level. A’(E) can be approximated by a smooth function f(s). Then the raw data for the energy levels E are transformed by defining E = &f’(e) (‘unfolding procedure’). Here, E denotes the unfolded level. From the unfolded energy eigenvalues is it possible to determine the energy level spacing distribution P(s), where s = AE/m is the difference AE of two successive eigenenergies divided by the mean spacing AE. The distribution P(s) has in principle also two delta peaks due to the degeneraties of levels for m + co. Otherwise, the distribution is exponentially decaying Fig. 3, as expected for an integrable system. Small spacings are most probable, large spacings are rare, similar to the distribution of the circle billiard13. We have also checked, that the distribution

QUANTUM

DOT

Vol. 96, No. 7

is independent of chioce of the parameters r-0 and m. In conclusion, the spectrum and corresponding eigenstates of an electron moving in a 2D magnetic quantum dot is analytically calculated. The eigenenergies are for positive and negative angular momenta decreased and increased, respectively, as compared to the Landau levels. In contrast to the earlier work4, we have used a different form of the wave function, which is also valid for energies between the Landau levels and which vanishes for r + M. The distribution of level spacings, calculated from about 10000 eigenvalues, is Poissonian as expected for an integrable system. The magnetic quantum dot considered here transforms for B + 00 into a classical circle billiard or a circular quantum well. For B < 00 the wave function penetrates deeper into the region of the magnetic field as in the case of the circular quantum well. It can also be localized outside of the dot. Acknowledgements - We thank Heinrich Heyszenau, Markus Batsch, Isa Zharekeshev, Wolfgang Hausler, Kristian Jauregui and Andrea Huck for fruitful discussions, and Tomi Ohtsuki for providing his program for the calculation of classical trajectories. This work was supported by the Freie und Hansestadt Hamburg and by the EU via SCIENCE SCC-CT90-0020 and HCM CHRX-CT93-0126.

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