# William C. Hall

UALR Department of Physics

REU-FOM/SILL

The existence
of extended states in quantum Hall type systems

Introduction

Studying extended states is important to physicists trying to understand
the transport properties of various materials, because the extended states
play an important role in determining these transport properties. Generally,
if the relative abundance of extended states in a system is high, then the
material in which the system exists will allow current or heat to flow through
it very well. Extended states are ones in which the electrons have a relatively
large probability of appearing across a large portion of a macroscopic sample.
This is the exact opposite of localized states, which have a large probability
of existing in a relatively small portion of a macroscopic sample.

In particular, we are interested in single electron states in a 2-dimensional
(2D) electron system, meaning the electrons are confined to move only parallel
to the surface of a sample material and not perpendicular to it. For example,
a 2D electron system can be “realized at the surface of a semiconductor like
silicon and or gallium arsenide where the surface is usually in contact with
a material which acts as an insulator (
for silicon field effect
transistors and, e.g.
for heterostructures). Electrons are confined close the surface
of the semiconductor by an electrostatic field
normal (pointing downward) to the interface, originating from
positive charges which cause a drop in the electric potential toward the
surface” (K. von Klitzing). In this 2D system, there is an electrical interaction
that varies across the surface, the “impurity” potential, whose physical
origin varies. For example, in the integer quantum Hall effect it is due
to atoms being substituted or displaced, whereas in the fractional quantum
Hall effect it is due to the other electrons. In addition, there is a uniform
magnetic field that is perpendicular to the plane of the surface. Our goal
is to understand, using computer simulations, the number and energy dependence
of extended states for electrons constrained to 2D motion in the presence
of a strong magnetic field perpendicular to the plane of motion and this
impurity potential.

We use the root-mean-square radius (
) and participation function (P) to quantify the extension of eigenstates
in these systems.
is a measure of the average radius of a state, while P is a
measure of an area. We first analyze
and P as
functions of eigenenergies for the single electron states. If either
or P is
equivalent to the size of the system, then we know that state is extended.
We also graph and analyze the ratio
, from which we can determine whether the states are disk-like (relatively
large area and radius) or ring-like (relatively large radius and a small
area). Finally, we will analyze the extended states and energy dependence
of
and P as
a function of the distribution of the impurities, their shape, and their
number relative to the strength of the magnetic field.

Methods

We study extended states by using two measures of extension: root-mean-square
radius
and the participation number (P). Roughly speaking,
measures the radius of an eigenstate, while P measures its area
(Horner, pg 17). We chose to use
and P because they do not measure the same things, and they
are both arguably relevant to extension: a large radius crosses the system
while a large area covers the system.
is given by
, where
is the expectation of the operator
, and is equal to
. P is given by
. In both cases,
is the
wave function of a single electron.
gives the
normalized probability of a wave function. For our methods, we can express
in terms
of
and
. We will look at two simple cases, which will give us a better idea of what
and P mean. We can first find
for a disk.
The idea is that we want a disk where the probability is constant inside
the disk and zero outside the disk and we require that if we integrate the
probability density over all space we will get 1. We want our single electron
to exist in the disk, but nowhere else. Thus, we have that

, where C is the normalization constant and
is the area of the disk. Since
C, then
(disk)
. We can now do the same steps for a thin ring, except we integrate
from 0 to
, and
from
to
, where
is the thickness of the ring. Thus have that
(thin ring). Now that we have calculated the values of
for a disk and a thin ring,
we can find
and P for a both a disk and
a thin ring. Thus, we have that

(disk)

P (disk)

(thin ring)

P (thin ring)

Comparatively speaking, a ring-like state at the edge of a system would show
a large radius but not a large area because of its thinness, while a disk-like
state would show a large radius and a large area as well. Both
and P show larger values for extended states than for localized
states.

Once we have calculated
the
and P
of the states, we graph the results of
and
, which are scaled to system size, as a function of their eigenenergies.
We scale to system size because it makes it easier to compare results for
different system radii. We use
instead
of P since we want to compare lengths instead of areas in our analyses. We
know that if the highest peak value of
goes to
1, then the radius of the state is the same as the radius of the system and
there is definitely an extended state. When we look at the peak of
, a value of approximately 1.77
would indicate
there is an extended state. We then graph the ratio
, which is a measure of the degree to which a state looks like a disk or
a ring (Horner, L. pg 19). In the near future, we will analyze the impurities
of
and
as a function of the number of impurities relative to the strength
of the magnetic field, the shape of the impurities, and the distribution
of the impurities.

Results

In my particular case, I am studying single electron states in 2D electron
systems where there is only one impurity. The following three graphs (figures
1, 2, and 3) are a sample of some of the results that I obtained for this
type of system.

Figure (1) is a graph of the
(scaled to system size) as a function of eigenenergy. Of particular
interest to us is the peak on the left side of the graph. It represents the
most de-localized state, which is crossing most of the system. Notice that
the peak value is 1. This means, as we explained earlier, that there is an
extended state since the radius of the state extends across the radius of
the system
. Other examples are less clear as to whether there are extended states,
since their peak values are large but not 1.

Figure (2) is a graph
of the
as a function of energy.
is scaled
to system size as well. The same thing is happening here as in figure 1,
only now the peak value measured (slightly higher that 0.75) is less than
the value of 1.77 needed for extension. We can now say that there are not
any disk-like states that cover the whole system in this particular system.

Figure (3) is a graph
of the ratio
. It is not completely clear to us at the present time if this graph will
prove directly useful to us in our current work. However, it does show us
that the
and
are different ways of measuring the extension of a state, since
the graph is not a straight line. This graph helps us to understand what
is meant by extended states by implying that if you look at extended states
at some level, you have to be more specific about how you define extended
states, since you now have a choice in what you want to call extended.

In the future, after
accumulating enough data from many different small system sizes, and extrapolating
to an infinitely large system, we will graph and analyze the
and
as functions of
the number of impurities relative to the strength of the magnetic field,
the distribution of the impurities, and their shape. We want to know if the
results we get will be any different from those obtained when we graphed
the
and
as functions of their energies for this system.

Summary

We are interested
in the existence and energy dependence of extended states in quantum Hall
type systems. We have computed the root-mean-square radius (
) and participation number (P) as a function of energy
for a few system sizes. Our next step is to analyze the peak
values of
and P as
a function of “impurity” distribution, shape, and number relative to the
strength of the magnetic field.

BIBLIOGRAPHY

Aoki, H., Computer simulation of two-dimensional disordered electron systems
in strong magnetic fields, J. Phys. C: Solid State Phys., vol. 10 (1977)
pp. 2583-2593.

Gedik, Z. and M. Bayindir, Disorder and localization in the lowest Landau
level in the presence of dilute point scatterers, Solid State Communications
vol. 112 (1999) pp. 157-160.

Horner, M.L., Transmutation of Electric into Magnetic Forces on a Planar
Electron: Impurity Spacing and Band Structure in Strong Magnetic Fields,
Disssertation (1999), State University of New York at Stony Brook

Jain, J. K., The Composite Fermion: A Quantum particle and its quantum Fluids,
Physics Today, vol. 53 no. 4 (April 2000) pp. 39-45.

von Klitzing, K., The quantized Hall effect, Nobel Lecture (1985), Max-Planck-Institut
für Festkörperforschung, D-7000 Stuttgart 80.

Störmer, H.L., D.C. Tsui and A.S. Gossard, The fractional Hall effect,
Reviews of Modern Physics, vol. 71 no. 2 (1999) pp. S298-S305.

Thouless, D.J., Electrons in disordered systems and the theory of Localization, Physics Reports, vol. 13 no. 3 (1974) pp. 93-142.