William C. Hall
UALR Department of Physics
The existence of extended states in quantum Hall type systems
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.
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
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.
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.
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.
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