The current percolation has been considered in the medium with boundaries under quantum Hall effect conditions. It has been shown that in that case the effective Hall conductivity has a nonzero value due to percolation of the Hall current through the finite number of singular points (in our model these are corners at the phase joints).
One of the basic issues of the study of current percolation in the heterogeneous medium is the problem to evaluate the effective conductivity
In two-dimensional case the exact results for the effective conductivity have been obtained due to the dual symmetry of equations for the constant current. The Keller’s [
However, the same reasoning is not applicable to a study of the effective Hall conductivity in the heterogeneous two-dimensional system under conditions of the quantum Hall effect:
The aim of present paper is to study the current percolation at the QHE conditions in the medium with boundaries. Values of Hall conductivity were found depending on conditions how the current percolates—along or across the layers. Also, the finite value of Hall conductivity at QHE conditions in media with boundaries was explained.
The structure of the paper is as follows. In the next section we consider a current percolation at QHE conditions in layered media. Section
In order to understand the features of current percolation at QHE conditions let us consider a simple model, in which two layered periodic media have different Hall conductivities
Layered media.
Layered media with shifting of layers at the beginning of coordinates.
Let us consider the case when the electric current flows perpendicularly to the phase boundary. At QHE conditions, such a current assures the electric field to be directed along layers. From the continuity of tangential components it follows that the field is equal to the average value
Accordingly, we obtain
Below, we consider the current percolation along the layers and seek for the solution with the
In this section the layered systems with layers shifted with respect to each other at the origin have been considered (see Figure The systems with the interphase boundaries under Quantum Hall effect conditions.
In order to find the local distribution of currents (fields) in layered media, in which regions are shifted in a chess order at the origin of coordinates, the methods of the complex variable function theory have been used. For this, let us consider the problem of conductivity in terms of the complex variable function theory [
Let us consider a plane of complex variable
The Ohm’s law in the magnetic field can also be presented in the following form:
Due to biperiodic symmetry of the layered media it is enough to consider an elementary cell consisting of two adjacent semi-infinite stripes with different conductivities. Below, we construct the conformal mapping of the internal areas of adjacent semistripes with conductivities
Here
The conformal mapping: correspondence of the points at mapping.
The boundary conditions are still valid for the electric current
Thus, the initial boundary conditions are reduced to the Riemann boundary problem for the piecewise continuous function
Considering the interval
Here the matrix
In the QHE case the eigen values have a simple form and are equal to
Let us introduce the effective Hall conductivity tensor:
Here
Firstly, it is easy to see that the components of the Hall conductivity tensor are expressed in terms of constants
As an example, let us consider two types of conditions at infinity.
We get then:
Therefore, it follows that
If these constants are equal to geometrical mean value
Therefore the following relation takes place:
If these constants are given by
This result corresponds to the solution with the
The obtained results for effective Hall conductivity are connected with unusual character of current percolation at QHE conditions. In this case, it follows from equation
As for nonzero value of the effective Hall conductivity at QHE conditions, we showed that it was due to the percolation of the Hall current through the finite number of singular points. We note that, as follows from our results, the conditions at infinity determine a character of percolation through singular points, which are corners at the phase joints in our model. There are two possibilities for the boundary Hall current percolation. First of them is the percolation of the boundary current from the first phase to the second one successively through the corner junctions. This result corresponds to the solution with the constant current at infinity (
Now the problem of construction of electron quantum interferometers is intensively studied due to a possibility of using these quantum devices for a study of many coherent inference phenomena [
The obtained results can also be used in the case of the strong magnetic field regime to find the Hall constant (which is determined by the carrier’s concentration). According to our results, the effective Hall concentration at strong magnetic fields and the two-phase case have been determined by the concentration of the phase, which forms a percolation cluster. If the concentration is distributed continuously in some interval, the answer is that the effective Hall concentration in strong magnetic field regime has been determined by the concentration of carriers at the percolation threshold.
The research has partially been supported by Grant of Russsian Foundation for Basic researchers (Grant 10-02-00573).