Abstract
A problem modeling Hall effect in a semiconductor film from an electrode of arbitrary shape is considered, which is a skew-derivative problem. Boundary Galerkin method for solving the problem in Sobolev spaces is developed firstly. The solution is represented in the form of the combined angular potential and single-layer potential. The final integral equations do not contain hypersingular integrals. Uniqueness and existence of the solution to the equations are proved. The weakly singular and Cauchy singular integral arising in these equations can be computed directly by truncated series of Chebyshev polynomials with their weighting function without approximation. The numerical simulation showing the high accuracy of the scheme is presented.
Highlights
Hall effect is the production of a voltage difference across a semiconductor, transverse to an electric current in the conductor and a magnetic field perpendicular to the current
We introduce our formulation of skewderivative problem and proceed by establishing the existence and uniqueness of a solution to integral equations
The problem is reduced to an integral equation with some additional conditions
Summary
Hall effect is the production of a voltage difference across a semiconductor, transverse to an electric current in the conductor and a magnetic field perpendicular to the current. The analysis in [4] is performed within a Sobolev space setting and a weak solution concept, but no numerical method is presented. These authors construct the solution, via a combination of a single- and a double-layer potential, which can not describe the singularity at the tips directly. Another more difficult problem is introduced, that is, hypersingularity. The solution which is a combination of a single-layer and angular potential may have weak singularity at the tips of arc if the density function has the form μ/√1 − σ2.
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