The Wigner equation describing stationary quantum transport has a singularity at the point \(k=0\). Deterministic solution methods usually deal with the singularity by just avoiding that point in the mesh (e.g., Frensley’s method). Results from such methods are known to depend strongly on the discretization and meshing parameters.
We propose a revised approach which explicitly includes the point \(k=0\) in the mesh. For this we give two equations for \(k=0\). The first condition is an algebraic constraint which ensures that the solution of the Wigner equation has no singularity for \(k=0\). If this condition is fulfilled we then can derive a transport equation for \(k=0\) as a secondary equation.
The resulting system with two equations for \(k=0\) is overdetermined and we call it the constrained Wigner equation. We give a theoretical analysis of the overdeterminacy by relating the two equations for \(k=0\) to boundary conditions for the sigma equation, which is the inverse Fourier transform of the Wigner equation.
We show results from a prototype implementation of the constrained equation which gives good agreement with results from the quantum transmitting boundary method. No numerical parameter fitting is needed.
Wigner function Sigma function Finite difference method Constrained equation Quantum transport Device simulation Resonant tunneling diode
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This work has been supported in parts by the Austrian Research Promotion Agency (FFG), grant 867997.