A Revised Wigner Function Approach for Stationary Quantum Transport

Kosik, Robert; Cervenka, J.; Thesberg, Mischa; Kosina, H.

doi:10.1007/978-3-030-41032-2_46

Cited by 4 publications

(5 citation statements)

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“…The constrained Wigner equation has first been introduced in [12]. Solving the constrained sigma equation in a least squares sense gives reasonable but not completely satisfactory results [10]. The natural way to incorporate double homogeneous boundary conditions is by employing an absorbing potential layer as recently introduced in [20].…”

Section: Simulation Resultsmentioning

confidence: 99%

“…Results showing a negative differential resistance when simulating IV-curves of resonant tunneling diodes are an artefact of numerical diffusion (upwinding) using a too coarse mesh. The method breaks down in the limit of a fine mesh [10].…”

Section: Discussionmentioning

confidence: 99%

“…The overall insight from these studies is that proper open boundary conditions should be employed on non-spatial boundaries when solving the stationary sigma equation (characteristic Neumann equation). Analysis of the two equations for k = 0 [10] has been a detour to arrive at this conclusion. An alternative formulation (not based on an absorbing potential and not overdetermined) of non-spatial open boundary conditions for the sigma equation has been suggested in [9].…”

Section: Discussionmentioning

confidence: 99%

“…The reason for the failure was not readily understood until further work was performed which is included in [10,12]. It turned out that simulation results which showed no negative differential resistance were actually numerically correct.…”

Section: Failure Of Frensley's Methodsmentioning

confidence: 99%

“…The only reasonable choice for the integration constant is to set on a s-domain symmetric around s = 0 , i.e., fully homogeneous boundary conditions. Fully homogeneous boundary conditions have been motivated by inverse Fourier transform of the integrability constraint (18) and the transport equation at k = 0 (19) in [10]. When solving the Wigner equation in k-space on a shifted k-grid, one has to incorporate the (18) which also results in an overdetermined system (constrained Wigner equation).…”

Section: The Integrability Constraint On a Finite Interval Becomesmentioning

confidence: 99%

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Numerical constraints and non-spatial open boundary conditions for the Wigner equation

2021

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We discuss boundary value problems for the characteristic stationary von Neumann equation (stationary sigma equation) and the stationary Wigner equation in a single spatial dimension. The two equations are related by a Fourier transform in the non-spatial coordinate. In general, a solution to the characteristic equation does not produce a corresponding Wigner solution as the Fourier transform will not exist. Solution of the stationary Wigner equation on a shifted k-grid gives unphysical results. Results showing a negative differential resistance in IV-curves of resonant tunneling diodes using Frensley’s method are a numerical artefact from using upwinding on a coarse grid. We introduce the integro-differential sigma equation which avoids distributional parts at $$k=0$$ k = 0 in the Wigner transform. The Wigner equation for $$k=0$$ k = 0 represents an algebraic constraint needed to avoid poles in the solution at $$k=0$$ k = 0 . We impose the inverse Fourier transform of the integrability constraint in the integro-differential sigma equation. After a cutoff, we find that this gives fully homogeneous boundary conditions in the non-spatial coordinate which is overdetermined. Employing an absorbing potential layer double homogeneous boundary conditions are naturally fulfilled. Simulation results for resonant tunneling diodes from solving the constrained sigma equation in the least squares sense with an absorbing potential reproduce results from the quantum transmitting boundary with high accuracy. We discuss the zero bias case where also good agreement is found. In conclusion, we argue that properly formulated open boundary conditions have to be imposed on non-spatial boundaries in the sigma equation both in the stationary and the transient case. When solving the Wigner equation, an absorbing potential layer has to be employed.

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Section: Simulation Resultsmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Failure Of Frensley's Methodsmentioning

confidence: 99%