Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Arnold, A., Lange, H., Zweifel, P.F.: A discrete-velocity, stationary Wigner equation. J. Math. Phys. 41(11), 7167–7180 (2000). https://doi.org/10.1063/1.1318732
Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge University Press, Cambridge (2004). https://doi.org/10.1017/cbo9780511543234
Frensley, W.R.: Boundary conditions for open quantum systems driven far from equilibrium. Rev. Mod. Phys. 62(3), 745–791 (1990). https://doi.org/10.1103/RevModPhys.62.745
Lent, C.S., Kirkner, D.: The quantum transmitting boundary method. J. Appl. Phys. 67(10), 6353–6359 (1990). https://doi.org/10.1063/1.345156
Li, R., Lu, T., Sun, Z.: Parity-decomposition and moment analysis for stationary Wigner equation with inflow boundary conditions. Front. Math. China 12(4), 907–919 (2017). https://doi.org/10.1007/s11464-017-0612-9
Suryanarayana, M.B.: On multidimensional integral equations of Volterra type. Pac. J. Math. 41(3), 809–828 (1972). https://doi.org/10.2140/pjm.1972.41.809
Tsuchiya, H., Ogawa, M., Miyoshi, T.: Simulation of quantum transport in quantum devices with spatially varying effective mass. IEEE Trans. Electron Devices 38(6), 1246–1252 (1991). https://doi.org/10.1109/16.81613
Wigner, E.P.: On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40(5), 749–759 (1932). https://doi.org/10.1103/PhysRev.40.749
Acknowledgement
This work has been supported in parts by the Austrian Research Promotion Agency (FFG), grant 867997.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Kosik, R., Cervenka, J., Thesberg, M., Kosina, H. (2020). A Revised Wigner Function Approach for Stationary Quantum Transport. In: Lirkov, I., Margenov, S. (eds) Large-Scale Scientific Computing. LSSC 2019. Lecture Notes in Computer Science(), vol 11958. Springer, Cham. https://doi.org/10.1007/978-3-030-41032-2_46
Download citation
DOI: https://doi.org/10.1007/978-3-030-41032-2_46
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-41031-5
Online ISBN: 978-3-030-41032-2
eBook Packages: Computer ScienceComputer Science (R0)