The Wigner transport equation can be solved stochastically by Monte Carlo techniques based on the theory of piecewise deterministic Markov processes. A new stochastic algorithm, without time discretization error, has been implemented and studied in the case of the quantum transport through a rectangular potential barrier.
1. H. Kosina Wigner function approach to nano device simulation International Journal of Computa- tional Science and Engineering vol. 2 no. 3-4 pp. 100-118 2006.
2. O. Morandi and L. Demeio A Wigner-function approach to interband transitions based on the multiband-envelope-function model Transport Theory and Statistical Physics vol. 37 no. 5-7 pp. 473-459 2008.
3. O. Morandi and F. Schürrer Wigner model for quantum transport in graphene Journal of Physics A: Mathematical and Theoretical vol. 44 no. 26 p. 265301 2011.
4. S. Shao T. Lu and W. Cai Adaptive conservative cell average spectral element methods for transient Wigner equation in quantum transport Communications in Computational Physics vol. 9 no. 3 pp. 711-739 2011.
5. A. Dorda and F. Schürrer A WENO-solver combined with adaptive momentum discretization for the Wigner transport equation and its application to resonant tunneling diodes Journal of Computational Electronics vol. 284 pp. 95-116 2015.
6. Y. Xiong Z. Chen and S. Shao An advective-spectral-mixed method for time-dependent many-body Wigner simulations SIAM Journal on Scientific Computing vol. 38 no. 4 pp. B491-B520 2016.
7. J.-H. Lee and M. Shin Quantum transport simulation of nanowire resonant tunneling diodes based on a Wigner function model with spatially dependent effective masses IEEE Transactions on Nan- otechnology vol. 16 no. 6 pp. 1028-1036 2017.
8. M. L. V. de Put B. Soree and W. Magnus Efficient solution of the Wigner-Liouville equation using a spectral decomposition of the force field Journal of Computational Physics vol. 350 pp. 314-325 2017.
9. L. Shifren and D. Ferry Particle Monte Carlo simulation of Wigner function tunneling Physics Letters A vol. 285 pp. 217-221 2001.
10. M. Nedjalkov R. Kosik H. Kosina and S. Selberherr A Wigner equation for nanometer and femtosecond transport regime in Proceedings IEEE Conference on Nanotechnology pp. 277-281 IEEE 2001.
11. D. Querlioz and P. Dollfus The Wigner Monte Carlo method for nanoelectronic devices. Wiley 2010.
12. P. Ellinghaus J. Weinbub M. Nedjalkov and S. Selberherr Analysis of lense-governed Wigner signed particle quantum dynamics Physica Status Solidi RRL vol. 11 no. 7 p. 1700102 2017.
13. M. Nedjalkov P. Ellinghaus J. Weinbub T. Sadi A. Asenov I. Dimov and S. Selberherr Stochastic analysis of surface roughness models in quantum wires Computer Physics Communications vol. 228 pp. 30-37 2018.
14. M. Nedjalkov H. Kosina S. Selberherr C. Ringhofer and D. K. Ferry Unified particle approach to Wigner-Boltzmann transport in small semiconductor devices Physical Review B vol. 70 p. 115319 2004.
15. W. Wagner A random cloud model for the Wigner equation Kinetic & Related Models vol. 9 no. 1 pp. 217-235 2016.
16. O. Muscato and W. Wagner A class of stochastic algorithms for the Wigner equation SIAM Journal on Scientific Computing vol. 38 no. 3 pp. A1438-A1507 2016.
17. O. Muscato A benchmark study of the signed-particle Monte Carlo algorithm for the Wigner equation Communications in Applied and Industrial Mathematics vol. 8 no. 1 pp. 237-250 2017.
18. O. Muscato and W. Wagner A stochastic algorithm without time discretization error for the Wigner equation Kinetic & Related Models vol. 12 no. 1 pp. 59-77 2019.
19. E. Wigner On the quantum correction for thermodynamic equilibrium Physical Review vol. 40 no. 2 pp. 749-759 1932.
20. V. F. Los and N. Los Exact solution of the one-dimensional time-dependent Schrödinger equation with a rectangular well/barrier potential and its applications Theoretical and Mathematical Physics vol. 177 no. 3 pp. 1706-1721 2013.
21. O. Muscato and V. Di Stefano Hydrodynamic modeling of silicon quantum wires Journal of Com- putational Electronics vol. 11 no. 1 pp. 45-55 2012.
22. O. Muscato and V. Di Stefano Hydrodynamic simulation of a n+ - n - n+ silicon nanowire Continuum Mechanics and Thermodynamics vol. 26 no. 2 pp. 197-205 2014.
23. O. Muscato and T. Castiglione Electron transport in silicon nanowires having different cross-sections Communications in Applied and Industrial Mathematics vol. 7 no. 2 pp. 8-25 2016.
24. O. Muscato and T. Castiglione A hydrodynamic model for silicon nanowires based on the maximum entropy principle Entropy vol. 18 no. 10 p. 368 2016.
25. O. Muscato T. Castiglione V. Di Stefano and A. Coco Low-field electron mobility evaluation in silicon nanowire transistors using an extended hydrodynamic model Journal of Mathematics in Industry vol. 8 p. 14 2018.