Wigner Function Modeling in Quantum Transport Systems

The Wigner function provides a quasi-probability distribution in phase space that encodes quantum coherence and interference within a framework analogous to classical Boltzmann transport theory. By representing electron dynamics in both position and momentum variables, this approach captures non-local quantum effects such as tunnelling and resonant transport in nanoscale heterostructures. The governing Wigner transport equation combines a drift term with a non-local integral operator that accounts for the quantum potential. Accurate numerical modelling must address the oscillatory nature of the Wigner function, enforce open boundary conditions, and manage high dimensionality. Deterministic schemes leverage high-order spatial discretisations and adaptive momentum grids, while stochastic particle methods generate and annihilate signed particles to sample phase-space evolution. Alternative formulations recast the centre-of-mass von Neumann equation into a Liouville-type form amenable to tight-binding Hamiltonians and domain decomposition, enabling detailed atomic-scale descriptions and efficient time propagation. Boundary treatments, including complex absorbing potentials, ensure stability by damping unphysical reflections. These advances support the simulation of resonant tunnelling diodes, field-effect transistors and emerging quantum devices, offering predictive insight into coherent transport, device optimisation and thermal management at the nanoscale.

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Recent studies have interrogated the impact of momentum-space discretisation on phase-space simulations. One investigation revealed that overly fine momentum grids introduce spurious regime changes at device contacts, leading to non-physical artefacts in current–voltage characteristics. By adopting non-rectangular window functions for the Wigner potential, the authors restored continuity and obtained reliable transport predictions. Concurrently, a tight-binding framework applied to the von Neumann equation has incorporated atomic lattice details into a Liouville-type model. The inclusion of a complex absorbing potential at device boundaries was shown to stabilise eigenvalue spectra and ensure accurate scattering and tunnelling profiles in semiconductor nanostructures. Finally, a discontinuous Galerkin approach has delivered high-order accuracy for the centre-of-mass Liouville–von Neumann equation, replacing traditional finite-volume discretisations. This method leverages local polynomial bases to achieve computational efficiency on high-performance platforms, promising scalable simulation of large-scale quantum transport systems.

Wigner Function Modeling in Quantum Transport Systems publication trend

The graph below shows the total number of articles in wigner function modeling in quantum transport systems across all publications each year (not limited to Nature Index journals).

Technical terms

Wigner function: A quasi-probability distribution in combined position and momentum space that represents quantum states and their statistical properties.

Phase space: The combined mathematical space of position and momentum variables used to describe dynamical systems.

Liouville–von Neumann equation: The quantum analogue of the classical Liouville equation governing the density matrix evolution in closed and open systems.

Tight-binding method: A Hamiltonian approximation that models electrons as hopping between discrete atomic sites, capturing band structure and lattice effects.

Complex absorbing potential (CAP): A non-Hermitian boundary term that attenuates outgoing waves, preventing unphysical reflections in open quantum simulations.

Wigner potential: The non-local integral term in the Wigner transport equation accounting for quantum interference and tunnelling effects.

References

1. A WENO-solver combined with adaptive momentum discretization for the Wigner transport equation and its application to resonant tunneling diodes. Journal of Computational Physics (2015).
2. Subdomain-based exponential integrators for quantum Liouville-type equations. Journal of Computational Electronics (2021).
3. Numerical constraints and non-spatial open boundary conditions for the Wigner equation. Journal of Computational Electronics (2021).
4. A stochastic algorithm without time discretization error for the Wigner equation. Kinetic and Related Models (2019).
5. On the momentum resolution limit in solving the discrete Wigner transport equation. AIP Advances (2023).
6. Application of the tight-binding method onto the Von Neumann equation. Journal of Computational Electronics (2024).
7. Efficiency analysis of discontinuous Galerkin approaches for the application onto quantum Liouville-type equations. Journal of Computational Electronics (2024).