Physics Background

What LWW Means

LWW stands for the lattice Weyl-Wigner formulation of quantum transport. It is a phase-space method: instead of propagating only a wave function or only a classical distribution, it solves for a Wigner distribution f(x, k, t) on a discrete lattice in position and momentum. The formulation was developed for open nonequilibrium quantum devices and connects Wigner transport ideas with nonequilibrium Green-function language [BuotJensen1990] [Buot2009] [KadanoffBaym1963].

For the one-dimensional model implemented here, the simulation region is a finite device between two contacts. The phase-space grid has nx spatial points and an even number n of momentum points. Positive and negative momenta are treated separately so that particles entering from each contact can be handled with the correct upwind/downwind boundary source.

The Wigner Distribution

The Wigner distribution is the central unknown. It provides a phase-space representation while retaining quantum behavior. The function itself can become negative, so it should not be interpreted as an ordinary probability density at each point. Its marginals and weighted sums, however, give physical observables such as carrier density and current density.

In this package:

  • wigstd solves the discretized Wigner equation.

  • curcalc(..., irj=1) evaluates carrier density from the Wigner function.

  • curcalc(..., irj=2) evaluates current density.

  • run_transient advances the Wigner distribution through a bias sweep and records the current trace.

Terms in the 1D LWW Equation

The implemented 1D equation contains three principal terms.

The drift or kinetic term transports the Wigner distribution across the spatial grid. The code uses a second-order upwind/downwind finite-difference stencil: the sign of momentum determines which spatial direction supplies the upstream values. This is why the boundary source has separate left-contact and right-contact entries.

The nonlocal potential term couples momentum points at the same spatial location. It is built from differences of the effective device potential at positions around each grid point. This is the term responsible for quantum features such as tunneling and interference in a barrier/well structure. In an RTD, oscillations and sign changes in the Wigner function are expected markers of coherent quantum behavior [JensenBuot1991Method] [Barraud2009].

The optional scattering term models relaxation toward a local equilibrium distribution. The original Jensen-Buot implementation used a modified relaxation-time approximation that preserves detailed balance rather than a plain -(f - f0) / tau decay [JensenBuot1991Method]. In this code path, irlx controls whether this relaxation contribution is included.

Self-Consistent Potential

The Wigner equation needs a device potential. The carrier density produced by the Wigner function also changes the electrostatic potential. The package therefore couples Wigner transport to a one-dimensional Poisson solve:

  • solve Wigner transport for the current potential,

  • compute density from the Wigner function,

  • update the electrostatic potential from Poisson’s equation,

  • repeat until the density/potential change is small enough.

This self-consistent loop is used for steady-state bias points and as part of the transient workflow.

What the Model Can Do

The current implementation targets one-dimensional resonant-tunneling structures, especially double-barrier RTDs. Within that scope it supports study of:

  • steady-state Wigner functions at a selected bias,

  • density and current density profiles through the device,

  • transient current traces during a bias sweep,

  • tunneling and quantum-interference structure in phase space,

  • negative differential resistance behavior in RTD-like geometries,

  • effects of barrier height, well width, spacer length, grid resolution, and relaxation settings.

The RTD use case follows the original LWW demonstration literature, where Wigner transport was used to model particle trajectories, intrinsic bistability, and high-frequency current oscillations in resonant tunneling structures [JensenBuot1991Method] [JensenBuot1991PRL].

The geometry plotting helper in lww_transport.visualization draws the barrier/well layout used by the simulation for pre-run geometry checks.

Current Limitations

The package is a deterministic one-dimensional effective-mass implementation. It supports the original LWW/Wigner-Poisson workflow, but it is not yet a full atomistic or multidimensional device simulator. Boundary conditions for open Wigner systems are a known subtle issue [Frensley1990] [Jiang2011] [Taj2006]. The present implementation keeps the conventional contact-source treatment used by the reference RTD calculations, with the code organized so more advanced contact/self-energy treatments, such as those common in modern open-device transport models, remain possible extensions.