The Wigner distribution function is a mathematical transform of the density matrix which approaches the classical distribution function as the system becomes classical (with large dimensions, slowly varying potentials, and/or high temperatures) [5,48,49,50,51]. This representation of the statistical state has proven to be useful in modeling quantum-effect devices such as the resonant-tunneling diode [52,53,54,55].
To derive the Wigner function from the density matrix defined in (73) one rewrites the arguments as and , and then Fourier transforms into a momentum variable p. Thus:
Applying the same procedure (which is known as the Wigner-Weyl transformation) to the Liouville-von Neumann equation (78) gives:
where the kernel of the potential operator is given by:
Let us examine the form of these equations. Because (90) is derived from (78) by a mathematical transformation, we would expect that it should also describe unitary time evolution. The condition for unitary evolution is that be an anti-Hermitian operator. The potential operator is anti-Hermitian [because ], and the drift term is anti-Hermitian if periodic boundary conditions are imposed. On the other hand, we have seen that if initial conditions are imposed, the drift term is a master operator, and the equation then describes irreversible time evolution. This is the origin of the usefulness of the Wigner representation for describing electron devices. One applies boundary conditions to so as to fix the distribution of electrons entering the domain:
where and are the distribution functions in the left- and right-hand contacts (reservoirs), respectively. Because these boundary conditions introduce irreversibility into the Liouville equation, one can now evaluate the time-evolution of a device, and observe an approach to steady-state . Inelastic processes such as phonon scattering may be included in a semi-classical way by adding the Boltzmann collision term [the integral expression in (88)] to the Liouville equation (78) [56,54], or by even simpler schemes such as the relaxation time approximation [53,57].
The open-system Wigner function approach has proved to be of use in understanding the behavior of resonant-tunneling diodes. This technique permits evaluation of steady-state behavior in the form of the curve, and calculations of the large-signal transient response and small-signal ac response . The curves derived from this model show the expected negative differential-conductance region, but the ratio of the peak to valley currents is always smaller than that obtained from the tunneling theory (38), and is often less than that observed experimentally. Recently, Tsuchiya and co-workers have developed an improved formulation of the Liouville equation (90) which takes the spatial variation of the effective mass into account, and which leads to larger peak-to-valley ratios than the simpler theory . These different formulations are compared in Figure 2.
Figure 2: Theoretical curves for a model resonant-tunneling diode structure . The solid line shows the results of the tunneling theory, equation (28). The dashed line shows the result of evaluating the Wigner function using a local model of the variation of the effective mass . The dotted line shows the result of a Wigner function calculation using the nonlocal effective-mass model of Tsuchiya and co-workers . (Calculations by C. Fernando, University of Texas at Dallas.)