Global coherent tunneling is usually simulated using
the transfer matrix method [Geh03].
Qualitatively, coherent
tunneling describes RTDs reasonably well.
However, comparing simulation with
experiment it has been found that the coherent tunneling model
may overestimate the peak to valley ratio by an order of magnitude.
In these circumstances we have to go beyond the global coherent
tunneling model to deal with dissipative tunneling processes.
Within the *NANOTCAD* project dissipation was modeled by
adding a classical Boltzmann scattering term to the
Wigner equation.

An alternative analytical model which incorporates scattering effects has been developed: incoherent (sequential) tunneling [FG01]. In this model, carriers tunnel through the first barrier into a quasi-bound state residing in the well and subsequently lose their phase memory through a scattering process there. The phase-randomized carriers tunnel out of the second barrier through a second uncorrelated tunneling process.

It has been argued that scattering that breaks phase-coherence results in a broadening of the transmission peaks at resonance and a dramatic degradation of the PVR of RTDs [MT95], see Figure 5.2. In addition, frequent scattering processes may have an influence on the distribution of electrons in the quantum well, which changes from a completely ballistic one to a well-thermalized one depending on the ratio of typical scattering time to electron dwell-time, the time which is required for an electron to tunnel through the barriers (or, equivalently, the time for which an electron stays between the barriers). The valley current is also increased at higher temperatures due to inelastic phonon-assisted tunneling.

Three-dimensional consideration of the scattering is required if one wishes to simulate the broadening of the transmission coefficient introduced by scattering processes in some modes of operation. Let us motivate it for the case of phonon scattering: For an incident electron with longitudinal energy for which resonant tunneling is not directly possible, there exists a family of phonons which will change its longitudinal and perpendicular wave-vector such that its final longitudinal energy is aligned with the virtual eigenvalue of the quantum well. A one-dimensional model does not account for all the possible scattered waves. Three-dimensional scattering is easily incorporated into a Wigner Monte Carlo simulation. A study of the influence of phonon scattering within the Wigner function approach can be found in [KKNS03].

For completeness we note that a comprehensive analytical model that unifies the sequential and resonant tunneling models in a wide range of transport regimes, from completely coherent to completely incoherent, has been developed [IP95].

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R. Kosik: Numerical Challenges on the Road to NanoTCAD