Novel device architecture and ultimate CMOS scaling require more rigorous modeling. For the latter partly quantum based and non-equilibrium (ballistic) device simulation is necessary. Comparison of approaches to the simulation of quantum transport using the resonant tunneling diode as a test case was the topic of the second half of this thesis.

The Wigner function method has been used to include time
dependence, inelastic phonon scattering and the self-consistent
potential in quantum transport modeling. Several interesting behaviors
of resonant tunneling diodes (RTDs) have been investigated through
numerical simulation [Bie98],
[ZCW^{+}00]: high frequency self-oscillations,
strong intrinsic hysteresis, and pronounced static bistability.
These are ``advanced'' features of RTDs in the sense that these
behaviors have been observed in RTDs, but the measured effects have
been slower (oscillations), weaker (hysteresis, bistability), or
required external inductance to occur (oscillations, hysteresis).
Moreover the physical origin of these effects has given rise to
heated controversy of the past few years and the experimental observation
of some of the reported effects is disputable ([MT95],
pp.113-117).

The strength of the WFM is its conceptual similarity with classical physics in phase space. At the same time this similarity bears the danger that classical features sneak into the model unnoticed. Contrary to wishful belief, the Wigner function does not solve all problems of quantization automagically. One such example was discussed in Section 6.2.5, where we show that a positive local quantum energy density is not given by the second moment of the Wigner distribution function, but a quantum correction term has to be added. This simple example shows that even in Wigner formalism one still has to take great care in finding the correct quantization for a classical quantity.

In practice the grief that we have with the Wigner function method is some lack of numerical soundness. Here by soundness we mean that the change in the result is sufficiently small if details in the discrete model (mesh size, coherence length, length of electrodes) are changed. Simulation can make use of this deficiencies by using these parameters for fitting. The danger here is that numerical effects interfere with the physics which was put into the model and this presents an abuse of TCAD.

Overall the future perspective of the finite difference Wigner method for TCAD may be limited, because one has to keep in mind that for many applications the method should also work efficiently in two spatial dimensions. In contrast the quantum transmitting boundary method is still feasible in two dimensions [LK90a], [LKF02].

So with respect to the Wigner and the Schrödinger QTBM method the status of fully quantum device simulation is an unsatisfactory one: For reliable and accurate simulation of coherent transport in resonant tunneling diodes one is better off with solving the von Neumann equation using the quantum transmitting boundary method. Models based on the Schrödinger equation are more robust and can be solved efficiently. The Wigner formulation on the other side is a much more versatile tool for modeling purposes, but its actual numerical performance is more cumbersome. One application domain is for the case of classical devices going ``quantum'' like ultrashort MOSFETS where QTBM will not work because scattering is still important (mean free path of - ).

In the case of the finite difference WFM no numerically feasible model for dissipation beyond the simple relaxation time approximation is known. Full classical electron phonon scattering can be included in the Monte Carlo WFM. Different from the finite difference Wigner method the Wigner Monte Carlo method can in principle also use a very large mesh size. However, due to the negative sign problem computational costs are several orders of magnitude higher than for all other discussed methods. Results from the Monte Carlo WFM were found to agree with other rigorous simulation tools [KKNS03]. The ultimate test is application to a variety of devices and comparison with experiment which is still pending.

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**Up:** 11. Resume
** Next:** 11.3 Closing Words

R. Kosik: Numerical Challenges on the Road to NanoTCAD