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7.2.5 Existence But Non-Uniqueness

There are reports in the literature about non-existence or existence of multiple solutions to Schrödinger-Poisson (or equivalently to Wigner-Poisson) problems. Reference [GMN93] discusses multiple self-consistent solutions at zero bias in selectively doped two-terminal quantum tunneling devices (that is, resonant tunneling diodes). References [JB91] and [Bie98] use a transient method to find the stationary solution for a Wigner-Poisson problem. Both papers report current oscillations at a fixed bias and failure to find a stationary solution in (parts of) the region of negative differential resistance.

However, from a theoretical point of view at least the existence of solutions can be guaranteed. (Please keep in mind that existence and uniqueness depends also on the boundary conditions which are used. Theorems for the closed von Neumann-Poisson system [Nie93] can not be applied.) For analysis of the existence problem it is instructive to consider the classical case first.

The classical analogon to the Wigner-Poisson problem is the Vlasov-Poisson system. Classical solutions always exist. They may be non-unique due to existence of ``bound states''. These represent particles which are bound to the device (e.g., trapped in a potential well). The boundary conditions have no influence on the number of such particles, hence the solution is non-unique. However, it is possible to add some ``damping'' to the equation which has the effect to suppress these bound states. Then in the limit of vanishing damping constant we get a well-defined and - in this sense - unique solution.

Existence of solutions is proved by use of Schauder's fixed-point principle [BA00], [BADM97]. However, this is a non-constructive proof and of little value for numerical solution. In the open quantum-case a similar proof as in the closed case can be carried through [BADM97], [BA00]. In the one-dimensional quantum transmitting boundary method bound states are not considered, while - at least in principle - they are present in the Wigner simulation. There is some probability for particles in quasi-bound states in a quantum well to tunnel through the barriers and eventually leave the device. Unless there are regions where the potential is smaller than in the leads, real bound states (which do not contribute to the current) as discussed in [CK85] are unlikely to occur.

What we observed in practice is consistent with observations from the literature: First there is non-uniqueness. In contrast to the classical case a multitude of solutions is possible, even when there are no bound states. Already in equilibrium there may be several solutions, as we discovered by chance, even if the doping is symmetric. For a certain example we found (at least) three solutions: a symmetric one and a pair of unsymmetric solutions.

Secondly, when using bias stepping to find the solution, in some cases it is not possible to step up any further. It seems that there are branches of solutions which cannot be extended beyond a certain value for the bias. For the simulation of hysteresis bifurcation methods (see [MRS84] and [Kel86]) have to be employed.

It is now clear that multi-branch I-V curves cannot be captured by stationary simulations alone, but transient simulations have to be applied. With absorbing boundary conditions this is difficult when using the QTBM, as the boundary conditions become non-local in time and the equation ``has a memory''.

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R. Kosik: Numerical Challenges on the Road to NanoTCAD