| For simulating the electronic transport through a ballistic semiconductor device quantum mechanically, one has to solve the stationary Schrödinger equation in a complex geometry with intricate boundary conditions. To this end, we directly compute the wave function of the device by adaptive higher order finite element methods using iso-parametric elements as it is necessary for curvi-linear boundaries. The semiconductor device itself is modeled as a finite domain containing the scatterers. The device is supposed to be connected to semi-infinite leads which model source and drain contacts. We are able to describe the infinite extent of the leads exactly by so-called Hardy-space infinite element methods (HSIEM) which allows us to choose a finite domain for the numerical simulation as it is mandatory for finite elements. Given the wave function it is easy to compute the S-matrix and thus, to get a comprehensive description of the transmission properties of the scattering states in the Landauer-Büttiker picture. As illustrative examples we first apply our approach to electron transport through Aharonov-Bohm rings with local defects and discuss the resulting S-matrix and second to magnetic focussing in billiard-like geometries. |
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