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Formulation of a Mesoscopic Electron Beam Splitter with Application in Semiconductor Based Quantum Computing

We aim to analytically arrive at a beam splitter formulation for electron waves. The electron beam splitter is an essential component of quantum logical devices. To arrive at the beam splitter structure, the electrons are treated as waves, i.e. we assume the transport to be ballistic. Ballistic electrons are electrons that travel over such short distances that their phase coherence is maintained. For mesoscopic devices with size smaller than the mean free path, the phase relaxation length and the Fermi wavelength of electrons in the medium, the transport can be considered to be coherent and hence ballistic. In such a case the electron motion can be completely described by the Schrodinger's wave equation with an effective mass assigned to the electron. To design a beam splitter, we draw inspiration from an electromagnetic beam splitter, where the Maxwell's equations (and subsequently the Helmholtz equation) are used to describe the splitting of light across a thin slab of dielectric material. The parameters of the media and the thickness of the slab can be manipulated to achieve a 50-50 beam splitter. We wish to achieve a similar 50-50 beam splitting for the electron wave. Using an analogy between the scalar Helmholtz equation and the Schrodinger's equation for a ballistic electron, we arrive at this desired beam splitter structure for the electron.