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Critical temperature of MgB$_2$ ultrathin superconducting films: BCS model calculations in tight-binding approximation

We develop the multi-band BCS model of superconductivity in the ultrathin films using the orthogonal tight-binding approximation for constructing the electron wavefunctions. This allows for relatively simple determination of the band structure near the Fermi level as well as the electron-electron interaction matrix elements of the BCS type. The model is applied to the ultrathin MgB$_2$(0001) films, for which the critical temperature values are calculated in the thickness range 2...10 MgB$_2$ layers. The importance of the boundary conditions is emphasised, as either boron or magnesium layers may cover the film. It is found that films thinner than 4 layers show substantial decrease in the critical temperature. The charge spillage outside the geometric boundaries of the film, which is allowed in our model, suppresses $T_c$ oscillations and weakens its dependence on film covering layer composition.