Spectral element modeling of semiconductor heterostructures

We present a fast and efficient spectral method for computing the eigenvalues and eigenfunctions for a one-dimensional piecewise smooth potential, as arises in the case of epitaxially grown semiconductor heterostructures. Many physical devices such as quantum well infrared photodetectors and quantum cascade lasers rely upon transitions between bound and quasi-bound or continuum states; consequently it is imperative to determine the resonant spectrum as well as the bound states. Instead of trying to approximate radiation boundary conditions, our method uses a singular mapping combined with deforming the coordinate system to a contour in the complex plane to construct semi-infinite elements of perfectly matched layers. We show that the PML elements need not be based on a smooth contour to absorb outward-propagating waves and that the resonant eigenvalues can be computed to machine precision. A fast means of computing inner products and expectations of quantum mechanical operators with quadrature accuracy in the spectral domain is also introduced.