Boundary-element method for the calculation of electronic states in semiconductor nanostructures

We hate developed a boundary-element method to treat the single-particle electronic properties of semiconductor nanostructures that consist of piecewise homogeneous materials of arbitrary shapes. Green's-function techniques are used to derive integral equations that determine these electronic properties. These equations involve integrals over the boundaries between the homogeneous regions, and they are discretized and solved numerically. In effect, this approach changes a partial differential equation with boundary conditions in d independent variables into an integral equation in d-1 independent variables, which leads to its efficiency. For bound states these methods are used to calculate eigenenergies, for scattering states to calculate differential cross sections, and for both bound and scattering states to calculate spectral density functions and wave functions. For such systems, are show that this method generally provides improved calculational efficiency as compared to alternative approaches such as plane-wave expansions, finite-difference methods, or finite-element methods and that it is more effective in treating highly excited states than are these methods. Illustrative examples are given here for several systems whose potentials are functions of two variables, such cis quantum wires or patterned two-dimensional electron gases.