Construction of a simple representation for the Green's function of a crystalline film

Abstract

For a one-dimensional Hamiltonian, $H=-\left(\frac{{\hslash }^2}{2m}\right)\left(\frac{d^2}{d{z}^2}\right)+V\left(z\right)\mathrm{}$, it is well known that the Green's function $G(z,z^{\prime };E)$ may be written in the simple form $G(z,z^{\prime };E)=\frac{{\psi }^{+}(z_{>};E){\psi }^{-}(z_{<};E)}{W\left(E\right)\mathrm{}}$, where ${\psi }^{\pm }\mathrm{}(z;E)\mathrm{}$ are the solutions to the Schrödinger equation, $\mathrm{}(E-H)\mathrm{}\psi \mathrm{}(z;E)=0\mathrm{}$, which satisfy outgoing boundary conditions, respectively, at $z\to \pm \infty$, and where $W\left(E\right)\mathrm{}$ is the Wronskian. Here, an analogous expression is derived for the Green's function of a crystalline film, i.e., of a solid whose one-electron potential $V(\overrightarrow{\rho },z)$ has the translational periodicity of a lattice in $\overrightarrow{\rho }\equiv \mathrm{}(x,y)\mathrm{}$, and vanishes as $z\to \pm \infty$.