Green's function and generalized phase shift for surface and interface problems

For electrons, phonons, etc., and regardless of symmetry, the Green's function in any mixed Wannier-Bloch representation is ${G}_{0}^{+}(z\ensuremath{-}{z}^{\ensuremath{'}}, \overline{\mathrm{k}}n\ensuremath{\omega})=\ensuremath{-}ia\ensuremath{\Sigma}{j}^{}\frac{{e}^{i{k}_{j}}(z\ensuremath{-}{z}^{\ensuremath{'}})}{v({k}_{j}\overline{\mathrm{k}}n)} \mathrm{sgn} (z\ensuremath{-}{z}^{\ensuremath{'}})+{G}_{\mathrm{BC}}$, where $\overline{\mathrm{k}}=({k}_{x},{k}_{y})$, $n$ is the branch index, and the values of $z$ correspond to lattice points. The ${k}_{j}$ are those values of ${k}_{z}$ for which the eigenvalue $\ensuremath{\epsilon}({k}_{z}\overline{\mathrm{k}}n)$ is equal to the parameter $\ensuremath{\omega}$, and for which $v({k}_{j}\overline{\mathrm{k}}n)\mathrm{sgn}(z\ensuremath{-}{z}^{\ensuremath{'}})g0$, if ${k}_{j}$ is real, or $\mathrm{Im}{k}_{j}\mathrm{sgn}(z\ensuremath{-}{z}^{\ensuremath{'}})g0$, if ${k}_{j}$ is complex. ${G}_{\mathrm{BC}}$ represents integrals around branch cuts, $a$ is the height of a unit cell, and $v({k}_{z}\overline{\mathrm{k}}n)\ensuremath{\equiv}\frac{\ensuremath{\partial}\ensuremath{\epsilon}({k}_{z}\overline{\mathrm{k}}n)}{\ensuremath{\partial}{k}_{z}}$. The above expression can be regarded as a generalization of the usual one-dimensional Green's function of quantum mechanics. ${G}_{0}^{+}(\ensuremath{\omega})$ diverges whenever $\ensuremath{\omega}$ is such that some $v({k}_{j}\overline{\mathrm{k}}n)$ goes to zero, and as a result the generalized phase shift $\ensuremath{\eta}(\ensuremath{\omega}\overline{\mathrm{k}})$ has discontinuities of $\ensuremath{-}\frac{\ensuremath{\pi}}{2}$ at these values of $\ensuremath{\omega}$. These discontinuities are present regardless of the strength of $V$, the perturbation associated with creating a pair of surfaces or interfaces. There is an exception: If det $\stackrel{\ensuremath{\leftrightarrow}}{M}=0$, where $\stackrel{\ensuremath{\leftrightarrow}}{M}$ is a matrix defined in terms of the matrix elements of $V$, then the discontinuity is eliminated. This condition is analogous to that for a "zero-energy resonance" in $s$-wave potential scattering, and it will ordinarily occur only at particular transitional strengths of $V$. The condition is always satisfied for acoustic phonons at $\ensuremath{\omega}=\overline{\mathrm{k}}=0$, however, because of a restriction on the force constants. The significance of $\ensuremath{\eta}(\ensuremath{\omega}\overline{\mathrm{k}})$ is that the surface or interface density of states $\ensuremath{\Delta}\ensuremath{\rho}(\ensuremath{\omega}\overline{\mathrm{k}})$ is given by ${\ensuremath{\pi}}^{\ensuremath{-}1}\frac{\ensuremath{\partial}\ensuremath{\eta}(\ensuremath{\omega}\overline{\mathrm{k}})}{\ensuremath{\partial}\ensuremath{\omega}}$. Each discontinuity of $\ensuremath{-}\frac{\ensuremath{\pi}}{2}$ in $\ensuremath{\eta}(\ensuremath{\omega}\overline{\mathrm{k}})$ at an extremum ${\ensuremath{\omega}}_{0}$ thus produces a contribution $\ensuremath{-}\frac{\ensuremath{\delta}(\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{0})}{2}$ in $\ensuremath{\Delta}\ensuremath{\rho}(\ensuremath{\omega}\overline{\mathrm{k}})$.