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- W2037845291 abstract "For electrons, phonons, etc., and regardless of symmetry, the Green's function in any mixed Wannier-Bloch representation is ${G}_{0}^{+}(zensuremath{-}{z}^{ensuremath{'}}, overline{mathrm{k}}nensuremath{omega})=ensuremath{-}iaensuremath{Sigma}{j}^{}frac{{e}^{i{k}_{j}}(zensuremath{-}{z}^{ensuremath{'}})}{v({k}_{j}overline{mathrm{k}}n)} mathrm{sgn} (zensuremath{-}{z}^{ensuremath{'}})+{G}_{mathrm{BC}}$, 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 $v({k}_{j}overline{mathrm{k}}n)mathrm{sgn}(zensuremath{-}{z}^{ensuremath{'}})>0$, if ${k}_{j}$ is real, or $mathrm{Im}{k}_{j}mathrm{sgn}(zensuremath{-}{z}^{ensuremath{'}})>0$, 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 $ensuremath{Delta}ensuremath{rho}(ensuremath{omega}overline{mathrm{k}})$." @default.
- W2037845291 created "2016-06-24" @default.
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- W2037845291 date "1979-01-15" @default.
- W2037845291 modified "2023-10-18" @default.
- W2037845291 title "Green's function and generalized phase shift for surface and interface problems" @default.
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- W2037845291 doi "https://doi.org/10.1103/physrevb.19.917" @default.
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