Green's function and generalized phase shift for surface and interface problems
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For electrons, phonons, etc., and regardless of symmetry, the Green's function in any mixed Wannier-Bloch representation is G0+(z-z, k»n)=-iajeikj(z-z)v(kjk»n) sgn (z-z)+GBC, where k»=(kx,ky), n is the branch index, and the values of z correspond to lattice points. The kj are those values of kz for which the eigenvalue (kzk»n) is equal to the parameter, and for which v(kjk»n)sgn(z-z)>0, if kj is real, or Imkjsgn(z-z)>0, if kj is complex. GBC represents integrals around branch cuts, a is the height of a unit cell, and v(kzk»n)(kzk»n)kz. The above expression can be regarded as a generalization of the usual one-dimensional Green's function of quantum mechanics. G0+() diverges whenever is such that some v(kjk»n) goes to zero, and as a result the generalized phase shift (k») has discontinuities of -2 at these values of. 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 M"=0, where 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 =k»=0, however, because of a restriction on the force constants. The significance of (k») is that the surface or interface density of states "(k») is given by -1(k»). Each discontinuity of -2 in (k») at an extremum 0 thus produces a contribution -(-0)2 in "(k»). © 1979 The American Physical Society.
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