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 WannierBloch representation is G0+(zz, k»n)=iajeikj(zz)v(kjk»n) sgn (zz)+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(zz)>0, if kj is real, or Imkjsgn(zz)>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 onedimensional 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 "zeroenergy resonance" in swave 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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