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Band Theory of Size‐Quantised Electron States in Thin Crystalline Films
Author(s) -
Cottey A. A.
Publication year - 1978
Publication title -
physica status solidi (b)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.51
H-Index - 109
eISSN - 1521-3951
pISSN - 0370-1972
DOI - 10.1002/pssb.2220880124
Subject(s) - brillouin zone , reciprocal lattice , degenerate energy levels , condensed matter physics , physics , scattering , multipole expansion , electron , quantum mechanics , mathematical analysis , mathematics , diffraction
The discrete electron states in a thin film with an arbitrary three‐dimensional periodic crystal potential are analysed. First the quantum size effect states of a symmetric film are found exactly for Dirichlet (ψ = 0) and Neumann (ψ′ = 0) boundary conditions, and the interpolation between these extremes is studied. Then a surface scattering matrix formalism is set up in the general (unsymmetric) case. This treats the general situation when there may be M up‐ and M down‐travelling Bloch waves with a given energy and surface‐parallel component of crystal momentum modulo surface reciprocal lattice. The equations are solved in the general 1 × 1 and symmetric 2 × 2 cases. The QSE solutions approximate gently undulating parallel surfaces in k ‐space. Usually the separation of these surfaces is roughly uniform. An exception occurs in the 2 × 2 case near a band edge which is not at the centre or edge of the Brillouin zone; the solutions are then in nearly degenerate pairs. The results agree with available experimental data.