Photoelectrons In Crystals Research Articles

Multipole wavefunctions for photoelectrons in crystals. IV. The irregular functions and the matching to an impurity

Extension of the method of partial waves to scattering by an impurity center in a crystal requires the construction of energy eigenfunctions of the unperturbed crystal that are irregular at the center. These irregular solutions of a nonseparable problem are constructed by utilizing the expansion of the Green’s function into the regular solutions symmetry-adapted about the impurity center, which have been obtained earlier. The wavefunction within an impurity can thus be continued throughout the surrounding crystal by matching it at the impurity boundary to a superposition of regular and irregular solutions. This procedure is compared to the current EXAFS treatment of photoabsorption.

Multipole wave functions for photoelectrons in crystals. III. The role of singular points in the band structure and the tails of the Wannier functions

Properties of multipole wave functions and of Wannier functions are studied jointly, because both are generated by unitary transformations of Bloch waves and because their symmetry and localization depend on the analytic behavior and phase normalization of Bloch waves. Fourier theory relates the amplitude and convergence of the tails of Wannier functions to the singularities of Bloch waves in $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ space, specifically to the contribution of singularities to integrals over $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$. These singularities may include branch cut surfaces bounded by curves of degeneracy. We discuss the convergence of the coefficients of a linear-combination-of-atomic-orbitals expansion in orbitals introduced recently by Kohn, which are Fourier coefficients of the eigenvectors ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{U}}}^{(\ensuremath{\mu})}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}})$ of the Hamiltonian in Kohn's basis. The phases of Bloch waves are fixed by a constraint on the Wannier functions. The symmetry of Bloch waves determines then both the species and the center of symmetry of the Wannier functions. An application to the conduction band of copper is presented. Contributions of neighborhoods of loci of degeneracy in the Brillouin zone to the multipole wave functions are singled out and the rate of convergence of their Wannier series is examined. The series converges faster than the corresponding Wannier series of Bloch waves.

Multipole wave functions for photoelectrons in crystals. II. Examples of constant-energy-surface harmonics. Application to thes−dbands of Cu

We construct a complete set of multipole wave functions for an electron in a crystal, appropriate to describe single-center phenomena. The set consists of superpositions of the Bloch waves of given energy $E$, weighted by a set of harmonics which is complete and orthonormal over the constant-energy surface ${E}_{\ensuremath{\mu}}(\stackrel{\ensuremath{\rightarrow}}{k})=E$. We give prescriptions to label these harmonics. Numerical examples are discussed for the $s\ensuremath{-}d$ bands of Cu. The case of a spherical constant-energy surface, whose harmonics are the spherical harmonics, reappears as a degenerate limit.