Abstract
We present a new efficient numerical approach for representing anisotropic physical quantities and/or matrix elements defined on the Fermi surface (FS) of metallic materials. The method introduces a set of numerically calculated generalized orthonormal functions which are the solutions of the Helmholtz equation defined on the FS. Noteworthy, many properties of our proposed basis set are also shared by the FS harmonics introduced by Philip B Allen (1976 Phys. Rev. B 13 1416), proposed to be constructed as polynomials of the cartesian components of the electronic velocity. The main motivation of both approaches is identical, to handle anisotropic problems efficiently. However, in our approach the basis set is defined as the eigenfunctions of a differential operator and several desirable properties are introduced by construction. The method is demonstrated to be very robust in handling problems with any crystal structure or topology of the FS, and the periodicity of the reciprocal space is treated as a boundary condition for our Helmholtz equation. We illustrate the method by analysing the free-electron-like lithium (Li), sodium (Na), copper (Cu), lead (Pb), tungsten (W) and magnesium diboride (MgB).
Highlights
The Fermi surface (FS) of a metal is a characteristic property of the crystal structure and the material itself
For comparison we have shown in the inset of figure 3 and 4 the degenerate eigenvalues of (3) when the Helmholtz Fermi surface harmonics (HFSH) are taken to be ordinary spherical harmonics ω where l denotes the angular momentum, vF is the mean velocity at the FS and kF is the mean radius of each of the nearly spherical Fermi surfaces displayed in figure 2
We propose a new functional set, the HFSH, which shows very interesting properties for efficiently representing physical quantities and/or integro-differential equations defined on the FS
Summary
The Fermi surface (FS) of a metal is a characteristic property of the crystal structure and the material itself. The standing waves calculated on this surface constitute the new proposed functional set As these functions are the solution of a second order partial differential equation, the generalized orthogonal property of the basis set is recovered by construction, and more importantly, the completeness of the set is automatically guaranteed. Another advantage of the present method, in comparison with the Allen FSH set, is that a definition of an energy cutoff Ec for the basis set appears naturally. For surfaces or multiple Fermi sheets intersecting the BZ boundaries, a numerically more elaborated procedure is needed
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