Brillouin zone unfolding of Complex Bands in a nearest neighbour Tight Binding scheme

TL;DR

This paper introduces a unified method for unfolding complex and real electronic band structures in semiconductors, ensuring invariance of wavefunction projections across energies, using a quadratic eigenvalue problem approach within a non-primitive lattice framework.

Contribution

It presents a novel, unified approach for unfolding complex bands in semiconductors, applicable to both real and complex bands, with an energy-invariant measure based on a quadratic eigenvalue formulation.

Findings

Method effectively unfolds complex bands onto the primitive Brillouin zone.

Ensures measure invariance of wavefunction projections with respect to energy.

Applicable to arbitrary directions in semiconductor crystals.

Abstract

Complex bands $\vec{k}^{\perp}(E)$ in a semiconductor crystal, along a general direction $\vec{n}$, can be computed by casting Schr\"odinger's equation as a generalized polynomial eigenvalue problem. When working with primitive lattice vectors, the order of this eigenvalue problem can grow large for arbitrary $\vec{n}$. It is however possible to always choose a set of non-primitive lattice vectors such that the eigenvalue problem is restricted to be quadratic. The complex bands so obtained need to be unfolded onto the primitive Brillouin zone. In this paper, we present a unified method to unfold real and complex bands. Our method ensures that the measure associated with the projections of the non-primary wavefunction onto all candidate primary wavefunctions is invariant with respect to the energy $E$.