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Theory

Band unfolding is an important technique to simplify and analyze the band structure characteristics in large, disturbed supercells (SC), such as when simulating materials with thermal vibrations, defects, dopants and interfaces. This section provides a brief introduction on the theoretical background of the band unfolding formalism in FHI-aims, for users to get familiarized with the methods before doing the exercise.

Band unfolding in a nutshell

In the following, we will discuss band unfolding by establishing relationships between electronic states in the primitive cell (PC) and SC. Notationwise, the difference between primitive and supercell properties is emphasised by using lower- and upper-case letters, respectively, and/or PC and SC subscripts.

In order to unfold energy state \(E_{{\bf K}N}\) at SC Brillouin zone (BZ) \({\bf K}\)-point to the \({\bf k}\)-point in PCBZ, we project the SC electronic wavefunction \(\ket{\Psi_{{\bf K}N}}\) to the subspace \({\bf k}\) in PCBZ:

\[\begin{equation} W^{\bf k}_{{\bf K}N} = \braket{ {\Psi_{{\bf K}N}} | {\bf P}_{\bf k} | \Psi_{{\bf K}N}} = \sum_j |\braket{{\bf k}j|\Psi_{{\bf K}N}}|^2 \ , \end{equation}\]

with the projection operator \(\bf P_k\) defined by:

\[ {\bf P_k} = \sum_j \ket{{\bf k}j}\bra{{\bf k}j} \ , \]

where { \(\ket{{\bf k}j}\) } is a set of basis to expand subspace \({\bf k}\), for example PC wavefunctions { \(\ket{\psi_{{\bf k}n}}\) } at \(\bf k\) ( but not necessarily be { \(\ket{\psi_{{\bf k}n}}\) } ). \(W^{\bf k}_{{\bf K}N}\) is called the unfolding weight of SC state \(E_{{\bf K}N}\) at PC \(\bf k\)-point. This allow us to define the electron spectral function \(A({\bf k}, E)\):

\[\begin{eqnarray} A({\bf k}, E) &=& \sum_{ {\bf K}N} W^{\bf k}_{{\bf K}N} \delta (E - E_{{\bf K}N}) \; . \end{eqnarray}\]

As a result, the key quantity in band unfolding is the unfolding weight \(W^{\bf k}_{{\bf K}N}\). For a 'perfect' SC, i.e., one consisting of identical PC replicas, these weights are either 0 or 1 since PC and SC eigenvectors are injective, but for a disturbed SC \(W^{\bf k}_{{\bf K}N}\) are fractional. In code implementation, \(W^{\bf k}_{{\bf K}N}\) must be caluclated with the corresponding basis set. While this may seems straightforward at first glance, calculating \(W^{\bf k}_{{\bf K}N}\) in a linear combination of atomic orbital (LCAO)-type of basis (including NAO basis used in FHI-aims) is actually quite challenging because:

  • Basis funcitons are non-orthogonal, we need to consider overlap matrix in the corresponding formula derivation.
  • Basis functions move associated with atomic positions, thus the basis function itself are changed in a disturbed SC, make it difficult to take inner product directly like the formula above.

In FHI-aims, we have derived and implemented an efficient band unfolding algorithm that overcome these difficulties.