Band Unfolding Theory#

The theoretical background of the methodology used in this package is available in the literature,[1][2] and also discussed in the JOSS paper. Here, only a summary is given for brevity.

Notation: $\vec{K}$ is the k-point of the supercell and $\vec{k}$ is that of the primitive cell.

Each $k$ in the primitive cell can be mapped to that in the supercell where:

\[ \vec{K} = \vec{k} - \vec{G}_0. \]

with $\vec{G}_0$ being a reciprocal lattice vector. Naturally, each $\vec{K}$ in the supercell can be unfolded into a set of $\vec{k}_i$:

\[ \vec{k_i} = \vec{K} + \vec{G}_i\ ; \ i=1,...,N_{\vec{K}} \]

The key implication is that for a given $\vec{k}_i$, there is a unique $\vec{K}$ that it folds to.

The goal of the band folding procedure is to obtain the $E(\vec{k})$ from the complicated $E(\vec{K})$, where $E(\vec{k})$ is the energy/eigenvalue of the Kohn-Sham electronic states at $\vec{k}$ in the Brillouin zone of the primitive cell. This can be achieved by projecting $\ket{\vec{K}m}$ on all of the primitive cell Bloch states $\ket{\vec{k}_i n}$ of a fixed $\vec{k}_i$, where $m$ and $n$ are band indices in the supercell and primitive cell respectively, and compute the spectral weight:

\[ P_{\vec{K}m}(\vec{k}_i) = \sum_n |\braket{\vec{K}m |\vec{k}_i n}|^2. \]

where $P$ represents the probability of finding a set of primitive cell states $\ket{\vec{k}_i n}$ contributing to the supercell state $\ket{\vec{K}m}$, or the amount of Bloch character $\vec{k}_i$ preserved in $\ket{\vec{K}m}$ at the same energy.

Based on this, one can further derive the spectral function $A(\vec{k}_i, E)$, which represents the probability of an electronic state with energy $E$ at $\vec{k}_i$ (i.e. a ‘band structure density’):

\[ A(\vec{k}_i, E) = \sum_m P_{\vec{K}m}(\vec{k}_i)\delta(E_m - E). \]

In practice, the $\delta$ function is replaced with a Gaussian or Lorentzian function to smear the contribution over a discretised energy grid.

Hence, the central quantity to be calculated is $P_{\vec{K}m}(\vec{k}_i)$. For a plane-wave basis, it can be shown that (equation 15 in Popescu et al.[2]):

\[ P_{\vec{K}m}(\vec{k}_j) = \sum_{\vec{g}} |C_{\vec{Km}}(\vec{g} + \vec{G_j})|^2, \]

where $C_{\vec{Km}}(\vec{g} + \vec{G_j})$ are the plane-wave coefficients of the supercell. Because the supercell is commensurate with the primitive cell, the vectors $\vec{g}$ are included in the basis set of the supercell calculations. Since $\vec{G_{j}}$ is the reciprocal lattice vector that wraps $\vec{k}$ to $\vec{K}$, all plane-wave coefficients that are needed to obtain $P_{\vec{K}_m}(\vec {k}_j)$ are present in the wavefunction output of the supercell calculation.

Symmetry Considerations#

The set of primitive cell k-points $\vec{k}_i$ are located on a given path in the Brillouin zone, going through multiple high symmetry points. The selection of the path is not unique, but often depends on the space group of the primitive cell. Only a limited set of paths are needed because of the presence of point group symmetry, as symmetrically equivalent $\vec{k}$ correspond to the same eigenstates.

However, the supercell to be unfolded does not necessarily contain the same point group symmetry due to imperfections (presence of defects, strain, distortions, disorder). The broken symmetry means that previously equivalent $\vec{k}$ are no longer equivalent.

We address this by first expanding each $\vec{k}$ based on the symmetry operations of the primitive cell, followed by a reduction using the symmetry of the supercell. The spectral weight at each $\vec{k}$ is then a weighted combination of that set of $\vec{k_s^\prime}$ points that are inequivalent under the symmetry of the supercell.

Cell and Transformation Matrix convention#

The cell matrix may be consisted of column or row lattice vectors. In this package we use the row vector convention as commonly found in many post-processing tools and DFT codes. The cell matrix is defined as:

\[\begin{split} \mathbf{C} = \begin{pmatrix} x_a & y_a & z_a \\ x_b & y_b & z_b \\ x_c & y_c & z_c \end{pmatrix} \end{split}\]

where $x_a$, $y_a$, $z_a$ are components of the lattice vector $\mathbf{a}$.

The cell matrix of the supercell $\mathbf{C_s}$ is obtained by (left) multiplying the original unit cell $\mathbf{C_u}$ by the transformation matrix $\mathbf{M}$:

\[ \mathbf{C_{s}} = \mathbf{M} \, \mathbf{C_u} \]

Note

Sometimes the cell matrix is defined by column vectors of the lattice parameters, e.g. $\mathbf{C_u^c} = \mathbf{C_u^T}$, and the relationship becomes: $$ \mathbf{C_u^c} = \mathbf{C_u^T} \, \mathbf{M^T} $$

Hence, when the column vector convention is used, the transformation matrix is the transpose of that used by the row convention.

One example of code using the column vector convention is Phonopy.

Band Unfolding Theory

Contents

Band Unfolding Theory#

Symmetry Considerations#

Cell and Transformation Matrix convention#