# Supercell (crystal)

An example of different supercell for 2D cubic crystal. Both diagonal and non-diagonal supercells presented.

In solid-state physics and crystallography, a crystal structure is described by unit cell. There are infinite number of unit cells with different shapes and sizes, which can describe the same crystal. Let's assume that crystal structure is described by unit cell U. The supercell S of unit cell U is a unit cell, which describe the same crystal, but has larger volume, than cell U. Many methods which use supercell perturbate it somehow to determine properties. which cannot be determined by initial cell. For example, during phonon calculations by small displacement method, phonon frequencies in crystals is calculated by forces values on slightly displaced atoms in supercell. Another very important example of supercell is conventional cell of body-centered (bcc) or face-centered (fcc) cubic crystals.

## Unit cell transformation

The basis vectors of unit cell U ${\textstyle ({\vec {a}},{\vec {b}},{\vec {c}})}$ can be transformed to basis vectors of supercell S ${\textstyle ({\vec {a}}',{\vec {b}}',{\vec {c}}')}$ by linear transformation[1]

${\displaystyle {\begin{pmatrix}{\vec {a}}'&{\vec {b}}'&{\vec {c}}'\\\end{pmatrix}}={\begin{pmatrix}{\vec {a}}&{\vec {b}}&{\vec {c}}\\\end{pmatrix}}{\hat {P}}={\begin{pmatrix}{\vec {a}}&{\vec {b}}&{\vec {c}}\\\end{pmatrix}}{\begin{pmatrix}P_{11}&P_{12}&P_{13}\\P_{21}&P_{22}&P_{23}\\P_{31}&P_{32}&P_{33}\\\end{pmatrix}}}$
where ${\textstyle {\hat {P}}}$ is a transformation matrix. All items ${\textstyle P_{ij}}$ should be integer numbers and ${\textstyle \det({\hat {P}})>1}$ (with ${\textstyle \det({\hat {P}})=1}$ the transformation preserve volume). For example, the matrix
${\displaystyle P_{P\rightarrow I}={\begin{pmatrix}0&1&1\\1&0&1\\1&1&0\\\end{pmatrix}}}$
transforms primitive cell to body centered. Another particular case of the transformation is a diagonal form (${\textstyle P_{i\neq j}=0}$) of the matrix. This called diagonal supercell expansion and can be represented as repeating of the initial cell over crystallographic axes of initial cell.

## Application

Supercells are also commonly used in computational models of crystal defects, in order to allow the use of periodic boundary conditions[2]. .