# POLARIZATION¶

## introduction¶

We calculate the spontaneous polarization of periodic solids by so-called Modern Theory of Polarization, namely Berry phase theory. The electric polarization \(\mathbf{P}\) is a modulo a quantum \(e\mathbf{R}/V_{c}\) multi-valued function, corresponding to the following implementation equation:

where \(\mathbf{A}_{n}(\mathbf{k})\) is the Berry connection of a single band.

## example¶

Here, we present an example (located in the `examples/PbTiO3`

folder) showcasing the calculation of the polarization of PbTiO\(_3\).

The `Input`

file is:

```
INPUT_PARAMETERS
{
nspin 1
package ABACUS
fermi_energy 13.38267075814371
fermi_energy_unit eV
HR_route data-HR-sparse_SPIN0.csr
SR_route data-SR-sparse_SPIN0.csr
rR_route data-rR-sparse.csr
HR_unit Ry
rR_unit Bohr
}
LATTICE
{
lattice_constant 7.3699
lattice_constant_unit Bohr
lattice_vector
1.0000000000 0.0000000000 0.0000000000
0.0000000000 1.0000000000 0.0000000000
0.0000000000 0.0000000000 1.0000000000
}
POLARIZATION
{
occ_band 22
nk1 10
nk2 10
nk3 10
atom_type 3
stru_file STRU
valence_e 14 12 6
}
```

`occ_band`

: The number of occupied energy bands of an insulator.

`nk1`

: This refers to the number of samples taken in the \(x\) direction of the reciprocal lattice vector \(\mathbf{G}\).

`nk2`

: This refers to the number of samples taken in the \(y\) direction of the reciprocal lattice vector \(\mathbf{G}\).

`nk3`

: This refers to the number of samples taken in the \(z\) direction of the reciprocal lattice vector \(\mathbf{G}\).

`stru_file`

: Specify the strucutre file. NAOs files are not required.

`atom_type`

: The number of element types in the system.

`valence_e`

: The number of valence electrons per element.

The parameters `nk1`

, `nk2`

, and `nk3`

correspond to the number of k-points used for the integration in the three lattice directions, and increasing their values leads to a more accurate and convergent result.

After completing the task, `polarization.dat`

appears in the `Out/Polarization`

folder which contains the electric polarization of the three lattice directions.