# 6. Berry-curvature

## 6.1. Introduction

RESCU has been used to predict the topological properties of heterostructures in a couple of studies [CMWG18, CMWG19]. A key element of the analysis is the Berry curvature, defined as follows

where the Berry connection \(\mathcal{A}_n\) is defined as

Eq. (6.1.2) is difficult to evaluate numerically because of the k-space gradient, and hence RESCU implements this equivalent sum-over-eigenstates formula [XYSV06]

where the velocity matrix elements are defined as

and

The Berry-curvature calculator was developed to study specific two-dimensional heterostructures (e.g. graphene deposited on boron nitride). Consequently, it is now subjected to the following constraints:

The system must be two-dimensional, oriented in the plane perpendicular to the z-direction.

The Berry curvature is calculated in the \(k_z = 0\) plane (hence Eq. (6.1.3) which gives \(\Omega_{n,xy}(\mathbf{k})\)).

The system must be solved by means of numerical atomic orbitals (

`LCAO.status = true`

).The implementation is tested for degenerate spin systems only, but the collinear-spin formalism is supported.

## 6.2. Example: Graphene/BN

In this section, we compute the Berry curvature of the first conduction
band of the graphene-boron-nitride heterostructure. Let’s define the
input file `gbn_lcao_scf.input`

as follows

```
LCAO.status = 1
info.calculationType = 'self-consistent'
info.savepath = './results/gbn_lcao_scf'
atom.element = [1 1 2 3]
da = 3.0425
atom.fracxyz = [2/3 1/3 0.5 + da/30
1/3 2/3 0.5 + da/30
2/3 1/3 0.5 - da/30
1/3 2/3 0.5 - da/30]
la = 4.648725932
domain.latvec = [[1/2 sqrt(3)/2 0]*la
[1/2 -sqrt(3)/2 0]*la
[0 0 30]]
domain.lowres = 0.3
element(1).species = 'C'
element(1).path = './C_ONCV_LDA.mat'
element(2).species = 'B'
element(2).path = './B_ONCV_LDA.mat'
element(3).species = 'N'
element(3).path = './N_ONCV_LDA.mat'
kpoint.gridn = [15,15,1]
```

This is a standard `self-consistent`

calculation input file which will
provide us with the ground state density (and Hamiltonian). You may
execute the program typing

```
rescu -i gbn_lcao_scf.input
```

Next, we define the input file `gbn_lcao_ber.input`

```
info.calculationType = 'berry-curvature'
info.savepath = './results/gbn_lcao_ber'
rho.in = './results/gbn_lcao_scf'
kpoint.gridn = [102,102,1]
berry.bandIndex = 9
```

Here are the key elements:

RESCU find the eigenvalues and eigenstates on a two-dimensional Monkhorst-Pack grid and then computes the Berry curvature when set to

`berry-curvature`

;Points RESCU to the ground state results;

Defines the size of the Monkhorst-Pack grid. The Berry curvature typically requires very high k-sampling, here we use more than \(10^{4}\) k-points. Alternatively, one can define a custom k-point mesh using the keyword

`kpoint.kdirect`

(see`inputDescription.m`

for more details);Determines the band index for which the Berry curvature is calculated (see Eq. (6.1.3)).

You may then run the program as follows

```
rescu -i gbn_lcao_ber
```

Upon return, the results will be in `./results/gbn_lcao_ber.mat`

. One
may find the Berry curvature in the structure component `berry.Omega`

.
The Berry curvature can also be visualize typing

```
rescu -p results/gbn_lcao_ber
```

The resulting graph should look like the one in Fig. 6.2.1.

Chen Hu, Vincent Michaud-Rioux, Wang Yao, and Hong Guo. Moiré Valleytronics: Realizing Dense Arrays of Topological Helical Channels. Physical Review Letters 121.18 (2018), p. 186403.

Chen Hu, Vincent Michaud-Rioux, Wang Yao, and Hong Guo. Theoretical Design of Topological Heteronanotubes. In: Nano Letters 19.6 (June 2019), pp. 4146–4150.

Xinjie Wang, Jonathan R. Yates, Ivo Souza, and David Vanderbilt. Ab initio calculation of the anomalous Hall conductivity by Wannier interpolation.. Phys. Rev. B 74 (19 Nov. 2006), p. 195118.