25: Gallium Arsenide — Nonlinear shift current¶

Outline: Calculate the nonlinear shift current of inversion asymmetric fcc Gallium Arsenide. In preparation for this tutorial it may be useful to read Ref. ¹
Directory: tutorials/tutorial25/ Files can be downlowaded from here
Input files:
- GaAs.scf The pwscf input file for ground state calculation
- GaAs.nscf The pwscf input file to obtain Bloch states on a uniform grid
- GaAs.pw2wan The input file for pw2wannier90
- GaAs.win The wannier90 and postw90 input file

Run pwscf to obtain the ground state of Gallium Arsenide
Terminal
```
pw.x < GaAs.scf > scf.out
```
Run pwscf to obtain the ground state of Gallium Arsenide
Terminal
```
pw.x < GaAs.nscf > nscf.out
```
Run Wannier90 to generate a list of the required overlaps (written into the GaAs.nnkp file)
Terminal
```
wannier90.x -pp GaAs
```
Run pw2wannier90 to compute:
- The overlaps $\langle u_{n\bf{k}}|u_{n\bf{k+b}}\rangle$ between spinor Bloch states (written in the GaAs.mmn file)
- The projections for the starting guess (written in the GaAs.amn file)
Terminal
```
pw2wannier90.x < GaAs.pw2wan > pw2wan.out
```
Run wannier90 to compute MLWFs
Terminal
```
wannier90.x GaAs
```
Run postw90 to compute nonlinear shift current
Terminal
```
postw90.x GaAs` # (1)! 

mpirun -np 8 postw90.x GaAs  # (2)! 
```
1. serial execution
2. example of parallel execution with 8 MPI processes

Shift current $\sigma^¶

The shift current tensor of GaAs has only one independent component that is finite, namely $\sigma^{xyz}$. For its computation, set

Input file

berry = true
berry_task = sc

Like the linear optical conductivity, the shift current is a frequency-dependent quantity. The frequency window and step is controlled by kubo_freq_min, kubo_freq_max and kubo_freq_step, as explained in the users guide.

The shift current requires an integral over the Brillouin zone. The interpolated k-mesh is controlled by berry_kmesh, which has been set to

Input file

berry_kmesh = 100 100 100

We also need to input the value of the Fermi level in eV:

Input file

fermi_energy = [insert your value here]

Due to the sum over intermediate states involved in the calculation of the shift current, one needs to consider a small broadening parameter to avoid numerical problems due to possible degeneracies (see parameter $\eta$ in Eq. (36) of Ref. ¹ and related discussion). This parameter is controlled by sc_eta. It is normally found that values between 0.01 eV and 0.1 eV yield an stable spectrum. The default value is set to $0.04$ eV.

Finally, sc_phase_conv controls the phase convention used for the Bloch sums. sc_phase_conv=1 uses the so-called tight-binding convention, whereby the Wannier centres are included into the phase, while sc_phase_conv=2 leaves the Wannier centres out of the phase. These two possible conventions are explained in Ref. ². Note that the overall shift-current spectrum does not depend on the chosen convention, but the individual terms that compose it do.

On output, the program generates a set of 18 files named SEED-sc_***.dat, which correspond to the different tensor components of the shift current (note that the 9 remaining components until totaling $3\times3\times3=27$ can be obtained from the 18 outputed by taking into account that $\sigma^{abc}$ is symmetric under $b\leftrightarrow c$ index exchange). For plotting the only finite shift-current component of GaAs $\sigma^{xyz}$ (units of A/V$^{2}$) as in the upper panel of Fig. 3 in Ref. ¹,

Terminal

gnuplot

Gnuplot shell

plot 'GaAs-sc_xyz.dat' u 1:2 w l

Julen Ibañez-Azpiroz, Stepan S. Tsirkin, and Ivo Souza. Ab initio calculation of the shift photocurrent by wannier interpolation. Phys. Rev. B, 97:245143, Jun 2018. URL: https://link.aps.org/doi/10.1103/PhysRevB.97.245143, doi:10.1103/PhysRevB.97.245143. ↩↩↩
T. Yusufaly, D. Vanderbilt, and S. Coh. Tight-Binding Formalism in the Context of the PythTB Package. \url http://physics.rutgers.edu/pythtb/formalism.html. ↩