30: Gallium Arsenide — Frequency-dependent spin Hall conductivity¶

Outline: Calculate the alternating current (ac) spin Hall conductivity of gallium arsenide considering spin-orbit coupling. To gain a better understanding of this tutorial, it is suggested to read Ref. ¹ for a detailed description of the theory and the berry_task=shc: spin Hall conductivity chapter of the User Guide.
Directory: tutorials/tutorial30/ Files can downloaded 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 Bloch states on a uniform \(k\)-point grid
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 between Bloch states and the projections for the starting guess (written in the GaAs.mmn and GaAs.amn files)
Terminal
```
pw2wannier90.x < GaAs.pw2wan > pw2wan.out
```
Run wannier90 to compute the MLWFs
Terminal
```
wannier90.x GaAs
```
Run postw90
Terminal
```
postw90.x GaAs # (1)!
mpirun -np 8 postw90.x GaAs # (2)!
```
1. serial execution
2. example of parallel execution with 8 MPI processes

ac spin Hall conductivity¶

The spin Hall conductivity is also dependent on the frequency \(\omega\) in this equation of the User Guide. The direct current (dc) SHC calculated in the previous tutorial corresponds to \(\sigma_{\alpha\beta}^{\text{spin}\gamma}\) in the limit \(\omega\rightarrow 0\) and it is a real number. At finite frequency \(\sigma_{\alpha\beta}^{\text{spin}\gamma}\) acquires an imaginary part.

To compute the ac spin Hall conductivity for \(\hbar\omega\) up to 8 eV, add the lines

Input file

shc_freq_scan = true
kubo_freq_min = 0.0
kubo_freq_max = 8.0
kubo_freq_step = 0.01

and re-run postw90. The file GaAs-shc-freqscan.dat contains the calculated ac SHC. Reasonably converged spectra can be obtained with a \(250\times 250\times 250\) \(k\)-point mesh. To plot the ac SHC, issue the following commands

Terminal

gnuplot

Gnuplot shell

plot 'GaAs-shc-freqscan.dat' u 2:3 w l title 'Re',
'GaAs-shc-freqscan.dat' u 2:4 w l title 'Im'

and then compare the result with Fig. 4 in Ref. ¹ or the Solution Booklet.

Notes¶

When calculating ac SHC, adaptive smearing can be used by add the following keywords in the GaAs.win,

Input file

kubo_adpt_smr = true
kubo_adpt_smr_fac = [insert here your smearing factor]
kubo_adpt_smr_max = [insert here your maximum smearing]

Adaptive kmesh refinement is not implemented for ac SHC calculation.
The first 10 semi-core states are excluded from the calculation by using the following keywords
Input file
```
exclude_bands = 1-10
```
and in the case of GaAs disentanglement is not adopted so
Input file
```
num_bands         =   16
num_wann          =   16
```
Since the band gap is often under estimated by LDA/GGA calculations, a scissors shift is applied to recover the experimental band gap by using the following keywords
Input file
```
shc_bandshift = true
shc_bandshift_firstband = 9
shc_bandshift_energyshift = 1.117
```
or by
Input file
```
num_valence_bands = 8
scissors_shift = 1.117
```

Junfeng Qiao, Jiaqi Zhou, Zhe Yuan, and Weisheng Zhao. Calculation of intrinsic spin hall conduc tivity by wannier interpolation. Phys. Rev. B, 98:214402, Dec 2018. URL: https://link.aps.org/doi/10.1103/PhysRevB.9 8.214402, doi:10.1103/PhysRevB.98.214402. ↩↩