18: Iron — Berry curvature, anomalous Hall conductivity and optical conductivity¶

Note: This tutorial requires a recent version of the pw2wannier90 interface.

Outline: Calculate the Berry curvature, anomalous Hall conductivity, and (magneto)optical conductivity of ferromagnetic bcc Fe with spin-orbit coupling. In preparation for this tutorial it may be useful to read Ref. ¹ and Ch. 11 of the User Guide.
Directory: tutorials/tutorial18/ Files can be downloaded from here
Input files
- Fe.scf The pwscf input file for ground state calculation
- Fe.nscf The pwscf input file to obtain Bloch states on a uniform grid
- Fe.pw2wan The input file for pw2wannier90
- Fe.win The wannier90 and postw90 input file

The sequence of steps below is the same of Tutorial 17. If you have already run that example, you can reuse the output files from steps 1 — 5, and only step 6 must be carried out again using the new input file Fe.win.

Run pwscf to obtain the ground state of iron
Terminal
```
pw.x < Fe.scf > scf.out
```
Run pwscf to obtain the Bloch states on a uniform k-point grid
Terminal
```
pw.x < Fe.nscf > nscf.out
```
Run wannier90 to generate a list of the required overlaps (written into the Fe.nnkp file)
Terminal
```
wannier90.x -pp Fe
```
Run pw2wannier90 to compute the overlaps between Bloch states and the projections for the starting guess (written in the Si.mmn and Si.amn files)
Terminal
```
pw2wannier90.x < Fe.pw2wan > pw2wan.out
```
Run wannier90 to compute the MLWFs
Terminal
```
wannier90.x Fe
```
Run postw90
Terminal
```
postw90.x Fe # (1)! 
mpirun -np 8 postw90.x Fe # (2)!
```
1. serial execution
2. example of parallel execution with 8 MPI processes

Berry curvature plots¶

The Berry curvature \(\Omega_{\alpha\beta}({\bf k})\) of the occupied states is defined in this equation of the User Guide. The following lines in Fe.win are used to calculate the energy bands and the Berry curvature (in bohr\(^2\)) along high-symmetry lines in \(k\)-space.

Input file

fermi_energy = [insert your value here]

berry_curv_unit = bohr2

kpath = true

kpath_task = bands+curv

kpath_bands_colour = spin

kpath_num_points = 1000

After executing postw90, plot the Berry curvature component \(\Omega_z({\bf k})=\Omega_{xy}({\bf k})\) along the magnetization direction using the script generated at runtime,

Terminal

python Fe-bands+curv_z.py

and compare with Fig. 2 of Ref. ¹.

In Tutorial 17 we plotted the Fermi lines on the (010) plane \(k_y=0\). To combine them with a heatmap plot of (minus) the Berry curvature set kpath = false, uncomment the following lines in Fe.win, re-run postw90, and issue

Terminal

python Fe-kslice-curv_z+fermi_lines.py

Compare with Fig. 3 in Ref. ¹. Note how the Berry curvature "hot-spots" tend to occur near spin-orbit-induced avoided crossings (the Fermi lines with and without spin-orbit were generated in Tutorial 17).

Anomalous Hall conductivity¶

The intrinsic anomalous Hall conductivity (AHC) is proportional to the BZ integral of the Berry curvature. In bcc Fe with the magnetization along \(\hat{\bf z}\), the only nonzero components are \(\sigma_{xy}=-\sigma_{yx}\). To evaluate the AHC using a \(25\times 25\times 25\) \(k\)-point mesh, set kslice = false, uncomment the following lines in Fe.win,

Input file

berry = true

berry_task = ahc

berry_kmesh = 25 25 25

and re-run postw90. The AHC is written in the output file Fe.wpout in vector form. For bcc Fe with the magnetization along [001], only the \(z\)-component \(\sigma_{xy}\) is nonzero.

As a result of the strong and rapid variations of the Berry curvature across the BZ, the AHC converges rather slowly with \(k\)-point sampling, and a \(25\times 25\times 25\) does not yield a well-converged value.

- Increase the BZ mesh density by changing berry_kmesh.

- To accelerate the convergence, adaptively refine the mesh around spikes in the Berry curvature, by adding to Fe.win the lines

This adds a \(5\times 5\times 5\) fine mesh around those points where \(\vert{\bm \Omega}({\bf k})\vert\) exceeds 100 bohr\(^2\). The percentage of points triggering adaptive refinement is reported in Fe.wpout.

Compare the converged AHC value with those obtained in Refs. ² and ¹.

The Wannier-interpolation formula for the Berry curvature comprises three terms, denoted \(D\)-\(D\), \(D\)-\(\overline{A}\), and \(\overline{\Omega}\) in Ref. ², and \(J2\), \(J1\), and \(J0\) in Ref. ³. To report in Fe.wpout the decomposition of the total AHC into these three terms, set iprint (verbosity level) to a value larger than one in Fe.win.

Optical conductivity¶

The optical conductivity tensor of bcc Fe with magnetization along \(\hat{\bf z}\) has the form

\[ \bm{\sigma}=\bm{\sigma}^{\rm S}+\bm{\sigma}^{\rm A}= \left( \begin{array}{ccc} \sigma_{xx} & 0 & 0\\ 0 & \sigma_{xx} & 0\\ 0 & 0 & \sigma_{zz} \end{array} \right)+ \left( \begin{array}{ccc} 0 & \sigma_{xy} & 0 \\ -\sigma_{xy} & 0 & 0\\ 0 & 0 & 0 \end{array} \right), \]

where "S" and "A" stand for the symmetric and antisymmetric parts and \(\sigma_{xx}=\sigma_{yy}\not=\sigma_{zz}\). The dc AHC calculated earlier corresponds to \(\sigma_{xy}\) in the limit \(\omega\rightarrow 0\). At finite frequency \(\sigma_{xy}=-\sigma_{yx}\) acquires an imaginary part which describes magnetic circular dichoism (MCD).

To compute the complex optical conductivity for \(\hbar\omega\) up to 7 eV, replace

Input file

berry_task = ahc

with

Input file

berry_task = kubo

add the line

Input file

kubo_freq_max = 7.0

and re-run postw90. Reasonably converged spectra can be obtained with a \(125\times 125\times 125\) \(k\)-point mesh. Let us first plot the ac AHC in S/cm, as in the lower panel of Fig. 5 in Ref. ¹,

Terminal

gnuplot

Gnuplot shell

plot 'Fe-kubo_A_xy.dat' u 1:2 w l

Comapare the \(\omega\rightarrow 0\) limit with the result obtained earlier by integrating the Berry curvature.

Note

The calculation of the AHC using berry_task = kubo involves a truncation of the sum over empty states in the Kubo-Greenwood formula: see description of the keyword kubo_eigval_max in the User Guide. As discussed around the formula for anomalous Hall conductivity of the User Guide, no truncation is done with berry_task = ahc.

Next we plot the MCD spectrum. Following Ref. ¹, we plot \({\rm Im}[\omega\sigma_{xy}(\hbar\omega)]\), in units of \(10^{29}\) sec\(^{-2}\). The needed conversion factor is \(9\times 10^{-18}\times e/\hbar\simeq 0.0137\) (\(e\) and \(\hbar\) in SI units),

Gnuplot shell

set yrange[-5:15]
plot 'Fe-kubo_A_xy.dat' u 1:(\$1)\*(\$3)\*0.0137 w l

Further ideas¶

- Recompute the AHC and optical spectra of bcc Fe using projected \(s\), \(p\), and \(d\)-type Wannier functions instead of the hybridrized MLWFs (see Tutorial 8), and compare the results.

- A crude way to model the influence of heterovalent alloying on the AHC is to assume that its only effect is to donate or deplete electrons, i.e., to shift the Fermi level of the pure crystal ⁴. Recalculate the AHC of bcc Fe for a range of Fermi energies within \(\pm 0.5\) eV of the true Fermi level. This calculation can be streamlined by replacing in Fe.win

Input file

fermi_energy = [insert your value here]

with

Input file

fermi_energy_min = [insert here your value minus 0.5]

fermi_energy_max = [insert here your value plus 0.5]

Use a sufficiently dense BZ mesh with adaptive refinement. To plot \(\sigma_{xy}\) versus \(\varepsilon_F\), issue

Terminal

gnuplot

Gnuplot shell

plot 'Fe-ahc-fermiscan.dat' u 1:4 w lp

Y. Yao, L. Kleinman, A. H. MacDonald, J. Sinova, T. Jungwirth, D.-S. Wang, E. Wang, and Q. Niu. Phys. Rev. Lett., 92:037204, 2004. ↩↩↩↩↩↩
X. Wang, J. R. Yates, I. Souza, and D. Vanderbilt. Ab initio calculation of the anomalous hall conductivity by wannier interpolation. Phys. Rev. B, 74:195118, 2006. ↩↩
M. G. Lopez, D. Vanderbilt, T. Thonhauser, and I. Souza. Phys. Rev. B, 85:014435, 2012. ↩
Y. Yao, Y. Liang, D. Xiao, Q. Niu, S.-Q. Shen, X. Dai, and Z. Fang. Phys. Rev. B, 75:020401, 2007. ↩