29: Platinum — Spin Hall conductivity¶
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Outline: Calculate spin Hall conductivity (SHC) and plot Berry curvature-like term of fcc Pt considering spin-orbit coupling. To gain a better understanding of this example, it is suggested to read Ref. 1 for a detailed description of the theory and Ch. 12.5 of the User Guide.
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1-6: Compute the MLWFs, spin Hall conductivity and
kpath,ksliceplots.
Spin Hall conductivity¶
SHC converges rather slowly with k-point sampling, and a \(25 \times 25 \times 25\) kmesh does not yield a well-converged value. To get a converged SHC value, increase the density of kmesh and then compare the converged result with those obtained in Refs. 1 and 2.
The file Pt-shc-fermiscan.dat contains the calculated SHC. The SHC for a
\(25\times25\times25\) kmesh are shown in the snippet below.
#No. Fermi energy(eV) SHC((hbar/e)*S/cm)
1 6.000000 0.00000000E+00
...
120 17.900000 0.17230482E+04
121 18.000000 0.17054542E+04
...
201 26.000000 0.22665760E+03
The calculated Fermi energy obtained from Quantum ESPRESSO is \(17.9919\) eV. It
may vary among different calculations due to the differences between versions of
Quantum ESPRESSO or compilers, and these may lead to deviations from the
following results. However, the difference should be acceptable and the
calculated SHC should be essentially the same.
The SHC at the Fermi energy is 1705 \((\hbar/e)\mathrm{S/cm}\). The converged results reported in Refs. 1 and 2 are around 2200 \((\hbar/e)\mathrm{S/cm}\). Hence, a \(25\times25\times25\) kmesh clearly gives an inaccurate value (\(\sim 22.5\%\) error).
Since these are quite demanding calculations, we only report the value of the SHC for a \(100\times100\times100\) kmesh (see snippet below). The value for the SHC at Fermi energy is 2207 \((\hbar/e)\mathrm{S/cm}\), which is in much closer agreement with the converged result from Refs. 1 and 2.
#No. Fermi energy(eV) SHC((hbar/e)*S/cm)
1 6.000000 0.00000000E+00
...
120 17.900000 0.21899191E+04
121 18.000000 0.22066678E+04
...
201 26.000000 0.24919920E+03
To complete the previous discussions, we also compare the Fermi energy scan plots of the two calculations as shown in Figure 1.
The seedname.wpout will print the percentage of \(k\)-points which have been
calculated at the moment, as well as the corresponding calculation time, as
shown in the following snippet.
Properties calculated in module b e r r y
------------------------------------------
* Spin Hall Conductivity
Fermi energy scan
Calculation started
-------------------------------
k-points wall diff
calculated time time
---------- ---- ----
0% 0.0 0.0
10% 22.7 22.7
20% 36.5 13.8
30% 50.4 14.0
40% 64.4 14.0
50% 78.4 14.0
60% 92.5 14.1
70% 106.5 14.0
80% 120.4 13.9
90% 134.2 13.8
100% 147.9 13.7
Interpolation grid: 25 25 25
Using adaptive smearing
adptive smearing prefactor 1.414
adptive smearing max width 1.000 eV
This might be helpful as you can roughly estimate the total computational time
of your calculation, or it might give credence to the code that it is actually
functioning :). Note this report is merely based on the "root" computation node.
It is accurate if the postw90 is run in serial, or the load on each node is
balanced if running in parallel. However, the estimation is rough if loads are
not balanced among nodes. This may happen if the performance of nodes in your
cluster are not identical, or adaptive kmesh refinements are triggered so some
nodes may compute much more \(k\)-points than others. Besides, if you are careful
enough, you may find the diff time of 10% is much larger than later ones. This
is caused by some done-once-and-for-all computations carried out at the
beginning, thus later computations are much faster.
Berry curvature-like term plots¶
The band-projected Berry curvature-like term \(\Omega_{n,\alpha\beta}^{\text{spin} \gamma}(\mathbf{k})\) is defined as Eq. (12.22) in the User Guide.
Plot the band structure of Pt and color it by the magnitude of its band-projected Berry curvature-like term \(\Omega_{n,xy}^{\text{spin}z}(\mathbf{k})\), and plot the k-resolved Berry curvature-like term \(\Omega_{xy}^{\text{spin}z}(\mathbf{k})\) along the same path in the BZ.
With Fermi energy set as 17.9919 eV we obtain the energy bands colored by the \(\Omega_{n,\alpha\beta}^{\text{spin} \gamma}(\mathbf{k})\) and the \(k\)-resolved Berry curvature-like term \(\Omega_{xy}^{\text{spin}z}(\mathbf{k})\) along high-symmetry lines as shown in Figure 2, which contains two plots calculated with different fixed smearing width.
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Combine the plot of the Fermi lines on the \((k_x,k_y)\) plane with a heatmap plot of the Berry curvature-like term of spin Hall conductivity.
The plots of the Fermi lines with a heatmap of \(\Omega_{xy}^{\text{spin}z}(k_x,k_y,0)\) are shown in Figure 3.
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Junfeng Qiao, Jiaqi Zhou, Zhe Yuan, and Weisheng Zhao. Calculation of intrinsic spin hall conductivity by wannier interpolation. Phys. Rev. B, 98:214402, Dec 2018. URL: https://link.aps.org/doi/10.1103/PhysRevB.98.214402, doi:10.1103/PhysRevB.98.214402. ↩↩↩↩
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G. Y. Guo, S. Murakami, T.-W. Chen, and N. Nagaosa. Intrinsic spin hall effect in platinum: first-principles calculations. Phys. Rev. Lett., 100:096401, Mar 2008. URL: https://link.aps.org/doi/10.1103/PhysRevLett.100.096401, doi:10.1103/PhysRevLett.100.096401. ↩↩↩