1\section{Iron---Berry curvature, anomalous Hall conductivity and optical conductivity}
2\label{sec18:IronBerry}
3
4\begin{itemize}
5	\item Outline: {\it Calculate the Berry curvature, anomalous Hall conductivity, and (magneto)optical conductivity of ferromagnetic bcc Fe with spin-orbit coupling. In preparation for this example it may be useful to read Ref.~\onlinecite{PhysRevLett92} and Ch. 11 of the User Guide.}
6\end{itemize}
7
8\begin{itemize}
9	\item[1-6] {\it Compute the MLWFs and compute the energy eigenvalues and spin expectation values.}
10
11	These are the same six steps of Ex.~\ref{sec17:IronSO} and therefore the results are not going to be showed here again.
12\end{itemize}
13
14\subsection*{Berry curvature plots}
15\begin{itemize}
16	\item {\it The Berry curvature $\Omega_{\alpha\beta}(\mathbf{k})$ of the occupied states is defined in Eq. (11.18) of the User Guide.}
17	{\it Plot the Berry curvature component $\Omega_z(\bfk) = \Omega_{xy}(\bfk)$ along the magnetization direction.}
18
19	The Fermi energy should be $12.6283$ eV. With this value we obtain the energy bands and the Berry curvature component $\Omega_z(\bfk) = \Omega_{xy}(\bfk)$ along high-symmetry points shown in \Fig{fig18.1} and \Fig{fig18.2}. Eq. (11.18) of the User Guide is reported below for completeness.
20
21\begin{equation}
22\Omega_{\alpha\beta}(\mathbf{k}) = \sum_{n}^{occ} f_{n\mathbf{k}}\Omega_{n,\alpha\beta},
23\end{equation}
24with
25\begin{equation}
26\Omega_{n,\alpha\beta} = \varepsilon_{\alpha\beta\gamma}\Omega_{n,\gamma} = -2\;\mathrm{Im}\braket{\nabla_{k_\alpha}\unk \vert \nabla_{k_\beta}\unk},
27\end{equation}
28where the Greek letters indicate Cartesian coordinates, $\varepsilon_{\alpha\beta\gamma}$ is the Levi-Civita antisymmetric tensor, and $\ket{u_{n\bfk}}$s
29are the cell-periodic Bloch functions.
30\end{itemize}
31
32\begin{figure}[b!]
33\centering
34\includegraphics[width=0.8\columnwidth,trim={50pt 80pt 50pt 80pt},clip]{figure/example18/Fe_bandstructure.pdf}
35\caption{Band structure of Fe along symmetry lines $\Gamma$-H-P-N-$\Gamma$-H-N-$\Gamma$-P-N.}
36\label{fig18.1}
37\end{figure}
38\clearpage
39
40\begin{figure}[t!]
41\centering
42\includegraphics[width=0.7\columnwidth]{figure/example18/Fe_Berry_phase.png}
43\caption{Berry curvature $\Omega_z(\bfk)$ in Fe along symmetry lines.}\label{fig18.2}
44\end{figure}
45
46\begin{itemize}
47	\item {\it Combine the plot of the Fermi lines on the $k_y$ plane with a heat-map plot of (minus) the Berry curvature}
48
49	The plot of the Fermi lines with a colour-map of $-\Omega_z(k_x,0,k_z)$ is shown in \Fig{fig18.2}.
50\end{itemize}
51
52\begin{figure}[b!]
53\centering
54\includegraphics[width=0.5\columnwidth]{figure/example18/Fe_Fermi_surface+Berry_phase.png}
55\caption{(Colour online) Calculated total Berry curvature $-\Omega_z(\bfk)$ in the plane $k_y=0$ (note log scale). Intersections of the Fermi surface
56with this plane are shown.}\label{fig18.3}
57\end{figure}
58
59\subsection*{Anomalous Hall conductivity}
60
61\begin{itemize}
62	\item {\it AHC converges rather slowly with k-point sampling, and a $25 \times 25 \times 25$ does not yield a well-converged value.
63	Compare the converged AHC value with those obtained in Refs.~\onlinecite{PhysRevB74} and \onlinecite{PhysRevLett92}.}
64
65	The {\it x,y,z}-components of the AHC for a $25\times25\times25$ BZ mesh are shown in the snippet below. The converged result reported in Refs.~\onlinecite{PhysRevB74} and \onlinecite{PhysRevLett92} for the \textit{z}-component is 756.76 ($(\Omega \mathrm{cm})^{-1}$). Hence, a $25\times25\times25$ BZ mesh clearly gives a very inaccurate value ($\sim 36.4\%$ error). Even with adaptive refinement the error is still very large ($\sim 31.7\%$). It is worth to note that the adaptive refinement slightly breaks the symmetry and gives non-zero values for the \textit{x}-component and \textit{y}-component, although these are opposite in sign.
66
67\begin{tcolorbox}[title=Without adaptive refinement,sharp corners,boxrule=0.5pt]
68{\small
69\begin{verbatim}
70 Properties calculated in module  b e r r y
71 ------------------------------------------
72
73   * Anomalous Hall conductivity
74
75 Interpolation grid: 25 25 25
76
77 Fermi energy (ev):   12.6283
78
79 AHC (S/cm)       x          y          z
80 ==========    -0.0000     0.0000   554.6437
81
82
83 Total Execution Time          59.112 (sec)
84
85\end{verbatim}
86}
87\end{tcolorbox}
88
89\begin{tcolorbox}[title=With adaptive refinement,sharp corners,boxrule=0.5pt]
90{\small
91\begin{verbatim}
92 Properties calculated in module  b e r r y
93 ------------------------------------------
94
95   * Anomalous Hall conductivity
96
97 Regular interpolation grid: 25 25 25
98   Adaptive refinement grid: 5 5 5
99       Refinement threshold: Berry curvature >100.00 bohr^2
100  Points triggering refinement:   42( 0.27%)
101
102 Fermi energy (ev):   12.6283
103
104 AHC (S/cm)       x          y          z
105 ==========     2.4602    -2.4602   574.2950
106\end{verbatim}
107}
108\end{tcolorbox}
109
110Since these are quite demanding calculations, we only report the value of the AHC for a $125\times125\times125$ BZ mesh with a $5\times5\times5$ adaptive refinement grid (see snippet below). The value for the \textit{z}-component is 729.8276 $(\Omega \mathrm{cm})^{-1}$, which is in much closer agreement with the converged result from Refs.~\onlinecite{PhysRevB74} and \onlinecite{PhysRevLett92}. Also, the magnitude of \textit{x,y}-component is greatly reduced as expected.
111
112\begin{tcolorbox}[title={$125\times125\times125$ BZ mesh with a $5\times5\times5$ adaptive refinement grid}]
113{\small
114\begin{verbatim}
115 Properties calculated in module  b e r r y
116 ------------------------------------------
117
118   * Anomalous Hall conductivity
119
120 Regular interpolation grid: 125 125 125
121   Adaptive refinement grid: 5 5 5
122       Refinement threshold: Berry curvature >100.00 Ang^2
123  Points triggering refinement: 1818( 0.09%)
124
125 Fermi energy (ev):   12.6283
126
127 AHC (S/cm)       x          y          z
128 ==========    -0.2775     0.2775   729.8276
129\end{verbatim}
130}
131\end{tcolorbox}
132\end{itemize}
133
134\begin{itemize}
135	\item {\it The Wannier-interpolation formula for the Berry curvature comprises three terms, denoted $J0$, $J1$, and $J2$ in Ref.~\onlinecite{PhysRevB85}.}
136
137From Ref.~\onlinecite{PhysRevB74}
138\begin{equation}
139-2\;\mathrm{Im} G_{\alpha\beta} = J0 + J1 + J2,
140\end{equation}
141where
142\begin{equation}
143G_{\alpha\beta} = Tr[(\partial_\alpha \hat{P})\hat{Q}\hat{H}\hat{Q}(\partial_\beta\hat{P})]
144\end{equation}
145
146The three components $J0, J1$ and $J2$ for the $k$-point sampling of $125\times125\times125$ and a $5\times5\times5$ adaptive refinement grid are shown in the snippet below
147\end{itemize}
148\begin{tcolorbox}[sharp corners,boxrule=0.5pt]
149{\small
150\begin{verbatim}
151 J0 term :      0.0002    -0.0002     2.8479
152 J1 term :      0.0004    -0.0004    18.4855
153 J2 term :     -0.2782     0.2782   708.4942
154 -------------------------------------------
155\end{verbatim}
156}
157\end{tcolorbox}
158
159\subsection*{Optical conductivity}
160\begin{itemize}
161	\item {\it The optical conductivity tensor of bcc Fe with magnetization along $\hat{\mathbf{z}}$ has the form }
162\begin{equation}
163\boldsymbol{\sigma} = \boldsymbol{\sigma}_\mathrm{S} + \boldsymbol{\sigma}_{\mathrm{A}} =
164\begin{pmatrix}
165\sigma_{xx} & 0                       & 0 \\
166 0          & \sigma_{yy}=\sigma_{xx}  & 0 \\
167 0          &  0                     & \sigma_{zz}
168\end{pmatrix} + \begin{pmatrix} 0 & \sigma_{xy} & 0 \\ -\sigma_{yx} & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}
169\end{equation}
170
171	\item {\it
172The DC AHC calculated earlier corresponds to $\sigma_{xy}$ in the limit $\omega \rightarrow 0$. At finite frequency $\sigma_{xy} = -\sigma_{yx}$ acquires
173an imaginary part which describes magnetic circular dichroism (MCD).
174Compute the complex optical conductivity for $\hbar\omega$ up to $7$ eV
175}
176
177The plot for the ac AHC is shown in \Fig{fig18.4}.
178
179\end{itemize}
180\begin{figure}[t!]
181\centering
182\includegraphics[width=0.7\columnwidth]{figure/example18/Fe-kubo_A_xy_125.pdf}
183\caption{Plot of the real part of the complex optical conductivity with a $50\times50\times50$ $k$-point mesh (black) and $125\times125\times125$ $k$-point mesh (red). The inset is a magnification of the region [0-0.1] eV.}\label{fig18.4}
184\end{figure}
185
186\begin{itemize}
187	\item {\it Compare the $\omega \rightarrow 0$ limit of $\sigma_{xy}$ with the result obtained earlier by integrating the Berry curvature.}
188
189	The result obtained by integrating the Berry curvature is 729.83 $(\Omega \mathrm{cm})^{-1}$ and the $\omega \rightarrow 0$ limit of the complex optical conductivity is $669.37$ $(\Omega \mathrm{cm})^{-1}$.
190
191    {\it Plot the MCD spectrum.}
192
193    The plot of the magnetic circular dichroism is shown in \Fig{fig18.5}.
194\end{itemize}
195
196\begin{figure}[t!]
197\centering
198\includegraphics[width=0.7\columnwidth]{figure/example18/Fe_MCD_xy_125_sp3d2_projections.pdf}
199\caption{The magnetic circular dichroism from interpolation of the Kubo-Greenwood formula.}
200\label{fig18.5}
201\end{figure}
202\clearpage
203\subsection*{Further ideas}
204\begin{itemize}
205	\item {\it Recompute the AHC and optical spectra of bcc Fe using projected s, p, and d-type Wannier
206functions instead of the hybridrised MLWFs (see Example 8), and compare the results.}
207
208First we have to modify the projection block in the input file {\tt Fe.win} as did in Ex.~\ref{sec8:Iron}
209
210{\tt
211\begin{quote}
212begin projections
213Fe:s;p;d
214end projections
215\end{quote}
216}
217
218Then we need to re-do points 3,4 and 6.
219
220Below there is the extract from the output file {\tt Fe.wpout}. The result obtained from $s,p$ and $d$ projections for the $z$ component, i.e. $\sigma_{xy}$, of the AHC is exactly the same as the one obtained from $sp_3d_2,d_{xy},d_{xz}$, and $d_{yz}$ projections. Plot of AHC and MCD are shown in \Fig{fig18.6}.
221	{\small
222	\begin{tcolorbox}[title=With adaptive refinement,sharp corners,boxrule=0.5pt]
223	\begin{verbatim}
224 Properties calculated in module  b e r r y
225 ------------------------------------------
226
227   * Anomalous Hall conductivity
228
229 Regular interpolation grid: 25 25 25
230   Adaptive refinement grid: 5 5 5
231       Refinement threshold: Berry curvature >100.00 bohr^2
232  Points triggering refinement:   42( 0.27%)
233
234 Fermi energy (ev):   12.6283
235
236 AHC (S/cm)       x          y          z
237 ==========
238 J0 term :      0.0006    -0.0006    -2.9033
239 J1 term :      0.0032    -0.0032    10.0566
240 J2 term :      2.4564    -2.4564   567.1417
241 -------------------------------------------
242 Total   :      2.4602    -2.4602   574.2950
243
244	\end{verbatim}
245	\end{tcolorbox}
246	}
247
248\begin{figure}[b!]
249\centering
250\subfloat[AHC]{\includegraphics[width=0.5\columnwidth]{figure/example18/Fe_optical_conductivity_xy_125_MLWFs_and_projections.pdf}}
251\centering
252\subfloat[MCD]{\includegraphics[width=0.5\columnwidth]{figure/example18/Fe_MCD_xy_125_MLWFs_and_projections.pdf}}
253\caption{Left panel: Anomalous Hall conductivity. Right panel: Magnetic circular dichroism for $\hbar\omega$ up to $7$ eV, starting from $s;p;d$ initial projections}\label{fig18.6}
254\end{figure}
255
256\end{itemize}
257