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