1\section{Iron---Orbital magnetization} 2\label{sec19:IronOM} 3 4\begin{itemize} 5 \item Outline: {\it Calculate the orbital magnetization of ferromagnetic bcc Fe by Wannier interpolation.} 6\end{itemize} 7 8\begin{itemize} 9 \item[1-6] These are the same steps performed for Ex.~\ref{sec17:IronSO} and Ex.~\ref{sec18:IronBerry}. Hence, they are not repeated here. 10 11 \item {\it The orbital magnetization is computed as the BZ integral of the quantity $\mathbf{M}^{\mathrm{orb}}(\bfk)$ defined in Eq. (12.20) 12of the User Guide.} 13 14Below we report Eq. (11.20) from the User Guide, and the total orbital magnetization as the integral of $\mathbf{M}^{\mathrm{orb}}(\bfk)$ over the BZ 15\begin{align} 16\mathbf{M}^{\mathrm{orb}}(\bfk) & = \sum_{n}\frac{1}{2}f_{n\bfk}\;\mathrm{Im} \braket{\nabla_{\bfk}\unk \vert \times (H_\bfk + \epsilon_\bfk - 2\epsilon_{\mathrm{F}}) \vert \nabla_\bfk \unk} 17\label{eq19.1} \\ 18\mathbf{M}^{\mathrm{orb}}_{\mathrm{tot}} & = V\int\,\frac{\diffk}{(2\pi)^3} \mathbf{M}^{\mathrm{orb}}(\bfk) 19\label{eq19.2} 20\end{align} 21 22The two snippets below show the components of the total orbital magnetization computed according to Eq.~(\ref{eq19.2}), and the spin magnetisation from the DFT calculation respectively 23{\small 24\begin{tcolorbox}[title=From Fe.wpout,sharp corners,boxrule=0.5pt] 25\begin{verbatim} 26 Properties calculated in module b e r r y 27 ------------------------------------------ 28 29 * Orbital magnetization 30 31 Interpolation grid: 25 25 25 32 33 34 Fermi energy (ev) = 12.628300 35 36 37 M_orb (bohr magn/cell) x y z 38 ====================== 39 Local circulation : 0.0000 -0.0000 0.0935 40 Itinerant circulation: 0.0000 0.0000 -0.0180 41 -------------------------------------------------------- 42 Total : 0.0000 -0.0000 0.0755 43 44\end{verbatim} 45\end{tcolorbox} 46} 47 48{\small 49\begin{tcolorbox}[title=From scf.out,sharp corners,boxrule=0.5pt] 50\begin{verbatim} 51 total magnetization = 0.00 -0.00 -2.22 Bohr mag/cell 52 absolute magnetization = 2.34 Bohr mag/cell 53\end{verbatim} 54\end{tcolorbox} 55 56} 57 58\item {\it Plot $\mathbf{M}^{\mathrm{orb}}(\bfk)$ along high-symmetry lines and compare the result with Fig.~2 of Ref.~\onlinecite{PhysRevB85}.} 59 60Before comparing the result of our calculation with the result in Fig.~2 of Ref.~\onlinecite{PhysRevB85}, 61we need to fix a unit-conversion problem in the python script {\tt Fe-bands+morb\_z.py}. In fact, the units of $\mathbf{M}^{\mathrm{orb}}(\bfk)$ are not Ry$\cdot\si{\angstrom}^2$ as stated in the python script but eV$\cdot\si{\angstrom}^2$ instead (as also stated in the User Guide). Moreover, in Ref.~\onlinecite{PhysRevB85} $\mathbf{M}^{\mathrm{orb}}(\bfk)$ is given in atomic units, i.e. Hartree$\cdot$bohr radii$^2$. In order to have a meaningful comparison we need to modify the python script accordingly. Open {\tt Fe-bands+morb\_z.py} and modify the following lines 62{\tt 63\begin{quote} 64data = np.loadtxt('Fe-morb.dat') 65 66x=data[:,0] 67 68y=data[:,3] 69\end{quote} 70} 71 72as 73 74{\tt 75\begin{quote} 76data = np.loadtxt('Fe-morb.dat') 77 78x=data[:,0] 79 80y=data[:,3] * 0.131234 81\end{quote} 82} 83where $0.131234$ is the conversion factor from eV$\cdot\si{\angstrom}^2$ to a.u. We also need to modify the label for the y-axis from 84 85\begin{quote} 86\begin{verbatim} 87 pl.ylabel(r'$M^{\rm{orb}}_z(\mathbf{k})$ [ Ry$\cdot\AA^2$ ]') 88\end{verbatim} 89\end{quote} 90 91to 92 93\begin{quote} 94\begin{verbatim} 95 pl.ylabel(r'$M^{\rm{orb}}_z(\mathbf{k})$ [ a.u. ]') 96\end{verbatim} 97\end{quote} 98 99Now we can run the python script 100 101{\tt 102\begin{quote} 103\$> python Fe-bands+morb\_z.py 104\end{quote} 105} 106 107and look at the plot, here shown in \Fig{fig19.1}. 108The difference between the quantities in the two plot is roughly the $-\frac{1}{2}$ factor due to the two different definitions of $\mathbf{M}^{\mathrm{orb}}$. 109\begin{figure}[t!] 110\centering 111\includegraphics[width=0.6\columnwidth]{figure/example19/Fe-morb_z.pdf} 112\caption{Plot of $\mathbf{M}^{\mathrm{orb}}(\bfk)$ calculated by Wannier 113interpolation along the path $\Gamma$--H--P in the Brillouin zone.}\label{fig19.1} 114\end{figure} 115 116{\it Plot $\mathbf{M}^{\mathrm{orb}}(\bfk)$ together with the Fermi contours on the (010) BZ plane} 117 118\begin{figure}[b!] 119\centering 120\includegraphics[width=0.7\columnwidth]{figure/example19/Fe-kslice-morb_z+fermi_lines.pdf} 121\caption{Plot of $\mathbf{M}^{\mathrm{orb}}(\bfk)$ together with the Fermi contours on the (010) BZ plane}\label{fig19.3} 122\end{figure} 123 124\end{itemize} 125