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