User:Xuwh02/Orbital magnetization

Orbital magnetization, $$\mathbf{M}_{\rm orb}$$, refers to the magnetization induced by orbital motion of charged particles, usually electrons, in solids. The term "orbital" distinguishes it from the contribution of spin degrees of freedom, $$\mathbf{M}_{\rm spin}$$, to the total magnetization.

A nonzero orbital magnetization requires the breaking of time reversal symmetry. For example, in ferromagnets, this can be due to spin-orbit interaction.

Definition
The magnetization of a macroscopic system is generally defined as magnetic moment per unit volume.


 * $$ \mathbf{M}_{\rm orb/\rm spin}=\frac{\mathbf{m}_{\rm orb/\rm spin}}{V}.$$

$$\mathbf{m}_{\rm orb/\rm spin}$$ is the total orbital/spin magnetic moment in the system and $$V$$ is the total volume of the system.

General Formulation
A general formula of orbital magnetization in solids is derived following a fundamental relation in thermodynamics :


 * $$ \mathbf{M}=-\left(\frac{\partial\Omega}{\partial\mathbf{B}}\right)_{T,\mu},$$

$$\mathbf{\Omega}$$ is the thermodynamic potential per unit volume. $$T$$ is temperature. $$\mu$$ is chemical potential. $$\mathbf{B}$$ is external magnetic field. In this formulation, the orbital magnetization can be written into two terms:


 * $$ \mathbf{M}_{\rm orb}=\mathbf{M}_{1}+\mathbf{M}_{2}, $$

with
 * $$ \mathbf{M}_{1}=\frac{e}{2(2\pi)^{3}\hbar}\sum_{n}\int_{BZ}d^{3}kf_{n\mathbf{k}}Im\langle

\frac{\partial u_{n\mathbf{k}}}{\partial{\mathbf{k}}}|\times(\hat{H}_{\mathbf{k}}-\epsilon_{n\mathbf{k}})|\frac{\partial u_{n\mathbf{k}}}{\partial{\mathbf{k}}} \rangle, $$

and


 * $$ \mathbf{M}_{2}=-\frac{e}{(2\pi)^{3}\hbar}\sum_{n}\int_{BZ}d^{3}k\frac{1}{\beta}\ln\left(1-f_{n\mathbf{k}}\right)\mathbf{\Omega}_{n}(\mathbf{k}). $$

SI units are adopted. The summation is over all band index $$n$$. The integrals are over the Brillouin zone. $$e=1.602\times10^{-19}C$$ is the elementary charge. $$\hbar$$ is the reduced Planck constant. $$\beta=1/k_{B}T $$ and $$k_{B}$$ is the Boltzmann constant. $$f_{n\mathbf{k}}$$ is the Fermi function,


 * $$f_{n\mathbf{k}}=\frac{1}{1+e^{\beta(\epsilon_{n\mathbf{k}}-\mu)}}.$$

$$H_{\mathbf{k}}$$ is the effective Hamiltonian for crystal momentum $$\mathbf{k}$$,


 * $$H_{\mathbf{k}}=e^{i\mathbf{k}\cdot\mathbf{r}}H e^{-i\mathbf{k}\cdot\mathbf{r}},$$

$$\epsilon_{n\mathbf{k}}$$ is the band energy and $$|u_{n\mathbf{k}}\rangle$$ is the periodic Bloch function, satisfying


 * $$H_{\mathbf{k}}|u_{n\mathbf{k}}\rangle=\epsilon_{n\mathbf{k}}|u_{n\mathbf{k}}\rangle$$.

$$\mathbf{\Omega}_{n}(\mathbf{k})$$ is Berry curvature,


 * $$\mathbf{\Omega}_{n}(\mathbf{k})=-Im\langle

\frac{\partial u_{n\mathbf{k}}}{\partial{\mathbf{k}}}|\times|\frac{\partial u_{n\mathbf{k}}}{\partial{\mathbf{k}}}\rangle. $$

The cross product $$\times$$ is the conventional vector product.

This formula is valid for normal insulators, metals as well as an exotic class of insulators, namely, anomalous Hall insulator or topological insulator.

Other Methodologies
Orbital magnetization has also been computed using other methods in the literature.


 * Cellular decomposition

In a classical picture that follows the current loop definition of the magnetic moment, the orbital magnetic moment of an atom is induced by the circulating currents of atomic orbitals. This picture is implemented in the muffin-tin approximation. In this approximation, atomic spheres are constructed in each unit cell, in which the microscopic circulating current density can be defined and calculated. The orbital magnetic moment in that atomic sphere is analogous to the magnetic moment of a current distribution in the classical case. The orbital magnetization is the total orbital magnetic moment in one unit cell divided by the cell volume.

This drawback of this cellular decomposition is that the construction of atomic spheres is artificial and in principle, the electron wavefunction does have contribution to the total orbital magnetic moment in the space between the atomic spheres.


 * Wannier function basis

For normal insulators, localized wavefunctions(Wannier functions) can be constructed in each unit cell. The orbital magnetization is then the total magnetic moment in the unit cell divided by the cell volume:

\mathbf{M}_{\rm orb}=\frac{-e}{2V_{cell}}\sum_{n}\langle w_{n}| \hat{\mathbf{r}}\times\hat{\mathbf{v}} |w_{n} \rangle. $$ The summation is over all occupied Wannier functions $$\{|w_n\rangle\}$$. $$\hat{\mathbf{r}}$$ and $$\hat{\mathbf{v}}$$ are quantum mechanical operator for position and velocity.


 * Semiclassical dynamics

Orbital magnetization is computed from electron density of states in physical phase space. The density of states follows the semiclassical equation of motion of Bloch wavepackets in weak electric and magnetic fields.

Literature