User:Dhv7/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 broken time-reversal symmetry, which can occur spontaneously in ferromagnetic and ferrimagnetic materials, or can be induced in a non-magnetic material by an applied magnetic field.

Definitions
The orbital magnetic moment of a finite system, such as a molecule, is given classically by

\mathbf{m}_{\rm orb}=\frac{1}{2}\int d^3r \; \mathbf{r}\times\mathbf{J}(\mathbf{r}) $$ where $$\mathbf{J}(\mathbf{r})$$ is the current density at point $$\mathbf{r}$$. (Here SI units are used; in Gaussian units, the prefactor would be $$1/2c$$ instead, where $$c$$ is the speed of light.) In a quantum-mechanical context, this can also be written as

\mathbf{m}_{\rm orb}=\frac{-e}{2m_e} \langle\Psi \vert\mathbf{L} \vert\Psi\rangle $$ where $$-e$$ and $$m_e$$ are the charge and mass of the electron, $$\Psi$$ is the ground-state wave function, and $$\mathbf{L}$$ is the angular momentum operator. The total magnetic moment is

\mathbf{m}=\mathbf{m}_{\rm orb}+\mathbf{m}_{\rm spin} $$ where the spin contribution is intrinsically quantum-mechanical and is given by

\mathbf{m}_{\rm spin}=\frac{-g_s\mu_{\rm B}}{\hbar} \, \langle\Psi \vert\mathbf{S} \vert\Psi\rangle $$ where $$g_s$$ is the electron spin g-factor, $$\mu_{\rm B}$$ is the Bohr magneton, $$\hbar$$ is the reduced Planck constant, and $$\mathbf{S}$$ is the electron spin operator.

The orbital magnetization $$\mathbf{M}$$ is defined as the orbital moment density i.e, orbital moment per unit volume. For a crystal of volume $$V$$ composed of isolated entities (e.g., molecules) $$j$$ having magnetic moments $$\mathbf{m}_{{\rm orb},j}$$, this is

\mathbf{M}_{\rm orb}=\frac{1}{V}\sum_{j\in V}\mathbf{m}_{{\rm orb},j} \;. $$ However, real crystals are made up out of atomic or molecular constituents whose charge clouds overlap, so that the above formula cannot be taken as a fundamental definition of orbital magnetization. Only recently have theoretical developments led to a proper theory of orbital magnetization in crystals, as explained below.

Difficulties in the definition of orbital magnetization
For a magnetic crystal, it is tempting to try to define

\mathbf{M}_{\rm orb}=\frac{1}{2V}\int_V d^3r \; \mathbf{r}\times\mathbf{J}(\mathbf{r}) $$ where the limit is taken as the volume $$V$$ of the system becomes large. However, because of the factor of $$\mathbf{r}$$ in the integrand, the integral has contributions from surface currents that cannot be neglected, and as a result the above equation does not lead to a bulk definition of orbital magnetization.

Another way to see that there is a difficulty is to try to write down the quantum-mechanical expression for the orbital magnetization in terms of the occupied single-particle Bloch functions $$\vert\psi_{n\mathbf{k}}\rangle$$ of band $$n$$ and crystal momentum $$\mathbf{k}$$:

\mathbf{M}_{\rm orb}=\frac{-e}{2m_e}\sum_n\int_{\rm BZ}\frac{d^3k}{(2\pi)^3}\,\langle\psi_{n\mathbf{k}}\vert\mathbf{r}\times\mathbf{p}\vert \psi_{n\mathbf{k}}\rangle \,, $$ where $$\mathbf{p}$$ is the momentum operator (note $$\mathbf{L}=\mathbf{r}\times\mathbf{p}$$) and the integral is carried over the Brillouin zone. However, because the Bloch functions are extended, the matrix element of a quantity containing the $$\mathbf{r}$$ operator is ill-defined, and this formula is actually ill-defined.

Atomic sphere approximation
In practice, orbital magnetization is often computed by decomposing space into non-overlapping spheres centered on atoms (similar in spirit to the muffin-tin approximation), computing the integral of $$\mathbf{r}\times\mathbf{J}(\mathbf{r})$$ inside each sphere, and summing the contributions. This approximation neglects the contributions from currents in the interstitial regions between the atomic spheres. Nevertheless, it is often a good approximation because the orbital currents associated with partially filled d and f shells are typically strongly localized inside these atomic spheres. It remains, however, an approximate approach.

Modern theory of orbital magnetization
A general and exact formulation of the theory of orbital magnetization was developed in the mid-2000s by several authors, first based on a semiclassical approach, then on a derivation from the Wannier representation, and finally from a long-wavelength expansion. The resulting formula for the orbital magnetization, specialized to zero temperature, is
 * $$ \mathbf{M}_{\rm orb}=\frac{e}{2\hbar}\sum_{n}\int_{\rm BZ}\frac{d^{3}k}{(2\pi)^{3}}\,f_{n\mathbf{k}}\;{\rm Im}\;\langle

\frac{\partial u_{n\mathbf{k}}}{\partial{\mathbf{k}}}|\times(H_{\mathbf{k}}+E_{n\mathbf{k}}-2\mu)|\frac{\partial u_{n\mathbf{k}}}{\partial{\mathbf{k}}} \rangle, $$ where $$f_{n\mathbf{k}}$$ is 0 or 1 respectively as the band energy $$E_{n\mathbf{k}}$$ falls above or below the Fermi energy $$\mu$$,

H_{\mathbf{k}}=e^{i\mathbf{k}\cdot\mathbf{r}}H e^{-i\mathbf{k}\cdot\mathbf{r}} $$ is the effective Hamiltonian at wavevector $$\mathbf{k}$$, and
 * $$u_{n\mathbf{k}}(\mathbf{r})=e^{-i\mathbf{k}\cdot\mathbf{r}}\psi_{n\mathbf{k}}(\mathbf{r})$$

is the cell-periodic Bloch function satisfying
 * $$H_{\mathbf{k}}|u_{n\mathbf{k}}\rangle=E_{n\mathbf{k}}|u_{n\mathbf{k}}\rangle\;.$$

A generalization to finite temperature is also available. Note that the term involving the band energy $$E_{n\mathbf{k}}$$ in this formula is really just an integral of the band energy times the Berry curvature. Results computed using the above formula have appeared in the literature. A recent review summarizes these developments.