Cycles of Time (book)

In quantum mechanics, orbital magnetization, M_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, M_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

${\mathbf{m}}_{\mathrm{o}\mathrm{r}\mathrm{b}}=\frac{1}{2}\int d^3\mathbf{r}\mathbf{r}\times \mathbf{J}(\mathbf{r})$

where J(r) is the current density at point 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

${\mathbf{m}}_{\mathrm{o}\mathrm{r}\mathrm{b}}=\frac{-e}{2m_e}⟨\mathrm{\Psi }\left|\mathbf{L}\right|\mathrm{\Psi }⟩$

where −e and m_e are the charge and mass of the electron, Ψ is the ground-state wave function, and L is the angular momentum operator. The total magnetic moment is

$\mathbf{m}={\mathbf{m}}_{\mathrm{o}\mathrm{r}\mathrm{b}}+{\mathbf{m}}_{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}$

where the spin contribution is intrinsically quantum-mechanical and is given by

${\mathbf{m}}_{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}=\frac{-g_s{\mu }_{\mathrm{B}}}{\hslash }⟨\mathrm{\Psi }\left|\mathbf{S}\right|\mathrm{\Psi }⟩$

where g_s is the electron spin g-factor, μ_B is the Bohr magneton, ħ is the reduced Planck constant, and S is the electron spin operator.

The orbital magnetization 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) labelled by an index j having magnetic moments m_{orb, j}, this is

${\mathbf{M}}_{\mathrm{o}\mathrm{r}\mathrm{b}}=\frac{1}{V}\sum _{j\in V}{\mathbf{m}}_{\mathrm{o}\mathrm{r}\mathrm{b},j}.$

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.

Theory

Difficulties in the definition of orbital magnetization

For a magnetic crystal, it is tempting to try to define

${\mathbf{M}}_{\mathrm{o}\mathrm{r}\mathrm{b}}=\frac{1}{2V}\underset{V}{\int }{d}^3\mathbf{r}\mathbf{r}\times \mathbf{J}(\mathbf{r})$

where the limit is taken as the volume V of the system becomes large. However, because of the factor of 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 Template:Ket of band n and crystal momentum k:

${\mathbf{M}}_{\mathrm{o}\mathrm{r}\mathrm{b}}=\frac{-e}{2m_e}\sum _n\underset{\mathrm{B}\mathrm{Z}}{\int }\frac{d^{3}k}{(2\pi {)}^3}⟨{\psi }_{n\mathbf{k}}|\mathbf{r}\times \mathbf{p}|{\psi }_{n\mathbf{k}}⟩,$

where p is the momentum operator, L = r × p, and the integral is evaluated over the Brillouin zone "BZ". However, because the Bloch functions are extended, the matrix element of a quantity containing the 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 r × J(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

${\mathbf{M}}_{\mathrm{o}\mathrm{r}\mathrm{b}}=\frac{e}{2\hslash }\sum _n\underset{\mathrm{B}\mathrm{Z}}{\int }\frac{d^{3}k}{(2\pi {)}^3}{f}_{n\mathbf{k}}\mathrm{I}\mathrm{m}⟨\frac{\partial u_{n\mathbf{k}}}{\partial \mathbf{k}}|\times (H_{\mathbf{k}}+E_{n\mathbf{k}}-2\mu )|\frac{\partial u_{n\mathbf{k}}}{\partial \mathbf{k}}⟩,$

where f_{n k} is 0 or 1 respectively as the band energy E_{n k} falls above or below the Fermi energy μ,

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

is the effective Hamiltonian at wavevector 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

$H_{\mathbf{k}}|u_{n\mathbf{k}}⟩=E_{n\mathbf{k}}|u_{n\mathbf{k}}⟩.$

A generalization to finite temperature is also available. Note that the term involving the band energy E_{n 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.

Experiments

The orbital magnetization of a material can be determined accurately by measuring the gyromagnetic ratio γ, i.e., the ratio between the magnetic dipole moment of a body and its angular momentum. The gyromagnetic ratio is related to the spin and orbital magnetization according to

$\gamma =1+\frac{M_{\mathrm{o}\mathrm{r}\mathrm{b}}}{(M_{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}+M_{\mathrm{o}\mathrm{r}\mathrm{b}})}$

The two main experimental techniques are based either on the Barnett effect or the Einstein–de Haas effect. Experimental data for Fe, Co, Ni, and their alloys have been compiled.