Orbital magnetization

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

${\displaystyle \mathbf {M} _{\rm {orb}}={\frac {1}{2V}}\int _{V}\mathbf {r} \times \mathbf {J} (\mathbf {r} )\ d^{3}\mathbf {r} }$

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.^[2]

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 |ψ_{n k}⟩ of band n and crystal momentum k:

${\displaystyle \mathbf {M} _{\rm {orb}}={\frac {-e}{2m_{e}}}\sum _{n}\int _{\rm {BZ}}{\frac {1}{(2\pi )^{3}}}\,\langle \psi _{n\mathbf {k} }\vert \mathbf {r} \times \mathbf {p} \vert \psi _{n\mathbf {k} }\rangle \,d^{3}k\,,}$

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.^[3]

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.^[4] 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.

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,^[5] then on a derivation from the Wannier representation,^[6][7] and finally from a long-wavelength expansion.^[8] The resulting formula for the orbital magnetization, specialized to zero temperature, is

${\displaystyle \mathbf {M} _{\rm {orb}}={\frac {e}{2\hbar }}\sum _{n}\int _{\rm {BZ}}{\frac {1}{(2\pi )^{3}}}\,f_{n\mathbf {k} }\;\operatorname {Im} \;\left\langle {\frac {\partial u_{n\mathbf {k} }}{\partial {\mathbf {k} }}}\right|\times \left(H_{\mathbf {k} }+E_{n\mathbf {k} }-2\mu \right)\left|{\frac {\partial u_{n\mathbf {k} }}{\partial {\mathbf {k} }}}\right\rangle \,d^{3}k\,,}$

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

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

is the effective Hamiltonian at wavevector k, and

${\displaystyle u_{n\mathbf {k} }(\mathbf {r} )=e^{-i\mathbf {k} \cdot \mathbf {r} }\psi _{n\mathbf {k} }(\mathbf {r} )}$

is the cell-periodic Bloch function satisfying

${\displaystyle H_{\mathbf {k} }\left|u_{n\mathbf {k} }\right\rangle =E_{n\mathbf {k} }\left|u_{n\mathbf {k} }\right\rangle \;.}$

A generalization to finite temperature is also available.^[3][8] 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.^[9] A recent review summarizes these developments.^[10]

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

${\displaystyle \gamma =1+{\frac {M_{\mathrm {orb} }}{(M_{\mathrm {spin} }+M_{\mathrm {orb} })}}}$

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.^[11]