Orbital magnetization

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[1]

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

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

${\displaystyle \mathbf {m} _{\rm {orb}}={\frac {-e}{2m_{e}}}\langle \Psi \vert \mathbf {L} \vert \Psi \rangle }$

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

${\displaystyle \mathbf {m} =\mathbf {m} _{\rm {orb}}+\mathbf {m} _{\rm {spin}}}$

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

${\displaystyle \mathbf {m} _{\rm {spin}}={\frac {-g_{s}\mu _{\rm {B}}}{\hbar }}\,\langle \Psi \vert \mathbf {S} \vert \Psi \rangle }$

where g_s is the electron spin g-factor, 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

${\displaystyle \mathbf {M} _{\rm {orb}}={\frac {1}{V}}\sum _{j\in V}\mathbf {m} _{{\rm {orb}},j}\;.}$

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.[2] 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

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

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.[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:

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

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]

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.[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.

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,[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

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

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]

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

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

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]

References

1. Jackson, John D. (1998). Classical Electrodynamics (3rd ed.). Wiley. ISBN 7-04-014432-8.
2. Hirst, L. L. (1997), "The microscopic magnetization: concept and application", Reviews of Modern Physics, 69 (2), pp. 607–628, Bibcode:1997RvMP...69..607H, doi:10.1103/RevModPhys.69.607
3. Resta, Raffaele (2010), "Electrical polarization and orbital magnetization: the modern theories", Journal of Physics: Condensed Matter, 22 (12), p. 123201, Bibcode:2010JPCM...22l3201R, doi:10.1088/0953-8984/22/12/123201, PMID 21389484
4. Todorova, M.; Sandratskii, M.; Kubler, J. (January 2001), "Current-determined orbital magnetization in a metallic magnet", Physical Review B, American Physical Society, 63 (5): 052408, Bibcode:2001PhRvB..63e2408T, doi:10.1103/PhysRevB.63.052408
5. Xiao, Di; Shi, Junren; Niu, Qian (September 2005), "Berry Phase Correction to Electron Density of States in Solids", Phys. Rev. Lett., 95 (13): 137204, arXiv:cond-mat/0502340, Bibcode:2005PhRvL..95m7204X, doi:10.1103/PhysRevLett.95.137204, PMID 16197171, S2CID 119017032
6. Thonhauser, T.; Ceresoli, D.; Vanderbilt, D.; Resta, R. (2005). "Orbital magnetization in periodic insulators". Phys. Rev. Lett. 95 (13): 137205. arXiv:cond-mat/0505518. Bibcode:2005PhRvL..95m7205T. doi:10.1103/PhysRevLett.95.137205. PMID 16197172. S2CID 11961765.
7. Ceresoli, D.; Thonhauser, T.; Vanderbilt, D.; Resta, R. (2006). "Orbital magnetization in crystalline solids: Multi-band insulators, Chern insulators, and metals". Phys. Rev. B. 74 (2): 024408. arXiv:cond-mat/0512142. Bibcode:2006PhRvB..74b4408C. doi:10.1103/PhysRevB.74.024408. S2CID 958110.
8. Shi, Junren; Vignale, G.; Niu, Qian (November 2007), "Quantum Theory of Orbital Magnetization and Its Generalization to Interacting Systems", Phys. Rev. Lett., American Physical Society, 99 (19): 197202, arXiv:0704.3824, Bibcode:2007PhRvL..99s7202S, doi:10.1103/PhysRevLett.99.197202, PMID 18233109, S2CID 7942622
9. Ceresoli, D.; Gerstmann, U.; Seitsonen, A.P.; Mauri, F. (Feb 2010). "First-principles theory of orbital magnetization". Phys. Rev. B. 81 (6): 060409 of 4 pages. arXiv:0904.1988. Bibcode:2010PhRvB..81f0409C. doi:10.1103/PhysRevB.81.060409. S2CID 118625623.
10. Thonhauser, T. (May 2011). "Theory of Orbital Magnetization in Solids". Int. J. Mod. Phys. B. 25 (11): 1429–1458. arXiv:1105.5251. Bibcode:2011IJMPB..25.1429T. doi:10.1142/S0217979211058912. S2CID 119292686.
11. Meyer, A.J.P.; Asch, G. (1961). "Experimental g' and g values for Fe, Co, Ni, and their alloys". J. Appl. Phys. 32 (3): S330. Bibcode:1961JAP....32S.330M. doi:10.1063/1.2000457.