Orbital magnetization

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.

Contents

Definitions

The orbital magnetic moment of a finite system, such as a molecule, is given classically by[1]


\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 me are the charge and mass of the electron, Ψ 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 gs is the electron spin g-factor, μB 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.[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


\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.[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 \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.[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 \mathbf{r}\times\mathbf{J}(\mathbf{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

 \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 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 \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.[3][8] 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.[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


  \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. ^ a b Hirst, L. L. (1997), "The microscopic magnetization: concept and application", Reviews of Modern Physics 69: 607–628, Bibcode 1997RvMP...69..607H, doi:10.1103/RevModPhys.69.607, http://rmp.aps.org/abstract/RMP/v69/i2/p607_1 
  3. ^ a b Resta, Raffaele (2010), "Electrical polarization and orbital magnetization: the modern theories", Journal of Physics: Condensed Matter 22 (12): 123201, doi:10.1088/0953-8984/22/12/123201, http://iopscience.iop.org/0953-8984/22/12/123201 
  4. ^ Todorova, M.; Sandratskii, M.; Kubler, J. (January 2001), "Current-determined orbital magnetization in a metallic magnet", Physical Reivew B (American Physical Society) 63 (5): 052408, 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. (American Physical Society) 95 (13): 137204, Bibcode 2005PhRvL..95m7204X, doi:10.1103/PhysRevLett.95.137204 
  6. ^ Thonhauser, T.; Ceresoli, D.; Vanderbilt, D.; Resta, R. (2005). "Orbital magnetization in periodic insulators". Phys. Rev. Lett 95: 137205. Bibcode 2005PhRvL..95m7205T. doi:10.1103/PhysRevLett.95.137205. 
  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: 024408. doi:10.1103/PhysRevB.74.024408. 
  8. ^ a b 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, Bibcode 2007PhRvL..99s7202S, doi:10.1103/PhysRevLett.99.197202 
  9. ^ Ceresoli, D.; Gerstmann, U.; Seitsonen, A.P.; Mauri, F. (Feb 2010). "First-principles theory of orbital magnetization". Phys. Rev. B 81 (6): 060409. doi:10.1103/PhysRevB.81.060409. 
  10. ^ Thonhauser, T. (May 2011). "Theory of Orbital Magnetization in Solids". Int. J. Mod. Phys. B 25 (11): 1429–1458. arXiv:1105.5251. doi:10.1142/S0217979211058912. 
  11. ^ Meyer, A.J.P.; Asch, G. (1961). "Experimental g' and g values for Fe, Co, Ni, and their alloys". J. Appl. Phys. 32: S330. Bibcode 1961JAP....32S.330M. doi:10.1063/1.2000457. 

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