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. 

Wikimedia Foundation. 2010.

Игры ⚽ Поможем решить контрольную работу

Look at other dictionaries:

  • Magnetism — This article is about magnetic materials. For information about objects and devices that produce a magnetic field, see magnet. For fields that magnets and currents produce, see magnetic field. For other uses, see magnetism (disambiguation).… …   Wikipedia

  • Magnetic moment — Electromagnetism Electricity · …   Wikipedia

  • magnetic resonance — Absorption or emission of electromagnetic radiation by electrons or atomic nuclei in response to certain magnetic fields. The principles of magnetic resonance are used to study the atomic and nuclear properties of matter; two common laboratory… …   Universalium

  • magnetism — /mag ni tiz euhm/, n. 1. the properties of attraction possessed by magnets; the molecular properties common to magnets. 2. the agency producing magnetic phenomena. 3. the science dealing with magnetic phenomena. 4. strong attractive power or… …   Universalium

  • geomagnetic field — Magnetic field associated with the Earth. It is essentially dipolar (i.e., it has two poles, the northern and southern magnetic poles) on the Earth s surface. Away from the surface, the field becomes distorted. Most geomagnetists explain the… …   Universalium

  • Nuclear magnetic resonance — This article is about the physical phenomenon. For its use as a method in spectroscopy, see Nuclear magnetic resonance spectroscopy. NMR redirects here. For other uses, see NMR (disambiguation). First 1 GHz NMR Spectrometer (1000 MHz,… …   Wikipedia

  • Moon — This article is about Earth s Moon. For moons in general, see Natural satellite. For other uses, see Moon (disambiguation) …   Wikipedia

  • Dynamical mean field theory — (DMFT) is a method to determine the electronic structure of strongly correlated materials. In such materials, the approximation of independent electrons, which is used in Density Functional Theory and usual band structure calculations, breaks… …   Wikipedia

  • Ferromagnetism — Not to be confused with Ferrimagnetism; for an overview see Magnetism A magnet made of alnico, an iron alloy. Ferromagnetism is the physical theory which explains how materials become magnets. Ferromagnetism is the basic mechanism by which… …   Wikipedia

  • Spintronics — (a neologism meaning spin transport electronics [1][2]), also known as magnetoelectronics, is an emerging technology that exploits both the intrinsic spin of the electron and its associated magnetic moment, in addition to its fundamental… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”