Antisymmetric exchange

α-Fe₂O₃ pictured as Haematite, the main source of iron for the steel industry

The discovery of antisymmetric exchange originated in the early 20th century from the controversial observation of weak ferromagnetism in typically antiferromagnetic α-Fe₂O₃ crystals.[1] In 1958, Igor Dzyaloshinskii provided evidence that the interaction was due to the relativistic spin lattice and magnetic dipole interactions based on Lev Landau's theory of phase transitions of the second kind.[2] In 1960, Toru Moriya identified the spin-orbit coupling as the microscopic mechanism of the antisymmetric exchange interaction.[1] Moriya referred to this phenomenon specifically as the "antisymmetric part of the anisotropic superexchange interacton." The simplified naming of this phenomenon occurred in 1962, when D. Treves and S. Alexander of Bell Telephone Laboratories simply referred to the interaction as antisymmetric exchange. Because of their seminal contributions to the field, antisymmetric exchange is sometimes referred to as the Dzyaloshinskii–Moriya interaction.[3]

The functional form of the DMI can be obtained through a second-order perturbative analysis of the spin-orbit coupling interaction, ${\displaystyle {\hat {\mathbf {L} }}\cdot {\hat {\mathbf {S} }}}$ between ions ${\displaystyle i,j}$ [1] in Anderson's superexchange formalism. Note the notation used implies ${\displaystyle {\hat {\mathbf {L} }}_{i}}$ is a 3-dimensional vector of angular momentum operators on ion i, and ${\displaystyle {\hat {\mathbf {S} }}_{i}}$ is a 3-dimensional spin operator of the same form:

${\displaystyle {\begin{aligned}\delta E=\sum _{m}&{\Biggl [}{\frac {\langle n|\lambda {\hat {\mathbf {L} }}_{i}\cdot {\hat {\mathbf {S} }}_{i}|m\rangle 2J(mn'nn'){\hat {\mathbf {S} }}_{i}\cdot {\hat {\mathbf {S} }}_{j}}{E_{n}-E_{m}}}\\&+{\frac {2J(nn'mn'){\hat {\mathbf {S} }}_{i}\cdot {\hat {\mathbf {S} }}_{j}\langle m|\lambda {\hat {\mathbf {L} }}_{i}\cdot {\hat {\mathbf {S} }}_{i}|n\rangle }{E_{n}-E_{m}}}{\Biggr ]}\\+\sum _{m'}&{\Biggl [}{\frac {\langle m'|\lambda {\hat {\mathbf {L} }}_{j}\cdot {\hat {\mathbf {S} }}_{j}|m\rangle 2J(m'nn'n){\hat {\mathbf {S} }}_{i}\cdot {\hat {\mathbf {S} }}_{j}}{E_{n'}-E_{m'}}}\\&+{\frac {2J(n'nm'n){\hat {\mathbf {S} }}_{i}\cdot {\hat {\mathbf {S} }}_{j}\langle m'|\lambda {\hat {\mathbf {L} }}_{j}\cdot {\hat {\mathbf {S} }}_{j}|n'\rangle }{E_{n'}-E_{m'}}}{\Biggr ]}\end{aligned}}}$

where ${\displaystyle J}$ is the exchange integral,

${\displaystyle J(nn'mm')=\int \int \phi _{n}^{*}(\mathbf {r_{1}} -\mathbf {R} )\phi _{n'}^{*}(\mathbf {r_{2}} -\mathbf {R'} ){\frac {e^{2}}{r_{12}}}\phi _{m}(\mathbf {r_{2}} -\mathbf {R} )\phi _{m'}(\mathbf {r_{1}} -\mathbf {R'} )\mathrm {d} \mathbf {r_{1}} \mathrm {d} \mathbf {r_{2}} }$

with ${\displaystyle \phi _{n}(\mathbf {r} -\mathbf {R} )}$ the ground orbital wavefunction of the ion at ${\displaystyle \mathbf {R} }$, etc. If the ground state is non-degenerate, then the matrix elements of ${\displaystyle \mathbf {L} }$ are purely imaginary, and we can write ${\displaystyle \delta E}$ out as

${\displaystyle {\begin{aligned}\delta E&=2\lambda \sum \limits _{m}{\frac {J(nn'mn')}{E_{n}-E_{m}}}\langle n|\mathbf {L_{i}} |m\rangle \cdot [\mathbf {S_{i}} ,(\mathbf {S_{i}} \cdot \mathbf {S_{j}} )]\\&+2\lambda \sum _{m'}{\frac {J(nn'nm')}{E_{n'}-E_{m'}}}\langle n'|\mathbf {L_{j}} |m'\rangle \cdot [\mathbf {S_{j}} ,(\mathbf {S_{i}} \cdot \mathbf {S_{j}} )]\\&=2i\lambda \sum \limits _{m,m'}\left[{\frac {J(nn'mn')}{E_{n}-E_{m}}}\langle n|\mathbf {L_{i}} |m\rangle -{\frac {J(nn'nm')}{E_{n'}-E_{m'}}}\langle n'|\mathbf {L_{j}} |m'\rangle \right]\cdot [\mathbf {S_{i}} \times \mathbf {S_{j}} ]\\&=\mathbf {D} _{ij}\cdot [\mathbf {S} _{i}\times \mathbf {S} _{j}].\end{aligned}}}$

The Dzyaloshinskii–Moriya interaction has proven difficult to experimentally measure directly due to its typically weak effects and similarity to other magnetoelectric effects in bulk materials. Attempts to quantify the DMI vector have utilized X-ray diffraction interference, Brillouin scattering, electron spin resonance, and neutron scattering. Many of these techniques only measure either the direction or strength of the interaction and make assumptions on the symmetry or coupling of the spin interaction. A recent advancement in broadband electron spin resonance coupled with optical detection (OD-ESR) allows for characterization of the DMI vector for rare-earth ion materials with no assumptions and across a large spectrum of magnetic field strength.[6]

Corundum crystal structure displaying the crystal forms of α-Fe₂O₃ and α-Cr₂O₃ (Metal ions in red, oxygen ions in blue)

The image at right displays a coordinated heavy metal-oxide complex that can display ferromagnetic or antiferromagnetic behavior depending on the metal ion. The structure shown is referred to as the corundum crystal structure, named after the primary form of Aluminum oxide (Al
₂O
₃), which displays the R3c trigonal space group. The structure also contains the same unit cell as α-Fe₂O₃ and α-Cr₂O₃ which possess D⁶_3d space group symmetry. The upper half unit cell displayed shows four M³⁺ ions along the space diagonal of the rhombohedron. In the Fe₂O₃ structure, the spins of the first and last metal ion are positive while the center two are negative. In the α-Cr₂O₃ structure, the spins of the first and third metal ion are positive while the second and fourth are negative. Both compounds are antiferromagnetic at cold temperatures (<250K), however α-Fe₂O₃ above this temperature undergoes a structural change where its total spin vector no longer points along the crystal axis but at a slight angle along the basal (111) plane. This is what causes the iron-containing compound to display an instantaneous ferromagnetic moment above 250K, while the chromium-containing compound shows no change. It is thus the combination of the distribution of ion spins, the misalignment of the total spin vector, and the resulting antisymmetry of the unit cell that gives rise to the antisymmetric exchange phenomenon seen in these crystal structures.[2]

• Exchange interaction
• Spin-orbit coupling
• Superexchange
• Landau theory
• Skyrmions
• Multiferroics