Antisymmetric exchange

Antisymmetric exchange, also known as the Dzyaloshinskii–Moriya interaction, is a contribution to the total magnetic exchange interaction between two neighboring magnetic spins, S i {displaystyle mathbf {S} _{i}} and S j {displaystyle mathbf {S} _{j}} . Quantitatively, it is a term in the Hamiltonian which can be written as H D M = D i j ⋅ ( S i × S j ) {displaystyle H_{DM}=mathbf {D} _{ij}cdot (mathbf {S} _{i} imes mathbf {S} _{j})} . In magnetically ordered systems, it favors a spin canting of otherwise (anti)parallel aligned magnetic moments and thus, is a source of weak ferromagnetic behavior in an antiferromagnet. The interaction is fundamental to the production of magnetic skyrmions and explains the magnetoelectric effects in a class of materials termed multiferroics. Antisymmetric exchange, also known as the Dzyaloshinskii–Moriya interaction, is a contribution to the total magnetic exchange interaction between two neighboring magnetic spins, S i {displaystyle mathbf {S} _{i}} and S j {displaystyle mathbf {S} _{j}} . Quantitatively, it is a term in the Hamiltonian which can be written as H D M = D i j ⋅ ( S i × S j ) {displaystyle H_{DM}=mathbf {D} _{ij}cdot (mathbf {S} _{i} imes mathbf {S} _{j})} . In magnetically ordered systems, it favors a spin canting of otherwise (anti)parallel aligned magnetic moments and thus, is a source of weak ferromagnetic behavior in an antiferromagnet. The interaction is fundamental to the production of magnetic skyrmions and explains the magnetoelectric effects in a class of materials termed multiferroics. The discovery of antisymmetric exchange originated in the early 20th century from the controversial observation of weak ferromagnetism in typically antiferromagnetic α-Fe2O3 crystals. In 1958, Igor Dzyaloshinskii provided evidence that the interaction was due to the relativistic spin lattice and magnetic dipole interactions based on Landau's theory of phase transitions of the second kind.. In 1960, Toru Moriya identified the spin-orbit coupling as the microscopic mechanism of the antisymmetric exchange interaction. 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. The functional form of the DMI can be obtained through a second-order perturbative analysis of the spin-orbit coupling interaction, L ^ ⋅ S ^ {displaystyle {hat {mathbf {L} }}cdot {hat {mathbf {S} }}} between ions i , j {displaystyle i,j} in Anderson's superexchange formalism. Note the notation used implies L ^ i {displaystyle {hat {mathbf {L} }}_{i}} is a 3-dimensional vector of angular momentum operators on ion i, and S ^ i {displaystyle {hat {mathbf {S} }}_{i}} is a 3-dimensional spin operator of the same form: where J {displaystyle J} is the exchange integral, with ϕ n ( r − R ) {displaystyle phi _{n}(mathbf {r} -mathbf {R} )} the ground orbital wavefunction of the ion at R {displaystyle mathbf {R} } , etc. If the ground state is non-degenerate, then the matrix elements of L {displaystyle mathbf {L} } are purely imaginary, and we can write δ E {displaystyle delta E} out as In an actual crystal, symmetries of neighboring ions dictate the magnitude and direction of the vector D i j {displaystyle mathbf {D} _{ij}} . Considering the coupling of ions 1 and 2 at locations A {displaystyle A} and B {displaystyle B} , with the point bisecting A B {displaystyle AB} denoted C {displaystyle C} , The following rules may be obtained : The orientation of the vector D i j {displaystyle mathbf {D} _{ij}} is constrained by symmetry, as discussed already in Moriya’s original publication. Considering the case that the magnetic interaction between two neighboring ions is transferred via a single third ion (ligand) by the superexchange mechanism (see Figure), the orientation of D i j {displaystyle mathbf {D} _{ij}} is obtained by the simple relation D i j ∝ r i × r j = r i j × x {displaystyle mathbf {D} _{ij}propto mathbf {r} _{i} imes mathbf {r} _{j}=mathbf {r} _{ij} imes mathbf {x} } . This implies that D i j {displaystyle mathbf {D} _{ij}} is oriented perpendicular to the triangle spanned by the involved three ions. D i j = 0 {displaystyle mathbf {D} _{ij}=0} if the three ions are in line. 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. 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 (Al2O3), which displays the R3c trigonal space group. The structure also contains the same unit cell as α-Fe2O3 and α-Cr2O3 which possess D63d 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.