Exchange and magnetic anisotropy energies

Borovik-Romanov, A. S.; Grimmer, H.

doi:10.1107/97809553602060000632

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 1.5, pp. 118-120

Section 1.5.3.2. Exchange and magnetic anisotropy energies

A. S. Borovik-Romanov^a ^‡ and H. Grimmer^b ^*

^a P. L. Kapitza Institute for Physical Problems, Russian Academy of Sciences, Kosygin Street 2, 119334 Moscow, Russia, and ^bLabor für Neutronenstreuung, ETH Zurich, and Paul Scherrer Institute, CH-5234 Villigen PSI, Switzerland
Correspondence e-mail: hans.grimmer@psi.ch

1.5.3.2. Exchange and magnetic anisotropy energies

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It is pertinent to compare the different kinds of interactions that are responsible for magnetic ordering. In general, all these interactions are much smaller than the electrostatic interactions between the atoms that determine the chemical bonds in the material. Therefore, if a crystal undergoes a transition into a magnetically ordered state, the deformations of the crystal that give rise to the change of its crystallographic symmetry are comparatively small. It means that most of the non-magnetic properties do not change drastically. As an example, the anisotropic deformation of the crystal that accompanies the transition into the ordered state (see Section 1.5.9.1) is mostly not larger than 10⁻⁴.

The formation of the ordered magnetic structures is due mainly to the exchange interaction between the spins $[{\bf S}_{\alpha}]$ (and corresponding magnetic moments $[{\boldmu}]$ of the atoms or ions). The expression for the exchange energy can contain the following terms [see formula (1.5.1.7)]: $[{\bf S}_{\alpha}{\bf S}_{\beta}, \ {\bf S}_{\alpha}[{\bf S}_{\beta}{\bf S}_{\gamma}]. \eqno(1.5.3.4)]$ The exchange interaction decreases rapidly as the distance between the atoms rises. Thus, it is usually sufficient to consider the interaction only between nearest neighbours. The exchange interaction depends only on the relative alignment of the spin moments and does not depend on their alignment relative to the crystal lattice. Therefore, being responsible for the magnetic ordering in the crystal, it cannot define the direction of the spontaneous magnetization in ferromagnets or of the antiferromagnetic vector. This direction is determined by the spin–orbit and magnetic spin–spin interactions, which are often called relativistic interactions as they are small, of the order of [v^2/c^2] , where v is the velocity of atomic electrons and c is the speed of light. The relativistic interactions are responsible for the magnetic anisotropy energy, which depends on the direction of the magnetic moments of the ions with regard to the crystal lattice. The value of the exchange energy can be represented by the effective exchange field H_e. For an ordered magnetic with a transition temperature of 100 K, H_e ≃ 1000 kOe. Thus the external magnetic field hardly changes the value of the magnetization $[\bf M]$ or of the antiferromagnetic vector $[\bf L]$ ; they are conserved quantities to a good approximation. The effective anisotropy field [H_a] in cubic crystals is very small: 1–10 Oe. In most non-cubic materials, [H_a] is not larger than 1–10 kOe. This means that by applying an external magnetic field we can change only the direction of $[\bf M]$ , or sometimes of $[\bf L]$ , but not their magnitudes.

The magnetic anisotropy energy [U_a] can be represented as an expansion in the powers of the components of the vectors $[\bf M]$ or $[\bf L]$ . The dependence of [U_a] on the direction of the magnetization is essential. Therefore, one usually considers the expansion of the spontaneous magnetization or antiferromagnetic vector in powers of the unit vector $[\bf n]$ . The anisotropy energy is invariant under time reversal. Therefore, the general expression for this energy has the form $[U_a = K_{ij}n_in_j + K_{ijk\ell}n_i n_j n_k n_\ell + K_{ijk\ell mn}n_in_{j}n_kn_\ell n_m n_n, \eqno(1.5.3.5)]$ where $[K_{ij}]$ , $[K_{ijk\ell}]$ , $[K_{ijk\ell mn}]$ are tensors, the components of which have the dimension of an energy density. The forms of the tensors depend on the symmetry of the crystal. There are at most two independent components in $[K_{ij}]$ . For a uniaxial crystal, the second-order term in the anisotropy energy expansion is determined by one anisotropy constant, K. Instead of using the components of the unit vector $[\bf n]$ , its direction can be described by two angles: polar $[\theta]$ and azimuthal $[\varphi]$ . Correspondingly, the anisotropy energy for a uniaxial crystal can be written as $[U_a = K(n^2_x + n^2_y) = K\sin ^2\theta. \eqno(1.5.3.6)]$ This relation is equivalent to $[U_a = K(1-n^2_z) = K-K\cos ^2\theta. \eqno(1.5.3.7)]$

The direction of the magnetization vector $[\bf M]$ in a ferromagnet or of the antiferromagnetic vector $[\bf L]$ in an antiferromagnet is called the direction or the axis of easy magnetization. The crystals in which this axis is aligned with a threefold, fourfold or sixfold axis of the magnetic point group are called easy-axis magnetics. The magnetic crystals with the main axis higher than twofold in the paramagnetic state in which, in the ordered state, $[\bf L]$ (or $[\bf M]$ ) is perpendicular to this axis are often called easy-plane magnetics. The anisotropy in this plane is usually extremely small. In this case, the crystal possesses more then one axis of easy magnetization and the crystal is usually in a multidomain state (see Section 1.5.4).

If the anisotropy constant K is positive, then the vector $[\bf n]$ is aligned along the z axis, and such a magnetic is an easy-axis one. For an easy-plane magnetic, K is negative. It is convenient to use equation (1.5.3.6) for easy-axis magnetics and equation (1.5.3.7) for easy-plane magnetics. In the latter case, the quantity K is included in the isotropic part of the thermodynamic potential $[\Phi]$ , and (1.5.3.7) becomes $[U_a=-K\cos^2\theta]$ . Instead we shall write $[U_a=K\cos^2\theta]$ in the following, so that K becomes positive for easy-plane ferromagnetics as well.

Apart from the second-order term, terms of higher order must be taken into account. For tetragonal crystals, the symmetry allows the following invariant terms in the anisotropy energy: $[\eqalignno{U_a(4) &= K_1(n^2_x + n^2_y) + K_2(n^2_x + n^2_y)^2 + K_{xxyy}n^2_x n^2_y &\cr &= K_1\sin^2\theta + K_2\sin^4\theta + K_{\perp}\sin^4\theta\sin^2 2\varphi\semi&\cr &&(1.5.3.8)}]$ the azimuthal angle $[\varphi]$ is measured from the twofold axis x in the basal plane and the constant $[K_{\perp}]$ determines the anisotropy in the basal plane.

Trigonal symmetry also allows second- and fourth-order invariants: $[\eqalignno{U_a(3) &= K_1(n^2_x + n^2_y) + K_2(n^2_x + n^2_y)^2 &\cr&\quad + K'_{\perp}{\textstyle{1\over 2}}n_z[(n_x + in_y)^3 + (n_x-in_y)^3]&\cr&= K_1\sin^2\theta + K_2\sin^4\theta + K'_{\perp}\cos\theta \sin^3\theta \cos 3\varphi, &\cr&&(1.5.3.9)}]$ where $[\varphi]$ is measured from the x axis, which is chosen parallel to one of the twofold axes. For easy-plane magnetics and $[K'_{\perp}\,\gt\,0]$ , the vector $[\bf n]$ is directed along one of the twofold axes in the basal plane. If $[K'_{\perp}]$ is negative, then $[\bf n]$ lies in a vertical mirror plane directed at a small angle to the basal plane. For the complete solution of this problem, the sixth-order term must be taken into account. This term is similar to the one that characterizes the anisotropy of hexagonal crystals. The expression for the latter is of the following form: $[\eqalignno{U_a(6) &= K_1(n^2_x + n^2_y) + K_2(n^2_x + n^2_y)^2 &\cr&\quad+ K''_{\perp}\, {\textstyle{1\over 2}}[(n_x + in_y)^6 + (n_x-in_y)^6]&\cr&= K_1\sin^2\theta + K_2\sin^4\theta + K''_{\perp} \sin^6\theta \cos 6\varphi, &\cr&&(1.5.3.10)}]$ where x and $[\varphi]$ have the same meaning as in (1.5.3.9).

The symmetry of cubic crystals does not allow any second-order terms in the expansion of the anisotropy energy. The expression for the anisotropy energy of cubic crystals contains the following invariants: $[{U_a}({\rm cub}) = K_1(n^2_x n^2_y + n^2_x n^2_z + n^2_y n^2_z) + K_2 n^2_x n^2_y n^2_z. \eqno(1.5.3.11)]$

In considering the anisotropy energy, one has to take into account spontaneous magnetostriction and magnetoelastic energy (see Section 1.5.9). This is especially important in cubic crystals. Any collinear cubic magnetic (being brought into a single domain state ) ceases to possess cubic crystallochemical symmetry as a result of spontaneous magnetostriction. If [K_1] is positive, the easy axis is aligned along one of the edges of the cube and the crystal becomes tetragonal (like Fe). If [K_1] is negative, the crystal becomes rhombohedral and can be an easy-axis magnetic with vector $[\bf n]$ parallel to one of the spatial diagonals (like Ni) or an easy-plane magnetic with $[\bf n]$ perpendicular to a spatial diagonal. We shall discuss this topic in more detail in Section 1.5.9.3.

The considerations presented above can be applied to all crystals belonging in the paramagnetic state to the tetragonal, trigonal or hexagonal system that become easy-plane magnetics in the ordered state. All of them, including the cubic crystals, may possess more than one allowed direction of easy magnetization. In the example considered in the previous section, these directions can be aligned along the three twofold axes for the structures [A_1^x, A_2^x, A_3^x, F^x] and can be parallel to the three mirror planes for [A_1^y, A_2^y, A_3^y, F^y] .

It is worth noting that in some applications it is more convenient to use an expansion of the anisotropy energy in terms of surface spherical harmonics. This problem has been considered in detail by Birss (1964).

References

Birss, R. R. (1964). Symmetry and magnetism. Amsterdam: North-Holland.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 1.5, pp. 118-120