Anisotropy and Crystal Field

Abstract

Magnetic anisotropy, imposed through crystal-field and magnetostatic interactions, is one of the most iconic, scientifically interesting, and practically important properties of condensed matter. This article starts with the phenomenology of anisotropy, distinguishing between crystals of cubic, tetragonal, hexagonal, trigonal, and lower symmetries and between anisotropy contributions of second and higher orders. The atomic origin of magnetocrystalline anisotropy is discussed for several classes of materials, ranging from insulating oxides and rare-earth compounds to iron-series itinerant magnets. A key consideration is the crystal-field interaction of magnetic atoms, which determines, for example, the rare-earth single-ion anisotropy of today’s top-performing permanent magnets. The transmission between crystal field and anisotropy is realized by spin-orbit coupling. An important crystal-field effect is the suppression of the orbital moment by the crystal-field, which is known as quenching and has a Janus-head effect on anisotropy: the crystal field is necessary to create magnetocrystalline anisotropy, but it also limits the anisotropy in many systems. Finally, we discuss some other anisotropy mechanisms, such as shape, magnetoelastic, and exchange anisotropies, and outline how anisotropy is realized in some exemplary compounds and nanostructures.

Appendices

Appendices

Appendix A: Spherical Harmonics

Separating radial (r) and angular (θ, ϕ) degrees of freedom, any function f(θ, ϕ) can be expanded into spherical harmonicsY_l^m(θ, ϕ). The present chapter uses this expansion to describe (i) atomic wave functions ψ(r), as in Figs. 4 and 11, (ii) atomic charge densities n(r), (iii) crystal-field potentialsV(r) and operator equivalents 𝒪_m^l, and (iv) magnetic anisotropy energies E_a(θ, ϕ). These quantities differ by radial part and physical meaning, but their angular dependences are all described by

$$ {Y_{\mathrm{l}}}^{\mathrm{m}}\!\left(\theta, \phi \right)={{\mathcal{N}}_{\mathrm{l}}}^{\mathrm{m}}\exp \left(\mathrm{i} m\phi \right){P_{\mathrm{l}}}^{\mathrm{m}}\!\left(\cos \theta \right) $$

(72)

where the P_l^m are the the associated Legendre polynomials. Concerning sign and magnitude of the normalization factor 𝒩_l^m, we use the convention

$$ {{\mathcal{N}}_{\mathrm{l}}}^{\mathrm{m}}=\sqrt{\frac{\left(2l+1\right)}{4\uppi}\frac{\left(l-m\right)!}{\left(l+m\right)!}} $$

(73)

It is sometimes useful to express Eq. (1) in terms of Cartesian coordinates or “direction cosines” x, y, and z. Last but not least, the complex functions exp.(imϕ) may be replaced by real functions, using exp.(±imϕ) = cos (mϕ) ± i sin(mϕ). These real spherical harmonics, also known as tesseral harmonics, are often convenient, because charge densities, crystal-field potentials, and anisotropy energies are real by definition. However, the distinction remains important in quantum mechanics, because complex and real spherical harmonics correspond to unquenched and quenched wave functions, respectively.

A very frequently occurring function is

$$ {Y_2}^0=\frac{1}{2}\sqrt{\frac{5}{4\pi }}\left(3\ {\cos}^2\theta \hbox{--} 1\right) $$

(74a)

or

$$ {Y_2}^0=\frac{1}{2}\sqrt{\frac{5}{4\pi }}\frac{3{z}^2-{r}^2}{r^2} $$

(74b)

Note that the Cartesian coordinates require a factor 1/r^l, which ensures that the Y_l^m are dimensionless and that the expansion is in terms of direction cosines x/r, y/r, and z/r. Up to the sixth order, there are Table 16 lists real and complex spherical harmonics up to the sixth order.

Table 16 Spherical harmonics in several representations. For m ≠ 0, the real representation requires an additional factor $1/\sqrt{2}$, because the normalization behavior of cos(mϕ) ± sin(mϕ) differs from that of exp.(imϕ). Furthermore, the minus sign in 𝒩_l^m is not used for m ≠ 0. The following formulae can be used to extract the full spherical harmonics from the table: Y_l^m = π^-1/2f_Nf_P exp.(imϕ), Y_l^m = π^-1/2f_Nf_R/r^l (m = 0), and Y_l^m = (2π)^-1/2f_Nf_R/r^l (m ≠ 0). Anisotropy energies involve even-order spherical harmonics only (gray rows)

Appendix B: Point Groups

Table 17 Less common space and point groups. The space groups in bold characters are frequently encountered in magnetism and separately considered in the main text of the chapter (Table 1)

Appendix C: Hydrogen-Like Atomic 3d Wave Functions

Hydrogen-like 3d wave functions are obtained by solving the Schrödinger equation for n = 3 (third shell) and l = 2 (d electrons). There are 2 l + 1 = 5 different orbitals, and each can be occupied by up to two electrons. Explicitly,

$$ {\mid} xy{>}={R}_{3\mathrm{d}}\!\left(\mathrm{r}\right)\!{\sin}^2\theta\ \sin 2\phi $$

(75)

$$ {\mid} {x}^2-{y}^2{>}=\mathcal{N}\;{R}_{3\mathrm{d}}\!\left(\mathrm{r}\right){\sin}^2\theta\ \cos 2\phi $$

(76)

$$ {\mid} xz{>}=2\mathcal{N}\;{R}_{3\mathrm{d}}\!\left(\mathrm{r}\right)\sin \theta \cos \theta\ \cos \phi $$

(77)

$$ {\mid} {z}^2{>}={R}_{3\mathrm{d}}\!\left(\mathrm{r}\right) \left(3\ {\sin}^2\theta -1\right) $$

(78)

$$ {\mid} yz\!>\,=2\mathcal{N}\;{R}_{3\mathrm{d}}\!\left(\mathrm{r}\right)\ \sin \theta\ \cos \theta \sin \phi $$

(79)

where 𝒩 = $\sqrt{15/16\uppi}$, a_o = 0.529 Å, and

$$ {R}_{3d}(r)=\frac{4{Z}^{5/2}{r}^2}{81{a}_o^2\sqrt{30{a}_o^3}}\kern0.5em \exp \left(-\frac{Zr}{a_o}\right) $$

(80)

Aside from the real set of wave functions, there exist complex wave functions of the type exp.(±imϕ). The two sets of wave functions are linear combinations of each other, and both are solutions of the Schrödinger equation. However, they are nonequivalent with respect to orbital moment and magnetic anisotropy.

More generally, Ψ(r, ϕ, θ) = R_n^l(r) Y_l^m(ϕ, θ), where it is convenient to express the radial wave functions in terms of the parameter r_o = a_o/Z:

$$ {R}_{1\mathrm{s}}=\frac{2}{\sqrt{{r_{\mathrm{o}}}^3}}\ \exp \left(-\frac{r}{r_{\mathrm{o}}}\right) $$

$$ {R}_{2\mathrm{s}}=\frac{1}{2\sqrt{2{r_{\mathrm{o}}}^3}}\left(2-\frac{r}{r_{\mathrm{o}}}\right)\ \exp \left(-\frac{r}{2{r}_{\mathrm{o}}}\right) $$

$$ {R}_{2\mathrm{p}}=\frac{1}{2\sqrt{6{r_{\mathrm{o}}}^3}}\frac{r}{r_{\mathrm{o}}}\ \exp \left(-\frac{r}{2{r}_{\mathrm{o}}}\right) $$

$$ {R}_{3\mathrm{s}}=\frac{2}{81\sqrt{3{r_{\mathrm{o}}}^3}}\ \left(27-18\frac{r}{r_{\mathrm{o}}}+2\frac{r2}{{r_{\mathrm{o}}}^2}\right)\ \exp \left(-\frac{r}{3{r}_{\mathrm{o}}}\right) $$

$$ {R}_{3\mathrm{p}}=\frac{4}{81\sqrt{6{r_{\mathrm{o}}}^3}}\ \left(6\frac{r}{r_{\mathrm{o}}}-\frac{r^2}{{r_{\mathrm{o}}}^2}\right)\ \exp \left(-\frac{r}{3{r}_{\mathrm{o}}}\right) $$

$$ {R}_{3\mathrm{d}}=\frac{4}{81\sqrt{30{r_{\mathrm{o}}}^3}}\ \frac{r^2}{{r_{\mathrm{o}}}^2}\ \exp \left(-\frac{r}{3{r}_{\mathrm{o}}}\right) $$

$$ {R}_{4\mathrm{f}}=\frac{1}{768\sqrt{35{r_{\mathrm{o}}}^3}}\ \frac{r^3}{{r_{\mathrm{o}}}^3}\ \exp \left(-\frac{r}{4{r}_{\mathrm{o}}}\right) $$

From the radial wave functions, the following averages are obtained:

$$ {<}{r}^2{>}=\frac{n^2{r_{\mathrm{o}}}^2}{2}\left(5{n}^2+1-3l\left(l+1\right)\right) $$

$$ {<}r{>}=\frac{r_{\mathrm{o}}}{2}\left(3\kern0.5em {\mathrm{n}}^2-l\left(l+1\right)\right) $$

$$ {<}1/r{>}\,=\frac{1}{n^2{r}_{\mathrm{o}}} $$

$$ {<}1/{r}^2{>}=\frac{2}{n^3{r_{\mathrm{o}}}^2\left(2\mathrm{l}+1\right)} $$

$$ {<}1/{r}^3{>}=\frac{2}{n^3{r_{\mathrm{o}}}^3\mathrm{l}\left(\mathrm{l}+1\right)\left(\mathrm{l}+2\right)} $$

These formulae have numerous applications. For example, <r> and the square root of <r²> are used to estimate shell radii, <1/r> gives the electronic energy, and <1/r³> determines the strength of the spin-orbit coupling on which magnetocrystalline anisotropy relies.