Quantitative microscopic theory of the temperature-dependent cubic magnetocrystalline anisotropy constants $K_1$ and $K_2$ of iron and nickel

The magnetocrystalline anisotropy of the cubic ferromagnetic metals iron and nickel is conventionally described by constants $K_1$ and $K_2$ in an expansion of the free energy in the direction cosines of the magnetization relative to the cube axes. Van Vleck (1937) traced the origin of this anisotropy to the interplay of anisotropic inter-atomic electrostatic coupling of the (largely quenched) orbital angular momenta with the spin-orbit interaction, obtaining the right order of magnitude, but his coupling constants entered phenomenologically and the temperature dependence was computed only in a Weiss molecular-field approximation. He could not account for the observed steep vanishing of $K_1$ in iron near the Curie point, for the strong low-temperature growth and sign change of $K_1$ in nickel, or for the magnitude of $K_2$, which experiments find comparable to $K_1$ although simple estimates make it $10^2$ to $10^3$ times smaller. Later literature settled the localized-moment version of the temperature-dependence problem (the Callen-Callen power laws) and achieved parameter-free zero-temperature anisotropy calculations for some metals, but nickel remains difficult for density-functional theory even at zero temperature; no first-principles calculation of the full $K_1(T)$ and $K_2(T)$ of fcc nickel has been reported; and finite-temperature ab initio methods have been demonstrated on bcc iron and on L1$_0$-ordered alloys (layered intermetallic compounds such as FePt) rather than on nickel. The open problem is a quantitative, material-specific, parameter-free theory of $K_1(T)$ and $K_2(T)$ for bcc iron and fcc nickel over the whole temperature range up to the Curie point.

Background

Magnetocrystalline anisotropy. In a ferromagnet the internal energy depends weakly on the direction in which the magnetization points relative to the crystal lattice. For a cubic crystal this energy is written as an expansion in the direction cosines $\alpha_1, \alpha_2, \alpha_3$ of the magnetization along the three cube axes,

$$F = K_0 + K_1(\alpha_1^2\alpha_2^2 + \alpha_2^2\alpha_3^2 + \alpha_3^2\alpha_1^2) + K_2\,\alpha_1^2\alpha_2^2\alpha_3^2 + \cdots,$$

with $K_1$ and $K_2$ the first and second cubic anisotropy constants. In iron (body-centered cubic lattice, positive $K_1$) the energetically preferred (easy) axes are the cube edges $\langle 100\rangle$; in nickel (face-centered cubic, negative $K_1$ at low temperature) they are the body diagonals $\langle 111\rangle$. These constants, of order $10^3$ to $10^5$ J/m$^3$, are central quantities of ferromagnetism: together with the exchange stiffness they set domain-wall widths and coercivity, and hence the usefulness of a material as a soft or a permanent magnet.

Physical origin. The exchange interaction that aligns neighboring spins is nearly rotation-invariant, so the directional dependence arises only through much weaker relativistic effects: the spin-orbit coupling (the interaction linking an electron's spin magnetic moment to its orbital motion) acting on orbital charge distributions tied to the lattice. In the 3d transition metals the orbital angular momentum is largely "quenched", i.e. averaged out by the electric field of the surrounding ions, which is why the anisotropy energy is only about $10^{-4}$ to $10^{-5}$ of the exchange energy per atom. Van Vleck's 1937 theory identified the dominant mechanism: anisotropic inter-atomic electrostatic coupling of the quenched orbital momenta ("orbital valence" in his terminology), acting in second order through the spin-orbit coupling, generates effective pseudo-dipole-dipole and pseudo-quadrupole-quadrupole couplings between the spins, about fifty times stronger than the true magnetic dipole-dipole interaction. The theory explained the observed order of magnitude; for a single-ion mechanism the spin-orbit parameter must enter to the fourth power to produce the $K_1$ angular dependence and to the sixth power for $K_2$.

Temperature dependence. Experiments going back to Akulov showed that near the Curie temperature $T_c$ (the critical temperature above which spontaneous magnetization vanishes) the anisotropy constant of iron falls off roughly as the tenth power of the magnetization $I$ — far faster than $I$ itself. Two theoretical traditions address this: Zener's classical theory, and the single-ion power laws of Callen and Callen, which for a magnetic ion whose moment is localized on the atom predict $K_l(T) \propto m^{l(l+1)/2}$ at low temperature (where $m(T)$ is the reduced magnetization), i.e. the tenth power of $m$ for the cubic $K_1$ and the twenty-first for $K_2$, crossing over to the $l$-th power near $T_c$. These laws are derived for localized moments; iron and nickel are itinerant (band) ferromagnets, in which the moments are carried by delocalized 3d conduction electrons, and whether the localized-moment framework applies to them is itself part of the problem.

What Van Vleck left open. Van Vleck's paper closed the question of the origin and order of magnitude of cubic anisotropy but explicitly left open: (i) the microscopic determination of his coupling constants, which enter phenomenologically and may themselves depend on temperature; (ii) the replacement of the Weiss molecular-field approximation (a mean-field treatment of the exchange interaction between spins) by a treatment of exchange correlations — his pair-exact calculation changed results little and the spin-wave (Bloch-wave) treatment "does not appear feasible"; (iii) the sign and temperature variation of $K_2$, whose observed magnitude exceeds simple dimensional estimates by factors of $10^2$ to $10^3$; and (iv) the detailed quenching mechanism in conducting ferromagnets, "a subject imperfectly understood at present". (A fifth point — that demagnetization, i.e. shape-dependent stray-field, corrections to measured anisotropy are rigorous only for ellipsoidal specimens — is now textbook material.) The open question is the quantitative, material-specific one: a theory that predicts the actual constants of the actual metals as functions of temperature, rather than their origin and order of magnitude.

Problem Statement

Construct a quantitative, material-specific microscopic theory that predicts the first two cubic magnetocrystalline anisotropy constants $K_1(T)$ and $K_2(T)$ of bcc iron and fcc nickel from the underlying electronic structure, without anisotropy constants fitted to anisotropy data, over the full temperature range $0 < T < T_c$, and that quantitatively reproduces the established experimental behavior: (i) the near-$T_c$ vanishing of $K_1$ in iron, observed to scale roughly as the tenth power of the reduced magnetization; (ii) the strong growth of $|K_1|$ in nickel at low temperature (about fifty times larger at 2 K than at room temperature) and the sign change of $K_1$ near 400 K; and (iii) the magnitude and sign of $K_2$ in both metals, which experiment finds comparable to $K_1$ although simple dimensional estimates make it $10^2$ to $10^3$ times smaller. Comparisons of the predicted curves with torque or magnetization measurements on single crystals require the standard demagnetization (shape-dependent stray-field) and magnetoelastic (magnetostriction-related) corrections.

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