Rare-earth/transition-metal magnets at finite temperature: Self-interaction-corrected relativistic density functional theory in the disordered local moment picture

Atomic-scale computational modeling of technologically relevant permanent magnetic materials faces two key challenges. First, a material's magnetic properties depend sensitively on temperature, so the calculations must account for thermally induced magnetic disorder. Second, the most widely used permanent magnets are based on rare-earth elements, whose highly localized $4f$ electrons are poorly described by standard electronic structure methods. Here, we take two established theories, the disordered local moment picture of thermally induced magnetic disorder and self-interaction-corrected density functional theory, and devise a computational framework to overcome these challenges. Using this approach, we calculate magnetic moments and Curie temperatures of the rare-earth cobalt (${\mathrm{RECo}}_5$) family for RE = Y-Lu. The calculations correctly reproduce the experimentally measured trends across the series and confirm that, apart from the hypothetical compound ${\mathrm{EuCo}}_5, {\mathrm{SmCo}}_5$ has the strongest magnetic properties at high temperature. An order-parameter analysis demonstrates that varying the RE has a surprisingly strong effect on the Co-Co magnetic interactions determining the Curie temperature, even when the lattice parameters are kept fixed. We propose the origin of this behavior is a small contribution to the density from $f$-character electrons located close to the Fermi level.

Figure 1

Schematic of the ${\mathrm{RECo}}_5$ crystal structure, showing the RE (yellow) and Co sites (gray). The $2c$ sites (referred to as ${\mathrm{Co}}_{\mathrm{I}}$ in the text) are in plane with the RE, and the $3g$ sites (${\mathrm{Co}}_{\mathrm{II}}$) lie above and below. An isosurface plot representing the Sm-$4f$ charge density, obtained as the sum of the squared spherical harmonics with $l=3,m=3,2,...,-1$ is also shown.

Figure 2

Scheme to correct RE-$4f$ states based on Hund's rules. Each triangle corresponds to a single spin and orbital angular momentum channel, i.e., $\sigma ,l(=3),m$. LSIC channels with $\sigma =\uparrow$ ($\downarrow$) are represented by yellow (blue) triangles. We also show the moments obtained simply by adding the expectation values of the spin and orbital operators acting on the individual corrected states.

Figure 3

Magnetic moments calculated at zero temperature with (squares) and without (crosses) the LSIC applied, compared to experimental values reported in Refs. [10] (circles) and [13] (stars). Faint symbols were calculated to be energetically unfavorable. The gray horizontal line at $8.78{\mu }_{\mathrm{B}}$ corresponds to the calculated ${\mathrm{YCo}}_5$ moment.

Figure 4

Magnetization per formula unit calculated at different temperatures for the (a) light and (b) heavy ${\mathrm{RECo}}_5$ compounds. Calculations were performed at the ${\mathrm{GdCo}}_5$ lattice parameters.

Figure 5

RE order parameters $m_{\mathrm{RE}}$ [Eq. (8)] from the calculations of Fig. 4 on the light (a) and heavy (b) ${\mathrm{RECo}}_5$ compounds.

Figure 6

(a) $T_{\mathrm{C}}$ calculated for ${\mathrm{RECo}}_5$ using the ${\mathrm{GdCo}}_5$ lattice parameters, compared to the experimental values reported in Ref. [10] (circles). Squares and crosses are calculated with and without the LSIC, respectively, and the faint square is the energetically unfavorable LSIC calculation for ${\mathrm{CeCo}}_5$. (b) Comparison of $T_{\mathrm{C}}$ calculated for ${\mathrm{GdCo}}_5$ using ${\mathrm{RECo}}_5$ lattice parameters (circles), ${\mathrm{RECo}}_5$ using ${\mathrm{GdCo}}_5$ lattice parameters (squares), and ${\mathrm{RECo}}_5$ using ${\mathrm{RECo}}_5$ lattice parameters (crosses). The faint gray lines separate light and heavy REs.

Figure 7

Different $J_{XY}$ parameters [cf. Eq. (17)] calculated for ${\mathrm{RECo}}_5$ on the ${\mathrm{GdCo}}_5$ lattice. Note that the Ce calculation was performed without the LSIC, i.e., assuming that the Ce $f$ electron is itinerant. We highlight $J_{XY}$ for ${\mathrm{YCo}}_5$ as crosses with horizontal dashed lines.

Figure 8

Schematic of antiferromagnetic RE-Co interaction, after Fig. 2 of Ref. [69]. Wide/narrow rectangles symbolize strong/weak RE-Co hybridization in a given spin channel.

Figure 9

DOS calculated just below $T_{\mathrm{C}}$ for ${\mathrm{SmCo}}_5$, resolved onto the Sm, ${\mathrm{Co}}_{\mathrm{I}}$, and ${\mathrm{Co}}_{\mathrm{II}}$ sublattices, shown (a) across a wide energy scale and (b) around the Fermi energy. The vertical dashed line intersects the center of the unoccupied $4f$ peak above the Fermi energy. (c) $T_{\mathrm{C}}$ plotted against the center of this unoccupied peak for the different ${\mathrm{RECo}}_5$ compounds. Note that here the value of $T_{\mathrm{C}}$ of ${\mathrm{CeCo}}_5$ was calculated with the LSIC applied.

Figure 10

Spin-resolved, zero-temperature DOS for ${\mathrm{SmCo}}_5$ (left) and ${\mathrm{DyCo}}_5$ (right), at energies corresponding to the majority-spin, SI-corrected states. The majority- and minority-spin contributions are plotted with positive and negative signs, respectively, and the lower plots zoom in on the minority contribution. Note the larger scale for ${\mathrm{DyCo}}_5$.

Figure 11

Magnetic moments calculated at zero temperature for ${\mathrm{RECo}}_5$ using ${\mathrm{GdCo}}_5$ lattice parameters (squares), and ${\mathrm{RECo}}_5$ using ${\mathrm{RECo}}_5$ lattice parameters (stars).