Jahn-Teller driven perpendicular magnetocrystalline anisotropy in metastable ruthenium

A metastable phase of body-centered-tetragonal ruthenium (bct Ru) is identified to exhibit a large perpendicular magnetocrystalline anisotropy (PMCA), whose energy $E_{\text{MCA}}$ is as large as $150 \mu \mathrm{eV}/\mathrm{atom}$, which is two orders of magnitude greater than those of $3d$ magnetic metals. Further investigation over the range of tetragonal distortion suggests that the appearance of magnetism in the bct Ru is governed by the Jahn-Teller spit $e_g$ orbitals. Moreover, from band analysis, MCA is mainly determined by an interplay between two $e_g$ states, $d_{x^2-y^2}$ and $d_{z^2}$ states, as a result of level reversal associated with tetragonal distortion.

Figure 1

Schematic presentation of the Jahn-Teller splitting of $d$ electrons. In the cubic symmetry, such as bcc $\left(c/a=1\right)$, the $d$ orbital splits into doublets $\left(e_g\right)$ and triplets $\left(t_{2g}\right)$. Tetragonal distortion further splits $e_g$ into $a_1$ and $b_1$; $t_{2g}$ splits into a singlet $b_2$ and a doublet $e$, where their relative order is shown depending on whether $c/a$ is greater or smaller than unity.

Figure 2

(a) Total energy with respect to the fcc structure $\left(c/a=1.41\right)$ of nonmagnetic bct Ru upon the tetragonal distortion $\left(c/a\right)$ in fixed volume of the bcc structure. The equilibria $c/a$ for bct, bcc, and fcc are denoted. (b) $N\left(E_F\right)$ of non-spin-polarized calculations (red squares), and the magnetic moment of bcc Ru as a function of the uniform lattice constant $a$ (black circles). The arrow denotes the equilibrium lattice constant of bcc Ru. (c) The energy difference $\Delta E=E_{\text{NM}}-E_{\text{FM}}$ (red dotted line) and magnetic moments (black solid line) as a function of $c/a$. The tetragonal distortion is classified into two regions, $A$ and $B$, by $c/a<1$ or $>1$. (d) $N\left(E_F\right)$ of NM bct Ru as a function of $c/a$. Total $N\left(E_F\right)$, those from $d_{z^2},d_{x^2-y^2}$, and the absolute value of the difference of the two $e_g$ orbitals, denoted as $\Delta e_g$, are shown as black solid circles, a black dotted line, a red dashed line, and a blue solid line, respectively.

Figure 3

Orbital-decomposed DOS of the $d$ orbital for spin-polarized calculations of bct Ru at $c/a=$ (a) 0.84, (b) 0.90, (c) 0.96, and (d) 1.06, respectively. The $d$-orbital states are shown in different colors: red ($d_{z^2})$, black $\left(d_{x^2-y^2}\right)$, blue $\left(d_{xy}\right)$, and green $\left(d_{xz,yz}\right)$, respectively.

Figure 4

(a) MCA energy dependence on $c/a$ for bct Ru, where $A$ and $B$ are defined as in Fig. 2. (b) Spin channel decomposed and total $E_{\text{MCA}}$ of bct Ru for various $c/a$. The black circles denote total MCA. The upper (lower) triangles denote the $\uparrow \uparrow \left(\downarrow \downarrow \right)$ channel, and squares denote the $\uparrow \downarrow$ channel.

Figure 5

Band structures of bct Ru for $c/a=0.84$, 0.90, 0.96, and 1.06 for the majority and minority spin states. The $d$-orbital states are shown in different colors: red $\left(d_{z^2}\right)$, black $\left(d_{x^2-y^2}\right)$, blue $\left(d_{xy}\right)$, orange $\left(d_{xz}\right)$, and green $\left(d_{yz}\right)$, respectively.