Effect of the valence state on the band magnetocrystalline anisotropy in two-dimensional rare-earth/noble-metal compounds

In intermetallic compounds with zero orbital momentum $(L=0)$ the magnetic anisotropy and the electronic band structure are interconnected. Here, we investigate this connection in divalent Eu and trivalent Gd intermetallic compounds. We find by x-ray magnetic circular dichroism an out-of-plane easy magnetization axis in two-dimensional atom-thick ${\mathrm{EuAu}}_2$. Angle-resolved photoemission spectroscopy and density-functional theory prove that this is due to strong $f\text{−}d$ band hybridization and ${\mathrm{Eu}}^{2+}$ valence. In contrast, the easy in-plane magnetization of the structurally equivalent ${\mathrm{GdAu}}_2$ is ruled by spin-orbit-split $d$ bands, notably Weyl nodal lines, occupied in the ${\mathrm{Gd}}^{3+}$ state. Regardless of the $L$ value, we predict a similar itinerant electron contribution to the anisotropy of analogous compounds.

Figure 1

Structure, Eu valence, and magnetic properties of ${\mathrm{EuAu}}_2$. (a) STM image ($U=-2$ V, $I=200$ pA). Inset: LEED pattern at $E$ = 40 eV energy; the red circles mark the Au(111) substrate spots prior to ${\mathrm{EuAu}}_2$ formation. (b) Resonant photoemission across the $4d\to 4f$ absorption edge determining the pure monolayer formation of divalent Eu atoms. The final state multiplets are taken from [50, 51]. (c) X-ray absorption at the Eu $M_{4,5}$ edge with left (${\mu }_{-}$) and right (${\mu }_{+}$) circularly polarized light at an applied field of ${\mu }_{0}H$ = 6T and at $T$ = 3 K in normal incidence geometry ($\theta$ = 0). The difference spectrum (XMCD) is shown for normal and grazing ($\theta ={70}^{\circ }$) incidences. (d) Magnetization curves taken at the Eu $M_5$ edge in these two geometries; the inset shows the corresponding Arrott plot analysis to determine $T_C$.

Figure 2

Band structures of ${\mathrm{GdAu}}_2$ and ${\mathrm{EuAu}}_2$. (a) Model of the $R{\mathrm{Au}}_2$ ML unit cell. (b) Path in the reciprocal space of the band structures measured by ARPES at photon energy $h\nu =21.2$ eV for (d) ${\mathrm{GdAu}}_2$ and (e) ${\mathrm{EuAu}}_2$. (c) Band structure from DFT calculations along the indicated high-symmetry direction for a ${\mathrm{EuAu}}_2$ monolayer, both freestanding (left) and on a 3 ML Au(111) slab (right). The Eu $4f$ contribution to the states is indicated by a color scale.

Figure 3

Band structure and MAE of ${\mathrm{GdAu}}_2$ as a function of band filling. (a) Orbital and spin characters of freestanding ${\mathrm{GdAu}}_2$ bands in the absence of SOC. Band structure of (b) a freestanding and (c) supported ${\mathrm{GdAu}}_2$ ML for spins aligned in the $X$ (violet) and $Z$ (green) directions in the force theorem approximation, with each calculation referred to its own Fermi level. The SOC-induced gaps at the band crossings relevant to the MAE are indicated with circles labeled $\alpha ,{\alpha }^{\prime }$, and $\beta$. The right-hand axes show the corresponding band filling, i.e., the number of electrons $N_e$ referred to charge neutrality. (d) MAE as a function of $N_e$ for freestanding (solid line) and supported (dashed line) ${\mathrm{GdAu}}_2$. Positive (negative) MAE values mean an OOP (IP) easy magnetization axis.

Figure 4

MAE of ${\mathrm{EuAu}}_2$ as a function of band filling. (a) MAE as a function of $N_e$ for freestanding (solid blue line) and supported (dashed red line) ${\mathrm{EuAu}}_2$ calculated with $U=5.5$ eV close to the neutrality condition $N_e=0$. Positive (negative) MAE values contribute to the OOP (IP) the IP easy axis. Additional solid curves show the results for ${\mathrm{EuAu}}_2$/3 ML Au(111) with other $U$ values. (b) Same curves in a wider range of $N_e$ to show the large contributions of $4f$ electrons, switching the easy axis of magnetization from IP (negative MAE) to OOP (positive MAE). The rectangle corresponds to the close-up shown in (a).