Orbital torques and orbital pumping in two-dimensional rare-earth dichalcogenides

Abstract

The design of spin-orbit torque properties in two-dimensional (2D) materials presents one of the challenges of modern spintronics. In this context, 2D layers involving rare-earth ions $-$ which give rise to robust magnetism, exhibit pronounced orbital polarization of the states, and carry strong spin-orbit interaction $-$ hold particular promise. Here, we investigate ferromagnetic Janus H-phase monolayers of 4$f$-Eu rare-earth dichalcogenides EuSP, EuSSe, and EuSCl using first-principles calculations. We demonstrate that all compounds exhibit significant spin-orbit torques which originate predominantly in the colossal current-induced orbital response on the Eu $f$-electrons. Moreover, we demonstrate that the corresponding orbital torques can be used to drive strong in-plane currents of orbital angular momentum with non-trivial direction of orbital polarization. Our findings promote $f$-orbital-based 2D materials as a promising platform for in-plane orbital pumping and spin-orbit torque applications, and motivate further research on educated design of orbital properties for orbitronics with 2D materials.

Figures (4)

• Figure 1: Orbital torque and orbital pumping in 2D Janus dichalcogenides. (a) In 2D magnetic dichalcogenides (magnetization out of the plane along $z$, indicated with a black arrow), application of an electric field $E_x$ along $x$ (red arrow) leads to generation of a current of orbital angular momentum $Q^L$ along $y$, polarized along $z$ (small red arrows and yellow circular arrows) via the mechanism of the orbital Hall effect (OHE). (b) At the same time, due to broken $z$-reflection symmetry in Janus geometry, the electric field will give rise to a current-induced orbital polarization $L_x$ (green arrow, as well as small red arrows and yellow circular arrows), which mediates an orbital torque on the magnetization along $y$ (depicted with fading arrows). (c) In a reciprocal process, perturbing the magnetization with a torque $T_y$ along $y$ (grey arrow) will result in a generation of an orbital current along $x$ polarized along $y$ and $z$ (shown with small yellow and red arrows), which constitutes the effect of in-plane orbital pumping. (d) For rare-earth dichalcogenides, the effect of orbital pumping is equivalent to the effect of orbital-to-orbital-current conversion, where the orbital current arises in response to an increasing linearly in time orbital field $B_L(t)$ applied along $x$ (pink arrow). While the OHE can be used to generate orbital currents in 2D materials, the effect of orbital pumping mediated by orbital torque can be used to achieve orbital current generation by magnetization dynamics in 2D geometry.
• Figure 2: Electronic structure of EuSX. (a) Top and side views of the Janus H-phase EuSX monolayers (X = P, Se, and Cl) are shown. The dark blue balls represent Eu atoms, and the yellow/red balls represent S and X atoms, respectively, with the axes shown with black arrows. The first Brillouin zone is shown at the bottom. Right: (b-d) The band structures and the corresponding spin-resolved density of states (DOS) are shown. The left and right parts of the DOS correspond to the majority and minority spin, respectively. The DOS of Eu-$f$, Eu-$d$, S-$p$ and X-$p$ states is shown with dark red, green, blue and pink, respectively.
• Figure 3: Orbital torque and orbital pumping in EuSX. (a-c) Band-filling dependence of the $y$-component of spin-orbital ($\langle T_{\rm SO}^{\mathbf{S}}\rangle$) and exchange ($\langle T_{\rm XC}^{\mathbf{S}}\rangle$) torque normalized to the strength of an electric field $E_x$ applied along $x$. The band-filling of the $x$-component of the normalized current-induced OAM ($\langle L_x\rangle$, d-e) and spin ($\langle S_x\rangle$, g-h), resolved into Eu and SX contributions, is shown for comparison. In (j-l), the distribution of orbital currents arising in response to the exchange torque along $y$, as given by tensor components $t^{L_z}_{xy}$ and $t^{L_y}_{xy}$ defined by Eq. \ref{['t']}, is shown. In (m-o), the respective tensors are shown for the case of spin current. The band filling dependence of the orbital-to-orbital-current conversion strength, as given by tensors $\tau_{x}^{xz}$ and $\tau_{x}^{xy}$ defined by Eq. \ref{['J']}, is shown for comparison in (p-r). The case of EuSP, EuSSe, and EuSCl corresponds to the left, middle, and right columns, respectively.
• Figure 4: Anatomy of orbital torques and orbital pumping in $k$-space. For each compound, the $k$-resolved distribution is shown for (a-c) exchange torque $\langle T_{\rm XC}^\mathbf{S}\rangle$ normalized to the field $E_x$, (d-f) current induced orbital moment $\langle L_x\rangle$ normalized to the field $E_x$, and pumped by the torque along $y$ orbital current density along $x$ polarized along $y$ ($t^{L_y}_{xy}$, g-i), and along $z$ ($t^{L_z}_{xy}$, j-l). For all cases, the colored circles represent the expectation value of the plotted quantity with the color code shown on the left.