Néel vector and Rashba SOC effects on RKKY interaction in 2D $d$-wave altermagnets

TL;DR

This work analyzes the RKKY interaction in two-dimensional $d$-wave altermagnets by independently tuning the Néel-vector orientation and Rashba SOC strength. Using a minimal model H = Dk^2 σ0 − α(z×k)·σ + βD(k_x^2−k_y^2)n·σ and numerically evaluating the real-space Green function, the authors derive a generalized RKKY Hamiltonian with collinear, DM, and frustrated terms. Without SOC, the Ising term alone links directly to the Néel vector, while with SOC, symmetry breaking activates DM components that reveal $n$-orientation; notably, a novel out-of-plane DM term $J_{DM}^z$ emerges only when altermagnetism and SOC coexist. As SOC strength grows, the RKKY spin model evolves through five mechanisms, ranging from altermagnetism-dominated to SOC-dominated, including competition and coincidence cases, with the impurity geometry and Néel-vector orientation crucial in determining the dominant mechanism. These findings offer a route to probe Néel-vector orientation and to engineer noncollinear spin textures in 2D altermagnets, supported by numerical verification and potential experimental implementations via spin-polarized probes or ESR techniques.

Abstract

Altermagnets possess two key features: non-relativistic alternating spin splitting (i.e., altermagnetism) and a material-dependent Néel vector. The former naturally coexists with Rashba spin-orbit coupling (SOC) in real materials on substrates, prompting the question of how SOC affects the magnetic properties of altermagnets. The latter is crucial for information storage, making it essential to determine its orientation. To address these issues, we study the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction in two-dimensional (2D) $d$-wave altermagnets by independently varying the Néel vector orientation and the SOC strength. Our results demonstrate that the Néel vector orientation can be accurately determined via the Ising term without SOC, or qualitatively inferred via the DM terms with SOC. Moreover, we observe a novel Dzyaloshinskii-Moriya (DM) component distinct from previous reports, whose emergence is attributed to the synergy between altermagnetism and SOC. Additionally, through tuning SOC strength, we reveal the evolution of the RKKY spin models governed by five distinct mechanisms: the spin model may be determined solely by altermagnetism, solely by SOC, or solely by the kinetic term; alternatively, altermagnetism may coincidentally yield the same moderately anisotropic spin model as SOC, or compete with SOC to produce a spin model with maximal anisotropy. Beyond SOC strength, which mechanism operates also relies on the Néel vector orientation and impurity configurations. All results are numerically verified. These findings -- which were inaccessible in prior studies due to the limitations of first-order SOC expansion and fixed Néel vector orientation -- provide important new insights into the magnetic properties of altermagnets.

Figures (5)

• Figure 1: (Color online) The Fermi surfaces and the spin expectation $s_i$ (whose value is evaluated by color) in altermagnet for different system parameters: (a) $\alpha=0$, $\mathbf{n}=(0,0,1)$; (b) $\alpha=0.2$${\rm eV}$${\rm \AA}$, $\mathbf{n}=(0,0,1)$. (c) $\alpha=0.2$${\rm eV}$${\rm \AA}$, $\mathbf{n}=(1,0,0)$. Other parameters are set as: $\beta=0.5$, $D=5.44$${\rm eV}$${\rm \AA^2}$, $u_F=0.05$${\rm eV}$.
• Figure 2: (Color online) The RKKY components $J_{ii}$ ($i=x,y,z$) versus the SOC strength $\alpha$ for different Néel vector orientations (fixed $\varphi_n=0$): (a-c) $\theta_n=0$; (d-f) $\theta_n=\pi/2$; (g-i) $\theta_n=\pi/3$. Impurities are aligned along $\theta_R=0$ with varying distances (see annotated values in the above figure). Other parameters are identical to Fig. 1(b).
• Figure 3: (Color online) The RKKY components $J_{ii}$ ($i=x,y,z$) versus the SOC strength $\alpha$ for different Néel vector orientations (fixed $\varphi_n=0$): (a-c) $\theta_n=0$; (d-f) $\theta_n=\pi/2$; (g-i) $\theta_n=\pi/3$. Impurities are aligned along $\theta_R=\pi/2$ with varying distances (see annotated values in the above figure). Other parameters are identical to Fig. 2.
• Figure 4: (Color online) The RKKY components $J_{ii}$ ($i=x,y,z$) versus the SOC strength $\alpha$ for different Néel vector orientations (fixed $\varphi_n=0$): (a-c) $\theta_n=0$; (d-f) $\theta_n=\pi/2$; (g-i) $\theta_n=\pi/3$. Impurities are aligned along $\theta_R=\pi/4$ with varying distances (see annotated values in the above figure). Other parameters are identical to Fig. 2.
• Figure 5: (Color online) The RKKY components $J_{ii}$ ($i=x,y,z$) versus the SOC strength $\alpha$ for different Fermi energies: (a) $u_F=0.05$${\rm eV}$; (b) $u_F=0.1$${\rm eV}$; (c) $u_F=0.15$${\rm eV}$. The Néel vector is along the $z$-axis ( $\theta_n=0$, $\varphi_n=0$). Impurities are aligned along $\theta_R=0$ with spacing $R=30$${\rm \AA }$.