Spin Currents in Rashba Altermagnets: From Equilibrium to Nonlinear Regimes

TL;DR

The paper addresses intrinsic spin-current generation in a two-dimensional Rashba spin-orbit coupled altermagnet by developing a semiclassical Boltzmann framework with a spin-current operator augmented by anomalous velocity from Berry curvature. It reveals a band-velocity–driven linear Spin Hall current unique to altermagnets, in addition to the conventional Berry-curvature–driven (anomalous-velocity) contribution, and shows that the background spin current originates from band velocity and scales with $t_j$ and $\lambda$. Nonlinear responses exhibit both longitudinal and transverse spin currents quadratic in the electric field, with distinct dependence on $ε_F$, tunable by $t_j$ and $\lambda$, enabling control of spin currents for spintronic applications. The results highlight the tunability of spin transport channels in Rashba-coupled altermagnets and suggest practical routes to engineer spin currents through the altermagnet and Rashba parameters for device functionality.

Abstract

We investigate equilibrium (background), linear, and nonlinear spin currents in two-dimensional Rashba spin-orbit coupled altermagnet systems, using a modified spin current operator that includes anomalous velocity from non-zero Berry curvature. The background spin current, stemming from spin-orbit coupling and modulated by the altermagnet term ($t_j$), exhibits in-plane polarization, increases linearly with Fermi energy ($ε_F$), and is enhanced by both the altermagnet ($t_j$) and the Rashba parameter ($λ$). Linear spin current is always transverse with out-of-plane polarization and can be viewed as Spin Hall current, primarily driven by band velocity, with $t_j$ enabling a band-induced contribution (previously absent in simple Rashba systems ($t_j=0$)). This highlights altermagnet system as a promising source of spin Hall current generation. For linear spin Hall current, its band contribution's magnitude increases linearly with $ε_F$, while the magnitude of anomalous component saturates at higher $ε_F$. Further, the magnitude of spin Hall current is enhanced by $t_j$ but reduced by $λ$. Nonlinear spin currents feature both longitudinal and transverse components with in-plane polarization. Both the nonlinear longitudinal spin current from band velocity and the nonlinear transverse spin current from anomalous velocity initially decrease with $ε_F$ before saturating at higher $ε_F$. Importantly, $t_j$ reduces these currents while $λ$ enhances them. Meanwhile, the nonlinear transverse current from band velocity increases and then saturates with $ε_F$, enhanced by $λ$ and showing non-monotonic variation with $t_j$. These findings highlight the tunability of spin current behavior through Rashba and altermagnet parameters, offering insights for spintronic applications.

Figures (8)

• Figure 1: (a) Band structure corresponding to the Hamiltonian in Eq. (\ref{['eq1']}) with parameters $\mathrm{t_0 = \hbar^2 / (2m^*) = 0.761}$$\mathrm{eV.nm^2}$, (with an effective mass of $\mathrm{m^* = 0.05m_e}$, where $\mathrm{m_e}$ is the free electron mass), $\mathrm{t_j = 0.2t_0}$ and $\lambda = 0.02$$\mathrm{eV.nm}$. The colormap indicates the out-of-plane spin polarization, ${\langle \sigma_z \rangle}_s$. Panel (b) shows a top-down (constant energy) view of the full band dispersion, while panel (c) plots all components of the spin vector $\langle \boldsymbol{\sigma} \rangle_+$ as a function of the azimuthal angle $\phi$. The anisotropic spin splitting reaches minima along azimuthal angles $\phi = n\pi/2$, where the spin vectors lie entirely within the $(k_x, k_y)$ plane. In contrast, maxima appear at $\phi = (2n + 1)\pi/4$, where the out-of-plane component $\langle\sigma_z\rangle$ is most pronounced.
• Figure 2: The background spin current $\mathcal{J}^{(0)}_{xy}$ (in $\mathrm{eV.nm^{-1}}$) as a function of Fermi energy $\epsilon_F$ (in $\mathrm{eV}$) (a) for different values of $t_j$ with $\lambda=0.04$$\mathrm{eV.nm}$ and (b) for different values of $\lambda$ with $t_j=0.2$$t_0$. The corresponding spin-split current is shown in the right panel. It should be noted that in the pure Rashba case ($t_j=0$, $\lambda \neq 0$, $M=0$) and in the gapped Rashba case ($t_j=0$, $\lambda \neq 0$, $M\neq0$), the background spin current becomes independent of the Fermi energy for $\epsilon_F > 0$ and $\epsilon_F > M$, respectively, and is given by $\mathcal{J}^{(0)}_{xy}=(\lambda^3{m^*}^2)/(6\pi\hbar^4)$. For $\lambda = 0.02, 0.04,$ and $0.06$$\mathrm{eV.nm}$, $\mathcal{J}^{(0)}_{xy}=(\lambda^3{m^*}^2)/(6\pi\hbar^4)$ takes the values $0.19 \times 10^{-6}$, $1.52 \times 10^{-6}$, and $5.13 \times 10^{-6}$$\mathrm{eV. nm^{-1}}$, respectively—significantly weaker than in the Rashba–altermagnet system.
• Figure 3: The plots for $\mathcal{J}^{(1),y}_{b,xz}/(e\tau E_y/\hbar)$ (in $\mathrm{eV}$), ($\mathcal{J}^{(1),y}_{b,xz}$: linear spin Hall current from band velocity) as a function of Fermi energy $\epsilon_F$ (in $\mathrm{eV}$) (a) for different values of $t_j$ with $\lambda=0.04$$\mathrm{eV.nm}$ and (b) for different values of $\lambda$ with $t_j=0.2$$t_0$. The right panel displays the corresponding spin-split current. It should be noted that, in both the pure and gapped Rashba cases, the band-velocity contribution to the linear spin Hall current vanishes.
• Figure 4: The plots for $\mathcal{J}^{(1),y}_{a,xz}/(eE_y)$ (dimensionless) ($\mathcal{J}^{(1),y}_{a,xz}$: linear spin Hall current from anomalous velocity) as a function of Fermi energy $\epsilon_F$ (in $\mathrm{eV}$) (a) for different values of $t_j$ with $\lambda=0.04$$\mathrm{eV.nm}$ and (b) for different values of $\lambda$ with $t_j=0.2$$t_0$. The corresponding spin-split current is shown in the right panel. It should be noted that, in the pure Rashba case, $\mathcal{J}^{(1),y}_{a,xz}/(eE_y)$ is independent of the Fermi energy for $\epsilon_F>0$ and takes a constant value: $-(1/8\pi) \approx -0.0398$. For gapped Rashba case with $\epsilon_F \gg M$, $\mathcal{J}^{(1),y}_{a,xz}/(eE_y)$ approaches the same constant value: $-(1/8\pi)$.
• Figure 5: The plots for $\mathcal{J}^{(1),y}_{b,xz}/(eE_y)$ (dimensionless) and $\mathcal{J}^{(1),y}_{a,xz}/(eE_y)$ (dimensionless) as a function of Fermi energy $\epsilon_F$ (in eV) with $\tau=2.5$ ps. This confirms that the band velocity plays the dominant role in realizing spin Hall currents in Rashba-coupled altermagnets. The above statement remains valid even under significant variations in material parameters.