Extrinsic Spin Splitter Currents in Altermagnets

Abstract

Altermagnets exhibit momentum-dependent spin splitting despite having zero net magnetization. This enables a spin-splitter effect in which an external electric field generates transverse spin currents by separating oppositely polarized carriers. Here, we develop a unified semiclassical theory of linear extrinsic spin-splitter currents, incorporating impurity-induced side-jump and skew-scattering contributions, and apply it to the $d$-wave altermagnet \ch{FeSb2}. We demonstrate that asymmetric impurity scattering provides a dominant channel for spin-splitter currents. Remarkably, the resulting extrinsic spin conductivity is time-reversal even, in contrast to previously studied spin-splitter responses arising from symmetric scattering.

Figures (6)

• Figure 1: Extrinsic spin-splitter current: Schematic illustration of intrinsic and extrinsic mechanisms contributing to the spin splitter current in the presence of asymmetric impurity scattering. (a) Intrinsic mechanism: An external electric field induces an anomalous velocity, transversely deflecting opposite-spin carriers via band structure topology. (b) Side-jump mechanism: Spin-up and spin-down electrons undergo equal and opposite vertical displacements at the impurity site, preserving total momentum. (c) Skew-scattering mechanism: asymmetric impurity scattering bends spin-up and spin-down trajectories in different directions, breaking wave-vector conservation. (d) Extrinsic processes originating from asymmetric impurity scattering rates $(w_{{\bm k}'{\bm k}}\neq w_{{\bm k}{\bm k}'})$. (e) Real-space schematic of a spin-splitter current arising from the momentum-dependent spin splitting of the Fermi surface in an altermagnet.
• Figure 2: Spin-split electronic structure of doped FeSb2: (a) Crystal structure and Brillouin zone (BZ) of FeSb2. (b) Spin-projected band-structure along the high symmetry path $\rm Z$-$\rm\Gamma$-$\rm S$-$\rm R$-$\rm T$-$\rm Y$-$\rm S$-$\rm X$-$\rm U$-$\rm R$-$\rm Z$, showing pronounced non-relativistic spin splitting along the $\rm \Gamma$-$\rm S$ and $\rm Z$-$\rm R$ directions. (c) Spin-projected Fermi surface on the $k_x-k_y$ plane at the chemical potential $\mu=0.16$ eV, exhibiting the characteristic $d$-wave anisotropy of the spin-split bands. (d) The band splitting, $\Delta\varepsilon$, between the two lowest spin-splitted bands on the $k_x-k_y$ plane within the BZ.
• Figure 3: Band-geometric quantities relevant for extrinsic spin-splitter transport in FeSb2: (a-c) Momentum-space distribution of Berry curvature components $\Omega^{yz}_{v2}$, $\Omega^{zx}_{v2}$, and $\Omega^{xy}_{v2}$ , and (d) the corresponding side-jump velocity component $v_{y,v2}^{\rm sj}$ , evaluated for the lowest valence band ($v2$) on the $k_x-k_y$ plane within the BZ. (e-g) Momentum-space distribution of Spin Berry curvature components $\Omega^{z,yz}_{v2}$, $\Omega^{z,zx}_{v2}$, and $\Omega^{z,xy}_{v2}$ , and (h) the spin-dependent side-jump velocity $j_{x,v2}^{z,\rm sj}$ for the same band.
• Figure 4: Extrinsic transverse spin conductivity and spin Hall angle in altermagnetic FeSb2: (a) Chemical-potential ($\mu$) dependence of different $\sigma_{xy}^z$ components, showing contributions from sk3, sk4, side-jump (sj), intrinsic mechanisms, and their total sum. (b) The corresponding spin Hall angle $\theta^{\rm SH}=\sigma^{z,\rm total}_{xy}/\sigma^e_{yy}$ as a function of $\mu$. Here, $\sigma^e_{yy}$ denotes the longitudinal charge conductivity. The spin Hall angle reaches values as large as $\theta^{\rm SH}_{\rm total}\sim 0.8$, significantly exceeding the intrinsic contribution. This indicates highly efficient spin–charge conversion driven by extrinsic scattering mechanisms in altermagnetic FeSb2.
• Figure 5: Role of spin–orbit coupling and Néel order in extrinsic spin-splitter current: (a) The variation of different extrinsic contributions to the transverse spin conductivity with SOC strength in doped-FeSb2 with chemical potential $\mu=-0.1~\rm eV$, (b) Chemical-potential dependence of the transverse spin conductivity for opposite orientations of the Néel order, parametrized by $J_z$ parameter of the Hamiltonian. These highlight the sensitivity of the spin splitter current to SOC strength and Néel vector direction.