Edge spin galvanic effect in altermagnets

Technical Summary: Edge Spin Galvanic Effect in Altermagnets

Problem and Motivation
The spin galvanic effect (SGE), the conversion of nonequilibrium electron spin polarization into an electric current, is a fundamental phenomenon in spintronics. However, the SGE is strictly forbidden in centrosymmetric systems due to symmetry constraints, as it requires a gyrotropic system to enable odd-in-momentum spin splitting. Altermagnets represent a rapidly developing class of condensed-matter systems characterized by large nonrelativistic spin splittings (exceeding traditional spin-orbit splittings by orders of magnitude). Despite these large splittings, altermagnets are centrosymmetric media where the bulk spin galvanic effect is symmetry-forbidden. Consequently, current-spin interconversion in altermagnets has previously been considered possible only in the nonlinear regime. This paper addresses the question of whether spin-to-current conversion can occur in altermagnets under linear conditions by exploiting boundary effects.

Methodology
The author proposes a theoretical framework for the Edge Spin Galvanic Effect (ESGE) in semi-infinite $d$-wave altermagnets. The study employs a phenomenological approach combined with microscopic kinetic theory.

• Model System: A two-dimensional $d$-wave altermagnet with an edge along the $y$-axis. The energy spectra for the two spin subbands are defined as $\varepsilon^\pm_k = \varepsilon_k \pm \beta(k_{x0}^2 - k_{y0}^2)$, where $\beta$ describes the altermagnetic order and $(x_0, y_0)$ are the main axes of the altermagnet.
• Mechanism: The ESGE arises from two microscopic ingredients: (1) the momentum alignment of spin-oriented carriers, where the distribution varies as a second angular harmonic in momentum space, and (2) the scattering of charge carriers by the sample edge. The edge breaks spatial inversion symmetry, rendering the spin galvanic current symmetry-allowed.
• Calculations: The electron distribution function $f(k, x)$ is calculated using the Boltzmann kinetic equation in the presence of steady-state spin pumping. The edge current is derived by integrating the current density $j_y(x)$ over the distance from the edge. The analysis considers both specular and diffuse edge scattering. Additionally, the paper extends the theory to the interaction with electromagnetic radiation to derive a pure spin edge photocurrent.

Key Contributions and Results

1. Proposal of the Edge Spin Galvanic Effect (ESGE):
   The paper demonstrates that an electric current flows along the edge of a $d$-wave altermagnet when a nonequilibrium spin polarization $S_N$ is present along the Néel vector $\mathbf{N}$. The current is described by the relation $J_{\text{edge}} = \Xi S_N$.

   • Dependence: The current coefficient $\Xi$ depends on the altermagnetic order parameter $\beta$, the Fermi wavevector $k_F$, and the angle $\theta$ between the edge and the main axes of the altermagnet ($\Xi \propto \sin 2\theta$).
   • Directionality: The current reverses direction upon reversal of the nonequilibrium spin direction or the Néel vector.
   • Spatial Profile: The current is localized near the edge, decaying rapidly within a width comparable to the electron mean free path ($v_F \tau$).
   • Magnitude: For realistic parameters ($\beta k_F^2 = 1$ eV, $\tau = 1$ ps), the estimated edge current is on the order of $1$ $\mu$A, comparable to currents measured in optical experiments.

2. Pure Spin Edge Photocurrent:
   The paper proposes that the absorption of linearly polarized radiation in $d$-wave altermagnets generates a pure spin edge photocurrent ($J^s_{\text{edge}}$), where carriers of opposite spins flow in opposite directions along the edge.

   • Polarization Dependence: This current is maximal when the radiation polarization is perpendicular to the edge and vanishes when parallel. It scales with $\sin 2\theta$.
   • Frequency Dependence: The spin current is nearly frequency-independent at low frequencies ($\omega \tau \ll 1$) but drops rapidly for $\omega > 1/\tau$.
   • Conversion to Electric Current: The application of an external magnetic field $\mathbf{B}$ parallel to the Néel vector converts this pure spin current into an electric edge current. This conversion occurs even in the absence of a Lorentz force (when $\mathbf{B}$ is in the $(x_0, y_0)$ plane), distinguishing it from mechanisms in non-magnetic systems.

3. Dynamic Response:
   The paper analyzes the time-dependent ESGE current under short-pulse spin excitation and in the presence of a magnetic field. The current exhibits damped oscillatory behavior determined by the Larmor precession frequency, interpreted as the coherent trembling motion (Zitterbewegung) of spin-polarized electrons.

Significance and Claims
The paper claims to establish a new mechanism for spin-to-current conversion in centrosymmetric altermagnets by utilizing edge scattering to break inversion symmetry. It posits that the ESGE and the associated pure spin edge photocurrent are intrinsic properties of $d$-wave altermagnets. The author suggests that these effects are experimentally accessible using existing techniques, specifically combining spin pumping (or optical excitation) with edge current measurements, similar to those performed with terahertz radiation. The work also notes that analogous surface phenomena are expected in three-dimensional $d$-wave altermagnets. The study provides a theoretical basis for detecting altermagnetic order and manipulating spin currents at the boundaries of these materials without requiring bulk symmetry breaking.