Spin Splitting Nernst Effect in Altermagnets

• Spin splitting Nernst effect is a phenomenon where a longitudinal temperature gradient induces a transverse pure spin current through intrinsic momentum-space spin splitting in altermagnets.
• It is modeled by a low-energy continuum Hamiltonian that incorporates parameters like Fermi energy, temperature, and crystal orientation to predict tunable quantum oscillations and spin current responses.
• This effect paves the way for advanced spin-caloritronic applications by offering a symmetry-controlled, band-structure-driven method for pure spin current generation without reliance on spin–orbit coupling.

The spin splitting Nernst effect, a variant of the spin Nernst effect (SNE), denotes the generation of a transverse pure spin current in response to a longitudinal temperature gradient, distinguished by the mechanism of spin-split band structures without net magnetization or spin–orbit coupling. Unlike the conventional SNE—which relies on spin–orbit interactions or magnonic Berry curvature—spin splitting Nernst effect manifests in altermagnets via intrinsic momentum-space spin splitting, resulting in transverse spin currents with symmetries and parameter dependence unique from established SNE systems (Yi et al., 4 Sep 2025). This effect deepens the understanding of spin-caloritronic transport in topologically nontrivial and symmetry-constrained quantum materials.

1. Conceptual Distinction: Conventional SNE vs. Spin Splitting Nernst Effect

The conventional spin Nernst effect describes the generation of a charge-neutral transverse spin current, $J_y^s$, when a longitudinal thermal gradient, $\nabla_x T$, is applied to a system with significant spin–orbit coupling or to magnonic insulators with Berry curvature (Meyer et al., 2016, Cheng et al., 2016). In these cases, the response is antisymmetric: $N_{s,xy} = -N_{s,yx}$, enforced by underlying time-reversal-even mechanisms.

In altermagnets, the spin splitting Nernst effect emerges from a symmetry-protected, momentum-dependent spin splitting in the absence of both net magnetization and spin–orbit coupling (Yi et al., 4 Sep 2025). The resulting Hamiltonian yields opposite transverse group velocities for spin-up and spin-down electrons under a thermal gradient, leading to a transverse spin current with $N_{s,xy}=N_{s,yx}$. This symmetric response directly reflects the time-reversal-odd, inversion-even character of altermagnetic order, in contrast to systems where spin Nernst currents are typically antisymmetric.

2. Microscopic Formalism and Model Hamiltonian

The essential physics in altermagnets is captured by a low-energy continuum Hamiltonian of the form: $$H(\mathbf{k}) = \frac{\hbar^2 k^2}{2m^*} - V_0 - \frac{\alpha_0}{2}\left[(k_x^2 - k_y^2)\cos2\theta + 2k_xk_y\sin2\theta\right]\,\sigma_z,$$ where $\sigma_z$ is the Pauli spin operator, $\theta$ the angle between crystalline and transport axes, and $\alpha_0$ quantifies $d$-wave spin splitting. For spin-$\uparrow$,

$$v_y^\uparrow = \frac{\hbar k_y}{m^*} + \frac{\alpha_0}{\hbar}k_y\cos2\theta - \frac{\alpha_0}{\hbar}k_x\sin2\theta,$$

with an analogous result for spin-$\downarrow$ with reversed sign.

Device-level implementation employs a four-terminal tight-binding Hamiltonian on a square lattice, with temperature biases applied longitudinally and spin-resolved particle currents calculated for each terminal. The spin-dependent transmission coefficients, $T_{p\sigma,q}(E)$, are computed using the nonequilibrium Green's function framework, and the spin Nernst coefficient $N_{s,ij}$ is extracted directly from the linear response to the imposed thermal bias (Yi et al., 4 Sep 2025).

3. Symmetry Analysis and Unique Response Tensor Structure

A key distinguishing feature of the spin splitting Nernst effect in altermagnets is the symmetry of the response tensor. Under $\pi/2$ rotation and spin exchange, the transmission functions transform such that: $N_{s,xy} = N_{s,yx},$ marking a departure from conventional SNE, where $N_{s,xy} = -N_{s,yx}$ due to time-reversal-even origins (Yi et al., 4 Sep 2025). This property arises because the spin splitting is odd under time-reversal but even under inversion, a combination unique to altermagnets.

This symmetry structure has direct consequences for device responses and protocols to distinguish spin splitting Nernst currents from their conventional counterparts. It also imposes constraints on the operation of spin-caloritronic devices utilizing altermagnet-based channels.

4. Parameter Dependence, Tuning, and Quantum Oscillations

The spin splitting Nernst coefficient $N_s$ in altermagnets is tunable by several experimentally controllable parameters:

• Fermi energy $E_F$: $N_s(E_F)$ displays pronounced quantum oscillations at low temperatures due to sensitivity near subband edges; these oscillations are smeared out at higher $T$.
• Temperature $T$: For $T\ll$ subband splitting, $N_s$ exhibits oscillations and little net growth; for $T\gtrsim$ subband splitting, $N_s\propto T$ and oscillations are suppressed.
• Crystal orientation $\theta$: $N_s\propto \alpha\sin2\theta$; the effect is zero for $\theta=0,\pi/2$ and maximal at $\theta=\pi/4$; $N_s(\theta)$ exhibits period $\pi$.
• System size $L$: Larger $L$ narrows subband spacing, densifies oscillations, and increases the absolute magnitude of $N_s$ proportionally to $L$.

These dependencies enable detailed experimental control and the development of spin splitting Nernst-based measurement protocols and applications (Yi et al., 4 Sep 2025).

5. Absence of Spin–Orbit Coupling and Magnetization: Intrinsic Origin

The spin splitting Nernst effect does not require spin–orbit coupling, magnetization, or magnonic Berry curvature. The altermagnetic band structure itself, protected by the specific symmetry constraints (time-reversal broken, inversion intact), supplies the necessary spin splitting—the $d$-wave form of the $\alpha_0\sigma_z$ term is sufficient to generate opposite transverse group velocities for each spin species under a thermal drift. The effect, therefore, represents an intrinsic, band-structure-driven avenue for generating transverse spin currents from temperature gradients (Yi et al., 4 Sep 2025).

This stands in strong contrast to both the conventional electronic SNE in heavy metals (which relies on spin–orbit interactions and often extrinsic scattering phenomena) (Wimmer et al., 2013, Akera et al., 2012) and to magnonic SNE in antiferromagnets/magnonics (requiring Berry curvature from Dzyaloshinskii–Moriya or other spin–orbit–driven terms) (Cheng et al., 2016).

6. Experimental and Theoretical Implications; Comparison with Other SNE Systems

The unique combination of zero net magnetization, absence of spin–orbit coupling, and electrically tunable spin-split transverse response positions the spin splitting Nernst effect as a novel channel for pure spin current generation in spin-caloritronics (Yi et al., 4 Sep 2025). In SNE devices based on heavy metals (Pt, W), the spin Nernst angle $\theta_{SN}$ is experimentally found to be substantial and often opposite in sign to the spin Hall angle (Meyer et al., 2016, Bose et al., 2017, Sheng et al., 2016). However, those effects always require intrinsic (Berry curvature) or extrinsic (side-jump, skew) spin–orbit interactions.

A comparative table of key differences is provided:

Feature	Conventional SNE (metals/magnons)	Spin Splitting Nernst (Altermagnet)
Requires SOC	Yes (electrons/magnons)	No
Net Magnetization	Not required, often zero	Strictly zero
Response tensor	$N_{s,xy} = -N_{s,yx}$	$N_{s,xy} = N_{s,yx}$
Underlying symmetry	Time-reversal even, inversion arbitrary	Time-reversal odd, inversion even
Intrinsic/Extrinsic	Both; intrinsic Berry or extrinsic skew	Intrinsic, band-structure only
Tunability	Doping, $T$, $E_F$, but symmetry fixed	$E_F$, $T$, orientation $\theta$, size $L$

The demonstration of this effect in both analytical and nonequilibrium Green's function calculations highlights the generality and microscopic viability of the spin splitting Nernst effect (Yi et al., 4 Sep 2025). It opens experimental routes distinct from those focused on materials with large spin–orbit coupling or topological magnon bands, and may enable spin-caloritronic applications in platforms where conventional SNE is symmetry-forbidden.

7. Outlook and Relation to Broader Spin-Caloritronic Phenomena

The spin splitting Nernst effect exemplifies how compensated magnetic order with momentum-dependent spin splitting can generalize spin-caloritronic transport beyond the canonical paradigms of the SNE and spin Seebeck effect. Its realization in altermagnets, with their time-reversal-odd and inversion-even symmetry, serves as a bridge connecting the fields of unconventional antiferromagnetism, topological band structure, and thermally-driven pure spin current generation. Future directions include synthesis and detection of the effect in candidate altermagnetic materials, leveraging its tunable tensor structure for device applications, and integrating altermagnet-based channels with conventional SNE systems for hybrid spin-caloritronic architectures (Yi et al., 4 Sep 2025).