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Intrinsic spin Nernst effect in spin-triplet superconductors

Taiki Matsushita, Youichi Yanase, Takeshi Mizushima, Satoshi Fujimoto, Ilya Vekhter

TL;DR

This work provides a microscopic, Berry-curvature–based account of the intrinsic spin Nernst effect (SNE) in spin-triplet superconductors, highlighting two intertwined contributions: a direct quasiparticle response and an indirect supercurrent backflow that ensures bulk charge-current screening. By employing a conserved spin current that includes the spin torque dipole, the authors show that the SNE can be finite even in cases where conventional spin-current definitions predict zero, and that nonunitary (spin-polarized) condensates yield sizable, comparable quasiparticle and supercurrent contributions. The analysis covers both helical (time-reversal invariant) and nonunitary (time-reversal broken) superconducting states within a Bogoliubov–de Gennes framework, linking the SNE to the d-vector–driven Berry curvature and to topological band structure (Chern numbers) of spin sectors. Although intrinsically small in the presence of impurities, the intrinsic SNE remains within experimental reach in ultraclean materials and offers a robust thermoelectric probe of spin-triplet pairing and its topological character, observable via inverse spin Hall detection schemes.

Abstract

We theoretically investigate the intrinsic (impurity-independent) spin Nernst effect (SNE), a spin current generation perpendicular to temperature gradients, in spin-triplet superconductors. We show that, in these systems, the SNE consists of two distinct contributions: a direct quasiparticle contribution and an indirect supercurrent contribution. The quasiparticle contribution originates from the momentum space Berry curvature generated by spin-triplet Cooper pairs. The indirect contribution arises from a compensating supercurrent that cancels the bulk thermoelectric charge current. While this contribution vanishes when the condensate has no spin-polarization in momentum space, it can be comparable in magnitude to the quasiparticle contribution in nonunitary superconductors. These results demonstrate that thermoelectric spin supercurrent must be explicitly accounted for when evaluating the SNE in nonunitary superconductors.

Intrinsic spin Nernst effect in spin-triplet superconductors

TL;DR

This work provides a microscopic, Berry-curvature–based account of the intrinsic spin Nernst effect (SNE) in spin-triplet superconductors, highlighting two intertwined contributions: a direct quasiparticle response and an indirect supercurrent backflow that ensures bulk charge-current screening. By employing a conserved spin current that includes the spin torque dipole, the authors show that the SNE can be finite even in cases where conventional spin-current definitions predict zero, and that nonunitary (spin-polarized) condensates yield sizable, comparable quasiparticle and supercurrent contributions. The analysis covers both helical (time-reversal invariant) and nonunitary (time-reversal broken) superconducting states within a Bogoliubov–de Gennes framework, linking the SNE to the d-vector–driven Berry curvature and to topological band structure (Chern numbers) of spin sectors. Although intrinsically small in the presence of impurities, the intrinsic SNE remains within experimental reach in ultraclean materials and offers a robust thermoelectric probe of spin-triplet pairing and its topological character, observable via inverse spin Hall detection schemes.

Abstract

We theoretically investigate the intrinsic (impurity-independent) spin Nernst effect (SNE), a spin current generation perpendicular to temperature gradients, in spin-triplet superconductors. We show that, in these systems, the SNE consists of two distinct contributions: a direct quasiparticle contribution and an indirect supercurrent contribution. The quasiparticle contribution originates from the momentum space Berry curvature generated by spin-triplet Cooper pairs. The indirect contribution arises from a compensating supercurrent that cancels the bulk thermoelectric charge current. While this contribution vanishes when the condensate has no spin-polarization in momentum space, it can be comparable in magnitude to the quasiparticle contribution in nonunitary superconductors. These results demonstrate that thermoelectric spin supercurrent must be explicitly accounted for when evaluating the SNE in nonunitary superconductors.
Paper Structure (18 sections, 128 equations, 7 figures)

This paper contains 18 sections, 128 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Definitions of spin current
  3. Spin Nernst conductivity
  4. Model Hamiltonian
  5. Intrinsic Spin Nernst effect in helical superconductors
  6. Intrinsic Spin Nernst effect in nonunitary superconductors
  7. Conclusion
  8. Hamiltonian and wave function of wave packet
  9. Effective Lagrangian of wave packet
  10. Equation of motion of wave packet
  11. Energy of wave packet
  12. Quadrupole moment of $\hat{\theta}^\mu=d\hat{\Theta}^\mu/dt_c$
  13. Positional shift when $\hat{\theta}^\mu=d\hat{\Theta}^\mu/dt_c$
  14. Field variation and local density
  15. Equilibrium conserved spin current
  16. ...and 3 more sections

Figures (7)

  • Figure 1: Quasiparticle spectrum in the helical superconducting state with the $d$-dvector in Eq. \ref{['eq: dvector_helicalstate']}. The parameters are set to $t=1.25$ meV, with $(\mu/t,\Delta_0(T_{\rm eq})/t)=(-2,0.05)$.
  • Figure 2: Berry curvature, along with the spin of Cooper pairs $s_{\rm pair}^z=\pm 1$ and the band-resolved Chern numbers ${\rm Ch}_n$, for the negative-energy quasiparticle states in the helical superconductor. The parameters are set to $t=1.25$ meV, with $(\mu/t,\Delta_0(T_{\rm eq})/t)=(-2,0.05)$.
  • Figure 3: Temperature dependence of $\tilde{\alpha}_{xy}^{s^z}$ with several values of $T_{\mathrm c}/t$ in the helical superconductor. The parameters are set to $t=1.25$ meV and $\mu=-2.0\times 10$ meV (bandwidth $E_{\mathrm B}=8t=10\;\mathrm{meV}$; equivalently $E_{\mathrm B}/k_{\mathrm B}=116\;\mathrm K$). The plotted values are obtained by converting the two-dimensional SNC to the three-dimensional bulk value by dividing by the film thickness $d=2\mu\mathrm{m}$.
  • Figure 4: The spin magnetic quadrupole moment $(d_x^{s^z}(\bm k))_n$ for the degenerate occupied states in the helical superconductor. The parameters are set to $t=1.25$ meV, with $(\mu/t,\Delta_0(T_{\rm eq})/t)=(-2,0.05)$.
  • Figure 5: Upper panels: Berry curvature and band-resolved Chern numbers ${\rm Ch}_n$ for the negative-energy quasiparticle states in the nonunitary superconductor, Eq. \ref{['eq:OP_nonun']}. Lower panels: Spin magnetic quadrupole moment and band-resolved Chern numbers ${\rm Ch}_n$ for the negative-energy quasiparticle states in the nonunitary superconductor. The parameters are set to $t=1.25$ meV, with $(\mu/t,\Delta_0(T_{\rm eq})/t,\eta)=(-2,0.05,0.4)$.
  • ...and 2 more figures