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Spin-valley physics in anomalous thermoelectric responses of the spin-orbit coupled $α$-$T_3$ system with broken time-reversal symmetry

Lakpa Tamang, Tutul Biswas

Abstract

We extract spin-valley physics in the anomalous Hall and Nernst responses of the spin-orbit coupled $α$-$T_3$ system in the presence of a time-reversal symmetry breaking staggered magnetization. We show that the interplay between the SOI, magnetization, and a model parameter $α$ for the $α$-$T_3$ lattice enables efficient tuning of spin- and valley-dependent Hall and Nernst signals. The spin-valley physics of the Hall and Nernst responses in the absence and presence of the magnetization are well explained. The peak-dip features of the Nernst responses are also understood from the corresponding Hall responses through the Mott relation. We find that the magnetization introduces highly tunable spin and valley polarizations, which are calculated from the spin- and valley-resolved Nernst conductivities. It is shown that both the spin and valley polarizations can attain nearly complete polarization over extended regions of the parameter space. Overall, our results highlight the $α$-$T_3$ lattice as a promising platform for spin and valley caloritronic applications.

Spin-valley physics in anomalous thermoelectric responses of the spin-orbit coupled $α$-$T_3$ system with broken time-reversal symmetry

Abstract

We extract spin-valley physics in the anomalous Hall and Nernst responses of the spin-orbit coupled - system in the presence of a time-reversal symmetry breaking staggered magnetization. We show that the interplay between the SOI, magnetization, and a model parameter for the - lattice enables efficient tuning of spin- and valley-dependent Hall and Nernst signals. The spin-valley physics of the Hall and Nernst responses in the absence and presence of the magnetization are well explained. The peak-dip features of the Nernst responses are also understood from the corresponding Hall responses through the Mott relation. We find that the magnetization introduces highly tunable spin and valley polarizations, which are calculated from the spin- and valley-resolved Nernst conductivities. It is shown that both the spin and valley polarizations can attain nearly complete polarization over extended regions of the parameter space. Overall, our results highlight the - lattice as a promising platform for spin and valley caloritronic applications.
Paper Structure (13 sections, 30 equations, 17 figures)

This paper contains 13 sections, 30 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Low-Energy description of the spin-orbit coupled $\alpha$-$T_3$ system
  3. Formalism for the Berry curvature mediated anomalous thermoelectric transport
  4. Results and discussions
  5. Band structure and Berry curvature
  6. Valley and spin Hall conductivities
  7. Case I: Hall conductivities in the absence of the staggered magnetization ($M = 0$)
  8. Case II: Hall Conductivities in the Presence of Magnetization ($M \neq 0$)
  9. Valley and spin Nernst conductivities
  10. Case I: Nernst Conductivities in the absence of Magnetization ($M = 0$)
  11. Nernst Conductivities in the Presence of Magnetization ($M \neq 0$)
  12. Spin and Valley Polarization
  13. Summary

Figures (17)

  • Figure 1: (Color online) Schematic illustration of the $\alpha$–$T_{3}$ lattice with hub (B) and rim (A, C) sites. The NN (NNN) hopping paths are indicated by solid (dashed) lines. The NN B-C hopping amplitude is $\alpha$ times to that of A-B hopping strength. The lattice translational vectors are ${\bm a_1}=(\frac{\sqrt{3}}{2},\frac{3}{2})\rm a$ and ${\bm a_2}=(-\frac{\sqrt{3}}{2},\frac{3}{2})\rm a$, where $\rm a$ is the NN distance.
  • Figure 2: (Color online) Low-energy dispersions in the $K$ and $K^\prime$ valleys for $\alpha=0.3$ and $\lambda=100$ meV. Upper panels [(a) and (b)] represent $M=0$, while lower panels [(c) and (d)] correspond to $M\neq 0$ case.
  • Figure 3: (Color online) Berry curvature distribution $\Omega^{n}_{K,\sigma}(\mathbf{k})$ in the $K$ valley for spin-up and spin-down bands. Left Panels [(a) and (c)] are for $M=0$, while right panels [(b) and (d)] represents $M\neq 0$ case.
  • Figure 4: (Color online) Same as Fig. \ref{['fig:BerryK']} but for the $K^{\prime}$ valley.
  • Figure 5: (Color online) (a) Valley Hall conductivity and (b) Spin Hall conductivity as a function of chemical potential $\mu$ for $M=0$, $\alpha=0.3$, $T=20$ K, and $\lambda=100$ meV.
  • ...and 12 more figures