The spin Hall conductivity in the hole-doped bilayer Haldane-Hubbard model with odd-parity ALM

Abstract

Spin current generated electrically is among the core phenomena of spintronics for driving high-performance spin device applications. Here, on the basis of systematic investigations for the hole doped single-layer Haldane-Hubbard(HH) model, we propose a new bilayer HH model to realize the compensated odd-parity spin splitting and the $T$-even spin Hall conductivity where the two layers are connected by the time reversal transformation. Our results show that the vanishing layer-dependent electric potential $V_{L}$ gives rise to odd-parity ALM protected by the combined symmetry $TM_{xy}$ with $T$ and $M_{xy}$ being the time reversal and mirror reflection perpendicular to $z$ axis, and the $T$-even spin Hall conductivity simultaneously. In addition, though the staggered magnetization within each layer is substantially impacted by the layer-dependent electric potential, small $V_{L}$'s only bring negligible changes to the net magnetization and the spin Hall conductivity, indicating that the alternating spin splitting in momentum space and the spin Hall conductivity are insusceptible to external elements. Most importantly, our work provides a general framework for the simultaneous realization of the compensated odd-parity spin splitting in momentum space and the spin Hall conductivity in collinear magnets, in terms of stacked multi-layer systems.

Figures (3)

• Figure 1: (Color online) The schematic illustration of the AA-stacking bilayer HH model with the lower and upper layer plotted at the left and right, respectively. Here the upward and downward spin polarizations are denoted by red and blue solid circles, and the arrows represent the sublattice currents. We note that the spins in the first and second layer are polarized in the reversed direction, and sublattice currents flow along opposite directions, reflecting the symmetry $TM_{xy}$ of this system.
• Figure 2: (Color online) (a)The magnitude of staggered magnetization $M_{A}/M_{B}$ on sublattice A and B with up- and down-spin polarizations, respectively, as a function of hole doping $\delta$ at $U=4.8$ and $\lambda=0.3$, where the black and red lines correspond to the sublattice potential $V_s=$ 0.1 and 0.3, respectively. Here the solid and dashed lines correspond to up- and down-spin electron quasiparticles, and also applies to other figures. The electron energy dispersion $E_{\bm{k}\sigma}$ as a function of momentum in the zigzag direction at (b)$\delta=0$ and (c)$\delta=0.2$ for $U=4.8$, $\lambda=0.3$, and $V_{s}=0.3$. The (d)longitudinal and (e)traverse optical conductivity, $\sigma_{xx}$ and $\sigma_{xy}$, coming from up- and down-spin electron quasiparticles as a function of energy $\Omega$ at $\delta=0$ with $U=4.8$ and $\lambda=0.3$ for $V_{s}=$ 0.1(black) and 0.3(red); (f)The up- and down-spin polarized anomalous Hall conductivity $\sigma_{\rm Hall}$ as a function of hole doping $\delta$ at $\lambda=0.3$ for $U=4.8$ with $V_{s}=$ 0.1(black), 0.3(red), and $U=5.0$ with $V_{s}=$ 0.1(blue), 0.3(magenta).
• Figure 3: (Color online) (a) The electron energy dispersion $E_{\bm{k}\sigma}$ as a function of momentum in the zigzag direction at (a)$V_{L}=0$ and (b)$V_{L}=0.1$ for $U=4.8$, $\lambda=0.3$, $V_{s}=0.3$, $t_{\perp}=0.1$, and $\delta=0.1$. Here the solid and dashed lines correspond to up- and down-spin electron quasiparticles, respectively, which also applies to other subfigures; (c) The spin-dependent Hall conductivity $\sigma_{\rm Hall}^{(\sigma)}$ as a function of hole doping $\delta$ calculated from Eq.\ref{['Hall-Cond']} for $U=4.8$ as well as $V_L=$ 0(black), 0.1(red), and $U=5.0$ as well as $V_L=$ 0(blue), 0.1(magenta); (d)The corresponding spin Hall conductivity $\sigma_{\rm spin \; Hall} = \sigma_{\rm Hall}^{(\uparrow)}-\sigma_{\rm Hall}^{(\downarrow)}$ as a function of $\delta$.