Influence of spin Nernst effect on continuum spin conductivity in antiferromagnets in the checkerboard lattice

Abstract

The influence of the spin Nernst effect on longitudinal spin conductivity, ${\sigma }_{\mathit{xx}}$ is investigated in the two-dimensional antiferromagnetic model in the checkerboard lattice and in the presence of Dzyaloshinskii-Moriya interaction employing spin waves. We present results of the influence of the Dzyaloshinskii-Moriya interaction as well as variation of the Nernst coefficient at finite temperature on continuum spin conductivity.

Introduction

The spin separation caused by the thermal flow of electrons in condensed matter or spin Nernst effect is a phenomenon that has been experimentally observed since 2016 where both spin-up and spin-down electrons are separated with the application of an external magnetic field [1], [2]. This effect is similar to the spin Hall effect where an electrical current leads to a spin separation. The interest in magnon spintronics as well as in the utilization of magnons to transport of information has increased a lot recently due to the fact that magnons allow the transport of information through spins without Joule effect [3], [4], [5], [6], [7], [8], [9], [10], [11]. The detection of the spin Nernst effect by applying a longitudinal temperature gradient has been performed in a Platinum thin film, where the spin Nernst effect and the quantum spin Hall effect are of comparable intensities however, differing in sign [2].

In the collinear easy-axis anisotropic antiferromagnet, the symmetry guarantees that the spin wave modes are doubly degenerate. The two modes may carry opposite spin angular momentum and therefore, exhibit opposite chirality. So, in a two-dimensional antiferromagnet in the honeycomb lattice the presence of a Dzyaloshinskii-Moriya interaction, a longitudinal temperature gradient $\nabla T$ may drive to two modes of opposite transverse directions, realizing a spin Nernst effect of magnons with the vanishing of thermal Hall current. Thus, the magnons around the $\mathrm{\Gamma }$ and K points of the first Brillouin zone contribute oppositely to transverse spin conductivity and the competition between them leads to a sign change of the spin Nernst coefficient at finite temperature.

The transmission of magnons through the antiferromagnetic and ferromagnetic chains can be altered by the Berry curvature generated by the Dzyaloshinskii-Moriya interaction (DM) which acts like an effective magnetic field leading so, to chiral magnons with nontrivial topological properties [12], [13], [14], [15]. This different kind of Hall effect arises naturally due to the coupling between spin-orbit interaction and the lattice being different from the Hall effect induced by the Lorentz force [16]. One of the firsts experimental studies of the magnon Hall effect were performed in the ferromagnetic insulator Lu₂V₂O₇, Ho₂V₂O₇, In₂Mn₂O₇ [17], [18]. Moreover, the magnon Hall effect has been observed in magnets Cu as well [19], [20]. In general, these studies display that the topology of the system leads to a sign change in the Hall spin conductivity ${\sigma }_{\mathit{xy}}$ as function of the temperature and magnetic field [14].

In general, the spin transport in two-dimensional antiferromagnetic materials have many applications in nano-devices where the spin transport by magnons has been experimentally and theoretically studied in [9], [10], [11], [21], [22], [23], [24], [25], [26]. The aim of this paper is to study the effect of not opening of the gap in the honeycomb lattice in the presence of DM interaction on spin transport or verify the influence of the magnon Nernst effect on longitudinal spin conductivity ${\sigma }_{\mathit{xx}}\left(\omega \right)$. This work is divided in the following way: In Section 2, we discuss about the spin Hall effect. In Section 3, we present results for longitudinal spin conductivity. In the last Section 4, ene presents our conclusions and final remarks.

We analyze the influence of the spin Nernst effect on spin transport in the model given by

$$\begin{array}{cc}\hfill & \mathcal{H}=J_1{\displaystyle \sum _{〈i,j〉}}\left(S_i^x{S}_j^x+S_i^y{S}_j^y+S_i^z{S}_j^z\right)\hfill \\ \hfill & +J_2{\displaystyle \sum _{〈〈i,j〉〉}}\left(S_i^x{S}_j^x+S_i^y{S}_j^y+\mathrm{\Delta }{S}_i^z{S}_j^z\right)\hfill \\ \hfill & +{\bar{J}}_2{\displaystyle \sum _{〈〈i,j〉〉}}\left(S_i^x{S}_j^x+S_i^y{S}_j^y+S_i^z{S}_j^z\right)\hfill \\ \hfill & +D{\displaystyle \sum _{〈i,j〉}}{v}_{\mathit{ij}}\left(S_i^x{S}_j^y-S_i^y{S}_j^x\right)\hfill \\ \hfill \end{array}$$

where $\mathrm{\Delta }$ is an anisotropy parameter and $v_{\mathit{ij}}=\pm 1$. $J_1,J_2$ and ${\bar{J}}_2$ are antiferromagnetic exchange interactions. The checkerboard lattice present two inequivalent sites where the symbols $〈i,j〉$ and $〈〈i,j〉〉$ in sums mean respectively nearest and next-nearest-neighbors interactions.