Spin Seebeck and Spin Nernst Effects of Magnons in Noncollinear Antiferromagnetic Insulators

TL;DR

This study demonstrates that noncollinear antiferromagnetic insulators can efficiently generate magnon spin currents via spin Seebeck and spin Nernst effects, with quantitative predictions for potassium iron jarosite.

Contribution

It provides a combined theoretical and computational analysis of heat-to-spin conversion in noncollinear antiferromagnetic insulators, highlighting their potential for spintronic applications.

Findings

Predicted spin Seebeck conversion factor of 0.2 μV/K at 20 K.

Explicit evaluation of spin Seebeck and Nernst effects in kagome lattice.

Identification of noncollinear antiferromagnets as promising materials for magnon spin current generation.

Abstract

Our joint theoretical and computer experimental study of heat-to-spin conversion reveals that noncollinear antiferromagnetic insulators are promising materials for generating magnon spin currents upon application of a temperature gradient: they exhibit spin Seebeck and spin Nernst effects. Using Kubo theory and spin dynamics simulations, we explicitly evaluate these effects in a single kagome sheet of potassium iron jarosite, KFe$_3$(OH)$_6$(SO$_4$)$_2$, and predict a spin Seebeck conversion factor of $0.2 \mu\mathrm{V}/\mathrm{K}$ at a temperature of $20 \mathrm{K}$.

Equations

$$\mathcal{H}=\frac{1}{2{\hslash }^2}\sum _{ij}\left(J_{ij}{\mathbf{S}}_i\cdot {\mathbf{S}}_j+{\mathbf{D}}_{ij}\cdot {\mathbf{S}}_i\times {\mathbf{S}}_j\right)+\frac{g{\mu }_{\mathrm{B}}}{\hslash }{B}_z\sum _i{S}_i^z$$

$${{\rm Y}}_{\mu \nu }^{\gamma }=\frac{\tau }{2VT}\sum _{\mathbf{k}}\sum _{n=1}^{2N}\mathrm{Re}\left[{\left(J_{\mathbf{k},\mu }^{\gamma }\right)}_{nn}\right]{\left(Q_{\mathbf{k},\nu }\right)}_{nn}\left(-\frac{\partial \rho }{\partial \epsilon }\right).$$

$${{\rm Y}}^x=\left(\begin{array}{cc}0& {{\rm Y}}_{xy}^x\\ {{\rm Y}}_{xy}^x& 0\end{array}\right),{{\rm Y}}^y=\left(\begin{array}{cc}{{\rm Y}}_{xy}^x& 0\\ 0& -{{\rm Y}}_{xy}^x\end{array}\right).$$

$${{\rm Y}}^x=\left(\begin{array}{cc}0& {{\rm Y}}_{xy}^x\\ {{\rm Y}}_{yx}^x& 0\end{array}\right),{{\rm Y}}^y=\left(\begin{array}{cc}{{\rm Y}}_{xx}^y& 0\\ 0& {{\rm Y}}_{yy}^y\end{array}\right).$$

Full text

Introduction.

Interconversion phenomena between physical quantities like sound, charge, spin, or heat Otani et al. (2017) are cornerstones in the solid-state research for next-generation alternatives to today’s CMOS technology. Two particularly active fields are those of spinelectronics (charge to spin and vice versa) Žutić et al. (2004) and spincaloritronics (heat to spin and vice versa) Bauer et al. (2012). While the former relies on electrons, the latter may disregard electrons as fundamental carriers in favor of collective magnetic excitations, i. e., magnons, thereby circumventing Joule heating.

A prominent magnonic heat-to-spin conversion phenomenon, which promises temperature control as well as waste-heat recovery, is the spin Seebeck effect (SSE) Uchida et al. (2010), comprising a spin current in a magnetic insulator as response to an applied temperature gradient. Magnons that “flow down” the gradient carry spin from the hot to the cold side of the sample. Accumulated at these ends, the spin diffuses into an adjacent heavy metal layer and gets converted into a transverse charge current by the inverse spin Hall effect Saitoh et al. (2006).

While the SSE is natural to ferromagnets, it does not show up in uniaxial collinear antiferromagnets, because of their spin-degenerate magnon bands. Only an external magnetic field, which causes a Zeeman splitting of the magnon bands, introduces nonzero spin Seebeck signals Ohnuma et al. (2013); Cheng et al. (2014); Brataas et al. (2015); Seki et al. (2015); Wu et al. (2016); Rezende et al. (2016). Thus, the status quo is that the heat-to-spin conversion by means of the SSE is possible in either ferromagnets (e. g., LaY2Fe5O12 Uchida et al. (2010)) or uniaxial collinear antiferromagnets or paramagnets (e. g., MnF2 Wu et al. (2016) and GGG Wu et al. (2015), respectively) in magnetic fields, or biaxial collinear antiferromagnets (e. g., NiO Holanda et al. (2017)) with nondegenerate magnon bands in zero field.

An alternative to the SSE is offered by the spin Nernst effect (SNE), which describes a transverse spin current as a response to a temperature gradient in magnetic insulators. It is found both in ferromagnets Kovalev and Zyuzin (2016); Han and Lee (2017); Wang and Wang (2018); Mook et al. (2018), collinear antiferromagnets Cheng et al. (2016); Zyuzin and Kovalev (2016); Shiomi et al. (2017a); Mook et al. (2018), and paramagnets Zhang et al. (2018). However, its proportionality to the strength of spin-orbit coupling (SOC) renders the heat-to-spin conversion rather inefficient. Therefore, it is about time to consider spin transport in a different material class, namely in noncollinear antiferromagnetic insulators (NAIs).

Herein, we show that NAIs are, in principle, materials for the generation of bulk magnon spin currents in zero magnetic field and without SOC. Taking a single kagome sheet of the NAI potassium iron jarosite KFe3(OH)6(SO4)2 as an example, we identify spin Seebeck and planar spin Nernst signals due to in-plane polarized bulk spin currents both within Kubo transport theory as well as atomistic spin dynamics simulations. Using superordinate symmetry arguments, these SSEs and planar SNEs are established as the magnonic version of spin-polarized electron currents in noncollinear antiferromagnetic metals Železný et al. (2017); Kimata et al. (2019).

Heat-To-Spin Conversion in Potassium Iron Jarosite.

We consider a single kagome sheet of potassium iron jarosite, KFe3(OH)6(SO4)2, which is an electrically insulating mineral built from Fe kagome planes [cf. Fig. 1(a)] stacked along the $c$ direction in ABC sequence Townsend et al. (1986). The almost classical $S=5/2$ spins order below $65\,\mathrm{K}$ in the positive vector chiral (PVC) phase Inami et al. (1998, 2000) depicted in Fig. 1(b). This phase is characterized by a positive $z$ component of the vector spin chirality $\boldsymbol{\kappa}=\boldsymbol{S}_{1}\times\boldsymbol{S}_{2}+\boldsymbol{S}_{2}\times\boldsymbol{S}_{3}+\boldsymbol{S}_{3}\times\boldsymbol{S}_{1}$, where $\boldsymbol{S}_{i}$ ($i=1,2,3$) are the three spins in the magnetic unit cell as indicated in Figs. 1(a) and (b).

The two-dimensional (2D) spin Hamiltonian Elhajal et al. (2002); Yildirim and Harris (2006)

[TABLE]

includes antiferromagnetic exchange $J_{ij}$ between nearest ($3.18\,\mathrm{meV}$ Matan et al. (2006); Yildirim and Harris (2006)) and next-nearest neighbors ($0.11\,\mathrm{meV}$ Matan et al. (2006); Yildirim and Harris (2006)), which—due to geometric frustration—favors any classical $120^{\circ}$ ground state. The Dzyaloshinskii-Moriya interaction (DMI) Dzyaloshinsky (1958); Moriya (1960) between nearest neighbors is described by the vector $\boldsymbol{D}_{ij}=\pm(D_{\parallel}\hat{\boldsymbol{n}}_{ij}+D_{z}\hat{\boldsymbol{z}})$ [positive (negative) sign for counterclockwise (clockwise) circulation], possessing out-of-plane ($D_{z}=-0.062J$ Matan et al. (2006); Yildirim and Harris (2006)) as well as in-plane components ($D_{\parallel}=0.062J$ Matan et al. (2006); Yildirim and Harris (2006)). The latter arise because the kagome planes are no mirror planes Elhajal et al. (2002). $\boldsymbol{D}_{ij}$ is orthogonal to the $ij$ bond and $\hat{\boldsymbol{n}}_{ij}=\hat{\boldsymbol{n}}_{ji}$ is an in-plane unit vector as shown in Fig. 1(a). A sign convention opposite to that of Ref. Elhajal et al., 2002 is used: $D_{z}<0$ stabilizes the PVC phase Elhajal et al. (2002) and $D_{\parallel}$ causes a tiny out-of-plane canting Elhajal et al. (2002) (canting angle $1.98^{\circ}$ Matan et al. (2006); Yildirim and Harris (2006)). Finally, we consider a magnetic field $\boldsymbol{B}$ along $z$ direction (with g-factor $g=2.13$ Grohol et al. (2005) and Bohr’s magneton $\mu_{\mathrm{B}}$).

At zero magnetic field the spin textures of adjacent kagome layers in bulk KFe3(OH)6(SO4)2 are exactly opposite Inami et al. (1998, 2000) due to weak interlayer coupling ($\approx-0.03\,\mathrm{meV}$ Matan et al. (2011)) that is neglected here. Upon application of a sufficiently large magnetic field of $B_{z}\approx-17\,\mathrm{T}$ Matan et al. (2011); Fujita et al. (2012) the spin orientations in every second layer flip and the kagome sheets exhibit identical magnetic orders Grohol et al. (2005); Matan et al. (2011); Fujita et al. (2012). As we show in the following, identical textures are important to ensure a finite spin current generation upon application of a temperature gradient. Thus, the results obtained for the 2D model at hand apply to the actual bulk material for $B_{z}\lesssim-17\,\mathrm{T}$.

Taking this magnetic field into account, we determine the resulting canting angle numerically ($\approx 2.85^{\circ}$) and carry out linear spin-wave calculations (cf. Sec. I A of the Supplemental Material (SM) Sup ). We obtain the magnon energies $\varepsilon_{n\boldsymbol{k}}$ (with $n=1,2,3$) shown in Fig. 1(d) along high symmetry lines depicted in Fig. 1(c). Following Ref. Okuma, 2017, we calculate the spin expectation values of magnons in the lowest band close to the Brillouin zone center [Fig. 1(e)]. We find the double winding of the magnon spin direction about the $\Gamma$ point known from Ref. Okuma, 2017. This spin-momentum locking suggests the possibility of net spin currents in nonequilibrium.

We are interested in the magnetothermal transport tensor $\varUpsilon_{\mu\nu}^{\gamma}$, that mediates between the temperature gradient $\nabla_{\nu}T$ in $\nu$ direction and the nonequilibrium spin current density $\langle j_{\mu}^{\gamma}\rangle$ of $\gamma$ spin in $\mu$ direction: $\langle j_{\mu}^{\gamma}\rangle=\varUpsilon_{\mu\nu}^{\gamma}(-\nabla_{\nu}T)$. This tensor comprises the SSE (diagonal elements, $\mu=\nu)$, the SNE (off-diagonal elements, $\mu\neq\nu$; antisymmetric part of $\varUpsilon^{\gamma}$), and the planar SNE (symmetric part of $\varUpsilon^{\gamma}$). Applying Kubo’s theory Kubo et al. (1957); Mahan (2000); Matsumoto et al. (2014); Zyuzin and Kovalev (2016) and considering only the intraband contributions proportional to a phenomenological magnon relaxation time $\tau$, we find (cf. Sec. I B of SM Sup for the derivation and a discussion of approximations)

[TABLE]

$(J_{\boldsymbol{k},\mu}^{\gamma})_{nn}$ and $(Q_{\boldsymbol{k},\mu})_{nn}$ denote diagonal matrix elements of the spin and heat current operators, respectively. $\rho=[\exp(\beta(G\mathcal{E}_{\boldsymbol{k}})_{nn})-1]^{-1}$ is the Bose-Einstein distribution function, $\beta=(k_{\mathrm{B}}T)^{-1}$ with $k_{\mathrm{B}}$ denoting Boltzmann’s constant, and $V$ is the sample’s volume. $G$ is the bosonic metric and $\mathcal{E}_{\boldsymbol{k}}$ the paraunitarily diagonalized Hamiltonian, containing the magnon energies (cf. Sec. I A of SM Sup ).

Eq. (2) describes the time-odd part of the full magnetothermal transport tensor (cf. Sec. I B of SM Sup ). To see so, recall that $J_{\boldsymbol{k},\mu}^{\gamma}$ is even but $Q_{\boldsymbol{k},\mu}$ odd under time reversal. Thus, reversal of the magnetic texture reverses the sign of $\varUpsilon_{\mu\nu}^{\gamma}$, which is well-known for the SSE in ferromagnets Uchida et al. (2010). This also explains the absence of the SSE in those antiferromagnets for which time reversal can be “repaired” by a sublattice swap: such antiferromagnets are effectively time-even and as such incompatible with a time-odd transport response. In zero field, bulk KFe3(OH)6(SO4)2 exhibits such a symmetry due to the opposite spin textures of adjacent kagome sheets and we expect zero $\varUpsilon$. This is why we consider the spin-flopped phase with $B_{z}<-17\,\mathrm{T}$ modeled by a single layer.

Applying Eq. (2) to the 2D model of KFe3(OH)6(SO4)2, we calculate the elements $\varUpsilon^{\gamma}_{\mu\nu}$ for $\mu,\nu,\gamma=x,y$ (in-plane transport of in-plane polarized spins); results are shown in Fig. 2 111The results in Fig. 2 were obtained for a realistic effective magnon relaxation lifetime $\tau=0.1\,\mathrm{ns}$. We divided the result of the two-dimensional integral in Eq. (2) by the kagome interlayer distance of $0.57\,\mathrm{nm}$ Townsend et al. (1986); Grohol et al. (2003) to obtain three-dimensional units.. There are several nonzero elements, which are identical in modulus (red line); the transport tensor assumes the form

[TABLE]

Consequently, when applying $\boldsymbol{\nabla}T$ in $x$ or $y$ direction, there is a longitudinal magnon particle current density, consisting of $y$ spin-polarized magnons (diagonal elements of $\varUpsilon^{y}$): there is a SSE. Moreover, there is a transverse $x$-polarized spin current (off-diagonal elements of $\varUpsilon^{x}$), i. e., a planar SNE. At $20\,\mathrm{K}$ we find a spin Seebeck coefficient of about $60\,\mathrm{keV/(Km)}$ (left ordinate in Fig. 2). Assuming an inverse-spin-Hall spin-to-charge current conversion factor of $1.3\times 10^{-4}\,\mathrm{Vs/(Am)}$ Basso et al. (2016) in an adjacent platinum layer, this corresponds to a spin Seebeck conversion factor (SSCF) of $0.2\,\mathrm{\muup V/K}$ (right ordinate in Fig. 2). This is to be compared with values of ferrimagnetic YIG ($\lesssim 5\,\mathrm{\muup V/K}$ Kikkawa et al. (2015)) or of collinear antiferromagnets in magnetic fields like Cr2O3 ($0.015\,\mathrm{\muup V/K}$ at $14\,\mathrm{T}$ and $35\,\mathrm{K}$ Seki et al. (2015)), MnF2 ($41.2\,\mathrm{\muup V/K}$ at $14\,\mathrm{T}$ and $15\,\mathrm{K}$ Wu et al. (2016)), or $\alpha$-Cu2V2O7 ($0.1\,\mathrm{\muup V/K}$ at $5\,\mathrm{T}$ and $2\,\mathrm{K}$ Shiomi et al. (2017b)). Hence, the SSE in KFe3(OH)6(SO4)2 is well within experimental range; the same arguments apply to the planar SNE.

In contrast to the aforementioned collinear antiferromagnets that exhibit the SSE only in magnetic fields, we also obtain a SSE in zero magnetic field (cf. Sec. II of SM Sup ), which would be experimentally accessible in a single kagome sheet.

Computer experiment.

Since spin is not conserved, spin currents are detected indirectly by measurement of the observable spin accumulation they bring about in samples with finite dimensions; standard means include the inverse spin Hall effect in an adjacent normal metal layer Saitoh et al. (2006), spin torque in an adjacent ferromagnet Locatelli et al. (2013), or magnetooptical Kerr microscopy Kato et al. (2004). We now demonstrate that the SSE and planar SNE in NAIs cause finite nonequilibrium spin accumulations.

The actual experimental situation, in which a temperature gradient is applied to the magnet, is simulated by relying on classical atomistic spin dynamics simulations based on the stochastic Landau-Lifshitz-Gilbert equation Evans et al. (2014). We set up a rectangular stripe of a single KFe3(OH)6(SO4)2 kagome sheet (built from about $50\,000$ spins) with finite width in $y$ direction and periodic boundary conditions along the longer $x$ direction. The orientation of the kagome triangles and the spin ordering is as indicated in Fig. 1(b). Each spin is coupled to its own heat bath at a spatially varying temperature as shown in Fig. 3(a); the temperature profile exhibits two opposite gradients in $x$ direction. Then, inspired by Ref. Ritzmann et al., 2014, we measure a position-resolved steady-state nonequilibrium spin accumulation $\langle\Delta\boldsymbol{S}_{i}\rangle$ (cf. Sec. III of SM Sup for technical details).

The position-resolved $x$, $y$, and $z$ components of $\langle\Delta\boldsymbol{S}_{i}\rangle$ are shown in panels (b), (c) and (d) of Fig. 3, respectively. In panel (b), an accumulation of $x$ spin is observed at the edges of the sample in those regions with a finite $\nabla_{x}T$ (cf. red and blue horizontal stripes), indicating a transverse spin current as expected from the off-diagonal elements of $\varUpsilon^{x}$ [cf. Eq. (3)]. In contrast, there is zero $x$ spin accumulation at the ends of the gradients, which is in agreement with zero diagonal elements of $\varUpsilon^{x}$. Overall, Fig. 3(b) proves the existence of the planar SNE with $x$-polarized transverse spin currents.

In Fig. 3(c), a finite $y$ spin accumulation is observed at the ends of the thermal gradients (cf. red and blue vertical stripes), which is in accord with the nonzero diagonal elements of $\varUpsilon^{y}$ [cf. Eq. (3)]. However, no $y$ spin accumulates at the edges of the sample (zero off-diagonal elements of $\varUpsilon^{y}$). These results demonstrate the existence of the SSE with $y$-polarized longitudinal spin currents.

We also observe a longitudinal accumulation of $z$ spin in Fig. 3(d). It is caused by the small out-of-plane canting induced by the in-plane DMI, due to which all magnons acquire a spin component in $z$ direction (cf. Sec. IV A of SM Sup ). This effect is just the usual SSE associated with ferromagnetism.

In Sec. III D of the SM Sup we show that reversal of the magnetic texture leads to a sign reversal of spin accumulations, unambiguously relating the accumulations with the time-odd part of $\varUpsilon$, which is in accordance with theory. This finding corroborates further that spin current responses of kagome lattices with opposite textures cancel out.

Origin of the SSE and SNE.

The qualitative results we have obtained here, namely the existence of both a SSE and SNE, are not limited to the material under consideration. They equally apply to any kagome antiferromagnet in the PVC phase as is evident from superordinate symmetry considerations.

The slightly out-of-plane tilted PVC phase has a three-fold rotational axis pointing out of the plane ($C_{3z}$) and a $M^{\prime}_{x}$ time-reversal mirror plane whose normal points along $x$ [Fig. 1(b)]. Applying Neumann’s principle Neumann and Meyer (1885), i. e., requiring the magnetothermal transport tensor to be invariant under the magnetic crystal symmetries, the shape of $\varUpsilon$ in Eq. (3) can be derived rigorously (cf. Sec. IV A of SM Sup ). In essence, the SSE and planar SNE are allowed to exist, because the symmetry of the noncollinear PVC texture is too low to forbid spin currents.

This finding is in accordance with the symmetry-restricted spin transport tensor shapes studied in Ref. Seemann et al., 2015 and the electronic spin-polarized currents in noncollinear antiferromagnetic metals Železný et al. (2017); Kimata et al. (2019) 222We recall that we have restricted our focus to the time-odd part of $\varUpsilon$ given in Eq. (2) (Sec. I B of SM Sup ); thus, the transport tensor expressions we list here do not appear in their full symmetry-allowed shape Seemann et al. (2015)..

In the Introduction we promised spin currents in zero field and without SOC. Armed with the above symmetry arguments, we construct a gedanken magnet that keeps these promises. Starting from a KFe3(OH)6(SO4)2 sheet, the limit of zero SOC is obtained by setting the DMI zero. This results in a perfectly coplanar PVC phase (zero magnetization) stabilized by the second-nearest neighbor exchange Harris et al. (1992); Chubukov (1992); Chernyshev and Zhitomirsky (2015). In addition to the $C_{3z}$ and $M^{\prime}_{x}$ symmetries, there are now a $M_{y}$ and a $M^{\prime}_{z}$ symmetry. The latter are still insufficient to forbid spin-polarized currents (Sec. IV B of SM Sup ) and the form of $\varUpsilon$ remains as in Eq. (3). Thus, assuming a single kagome sheet with a single magnetic domain, we obtain both an SSE and SNE in zero field 333In actual experiments, one might need a tiny in-plane magnetic field to energetically favor one particular domain..

We have numerically verified the SSE and planar SNE in the gedanken magnet by calculating $\varUpsilon$ by Eq. (2) (Sec. IV B of SM Sup ). There are three Goldstone modes (cf. Fig. S3 of SM Sup ) Harris et al. (1992); Chubukov (1992), associated with a global rotation of the texture; all three magnon branches contribute to transport (cf. Fig. S4 of SM Sup ). Unfortunately, we are not aware of a material which strictly realizes this model, owing to the inevitibility of nearest-neighbor DMI on the kagome lattice. Since a single KFe3(OH)6(SO4)2 kagome sheet deviates only slightly from the gedanken magnet, single-crystalline bulk KFe3(OH)6(SO4)2 Grohol et al. (2005) serves as a candidate material on which a proof-of-principle experiment can be performed. Other iron jarosites Grohol et al. (2003) are also candidates, e. g., silver iron jarosite AgFe3(OH)6(SO4)2, which orders below $59\,\mathrm{K}$ Matan et al. (2011) and takes $B_{z}<-10\,\mathrm{T}$ Matan et al. (2011) to ensure identical layer spin textures.

Other antiferromagnetic textures.

By extending the symmetry considerations to other noncollinear antiferromagnetic textures, it becomes evident that the SSE and planar SNE are inherent phenomena in NAIs.

In contrast to the PVC phase, the negative vector chiral (NVC) phase (cf. Fig. S5 of SM Sup ), which is stabilized for $D_{z}>0$ Elhajal et al. (2002) (recall opposite sign convention) and characterized by $\kappa_{z}<0$, does not have a $C_{3}$ axis. Thus, symmetry considerations are restricted to an $M^{\prime}_{x}$ time-reversal mirror plane, yielding (Sec. IV C of SM Sup )

[TABLE]

Since in general $\varUpsilon^{y}_{xx}\neq\varUpsilon^{y}_{yy}$ and $\varUpsilon^{x}_{xy}\neq\varUpsilon^{x}_{yx}$, there is on top of the planar SNE of $x$-polarized spin (symmetric part of $\varUpsilon^{x}$), also a “magnetic” 444The adjective “magnetic” is inspired by the magnetic spin Hall effect studied in Refs. Železný et al., 2017, Chen et al., 2018 and Kimata et al., 2019. SNE (antisymmetric part of $\varUpsilon^{x}$). A $90^{\circ}$ rotated version of the NVC phase appears in cadmium kapellasite due to local anisotropies Okuma et al. (2017) (Sec. IV D of SM Sup ).

Concerning three-dimensional NAIs, we note that pyrochlores with the all-in–all-out texture as, e. g., Sm2Ir2O7 Donnerer et al. (2016), are expected to exhibit a planar SNE with the property that the force, the current, and the transported spin are mutually orthogonal to each other (Sec. IV E of SM Sup ).

Conclusion.

We showed that NAIs can exhibit bulk magnon spin currents, thus complementing the recent proposal of an interfacial SSE Flebus et al. (2018). As results, NAIs offer the combined advantages of nonelectronic spin transport and antiferromagnetism. They may replace ferromagnets as spin-active components of next-generation spin(calori)tronic devices and introduce a paradigm of an “antiferromagnetic insulator spincaloritronics”, as the magnonic pendant to the thriving field of “antiferromagnetic spinelectronics” Baltz et al. (2018); Jungwirth et al. (2018); Gomonay et al. (2018); Duine et al. (2018); Železnỳ et al. (2018); Němec et al. (2018); Šmejkal et al. (2018); Jungfleisch et al. (2018).

Besides an experimental proof of principle of our quantitative predictions for KFe3(OH)6(SO4)2 our work calls for the development of a theory of magnon spin and heat diffusion Cornelissen et al. (2016) in NAIs, an investigation of the influence of noncollinearity-induced magnon-magnon interactions Chernyshev and Zhitomirsky (2015) on spin transport, of the dynamics of noncollinear antiferromagnetic domain walls Lhotel et al. (2011) in temperature gradients, and a material search for NAIs with ordering temperatures above room temperature.

By virtue of Onsager’s reciprocity relation Onsager (1931), the existence of SSEs and SNEs in noncollinear antiferromagnets, immediately implies that of the spin Peltier and spin Ettingshausen effects. A recent experimental proof Sola et al. (2018) of the reciprocity between the spin Seebeck and spin Peltier effect in a YIG/Pt bilayer could be conceptually carried over to noncollinear antiferromagnets.

Acknowledgements.

This work is supported by SFB 762 of Deutsche Forschungsgemeinschaft (DFG).