Abstract
Spin current–a flow of electron spins without a charge current–is an ideal information carrier free from Joule heating for electronic devices. The celebrated spin Hall effect, which arises from the relativistic spinorbit coupling, enables us to generate and detect spin currents in inorganic materials and semiconductors, taking advantage of their constituent heavy atoms. In contrast, organic materials consisting of molecules with light elements have been believed to be unsuited for spin current generation. Here we show that a class of organic antiferromagnets with checkerplate type molecular arrangements can serve as a spin current generator by applying a thermal gradient or an electric field, even with vanishing spinorbit coupling. Our findings provide another route to create a spin current distinct from the conventional spin Hall effect and open a new field of spintronics based on organic magnets having advantages of small spin scattering and long lifetime.
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Introduction
Organic metals and semiconductors^{1} possess a variety of features not shared by inorganic materials, e.g., light, flexible, and toxicelementfree. They have been rapidly developed over the past decades for use in consumer electronic devices, such as organic transistors, lightemitting diodes, and piezo actuators. These accomplishments, in combination with recent evolutions of inorganic spintronics based on spin current physics, have promoted a new field, i.e., organic spintronics. Now significant efforts are being made to elucidate spin transport phenomena in organic semiconductors^{2,3,4}. However, organic spintronics devices are actually not purely organic but are hybrid with inorganic materials, because the generation of spin current basically requires an inorganic magnetic electrode. In fact, attempts for exploiting organic materials as the spin current generator are quite limited^{5,6}.
Here, we theoretically propose a microscopic mechanism of spin current generation in organic materials utilizing an archetypal antiferromagnet. Figure 1a provides a schematic illustration of the present spin current generation in the antiferromagnetic (AFM) state, where the up and down spins aligned on the molecular checker plate play a role of a spinrectifier converting a heatcurrent driven by a thermal gradient, or an electron current by an electric field, into the spin current. When we rotate the external field with respect to the crystal axes in the twodimensional plane, the direction of the generated spin current rotates in the opposite direction as shown in Fig. 1b. The directional dependence is strikingly different from the conventional spin Nernst and spin Hall effects^{7,8,9,10,11,12}, in which the spin current always flows perpendicular to the field direction. As a platform of this phenomenon, we focus on an organic antiferromagnet \(\kappa\)(BEDTTTF)\({}_{2}\)Cu[N(CN)\({}_{2}\)]Cl (abbreviated as \(\kappa\)Cl).
Results
Crystal structure and model
\(\kappa\)Cl is a wellstudied insulator, showing a variety of cooperative phenomena, e.g., AFM ordering, insulatortometal transition, and superconductivity, at low temperatures and/or under pressures^{13,14,15,16,17,18}. The crystal structure is composed of an alternate stacking of twodimensional conducting BEDTTTF (abbreviated as ET) layers and insulating Cu[N(CN)\({}_{2}\)]Cl layers. Figure 2a shows the molecular arrangement (called \(\kappa\)type) in the conducting layer, where four ET molecules in the unit cell form two kinds of dimers with different orientations, termed \(A\) and \(B\), connected by a glide operation (mirror and half translation).
This class of organic materials is known to show a simple electronic structure composed of frontier molecular orbitals^{19,20}. In the \(\kappa\)type materials, the frontier orbitals in each ET dimer become strongly hybridized by the intradimer transfer integral shown in Fig. 2b, and constitute bonding and antibonding orbitals. They result in four bands as there are two dimers in the unit cell: two lower(higher)energy bands are from the (anti)bonding orbitals, as shown in Fig. 2c. The system has three electrons per two dimers on average, and hence, the four bands are threequarter filled.
In the last few decades, extensive studies have been made for understanding the cooperative phenomena in this system^{21,22,23}. Most of them, however, are based on the singleband picture, where the two fully occupied bands are disregarded (see the broken lines in Fig. 2c). This approach is justified in the large dimerization limit^{19}, where the crystallographic distinction of the \(A\) and \(B\) dimers is lost. In other words, the glide symmetry in the molecular arrangement in the conducting layer was disregarded in the previous studies. In the following, we will discuss that the breaking of the glide symmetry by the AFM ordering plays an essential role in a peculiar spin current generation.
We investigate electronic structures and spin current transport properties of \(\kappa\)Cl based on the Hubbard model, taking into account the distinct two types of dimers and the anisotropy in the transfer integrals between them^{19}, \(({t}_{{\rm{a}}},{t}_{{\rm{p}}},{t}_{{\rm{q}}},{t}_{{\rm{b}}})=(0.207,0.102,0.043,0.067)\) eV, evaluated by a firstprinciples calculation^{24} (see Fig. 2b). At threequarter filling where the number of electrons in the unit cell is equal to \(6\), the ground state exhibits a metaltoinsulator transition from a paramagnetic (PM) phase to an AFM phase on increasing the intramolecular Coulomb interaction \(U\)^{19,25}.
Spin splitting
A crucial feature in the AFM state of \(\kappa\)Cl is that up and down spins are situated on the dimers with the different orientations as shown in Fig. 2b, resulting in the glide symmetry breaking with respect to the \(yz\) plane. Here we consider the glide operation not acting on the spins. The molecular orientation makes the AFM state not invariant under the combination of time reversal and spatial translation operations, unlike simple N\(\acute{{\rm{e}}}\)eltype AFM state, e.g., on the square lattice. This situation gives rise to an energy band splitting depending on the spins, which has been overlooked previously. Figure 2d shows the band structure in the AFM state, calculated within the selfconsistent meanfield theory (see Methods). The spin splitting appears in the whole Brillouin zone except on the \({k}_{x}\), \({k}_{y}\)axes and the zone boundary as shown in Fig. 2e.
The origin of the spin splitting is understood from the realspace anisotropy induced by the AFM ordering as follows. Figure 3 shows the effective interdimer transfer integrals between the antibonding orbitals, calculated by the secondorder perturbation with respect to the interorbital hybridizations (see Methods). In the PM phase, as shown in Fig. 3a, b, the \(A\) and \(B\) dimers show different realspace anisotropies owing to the molecular orientations, but the anisotropies are symmetric with respect to the glide operation and do not depend on the spin degree of freedom. In the AFM phase, in contrast, the transfer integrals for upspin electrons on the \(A\) dimer (Fig. 3c) and downspin electrons on the \(B\) dimer (Fig. 3f) are enhanced, whereas their counterparts (Fig. 3d, e) are reduced. This spindependent anisotropy leads to the spin splitting.
The realspace anisotropies also show up in the effective spin exchange interactions in the Heisenberg model, derived from the above Hubbard model. Note that the system retains \(SU(2)\) symmetry because of the absence of the spinorbit coupling. Figure 4a shows the spatial distributions of the nearestneighbor (NN) exchange interactions \(J\) and \(J^{\prime}\), and the nextnearestneighbor (NNN) interactions \(K\) and \(K^{\prime}\). \(K\) and \(K^{\prime}\) arise from fourthorder perturbation processes with respect to the NN transfer integrals (see Methods). As shown in Fig. 4b, \(K^{\prime}\) becomes much smaller than \(K\) for realistic parameters. Then, the AFM magnon dispersion of the Heisenberg model exhibits a spin splitting as shown in Fig. 4c, where we take \(K=2\) meV and \(K^{\prime} =0\) for simplicity, and \(J=80\) meV and \(J^{\prime} =20\) meV^{14} (see Methods). Similar spin splitting was reported in noncentrosymmetric systems with the spinorbit coupling^{26,27}, but the present mechanism requires neither noncentrosymmetry nor the spinorbit coupling.
Spin current by a thermal gradient
The spinsplit magnon excitations lead to a spin current generation. Figure 4d shows the offdiagonal spin current conductivity, along the \(x\)axis with respect to the thermal gradient along the \(y\)axis, \({\chi }_{xy}^{{\rm{SQ}}}\), as a function of temperature \(T\) and the exchange interaction \(K\), calculated by the linear response theory (see Methods). The range of \(T\) is chosen well below the N\(\acute{{\rm{e}}}\)el temperature of \(\kappa\)Cl, \(23\) K. The polarization of the spin current is parallel to the AFM moment, and the damping factor \(\eta\) is fixed at \(1\) meV. We obtain nonzero \({\chi }_{xy}^{{\rm{SQ}}}\) for \(T> 0\) and \(K> 0\), which monotonically increases in proportion to \({T}^{2}\) and \(K\). Remarkably, the conductivity tensor \({{\boldsymbol{\chi }}}^{{\rm{SQ}}}\) is symmetric, \({\chi }_{xy}^{{\rm{SQ}}}={\chi }_{yx}^{{\rm{SQ}}}\), with vanishing diagonal elements, \({\chi }_{xx}^{{\rm{SQ}}}={\chi }_{yy}^{{\rm{SQ}}}=0\). This leads to the peculiar fieldangle dependence that we showed in Fig. 1b, which is distinct from the conventional spin Nernst effect where the spin current is always perpendicular to the thermal gradient.
This spin current generation is a direct consequence of the magnon dispersion in Fig. 4c which indicates that the upspin magnon has high mobility along the \((1,1)\) and \((1,1)\) directions, while the downspin magnon has along the \((1,1)\) and \((1,1)\) directions. When the temperature gradient is applied along the \(y\)axis as shown in Fig. 1a, the up and downspin magnons are rectified toward the \((1,1)\) and \((1,1)\) directions, respectively, in a symmetric way. Accordingly, a pure spin current, where the net magnon current is canceled out, is generated along the \(x\)axis. This gives rise to the positive transverse \({{\boldsymbol{\chi }}}^{{\rm{SQ}}}\) in Fig. 4d. On the other hand, if the temperature gradient is parallel to the \((1,1)\) direction, while the transverse component disappears, a net upspin magnon current is generated parallel to the field as a result of the incomplete cancellation (see Fig. 1b). This provides a finite longitudinal component of \({{\boldsymbol{\chi }}}^{{\rm{SQ}}}\) in the rotated coordinate, which is consistent with the symmetric form of the conductivity tensor.
We find that \({\chi }_{xy}^{{\rm{SQ}}}\) is inversely proportional to the damping factor \(\eta\) and diverges in the clean limit (\(\eta =0\)), in analogy with the diagonal thermal conductivities \({\kappa }_{xx}\) and \({\kappa }_{yy}\) (see Supplementary Fig. 1a). This indicates that the ratio \(\alpha \equiv 2J{\chi }_{\mu \nu }^{{\rm{SQ}}}/\hslash {\kappa }_{\nu \nu }\), which is used in the literatures as the conversion rate from the heatcurrent to the spin current, does not depend on \(\eta\), however, depends on the field angle. Therefore, we choose \(\mu =x\) and \(\nu =y\) since \({\kappa }_{\nu \nu }\) is largest in this direction, considering its implication as the conversion rate. Figure 4e shows \(K\) dependences of \(\alpha\) at \({k}_{{\rm{B}}}T=0.5\) meV and \(1\) meV linearly increasing with \(K\), but almost independent of \(T\). The heatspin current conversion efficiency reaches \(\sim 5\)\(\%\) for the case of \(\kappa\)Cl, which is close to onequarter of that in Pt due to the strong spinorbit coupling^{28}.
Spin current by an electric field
Now we propose another way of a spin current generation, in carrier doped metallic regions. The carrier doping has recently been realized experimentally^{17,18}. We here focus on the electrondoping case where the AFM metallic state is stable in our model. Figure 5a shows the offdiagonal spin current conductivity induced by the electric field, \({\chi }_{xy}^{{\rm{SC}}}(={\chi }_{yx}^{{\rm{SC}}})\), as a function of the Coulomb interaction \(U\) and the number of electrons in the unit cell \(n\) in the ground state (see Methods). \({\chi }_{xy}^{{\rm{SC}}}\) is zero in the PM metallic and the AFM insulating phases, while it turns finite in the AFM metallic phase where the Fermi energy lies in the top band in Fig. 2d, whose spin splitting was shown in Fig. 2e. We note that the sign of \({\chi }_{xy}^{{\rm{SC}}}\) changes around \(n=6.2\), associated with the change in the Fermi surface topology as shown in the insets of Fig. 5a. The conductivity tensor is also symmetric with zero diagonal components and inversely proportional to the damping factor (see Supplementary Fig. 1b). This means that the fieldangle dependence is the same as that of \(\chi_{xy}^{\mathrm{{SQ}}}\) in the insulating case. It is comprehensible from the spindependent anisotropy of the electron transfers in Fig. 3c–e, reflected in the anisotropy in the spinsplit band in Fig. 2e with the same character as the magnon band in Fig. 4c. We define the chargespin current conversion rate by \(\beta \equiv 2e{\chi }_{yx}^{{\rm{SC}}}/\hslash {\sigma }_{xx}\), in analogy with \(\alpha\) above (the electrical conductivity \({\sigma }_{\nu \nu }\) becomes largest in the \(\nu =x\) direction due to the quasionedimensional Fermi surfaces). As shown in Fig. 5b, in the lightly doped region with small \({\sigma }_{xx}\), \(\beta\) becomes relatively large and approaches \(7\)\(\%\), comparable to the spin Hall effect in Pt^{29}, while in the highly doped region, it decreases because of the suppression of the AFM ordering and the spin splitting.
The spin current generation in our mechanism is expected to be observed at sufficiently low temperature compared to the N\(\acute{{\rm{e}}}\)el temperature, which is not determined for doped \(\kappa\)Cl. We anticipate it to be lower than the undoped case of \(23\) K, but to remain the same order in the lightly doped region^{30}.
Discussion
The present spin current generation is strikingly different from the conventional spin Nernst and spin Hall effects. In the conventional mechanisms, a spin current is activated by the spin–orbit coupling in noncentrosymmetric lattice structures. The conductivity tensor is antisymmetric, namely, the generated spin current is always perpendicular to the applied field direction and the conversion rate is invariant under the rotation of the field. The transverse conductivity converges to a finite value in the clean limit because of the dominant interband contributions^{7,8}. However, the strong spin–orbit coupling also disturbs the spin polarization of carriers via the spinflipping process.
In stark contrast, the present mechanism requires neither the spinorbit coupling nor spatial inversion symmetry breaking. The spin current conductivity is described by the symmetric tensor which results in the peculiar fieldangle dependence shown in Fig. 1b, and diverges in the clean limit due to the intraband contributions. In \(\kappa\)type ET systems, the DzyaloshinskiiMoriya interaction due to the spinorbit coupling is estimated to be a few Kelvin^{15,31}, which is much smaller than the NNN exchange interaction \(K\). Furthermore, this class of organic charge transfer salts is known to have relatively less impurities and lattice disorders compared to inorganic crystals and organic polymers. Indeed, the specific heat and thermal transport measurements^{32,33} suggest that the low temperature properties are well described by intrinsic contributions from electronic charge and spin degrees of freedom. These facts ensure a long spin lifetime in \(\kappa\)Cl, which facilitates the experimental detection. Although the phenomenon has a similarity with the spin current generation in ferromagnetic metals in the sense that the time reversal symmetry is lost, the net magnetization is absent in our system; this enables us to generate a pure spin current in contrast to the spinpolarized current in ferromagnets and has the advantage of small field leakage as discussed in AFM spintronics. These considerations lead us to conclude that our proposal provides a new type of spin current generation essentially distinct from the other existing mechanisms.
As a recent experimental progress relevant to our proposal, the threedimensional AFM structures in several \(\kappa\)type ET systems have been determined by the detailed analyses of the magnetization processes^{15}. It was found that \(\kappa\)Cl and \(\kappa\)Br show the same intralayer AFM structure as shown in Fig. 2b, but different interlayer stackings; the “inphase” stacking, where the interlayer NN spins are ferromagnetically aligned, is realized in \(\kappa\)Cl, while \(\kappa\)Br shows the “antiphase” stacking. This difference will give an effective way to verify the present spin current generation because in our mechanism the sign of generated spin current is reversed by the reversal of the AFM moment. This means that a net spin current is expected in \(\kappa\)Cl while it will be canceled out in \(\kappa\)Br. In addition, our mechanism also has the inverse effect similar to the inverse spin Nernst or spin Hall effect, i.e., the generation of a thermal gradient or an electrical voltage by spin current injection parallel to the AFM ordered ET layers, which will give another experimental approach.
The present spin current generation arises from AFM ordering in spatiallyoriented molecular orbitals. The molecular orbital degrees of freedom are fundamental and ubiquitous in organic materials. Meanwhile, similar orbital degrees of freedom are also found in inorganic materials, such as transition metal and rareearth compounds. Thus, our new mechanism can be applied to a wide range of AFM materials. In this perspective, therefore, our finding strikes out a new direction of materials exploration for spintronics without relying on the spinorbit coupling.
Methods
Meanfield approximation
The Hamiltonian of the Hubbard model on the \(\kappa\)type lattice is given by
where \({c}_{i\mu \sigma }\) and \({n}_{i\mu \sigma }(={c}_{i\mu \sigma }^{\dagger }{c}_{i\mu \sigma })\) are the annihilation operator and the number operator of an electron with a spin \(\sigma (=\uparrow ,\downarrow )\), on the frontier orbital of molecular site \(\mu (=a,b)\) in the \(i\)th dimer, respectively, \(U\) is the intramolecular Coulomb interaction, and \({t}_{{\rm{a}}}\) and \({t}_{ij}^{\mu \mu ^{\prime} }\) are the intermolecular transfer integrals shown in Fig. 2b. We treat the Coulomb interaction term within the meanfield approximation as \({n}_{i\mu \uparrow }{n}_{i\mu \downarrow }\simeq {n}_{i\mu \uparrow }\langle {n}_{i\mu \downarrow }\rangle +\langle {n}_{i\mu \uparrow }\rangle {n}_{i\mu \downarrow }\langle {n}_{i\mu \uparrow }\rangle \langle {n}_{i\mu \downarrow }\rangle\), and determine the expectation values selfconsistently so as to minimize the total energy of the system.
Effective electron transfer integrals
We divide the meanfield Hamiltonian in the AFM phase into three terms as \({{\mathcal{H}}}_{{\rm{MF}}}={{\mathcal{H}}}_{{\rm{intra}}}+{{\mathcal{H}}}_{{\rm{inter}}}+{{\mathcal{H}}}_{{\rm{AFM}}}\), where the first and second terms represent the intraorbital and interorbital transfer integrals, respectively, and the third term is the local AFM field. By taking the linear combinations of the original electron operators, we define the annihilation operator of an electron in the antibonding (bonding) orbital on the \(i\)th dimer as \({\tilde{c}}_{i\alpha (\beta )\sigma }=({c}_{ia\sigma }(+){c}_{ib\sigma })/\sqrt{2}\), and the three terms are given by
where the number operator is given by \({\tilde{n}}_{i\nu \sigma }={\tilde{c}}_{i\nu \sigma }^{\dagger }{\tilde{c}}_{i\nu \sigma }\), and \(\bar{\nu }=\beta\) (\(\alpha\)) for \(\nu =\alpha\) (\(\beta\)). The transfer integral between the neighboring dimers is given by \({\tau }_{ij}={\cal{U}}{t}_{ij}{\cal{U}}^{{\rm{T}}}\), by using the twobytwo unitary matrix \(\cal U\) satisfying \({({\tilde{c}}_{\alpha \sigma },{\tilde{c}}_{\beta \sigma })}^{{\rm{T}}}={\cal{U}}{({c}_{a\sigma },{c}_{b\sigma })}^{{\rm{T}}}\). The amplitude of the local AFM field is given by \(\delta =\langle {\tilde{n}}_{i\in A\uparrow }\rangle \langle {\tilde{n}}_{i\in A\downarrow }\rangle =\langle {\tilde{n}}_{i\in B\downarrow }\rangle \langle {\tilde{n}}_{i\in B\uparrow }\rangle\), where \({\tilde{n}}_{i\sigma }={\sum }_{\nu }{\tilde{n}}_{i\nu \sigma }\).
We treat \({{\mathcal{H}}}_{{\rm{inter}}}\) as the perturbation term and calculate the effective transfer integrals for the bonding and antibonding orbitals up to \({\mathcal{O}}({{\mathcal{H}}}_{{\rm{inter}}}^{2})\). In the \({\bf{k}}\) space, the meanfield Hamiltonian is described by the matrix form as \({{\mathcal{H}}}_{{\rm{MF}}}={\sum }_{{\bf{k}}\sigma }{{\bf{d}}}_{{\bf{k}}\sigma }^{\dagger }({H}_{{\bf{k}}\sigma }^{(0)}+{V}_{{\bf{k}}\sigma }){{\bf{d}}}_{{\bf{k}}\sigma }\), where \({H}_{{\bf{k}}\sigma }^{(0)}\) and \({V}_{{\bf{k}}\sigma }\) are the unperturbed and perturbed terms, respectively, given by \(4\times 4\) matrices. \({{\bf{d}}}_{{\bf{k}}\sigma }\) is the vector of the annihilation operators of the Bloch states, which is chosen so as to diagonalize the unperturbed term as \({\hat{H}}_{{\bf{k}}\sigma }^{(0)}{\bf{k}}{\nu }_{\sigma }^{\xi }\rangle ={\varepsilon }_{{\bf{k}}\nu {\sigma} }^{\xi }{\bf{k}}{\nu }_{\sigma }^{\xi }\rangle\), where \(\xi\)(\(=1,2\)) indicates the two bands in the bonding and antibonding bands each originating from the two dimers in the unit cell. The secondorder perturbation term \({H}_{{\bf{k}}\sigma }^{(2)}\) is decomposed into two \(2\times 2\) matrices for the antibonding (\(\alpha\)) and bonding bands (\(\beta\)) as \({H}_{{\bf{k}}\sigma }^{(2)}={h}_{{\bf{k}}\alpha \sigma }^{(2)}\oplus {h}_{{\bf{k}}\beta \sigma }^{(2)}\). The matrix element of \({h}_{{\bf{k}}\nu \sigma }^{(2)}\) is given by
where the indices \({\bf{k}}\) and \(\sigma\) are omitted for simplicity. By the Fourier transformation of \({H}_{{\bf{k}}\sigma }^{(2)}\), we obtain the effective transfer integrals shown in Fig. 3.
Nextnearestneighbor exchange interactions
From the Hubbard model in Eq. (1), we derive the effective NNN exchange interaction in the restricted space where each antibonding orbital is occupied by one hole due to the strong Coulomb interaction \(U\). The NNN exchange interaction is derived from the fourthorder perturbation process with respect to the interdimer transfer integrals, which is given by
where \({\mathcal{P}}\) and \({\mathcal{Q}}\) are the projection operators onto inside and outside of the restricted space, respectively, and satisfy \({\mathcal{P}}+{\mathcal{Q}}=1\), \({\mathcal{V}}\) is the perturbation given by the third term in Eq. (1), \({{\mathcal{H}}}^{(0)}\) is the unperturbed Hamiltonian given by the first and second terms in Eq. (1), and \({E}_{I}\) is the energy of the initial eigenstate \(I\rangle\) of \({{\mathcal{H}}}^{(0)}\). The resultant exchange interaction on the NNN bond between the \(A\) dimers denoted by \(K\) in the middle panel of Fig. 4a is given by
where the indices \(ijk\) denote the three neighboring dimers, \({{\bf{S}}}_{i}\) is the spin operator of the \(i\)th dimer, and \(K\) is the NNN exchange constant. \(\tilde{J}\) is the NN exchange constant arising from the fourthorder perturbation process, which does not contribute to the magnon splitting. The details of the fourthorder process and the explicit form of \(K\) (and \(K^{\prime}\)) are given in Supplementary Note 2.
Linear spinwave approximation
The effective Heisenberg model involving the NNN exchange interaction is given by
where \(\langle ij\rangle\) and \(\langle ij\rangle ^{\prime}\) stand for the diagonal and horizontal NN bonds on the equilateral triangular lattice, \(\langle \langle ij\rangle \rangle\) and \(\langle \langle ij\rangle \rangle ^{\prime}\) are the NNN bonds shown in Fig. 4a. By using the HolsteinPrimakoff transformation, we obtain the linear spinwave Hamiltonian given by
where \({a}_{{\bf{k}}}\) and \({b}_{{\bf{k}}}\) are the Fourier transforms of the annihilation operators of magnons on the \(A\) and \(B\) dimers, respectively. The coefficients are given by
and
where \({{\bf{a}}}_{x}\) and \({{\bf{a}}}_{y}\) are the primitive translational vectors. \({{\mathcal{H}}}_{{\rm{LSW}}}\) is easily diagonalized by the standard Bogoliubov transformation, and the magnon energy dispersion shown in Fig. 4c is obtained.
Spin current conductivity to a thermal gradient
The spin current and energy current operators in the magnon system^{34} are given by
and
respectively. \({{\bf{P}}}_{{S}^{z}}\) and \({{\bf{P}}}_{E}\) are the spin polarization and the energy polarization operators defined by \({{\bf{P}}}_{{S}^{z}}=\hslash {\sum }_{i}{S}_{i}^{z}{{\bf{R}}}_{i}\) and \({{\bf{P}}}_{E}={\sum }_{i}{{\mathcal{H}}}_{i}{{\bf{R}}}_{i}\), respectively, where \({{\bf{R}}}_{i}\) is the position vector of the center of the \(i\)th dimer and \({{\mathcal{H}}}_{i}\) is the local energy density defined by \({{\mathcal{H}}}_{{\rm{LSW}}}={\sum }_{i}{{\mathcal{H}}}_{i}\), by the Fourier transformation of Eq. (9). In the magnon system where the chemical potential is zero, the heatcurrent operator \({{\bf{J}}}_{Q}\) is identical to the energy current operator \({{\bf{J}}}_{E}\). We note that the spin is a conserved quantity and the spin current is well defined here since our model does not include the spinorbit coupling. In the linear response theory, the spin current conductivity to a static thermal gradient is given by
where \(\mu\) and \(\nu\) represent the spatial axes \(x\) and \(y\). The spincurrentheatcurrent response function \({Q}_{\mu \nu }^{{\rm{SQ}}}(\omega )\) is given by the Kubo formula
where \({{\bf{J}}}_{{S}^{z}}(t)\) is the Heisenberg representation of the spin current operator, \(\eta\) is the damping factor, \(V\) is the volume of the system, and \({\langle \cdots \rangle }_{{\rm{eq}}}\) represents the thermal average under the temperature \(T\).
Spin current conductivity to an electric field
The spin current and charge current operators are defined by
and
respectively. \({{\bf{P}}}_{{s}^{z}}\) and \({\bf{P}}\) are the spin \({s}^{z}\) polarization and the electric polarization operators defined by \({{\bf{P}}}_{{s}^{z}}=\hslash {\sum }_{i}{s}_{i}^{z}{{\bf{r}}}_{i}\) and \({\bf{P}}=e{\sum }_{i}{{\bf{r}}}_{i}\), respectively, where \({s}_{i}^{z}=\frac{{n}_{i\uparrow }{n}_{i\downarrow }}{2}\) is the spin operator of the \(i\)th molecule at the position vector \({{\bf{r}}}_{i}\). The spin current conductivity to an static electric field is given by
The spincurrentchargecurrent response function \({Q}_{\mu \nu }^{{\rm{SC}}}(\omega )\) is given by the Kubo formula
where \({{\bf{J}}}_{{S}^{z}}(t)\) is the Heisenberg representation of the spin current operator, \(\gamma\) is the damping factor, and \({\langle \cdots \rangle }_{0}\) represents the average with respect to the ground state.
Data availability
Data are available from the corresponding author upon reasonable request.
Code availability
Computer codes used in this study are available from the corresponding author upon reasonable request.
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Acknowledgements
This work is supported by GrantinAid for Scientific Research, No. JP16K17731, No. JP19K03723, No. JP18H04296 (JPhysics), No. JP18K13488, No. JP15H05885 (JPhysics), No. JP19K03752, and No. JP26400377 from MEXT (Japan).
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M.N., S.H., H.K., Y.Y., Y.M., and H.S. contributed to conception, execution, and writeup this project. The numerical and analytical calculations were performed by M.N.
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Naka, M., Hayami, S., Kusunose, H. et al. Spin current generation in organic antiferromagnets. Nat Commun 10, 4305 (2019). https://doi.org/10.1038/s4146701912229y
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DOI: https://doi.org/10.1038/s4146701912229y
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