Vector spin Seebeck effect and spin swapping effect in antiferromagnetic insulators with non-collinear spin structure

Antiferromagnets (AFs) are prospective for next-generation high-density and high-speed spintronic applications due to their negligible stray field and ultrafast spin dynamics, notwithstanding the challenges in detecting and manipulating AF order with no magnetization (M = 0). Among the AFs, non-collinear AFs are of particular interest because of their unique properties arising from the non-collinear spin structure and the small magnetization M. In this work, we describe the recently observed vector spin Seebeck effect in non-collinear LuFeO$_3$, where the magneto-thermovoltage under an in-plane temperature gradient, not previously observed, is consistent with the predicted spin swapping effect. Our results shed light on the importance of the non-collinear spin structure in the emerging spin phenomena in non-collinear AFs and offer a new class of materials for AF spintronics and spin caloritronics.


I. INTRODUCTION
Ferromagnets (FMs) with collinear moments and a large magnetization M have dominated spintronics and spin caloritronics [1,2]. The two directions of ± M of FM have been the basis for magnetic storage. A variety of magneto-transport effects (anomalous Hall effect, anisotropic magnetoresistance, tunnelling magnetoresistance, spin Hall magnetoresistance, among others) in addition to magnetometry can detect the large magnetization M, which can be manipulated by spin-transfer torque, spin-orbit torque and magnetic field. However, the large M is also susceptible to external magnetic field and stray field interference. In contrast, antiferromagnetic (AF) materials with zero or nearly zero magnetization are immune to the stray field issues. Some AFs also harbor high frequency excitations beneficial for high-speed devices. However, it is challenging to detect and manipulate AFs with M = 0 [3][4][5][6][7][8][9]. There have been reports of electrical switching and detection of the AF Néel vector in collinear AF metals and insulators [10][11][12][13], where the origin of the electrical signals remains controversial, including some nonmagnetic thermal contributions and electromigration [14][15][16][17][18][19].
Spin Seebeck effect (SSE) and spin pumping, the two well-established spin injection methods in FMs, are not feasible in collinear AF insulators with M = 0, unless acquiring a small induced M by a large magnetic field or beyond the spin-flop field of some AFs [20][21][22][23]. In contrast, noncollinear AF with an inherent small M have revealed new spintronics phenomena. For example, in non-collinear AF semimetals of Kagome AF Mn3Sn, the Berry curvature associated with the non-collinear spin structure enables spin-orbit torque switching as evidenced by the anomalous Hall effect, anomalous Nernst effect, and AF tunnelling magnetoresistance [24][25][26][27][28][29][30][31]. Shortwavelength coherent magnon propagation has been demonstrated in non-collinear AF insulating orthoferrite DyFeO3 [32]. Long-distance spin transport has also been reported in the same class of material YFeO3 [33]. In non-collinear AF insulating orthoferrites of LaFeO3 and LuFeO3 [34,35], vector SSE and spin swapping effect have been discovered, as described in the following. The noncollinear AFs offer a unique platform to explore new spin current phenomena that are not viable in collinear FMs and AFs [34][35][36][37][38], and open new grounds for AF spintronic and spin caloritronic applications without the need for a large magnetic field.
Within the SSE, it has been well established in FM insulators with collinear moments that there is only longitudinal SSE but no transverse SSE, with spin injection administered via a temperature gradient in the out-of-plane and the in-plane direction, respectively. In contrast, vector SSE has been observed in non-collinear AF LuFeO3 and LaFeO3, where spin injection via temperature gradient may be administered in any direction, out-of-plane as well as in-plane. The unusual characteristics of the vector SSE are intimately linked to the non-collinear spin structure leading to a small and spontaneous M. The characteristics of the unprecedented magneto-thermovoltage under an in-plane temperature gradient in LuFeO3 and LaFeO3 are consistent with those of the predicted spin swapping effect [39]. The magnitude of the vector SSE in LuFeO3 increases greatly by 10-fold from 300 K and peaks at about 400 K. These results highlight the unique role of a new class of non-collinear materials for AF spintronics and spin caloritronics without the limitations of ferromagnets and antiferromagnets with collinear moments. Spin Seebeck effect (SSE) measurements: We used two SSE setups for longitudinal SSE and transverse SSE measurements, in which the temperature gradient has been applied exclusively in the out-of-plane and in-plane directions, respectively. We thermally anchored the W/LuFeO3 devices to two Cu blocks via silicone thermal pads for better thermal contact, and one of the Cu blocks was heated by a resistive heater. The temperatures of the two Cu blocks (Thot and Tcold), as monitored by two Cernox temperature sensors, give the temperature difference ∆ ≡ hot − cold , and the sample temperature avg ≡ ( hot + cold )/2 . We express the temperature gradient as ∇ ≡ ∆ / T , where T is the sample length between the two temperatures with typical values of ∇ on the order of 5 K/mm. We express the magneto-thermal voltages as ∆ / V , where LV is the separation of the voltage leads. Since the SSE voltage is proportional to the temperature gradient, we display the results as (∆ / V )/(∆ / T ), taken into account the dimensions of the samples and thermal gradients.

Longitudinal spin Seebeck effect in collinear ferromagnetic insulators
We first illustrate the well-known longitudinal SSE results in FM insulators using the W/YIG devices, where an out-of-plane temperature gradient injects a pure spin current Js into the W layer.
The inverse spin Hall effect (ISHE) in the W layer generates an electric field of and detected as an ISHE voltage V, where SH and are the spin Hall angle and resistivity of W, respectively. It has been well established that there is only longitudinal SSE but no transverse SSE in collinear FMs and AFs [40][41][42][43][44][45][46]. In W/YIG, under an out-of-plane temperature gradient ∇ and an in-plane magnetic field Hy, which aligns the spin index σ of YIG along the y-direction, one detects the ISHE voltage in the x-direction but not in the y-direction, according to Eq.(1) as shown in Fig. 1a. In FM insulators, such as YIG, a modest magnetic field exceeding the anisotropy field can readily align the magnetic moments to any direction and establish σ in that direction.
On the other hand, as shown in Fig

Non-collinear antiferromagnetic spin structure in LuFeO 3
LuFeO3 belongs to the orthoferrite family RFeO3 (R = rare earth) with an orthorhombic Pbnm Because of the strong spin-spin exchange interaction, Fe 3+ moments are aligned antiparallel along the a-axis with TN = 620 K. Due to the Dzyaloshinskii-Moriya interaction (DMI) and/or single ion anisotropy (SIA), a non-collinear spin structure exists in these orthoferrites, exhibiting a weak ferromagnetism [47]. When the rare earth element carries a magnetic moment, the orthoferrites show more complicated and exotic magnetic phases and phenomena, such as spin-reorientation due to the interaction between R 3+ and Fe 3+ moments [48][49][50].  (Fig.2c) and Hx (Fig.2d). Likewise, the magnetizations My and Mz were measured with field Hz (Fig.2f) and Hy (Fig.2g). With Hz, a square hysteresis loop shows ± Mz with a sharp switching at ±150 Oe with Mz ≈ 0.05 µB/Fe (Fig. 2c and 2f), which is only about 1% of the full Fe moment. In addition, there is also simultaneous switching in the much smaller Mx and My, due to the small misalignment between c-axis and z-direction. Similar behaviors are observed with the applied fields of Hx and Hy, but with a much larger switching field, as shown in

Vector spin seebeck effect in LuFeO 3
The vector SSE results in c-oriented LuFeO3 with field applied along z-direction are shown in Fig. 4. Firstly, we observed longitudinal SSE, a magneto-thermovoltage under an out-of-plane temperature gradient ∇ (Fig. 4a and 4b). The longitudinal SSE signal, on the order of ~ 0.5 nV/K at Tavg = 325 K, exists in the x-direction (Fig. 4a) and also in the y-direction (Fig. 4b), different from those in W/YIG shown in Fig. 1a. Even more surprising, we have observed a magnetothermovoltage under an in-plane temperature gradient ∇ (Fig. 4c) and also ∇ (Fig. 4d) with a much larger signal of ~ 4.5 nV/K at Tavg = 320 K, instead of the null signals in W/YIG as shown Unlike those in collinear AFs, where a large magnetic field (on the order of several Tesla) is required for SSE, the vector SSE in c-oriented LuFeO3 is readily visible even at zero magnetic field due to the small spontaneous magnetization, which can be reversed by a small switching field of 150 Oe. These non-collinear AFs may provide a promising playground for low-field AF spintronics and spin caloritronics.

Spin swapping effect
In c-oriented W/LuFeO3, the small misalignment between the spin index direction along c, and the our-of-plane direction along z, gives rise to the small longitudinal SSE results as shown in Fig.   4a and 4b. Even more fascinating is the presence of much larger magneto-thermovoltage under an in-plane temperature gradient ∇ (Fig. 4c) and ∇ (Fig. 4d), which are completely absent in collinear FMs (e.g., YIG) or AFs (e.g., Cr2O3), and with a magnitude of 4.5 nV/K at Tavg  Specifically, the injected Jy with z has been converted into Jz with y in c-oriented LuFeO3 and/or its interface with W. This is consistent with the spin swapping effect, where the directions of the spin index and the spin current interchanged. This interchange of directions between spin current and spin index can arise from magnon-magnon or magnon-phonon scattering in the presence of spin-orbit coupling [51][52][53]. Experimentally, the spin swapping effect has been realized in non-collinear AF of LuFeO3 and LaFeO3 [34,35], but not in collinear FMs or AFs.
Indeed, the spin swapping effect has been proposed in non-magnetic metals with strong spin-orbit coupling [39]. Experimentally, spin currents injected from collinear FMs and AFs do not, and thus far those from non-collinear AFs do exhibit the spin swapping effect.

Temperature dependence of vector spin Seebeck effect
In SSE, the temperature gradient injects the spin current from the FM (e.g., YIG) or the noncollinear AF with a net M (e.g., LuFeO3) into the adjacent heavy metal (e.g., W) and be detected. in c-oriented LuFeO3 at several temperatures are shown in Fig. 6. Representative curves at three temperatures illustrate the strong temperature dependence of magneto-thermovoltages under outof-plane temperature gradient (Fig. 6a) and in-plane temperature gradient (Fig. 6b). As shown in Fig. 6c, from 300 K to 400 K, magneto-thermovotlages under out-of-plane and in-plane temperature gradient both increase by more than 10-fold and peaks at 400 K. As shown in Fig. 3a, the values of Mx, My, and Mz actually decrease slightly from 300 K to 400 K. Thus, the sharp increase of the vector SSE, despite the decrease of M, comes from the strong temperature dependence of the factors a and b, due to the strong temperature dependence of magnon properties, magnon-phonon interaction, and spin mixing conductance at the interface [21,22,[54][55][56][57]. Due to the entropy, the transport of thermal magnons is known to be vanishing towards zero temperature and above the Néel temperature, and thus peaks at a temperature in between. Further studies are required to account for the rapidly increasing vector SSE that peaks at about 400 K.

IV. CONCLUSION
Previously