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Anomalous Thermal Hall Effect in an Insulating van der Waals Magnet VI3

Two-dimensional (2D) van der Waals (vdW) magnets have been a fertile playground for the discovery and exploration of physical phenomena and new physics. In this Letter, we report the observation of an anomalous thermal Hall effect (THE) with \k{appa}_xy ~ 1x10^(-2) W K^(-1) m^(-1) in an insulating van der Waals ferromagnet VI3. The thermal Hall signal persists in the absence of an external magnetic field and flips sign upon the switching of the magnetization. In combination with theoretical calculations, we show that VI3 exhibits a dual nature of the THE, i.e., dominated by topological magnons hosted by the ferromagnetic honeycomb lattice at higher temperatures and by phonons induced by the magnon-phonon coupling at lower temperatures. Our results not only position VI3 as the first ferromagnetic system to investigate both anomalous magnon and phonon THEs, but also render it as a potential platform for spintronics/magnonics applications.

Recently, the thermal Hall effect (THE), i.e., the generation of a transverse heat current in the presence of a longitudinal temperature gradient, has been gaining increasing attention as a powerful tool to probe charge neutral quasiparticles in insulating quantum materials 1- 14 . In particular, the THE has been predicted in magnets possessing nontrivial spin excitations, e.g., magnons [15][16][17][18][19][20][21][22] . Distinct from the ordinary charge Hall effect in which charge carries are deflected by the Lorentz force, the transverse flow of magnon current [ Fig. 1(a)] is driven by the Berry curvature resulting from the nontrivial topology of the magnon bands. The Berry curvature of magnons usually arises from the Dzyaloshinskii-Moriya interaction (DMI), which depends sensitively on the lattice geometry 8 . Prototypical lattice structures capable of exhibiting the magnon THE include pyrochlore lattice in 3D 1 and kagome lattice in 2D 7 , in both of which a nearest neighbor DMI is responsible for the appearance of the Berry curvature. Nevertheless, magnon THE has only been reported in a few ordered magnets to date and the observed thermal Hall conductivity in these materials is rather weak, in the order of 1 × 10 −3 −1 −1 1, 7,8 .
The recent emergence of 2D van der Waals (vdW) magnets 23 provide a new platform in pursuit of a large THE. Specifically, in many of these layered materials, the magnetic ions form a honeycomb lattice in each layer. It has been predicted that magnon bands of such a system could possess non-zero Berry curvature due to a next-nearest-neighbor DMI and can be characterized by non-trivial topological invariants 24 . As a result, the THE of magnons is expected 21,25 . The DMI in these materials can be substantial [26][27][28] , and the DMI -as opposed to Kitaev interactions 29 -has been demonstrated to be the microscopic origin of the Dirac gap revealed in inelastic neutron scattering studies 28 . However, direct experimental evidence of the magnon THE in 2D vdW magnets is still lacking.
In this Letter, we report the observation of an anomalous THE in a vdW ferromagnetic insulator VI3 (Tc = 50 K). The thermal Hall conductivity reaches a magnitude of ~ 1 × 10 −2 −1 −1 in a wide temperature range (10 K to 45 K) and the maximum thermal Hall angle ( = / ) is about 4.3 × 10 −3 . We find that the thermal Hall conductivity tracks nicely the magnetization hysteresis loop and persists at zero magnetic field. In conjunction with theoretical calculations, we show that the observed thermal Hall signal at higher temperatures is mainly determined by the non-trivial magnon band topology while, interestingly, at lower temperatures the THE is driven by the magnon-phonon coupling and is of phononic nature.
VI3 crystalizes in the trigonal space group 3 ̅ (No. 148) at room temperature with lattice constants a = b = 6.8325 Å, c = 19.6776 Å and crystalline angles of = = 90°, = 120°. The vanadium atoms form a honeycomb structure in the ab-plane, as illustrated in Figure 1(b). VI3 is a highly resistive semiconductor with a bandgap of ~ 0.67 eV 30,31 . It undergoes a subtle structural phase transitions at Ts ~ 77 K, followed by a paramagnetic-ferromagnetic phase transition at Tc ~ 50 K [30][31][32] . Figure 2(a) shows the temperature dependence of magnetic susceptibility ( ) of a VI3 single crystal measured under the field-cooled condition. A magnetic field of 0.1 T is applied along the c-axis. It is clearly seen that the sample exhibits a ferromagnetic phase transition at Tc = 50 K as determined by the peak in / (black curve, arbitrary unit). The magnetic easy axis is found to be along the c-axis, which is supported by the isothermal magnetization measurements shown in Supplementary Figure S2 33 . These features are consistent with previous reports 30-32,34 , affirming the high quality of our sample. below Tc in the absence of magnetic field. As will be discussed later, this feature suggests that there is strong magnon-phonon scattering in VI3 which leads to a reduction of thermal conduction by phonons. Such a reduction even surpasses the thermal conductivity contributed by magnon carriers, resulting in the decrease of total seen in Figure 2(b).
The isothermal magnetization (M), , and of VI3 as a function of the applied magnetic field along the c-axis are presented in Figure 3. As illustrated in Figure 1 This is also seen from the comparison of ( ) measured at 0 T and 7 T shown in Figure 2(b). In VI3, the total longitudinal thermal conductivity increases with magnetic field, which suggests that the longitudinal heat current is mainly carried by phonons and the phonons are strongly scattered by magnons.
The lower panel of Figure 3(a) shows the field dependent measured at T = 15 K. For reference, two vertical dashed lines at the coercive field are plotted across the three panels in Figure   3 (a). It is clearly seen that the ( ) curve tracks the ( ) curve very well and saturates right above the coercive field. Remarkably, the magnitude of reaches ~ 0.01 −1 −1 , nearly an order of magnitude larger than the thermal Hall signal observed in other ordered magnets 1,7,8 . Similar behavior is also observed at higher temperatures up to Tc, as shown in Figure 3(b) and Figure S4 33 . At lower temperatures, due to the strong mixing between the longitudinal and transverse components, an antisymmetrization procedure needs to be carried out 7,35 , which allows us to extract saturated value of (see Figure S5 and S6 in the Supplemental Material 33 ). The thermal Hall angle, which is defined as = / , reaches a maximum value of ~ 4.3 × 10 −3 at T = 44 K. With a stronger magnetic field applied, as shown in Fig. S7, the magnitude of decreases while increases and the magnetization M remains nearly saturated. Thus, the ( ) curves suggest that the mechanism responsible for THE in VI3 should be closely related to its spin degrees of freedom, i.e., magnons, since the suppression of magnon population at high field leads to a reduction in and an increase in due to the suppression of phonon scattering by magnons. Furthermore, the thermal Hall signal persists in the absence of an external magnetic field with being nearly the same as the saturated value, indicating an anomalous THE, and flips sign upon the switching of the magnetization. These merits, together with its quasi-2D nature, demonstrate that VI3 could be a promising platform for future spintronics/magnonics applications [36][37][38][39][40] .
To understand the physical origin of the anomalous THE observed in VI3, we examine the corresponding 2D Heisenberg model of spins on a honeycomb lattice. Since VI3 is a 2D vdW ferromagnet with a large spacing between layers along the c-axis and a much weaker interlayer magnetic exchange interaction than the intralayer exchange interactions, the magnon band structure is mainly determined by the intralayer interactions. This has been elaborated in recent inelastic neturon scattering studies of sister compounds CrI3 and CrBr3 [26][27][28] . It has previously been predicted that a second-nearest neighbor DMI should be allowed in such a system 24 , and this interaction gives rise to topologically-driven response functions in honeycomb ferromagnets 21,25,41 . The Hamiltonian is given by where the symmetric exchange parameters 1 , 2 , and 3 describe interactions between nearest, second-nearest, and third-nearest neighbors, respectively, such that < 0 leads to ferromagnetic The Berry curvature projection along the c-axis (|Ω |) of the lower magnon band is presented in Figure 4(b). We see that non-zero Ω is found along the K − M − K lines in the reciprocal space.
By integrating Ω over the first Brillouin zone, the Chern number for the upper and lower magnon bands are found to be C = +1 and C = -1, respectively. Due to the bulk-boundary correspondence, edge states are anticipated to connect these two bands with different topological indices.
To further verify that the observed in VI3 is indeed driven by magnons, we have calculated the temperature dependence of the thermal Hall conductivity of the model introduced in Eq. 1 and compare it with the experimental results. It is standard practice to deduce the 3D thermal conductivity from the calculated 2D "sheet conductivity" through division by the interlayer distance of the material. To cover the entire temperature range, we used a mean-field treatment of the magnetization in the Holstein-Primakoff transformation 44 Fig. 4(d). For comparison, the temperature dependence of the longitudinal thermal conductivity measured at zero field, which is dominated by phonons as heat carriers, is also presented in Fig. 4 First, we can exclude the possibility that the THE observed in VI3 is purely driven by phonons that directly couple to the magnetic field as in the case of nonmagnetic SrTiO3 13 . In this scenario, the THE should vanish at zero field and increase with magnetic field, but in VI3 we observe a large thermal Hall signal at zero field, i.e., anomalous THE. This suggests that the thermal Hall signal observed in VI3 is associated with the magnetism. Second, we can also exclude the possibility that the observed thermal Hall signal in VI3 is mainly driven by the phonon THE x-ray scattering. Nevertheless, it is worth pointing out that when D = 0 the calculated due to magnon-phonon interaction alone is much smaller [ Figure S9(f) 33 ] and has a qualitatively different temperature dependence than that observed experimentally. Interestingly, the calculated THE induced by magnon-phonon coupling shows a peak at lower temperature, as plotted in Fig. S9(f), which is consistent with the Δ presented in Fig. 4(e). This suggests the dominant phononic origin of the THE observed at lower temperatures, which is mainly induced by the magnon-phonon coupling. In the presence of a strong magnetic field, the magnetic field shifts the magnon band upwards in energy, which in turn moves the anticrossing point upwards, leading to a reduction of , as presented in Fig. S10. This is consistent with the experimental observation shown in Fig.   S7.