Recent studies have demonstrated that the thermal Hall effect (THE) can originate from magnons (magnon Hall effect), phonons (phonon Hall effect), or their combination (magnon–polaron Hall effect). The magnon–polaron Hall effect, first observed in Fe2Mo3O8, is particularly intriguing as its thermal Hall signal can be remarkably large. In this study, we explore the THE in MnPS3, an insulating antiferromagnetic material exhibiting a spin-flop (SF) transition and significant magnetoelastic coupling, making it a strong candidate for studying the THE originating from spin–lattice coupling. We report an exceptionally large thermal Hall angle down to 4 K and show that it cannot be accounted for by standard calculations based on the intrinsic magnon–polaron Berry curvature. Our findings provide an in-depth analysis of the role of the SF transition in the thermal properties of MnPS3and call for further theory development on magnon–phonon coupling and scattering to reveal their influence on transverse heat transport.
]]>The search of materials hosting charge-neutral topological excitations has gained significant interest in recent years [1,2]. Experimental investigations of electronic topological excitations are facilitated by high-sensitivity techniques like voltage measurements. Measurements such as the electrical Hall effect and angle-resolved photoemission spectroscopy have been instrumental in confirming electronic topological states [3].
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However, detecting charge-neutral topological quasiparticles like magnons has proven far more challenging, as their detection relies on techniques such as thermal Hall effect (THE) measurements and inelastic neutron scattering (INS) [4]. Early studies of THE in ferromagnetic insulators suggested the presence of topological magnons and attributed the non-zero Berry curvature responsible for these effects to the Dzyaloshinskii-Moriya interaction (DMI) [5][6][7].
Subsequent research revealed that magnetic long-range order is not necessary for the emergence of THE, indicating that magnetic quasiparticles are not the only source of transverse heat transport. Evidence for phonons contributing to THE grew stronger as THE was observed in the paramagnetic regime of insulating magnets and in non-magnetic insulators [8][9][10][11]. In addition to THE, the coupling between magnons and phonons has been shown to impact the spin Seebeck effect in Yttrium Iron garnet (YIG) [12]. These findings showed that the role of phonons cannot be ignored when analyzing THE in insulating magnets and that the interplay between magnons and phonons is crucial.
Studying materials with strong spin-lattice coupling was essential to understand the topological nature of this interplay. Thermal Hall measurements on Fe 2 Mo 3 O 8 , a multiferroic material with a strong spin-lattice coupling, demonstrated a record-high THE at the time [13]. Detailed INS studies confirmed the anti-crossing between phonon and magnon bands, providing experimental evidence of the topological gap [14]. The resulting non-zero Berry curvature from this hybridization mechanism was proposed to contribute to the observed THE [15].
MnPS 3 , a two-dimensional insulating magnet, emerges as a promising candidate for exploring magnon-phonon hybridization due to its significant magnetoelastic coupling and Néel AFM order [16], which can be manipulated with magnetic fields. MnPS 3 has been shown to host the spin Nernst effect, which has been attributed to the non-zero Berry curvature of the magnon bands [17][18][19]. Moreover, it has been theoretically proposed that the coupling between its magnon and phonon bands creates a topologically non-trivial gap, resulting in a finite THE [20]. The material also undergoes a spin-flop (SF) transition, altering its spin structure and magnon bands [21]. Computational studies have shown the effect of the SF transition on the expected THE [22,23]. A SF transition potentially alters the magnon-phonon hybridization in the material and may have a strong impact on the THE. NiPS 3 , a sister compound to MnPS 3 , was shown to host a large THE (∼2 W m -1 K -1 ), but no SF transition was observed up to 50 T. The authors attributed the large THE to magnon-phonon hybridization [24]. These properties make MnPS 3 a compelling candidate to further study THEs induced by magnonphonon interactions.
In this letter, we report our investigation of thermal transport properties in MnPS 3 . We measured thermal conductivity (κ xx ), thermal Hall conductivity (κ xy ), and heat capacity as functions of magnetic field across a range of temperatures, down to 4 K. The thermal transport data exhibits distinct behavior directly linked to the SF transition in MnPS 3 . Notably, the thermal Hall conductivity (κ xy ) demonstrates a non-monotonic dependence on the magnetic field and undergoes a sign change around 10 K, reaching an astounding value of -14 W m -1 K -1 . We present multiple observations that point toward significant magnon-phonon interactions, but a magnon-phonon hybridization model cannot account for the magnitude of the observed effect. This observation suggests that instead of the intrinsic magnonpolaron Berry curvature, magnon-phonon scattering might be more relevant for the explanation of the measurements. We discuss experimental and theoretical challenges with respect to MnPS 3 and thermal Hall measurements more generally.
The crystal structure of MnPS 3 belongs to the monoclinic space group C2/m (figure 1(a)) [25]. It is an insulating antiferromagnet with a Néel temperature of 78 K and an electronic band gap of 2.79 eV [26,27]. Mn 2+ ions are arranged on a honeycomb lattice in a Néel antiferromagnetic configuration, where spins are oriented along the out-of-plane direction (figure 1(a)) [21]. The spins exhibit a canting angle of approximately 8 • toward the ab-plane [28].
The Néel temperature was determined by identifying the peak in the derivative of magnetization with respect to temperature (dM/ dT), yielding a value of 78 K, consistent with previous findings (figure 1(b)) [21]. A small separation between the zero-field cooled (ZFC) and field cooled (FC) curves was observed at low temperatures ∼20 K. This behavior is attributed to spin canting in Mn 2+ ions, which can induce a weak ferromagnetic moment at a low magnetic field [21], a phenomenon further confirmed in our magnetization measurements (figure S1). Additionally, no significant difference was observed between the magnetization along the a and b axes (figure 1(d)).
Upon applying a strong magnetic field along the c axis, the sample undergoes a broad SF transition (figure 1(c)). The sharpness of the transition increases with lower temperatures. At 5 K, the SF transition is the sharpest, beginning at 3.5 T, where the magnetization deviates from its linear behavior, and ending at 5 T, where the magnetization re-enters the linear regime. The magnetic transition region broadens, and the SF field increases with increasing temperature, as clearly illustrated in figure S1(d).
The thermal conductivity (κ xx ) of MnPS 3 (sample 1) at 0 T (figure 2(b)) is consistent with a previous report [29], reaching a peak value of 290 W m -1 K -1 at 14 K. In the lowtemperature regime (<14 K), κ xx follows a T 1.5 dependence. Above 14 K, the thermal conductivity decreases due to Umklapp scattering. Thermal transport data was consistently reproduced across four different samples, each using varied thermometry methods (figure S4). Unlike thermal conductivity, the heat capacity (C v ) in MnPS 3 follows the conventional T 3 dependence at low temperatures as shown in figure 2(c).
Figure 2(d) highlights the magnetic field dependence of thermal conductivity. Below the Néel temperature (78 K), κ xx (B z ) exhibits a distinct trend: it remains nearly constant until approximately 4 T, then sharply decreases and stabilizes again at a lower value. This behavior is strongly correlated with At temperatures below the Néel temperature, the observed thermal Hall signal exhibits a non-monotonic dependence on the magnetic field, characterized by an increase up to a maximum value, followed by a decrease, forming a pronounced peak. This behavior aligns closely with the SF transition observed in the sample. Due to this non-monotonic nature of κ xy (B z ), the peak value was extracted manually and used to analyze its temperature dependence (figure 3(d)). κ xy (B z ) data for all temperatures and samples are provided in supplementary materials.
The temperature dependence of κ xy exhibits two distinct regimes with opposite signs: a low-temperature regime (<10 K) exhibiting a positive sign and a high-temperature regime (>10 K) with a negative sign. This behavior was successfully reproduced across multiple samples (figure 3(c)). This temperature-induced sign reversal coincides with the peak in magnon-phonon scattering, as reflected in the difference between zero-field and high-field thermal conductivity (κ xx (0 T)-κ xx (9 T)). The magnitude of these effects suggests that magnon-phonon interactions play an essential role in the THE (figure S5).
Figure 4(a) shows the magnetization curve at 5 K, revealing that the SF transition field begins around 3.5 T and ends around 5 T. Above 5 T, the sample enters a state where the spins are forced to align increasingly along the external magnetic field; however, full saturation is not observed up to 7 T. This behavior is reflected in the heat capacity curve (figure 4(b)) which increases linearly up to about 4 T, where the system enters the SF transition. Between 4 T and 5 T, the heat capacity exhibits a slight decrease, followed by a more pronounced drop above the completion of the SF transition. Heat capacity conveys a thermal picture of the occupied magnon and phonon density of states. Below the SF transition, the magnetic field shifts the two magnon branches upwards and downwards due to Zeeman energy (figure 4(e)). According to Bose-Einstein statistics, the thermal population in the upper branch decreases marginally, while the lower branch occupation increases exponentially, resulting in an overall rise in heat capacity. At the SF transition, the lower magnon branch becomes a zero-energy Goldstone mode (figure 4(f)), leading to maximum thermal occupation and, consequently, maximum heat capacity. Above the SF transition, the upper branch shifts further upward, reducing thermal occupation and causing a decline in heat capacity.
Thermal conductivity (κ xx ) and the THE (κ xy ) exhibit a strong correlation with the SF transition as well (figures 4(c) and (d)). The interplay between the magnon and phonon modes provides a plausible explanation for both behaviors. In the case of thermal conductivity κ xx , the primary contributions are specific heat capacity and the scattering of the heat carriers. However, κ xx exhibits a distinct and more pronounced behavior compared to specific heat. While specific heat varies by only ∼4%, thermal conductivity remains nearly constant at low fields and drops sharply by approximately 40% around the SF transition. The thermal conductivity is directly related to heat capacity (C) through the relationship, κ = 1 3 Cv 2 τ , where v is the sound velocity and τ is the relaxation time. Since sound velocity is constant with the applied magnetic field, and C varies only slightly, we conclude that magnon-phonon scattering (τ ) is the dominant term in κ xx (B z ).
These observations suggest that magnon-phonon scattering plays a vital role in thermal conductivity, as the field-induced reduction of the magnon energies provides more efficient scattering channels during the SF transition, thereby enhancing scattering.
To explore the origin of the observed thermal Hall signal and its non-monotonic behavior, we calculated the intrinsic Berry curvature arising from the hybridization of magnon and phonon bands that leads to hybrid quasiparticles known as magnon polarons [20,[32][33][34][35][36]. We modeled a monolayer of MnPS 3 by a Hamiltonian that incorporates magnetic, elastic, and magnetoelastic components. The magnetic component of the Hamiltonian includes the Heisenberg exchange parameters (J 1 , J 2 , J 3 ), the DMI (D), an easy-axis anisotropy (K), and a Zeeman term (with Lande´factor g). The lattice component is represented by a spring model for the out-of-plane displacements of the Mn ions between nearest neighbors (elastic constant C). The magnetoelastic component couples the spins to the out-of-plane displacements (magnetoelastic constant λ).
The parameters used for this model were obtained from ab-initio calculations [16], except for the elastic constant, which was obtained from a separate experiment that we conducted using resonant ultrasound spectroscopy. For calculating the thermal Hall conductivity, we exclusively considered the Berry curvature-driven intrinsic contribution and neglected anharmonicities such as magnon-magnon interactions and any additional scattering effects. Further details are given in methods.
Figures 5(a higher temperatures (relative to T N ), magnon-magnon interactions are expected to suppress the intrinsic thermal Hall. Our model based on linear spin-wave theory is not reliable in this limit.
To scrutinize the behavior of κ xy and its correlation with the SF transition, we calculated the evolution of the magnon band structure and the Berry curvature of the lowest band with the magnetic field as shown in figure 6. At zero field and without magnetoelastic coupling, the spin-up and spindown magnon bands are degenerate. By introducing magnetoelastic coupling, the degeneracy is lifted, resulting in hybridization between magnon and phonon bands (figure 6(a)); however, the bands are no longer spin-polarized. Furthermore, the Berry curvature of the lowest band is zero (figure 6(b)). By increasing the magnetic field, the two magnon bands are split due to the Zeeman interaction, with one band shifted upwards and one downwards. The downward shift of the lower magnon band results in a second avoided crossing at low energies with the acoustic phonon band (figure 6(c)). These two avoided crossings create pronounced Berry curvature rings centered around the Γ point with different radii and opposite signs (figure 6(d)). At B z = 4.26 T, the lower magnon band reaches zero energy, and the antiferromagnetic phase becomes unstable and transitions to the SF phase. Consequently, the Néel vector is rotated from its out-of-plane to an in-plane orientation, and a ferromagnetic moment is developed along the externally applied magnetic field. As an effect of the SF transition, the magnon bands approach each other (figure 6(e)). The lower magnon band forms a Goldstone mode with linear dispersion, whose velocity exceeds that of the transverse phonons, resulting in the disappearance of the low-energy avoided crossing (figure 6(f)). This eliminates the low-energy Berry curvature ring observed in the antiferromagnetic phase and explains the suppression of κ xy in the SF phase as observed in figure 5(a).
As the ferromagnetic moment increases with the applied magnetic field, the upper magnon band gradually changes its curvature, while the lower one is pinned at zero energy due to the isotropy of the ground state energy with respect to the in-plane Néel vector, preserving the Goldstone mode (figure 6(g)). Because the bands are mostly modified at the Γ point, the location of the high-energy avoided crossing remains constant over a large range of magnetic field (figure 6(h)). Hence, κ xy does not strongly depend on B z above the SF phase.
Although the experimental and the calculated κ xy share some qualitative similarities, there is a large difference in magnitude. The discrepancy can be related to complications in the experimental measurements as well as assumptions of the model. The measurement of thermal Hall conductivity, which corresponds to the odd part of the thermal conductivity tensor, is challenging because it relies on several assumptions. It is commonly known that the transverse thermocouples can pick up signals from the longitudinal thermal resistivity due to small misalignments. These can be eliminated by measuring the transverse temperature gradient at positive and negative magnetic fields and antisymmetrizing them because only the thermal Hall conductivity is odd under magnetic field, according to the Onsager relation [37]. This, however, assumes that an applied magnetic field also reverses the microscopic magnetic structure. This assumption is violated in materials with a magnetic hysteresis in M(B). Although our data suggests a finite ferromagnetic component, we have verified that the hysteresis does not persist above 0.3 T (figure S1), which is below the smallest measured magnetic field data point in the thermal transport experiment. Nevertheless, we cannot exclude that the magnetic components hidden in M(B) do not follow the magnetic field, which is a potential issue for antiferromagnets in general.
Another complication arises from the anisotropy of the material. Because of the monoclinic structure and the tilting of the Néel vector in MnPS 3, κ xx and κ yy do not have to be equal. In the experiment, the thermal resistivity w xx is measured. In order to convert to conductivities, both w xx and w yy would be required, which cannot be obtained in a single measurement geometry. Hence, we had to assume w xx = w yy .
On the side of the model, we have focused on one particular mechanism, which originates from the intrinsic Berry curvature of magnon-phonon hybrids. For the magnetoelastic coupling, we have focused on a particular interaction derived from spin-orbit coupling that exclusively hybridizes magnons and out-of-plane phonons [16,20,32,[38][39][40]. In the SF phase, other symmetry-allowed interactions can also lead to hybridizations, including in-plane phonons [23,41]. However, these would introduce more parameters which have not been calculated for this material yet. They could give rise to a nonzero κ xy in the SF phase if the longitudinal acoustic phonon bands cross with the magnon bands. Within the antiferromagnetic phase, the considered form of magnetoelastic coupling is dominant [32], which is why we do not expect other forms of magnetoelastic couplings to strongly modify our results below ∼4 T.
Another source of transverse currents due to time-reversal symmetry breaking may arise from many-body magnonmagnon, phonon-phonon, and magnon-phonon interactions, which not only can renormalize the Berry-curvature driven contribution studied here [42][43][44][45][46][47], but can give rise to skew scattering and side jump as an alternative mechanism [48][49][50][51]. Scattering off defects can further contribute to the THE [52]. These theories exceed the scope of this project.
Considering the model, it should be noted that experimental findings that challenge the out-of-plane Néel order in MnPS 3 have started a debate on the magnetic structure. Our model does not reproduce the complex temperature-dependent magnetic phase diagram reported here and elsewhere [21]. In particular, the ferromagnetic moment below 20 K cannot be accounted for. This should be addressed in the future by constructing a magnetic Hamiltonian capturing the magnetic ground states.
While the observed values of κ xy in our study are relatively high, it is worth noting that comparable magnitudes have been reported in another material of the MPX 3 system, NiPS 3 [24]. The authors of the study reported values of κ xx up to 300 W mK -1 and κ xy reaching 2 W mK -1 , which are of the same order as those observed in our measurements. Notably, NiPS 3 does not undergo a SF transition up to 50 T and exhibits a different spin structure compared to MnPS 3 . Consequently, κ xy in NiPS 3 displays a linear field dependence, in contrast to the non-monotonic behavior observed in our system. Additionally, κ xx did not exhibit any major reduction up to 14 T, which could also be attributed to the lack of SF transition. The authors of that study also highlighted the role of crystal twinning in affecting the magnitude of κ xy and the magnetic anisotropy. However, in our crystals, we did not observe a significant reduction of κ xy across different crystals, and optical microscopy inspection revealed no prominent twinning features similar to those reported in NiPS 3 . Therefore, we consider the effect of crystal twinning to be negligible in our samples and have not included it as a contributing factor in our analysis.
During the preparation of this manuscript, we became aware of a similar study on MnPS 3 that reached a different conclusion [41]. Our sample exhibited κ xy values that are two orders of magnitude larger and displayed a pronounced nonmonotonic behavior. In contrast, the other study reported a monotonic κ xy with a small hump near the SF transition. We suspect that this discrepancy could stem from sample quality as our thermal conductivity data is three times higher than the one they reported. We also note that our thermal conductivity values agree with a separate study [29].
In this study, we investigated the thermal transport properties of MnPS 3 , focusing on the magnetic field dependence of thermal conductivity (κ xx ) and thermal Hall conductivity (κ xy ). The observed behavior of κ xx points to significant magnonphonon scattering above the SF transition. The thermal Hall conductivity (κ xy ) exhibited a non-monotonic dependence on the magnetic field.
Within our model, the anti-crossing between magnon and phonon bands is maximized below the SF transition, generating a non-zero Berry curvature near the Γ-point. As the applied magnetic field increases, the slope of the lower magnon band steepens, causing the hybridization point and the associated Berry curvature to vanish, thus generating a non-monotonic κ xy . Furthermore, the model produces a low-temperature sign reversal in κ xy through the calculated intrinsic Berry curvature, which features contributions of opposite signs around the Γpoint.
While our magnon-phonon hybridization model successfully reproduces some qualitative features of the observed THE, it does not account for the exceptionally large magnitude of κ xy . Therefore, our findings call for a better understanding of the magnetic structure of MnPS 3 and the exploration of alternative mechanisms, such as magnon-phonon skew scattering, that give rise to a THE.
MnPS 3 single crystals were grown using chemical vapor transport technique. Stoichiometric amounts of elemental powders and 0.03 g of iodine (transport gas) were mixed together in an Ar-filled glovebox and then loaded into a fused silica tube and sealed under vacuum (<1 Torr). During the sealing, the precursors were immersed in liquid nitrogen to prevent the evaporation of iodine. The sample was then placed in a twozone furnace with T hot at 680 • C and T cold at 650 • C for one week with a ramp rate of 5 • min -1 and cooling rate of 0.1 • min -1 . The resulting crystals were green and shown in figure S2. Sample purity was confirmed using x-ray diffraction (figure S2). Magnetization measurements were conducted using a Quantum Design Magnetic Property Measurement System with maximum field of 7 T. A single crystal was mounted on a quartz tube using GE varnish and measured in DC mode. Heat capacity was conducted on the Quantum Design Physical Property Measurement System (PPMS) with a maximum field of 9 T using the heat capacity mode and puck; the sample was mounted with the magnetic field pointed along the c-axis.
Thermal transport measurements were conducted using Quantum Design PPMS. The thin fragile sample was mounted on our home-made setup for thermal transport (figure S3). The sample was mounted such that the magnetic field is pointing out-of-plane, along the c-axis, and the heat flux is sent across the ab-plane. The whole mount was made from Kapton tape, which has a very small thermal conductivity so that it would be shorted out by the high thermal conductivity of the sample. We used GE varnish to connect the sample to the heat sink and the heater. Thermocouples were connected to the sample using silver epoxy.
We used a 120 Ω resistor to send heat along the x-axis and a heat sink made from LiF. The measurements were conducted in a steady-state mode, where we waited for over 10 min after the heater was turned on to ensure thermal equilibrium across the sample. Additionally, we waited for 5 min after the magnetic field was applied at each step to avoid any extrinsic magnetocaloric effects. All voltage measurements across thermocouples were conducted using a 2182A Keithley nanovoltmeter. At low temperatures, we averaged the measured voltage for a few minutes (5-10 min) to lower the error as much as possible.
We used different thermometry across the samples. In samples 1 and 2, we used type E thermocouples, while with samples 3 and 4, type T thermocouples were used. Sample 1, three thermocouples were connected differentially, while a separate fourth thermocouple was mounted on the sample to measure sample temperature. Samples 2, 3 and 4 used three separate thermocouples. The longitudinal (∆T x ) and transverse (∆T y ) temperature differences were calculated by directly converting the measured voltages using the thermocouples standardized Seebeck polynomials. ∆T y was then anitsymmetrized to remove the contact misalignment. ∆T x and ∆T y were then used to calculate κ xx and κ xy using Fourier's law of heat conduction. We describe a monolayer of MnPS 3 by the Hamiltonian H = H s + H l + H sl , which comprises a spin H s , a lattice H l , and a spin-lattice Hamiltonian, H sl . The magnetic interactions, given by [16,17]
(S i spin operator of site i, h Planck constant, g Landé factor, µ B Bohr magneton) encompass Heisenberg exchange interactions up to third-nearest neighbors (J r , r = 1, 2, 3), outof-plane DMI between second-nearest neighbors (D ij = ± Dẑ) an easy axis anisotropy (K), and a Zeeman term that couples the spins to the external magnetic field B z . For the lattice Hamiltonian we use [16]
We focus on the out-of-plane vibrations as the in-plane phonon modes do not couple to the magnons to leading order (see below). Here, M is the mass of the Mn 2+ ions, p z i is the z component of the momentum operator of site i, and u z i is the displacement of site i along z. The displacements are coupled to the spins by [16] H sl = λ h2
where λ is the strength of the spin-lattice coupling and rij is the normalized nearest-neighbor bond vector. We employ the following parameters specific to MnPS 3 : J 1 1.054 meV, J 2 = 0.048 meV, J 3 = 0.3 meV, |D| = 0.78 µeV, K = -2 µeV, g = 1.824, and λ = 0.0292 meV Å -1 [16]. The spin quantum number of the local Mn 2+ spins is S = 5/2 [53]. For the elastic constant, we experimentally determined the sound velocity for the transverse phonons as 3238 m s -1 , which implies C = 690.7 m Å -2 assuming a lattice constant of a = 5.88 Å.
In order to describe the spin degrees of freedom (S i ) by bosonic creation and annihilation operators
, the truncated Holstein-Pimakoff transformation [54]
is employed using the classical ground state spin orientations ẑi , which play the role of local quantization axes. Here, ê± i = (x i ± iŷ i ) / √ 2. The local axes xi , ŷi , and ẑi are chosen to constitute a right-hand coordinate system. Relabeling the sites i For the vibrational degrees of freedom (u z i , p z i ), we apply the transformation
where b i , b i † are bosonic annihilation and creation operators, and ω = √ 6C M is the local eigenfrequency.