Planar thermal Hall effect from phonons in a Kitaev candidate material

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by phonons.The phonon contributed planar   also shows a strong sample dependence, which indicates an extrinsic origin of the mechanism.By conducting a complete study with different in-plane configurations of heat current J and magnetic field H, i.e.H // J and H ⊥ J, we observe a large difference in   between these two configurations, which reveals that the direction of the heat current J may play an important role in determining the planar thermal Hall effect.Our observation calls for a re-evaluation of the planar thermal Hall effect observed in -RuCl 3 .
The quest for quantum spin liquids (QSLs) has attracted tremendous interest due to the potential realization of non-Abelian statistics and novel exotic excitations [9].A promising platform for the realization of QSLs is the Kitaev model, which features bond-dependent Ising interactions between spin-1/2 degrees of freedom on a honeycomb lattice [10].The Kitaev model is exactly solvable, and it predicts the existence of itinerant Majorana fermions that carry heat and should therefore contribute to thermal transport [5].A topologically protected edge current can emerge from the bulk Majorana bands under an external magnetic field and be detected by the thermal Hall effect as a half-quantized thermal Hall conductivity  #$ [11,12].
The search for Kitaev QSLs in real materials has focused on 5d iridium [13] and 4d ruthenium compounds [14], of which the quasi-2D Mott insulator -RuCl3 has been the most intensively studied.In -RuCl3, antiferromagnetic (AF) order sets in below a temperature TN ≃ 7 K, with a spin configuration called "zigzag" order, but the application of a magnetic field H parallel to the honeycomb layers suppresses this order for H ≳ 7 T, thereby raising the possibility of a field-induced QSL state at low temperature when H ≳ 7 T.A half-quantized  #$ (i.e.,  #$ %& / =  ' % /6ℏ) was reported in -RuCl3 -for an in-plane field in excess of 7 T -and interpreted as evidence of itinerant Majorana fermions [1,2].The half-quantized  #$ plateau appears even for a "planar" Hall configuration [8], i.e., when the magnetic field is applied within the 2D plane and parallel to the heat current J, specifically for H // a, where a is the crystal direction perpendicular to the Ru-Ru bond (the so-called zigzag direction).Subsequently, Czajka et al. reported that the planar  #$ in -RuCl3 shows no sign of half-quantization, and they instead attributed its smooth growth with temperature for H // J // a to chiral magnons [3].
Theoretical work has shown that Majorana fermions [5] and topological magnons [6,7] are both able to generate a planar  #$ in -RuCl3, when H // a.
In contrast to these two scenarios of exotic topological excitations, it has also been argued that phonons are the main carriers responsible for the thermal Hall effect in -RuCl3 -at least for a field normal to the 2D planes (H // c and J // a) [4].The argument is based on the striking similarity of  #$ (T) to  ## (T), the phonon-dominated longitudinal thermal conductivity.
However, it remains unknown whether phonons can also generate a planar  #$ , where H // J // a.
Note that a non-zero planar Hall effect -i.e. a non-zero ΔTy for H // x in Fig. 1c and 1d is in principle only allowed if the crystal structure of a material breaks three symmetries: the xy and yz planes are not mirror planes, and the C2 rotational symmetry is broken along the x direction.In the monoclinic -RuCl3 (space group C2/m), the honeycomb (ab) plane is not a mirror plane, nor is the plane normal to the a axis.Furthermore, the C2 rotational symmetry is broken along the a axis, so a planar  #$ is allowed by symmetry for H // a, and is indeed observed [3,8].However, the plane normal to the b direction is a mirror plane and the C2 rotational symmetry is also preserved along the b direction; consistently, measurements report Here we turn to another Kitaev magnet candidate, the insulating material Na2Co 2 TeO6 [15,16], and present a study of its planar thermal Hall effect.Na2Co 2 TeO6 is a honeycomblayered insulator (Fig. 1a) that develops long-range AF order below TN ≃ 27 K [17] -which resembles the low-temperature formation of AF order in -RuCl3.It has been theoretically predicted that the Kitaev model can also be realized in materials with d 7 ions such as Co 2+ [18-20] and magnetic excitations in Na2Co 2 TeO6 indeed resemble calculations based on extended Kitaev-Heisenberg models [21][22][23][24][25][26].In our thermal transport study, the magnetic field H and heat current J are both applied in the ab plane, either parallel to each other (Fig. 1c) or perpendicular to each other (Fig. 1d).[27,28].A similar behaviour is observed in -RuCl3 [29,30], with a sudden increase of  ## when  > 7 T.In both materials,  ## is attributed to phonons that are strongly scattered by spin fluctuations.When a field large enough to suppress AF order is applied in the 2D plane, a spin gap opens in the field-polarized state [31], and so the spin scattering is reduced at low T, leading to an increase in  ## [27,29,30].
In Fig. 2b, we show the thermal Hall conductivity of Na2Co 2 TeO6, measured on the same sample (A) in the same configuration (H // J // a), plotted as  #$ vs T. Surprisingly, we observe a non-zero  #$ , which is supposed to be forbidden by the two-fold rotational symmetry along this direction. #$ () mirrors the evolution of  ## () at different fields.At H = 5 T and 10 T,  #$ () and  ## () both show a broad hump around 40 K, while at H = 15 T, both display the same dramatic increase at low T, peaking at T ~ 10 K.This striking similarity between  #$ () and  ## () is compelling evidence that  #$ is carried predominantly by phonons in Na2Co 2 TeO6.Evidence from other insulators has indeed shown that for phonons  #$ and  ## both increase in tandem [32][33][34][35].By contrast, if  #$ were caused by Majorana fermions or magnons (or any other spin-based excitation), there would be no reason for it to mimic the phonon-dominated  ## ().
In Figs.2c and 2d, we report the equivalent study for the planar-parallel configuration H // J // a*, performed on a second sample (B).Again, we observe a non-zero  #$ signal with H // J // a*, which is also supposed to be forbidden by the two-fold rotational symmetry along this direction.This observation shows that the actual mechanism behind the planar thermal Hall effect in this material remains effective even though the pristine lattice has two-fold rotational symmetry in the a* direction.We see again a dramatic increase of both  #$ and  ## when a field of 15 T is applied, reinforcing the close correlation between  #$ and  ## seen in the first configuration.Interestingly, we find that in the second configuration (H // a*) the parallel increase of  #$ and  ## at low T even begins at 10 T, further confirming that  #$ mimics  ## .
We infer that the critical field for suppressing the AF order in Na2Co 2 TeO6 is slightly less than 10 T for H // a* (and more than 10 T for H // a).
To check the reproducibility of our data, we performed the same measurements on another two samples (C and D) that were cut from the same mother sample. ## and  #$ measured on sample C with H // J // a and on sample D with H // J // a* are plotted in Extended Data Fig. 2.
Similar behavior of  ## and  #$ are observed in samples C and D. However, the magnitude of  #$ shows a clear sample dependence, especially when comparing samples B and D, which points to an extrinsic origin of the planar thermal Hall effect in Na2Co 2 TeO6.This sample dependence may also explain the much smaller magnitude of  #$ reported in a prior study by Takeda et al. [28].
Based on the close similarity we observe -for the two distinct field directions -between the temperature and field dependence of the planar  #$ and that of the phonon-dominated  ## , we conclude that phonons are responsible for the planar thermal Hall conductivity  #$ in Na2Co 2 TeO6 -where field and current are both in the plane and parallel to each other.
This shows it is possible -and makes it likely -that the planar  #$ observed in -RuCl3 is also carried by phonons.
In Fig. 3a, we compare the ratio of  #$ over  ## , plotted as  #$ /  ## vs T, in all four samples.With a configuration of H // J // a, the ratio of sample C is about two times larger than that of sample A, at T = 20 K.With a configuration of H // J // a*, the ratio of sample B is about five times larger than that of sample D. Although a clear sample dependence is observed, the magnitude of |  #$ /  ## | in all cases is typical of the phonon thermal Hall effect found in various insulators (albeit for H // z) [33,34,35], where 0.05% ≲ ; ( !" ; ≲ 0.5% at T = 20 K and H = 15 T. In Fig. 3b, we also see that the planar thermal Hall effect in Na2Co 2 TeO6 is comparable -in both magnitude and temperature dependence -to that seen in -RuCl3 for H // J // a [36].This striking similarity between Na2Co 2 TeO6 and -RuCl3 points to a common underlying mechanism. After measuring both  ## and  #$ with the heat current and magnetic field parallel to each other, we conduct the same measurements with H and J both in plane but perpendicular to each other (H ⊥ J).In Fig. 4a, we show the thermal conductivity  ## of Na2Co 2 TeO6 at H = 15 T as a function of temperature with H // a, for two current directions: J // H (sample C, J // a, red) and J ⊥ H (sample D, J // a * , blue).In Fig. 4c, we show the same comparison of current directions for H // a * .We see that with the same field direction,  ## for J ⊥ H is very similar in magnitude and temperature dependence to  ## for J // H.In other words, the current direction matters very little for  ## .However, as shown in Figs.4b and 4d,  #$ decreases dramatically when H changes from parallel to J to perpendicular to J.This observation clearly shows that the magnitude of the planar  #$ strongly depends on the direction of the heat current relative to the magnetic field, i.e. whether H // J or H ⊥ J.A similar behavior is also observed in sample A and sample B (Extended Data Fig. 3).
Three main questions arise.First, what makes phonons chiral in Na2Co 2 TeO6?The sample dependence we observe suggests an extrinsic origin for the phonon thermal Hall effect, e.g. from scattering of phonons by defects or impurities.In a recent model, it was shown that defects embedded in an insulator with AF order can scatter phonons in a way that produces a thermal Hall effect in a magnetic field [37] -a mechanism that may well explain the dependence of  #$ on impurity concentration in the AF insulator Sr2IrO4 [38].
The second question is: how can there be a non-zero  #$ signal in Na2Co 2 TeO6 when H // J // a or H // J // a*, two field directions for which a non-zero  #$ is in principle forbidden by the C2 rotational symmetry?Clearly, this planar Hall effect cannot originate from some intrinsic scenario controlled entirely by the underlying crystal symmetry, such as the scenario of Majorana fermions or topological magnons.Instead, we suggest that it may be due to a local symmetry breaking induced by some extrinsic effects.For example, by structural defects like stacking faults or domains, reminiscent of the proposal that structural domains play a role in generating a phonon thermal Hall effect in SrTiO3 [32].Indeed, it has been reported that the Na layers in Na2Co 2 TeO6 are highly disordered [16], which could possibly break the local crystal symmetry.
The third question is: how to understand the large difference in the magnitude of  #$ between H // J and H ⊥ J? Our results indicate that when putting both H and J in the plane, the planar  #$ can be dramatically reduced when current and field are perpendicular to each other.
In previous theoretical explanations [6][7][8] for the planar thermal Hall effect observed in RuCl3, whether a non-zero planar  #$ can arise only depends on the underlying crystal symmetry and the direction of magnetic field, regardless of the direction of heat current.Our findings reveal that the direction of heat current also plays an important role in producing the planar thermal Hall effect.This calls for a re-evaluation of the mechanism responsible for the planar  #$ observed in RuCl3.
Note that in addition to Na2Co 2 TeO6, we have also observed a phononic planar thermal Hall signal (comparable in magnitude to the conventional thermal Hall signal) in both cuprates [39] and in the frustrated antiferromagnetic insulator Y-kapellasite [40], thereby further validating the existence of a planar thermal Hall signal coming from phonons.[27] (see Extended Data Fig. 1). $$ and  ## reported in Ref. [27] are measured on two samples that are cut from the same mother sample, which reflects the intrinsic anisotropy of the longitudinal thermal conductivity when the heat current is applied along a or a * direction.This anisotropy is also consistent with what we get from sample C and sample D, which are cut from the same mother samples.
The experimental technique used here was the same as described in Refs.[39,40].The heat current is generated by a resistive heater connected to one end of the sample (Fig. 1c and 1d).
The other end of the sample is glued to a copper block with silver paint that acts as a heat sink.
The longitudinal and transverse temperature differences ∆ # and ∆ $ are measured using type-E thermocouples.All the measurements are conducted with a steady-state method in a variable temperature insert (VTI) system up to H = 15 T. The data was taken by changing temperature in discrete steps at a fixed magnetic field.After the temperature is stabilized at each temperature, the background value of the thermocouple is eliminated by subtracting the heater-off value from the heater-on value.When measuring  #$ , the contamination from  ## due to a slight misalignment of contacts for ∆ $ is removed by doing field anti-symmetrization to the transverse temperature difference.That is to say, we measure ∆ $ with both positive and negative magnetic fields exactly in the same conditions, then the transverse temperature difference used to obtain

. 2 |
Thermal transport in samples C and D. Thermal conductivity  ## vs T in Na2Co2TeO6, measured on sample C (a), measured with H // J // a, and sample D (c), measured with H // J // a * , at H = 0, 5, 10 and 15 T. Corresponding thermal Hall conductivity  #$ for sample C (b) and sample D (d).

. 3 |Na 2
Comparing H // J and H ⊥ J in samples A and B. a) Thermal conductivity  ## vs T and b) thermal Hall conductivity  #$ vs T in Na2Co2TeO6 measured on sample A with J // a for H // a (red) and H // a * (blue), at H = 15 T. c, d) Corresponding data for sample B, measured with J // a * .We observe that  #$ measured in the configuration H ⊥ J is much smaller than in the configuration H // J. Co 2 TeO 6 sample B J // a* xy (mW / Km) T (K) ## and  #$ are measured simultaneously, for four Note that the structure of Na2Co 2 TeO6 is such that a non-zero  #$ is not allowed for either H // a or H // a * because its crystal structure (space group P6322) has C2 rotational symmetry along both a and a * directions.
* , and J // a * & H // a. however, produces a dramatic enhancement of  ## at low T, in agreement with prior data

Fig. 4 | Thermal transport data in Na2Co2TeO6 for H // J and H ⊥ J. a)
Na 2 Co 2 TeO 6 Thermal conductivity  ## and b) thermal Hall conductivity  #$ vs T in Na2Co2TeO6 s, measured at H = 15 T for H // a: on sample C with J // a (red) and sample D with J // a * (blue).c,d)Correspondingdata for H // a * .In both field directions,  #$ measured with H ⊥ J is much smaller than that measured with H // J.The thermal Hall conductivity is defined as  #$ = − $$ (∆ $ / ∆ # )(/).For sample A, the current and field directions are J // H // a or J // a & H // a * , where a is perpendicular to the Co-Co bond direction in the lattice and a* is parallel to the Co-Co bond direction.For sample B, J // H // a* or J // a * & H // a.In a honeycomb lattice,  ## ≠  $$ .For sample A and B, we obtain  $$ by multiplying the  ## measured on the same sample by the anisotropy factor  $$ / ## reported in Ref.

THERMAL CONDUCTIVITY AND THERMAL HALL CONDUCTIVITY MEASURED IN FOUR Na2Co2TeO6 SAMPLES
.  ## and  #$ measured in sample C with H // J // a and sample D with H // J // a* are plotted in Extended Data Fig. 2.
[27]nd  #$ measured in sample A with H // J // a or J // a & H // a * and sample B with H // J // a * or J // a * & H // a are plotted in Extended Data Fig.3.Extended DataFig. 1 | Thermal conductivity data in Na2Co2TeO6 with J // a.Red curve is the thermal conductivity  ## vs T in Na2Co2TeO6 sample A measured at H = 0 T with J // a.The thermal conductivity  $$ at H = 0 T with J // a * in that sample (blue curve) is estimated by multiplying the  ## data by the anisotropy factor  $$ / ## reported in Ref.[27].