## Abstract

Breaking of spatial inversion symmetry induces unique phenomena in condensed matter. In particular, by combining this symmetry with magnetic fields or another type of time-reversal symmetry breaking, noncentrosymmetric materials can be made to exhibit nonreciprocal responses, which are responses that differ for rightward and leftward stimuli. However, the effect of spatial inversion symmetry breaking on thermal transport in uniform media remains to be elucidated. Here, we show nonreciprocal thermal transport in the multiferroic helimagnet TbMnO_{3}. The longitudinal thermal conductivity depends on whether the thermal current is parallel or antiparallel to the vector product of the electric polarization and magnetization. This phenomenon is thermal rectification that is controllable with external fields in a uniform crystal. This discovery may pave the way to thermal diodes with controllability and scalability.

## INTRODUCTION

Unconventional physical properties in noncentrosymmetric materials have been investigated for long (*1*, *2*). Besides the classic examples such as natural optical activity and piezoelectricity, the spin-orbit interaction further enriches the effect of spatial inversion symmetry (SIS) breaking, as exemplified by the Rashba effect (*3*). When SIS and time-reversal symmetry (TRS) are broken simultaneously, there emerge distinct phenomena denoted as nonreciprocal responses, which are responses that differ for rightward and leftward stimuli (*4*). Nonreciprocal responses were first studied in terms of optical properties (*5*, *6*). When TRS and SIS are simultaneously broken, the optical properties such as absorption and luminescence become dependent on the sign of the optical wave vector irrespective of the polarization. Similar directional dependences were observed for magnon and phonon propagation (*7*–*10*). Regarding electronic transport, resistivity becomes directionally dependent; that is to say, rectification appears in materials without TRS and SIS (*11*–*14*). The symmetrical rule for the unidirectional response is common among these nonreciprocities. In chiral materials, in which any mirror symmetry is broken, directional dependence is observed along the magnetization **M** for any nonreciprocities. On the other hand, for polar materials, which have finite electric polarization **P**, the directional dependence is observed along **P** × **M**. In this paper, we study nonreciprocal thermal transport in a polar system.

We chose the multiferroic spiral magnet TbMnO_{3} as a sample (*15*). TbMnO_{3} shows transverse spiral magnetic order with the propagation vector along the *b* axis, accompanied by **P** along the *c* axis. We measured nonreciprocal thermal transport in magnetic fields parallel to the *a* axis, in which a conical magnetic structure is realized (Fig. 1A). Let us discuss the phononic state in this magnetic structure to study the thermal transport microscopically. As Hamada and co-workers theoretically suggested, the breaking of SIS induces phonon angular momentum **L** (*16*), similarly to the Rashba and Dresselhaus effects in electronic states (*17*). In polar systems, the induced **L** at momentum **k** is along **k** × **P**. Because **L** is caused by circular or ellipsoidal phonon polarization in the plane perpendicular to **L**, it originates from the mixing of two linearly polarized phonon modes in the plane. In the low-energy acoustic branches, these two modes respectively correspond to the longitudinal mode and one of the transverse modes with phonon polarization along **P**. The solid lines in Fig. 1B show the expected acoustic phonon modes for ferroelectric TbMnO_{3} at zero field. The longitudinal mode and one of the transverse modes have the phonon angular momenta *L*_{0} and −*L*_{0}, respectively, along **k** × **P** as a result of mixing. On the other hand, the other transverse mode has linear polarization along **k** × **P**, and therefore, the phonon angular momentum is zero. When **k** changes its sign, **L** is reversed, but the energy is unchanged. By the application of the magnetic field along the *a* axis, the energies of phonon modes shift depending on **L** (dotted lines in Fig. 1B), owing to the so-called Raman-type spin-phonon interaction H_{Raman} = K **m∙L** (*18*, *19*), where **m** is a magnetic moment. Thus, the phonon energy dispersion becomes asymmetric in the magnetic field. This asymmetry of phononic dispersion is the origin of thermal nonreciprocity.

## RESULTS

### Experimental setup

Thermal conductivity is usually measured in an experimental setup where one side of the sample is attached to a thermal bath and the other side to a heater (Fig. 1C). Because one needs to modify the experimental setup to reverse the heat current, it is difficult to estimate small rectification precisely. Here, we demonstrate thermal nonreciprocity in the multiferroic spiral magnet TbMnO_{3} by a comparison of the thermal conductivities for ±*P* and ±*M* states, without changing the direction of the thermal current. The thermal conductivity is expected to be rectified along the *b* axis (∣∣**P** × **M**). Reversal of thermal current is equivalent to **P** (**M**) reversals and 180° rotation of experimental setup around **M** (**P**). The 180° rotation does not change any physical responses, and therefore, the reversal of **P** or **M** corresponds to reversal of the thermal current. Thus, we can estimate a small thermal rectification by comparative measurements with positive and negative **P** and **M**. For this purpose, we attached a pair of electrodes to control the electric polarization in addition to the conventional thermal transport setup, as shown in Fig. 1C.

### Magnetic, electric, and thermal transport properties of TbMnO_{3}

TbMnO_{3} shows several magnetic phase transitions in the low-temperature region. In Fig. 2A, we show the temperature dependence of magnetization at 0.5 T for TbMnO_{3}. The anomalies at 7, 27, and 43 K indicate the magnetic transitions (*15*, *20*). The manganese magnetic moments order incommensurately below 43 K. While the magnetic moments are almost collinear above 27 K, a cycloidal magnetic structure emerges, and ferroelectric polarization is induced along the *c* axis below 27 K (Fig. 2B). The increase at 7 K corresponds to the incommensurate order of terbium moments. Note that terbium oxides frequently show gigantic magnetoelastic phenomena such as phonon Hall effect (*21*, *22*) and acoustic Faraday effect (*23*). The thermal nonreciprocity in the present study also seems to be related to the strong magnetoelastic coupling of terbium moments, as discussed later. The temperature dependence of thermal conductivity is shown in Fig. 2C. The thermal conductivity is relatively small and monotonically decreases with lowering temperature. A previous study suggests that these are caused by strong scattering due to terbium moments (*24*).

### Nonreciprocal thermal transport

Next, let us discuss the magnetic field dependence of the magnetic and thermal properties at 4.2 K to demonstrate the thermal nonreciprocity. Figure 3A shows the magnetization as a function of the magnetic field parallel to the *a* axis at 4.2 K. The magnetization shows a step-like increase around 2 T. Because the *c*-axis polarization due to the spiral spin structure of manganese moments is robust in this magnetic field range (*25*), the transition is relevant to the ferromagnetic ordering of terbium moments. In Fig. 3B, we show the magnetic field dependence of thermal conductivity at 4.2 K. The thermal conductivity shows spike-like anomalies at the magnetic transition fields. A small peak is also observed at 0 T. Small hysteresis is observed in the low-field region below the transition field. In Fig. 3C, we plot the magnetic field dependence of thermal conductivity above the metamagnetic transition field on a magnified scale for the ±*P* states. It shows an asymmetric magnetic field dependence, and the asymmetry is reversed when the electric polarization is reversed. To scrutinize this phenomenon, we show in Fig. 3D the asymmetric part of the magnetic field–dependent thermal conductivity*P* states. Here, κ_{+}(*H*) and κ_{−}(*H*) are the thermal conductivity at a magnetic field *H* measured in field-increasing and field-decreasing runs, respectively. The magnetization measured after decreasing the magnetic field from a high enough positive field to a low magnetic field +*H* and that after increasing the magnetic field from a large-magnitude negative field to a magnetic field −*H* should have the same magnitude, but the signs are opposite to each other. Therefore, the difference κ_{+}(*H*) – κ_{−}(−*H*) reflects the effect of sign difference of magnetization, being free from the effect of magnetic hysteresis. Δκ/κ for +*P* shows an abrupt decrease around the magnetic transition and is almost constant in the high field region. When the polarization is reversed, Δκ/κ changes its sign. These observations indicate that the thermal conductivity depends on whether the thermal current is parallel or antiparallel to **P** × **M**. As mentioned above, various kinds of nonreciprocity were observed along **P** × **M** for samples with finite **P** and **M**. The present observation seems to correspond to the thermal version of nonreciprocal response. The thermal nonreciprocity is particularly enhanced when the terbium moments are fully polarized.

In Fig. 3E, we show the thermal current dependence of *P* states. Here, Δκ_{av}/κ is the average of Δκ/κ in the region 3T ≤ *H* ≤ 5T and −Δκ/κ in the region −5T ≤ *H* ≤ − 3T. When the thermal current is too large, we cannot maintain the sample temperature. When it is too small, we cannot measure the thermal conductance accurately. At present, we measured thermal conductivity in the thermal current range between 1.6 × 10^{2} Wm^{−2} and 3.2 × 10^{2} Wm^{−2}. The variation of Δκ_{av}/κ is within the error bar in this range. Nevertheless, it is reversed by the reversal of **P**, which corresponds to reversal of the thermal current as mentioned above. Therefore, these observations indicate that the thermal conductivity depends on the sign of the thermal current but not on the magnitude.

The nonreciprocity of thermal conductivity _{av}/κ. It emerges around 8 K and increases with decreasing temperature. The onset temperature is close to the transition temperature of terbium. In the high-temperature region above 8 K, the terbium moments with large magnetoelastic coupling is decoupled from the incommensurate magnetic structure, which is the source of symmetry breaking. As a result, the effect of magnetic order on the phononic system becomes much smaller. The magnetic field variation of thermal conductivity owing to the terbium moments is quite small above 8 K (see fig. S1). These characteristics also imply that the terbium moments are related to the origin of the nonreciprocal thermal transport.

## DISCUSSION

Last, let us discuss the origin of the nonreciprocal thermal transport. As mentioned above, the nonreciprocal signal correlates with the terbium magnetic state. Nevertheless, the terbium magnetic excitation hardly contributes to the thermal transport because it has a dispersionless local nature (*26*) and cannot carry heat. Therefore, inelastic scattering of phonons induced by the terbium moments seems to be the most plausible origin. When a phonon with angular momentum *L* is scattered by a terbium moment with total angular momentum *J*, the sum of *L* and *J* must be conserved through the entire scattering process. Therefore, the transition matrix is expressed as*H*′ is the interaction between the phonon and magnetic moment, and *q* is the angular momentum transferred from the terbium magnetic moment to the phonon. As shown in Fig. 3D, the nonreciprocal thermal transport is notable when the terbium moment is ferromagnetically polarized. In this case, the initial state of *J* is maximized. In other words, *J* should be decreased and *L* should be increased in the course of scattering. Therefore, the scattering by terbium moments is forbidden if the initial state is *L* = + 1 or the final state is *L* = − 1. As mentioned above, when SIS is broken, some phonon modes have finite angular momentum. Therefore, the effect of the selection rule should appear in the phononic properties.

To understand how the terbium magnetic scattering affects the thermal transport properties, let us calculate the nonreciprocal thermal transport for a simplified model. This is just for a qualitative understanding of nonreciprocal thermal transport in this system, while a more elaborate theory is needed for the quantitative understanding. We assume one-dimensional asymmetric phonon bands as indicated by the dotted lines in Fig. 1B and an induced phonon momentum magnitude ∣*L*_{0}∣ = 1. Therefore, the phonon angular momenta for three acoustic branches are *L* = 0, ± 1. The phonon density for the angular momentum *L* at momentum *k* under a temperature gradient is expressed as *g*(*k*, *L*) = *g*_{0}(*k*, *L*) + *g*′(*k*, *L*), where *g*_{0} and *g*′ are the phonon density in thermal equilibrium and the deviation, respectively. Assuming the relaxation time approximation, we get *v*, and *x* are the relaxation time, phonon velocity, and position, respectively. When *g*′(*k*, *L*) > 0 (*g*′(*k*, *L*) < 0), the relaxation process is caused by the scattering whose initial state (final state) is ∣*k*, *L*>. Therefore, for *L* = 1 (*L* = −1) mode in the range *k* > 0 and for the *L* = − 1 (*L* = 1) mode in the range *k* < 0. For simplicity, we assume linear dispersion relations in the phonon band structure. The velocities for the *L* = +1, 0, −1 modes are *v*_{+,+}, *v*_{0}, and *v*_{−,+} for *k* > 0 and −*v*_{+,−}, −*v*_{0}, and −*v*_{−,−} for *k* < 0. The scattering rate is assumed to be

Here, ϵ is phonon energy *vk*. Similarly, for

Then, the nonreciprocity is

This relation is finite when *v*_{+, +} ≠ *v*_{+, −} or *v*_{−, +} ≠ *v*_{−, −}. Thus, the scattering by ferromagnetic terbium moments and simultaneous breaking of SIS and TRS induce the nonreciprocal thermal transport. This is independent of the magnitude of the thermal current density. If the magnitude of the induced phonon angular momentum *L*_{0} is less than 1, the nonreciprocity is reduced but still finite unless *L*_{0} becomes zero. This simple model is consistent with the experimental observations; the nonreciprocity is fairly enhanced by the ferromagnetic alignment of terbium moments and Δκ/κ is independent of thermal current density.

In summary, we have demonstrated nonreciprocal thermal transport in multiferroic TbMnO_{3}; the thermal transport is rectified along the vector product of the polarization and magnetization. Similar thermal rectification phenomena were already studied theoretically and experimentally. Zhang *et al*. (*27*) theoretically suggest that thermal rectification can be induced by a magnetic field in a nanoscale three terminal junctions. Some theories also suggest that nonlinear lattice dynamics induces the thermal rectification (*28*, *29*). Experimentally, thermal rectifications were previously realized by using combinations of two different materials or asymmetrically shaped media (*30*–*35*). On the other hand, we achieved the thermal rectification in a uniform bulk crystal that is induced by the phononic band asymmetry that originates from the symmetry breaking. The present result suggests that the thermal rectification should be emergent in any materials without both the time-reversal and spatial inversion symmetries. In some of these materials, the symmetries are broken even at room temperature. Therefore, thermal rectification at room temperature is also expected. Much larger nonreciprocity may also show up in some of these materials. A promising way to increase the magnitude of rectification is to use the material with the large electric polarization. The electric polarization in TbMnO_{3} is orders of magnitude smaller than that in typical ferroelectrics because the ferroelectricity in TbMnO_{3} is induced by the magnetic structure. Materials with larger polarization are expected to show larger thermal nonreciprocity. The thermal rectification seems useful for the efficient management of heat in various circumstances (*30*). In the cases of previously realized thermal rectification (*30*–*35*), the forward and reverse directions are determined by structural asymmetries, and the directionality cannot be reversed by external fields. In addition, an asymmetric structure limits the scale of the device. In the present case, the thermal rectification is realized in a uniform crystal and can be reversed by the external electric or magnetic fields. If the magnitude of thermal nonreciprocity becomes large enough to make practical applications feasible, the scalability of the uniform crystal and the ability to control the phenomenon with the external fields are substantial advantages over the previous approaches. While thermal current can be controlled by the application of electric current with the use of thermoelectric effect as exemplified by a recent paper (*36*), constant application of electric fields is not needed after the alignment of *P*, and the energy consumption is much less in the present case. Thus, the present results suggest that the exploration of thermal properties in symmetry-broken materials is a new research strategy that should be investigated toward the realization of thermal diode devices.

## MATERIALS AND METHODS

A TbMnO_{3} single crystal was grown by means of the floating zone technique. The conditions were almost the same as those in previous reports (*15*, *25*). Magnetization was measured with the use of a superconducting quantum interference device. Thermal conductivity measurements were performed by means of the steady-state method in a superconducting magnet. The measurement atmosphere was evacuated down to 3 × 10^{−3} Pa. Heat current was generated by chip resistance. The thermal gradient was measured by two thermometers (CERNOX, Lake Shore Cryotronics Inc.). The sample temperature is estimated as the average of the measured temperatures. To control the electric polarization, a pair of Au/Ti electrodes were formed on the surface of the thermal transport sample. The thicknesses of the Au and Ti were 20 and 2 nm, respectively. Before the thermal conductivity measurement, we performed the following poling procedure to obtain the full alignment of ferroelectric polarization (*15*, *25*). We applied an electric field between the electrodes as large as 750 V/mm in high-temperature paraelectric state and slowly decreased the temperature to the measurement temperature. Before the thermal conductivity measurement, we turned off the external electric field and wait a long time so that the relative temperature fluctuation δ*T*/*T* is comparable to 10^{−4}.

## SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/6/40/eabd3703/DC1

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is **not** for commercial advantage and provided the original work is properly cited.

## REFERENCES AND NOTES

**Acknowledgments:**We thank S. Murakami for fruitful discussions.

**Funding:**This work was supported in part by JSPS KAKENHI (grant numbers JP16H04008, JP17H05176, JP18K13494, and 20K03828), PRESTO (grant number JPMJPR19L6), the Murata Science Foundation, and the Mitsubishi foundation.

**Author contributions:**Y. H. carried out the crystal growth, magnetization measurement, and thermal transport measurement with assistance from Y.N. and H.M. Y.O. conceived and supervised the project. Y.O. wrote the paper through the discussion and assistance from Y. H., Y. N., and H. M.

**Competing interests:**The authors declare that they have no competing interests.

**Data and materials availability:**All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.

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