Angle-dependent planar thermal Hall effect by quasi-ballistic phonons in black phosphorus

The thermal conductivity is a second rank tensor linking the heat current density and the temperature gradient vectors. The thermal Hall effect (THE) refers to non-zero off-diagonal components of this tensor, $\overline{\kappa}$. Its origin in insulators has attracted much recent experimental [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17] and theoretical [18, 19, 20, 21, 22, 23, 24, 25, 26, 27] interest.

The planar thermal Hall effect refers to a configuration in which the three relevant vectors (heat current, temperature gradient and magnetic field) lie in the same plane. This was first observed in the Kitaev spin liquid candidate $\alpha$-RuCl₃ [4, 28, 29, 30]. More recently, it was also observed in other solids, where the thermal Hall signal is attributed to phonons [31, 32, 33]. The persistence of the signal, when it is forbidden by crystal symmetry, was attributed to defects breaking the local symmetry [32].

Here, we report on the observation of a planar Hall effect in black phosphorus(P), an elemental insulator with ballistic phonons [34]. The thermal conductivity tensor has unequal diagonal components ($\kappa_{zz}\simeq 5\kappa_{xx}$). Nevertheless, the off-diagonal components match each other ($\kappa_{xz}(B)\simeq\kappa_{zx}(-B)$), as expected by Onsager reciprocity. We quantify the variation of the thermal Hall signal as a function of the in-plane orientation of the magnetic field and find that it displays a twofold oscillation with minimum and maximum along one of the two diagonals of the $xz$ plane of the puckered honeycomb lattice. This indicates that the signal is the sum of two sinusoidal contributions along two high-symmetry axes. A quantitative account of our observation is missing. Nevertheless, we argue that because of the finite thermal expansion, a temperature gradient allows the emergence of a signal forbidden in thermodynamic equilibrium. Electric dipole moments travelling with sound can couple to magnetic field and generate a thermal Hall signal in planar configuration.

Detailed materials and methods can be found in the supplementary materials, including sample details, thermal transport measurement methods.

Fig. 1a shows the crystal structure of black P. Identical phosphorus atoms located on two distinct sites are marked in blue and red. Each layer is a puckered honeycomb lattice in the $xz$ crystallographic plane, where the $x$ and $z$ axes correspond to the armchair and zigzag orientations. As seen in Fig. 1b, the longitudinal thermal conductivity, as found previously [34, 35], is significantly different along the two orientations. Along the zigzag orientation, thermal conductivity is much larger than along the armchair orientation in the low temperature limit. The sound velocity shows a similar but attenuated anisotropy (9.6 km/s along $z$-axis vs. 4.6 km/s along the $x$-axis) [36]. As a consequence, the phonon mean free path $\ell_{\mathrm{ph}}(T)$, shown in Fig. 1c, is also anisotropic. It is twice longer along the $z$-axis and approaches the sample thickness (30 $\mu$m) around 20 K. Thus, at this temperature phonons become quasi-ballistic and accordingly thermal conductivity becomes size dependent [34].

A Hall response refers to a signal odd in magnetic field and oriented perpendicular to the applied current. Designating the transverse temperature gradient by $(\nabla T)_{\perp}$, the applied longitudinal heat current by $J_{\mathrm{q}}$ and the magnetic field by $B$, the standard thermal Hall configuration corresponds to $B\perp J_{\mathrm{q}}\perp(\nabla T)_{\perp}$. It is sketched in panel d and was the subject of our previous study [15]. Panel e shows the configuration for planar thermal Hall effect, which corresponds to $B~{}||~{}(J_{\mathrm{q}}\perp(\nabla T)_{\perp})$. We use $\kappa_{ij}^{k}$ to designate the $\overline{\kappa}$ component corresponding to $J_{\mathrm{q}}$ along the $i$-axis, $(\nabla T)_{\perp}$ along the $j$-axis and the magnetic field along the $k$-axis.

Fig. 2a–c shows the planar thermal Hall data for three different configurations. In each panel, the orientations of the three relevant vectors ($B$, $J_{\mathrm{q}}$ and $(\nabla T)_{\perp}$) are sketched and the field dependence of the ratio of the transverse to longitudinal temperature gradients ($\nabla_{j}T/\nabla_{i}T$) is shown.

Fig. 2d compares the temperature dependence of the thermal Hall angle in three planar configurations at 12 T. It is finite in the three planar configurations, but there is a large difference in amplitude when one permutes the orientations of $J_{\mathrm{q}}$ and $(\nabla T)_{\perp}$. Since the longitudinal thermal conductivity is anisotropic by the same factor, this difference warrants an equality between the absolute values of $\kappa_{xz}^{z}$ and $\kappa_{zx}^{z}$, as expected by Onsager reciprocity.

The thermal Hall angle (Fig. 2d) combined with the longitudinal thermal conductivity (Fig. 1b) leads to the planar thermal Hall conductivity, which is shown in Fig. 2e and f. The thermal Hall response is finite in four distinct configurations. We can see not only the validity of the Onsager reciprocity (Fig. 2e), but also the very small influence of the orientation of magnetic field on the amplitude of $\kappa_{xz}$ (Fig. 2f).

We then proceeded to quantify the angle dependence of this planar Hall signal at fixed temperature as the magnetic field rotated in the $xz$ plane. The raw data, shown in the Supplementary material, shows a field-linear response for all explored angles. The measurements were performed at 44.5 K.This temperature was chosen to strike a balance between proximity to the peak temperature and the optimal sensitivity of the thermocouples. As seen in Fig. 3a, the signal exhibits a twofold oscillation. It peaks (with opposite signs) at $\phi=-\pi/4$ and $\phi=3\pi/4$, i.e., along one of the diagonals of the $xz$ plane, and vanishes along the other diagonal.

This intriguing feature would find a straightforward interpretation if the signal is the sum of two contributions of equal amplitude shifted by $\pi/2$, one following cos$\phi$ (peaking along the $x$-axis and vanishing along the $z$-axis) and another following cos$(\phi+\pi/2)$ (peaking along the $z$-axis and vanishing along the $x$-axis), indicated by dashed lines in Fig. 3a. As sketched in Fig. 3b, each odd-field contribution would be bounded to one mirror plane.

Thus, we detect in a crystal belonging with the $mmm$ point group symmetry, there is a planar thermal Hall signal with bi-axial symmetry. Let us note that at peak temperature of this thermal Hall signal, the longitudinal thermal conductivity of black P is as large as $\kappa_{\mathrm{p}}\simeq 2000$ WK^-1m^-1. This is to be compared with cases such as $\alpha$-RuCl₃ ($\kappa_{\mathrm{p}}\simeq$ 3 WK^-1m^-1) [14], Na₂Co₂TeO₆ ($\kappa_{\mathrm{p}}\simeq$ 10 WK^-1m^-1) [31] or Y-kapellasite ($\kappa_{\mathrm{p}}\simeq$ 2 WK^-1m^-1) [32]. The thermal conductivity of black phosphorus is three orders of magnitude larger, with a phonon mean free path close to the sample thickness. These features strongly suggest that the thermal transport mechanism in black phosphorus is dominated by boundary scattering and phonon-phonon scattering, rather than being influenced by local defect scattering. However, we cannot totally exclude a minor role played by local defects.

In anisotropic dense medium, thermal diffusivity is a tensor, $\overline{D}$. The heat equation becomes:

$$\frac{\partial T}{\partial t}=\overrightarrow{\nabla}\cdot(\overline{D}\overrightarrow{\nabla}T)$$

(1)

Consider now the energy continuity equation, which relates the energy flux to specific heat per volume, $C$:

$$\overrightarrow{\nabla}\cdot\overrightarrow{J_{\mathrm{q}}}+C\frac{\partial T}{\partial t}=0$$

(2)

The combination of the two previous equations leads to:

$$\overrightarrow{J_{\mathrm{q}}}=-C\overline{D}\overrightarrow{\nabla}T$$

(3)

The thermal conductivity tensor $\overline{\kappa}$, is thus the thermal diffusivity tensor multiplied by a scalar: $\overline{\kappa}=\overline{D}C$. Therefore, off-diagonal components in $\overline{\kappa}$ are proportional to their counterparts in $\overline{D}$. The latter is the product of carrier velocity and carrier mean free path. Sound velocity, $v_{\mathrm{s}}$, and phonon mean free path $\ell_{\mathrm{ph}}$ (see Fig. 1c) are both anisotropic at zero magnetic field. The issue is to find a way for magnetic field to skew one or both of these tensors off the symmetry axes.

Recent theoretical studies have scrutinized intrinsic phonon band [37] incorporated with energy magnetization contribution [38, 21] and phonon angular momentum acquired through interaction with the spins [39, 40] or with orbital motion of ions [41].

Absent magnetism, the interplay between electric polarization and phonons deserves scrutiny. The computed map of charge distribution is strongly orientational [42]. Dipole-active phonon modes has been detected by infrared spectroscopy [43]. These are optical phonons and not heat-carrying acoustic modes which interest us. The lowest optical phonon frequency of black phosphorus is around 140 cm^-1 [43] (see the Supplementary material for more details), much larger than the acoustic phonons. However, for the high temperature signals, the indirect contribution of optical phonons through coupling with acoustic phonons may not be negligible. Nevertheless, any phonon breaking the inversion symmetry will generate an electric dipolar wave. This feature highlighted in the context of ferrons, the elementary excitation of a ferroelectric solid [44, 45], persists even in a paraelectric insulator.

Black P has an anisotropic thermal expansion with opposite signs along armchair and zigzag crystalline orientations [46]. In presence of a temperature gradient, each unit cell is slightly distorted (See Fig. 4a, b and d). This feature, driven by unavoidable anharmonicity in any solid [47], is the fundamental reason that a signal forbidden in thermodynamic equilibrium is allowed in presence of a temperature gradient. It also paves the way for chirality in presence of the magnetic field (See Fig. 4c, e and f).

Let us now draw a tentative picture of how the magnetic field may couple to heat-carrying phonons. The inversion center of the pristine unit cell is not an atomic site and atomic displacements associated with phonons generate electric dipole moments (Fig. 5a). Therefore, acoustic phonons breaking inversion centers are traveling electric dipolar moments capable of coupling to a static magnetic field (Fig. 5b). Electric dipole wave and its evolution under magnetic field can be detected via methods such as $P$-$E$ response and second harmonic generation directly in the future. The torque exerted by a static magnetic field, $\overrightarrow{B}$, on an electric dipole of $\overrightarrow{P}$ moving with a velocity of $\overrightarrow{v}$ is : $\overrightarrow{\tau}=\overrightarrow{P}\times(\overrightarrow{B}\times\overrightarrow{v})$ [48]. Let us rewrite it as:

$$\overrightarrow{\tau}=(\overrightarrow{P}\cdot\overrightarrow{v})\overrightarrow{B}-(\overrightarrow{P}\cdot\overrightarrow{B})\overrightarrow{v}$$

(4)

Thus, the torque due to the Lorenz force exerted on each of the two poles is finite when the magnetic field is oriented perpendicular to the orientation of the dipole propagation (Fig. 5c). These are ingredients for a scenario for planar thermal Hall signal in a non-magnetic insulator (see the Supplementary material for a discussion of the expected angular variation). A rigorous treatment is beyond the scope of this paper, but torque is known to be responsible for changes in angular momentum, so a quantitative theoretical analysis could involve either the phonon angular momentum [39, 49] or the phonon Berry curvature [20]. However, let us compare the order of magnitude of the expected and the measured signals.

Experimentally, the thermal Hall angle at $B$ = 10 T peaks to $\approx 10^{-2}$ in black P. The length scale extracted from this angle and fundamental constants ($\lambda_{\mathrm{tha}}$ = $\ell_{B}\sqrt{\kappa_{ij}/\kappa_{jj}}$) is about 5 Å. Intriguingly, in all insulators displaying a thermal Hall signal, this length scale lies in a narrow range  [15, 17] (2 Å$~{}<\lambda_{\mathrm{tha}}<7$ Å). As seen in Fig. 6, this remains true for recently studied cases of Si, Ge and quartz [33].

Suppose an electric dipole moment of $\delta p$ traveling with the velocity of sound, $v_{\mathrm{s}}$ (loosely linked to the Debye frequency by $v_{\mathrm{s}}\sim a\omega_{\mathrm{D}}$, where $a$ is the interatomic distance). The exerted torque, that is the angular derivative of the energy, would be $Ba\omega_{D}\delta p$. Divided by the Debye energy, it yields $\theta_{H}\simeq\frac{a\delta p}{\hbar}B$, the order of magnitude of the expected rotation angle. With $\ell_{B}^{2}=\frac{\hbar}{eB}$, assuming $\delta p\approx ea$, leads to:

$$\theta_{H}\approx\frac{a^{2}}{\ell_{B}^{2}}$$

(5)

This is of the order of magnitude of the measured signal. A rigorous treatment would presumably include the Grüneisen parameter [47] and the atomic electric dipole polarizability. The former, which is dimensionless, remains of the order of unity and the latter introduces a length scale which does not vary widely among different solids [52]. This admittedly hand-waving scenario, attributing the thermal Hall signal to the interaction between a static magnetic field and traveling electric dipolar waves, was not considered before. Moreover, its influence merits in-depth consideration when applied to other elemental insulators.

The authors declare that they have no conflict of interest.

We thank Benoît Fauqué for stimulating discussions. This work was supported by the National Key Research and Development Program of China (2022YFA1403500), the National Natural Science Foundation of China (12004123, 51861135104 and 11574097 ), and the Fundamental Research Funds for the Central Universities (2019kfyXMBZ071). X. L. was supported by The National Key Research and Development Program of China (2023YFA1609600) and the National Natural Science Foundation of China (12304065).

Xiaokang Li, Zengwei Zhu, and Kamran Behnia proposed and supervised the project. Xiaokang Li and Xiaodong Guo prepared the samples, designed and performed the measurements. Xiaokang Li, Zengwei Zhu, and Kamran Behnia analyzed data and wrote the manuscript with the contributions of all authors.

Black phosphorus crystals used in this work are same to our previous report [15]. They were synthesized under high pressure and came from two different sources. Sample #1 was obtained commercially and sample #2 was kindly provided by Prof. Yuichi Akahama (University of Hyogo).

All thermal transport experiments were performed in a commercial measurement system (Quantum Design PPMS) within a stable high-vacuum sample chamber. An one-heater-four-thermocouples (type E) techniques was employed to simultaneously measure the longitudinal and transverse thermal gradient. The thermal gradient in the sample was produced through a 4.7 k$\Omega$ chip resistor alimented by a current source (Keithley6221). The DC voltage on the heater and thermocouples (thermometers) was measured through the DC-nanovoltmeter (Keithley2182A). The thermocouples, the heat-sink, and the heater were connected to samples directly or by gold wires with a 50 microns diameter. All contacts on the sample were made using silver paste. In the angular dependent measurements, a thermal transport rotation probe was used.

The longitudinal ($\nabla_{i}T$) and the transverse ($\nabla_{j}T$) thermal gradient generated by a longitudinal thermal current $J_{q}$ were measured. They lead to the longitudinal ($\kappa_{ii}$) and the transverse ($\kappa_{ij}$) thermal conductivity, as well as the thermal Hall angle ($\nabla_{j}T/\nabla_{i}T$):

$$\kappa_{ii}=\frac{Q_{i}}{\nabla_{i}T}$$

(S1)

$$\frac{\nabla_{j}T}{\nabla_{i}T}=\frac{\kappa_{ij}}{\kappa_{jj}}$$

(S2)

$$\kappa_{ij}=\frac{\nabla_{j}T}{\nabla_{i}T}\cdot\kappa_{jj}$$

(S3)

Here $Q$ is the heat power.

Fig. S1 shows the field dependent thermal Hall angles ($\nabla_{j}T/\nabla_{i}T$) with the $J_{q}$ alongs $x$ axis and the $(\nabla T)_{\perp}$ along $z$ axis at six different field orientations. $\phi$ is the angle of the magnetic field with respect to the $J_{q}$ ($x$ axis). $\phi=0^{\circ}$ and $\phi=90^{\circ}$ represent the orientations along two high-symmetry axes, and their $\nabla_{j}T/\nabla_{i}T$ signal have the same amplitude but opposite signs. $\phi=-45^{\circ}$ and $\phi=45^{\circ}$ represent two diagonal orientations of the puckered honeycomb plane that have the maximum and vanished responses respectively, suggesting that the planar thermal Hall signal consists of two distinct contributions each along one high-symmetry axis. $\phi=-45^{\circ}$ and $\phi=135^{\circ}$ have signals with opposite signs, as expected from the antisymmetric operation ($\kappa_{ij}$(B) = -$\kappa_{ij}$(-B)).

The torque exerted by a static magnetic field, $\overrightarrow{B}$, on an electric dipole of $\overrightarrow{P}$ moving with a velocity of $\overrightarrow{v}$ is :

$$\begin{split}\overrightarrow{\tau}&=\overrightarrow{P}\times(\overrightarrow{B}\times\overrightarrow{v})\\ &=(\overrightarrow{v}\times\overrightarrow{B})\times\overrightarrow{P}\\ &=(\overrightarrow{P}\cdot\overrightarrow{v})\overrightarrow{B}-(\overrightarrow{P}\cdot\overrightarrow{B})\overrightarrow{v}\end{split}$$

(S4)

The electric dipole $\overrightarrow{P}$, field $\overrightarrow{B}$, and the velocity $\overrightarrow{v}$ can be expanded as:

$$\begin{split}\overrightarrow{P}=P_{x}\overrightarrow{e_{x}}+P_{y}\overrightarrow{e_{y}}+P_{z}\overrightarrow{e_{z}};\\ \\ \overrightarrow{B}=B_{x}\overrightarrow{e_{x}}+B_{y}\overrightarrow{e_{y}}+B_{z}\overrightarrow{e_{z}};\\ \\ \overrightarrow{v}=v_{x}\overrightarrow{e_{x}}+v_{y}\overrightarrow{e_{y}}+v_{z}\overrightarrow{e_{z}}.\end{split}$$

(S5)

Thus, the S4 can be rewritten as:

$$\begin{split}\overrightarrow{\tau}=(P_{x}v_{x}+P_{y}v_{y}+P_{z}v_{z})\cdot(B_{x}\overrightarrow{e_{x}}+B_{y}\overrightarrow{e_{y}}+B_{z}\overrightarrow{e_{z}})\\ \\ -(P_{x}B_{x}+P_{y}B_{y}+P_{z}B_{z})\cdot(v_{x}\overrightarrow{e_{x}}+v_{y}\overrightarrow{e_{y}}+v_{z}\overrightarrow{e_{z}})\\ \\ =(P_{y}v_{x}B_{x}+P_{z}v_{z}B_{x}-P_{y}B_{y}v_{x}-P_{z}B_{z}v_{x})\cdot\overrightarrow{e_{x}}\\ \\ +(P_{x}v_{y}B_{y}+P_{z}v_{z}B_{y}-P_{x}B_{x}v_{y}-P_{z}B_{z}v_{y})\cdot\overrightarrow{e_{y}}\\ \\ +(P_{x}v_{x}B_{z}+P_{y}v_{y}B_{z}-P_{x}B_{x}v_{z}-P_{y}B_{y}v_{z})\cdot\overrightarrow{e_{z}}\end{split}$$

(S6)

In our experiment, the magnetic field was rotated in the $xz$ plane, the longitudinal thermal gradient was applied along the $x$-axis. Putting $B_{y}=0$ and $P_{y}=0$, Equation S6 becomes:

$$\begin{split}\overrightarrow{\tau}=(P_{z}v_{z}B_{x}-P_{z}B_{z}v_{x})\cdot\overrightarrow{e_{x}}\\ \\ -(P_{x}B_{x}v_{y}+P_{z}B_{z}v_{y})\cdot\overrightarrow{e_{y}}\\ \\ +(P_{x}v_{x}B_{z}-P_{x}B_{x}v_{z})\cdot\overrightarrow{e_{z}}\end{split}$$

(S7)

We measured the transverse thermal gradient along the $z$-axis. The torque along the z-axis is equal to :

$$\tau_{z}=(P_{x}v_{x})B_{z}-(P_{x}v_{z})B_{x}$$

(S8)

One can see that the expected torque along the $z$-axis has two components. Each of them vanishes when the field is along either the $x$-axis or the $z$-axis. This is in agreement with the experimental data seen in Figure 3 of the main text.

Table S1 lists the velocity of longitudinal (LA) and transverse acoustic (TA) modes in black P according to the calculated phonon spectrum [34] and the neutron scattering experiments [53]. Note the good agreement between theory and experiment. Remarkably, the longitudinal mode is faster than the transverse mode.

Fig. S2 shows the phonon spectrum up to 500 cm^-1. The lowest optical phonon frequency of black phosphorus is around 140 cm^-1, much larger than the acoustic phonons below 40 cm^-1 which are the focus of this work.