Thermal Hall Effects of Spins and Phonons in Kagome Antiferromagnet Cd-Kapellasite

We investigate the thermal-transport properties of the kagome antiferromagnet Cd-kapellasite (Cd-K). We find that a field-suppression effect on the longitudinal thermal conductivity ${\kappa }_{xx}$ sets in below approximately 25 K. This field-suppression effect at 15 T becomes as large as 80% at low temperatures, suggesting a large spin contribution ${\kappa }_{xx}^{\mathrm{sp}}$ in ${\kappa }_{xx}$. We also find clear thermal Hall signals in the spin liquid phase in all Cd-K samples. The magnitude of the thermal Hall conductivity ${\kappa }_{xy}$ shows a significant dependence on the sample’s scattering time, as seen in the rise of the peak ${\kappa }_{xy}$ value in almost linear fashion with the magnitude of ${\kappa }_{xx}$. On the other hand, the temperature dependence of ${\kappa }_{xy}$ is similar in all Cd-K samples; ${\kappa }_{xy}$ shows a peak at almost the same temperature of the peak of the phonon thermal conductivity ${\kappa }_{xx}^{\mathrm{ph}}$ which is estimated by ${\kappa }_{xx}$ at 15 T. These results indicate the presence of a dominant phonon thermal Hall ${\kappa }_{xy}^{\mathrm{ph}}$ at 15 T. In addition to ${\kappa }_{xy}^{\mathrm{ph}}$, we find that the field dependence of ${\kappa }_{xy}$ at low fields turns out to be nonlinear at low temperatures, concomitantly with the appearance of the field suppression of ${\kappa }_{xx}$, indicating the presence of a spin thermal Hall ${\kappa }_{xy}^{\mathrm{sp}}$ at low fields. Remarkably, by assembling the ${\kappa }_{xx}$ dependence of ${\kappa }_{xy}^{\mathrm{sp}}$ data of other kagome antiferromagnets, we find that, whereas ${\kappa }_{xy}^{\mathrm{sp}}$ stays a constant in the low-${\kappa }_{xx}$ region, ${\kappa }_{xy}^{\mathrm{sp}}$ starts to increase as ${\kappa }_{xx}$ does in the high-${\kappa }_{xx}$ region. This ${\kappa }_{xx}$ dependence of ${\kappa }_{xy}^{\mathrm{sp}}$ indicates the presence of both intrinsic and extrinsic mechanisms in the spin thermal Hall effect in kagome antiferromagnets. Furthermore, both ${\kappa }_{xy}^{\mathrm{ph}}$ and ${\kappa }_{xy}^{\mathrm{sp}}$ disappear in the antiferromagnetic ordered phase at low fields, showing that phonons alone do not exhibit the thermal Hall effect. A high field above approximately 7 T induces ${\kappa }_{xy}^{\mathrm{ph}}$, concomitantly with a field-induced increase of ${\kappa }_{xx}$ and the specific heat, suggesting a coupling of the phonons to the field-induced spin excitations as the origin of ${\kappa }_{xy}^{\mathrm{ph}}$.

Popular Summary

The trajectory of an electron bends as it moves through a magnetic field. For electrons in a metal, the same phenomenon is known as the Hall effect. It lays the foundation for characterizing the fundamental properties of metals and has applications in magnetic sensors in smartphones. Recently, researchers have reported a version of the Hall effect in insulators as well. To better understand the origin of this phenomenon, we investigate the Hall effect in a magnetic insulator.

Traditionally, the Hall effect has been thought to not appear in insulators because of the apparent absence of mobile electrons. But recent reports of a thermal version of this effect, known as the thermal Hall effect, in numerous insulators has led to broad attention from researchers to understand its origin as well as potential applications for thermal current control.

Thermal current in an insulator is carried by electron spins and phonons, the fundamental quanta of lattice vibrations. This leads us to ask how these charge-neutral carriers can be bent by magnetic fields and how one can separate these two effects. We investigate the magnetic insulator cadmium kapellasite and find that phonons and spins contribute to thermal Hall effects in this compound. Remarkably, detailed studies of the field dependence allow us to separate these two thermal Hall effects, leading us to conclude that these two effects are intimately coupled to each other.

Our findings not only provide new insight for unknown thermal Hall effects in an insulator but also pave the way for developing a device that can control thermal current with a magnetic field.

Figure 1

(a) Crystal structure and (b) top view of a kagome layer of Cd-K [48]. The magnetic interactions between nearest-neighbor, next-nearest-neighbor, and diagonal ${\mathrm{Cu}}^{2+}$ spins are denoted by $J$ (solid black line), $J_2$ (dotted line), and $J_d$ (dashed line), respectively. (c) Schematic illustration of ${\kappa }_{xx}$ and ${\kappa }_{xy}$ measurements. A heater and three thermometers ($T_{\mathrm{high}}$, $T_{L1}$, and $T_{L2}$) are attached to the sample fixed on the LiF heat bath. A heat current $Q$ is applied within the kagome layer, and a magnetic field $B$ is applied along the $c$ axis. (d) A typical crystal of Cd-K. The direction of the heat current for samples 1, 2, and 3-1 (3-2) is shown by arrow 1 (2).

Figure 2

(a) Temperature dependence of the longitudinal thermal conductivity (${\kappa }_{xx}$) of all samples of Cd-kapellasite (Cd-K) and that of Ca-kapellasite (Ca-K) at 0 T. The longitudinal thermal conductivity of Ca-K is taken from Ref. [26]. (b)–(f) The data of each sample under magnetic fields. In sample 2, ${\kappa }_{xx}$ is measured down to 0.1 K. An enlarged view of the low-temperature region (0.1–3 K) of (c) is shown in (d). The filled and open circles in (c) and (d) show ${\kappa }_{xx}$ measured by a VTI ($2\mathrm{K}<T$) and a DR ($0.1<T<4\mathrm{K}$), respectively. The slight difference between the two data might be caused by a thermal cycle effect and/or different setups between the VTI and the DR measurements.

Figure 3

Magnetic field dependence of the longitudinal thermal conductivity normalized by the zero-field value $\{[{\kappa }_{xx}(B)-{\kappa }_{xx}(0)]/{\kappa }_{xx}(0)\}$ of sample 2 above 30 K (a), for 2–20 K (b), and below 2 K (c).

Figure 4

(a) Magnetic field dependence of the transverse temperature difference divided by the heat current $(\mathrm{\Delta }{T}_y/Q)$. The zero-field value of $\mathrm{\Delta }{T}_y/Q$, which is caused by the misalignment effect, is subtracted for clarity. (b) Magnetic field dependence of the asymmetrized $\mathrm{\Delta }{T}_y/Q$ of (a) with respect to the field direction. See the text for details. The solid lines represent a linear fitting to $\mathrm{\Delta }{T}_y^{\mathrm{asym}}/Q$ for each temperature. (c) Magnetic field dependence of $\mathrm{\Delta }{T}_y^{\mathrm{asym}}$ below 2 K. Error bars represent the standard error of the data, which are smaller than the symbol size except for the 5 K data in (b).

Figure 5

(a–d) The field dependence of ${\kappa }_{xy}$ of all Cd-K samples for 4–20 K. The data above 4 K are shifted for clarity. The offsets for the shifted data are indicated by the dashed lines. The solid lines are drawn to show a deviation from the linear increase of ${\kappa }_{xy}$ to that at 15 T. Error bars estimated by the standard error are smaller than the symbol size for all the measurements.

Figure 6

Magnetic field dependence of the asymmetrized transverse temperature difference divided by the heat current $(\mathrm{\Delta }{T}_y^{\mathrm{asym}}/Q)$ at 2 K. Error bars represent the standard error of the measurements.

Figure 7

Temperature dependence of ${\kappa }_{xy}/TB$. (a) Comparison of ${\kappa }_{xy}/TB$ of three Cd-K samples. The filled (open) symbols represent ${\kappa }_{xy}/TB$ at 15 T in the VTI measurements (14 T in the DR measurements). (b) Comparison of ${\kappa }_{xy}/TB$ of Cd-K sample 2 (green circles), Ca-K [26] (open gray pentagon), and volborthite [25] (open gray hexagon). For clarity, ${\kappa }_{xy}/TB$ of volborthite is multiplied by $-1$. The inset shows an enlarged view of the low-temperature data of ${\kappa }_{xy}/TB$ of sample 2. The dashed line is a guide to the eye.

Figure 8

(a) Temperature dependence of the specific heat divided by the temperature ($C/T$) measured in PPMS. The measurements are done for the same set of multiple single crystals used in Ref. [47]. The peak around 4 K corresponds to the AFM transition. The dashed line shows the phonon contribution $C_{\mathrm{ph}}/T$ estimated by a fitting for the data at high temperatures (see Supplemental Material [51] for details). (b) Temperature dependence of $C/T$ of another set of multiple single crystals measured in DR. The zero-field data from the previous report [47] are also shown by open circles. (b) Magnetic field dependence of $C$ at 0.5 K. The red dotted line shows an estimation of the magnetic field dependence of the nuclear Schottky specific heat ($C_{\mathrm{ncl}}$) at 0.5 K. The data of $C_{\mathrm{ncl}}$ are shifted to compare the amount of the field increase.

Figure 9

$B\text{−}T$ phase diagram of Cd-K. The boundary of the AFM phase is determined by the peak temperature of $C/T$ (black diamonds) shown in Fig. 8. The threshold fields where the onset of the finite $\mathrm{\Delta }{T}_y^{\mathrm{asym}}$ is observed in the field dependence [Fig. 4] are shown by gray circles.

Figure 10

Normalized thermal Hall conductivity $f_{\exp }$ of kagome lattice antiferromagnets fitted by the parameters listed in Table 1. The solid line shows a numerical calculation of $f_{\mathrm{SBMFT}}$ at $D/J=0.1$ by the SBMFT [26]. The data of Ca-K and those of volborthite are taken from Refs. [26, 25], respectively.

Figure 11

Temperature dependence of ${\kappa }_{xx}/T$ (left axis) and that of ${\kappa }_{xy}/TB$ (right axis) at 15 T of all Cd-K samples (a)–(d) and Ca-K (e),(f). The data of Ca-K are taken from Ref. [26].

Figure 12

Temperature dependence of $\delta {\kappa }_{xy}/TB$ at 6 T of all Cd-K samples.

Figure 13

The spin thermal Hall conductivity per the 2D kagome layer $|{\kappa }_{xy}^{\mathrm{sp},2\mathrm{D}}|/TB$ of Cd-K, Ca-K [26], and volborthite [25] plotted as a function of the longitudinal thermal conductivity ${\kappa }_{xx}/T$ at 0 T. The value of ${\kappa }_{xx}/T$ is taken at the peak temperature of $\delta {\kappa }_{xy}/TB$ for Cd-K (4–6 K; see Fig. 12) and that of $|{\kappa }_{xy}^{2\mathrm{D}}|/TB$ for Ca-K and volborthite (16 K). The error bars represent the ambiguity in estimating the absolute value of ${\kappa }_{xx}$ and ${\kappa }_{xy}$ owing to the nonrectangular shape of the samples (see Supplemental Material [51] for details). The blue and pink dashed lines are guides to the eye for the intrinsic and extrinsic contributions, respectively (see the main text).