Phonon Thermal Hall Effect in quartz and its absence in silica

Abstract

The observation of a misalignment between the applied heat flux and the measured temperature gradient in insulating solids induced by magnetic field has become a subject of experimental investigation, theoretical speculation, and unsettled controversy. To identify the origin of this phonon thermal Hall effect, we performed a comparative study of longitudinal and transverse heat transport in crystalline (quartz) and vitreous (silica) SiO$_2$ using identical experimental set-ups and thermometers. A finite signal was detected in the crystalline samples and none in the amorphous sample, within our resolution. The cleaner crystal exhibited a larger thermal Hall conductivity than the dirtier one, ruling out disorder as the driver of the effect. On the other hand, the amplitude of the transverse thermal resistivity is almost identical in the two crystalline samples (W$_{\perp}$/B$\approx 10^{-6}$ m.K.W$^{-1}$.T$^{-1}$). We show that in a phonon gas, as in a molecular gas displaying the Senftleben-Beenakker effect, heat is conducted through two channels, and argue that a thermal Hall response is unavoidable whenever these channels differ both in entropy production and in their coupling to the magnetic field. Under such conditions, the conserved energy current and the non-conserved entropy current cease to be parallel. Finally, the magnitude of the transverse thermal resistivity can be accounted for by a surprisingly simple picture. The heat flux induces a tiny drift velocity of the lattice nuclei, the magnetic field exerts a transverse Berry force on this drift, and this force is balanced by an entropic restoring force.

Figures (7)

• Figure 1: Comparison of longitudinal thermal conductivity and atomic structure of quartz and silica (a) Measured $\kappa_{xx}$ for the two quartz samples is shown by red circles (Quartz#1) and pink triangles (Quartz#2). Blue squares represent the measured thermal conductivity of the silica sample. Gray solid lines represent the $\kappa_{xx}$ values reported in Zeller1971 for quartz and silica (b) Side view (from a-axis) of the periodic lattice of quartz. Silicon atoms and oxygen atoms are are shown as blue and red spheres. SiO$_4$ tetrahedra are highlighted in light blue. (c) Schematic diagram of the disordered network of corner-sharing, distorted SiO$_4$ tetrahedra in silica. Curved orange arrows indicate possible motions of oxygen atoms and green arrows indicate stretching motions between oxygen and silicon atoms taraskin1997silica.
• Figure 2: Comparison of raw data in quartz and in silica. (a) Sketch of the experimental set-up with the sample sandwiched between a heater and a cold finger supporting three Cernox thermometers. (b) A photograph of a sample with thermometers. (c-e) Raw data for quartz#1 (c), quartz#2 (d) and silica (e) at 15 K. $R_1$ and $R_3$ are the resistance of two Cernox thermometers labeled 1 and 3 and monitoring the transverse temperature difference. The dashed lines highlight the presence of asymmetry due to an odd response in quartz and its absence in silica.
• Figure 3: The transverse temperature difference and the thermal Hall conductivity. (a-d)$\Delta T_y$ extracted from the raw data. Light red and light blue lines represent $\Delta T_y$ in quartz and silica. Red circle and blue square symbols represent the average of 200 measurements. An odd response is detectable in quartz and absent in silica. (e) Temperature-dependence of $-\kappa_{xy}$ in Quartz#1 (red circles) and in Quartz#2 (pink triangles), plotted on a logarithmic scale. (f) Temperature-dependence of $-\kappa_{xy}$ of silica, plotted in a linear scale. The plateau region of $\kappa_{xx}$ is highlighted in light blue.
• Figure 4: Thermal Hall angle and thermal Hall resistivity (a) The thermal Hall angle, |$\kappa_{xy}/\kappa_{xx}$| (red circles), and the longitudinal thermal conductivity, $\kappa_{xx}$ (black diamonds), near their maximum values. They peak at 15.4 K and 16 K, respectively. Light red and gray vertical stripes highlight the vicinity and the partial overlap of the two peaks.(b) Maximum $|\kappa_{ij}|/B$ as a function of maximum $\kappa_{jj}$ in different insulators. Our data points for quartz#1 (quartz#2) sample are shown by red and pink circles. Data points for silica at different temperatures are also shown as blue triangles in the light-blue region. They remain below our margin of error. (c) Thermal Hall resistivity as a function of temperature in the two quartz samples. Within our experimental uncertainty, there is no detectable temperature or sample dependence.
• Figure 5: Field-induced twist angle between thermal energy flux and entropy flux In both a molecular gas and a phonon gas, there are two channels of heat flow between the hot and the cold sides. Each channel is associated with an entropy flux with different proportionality factors, $\beta$. If the application of an external magnetic field does not affect both channels identically, a misalignment between the entropy flux and heat flux vectors will emerge.