Phenomenological Evidence for the Phonon Hall Effect

In the electrical Hall effect, a magnetic field, applied perpendicular to an electrical current, induces through the Lorentz force a voltage perpendicular to the field and the current. It is generally assumed that an analogous effect cannot exist in the phonon thermal conductivity, as there is no charge transport associated with phonon propagation. In this Letter, we argue that such a magnetotransverse thermal effect should exist and experimentally demonstrate this “phonon Hall effect” in ${\mathrm{Tb}}_3{\mathrm{Ga}}_5{\mathrm{O}}_{12}$.

Figure 1

(a) Setup and geometry of the magnetotransverse phonon transport. (b) Phenomenology: Isotherms without and with a magnetic field.

Figure 2

Measurement protocol. (a) Output of the differential bridge at positive field (typical). (b) Mean values of bridge output for positive and reversed field directions. Up triangles: positive field direction. Down triangles: reversed field direction. The error bars result from averaging measurements as in (a) at positive and negative field values. (c) As in (b), but for a case in which the first value is distinct from the long term average. (d) Magnetotransverse temperature difference, obtained by subtracting the upper and lower curves obtained in a measurement cycle like (b) and converting the voltage difference to a temperature difference, as a function of magnetic field.

Figure 3

Magnetotransverse temperature difference in a nonoriented sample of ${\mathrm{Tb}}_3{\mathrm{Ga}}_5{\mathrm{O}}_{12}$ (size $15.7\times 5.7\times 0.67{\mathrm{mm}}^3$) for heat currents perpendicular (circles) and parallel to the field (squares). The mean sample temperature was adjusted to 5.45 K at $B=0$ for all points. The heater power amounts to 0.14 mW, generating a longitudinal gradient of about $1\mathrm{mK}/\mathrm{cm}$. The straight lines are linear fits through the origin.

Figure 4

The magnetotransverse temperature difference of ${\mathrm{Tb}}_3{\mathrm{Ga}}_5{\mathrm{O}}_{12}$ as a function of magnetic field for different heater powers (squares: 0 mW; circles: 0.05 mW; up triangles, 0.09 mW; down triangles: 0.14 mW). The mean sample temperature was adjusted to 5.45 K at $B=0$ for all points. The straight lines are linear fits through zero. The upper curve is identical with the upper curve in Fig. 2. Inset: The magnetotransverse temperature difference normalized to the magnetic field (slopes of the lines in the main graph) as a function of heater power. The straight line is a linear fit. Errors are slightly larger than the symbols