Low-temperature specific heat and heat transport of Tb$_2$Ti$_{2-x}$Zr$_x$O$_7$ single crystals

We report a study on the specific heat and heat transport of Tb$_2$Ti$_{2-x}$Zr$_x$O$_7$ ($x =$ 0, 0.02, 0.1, 0.2, and 0.4) single crystals at low temperatures and in high magnetic fields. The magnetic specific heat can be described by the Schottky contribution from the crystal-electric-field (CEF) levels of Tb$^{3+}$, with introducing Gaussian distributions of the energy split of the ground-state doublet and the gap between the ground state and first excited level. These crystals has an extremely low phonon thermal conductivity in a broad temperature range that can be attributed to the scattering by the magnetic excitations, which are mainly associated with the CEF levels. There is strong magnetic field dependence of thermal conductivity, which is more likely related to the field-induced changes of phonon scattering by the CEF levels than magnetic transitions or spin excitations. For magnetic field along the [111] direction, there is large thermal Hall conductivity at low temperatures which displays a broad peak around 8 T. At high fields up to 14 T, the thermal Hall conductivity decreases to zero, which supports its origin from either the spinon transport or the phonon skew scattering by CEF levels. The thermal Hall effect is rather robust with Zr doping up to 0.2 but is strongly weakened in higher Zr-doped sample.