Thermal conductivity of bulk ${\mathrm{In}}_2{\mathrm{O}}_3$ single crystals

The transparent semiconductor ${\mathrm{In}}_2{\mathrm{O}}_3$ is a technologically important material. It combines optical transparency in the visible frequency range and sizable electric conductivity. We present a study of thermal conductivity of ${\mathrm{In}}_2{\mathrm{O}}_3$ crystals and find that around 20 K, it peaks to a value as high as 5000 $\mathrm{W}{\mathrm{K}}^{-1}{\mathrm{m}}^{-1}$, comparable to the peak thermal conductivity in silicon and exceeded only by a handful of insulators. The amplitude of the peak drastically decreases in the presence of a type of disorder, which does not simply correlate with the density of mobile electrons. Annealing enhances the ceiling of the phonon mean free path. Samples with the highest thermal conductivity are those annealed in the presence of hydrogen. Above 100 K, thermal conductivity becomes sample independent. In this intrinsic regime, dominated by phonon-phonon scattering, the magnitude of thermal diffusivity, $D$, becomes comparable to many other oxides, and its temperature dependence evolves towards $T^{-1}$. The ratio of $D$ to the square of sound velocity yields a scattering time which obeys the expected scaling with the Planckian time.

Figure 1

Thermal conductivity of ${\mathrm{In}}_2{\mathrm{O}}_3$ and its sample dependence. (a) Thermal conductivity $\kappa$ of four different samples of ${\mathrm{In}}_2{\mathrm{O}}_3$. Note the strong variation in the magnitude of maximum $\kappa$ among the samples. Purple bars represent the margin of experimental reproducibility in repeated measurements on sample 6. Because of its large thermal conductance, the temperature difference caused by the application of the heat current was small, leading to a larger experimental uncertainty. (b) Comparison of thermal conductivity in three samples having different plane orientations. The heat current was always applied along the longest axis. No anisotropy is expected in this cubic system and the difference is due to different levels of disorder. The inset shows the measurement setup with thermocouples.

Figure 2

Phonon mean free path in ${\mathrm{In}}_2{\mathrm{O}}_3$, extracted from thermal conductivity, specific heat, and the sound velocity. The electric resistivity and the carrier density of these samples are listed in Table 1. There is no visible correlation between the electronic properties and phonon mean free path. Note that in the best sample (6), the mean free path approaches the sample thickness of 0.5 mm at low temperature.

Figure 3

Comparison with other solids. Comparison of the thermal conductivity in the best ${\mathrm{In}}_2{\mathrm{O}}_3$ sample with diamond [13], silicon [38], ${\mathrm{Al}}_2{\mathrm{O}}_3$ (sapphire) [14], and a perovskite ${\mathrm{SrTiO}}_3$ [15]. Thermal conductivity in ${\mathrm{In}}_2{\mathrm{O}}_3$ is comparable to a ternary oxide such as ${\mathrm{SrTiO}}_3$ at room temperature, but peaks to a value comparable to silicon's peak at low temperature.

Figure 4

Thermal diffusivity in the intrinsic regime: (a) Temperature dependence of thermal diffusivity, $D$, in ${\mathrm{In}}_2{\mathrm{O}}_3$, compared to a few representative solids (Si, ${\mathrm{SrTiO}}_3$, PbTe) [15] and ${\mathrm{Al}}_2{\mathrm{O}}_3$[14, 43]. The dashed line represents $T^{-1}$. At room temperature, $D$ in ${\mathrm{In}}_2{\mathrm{O}}_3$ (as well as in ${\mathrm{Al}}_2{\mathrm{O}}_3$ and in Si) approaches this behavior without fully attaining it.

Figure 5

Scattering time, Planckian time, sound velocity, and melting velocity: The $\tau /{\tau }_P$ ratio vs the $v_M/v_s$ ratio in a different system adopted from Ref. [17] with the new data point for ${\mathrm{In}}_2{\mathrm{O}}_3$. Note that $\tau$ in ${\mathrm{In}}_2{\mathrm{O}}_3$ was derived at room temperature. Its value at higher temperature may be slightly lower. The blue region contains column IV elements and III-V semiconductors, which do not participate in this trend. Most other systems lie in a green stripe whose boundaries are set by $0.6\pm 0.4v_M/v_s$. Comparisons between ${\mathrm{In}}_2{\mathrm{O}}_3$ and ${\mathrm{Al}}_2{\mathrm{O}}_3$ and between LiF and RbI reveal the role played by atomic mass in setting the sound velocity (see text).