Thermal Transport and Phonon Hydrodynamics in Strontium Titanate

We present a study of thermal conductivity, $\kappa$, in undoped and doped strontium titanate in a wide temperature range (2–400 K) and detecting different regimes of heat flow. In undoped ${\mathrm{SrTiO}}_3$, $\kappa$ evolves faster than cubic with temperature below its peak and in a narrow temperature window. Such behavior, previously observed in a handful of solids, has been attributed to a Poiseuille flow of phonons, expected to arise when momentum-conserving scattering events outweigh momentum-degrading ones. The effect disappears in the presence of dopants. In ${\mathrm{SrTi}}_{1-x}{\mathrm{Nb}}_x{\mathrm{O}}_3$, a significant reduction in lattice thermal conductivity starts below the temperature at which the average inter-dopant distance and the thermal wavelength of acoustic phonons become comparable. In the high-temperature regime, thermal diffusivity becomes proportional to the inverse of temperature, with a prefactor set by sound velocity and Planckian time (${\tau }_p=(\hslash /k_{B}T)$).

Figure 1

(a) Crystal structure of strontium titanate. (b) The cubic Brillouin zone and its high-symmetry points. (c) The temperature dependence of the two soft modes according to neutron scattering studies [23] and hyper-Raman [25] and Brillouin scattering spectroscopy [26]. (d) Thermal conductivity of a ${\mathrm{SrTiO}}_3$ crystal (closed red squares) in a log-log plot [for a linear plot, see Fig. 3]. Different regimes of thermal transport are identified. Solid lines represent the expected behaviors in these regimes. An additional window due to enhanced umklapp scattering opens up in the vicinity of the antiferrodistortive (AFD) transition.

Figure 2

Thermal conductivity, $\kappa$, as a function of $T^3$, (a) in silicon (after Ref. [36]) and (b) in ${\mathrm{KTaO}}_3$. In both, $\kappa$ deviates downward from the $T^3$ line. (c) In bismuth (after Ref. [37]), it deviates upward. (d–f) In three different crystals of ${\mathrm{SrTiO}}_3$, the deviation is upward. (g) The thermal conductivity and specific heat of ${\mathrm{SrTiO}}_3$ evolve faster than cubic in this temperature range. But in a narrow window, thermal conductivity increases more rapidly. (h) The apparent mean free path in both Bi and ${\mathrm{SrTiO}}_3$ present a local peak, the hallmark of Poiseuille flow.

Figure 3

(a) $\kappa$ as a function of temperature in ${\mathrm{SrTi}}_{1-x}{\mathrm{Nb}}_x{\mathrm{O}}_3$. (b) Electronic, ${\kappa }_e$ and phononic, ${\kappa }_{\mathrm{ph}}$, components of the thermal conductivity in three doped samples compared to undoped strontium titanate. Note the persistence of a $T^3$ behavior over a wide temperature window with a drastically reduced magnitude. (c) Relative attenuation in phonon thermal conductivity, $\mathrm{\Delta }{\kappa }_{\mathrm{ph}}=1-\mathbf{(}{\kappa }_{\mathrm{ph}}(x\ne 0)/{\kappa }_{\mathrm{ph}}(x=0)\mathbf{)}$ in ${\mathrm{SrTi}}_{1-x}{\mathrm{Nb}}_x{\mathrm{O}}_3$ (top) and in ${\mathrm{Sr}}_{1-x}{\mathrm{Ca}}_x$ ${\mathrm{TiO}}_3$ ($x=0.0045$) and in ${\mathrm{SrTiO}}_{3-\delta }$ ($n=7\times {10}^{17}{\mathrm{cm}}^{-3}$). Small arrows represent $T_{qn}$ (See text).

Figure 4

(a) Thermal conductivity at high temperatures compared to previous experimental reports [11, 50, 51] and theoretical calculations [48, 49]. (b) Thermal diffusivity, $D$, extracted from thermal conductivity and specific heat data as a function of temperature in ${\mathrm{SrTiO}}_3$ (solid blue circles) together with data from Ref. [52] (open circles), compared to silicon and PbTe. Solid lines represent $D=s{v}_s^2{\tau }_P$ (see text).