Reduced phase space of heat-carrying acoustic phonons in single-crystalline InTe

Chalcogenide semiconductors and semimetals are a fertile class of efficient thermoelectric materials, which, in most cases, exhibit very low lattice thermal conductivity ${\kappa }_{\mathrm{ph}}$ despite lacking a complex crystal structure such as the tetragonal binary compound InTe. Our measurements of ${\kappa }_{\mathrm{ph}}\left(T\right)$ in single-crystalline InTe along the $c$ axis show that ${\kappa }_{\mathrm{ph}}$ exhibits a smooth temperature dependence upon cooling to about 50 K, the temperature below which a strong rise typical for dielectric compounds is observed. Using a combination of first-principles calculations, inelastic neutron scattering (INS), and low-temperature specific heat and transport properties measurements on single-crystalline InTe, we show that the phonon spectrum exhibits well-defined acoustic modes, the energy dispersions of which are constrained to low energies due to distributions of dispersionless, optical modes, which are responsible for a broad double peak structure in the low-temperature specific heat. The latter are assigned to the dynamics of ${\mathrm{In}}^{+}$ cations in tunnels formed by edge-sharing ${\left({\mathrm{In}}^{3+}{{\mathrm{Te}}_4}^{2-}\right)}^{-}$ tetrahedra chains, the atomic thermal displacement parameters of which, probed as a function of temperature by means of single-crystal x-ray diffraction, suggest the existence of a complex energy potential. Indeed, the ${\mathrm{In}}^{+}$-weighted optical modes are not observed by INS, which is ascribed to the anharmonic broadening of their energy profiles. While the low ${\kappa }_{\mathrm{ph}}$ value of $1.2\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$ at 300 K originates from the limited energy range available for acoustic phonons, we show that the underlying mechanism is specific to InTe and argue that it is likely related to the presence of local disorder induced by the ${\mathrm{In}}^{+}$ site occupancy.

Figure 1

(a) Perspective view of the crystal structure of InTe. In1 $\left(4a\right)$, In2 $\left(4b\right)$, and Te $\left(8h\right)$ atoms in the space group I4/mcm are shown in blue, green, and red, respectively. The In1 atoms reside in large tunnels created by the octahedra, forming infinitelike, one-dimensional chains. The thermal ellipsoids are drawn at the 97% probability level. (b) Crystal structure of InTe projected along the $c$ direction.

Figure 2

Phonon-dispersion curves and projected partial phonon density of states of InTe as a function of energy calculated by considering a unit cell pressurized under 3 GPa.

Figure 3

Temperature dependence of the equivalent ADP values $U_{iso}$ that characterize the thermal vibrations of the In1 atoms in the tunnels. For comparison purposes, literature data obtained on the type-I clathrate ${\mathrm{Sr}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$, the pyrochlore $\mathrm{K}{\mathrm{Os}}_2{\mathrm{O}}_6$, the tetrahedrite ${\mathrm{Cu}}_{11}\mathrm{Ni}{\mathrm{Sb}}_4{\mathrm{S}}_{13}$ and the cagelike compound $\mathrm{V}{\mathrm{Al}}_{10.1}$ have been added (Refs. [51, 47, 35, 32], respectively). In all these compounds, the rattling atoms (Sr on the $6d$ site of the tetrakaidecahedra cages, K, Cu2 with a quasiplanar coordination and Al on the $8a$ site of the CN16 Frank-Kaspar coordination polyhedra) have been shown to experience a complex, multiple-well potential. The solid curves are guides to the eye.

Figure 4

(a) $C_p/T^3$ as a function of T for single-crystalline InTe measured down to 0.35 K (blue filled squares). The electronic contribution defined as $\gamma T$, where $\gamma$ is the Sommerfeld coefficient responsible for the upturn upon cooling below 1 K, has been subtracted to obtain the lattice contribution to the specific heat $C_{\mathrm{ph}}$ (orange filled circles). The theoretical temperature dependence of $C_v/T^3$ (green solid curve) has been derived for a unit cell pressurized under 3 GPa. (b) Temperature dependence of the lattice thermal conductivity of single-crystalline InTe measured along the $c$ axis. In both panels, the solid curves are guides to the eye.

Figure 5

(a) Low-frequency Raman spectrum of InTe measured at 300 K. The $x$ axis is given in $\mathrm{c}{\mathrm{m}}^{-1}$ (bottom) and in meV (top). The inset shows a magnification of the spectrum for frequencies lower than 130 $\mathrm{c}{\mathrm{m}}^{-1}$. (b) Blue solid lines correspond to the ab initio calculated dispersions as shown in Fig. 2 along the [100] (Γ-M) and [110] (Γ-X) directions, respectively. The red squares at the Γ point show the energies of the Raman peaks observed in the spectrum (a). The black filled circles give the dispersions of the transverse acoustic phonon as extracted from our INS measurements.

Figure 6

(a) Inelastic neutron intensity as function of energy recorded at 300 K for a series of energy scans through the transverse acoustic (TA) dispersion, polarized along the [010] direction $\left({\mathrm{TA}}_{100}^{010}\right)$, at constant wave vectors along the [100] direction labeled from $q_1$ to $q_4$. (b) Left panel: Dispersion of ${\mathrm{TA}}_{100}^{010}$ along [100] and polarized along [010]. The wave vectors $q_{1–4}$ are indicated. Right panel: TA dispersion along [110] and polarized along [1–10], ${\mathrm{TA}}_{110}^{1-10}$. Blue solid lines correspond to the ab initio calculated dispersions as shown in Fig. 2 along the [100] (Γ-M) and [110] (Γ-X) directions, respectively. The computed energies have been multiplied by a factor 1.12 for phonons propagation along (Γ-M) and 0.9 for those propagating along (Γ-X). The error bars of the extracted phonon energies are lower than the size of the symbols used. (c) Inelastic neutron intensity as a function of energy at a constant wave vector $Q=\left(6,0.2,0\right)$ [$q_2$ in panels (a) and (b)], which belongs to the TA dispersion shown in the left panel recorded at 300 K (*), 200 K (▲), 150 K (○), 100 K (□), 55 K (•), and 4 K (×). Data shown in (a) and (c) were collected on the spectrometer 1T@LLB.

Figure 7

Fit of the specific-heat data of single-crystalline InTe plotted as $C_p/T^3$ vs T in log scale. The orange solid curve stands for the best fit to the data according to Eq. (2).

Figure 8

(a) Phonon-dispersion curves of InTe calculated for several pressurized unit cells: models (i) from experimental parameters, (ii) from DFT relaxed structure, (iii) under 3 GPa, and (iv) under 13 GPa. (b) Isothermal free-energy curves as a function of volume (blue dots and curves) and behavior of the free-energy minimum as a function of volume (red dots and curves). The curves calculated for the volume of $0.73V_0$ have been shifted towards a volume of $0.88V_0$ used to calculate the phonon-dispersion curves and phonon density of states shown in the main text.

Figure 9

(a) Comparison of the experimental and theoretical temperature dependencies of the specific heat $C_p$ of InTe. The horizontal solid green line represents the Dulong-Petit limit of $3NR$ where $N$ is the number of atoms per formula unit and $R$ is the ideal gas constant. (b) Temperature dependence of the electrical resistivity $\rho$ measured along the $c$ axis. (c) Comparison of the temperature dependence of the total and lattice thermal conductivities $\kappa$ and ${\kappa }_{\mathrm{ph}}$, respectively, measured along the $c$ axis. The Lorenz number $L$ used to calculate the electronic contribution ${\kappa }_e=LT/\rho$ has been calculated using a single-parabolic band model.

Figure 10

Electronic band structure of InTe calculated (a) using a unit-cell volume of $0.88V_0$ used in phonon calculations and (b) using the experimental lattice parameters. (c) Brillouin zone of the body-centered-tetragonal unit cell of InTe. The high-symmetry points are indicated.

Figure 11

Elastic scans performed across (a) the (600) and (b) (330) Bragg peaks along the [010] and [−110] directions, respectively. The solid lines stand for the best fit as described in the text.

Figure 12

Energy scans at constant wave vectors across the transverse-acoustic (TA) dispersion measured on the three-axis spectrometer 1T@LLB around the Bragg peak (600). The total transferred wave vectors are indicated in the subfigures, with the phonon wave vector corresponding to $q=Q-\left(600\right)$. The solid black lines are fits as explained in the text. The dotted red lines in panels (e) and (c) show the fits for which a finite energy width of 500 μeV has been imposed.

Figure 13

Energy scans at constant wave vectors across the transverse-acoustic (TA) dispersion measured on the three-axis spectrometer 1T@LLB around the Bragg peak (330). The total transferred wave vectors are indicated in the subfigures, with the phonon wave vector corresponding to $q=Q-\left(330\right)$. The solid black lines are fits as explained in the text. The dotted red line in panel (b) shows the fits for which a finite-energy width of 500 μeV has been imposed.