Phonon density of states and heat capacity of ${\text{La}}_{3-x}{\text{Te}}_4$

The phonon density of states (DOS) of ${\text{La}}_{3-x}{\text{Te}}_4$ compounds $\left(x=0.0,0.18,0.32\right)$ was measured at 300, 520, and 780 K, using inelastic neutron scattering. A significant stiffening of the phonon DOS and a large broadening of features were observed upon introduction of vacancies on La sites (increasing $x$). Heat-capacity measurements were performed at temperatures $1.85\le T\le 1200\text{K}$ and were analyzed to quantify the contributions of phonons and electrons. The Debye temperature and the electronic coefficient of heat capacity determined from these measurements are consistent with the neutron-scattering results, and with previously reported first-principles calculations. Our results indicate that La vacancies in ${\text{La}}_{3-x}{\text{Te}}_4$ strongly scatter phonons and this source of scattering appears to be independent of temperature. The stiffening of the phonon DOS induced by the introduction of vacancies is explained in terms of the electronic structure and the change in bonding character. The temperature dependence of the phonon DOS is captured satisfactorily by the quasiharmonic approximation.

Figure 1

(Color online) (a) Neutron-weighted phonon DOS of ${\text{La}}_{3-x}{\text{Te}}_4$ at 300 K, measured by inelastic neutron scattering $\left(E_i=40\text{meV}\right)$. Error bars represent uncertainties from counting statistics of the measurement. (b) The thick curves labeled $\mathcal{Q}=5$ and $\mathcal{Q}=10$ were obtained by convolving the DOS for ${\text{La}}_3{\text{Te}}_4$ with a damped oscillator linewidth of corresponding quality factor $\mathcal{Q}$ and applying a shift of the energy axis $E^{\prime }=\left(1.15+\left(1.05–1.15\right)\times E/16.0\right)\times E$. The thin curves at the bottom illustrate the phonon line shapes ${\mathcal{L}}_{\mathcal{Q},E_0}\left(E\right)$ at $E_0=8.5$ and 15 meV, for $\mathcal{Q}=5$ and 10 meV.

Figure 2

(Color online) Neutron-weighted phonon DOS curves measured at 300, 520, and 780 K for (a) ${\text{La}}_3{\text{Te}}_4$, (b) ${\text{La}}_{2.82}{\text{Te}}_4$, and (c) ${\text{La}}_{2.68}{\text{Te}}_4$. Dashed red lines in all panels (QH 780 K) correspond to the quasiharmonic prediction for the phonon DOS at 780 K, given the phonon DOS at 300 K, and a softening of phonon energies $E_{780}=0.96\times E_{300}$.

Figure 3

(Color online) Temperature dependence of the average phonon energy, $⟨E⟩$, of ${\text{La}}_{3-x}{\text{Te}}_4$, determined from the phonon DOS. Closed and open markers are for measurements performed with incident neutron energies $E_i=40$ and 80 meV, respectively. The thick dark line is the expected quasiharmonic dependence of $⟨E⟩$ for ${\text{La}}_{2.82}{\text{Te}}_4$ based on our dilatometry measurement and a Grüneisen parameter ${\overline{\gamma }}_{\text{G}}=1.76$.

Figure 4

(Color online) (a) Heat-capacity data for ${\text{La}}_3{\text{Te}}_4$, ${\text{La}}_{2.82}{\text{Te}}_4$, and ${\text{La}}_{2.68}{\text{Te}}_4$, measured with Quantum Design PPMS apparatus. The inset shows $C_P/T$ vs $T^2$ for $T<4.5\text{K}$. (b) Fits of $C_P/T$ vs $T^2$ to obtain ${\gamma }_{\text{el}}$ and ${\theta }_{\text{D}}$ in Table (see text).

Figure 5

(Color online) (a) ${\text{La}}_3{\text{Te}}_4$ $C_P$ curves, measured at low (PPMS) and high temperatures (DSC), and calculated $C_P$ components from harmonic phonons $\left(C_{\text{ph},\text{H}}\right)$, electrons $\left(C_{\text{el}}\right)$, and dilation of the lattice $\left(C_{\text{D}}\right)$ (see text). (b) Detail of electronic heat capacity calculated from density-functional theory results reported in Ref. 27, and Sommerfeld approximation $C_{\text{el}}={\gamma }_{\text{el}}T$, with experimental ${\gamma }_{\text{el}}$ from PPMS measurements. (c) Measured thermal expansion $dL/L_0$ of ${\text{La}}_{2.86}{\text{Te}}_4$ (markers), extrapolated at low temperatures using measured heat capacity (see text), and linear thermal-expansion coefficient, $\alpha$.