Lattice dynamics in Bi${}_2$Te${}_3$ and Sb${}_2$Te${}_3$: Te and Sb density of phonon states

The lattice dynamics in Bi${}_2$Te${}_3$ and Sb${}_2$Te${}_3$ were investigated both microscopically and macroscopically using ${}^{121}$Sb and ${}^{125}$Te nuclear inelastic scattering, x-ray diffraction, and heat capacity measurements. In combination with earlier inelastic neutron scattering data, the element-specific density of phonon states was obtained for both compounds and phonon polarization analysis was carried out for Bi${}_2$Te${}_3$. A prominent peak in the Te specific density of phonon states at $13\mathrm{meV}$, that involves mainly in-plane vibrations, is mostly unaffected upon substitution of Sb with Bi revealing vibrations with essentially Te character. A significant softening is observed for the density of vibrational states of Bi with respect to Sb, consistently with the mass homology relation in the long-wavelength limit. In order to explain the energy mismatch in the optical phonon region, a $\sim$$20\%$ force constant softening of the Sb-Te bond with respect to the Bi-Te bond is required. The reduced average speed of sound at $20\mathrm{K}$ in Bi${}_2$Te${}_3$, $1.75\left(1\right)\mathrm{km}/\mathrm{s}$, compared to Sb${}_2$Te${}_3$, $1.85\left(4\right)\mathrm{km}/\mathrm{s}$, is not only related to the larger mass density but also to a larger Debye level. The observed low lattice thermal conductivity at $295\mathrm{K}$, $2.4{\text{Wm}}^{-1}{\text{K}}^{-1}$ for Sb${}_2$Te${}_3$ and $1.6{\text{Wm}}^{-1}{\text{K}}^{-1}$ for Bi${}_2$Te${}_3$, cannot be explained by anharmonicity alone given the rather modest Grüneisen parameters, $1.7\left(1\right)$ for Sb${}_2$Te${}_3$ and $1.5\left(1\right)$ for Bi${}_2$Te${}_3$, without accounting for the reduced speed of sound and more importantly the low acoustic cutoff energy.

Figure 1

X-ray diffraction pattern of Bi${}_2$${}^{125}$Te${}_3$ obtained at $295\mathrm{K}$ using synchrotron radiation (red dots), expected peak positions (black tics), corresponding refinement (black line) obtained using fullprof,[22] and difference plot (green line). (Inset) (Left) Quarter of the corresponding detector image and (right) intensity distribution along the $\varphi$ angle for the $\left(015\right)$, $\left(110\right)$, and $\left(0015\right)$ reflections in the pseudohexagonal setting.

Figure 2

The temperature dependence of the $a$ (circles) and $c$ (squares) lattice parameters for Sb${}_2$Te${}_3$ (open tics) and Bi${}_2$Te${}_3$ (filled tics) between 50 and $300\mathrm{K}$. Point sizes define error bar. The linear thermal expansion above $200\mathrm{K}$ in both orientations is indicated by red lines.

Figure 3

Nuclear inelastic scattering, NIS, spectra (lines) and the time integrated nuclear forward scattering, NFS-instrumental function, (dashed lines) obtained with the ${}^{125}$Te (blue lines) and the ${}^{121}$Sb (red line) resonances on Bi${}_2$${}^{125}$Te${}_3$ and Sb${}_2$${}^{125}$Te${}_3$.

Figure 4

Density of phonon states measured with the ${}^{125}$Te (blue points) and ${}^{121}$Sb (red points) resonances at $20\mathrm{K}$ on Sb${}_2$Te${}_3$, for clarity a line is drawn between points and only one typical error bar is given. Comparison of the neutron weighted DPS[36] of Sb${}_2$Te${}_3$ obtained with NIS (highlighted area), see text, with the one obtained at $77\mathrm{K}$ using neutrons[20] (dashed line). Calculated[37] $\Gamma$-point phonon mode energies and symmetries are indicated by the labeled tics.

Figure 5

Element specific, Sb or Bi (in red) and Te (in blue), density of phonon states in Sb${}_2$Te${}_3$ (open circles) and Bi${}_2$Te${}_3$ (filled squares). The Bi specific DPS was extracted from the combination of NIS and reference neutron data[20] obtained at $77\mathrm{K}$ (black line) and renormalization according to a homology relation, see text, was applied (red line). (Inset) Debye level calculated from the Te specific DPS measured in Sb${}_2$Te${}_3$ and Bi${}_2$Te${}_3$.

Figure 6

Orientation-dependent Te DPS measured on a Bi${}_2$Te${}_3$ single crystal parallel (green squares) and perpendicular (blue circles) to the crystallographic $c$ axis. A comparison of the measured Te DPS in a Bi${}_2$Te${}_3$ polycrystalline sample (black curve) with the isotropic average obtained from data measured on single crystal (red curve) is given. (Inset) Debye level calculated in the two orientations (solid lines) from the measured Te specific DPS.

Figure 7

Specific heat at constant pressure $C{}_{\mathrm{P}}$ data measured on Sb${}_2$Te${}_3$ (circles) and Bi${}_2$Te${}_3$ (triangles) between 3 and 300 K and heat capacity at constant volume $C$${}_{\mathrm{V}}$ calculated from low-temperature DPS (red line). A typical error bar is given. (Inset) the Debye representation $C_{\mathrm{P}}/T^3$ of both measurements. A fit of the experimental data was applied using a two-level Schottky model superimposed to a Debye model with embedded Einstein oscillators (solid lines). The contribution of the Schottky model is given in the lower part of the inset. Specific heat measurements under $4-\mathrm{T}$ magnetic field are shown as red symbols.