Understanding the evolution of anomalous anharmonicity in ${\mathrm{Bi}}_2{\mathrm{Te}}_{3-x}{\mathrm{Se}}_x$

The anharmonic effect in thermoelectrics has been a central topic for decades in both condensed matter physics and material science. However, despite the long-believed strong and complex anharmonicity in the ${\mathrm{Bi}}_2{\mathrm{Te}}_{3-x}{\mathrm{Se}}_x$ series, experimental verification of anharmonicity and its evolution with doping remains elusive. We fill this important gap with high-resolution, temperature-dependent Raman spectroscopy in high-quality single crystals of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$, ${\mathrm{Bi}}_2{\mathrm{Te}}_2\mathrm{Se}$, and ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ over the temperature range from 4 to 293 K. Klemens's model was employed to explain the renormalization of their phonon linewidths. The phonon energies of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ and ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ are analyzed in detail from three aspects: lattice expansion, cubic anharmonicity, and quartic anharmonicity. For the first time, we explain the evolution of anharmonicity in various phonon modes and across the series. In particular, we find that the interplay between cubic and quartic anharmonicity is governed by their distinct dependence on the phonon density of states, providing insights into anomalous anharmonicity designing of new thermoelectrics.

Figure 1

(a) Temperature-dependent Raman spectra of ${\mathrm{Bi}}_2{\mathrm{Te}}_{3-x}{\mathrm{Se}}_x$. From top to bottom: ${\mathrm{Bi}}_2{\mathrm{Te}}_3$, ${\mathrm{Bi}}_2{\mathrm{Te}}_2\mathrm{Se}$, and ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. (b) Phonon linewidth of the ${\mathrm{A}}_g^1$ mode of ${\mathrm{Bi}}_2{\mathrm{Te}}_{3-x}{\mathrm{Se}}_x$. Red lines represent the anharmonic prediction. The temperature-dependent renormalization of the phonon linewidth is relatively simple, originating from the cubic anharmonicity and free of quartic anharmonicity to the lowest order.

Figure 2

Temperature dependence of the phonon frequency of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ (a–c) and ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ (d–f). All data points are offset by the phonon frequency at the lowest temperature. The dashed line indicates zero offset. We note that the anharmonicity-induced phonon energy shifts are dramatically different between the ${\mathrm{A}}_g^1$ mode of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ (hardening) and that of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ (softening). This is explained by the anomalous quartic anharmonicity in ${\mathrm{Bi}}_2{\mathrm{Te}}_3$, which is absent in ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. Besides, as the phonon energy increases in ${\mathrm{Bi}}_2{\mathrm{Te}}_3$, the trend switches sign from hardening to softening due to the relative contributions from the phonon joint density of states and density of states. Detailed discussion is given in the text.