Origin of the thermal expansion anomaly in layered ${\mathrm{Bi}}_2{\mathrm{X}}_3$ topological insulators: Ultrafast time-resolved pump-probe experiments and theory

Recent experiments on the thermal expansion of ${\mathrm{Sb}}_2{\mathrm{Te}}_3$, a prototypical example of strong three-dimensional topological insulators, have shown an intriguing anomaly in the thermal expansion coefficient along the hexagonal axis $\left({\alpha }_{{}_{\parallel }}\right)$, which drops sharply to almost zero in a narrow range of temperature around $\sim 225$ K. With no accompanying signatures in other properties, the origin of this anomaly is not understood. We present here femtosecond pump-probe differential reflectivity measurements on single crystals of ${\mathrm{Sb}}_2{\mathrm{Te}}_3$ as a function of temperature from 3 to 300 K to determine the temperature dependence of coherent optical and acoustic phonons along with the dynamics of the photoexcited carriers. We find clearly anomalous temperature dependence of the parameters associated with vibrational and electronic relaxation in the narrow temperature range of 200–250 K. Within first-principles density functional theoretical analysis, we show that the observed anomalies can be explained with a mechanism of formation of stacking faults stabilized by vibrational entropy above 200 K. As a similar anomaly in the thermal expansion is also observed in other chalcogenides in the same family, the proposed mechanism may also be applicable to these layered strong topological insulators.

Figure 1

(a) $\mathrm{\Delta }R/R_0$ signal at 300 K showing different processes: excited carrier relaxation, coherent optical phonon $A_{1g}^1$, and acoustic phonon. Solid line is the fit as discussed in text. Inset shows zoomed view of $A_{1g}^1$ optical phonon oscillations with solid lines (red color) as fit. Temperature dependence of (b) initial amplitude $A_e$ and (c) carrier relaxation time ${\tau }_e$ (dashed lines in red color joining through data points are guide to eye). Optical phonon $A_{1g}^1$ (d) oscillation amplitude $B_{{}_{OP}}$ (e) scattering rate $\mathrm{\Gamma }$ and (f) frequency ${\nu }_{{}_{OP}}$ as a function of temperature. Solid lines (red color) in (d) and (e) are fit with cubic anharmonicity model for $B_{{}_{OP}}$ and $\mathrm{\Gamma }$. Solid line (red) in (f) is fit to frequency shift with temperature by cubic anharmonic model together with thermal expansion contribution. Error bars are smaller than symbol size.

Figure 2

Temperature dependence of (a) reflectivity $R$, longitudinal acoustic phonon (b) initial amplitude $B{}_{LA}$, (c) dephasing time $\tau {}_{LA}$, and (d) frequency of oscillations $\nu {}_{LA}$ obtained by fitting differential reflectivity signals $\mathrm{\Delta }R/R_0$ at each temperature. Obtained temperature dependence of optical constants (e) refractive index $n$, (f) extinction coefficient $k$, (g) sound velocity $v_{{}_{LA}}$, and (h) longitudinal modulus of elasticity $Y_{{}_{LA}}$. Dashed lines joining through data points are guide to eye. Error bars are smaller than symbol size.

Figure 3

(a) Hexagonal crystal structure of pristine ${\mathrm{Sb}}_2{\mathrm{Te}}_3$, its side view and stacking faulted configuration as seen from a side view. (b) Free energy difference $\left(\mathrm{\Delta }F\right)$ of faulted and pristine structures as a function of temperature. (c) Electronic structures of the pristine and faulted configurations of ${\mathrm{Sb}}_2{\mathrm{Te}}_3$, calculated with spin orbit coupling (SOC). (d) Total energy (setting zero as the energy of the pristine structure) (e) band gap (determined with SOC), (f) calculated refractive index, and (g) longitudinal modulus of elasticity (calculated using ${\mathrm{Y}}_{LA}=\frac{1}{V}\frac{{\partial }^2{E}_{\mathrm{tot}}}{\partial {\epsilon }_{zz}^2}{|}_{{\epsilon }_{zz}=0}$, where $V$ is volume of the unit cell, $E_{\mathrm{tot}}$ is the total energy obtained with SOC, and ${\epsilon }_{zz}$ is the uniaxial strain) is plotted for the pristine, faulted, and their intermediate configurations along the path parameterized by $f$. $\text{P}$ and $\text{F}$ represent the positions of pristine and faulted configurations, respectively.