Nonlinear phononics in Bi$_2$Te$_3$ nanoscale thin films: A theoretical approach

TL;DR

This work combines first-principles energy-surface calculations with a minimal dynamical model to explain nonlinear phononics in Bi$_2$Te$_3$ nanoscale films. It shows that a THz pump nonresonantly excites the infrared E$_u^1$ mode, which, via cubic phonon–phonon couplings, drives the Raman-active A$_{1g}^1$ mode (and to a lesser extent E$_g^1$), with a robust agreement to observed transmittance oscillations around $ u_{A_{1g}} oughly 1.86$ THz. The detection process is modeled through the THz–probe optical response, validating that the transmittance changes arise primarily from lattice anharmonicity rather than Raman-sum-frequency processes, and enabling quantitative inference of ionic displacements from ab initio data. The study also predicts that resonant excitation of E$_u^1$ could transiently lower Bi$_2$Te$_3$ symmetry, offering a route to selectively control Eg modes and tailor ultrafast lattice dynamics. Overall, the paper provides a predictive, parameter-free framework linking THz driving, phonon couplings, and optical detection in a topological insulator film, with implications for ultrafast control of crystal symmetry.

Abstract

Density Functional Theory (DFT) calculations not only allow to predict the vibrational and optical properties of solids but also to understand and disentangle the mechanisms playing a key role in the generation of coherent optical phonons. Recent experiments performed on a Bi$_2$Te$_3$ nanofilm have shown that a THz pulse launches at least a coherent $A_{1g}^1$ phonon as the transient transmittance measured using an isotropic detection scheme displays oscillations with a frequency matching the frequency of the $A_{1g}^1$ mode measured in Raman experiments. Such an observation can be explained by invoking either a sum frequency process or cubic/quartic phonon-phonon couplings as considered for Bi$_2$Se$_3$, a parent compound of Bi$_2$Te$_3$. By resorting to group theory and calculating energy surfaces from first-principles, the main phonon-phonon couplings can be identified. Furthermore, a minimal model can be built to compute the dynamics of the Raman active modes coupled to the infrared active mode driven by the experimental THz pulse. Our model firmly establishes that cubic phonon-phonon interactions are relevant as the agreement between the computed and experimental transmittance is noteworthy.

Figures (11)

• Figure 1: (a) The THz electric field (open circles) measured using an electro-optic methodBrunner_2014Johnson2_2014, is compared to a fit based on the analytical function given by Eq. \ref{['E0_vs_t']} (solid black line). (b) Fourier transforms of both the experimental (open circles) and analytical (solid black line) THz waveform together with the computed zone center frequencies at the LDA level shown as vertical arrowsbusselez_2023.
• Figure 2: (a) Transient experimental relative transmittance $\Delta T/T$ (black line) measured as a function of the time delay $t$ between the optical pulse and the THz pulse compared to the fitting curve (red dashed line) obtained from Eq. \ref{['fit_eq']} with the parameters reported in Table \ref{['fit_expt_tab']}. The difference between the experimental and the fitting curve corresponds to the blue dotted curve. (b) Experimental curve (black line) compared to the background curve (red dashed line) obtained by setting $a_4=0$ in Eq. \ref{['fit_eq']} while keeping the values of the other parameters found in Table \ref{['fit_expt_tab']}. The difference between the experimental and the background curve provides the high frequency part of the signal (blue dotted curve) related to the coherent optical phonon oscillations. The signal to noise ratio of the experimental relative transmittance is $\sim 200$.
• Figure 3: (a) Hexagonal structure of Bi$_2$Te$_3$ spanned by the lattice vectors ${\bf{a}}_{1, \perp}$, ${\bf{a}}_{2, \perp}$ and ${\bf{a}}_{\parallel}$. The Te$_1$ atoms, Te$_2$ atoms and Bi atoms are respectively colored in red, purple and blue. The double-headed arrow delineates one quintuple layer (QL) with a thickness of approximately $0.76$ nm. (b) Top view of the hexagonal structure with the three $\sigma_d$ planes and $C_2$ axis respectively represented by straight and dotted lines.
• Figure 4: $f_m$ (See Eq. \ref{['Eq_fm']}) as a function of $\alpha=\overline{V}/V$, where $\overline{V}$ ($V$) is the longitudinal sound velocity inside the oxidized layer (Bi$_2$Te$_3$ film). Here $m=1, \cdots, 4$ denotes the number of oxidized blocks. The horizontal lines show the only allowed integer value of $n$ (number of unit cells) and the vertical lines show that the only realistic value of $m$ compatible with the values of $n$ are one and four.
• Figure 5: (a) Real and imaginary parts of the optical index of Bi$_2$Te$_3$ computed at the RPA levelarnaud_2001yambo_2019busselez_2023 for an electric field perpendicular to the trigonal axis as a function of the wavelength $\lambda$ (in nm). (b) Experimental real part of the optical index of Bi$_2$O$_3$dolocan_1981, TeO$_2$uchida_1971 and micanitsche_2004 as a function of $\lambda$ (in nm). (c) Computed transmittance of the heterostructure corresponding to $n=5$ ($d\sim 15.24$ nm) and $m=1$ ($d_{ox}\sim 1.01$ nm) as a function of $\lambda$ (in nm) for an oxide layer made either of Bi$_2$O$_3$ (purple curve) or TeO$_2$ (red curve) compared to the experimental transmittance (black curve) measured with a white lamp (black curve). (d) Same as (c) but for $n=4$ ($d\sim 12.19$ nm) and $m=4$ ($d_{ox}\sim 4.06$ nm). The two possible heterostructures are schematically represented in (c) and (d). The scale for the mica layer thickness ($d_{mica}\sim 17.1 \mu$m) is not respected as it is much larger than the oxide and nanofilm thicknesses.