Lattice dynamics of Bi$_2$Te$_3$ and vibrational modes in Raman scattering of topological insulators MnBi$_2$Te$_4\cdot n$(Bi$_2$Te$_3$)

This work is devoted to the experimental study and symmetry analysis of the Raman-active vibration modes in MnBi$_2$Te$_4\cdot n$(Bi$_2$Te$_3$) van der Waals topological insulators, where $n$ is the number of Te–Bi–Te–Bi–Te quintuple layers between two neighboring Te–Bi–Te–Mn–Te–Bi–Te septuple layers. Confocal Raman spectroscopy is applied to study Raman spectra of crystal structures with $n = 0,1,2,3,4,5,6$, and $\infty$. The experimental frequencies of vibration modes of the same symmetry in the structures with different n are compared. The lattice dynamics of free-standing one, three, and four quintuple layers, as well as of bulk Bi$_2$Te$_3$ ($n = \infty$) and MnBi$_2$Te$_4$ ($n = 0$), is considered theoretically. Vibrational modes of the last two systems have the same symmetry, but different displacement fields. These fields in the case of a Raman-active mode do not contain displacements of manganese atoms for any finite $n$. It is shown that two vibrational modes in the low-frequency region of the spectrum ($35$–$70$ cm$^{-1}$) of structures with $n = 1, 2, 3, 4, 5$, and $6$ practically correspond to the lattice dynamics of $n$ free quintuple Bi$_2$Te$_3$ layers. For this reason, the remaining two vibration modes, which are observed in the high-frequency region of the spectrum ($100$–$140$ cm$^{-1}$) and are experimentally indistinguishable in the sense of belonging to quintuple or septuple layer or to both layers simultaneously, should also be assigned to vibrations in quintuple layers under immobile atoms of septuple layers.