An intrinsic antiferromagnetic topological insulator $mathrm{MnBi_2Te_4}$ can be realized by intercalating Mn-Te bilayer chain in a topological insulator, $mathrm{Bi_2Te_3}$. $mathrm{MnBi_2Te_4}$ provides not only a stable platform to demonstrate exotic physical phenomena, but also easy tunability of the physical properties. For example, inserting more $mathrm{Bi_2Te_3}$ layers in between two adjacent $mathrm{MnBi_2Te_4}$ weakens the interlayer magnetic interactions between the $mathrm{MnBi_2Te_4}$ layers. Here we present the first observations on the inter- and intra-layer phonon modes of $mathrm{MnBi_{2n}Te_{3n+1}}$ (n=1,2,3,4) using cryogenic low-frequency Raman spectroscopy. We experimentally and theoretically distinguish the Raman vibrational modes using various polarization configurations. The two peaks at 66 cm$^{-1}$ and 112 cm$^{-1}$ show an abnormal perturbation in the Raman linewidths below the magnetic transition temperature due to spin-phonon coupling. In $mathrm{MnBi_4Te_7}$, the $mathrm{Bi_2Te_3}$ layers induce Davydov splitting of the A$_{1g}$ mode around 137 cm$^{-1}$ at 5 K. Using the linear chain model, we estimate the out-of-plane interlayer force constant to be $(3.98 pm 0.14) times 10^{19}$ N/m$^3$ at 5 K, three times weaker than that of $mathrm{Bi_2Te_3}$. Our work discovers the dynamics of phonon modes of the $mathrm{MnBi_2Te_4}$ and the effect of the additional $mathrm{Bi_2Te_3}$ layers, providing the first-principles guidance to tailor the physical properties of layered heterostructures.
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