Linear chain model for the low-frequency phonon modes in van der Waals layered materials

One-dimensional linear atomic chain model is applied to understand the atomic vibrations in the low-frequency modes in van der Waals layered materials. We present a systematic study of different methods (analytical, numerical, and electrical circuit analogy) to calculate the atomic vibrational modes and their frequencies, and discuss the effects of different boundary conditions on the results. We also show that for a finite atomic chain with N atoms, the vibrational modes are standing waves, and their vibrational frequencies can be obtained through zone folding of an infinite atomic chain. Finally, the one-atom chain model is extended to a two-atom chain model to study the low-frequency modes in InSe multilayers.