Frequency filtering and vibrational spectrum of multiple-valley one-dimensional phononic crystals

In the present study, we have fixed the number of atoms in a one-dimensional phononic chain and changed the number of the valleys (or numbers of wells NOW) in it (by adding new terms to the system Hamiltonian) and studied the transmission properties of his system. Then, we have obtained the phononic transmission coefficient by using a one-dimensional transfer matrix method. We have also investigated the vibrational modes of the assumed system by using a direct diagonalization technique. We show that in our assumed system, there is an Omni-k bandgap within which bandgap exists for all values of the studied spring constant k . In our assumed system, we have also obtained some bi-convex lens-shaped phononic bandgaps as a function of the parameter V conf which their surface areas can be tuned by changing the number of valleys NOW. An interesting fact is that the transmission coefficient is found to be quantized versus variation of the number of wells NOW which may help experimentalists to use lower values of the material to fabricate their desired system (see the text). We observe that the transmission coefficients as a function of the NOW are some Cantor-like ones that obey self-similar bandgap structure patterns. Besides, we show that NOW and V conf are good tuning parameters that can localize the eigenmodes of the system in any part of the chain. We have provided some animation to more clearly illustrate the effect of different parameters on the transmission and vibrational properties of the system. Finally, by using a few parameters, which we have presented in this paper, we can do bandgap engineering and determine the bandgap widths, as well as their places on the frequency axis.