An exploration of the phonon frequency spectrum and Born-von Karman periodic boundary conditions in 1D and 2D Lattice systems using a computational approach

TL;DR

This paper demonstrates how to computationally analyze phonon spectra in 1D and 2D lattice systems using Born-von Karman periodic boundary conditions, with practical implementations suitable for undergraduate education.

Contribution

It provides a computational method employing Python, FFT, and Runge-Kutta to calculate phonon spectra with PBCs, filling a gap in undergraduate lattice dynamics education.

Findings

Validated computational phonon spectra against analytical solutions.

Extended methods to 2D lattices with second nearest neighbor interactions.

Provided accessible educational tools for lattice dynamics analysis.

Abstract

The concept of periodic boundary conditions (PBCs) is immensely significant in treating an ideal lattice of infinite extent as a finite lattice. An explicit usage of PBCs is often found missing in undergraduate texts on analytical treatment of lattice dynamics. The aim of the present work is to cover this gap by illustrating the application of Born-von Karman PBCs in lattice dynamical calculations using a computational approach. The equations of motion are set up for a linear diatomic lattice with a basis, using the nearest neighbour approximation. The solution is obtained by implementing fourth order Runge-Kutta algorithm in python. Fast Fourier Transform (FFT) technique is then used to obtain the phonon frequency spectrum corresponding to the computed solutions. Similar computations are extended to obtain the phonon spectrum for monatomic square and honeycomb lattices under the second nearest neighbour approximation. The computed results are validated against the analytical ones in each case. The target group of the present work are the students and educators at the undergraduate level.