Zeroes of combinations of Bessel functions and mean charge of graphene nanodots

TL;DR

This paper explores mathematical properties of Bessel function zeroes and applies these findings to determine energy levels and mean charge in graphene nanodots under specific boundary conditions.

Contribution

It introduces new mathematical insights into Bessel function zeroes and applies them to analyze physical properties of graphene nanodots.

Findings

Zeroes of Bessel function combinations have specific interlacing properties.

Allowed energy levels of graphene nanodots are determined.

Mean charge depends on gate potential and is calculated at zero temperature.

Abstract

We establish some properties of the zeroes of sums and differences of contiguous Bessel functions of the first kind. As a byproduct, we also prove that the zeroes of the derivatives of Bessel functions of the first kind of different orders are interlaced the same way as the zeroes of Bessel functions themselves. As a physical motivation, we consider gated graphene nanodots subject to Berry-Mondragon boundary conditions. We determine the allowed energy levels and calculate the mean charge at zero temperature. We discuss in detail its dependence on the gate (chemical) potential.