Energy spectrum of massive Dirac particles in gapped graphene with Morse potential

TL;DR

This paper derives the energy spectrum of massive Dirac particles in gapped graphene influenced by a Morse potential, using confluent Heun functions to solve the Dirac equation and analyzing the band structure.

Contribution

It introduces a novel analytical approach to solve the Dirac equation with Morse potential in gapped graphene, providing explicit eigenvalues and wavefunctions.

Findings

Energy spectrum expressed in terms of quantum numbers N and k

Wavefunctions calculated using confluent Heun functions

Graphical analysis of energy bands and wavefunctions

Abstract

In this paper, we study the massive Dirac equation with the presence of the Morse potential in polar coordinate. The Dirac Hamiltonian is written as two second-order differential equations in terms of two spinor wavefunctions. Since the motion of electrons in graphene is propagated like relativistic fermionic quasi-particles, then one is considered only with pseudospin symmetry for aligned spin and unaligned spin by arbitrary $k$. Next, we use the confluent Heun's function for calculating the wavefunctions and the eigenvalues. Then, the corresponding energy spectrum obtains in terms of $N$ and $k$. Afterward, we plot the graphs of the energy spectrum and the wavefunctions in terms of $k$ and $r$, respectively. Moreover, we investigate the graphene band structure by a linear dispersion relation which creates an energy gap in the Dirac points called gapped graphene. Finally, we plot the graph of the valence and conduction bands in terms of wavevectors.