Tight-binding implementation of the quantum kinetic equation for graphene

Abstract

Background and Objectives: Progress in the development of pulsed radiation sources with high energy density makes it possible to study the nonlinear response of condensed matter to the disturbing influence of high-intensity electromagnetic fields. To understand the processes occurring in this case, adequate models are needed that qualitatively and quantitatively reproduce the characteristics of the materials under study. In this area, graphene is considered one of the most promising materials due to the specificity of its band structure. The purpose of the work is to present and test a new model based on the quantum kinetic equation, free from restrictions on such parameters as the frequency and strength of the electric field of the disturbing influence. Materials and Method: The approach used in the work is based on the quantum kinetic equation for the distribution function of charge carriers in the state space. It makes it possible, in the one-electron approximation, to nonperturbatively reproduce the ultrafast dynamics of carriers in an external classical electric field. The system under consideration is specified by the electron dispersion law. The approach was developed and implemented for the pseudo-relativistic approximation of massless fermions, successfully used in describing the features of graphene. However, by its definition, this approximation quite accurately reproduces the real dispersion law only in the low-energy region in the vicinity of the Dirac points. Therefore, the direct use of this version of the model to describe processes in which electronic states with high excitation energies are known to participate raises questions about the accuracy of the results obtained. The problem can be resolved by moving to an exact definition of the dispersion law through the parameters of the tight-binding model of nearest neighbors in the crystal lattice of the graphene. The presented work proposes an implementation option for such a procedure and verifies the results obtained. A generalization of the formalism for a two-level system with a massless Hamiltonian of general form is used, which universally defines the explicit form of the quantum kinetic equation and expressions for macroscopic observable parameters. Results: A computational model based on the exact tight-binding model Hamiltonian has been determined, which strictly takes into account the real law of graphene dispersion in reciprocal space. The new model has been verified. For this purpose, the results of its use are compared with the results of a similar model based on the massless fermion approximation. Under conditions of limiting the parameters of the perturbing influence, ensuring the generation of excited states with only low energies in the immediate vicinity of the Dirac points, an exact coincidence has been demonstrated both at the stage of determining the values of the distribution function and for the observed parameters. It has been shown that going beyond the applicability limits of the massless fermion approximation is accompanied by the appearance of qualitative and quantitative differences in the results obtained. Conclusion: The results of the work provide new opportunities for studying the behavior of graphene under extreme conditions of strong high-frequency fields, modeling and searching for new nonlinear effects, and accurately reproducing the ultrafast quantum dynamics of its electrons for states with high energy values.

Keywords

• graphene
• quantum kinetic equation
• tight-binding model
• ultrafast electron dynamics
• nonlinear effects