Abstract
In this paper, we review the tight-binding model in the first and second quantization and show how it can be used to calculate the energy spectrum of some crystals. From an approach based on the Schrödinger equation (first quantization), we demonstrate the procedure for writing a generic Hamiltonian in the second quantization formalism. The connection between these two formalisms is generally not discussed in technical and applied works. As application examples, we use both methodologies to calculate the energy spectrum of a linear chain and a square lattice analytically, initially considering only one site per unit cell and later taking two sites per unit cell. Next, we apply the tight-binding model to graphene and compare such description with the brick lattice, showing that graphene lattice can be mapped as a square lattice with some hopping parameters being neglected. Finally, we apply the model to the \(\tau _3\)-lattice, a three-band system. In all cases, we present the energy spectrum and the density of states.
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Acknowledgements
Discussions with André J. Chaves are gratefully acknowledged. The authors are grateful to Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), to Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) of Brazil for financial support, and to Fundação Cearense de Apoio ao Desenvolvimento Científico e Tecnológico (FUNCAP). D.R.C is supported by CNPq grant numbers 310019/2018-4 and 437067/2018-1. J.M.P.Jr. and D.R.C are supported by bilateral ITA-UFC CNPq project number 400879/2019-0.
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Lima, W.P., Araújo, F.R.V., da Costa, D.R. et al. Tight-binding Model in First and Second Quantization for Band Structure Calculations. Braz J Phys 52, 42 (2022). https://doi.org/10.1007/s13538-021-01027-x
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DOI: https://doi.org/10.1007/s13538-021-01027-x