Issues $\rightarrow$ Volume 26, Issue 3 (2025)

Green’s Function Technique in the Theory of Disordered Crystals: Application to Potassium-Doped Graphene

REPETSKY S.P.$^{1,2}$, VYSHYVANA I.G.$^{3}$, LIZUNOV V.V.$^{1}$, MELNYK R.M.$^{2}$, REZNIKOV M.I.$^{3}$, RADCHENKO T.M.$^{1}$, and TATARENKO V.A.$^{1}$

$^1$G.V. Kurdyumov Institute for Metal Physics of the N.A.S. of Ukraine, 36 Academician Vernadsky Blvd., UA-03142 Kyiv, Ukraine
$^2$National University of Kyiv-Mohyla Academy, 2 H. Skovoroda Str., UA-04070 Kyiv, Ukraine
$^3$Taras Shevchenko National University of Kyiv, 60 Volodymyrs'ka Str., UA-01033 Kyiv, Ukraine

Received / Final version: 02.05.2025 / 16.08.2025 Download PDF logo PDF

Abstract
The method of describing the energy spectrum, free energy, and electrical conductivity of disordered crystals based on the use of the Hamiltonian of electrons and phonons is reviewed, analysed, and developed. The electron states of a system are described through the tight-binding model. A simple procedure for calculating the matrix elements of the Hamiltonian within the Wannier’s representation is proposed. Expressions for the Green’s functions, free energy, and electrical conductivity are derived using the diagram method. Using this procedure, the vertex parts of the mass operators of the electron–electron and electron–phonon interactions are renormalized. A set of exact equations is obtained for the spectrum of elementary excitations in a crystal. This enables the performance of numerical calculations on the energy spectrum and the prediction of system properties with predetermined accuracy. Expressions are obtained for the static waves of concentrations, charge and spin densities, which determine the phase state of a disordered crystal. In contrast to other approaches, which account for electron correlations only within the limiting cases of infinitely large and infinitesimal electron densities, this method describes electron correlations in the general case of an arbitrary density. In addition to the theory, the results of a numerical calculation of the energy spectrum of a graphene layer with adsorbed potassium (K) atoms are presented. As established, at the K-atoms’ concentration such that the unit cell includes two carbon (C) atoms and one K atom, the latter being located (adsorbed) on the graphene layer surface 0.286 nm above the C atom, the energy gap is ≅ 0.25 eV. The location of the Fermi level (εF) in the energy spectrum depends on the potassium-atoms’ concentration and is in the energy interval −0.36 Ry ≤ εF ≤ −0.23 Ry.

Keywords: disordered crystals, electronic structure, conductivity, Green’s functions, the mass operator, density of states, free energy.

DOI: https://doi.org/10.15407/ufm.26.03.460

Citation: S.P. Repetsky, I.G. Vyshyvana, V.V. Lizunov, R.M. Melnyk, M.I. Reznikov, T.M. Radchenko, and V.A. Tatarenko, Green’s Function Technique in the Theory of Disordered Crystals: Application to Potassium-Doped Graphene, Progress in Physics of Metals, 26, No. 3: 460–*** (2025)

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