Thermodynamical properties of a superconducting quantum cylinder (Q455511)

The paper is devoted to studying thermodynamic properties of a superconducting quantum cylinder. In their investigation, the authors use the Bardeen-Cooper-Schrieffer (BCS) model. The main interaction in the BCS Hamiltonian in the 3D case is the interaction of pairs of electrons with opposite momenta and spins. First, the energy of the stationary states of an electron is defined and it is assumed that the orbital angular momenta of the electrons in a Cooper pair on the surface of the cylinder are directed oppositely, as well as the momenta of the longitudinal motion and the projections of the electron spins to the cylinder axis. In the representation of second quantization, the simplest model Hamiltonian in the superconductivity theory of a quantum cylinder (in which the interaction of electrons with values of quantum numbers of opposite signs is preserved) is represented. Then, by using the (\(u\)-\(v\)) Bogolyubov transformation for Fermi operators and the Bogolyubov statistical variational principle, the thermodynamic potential of the superconducting electron gas of the quantum cylinder is written via the width of the energy gap, for which is stated the temperature dependence of the gap value. The obtained formula of the potential presents its dependence on the geometric sizes of the nanotube and on the concentration of electrons determined by the width of the gap under zero temperature. The heat capacity of the system is found by double differentiating the thermodynamic potential with respect to the temperature. Then, by using the asymptotic expressions for the energy gap in the domain of low temperatures and near the critical temperature, it is found the heat capacity of the superconducting electron gas of the quantum cylinder in these limit cases. It is shown that the system is subjected to a phase transition of the second kind from the superconducting state to the normal one of the electron gas with the finite jump of heat conductivity.