Remarks on thermodynamic properties of a double ring-shaped quantum dot at low and high temperatures

Abstract

In a recent paper published in this Journal, Khordad and collaborators [J Low Temp Phys (2018) 190:200] have studied the thermodynamics properties of a GaAs double ring-shaped quantum dot under external magnetic and electric fields. In that meritorious research the energy of system was obtained by solving the Schrödinger equation. The radial equation was mapped into a confluent hypergeometric differential equation and the differential equation associated to $z$ coordinate was mapped into a biconfluent Heun differential equation. In this paper, it is pointed out a misleading treatment on the solution of the biconfluent Heun equation. It is shown that the energy $E_z$ can not be labeled with $n_z$ and this fact jeopardizes the results of this system. We calculate the partition function with the correct energy spectrum and recalculate the specific heat and entropy as a function of low and high temperatures.

11 pages, 4 figures

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