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

TL;DR

This paper critiques previous work on the thermodynamics of a double ring-shaped quantum dot, corrects the energy spectrum calculation, and reevaluates the specific heat and entropy at various temperatures.

Contribution

It identifies and corrects a mistake in the energy spectrum calculation of the quantum dot system, impacting thermodynamic property evaluations.

Findings

Corrected the energy spectrum of the quantum dot system.

Recalculated the partition function, specific heat, and entropy.

Provided accurate thermodynamic properties 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 Schr\"{o}dinger 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.