Remarks on Thermodynamic Properties of a Double Ring-Shaped Quantum Dot at Low and High Temperatures

In a recent paper published in this Journal, Khordad and Sedehi (J Low Temp Phys 190(3):200, 2018) have studied the thermodynamic properties of a GaAs double ring-shaped quantum dot under external magnetic and electric fields. In that meritorious research the energy of the 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 Ez can not be labeled with nz 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.