We study the relationship between the critical temperature Tc and Debye temperature θD for 23 single-element and compound superconductors and find three useful models to fit their products TcθD.
The first model gives a constant Debye temperature of 288 K for the 23 superconductors, but it is not a good enough model to fit all 23 data. Its benefit is at low Tc < 19 K. The second model, described by five parameters, is the best one at Tc ≥ 19 K. It uses all 23 data to find out the optimized parameters. The third model, described by six parameters, only considers the fittings of the 12 superconductors at Tc ≥ 19 K. Both the second and third models are from the same mathematical expression, but with different parameters.
The special thing is that the two parameters of ω0 = 1.79 and γ = 0.261 are suitable for the second and third models, no matter whether the data are less than 19 K. The additional 11 superconductors of Tc < 19 K do not change these two parameters, so the two parameters seem to be the fingerprint of all these 23 superconductors. The mathematical expression can be divided into five terms and seems to permit us to explain this fitting model by the average Debye temperature, the change rate of the Debye temperature with respect to the critical temperature and the electron-phonon scattering.
The second model can be extended to fit the nine data of high-temperature hydride superconductors at ultra-high pressure. The calculated absolute errors of the Debye temperatures show that 4 out of 9 are less than 5.0%, and 7 out of 9 are less than 10.0%. Our extended model is well-workable for the nine data.
According to these possible explanations, this mathematical expression reflects that Tc has something to do with θD in superconductors. So, this research would help us to know more about the relationship between Tc and θD as well as the possible factors inside the superconductors.
