Superconducting critical temperature and the isotope exponent versus total electron concentration for two-band superconductors: Effect of the band structure

Abstract

We consider the superconducting critical temperature ${T}_{c}$ and the isotope exponent $\ensuremath{\alpha}$ versus carrier concentration $\ensuremath{\rho}$ using the one-particle Green functions of the simplest coupled Hamiltonian for the two-band model with a single ${T}_{c}$ [N. Kristoffel and P. Rubin, Physica C $356,$ 171 (2001)]. One of the bands, ${\ensuremath{\varepsilon}}_{2}(\stackrel{\ensuremath{\rightarrow}}{k}),$ is taken to be two dimensional, while the second band ${\ensuremath{\varepsilon}}_{3}(\stackrel{\ensuremath{\rightarrow}}{k})$ is taken to be three dimensional. Taking only nearest-neighbor hopping for each one of the bands, we obtain that ${T}_{c}$ versus $\ensuremath{\rho}$ is symmetric around half filling in agreement with a general theorem of quantum mechanics. We also obtained the chemical potential $\ensuremath{\mu}$ versus $\ensuremath{\rho}$ for several values of attractive interaction. We find that (1) the superconducting critical temperature ${T}_{c}$ is maximum at $\ensuremath{\rho}=1/2;$ (2) the chemical potential remains basically inside the smaller band, for the range of carrier concentration and Debye frequencies considered; (3) the isotope exponent has several symmetric minima around half filling depending on the anisotropy of the three-dimensional tight-binding band, namely, $\ensuremath{\gamma}\ensuremath{\equiv}{t}_{3,x}{/t}_{3,z}\ensuremath{\ne}1;$ (4) the chemical-potential curves, namely, $\ensuremath{\mu}$ versus $\ensuremath{\rho}$ are completely independent of pairing interaction and Debye frequency; and (5) the number of symmetric minima and ${T}_{c}^{\mathrm{max}}$ goes down with increasing ${t}_{3}{/t}_{2}$ ratio. We have briefly discussed the possible relevance of our results with recent experimental data of the two-band compound ${\mathrm{MgB}}_{2}.$