Critical temperature of ${\mathrm{MgB}}_2$ ultrathin superconducting films: BCS model calculations in the tight-binding approximation

We develop the multiband BCS model of superconductivity in the ultrathin films using the orthogonal tight-binding approximation for constructing the electron wave functions. This allows for relatively simple determination of the band structure near the Fermi level as well as the electron-electron interaction matrix elements of the BCS type. The model is applied to the ultrathin ${\mathrm{MgB}}_2\left(0001\right)$ films, for which the critical temperature values are calculated in the thickness range $2–10{\mathrm{MgB}}_2$ layers. The importance of the boundary conditions is emphasized, as either boron or magnesium layers may cover the film. It is found that films thinner than four layers show a substantial decrease in the critical temperature. The charge spillage outside the geometric boundaries of the film, which is allowed in our model, supresses $T_c$ oscillations and weakens its dependence on film covering layer composition.

Figure 1

(a) Schematic side view of the (0001)-axis oriented thin ${\mathrm{MgB}}_2$ film of $\mathrm{Mg}\cdots \mathrm{B}$ covering type, containing $N$ boron layers. Solid double lines are the geometric boundaries of the film while dashed ones are the boundaries of the assumed infinite quantum well for the electrons in the $z$ direction. The dashed curve on the right depicts the behavior of $\Gamma$ amplitude. (b) The view of the $\mathrm{B}\cdots \mathrm{B}$ type film and (c) the view of the $\mathrm{Mg}\cdots \mathrm{Mg}$ type film, both for $N=2$.

Figure 2

The Fermi level change computed in the tight binding model for ${\mathrm{MgB}}_2$ ultrathin film, for $\delta =0.0\mathrm{\AA }$ (a) and $\delta =0.5\mathrm{\AA }$ (b).

Figure 3

Temperature dependence of bulk energy gap values. The experimental values are taken from Ref. 19. The solid line is a result of our calculations in the BCS model with the TBA-obtained DOS.

Figure 4

Critical temperature of ${\mathrm{MgB}}_2$ ultrathin films calculated according to the model developed, for $\delta =0.0\mathrm{\AA }$ (a) and $\delta =0.5\mathrm{\AA }$ (b). The dashed line is bulk value.

Figure 5

Superconducting gap parameters at $T=0$ for ${\mathrm{MgB}}_2$ ultrathin films, $\delta =0.5\mathrm{\AA }$. The integer thicknesses correspond to $\mathrm{Mg}\cdots \mathrm{B}$ covering, while the half integers are for $\mathrm{Mg}\cdots \mathrm{Mg}$ structures. The $\mathrm{B}\cdots \mathrm{B}$ type covering is omitted for clarity. The dashed lines are the bulk gaps for $\pi$ and $\sigma$ bands, according to Ref. 19.