Microscopic description of surface superconductivity

In this work, we revisit the problem of superconductivity under the influence of boundary effects. By solving the Bogoliubov-de Gennes (BdG) equations for the tight-binding model, we demonstrate that the critical temperature of the nucleation of superconductivity near a sample surface can be considerably enhanced as compared to its bulk value. To bring to light this effect, we investigate different methods to solve numerically the BdG equations, including the continuous and Anderson approximations, and perform the calculations for a wide range of the system parameters. We obtain that all the self-consistent BdG eigenstates are delocalized and occupy the entire volume of the sample. Our results reveal that the enhancement of the surface critical temperature originates from the quantum interference of different BdG states contributing to the order parameter. We also find that the surface enhancement is the largest when the conduction band is symmetric with respect to the Fermi level, particularly, the half filling is an important proviso for the pronounced surface effect on the critical temperature. The approximate continuous model as well as the Anderson approximation do not capture the main feature of the surface effect. In addition, our study of this effect versus surface roughness reveals its fragile character.

Figure 1

The order parameter, ${\mathrm{\Delta }}_j$ as a function of $j$ and $T$. The results are calculated within the $t\text{−}\mathrm{BdG}$ for the half-feeling case $n_e=1$ with the coupling $g=2$, the cutoff energy $\hslash {\omega }_D=5$, and $N=128$.

Figure 2

The gap function ${\mathrm{\Delta }}_j$ versus $j$, as calculated within $t$-BdG for the electron densities $n_e=1$(a), 0.6 (b), and 0.1 (c) at different temperatures. Other microscopic parameters are the same as in Fig. 1.

Figure 3

The ratio $T_{cS}/T_{cB}$ calculated within the $t\text{−}\mathrm{BdG}$ as a function of $n_e$ for the different coupling constants $g$. The cutoff energy is $\hslash {\omega }_D=5$.

Figure 4

Typical quasiparticle amplitudes $u_j^{\left(\alpha \right)}$ (first column), quasihole amplitudes $v_j^{\left(\alpha \right)}$ (second column), and the partial order parameter ${\mathrm{\Delta }}_j^{\left(\alpha \right)}=u_j^{\left(\alpha \right)}{v}_j^{\left(\alpha \right)*}(1-2f_{\alpha })$ (third column) vs $j$. The rows are for $\alpha =1,2,127$, and 128 (from above to below). Obtained within $t\text{−}\mathrm{BdG}$ at the zero temperature for the half-filling case $n_e=1$. The other model parameters are the same as in Fig. 1.

Figure 5

The same as in Fig. 4 but for the occupation $n_e=0.1$.

Figure 6

The excitation energies $E_{\alpha }$ versus the quasimomentum $k_{\alpha }>0$ calculated for $n_e=0.1$, $n_e=0.6$, $n_e=1$. The calculations are done within $t\text{−}\mathrm{BdG}$ for $N=32$.

Figure 7

The cumulative order parameter ${\mathrm{\Delta }}_j^{\left(E\right)}$ versus $j$ for different ascending values of $0<E<\hslash {\omega }_D$. Calculated within $t\text{−}\mathrm{BdG}$ for $n_e=1$ at $T=0$ (a) and 1.05 (c), and for $n_e=0.1$ at $T=0$ (b) and 1.05 (d). The other microscopic parameters are the same as in Fig. 1. The evolution of ${\mathrm{\Delta }}_j^{\left(E\right)}$ with increasing $E$ allows one to trace the constructive interference of the contributing BdG states.

Figure 8

Contour plot of the amplitude spectral density, $\left|{\mathrm{\Delta }}_f^{\left(\alpha \right)}\right|$, shown on $f-\alpha$ plane, calculated for $n_e=1$ [(a) and (c)] and $n_e=0.1$ (b) to illustrate the interference in the order parameter. Calculations are performed within $t\text{−}\mathrm{BdG}$ [(a) and (b)] and within $a\text{−}\mathrm{BdG}$ [(b) and (c)].

Figure 9

Excitation spectrum $E_{\alpha }$ as a function of the quasimomentum $k_{\alpha }$, calculated at the zero temperature within $t\text{−}\mathrm{BdG}$ (circles), $a\text{−}\mathrm{BdG}$ (rhombus) and $c\text{−}\mathrm{BdG}$ (stars). The calculations are done for $n_e=1$ (a) and $n_e=0.1$ (b), the other microscopic parameters are the same as in the previous figure.

Figure 10

${\mathrm{\Delta }}_j$ versus $j$, as obtained with the Anderson approximation for different selected temperatures at $n_e=1$. Other parameters are taken the same as in Fig. 2.

Figure 11

The zero-temperature cumulative order parameter ${\mathrm{\Delta }}_j^{\left(E\right)}$ versus $j$, plotted for different ascending values of $0<E<\hslash {\omega }_D$ in the same manner as in Fig. 7. (a) is for $n_e=1$ (a) and (b) corresponds to $n_e=0.1$, the other microscopic parameters are the same as in the previous figure. Obtained within the $a\text{−}\mathrm{BdG}$ approximation.

Figure 12

Spatial profile of the order parameter ${\mathrm{\Delta }}_j$, calculated within the $c\text{−}\mathrm{BdG}$ model for different temperatures at $n_e=1$ (the other model parameters are the same as in Fig. 2.

Figure 13

The cumulative order parameter ${\mathrm{\Delta }}_j^{\left(E\right)}$ for different ascending values of $0<E<\hslash {\omega }_D$ (in the manner of Fig. 7). Obtained within the zero-temperature $c\text{−}\mathrm{BdG}$ for $n_e=1$, the other model parameters are the same as in the previous figure.

Figure 14

(a) The ratio $T_{cS}/T_{cB}$ as a function of the disorder strength $\lambda$, as calculated for the model with the surface disorder at different values of coupling constant. (b) Contour plot of the amplitude spectral density, $\left|{\mathrm{\Delta }}_f^{\left(\alpha \right)}\right|$, shown on $f-\alpha$ plane, calculated for $n_e=1$ and the disorder strength $\lambda =0.8$. Calculations are performed within $t\text{−}\mathrm{BdG}$.