Interference-induced surface superconductivity: Enhancement by tuning the Debye energy

In the usual perception, surface superconductivity is associated with the surface nucleation of a superconducting condensate above the upper critical field in type-II superconductors or with a rearrangement of phonon properties and the electron-phonon coupling near surfaces/interfaces. Recently, it has been found that there is another example when the surface superconducting temperature is increased up to $20–25\%$ as compared to the bulk one due to constructive interference of superconducting pair states. In the present work, we demonstrate that in fact, such an interference-induced enhancement can be much more pronounced, up to nearly $70\%$. Furthermore, here it is shown that such an interference enhancement persists over a wide range of microscopic parameters.

Figure 1

(a),(b), and (c) The pair potential ${\mathrm{\Delta }}_i$ versus the site number $i$, calculated at the bulk critical temperature. (d),(e), and (f) The pair potentials at the edge (surface) ${\mathrm{\Delta }}_1$ and in the center of the chain (bulk) ${\mathrm{\Delta }}_{(N+1)/2}$ versus the temperature. The calculations are done at $\hslash {\omega }_{\mathrm{D}}=1.5$ (a),(d); 1.8 (b),(e); and 2.1 (c),(f) for $g=2$ and $n_e=1$, other parameters are discussed in the text.

Figure 2

(a) The difference of $T_{\mathrm{cs}}$ and $T_{\mathrm{cb}}$ in units of $T_{\mathrm{cb}}$ as a function of $\hslash {\omega }_{\mathrm{D}}$. (b) $T_{\mathrm{cs}}$ and $T_{\mathrm{cb}}$ versus $\hslash {\omega }_{\mathrm{D}}$. The microscopic parameters are the same as in Fig. 1.

Figure 3

The cumulative pair potentials ${\mathrm{\Delta }}_s^{\left(E\right)}\equiv {\mathrm{\Delta }}_1^{\left(E\right)}$ and ${\mathrm{\Delta }}_b^{\left(E\right)}\equiv {\mathrm{\Delta }}_{(N+1)/2}^{\left(E\right)}$ calculated for $\hslash {\omega }_{\mathrm{D}}=1.5$, 1.8, and 2. Panel (a) corresponds to $T=0.5T_{\mathrm{cb}}$, and panel (b) is for $T=1.0T_{\mathrm{cb}}$. Other parameters are the same as in Figs. 1 and 2.

Figure 4

$(T_{\mathrm{cs}}-T_{\mathrm{cb}})/T_{\mathrm{cb}}, T_{\mathrm{cs}}$, and $T_{\mathrm{cb}}$ as functions of $\hslash {\omega }_{\mathrm{D}}$ at $g=1.5$ (a),(d); 2.5 (b),(e); and 3.5 (c),(f) for the half-filling case.

Figure 5

Beyond the half-filling: $(T_{\mathrm{cs}}-T_{\mathrm{cb}})/T_{\mathrm{cb}}, T_{\mathrm{cs}}$, and $T_{\mathrm{cb}}$ as functions of $\hslash {\omega }_{\mathrm{D}}$ at $n_e=0.9$ (a),(c) and $n_e=0.95$ (b),(d). The coupling is chosen as $g=2$.

Figure 6

(a) The site-dependent order parameter and local electron density at $T=1.1T_{\mathrm{cb}},n_e=1,\hslash {\omega }_D=1.5$, and $g=2$; (b) and (c) demonstrate the zero-bias and energy dependent normal LDOS ($g=0$) for the same $T$ and $n_e$ as in panel (a).