Number theory, periodic orbits, and superconductivity in nanocubes

We study superconductivity in isolated superconducting nanocubes and nanosquares of size $L$ in the limit of negligible disorder $\delta /{\Delta }_0\ll 1$ and $k_{F}L\gg 1$ for which mean-field theory and semiclassical techniques are applicable, with $k_F$ the Fermi wave vector, $\delta$ the mean level spacing, and ${\Delta }_0$ the bulk gap. By using periodic orbit theory and number theory we find explicit analytical expressions for the size dependence of the superconducting order parameter. Our formalism takes into account contributions from both the spectral density and the interaction matrix elements in a basis of one-body eigenstates. The leading size dependence of the energy gap in three dimensions seems to be universal as it agrees with the result for chaotic grains. In the region of parameters corresponding to conventional metallic superconductors, and for sizes $L\gtrsim 10$ nm, the contribution to the superconducting gap from the matrix elements is substantial $(\sim$$20\%)$. Deviations from the bulk limit are still clearly observed even for comparatively large grains $L\sim 50$ nm. These analytical results are in excellent agreement with the numerical solution of the mean-field gap equation.

Figure 1

The matrix elements Eq. (8) as a function of the square size $L$. Blue dots correspond to the exact numerical calculation, the blue solid line shows the numerical average of the exact results, and the red dashed line corresponds to the analytic prediction, Eq. (16). For sufficiently large grains $L>20$ nm the agreement is excellent. For small sizes deviations are expected since our results neglect higher order terms in the expansion parameter ${\left(k_{F}L\right)}^{-1}$.

Figure 2

The matrix elements Eq. (8) as a function of the square size $L$. Blue dots correspond to the exact numerical calculation, the blue solid line shows the numerical average of the exact results, and the red dashed line corresponds to the analytic prediction, Eq. (23). Since the analytical calculation does not neglect any term in the expansion the agreement is much better than for the cube across the whole size range.

Figure 3

Comparison of the numerical and analytical calculation of ${\Delta }_{\mathrm{Diff}}$, the difference between the superconducting gap with and without matrix elements. The upper plot shows the mean value and the lower plot the standard deviation taken over consecutive intervals of size $0.4$ nm. The solid line shows numerical results and the dashed line the results from the periodic orbit calculation, Eq. (31). From top to bottom the line pairs correspond to $\lambda =0.2$ (green), $0.3$ (red), and $0.4$ (blue). We note the extremely good agreement not just in the line shape but also in the fine structure of the standard deviation. The contribution of the matrix elements to the gap size dependence is substantial in the region $L\sim 10$ nm and where the BCS formalism is still applicable. Deviations are still noticeable even for much larger grains $L\le 50$ nm. As was expected (see text) the expansion begins to breakdown for the case of $\lambda =0.2,L\sim 10$ nm.