Discrete transverse superconducting modes in nanocylinders

Spatial variation in the superconducting order parameter becomes significant when the system is confined at dimensions well below the typical superconducting coherence length. Motivated by recent experimental success in growing single-crystal metallic nanorods, we study quantum confinement effects on superconductivity in a cylindrical nanowire in the clean limit. For large diameters, where the transverse level spacing is smaller than the superconducting order parameter, the usual approximations of Ginzburg-Landau theory are recovered. However, under external magnetic field the order parameter develops a spatial variation much stronger than that predicted by Ginzburg-Landau theory, and gapless superconductivity is obtained above a certain field strength. At small diameters, the discrete nature of the transverse modes produces significant spatial variations in the order parameter with increased average magnitude and multiple shoulders in the magnetic response.

Figure 1

Averaged superconducting order parameter in a cylindrical sample of large diameter $\left(D=200\mathrm{nm},D∕{\xi }_0=0.57\right)$ with axial external magnetic fields. Normalized order parameter to the zero field value $\Delta ∕{\Delta }_0$ is plotted as a function of dimensionless field $HR∕\left(H_{\xi }{\xi }_0\right)$ with $H_{\xi }=2c{\Delta }_0∕\left(e{v}_F{\xi }_0\right)$. Overall agreement of the analytic formula (18) and the numerical results are good. The solution of the Bogoliubov–de Gennes (BdG) equation in the clean limit remains constant until $H_1=2c{\Delta }_0∕e{v}_{F}R$. The full solution of the BdG equation (solid circles) has larger order parameters near the critical field than those constrained to a spatially uniform $\Delta$ (open circle), due to spatial adjustments of the superconducting wave function.

Figure 2

Density of states as a function of external field. The superconductor becomes gapless at $H=H_1$ (dashed line) in the clean limit well before the critical field $H_c$, due to the coupling of orbital angular momenta to external field. The external fields in the plot are from $HR∕\left(H_{\xi }{\xi }_0\right)=0$ (thick line) to 1.83 with equal intervals between the curves.

Figure 3

Spatial variation of order parameter $\Delta$ in the large diameter limit $\left(D∕{\xi }_0=0.57\right)$. For external fields of $0<H<H_1$, $\Delta$ remains flat. As the field increases from $H_1$, $\Delta$ drops with a knee which progresses towards $\mathbf{r}=0$.

Figure 4

The order parameter $\Delta$ as a function of the radius at small diameter $D∕{\xi }_0=0.057$ $\left(\delta R=2\mathrm{\AA }\right)$. The spatial variation of $\Delta$ is much stronger than for the large diameter wires of Fig. 3.

Figure 5

Schematic two-dimensional phase space with zero axial wave vector $\left(k_z=0\right)$. Transverse modes within the shell of thickness $\delta k_{\perp }$ contribute strongly to the order parameter. The area of the shell $2\pi k_F\delta k_{\perp }\sim 4\pi m^{*}{\omega }_D$ does not depend on the choice of Fermi energy.

Figure 6

Average order parameters $\Delta$ as a function of diameter for different radius smearings $\delta R$. Filled circles are the full BdG solution while empty circles represent the constrained case where spatial variation of $\Delta$ is disallowed. Confinement effects appear at $D∕{\xi }_0\sim 0.1$, i.e., when the transverse energy level spacing $\delta E\approx \Delta$. $\Delta$ converges to about $4\mathrm{K}$ as the diameter increases, regardless of $\delta R$. In the full solution, $\Delta$ increases slightly as $D∕{\xi }_0$ drops below about 0.1. Compared to the constant-$\Delta$ behavior, this enhanced order parameter takes advantage of the spatial variation in the BdG solution.

Figure 7

The order parameter $\Delta$ as a function of the external field at a small diameter $D∕{\xi }_0=0.057-0.086$ ($D=20–30\mathrm{nm}$, $\delta R=2\mathrm{\AA }$). $\Delta$ displays several shoulders as the field increases and then vanishes abruptly, unlike the large diameter case of Fig. 1. This structure reflects the discrete nature of the transverse modes. The sudden cutoff of $\Delta$ at the critical field is due to an absence of small angular momentum transverse states near the chemical potential. The dashed line is Eq. (18).

Figure 8

The critical field vs rod diameter. The critical field $H_c$ varies inversely proportional to the diameter $D$. The solutions of the full BdG equation follow the general trend of Eq. (19), which was obtained from the BCS equation under the constraint of a spatially constant order parameter. The full BdG solution satisfies the $1∕R$ prediction down to fairly small diameters, far smaller than coherence length, although they fluctuate strongly about the relation (19).