Anisotropy of the critical temperature of a superconducting niobium thin film with an array of nanoscale holes in an external magnetic field

We report on magnetotransport experiments investigating the effect of a regular array of nanoscale holes on the anisotropic response in the transition temperature of a superconducting niobium thin film. We find that the angle dependence of the critical temperature exhibits two strong anisotropic effects: Little-Parks oscillations whose period varies with field direction and a smooth background arising from one-dimensional confinement by the finite lateral space between neighboring holes. The two components of the anisotropy are intrinsically linked and appear in concert with one superimposed on top of the other.

Figure 1

SEM micrograph of sample A. The inset shows definitions of the magnetic field direction $\theta$, film thickness $t$, and lattice constant $D$ of the hole array.

Figure 2

$T_c\left(\theta \right)$ obtained at various fixed magnetic fields for sample A (upper panel) and sample B (lower panel). The curves are measured values and the symbols are calculated based on an analysis procedure described in the text. For clarification we showed only the calculated data at 900 G for sample B. Inset (a) presents $T_c\left(\theta \right)$ at 1 T for sample A, demonstrating the failure of the scaling at low $T$ $(<T^{*})$. Inset (b) shows comparisons of $T_c\left(\theta \right)$ data from sample B and the reference film measured at $H=900$ G.

Figure 3

(a) $T_c\left(H\right)$ data of sample A for $\theta =0^{\circ }$ (open circles) and ${90}^{\circ }$ (solid squares). The open squares delineate an upper bound for the $T_c\left(H_{\perp }\right)$ curve and were obtained by multiplying the $H_{\parallel }$ values with a scaling factor of $0.587$. The green line represents the 1D fit to this upper-bound background data. Inset shows $\Delta T_c\left(H_{\perp }\right)$ by subtracting the green line from the experimental $T_c\left(H_{\perp }\right)$ curve. (b) $T_c\left(H\right)$ data for $\theta =0^{\circ }$, ${30}^{\circ }$, ${45}^{\circ }$, ${60}^{\circ }$, ${75}^{\circ }$, and ${90}^{\circ }$ for sample A. The inset shows the angular dependence of the oscillation period $H_1$.

Figure 4

(a) Angle dependence of $T_c\left(\theta \right)$ and $\Delta T_c\left(\theta \right)$ obtained in a magnetic field of 2500 G: The red circles represent $\Delta T_c\left(\theta \right)$ derived from $\Delta T_c\left(H_{\perp }\right)$ by converting $H_{\perp }$ to $\theta$ through a relation $\theta =\arccos (H_{\perp }/H)$; the blue curve represents the calculated $T_c\left(\theta \right)$ curve without the hole contribution; the blue circles are the sum of the red circles and the blue curve while the open purple squares are the experimentally measured values. (b) Scaling behavior of the $T_c\left(H\right)$ data in Fig. 3b (the same symbols). Inset shows the angular dependence of the scaling factor and fits with formulas for thin film (dashed line) and 1D strip (solid lines).