Critical field behavior of a multiply connected superconductor in a tilted magnetic field

We report magnetotransport measurements of the critical field behavior of thin Al films deposited onto multiply connected substrates. The substrates were fabricated via a standard electrochemical process that produced a triangular array of 66-nm-diameter holes having a lattice constant of 100 nm. The critical field transition of the Al films was measured near $T_c$ as a function of field orientation relative to the substrate normal. With the field oriented along the normal ($\theta =0$), we observe reentrant superconductivity at a characteristic matching field $H_m=0.22$ T, corresponding to one flux quantum per hole. In tilted fields, the position $H^{*}$ of the reentrance feature increases as $sec\left(\theta \right)$, but the resistivity traces are somewhat more complex than those of a continuous superconducting film. We show that when the tilt angle is tuned such that $H^{*}$ is of the order of the upper critical field $H_c$, the entire critical region is dominated by the enhanced dissipation associated with a submatching perpendicular component of the applied field. At higher tilt angles a local maximum in the critical field is observed when the perpendicular component of the field is equal to the matching field.

Figure 1

Plot of the critical field transition of a 9-nm-thick Al film on glass as a function of the angle between the applied field and the normal to the film surface.

Figure 2

Resistive critical field transitions of a 9-nm-thick Al film on a nanohoneycomb substrate near $T_c=2.27$ K. The dips in the traces occur when the perpendicular component of the applied field equals the matching field $H_m=0.22$ T. The dip location is a function of angle and is denoted by $H^{*}$. The figure legend represents the angle between the magnetic field and the normal to the face of the substrate.

Figure 3

Resistance as a function of perpendicular field of a 9-nm-thick Al film on glass and a 9-nm-thick Al film on an AAO substrate.

Figure 4

Black symbols: Midpoint critical field values for a 9-nm film on glass (see data in Fig. 1) as a function of tilt angle. The dashed line represents the predicted angular dependence of the Tinkham formula, Eq. (2). Orange symbols: Midpoint critical field values for a 9-nm Al film on an AAO substrate (see data in Fig. 2). The dashed line represents the predicted angular dependence of the Tinkham formula.

Figure 5

The matching fields from the data in Fig. 2 as a function of $sec\left(\theta \right)$. The dashed line represents a linear least-squares fit to the data. The slope of the fit corresponds to a matching field of $H_m=0.24$ T. Inset: Critical field transitions of Fig. 2 plotted as a function of the perpendicular component of the magnetic field.

Figure 6

Red symbols: Midpoint critical field values for the 9-nm film on AAO used in Fig. 2. These data were taken at 1.83 K. Blue symbols: Critical field values defined by the field at which the resistance is 10% of the normal-state resistance. The dashed lines represent the predicted angular dependence of the 1D and 2D Tinkham formulas; see text. Inset: Resistive critical field transitions of the AAO sample from Fig. 2 taken at $T=1.83$ K. The reentrant features are no longer present at this lower temperature.