Magnetoresistance Anisotropy of a One-Dimensional Superconducting Niobium Strip

We investigated confinement effects on the resistive anisotropy of a superconducting niobium strip with a rectangular cross section. When its transverse dimensions are comparable to the superconducting coherence length, the angle dependent magnetoresistances at a fixed temperature can be scaled as $R(\theta ,H)=R(H/H_{c\theta })$ where $H_{c\theta }=H_{c0}({cos}^2\theta +{\gamma }^{-2}{sin}^2\theta {)}^{-1/2}$ is the angular dependent critical field, $\gamma$ is the width to thickness ratio, and $H_{c0}$ is the critical field in the thickness direction at $\theta =0°$. The results can be understood in terms of the anisotropic diamagnetic energy for a given field in a one-dimensional superconductor.

Figure 1

Magnetic field versus temperature ($H\mathrm{\text{−}}T$) phase diagrams of sample $A$. The symbols are experimental data and solid lines represent the parabolic fits (see text). Lower inset is a SEM micrograph of the strip and upper inset presents schematic of the width $\mathit{w}$ and thickness $\mathit{d}$ directions, the coordinates and the angle $\theta$ for the field direction. $\theta =0°$ and 90° corresponds to $\mathit{H}\parallel \mathit{d}$ and $\mathit{H}\parallel \mathit{w}$, respectively.

Figure 2

Scaling behavior of the anisotropic resistance of sample $A$ at 6.90 K. (a) for $R(H)$ curves at various field directions and (b) for $R(\theta )$ curves in various fixed fields. The main panels present the data of $R$ versus $H({cos}^2\theta +{\gamma }^{-2}{sin}^2\theta {)}^{1/2}$ with $\gamma =1.52$, and the original data are presented in the insets.

Figure 3

Scaling behavior of the anisotropic resistance of sample $A$. (a) $R(\theta )$ curves at a fixed temperature (6.90 K) and in various fixed fields $H$. (b) $R(\theta )$ curves in a fixed field $H$ (0.4 T) and at various fixed temperatures. The solid lines are directly measured data and symbols are values derived from the $R(H_0)$ curves obtained at $\theta =0°$ (see text).

Figure 4

Angle dependence of the critical field $H_{c\theta }$. (a) at 6.90 K for sample $A$ and 7.60 K for sample $B$. (b) at 6.0 K for sample $A$. The inset of (b) compares $H_{c\theta }$ curves of sample $A$ and the reference film at the same reduced temperature ($T/T_{c0}=0.98$). $H_{c\theta }$ is normalized to $H_{c0}$, the value at $\theta =0°$. The dashed lines and solid lines are fits with Eqs. (1, 2) (for 2D and 1D), respectively.