Phase slips in superconducting films with constrictions

A system of two coplanar superconducting films seamlessly connected by a bridge is studied. We observe two distinct resistive transitions as the temperature is reduced. The first one, occurring in the films, shows some properties of the Berezinskii-Kosterlitz-Thouless (BKT) transition. The second apparent transition (which is in fact a crossover) is related to freezing out of thermally activated phase slips (TAPS) localized on the bridge. We also propose a powerful indirect experimental method allowing an extraction of the sample’s zero-bias resistance from high-current-bias measurements. Using direct and indirect measurements, we have determined the resistance $R\left(T\right)$ of the bridges within a range of eleven orders of magnitude. Over such broad range the resistance follows a simple relation $R\left(T\right)=R_N\exp [-(c∕t){(1-t)}^{3∕2}]$, where $c=\Delta F\left(0\right)∕k{T}_c$ is the normalized free energy of a phase slip at zero temperature, $t=T∕T_c$ is normalized temperature, and $R_N$ is the normal resistance of the bridge.

Figure 1

(a) Sample schematic. MoGe film (black) of thickness $d=2.5–4.5\mathrm{nm}$ is deposited over a SiN membrane (gray) substrate with a constriction of width $w$. (b) An SEM micrograph of a typical sample. The MoGe coated SiN bridge (gray) is suspended over a deep trench (black).

Figure 2

Low-bias resistance versus temperature dependence (open circles), measured on a thin film $(d=3.5\mathrm{nm})$ without constriction. The solid line is a fit to the Halperin-Nelson theory [Eq. (1)]. Inset: Resistance of the film without constriction (multiplied by a constant factor), shown as open circles, is compared to the sample with a hyperbolic bridge (B2), shown by the solid line. The only qualitative difference is the presence of a “resistive tail,” observed on all samples with constrictions.

Figure 3

Low-bias resistance for five different samples with bridges. The parameters of the samples are given in Table . The data points are shown by open symbols. Solid lines are fits to the “bridge” phase slip model given by Eqs. (7, 9).

Figure 4

Differential resistance as a function of the dc bias current for sample B2. Experimental data are denoted by open symbols and the solid lines are fits to $dV∕dI=R\left(T\right)\cosh (I∕I_0)$. Temperatures from left to right are 4.12, 3.92, 3.80, 3.64, 3.45, 3.36, 3.26, 3.16, 3.07, 2.80, and $2.68\mathrm{K}$.

Figure 5

Resistance vs temperature curve for sample B2. Open circles represent data that have been directly measured while filled boxes give the resistance values determined by fitting the $dV∕dI$ curves of Fig. 4 using the formula $dV∕dI=R\left(T\right)\cosh (I∕I_0)$. The solid and the dashed curves give the best fits generated by the $R_{\mathrm{WL}}\left(T\right)$ $(T_c=4.81\mathrm{K})$ and $R_{\mathrm{LAMH}}\left(T\right)$ $(T_c=5.38\mathrm{K})$ formulas, respectively.