Local destruction of superconductivity by non-magnetic impurities in mesoscopic iron-based superconductors

The determination of the pairing symmetry is one of the most crucial issues for the iron-based superconductors, for which various scenarios are discussed controversially. Non-magnetic impurity substitution is one of the most promising approaches to address the issue, because the pair-breaking mechanism from the non-magnetic impurities should be different for various models. Previous substitution experiments demonstrated that the non-magnetic zinc can suppress the superconductivity of various iron-based superconductors. Here we demonstrate the local destruction of superconductivity by non-magnetic zinc impurities in Ba0.5K0.5Fe2As2 by exploring phase-slip phenomena in a mesoscopic structure with 119 × 102 nm2 cross-section. The impurities suppress superconductivity in a three-dimensional ‘Swiss cheese'-like pattern with in-plane and out-of-plane characteristic lengths slightly below ∼1.34 nm. This causes the superconducting order parameter to vary along abundant narrow channels with effective cross-section of a few square nanometres. The local destruction of superconductivity can be related to Cooper pair breaking by non-magnetic impurities.

Supplementary Figure 9. Electrical potential as function of the applied current for a one dimensional superconductor with length, L = 150ξ 0 . Right to left (left to right) arrows indicate that the I-V curve below the arrows was taken starting at a high (low) value of current, slightly above critical current (much below critical current).

Supplementary Note 1: phase-slips in superconducting BKZn microbridges
At 21 K, the voltage state of the 141.4 nm thick microbridge (see Supplementary   Figure 7(a)) is switched from the superconducting to the normal state directly with absence of flux-flow process, and the critical current density J c shows a high value of 4.3 MA/cm 2 , which is close to the Ginzburg-Landau depairing current density GL dp J [1]. When the current is swept down, the voltage retraps from the normal to superconducting states, followed by a hysteretic resistance state. After slightly increasing T to 22 K, current sweep-up leads to voltage jumps from superconducting to the hysteretic resistance state with a slight flux-flow [4], and then to the normal state. The returning branch locates on the hysteretic resistance  [5] and Pekker et al. [6]).

Supplementary Note 2: simulations on the phase-slips in superconducting nanowires
A nanowire with the width (W) and height (h) of the order of ξ(T) and length L >> ξ, can be consider as nearly one-dimensional object. If we apply to it a current slightly below critical current we may obtain a phase slippage that can be observed as a jump in the I-V curves [7]. This is called a phase slip center since it is located on a point of the nearly one-dimensional nanowire [7]. In thicker nanowires, where W >> ξ(T), phase slips were also observed when W<< λ eff [8] which present similar I-V curves to the ones of phase slip centers.
These are phase slips lines along which the order parameter is zero. One of the most This creates a geometry/topology that resembles a "Swiss cheese" which gave the name to this model. In case of an applied current, the superconducting electrical charges are transported along 1D percolation channels shaped by the impurities in the samples.
To study phase slips, we considering a similar model to the one presented in Ref. [10], that we will further describe. We solved the dimensionless generalized time dependent Ginzburg-Landau [11], in a one-dimensional line of length L. In this equation Ψ is the order parameter, φ is the electrical potential,  Ais the magnetic field, i is the imaginary constant, and γ and u are two phenomenological parameters related with the electron-phonon relaxation time and the cleanness/dirtiness of the sample, respectively. The generalized time-dependent Ginzburg-Landau equation is complemented by the equation for the electric field: In these equations the length is given in units of  (0)  8k B T c  D , where, T in units of T c , Ψ in units of (0)  4k B T c u , (bulk value at zero temperature), φ in units of Additionally "bridge" boundary conditions were applied to the ends of the one-dimensional line, i.e.
and initial conditions were set  x,0    1 T and (x,0)  0 .
Simulations were made for parameters, L = 150ξ 0 , u = 5.79, T = 0, γ= 40. The value of γ is correlated with the relaxation time of the pairing mechanism. For this iron-based superconductor γ = 40 corresponds to a realistic order of magnitude for this parameter. Figure 8 presents a snapshot the order parameter distribution and the electrical potential along the domain. The order parameter is highly depleted in a fix number of points, and as times passes the value of the order parameter in these points diminishes until it reaches zero. At this time a phase slip center is formed and later the value of the order parameter becomes finite and increases until it reaches a maximum value, marking the end of another periodic cycle. In Supplementary Figure 9, we can observe I-V curves taken from simulation for a fixed applied current. We note that in these units j c = 0.38j 0 . If we apply a current above critical current the order parameter starts to be spatially modulated and then later phase slips centers nucleate in the middle of the sample (due to symmetry reasons), and afterwards these centers become equally spaced and appear periodically in time. As we decrease current the period of oscillation becomes longer and the number of phase slips becomes lower until it reaches a critical value where all the phase slips disappear. However,

Supplementary
if we start by applying a low current (much below critical current) no signs of phase slips are shown and as we increase current, at some critical value, phase slip centers start to nucleate near the boundary of the sample. As the phase slip centers enter the sample a sudden increase on voltage is observed, as in an avalanche process, in opposition to what is observed in the process going from high to lower currents where the voltage drop with the current decrease is a much smoother process. We also experimented coupling the set of equations to the temperature diffusion equation, in order to observe the influence of the thermal conductivity. In these equations, where h c is the heat transfer coefficient, T 0 is the applied temperature, C s , C f , k s and k f are the heat capacity and heat conductivity of the substrate and of the sample, respectively. The initial and boundary conditions applied to this equation were T(x, 0) = T 0 and T(±L/2, t) = T 0 .
In preliminary results, we observed an increase in the jumps when considering this equation.
Furthermore, ref. [10] and Ref. [12] present studies on the behavior of phase slip centers when considering: coupling with the heat diffusion equation, non-uniform values of T c and of width of the wire, the presence of defected. Fluctuation processes, such as thermal fluctuations, should be also taken into account on later studies, yet it is expected that they help nucleate phase slips centers that are, then, maintained by the applied electrical current.
Comparing the theoretical study with the experimental curves, we conclude that the hysteretic and stair like behavior presented in the experimental I -V curves are watermark indications for phase slips centers/lines. However, from the simulation we could not reproduced the big steps in the I -V curves of Figure 2 of the main text. These big steps in the low to higher current and high to lower current curves are likely related to pinning of phase slip centers that increase the step characteristic of the I -V (with higher steps and larger plateaus) due the avalanche process of entering and leaving of phase slip centers.