Geometric quenching of orbital pair breaking in a single crystalline superconducting nanomesh network

In a superconductor Cooper pairs condense into a single state and in so doing support dissipation free charge flow and perfect diamagnetism. In a magnetic field the minimum kinetic energy of the Cooper pairs increases, producing an orbital pair breaking effect. We show that it is possible to significantly quench the orbital pair breaking effect for both parallel and perpendicular magnetic fields in a thin film superconductor with lateral nanostructure on a length scale smaller than the magnetic length. By growing an ultra-thin (2 nm thick) single crystalline Pb nanowire network, we establish nm scale lateral structure without introducing weak links. Our network suppresses orbital pair breaking for both perpendicular and in-plane fields with a negligible reduction in zero-field resistive critical temperatures. Our study opens a frontier in nanoscale superconductivity by providing a strategy for maintaining pairing in strong field environments in all directions with important technological implications.


Supplementary Note 1. Atomic images on Pb wires and in void area
The atomic image shown in Supplementary Fig. 1a for Pb (111) surface always has the same orientation with respect to the Si(111) substrate (albeit with a 10% lattice mismatch). Since the Si substrate being used is single crystalline, single crystallinity of the nanomesh is well maintained.

Supplementary Note 2. The saturation behavior of Im( ) and the quantification of superfluid density
To the zeroth order, the imaginary part of the corresponds to the imaginary part of sheet conductivity (σ1+iσ2) which has been used to determine the superfluid density (SFD). Quantification of SFD requires accurate measurements of σ2(T) over a large temperature range. The saturation behavior of due to the limitation of the instrument response, prevents direct quantification of SFD. Nevertheless, we found that − ( )⁄ can be used to infer SFD. This is based on the observation that as the SFD increases, the σ2 signal reaches saturation region within a smaller temperature window below Tc. This suggest that the slope of can be used to estimate SFD. Indeed, measurement on atomically smooth Pb film, show linear dependence of − ( )⁄ with respect to the film thickness (shown in Supplementary Fig. 2a Since Ge has a small energy gap of 0.7 eV and a low effective mass, the effective tunneling length should be significantly longer than the tunneling decay length in vacuum. Since we have not found the quantitative estimate in the literature, we can estimate the tunneling decay length based on the barrier height (taken at value of 0.35 eV, half of the gap) and an estimated effective mass of 0.3 which will yield a decay constant of ~ 0.16 Å -1 . This is about 1/6 of the decay length in vacuum tunneling. Thus, the tunneling probability across a 3 nm Ge barrier would be similar to the tunneling probability across a 0.5 nm vacuum barrier. Taking a typical vacuum tunneling gap distance of 1 nm, tunneling through a 3 nm Ge-capped sample would not be a problem in an STM step up. We further experimentally verify this conclusion by carrying out STS of a 5 ML-Pb film before and after in-situ capping in UHV. Shown in Supplementary Fig. 3 is a comparison of the STS measured at 4.2 K before and after Ge-capping. As can be seen, the tunneling spectra are nearly identical.

Supplementary Note 4. Superconducting gap fitting
We fit our superconducting spectrum by the tunneling equation, which includes the Fermi-Dirac distribution function. We also include a phenomenological broadening parameter, which is introduced by Dyne et al. 1 so that E E+i. Here,  is for the pairing breaking effect due to either the magnetic field or temperature as shown in our fitting and other systems under magnetic field 2, 3 . A rigorous theoretical description on "Dynes formula" in the presence of magnetic field has been discussed in Ref. 4. The values Δ and Γ as a function of magnetic field are listed in the Supplementary Table 1.

Supplementary Note 5. Generation of a 2D Gaussian random field 5
For a Gaussian random field the joint distribution function is where = ( 1 , 2 , … , ) represents the values of the random field at all spatial positions. The vector is the mean and the matrix is the covariance: Note that a Gaussian random field is completely determined by and . Moreover, for a 2D stationary Gaussian field with zero mean is identically zero, and is only dependent on the distance between and , = − . However, it should be noted that ≡ ( ) can take various forms (i.e., it is not necessarily a Gaussian function). The common covariance functions are White noise : Since we need a spatial correlation length we are going to use the Gaussian covariance function. Using this covariance function ( ) we can get the spectral density ( ) as which is a Gaussian function with variance −2 . Therefore if a random, uncorrelated, zero mean ( ) are generated with the variance ( ) above, their inverse Fourier transform will give a correlated random Gaussian field ( ) with correlation length .
In numerical generation of random fields one has to stick with finite random fields on a discrete grid. Moreover, since the spectral representation of a stationary random field is only defined for an infinite domain, it is assumed that the random field is periodic. A × grid in real space will correspond to a × grid in the frequency space. On this grid we have a Fourier series where is the normal state free energy, is the magnetic field induced by the supercurrents and is the vector potential with ∇ × = + , where is the applied By letting the variation of vanish we obtain the Ginzburg-Landau equations: Using the definition of above equation can be rewritten as which is still an eigenvalue problem but with a spatially dependent potential term

Supplementary Note 7. Numerical methods 6
Assuming a 2D square lattice, we introduce the gauge variables At the boundary of a finite system the net supercurrent must not have any component perpendicular to the boundary, which indicates where ̂ is a unit vector normal to the surface at the position labeled by subscript . In terms of the gauge variables introduced above this can be written as To make direct comparison with experimental results, we choose a critical value c of the disorder potential, so that the total area of the regions with > c is equal to the desired coverage of Pb. Moreover, we note that according to experiments the SIC phase has a c ≈ 1.85 K, which is about 31% of the c of a homogeneous Pb film. Therefore we replace the disorder potential by ̃, defined by This approximation is valid as long as the thickness of the interface between the Pb and the SIC regions is much smaller than the disorder correlation length l and the coherence length 0 , which is the case in the experiments.

Supplementary Note 8. For a regular network formed by periodically modulated pairing potential
We emphasize here again that the mechanisms for quenching the orbital pairbreaking effect of a perpendicular magnetic field are different at different length scales.
When the magnetic length is much larger than the size of the voids, the magnetic field still sees the whole mesh as a percolated 2D system, and the orbital-pair breaking is through formation of circulating supercurrents in the mesh network. For a regular network formed by periodically modulated pairing potential, the long-wavelength behavior of the linearized Ginzburg-Landau equation is still similar to that of an ordinary 2D electron gas. Therefore the pair-breaking parameter for a perpendicular field is linear in field which leads to the linear dependence of Tc on ⊥ . This is shown explicitly in Supplementary Figure 4. At small fields when the magnetic length is much larger than the modulation period, one still has the expected linear behavior. The nontrivial dependence of Tc on ⊥ only appears when the magnetic length is comparable to the period. Thus at the superconducting-normal phase boundary in the H-T phase diagram, quenching of the orbital pair-breaking effect at high temperatures must be due to deviation of the long-wavelength behavior of the network from that of ordinary 2DEG. This is the reason why the randomness of the mesh network is critical in this regime. As one moves along the phase boundary to lower temperatures, the magnetic length gradually becomes shorter than the voids, but still larger than the wire width, so that closed loops of supercurrents cannot form by going through several links surrounding a void. This is when the narrow width of the wires quenches the orbital pairbreaking effect, similar to that of a parallel magnetic field in a thin film superconductor.