Magnetic field induced emergent inhomogeneity in a superconducting film with weak and homogeneous disorder

Abstract: When a magnetic field is applied, the mixed state of a conventional Type II superconductor gets destroyed at the upper critical field Hc2, where the normal vortex cores overlap with each other. Here, we show that in the presence weak and homogeneous disorder the destruction of superconductivity with field follows a different route. Starting with a weakly disordered NbN thin film ( Tc ~ 9K ), we show that under the application of magnetic field the superconducting state becomes increasingly granular, where lines of vortices separate the superconducting islands. Consequently, phase fluctuations between these islands give rise to a field induced pseudogap phase, which has a gap in the electronic density of states but where the global zero resistance state is destroyed.

• The paper demonstrates that weak, homogeneous disorder combined with an applied magnetic field nucleates nanoscale granularity, fragmenting superconductivity in NbN films.
• Experimental STS measurements reveal suppressed coherence peaks and a persistent soft gap in vortex cores, which challenges the classical Abrikosov vortex picture.
• Self-consistent BdG simulations correlate disorder with an emergent pseudogap regime, offering fresh insights into the superconductor-insulator transition.

Magnetic Field-Induced Emergent Inhomogeneity in Weakly Disordered Superconducting Films

Introduction

The investigated work addresses the evolution of superconductivity in conventional Type II NbN thin films with weak, homogeneous disorder as a function of applied magnetic field. Historically, the destruction of superconductivity with field has been described by the overlap and percolation of normal vortex cores, leading to a loss of global superconductivity at $H_{c2}$. This study demonstrates that, contrary to the classical picture, even weak disorder fundamentally alters the mechanism: The destruction of superconductivity proceeds through the nucleation of emergent nanoscale granularity under an external field, resulting in a fragmented superconducting landscape and a field-induced pseudogap regime.

Experimental Approach

Homogeneously disordered epitaxial NbN films were fabricated by pulsed laser deposition, resulting in $T_c \approx 9\,\mathrm{K}$. The level of disorder was tuned such that the zero-field superconducting transition coincides with the emergence of resistance, distinguishing the regime from strongly disordered samples which exhibit a pseudogap above $T_c$. Scanning tunneling spectroscopy (STS) at temperatures down to 350 mK and in fields up to 90 kOe enabled spatially resolved mapping of the local density of states (LDOS) and the superconducting gap structure. Complementary transport and magnetization studies established macroscopic superconducting properties and flux penetration characteristics.

Results: Superconducting State and Granularity

Zero-Field Homogeneity and Mesoscale Fluctuations

STS in zero field confirmed a spatially homogeneous gap with strong coherence peaks. However, the height of the coherence peaks (a proxy for local order parameter amplitude) showed smooth nanoscale inhomogeneity ($\mathcal{O}(10\,\mathrm{nm})$), in accord with theoretical predictions and Quantum Monte Carlo studies of disordered s-wave systems. The energy gap followed BCS-like temperature dependence, vanishing at $T_c$.

Field-Driven Inhomogeneity

Upon field application, the superconducting state exhibited emergent inhomogeneity not attributed simply to the overlap of regular Abrikosov vortex cores. Instead, vortices, as resolved by suppressed coherence peaks in the conductance maps, preferentially nucleated in areas of reduced local order parameter, further depressing superconducting correlations along their lines. The vortex structure was aperiodic, lacking the regularity of the Abrikosov lattice seen in cleaner systems; regions of suppressed superconductivity coalesced along the vortex paths, fragmenting the global superconducting manifold into weakly coupled "islands" separated by vortex lines.

Spectroscopically, the core regions did not revert to conventional metallic-like behavior with Caroli-de Gennes-Matricon (CDM) states, but instead maintained a soft gap, indicating the survival of local Cooper pairing amplitude despite loss of phase coherence.

Field-Induced Pseudogap Regime

Temperature-dependent STS revealed that in finite fields, the local gap survives to $T^* > T_c(H)$: a pseudogap regime emerges where the LDOS remains suppressed (indicative of local pairing), but zero resistance is lost due to phase decoherence across the fragmented superconducting network. The width of this pseudogap regime in the $H$-$T$ plane broadens with increasing field strength, analogous to the disorder-induced pseudogap that appears in the zero-field phase diagram of strongly disordered films.

These phenomena cannot be explained by simple local variation in $T_c$ due to disorder; the transition to $GN(0) \sim 1$ (indicative of gap closure) occurred concurrently in all spatial locations, suggesting a collective effect tied to the emergent mesoscale granularity and inter-island phase fluctuations.

Theoretical Modeling

Self-consistent Bogoliubov–de Gennes (BdG) calculations on an attractive Hubbard model with random onsite disorder were used to capture qualitative features. Simulations—even though employing higher interaction strengths for numerical tractability—reproduced the observed fragmentation of the order parameter, the preferential location of vortices in regions of low pairing amplitude, the survival of the soft gap within vortex cores, and the field-driven broadening of the inhomogeneity distribution. The simulated LDOS in clean versus disordered systems highlighted the absence of zero-bias peaks (CDM states) at the core in the latter, further corroborating experimental results.

Implications and Future Directions

Contradictions and Advances over Prior Understanding

The results underscore that even weak, homogeneous disorder causes a dramatic departure from the standard picture of vortex-driven destruction of superconductivity. The classical view—of superconductivity vanishing solely due to core overlap at $H_{c2}$—is rendered incomplete. Instead, field and disorder act synergistically to induce an emergent granularity, manifesting as a field-induced pseudogap regime where local pairing persists despite the destruction of long-range phase order.

This finding aligns the magnetic-field-driven superconductor-insulator transition (SIT) with the disorder-driven SIT, supporting the hypothesis that both are governed by the same underlying physics: the proliferation of granularity, fragmentation of the superconducting order, and subsequent localization of Cooper pairs due to Coulomb blockade or other mechanisms.

Numerical Results and Bold Claims

The persistence of the gap within vortex cores under weak disorder, and field-induced spatial inhomogeneity at much weaker disorder than typically required for pseudogap physics, are highlighted in this work. The assertion that phase decoherence triggers resistive behavior long before the pairing amplitude vanishes represents a key contradiction to mean-field-based expectations for weakly disordered superconductors.

Theoretical and Practical Impact

These findings have direct implications for the analysis of quantum phase transitions in two-dimensional superconducting films and may inform the design and interpretation of experiments on SIT phenomena in other material families, including high-$T_c$ cuprates, disordered conventional superconductors, and engineered Josephson junction arrays.

The methodology, combining high-resolution STS with controlled disorder tuning and detailed BdG simulations, establishes a template for future studies aimed at elucidating the interplay between disorder, magnetic field, and quantum criticality in low-dimensional systems.

Conclusion

This work demonstrates that applied magnetic fields induce emergent granularity and a robust pseudogap regime in weakly disordered NbN films, fundamentally contradicting the classical view of vortex dynamics in superconductors. Both experiment and theory reveal field and disorder as cooperative drivers of spatially inhomogeneous superconductivity, leading to new understanding of the SIT landscape. These insights are expected to influence both applied and fundamental studies of correlated electron systems exhibiting intricate interplay between pairing, disorder, and magnetic field (1703.04667).