Evolution of inhomogeneities in two-dimensional disordered superconductors in a magnetic field

TL;DR

The paper investigates how non-magnetic disorder $V$ and an orbital magnetic field $H$ jointly affect superconductivity in a two-dimensional film. It uses self-consistent Bogoliubov–de Gennes calculations on a 2D attractive Hubbard model with impurities and Peierls phases to predict inhomogeneities in the local pairing amplitude $\Delta_i$, coherence peak height $h_i$, local density of states $N(i,\omega)$, and supercurrents, and it validates these predictions against scanning tunneling spectroscopy (STS) measurements on Sr$_2$VO$_{3-\text{x}}$FeAs structures. The results show that $V$ and $H$ induce island-like superconducting regions, weaken correlations between $\Delta_i$ and $h_i$, and promote the appearance of subgap states, with a crossover from Abrikosov to coreless Josephson vortices as disorder grows. This work provides spectroscopic and current-mmapping fingerprints for identifying vortex type in disordered superconductors and offers a framework for interpreting STS observations in complex superconductors under simultaneous disorder and magnetic field.

Abstract

Emerging granularity in superconducting films by tuning disorder is a well-studied topic, both theoretically and experimentally. However, the orbital magnetic field generates a vortex lattice and contributes to the formation of periodic inhomogeneities. Here, we study superconducting films in the simultaneous presence of disorder and a magnetic field, examining how inhomogeneities in various superconducting correlations evolve under these two perturbations. By performing scanning tunneling spectroscopy (STS) on thin films of \ce{Sr2VO_{3-\text{x}}FeAs} layer structures under both zero and finite orbital magnetic fields, we report impressive similarities between our theoretical results and the experimental findings. Our results have strong implications for identifying the nature of vortices in disordered superconductors, demonstrating a crossover from Abrikosov to Josephson character with increasing disorder, and provide predictive guidance for interpreting STS and current mapping data in complex superconductors.

Figures (7)

• Figure 1: Panel (a) shows normalized distributions of SC pairing amplitude $P(\tilde{\Delta})$: The red trace is for $\tilde{\Delta}^{\rm theo}$ simulated using the BdG method on a $50 \times 50$ system for $V=0.5$ in the absence of a magnetic field. The violet trace represents $P(\tilde{\Delta}^{\rm expt})$ obtained from the experiments with Low disorder at $5\,\text{K}$ in zero field. $P(\tilde{\Delta}^{\rm theo})$ has a standard deviation, $\sigma_{\tilde{\Delta}^{\rm theo}} = 0.1534$, whereas $P(\tilde{\Delta}^{\rm expt})$ yields a similar value for the standard deviation, $\sigma_{\tilde{\Delta}^{\rm expt}} = 0.1518$, confirming that the extent of disorder in these two complementary methods are broadly similar. Panel (b) displays the evolution of the average density of states (AvDoS), $N(\omega)$, of the 2D SC (for system size $50 \times 50$). Exploiting the particle–hole symmetry of the SC state, we plot only one side of $\omega$ for the spectrum. The left part of the panel shows $N(\omega)$ for fixed disorder strength $V=0.5t$ at different magnetic fields $\phi/\phi_0 = 0, 2, 6$ over the range $\omega \in [-0.15, 0]$. In contrast, the right part presents $N(\omega)$ for different $V$ at zero field ($\phi/\phi_0=0$) over the range $\omega \in [0, 0.15]$.
• Figure 2: The spatial correlation of SC pairing amplitude $\Delta_i$ in panel (a), and the coherence peak height $h_i$ in panel (b) are demonstrated at a weak disorder strength $V=0.5$. Similar results for stronger disorder ($V=1.25$) are also shown in panels (c) and (d). The magnetic field is turned off ($\phi=0$) for all the results in this figure. In panels (e) and (f), we plotted the frequency (probability) distributions of $\Delta_i$ and $h$, respectively, where both $P(\Delta)$ and $P(h)$ broaden with increasing $V$, shifting the mode of these distributions toward zero, producing skewed distributions for larger $V$'s.
• Figure 3: Scatter plots of the local order parameter $\Delta_i$, versus local coherence peak height $h_i$ with different $V$'s and $\phi$'s. Panels (a–c) present our theoretical predictions, while panels (d–f) show the results from our STS measurements. In both cases, the correlations between $\Delta_i$ and $h_i$ decrease systematically with increasing either $V$ or $\phi$, thereby establishing a close match between predictions and observations. To quantify this observation, we calculated the Pearson correlation coefficient $r$ between $\Delta$ and $h$, and obtained $r\sim 0.78$ for $V=0.5$ and $r\sim 0.72$ for $V=1.25$ when $\phi=0$, and $r\sim 0.66$ for $V=0.5$, $\phi/\phi_0=6$. Here, $r = \mathrm{cov}(\Delta, h)/\sigma_\Delta \sigma_h$ -- for details, see note. From experimental data, we find $r \approx 0.756$ at low $V$ and $0$ T, $r \approx 0.73$ at high $V$ and $0$ T, and $r \approx 0.70$ at low $V$ and $7$ T, which are in close agreement with the theoretical trends.
• Figure 4: Panels (a, c, e) reflect the spatial map of $\Delta_i$, $h_i$, and $N(r_i, E_g/2)$ respectively, for $\phi_/\phi_0 = 2$, whereas, the same observables are presented in panels (b, d, f) for a larger $\phi_/\phi_0 = 6$. All panels show results for $V=0.5$. In (a, b), we observe a significant suppression of SC pairing amplitude in the dark regions, due to the formation of two and six vortices, respectively. Outside the vortex cores, disorder induces weaker spatial fluctuations in $\Delta_i$. Similarly, in (c,d), the local coherence peaks are fully suppressed, while in (e,f), normal electronic states develop within the subgap region at and around the vortex cores.
• Figure 5: Distributions of local superconducting properties under varying magnetic fields. Panels (a–c) show the evolution of the probability distributions of the local SC pairing amplitude $\Delta_i$, the local coherence peak height $h_i$, and the subgap (actually, at $\omega=E_g/2$) LDoS $N(i, E_g/2)$, respectively, for our calculated estimates in the absence ($\phi/\phi_0=0$) and presence ($\phi/\phi_0=2,6$) of an orbital magnetic field for $V=0.5$. The distributions reveal enhanced spatial inhomogeneity and spectral broadening under field, reflecting vortex-induced suppression of coherence peaks and the emergence of low-energy states. Panels (d–f) show the corresponding distributions from the measured STS data obtained in the absence ($0$ T) and presence ($7$ T) of an orbital magnetic field.