The Low-Temperature Phenomenology of Gap Inhomogeneity in the Cuprate Superconductors: High-Energy Granularity, Low-Energy Homogeneity, and Spectral Kinks

TL;DR

This work investigates how the length scale of gap inhomogeneity, modeled as square patches with a broadly distributed $d$-wave order parameter, shapes the local density of states in cuprate superconductors. Using a large-scale Bogoliubov–de Gennes framework and Green's function techniques, it identifies three regimes determined by the patch size $l$ relative to the coherence length $ξ$: (i) $l<ξ$ yields a effectively homogeneous low-energy state with a narrowly distributed spectral gap, (ii) $l≈ξ$ produces a low-energy homogeneous sector and a high-energy inhomogeneous sector with pronounced kinks at a characteristic energy $Δ_K$, and (iii) $l>ξ$ approaches bulk-like behavior with strong proximity effects but residual inhomogeneity at boundaries. The study shows that $Δ_S$, $Δ_L$, and $Δ_K$ are emergent energy scales that separate as $l$ grows, and that many STM observables (gap maps, kink energies, nodal gaps) can be understood within this mean-field, non-coexisting-order framework, while highlighting where beyond-mean-field correlations may be essential. Overall, the results offer a coherent, quantitative route to interpret STS phenomenology in the cuprates and suggest that much of the observed low-energy–high-energy dichotomy can arise from gap inhomogeneity alone, with kinks arising from a proximity-driven mechanism rather than a separate order parameter.

Abstract

Scanning tunneling spectroscopy experiments on a number of cuprate superconductors have revealed that these materials are highly inhomogeneous. However, even though this inhomogeneity is well-characterized experimentally, a theoretical understanding of the effect of an inhomogeneous superconducting $d$-wave order parameter on various observables is still not complete. Here, we focus on the particular role played by the length scale of superconducting order-parameter inhomogeneity. We make use of a model involving square patches tiling the system, with each patch hosting a broadly distributed random value of the $d$-wave parameter. By using large-scale simulations, we are able to study how the size of the patches affects the correspondence between various measures of the superconducting gap and the underlying order parameter. If the length scale of the inhomogeneity is smaller than the average superconducting coherence length, the resulting $d$-wave superconductor is homogeneous. However, when the order parameter varies on the scale of the coherence length, we find the emergence of a striking low-/high-energy dichotomy, in which the low-energy regime is homogeneous while the high-energy states are strongly inhomogeneous. Kinks in the local spectra are found at the energy demarcating the homogeneous-inhomogeneous transition. We also observe that the gap extracted from the low-energy slope of the LDOS is extremely uniform. We find in both of these regimes that the distribution of the spectral gap is narrower than that of the order parameter; these start to match only when the size of the patches becomes parametrically larger than the coherence length. We comment on the applicability of these results to the cuprates, discuss the limitations of the inhomogeneous $d$-wave model, and point out where beyond-mean-field correlation effects are likely to be present in addition to inhomogeneity.

Figures (28)

• Figure 1: Plots of $\Delta_T$ (a), $\Delta_S$ (c), and $\Delta_L$ (e), and the absolute value of their respective Fourier transforms (b, d, and f) for patch size $l = 1$. Only one disorder realization is presented. Shown here are quantities taken from the middlemost $256 \times 256$ segment of the full $1024 \times 256$ system.
• Figure 2: Left: Plots of the binned average local density of states for $l = 1$. Each LDOS spectrum is binned according to its spectral gap (with four bins in all), and all spectra in a given bin are then averaged over. Right: Plot of the distribution of the local density of states as a function of energy (heat map) and the average local density of states (dashed white line) for $l = 1$. The spectra are binned according to the energy, and histograms for each energy are taken. The counts per energy bin are shown as a heat map. For both these plots, 4 disorder realizations and a total of 1,048,576 individual spectra are used in this plot.
• Figure 3: Plots of the LDOS at different frequencies for patch size $l = 1$. The same disorder realization and field of view as in Fig. \ref{['fig:centeredplots_1']} are used here.
• Figure 4: Left: Plots of the local density of states along a straight line through the middle of the sample from $(516,64)$ to $(516,192)$ for $l = 1$. The filled circles indicate the position of the underlying order parameter $\Delta_T$, while open circles and asterisks indicate the spectral gap $\Delta_S$ and low-energy gap $\Delta_L$, respectively. Only a single disorder realization is shown here. The plots are colored according to the local value of $\Delta_T$. Right: Plots of $\Delta_T$ (blue), $\Delta_S$ (red), and $\Delta_L$ (green) along the same linecut.
• Figure 5: Plots of the two-dimensional histograms for a) $\Delta_T$ and $\Delta_S$; b) $\Delta_T$ and $\Delta_L$; and c) $\Delta_S$ and $\Delta_L$ for $l = 1$. Included here are data points from four disorder realizations corresponding to 1,048,576 individual LDOS spectra.