Short-Range Modulated Electron Lattice and d-Wave Superconductivity in Cuprates: A Phenomenological Ginzburg-Landau Framework

TL;DR

This work proposes a phenomenological Ginzburg-Landau framework where a short-range, coherence-linked modulated electron lattice (MEL) couples to a $d$-wave superconducting condensate in cuprates. The MEL is selected by a momentum-dependent kernel $\alpha(q)$ with a preferred wave vector $q^{\ast}$ near $0.30$ r.l.u. along Cu–O bonds, arising from electronic susceptibility and bond-stretching phonons, and is stabilized by couplings $\gamma_1$, $\gamma_2$ to the superconducting order parameter. Monte Carlo simulations reveal an MEL enhancement window in doping, temperature, MEL correlation length $\xi_{\mathrm{MEL}}$ and disorder where in-plane stiffness increases by about $\sim10\%$, reducing $\lambda_{ab}$ from $\sim150$ nm to $\sim140$ nm; this prediction is to be checked by self-consistent BdG calculations. The framework yields falsifiable STM/STS signatures, such as a sharpened $q^{\ast}$ peak in Fourier-transformed LDOS that grows below $T_c$ and a positive correlation between the local gap $\Delta(\mathbf{r})$ and MEL amplitude, and it connects MEL behavior to vortex pinning and percolation phenomena, offering a coherent organization of cuprate phenomenology within an experimentally testable, albeit approximate, GL description.

Abstract

We develop a phenomenological Ginzburg-Landau (GL) framework for high-$T_c$ cuprates in which a short-range modulation of the electronic charge density couples to a $d$-wave superconducting condensate. The resulting modulated electron lattice (MEL) state is distinct from long-range static charge density wave order: it is short range, partially phase coherent, and linked to superconducting coherence. A preferred wave vector $q^{\ast} \approx 0.3$ reciprocal lattice units along the Cu-O bond direction emerges from the interplay between a momentum-dependent susceptibility and bond-stretching phonons, consistent with neutron and x-ray data on YBa$_2$Cu$_3$O$_{7-δ}$ and related cuprates. The GL free energy contains coupled $d$-wave superconducting and charge sectors with parameters constrained by optimally doped YBa$_2$Cu$_3$O$_{7-δ}$. We identify an MEL enhancement window in doping, temperature, MEL correlation length, and disorder where a coherence linked modulation enhances the superfluid stiffness. Classical Monte Carlo simulations yield an in-plane stiffness enhancement of order ten percent, which we treat as a qualitative prediction to be tested by self-consistent Bogoliubov de Gennes calculations. The MEL framework yields falsifiable experimental signatures. For scanning tunneling spectroscopy in Bi-based cuprates we highlight two predictions: the Fourier-transformed local density of states should exhibit a $q^{\ast} \approx 0.3$ peak whose spectral weight sharpens as superconducting phase coherence develops below $T_c$, in contrast to static charge scenarios, and the local gap magnitude $Δ(r)$ should correlate positively with the local MEL amplitude. The framework implies correlations between MEL correlation length, superfluid stiffness, disorder, and vortex pinning, and organizes cuprate observations into testable STM/STS predictions.

Figures (6)

• Figure 1: Temperature dependence of the in-plane penetration depth $\lambda_{ab}(T)$ obtained from the MEL GL framework in the enhancement window. The low-temperature value is set by the tuned superfluid density to match optimally doped YBa$_2$Cu$_3$O$_{7-\delta}$, and the overall magnitude of $\lambda_{ab}(0)\sim 140$ nm is consistent with experimental values. Within the MEL window the effective stiffness is modestly enhanced, leading to a correspondingly smaller $\lambda_{ab}$ than in the absence of MEL, while outside this window the MEL contribution is negligible.
• Figure 2: Field dependence of the vortex pinning energy and supercurrent density in the MEL framework. The solid curve shows the characteristic pinning energy $E_{\mathrm{pin}}(B)$ extracted from the GL simulations, while the dashed curve shows the corresponding critical supercurrent density $J_s(B)$. The overall scales are chosen to be consistent with strong pinning and large $J_s$ in high-quality YBa$_2$Cu$_3$O$_{7-\delta}$. Within the MEL enhancement window the stiffness and pinning are both enhanced relative to a purely homogeneous superconducting background.
• Figure 3: Percolation of MEL active domains under disorder and its connection to oxygen stoichiometry. The lower horizontal axis shows the dimensionless disorder strength $\sigma/A_0$, defined as the ratio of the standard deviation of the local MEL amplitude to its clean-limit scale. The left vertical axis (solid curve) gives the domain continuity fraction $f_c$, i.e. the fraction of the system participating in a system-spanning MEL enhanced superconducting network. For weak disorder $f_c \approx 1$, while above a threshold $\sigma/A_0 \sim 0.3$--$0.4$ the connectivity collapses rapidly, signaling a percolation driven loss of global phase stiffness. The upper horizontal axis indicates a representative mapping to local oxygen concentration $x$ in YBa$_2$Cu$_3$O$_{7-\delta}$, and the right vertical axis (dashed curve) shows the corresponding MEL domain formation probability or normalized local conductance, $G(x)$, extracted from the same simulations. The coincidence of the sharp drop in $f_c$ with the rapid change in $G(x)$ highlights that MEL enhanced superfluid stiffness is controlled by the connectivity of MEL domains rather than by local order parameter amplitude alone.
• Figure 4: Schematic Fourier transform STS intensity $|g(q,E)|$ near the bond direction wave vector $q^\ast \simeq 0.3$ r.l.u. across the superconducting transition. Solid curves show the MEL scenario: above $T_{\mathrm{c}}$ ($T > T_c$) the $q^\ast$ peak is broad and relatively weak, at $T \approx T_{\mathrm{c}}$ it is moderately enhanced, and below $T_{\mathrm{c}}$ ($T < T_c$) it becomes both sharper and higher in intensity as the MEL envelope $\Phi(\mathbf{r})$ locks to the superconducting condensate $\psi(\mathbf{r})$. The vertical dotted line marks $q^\ast$. For comparison, the dashed curve sketches a conventional CDW competition scenario in which the $q^\ast$ peak is reduced below $T_{\mathrm{c}}$ as the competing superconducting gap opens. The qualitative trend in the $q^\ast$ peak height and width across $T_{\mathrm{c}}$ provides a decisive discriminator between MEL and pure CDW competition.
• Figure 5: Real space correlation between the local superconducting gap and MEL related modulation strength. The upper panel shows the simulated gap magnitude $\Delta(x)$ along a one-dimensional cut through the sample, and the lower panel shows the corresponding Fourier transform STS coherence peak intensity or local MEL amplitude along the same cut. Regions with larger $\Delta(x)$ coincide with stronger modulation, demonstrating the positive spatial correlation characteristic of a coherence linked MEL state. Such a trend is difficult to reconcile with simple CDW competition scenarios, which typically produce anti correlation between charge order and superconducting gap strength.