Inhomogeneity, Fluctuations, and Gap Filling in Overdoped Cuprates

TL;DR

This work tackles why overdoped cuprates defy conventional Landau-BCS expectations by proposing nm-scale inhomogeneity in the pairing interaction combined with thermal phase fluctuations. Using Bogoliubov-de Gennes mean-field theory and time-dependent Ginzburg-Landau simulations, the authors reproduce key tunneling spectra features: a low-energy, homogeneous gap coexisting with a highly inhomogeneous larger gap, and a temperature evolution where subgap states fill rather than close. Thermal fluctuations broaden the spectra, cause a smeared BKT-like transition, and generate robust superconducting islands that persist above $T_c$, while the local gap remains inhomogeneous at $T_c$. The results imply that overdoped cuprates cannot be described by homogeneous BCS theory; instead, a spatially random pairing interaction with phase fluctuations explains spectroscopic observations and links to strange-metal behavior observed in the normal state.

Abstract

Several recent experiments have challenged the premise that cuprate high-temperature superconductors approach conventional Landau-BCS behavior in the high-doping limit. We argue, based on an analysis of their superconducting spectra, that anomalous properties seen in the most-studied overdoped cuprates require a pairing interaction that is strongly inhomogeneous on nm length scales. This is consistent with recent proposals that the "strange-metal" phase above $T_c$ in the same doping range arises from a spatially random interaction. We show, via mean-field Bogoliubov-de Gennes (BdG) calculations and time-dependent Ginzburg-Landau (TDGL) simulations, that key features of the observed tunneling spectra are reproduced when both inhomogeneity and thermal phase fluctuations are accounted for. In accord with experiments, BdG calculations find that low-$T$ spectra are highly inhomogeneous and exhibit a low-energy spectral shoulder and broad coherence peaks. However, the spectral gap in this approach becomes homogeneous at high $T$, in contrast to experiments. This is resolved when thermal fluctuations are included; in this case, global phase coherence is lost at the superconducting $T_c$ via a broadened BKT transition, while robust phase-coherent superconducting islands persist well above $T_c$. The local spectrum remains inhomogeneous at $T_c$, and the gap is found to fill instead of close with increasing temperature.

Figures (7)

• Figure 1: Spatial maps of the $d$-wave order parameter $\Delta_{OP}$ (top row) and spectral gap $\Delta_{SG}$ (middle row) for a single realization of the pairing interaction. The LDOS (bottom row) is calculated for an ensemble of 50 random pairing realizations and for $15\times 15$ supercells. Sites are binned by the value of $\Delta_{SG}$ at $T=0.001$, and the average LDOS is shown for each bin (bottom row). The same sites are in each bin at all temperatures. Ten equal-width bins spanning $0.085 < \Delta_{SG} < 0.605$ are used; the legend shows the fraction of sites in each bin and darker colors correspond to lower-energy bins. Results are shown at five temperatures ranging from $T\ll T_\mathrm{mf}$ to $T \approx T_\mathrm{mf}$, for an impurity concentration of 7.5%. The red arrow in (k) indicates the subgap shoulder.
• Figure 2: The binned spectral gap (a) and order parameter (b) as functions of temperature. At the lowest measured temperature, each site is sorted its local value of $\Delta_{SG}$ and put into one of ten bins; the values of $\Delta_{SG}$ and $\Delta_{OP}$ are then averaged over each bin at each temperature.
• Figure 3: Effect of thermal fluctuations on the density of states. Temperature dependence of (a) the root-mean-square correlation function, (b)-(f) the GL local density of states, and (g)-(l) the TDGL local density of states. Insets show the time-averaged phase. In (a), $T_c$ is determined from the scaling of the spatial correlation function (see Section \ref{['sec_length_scale']}). The legend shows the spectral-gap ranges for each bin at $T=0.010$ and the number of sites within each bin.
• Figure 4: (a) Pair-phase correlation function $C_\theta({\bf r})$ along ${\bf r}=(x,0)$ at different temperatures. Symbols show numerical results for Eq. (\ref{['eq:Ctheta']}) taken from TDGL data for cuts along the $x$-axis, while solid curves are fits to the asymptotic forms in Eq. (\ref{['eq:BKT']}). Fits are to the power-law form for $T\leq 0.045$ and the exponential form for $T\geq 0.55$. The data at $T=0.050$ cannot be fitted by either form. (b) Temperature dependence of the spatially averaged phase correlation function. The average is performed over sites separated by $50 < |{\bf r}| < 150$ lattice constants.
• Figure 5: Density of states at $T=0.045$ for (a), (b) the $t$-$t'$ model shown in the main text, and (c), (d) a $t$-$t'$-$t"$ model. Results are shown for (a), (c) static GL calculations and (b), (d) thermally fluctuating TDGL calculations. The calculations use the same inhomogeneous GL parameters, $T_{i\alpha}^\ast$, as in the main text but different band structures for the calculated LDOS. The LDOS is binned such that the same sites are allocated to each bin for the two models.