A unified model for lunderdoped and overdoped cuprate superconductors based on a spinodal transition

TL;DR

This paper presents a unified theory of cuprate superconductivity across the doping range by invoking a spinodal decomposition or charge-separation transition that originates near the pseudogap temperature $T^*(p)$ and seeds CDW domains. It combines a Cahn–Hilliard description of phase separation with self-consistent Bogoliubov–de Gennes calculations for local pairing and an XY-like Josephson-coupled network to capture phase coherence, predicting a global $T_c(p)$ set by the balance with thermal fluctuations. The model reproduces key experimental observations, including CDW patterns, mesoscopic superconducting puddles in the overdoped regime, and the persistence of local gaps up to temperatures well above $T_c$, thereby unifying underdoped and overdoped cuprates under a single framework. The approach links Hall effect anomalies, pseudogap physics, and superconductivity through a common spinodal mechanism that yields a mesoscopic granular superconductor with density- and temperature-dependent coupling. The findings provide a mechanistic bridge between insulating, CDW-ordered, and superconducting phases across the cuprate phase diagram with potential implications for interpreting nanoscale inhomogeneity and high-$T_c$ phenomena.

Abstract

Many years of intense research on cuprate superconductors have led to several discoveries, such as the pseudogap and charge density waves (CDW), yet a complete theory is still lacking. By analyzing some experiments and performing calculations, we provide a full interpretation of their properties; from the undoped insulator to the overdoped metallic compounds. The variation of the anomalous Hall coefficient ($R_{\rm H}(T)$) with temperature at half-filling ($n = 1$) and, combinations of undoped ($p = 0$) insulators and metallic films, which, among other things, are indicative of a thermodynamic transition. On the overdoped side, recent experiments near the superconducting-to-metal transition detecting superconducting puddles and a considerable degree of charge disorder, suggest that a similar thermodynamic transition operates at all doping levels. We propose a spinodal or charge-separation transition starting near the pseudogap temperature $T^*(p)$, which among other things generates the CDW domains with a typical double-well Landau free-energy functional. Thus, from the half-filled to the overdoped region, the free energy forms an array of wells with $n = 1$ {\it static} holes. With doping, {\it mobile} holes tend to occupy these wells with alternating high and low densities, generating the CDW pattern. The confined holes in small regions develop local superconducting amplitudes, giving rise to a mesoscopic granular superconductor. Similar to the XY model, the grains develop correlation effects mediated by Josephson coupling, which is proportional to the local superfluid density. This approach yields a unified theory of cuprate superconductors.

Figures (4)

• Figure 1: The CH simulations of the phase separation GL free energy potential. The $V_{\rm GL}(u({\bf r}))$, that leads to the bimodal distribution. We show in c) a central portion of a $N \times N$ array of unit cells that are square-shaped. In a) and b) $V_{\rm GL}(r_i)$ along the $x$ direction of the CuO plane. These insets show the tendency to form pair-density waves (PDW)Mello2021 at low temperature.
• Figure 2: The local charge density and the superconducting amplitude. a) Charge distribution $p(r_i)$ of a compound with average doping $p = 0.25$. b) The two-dimensional BdG $\Delta_d(p=0.25,0)$ calculations on the same location. It demonstrates that, everything being the same, regions with local higher densities have smaller local superconducting amplitudes, and this deminstrate the suppression of the superconducting phase in strongly overdoped regions. The results in b, with gaps everywhere, is consistent with the break down of superconductivity by gap fillingOver2023.
• Figure 3: The average $d$-wave superconducting amplitude for selected compounds$\left <\Delta_{\rm sc}(p, T)\right>$ as a function of temperature for some overdoped compounds. The zero temperature results match the experimental valuesKato2008 and we derive the onset of superconducting fluctuations $T_{\rm c}^{on}(p)$ when $\left <\Delta_{\rm sc}(p, T)\right> \rightarrow 0$ that coincides with the onset of susceptibility signal of Ref. OverJJ2022 and the Nernst effect measurementsNernst2010Rourke2011NernstPRB2018.
• Figure 4: The average Josephson coupling as a function of temperature.$\left < E_{\rm J}(p,T_{\rm c}) \right >$ curves and the thermal fluctuation energy $k_{\rm B}T$. The superconducting long-range phase-ordering transition at $T_{\rm c}(p)$ occurs when the straight line $k_{\rm B}T$ crosses the $\left < E_{\rm J}(p,T_{\rm c}) \right >$ curves.