Scaling between Superfluid Density and T_c in Overdoped La_2−xSr_xCuO₄ Films

Abstract

We used an electronic phase separation approach to interpret the scaling between the low-temperature superfluid density average ${\rho }_{\mathrm{sc}}\left(0\right)$ and the superconducting critical temperature $T_{\mathrm{c}}$ on overdoped La${}_{2-x}$Sr${}_x$CuO${}_4$ films. Guided by the observed nematic and incommensurate charge ordering (CO), we performed simulations with a free energy that reproduces charge domains with wavelength ${\lambda }_{CO}$ and provides a scale to local superconducting interactions. Under these conditions a complex order parameter with amplitude ${\Delta }_d\left(r_i\right)$ and phase $\theta \left(r_i\right)$ may develop at a domain i. We assumed that these domains are coupled by Josephson energy $E_{\mathrm{J}}\left(r_{ij}\right)$, proportional to the local superfluid density ${\rho }_{\mathrm{sc}}\left(r_{ij}\right)$. Long-range order occured when the average $E_{\mathrm{J}}\left(T_{\mathrm{c}}\right)$ is $\sim k_{\mathrm{B}}{T}_{\mathrm{c}}$. The linear ${\rho }_{sc}\left(0\right)$ vs. $T_{\mathrm{c}}$ relation was satisfied whenever CO was present, even with almost vanishing charge amplitudes.

1. Introduction

Overdoped high-temperature cuprate superconductors have been widely believed to be described by the physics of d-wave BCS-like superconductivity. However, recent measurements [1] indicate that as the doping is increased, the superfluid density decreases smoothly to zero rather than increasing as expected by BCS theory in the absence of disorder. The authors in [1] developed a technique to grow homogeneous overdoped La${}_{2-x}$Sr${}_x$CuO${}_4$ (LSCO) films and measured the penetration depth from which ${\rho }_{sc}\left(0\right)$ is derived, establishing a new unforeseen scaling law: ${\rho }_{\mathrm{sc}}(p,T\sim 0 \mathrm{K})$ is directly proportional to $T_c\left(p\right)$, both being maximum at $p=0.16$ and vanishing near the superconducting (SC) limiting phase at $p=0.26$. Their films displayed homogeneous properties with less than 1% variations in $T_{\mathrm{c}}$ [1,2], but the induction of an anisotropic transverse voltage [3] indicates some intrinsic electronic disorder.

This anisotropic transport is compatible with a nematic order, and this is another manifestation of the ubiquitous CO in cuprates [4]. In fact, the presence of bulk magnetic excitation from low to high doping [5,6,7] suggests the presence of either static or correlated charge fluctuations in time and space with local antiferromagnetic (AF) order that decreases with doping [6,7]. Concomitantly, the very low normal state residual resistance measured in the same experiment [1] and subsequent time-domain THz spectroscopy [8] ruled out an approach based on the dirty BCS scenario [9].

2. Results

Competing broken symmetry states in cuprates arising from a nanoscale electronic phase separation have been predicted by many different microscopic models using Hubbard, Holstein, t-J Hamiltonians, and with many different techniques [10,11,12,13]. All these models predict different forms of charge disorder such as stripes, CO, or charge density waves (CDWs) under distinct phase diagram parameters. These fundamental calculations are essential to demonstrate that phase separation is a real phenomenon of high-$T_{\mathrm{c}}$ superconductors, but they do not reproduce the fine structure of the observed charge wavelength ${\lambda }_{\mathrm{CO}}$ as a function of hole-doping p that has been generally observed [4,14,15]. For exactly this purpose, we performed electronic phase separation calculations [16,17,18,19,20] based on the Cahn–Hilliard (CH) nonlinear differential equation [21]. The main achievement of our work lies in almost exactly mimicking the CO structure details, that is ${\lambda }_{\mathrm{CO}}\left(p\right)$, and this was accomplished through the parameters involved in the CH equation and by stopping the simulation at fixed times. Although this is artificial, it is the only way to obtain the charge density maps $p\left(\mathbf{r}\right)$ on the CuO plane close to the observed modulations. The approach also has the great advantage of concomitantly providing the free energy map, which leads to the superconducting interaction, as will be explained below.

Concerning the superconducting (SC) properties, there is evidence [22,23,24] in favor of a model exploring the similarities between the CDW alternating charge domains with a granular superconductor in which SC grains are coupled via Josephson tunneling [25,26]. We used this approach, developed mainly in [18,26], to study the superfluid and $T_{\mathrm{c}}$ scaling relation.

Therefore, we started with the CH equation, which is based on the Ginzburg–Landau (GL) free energy expansion in terms of a phase separation local order parameter $u(\mathbf{r},T)$, which is a function of position and temperature T:

$$f\left(u\right)=\frac{1}{2}\epsilon {|\nabla u|}^2+V_{\mathrm{GL}}(u,T),$$

(1)

where $V_{\mathrm{GL}}(u,T)=-\alpha [T_{\mathrm{PS}}-T]u^2/2+B^2{u}^4/4+\dots$ is a double-well potential that characterizes the electronic phase separation below $T_{\mathrm{PS}}\approx T^{*}\left(p\right)$ and $T^{*}\left(p\right)$ is the pseudogap temperature [27,28]. We used $\alpha$ and $B=1$ here but they are usually used to control the CO pattern, and $\epsilon$ controls the spatial separation of the charge-segregated patches. The CH equation can be written in the form of the following continuity equation for the local GL free energy current density $\mathbf{J}=-M{\nabla }^2((\partial f/\partial u)$ [29]:

$$\frac{\partial u}{\partial t}=-\nabla \mathbf{J}=-M{\nabla }^2\left[\epsilon {\nabla }^{2}u+\frac{\partial V_{\mathrm{GL}}}{\partial u}\right],$$

(2)

where M is the order parameter mobility that sets the time scale. To solve this equation, we used a time-dependent [17,18,20] conserved order parameter associated with local electronic density, $p\left(\mathbf{r}\right)=$ A$u\left(\mathbf{r}\right)+p$, where A controls the CO amplitude of oscillation. A is 0.00005 to $\Delta p\approx {10}^{-3-4}$ variations around p, like in typical YBCO systems [30]. All the simulations here used $\delta t=700$ time steps, and we dropped the dependence of t on $u\left(\mathbf{r}\right)$ and on $V_{\mathrm{GL}}(u,T)\equiv V_{\mathrm{GL}}(\mathbf{r},T)$. We assumed that overdoped compounds also have such small charge oscillations $\Delta p$ that, together with metallic conductivity, they would be almost undetectable.

Figure 1a shows the density distribution $p\left(\mathbf{r}\right)$ for $p=0.16$ with hole-rich (red domains) and hole-poor (blue domains) at $T\sim 0$ K. The anisotropic checkerboard-like pattern also favors anisotropic conduction along the easy axis at $\pm 45{}^{\circ }$ or $\pm 135{}^{\circ }$, which may be an approximated explanation of the transverse voltages measured in [3] from low temperature to room temperature, suggesting that the CO transition may start much above $T^{*}$. Figure 1b shows the $T\sim 0$ K free energy map $V_{\mathrm{GL}}\left(\mathbf{r}\right)$ derived from the same $u\left(\mathbf{r}\right)$ simulation. It is important to note that the domains in blue correspond to both hole-rich and hole-poor regions. To model the observed broadening and the intensity loss of the modulated dynamical spin correlations [6,7,31], we slightly increased $\epsilon$ with p.

Accordingly, Figure 1c shows $V_{\mathrm{GL}}\left(x\right)$ along the x-direction for $p=0.16,0.21$, and 0.25, where the amplitudes decreased with p to mimic the broader peaks in the dynamical spin measurements [6,7,31]. The main idea of our model is that the $V_{\mathrm{GL}}(\mathbf{r},T)$ modulations shown in Figure 1c create regions with alternating charge probability domains or CDWs that promote local SC amplitudes. It was shown by high-energy X-ray diffraction [32] that the observed CDWs also produce atomic displacements with the same modulations, which is a direct experimental demonstration of the electron–phonon coupling. This experiment on single-crystal YBCO, considered the cleanest cuprate, validates the early proposal by A. Muller that strong electron lattice interaction with the formation of polaronic states plays a key role in the high superconductivity of doped perovskites [33]. In La-based superconductors, the connection between inhomogeneous charge states and lattice fluctuations was observed by extended X-ray absorption fine structure (EXAFS) through the Debye–Waller factor of the Cu–O bonds. From these data, the polaronic distortion order parameter across the charge stripe ordering temperature was extracted [34,35]. The results are supported by a temperature-dependent Cu K-edge X-ray absorption near edge structure (XANES) revealing a particular change in the local lattice displacements at the charge stripe ordering. The experimental Cu K-edge XANES results are well reproduced by full multiple scattering calculations including different distortions of the CuO plane showing particular lattice displacements which are consistent with the EXAFS results [36,37].

The isotopic effect at the ordering temperature [38,39,40] of polaron stripes [41] has also provided compelling evidence that local lattice fluctuations and given experimental support for the polaronic proposal of Alex Muller [33]. The fact that the size of the SC coherence lengths are generally less than ${\lambda }_{\mathrm{CO}}$ also suggests that valence electrons form lattice-induced Cooper pairs in CO domains.

Therefore, the strength of the attractive two-body SC interaction is assumed to be scaled by the spatial average $〈V_{\mathrm{GL}}(p,T)〉\equiv {\sum }_i^N{V}_{\mathrm{GL}}(r_i,p,T)/N$, where N is the number of unit cells in the CuO plane. An indication of this proportionality was extracted by comparing the results of the experiment of Poccia et al. [42]. They controlled the degree of oxygen ordering by tuning the time t of X-ray irradiation in optimally doped La${}_2$CuO${}_{4+y}$ and subsequently measured the SC critical temperature $T_{\mathrm{c}}\left(t\right)$, which increased with t up to a saturation limit. We interpreted their results with n$\delta t$ simulation time and $〈V_{\mathrm{GL}}(p,n\delta t)〉$ proportional to the SC interaction in the self-consistent calculations [18]. We found that taking time steps of 2000 $\delta t$ in the CH simulations was equivalent to $0.1$ h of X-ray irradiation, and we could closely reproduce the $T_c\left(t\right)$ evolution with time, as described in [18].

The local SC amplitude map ${\Delta }_d\left(\mathbf{r}\right)$ was calculated via self-consistent Bogoliubov–deGennes (BdG) calculations [17,18,19,20] based on a Hubbard Hamiltonian with an attractive potential scaled by $〈V_{\mathrm{GL}}(p,T)〉$. The GL potential average $〈V_{\mathrm{GL}}(p,T)〉$ has no dimension and it was necessary to multiply it by a constant in order to define the attractive pairing potential $〈V_{\mathrm{GL}}(p,T)〉$ in eV units. We adjusted the $T\sim 0$ K $〈V_{\mathrm{GL}}(p_0=0.16,T)〉$ to reproduce the measured SC gap ${\Delta }_{\mathrm{sc}}(p_0=0.16)$. Once this was done, all the other $〈V_{\mathrm{GL}}\left(p\right)〉$ followed from their $〈V_{\mathrm{GL}}\left(p\right)〉/〈V_{\mathrm{GL}}\left(p_0\right)〉$ ratio. The $T\sim 0$ K calculation with $〈V_{\mathrm{GL}}\left(p\right)〉=0.234$ eV yielded $〈{\Delta }_d(\mathbf{r},p_0=0.16)〉\sim 16.9$ meV, which is close to the measured LSCO nodal gap $(〈{\Delta }_0(p_0=0.16)〉\sim 16-17$ meV) [43].

A typical SC ${\Delta }_d(\mathbf{r},p=0.16)$ result in contour map format is shown in Figure 1d. ${\Delta }_d(\mathbf{r},p)$ and $p\left(\mathbf{r}\right)$ are plotted with different scales, but the charge and pairing amplitude density modulations had the same ${\lambda }_{\mathrm{CO}}$. Figure 2a shows $〈{\Delta }_d(p,T)〉$, and it is clear that all SC amplitudes remained finite above $T_{\mathrm{c}}$, in agreement with measurements of persisting SC correlations above $T_{\mathrm{c}}$ [22,23,28,44].

3. Discussion

Since our calculations rely heavily on the existence of CO or CDW, which is not widely observed in overdoped samples (other than in [6,7]), we studied the case of very small (i.e., $\Delta p={10}^{-4}$) relative electronic modulations, as well as large oscillations for comparison. The calculations with only $\Delta p=0.0005$% variation in the charge density yielded essentially the same $〈{\Delta }_d(p,T)〉$ of the case studied here with 0.5% charge oscillation. The reason for this is that the free energy average $〈V_{\mathrm{GL}}〉$ is independent of the charge amplitude. Therefore, we draw the important conclusion that the SC properties depend very little on the charge amplitude but—as shown elsewhere [20]—they depend strongly on the wave vector of the charge order ${\lambda }_{\mathrm{CO}}$.

When the temperature decreases, the two-component SC order parameter (${\Delta }_d\left(r_i\right)$, ${\theta }_i$) develops at each charge domain like in granular superconductors [26]. The ${\Delta }_d\left(\mathbf{r}\right)$ distribution in Figure 1d resembles a multiple-grain superconductor. In this case, the interface between two neighbor grains gives rise to a Josephson junction with an energy $E_{\mathrm{J}}\left(r_{ij}\right)$ between grains i and j that is proportional to the local superfluid density ${\rho }_{\mathrm{sc}}$ [24]. Therefore, cuprates may be regarded as an array of these junctions. For a d-wave superconductor in single crystals, it is sufficient to use the s-wave relation for the average Josephson coupling energy [17,18,20]:

$$〈E_{\mathrm{J}}(p,T)〉=\frac{\pi h〈{\Delta }_d(p,T)〉}{2e^2{R}_{\mathrm{n}}\left(p\right)}\tanh \left[\frac{〈{\Delta }_d(p,T)〉}{2k_{\mathrm{B}}T}\right].$$

(3)

$R_{\mathrm{n}}\left(p\right)$ is proportional to the normal-state in-plane resistivity ${\rho }_{ab}\left(p\right)$ that was also measured in [1], which is crucial to our approach. We used the values of $〈{\Delta }_d(p,T)〉$ plotted in Figure 2a to derive $〈E_{\mathrm{J}}(p,T)〉$. Figure 2b shows $〈E_{\mathrm{J}}(p,T)〉$ vs. T. $R_{\mathrm{n}}(p_0=0.16)$ was scaled to yield $T_c\sim 42$ K and all the other $R_{\mathrm{n}}\left(p\right)$ follow from their experimental ratio. The derived $R_{\mathrm{n}}\left(p\right)$ by this procedure closely follows the measured [1] ${\rho }_{\mathrm{n}}\left(p\right)\sim {\rho }_{ab}\left(p\right)$ plotted in [1] (extended data Figure 8). We emphasize that $R_{\mathrm{n}}(p_0=0.16)$ and $〈{\Delta }_d(\mathbf{r},p_0=0.16)〉$ contain the only two adjustable parameters of our theory, from which we derived the entire phase diagram; that is, ${\rho }_{\mathrm{sc}}\left(0\right)$, $〈{\Delta }_d\left(\mathbf{r}\right)〉$, and $〈E_{\mathrm{J}}〉$ to any hole doping p and T, and especially $T_c\left(p\right)$.

Long-range SC phase coherence between the CO or CDW nano-domains occurs when $k_{\mathrm{B}}T\approx 〈E_{\mathrm{J}}(p,T)〉$. Figure 2b shows the calculations of $k_{\mathrm{B}}T\approx 〈E_{\mathrm{J}}(p,T)〉/k_{\mathrm{B}}$, and the bulk critical temperature $T_c\left(p\right)$ is given by the intersection with the temperature T black line. The Josephson coupling yields the superfluid density [24] because ${\rho }_{\mathrm{sc}}(p,0)\sim 〈E_{\mathrm{J}}(p,T\sim 0\mathrm{K})〉$. Thus, from the curves $〈E_{\mathrm{J}}(p,T)〉$ vs. T we simultaneously derived $T_c\left(p\right)$ and ${\rho }_{\mathrm{sc}}(p,0)$.

We plot our main result, the calculated $T_c\left(p\right)$ vs. ${\rho }_{\mathrm{sc}}(p,0)$ in Figure 2c together with the measurements of Bozovic et al. [1]. We also plot three points with large charge amplitude ($\Delta p=100$% modulations with A = 1), which demonstrate that increasing the charge amplitude $\Delta p$ had very little effect on $T_{\mathrm{c}}$ and ${\rho }_{\mathrm{sc}}$. The linear relation between ${\rho }_{\mathrm{sc}}\left(0\right)$ and $T_{\mathrm{c}}$ was robust to all the different possible charge modulations. Beyond $p=0.24$, the measured residual resistivity became very small [1], and the weak links broke down, allowing quasiparticle current. The Josephson coupling energy vanished, and the charge domains all had the same phase. In this regime, the system crossed over from a granular superconductor to a single disordered superconductor, and the resistivity transition occurred by percolation and proximity effect. Under these conditions, $T_c\left(p\right)$ coincided with the onset of $〈{\Delta }_d(p,T)〉$ shown in Figure 2a and represented by the blue points and line in Figure 2c for the range of $p=0.24$–0.26.

4. Conclusions

We verified that LSCO films are well-described by mesoscopic grains with charge modulation ${\lambda }_{\mathrm{CO}}$, similar to a granular superconductor. Our calculations showed that the linear ${\rho }_{\mathrm{sc}}\left(0\right)$ vs. $T_c$ scaling relation was robust even when the relative charge variations in CO or CDW were as small as ∼${10}^{-4}$. The decrease of ${\rho }_{\mathrm{sc}}\left(0\right)$ with p is also consistent with another interpretation derived from EXAFS studies [36,37], suggesting that electron–lattice interaction decreases with increasing doping. This behavior was argued to be closely related to the evolving anomalous local CuO distortion and charge inhomogeneity with doping, even into the overdoped region of LSCO compounds [45]. These experiments and our calculations on the ${\rho }_{\mathrm{sc}}\left(0\right)$ vs. $T_c$ experiment provide more support for the polaronic proposal of Alex Muller [33].

Recently, this scenario has also been supported by the investigations on new functional solids by new advanced methods [46,47]. The complexity is driven by cooperative effects of the self organization of dopants discussed in detail in a Special Issue [48], the pseudo Jahan Teller effect involving multiple orbitals [49,50], and local lattice tilts [51]. These results are currently a hot topic, since the strong electron–phonon interaction in complex multi-band materials in extreme conditions near lattice instabilities is emerging as a key mechanism in high-temperature superconductor hydrides [52,53] showing anomalous isotope effect [54,55].