Static Charge Density Wave Order in the Superconducting State of La2-xBaxCuO4

TL;DR

This study demonstrates that static charge density wave order coexists with superconductivity in La$_{2-x}$Ba$_x$CuO$_4$, persisting at low temperatures and up to high fractions of the transition temperature, challenging the notion of purely competing orders.

Contribution

First direct observation of static CDW order coexisting with superconductivity in La$_{2-x}$Ba$_x$CuO$_4$ using resonant x-ray photon correlation spectroscopy.

Findings

Static CDW order exists in the superconducting state.

CDW correlations persist up to at least 85% of the transition temperature.

Implications for coexistence of competing order parameters in cuprates.

Abstract

Charge density wave (CDW) correlations feature prominently in the phase diagram of the cuprates, motivating competing theories of whether fluctuating CDW correlations aid superconductivity or whether static CDW order coexists with superconductivity in inhomogeneous or spatially modulated states. Here we report Cu $L$-edge resonant x-ray photon correlation spectroscopy (XPCS) measurements of CDW correlations in superconducting La$_{2-x}$Ba$_x$CuO$_4$ $x=0.11$. Static CDW order is shown to exist in the superconducting state at low temperatures and to persist up to at least 85\% of the CDW transition temperature. We discuss the implications of our observations for how \emph{nominally} competing order parameters can coexist in the cuprates.

Figures

Equations

$$g_2(\text{emq},\tau )=\frac{⟨I(\text{emq},t)I(\text{emq},t+\tau )⟩}{⟨I(\text{emq},t){⟩}^2}=1+\beta |F(\text{emq},\tau ){|}^2.$$

$$\beta ={\left(\frac{I_{\text{s}}^{\text{max}}-I_{\text{s}}^{\text{min}}}{I_{\text{s}}^{\text{max}}+I_{\text{s}}^{\text{min}}}\right)}^2,$$

Full text

The properties of a complex material, such as a high temperature cuprate superconductor, are determined by its ground state configuration and spectrum of low energy fluctuations. For this reason, the discovery of charge density wave (CDW) correlations in various cuprates has attracted considerable attention Tranquada et al. (1995); Ghiringhelli et al. (2012); Chang et al. (2012); da Silva Neto et al. (2014); Comin et al. (2014); Fujita et al. (2014); Thampy et al. (2014); Tabis et al. (2014). Static charge order that develops from a conventional metallic state, as in various transition metal chalcongenides, tends to reduce the electronic density of states at the Fermi level and would be expected to suppress the superconducting transition temperature, $T_{\mathrm{SC}}$, of a standard BCS type superconductor Rossnagel (2011). Fluctuations associated with incipient ordering tendencies, on the other hand, are often invoked in quantum critical theories of superconductivity Varma (1997); Castellani et al. (1996); Sachdev (1999); Vojta et al. (2000); Chakravarty et al. (2001); Sachdev (2003); Abanov et al. (2003) and some theories posit CDW Castellani et al. (1996); Vojta et al. (2000); Wang and Chubukov (2015); Caprara et al. (2016) or nematic Maier and Scalapino (2014); Lederer et al. (2015); Metlitski et al. (2015) fluctuations as the key modes. The motivation for such theories is clear in the context of the phase diagram of La2-xBaxCuO4 plotted in Fig. 1. At $x=0.125$ bulk superconductivity is almost completely suppressed coincident with the strongest CDW correlations Moodenbaugh et al. (1988); Hücker et al. (2011), but at other dopings bulk superconductivity coexists with CDW correlations.

We recently implemented Cu $L$-edge resonant x-ray photon correlation spectroscopy (XPCS) as a means to determine whether CDW correlations in the cuprates are static, fluctuating, or a combination of both Chen et al. (2016). This technique has the advantage of a bulk probing depth, the ability to isolate the wavevector of charge (rather than spin) correlations and an excellent sensitivity to even very slow fluctuations. Our first experiment showed that the CDW in La2-xBaxCuO4 $x=0.125$ is robustly static up to 90% of the CDW transition temperature, but this represents a special case in the LBCO phase diagram (Fig.1) in which bulk superconductivity is almost completely suppressed Chen et al. (2016).

In this Rapid Communication we demonstrate that static CDW correlations also exist in the superconducting state of La2-xBaxCuO4 $x=0.11$ and that these static correlations extend above $T_{\mathrm{SC}}$ up to 40 K (85% of the CDW transition temperature). To the extent that the superconducting mechanism in La2-xBaxCuO4 is the same as in other cuprates, this sets important constraints on theories that suggest CDW fluctuations are crucial to superconductivity and how spatially-modulated states can be reconciled with superconductivity.

La2-xBaxCuO4 crystals were prepared using the floating zone method and cleaved ex-situ to produce a scattering face with an approximate $c$-axis surface normal. Extensive previous studies on samples prepared in the same way demonstrated high sample quality with typical crystal mosaics of $\sim 0.02^{\circ}$ Wilkins et al. (2011); Dean et al. (2013); Thampy et al. (2013); Dean (2015); Chen et al. (2016); Miao et al. (2017). XPCS experiments were performed at the 23-ID-1 beamline at the National Synchrotron Light Source II using a $[H,0,L]$ scattering plane as indexed in the high-temperature-tetragonal (HTT) unit cell with lattice constants $a=b=3.78$ Å and $c=13.28$ Å. In this notation, the CDW occurs at $(2\delta,0,0.5)$ where the incommensurability $\delta\approx x$ Fujita et al. (2004). X-rays were focused onto a 10 $\mathrm{\mu}$m pinhole located $\sim$ 10 mm in front of the sample. Due to the diffraction-limited source and the beamline design, this leads to very high coherent flux ($\sim∼10^{13}$ photons/s) at the sample with a longitudinal coherence length of 2 $\mathrm{\mu}$m and a transverse coherence length of 10 $\mathrm{\mu}$m (set by the pinhole). A CCD Doering et al. (2011) with a 30$\times$30 $\mathrm{\mu}$m2 pixel size situated 340 mm from the sample was used to measure the CDW Bragg peak. All data were collected with horizontal ($\sigma$)-polarized incident x-rays at the Cu $L_{3}$-edge (931 eV) and an incident x-ray angle of approximately $33^{\circ}$.

Figure 2(a) plots a detector image for La2-xBaxCuO4 $x=0.11$ at the CDW wavevector $\textbf{\em Q}=(-0.225,0,1.5)$ HKL summed over 120 minutes at 10 K. Speckle intensity modulations coming from coherent interference between different domains of CDW order appear throughout the image, modulating the overall peak shape. We compare this image to that obtained in La2-xBaxCuO4 $x=0.125$ collected during previous measurements under similar conditions in Fig. 2(b). $x=0.11$ shows a factor of 2 drop in the peak intensity on the detector relative to $x=0.125$ and a decrease in correlation length from 230 to 150 Å, consistent with previous incoherent x-ray scattering results Hücker et al. (2011). The elongation direction of the speckles is also different: being diagonal in Fig. 2(a) and vertical Fig. 2(b), which we assign to a deviation between the sample surface normal and the $c$-axis in the current $x=0.11$ sample Pitney et al. (2000).

Having measured the base temperature behavior, we explore possible variations with temperature. To provide context for this, we plot the magnetic susceptibility and relative CDW peak intensities at both doping levels in Fig. 3. As seen in Fig. 3(a), La2-xBaxCuO4 $x=0.125$ becomes superconducting only below 5 K Li et al. (2007), whereas for La2-xBaxCuO4 $x=0.11$, the superconductivity occurs around 20 K with the magnetic susceptibility leveling off below 11 K Hücker et al. (2011). Figure 3(b) plots the relative CDW peak intensity $r$, defined as the ratio of the CDW scattering peak intensity to the total scattering intensity (peak + background). We choose this means of quantifying intensity as it will prove useful in evaluating the speckle visibility later in this work. The CDW order sets in at lower temperature for lower dopings: 48 K for $x=0.11$ versus 54 K for $x=0.125$ and grows in intensity as the temperature is reduced consistent with Ref. Hücker et al. (2013).

The time dependence of the CDW domain configurations was then tested at different temperatures by preparing “waterfall” plots, or kymographs, as shown in Fig. 4. Each panel is constructed by taking the same horizontal single-pixel-wide line-cut through the CDW peak at different times and stacking them on top of one another. Vertical lines in these plots come from speckles persisting at the same pixel and indicate static CDW domains. Despite the signal weakening with increasing temperature, these plots indicate static CDW behavior up to 40 K. We further quantified the statistical speckle behavior using the normalized one-time correlation function

[TABLE]

This correlates $I(\textbf{\em q},t)$, the intensity in a pixel at wavevector q and time $t$, with the same quantity at lag time $\tau$ later Sutton (2008); Shpyrko (2014). $\langle\dots\rangle$ indicates averaging over equivalent q and different start times. $F(\textbf{\em q},\tau)$ is called the intermediate scattering function, which describes the time-dependent behavior of the sample and is defined as equal to 1 in the $\tau\rightarrow 0$ limit. Therefore, any reduction in $F(\textbf{\em q},\tau)$ at finite $\tau$ implies the presence of dynamics. $\beta$ denotes the speckle contrast factor, the square of the optical visibility, which describes the magnitude of the speckle-modulated intensity

[TABLE]

where $I_{\text{s}}^{\text{max}}$ and $I_{\text{s}}^{\text{min}}$ are the maximum and minimum speckle-modulated intensities. Here we use the full measured intensity (from both the CDW and the fluorescence background) in order to avoid issues with subtracting background, which can be ambiguous when dealing with the weak CDW intensity at temperatures near the transition. $\beta$ can be reduced from its maximum value of 1 by the coherence properties of the incident beam, the background strength and intrinsic fast-timescale sample dynamics. As we go on to explain, the reduction in $\beta$ observed here can be assigned to the beam coherence and the signal-to-background ratio. In these measurements, the x-ray beam at the sample had horizontal and vertical transverse coherence lengths of $\xi_{h}=\xi_{v}=10$ $\mathrm{\mu}$m (as set by the pinhole) and energy resolution $\Delta\lambda/\lambda\approx 1/1900$ giving a longitudinal coherence length $\xi_{\parallel}=2.5$ $\mathrm{\mu}$m. Without taking into account the background coming primarily from x-ray fluorescence, the expected $\beta$ calculated based on the scattering angles and x-ray penetration depth is $\beta\sim 0.04$ 111The details on the calculations are available in the Supplementary Materials of Ref. Chen et al. (2016).. A constant background further suppresses the speckle visibility by $r(T)^{2}$, where $r(T)$ was previously defined for Fig. 3(b) to yield a $\beta$ value of approximately 0.002 at our base temperature of 10 K. Empirically we measure $\beta=0.003(1)$ consistent with all of the measured CDW intensity being static, otherwise we would expect to see reduced visibility.

Figure 5 plots $g_{2}(\tau)-1$ where we have explored low temperatures in the superconducting state, through the superconducting transition, up to temperatures approaching the CDW transition at 48 K. The base temperature data at 10.5 K is independent of timescale up to around $\tau\approx 100$ minutes. As the sample is heated up, the $g_{2}(\tau)-1$ traces are roughly similar in shape, but with a reduction in $g_{2}(\tau\rightarrow 0)-1=\beta$, which comes from the drop in $r(T)$ near the transition 222See the Supplemental Material at XXX for further details on the temperature scaling of the speckle contrast factor. We consequently conclude that the CDW in La2-xBaxCuO4 $x=0.11$ is static up to at least 100 minutes up to a temperature of 45 K. At higher temperatures, the signal to noise ratio proved insufficient for a definitive statement.

These results show that static CDW order exists in the superconducting state, which has an onset temperature of 20 K and which saturates below 11 K. We believe this to be the first demonstration of static CDW order, on the timescale of hours, in the superconducting state of a ‘214’-type cuprate. We furthermore find that the speckle contrast is consistent with perfectly coherent scattering from the sample, with no evidence for significant fast fluctuations. Other techniques capable of detecting CDW order on similar macroscopic timescales to those studied here include scanning tunneling spectroscopy (STS), which has been applied to cleavable cuprates such as Bi2Sr2Can-1CunO2n+4+x and Ca2-xNaxCuO2Cl2 Hoffman et al. (2002); Hanaguri et al. (2004); Parker et al. (2010); Comin et al. (2014); da Silva Neto et al. (2015). STS has demonstrated robustly static charge order, but is surface sensitive. Indeed, the difficulty of producing large, atomically flat surface areas has prevented STS imaging studies for various non-cleavable cuprates including La2-xBaxCuO4 Valla et al. (2006). CDWs in ‘214’-type cuprates are also highly challenging to study with nuclear magnetic resonance (NMR). Due to the lack of a spin gap, the signal tends to be dominated by spin fluctuations Julien et al. (2001); Pelc et al. (2017). To date, many of the most compelling NMR studies have focused on YBa2Cu3Oy Wu et al. (2011, 2013, 2015), reporting static CDW order at low temperatures. It has been inferred that disorder-free YBa2Cu3Oy Wu et al. (2015) might be dynamic, similar to classical CDW systems such as chromium Shpyrko et al. (2007) and TaS2 Su et al. (2012).

How can we reconcile the existence of static CDW order below the superconducting transition in light of classical ideas that these two order parameters should compete? Trivial, macroscopic inhomogeneity is, a priori, possible, but as shown in Fig. 3(a) the magnetic susceptibility evinces a bulk superconductor similar to previously reported results Hücker et al. (2013). Volume-sensitive probes of the CDW order parameter such as thermopower Tranquada et al. (2008), muon spin rotation Savici et al. (2005) and nuclear magnetic/quadrapole resonance Hunt et al. (2001); Pelc et al. (2017) also do not find evidence for significant inhomogeneity. More compellingly still, diffraction measurements show that CDW order is enhanced when superconductivity is suppressed by the application of magnetic field ruling out the idea that the ordering parameters are completely separated from one-another Hücker et al. (2013).

One particular strength of XPCS is that it is sensitive to all the fluctuations within its measured range of wavevectors, in this case $\Delta Q\approx$ 0.005 r.l.u. Sutton (2008); Shpyrko (2014). Within this range, our data implies fully static CDW correlations on a timescale of hours and as we note above, we can exclude any significant fast fluctuations in terms of speckle visibility considerations. Our results therefore argue against any important role for CDW fluctuations around this wavevector for mediating superconductivity Castellani et al. (1996); Vojta et al. (2000); Wang and Chubukov (2015); Caprara et al. (2016). An interesting recent development is the discovery of finite charge correlations even at wavevectors very far from the central ordering wavevector, consistent with the idea that the CDW peak arises from a small fraction of pinned precursor correlations Miao et al. (2017); Bozin et al. (2015); Reznik et al. (2006). A role for such precursor correlations remains possible, but this is beyond the scope of the current study as the intensity of such correlations is too low to be studied, despite the exceptionally high sensitivity of the 23-ID-1 setup.

A final consideration relates to pair-density-wave states Berg et al. (2007); Corboz et al. (2014); Lee (2014); Fradkin et al. (2015). In such a state, the superconducting pair wave function intertwines with the CDW to form a spatially modulated state that is commensurate with the CDW but with twice the period. STS has recently provided direct evidence for such a state in Bi2Sr2CaCu2O8+x Hamidian et al. (2016). Transport and neutron scattering measurements are also suggestive of such a state in La2-xBaxCuO4 Li et al. (2007); Xu et al. (2014). This state is also compatible with our observation of static coexistence of CDW and superconducting order parameters Berg et al. (2007); Corboz et al. (2014); Lee (2014); Fradkin et al. (2015).

In conclusion, we have used Cu $L_{3}$ edge resonant XPCS to measure the nature of the CDW correlations in La2-xBaxCuO4 $x=0.11$. We demonstrate that static CDW correlations exist over a timescale of at least 100 minutes within the superconducting state and persist up to at least 85% of the CDW transition temperature. The apparent dichotomy of competing but coexisting order parameters can be possibly reconciled by invoking pair density wave states.