Evolution of Charge and Pair Density Modulations in Overdoped Bi2Sr2CuO6+delta

One of the central issues concerning the mechanism of high temperature superconductivity in cuprates is the nature of the ubiquitous charge order and its implications to superconductivity. Here we use scanning tunneling microscopy to investigate the evolution of charge order from the optimally doped to strongly overdoped Bi2Sr2CuO6+{\delta} cuprates. We find that with increasing hole concentration, the long-range checkerboard order gradually evolves into short-range glassy patterns consisting of diluted charge puddles. Each charge puddle has a unidirectional nematic internal structure, and exhibits clear pair density modulations as revealed by the spatial variations of superconducting coherence peak and gap depth. Both the charge puddles and the nematicity vanish completely in the strongly overdoped non-superconducting regime, when another type of short-range order with root2 * root2 periodicity emerges. These results shed important new lights on the intricate interplay between the intertwined orders and the superconducting phase of cuprates.

One of the central issues concerning the mechanism of high temperature superconductivity in cuprates is the nature of the ubiquitous charge order and its implications to superconductivity.
Here we use scanning tunneling microscopy to investigate the evolution of charge order from the optimally doped to strongly overdoped Bi2Sr2CuO6+δ cuprates. We find that with increasing hole concentration, the long-range checkerboard order gradually evolves into short-range glassy patterns consisting of diluted charge puddles. Each charge puddle has a unidirectional nematic internal structure, and exhibits clear pair density modulations as revealed by the spatial variations of superconducting coherence peak and gap depth. Both the charge puddles and the nematicity vanish completely in the strongly overdoped non-superconducting regime, when another type of short-range order with √2 × √2 periodicity emerges. These results shed important new lights on the intricate interplay between the intertwined orders and the superconducting phase of cuprates.

I. INTRODUCTION
A key characteristic of the cuprate high temperature superconductors is the strong tendency for the charge carriers to form symmetry-breaking ordered states [1]. In particular, the charge order phenomena have been found ubiquitously by various experimental techniques in different cuprate families [2][3][4]. Earlier scanning tunneling microscopy (STM) experiments revealed a checkerboard charge order on the surface of Bi2Sr2CaCu2O8+δ (Bi-2212) and Ca2-xNaxCuO2Cl2 (Na-CCOC), especially when the superconductivity is suppressed by magnetic field, reduced hole density, and elevated temperatures [2,[5][6][7]. More recently, elastic and inelastic x-ray scattering experiments confirmed that the charge order is a bulk phenomenon, and covers a wide range of the cuprate phase diagram [8][9][10]. Currently, there are still heated debates regarding the nature and implications of the charge order, such as whether it is a nematic order that breaks the C4 rotational symmetry [11][12][13][14][15][16][17][18][19], and whether it competes with superconductivity [7,14,20,21].
Most previous STM studies on the charge order focused on underdoped and optimally doped cuprates with a well-defined pseudogap phase. It was found that the checkerboard pattern with 4 a0 periodicity (a0 is the lattice constant ~3.8 Å) is the first electronic order that emerges when doping holes into the parent Mott insulator [22], and it persists to hole density close to optimal doping [23]. With further doping into the overdoped regime, resonant x-ray scattering [9] and STM experiments [23] reported the persistence of checkerboard order with increasing wavelength up to 10 a0. However, it remains unknown when and how the checkerboard order eventually vanishes in the strongly overdoped regime. Another important recent progress is the observation that the charge order is closely related to the pair density wave (PDW) order [24][25][26][27][28][29], i.e., a periodic modulation of the Cooper pairing amplitude. Both scanning Josephson tunneling microscopy on optimally doped Bi-2212 [30] and spectroscopic imaging STM on underdoped Bi-2212 [31] illustrate that the PDW has the same periodicity as the charge order, and they are positively correlated and in phase with each other. However, so far there is no experimental study about the PDW order in the overdoped regime, where superconductivity is severely weakened. Therefore, it is highly desirable to elucidate the doping evolution of both the charge and pair density orders in the overdoped regime of cuprates.
In this article, we report STM studies on Bi2Sr2CuO6+δ (Bi-2201) cuprates with La or Pb substitutions from the optimally doped to strongly overdoped regime. We find that with increasing hole density, the long-range checkerboard charge order gradually evolves into short-range glassy patterns consisting of diluted charge puddles. More interestingly, the unidirectional stripe-like internal structure of the charge puddle pervades in the entire superconducting (SC) phase, and exhibit clear PDW features as revealed by the spatial variations of SC coherence peak and gap depth. Both the charge puddles and the intra-puddle structure vanish completely in the strongly overdoped non-SC regime, when another type of short-range order with √2 × √2 periodicity emerges. These results shed important new lights on the intricate interplay between the intertwined orders and the SC phase of cuprates.
High-quality Bi-2201 single crystals with La or Pb substitutions are grown by the traveling solvent floating zone method and post annealed in O2 for extended period, as described in a previous report [32,33]. The hole densities of the four samples studied here are p = 0. 16 surface [34], and dI/dV spectra are collected by using standard lock-in technique with a modulation bias voltage with frequency f = 423 Hz and 3 mV RMS amplitude.

A. Checkerboard charge order in the optimally doped sample
We first investigate the optimally doped Bi2Sr1.63La0.37CuO6+δ with hole density p = 0.16 and Tc = 32.5 K (denoted as OP-32K). The topographic image in Fig. 1(a) shows that the Bi atoms and the structural super-modulations of the exposed BiO surface can be clearly resolved. Figure 1(b) displays the differential conductance map dI/dV(r, V) at a representative bias voltage V = 10 mV, which directly visualizes the spatial distribution of local electron density of state (DOS) in this field of view. The most pronounced feature is the well-known checkerboard pattern, in which granular charge puddles with a typical diameter ~ 2 nm form a long-range order along the Cu-O 4 bond direction. These results are highly consistent with previous STM studies on underdoped and optimally doped Bi-2201 [22,35].
To quantitatively characterize the checkerboard order, in Fig. 1(c) we plot the Fourier transform (FT) of the dI/dV map in Figs. 1(b), which was widely utilized to determine the modulation wavevector of charge orders in cuprates [5,7,23,36].

B. Charge order evolution in the overdoped regime
As the hole density is increased to p = 0.19, 0.21 and 0.23, Tc is decreased to 24 K (OD-24K), 15 K (OD-15K) and < 2 K (OD-0K), respectively. The charge order phenomena exhibit significant and systematic variations with doping. Figures 2(a) displays the topographic image of the OD-24K sample with chemical formula Bi1.62Pb0.38Sr2CuO6+δ, in which the surface Bi atoms form a regular square lattice as the structural supermodulation is removed by Pb substitutions [38].
is the dI/dV map measured at V = 10 mV, in which granular charge puddles with typical size ~2 nm are still evident, but the inter-puddle pattern is less close-packed and less ordered than the optimally doped sample shown in Fig. 1(b). This is directly revealed by the FTmap in Fig. 2(c), in which the wavevector corresponding to the checkerboard order becomes barely visible. The AC(R) map in Fig. 2(d) still has four bright spots (circled in red), but is much weaker and less symmetric than that in Fig. 1(d). By averaging the first peak position along the two directions, the inter-puddle distance is obtained to be CO  5.4 a0. The first peak in the AC(R) line-cuts shown in Fig. 2(e) is non-dispersive in the bias range from 0 to 30 mV, so that the observed patterns are also static charge density modulations with short-range correlations.
In the more overdoped OD-15K sample, the checkerboard pattern is further weakened. order, which can be seen more clearly by the cross feature (circled by black dash lines) at the center of the AC(R) map in Fig. 4(d).
The comparison between the FT map and AC(R) map in the overdoped samples clearly demonstrates the advantage of the autocorrelation analysis. When the charge density modulation becomes short-ranged, its feature in the FT map becomes very fuzzy. Instead, the autocorrelation maps are more sensitive to short-range correlations, thus the glassy patterns or small patches of orders can still be revealed.
To quantify the charge puddle density for each sample, we identify the non-dispersive local maxima on dI/dV maps as the center of charge puddles (indicated by yellow dots in supplementary Figs. S1(a)-(d)) and count the total number in the measured area. Figure 5 density. On the contrary, the √2 × √2 charge order that prevails in the OD-0K sample displays an opposite trend. The percentage of the area with √2 × √2 charge order (see the estimation in supplementary section 2) grows with overdoing and increases steeply upon entering the non-SC regime. Therefore, in the overdoped regime of Bi-2201, the checkerboard order consisting of granular charge puddles along the Cu-O bond direction becomes more dilute and irregular, and is eventually replaced by the √2 × √2 order along the Cu-Cu direction.

C. The internal structure of charge puddles
Another intriguing observation is that in the dI/dV maps of all three SC samples shown in  [18,[39][40][41]. In the OD-15K sample, the stripy pattern is still so pronounced that it can be directly visualized by the maze-like topography in Fig. 3(a), also similar to that observed previously in lightly doped CCOC [16]. The granular charge puddles and the constituent stripes disappear altogether in the overdoped non-SC sample OD-0K ( Fig. 4(b)).

A. Pair density wave in the optimally doped regime
The evolution of charge order is also closely entangled with the PDW order, which has become a focus topic in the debates of intertwined orders in cuprates [24][25][26][27][28][29][30][31]42,43]. Traditionally, the charge order is revealed by the periodic modulation of dI/dV at specific energies, as shown above in the dI/dV maps. In order to unveil the PDW order, one must extract the spectral information directly related to the SC properties. In a previous report [31], we have demonstrated that there are two quantities in the dI/dV spectrum that directly characterize the strength of local superconductivity. The first is the depth of the SC gap, which reflects the depletion of low energy quasiparticle spectral weight by Cooper pairing. This quantity can be represented by the height difference H = dI/dV(VSC) -dI/dV(0) between the dI/dV values at the SC gap edge and zero bias.
The second is the amplitude of the SC coherence peak, which has been shown by various experimental techniques to be related to the local superfluid density [44][45][46][47][48][49]. This quantity can be characterized by the minus second derivative D(V) = −d 3 I/dV 3 at the SC gap edge of the dI/dV spectrum, which is proportional to the sharpness of the coherence peak. These data analysis methods on Bi-2212 have revealed similar PDW features to that obtained by scanning Josephson tunneling microscopy, which directly probes the local SC order parameter.
We first apply these spectral analyses to the optimally doped Bi-2201 sample. Figure 6(a) displays dI/dV curves along a line in OP-32K, which crosses two charge puddles as indicated in the inset. All the spectra possess two separated gaps: a large pseudogap with typical size PG ~ 40 meV and a small SC gap SC  10 meV. Even from the raw data, it can be easily seen that the SC coherence peaks manifest spatial modulations following the charge order, and are much more pronounced on the granular puddles. Three representative dI/dV spectra are shown in Fig. 6(b) with the corresponding colored spots marked in the inset. The spectrum taken on top of the bright stripe within a puddle (green curve) has very sharp SC coherence peaks. The blue curve taken on a less bright stripe has relatively weaker coherence peaks, whereas the magenta curve taken between two puddles has shoulders instead of peaks at VSC = 10 mV. The depth of the SC gap as represented by the height difference H = dI/dV(VSC) -dI/dV(0) for the three curves are marked by the dashed lines in Fig. 6(b), which clearly reveals that the green, blue and magenta curves have the strongest, intermediate and weakest superconductivity. To directly visualize the spatial distribution of pair density, the SC gap depth H and coherence peak sharpness D are extracted from each dI/dV curve acquired on the area in Fig. 1(a), and the H-map and D-map at VSC ~ 10 mV are displayed in Figs. 6(d) and 6(e), respectively. Both images reveal similar long-range checkerboard-like patterns, which unveil the periodic modulation of SC properties, i.e., a well-defined PDW order. More interestingly, in both images the pair density modulation also has the stripe-like internal structure within each puddle, which can already be anticipated from the line-cut in Fig. 6(a). To elucidate the relationship between charge and pair 8 density modulations, the cross correlations of H(r) and D(r, VSC) with the corresponding dI/dV(r, V = 10 mV) are shown in Figs. 6(f) and 6(g). Both cross-correlation maps exhibit a maximum at the center, indicating a positive correlation between the charge and pair density modulations.

B. Pair density modulations in the overdoped regime
We extend the same measurement and analysis procedures regarding the pair density modulations to the overdoped OD-24K sample with p = 0.19. In the inset of Fig. 7(a), a line-cut of dI/dV curves is obtained across a single puddle. For each curve, there are still well-defined SC coherence peaks with gap size SC  10 meV, while the pseudogap size is reduced to PG ~ 25 meV and the features become much weaker. Similar to that in the OP-32K sample, the SC coherence peaks exhibit periodic spatial variations that are strongly correlated to the charge puddle and the stripe-like internal structure. The three representative colored curves in Fig. 7(a) demonstrate that the SC coherence peaks are most pronounced at the bright stripe within a puddle (green curve), become slightly weakened (blue curve) at a less-bright stripe, and evolve into broad shoulders at locations between charge puddles (magenta curve). The H-map and D-map of the OD-24K sample on the same area as Fig. 2(a)  illustrate the cross-correlation between the H-map and D-map with the dI/dV map in Fig. 2(b), and the maximum at the center demonstrates that the pair density modulations are positively correlated with the charge density modulations When the hole density is increased to p = 0.21, the pseudogap of the OD-15K sample becomes too small to be distinguished from the SC gap, as revealed by the dI/dV curves in Fig.   7(d). Nevertheless, the SC coherence peak features still closely follow the granular puddles and stripes, as revealed by the three representative colored curves in the linecut. Even though the longrange inter-puddle ordering disappears, the local puddles and internal stripes are still evident in both the H-map (Fig. 7(e)) and D-map (Fig. 7(f)). The insets display the cross-correlation maps between them and the charge order map in Fig. 3(b), and the central maximum reveals a positive correlation between the intertwined pair and charge density modulations.

IV. DISCUSSION
The origin of the various intertwined orders in cuprates, the intricate relationship between them and implications to superconductivity have been core issues in the mechanism problem. Our systematic STM studies on Bi-2201 cover a broad range of phase diagram from the optimally doped to overdoped non-SC regimes (Fig. 5(a)), which was much less explored compared to the underdoped regime. In particular, the quantitative analysis of the overall dI/dV lineshape enables us to extract the characteristic features that are selectively sensitive to local superconductivity, and the correlation analysis between various spatial patterns reveal the relationship between the intertwined orders. Our results clarify several important issues concerning the fate of charge and pair density modulations in overdoped cuprates, as will be discussed below.
Firstly, we show that with increasing doping, the long-range checkerboard orders in the underdoped and optimally samples gradually evolve into a glassy state with diluted and disordered charge puddles in the overdoped regime. This picture is distinctively different from the scenario of expanded checkerboard wavelength with overdoping while keeping the ordered structure intact [9,23], despite the deceptive agreement of increased average distance between neighboring puddles in both pictures. The interpretation for the superlattice expansion involves the weak coupling theory of Fermi surface nesting with enlarged hole pocket [4,35], whereas our results indicate that the strong local interaction is still essential even in the overdoped regime.
Secondly, we show that the nematic electronic structure is highly robust against overdoping, and exist in all SC samples studied so far in the Bi-2201 system. As illustrated by the cartoon in Fig. 5(c), the C4-symmetry-breaking stripes form a charge puddle with size ~ 4 a0, or 2 nm, and the puddles are subsequently organized into long-range checkerboard order in the optimally doped sample and diluted glassy state in the overdoped regime (Fig. 5(d)). This observation provides another evidence for the relevance of residual correlation effect between doped holes in overdoped cuprates [11,15,18,[50][51][52].
Thirdly, we show that the charge and pair density modulations are closely entangled, and coexist with the SC phases at least up to p = 0.21. More importantly, they vanish altogether in the strongly overdoped limit when superconductivity disappears. This observation suggests that these incipient charge and pair density modulations are not simple competing orders to superconductivity, but may instead share some common origins with Cooper pairing. In the 10 strongly overdoped non-SC regime, they are eventually replaced by the √2 × √2 charge order ( Fig. 5(e)), which may represent a mechanism for suppressing Cooper pairing [38].
In summary, our STM studies on Bi-2201 cuprates reveal the evolution of charge and pair density modulations in the overdoped regime. We find a gradual deformation of the checkerboard charge order, but the granular puddles with internal nematicity are more robust and exhibit close correlations with local pair density modulations. The dichotomy between the intra-puddle structure and inter-puddle order, plus the complete disappearance of both features in the non-SC regime, shed important new lights on the nature of these intertwined orders in cuprates.