Visualizing the charge order and topological defects in an overdoped (Bi,Pb) 2 Sr 2 CuO 6+x superconductor
Ying Fei, Yuan Zheng, Kunliang Bu, Wenhao Zhang, Ying Ding, Xingjiang Zhou, Yi Yin
VVisualizing the charge order and topological defects in anoverdoped (Bi,Pb) Sr CuO x superconductor Ying Fei, Yuan Zheng, ∗ Kunliang Bu, WenhaoZhang, Ying Ding, Xingjiang Zhou,
2, 3 and Yi Yin
1, 4, † Department of Physics, Zhejiang University, Hangzhou 310027, China Beijing National Laboratory for Condensed Matter Physics,Institute of Physics, Academy of Sciences, Beijing 100190, China Collaborative Innovation Center of Quantum Matter, Beijing 100871, China Collaborative Innovation Center of Advanced Microstructures,Nanjing University, Nanjing 210093, China
Abstract
Electronic charge order is a symmetry breaking state in high- T c cuprate superconductors. Inscanning tunneling microscopy, the detected charge-order-induced modulation is an electronic re-sponse of the charge order. For an overdoped (Bi,Pb) Sr CuO x sample, we apply scanningtunneling microscopy to explore local properties of the charge order. The ordering wavevectoris non-dispersive with energy, which can be confirmed and determined. By extracting its order-parameter field, we identify dislocations in the stripe structure of the electronic modulation, whichcorrespond to topological defects with an integer winding number of ±
1. Through differentialconductance maps over a series of reduced energies, the development of different response of thecharge order is observed and a spatial evolution of topological defects is detected. The intensity ofcharge-order-induced modulation increases with energy and reaches its maximum when approach-ing the pseudogap energy. In this evolution, the topological defects decrease in density and migratein space. Furthermore, we observe appearance and disappearance of closely spaced pairs of defectsas energy changes. Our experimental results could inspire further studies of the charge order inboth high- T c cuprate superconductors and other charge density wave materials. ∗ [email protected] † [email protected] a r X i v : . [ c ond - m a t . s up r- c on ] J u l NTRODUCTION
The electronic charge order, accompanying the pseudogap (PG) state, is an interestingphenomenon in high- T c cuprate superconductors [1–3]. In the electronic phase diagram, thePG state and the charge order emerge when a parent Mott insulator is doped with chargecarriers [4]. With the increase of doping level, superconductivity is developed and coexistswith both the PG and charge order states over a broad doping range. The study of the PGand charge order states may help to unravel the mechanism of high- T c superconductivity [3–26].As a powerful tool of detecting the electronic structure with atomic resolution [27], scan-ning tunneling microscopy (STM) has been extensively applied to investigate the chargeorder of cuprates [4, 6–20]. In an STM conductance map , the response of the chargeorder is usually identified by a checkerboard-like modulation along two perpendicular CuObond directions. In the Fourier-transformed data, the checkerboard-like modulation is rep-resented by four peaks centered at ± q ∗ x and ± q ∗ y . The absolute values of the charge orderpeaks are inversely proportional to the periodicity a /δ of the real-space modulation, givenby | q ∗ x | = | q ∗ y | = 2 πδ/a with a the lattice constant. The checkerboard-like modulationwith δ ≈ / δ ≈ / d -form factor [12–14], compatible with the charge order of δ ≈ / Sr CaCu O x (Bi-2212) and Bi Sr CuO x (Bi-2201). With respect to the crystal struc-ture, two CuO layers exist in a unit cell of Bi-2212, while only one in that of Bi-2201. Theelectronic structures of these two cuprates are different as well. The PG state vanishes inthe middle of the superconducting regime in Bi-2212 [10], while it extends to the overdopedregime in Bi-2201 [28, 29]. Therefore the charge order in Bi-2201 can be explored in a broadrange of doping. The determination of the ordering wavevector has been a main focus inprevious studies [14, 20]. With the increase of doping, a commensurate to incommensuratetransition of the wavevector has been discovered [20].The structural disorder in doped cuprates can induce spatial fluctuations in the electronicorders. In the Ginzburg-Landau theory, the charge order can be described by an order-2arameter field ψ ( r ) = A ( r ) exp[ iφ ( r )], where the amplitude A ( r ) and the phase φ ( r ) varyspatially. In general, charge order is a long-range order with local fluctuations resultingfrom perturbations. Furthermore, dislocations may interrupt the long range order, withdislocation cores represent singularities in the order parameter phase φ ( r ). A nonzero integerwinding number is obtained along any path enclosing such a singular point, thus nameda topological defect. Previous work examining underdoped Bi-2212 extracted the spatialstructure of local fluctuations in the charge-order-induced modulation, revealing a closeconnection to nematicity [9]. A similar study examining the local structure of the charge-order-induced signal in Bi-2201 is a natural extension.In this paper, we apply STM to study the charge order in overdoped Bi-2201. A non-dispersive and incommensurate wavevector is determined to be at δ ≈ .
83. From thelocal amplitude and phase of the modulation, we identify singular topological defects in theorder parameter field, corresponding to dislocation cores in the filtered stripe structure of thecharge-order-induced modulation. Around a single defect, the data analysis leads to a genericphase slip picture. As energy increases away from the Fermi level, the response of the chargeorder is gradually enhanced, and the topological defects migrate in space. The appearanceand disappearance of defect pairs are also observed. The defect ‘movement’ can be relatedto the transfer of dislocation cores in regions of bent stripes. The total number of defectsdecreases with energy until it saturates, which implies that the intensity of modulation canaffect how robust the response of charge order is against local perturbations. This energy-dependent phenomenon can be further investigated in other cuprate superconductors andcharge density wave materials.
MATERIALS AND METHOD
The high-quality (Bi,Pb) Sr CuO x single crystal in our experiment is grown by thetraveling-solvent floating-zone method [30]. With a size of 2 mm × × T c ≈
13 K. The hole doping for this overdoped(OD13K) sample is estimated to be p ≈ .
21 according to the Ando formula [31, 32].All the data in this paper are taken in an ultra-high-vacuum STM [29]. The Bi-22013ample is cleaved at liquid-nitrogen temperature and immediately inserted into the STMhead. The measurement is performed at liquid-helium temperature ( T ≈ . V b = 100 mV and a setpoint current of I s = 100pA. The local differential conductance ( dI/dV ) spectra are acquired simultaneously withthe topography by a standard lock-in technique with a modulation frequency of f = 983 . RESULTS
Topography and charge order of overdoped Bi-2201.
We work with a high-qualityoverdoped (Bi,Pb) Sr CuO x single crystal sample [29, 30] whose critical temperature T c ≈
13 K and hole doping p ≈ .
21 [31, 32]. The data is taken in an ultra-high-vacuum STM at T ≈ . ×
27 nm taken on a cleavedBiO surface, showing a square lattice of Bi atoms with the lattice constant a ≈ . V , in whichthe PG magnitudes (∆ PG ) can be extracted from coherence-peak positions. The local elec-tronic structure is probed by a differential conductance map, g ( r , E = eV ) = dI/dV ( r , V ).In our Pb-doped Bi-2201 sample, the distribution of both the differential conductance andthe PG magnitude is spatially inhomogeneous (see Supplementary I). To exclude the spa-tial inhomogeneity of the PG magnitude, the conductance map is rescaled as g ( r , ε ) witha reduced energy ε = E/ ∆ PG ( r ). Subsequently, we calculate a ratio map, Z ( r , ε ) = g ( r , ε ) /g ( r , − ε ), which reduces the systematic error known as the setpoint effect in STMmeasurement [6, 8, 28]. In this field of view (FOV), we also find some zero gap patches whichcan be attributed to the van Hove singularity (VHS) states [27, 29, 34]. A spatially averagedvalue of Z ( r , ε ) is applied to fill in the VHS regions [29]. As the differential conductance isproportional to the electronic density of states, the response of charge order is reflected inthe Z ( r , ε )-maps at different reduced energies, which is the main focus of data analysis inthis paper.Figure 1(b) displays a ratio map, Z ( r , ε = 1 . x yx q x q y nm 5 nmLow High Low High (a) (b) (c) Low High(0, -2 π /a ) (2 π /a , 0) (d) I n t en s i t y ( a . u . ) q y / Q y * ε FIG. 1. Topography and electronic properties of an overdoped Bi-2201 sample. ( a ) A 27 nm ×
27 nm topographic image taken on a cleaved BiO surface. ( b ) The ratio map Z ( r , ε = 1 .
05) inthe same FOV as in panel ( a ), with the VHS regions filled in the averaged value. ( c ) The Fouriertransform of panel ( b ). The four Bragg peaks are highlighted by yellow circles, while the chargeorder peaks are highlighted by white circles. ( d ) In the top and bottom panels, the position andintensity of the charge order peak are extracted as a function of the reduced energy ε , respectively. same FOV as in Fig. 1(a). A small deviation of ε from unity is due to our technique of binningthe reduced energies. A checkerboard-like spatial modulation is observed, which representsa response of the static charge order in recorded STM results. To maintain an atomicregistry across the FOV, drift correction of the lattice is applied to both topographic andelectronic data [11]. To quantify the spatial periodicity of charge-order-induced modulationat the PG energy, the Fourier transform of Fig. 1(b), ˜ Z ( q , ε ) = FT[ Z ( r , ε )], is shown inthe momentum q -space in Fig. 1(c). Four sharp Bragg peaks located at ± Q x and ± Q y are observed, each collapsed into a single pixel due to the lattice drift correction [11, 29].The Bragg wavevectors, Q x = (2 π/a ,
0) and Q y = (0 , π/a ), are consistent with thesquare lattice structure of Bi atoms. In addition, there exist four peaks near the Braggpeaks, which correspond to the checkerboard-like spatial modulation induced by the chargeorder. The centers of the four charge order peaks are estimated to be at ± q ∗ x ≈ ± . Q x and ± q ∗ y ≈ ± . Q y from Gaussian fitting (Supplementary II). The same analysis can beapplied to ˜ Z ( q , ε )-maps at different reduced energies. In the top panel of Fig. 1(d), we plotthe amplitude | q ∗ x/y | of the charge-order wavevectors as a function of reduced energy ε . As ε changes, the wavevectors q ∗ x/y are nearly invariant, suggesting a non-dispersive and staticcharge order. The wavevectors are incommensurate with the lattice, consistent with theresult for overdoped Bi-2201 in Ref. [20]. In the bottom panel of Fig. 1(d), we also plot the5 a) (b) (c)(d) (e) (f) Low High Low High - π + π Z A φ x x x Z A φ y y yff nm 5 nm 5 nm FIG. 2. Maps of the charge order after a Gaussian filtering. ( a ) Z xf ( r , ε = 1 . b ) A x ( r , ε = 1 . c ) φ x ( r , ε = 1 .
05) describe x -direction modulations. ( d ) Z yf ( r , ε = 1 . e ) A y ( r , ε = 1 . f ) φ y ( r , ε = 1 .
05) describe y -direction modulations. In ( c ) and ( f ), topological defects withpositive and negative polarities are shown with white and black circles, respectively. intensity of the charge-order peak as a function of ε , showing that the response of the chargeorder is weak at small values of ε and becomes intensified around the PG energy. A moredetailed discussion of the ε -dependence will be provided later. Identification of topologaical defects.
To explore the spatial variation of thecheckerboard-like modulation, we first separate the x - and y -components with a Gaus-sian filtering technique. The x -component of the modulation in the momentum space isextracted as ˜ Z xf ( q , ε ) = ˜ Z ( q , ε )[ ˜ f ( q + q ∗ x ) + ˜ f ( q − q ∗ x )] , (1)where ˜ f ( q ) = exp( − q Λ /
2) is a Gaussian filtering function. In practice, the cutoff size ischosen at Λ = 1 . Z xf ( r , ε ) = FT − [ ˜ Z xf ( q , ε )], weobtain the checkerboard-like modulation along the x -direction in the real space. The same6echnique is applied to extract the y -components, ˜ Z yf ( q , ε ) and Z yf ( r , ε ). The final resultsof Z xf ( r , ε = 1 .
05) and Z yf ( r , ε = 1 .
05) at the PG energy are plotted in Figs. 2(a) and 2(d),from which we observe stripes along the y - and x -directions, respectively. In general, the x -component of the charge order can be re-expressed as Z xf ( r , ε ) = A x ( r , ε ) cos[ q ∗ x · r + φ x ( r , ε )] , (2)where A x ( r , ε ) and φ x ( r , ε ) are the amplitude and phase of the order-parameter field atlocation r . Here another Gaussian filtering process is applied to extract φ x ( r , ε ) around thecharge order wavevector [9], given bytan φ x ( r , ε ) = R d r Z xf ( r , ε ) sin[ q ∗ x · r ] f ( r − r ) R d r Z xf ( r , ε ) cos[ q ∗ x · r ] f ( r − r ) , (3)with f ( r ) = FT − [ ˜ f ( q )]. The amplitude A x ( r , ε ) is then calculated using the definitionin Eq. (2). Accordingly, the order-parameter field associated with the x -component ofthe charge order is given by ψ x ( r ) = A x ( r ) exp[ iφ x ( r )]. In Figs. 2(b) and 2(c), we show A x ( r , ε = 1 .
05) and φ x ( r , ε = 1 .
05) extracted from Fig. 2(a). The amplitude and phaseare inhomogeneous, unlike the uniform order parameter of ideal long range order. Thethree spatially resolved quantities, Z xf ( r , ε = 1 . A x ( r , ε = 1 .
05) and φ x ( r , ε = 1 . y -component of the charge-order-induced modulation ( Z yf ( r , ε = 1 .
05) in Fig. 2(d)), and theresulting A y ( r , ε = 1 .
05) and φ y ( r , ε = 1 .
05) are plotted in Figs. 2(e) and 2(f), respectively.From Figs. 2(a) and 2(d), we observe that the stripes of charge-order-induced modulationare distorted in space, including weak fluctuations, as well as strong disruptions. As anillustration, we select a small area in the solid box in Fig. 2(a) and plot magnified maps of Z xf ( r , ε = 1 . A x ( r , ε = 1 .
05) and φ x ( r , ε = 1 .
05) in Figs. 3(a)-3(c). Near the middle ofthe image, a single stripe splits into two stripes along the y -direction, analogous to an edgedislocation in a crystal lattice. The manifestation of the branch point, or dislocation core,in A x ( r ) and φ x ( r ) is associated with the presence of a singularity: (1) The phase φ x ( r )is undefined at the dislocation core, while a winding phase of 2 π is acquired for φ x ( r ) ifa clockwise cycle is taken around this singularity. (2) To generate a physically meaningfulorder parameter, the corresponding amplitude A x ( r ) is suppressed to zero, as shown by adark region near the middle of Fig. 3(b). To further visualize the behavior of the singularity,seven linecuts along the y -direction are chosen around the dislocation in Fig. 3(a). For7 i s p l a c e m en t a l ong t he li ne e i q r y * . { } e i q r y * . { }(a) (b) (c)(d) Low HighLow High - π + π nm 1 nm 1 nm FIG. 3. Enlarged maps of ( a ) Z xf ( r , ε = 1 . b ) A x ( r , ε = 1 . c ) φ x ( r , ε = 1 .
05) around aselected topological defect in the solid box in Figs. 2(a)-2(c). In ( a )-( c ), seven lines perpendicularto the stripe direction are chosen to demonstrate the phase slip picture. ( d ) Along the seven chosenlinecuts in ( a )-( c ), the real and imaginary parts of ψ x ( r ) exp[ i q ∗ x · r ] are displayed as a function ofthe displacement along each linecut. The experimentally extracted data are shown in solid dotsand the spirals (solid lines) are obtained by the cubic interpolation method. each linecut, the real and imaginary parts of ψ x ( r ) exp[ i q ∗ x · r ] are plotted as a function ofthe spatial location r (see Fig. 3(d)). Along the right top linecut (in blue color), a spiral of ψ x ( r ) exp[ i q ∗ x · r ] propagates with a roughly constant amplitude and four turns are generatedwithin the selected distance. As the linecut approaches the dislocation core (the green line),the spiral is disrupted and the middle turn is broken into two very small turns. An extraphase of 2 π is acquired through φ x ( r ) so that the number of the spiral turns is increasedfrom 4 to 5 along the linecuts below the green one, similar to the phase slip picture for otherordered states [35, 36]. Through a combined analysis of Z x/yf ( r , ε ), A x/y ( r , ε ) and φ x/y ( r , ε ),the dislocation cores behave as topological defects in the charge-order-induced modulation,8nd almost do not affect the wavevector of the modulation (see Fig. S3 in SupplementaryIII).With φ x/y ( r ) defined modulo 2 π , a branch cut following a curved line can be emanatingfrom each defect (see Figs. 2(c) and 2(f)). Crossing a branch cut generates an artificial jumpof ± π , which however does not affect the value of the order parameter ψ x/y ( r ). In this FOV,most of these branch cuts connect two defects with opposite polarities defined by the signs oftheir winding phases. In general, topological defects are emergent excitations in an orderedstate, such as quantized vortices in superfluids and superconductors [35, 36]. Compared toa single unbounded defect, a pair of defects with opposite polarities have a lower excitationenergy associated with an attractive interaction. In our system, defect pairing is observedin Figs. 2(c) and 2(f), except for those around the corners of the FOV. Evolution of the charge-order-induced modulation and topological defects.
Incuprates such as Bi-2201 superconductors, various electronic orders coexist and can competewith each other. In Fig. 1(d), we find that response of the charge order appears at a low ε and is intensified at the PG energy, consistent with a consensus that the charge order iscorrelated with the PG state [7, 10, 16, 19, 21, 28]. Next we explore the spatial variationof the charge-order-induced modulation at different reduced energies. The above analysisprocedure is applied to Z ( r , ε ) maps of different ε to obtain the x - and y -components ofthe modulation, as well as their amplitudes and phases. From a series of the filtered mapsof Z xf ( r , ε ) and Z yf ( r , ε ) (see Supplementary Fig. S6 and S8), we observe that the stripestructure of the modulation is maintained when ε increases. However, we also find visiblechanges at certain regions. The one-dimensional structure of a stripe is easily bent by localstress. Neighboring stripes are bent similarly but with gradually decaying strengths. Underlarge stress, dislocations can be created among the bent stripes. As ε changes, this localstress may be released or enhanced, causing ‘movement’ of defects and even their appearanceand disappearance. We emphasize that the charge order itself is a static electronic orderwithout energy dependence. The ε -dependent phenomenon discussed here is the differentresponse of the charge order represented in the STM spectroscopy measurement.To visualize the ε -dependence of the detailed stripe structure, we inspect the phase mapsof φ x/y ( r , ε ). As shown in Figs. 4(a) and 4(b), the overall structure of two phase maps, φ x ( r , ε = 0 .
45) and φ x ( r , ε = 1 . .60.750.91.05 1.050.90.60.75 ε = 0.45 ε = 1.05 − π π − π π ε r r − π π (a) (b)(d) (c)(e) N u m be r ε r r r r FIG. 4. Reduced-energy dependence of the distribution of topological defects. ( a ) and ( b ) arethe two phase maps, φ x ( r , ε = 0 .
45) and φ x ( r , ε = 1 . c ) The total number of defects as a function of the reducedenergy. ( d ) The spatial distribution of defects as a function of the reduced energy. Here r and r represent two perpendicular axes of the FOV. ( e ) Enlarged phase maps of showing how a pair ofdefects appear (right column) or disappear (left column) when the reduced energy increases. Thearea of these maps are shown by white boxes in ( b ). the charge-order-induced modulation but with subtle changes. As shown in Fig. 4(c), thetotal number of defects, extracted from both φ x ( r , ε ) and φ y ( r , ε ), gradually decreases with ε and is stabilized around 30 when ε is larger than 0.5. We record the spatial locations of de-fects at different ε and present a three-dimensional map in Fig. 4(d), showing the movementof defects. The appearance and disappearance of defect pairs are observed, and can evenoccur around the PG energy (see Supplementary VI for more details). Two examples areshown in Fig. 4(e). In the left column, the distance between two defects gradually decreaseswith the increasing ε . For ε = 0 . ε . The ‘movement’ of defects corresponds to transferof dislocation cores within a region of bent stripes (see Supplementary Fig. S10). Further-more, we find a negligible correlation between the topological defects and inhomogeneous PG(see Supplementary IV). Although the spatial resolution of detecting paired defects changeswith the cutoff size of the filtering function (or the coarse grained length equivalently), theevolution of defects qualitatively holds for various cutoff sizes (see Supplementary V). CONCLUSIONS
In this paper, we investigate the charge order in an overdoped (Bi,Pb) Sr CuO x (Bi-2201) sample with STM. In real space, the charge-order-induced modulation is identified inthe ratio Z -map of Z ( r , ε ). In momentum space, the modulation is represented by four peaksaround incommensurate wavevectors ± q ∗ x ≈ ± . π/a ,
0) and ± q ∗ y ≈ ± . , π/a ).The charge-order-induced modulation starts to appear at low reduced energies. With in-creasing ε , the modulation is gradually intensified and reaches its maximum strength aroundthe PG energy. The incommensurate wavevector is not dispersive with ε , consistent withthe static charge order or charge density wave order. The spatially-varied order-parameterfields enable identification of singular points, i.e. topological defects. In consecutive order-parameter maps with changing ε , the positions of defects gradually change, and pairs ofdefects appear and disappear. As energy approaches the PG energy, the number of defectsdecreases and saturates. This phenomenon uncovers a new aspect of the charge-order re-sponse in STM measurement. We expect to investigate whether the observed behavior isgeneric in cuprate superconductors or other charge density wave materials, which requiresfuture experimental and theoretical efforts. ACKNOWLEDGEMENTS
This work is supported by the National Basic Research Program of China (2015CB921004),the National Natural Science Foundation of China (NSFC-11374260), and the FundamentalResearch Funds for the Central Universities in China. XJZ thanks financial support from11he National Natural Science Foundation of China (NSFC-11334010), the National Key Re-search and Development Program of China (2016YFA0300300), and the Strategic PriorityResearch Program (B) of the Chinese Academy of Sciences (XDB07020300).
REFERENCES [1] B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida, and J. Zaanen, Nature (London) ,179 (2015).[2] T. Timusk and B. Statt, Rep. Prog. Phys. , 61 (1999).[3] M. Vojta, Adv. Phys. , 699 (2009).[4] P. Cai, W. Ruan, Y. Y. Peng, C. Ye, X. T. Li, Z. Q. Hao, X. J. Zhou, D.-H. Lee, and Y. Y.Wang, Nat. Phys. , 1047 (2016).[5] G. Gr¨uner, Density Waves in Solids (Perseus Publishing, Cambridge, Massachusetts, 1994).[6] Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C. Lupien, T. Hanaguri, M. Azuma, M. Takano,H. Eisaki, H. Takagi, H. Uchida, and J. C. Davis, Science , 1380 (2007).[7] C. V. Parker, P. Aynajian, E. H. da Silva Neto, A. Pushp, S. Ono, J. S. Wen, Z. J. Xu, G. D.Gu, and A. Yazdani, Nature (London) , 677 (2010).[8] Y. Kohsaka, C. Taylor, P. Wahl, A. Schmidt, J. Lee, K. Fujita, J. W. Alldredge, K. McElroy,J. Lee, H. Eisaki, S. Uchida, D.-H. Lee, and J. C. Davis, Nature (London) , 1072 (2008).[9] A. Mesaros, K. Fujita, H. Eisaki, S. Uchida, J. C. Davis, S. Sachdev, J. Zaanen, M. J. Lawler,and E.-A. Kim, Science , 426 (2011).[10] K. Fujita, C. K. Kim, I. Lee, J. H. Lee, M. H. Hamidian, I. A. Firmo, S. Mukhopadhyay, H.Eisaki, S. Uchida, M. J. Lawler, E. A. Kim, S. Sachdev, and J. C. Davis, Science , 612(2014).[11] M. J. Lawler, K. Fujita, J. Lee, A. R. Schmidt, Y. Kohsaka, C. K. Kim, H. Eisaki, S. Uchida,J. C. Davis, J. P. Sethna, and E.-A. Kim, Nature (London) , 347 (2010).[12] K. Fujita, C. K. Kim, I. Lee, J. H. Lee, M. H. Hamidian, I. A. Firmo, S. Mukhopadhyay, H.Eisaki, S. Uchida, M. J. Lawler, E. A. Kim, S. Sachdev, and J. C. Davis, Proc. Natl. Am. Sci. , E3026 (2014).
13] M. H. Hamidian, S. D. Edkins, C. K. Kim, J. C. Davis, A. P. Mackenzie, H. Eisaki, S. Uchida,M. J. Lawler, E. -A. Kim, S. Sachdev and K. Fujita, Nat. Phys. , 150 (2016).[14] A. Mesaros, K. Fujita, S. D. Edkins, M. H. Hamidian, H. Eisaki, S. Uchida, J. C. Davis, M.J. Lawler, and E. -A. Kima, Proc. Natl. Am. Sci. , 12661 (2016).[15] J. E. Hoffman, E. W. Hudson, K. M. Lang, V. Madhavan, H. Eisaki, S. Uchida, and J. C.Davis, Science , 466 (2002).[16] M. Vershinin, S. Misra, S. Ono, Y. Abe, Y. Ando, and A. Yazdani, Science , 1995 (2004).[17] E. H. da Silva Neto, P. Aynajian, A. Frano, R. Comin, E. Schierle, E. Weschke, A. Gyenis, J.S. Wen, J. Schneeloch, Z. J. Xu, S. Ono, G. D. Gu, M. Le Tacon, and A. Yazdani, Science , 393 (2014).[18] W. D. Wise, M. C. Boyer, K. Chatterjee, T. Kondo, T. Takeuchi, H. Ikuta, Y. Y. Wang, andE. W. Hudson, Nat. Phys. , 696 (2008).[19] W. D. Wise, K. Chatterjee, M. C. Boyer, T. Kondo, T. Takeuchi, H. Ikuta, Z. J. Xu, J. S.Wen, G. D. Gu, Y. Y. Wang, and E. W. Hudson, Nat. Phys. , 213 (2009).[20] T. A. Webb, M. C. Boyer, Y. Yin, D. Chowdhury, Y. He, T. Kondo, T. Takeuchi, H. Ikuta,E. W. Hudson, J. E. Hoffman, and M. H. Hamidian, Phys. Rev. X , 021021 (2019).[21] R. Comin, A. Frano, M. M. Yee, Y. Yoshida, H. Eisaki, E. Schierle, E. Weschke, R. Sutarto, F.He, A. Soumyanarayanan, Y. He, M. Le Tacon, I. S. Elfimov, J. E. Hoffman, G. A. Sawatzky,B. Keimer, and A. Damascelli, Science , 390 (2014).[22] Y. Y. Peng, R. Fumagalli, Y. Ding, M. Minola, D. Betto, G. M. De Luca, K. Kummer, E.Lefrancois, M. Salluzzo, H. Suzuki, M. Le Tacon, X. J. Zhou, N. B. Brookes, B. Keimer, L.Braicovich, M. Grilli, and G. Ghiringhelli, Nat. Mater. , 697 (2018).[23] Y. Y. Peng, M. Salluzzo, X. Sun, A. Ponti, D. Betto, A. M. Ferretti, F. Fumagalli, K. Kummer,M. Le Tacon, X. J. Zhou, N. B. Brookes, L. Braicovich, and G. Ghiringhelli, Phys. Rev. B , 184511 (2016).[24] R. Comin, R. Sutarto, F. He, E. H. da Silva Neto, L. Chauviere, A. Frano, R. Liang, W. N.Hardy, D. A. Bonn, Y. Yoshida, H. Eisaki, A. J. Achkar, D. G. Hawthorn, B. Keimer, G. A.Sawatzky, and A. Damascelli, Nat. Mater. , 796 (2015).[25] J. Chang, E. Blackburn, A. T. Holmes, N. B. Christensen, J. Larsen, J. Mesot, R. X. Liang,D. A. and Bonn, W. N. Hardy, A. Watenphul, M. V. Zimmermann, E. M. Forgan, and S. M.Hayden, Nat. Phys. , 871 (2012).
26] T. Wu, H. Mayaffre, S. Kr¨amer, M. Horvati´c, C. Berthier, W. N. Hardy, R. X. Liang, D. A.Bonn, and M.-H. Julien, Nature (London) , 191 (2011).[27] Ø. Fischer, M. Kugler, I. Maggio-Aprile, C. Berthod, and C. Renner, Rev. Mod. Phys. ,353 (2007).[28] Y. He, Y. Yin, M. Zech, A. Soumyanarayanan, M. M. Yee, T. Williams, M. C. Boyer, K.Chatterjee, W. D. Wise, I. Zeljkovic, K. Takeshi, T. Takeuchi, H. Ikuta, P. Mistark, R. S.Markiewicz, A. Bansil, S. Sachdev, E. W. Hudson, and J. E. Hoffman, Science , 608(2014).[29] Y. Zheng, Y. Fei, K. L. Bu, W. H. Zhang, Y. Ding, X. J. Zhou, J. E. Hoffman, and Y. Yin, Sci. Rep. , 8059 (2017).[30] L. Zhao, W. T., Zhang, H. Y. Liu, J. Q. Meng, G. D. Liu, W. Lu, X. L. Dong, and X. J. Zhou,Chin. Phys. Lett. , 087401 (2010).[31] Y. Ando, Y. Hanaki, S. Ono, T. Murayama, K. Segawa, N. Miyamoto, and S. Komiya, Phys.Rev. B. , 14956 (2000).[32] Y. Fei, K. L. Bu, W. H. Zhang, Y. Zheng, X. Sun, Y. Ding, X. J. Zhou, and Y. Yin, Sci.China-Phys. Mech. Astron. , 127404 (2018).[33] M. H. Hamidian, I. A. Firmo, K. Fujita, S. Mukhopadhyay, J. W. Orenstein, H. Eisaki, S.Uchida, M. J. Lawler, E.-A. Kim, and J. C. Davis, New J. Phys. , 053017 (2012).[34] X. T. Li, Y. Ding, C. C. He, W. Ruan, P. Cai, C. Ye, Z. Q. Hao, L. Zhao, X. J. Zhou, Q. HWang and Y. Y. Wang, New J. Phys. , 063041 (2018).[35] P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics, 1st ed. (Cam-bridge University Press, Cambridge, 1995).[36] M. Tinkham,
Introduction to Superconductivity, 2nd ed. (Courier Corporation, New York,1996). upplementary Information for“Visualizing the charge order and topological defects in anoverdoped (Bi,Pb) Sr CuO x superconductor ” Ying Fei, Yuan Zheng, ∗ Kunliang Bu, WenhaoZhang, Ying Ding, Xingjiang Zhou,
2, 3 and Yi Yin
1, 4, † Department of Physics, Zhejiang University, Hangzhou 310027, China Beijing National Laboratory for Condensed Matter Physics,Institute of Physics, Academy of Sciences, Beijing 100190, China Collaborative Innovation Center of Quantum Matter, Beijing 100871, China Collaborative Innovation Center of Advanced Microstructures,Nanjing University, Nanjing 210093, China ∗ [email protected] † [email protected] a r X i v : . [ c ond - m a t . s up r- c on ] J u l
40 -20 0 20 40 d I/ d V ( a . u . ) Bias (mV)
10 20 30 40 5000.00.050.1 PG (meV) P e r c en t age ( % ) -50 0 50 with VHSwithout VHS d I/ d V ( a . u . ) Bias (mV)-25 25 (a) (b) (c)(d)
FIG. S1. (a) A 27 nm ×
27 nm pesudogap map in the same field of view as in Fig. 1(a) in themain text. (b) The average dI/dV spectra over a bin size of δ ∆ PG = 2 . PG ranging from 2.5 meV to 45 meV. (c) Histogram of the PG magnitudes. (d) Theaverage dI/dV spectra with and without the van Hove singularity states. I. Spatially inhomogeneous differential conductance spectra
The sample studied in this paper is an overdoped (OD) (Bi,Pb) Sr CuO x (Bi-2201)single crystal with the superconducting critical temperature of T c ≈
13 K and the holedoping of p ≈ .
21. For the field of view (FOV) shown in Fig. 1(a) of the main text, a seriesof dI/dV spectra have been taken simultaneously with the topographic image at a densearray of locations. The pseudogap (PG) magnitude (∆ PG ) at each location is determinedby extracting the bias voltage of the positive coherence peak. The symmetric negativecoherence peaks are hidden in the filled-state spectra [1–3]. The resulting PG map is shownin Fig. S1(a), from which a nanoscale inhomogeneity can be observed. This inhomogeneity isalso illustrated in spatially averaged spectra binned by the PG magnitudes (see Fig. S1(b)).The broad distribution of PG, varying from 2.5 meV to 45 meV, is represented by a histogramin Fig. S1(c). With a peak value at ∆ PG ≈
19 meV, the distribution of PG in this overdopedBi-2201 is relatively broader than the one in another overdoped Bi-2201 with a similar T c of15 K [4]. In this FOV, we also find around 10 .
9% area with zero gap, which is attributed tothe van Hove singularity (VHS) state [3, 5]. Figure S1(d) presents the average spectra withand without van Hove singularity states. The two spectra are similar to each other, except2 y / Q y R edu c ed E ne r g y chargeorder Braggpeak (0, -2 π /a ) (2 π /a , 0) q y (a) (b) (c) Low High I n t en s i t y ( a . u . ) q y / Q y ε= ε= ε= FIG. S2. (a) The symmetrized ratio map Z ( r , ε = 1 . Z ( q , ε ) are extracted as a function of the reduced energy ε . (c) Three linecuts are shown asexamples, with ε = 1 .
05, 0.45 and 0.375. Results of the Gaussian function fitting are also shown,from which the charge order peak positions are determined. for a small difference of the zero bias conductance.
II. Determining the wavevectors of the charge-order-induced modulation
In Figs. 1(b) and 1(c) of the main text, we present a ratio map Z ( r , ε = 1 .
05) and itsFourier transform ˜ Z ( q , ε = 1 .
05) at the PG magnitude. To quantify the wavevector q ∗ x/y ofthe charge-order-induced modulation, we symmetrize ˜ Z ( q , ε = 1 .
05) with a four-fold rotationsymmetry and a mirror symmetry [4]. The signal to noise ratio is enhanced accordingly andthe result is plotted in Fig. S2(a). A line of q is drawn along the q y -axis from the center(0 ,
0) to one Bragg peak Q y = (0 , π/a ) in Fig. S2(a). The same operation is applied tothe Fourier transform ˜ Z ( q , ε ) at various reduced energies. In Fig. S2(b), collected linecutsof ˜ Z ( q y , ε ) are shown as a function of the reduced energy ε and the relative amplitude ofthe wavevector | q y / Q y | . To increase the signal to noise ratio, the pixels of ˜ Z ( q , ε ) close tothe q y -axis are partially added. In addition to the sharp Bragg peak at Q y , a bright stripeis observed, which trace the peaks induced by charge order. At each reduced energy, thecharge-order-induced peak in ˜ Z ( q y , ε ) is fitted with a Gaussian function, as shown by theexamples of ε = 1 .
05, 0.45 and 0.375 in Fig. S2(c). Centers of the the peak are located3 a) Low High I n t en s i t y ( a . u . ) (b) q y / Q y FIG. S3. (a) The modified map of Z ( r , ε = 1 . Z ( q , ε = 1 . q y -axis is extracted and comparedwith that for the full FOV. at q ∗ y ≈ . Q y , nearly invariant as the reduced energy changes (see Fig. 1(d) of the maintext). This result indicates a static and non-dispersive order. III. Influence of topological defects in analysis of the charge-order-inducedmodulation
Each ‘topological defect’ of the charge-order-induced modulation corresponds to a localdisruption of stripe-like modulation along x/y -direction, which also causes a local change tothe spatial periodicity. Whether the average wavevector of modulation in ˜ Z ( q , ε ) is affectedby topological defects is an interesting question to be addressed [7]. In the real-space mapof Z ( r , ε = 1 . Z ( r , ε = 1 .
05) = 0, is filled in the selected area as shown in Fig. S3(a).Through this modification, the influence of defects is almost completely removed. Followingthe treatment in Sec. I, we extract the signal of ˜ Z ( q y , ε = 1 .
05) along q y -axis for the modifiedmap of Z ( r , ε = 1 . q ∗ y . Thus, the average wavevector is almost unaffected bytopological defects. 4 nm Low High 5 nm Low High5 nm 0 50 meV (a) (b) (c) (d) C r o ss C o rr e l a t i on Displacement (nm) = 0.300 = 0.975 = 0.300 = 0.975 = 0.300 = 0.975 FIG. S4. (a) The map of the PG magnitude. The topological defects at two typical reducedenergies are superimposed by white and black dots, respectively. (b-c) The distance map of D ( r )for (c) ε = 0 .
300 and (d) ε = 0 . IV. Relation between the topological defects and PG
As discussed previously, the PG magnitude ∆( r ) can be determined for each spatial lo-cation to produce a two-dimensional PG-map. In Fig. S4(a), the topological defects at twotypical reduced energies are superimposed on top of the PG-map. No obvious correlationbetween the topological defects and PG can be directly observed. We need a further quan-titative analysis of the correlation. With the distribution of topological defects, a distancefunction D ( r ) is defined as the distance between the spatial position r and the nearest defect(Fig. S4(b) and S4(c)), similar as in the oxygen defect analysis [8]. The cross correlationbetween the PG map and the topological defect distribution is defined by C ( R ) = − R d r [ D ( r ) − ¯ D ][∆( r + R ) − ¯∆] qR d r [ D ( r ) − ¯ D ] R d r [∆( r ) − ¯∆] (1)where ¯ D is the average distance from the nearest topological defect. An integration overthe angular coordinate is further applied to obtain a cross correlation C ( R ). As shown inFig. S4(d), a negligible cross correlation can be observed in results of C ( R ) for both the highand low reduced energies. 5 = 1.2 nm Λ = 1.4 nm Λ = 1.0 nm (a) (b) (c) nm - π π - π π nm - π π nm FIG. S5. Phase φ -maps ( x -component) extracted from Z ( r , ε = 1 . . V. Effect of the cutoff size in the filtering process
As shown in Eq. (1) of the main text, a Gaussian function ˜ f ( q ) = exp[ − ( q − q ) / − ]is applied to the ratio ˜ Z ( q , ε )-map, which leads to the x - or y -component of the modulationwith q = ± q ∗ x or ± q ∗ y . The cutoff size Λ is thus a variable parameter in our analysis.In practice, we choose Λ = 1 . / Λ ≈ . × π/a is relatively largerthan the standard deviation of the charge-order-induced peak ( ≈ . × π/a ) shown inFig. S2(c). As a result, the majority of modulation information is preserved in our filteredmaps of Z xf ( r , ε ) and Z yf ( r , ε ). To further confirm the reliability of the chosen cutoff size, weconsider two other values, Λ = 1 . φ x ( r , ε = 1 . φ x -patterns and the number of topological defects, themain feature of the spatial distribution is same in these three maps. For a relatively smallercutoff size (or coarsening length), a pair of closely spaced defects is easier to be identified.The smaller the cutoff size, the larger the number of identified defects. The ε -dependenttrend is still kept the same. Thus, the conclusion from our data analysis is reliable from aqualitative point of view. 6 I. Reduced-energy dependence of the spatial distribution of charge-order-induced modulation
In the main text, we discuss the ε -dependence of the spatial distribution of charge-order-induced modulation. Although the modulation is non-dispersive and the STM is a staticmeasurement tool [9, 10], its spatial distribution is not necessarily fixed as ε changes. InFigs. S6 and S7, we present Z xf ( r , ε ) and φ x ( r , ε ) at 11 different values of ε . In Figs. S8and S9, we present Z yf ( r , ε ) and φ y ( r , ε ) at the same values of ε .With the determination of defect locations in φ x/y ( r , ε ), we can map defects back to Z x/yf ( r , ε ) to visualize the defect (or dislocation core) ‘movement’ in the stripe structure.Around each defect, the modulation amplitude is close to zero, which obscures the signalin Z x/yf ( r , ε ). In Fig. S10, we enhance the signal by a normalization of Z x/yf ( r , ε ) /A x/y ( r , ε )together with a cubic interpolation processing. Figure S10(a) is a typical example of singledefect ‘movement’ in Z xf ( r , ε ) from ε = 0 .
675 to 0.750 and 0.825. Figure S10(b) is anexample of how two closely spaced defects ‘annihilate’ through stripe bending in Z yf ( r , ε )from ε = 0 . [1] M. C. Boyer, W. D. Wise, K. Chatterjee, M. Yi, T. Kondo, T. Takeuchi, H. Ikuta and E. W.Hudson, Nat. Phys. , 802 (2007).[2] P. Cai, W. Ruan, Y. Y. Peng, C. Ye, X. T. Li, Z. Q. Hao, X. J. Zhou, D.-H. Lee, and Y. Y.Wang, Nat. Phys. , 1047 (2016).[3] Ø. Fischer, M. Kugler, I. Maggio-Aprile, C. Berthod, and C. Renner, Rev. Mod. Phys. ,353 (2007).[4] Y. He, Y. Yin, M. Zech, A. Soumyanarayanan, M. M. Yee, T. Williams, M. C. Boyer, K.Chatterjee, W. D. Wise, I. Zeljkovic, K. Takeshi, T. Takeuchi, H. Ikuta, P. Mistark, R. S.Markiewicz, A. Bansil, S. Sachdev, E. W. Hudson, and J. E. Hoffman, Science , 608(2014).[5] Y. Zheng, Y. Fei, K. L. Bu, W. H. Zhang, Y. Ding, X. J. Zhou, J. E. Hoffman, and Y. Yin,Sci. Rep. , 8059 (2017).
6] T. A. Webb, M. C. Boyer, Y. Yin, D. Chowdhury, Y. He, T. Kondo, T. Takeuchi, H. Ikuta,E. W. Hudson, J. E. Hoffman, and M. H. Hamidian, Phys. Rev. X , 021021 (2019).[7] I. E. Baggari, B. H. Savitzky, A. S. Admasu, J. Kim, S. -W. Cheong, R. Hovden, and L. F.Kourkoutis, Proc. Natl. Am. Sci. , 1445 (2018).[8] Y. Fei, K. L. Bu, W. H. Zhang, Y. Zheng, X. Sun, Y. Ding, X. J. Zhou, and Y. Yin, Sci.China-Phys. Mech. Astron. , 127404 (2018).[9] M. J. Lawler, K. Fujita, J. Lee, A. R. Schmidt, Y. Kohsaka, C. K. Kim, H. Eisaki, S. Uchida,J. C. Davis, J. P. Sethna, and E.-A. Kim, Intra-unit-cell electronic nematicity of the high-Tccopper-oxide pseudogap states. Nature , 347 (2010).[10] K. Fujita, C. K. Kim, I. Lee, J. H. Lee, M. H. Hamidian, I. A. Firmo, S. Mukhopadhyay, H.Eisaki, S. Uchida, M. J. Lawler, E. A. Kim, S. Sachdev, and J. C. Davis, Science , 612(2014). = 0.300 ε = 0.375 ε = 0.450 ε = 0.525 ε = 0.600 ε = 0.675 ε = 0.750 ε = 0.825 ε = 0.900 ε = 0.975 ε = 1.050 -0.05 0.05 -0.05 0.05 -0.08 0.08 -0.09 0.09 -0.09 0.09 -0.10 0.10 -0.11 0.11 -0.11 0.11 -0.11 0.11 -0.11 0.11 -0.11 0.11 FIG. S6. Maps of the charge-order-induced modulation after a Gaussian filtering. For ε from 0.3to 1.105, Z xf ( r , ε ) are sequentially shown. = 0.300 ε = 0.375 ε = 0.450 ε = 0.525 ε = 0.600 ε = 0.675 ε = 0.750 ε = 0.825 ε = 0.900 ε = 0.975 ε = 1.050 − ππ FIG. S7. Maps of the charge-order-induced modulation after a Gaussian filtering. For ε from 0.3to 1.105, φ x ( r , ε ) are sequentially shown. = 0.300 ε = 0.375 ε = 0.450 ε = 0.525 ε = 0.600 ε = 0.675 ε = 0.750 ε = 0.825 ε = 0.900 ε = 0.975 ε = 1.050 -0.05 0.05 -0.05 0.05 -0.08 0.08 -0.09 0.09 -0.09 0.09 -0.10 0.10 -0.11 0.11 -0.11 0.11 -0.11 0.11 -0.11 0.11 -0.11 0.11 FIG. S8. Maps of the charge-order-induced modulation after a Gaussian filtering. For ε from 0.3to 1.105, Z yf ( r , ε ) are sequentially shown. = 0.300 ε = 0.375 ε = 0.450 ε = 0.525 ε = 0.600 ε = 0.675 ε = 0.750 ε = 0.825 ε = 0.900 ε = 0.975 ε = 1.050 − ππ FIG. S9. Maps of the charge-order-induced modulation after a Gaussian filtering. For ε from 0.3to 1.105, φ y ( r , ε ) are sequentially shown. = 0.675 ε = 0.750 ε = 0.825 ε = 0.300 ε = 0.375 ε = 0.525 (b)(a) FIG. S10. (a) An example of how a defect changes position in the stripe structure at different ε .(b) An example of how two closely spaced defects disappear in the stripe structure at different ε .In both (a) and (b) we present the normalized maps of Z x/yf ( r , ε ) /A x/y ( r , ε ) to enhance the signal.) to enhance the signal.