Simultaneous Transitions in Cuprate Momentum-Space Topology and Electronic Symmetry Breaking

See allHide authors and affiliations

Science  09 May 2014:
Vol. 344, Issue 6184, pp. 612-616
DOI: 10.1126/science.1248783

Under the Dome

The superconducting transition temperature Tc of copper oxides has a dome-shaped dependence on chemical doping. Whether there is a quantum critical point (QCP) beneath the dome, and whether it is related to the enigmatic pseudogap, has been heavily debated. Two papers address this question in two different families of Bi-based cuprates. In (Bi,Pb)2(Sr,La)2CuO6+δ, He et al. (p. 608) found that the Fermi surface (FS) undergoes a topological change as doping is increased, which points to the existence of a QCP at a doping close to the maximum in Tc, seemingly uncorrelated with the pseudogap. Fujita et al. (p. 612) studied a range of dopings in Bi2Sr2CaCu2O8+δ to find an FS reconstruction simultaneous with the disappearance of both rotational and translational symmetry breaking, the latter of which has been associated with the pseudogap. These findings point to a concealed QCP.


The existence of electronic symmetry breaking in the underdoped cuprates and its disappearance with increased hole density p are now widely reported. However, the relation between this transition and the momentum-space Embedded Image electronic structure underpinning the superconductivity has not yet been established. Here, we visualize the Embedded Image (intra–unit-cell) and Embedded Image (density-wave) broken-symmetry states, simultaneously with the coherent Embedded Image topology, for Bi2Sr2CaCu2O8+δ samples spanning the phase diagram 0.06 ≤ p ≤ 0.23. We show that the electronic symmetry-breaking tendencies weaken with increasing p and disappear close to a critical doping pc = 0.19. Concomitantly, the coherent Embedded Image topology undergoes an abrupt transition, from arcs to closed contours, at the same pc. These data reveal that the Embedded Image topology transformation in cuprates is linked intimately with the disappearance of the electronic symmetry breaking at a concealed critical point.

The highest known superconducting critical temperature Tc (13) occurs atop the Tc(p) (where p is hole density) “dome” of hole-doped cuprates (Fig. 1A). In addition to the superconductivity, electronic broken-symmetry states (4) have also been reported at low p in many such compounds. Wave vector Embedded Image (intra–unit-cell) symmetry breaking, typically of 90°-rotational (C4) symmetry, is reported in YBa2Cu3O6+δ, Bi2Sr2CaCu2O8+δ, and HgBa2CuO4+x (514). Finite wave vector Embedded Image (density-wave) modulations breaking translational symmetry, long detected in underdoped La2–xyNdySrxCuO4 and La2–xBaxCuO4 (15, 16), are now also reported in underdoped YBa2Cu3O7–δ, Bi2Sr2CuO6+δ, and Bi2Sr2CaCu2O8+δ (1725). Summarizing all such reports in Fig. 1A reveals some stimulating observations. First, although the Embedded Image and Embedded Image states are detected by widely disparate techniques and are distinct in terms of symmetry, they seem to follow approximately the same phase-diagram trajectory (shaded band Fig. 1A), as if facets of a single phenomenon (26). The second implication is that a critical point (perhaps a quantum critical point) associated with these broken-symmetry states may be concealed beneath the Tc(p) dome. Numerous earlier studies reported sudden alterations in many electronic and magnetic characteristics near p = 0.19 (2, 3, 27), but whether these phenomena are caused by electronic symmetry changes (28) at a critical point was unknown.

Fig. 1 Hole-density dependence of cuprate broken-symmetry states.

(A) Phase diagram of hole-doped cuprates showing the Tc(p) “dome” (blue curve). Symbols denote the onset temperature (T) of two types of symmetry breaking reported in Bi2Sr2CuO6+δ(B1), Bi2Sr2CaCu2O8+δ(B2), YBa2Cu3O7–δ(Y), and HgBa2CuO4+x(H). Circles, Embedded Image (intra–unit-cell) symmetry breaking; diamonds, Embedded Image broken translational symmetry (density waves). Error bars indicate the uncertainties in estimated hole concentration (horizontal) and the onset temperatures (vertical). (B) The dashed black curve denotes the typical differential conductance spectrum g(E) of underdoped Bi2Sr2CaCu2O8+δ here at p = 0.06. The pale blue shaded region indicates where the dispersive Embedded Image Bogoliubov QPI wave vectors indicative of Cooper pair breaking exist (color coded in inset); they disappear at energy Δ0. Solid curves, simulation for QPI in a Embedded Image symmetry superconductor; pink shaded regions, quasi-static conductance modulations (density waves) exhibiting wave vectors Embedded Image and Embedded Image.

In momentum space Embedded Image, the hole-doped cuprates also exhibit an unexplained transition in electronic structure with increasing hole density. Open contours, or “Fermi arcs” (2932), are reported at low p in all compounds studied, whereas at high p closed holelike pockets surrounding Embedded Image (where a0 is the unit cell dimension) are observed (33, 34). One possibility is that such a transition could occur due to the disappearance of an electronic ordered state, with the resulting modifications to the Brillouin zone geometry altering the topology of the electronic bands (28).

Our strategy, therefore, is a simultaneous examination of both the Embedded Image electronic structure and the Embedded Image broken-symmetry states, over a sufficiently wide range of p to include any concealed critical point. The objective is to search for a relation between the broken-symmetry states (525) and the Fermi surface (FS) topology (2934). Fourier transform analysis of spectroscopic imaging scanning tunneling microscopy (SI-STM) is distinctive in that it allows access simultaneously to the Embedded Image broken-symmetry states (26, 35) and to the Embedded Image electronic structure by using quasi-particle scattering interference (QPI) (35). The SI-STM tip-sample differential tunneling conductance Embedded Image (e, electron charge; V, tip-sample bias voltage) at location Embedded Image and energy E relates to density of electronic states Embedded Image as Embedded Image (IS and VS are arbitrary parameters) (35). In cuprates, it is necessary to study the ratio of the differential conductances Embedded Image because it suppresses the severe systematic errors due to the unknown denominator Embedded Image by division, thereby allowing Embedded Image and symmetries to be measured correctly (35).

The Embedded Image structure of coherent Bogoliubov quasi-particles is then detectable, because scattering between the eight density-of-states maxima at Embedded Image (circles in inset Fig. 1B) produces interference patterns with wave vectors Embedded Image in Embedded Image, the Fourier transform of Embedded Image. At low p, one finds that the predicted set of seven inequivalent Bogoliubov QPI wave vectors Embedded Image (inset Fig. 1B) exist only below an energy Δ0 (Fig. 1B), indicating that it is the limiting binding energy of a Cooper pair (35). At |E| > Δ0, the dispersive Embedded Image disappear (Fig. 1B and movie S1) and are replaced by broken-symmetry states consisting of (i) spatial modulations with energetically quasi-static wave vectors Embedded Image and Embedded Image breaking translational symmetry (24, 25, 35, 36) and (ii) Embedded Image (intra–unit-cell) breaking of C4 symmetry detectable either directly in Embedded Image (11, 36) or at the Bragg wave vectors (11, 26, 35). But the complete doping dependence of these broken-symmetry signatures was unknown.

To determine the Embedded Image topology of coherent states across the phase diagram of Bi2Sr2CaCu2O8+δ, we use a recently developed approach (37) that requires measurement of only a single QPI wave vector, Embedded Image (blue arrow in Fig. 1B, inset). Brillouin zone geometry (Fig. 2A) means that Embedded Image is simply related to the FS states Embedded Image as Embedded Image (1)Here Embedded Image is the Embedded Image superconducting energy gap whose energy-minimum follows the trajectory of Embedded Image. An image of the locations of Embedded Image for all 0 < |E| < Δ0 (pale blue region in Fig. 1B) then yields the FS location of coherent states that contribute to Cooper pairing. An efficient way to locate these states is to sum all the Embedded Image images to the energy Δ0Embedded Image (2)and then to plot the contour of Embedded Image within these Embedded Image images (see supplementary text section I and figs. S1 and S2) (38). The power of this procedure is demonstrated in the determination of the FS in Fig. 2B (supplementary text section II) (38). Applying this Embedded Image approach to determine the doping dependence of Embedded Image topology, we find very different results at low and high p. Figure 2C shows Embedded Image at p = 0.14, whereas Fig. 2D shows Embedded Image at p = 0.23. The most prominent difference between the two is that the contour of Embedded Image only spans four arcs in Fig. 2C, whereas it completes four closed curves surrounding Embedded Image in Fig. 2D. He et al. report similar phenomena in Bi2Sr2CuO6+δ (37). In Fig. 2E, we show the complete doping dependence of measured Embedded Image over the full range of p (supplementary text section II) (38). A notable transition in Embedded Image topology is observed within the narrow range p ≈ 0.19 ± 0.01, wherein the arc of coherent Bogoliubov states typical of low p suddenly switches to the complete closed contour surrounding Embedded Image (supplementary text section II, fig. S3, and movie S2) (38).

Fig. 2 Momentum-space topology transition from Bogoliubov QPI.

(A) Schematic of d-wave superconducting energy gap Embedded Image on two opposing segments of a FS. Embedded Image is indicated for several Embedded Image by colored arrows. (B) The location of the FS is identified from the measured wave vectors Embedded Image. Shown are the resulting Embedded Image for a sample with p = 0.23; Embedded Image meV. (C) The measured Embedded Image for p = 0.14 sample; Embedded Image meV. No complete contour for Embedded Image can be detected; instead the coherent Bogoliubov quasi-particles are restricted to four arcs terminating at lines joining Embedded Image to Embedded Image. (D) The measured Embedded Image for p = 0.23 sample; Embedded Image meV. A complete closed contour for Embedded Image surrounding Embedded Image can be identified immediately. (E) The measured doping dependence of the Embedded Image topology of coherent Bogoliubov quasi-particles using the Embedded Image technique. The transition from arcs terminating at the lines Embedded Image to Embedded Image to complete hole-pockets surrounding Embedded Image at p ≈ 0.19 is evident.

Next we study the broken-symmetry states by examining Embedded Image measured simultaneously with the Embedded Image data in Fig. 2, but now for Δ0 < |E| < Δ1 (pink regions in Fig. 1B), where Δ1 is the maximum detectable gap [pseudogap at low p and maximum superconducting gap at high p (35)]. These images exhibit several distinct broken spatial symmetries whose evolution with p we explore. Figure 3A shows Embedded Image for p = 0.08, whereas Fig. 3B shows Embedded Image for p = 0.23, with their Fourier transforms Embedded Image shown in Fig. 3, C and D, respectively. The former exhibits the widely reported (2426, 35, 36) quasi-static wave vectors Embedded Image and Embedded Image of states with local symmetry breaking along with the Bragg peaks (red circle), whereas in the latter, the quasi-static wave vectors Embedded Image and Embedded Image have disappeared. The Embedded Image broken C4-symmetry states can be detected by using the lattice-phase–resolved nematic order parameter (11) Embedded Image (3)The Embedded Image and Embedded Image are the Bragg vectors after the necessary transformation to nearly perfect lattice periodicity in Embedded Image so that real and imaginary components of the Bragg amplitudes, Embedded Image and Embedded Image, are well defined (11, 35). The measured Embedded Image for p = 0.06 and Embedded Image for p = 0.23 are shown in Fig. 3, E and F, respectively (supplementary text section IV) (38). Here we see that the extensive order in Embedded Imageobserved at low p (11) has disappeared at high p, leaving nanoscale domains of opposite nematicity (26) probably nucleated by disorder. The doping dependence of Embedded Image, the intensity of the Embedded Image modulations in Embedded Image, is shown in Fig. 3G (supplementary text section III and figs. S4 and S5) (38), whereas the dependence of the spatially averaged magnitude Embedded Image of the Embedded Image C4 breaking is shown in Fig. 3H (supplementary text section IV and figs. S7 and S8) (38). These plots reveal that the more extended Embedded Image broken symmetry and the shorter-range ordering tendencies in Embedded Image modulations (11, 26, 35) disappear near a critical doping pc ≈ 0.19.

Fig. 3 Measurements of hole-density dependence of Embedded Image and Embedded Image ordering.

(A) Embedded Image for p = 0.08. Incommensurate conductance modulations are clearly seen; Embedded Image meV. (B) Embedded Image for p = 0.23 is shown; Embedded Image meV. No specific Embedded Image for modulations is seen, although the QPI signature of Bogoliubov quasi-particles does produce a jumbled standing wave pattern. (C) Embedded Image for p = 0.08 from (A); Embedded Image and Embedded Image wave vectors are indicated by purple and orange circles, respectively. Bragg peaks in (C) and (D) are denoted by red circles. (D) Embedded Image for p = 0.23 from (B). No specific broken-symmetry state wave vectors are apparent, whereas the residual dispersive effects of Bogoliubov quasi-particles are seen. (E) Embedded Image C4-symmetry order parameter Embedded Image for p = 0.06; Embedded Image meV. This whole field of view is a single color, indicating that long-range Embedded Image intra–unit-cell C4 symmetry breaking exists. (F) Intra–unit-cell broken C4 symmetry Embedded Image for p = 0.22; Embedded Image meV. Long-range order has been lost, but nanoscale domains of opposite nematicity persist. The dashed circle represents the spatial resolution of the analysis. (G) Intensity of incommensurate modulations with wave vector Embedded Image, Embedded Image initially increases upon doping peaking near p ~ 1/8, and then diminishes to reach zero at p ≈ 0.19. (H) Squares indicate measured spatial average value of the Embedded Image broken C4 symmetry Embedded Image, which diminishes steadily with increasing p, to reach zero near p = 0.19. Triangles indicate measured standard deviation of Embedded Image.

Figure 4A is a schematic summary of our findings, from Bogoliubov QPI techniques (32, 35, 37), on the dependence of Embedded Image electronic structure with increasing p. Fig. 4B shows that the wave vectors Embedded Image of states at which Bogoliubov QPI disappears (circles) evolve along the Embedded Image lines Embedded Image with increasing p. Concomitantly, the quasi-static wave vectors Embedded Image and Embedded Image of broken-symmetry states also evolve on the same trajectory (squares). Thus, the Embedded Image and Embedded Image wave vectors of incommensurate (density-wave) modulations evolve with doping, as shown in Fig. 4C (35). Figure 4D shows the area of Embedded Image between the arc and the line Embedded Image (left inset) increasing proportional to hole density p (32, 35); at p = 0.19, there is a transition to a diminishing area of electron count as 1-p for the closed-contour FS topology. Finally, we show in Fig. 4, E to G, that the critical point pc ≈ 0.19 is associated microscopically with a transition to conventional d-wave Bogoliubov QPI on a complete FS (simulated in Fig. 4E and measured at p > pc in 4G) from a highly distinct form of scattering (Fig. 4F) of unknown cause (39).

Fig. 4 Interlinked Embedded Image structure of Bogoliubov QPI and Embedded Image symmetry breaking at lines joining Embedded Image to Embedded Image.

(A) Schematic of Embedded Image locus of states generating Bogoliubov QPI with increasing hole density in Bi2Sr2CaCu2O8+δ. An abrupt transition occurs at p ≈ 0.19. (B) Measured wave vectors Embedded Image of states at which Bogoliubov QPI disappears Embedded Image (circles), and those of the quasi-static broken-symmetry modulations with Embedded Image and Embedded Image (squares). (C) Measured doping dependence of wave vectors of incommensurate conductance modulations (density waves) Embedded Image and Embedded Image derived from (B) (35). The vertical dashed line indicates pc. (D) The Embedded Image area between the arc and the lines joining Embedded Image to Embedded Image is proportional to p. With the appearance of the closed FS at p ≈ 0.19, there is a transition to a diminishing area of electron count 1-p. Insets, Embedded Image areas used. (E) T-matrix scattering interference simulation for Embedded Image for a complete FS and d-wave gap with conventional (time-reversal preserving) scattering. (F) Measured Embedded Image for p = 0.14 Embedded Image is in sharp contrast to the simulation result in (E). (G) For p = 0.23, Embedded Image conforms closely to the conventional d-wave Bogoliubov scattering scheme, as anticipated in (E).

To recapitulate: With increasing hole density, the Embedded Image modulations (density waves) weaken and disappear at pc ≈ 0.19 (Fig. 1A and Fig. 3, A to D and G). Concurrently, the Embedded Image broken-symmetry (intra–unit-cell nematic) states become progressively more disordered (13) and reach a zero average value at approximately the same pc (Fig. 1A and Fig. 3, E to F and H). Simultaneously, the Embedded Image topology of coherent Bogoliubov quasi-particles (or the FS supporting their superconducting gap) undergoes an abrupt transition from arcs to closed contours (Figs. 2 and 4 and movie S2) (38). This key transformation of cuprate electronic structure is therefore linked directly with disappearance of electronic symmetry breaking. However, this phenomenology also exhibits many peculiar components unexpected within a simple FS reconstruction scenario. First, the coevolution and contiguous disappearance at pc of the signatures of two distinct broken symmetries (Figs. 1A and 3) reinforce the deductions that they are microscopically closely related (11, 26, 35, 36, 40). Second, because the Embedded Image modulations exhibit wave vectors generated by scattering regions (hot spots) moving along the Embedded Image lines Embedded Image (Fig. 4, A to C) (32, 35), FS nesting provides an inadequate explanation for the cuprate density waves. Third, the abrupt Embedded Image topology change at pc (Fig. 2E and Fig. 4, A, F, and G) exhibits characteristics more reminiscent of an antinodal coherence recovery transition (41) than of a conventional band reorganization. Fourth, because the disappearance of the pseudogap is associated axiomatically with the reappearance of coherent antinodal states, and because the latter is precisely what occurs at pc (Figs. 2E and 4A), the pseudogap (13) and the electronic symmetry breaking (525) must be intimately linked (Fig. 1A). Finally, as neither long-range Embedded Image order nor any associated quantum critical point can exist with quenched disorder (40), a nematic critical point, at which the electronic symmetry breaking between the two oxygen sites within the CuO2 unit cell (11, 26, 35, 36) disappears, seems most consistent with our observations.

Supplementary Materials

Materials and Methods

Supplementary Text

Figs. S1 to S9

References (4245)

Movies S1 and S2

References and Notes

  1. Materials and methods are available as supplementary materials on Science Online.
  2. Acknowledgments: We are particularly grateful to S. Billinge, J. E. Hoffman, S. A. Kivelson, D.-H. Lee, and A. P. Mackenzie for key scientific advice. We thank K. Efetov, E. Fradkin, P. D. Johnson, J. W. Orenstein, C. Pepin, S. Sachdev, and K. M. Shen for helpful discussions and communications. Experimental studies were supported by the Center for Emergent Superconductivity, an Energy Frontier Research Center, headquartered at Brookhaven National Laboratory (BNL) and funded by the U.S. Department of Energy under grant DE-2009-BNL-PM015, as well as by a Grant-in-Aid for Scientific Research from the Ministry of Science and Education (Japan) and the Global Centers of Excellence Program for Japan Society for the Promotion of Science. C.K.K. acknowledges support from the FlucTeam program at BNL under contract DE-AC02-98CH10886. J.L. acknowledges support from the Institute for Basic Science, Korea. I.A.F. acknowledges support from Fundação para a Ciência e a Tecnologia, Portugal, under fellowship number SFRH/BD/60952/2009. S.M. acknowledges support from NSF grant DMR-1120296 to the Cornell Center for Materials Research. Theoretical studies at Cornell University were supported by NSF grant DMR-1120296 to Cornell Center for Materials Research and by NSF grant DMR-0955822. The original data are archived by Davis Group, BNL, and Cornell University.
View Abstract

Navigate This Article