Abstract
Research on high-temperature superconducting cuprates is at present focused on identifying the relationship between the classic ‘pseudogap’ phenomenon1,2 and the more recently investigated density wave state3,4,5,6,7,8,9,10,11,12,13. This state is generally characterized by a wavevector Q parallel to the planar Cu–O–Cu bonds4,5,6,7,8,9,10,11,12,13 along with a predominantly d-symmetry form factor14,15,16 (dFF-DW). To identify the microscopic mechanism giving rise to this state17,18,19,20,21,22,23,24,25,26,27,28,29, one must identify the momentum-space states contributing to the dFF-DW spectral weight, determine their particle–hole phase relationship about the Fermi energy, establish whether they exhibit a characteristic energy gap, and understand the evolution of all these phenomena throughout the phase diagram. Here we use energy-resolved sublattice visualization14 of electronic structure and reveal that the characteristic energy of the dFF-DW modulations is actually the ‘pseudogap’ energy Δ1. Moreover, we demonstrate that the dFF-DW modulations at E = −Δ1 (filled states) occur with relative phase π compared to those at E = Δ1 (empty states). Finally, we show that the conventionally defined dFF-DW Q corresponds to scattering between the ‘hot frontier’ regions of momentum-space beyond which Bogoliubov quasiparticles cease to exist30,31,32. These data indicate that the cuprate dFF-DW state involves particle–hole interactions focused at the pseudogap energy scale and between the four pairs of ‘hot frontier’ regions in momentum space where the pseudogap opens.
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Main
A conventional ‘Peierls’ charge density wave (CDW) in a metal results from particle–hole interactions which open an energy gap at specific regions of k-space that are connected by a common wavevector Q. This generates a modulation in the density of free charge at Q along with an associated modulation of the crystal lattice parameters. Such CDW states are now very well known33. In principle, a density wave modulating at Q can also exhibit a ‘form factor’ (FF) with different possible symmetries34,35 (see Supplementary Section 1). This is relevant to the high-temperature superconducting cuprates because numerous researchers have recently proposed that the ‘pseudogap’ regime1,2 (PG in Fig. 1a) contains an unconventional density wave with a d-symmetry form factor17,18,19,20,21,22,23,24,25,26,27,28,29. The basic phenomenology of such a state is that intra-unit-cell (IUC) symmetry breaking renders the Ox and Oy sites within each CuO2 unit-cell electronically inequivalent, and that this inequivalence is then modulated periodically at wavevector Q parallel to (1,0);(0,1). The real-space (r-space) schematic of such a d-symmetry FF density wave (dFF-DW) at Qx, as shown in Fig. 1b, exemplifies periodic modulations at the Ox sites that are π out of phase with those at the Oy sites. Such a state is then described by A(r) = D(r)cos(φ(r) + φ0(r)), where A(r) represents whatever is the modulating electronic degree of freedom, φ(r) = Qx ⋅ r is the DW spatial phase at location r, φ0(r) represents disorder related spatial phase shifts, and D(r) is the magnitude of the d-symmetry form factor14,21,23. To distinguish between the various microscopic mechanisms proposed for the Q = (Q, 0); (0, Q) dFF-DW state of cuprates17,18,19,20,21,22,23,24,25,26,27,28,29, it is essential to establish its atomic-scale phenomenology, including the momentum space (k-space) eigenstates contributing to its spectral weight, the relationship (if any) between modulations occurring above and below the Fermi energy, whether the modulating states in the DW are associated with a characteristic energy gap, and how the dFF-DW evolves with doping.
To visualize such phenomena directly as in Fig. 1c, we use sublattice-phase-resolved imaging of the electronic structure14 of the CuO2 plane. Both the scanning tunnelling microscope (STM) tip–sample differential tunnelling conductance dI/dV (r, E = eV) ≡ g(r, E) and the tunnel-current I(r, E) are measured at bias voltage V = E/e and with sub-unit-cell spatial resolution. Because the density of electronic states N(r, E) is related to the differential conductance as g(r, E) ∝ [eIs/N(r, E′)dE′]N(r, E), with Is and Vs being arbitrary parameters and the denominator N(r, E′)dE′ unknown, valid imaging of N(r, E) is challenging (Supplementary Section 2). However, one can suppress these serious systematic ‘set-point’ errors by using R(r, E) = I(r, E)/I(r, −E) or Z(r, E) = g(r, E)/g(r, −E); this allows distances, wavelengths and spatial phases of electronic structure to be measured accurately. The unprocessed g(r, E) acquired for and analysed in this paper are measured over very large fields of view (to achieve high phase precision in Fourier analysis), simultaneously maintain deeply sub-unit-cell precision measurements in r (to achieve high precision in sublattice segregation), and are taken over a wide range of energies E with fine energy spacing, so that energy dependences of d-symmetry FF modulations may be accurately determined. We then calculate each sublattice-phase-resolvedZ(r, E) image and separate it into three: the first, Cu(r), contains only the measured values of Z(r) at Cu sites, whereas the other two, Ox(r) and Oy(r), contain only the measurements at the x/y-axis oxygen sites. Phase-resolved Fourier transforms of the Ox(r) and Oy(r) sublattice images14, ; , are used to determine the form factor symmetry for modulations at any q
where the superscript Z identifies the type of sublattice-resolved data used. Specifically for a DW occurring at Q, one can then evaluate the magnitude of its d-symmetry form factor and its s′- and s-symmetry form factors and , respectively. Studies of non-energy-resolved R(r, E) images using this approach have revealed that the DW modulations in the Ox(r) and Oy(r) sublattice images of electronic structure in underdoped Bi2Sr2CaCu2O8+x and Ca2−xNaxCuO2Cl2 consistently exhibit a relative phase of π and therefore a predominant d-symmetry form factor14; X-ray scattering studies15,16 yield the same conclusion for two other cuprates, YBa2Cu3O7−x and Bi2Sr2−xLaxCuO6+δ.
Such X-ray scattering studies now generally report a short-ranged density wave with wavevector centred around Q = (Q, 0); (0, Q) occurring approximately in the pink shaded regions11,12,13 of the schematic phase diagram in Fig. 1a. Figure 1b, c exemplifies the predominately d-symmetry form factor14,15,16 of this DW when imaged directly. One obstacle to understanding this dFF-DW state is that large-field-of-view sublattice-resolved images of cuprate electronic structure14 never look like an ideal long-range ordered version of Fig. 1b. Instead, Fig. 2a shows a typical Z(r, 150 meV) image of p = 8% Bi2Sr2CaCu2O8+x, for T ≪ Tc in the superconducting phase, whereas Fig. 2b shows the equivalent Z(r, 150 meV) for T > Tc in the cuprate pseudogap phase. Although elements indistinguishable from Fig. 1c can be seen in 2a, b, no long-range order is obvious. Therefore, to explore the spatial arrangements of the dFF-DW in such electronic-structure images, we analyse , which is a robust sublattice-phase-resolved measure of the d-symmetry form factor (Supplementary Section 3). Analysis of Fig. 2a, b in this fashion yields Fig. 2c, d; both clearly exhibit the dFF-DW maxima at the two inequivalent wavevectors Qx and Qy. Fourier filtering these two from Fig. 2a, b for only those regions surrounding Qx and Qy (within dashed circles) generates two complex-valued r-space images Dx(r), Dy(r)
where Λ−1 is the characteristic length scale over which variations in Dx(r), Dy(r) can be resolved, and is set by the filter width in Fourier space.
Their magnitudes
represent the local amplitudes of dFF-DW modulations along Qx and Qy, respectively. Any unidirectional domain arrangements of the dFF-DW state can then be determined by introducing
which is designed to identify regions where the dFF-DW modulation is primarily along the x-axis or the y-axis, depending on the sign of F(r) (Supplementary Section 4). Figure 2e, f shows how regions of −1.0 < F(r) < −0.3 (shaded blue) are primarily modulating along y-axis whereas regions +0.3 < F(r) < +1.0 (shaded orange) are primarily modulating along x-axis (those with −0.3 < F(r) < +0.3 shaded white appear at boundaries). Figure 2g, h reveals the results of this analysis for the data in Fig. 2a, b respectively. Overall, the system is configured into spatial regions within which the dFF-DW along only one direction is dominant. By overlaying the colour scale for F(r) on the data in Fig. 2a, b to create Fig. 2g, h, one can see directly the unidirectional region configurations derived from equation (7). These observations of coexisting nanoscale unidirectional regions are in reasonable agreement with deductions from related X-ray studies36 of YBa2Cu3O7−x. Finally, because the data in Fig. 2b and Fig. 2h were measured at T > Tc (pink region Fig. 1a), this demonstrates directly that the cuprate dFF-DW appears first in the non-superconducting ‘pseudogap’ regime.
A conventional CDW state opens a gap in the energy spectrum of k-space electronic eigenstates with the maximum spectral weight of modulating states occurring at the edges of this energy gap33. But which energy gap (if any) is associated with the dFF-DW state found in underdoped cuprates is unknown. Figure 3a shows how a typical tunnelling conductance spectrum representative of strongly underdoped cuprates exhibits two characteristic energies30,31,32. Whereas the lower-energy scale Δ0 represents the maximum energy at which Bogoliubov quasiparticle excitations exist30,31,32 (see Fig. 3b), the higher-energy scale (dashed blue line) is the cuprate ‘pseudogap’ as determined from its comparison with the doping dependence of the pseudogap scale in tunnelling and photoemission. To identify the energy dependence of the cuprate dFF-DW states, we measure Z(r, |E|), and from it calculate , and . Figure 3c shows the measured power spectral density of the d-symmetry FF modulations , with the wavevectors near Qx and Qy indicated by red circles. Adopting the common convention in X-ray studies9,10,11,16 for estimating the DW wavevector magnitude |Q|, we measure along a line in the high-symmetry directions (1,0):(0,1) passing through the region of the dFF-DW peak and fit these data to a background plus Gaussian; the peak positions of the two Gaussians are then assigned to be the values of Qx and Qy. It is important to appreciate, however, that a wide range of Q values can actually be detected under each peak in and in X-ray scattering intensities; it remains to be determined whether these broad peaks with incommensurate maxima are due to domains of continuously incommensurate dFF-DW or domains of commensurate dFF-DW separated by discommensuration37. Nevertheless, Fig. 3b plots the energy dependence of the dFF-DW wavevectors (blue line) determined in this way for a p = 0.06 sample. Such information was not previously available from measurements of the modulation wavevectors from STM images lacking sublattice-phase-resolved segregation14 into and . Figure 3e shows the measured k-space locus where Bogoliubov quasiparticles exist30,31,32 as a function of hole density. When the dispersive ‘octet’ of Bogoliubov scattering interference disappears, a transition occurs to an ultra-slow dispersing density wave modulation (Fig. 3b). In Fig. 3d, the doping dependence of the conventional Qx, Qy of the d-symmetry form factor modulations is shown using blue symbols, while the shortest wavevectors interconnecting the measured k-space regions where Bogoliubov scattering interference disappears (Fig. 3e) are indicated by using coloured symbols referring to each hole density in Fig. 3e. These data demonstrate directly that the conventionally determined Qx, Qy of the dFF-DW state correspond to the locations in k-space of these arc tips. Finally, in Fig. 3f we show the measured energy dependences of the amplitudes of the s-, s′- and d-form factor modulations, SZ(E), S′Z(E) and DZ(E), determined by integrating over the region of q-space enclosed by solid red circles in Fig. 3c (Supplementary Section 5). The d-symmetry form factor is negligible for modulations in the low-energy range that contains only Bogoliubov quasiparticles (and which we now see is dominated by s′-symmetry form factors), but it rapidly becomes intense at higher energy and reaches maximum at the pseudogap energy scale, which for this sample is Δ1 ∼ 90 meV. This reveals that the characteristic energy of electronic-structure modulations in the cuprate d-symmetry FF density wave is actually the pseudogap energy.
As a function of energy, the transition from Bogoliubov quasiparticle interference (QPI) modulations to dFF-DW modulations occurs in an unusual fashion. Although Bogoliubov QPI is observed as expected everywhere on the Fermi surface in overdoped cuprates32, in underdoped samples it evolves as expected only until the energy E ≈ Δ0, at which the terminations of the Bogoliubov coherent k-space arcs (Fig. 3e) are observed30,31,32. Here, the set of seven dispersive scattering interference modulations q1, q2, …q7 signifying Bogoliubons31 (Supplementary Section 5) disappears in a narrow energy window during which dispersion of the two surviving modulations q1(E) and q5(E) comes to a halt, leaving the ultra-slow dispersing dFF-DW modulations with q1∗ ≈ q1(Δ0) and q5∗ ≈ q5(Δ0) (see Fig. 3b and refs 30,31,32). The intensity of these non-dispersive modulations first becomes detectable at Δ0 and, as we show below, reaches an intense maximum at Δ1, all the while maintaining the same wavevectors Qxd and Qyd, as shown in Fig. 3b. We refer to this k-space region where Bogoliubov quasiparticles yield to modulations of a dFF-DW as the ‘hot frontier’ (we thank S. A. Kivelson for proposing the term ‘hot frontier’ to describe the k-space phenomenology of the cuprate dFF-DW as observed using spectroscopic imaging STM) to distinguish it from the colloquial ‘hot spots’ beyond which, in a conventional density wave, dispersive quasiparticle states would reappear. In cuprates, this does not occur and, instead, the ‘hot frontiers’ define the k-space limit beyond which only dFF-DW modulations are detected30,31,32 using spectroscopic imaging STM (blue in Fig. 3b).
Key information on the microscopic cause of any DW state is also contained in the relationship between modulations of states above and below the Fermi energy. For example, mixing via interactions of states with momenta k1 and k2 generates modulations at wavevector Q = k1–k2. The wavefunctions of any resulting DW would then form bonding/anti-bonding states below/above the Fermi level which are proportional to ± . The related densities of these states would then exhibit modulations governed by | ± = 2(1 ± cos(Q ⋅ r)). In such scenarios, the DW modulations above the Fermi energy should always be π out of phase with the equivalent ones below. To explore this issue in Bi2Sr2CaCu2O8+x, we show in Fig. 4a, b the measured g(r, +94 meV) from filled states and g(r, −94 meV) from empty states, respectively, each at the characteristic energy of the dFF-DW (Fig. 3f and Supplementary Section 6); avoidance of the set-point error is discussed in Supplementary Section 6. For these two images the sublattice-phase-resolved equation (1) are calculated and reveal a predominantly d-symmetry form factor modulation with wavevectors near Qx and Qy in Fig. 4a, b. Next, by Fourier filtering these two for regions surrounding Qy, we determine the complex-valued Dy(r), and thus the spatial phase of dFF-DW modulation along Qy as
This is shown for E = +94 meV in Fig. 4c and for E = −94 meV in Fig. 4d. Visual comparison reveals that these two φy(r, ±E) images are out of phase with each other by π. And, indeed, the spatial-average value of φy(r, +E) − φy(r, −E) as a function of E (over the whole field of view of A and B) is shown in Fig. 4e. It reveals that, whereas the relevant Qx and Qy components of g(r, +E) and g(r, −E) images are in phase with each other at low energy, they rapidly evolve at |E| > 70 meV and become globally π out of phase at |E| ∼ Δ1 (Fig. 4a, b). The shaded region indicates evolution through a situation where some areas exhibit φ ∼ 0 and some φ ∼ π, but this is quickly resolved on reaching the pseudogap energy Δ1. Similar analysis for the particle–hole symmetry in phases φx(r, ± E) of Qx modulations
yields a virtually identical result (Fig. 4f). These phenomena also occur throughout the underdoped regions of the phase diagram (Supplementary Section 6.III), demonstrating that, in the cuprate dFF-DW state, a phase difference of π exists between spatial modulations of the filled states below energy E ∼ −Δ1 and the empty states above E ∼ +Δ1.
To summarize: by introducing techniques to determine the energy/momentum and doping dependence of modulation form factor symmetry, we find that the predominantly d-symmetry form factor density wave exists throughout the underdoped region of the Bi2Sr2CaCu2O8 phase diagram (Fig. 3d), including in the pseudogap regime T > Tc (Figs 1c and 2b). The spatial arrangements are primarily in the form of nanoscale regions, each containing a primarily unidirectional dFF-DW (Fig. 2g, h). The conventionally defined wavevectors Qx and Qy of the dFF-DW state evolve with doping as determined by the four shortest scattering vectors linking the k-space regions beyond which Bogoliubov quasiparticle excitations are non-existent (Fig. 3d, e) and at which the pseudogap emerges. Further, we demonstrate that, as determined in terms of tunnelling probabilities, the dFF-DW state is particle–hole antisymmetric, in the sense that a phase difference of π exists between spatial modulations of the filled states (E ∼ −Δ1) and the empty states (E ∼ +Δ1) (Fig. 4e, f). Perhaps most significantly, we show that the characteristic energy of the cuprate dFF-DW state is actually the pseudogap energy Δ1 (Fig. 3f).
These data provide clear evidence that the cuprate d-symmetry form factor density wave state involves particle–hole interactions, and that these occur primarily very near wavevectors interconnecting the ‘hot frontiers’ in k-space at which the pseudogap emerges30,31,32. Moreover, the dFF-DW electronic-structure modulations have a characteristic energy scale which is the pseudogap energy. This intimate connection of the dFF-DW state with the pseudogap electronic structure is consistent with the fact that this state is found only within the pseudogap regime11,12,13. Of course, electron–lattice interactions can also play a significant role, with the coupling to the B1g modes long being of foremost interest19,20,38. Strong interactions of this mode with the electrons39 ultimately leading to static, finite-Q lattice distortions with d-symmetry form factor15 have recently been discovered in association with the cuprate dFF-DW state. Nevertheless, electron–lattice interactions are not by themselves sufficient to explain the phase diagram of the dFF-DW (refs 11,12,13) because, for example, they also exist in the overdoped regime where the dFF-DW is absent. Moreover, theoretical models involving k-space instabilities26,27,29,40 which are consistent with the results herein, emphasize that a density wave with this Q and form factor symmetry cannot emerge from a large Fermi surface; instead, a pre-existing reorganization of k-space due to the pseudogap would be required. Overall, our data support a microscopic picture in which the exotic electronic structure of the pseudogap is parent to the dFF-DW state and not vice versa, where the energy scale and wavevectors of the dFF-DW are intimately linked to those of the pseudogap, and in which the d-symmetry DW competes directly for spectral weight with the d-symmetry superconductor at the k-space ‘hot frontier’ between superconductivity and the pseudogap.
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Acknowledgements
We acknowledge and thank H. Alloul, D. Chowdhury, R. Comin, A. Damascelli, E. Fradkin, D. Hawthorn, S. Hayden, J. E. Hoffman, M.-H. Julien, D. H. Lee, M. Norman and C. Pepin for helpful discussions and communications. We are especially grateful to S. A. Kivelson for key scientific discussions and advice. Experimental studies were supported by the Center for Emergent Superconductivity, an Energy Frontier Research Center, headquartered at Brookhaven National Laboratory and funded by the US Department of Energy under 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 under the FlucTeam Program at Brookhaven National Laboratory (Contract DE-AC02-98CH10886). S.D.E., J.C.D. and A.P.M. acknowledge the support of EPSRC through the Programme Grant ‘Topological Protection and Non-Equilibrium States in Correlated Electron Systems’. Theoretical studies at Cornell University were supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Science and Engineering under Award DE-SC0010313. Theoretical studies at Harvard University were supported by NSF Grant DMR-1103860 and by the Templeton Foundation. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. The data and/or materials supporting this publication can be accessed at http://dx.doi.org/10.17630/f17227bc-3045-40d6-b289-30a4c1a8966c.
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M.H.H., S.D.E., C.K.K. and K.F. carried out the experiments; K.F., H.E. and S.U. synthesized and characterized the samples; M.H.H., S.D.E. and K.F. developed and carried out analysis; E.-A.K., M.J.L. and S.S. provided theoretical guidance; A.P.M. and J.C.D. supervised the project and wrote the paper with key contributions from M.H.H., S.D.E., C.K.K. and K.F. The manuscript reflects the contributions and ideas of all authors.
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Hamidian, M., Edkins, S., Kim, C. et al. Atomic-scale electronic structure of the cuprate d-symmetry form factor density wave state. Nature Phys 12, 150–156 (2016). https://doi.org/10.1038/nphys3519
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DOI: https://doi.org/10.1038/nphys3519
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