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Local Ordering in the Pseudogap State of the High-Tc Superconductor Bi2Sr2CaCu2O8+δ

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Science  26 Mar 2004:
Vol. 303, Issue 5666, pp. 1995-1998
DOI: 10.1126/science.1093384

Abstract

We report atomic-scale characterization of the pseudogap state in a high-Tc superconductor, Bi2Sr2CaCu2O8+δ. The electronic states at low energies within the pseudogap exhibit spatial modulations having an energy-independent incommensurate periodicity. These patterns, which are oriented along the copper-oxygen bond directions, appear to be a consequence of an electronic ordering phenomenon, the observation of which correlates with the pseudogap in the density of electronic states. Our results provide a stringent test for various ordering scenarios in the cuprates, which have been central in the debate on the nature of the pseudogap and the complex electronic phase diagram of these compounds.

Adding charge carriers to antiferromagnetic Mott insulators produces some of the most fascinating and unconventional electronic states of matter. In cuprates, samples with low hole concentration exhibit, in addition to high-temperature superconductivity, a conducting state characterized by anomalous transport, magnetic, and optical properties at temperatures above the superconducting transition temperature Tc (1). This regime, the pseudogap state, is characterized by partial gapping of the low-energy density of states (DOS), which is detected below a characteristic pseudogap temperature T*, in optical, photoemission, and tunneling spectroscopy measurements (2). It has been argued that the pseudogap state represents a departure from the Fermi liquid picture, and thus requires a fundamentally different electronic description from that used to describe most metals (3). Many of the proposed scenarios involve some form of static or fluctuating ordering phenomena, such as charge, spin (47), or orbital current order (8, 9), as a distinguishing feature of the pseudogap phase. Others have focused on the observation that the pseudogap in underdoped samples evolves continuously into the superconducting gap (10), on unusual high-frequency conductivity (11), or on the Nernst effect in underdoped samples (12); these findings suggest that the pseudogap state has local pairing correlations required for superconductivity but lacks long-range phase coherence (13). Currently, however, there is no prevailing view on the pseudogap problem.

We used scanning tunneling microscopy (STM) to examine at atomic scale the electronic states of a cuprate superconductor in its pseudogap state. Our results show that the electronic excitations with energies below the pseudogap energy exhibit incommensurate spatial modulations that are oriented along the Cu-O bond directions. Among the possible scenarios for the observed ordering, the period of our modulations corresponds most closely to that expected from the pinning of a fluctuating incommensurate spin-density wave (14, 15), which is commonly observed in neutron-scattering measurements of underdoped cuprates (16, 17). We also consider scenarios other than spin ordering as potential explanations for our findings. Regardless of its specific origin, the results establish that the electronic states affected by the pseudogap exhibit real-space organization.

We also contrast our observation of local ordering in the pseudogap phase with the modulated patterns in the superconducting state. In agreement with previous reports (18), and in strong contrast with our observations in the pseudogap regime, we find the presence of electronic modulations with energy-dependent periodicity in the superconducting state. These modulations in the superconducting state appear for the most part to be described by models of quantum interference of quasiparticle states caused by scattering from random defects (1823). However, despite the success of the quantum interference picture in describing STM data in the superconducting state, there might be a propensity for ordering (24, 25); previous reports on superconducting samples in a magnetic field have shown incommensurate electronic modulations near vortices (26). Currently, it is unclear whether the vortex-induced modulations show any energy dispersion; however, it is conceivable that the local ordering phenomena we have found in the pseudogap state may also nucleate near vortices in the superconducting state (27).

To perform STM measurements, we cleaved the Bi2Sr2CaCu2O8+δ samples in situ in ultrahigh vacuum at room temperature before inserting them into the cooled microscope stage, where they were stabilized at various temperatures. The measurements reported here were carried out on slightly underdoped Bi2Sr2CaCu2O8+δ single crystals, which contained 0.6% of Zn impurities (Tc = 80 K with pseudogap energy Δps ∼ 35 to 40 meV at 100 K; Fig. 1A). Similar results have been observed in Zn-free samples (28).

Fig. 1.

Energy and spatial dependence of the DOS at 100 K. All conductance (DOS) measurements at 100 K were taken using a standard ac lock-in technique with an initial tunneling current IT, initial sample bias VS, and bias modulation of 4 mV rms. (A) A typical conductance spectrum (IT = 100 pA, VS = –150 mV) shows a pseudogap in the DOS at the Fermi energy. (B) A typical topograph, taken at a constant IT = 40 pA and VS = –150 mV, over a 450 Å by 195 Å field of view shows atomic corrugation and the incommensurate supermodulation along the b axis. (C to F) Real-space conductance maps recorded simultaneously at 41 mV (C), 24 mV (D), 12 mV (E), and 6 mV (F) show the appearance and energy evolution of DOS modulation along the Cu-O bond directions. Also evident from these maps is the presence of electronic variations associated with defects (Zn and others) and the dopant inhomogeneity of the material system. The atomic-scale contributions of defects to the DOS in the pseudogap state are of great interest; however, we focus our discussion on the periodic modulations.

The spatial modulation of the pseudogap's electronic state manifests itself when we map the variation of the local DOS at energies below Δps. Unprocessed maps of the differential conductance measured simultaneously at different voltage biases are shown in Fig. 1, C to F, corresponding to maps of the DOS at specific energies. The DOS at low energies shows periodic patterns oriented along the Cu-O bond directions, which are at a 45° angle relative to the structural supermodulation. To characterize these periodic electronic patterns and determine their connection with the pseudogap in the DOS, we performed a two-dimensional Fourier analysis of the unprocessed conductance maps and tracked the Fourier amplitude as a function of in-plane wave vector and energy. We show an example of such a Fourier map in Fig. 2A, and a schematic labeling of various peaks in this map in Fig. 2B. An important distinguishing feature between the Fourier peaks arising from the crystal structure (atomic lattice at peaks labeled A; b-axis supermodulation at peaks labeled S) and those due to the electronic patterns of interest (centered at peaks labeled Q) is their energy dependence (Fig. 2C). The intensities of the structural peaks in the Fourier maps measured at different energies correspond to the overall electronic DOS—a common behavior observed with STM on various surfaces. However, the Fourier peaks centered at Q show the opposite trend, as they can be detected in the DOS maps only for |eV| values approximately less than Δps and, more important, they increase in intensity with lowering of energy below Δps. The appearance and enhanced intensity of bond-oriented electronic modulations at low energies, and their close connection with the pseudogap energy, constitute our main experimental findings.

Fig. 2.

Fourier analysis of DOS modulations. (A) Fast Fourier transform (FFT) of an unprocessed conductance map (IT = 40 pA, VS = –150 mV) acquired over a 380 Å by 380 Å field of view (on a 200-square grid) at 15 mV. See (28) for the unprocessed conductance map. (B) The FFT has peaks corresponding to atomic sites (colored black and labeled A), primary (at 2π/6.8a0) and secondary peaks corresponding to the b-axis supermodulation (colored cyan and labeled S), and peaks at ∼2π/4.7a0 along the 〈π,0〉 and 〈0,π〉 directions (colored red and labeled Q). The point (0, 2π/4a0) is also labeled for reference. (C) The energy evolution of the peaks in (B), scaled by their respective magnitudes at 41 mV (Q, 70.3 pS; S, 227.7 pS; A, 35.5 pS). (D) Two-pixel-averaged FFT profiles taken along the dashed line in (B) for the DOS measurement at 15 mV shown in (A) and measurements acquired simultaneously at 0 mV and –15 mV. The positions of key peaks are shown by dashed lines and labeled according to their location in (B).

Quantitative analysis of the Fourier maps shows that the bond-oriented modulations have an incommensurate periodicity of 4.7 ± 0.2 a0 (where a0 is the Cu-Cu distance) for samples containing Zn (Fig. 2, A and D). Although the intensity of the peaks centered at Q varies with energy, their wavelength remains independent of energy within our experimental resolution (±0.2a0) (Fig. 2D). Measurements on different samples suggest that the periodicity of the modulations corresponding to points Q varies between 4.5 and 4.8 a0 among the different samples, perhaps because of differences in doping or Zn concentration, the systematic variation of which remains to be investigated. Furthermore, examining the width of the peaks centered at Q shows that the electronic modulations have a correlation length of four to five periods (∼90 Å)—a behavior consistent with the locally disordered or glassy nature of the modulations in the conductance maps (Fig. 1).

The fundamental importance of the modulations of the local DOS in the pseudogap state becomes more evident when we compare them to those observed in the superconducting state. As previously reported (18, 19, 24), there are several different modulated patterns in the superconducting state; however, unlike those in the pseudogap state, they all display energy-dependent periods. To illustrate this important property, and for the purpose of comparison with the pseudogap patterns, we focus our attention on the bond-oriented modulations. Figure 3 shows our measurements of the wave vectors associated with these modulations on Zn-free samples in the superconducting state at 40 K, together with measurements at 4.2 K (18). All data in the superconducting state show an agreement in their overall trends and display similar dispersions with energy, which is remarkable given that they result from measurements made at different temperatures and on samples having different concentrations of defects and doping. From these results, we conclude that energy dispersion is a robust feature of bond-oriented modulations in the superconducting state, hence distinguishing them from those in the pseudogap phase.

Fig. 3.

Dispersion of bond-oriented modulations in the superconducting state. A set of conductance maps at several energies, with a 380 Å by 380 Å field of view (IT = 75 pA, VS = –150 mV, on a 128-square grid), was taken simultaneously on a Bi2Sr2CaCu2O8+δ (Tc = 88 K) sample at 40 K. Here we show the energy dependence of the DOS modulation peak in the Fourier maps of the DOS in the superconducting state. The peak position clearly changes as a function of energy (shown in red) and is in good agreement with the previous measurements at 4.2 K (18). Also shown is the simple evolution of scattering wave vectors from band structure considerations.

To interpret our experimental results in the pseudogap state, we consider various possible scenarios for the observation of a periodic modulation in the local DOS recorded by STM. We believe it unlikely that the wave vectors centered at Q are due to some form of lattice effect, as they behave quantitatively differently from those associated with the crystal structure (Fig. 2C). Another possibility is that the observed patterns are associated with the formation of standing waves caused by the quantum interference of quasiparticles elastically scattered from defects, as previously proposed for the modulations in the superconducting state (1823). In this scenario, the wavelengths of the electronic patterns are directly related to the wave vectors that connect points on curves of constant electron energy in k-space. This relation has been previously verified by demonstrating that analysis of low-temperature STM and angle-resolved photoemission spectroscopy (ARPES) data in the superconducting state produce approximately the same Fermi surface for Bi2Sr2CaCu2O8+δ (18, 19) (see also comparison in Fig. 3). To evaluate whether a similar scenario could apply to our results above Tc, we have performed model calculations using various electron Green functions that approximate the shape of the constant energy surfaces as measured by ARPES in the normal and pseudogap states (29, 30). Qualitatively, none of the calculated interference patterns, some of which are shown in Fig. 4, resemble the experimental patterns (Fig. 2). In general, the features in all of the calculated patterns disperse through a range of wavelengths that should be easily resolved in our data (∼0.4 to 1.0 a0 over 40 mV). The underlying reason for this dispersion is described by the schematic in Fig. 4D, which shows how one wave vector potentially relevant for quantum interference changes with energy. The experimental findings in the pseudogap state reported here are thus also quantitatively inconsistent with the quantum interference scenario, simply because the observed modulations' wavelength is independent of energy (to within 0.2a0) (28).

Fig. 4.

The quasiparticle scattering picture, in which electron waves scattering off of random defects interfere coherently to form standing waves, qualitatively disagrees with the data in Fig. 2. Using a T-matrix formalism (2123) for scattering quasiparticles from a point-like nonmagnetic defect of strength 160 mV, we first reproduced the dispersion of the peaks in q-space seen in low-temperature STM data (not shown). We next calculated the scattering DOS for the same impurity in the first Brillouin zone in q-space for Fermi liquid quasiparticles (with broadening of 3 mV) with the Bi2Sr2CaCu2O8+δ band structure (29) at (A) –20 meV, (B) +20 meV, and (C) +60 meV. The insets show the same calculation at the respective energies using an electronic state that phenomenologically approximates the Fermi arcs seen in ARPES data at T = 100 K [Δ0 = 40 mV, decoherence = 10 mV, broadening = 3 mV; the extent of the arcs in k-space was chosen such that a vector q = (0, 2π/4.7a0) joins the tips of the arcs at the Fermi energy] (30). The features seen in these calculated diagrams do not change appreciably for different choices of the broadening parameters, or the strength or nature of the scattering center; here, unusually small values for broadening have been used to produce sharper q-space pictures that more closely resemble the data. (D) Qualitatively, the scattering DOS can be thought of as a statistical weight assigned to the vector (e.g., yellow arrows) that joins two points on curves of constant electron energy in k-space (blue curves), which we show in the main part of the figure for –20 mV (light blue, light yellow), +20 mV (blue, yellow), and +60 mV (dark blue, dark yellow) electrons in the Fermi liquid picture. Even if we assume that scattering occurs only between the tips of the Fermi arcs from the ARPES data (30), the length of the scattering wave vector changes between 2π/4.7a0 at the Fermi energy (light blue, light yellow) up to 2π/5. 1a0 at 36 meV (dark blue, dark yellow), as shown in the inset.

Next, we consider how the observed incommensurate modulation in the pseudogap state might be related to the various proposals for electronic ordering in the cuprates. On general theoretical grounds, the static electronic patterns we observed could be caused by one of the proposed ordered states or could be a consequence of electronic scattering from a fluctuating order that is pinned by disorder in the sample. The first candidate is one-dimensional charge ordering (i.e., stripes), which is predicted to have a spacing of four lattice constants— close to that reported here. However, the electronic patterns that we observe in the pseudogap state appear to be inherently two-dimensional, which either suggests the absence of stripes or requires fluctuating or highly disordered stripes in two orthogonal directions.

The second possibility is that of local ordering of spins (4, 5, 7), fluctuations of which are commonly observed in neutron scattering experiments on underdoped samples both above and below Tc (16, 17). Recent theoretical work, in connection with both neutron scattering and STM experiments, has yielded the idea that pinning of spin fluctuations by imperfections can give rise to modulations in the local DOS having a period corresponding to half of that of spin correlations (15). This connection between spin and charge modulations appears to be approximately observed if the magnetic modulation wave vectors from neutron scattering are compared with the STM ordering wave vectors reported here for the pseudogap state. Similar arguments have been made in connection with experimental measurements of electronic modulations in the vicinity of vortices (15, 26). Despite the attractive features of this scenario, at this time there is no consensus on whether spin fluctuations alone can explain the opening of the pseudogap, which we find to be directly correlated with the observation of the modulations.

In addition to the variety of ordering proposals, a view of the pseudogap state that has received considerable attention is that of a fluctuating superconducting state, with pair correlation persisting well above the resistive transition (13). Within this and related models, it has been shown that a pair-liquid having a fluctuating phase can also show modulated patterns in its local DOS. These patterns are predicted to have energy-dependent dispersions related to the shape of the Fermi surface (31). This prediction contradicts our finding of dispersionless modulations in the pseudogap state, hence posing a difficulty for the preformed-pairs scenario. However, as an alternative to a phase-disordered pair-liquid, it might be possible for the preformed pairs to localize and form a disordered static lattice. In this picture, assuming that the pseudogap is related to pairing, the pair-lattice modulation could only be detected when probing electronic states at energies smaller than the pseudogap energy. Such a scenario, which combines elements from different proposals on the pseudogap state, has in fact been discussed with regard to vortex cores in a doped Mott insulator (32).

Overall, the connection between spatially organized electronic patterns and the pseudogap reported here supports the theme that the complex cuprate phase diagram is a result of a competition between various types of ordering phenomena. Although we have not clearly identified an order parameter in the pseudogap regime, our findings together with those in the vortex state (26, 27) suggest that such an order parameter competes or is incompatible with superconductivity.

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