Evidence for Quantum Critical Behavior in the Optimally Doped Cuprate Bi2Sr2CaCu2O8+δ

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Science  24 Sep 1999:
Vol. 285, Issue 5436, pp. 2110-2113
DOI: 10.1126/science.285.5436.2110


The photoemission line shapes of the optimally doped cuprate Bi2Sr2CaCu2O8+δ were studied in the direction of a node in the superconducting order parameter by means of very high resolution photoemission spectroscopy. The peak width or inverse lifetime of the excitation displays a linear temperature dependence, independent of binding energy, for small energies, and a linear energy dependence, independent of temperature, for large binding energies. This behavior is unaffected by the superconducting transition, which is an indication that the nodal states play no role in the superconductivity. Temperature-dependent scaling suggests that the system displays quantum critical behavior.

Understanding transport in the cuprate superconductors and other synthetic metals is at the heart of current research in solid-state physics. There is speculation that the elementary excitations in these materials might well be drastically different from those in traditional metals where Fermi liquid theory has proved most successful (1). Angle-resolved photoemission (ARPES) has been an important experimental probe of the electronic excitations, providing insight into the fundamental interactions and quantities, such as the single-particle self energy Σ. The self energy reflects the interaction of the excitation with its local environment: the real part describing the screening of the excitation and the imaginary part describing the inverse lifetime. These microscopic properties are reflected in macroscopic behavior such as conductivity and the optical response. In two-dimensional (2D) systems, the self-energy effects measured in photoemission reflect solely the response to the photohole created in the emission process. As such, ARPES has provided a number of important insights into the cuprates, where the superconductivity is believed to be associated with the 2D CuO2 planes. Highlights from ARPES include mapping of the Fermi surface (2), the identification of d-wave symmetry of the order parameter in the superconducting state (3), and the detection of a pseudogap for underdoped compounds above the transition temperature T c (4, 5). However, the limited energy and momentum resolution of ARPES to date have meant that a quantitative determination of the Σ underlying the ARPES spectra has remained a considerable challenge.

Here we present ARPES data obtained from optimally doped Bi2Sr2CaCu2O8+δ, with excellent energy and momentum resolution, and employ a method of analysis allowing a detailed evaluation of the self energy along the (0,0) → (π,π) direction as a function of both binding energy and temperature. Overall, the imaginary component of Σ measured here is very different from that expected for a Fermi liquid or indeed that measured in normal metals (6). In the former, the width or scattering rate as determined by electron-electron interactions should have an ω2 andT 2 dependence (where ω is the binding energy). In most metals, the electron-phonon interaction dominates. The self energy due to this interaction displays a rapid change in the scattering rate in the vicinity of the Fermi level and saturation at higher energies (7). However, in the present study of a high-temperature superconductor, we find possible evidence for quantum critical behavior. The possibility of such behavior has recently been discussed in several models of high-T c superconductivity (8). In a quantum critical system, a phase transition can occur at zero temperature. As such, the transition is driven by quantum fluctuations rather than thermal fluctuations.

The present measurement can also be tied to conductivity measurements of the bulk material. Transport indicates that for optimally doped compounds, the resistivity in the normal state, and thus the inverse lifetime at the Fermi level, is linear with temperature over a wide range (9). Furthermore, infrared studies indicate that the scattering rate is linear in both temperature and frequency (10).

The photoemitted intensity in the (0,0) → (π, π) direction of the Brillouin zone (Fig. 1), as indicated by the arrow in the upper left inset, represents a 2D intensity map with binding energy in one dimension and momentum in the other (11). The intensity at any point is given by a false color scale, with white being the most intense. The two other insets show 1D scans obtained by taking cross-section cuts through the data in orthogonal directions. The lower right inset shows a cross section of the intensity as a function of energy at a constant in-plane momentum. Such a scan represents an energy distribution curve (EDC), which is the standard format for acquiring and displaying photoemission data. The inset in the upper right is an alternative method of displaying ARPES information. It shows the intensity as a function of parallel momentum at constant energy—a momentum distribution curve (MDC).

Figure 1

Two-dimensional spectral plot showing the intensity of emission in the (π,π) direction of the Brillouin zone as a function of ω, the binding energy, and k , the parallel momentum. The photon energy is 21.2 eV and the sample temperature is 48 K. Clockwise from upper left, the insets show the region of the Brillouin zone sampled in the experiment, a cross section through the intensity at constant energy (ω = 0) as a function of momentum (an MDC), and a cross section through the intensity at constant angle or momentum (k = k F) as a function of ω (an EDC).

The 2D intensity plot I(k,ω) represents a direct visualization of the spectral function of the photohole,A(k,ω), weighted by the Fermi distribution function. A(k,ω) is given by (12)Embedded Image Embedded Image(1)where ReΣ(k,ω) and ImΣ(k,ω) represent the real and imaginary components of the self energy, respectively. The width of the photoemission peak in an EDC reflects the inverse lifetime, ℏ︀/τ and is given by |2ImΣ(k,ω)|. Similarly, the width of the peak in an MDC reflects the mean free path ℓ = 1/Δk such thatEmbedded Image(2)Here νk represents the velocity of the excitation as determined by the band dispersion. Previous work analyzing ARPES line shapes from the cuprates concentrated on EDCs and attempted to fit the line shape to some form ofA(k,ω). Several difficulties arise with this procedure. In any EDC, there is a background component that is not included within A(k,ω). Determining which fraction should be regarded as this component is not straightforward. Furthermore, the line shapes of EDCs in these materials are complex, with no unique theoretical description to serve as a guide. The use of MDCs has several benefits. At the Fermi level, the peaks are symmetric with a simple background. Away from the Fermi level, the peaks develop a slight asymmetry reflecting the binding energy dependence of ImΣ. Ambiguities are not introduced by the Fermi function cutoff. The background in a MDC is constant because the step-like distribution in the vicinity of the Fermi level, which appears to exist at all momenta, is sampled at a constant energy. Finally, it is possible to take MDC cuts that are closely spaced in energy. These combined advantages allow an easier determination of the positions and widths of peaks. The resulting dispersion obtained from such cuts is more detailed than, but in general agreement with, previously published results. In agreement with calculation (13), the wave vector corresponding to the Fermi surface crossing is equal to 0.446 Å−1 or 0.391(π,π). In addition, the velocity or rate of dispersion changes in the vicinity of the Fermi level, resulting in an increased effective mass m* such thatm*/m b ≈ 1.6, wherem b represents the mass at higher binding energies.

The MDCs taken at ω = 0.0 ± 2.5 meV for several temperatures (Fig. 2A) represent the ARPES intensity at E F as a function of momentum along the (0,0) → (π,π) direction. Fitting the scans with a Lorentzian, we obtain momentum widths Δk for several samples over a range of temperatures (Fig. 2B). The offset between the different sets of data represents a small broadening, which varies from one cleave to another. However, the slopes are all identical within error bars. Also shown is the resistivity of a sample from the same crystal growth with the superconducting transition at 91 K. In simple models of conductivity, the resistivity is proportional to the inverse lifetime, or from Eq. 2, to the inverse mean free path (14). Although inverse lifetimes measured in conductivity are not identical to those measured in photoemission, in most solid-state systems they are approximately the same (15), and we see (Fig. 2B) that a proportionality does exist between the two measurements in the normal state. This observation is surprising because our own data, as discussed below, and a preponderance of previous results have demonstrated that optimally doped cuprates are not well described by Fermi liquid theory. However, our observation can be taken as an indication that the same physical mechanisms underlie both measurements.

Figure 2

(A) Momentum distribution curves as a function of temperature. The cuts are made at a binding energy corresponding to the Fermi surface. (B) Δk obtained from MDCs of the type shown in (A) for three samples indicated by the circles, squares, and diamonds. The error bars represent the statistical uncertainties from the fits to Lorentzian line shapes The thin lines represent linear fits to the data. The solid black line shows the temperature dependence of the resistivity ρ measured on an identical sample. The double-headed arrow indicates the combined energy and angular contributions to the momentum resolution of the experiment.

In the superconducting state, any such correlation disappears. There is no indication in Fig. 2B of any deviation in the linearity of the temperature dependence of the MDC peak widths at the transition temperature. Nor is any change observed in the peak dispersion near the Fermi level. It appears therefore that states in the (0,0) → (π,π) direction are not affected by the formation of the superconducting state. This is different from the behavior observed in other parts of the zone. The gap that opens atT c has its maximum value in the (π,0) direction, and in the same direction a sharp peak appears in the spectra reflecting the superconducting state (3,16). Thus, the overall picture that emerges is that in the normal state the conductivity is related to the excitations probed by photoemission along (0,0) → (π,π) and that the property of the superconducting state is determined by states in the (π,0) region of the Brillouin zone.

A series of EDC cuts through the 2D data array of Fig. 1 (Fig. 3A) is presented after background subtraction. Here we follow a procedure used in our earlier study of emission in the (π,0) direction (16), where we subtract a representative background taken at a momentum kfar from the region where any peak is found. Because there should nominally be no intensity at such a point, we assume that it represents an angle-independent background. In Fig. 3B, energies are rescaled so that the maximum intensity appears at the same energy for all EDCs, and the intensities are normalized to the same peak value. This procedure corresponds to a linear scaling of ω′ = ω/E P, with ω the measured binding energy andE P the position of the peak in an EDC. Despite their asymmetric shape, the EDCs at higher binding energies all scale linearly with binding energy, a direct indication that the width is proportional to ω. We also note from the scaling procedure that the width of the peak at binding energy ω is approximately 1.5 ω.

Figure 3

(A) EDCs obtained in the (0,0) → (π,π) direction after background subtraction. (B) EDCs scaled to the same peak position, showing that the overall shape scales linearly with binding energy. The peak width is approximately 1.5 times the binding energy in all spectra (double-headed arrow).

Summarizing ImΣ versus energy, data recorded at three temperatures from the same cleave along the (0,0) → (π,π) direction are shown (Fig. 4). Values derived from fits to MDC scans (converted to ImΣ using Eq. 2) are shown for the regions in which ν = νF. Values obtained directly from EDCs normalized to the Fermi function are also shown, along with a dashed line indicating |ImΣ| = 0.75 ω as determined from the scaling analysis in Fig. 3. The same behavior is obtained from analysis of either EDCs or MDCs. For a given temperature, ImΣ is constant up to some energy ω ∼ 2.5k B T. At larger binding energies, ImΣ scales linearly with binding energy and is independent of temperature, which is behavior very different from that found in metallic systems (17). Evidence for a linear dependence of ImΣ on ω has been presented in earlier ARPES studies (18). The crossover energy between the two regions is temperature-dependent. This energy/temperature dependence for ImΣ, embodied in the Marginal Fermi Liquid model (MFL) (19) and earlier non-Fermi liquid theories (20) for the normal state, continues smoothly through the superconductingT c. The observation that the behavior in the normal state is completely unaffected by the transition into the superconducting state suggests that the excitation is not decaying via scattering as would be the case for quasiparticles in a Fermi liquid. Such scattering would be sensitive to the change in available phase space upon formation of a gap. Rather the lack of sensitivity atT c suggests that the excitation decays through some other mechanism. The observation in the present study that the energy sets the temperature scale and the temperature sets the energy scale is consistent with quantum critical scaling (8). Indeed, as shown explicitly in the inset of Fig. 4, all of the data collapse onto a single curve when both ImΣ and ω are scaled byT, which is suggestive of quantum critical behavior. The demonstration of quantum critical behavior in the magnetic fluctuations has recently been reported in a study of the normal state of near optimally doped La2-xSrxCuO4(21). The present study suggests that single particle excitations also show behavior consistent with the existence of a nearby quantum critical point and that this is in fact a general property of the cuprates. We believe that the detailed information on the k and ω dependence of the single-particle excitations presented here, as well as their connection to transport properties, provides both a challenge to and a constraint on any theory of these complex materials.

Figure 4

A compilation of ImΣ obtained from Δkcuts or MDCs (solid symbols) and from peak widths in EDCs (open symbols) as a function of binding energy for 48 temperatures of K (diamonds), 90 K (triangles), and 300 K (circles). The inset shows the same data plotted in dimensionless units confirming scaling behavior. Error bars indicate uncertainties from the fits of MDCs to Lorentzian line shapes.


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