Abstract
Angleresolved photoemission spectroscopy (ARPES) probes the momentumspace electronic structure of materials and provides invaluable information about the hightemperature superconducting cuprates^{1}. Likewise, scanning tunnelling spectroscopy (STS) reveals the cuprates’ realspace inhomogeneous electronic structure. Recently, researchers using STS have exploited quasiparticle interference (QPI)—wavelike electrons that scatter off impurities to produce periodic interference patterns—to infer properties of the quasiparticles in momentum space. Surprisingly, some interference peaks in Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} (Bi2212) are absent beyond the antiferromagnetic zone boundary, implying the dominance of a particular scattering process^{2}. Here, we show that ARPES detects no evidence of quasiparticle extinction: quasiparticlelike peaks are measured everywhere on the Fermi surface, evolving smoothly across the antiferromagnetic zone boundary. This apparent contradiction stems from differences in the nature of singleparticle (ARPES) and twoparticle (STS) processes underlying these probes. Using a simple model, we demonstrate extinction of QPI without implying the loss of quasiparticles beyond the antiferromagnetic zone boundary.
Main
Recently, STS has been used to infer momentumspace information from the Fourier transform of the position (r) and energy (ω)dependent local density of states (LDOS), ρ(r,ω) (refs 3, 4, 5, 6). Conventionally, a superconductor has welldefined momentum eigenstates (that is, Bogoliubov quasiparticles), so ρ(r,ω) is spatially homogeneous. However, scattering from inhomogeneities often present in the cuprates^{7} mixes momentum eigenstates and QPI manifests itself as a spatial modulation of ρ(r,ω) with welldefined wave vector q, appearing in the Fourier transform, ρ(q,ω). QPI experiments are interpreted using the octet model^{3,4,5,6}, positing that wave vectors q_{1−7} connecting the ends of ‘banana shaped’ contours of constant energy (CCEs) dominate ρ(q,ω). Dispersing wave vectors are associated with coherent superconducting quasiparticles and the evolution of q_{1−7} as a function of bias voltage is used to infer the Fermi surface and the magnitude of the dwave superconducting gap. QPI experiments have found that the intensity of some of the peaks in ρ(q,ω) vanishes on approaching the diagonal line between (0,π) and (π,0) (corresponding to the antiferromagnetic zone boundary)^{5}. These results lead naturally to speculation about the disappearance of QPI and possible extinction of quasiparticles themselves near the boundary of the Brillouin zone (antinodal region).
Notably, ARPES has long shown antinodal quasiparticles below the critical temperature T_{c} in Bi2212 over a wide doping range (p>0.08; refs 8, 9, 10), although this point has not been a focus in the literature. In Fig. 1, we report data on single crystals of Bi2212 at four dopings spanning much of the same doping range as reported in recent STS studies^{5}, from slightly underdoped (T_{c}=92 K, denoted UD92) to substantially underdoped (UD50). ARPES measurements were carried out at Beamline 5–4 of the Stanford Synchrotron Radiation Lightsource using a Scienta R4000 electron analyser. Samples were cleaved in situ at a pressure better than 5×10^{−11} torr and measured at 10 K with an energy resolution of 8 meV. UD50 was measured with 19eV photons with cuts parallel to Γ Y and the other samples were measured with 22.7eV photons with cuts parallel to Γ M. These data demonstrate sharp peaks at low binding energy, associated with coherent quasiparticles, in all of the energy distribution curves (EDCs) around the Fermi surface, strongly arguing against extinction in the singleparticle spectrum. On the other hand, Fouriertransform STS studies report dispersing QPI over only a limited energy range, corresponding to the region of the Fermi surface intermediate between the node and the antinode. In this energy range, both ARPES and Fouriertransform STS show a similar Fermi surface for samples with p∼11% (Fig. 1f), but quasiparticles are clear in ARPES both in the nodal region and in the antinodal region, the latter being the focus of our study. Although the antinode is associated with pseudogap physics, antinodal quasiparticles seem to ‘know’ about superconductivity: for underdoped Bi2212, the antinodal quasiparticle weight has been shown to scale with the superfluid density^{2,8} and T_{c}. The inset of Fig. 1f illustrates that the quasiparticle peak emerges near the antinode only below T_{c} (ref. 11), whereas the antinodal gap seems to smoothly evolve into the pseudogap above T_{c}.
The EDCs in Fig. 1 were fitted to a simple model to assess whether there is anomalous scattering near the antiferromagnetic zone boundary. Although many factors contribute to the amplitude and width of the quasiparticle peaks, including matrix element and k_{z} effects^{12}, qualitative conclusions about the singleparticle scattering rate can be deduced by analysing the momentum dependence of the peak width. Symmetrized EDCs were fitted to a spectral function with an energydependent scattering rate, Γ(ω)=α ω, convolved with the experimental energy resolution. A similar model was found to provide a robust fit for the spatially inhomogeneous STS conductances seen in Bi2212 (ref. 13). At k=k_{F}, the only free parameters are the prefactor, α, and the gap energy, Δ(k); together, these define a characteristic scattering rate, Γ_{2}^{*}(k)=α Δ(k). Figure 2 shows Γ_{2}^{*} as a function of Fermi surface angle for UD75 and UD92, plotted together with the fitted gap. Near the antinode, the peak width is smaller than the gap, implying that the peaks are quasiparticlelike in this region of interest. For UD92, the quasiparticle width changes little from the node to the antinode, a result similar to earlier work on overdoped samples^{14}. The UD75 scattering rate shows a stronger momentum dependence, but the overall variation is still only a factor of three. For both dopings, there is no anomaly in the scattering at the antiferromagnetic zone boundary, ruling out proposals for quasiparticle extinction that invoke a sudden increase in singleparticle scattering there^{5}.
What could plausibly explain the apparent contradiction between ARPES and Fouriertransform STS results? One intriguing observation is that not all QPI wave vectors in ρ(q,ω) vanish across the antiferromagnetic zone boundary: q_{1,4,5} survive, whereas the others fall below the Fouriertransform STS noise floor^{5}. Here, we study the effects of quasiparticle scattering from impurities on the basis of a weakcoupling approach^{15}. Although this neglects the large, relevant, spatial inhomogeneity in the LDOS shown with STS (ref. 7), it places a simple focus on the differences between measurements, contrasting electron removal spectra in ARPES against twoparticle QPI mechanisms in Fouriertransform STS.
On the basis of ARPES results, the electron propagators are described by the Nambu Green’s function in the superconducting state and the momentumdependent matrix determines the nonuniform part of ρ(q,ω) (ref. 16):
where is written in terms of Pauli matrices , the unit matrix , the band structure ξ(p) and the dwave superconducting gap Δ(p)=Δ_{0}[cos (p_{x}a)−cos(p_{y}a)]/2. Contributions to the matrix can be classified according to how they modify electron parameters: conventional impurity scattering occurs in the channel, whereas local superconducting gap modification occurs in the channel.
A single impurity at site (0, 0) locally modifies hopping and the dwave superconducting gap through an impurity contribution to the Hamiltonian^{17,18,19}: , where spin indices have been suppressed. The hopping and dwave gap modulations, δ t(r) and δΔ(r), are proportional to δ(r−a x)±δ(r−a y)+δ(r+a x)±δ(r+a y), where a is the square lattice spacing and the upper (lower) sign is for hopping (dwave gap) modulation. Given this form of the impurity Hamiltonian, the Fourier transform, , has a simple momentumspace form: for hopping and dwave gap modulated scattering, respectively. In the channel (gap modulation), scattering vanishes between points with opposite orderparameter phases. Thus, q_{2,3,6,7} vanish identically, whereas q_{1,4,5} persist. In the channel (hopping modulation), scattering between equivalent points leads to no such cancellation, but the momentum dependence of the matrix implies a loss of intensity for q vectors connecting points p and p+q, which both lie on the antiferromagnetic zone boundary. Thus, the disappearance of QPI peaks associated with q_{2,3,6,7} at the antiferromagnetic zone boundary for the types of disorder considered here may be associated simply with the momentum dependence of the matrix, rather than implying the ‘extinction’ of quasiparticles. This also may reconcile the observation that the QPI peaks vanish even at large dopings (p≈0.19), a regime with diminished influence from the antiferromagnetic Mott insulating state.
The analysis presented above—a single impurity scatterer embedded in an infinite host—provides a simple view of the momentum dependence of the matrix and a plausible explanation for the loss of QPI peak intensity at the antiferromagnetic zone boundary. However, it cannot generally describe the behaviour of Fouriertransform STS QPI peak intensity as a function of bias voltage or the implications associated with the Fourier transform of the Z map, a ratio of the LDOS at positive and negative bias, rather than the LDOS itself. A better comparison with Fouriertransform STS results from considering the effects of an extended ‘patch’ impurity embedded in a finitesize, periodic host with the matrix determined selfconsistently. Finitesize effects are partially mitigated by smoothing the impurity with a Gaussian envelope, thus decreasing the influence of the impurity on electronic parameters further from the centre of the patch. In addition, the QPI intensity is determined from a Fourier transform of the Z map, as done in Fouriertransform STS experiments^{4,5}. More details of the calculation including the exact patch geometry and modulation parameters can be found in the Supplementary Information.
Figure 3c–h shows results for 5%, 10% and 15% hole doping. The dashed line in each panel indicates the energy associated with the antiferromagnetic zone boundary. Note that the general trends agree with the initial analysis from a single impurity scatterer. Intensities for peaks q_{2,3,6,7}, shown in Fig. 3c,e,g, initially rise on moving away from the nodal point and then begin to fall on approaching the antiferromagnetic zone boundary. The intensity for QPI peaks q_{1,5} is small initially and rises abruptly as the bias voltage approaches and crosses the energy associated with the antiferromagnetic zone boundary, both trends in agreement with the single impurity analysis. This behaviour is in qualitative agreement with the experimental intensity profiles as a function of energy for all QPI peaks^{5}. Although there are general quantitative changes with bandstructure parameters, impurity ‘patch’ size and shape and degree of modulation, the qualitative agreement with the singleimpurity analysis and experimental findings remains robust.
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Acknowledgements
We thank N. Nagaosa and J. Zaanen for helpful discussions. SSRL is operated by the US Department of Energy, Office of Basic Energy Science, Division of Chemical Science and Material Science. This work is supported by the US Department of Energy, Office of Science, Division of Materials Science, with contracts DEAC0276SF00515, DEFG0301ER45929A001 and NSF grant DMR0604701.
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I.M.V. analysed ARPES data and wrote the portions of the manuscript and Supplementary Information relevant to ARPES results. W.S.L. carried out ARPES experiments on samples grown by T.S. and K.T. carried out experiments on samples grown by T.F. B.M. and E.A.N. wrote and modified the computer codes and carried out the computational simulations. T.P.D., B.M. and E.A.N. wrote and edited those portions of the main text and Supplementary Information relevant to the simulations. T.P.D. and Z.X.S. are responsible for project direction, planning and infrastructure.
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Vishik, I., Nowadnick, E., Lee, W. et al. A momentumdependent perspective on quasiparticle interference in Bi_{2}Sr_{2}CaCu_{2}O_{8+δ}. Nature Phys 5, 718–721 (2009). https://doi.org/10.1038/nphys1375
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DOI: https://doi.org/10.1038/nphys1375
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