Focus on Fermiology of the Cuprates

Almost three decades after the discovery of high temperature cuprate superconductors, the origin of their high critical temperature remains an enigma. But it is now possible to discuss the 'fermiology' of the cuprates thanks to the discovery of quantum oscillations, which give valuable information about the Fermi surface and therefore the physical properties of a metal. The aim of this 'focus on' collection is to bring together experimentalists and theorists in order to provide new insights into the Fermi surface of cuprate superconductors. This new direction of research has definitely stimulated further progress in the field of high temperature superconductivity.

We show that a Fermi surface in underdoped YBa₂Cu₃O_6+x yielding the distribution of quantum oscillation frequencies observed over a broad range of magnetic field can be reconciled with the wavevectors of charge modulations found in nuclear magnetic resonance and x-ray scattering experiments within a model of biaxial charge ordering occurring in a bilayer CuO₂ planar system. Bilayer coupling introduces the possibility of different period modulations and quantum oscillation frequencies corresponding to each of the bonding and antibonding bands, which can be reconciled with recent experimental observations.

We survey the use of spectroscopic imaging scanning tunneling microscopy (SI-STM) to probe the electronic structure of underdoped cuprates. Two distinct classes of electronic states are observed in both the d-wave superconducting (dSC) and the pseudogap (PG) phases. The first class consists of the dispersive Bogoliubov quasiparticle excitations of a homogeneous d-wave superconductor, existing below a lower energy scale E=Δ₀. We find that the Bogoliubov quasiparticle interference (QPI) signatures of delocalized Cooper pairing are restricted to a k-space arc, which terminates near the lines connecting k=±(π/a₀,0) to k=±(0,π/a₀). This arc shrinks continuously with decreasing hole density such that Luttinger's theorem could be satisfied if it represents the front side of a hole-pocket that is bounded behind by the lines between k=±(π/a₀,0) and k=±(0,π/a₀). In both phases, the only broken symmetries detected for the |E|<Δ₀ states are those of a d-wave superconductor. The second class of states occurs proximate to the PG energy scale E=Δ₁. Here the non-dispersive electronic structure breaks the expected 90°-rotational symmetry of electronic structure within each unit cell, at least down to 180°-rotational symmetry. This electronic symmetry breaking was first detected as an electronic inequivalence at the two oxygen sites within each unit cell by using a measure of nematic (C₂) symmetry. Incommensurate non-dispersive conductance modulations, locally breaking both rotational and translational symmetries, coexist with this intra-unit-cell electronic symmetry breaking at E=Δ₁. Their characteristic wavevector Q is determined by the k-space points where Bogoliubov QPI terminates and therefore changes continuously with doping. The distinct broken electronic symmetry states (intra-unit-cell and finite Q) coexisting at E∼Δ₁ are found to be indistinguishable in the dSC and PG phases. The next challenge for SI-STM studies is to determine the relationship of the E∼Δ₁ broken symmetry electronic states with the PG phase, and with the E<Δ₀ states associated with Cooper pairing.

We report on the study of the Fermi surface of the electron-doped cuprate superconductor Nd_2−xCe_xCuO₄ by measuring the interlayer magnetoresistance as a function of the strength and orientation of the applied magnetic field. We performed experiments in both steady and pulsed magnetic fields on high-quality single crystals with Ce concentrations of x=0.13–0.17. In the overdoped regime of x>0.15, we found both semiclassical angle-dependent magnetoresistance oscillations (AMROs) and Shubnikov–de Haas (SdH) oscillations. The combined AMROs and SdH data clearly show that the appearance of fast SdH oscillations in strongly overdoped samples is caused by magnetic breakdown. This observation provides clear evidence for a reconstructed multiply connected Fermi surface up to the very end of the overdoped regime at x≃0.17. The strength of the superlattice potential responsible for the reconstructed Fermi surface is found to decrease with increasing doping level and likely vanishes at the same carrier concentration as superconductivity, suggesting a close relation between translational symmetry breaking and superconducting pairing. A detailed analysis of the high-resolution SdH data allowed us to determine the effective cyclotron mass and Dingle temperature, as well as to estimate the magnetic breakdown field in the overdoped regime.

A systematic angle-resolved photoemission study of the electronic structure of La_2−xSr_xCuO₄ in a wide doping range is presented in this paper. In addition to the main energy band, we observed a weaker additional band, the (π, π) folded band, which shows unusual doping dependence. The appearance of the folded band suggests that a Fermi surface reconstruction is doping dependent and could already occur at zero magnetic field.

After a brief review of current ideas on stripe order in cuprate high-temperature superconductors, we discuss the quasiparticle Nernst effect in cuprates, with focus on its evolution in non-superconducting stripe and related nematic states. In general, we find the Nernst signal to be strongly enhanced by nearby van-Hove singularities and Lifshitz transitions in the band structure, implying that phases with translation symmetry breaking often lead to a large quasiparticle Nernst effect due to the presence of multiple small Fermi pockets. Open orbits may contribute to the Nernst signal as well, but in a strongly anisotropic fashion. We discuss our results in the light of recent proposals for a specific Lifshitz transition in underdoped YBa₂Cu₃O_y and make predictions for the doping dependence of the Nernst signal.

We show that a variety of spectral features of high-T_c cuprates can be understood on the basis of the coupling of charge carriers to some kind of dynamical order that we exemplify in terms of fluctuating charge and spin density waves. Two theoretical models that capture different aspects of such a dynamical scattering are investigated. The first approach leaves the ground state in the disordered phase but couples the electrons to bosonic degrees of freedom, corresponding to the quasi-singular scattering associated with closeness to an ordered phase. The second, more phenomological approach starts from the construction of a frequency-dependent order parameter that vanishes for small energies. Both theories capture scanning tunneling microscopy and angle-resolved photoemission experiments that suggest the protection of quasi-particles close to the Fermi energy but the manifestation of long-range order at higher frequencies.

We report a detailed quantum oscillation study of the overdoped cuprate Tl₂Ba₂CuO_6+δ at two different doping levels (T_c=10 and 26 K). The derived Fermi surface size and topology complement earlier angle-dependent magnetoresistance studies and confirm the existence of a large quasi-cylindrical hole-doped Fermi surface with a small, but finite, c-axis warping. An accurate determination of the hole concentration reveals that superconductivity in Tl₂Ba₂CuO_6+δ does not follow the universal T_c(p) parabola for cuprate families and survives up to a larger doping of p_c=0.31. The observation of quantum oscillations for both dopings demonstrates that Fermi liquid behaviour is not confined to the edge of the superconducting dome, but is robust up to at least 0.3T^max_c. Moreover, the observation of such well-resolved oscillations implies that the physical properties of overdoped Tl₂Ba₂CuO_6+δ are determined by a single, spatially homogeneous electronic ground state. Finally, analysis of the different quasiparticle masses points towards a purely magnetic or electronic pairing mechanism.

We present our angle-resolved photoemission spectroscopy (ARPES) studies of the cuprate high-temperature superconductors; these studies elucidate the relation between superconductivity and the pseudogap and highlight low-energy quasiparticle dynamics in the superconducting state. Our experiments suggest that the pseudogap and the superconducting gap represent distinct states that coexist below T_c. Studies of Bi-2212 demonstrate that the near-nodal and near-antinodal regions behave differently as a function of temperature and doping, implying that different orders dominate in different momentum-space regions. However, the ubiquity of sharp quasiparticles all around the Fermi surface in Bi-2212 indicates that superconductivity extends into the momentum-space region dominated by the pseudogap, revealing subtlety in this dichotomy. In Bi-2201, the temperature dependence of antinodal spectra reveals particle–hole asymmetry and anomalous spectral broadening, which may constrain the explanation for the pseudogap. Noting that electron–boson coupling is an important aspect of cuprate physics, we end the paper with a discussion of the multiple 'kinks' in the nodal dispersion. Understanding these will be useful in establishing which excitations are important for superconductivity.

We discuss the low-energy theory of two-dimensional metals near the onset of spin density wave order. It is well known that such a metal has a superconducting instability induced by the formation of spin-singlet pairs of electrons, with the pairing amplitude changing sign between regions of the Fermi surface connected by the spin density wave ordering wavevector. Here, we review recent arguments that there is an additional instability that is nearly as strong: towards the onset of a modulated bond order that is locally an Ising-nematic order. This new instability is a consequence of an emergent 'pseudospin' symmetry of the low-energy theory—the symmetry maps the sign-changing pairing amplitude to the bond order parameter.

We present a neutron triple-axis and resonant spin-echo spectroscopy study of the spin correlations in untwinned YBa₂Cu₃O_6+x single crystals with x=0.3, 0.35 and 0.45 as a function of temperature and magnetic field. As the temperature T→0, all samples exhibit static incommensurate magnetic order with propagation vector along the a-direction in the CuO₂ planes. The incommensurability δ increases monotonically with hole concentration, as it does in La_2−xSr_xCuO₄ (LSCO). However, δ is generally smaller than in LSCO at the same doping level, and there is no sign of a reorientation of the magnetic propagation vector at the lowest doping levels. The intensity of the incommensurate Bragg reflections increases linearly with magnetic field for YBa₂Cu₃O_6.45 (superconducting T_c=35 K), whereas it is field independent for YBa₂Cu₃O_6.35 (T_c=10 K). These results fit well into a picture in which superconducting and spin-density wave order parameters coexist, and their ratio is controlled by the magnetic field. They also suggest that YBa₂Cu₃O_6+x samples with x∼0.5 exhibit incommensurate magnetic order in the high fields used for the recent quantum oscillation experiments on this system, which likely induces a reconstruction of the Fermi surface. We present neutron resonant spin-echo measurements (with energy resolution ∼1 μeV) for T≠0 that demonstrate a continuous thermal broadening of the incommensurate magnetic Bragg reflections into a quasi-elastic peak centered at excitation energy E=0, consistent with the zero-temperature transition expected for a two-dimensional spin system with full spin–rotation symmetry. Measurements on YBa₂Cu₃O_6.45 with a conventional triple-axis spectrometer (with energy resolution ∼100 μeV) yield a characteristic crossover temperature T_SDW∼30 K for the onset of quasi-static magnetic order. Upon further heating, the wavevector characterizing low-energy spin excitations progressively approaches the commensurate antiferromagnetic wavevector, and the incommensurability vanishes in an order-parameter-like fashion at an 'electronic liquid crystal' onset temperature T_ELC∼150 K. Both T_SDW and T_ELC increase continuously as the Mott-insulating phase is approached with decreasing doping level. These findings are discussed in the context of current models of the interplay between magnetism and superconductivity in the cuprates.

Experiments have revealed multiple quantum oscillation frequencies in underdoped high-temperature superconductor YBa₂Cu₃O_6+δ, corresponding to approximately 10% doping, which contains CuO bilayers in the unit cell. These unit cells are further coupled along the c-axis by a tunneling matrix element. A model of the energy dispersion that has its roots in the previously determined electronic structure, combined with twofold commensurate density waves, reveals multiple electron and hole pockets. To the extent that quasiparticles of the reconstructed Fermi surface have finite residues, however small, the formation of Landau levels is the cause of these oscillations, and the bilayer splitting and warping of the electronic dispersion along the direction perpendicular to the CuO-planes are firm consequences. The goal here is to explore this possibility from various directions and provide a better understanding of the rapidly developing experimental situation involving multiple frequencies. An important conclusion is that bilayer splitting is considerably renormalized from the value obtained from band structure calculations. It would be extremely interesting to perform these experiments for higher values of doping. We roughly expect the splitting of the frequencies to increase with doping, but the full picture may be more complex because the density wave order parameter is also expected to decrease with doping, vanishing around the middle of the superconducting dome.