Strongly correlated systems: high-Tc superconductors: cuprates

  • P. D. JohnsonEmail author
Part of the Condensed Matter book series (volume 45B)


This chapter provides a brief introduction to high temperature superconductivity and also high temperature superconductivity in cuprates is investigated using different techniques.

In 1986, high-Tc superconductivity was discovered in a class of materials known as the cuprates [86B1]. For the first time, superconductivity was achieved at temperatures above those thought to be enabled through the electron-phonon interaction. In short order materials were identified with transition temperatures, Tc, above liquid nitrogen temperatures [87W1]. These discoveries prompted some major challenges for condensed matter physics research, namely, how do you explain the high-transition temperatures and how do we develop new theories that calculate the properties of strongly correlated materials. An excellent review of ARPES studies of the cuprates has been provided by Damascelli and coworkers [03D1]. While there has been very little study of surface-related phenomena in the cuprates, we include a discussion here because in that the cuprate materials are essentially two dimensional; they have proven to be perfect materials for study by photoemission. The 2D property plus new developments in the technology associated with ARPES pushed the photoemission technique to the very forefront in the study of condensed matter systems.

In the area of high-Tc superconductivity with the transition temperatures sufficiently high that the superconducting gap was visible in the ARPES spectra, it became possible to demonstrate the order parameter in the gap function, both in the superconducting phase and in the so-called pseudogap phase that exists in the underdoped region of the phase diagram in the normal state. The comparison of gap measurements along the copper-oxygen bond direction (A) and the zone diagonal (B) provided a clear demonstration of a superconducting order parameter having d-wave symmetry described by
$$ \Delta (k)={\Delta}_0\left(\cos {k}_xa-\cos {k}_ya\right) $$
where Δ0 represents the gap measured in the copper-oxygen bond direction at the zone boundary. This function is clearly very different from the isotropic s-wave symmetry associated with phonon-mediated superconductivity. Subsequent measurements found evidence of a gap, the so-called pseudogap, showing an angular dependence in the underdoped regime, even above Tc [96D2, 96M1]. There are now several studies that indicate that while in the overdoped regime the Fermi surface represents a full Fermi surface with area equal to 1 + p with p the number of doped holes, in the underdoped or pseudogap regime the Fermi surface is not simply a truncated full Fermi surface but rather deviates form that shape indicative of the presence of smaller pockets with area proportional to p (Figs. 125.1, 125.2, 125.3, 125.4, 125.5, 125.6, and 125.7).
Fig. 125.1

Crystal structure, Fermi surface, and low-energy electronic configuration of La2-xSr2CuO4 (LSCO): top, crystal structure, 3D Brillouin zone (body-centered tetragonal) and its 2D projection; diamond, Fermi surface at half filling calculated with only the nearest-neighbor hopping; gray area, Fermi surface obtained including also the next-nearest-neighbor hopping. Note that is the midpoint along Γ-Z and not a true symmetry point (From [03D1])

Fig. 125.2

(Left) measured superconducting gap anisotropy in underdoped superconductor Bi2Sr2CaCu2O8. The gap is shown in two key directions (From Shen [93S2]). (Right) measured superconducting gap as a function of angle (From [96D1])

Fig. 125.3

A spectroscopic measurement of the low-lying excitations around the Fermi surface, typical of the underdoped regime of the cuprate superconductors. The measurement is shown in one quadrant of the Fermi surface. The pseudogaps are observed at the end of the arcs in the (π,0) and (0,π) directions (This particular image is taken from [07N1])

Fig. 125.4

Spectral plots for optimally doped (Tc = 91 K) and underdoped (Tc = 65 K) BSCCO in the normal state after analysis allowing observation of the states above the chemical potential. The incident photon energy in the studies was 16.5 eV, and in all cases the spectra were recorded at a temperature of 140 K. (ac )Plots recorded from the optimally doped material in the nodal direction and away from the nodal direction, as indicated in (d). (eg) Same as (ac), but for the underdoped material, as indicated in (h). In (e) and (f), the vertical black dashed lines indicate Fermi surface crossings. In (f) and (g), the vertical blue dashed line indicates the turning point at the top of the dispersion. These are indicated in (h) by the open circles, black indicating a turning point above the Fermi level and red a turning point below the Fermi level. The filled black circles indicate the position of Fermi crossings. The possible pocket is indicated by the area enclosed by the blue dashed line in h (From [08Y1]). More details of the analysis are presented in [08Y1]

Fig. 125.5

Measured Fermi surfaces in the cuprates. Left panel. Measurements of underdoped and slightly overdoped Bi2Sr2CaCu2O8+δ (TCs indicated) (From [96M1]). Center panel. The Fermi surface crossings determined for a Bi2Sr2CaCu2O8+δ Tc = 45 K sample at three different temperatures. The triangles indicate measurements at a sample temperature of 140 K, the circles measurements at 90 K, and the diamonds measurements at 60 K (From [11Y1]). Right panel. ARPES measurements of the overdoped Tl2Ba2CuO6+δ system showing a full FS (From [07P1])

Fig. 125.6

(a) Phase diagram of Bi2201 with superconducting, pairing, and pseudogap temperatures. In the blue area, only the pseudogap is present in the samples, and real, gapless Fermi exist. In the red area, a pseudogap coexists with pairing and a d-wave-order parameter gaps Fermi surface. Panels (bd) schematically show gapless Fermi surface in three key areas (From [15K1])

Fig. 125.7

(a) The pseudopockets determined for three different doping levels. The black data correspond to the Tc = 65 K sample, the blue data correspond to the Tc = 45 K sample, and the red data correspond to the nonsuperconducting Tc = 0 K sample. The area of the pockets xARPES scales with the nominal of doping level xn, as shown in the inset. (b) The Fermi pockets derived from YRZ ansatz with different doping level (From [11Y1])

There have also been a number of studies that have targeted the self-energy effects in these materials. Studies of the imaginary part of the self-energy, ImΣ, as a function of binding energy and temperature [99V2, 06V1] confirmed a so-called marginal Fermi liquid picture [89V1] whereby ImΣ scaled linearly with ω and T in the normal state. These studies have also revealed a mass renormalization characteristic of the coupling to some form of boson. From its very first observation, i.e., Fig.  90.2 [99V2], through to the present time, the origin of the renormalization has been heavily disputed between on the one hand coupling to spin excitations [01J1] or to phonons [01L2]. Aside from structure in the measured dispersion in the vicinity of 60–70 meV binding energy, other structure has also been identified in the region of 10 meV [09R1, 10P1] binding energy and also at the higher binding energy of 300 meV [07V1]. Lifetime effects have also been investigated in pump-probe studies or two-photon photoemission (Figs. 125.8, 125.9, 125.10, 125.11, 125.12, 125.13, 125.14, 125.15, and 125.16).
Fig. 125.8

Quasiparticle dispersion of Bi2212, Bi2201, and LSCO along the nodal direction, plotted vs the rescaled momentum for (ac) different dopings and (d, e) different temperatures; black arrows indicate the kink energy; the red arrow indicates the energy of the q = (π,0) oxygen-stretching phonon mode; inset of (e): temperature-dependent Σ′ for optimally doped Bi2212; (f) doping dependence of the “coupling constant” λ′ along (0,0)-(π,π) for the different cuprates (From [01L2])

Fig. 125.9

MDC-derived quasiparticle dispersion along (0,0)-(π,π) for Bi2212: (a) underdoped; (b) optimally doped; (c) overdoped; (red) above Tc; (blue) below Tc; (df) corresponding Σ′, green symbols, difference between superconducting- and normal-state results; red lines, marginal-Fermi-liquid fits to the normal state data (From [01J1])

Fig. 125.10

Plot of ω0, the energy of the maximum value of ReΣ in the superconducting state (open circles), and (gray circles) the energy of the maximum in difference between the superconducting and normal state values plotted as a function of Tc referenced to the maximum (~91 K). The coupling constant λ (black triangles) is referenced to the right-hand scale (From [01J1])

Fig. 125.11

6.0 eV laser-based ARPES studies of optimally doped BSCCO in the vicinity of the nodal region. False color intensity maps of the raw [panels (ac)] and deconvolved [panels (df)] ARPES spectra. The color scaling is depicted at the top of panel (c). The intensity scales of the raw spectra are normalized to those of their deconvolved counterparts (From [09R1])

Fig. 125.12

6.0 eV laser-based ARPES studies of BSCCO for different doping levels showing the “kinks” in the dispersion in the vicinity of 70 meV and the low-energy kink in the vicinity of 10 meV. The inset highlights the dispersion in the vicinity of the Fermi level for the UD92 sample (From [10V1])

Fig. 125.13

Laser-based studies reveal the ReΣ as a function of temperature in the lower portion of the figure. Besides the well-known peak at about 70 meV, there is a low-energy (<10 meV) feature in ReΣ occurring below Tc (black arrow). The low-energy feature is especially evident in the change in ReΣ from 130 to 70 K (upper portion of figure) (From [10P1])

Fig. 125.14

Pump-probe AREPES studies of optimally doped BSCCO in the normal state. (ac) Photoelectron intensities as a function of electron momentum k-kF and binding energies with respect to EF before, during, and after optical excitation, respectively. The incident pump laser fluence in these studies was 300 μJ/cm2. Panel (d) shows the momentum-dependent intensity at EF integrated within +/− 10 meV before and during optical excitation as well as the respective differences. Solid black lines are Lorentzian fits. MDC widths as a function of binding energy are shown in panel (e) for 300 μJ/cm2 fluence, t = −5 psec (blue circles) and t = 0 fsec. (red squares). The solid line represents a linear-squares fit to E-EF< −70 meV, (t = −5 ps), MDC widths extrapolated to EF. The dashed line has the same slope and is shifted upward by 6 meV. The inset serves to highlight the pump-induced changes (From [14R1])

Fig. 125.15

Pump-probe studies of the fluence dependence of the nonequilibrium gap dynamics inside the Fermi arc of optimally doped BSCCO. (ad) Symmetrized EDCs at kF and fit curves for a gapped k-space cut at φ = 30° and T = 18 K (T < Tc). Bold curves respond to t = 0 ps. (e) Normalized gap magnitude versus pump-probe delay. (f) Gap recovery rates γΔ extracted from fitting the data as indicated in the ref. between 2–4 ps, 4–6 ps, and 6–10 ps (From [14S2])

Fig. 125.16

Laser-based photoemission studies of the temperature evolution of ARPES spectra in the nodal region of OP92. (a) Dispersion maps at several temperatures measured along a momentum cut close to the node (a red line in the inset of d). Each map is divided by the Fermi function at the measured temperature. (b) Temperature evolution of EDCs at kF (a circle in the inset of d) from deep below (10 K) to much higher than Tc (130 K). Each spectrum is divided by the Fermi function at the measured temperature. (c) The same data as in (b), but symmetrized about EF. (d) The same data as in (c) plotted without an offset. The inset represents the Fermi surface. The bold orange line indicates the momentum region where the Fermi arc was previously claimed to emerge at Tc. (e) Peak energies of spectra in (b, c) plotted as a function of temperature, εpeak. The solid and dashed blue curves show the BCS gap function with an onset at Tc (92 K) and slightly above Tc, respectively (From [15K2])

Symbols and abbreviation

Short form

Full form


angle-resolved photoelectron spectroscopy


critical temperature




Yang, Rice, and Zhang


momentum distribution curve


energy distribution curve


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Copyright information

© Springer-Verlag GmbH Germany 2018

Authors and Affiliations

  1. 1.Condensed Matter Physics and Materials Science DepartmentBrookhaven National LabUptonUSA

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