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Cuprate Superconductors

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Phase diagram of the hole-doped cuprate superconductor Bi—2212.

Discovered in 1986, the cuprate superconductors hold the record for highest superconducting transition temperature (Tc) under ambient pressure to date. The high Tc's and critical magnetic fields of these materials have been crucial for technological applications. Moreover, their intriguing superconducting and normal state properties have continually challenged our conventional understanding of solids, stimulating research on strongly correlated electron systems.

The cuprate superconductors share a layered crystal structure consisting of copper–oxygen planes and intervening charge reservoir layers. In a simplified picture, a copper oxide plane can be viewed as a square lattice where every site contains one hole with dx2y2 orbital character. Hole concentration can be altered by hole doping (p) or electron doping (n) from the charge reservoir layers. There are two relatively well understood limits. Without doping, the system is a Mott insulator. Holes cannot move around because two holes occupying the same site incurs a large energy penalty (U). On the other hand, at very high doping, correlation effects weaken and the system approaches a normal metal well-described by band theory. It is between these limits where superconductivity and a veritable potpourri of other fascinating phenomena emerge — a representative phase diagram is shown above. A subtle balance of different interactions presents significant challenges for theoretical treatment.

The cuprate problem is an ideal subject for angle-resolved photoemission spectroscopy (ARPES) studies [1–3]. Firstly, from a material perspective, several families of cuprates are weakly bonded along the out-of-plane direction. This leads to flat cleavage planes and two-dimensional electronic structures, both of which greatly enhance the ability of ARPES to precisely probe the electron spectral function. Furthermore, the physics of cuprates features a cascade of important energy scales, from several electron volts (eV) to less than one micro electron volt (meV), which have motivated the pursuit for better resolution in ARPES and consistently provided positive feedback for instrumentation improvement. The study of the cuprates has provided a major impulse that has propelled ARPES to the forefront of quantum material research.

Evolution of ARPES. Top: approximate number of publications per year utilizing ARPES. Bottom: improvement of ARPES energy resolution with time.

The physics of the cuprate superconductors is one of the central research topics in the Shen group. A selection of our previous highlights is presented in the sections below. Current projects include: the study of heavily hole-doped compounds, with the goal to better understand high–Tc superconductivity (we are the only major player in this doping range); design and study of cuprate thin films, using our unique setup which combines oxide molecular beam epitaxy (MBE), synchrotron ARPES, and scanning tunneling microscopy; and study of electron-doped cuprates and multilayer cuprates, which were less explored in the past.


  1. A. Damascelli, Z. Hussain, and Z.-X. Shen, Review of Modern Physics 75, 473 (2003).
  2. J. Sobota, Y. He, and Z.-X. Shen, Review of Modern Physics 93, 025006 (2021).
  3. S. D. Chen, M. Hashimoto et al., Science 366, 1099 (2019).

Previous Highlights

d-wave superconductivity 
Coupling between electrons and bosonic modes
Pseudogap critical doping
Undoped and lightly doped Mott insulators


d-wave superconductivity 

Superconducting gap in Bi2Sr2CaCu2O8+δ. Left: ARPES data from 1993 showing the anisotropy of the superconducting gap. Right: Detailed electronic structure near the d-wave node measured by laser ARPES in 2010.

In the elementary example of a hydrogen atom, the symmetry of the (proton-electron) bound state is dictated by its angular quantum number l. In a superconductor, the same angular quantum number is carried by Cooper pairs of electrons and reflects the symmetry of the superconducting gap. Following the spectroscopic convention for atomic orbitals, l = {0, 1, 2, 3} superconducting pairing states are also labeled s, p, d, and f-wave respectively.

In conventional BCS superconductors, whose study dominated the field before 1986, the structure of the pairing interaction favors a simple s-wave (l = 0) state where the superconducting gap preserves the same amplitude across different momenta. In contrast, complex interactions in the high–Tc cuprates admits new possibilities in gap structure. In 1993, we used ARPES to discover a large in-plane anisotropy in the superconducting gap of Bi2Sr2CaCu2O8+δ (Bi2212) [1]. This was one of the earliest experiments that suggested a d-wave pairing symmetry. Various other experimental probes later confirmed this, and further ARPES experiments also established that the gap indeed follows the d-wave functional form. As ARPES technique evolved, we have measured the d-wave gap to very high precision [2]. Measuring the detailed profile of the gap has also allowed us to track the underlying electronic phases as a function of doping [3]. The study of the superconducting gap is a classic textbook example for how ARPES can facilitate our understanding of correlated electron systems.

Full momentum dependence of the superconducting gap in Bi2212. Left: Schematics of the band structure. Middle: ARPES energy distribution curves measured along the Fermi surface. Right: Extracted gap size as a function of Fermi surface angle.


  1. Z. X. Shen et al., PRL 70, 1553 (1993).
  2. I. M. Vishik et al., PRL 104, 207002 (2010).
  3. I. M. Vishik et al., PNAS 109, 18332 (2012).


ARPES intensity mappings at the Fermi energy, showing the Fermi arcs in Ca2-xNaxCuO2Cl2 (CCOC, left), Bi2212 (middle), and Bi2Sr2CuO6+δ (Bi2201, right).

In conventional superconductors, the superconducting gap vanishes in the normal state, the region of the phase diagram above Tc. In the cuprates, however, we found that a certain portion of the otherwise-large metallic Fermi surface is truncated and gapped in the normal state at low temperatures. This feature sets the cuprates apart from conventional metals.

Temperature evolution of the antinodal electronic structure in Bi2212 with Tc ~ 86 K. A. Energy distribution curves at (π, 0). The red arrow and blue stripe mark the development of the pseudogap (PG) and superconducting peak, respectively. B. Difference between the ARPES spectra near (π, 0) at 150 K and 250 K, highlighting the pseudogap effect. C. Difference between the ARPES spectra at 90 K and 150 K, showing the build-up of Bogoliubov quasiparticle (BQP) peak above Tc.

Along with the suppression of low energy states around the Cu–O bond direction (antinode), this feature was already evident in our early studies of global electronic structure evolution from an undoped insulator to a superconductor [1, 2]. Even for slightly underdoped samples, we found that low energy spectral weight remains suppressed in the normal state [3], leaving only a Fermi arc [4]. This peculiar phenomenon has since become a central topic in high–Tc research.

One of the most debated questions is whether this low-energy spectral weight suppression is due to pre-formed Cooper pairs, or something not directly related to superconductivity. Through a series of ARPES studies, we have shown that the spectral weight suppression contains a component which has different momentum [5], temperature [6], and doping [7] dependencies from those of the superconducting gap. In this context, we refer to this component as the pseudogap. We further discovered that the pseudogap does not obey particle-hole symmetry [8, 9], a symmetry which is usually preserved in a superconductor. Moreover, we found spectroscopic evidence suggesting competitive behavior between the pseudogap and superconductivity [10]. All these observations indicate that the pseudogap and superconductivity are distinct phenomena. On the other hand, we found that in both the presence and absence of the pseudogap, spectral features associated with superconductivity survive in a moderate temperature range above Tc [11]. This suggests that the low-energy spectral weight suppression is a combined effect of the pseudogap and superconductivity fluctuations.


  1. B. O. Wells et al., PRL 74, 964 (1995).
  2. D. M. King et al., Journal of Physics and Chemistry of Solids 56, 1865 (1995).
  3. A. G. Loeser et al., Science 273, 325 (1996).
  4. D. S. Marshall et al., PRL 76, 4841 (1996).
  5. K. Tanaka et al., Science 314, 1910 (2006).
  6. W. S. Lee et al., Nature 450, 81 (2007).
  7. I. M. Vishik et al., PNAS 109, 18332 (2012).
  8. M. Hashimoto, R. H. He et al., Nature Physics 6, 414 (2010).
  9. R. H. He, M. Hashimoto et al., Science 331, 1579 (2010).
  10. M. Hashimoto et al., Nature Materials 14, 37 (2015).
  11. S. D. Chen, M. Hashimoto et al., Science 366, 1099 (2019).

Coupling between electrons and bosonic modes

A pairing interaction is required to form Cooper pairs. In earlier conventional superconductors, and in BCS theory, the pairing interaction is phonon-mediated. In the hole-doped cuprates, however, the d-wave nature of the pairing state suggests pairing interactions are different and more complex. To find these interactions, it is crucial to examine the roles of various bosonic modes, especially those strongly coupled to electrons.

When bosonic modes couple to electrons, they leave telltale signs in the electron spectral function, most famously a kink in the dispersion. In general, electron-boson coupling strength g is a function of both electron momentum k and mode wave vector q. ARPES can provide important insights on coupling, owing to its ability to probe the spectral function with momentum resolution.

Intertwined growth of superconductivity and electron-phonon coupling. The red line is an illustration of Tc in Bi2212. The blue shaded region and line represent the single-layer Bi2201 system, in which the coupling to the B1g mode is weak. The yellow ball represents the optimally doped trilayer Bi2Sr2Ca2Cu3O10+δ.

Using ARPES, we discovered distinct signatures of mode coupling with mode energies around 70 meV [1], 35 meV [2] and 10 meV [3]. We systematically studied their momentum, temperature, doping, and material family dependencies, and found that these modes are best matched to three respective phonons: the in-plane half-breathing Cu–O bond stretching phonon [4], out-of-plane B1g oxygen buckling phonon [4], and an in-plane small-q acoustic phonon [5]. We further observed that for the B1g phonon, doping evolution of its coupling strength correlates with that of the superconducting gap size [6], which is consistent with the theoretical prediction that this phonon enhances d-wave pairing [7]. This mechanism is echoed in our understanding of interface-enhanced superconductivity in monolayer FeSe films [8].

Recently, we utilized time-resolved ARPES to measure the real-time quasiparticle energy shift due to coherent lattice vibrations. This complements equilibrium ARPES experiments due to its improved sensitivity to low-energy phonons. Furthermore, when combined with ultrafast diffraction techniques, this approach provides a direct measure of the electron-phonon coupling strength in correlated electron systems. Using this method, we have shown that electron-electron and electron-phonon interactions can cooperatively enhance each other [9]. We have also demonstrated a measurement of state- and mode-specific electron-phonon interactions in the cuprates [10].


  1. A. Lanzara et al., Nature 412, 510 (2001).
  2. T. Cuk et al., PRL 93, 117003 (2004).
  3. I. M. Vishik et al., PRL 104, 207002 (2010).
  4. T. P. Devereaux et al., PRL 93, 117004 (2004).
  5. S. Johnston et al., PRL 108, 166404 (2012).
  6. Y. He, M. Hashimoto et al., Science 362, 62 (2018).
  7. S. Johnston et al., PRB 82, 065513 (2010).
  8. J. J. Lee, F. T. Schmitt, R. G. Moore et al., Nature 515, 245 (2014).
  9. S. Gerber, S. L. Yang et al., Science 357, 71 (2017).
  10. S. L. Yang et al., PRL 122, 176403 (2019).

Pseudogap critical doping

ARPES spectra (top) and their second energy derivatives (bottom) for Pb-Bi2212 crystals with hole doping below (left) and above (right) pc ~ 0.19. Data taken along the Brillouin zone boundary at 250 K. Quasiparticle dispersions are clear for p > pc and absent for p < pc.

In several heavy fermion materials, superconductivity emerges in the vicinity of a quantum critical point, a point in the phase diagram where a continuous phase transition occurs at zero temperature upon varying a tuning parameter. It is often conjectured that similar physics happens in the cuprates, with doping as the tuning parameter, and the pseudogap phase as the manifestation of underlying order.

We performed a set of experiments on Bi2212 to test this scenario. To locate the zero-temperature point at which the pseudogap ends, we measured the momentum profile of the energy gap inside the superconducting state. By tracing the doping evolution of this profile, we identified pseudogap critical doping at pc ~ 0.19, defined as the point where the d-wave nodal gap suddenly changes behavior [1]. This was consistent with several other experiments. We then systematically mapped out the phase diagram as a function of temperature and doping, near this critical doping, by precisely measuring the spectral function in a representative momentum range. Surprisingly, we found that the critical doping is not associated with a V-shaped quantum critical region, as one would have expected. Instead, a temperature-independent boundary was discovered, across which the pseudogap disappears, and conventional quasiparticle dispersions recover [2]. We also noted a rapid change of electron-phonon coupling across this boundary [3]. These results indicate that the pseudogap observed by ARPES in Bi2212 is not linked to a quantum critical point. In lieu of this, a close connection between the pseudogap and quasiparticle decoherence is suggested.


  1. I. M. Vishik et al., PNAS 109, 18332 (2012).
  2. S. D. Chen, M. Hashimoto et al., Science 366, 1099 (2019).
  3. Y. He, M. Hashimoto et al., Science 362, 62 (2018).

Undoped and lightly doped Mott insulators

The high–Tc cuprates are doped Mott insulators. A first step towards understanding these materials, then, is to study the behavior of a single doped hole in the Mott insulating state. To do so, we performed ARPES experiments on undoped cuprates. The single-hole states are easily realized in such experiments, as each photoemission event creates one hole inside the sample. By measuring the angles and energies of the outgoing photoelectrons, the energy distribution curves (EDC) of single-hole states at different momenta can be obtained.

ARPES results on Ca2-xNaxCuO2Cl2 (CCOC). Left: Quasiparticle dispersion and energy-integrated spectral weight as a function of momentum in undoped CCOC. Right: Change of EDC lineshape with hole doping.

Building on our early study of undoped insulators [1], we found that while the overall bandwidth of the single-hole state is consistent with tJ model predictions, the detailed dispersion only agrees with theory when hopping terms up to the third nearest neighbor are included [2]. Moreover, we observed that spectral intensity drops rapidly with increasing momentum across the antiferromagnetic zone boundary [2]. This is expected in a small-U spin-density-wave scenario, but not the tJ model or its variants. These observations suggest that a Hubbard model with intermediate U and longer-range hopping terms are likely needed.

Moreover, we discovered that the EDC line shape is always much broader than those predicted in purely electronic models. We proposed that this additional broadening is caused by polaronic electron-phonon interaction [3]. Due to the lack of metallic screening, the charged carriers are strongly coupled to lattice deformations and thus exhibit suppressed quasiparticle weights. As a result, the apparent maximum of an EDC can come from the incoherent part of the spectral function and occur at a binding energy higher than the real quasiparticle energy. We also found that this polaronic line shape persists to finite doping with strong momentum anisotropy [3, 4]. These results indicate that the physics of the cuprates—as realizations of doped Mott insulators—is governed by not only electron-electron, but also electron-phonon interactions.


  1. B. O. Wells et al., PRL 74, 964 (1995).
  2. F. Ronning et al., Science 282, 2067 (1998).
  3. K. M. Shen et al., PRL 93, 267002 (2004).
  4. K. M. Shen et al., Science 307, 901 (2005).