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Cuprate Superconductors

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

Phase diagram of the hole-doped cuprate superconductor Bi2Sr2CaCu2O8+δ.

Discovered in 1986, the cuprate superconductors to date hold the record for superconducting transition temperature (Tc) under ambient pressure. The high Tc and critical magnetic field of these materials have been crucial for many technological applications. Moreover, the intriguing superconducting and normal state properties of the cuprates have continuously challenged the conventional understanding of solids, galvanizing the exciting research field known as the strongly correlated electron systems.

The cuprate superconductors have a layered crystal structure consisting of copper oxygen planes and 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 dx2-y2 orbital character. The hole concentration can be altered by hole doping (p) or electron doping (n) from the charge reservoir layers. With no doping, the system is a Mott insulator, where holes cannot move around because of the large energy penalty (U) associated with two of them occupying the same site. At very high doping, correlation effects weaken, and the system approaches a normal metal described by band theory. While these two limits are relatively well understood, superconductivity and many other fascinating phenomena emerge in between, where a subtle balance of different interactions presents significant challenges for theory.

It turned out that the cuprate problem is an ideal subject for angle-resolved photoemission spectroscopy (ARPES) studies [1-3]. 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. On the other hand, the physics of cuprates contains a cascade of important energy scales from several electron volts to a fraction of one micro electron volt, which have motivated the pursuit for better resolution in ARPES and consistently provided positive feedback for instrumentation improvements. The study of cuprates provided a major imputation to propel 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.

Physics of the cuprate superconductors is one of the central research topics in the Shen group. Summaries of our previous highlights can be found in the sections below. Current projects include the study of heavily hole-doped compounds, with the goal to better understand high-Tc superconductivity, 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, synchrotron ARPES, and scanning tunneling microscopy; study of electron-doped cuprates and multilayer cuprates, which are 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, submitted.
  3. S. D. Chen, M. Hashimoto et al., Science 366, 1099 (2019).

Previous Highlights

d-wave superconductivity

d-wave superconducting gap

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 a hydrogen atom, the symmetry of the proton-electron bound state is dictated by the angular quantum number l. In a superconductor, the same quantum number is carried by Cooper pairs of electrons and reflects the symmetry of the superconducting gap. For l = 0, 1, 2, 3, the pairing states are often labeled as s, p, d, f-wave, respectively.

In conventional BCS superconductors, the structure of the pairing interaction prefers a simple s-wave state, where the superconducting gap has the same size at different momenta. In contrast, the intricate interactions in the high-Tc cuprates give rise to new possibilities. In 1993, we observed with ARPES that the superconducting gap in Bi2Sr2CaCu2O8+δ (Bi2212) has a large in-plane anisotropy [1]. This is one of the earliest experiments suggesting a d-wave pairing symmetry, which is later on confirmed by various other probes. Updated ARPES experiments further revealed that the gap follows the d-wave functional form. As the ARPES technique evolves, we have measured the d-wave gap to very high precision [2]. 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 provides a 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 above Tc. In the cuprates, we found that certain portion of the otherwise large metallic Fermi surface is truncated and gapped in the normal state. This is a defining feature that set apart the cuprates from conventional metals.

temperature evolution

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.

This effect, with the low energy states around the Cu-O bond direction (antinode) being suppressed, was already evident in our early studies of the global electronic structure evolution from undoped insulator to superconductor [1, 2]. Even for slightly underdoped samples, we found that the 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 preformed 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] dependences 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 the particle-hole symmetry [8, 9], a symmetry which is usually preserved in a superconductor. Moreover, we found spectroscopic evidence suggesting competition 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 both in 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 overall 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

In the BCS theory, the pairing interaction is mediated by phonons. In the hole-doped cuprates, the d-wave nature of the paring state suggests that the pairing interactions are different and more complex. To find the interactions which enable the Cooper pairing, 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 fingerprints in the electron spectral function. In general, the strength of coupling is a function of both the electron momentum and the mode wave vector. As such, ARPES can provide important insights on this subject, thanks to its ability to probe the spectral function in a momentum-resolved fashion.

mode coupling

Intertwined growth of superconductivity and electron-phonon coupling. The red line is an illustration of the Tc in Bi-2212. The blue shaded region and line represent the single-layer Bi-2201 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 dependences, and found that these modes are best matched to the in-plane half-breathing Cu-O bond stretching phonon [4], the out-of-plane B1g oxygen buckling phonon [4] and an in-plane small-q acoustic phonon [5], respectively. We further observed that for the B1g phonon, the doping evolution of coupling strength correlates with that of the superconducting gap size [6], which is consistent with the theory 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 also utilized time-resolved ARPES to measure the real-time quasiparticle energy shift due to coherent lattice vibrations. This is complementary to 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 system. Using this method, we have shown that electron-electron and electron-phonon interactions can cooperatively enhance each other [9]. We have also demonstrated the measurement of state-and-mode-specific electron-phonon interactions in 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

critical doping in Bi2212

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, where with the change of a tuning parameter, a continuous phase transition occurs at zero temperature. It is often conjectured that similar physics happens in the cuprates, with doping being the tuning parameter and the pseudogap being the manifestation of the underlying order.

We performed a set of experiments on Bi2212 to test this scenario. To locate the zero-temperature pseudogap ending point, we measured the momentum profile of the energy gap inside the superconducting state. By tracing the doping evolution of this profile, we identified the pseudogap critical doping at pc ~ 0.19 where the d-wave nodal gap changes behavior suddenly [1], which is consistent with several other experiments. We then systematically mapped out the temperature-doping phase diagram 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, but instead a temperature-independent boundary across which the pseudogap disappears and the 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. On the other hand, 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. From this perspective, a first step towards understanding these materials is to study the behavior of a single doped hole in the Mott insulating state. To accomplish this task, 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 the 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 insulator [1], we found that while the overall bandwidth of the single-hole state is consistent with t-J 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 the spectral intensity drops rapidly with increasing momentum across the antiferromagnetic zone boundary [2], which is expected in a small-U spin-density-wave scenario but not the t-J model or its variations. 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 be 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 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).