Cuprate superconductors as viewed through a striped lens

Abstract

Understanding the electron pairing in hole-doped cuprate superconductors has been a challenge, in particular because the “normal” state from which it evolves is unprecedented. Now, after three and a half decades of research, involving a wide range of experimental characterizations, it is possible to delineate a clear and consistent cuprate story. It starts with doping holes into a charge-transfer insulator, resulting in in-gap states. These states exhibit a pseudogap resulting from the competition between antiferromagnetic superexchange J between nearest-neighbor Cu atoms (a real-space interaction) and the kinetic energy of the doped holes, which, in the absence of interactions, would lead to extended Bloch-wave states whose occupancy is characterized in reciprocal space. To develop some degree of coherence on cooling, the spin and charge correlations must self-organize in a cooperative fashion. A specific example of resulting emergent order is that of spin and charge stripes, as observed in La${}_{2-x}$Ba${}_x$CuO${}_4$. While stripe order frustrates bulk superconductivity, it nevertheless develops pairing and superconducting order of an unusual character. The antiphase order of the spin stripes decouples them from the charge stripes, which can be viewed as hole-doped, two-leg, spin-$\frac{1}{2}$ ladders. Established theory tells us that the pairing scale is comparable to the singlet-triplet excitation energy, $\sim J/2$, on the ladders. To achieve superconducting order, the pair correlations in neighboring ladders must develop phase order. In the presence of spin stripe order, antiphase Josephson coupling can lead to pair-density-wave superconductivity. Alternatively, in-phase superconductivity requires that the spin stripes have an energy gap, which empirically limits the coherent superconducting gap. Hence, superconducting order in the cuprates involves a compromise between the pairing scale, which is maximized at $x\sim \frac{1}{8}$, and phase coherence, which is optimized at $x\sim 0.2$. To understand further experimental details, it is necessary to take account of the local variation in hole density resulting from dopant disorder and poor screening of long-range Coulomb interactions. At large hole doping, kinetic energy wins out over J, the regions of intertwined spin and charge correlations become sparse, and the superconductivity disappears. While there are a few experimental mysteries that remain to be resolved, I believe that this story captures the essence of the cuprates.

Keywords:

• Superconductivity
• hole-doped cuprates
• charge stripes
• spin stripes
• charge density wave
• pair density wave
• pairing mechanism
• disorder

PACS:

• 74.72.Gh
• 75.50.Ee
• 71.10.Li

Acknowledgments

The author has benefited from discussions with and comments by I. Bozovič, E. Dagotto, M. P. M. Dean, E. Fradkin, K. Fujita, C. C. Homes, M. Hücker, P. D. Johnson, S. A. Kivelson, D.-H. Lee, P. A. Lee, Q. Li, Y. Li, H. Miao, N. J. Robinson, D. J. Scalapino, Senthil, A. Tsvelik, T. Valla, I. Zaliznyak, and many others. The author is especially grateful to T. Egami for challenging me to tell an interesting story.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 La${}_2$CuO${}_4$ can easily pick up a small amount of excess oxygen during synthesis, and phase separation can result in an impurity phase that is metallic and superconducting [12].

2 Earlier Hartree-Fock and related calculations had provided solutions of charge and spin stripes [36–39]; however, the charge stripes were always insulating.

3 Another challenge in the superconducting stripe model of [41] is that charge-density-wave order within the stripes would compete with pairing. To avoid this possibility, the idea of fluctuating stripes in the form of nematic and smectic orders was introduced [44]. Again, this has a bias of static stripe order being bad for hole pairing.

4 This is supported by a recent phenomenological analysis [54].

5 There is some hybridization of the $3d_{x^2-y^2}$ orbital to the neighboring O $2p_{\sigma }$ orbitals [65], where the degree of hybridization can vary among cuprate families [66].

6 If one ignores the O sites and uses a single-band Hubbard model, then $J=4t^2/U$, where t is the Cu-Cu hopping parameter. Quantitative agreement with experiment requires consideration of the O, which leads to a more complicated formula, proportional to $t_{pd}^4$, where $t_{pd}$ is the hopping energy from O to Cu [68,69].

7 In this notation, the nearest-neighbor Cu spacing a is set equal to 1. In reciprocal lattice units based on $a^{\ast }=2\pi /a$, one would write ${\mathbf{Q}}_{\mathrm{A}\mathrm{F}}=(\frac{1}{2},\frac{1}{2})$.

8 Anderson speculated that the quantum spin fluctuations for the 2D $S=\frac{1}{2}$ Heisenberg model might be great enough to prevent order, resulting in an RVB state [13]. This stimulated a great deal of theoretical effort on quantum spin liquids, and speculation that it was the RVB character of the initial state that led to superconductivity on doping [30,73,74]. In contrast, as we will see, competition been electronic kinetic energy and AF order is a key factor in the present story.

9 For a review of techniques for measuring optical conductivity and applications to high-$T_c$ superconductors, see [100,101]

10 For a review of scanning tunneling spectroscopy and applications to high-temperature superconductors, see [103].

11 This is essentially the Emery model [59], but with the onsite Coulomb repulsion U set to zero.

12 Note that, for this section, the lattice parameter a is set equal to 1.

13 The imaginary part of the Bloch wave forms a degenerate state, decoupled from the real part.

14 For a proper analysis of the problem of one hole in a two-dimensional (2D) antiferromagnet, see [110,111].

15 For a review of angle-resolved photoemission and applications to cuprates, see [14].

16 The picture of an antinodal pseudogap defined by scattering from spin fluctuations is supported by a variety of advanced numerical calculations applied to the Hubbard model [113–116].

17 For reviews of neutron scattering, with applications to cuprates and related systems, see [118,119].

18 Another factor for the STM results concerns the nature of the tunneling process from the probe tip to the sample surface and along the c axis, typically through an apical O site, to a Cu site in the CuO${}_2$ plane nearest the surface. The $2p$ orbital on the apical site has s symmetry relative to the Cu, so that it cannot couple to the $3d_{x^2-y^2}$ but may couple to the in-plane O $2p_{\sigma }$ states. From Figure 4(b) and 4(d), one can see that no coupling is possible at $\mathbf{k}=(\pi /2,\pi /2)$ because the O orbitals are all in phase with the nearest Cu $3d_{x^2-y^2}$ orbital, but there is a finite coupling at $(\pi ,0)$ as the phasing is different. Such effects were originally noted in analyses of c-axis conduction and planar tunneling [121,122] and are discussed for STM in [123].

19 This is different from the proposal of the Zhang-Rice singlet [127]. That picture assumes that a hole is bound symmetrically about a Cu site, resulting in no net moment. This would tend to act as a dilution of the AF lattice. But we know from experiment that dilution with Zn only destroys long-range order at the percolation limit (41% Zn concentration) [128].

20 For a review of Raman scattering and applications to cuprates, see [133].

21 Magnetic excitations can also be probed by resonant inelastic x-ray scattering (RIXS) at the Cu $L_3$ edge. For example, measurements of magnetic excitations for Q from zero to the AF zone boundary performed on thin films of LSCO for doping extending out to the highly overdoped regime suggest less doping dependence of the measured excitations [144]. However, there are several issues to consider in evaluating these results, the most important being that the measurements cannot reach the vicinity of the ${\mathbf{Q}}_{\mathrm{A}\mathrm{F}}$, where inelastic neutron scattering studies clearly demonstrate large changes with doping in the definitive AF correlations that theory considers most relevant to superconductivity [145–147].

22 A study using terahertz spectroscopy in pulsed magnetic fields has determined a cyclotron mass of $4.9\pm 0.8m_e$ for LSCO with x = 0.16 [151].

23 Related data for Bi2212 are reported in [152].

24 This was estimated by suppressing the superconductivity by Zn substitution [155].

25 In the system La${}_{1.6-x}$Nd${}_{0.4}$Sr${}_x$CuO${}_4$, measurements of the Hall effect at low temperature and high magnetic field (to suppress superconductivity) have been interpreted as indicating a rapid rise of the carrier concentration to a level of 1 + p at $p^{\ast }\approx 0.23$ [158] (with analogous behavior in YBCO at $p^{\ast }\approx 0.19$ [159]). The analysis here is not clear cut, as ARPES measurements [160] indicate that a Lifshitz transition, along with the closing of the antinodal pseudogap, occur at or near $p^{\ast }$. These latter features also appear to be consistent with the observation of a peak in the electronic specific heat at $p^{\ast }$ [161] (where analysis is complicated by a Schottky anomaly due to the magnetic Nd ions [162]). Hall effect measurements on LSCO and Bi2201 yielded sharp cusps in the vicinity of $p^{\ast }$ [163]. Theoretical analyses indicate that anomalous behavior can occur near a Lifshitz transition [164], especially when strong-correlation effects are important [165,166].

26 A related figure of $p_s$ in LSCO, including detailed data from [169], is presented in [170].

27 Note that there is no sign of enhanced scattering at any unique wave vector, as proposed in “hot spot” models, where the interaction is assumed to be restricted to k points nested by ${\mathbf{Q}}_{\mathrm{A}\mathrm{F}}$.

28 One of the first identifications of the pseudogap was based on Knight-shift measurements using Y nuclear magnetic resonance (NMR) in YBCO [191]. The Knight shift is generally proportional to the bulk spin susceptibility, and the results for YBCO are similar to bulk susceptibility results for the CuO${}_2$ planes, after correction for a chain contribution [192,193]. In the original interpretation of the temperature dependence of the Knight shift data [191], the role of Cu moments (clearly detected by neutron scattering [194]) was ignored; instead, it was interpreted as the response of electronic quasiparticles, with a decrease in carrier density on cooling. While other measurements, such as the Hall effect, do indicate a temperature-dependent carrier density, the spin susceptibility is dominated by the Cu moments (at least for $p\lesssim 0.2$). Perhaps the clearest demonstration of this is from anisotropic susceptibility measurements on single crystals of LBCO [195].

29 For an extended discussion of scattering measurements of stripe order, see [118]

30 There were early theoretical considerations of a spin-spiral order in a mean-field description of the hole-doped antiferromagnet [227–229]. Such a spiral might, in principle, yield the incommensurate magnetic peaks seen with neutrons; however, it was soon shown that the spiral state is unstable to charge modulation [230–232], which is confirmed by the direct detection of charge-stripe order.

31 The sinusoidal modulation shown is consistent with the experimental observation of a single modulation wave vector. Deviations from sinusoidal require higher harmonics, with consequent higher-harmonic superlattice peaks. Examples of stripe order with multiple harmonics occur in the case of La${}_2$NiO${}_{4.133}$ [233]. As the diffraction intensity is proportional to the square of the amplitude, the relative intensities tend to be quite weak, at best, and would be reduced by disorder and fluctuations. In any case, bond-centered stripes should minimize harmonic content.

32 For LSCO, commensurate order disappears at x = 0.02, with neutron scattering detecting only incommensurate peaks at low energy for x>0.02 [241]. Optical conductivity measurements on detwinned crystals with x = 0.03 and 0.04 suggest that the spin order is accompanied by modulated charge order [242]. For x<0.02, phase separation occurs at $T\lesssim 30\mathrm{K}$, with loss of commensurate intensity and corresponding appearance of incommensurate peaks characteristic of x = 0.02 [243].

33 For YBa${}_2$Cu${}_3$O${}_{6+x}$, there is a jump in the hole concentration in the planes from $p\lesssim 0.02$ to $\sim 0.05$ at the tetragonal-to-orthorhombic transition which occurs at $x\approx 0.3$ [248]. Hence, magnetic incommensurability appears [249] as soon as there is a meaningful hole density in the planes.

34 The insulating stripes have an increased hole density (one per lattice spacing along the length of a stripe) compared to metallic stripes, with a corresponding increase in the stripe period. In particular, the charge-stripe period predicted by Hartree-Fock is inconsistent with experiment.

35 If one starts with a single-band Hubbard model, calculations at large U/t indicate that it is very unlikely that two electrons will sit on the same site, while electron spins on neighboring sites are coupled by $J=4t^2/U$. An approximate version of this, called the t-J model, keeps the hopping kinetic energy t, includes the superexchange J, and excludes any states with electron double-occupancy [127]. To tune the calculated electronic structure near the chemical potential, variants may include next- or next-next-nearest-neighbor hoppings $t^{\prime }$ and $t^{″}$, respectively.

36 The incommensurability of the charge-stripe fluctuations appears to increase with temperature, opposite to the spin correlations [265]. This might be a consequence of the apparent strong sensitivity of the RIXS measurements to electron-phonon coupling [266–268], which, for the relevant Cu-O bond stretching response, increases greatly toward the Brillouin-zone boundary [269].

37 The name “ultra-quantum metal” is meant to convey the idea that we have a metallic phase that cannot be understood within a semi-classical model.

38 It is worth noting that it has been proven that one can obtain superconductivity from a model with only repulsive interactions [294]; however, the proof applies in the weak-coupling limit, where $T_c$ is quite small. The remaining challenge is to show that one can get high transition temperatures from repulsive interactions.

39 One can get dynamic correlations from quantum Monte Carlo (QMC) calculations [271], as we will consider below; however, it is challenging to extend QMC calculations down to temperatures comparable to $T_c$ because of the “sign problem” [295].

40 An independent 2-component analysis of the spin excitations in LSCO, more in the spirit of the analysis below, is reported in [318].

41 Figure 7 shows results for YBCO; however, the Raman data show comparable results for other cuprates, including LSCO [137].

42 Evidence for incompatibility of SC and AF orders is also provided by a study of La${}_{2-x}$Sr${}_x$CuO${}_4$/La${}_2$CuO${}_4$ heterostructures [329].

43 This behavior has been observed in LSCO [306,341], YBCO [342,343], Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$ [331], HgBa${}_2$CuO${}_{4+\delta }$ [339,344], and La${}_{2-x}$Ca${}_{1+x}$Cu${}_2$O${}_6$ [285].

44 The dispersion in HgBa${}_2$CuO${}_{4+\delta }$ has been characterized as having the form of a “wine glass” rather than an hourglass [339]; however, given the large damping, the data appear roughly consistent with a parabolic upward dispersion where the lower part of the dispersion is not resolved due to the damping. A downwardly-dispersing component that was resolvable at $T<T_c$ has now been observed in a sample with $T_c=88\mathrm{K}$ [344]. Such a change in damping across $T_c$ was previously observed in YBCO [335,342].

45 SQUID = superconducting quantum interference device.

46 This averaging effect was recently taken into account to explain the anomalous temperature dependence near $T_c$ of the antinodal spectral function in slightly-overdoped Bi2212 crystals [372]; it was used previously to simulate ARPES results for LSCO [373].

47 No neutron scattering studies of underdoped Bi2212 crystals have been reported yet.

48 As already discussed in Section 4.2, the concept of a spin resonance comes from the weak coupling perspective, in which the same electrons are responsible for both the spin correlations and the superconductivity [394,395]. In the superconducting state, where the antinodal electronic states should be gapped by $2{\mathrm{\Delta }}_0$, a spin excitation at $E_r<2{\mathrm{\Delta }}_0$ requires a resonant enhancement relative to the bare Lindhard spin susceptibility. The “resonance” language is misleading when applied to the experimental results, because a weak-coupling description is not relevant.

49 Comparisons of $2{\mathrm{\Delta }}_{\mathrm{A}\mathrm{N}}$ from a broad range of techniques have been presented elsewhere [398]; the other measures are consistent with the Raman results. For the families YBCO, Bi2212, Tl${}_2$Ba${}_2$CuO${}_{6+\delta }$, and HgBa${}_2$CuO${}_{4+\delta }$, it has been observed that $2{\mathrm{\Delta }}_c\approx 6k{T}_c$ [214,215].

50 ${\mathrm{\Delta }}_c$ is also detected by Andreev reflection in tunneling spectroscopy [399].

51 The Nernst effect is the transverse voltage measured in response to a longitudinal temperature gradient in the presence of a perpendicular magnetic field; it is sensitive to vortex fluctuations for temperatures near $T_c$ [382].

52 This small pseudogap is distinct from the large pseudogap associated with scattering by spin fluctuations.

53 The stripe spacing varies with p. When p is small and charge stripes are farther apart, they are likely to be wider. Correspondingly, the spin gap on a ladder with an even number of legs decreases as that number increases [409]. Thus, the pairing scale will be reduced for holes in wider ladders.

54 In particular, there is no evidence of a collapsing of the excitation spectrum towards zero frequency, as one might expect in the case of a quantum critical point.

55 Alternatively, it has been argued that some of the phenomenology observed in overdoped cuprates can be understood in terms of “dirty d-wave theory”, which is based on a conventional BCS picture [423,424]. While that may be the case, it does not provide a specific understanding of the pairing mechanism, especially for $p<p^{\ast }$.

56 There are experimental technique issues that may qualify the identification of nematic order by STM [430].

57 In YBCO, ordering of Cu-O chains provides a broken symmetry that enables the detection of a nematic effect that turns on well below the chain-ordering temperature.

58 Charge-density waves and charge stripes are both names for a periodic modulation of the charge density.

59 In bilayer cuprates, the AF coupling between neighboring CuO${}_2$ layers may enhance the tendency to developing a spin gap [397].

60 While there is no static spin order to conflict with the uniform SC, it may be that defects locally make the CDW/PDW orders energetically competitive, as proposed in the case of Zn-doped LBCO [278].

61 There is some connection here with the “spin-bag” model [520], which assumed the frustrated interactions of doped holes and AF spins would lead to inhomogeneity and pairing.

62 I have been encouraged to make some predictions based on the picture painted here. I am uncomfortable with the “p” word, but am happy to speculate, which is more in line with the title metaphor of a striped lens. Of course, one scientist's speculation may be another's prediction.

63 P. A. Lee [522] made the argument that, because the energetic cost of creating a vortex is empirically small, there must be some sort of competing order induced locally with a vortex.

Additional information

Funding

This work was supported at Brookhaven National Lab by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division under Contract No. DE-SC0012704.