Cuprate superconductors as viewed through a striped lens
FFebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3
To appear in
Advances in Physics
Vol. 00, No. 00, Month 20XX, 1–62
REVIEW ARTICLECuprate superconductors as viewed through a striped lens
J. M. Tranquada
Condensed Matter Physics and Materials Science Division, Brookhaven National Laboratory,Upton, New York 11973, USA ( Received 00 Month 20XX; final version received 00 Month 20XX ) Understanding the electron pairing in hole-doped cuprate superconductors has been a chal-lenge, 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 charac-terizations, it is possible to delineate a clear and consistent cuprate story. It starts withdoping holes into a charge-transfer insulator, resulting in in-gap states. These states exhibita pseudogap resulting from the competition between antiferromagnetic superexchange J be-tween nearest-neighbor Cu atoms (a real-space interaction) and the kinetic energy of thedoped holes, which, in the absence of interactions, would lead to extended Bloch-wave stateswhose occupancy is characterized in reciprocal space. To develop some degree of coherenceon cooling, the spin and charge correlations must self-organize in a cooperative fashion. Aspecific example of resulting emergent order is that of spin and charge stripes, as observedin La − x Ba x CuO . While stripe order frustrates bulk superconductivity, it nevertheless de-velops pairing and superconducting order of an unusual character. The antiphase order ofthe spin stripes decouples them from the charge stripes, which can be viewed as hole-doped,two-leg, spin- ladders. Established theory tells us that the pairing scale is comparable to thesinglet-triplet excitation energy, ∼ J/
2, on the ladders. To achieve superconducting order,the pair correlations in neighboring ladders must develop phase order. In the presence of spinstripe 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 inthe cuprates involves a compromise between the pairing scale, which is maximized at x ∼ ,and phase coherence, which is optimized at x ∼ .
2. To understand further experimentaldetails, it is necessary to take account of the local variation in hole density resulting fromdopant 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 becomesparse, and the superconductivity disappears. While there are a few experimental mysteriesthat remain to be resolved, I believe that this story captures the essence of the cuprates. Contents
CONTACT J. M. Tranquada. Email: [email protected] a r X i v : . [ c ond - m a t . s up r- c on ] F e b ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3
1. Introduction
The discovery of high-temperature superconductivity in La − x Ba x CuO by Bednorz andM¨uller in 1986 [1] came as a tremendous and stupefying shock. Transition-metal oxideswere out of fashion at the time, and they were certainly not viewed by most researchersas a good place to search for new superconductors. In terms of theoretical concepts thatwould become relevant, Mott’s conclusion that NiO is an insulator because of strongonsite Coulomb repulsion experienced by Ni 3 d electrons [2] was still hotly debated[3, 4]. Thus, it should not be surprising that it has taken us a long time to make senseof the layered cuprates.The success of solid-state physics in the mid-20 th century was largely due to the bandtheory of solids. With the assumption that atomic states combine to form Bloch states[5], coherent in every unit cell and characterized by wave vector k , one obtains a set oflevels that can be filled, two at a time, with fermionic electrons. In a metal, the top ofthe distribution defines a Fermi surface. With Landau, the picture is modified to allowinteractions that modify the Bloch states, but one nevertheless ends up with well-definedquasiparticles close to the Fermi energy. The resulting Fermi liquid is the starting pointfor the Bardeen-Cooper-Schrieffer theory of superconductivity [6], which, in turn, is theinevitable starting point for the consideration of any new superconductor.Of course, not all solids are metals. If the highest-energy occupied band of states iscompletely filled and there is an energy gap to the next (empty) band, one has a bandinsulator. In contrast, there are also cases of insulators that cannot be explained by havingan even number of electrons to fill a band. As emphasized by Mott [2], there are caseswhere local Coulomb repulsion overwhelms kinetic energy. This is most easily picturedin a tight-binding model based on atomic orbitals; if the highest-energy, partially-filledorbital has a strong Coulomb repulsion U between two electrons on the same site, itis possible to end up with an insulator where band theory would predict a metallicconductor. 2 ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 Hubbard [7] provided a convenient parameterization of this picture involving compet-ing kinetic energy, represented by intersite hopping energy t , and onsite repulsion U .The Hubbard model is especially useful for treating insulating transition-metal oxidecompounds, many of which exhibit antiferromagnetic order. Anderson [8] showed thatthe superexchange interaction between nearest-neighbor magnetic moments can be un-derstood by applying second-order perturbation theory to a Hubbard-like model. For thecase of a single half-filled orbital, the kinetic energy is lowered when one electron hopsto a neighboring site and back home again; however, because of the Pauli exclusion prin-ciple, this can only happen if the neighboring spins are antiparallel. The net magneticinteraction, termed superexchange, is given by J = 4 t /U . Hence, antiferromagnetismcan be a symptom of strong onsite Coulomb repulsion.When Bednorz and M¨uller discovered high-temperature superconductivity, the spe-cific compound that was superconducting came as a surprise even to them. They wereattempting to explore the perovskite LaCuO doped with Ba; however, follow-on workdetermined that the superconducting compound is La − x Ba x CuO (LBCO), a layeredperovskite with the K NiF structure [9, 10]. It took further investigations to establishthat the parent compound, La CuO , is an antiferromagnetic insulator [11]. Anderson quickly recognized that La CuO should be a Mott-Hubbard insulator withantiferromagnetic correlations due to superexchange [13]. This set up the initial theoret-ical quandary. The doped compounds are superconductors, and the extremely successfulBCS theory of superconductors is based on Fermi liquid theory. At the same time, theparent compounds are antiferromagnetic insulators due to strong electron-electron inter-actions, as eventually confirmed by experiment (see Sec. 2.1). Something would have togive.There is a great deal of mathematical machinery that has been developed within thecontext of the Fermi liquid model. All types of ordering (magnetism, charge density waves,etc.) are typically described in terms of interactions between quasiparticles near the Fermilevel. From this perspective, antiferromagnetism (or spin-density-wave order) should bedescribed by an interaction that causes a scattering between quasiparticles at the Fermisurface that are separated by the AF wave vector. In the absence of actual order, suchscattering should result in “hot” spots on the Fermi surface. It took quite some timebefore this proposal could be tested by experiments using angle-resolved photoemissionspectroscopy (ARPES). Now we have many results [14], and none of them show evidenceof hot spots. To be clear, there are strong anomalies, but they are not restricted to uniquepoints on the Fermi surface.As experiments on cuprates have shown, doping a sufficient density of holes p (normal-ized per Cu) into the parent antiferromagnetic insulator leads to unusual metallic and su-perconducting states, with a density of states near the Fermi energy that is strongly sup-pressed compared to predictions based on conventional band calculations (see Sec. 2.2).Within the Fermi-liquid approach, the depressed density of states is typically analyzedin terms of some sort of competing order. With increasing hole concentration, the ef-fects attributed to the competing order decrease, eventually disappearing at an assumedquantum critical point (QCP) below the superconducting dome, at the critical hole den-sity p ∗ . This perspective draws parallels to a number of experimental systems wheresuperconductivity is observed around a QCP associated with ferromagnetism [15–17],charge-density-wave order [18–20], or even antiferromagnetism due to RKKY couplingof f -electron moments [15, 16]. It is the soft quantum fluctuations of the dying order La CuO can easily pick up a small amount of excess oxygen during synthesis, and phase separation can resultin an impurity phase that is metallic and superconducting [12]. ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 parameter that drive the electron pairing in the vicinity of a QCP [21]. For cuprates,there have been theoretical proposals for a QCP associated with antiferromagnetic [22],charge-density-wave [23], current-loop [24], d -density-wave [25] and nematic [26, 27] or-ders.The problem with the QCP scenario is that it tends to imply a large density of coherentcarriers at high temperature, with a transition at a doping-dependent temperature T ∗ toa “pseudogapped” state. There is no experimental support for such a high-temperaturephase; the depressed density of states occurs over a large energy scale and involves largelyincoherent states, as will be discussed. Furthermore, there is no evidence for a phase-transition-like onset of a pseudogap on reducing temperature.As emphasized by Anderson [28], the superexchange interaction J underlying the an-tiferromagnetism in the parent cuprates is a very short-range effect driven by U ; it hasnothing to do with Bloch states. He also proposed that the two large interactions J and U should dominate the electronic and magnetic properties of cuprates [29]. While I agreewith this idea, there is still the trick of how to explain the details. Anderson proposedthat the two-dimensionality and minimal spin S = 1 / t - J model)of the Hubbard model; such large-scale segregation is possible because of the neglect oflong-range Coulomb interactions. Inclusion of those interactions in an effective modelprovided evidence for periodic patterns of hole-rich and hole-poor (antiferromagnetic)patches [33, 34], including charge and spin stripes. With the experimental discoveryof stripe order in one specific cuprate family [39], a model of superconductivity basedon stripes was proposed [40]. An assumption of this model was that the magnetic spingap needed to induce pairing correlations within the charge stripes would come fromthe hole-poor spin stripes [41]; that assumption appears to be incompatible with staticspin stripes (which exhibit no significant spin gap, as I will discuss), so that stripe orderwould then appear to compete with superconductivity [42]. The situation changed a decade later, when experiments exploring the anisotropicresponse of LBCO single crystals provided evidence that stripe order is compatible withpairing and two-dimensional (2D) superconductivity [44, 45]. This discovery has beenrationalized in terms of a proposed pair-density-wave superconducting state [46–48]. Itshifted the narrative from one of competing orders to a picture of intertwined orders [49].The concept of intertwined orders, while providing a framework for interpreting ex-periments on cuprates, does not provide a specific explanation for the pairing withincharge stripes. This is where I propose a new synthesis that exploits pre-existing ideas Earlier Hartree-Fock and related calculations had provided solutions of charge and spin stripes [35–38]; however,the charge stripes were always insulating. Another challenge in the superconducting stripe model of [40] is that charge-density-wave order within the stripeswould compete with pairing. To avoid this possibility, the idea of fluctuating stripes in the form of nematic andsmectic orders was introduced [43]. Again, this has a bias of static stripe order being bad for hole pairing. ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3Table 1. List of frequently men-tioned cuprate (and nickelate) fam-ilies and associated acronyms.Formula AcronymLa − x Ba x CuO LBCOLa − x Sr x CuO LSCOYBa Cu O x YBCOBi Sr − x La x CuO δ Bi2201Bi Sr CaCu O δ Bi2212HgBa CuO δ Hg1201La − x Sr x NiO LSNO (see Sec. 4). The pairing is a consequence of singlet correlations of Cu moments, butthese singlets are on hole-doped stripes, not the spin stripes. Here I invoke the theoret-ical analysis [50, 51] and experimental evidence [52] that hole-doped S = 1 / Indeed, exper-iments indicate that the coherent superconducting gap is limited by the incommensuratespin gap [54].To appreciate why there is a coherent superconducting gap that is smaller than themaximum of the d -wave gap function, we have to consider the role of a second formof inhomogeneity, which is quenched dopant disorder that results in local variations inthe average charge density on a scale coarser than that of the stripes. This leads tobehavior similar to that of a granular superconductor [55], such that the development ofsuperconducting order tends to be limited by phase fluctuations [56, 57].The normal state from which the stripes and pairing develop involves both antifer-romagnetic correlations based on superexchange and a low density of doped holes thatwant to minimize their kinetic energy by delocalizing. The superexchange interactionis only nearest-neighbor, so it essentially lives in real space, whereas electronic quasi-particles live in reciprocal space. The doped holes strongly damp the antiferromagneticspin correlations, resulting in a short spin-spin correlation length. Where antiparallelspin correlations overlap with Bloch states, they cause a large k -dependent electronicself energy, corresponding to a pseudogap. The effects of superexchange dominate atlow doping, but with increasing hole concentration, the energy-scale of the pseudogapdecreases. The holes and spins initially find a compromise in the form of stripes, butat higher hole concentrations, the pseudogap eventually disappears, indicating that re-gions with strong local AF correlations no longer percolate across the CuO planes. Sincethe Cu singlet correlations are crucial to pairing, the strong decrease in spin-correlationweight in highly-overdoped samples leads to a rapid decrease in superfluid density.I have made some bold claims in this introduction. The rest of this perspective is aimedat backing these up, largely based on experimental evidence, relying on relevant theoret- This is supported by a recent phenomenological analysis [53]. ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 ical results when available. We start in Sec. 2 by reviewing the experimental observationson the normal and superconducting states. Section 3 covers the evidence for stripe orderand associated superconductivity, followed by a description of the stripe-based pairingmechanism in Sec. 4. This mechanism is used to explain the spatially-uniform supercon-ducting state in Sec. 5. Some remaining experimental aspects are discussed in Sec. 6,with a brief conclusion in Sec. 7.
2. Cuprate phenomenology
To appreciate the challenges in understanding the cuprates, it is necessary to start withthe parent compounds, such as La CuO . We then look at what happens when holesare introduced, considering first the correlated evolution of electronic and magnetic re-sponses, and then measured features of the superconducting order. Parent correlated insulators
Within the CuO planes, the states near the chemical potential are dominated by con-tributions from the Cu 3 d x − y and O 2 p σ orbitals (where the σ orbitals are the onesthat have their lobes pointing at the neighboring Cu sites) [58]. If one considers onlythe nearest-neighbor hopping energy t pd , then, since there is just one hole per Cu site,one would expect to find a half-filled hybridized band. From experiment, we know thatthe parent cuprates are insulators with an optical gap of 1.5–2 eV [59], and x-ray spec-troscopies confirm that the Cu 3 d hole is of x − y character [60–62], with the other3 d orbitals separated by at least 1.5 eV [63]. To capture this behavior, one must takeaccount of the strong Coulomb repulsion U that is experienced by two electrons in thesame 3 d x − y orbital, as treated by the theory of Mott-Hubbard insulators [2, 7]. Theoptical gap does not correspond to U , however; the orbital energies are such that thevalence band has strong O 2 p character, while the lowest excited state corresponds tofilling the 3 d x − y orbital (upper Hubbard band). This arrangement corresponds to acharge-transfer insulator [66].As illustrated in Fig. 1, the electron in a half-filled 3 d x − y acts as a local moment,even though it sits below the filled O 2 p σ level, since filling it with an electron from Oraises its energy by U, to above ε p . The large U acts to frustrate the kinetic energy inthe ground state; nevertheless, the energy can be lowered if virtual hops from O to Cuand back occur. Of course, due to the Pauli exclusion principle, hops to both neighboringsites can only occur if the spins on neighboring Cu sites are antiparallel. Thus, as shownby Anderson [8, 13], these effects lead to a net superexchange interaction J betweenneighboring Cu spins that drives antiferromagnetic (AF) correlations. The superexchange mechanism of the Hubbard model translates to an effective Heisen-berg Hamiltonian for spins S = . If the AF correlations were purely two dimensional,then the system would be disordered at finite temperature due to the Mermin-Wagnertheorem [69]. There is also the fact that the ordered moment is reduced by almost 40% dueto zero-point fluctuations [70]. Nevertheless, the spin-spin correlation length is observed There is some hybridization of the 3 d x − y orbital to the neighboring O 2 p σ orbitals [64], where the degree ofhybridization can vary among cuprate families [65]. If one ignores the O sites and uses a single-band Hubbard model, then J = 4 t /U , where t is the Cu-Cuhopping parameter. Quantitative agreement with experiment requires consideration of the O, which leads to amore complicated formula, proportional to t pd , where t pd is the hopping energy from O to Cu [67, 68]. Anderson speculated that the quantum spin fluctuations for the 2D S = Heisenberg model might be great ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 excited stateground state ε d ε d + U ε p Figure 1. Energy level diagrams illustrating states relevant to the superexchange mechanism. In the ground state,there is one electron in each Cu 3 d x − y orbital, and the O 2 p is filled (with orbitals illustrated schematicallybelow). In an intermediate excited state, one electron has hopped from O to fill the Cu level, raising its energy by U , which also allows the electron on the other Cu site to hop and refill the O site. Because of the Pauli exclusionprinciple, these hops are only possible when the ground state involves antiparallel spins on neighboring Cu atoms.(Obviously, electrons do not have a color; the color is used to indicate schematically how hops between sites canoccur.) to grow exponentially with cooling [73–75] (theoretically, ξ ∼ exp( αJ/k B T ) [76, 77]), andonly a small interlayer exchange interaction is necessary to produce three-dimensionalorder [78]. The excitations of the the ordered antiferromagnet are spin waves, with abandwidth of 2 J ; neutron scattering measurements on various layered cuprate insulatorsfind J in the range of 100 to 150 meV [79–81].Detailed studies of the spin-wave dispersion in La CuO [81, 82] reveal modest de-viations from the predictions based on linear-spin-wave theory for the nearest-neighborHeisenberg model. The superexchange interaction is the lowest-order spin-spin couplingterm obtained in a perturbation expansion of the Hubbard model [8], and one can findcorrections with higher-order terms, such as 4-spin cyclic exchange [83], as well as a mul-timagnon continuum [84], that can account for much of the deviation. A recent analysisin terms of a single-band Hubbard model found that fitting the spin-wave dispersion ofLa CuO [82] gave U ≈ t [85]. Since 8 t is the electronic bandwidth in a noninteractingsquare lattice, one can see that the cuprates have a strong competition between Coulomband kinetic energies.Anderson [28] argued that the superexchange interaction cannot be described in termsof a conventional band structure picture as an interaction between quasiparticles. Start-ing with a mean-field approach such as density-functional theory (DFT) or Hartree-Focktheory, it is certainly possible to find a solution with antiferromagnetic order and anassociated insulating gap [86, 87]; however, that gap disappears in the absence of AForder [88], in contrast to experiment. It is necessary to account for the local dynamicalelectronic correlations, and calculations on the Hubbard model using a version of Dy-namical Mean Field Theory (DMFT) confirm that the insulating gap does not requireAF order [89]. It was only in 2016 that a first-principles calculation, employing dynamicalmean-field theory (DMFT), was finally able to reproduce the charge-transfer gap in theparamagnetic phase of La CuO [90]. enough to prevent order, resulting in an RVB state [13]. This stimulated a great deal of theoretical effort on quan-tum spin liquids, and speculation that it was the RVB character of the initial state that led to superconductivityon doping [71, 72]. In contrast, as we will see, competition been electronic kinetic energy and AF order is a keyfactor in the present story. ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3Figure 2. Tunneling curves at various points on the surface of Bi Sr − x La x CuO δ with p = 0 .
07 (charge-orderedinsulator) measured by STM. Reprinted with permission from [95], Springer Nature © Hole-doped antiferromagnet
To obtain superconductivity, one must dope charge carriers into the CuO planes. Fora given compound, one generally can dope either holes or electrons. Switching from oneto the other requires a large jump in chemical potential, and I know of only one com-pound, Y . La . (Ba . La . ) Cu O y , in which ambipolar doping has been reported[91]. There is considerable asymmetry between electron and hole doping, with muchhigher T c ’s achieved with hole doping. Here we consider only the case of holes.The character of the dopant-induced holes has been characterized by x-ray absorptionspectroscopy and electron-energy-loss spectroscopy at the O K edge (probing 1 s → p transitions) and Cu L edge (probing 2 p → d ) [62, 92, 93]. The unanimous conclusionis that the holes are dominantly associated with O 2 p σ states, as expected for the caseof a large U on Cu [94].In a conventional, mean-field band model, one would expect doping to cause the chem-ical potential to drop into the top of the valence band. On the other hand, if thedoped charges segregate into stripes, the chemical potential should be pinned withinmid-gap states induced by the doping [96, 97]. The presence of mid-gap states was ini-tially inferred from optical conductivity measurements on LSCO [98]. Their existence hasbeen confirmed by a recent scanning tunneling microscopy (STM) study of lightly-dopedBi Sr − x La x CuO δ (Bi2201) [95]. Figure 2 shows tunneling conductance measurementsmade at several different points within 200 ˚A of one another on the same atomically-flatsample surface [95]. (The hole concentration is estimated to be p = 0 .
07, but the sam-ple is not superconducting.) A crucial feature of these spectra is that the range of biasvoltage spans the charge-transfer gap [98, 99]. Looking at curve 1, we see an examplewhere, at that particular spot, there are essentially no states within the gap and thecharge transfer gap is about 2 eV. For the other curves, we see that, locally, weight fromboth the lower and upper edges of the gap has moved into the gap, and the chemicalpotential is pinned within the gap. Furthermore, the transferred weight is spread over alarge energy scale ( ∼ . ion with a Sr . The hole goes into a neighboring CuO plane, but it is not free to8 ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 k x k y -0.500.5 E ne r g y ( e V ) k x k y E n e r g y ( e V ) k = ( π /2, π /2 ) k = ( π ,0 )(a) (b)(c) (d)(e) Figure 3. (a) Dispersion of the antibonding band for a non-interacting 3-band model with only nearest-neighborhopping, projected onto a quadrant of the first Brillouin zone with energy referenced to the chemical potential athalf-filling. Circles denote the points on the Fermi surface at ( π/ , π/
2) and ( π, k = ( π/ , π/
2) [104]. (c) Antiferromagnetic spin structure with spin size proportional to the d x − y weights inthe ( π/ , π/
2) wave function. (d) Schematic wave function at k = ( π,
0) [104], and (e) the corresponding weightedspin components. move away. It feels a substantial Coulomb attraction to the dopant site, and in the limit ofvery small doping, the screening comes mainly from phonons [100]. As reviewed in [100],a single hole may be localized on a scale of a few lattice spacings. Evidence for variable-range hopping in a Coulomb potential has been provided by transport measurements onLa − x Sr x CuO with a hole concentration p (cid:46) .
05 [100, 101].It is clear that the real materials are quite different from the predictions of conven-tional band theory [102, 103]; nevertheless, from comparison of such predictions withthe antiferromagnetic structure of the undoped system, one can gain some appreciationfor how the electronic quasiparticles and the antiferromagnetic correlations compete. Forthis purpose, we return to the “3-band” model involving the Cu 3 d x − y and O 2 p σ orbitals with only nearest-neighbor hopping, and look at a tight-binding calculation. The relevant formulas have been given in detail by Andersen et al. [104]. Using commonparameter values (3 eV for the energy separation between the Cu and O levels and hop-ping energy t pd = 1 . µ and be half-filled for an undoped CuO layer, is easily calculated. The dis-persion, projected onto a quadrant of the first Brillouin zone, is shown in Fig. 3(a). Inthis simple model, the Fermi surface at half filling runs diagonally from ( π,
0) to (0 , π )through ( π/ , π/ d -wave superconductinggap (the nodal point). The schematic wave functions at two of these points [indicated by circles in Fig. 3(a)]are also given in [104], and we reproduce them in Fig. 3(b) and (d). At each wave vector,the variation of the real part of the Bloch wave in real space follows cos( k · r n ), where r n isthe center of the n th unit cell. For k = ( π/ , π/ d orbitalsis identical for all sites along the diagonal, but is staggered for second neighbors alongthe Cu-O bond directions; for nearest neighbors, the amplitude is zero. For k = ( π, d orbitals are in-phase along [0,1] and antiphase along [1,0]. Again, these states haveidentical energies in this noninteracting model, where there is no magnetic order. This is essentially the Emery model [58], but with the onsite Coulomb repulsion U set to zero. Note that, for this section, the lattice parameter a is set equal to 1. The imaginary part of the Bloch wave forms a degenerate state, decoupled from the real part. ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 Q = (H,0.5,0) E ne r g y ( m e V ) (a) spin wavesLa CuO dI/dV (pS) (b) pointtunnelingBi2201p = 0.03 |cos(k x a)-cos(k y a)|/2 (c) electronicdispersionCa CuO Cl
200 k x k y Figure 4. (a) Spin-wave dispersion in La CuO measured by neutron scattering [81, 82]. (b) Schematic versionof typical conductance curves measured by STM at typical locations in Bi Sr − x La x CuO δ with p = 0 .
03 [95],plotted as binding energy vs. conductance; black: region with no in-gap states; magenta: region with significantin-gap states. (c) Effective gap or dispersion along the nominal weak-coupling Fermi surface measured by ARPESin Ca CuO Cl [112]. Now consider the impact of antiferromagnetism. At the nodal wave vector, the Cu sitesthat contribute to the wave function all have weight on a sublattice of the AF order withunique spin direction, as shown in Fig. 3(c), so that the electronic state is compatiblewith the spin correlations. On the other hand, at the antinodal (AN) wave vector ( π, π,
0) is incompatible with AF spincorrelations. Note that the incompatibility does not depend on static order; if the spinson neighboring sites are instantaneously antiparallel, then they cannot be part of thesame extended state. The electronic spectral function can be probed directly by angle-resolved photoemissionspectroscopy (ARPES). While much of the discussion of ARPES results on cuprates hasfocused on the difference between nodal and antinodal responses (the “nodal-antinodaldichotomy” [107]), we are interested in the points in between, as well. For a particular k along the noninteracting Fermi surface, a practical measure is the weight of the wavefunction on nearest-neighbor Cu sites, w AF ( k ) = 12 ( | cos( k x a ) | + | cos( k y a ) | ) . (1)In fact, this has the same k dependence as the absolute magnitude of the d -wave gap,∆( k ) = 12 ∆ [cos( k x a ) − cos( k y a )] . (2)Both functions go to zero at the nodal point and have extrema at the antinodes.To estimate the energy scale of the pseudogap, note that, in order to transform theAF state in Fig. 3(e) to a single-spin state, it is necessary to flip half of the Cu spins.The energy to locally flip a spin corresponds to the maximum spin-wave energy, ∼ J ∼
300 meV. Hence, it is plausible that, for a hole moving in an AF, the energy differencebetween the antinodal (AN) and nodal points is ∼ J . Figure 4 compares the energy scales of spin fluctuations in a parent insulator, in (a),with the effective dispersion of one hole in an antiferromagnet, in (c). Here, the spin-wave For a proper analysis of the problem of one hole in a two-dimensional (2D) antiferromagnet, see [105, 106]. The picture of an antinodal pseudogap defined by scattering from spin fluctuations is supported by a variety ofadvanced numerical calculations applied to the Hubbard model [108–111]. ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3Figure 5. Hole on O favors parallel alignment on neighboring Cu sites. results are from neutron scattering measurements on La CuO [82], while the ARPESwere obtained on Ca CuO Cl [112]. It may at first seem surprising that one can measureelectronic dispersion in an insulating compound; however, the photoemission processinvolves the absorption of a photon and emission of an electron, leaving a final state withone hole, whose energy varies with wave vector. The results shown are roughly alongthe direction of the anticipated Fermi surface for noninteracting electrons, so that thedispersion corresponds to the pseudogap. The pseudogap has a k dependence similar to∆( k ) and an energy scale essentially equal to 2 J [113], as proposed above.STM effectively measures the density of states. For noninteracting band structure,there is a Van Hove singularity at the ( π, , π ) points [103], so we expect the largestcontributions to the density of states to come from the antinodal regions. Figure 4(b)shows representative tunneling conductance curves for Bi2201 with small-but-finite p ∼ . J , consistentwith the pseudogap seen in ARPES. Of course, we have already seen that the STMconductance curves in Fig. 2 show similar gapping effects for states both above and belowthe Fermi level, which is also consistent with the pseudogap due to AF correlations. While the AF correlations have a big impact on the states associated with the dopedholes, we also have to consider the back action of the holes on the AF correlations, which isdrastic. For example, as one introduces holes into La − x Sr x CuO (LSCO) by increasing x from zero, the AF ordering temperature, T N , decreases rapidly, with commensurate orderdisappearing at x ≈ .
02 [80]. In the case of La CuO δ , where interstitial oxygen atomsgrab electrons and induce holes, there is actually phase separation into distinct antifer-romagnetic and doped superconductor phases [117]. The doped holes at low temperatureonly become delocalized as the the static magnetic order disappears [118]; nevertheless,dynamic AF correlations survive [119], as we will discuss. It is this competition betweenthe kinetic energy of the holes and superexchange between local Cu moments that is thedominant feature for understanding the cuprates.While superexchange tends to cause neighboring Cu spins to be antiparallel, a hole onan O site will have the best chance to hop and reduce its kinetic energy if its Cu neighborshave parallel spins [68], as indicated in Fig. 5. This alignment clearly frustrates commen-surate AF order. The motion of a hole through the lattice causes further disruption. Another factor for the STM results concerns the nature of the tunneling process from the probe tip to thesample surface and along the c axis, typically through an apical O site, to a Cu site in the CuO plane nearestthe surface. The 2 p orbital on the apical site has s symmetry relative to the Cu, so that it cannot couple to the3 d x − y but may couple to the in-plane O 2 p σ states. From Fig. 3(b) and (d), one can see that no coupling ispossible at k = ( π/ , π/
2) because the O orbitals are all in phase with the nearest Cu 3 d x − y orbital, but thereis a finite coupling at ( π,
0) as the phasing is different. Such effects were originally noted in analyses of c -axisconduction and planar tunneling [114, 115], and are discussed for STM in [116]. This is different from the proposal of the Zhang-Rice singlet [120]. That picture assumes that a hole is boundsymmetrically about a Cu site, resulting in no net moment. This would tend to act as a dilution of the AF lattice. ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 p E ne r g y ( m e V ) YBa Cu O (a)2-magnon Raman0 0.05 0.1 0.15 0.2 0.25 0.3 p E ne r g y ( m e V ) (b) mid IR Figure 6. (a) Two-magnon response measured by Raman scattering in YBa Cu O x as a function of estimatedhole concentration p [130]. (b) Mid-infrared optical conductivity in YBa Cu O x [131], with the energy scaledivided by 2. In both (a) and (b), the data have been interpolated, and circles denote measured peak positions.Dashed lines are quadratic extrapolations of the peak positions. This can be seen easily in the case of a single hole introduced into an AF array of Cuatoms, ignoring the O sites. For the hole to move from one Cu site to the next, a Cu spinmust move in the opposite direction. This means that as the hole moves, a string of sitesdevelops with ferromagnetic correlations to nearest neighbors [122, 123]. Such behaviorin a model of one hole in an Ising AF has been simulated with cold atoms, confirmingthe string-like defects [124, 125].
Intertwined spin and charge fluctuations
While superexchange interactions among Cu spins and the kinetic energy of doped holescompete with one another, the charge and spin fluctuations of hole-doped cuprates dis-play a correlated evolution with doping and with temperature. Here we will first considerfluctuations on the scale of J and then on the scale of k B T .One useful measure of the high-energy AF excitations is given by Raman scattering[126]. While a photon does not couple directly to a spin, it can cause a pair of spinsto flip; the corresponding inelastic scattering feature is known as two-magnon scattering[127]. Recent theoretical analysis for 2D antiferromagnets [128] gives a peak energy of2 . J and an asymmetric peak shape that is in quantitative agreement with experimentalmeasurements of cuprates [129].Figure 6(a) shows how the two-magnon scattering evolves with doping inYBa Cu O x [130]. As one can see, the well-defined peak in the lightly-doped sys-tem softens in energy as holes are added. Sugai et al. [130] have similar results forLa − x Sr x CuO (LSCO), Bi Sr CaCu O δ (Bi2212), and Bi2201, extending to higher But we know from experiment that dilution with Zn only destroys long-range order at the percolation limit (41%Zn concentration) [121]. ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 doping, and in all cases the two-magnon peak appears to become overdamped in thevicinity of p ∼ .
2. If we assume that the excitation mechanism remains the same inthe doped systems as in the parent AF, then the peak energy will depend both on thestrength of J and the degree to which a finite patch of Cu sites retains AF correlations.While the AF correlations restrict the hole motion and spatial distribution, the increasingdensity of holes must reduce the spatial areas in which AF correlations among Cu spinscan survive. Neutron scattering measurements have shown that the instantaneous spincorrelation length is reduced to approximately one lattice spacing near optimal doping[132, 133], so a decrease in correlation length can describe much of the decay in the peakenergy with p . Nevertheless, it can also be convenient to think of an average reductionin J . For example, Johnston [134] pointed out that the temperature dependence of thebulk magnetic susceptibility in La − x Sr x CuO can be scaled by doping-dependent val-ues of J and average moment per Cu, with both trending toward zero at p = x ∼ . σ ( ω )[137]. At energies below that of the charge-transfer gap, optical measurements probe alldirect transitions from filled to empty mid-gap states; from the STM results in Fig. 2,we already saw that, for a very underdoped cuprate, the mid-gap states extend over asignificant energy range with a pseudogap about µ . For moderately underdoped cuprates,it is possible at low temperature to distinguish the corresponding broad “mid-IR” con-ductivity from the more coherent Drude peak, centered at zero energy and correspondingto the quasiparticles that contribute to the metallic transport [131]. Figure 6(b) showsa rough version of the mid-IR feature observed in YBCO at temperatures close to butabove the superconducting T c [131]. Here the energy scale has been divided by two tomake it comparable to electron binding energies. The energy scale is virtually the sameas, and decreases in a fashion similar to, the two-magnon energies in Fig. 6(a), while thespectral weight grows with doping.The variation of the effective carrier concentration is considered more quantitativelyin Fig. 7. Integrating the optical conductivity from zero to a cutoff energy gives a resultproportional to the effective hole density p eff divided by an effective mass m ∗ . Padilla et al. [140] compared optical data with measures of p from the Hall coefficient to obtain m ∗ /m e ≈ m e is the electron mass. (A later studyhas suggested that m ∗ decreases by ∼
50% with doping in the range 0 . < p < . σ ( ω ) data for LSCO and YBCO to 80 meV, which yields the p eff values indicated byfilled squares in Fig. 7, plotted against the estimated dopant-induced carrier density p [131, 140]. At small p , the effective Drude carrier density is close to p , while itbegins to rise above it for p (cid:38) .
1; such behavior in LSCO was originally noted basedon measurements of the Hall coefficient [143], and similar behavior is seen in the nodalweight detected by ARPES [144]. Integrating up to 1.5 eV, an energy comparable to thecharge-transfer gap of the parent insulators, one captures the p eff associated with boththe Drude peak and the mid-IR signal, indicated by the filled circles. This value is severaltimes larger than the Drude weight alone, and approaches a maximum of ∼ p ≈ . p eff is consistent with other measurements. The T -linear electronic A study using terahertz spectroscopy in pulsed magnetic fields has determined a cyclotron mass of 4 . ± . m e for LSCO with x = 0 .
16 [141]. Related data for Bi2212 are reported in [142]. ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 p p s , p e ff LSCOYBCOp eff = pp s DrudeDrude + mid-IR
Figure 7. Comparison of various estimates of carrier density in La − x Sr x CuO (violet) andYBa Cu O x (green). Triangles: superfluid density, p s , from measurements of the magnetic penetration depthon Y . Ca . Ba Cu O x by muon spin rotation [138] and on LSCO films by mutual inductance [139]; squares:effective carrier density, p eff , from integrating in-plane optical conductivity to 80 meV [140]; circles: p eff fromintegrating to 1.5 eV [98, 140]. In evaluating p eff , the effective masses of 4 m e and 3 m e for LSCO and YBCO,respectively, determined in [140] were used. specific heat coefficient for the normal state of LSCO shows a maximum at p ∼ . π, , π ) points in the range 0 . < p < .
21; as alreadymentioned, a noninteracting system would have a Van Hove singularity that hits E F atthis point [103]. That crossing would also represent a Lifshitz transition, where the Fermisurface changes from hole-like to electron-like. In a complementary fashion, one can obtain the superfluid density p s from measure-ments of the magnetic penetration depth λ at T (cid:28) T c , using [157]1 λ = 4 πp s e m e c . (3)Two ways to measure λ are by muon spin relaxation ( µ SR) in an applied magnetic field[158] and by mutual inductance [139]. The open triangles in Fig. 7 indicate p s values forY . Ca . Ba Cu O x (determined by µ SR [138]) and for LSCO thin films (from mutual This was estimated by suppressing the superconductivity by Zn substitution [145]. In the system La . − x Nd . Sr x CuO , measurements of the Hall effect at low temperature and high magneticfield (to suppress superconductivity) have been interpreted as indicating a rapid rise of the carrier concentrationto a level of 1 + p at p ∗ ≈ .
23 [148] (with analogous behavior in YBCO at p ∗ ≈ .
19 [149]). The analysis hereis not clear cut, as ARPES measurements [150] indicate that a Lifshitz transition, along with the closing of theantinodal pseudogap, occur at or near p ∗ . These latter features also appear to be consistent with the observation ofa peak in the electronic specific heat at p ∗ [151] (where analysis is complicated by a Schottky anomaly due to themagnetic Nd ions [152]). Hall effect measurements on LSCO and Bi2202 yielded sharp cusps in the vicinity of p ∗ [153]. Theoretical analyses indicate that anomalous behavior can occur near a Lifshitz transition [154], especiallywhen strong-correlation effects are important [155, 156]. ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 (|cos(k x a)|+|cos(k y a)|)/2 ( m e V ) Figure 8. Scattering rate Γ for electron spectral function obtained on optimally doped Bi2212 from ARPESmeasurements (a) at 100 K (violet squares) [161]; (b) at 140 K (purple circles) [162]. inductance [139]). Much of the Drude weight from
T > T c goes into the superfluid,while p s in these materials reaches a maximum near p ∼ . An exception to this trend occurs when incommensurate AF order is present. Forexample, neutron scattering shows static or quasi-static incommensurate spin correlationsin LBCO with x = 1 / T (cid:46)
50 K [166, 167], and ARPES finds a pseudogap consistentwith Eq. (1), reaching an antinodal gap of ∼
20 meV [168, 169]. A similar relationshipexists for quasi-static spin correlations in LSCO with x = 0 .
07 [170] and ARPES onLSCO with x = 0 .
08 [171].Given the minimum in scattering rate in the near nodal region, it is plausible thatthe near-nodal states make the dominant contribution to the in-plane resistivity and theDrude peak in optical conductivity. Indeed, an analysis of the self energy of near-nodalstates as a function of temperature and doping in Bi2212 supports such a conclusion[172]. On the other hand, the conductance between layers depends on antinodal states[114], so that the evolution of the pseudogap is reflected in the T dependence of the c -axisresistivity [173, 174] and c -axis polarized optical conductivity [175, 176]. Temperature dependence
We have already seen that the magnetic and electronic excitations have correlated energyscales that vary with doping in a parallel fashion. There are related correlations in behav- A related figure of p s in LSCO, including detailed data from [159], is presented in [160]. 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 Q AF . ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 p H / x La Sr x CuO x = 0.170.140.110.08(a)0 100 200 300 400 500 Temperature (K) ( - e m u / g ) (b) Figure 9. (a) Ratio of the hole concentration p H determined from the Hall effect to the Sr fraction x inLa − x Sr x CuO ; data from [186] have been interpolated. T ∗ ∼
410 K and 535 K for x = 0 .
17 and 0.14, respectively[187]. (b) Bulk magnetic susceptibility in La − x Sr x CuO ; data from [135] have been interpolated. T max ∼
280 Kand 500 K for x = 0 .
17 and 0.14, respectively. ior with temperature. From the temperature dependence of various physical quantitiessuch as the bulk spin susceptibility χ and the in-plane Hall coefficient R H , a temperature T ∗ (which decreases with p ) is commonly identified which behaves as a crossover to thepseudogap phase [40, 177]. It is instructive to consider the measurements and analysisby which T ∗ has been determined. χ ( T ) evolves continuously with doping from the antiferromagnetic parent state. Fora S =
2D Heisenberg model, as appropriate to a system such as La CuO , χ ispredicted to have a maximum at k B T max ≈ J [178]. The maximum occurs when thespin-spin correlation length reaches ∼ . a [179], and χ decreases as the AF correlationlength grows further. T max for La CuO is estimated to be comparable to the meltingtemperature; nevertheless, χ ( T ) has been measured to decrease on cooling below 1200 K[180]. For La − x Sr x CuO , the χ ( T max ) becomes observable for x (cid:38) .
09 as T max dropsbelow 800 K [135, 180]. It has been demonstrated that the temperature dependence of χ can be scaled with doping, with the temperature scaled by T max [134, 135]. Representative(interpolated) results for several values of x are shown in Fig. 9(b). It happens that R H is also temperature dependent [186, 187], and it can be scaled interms of a temperature T ∗ , which, within error bars, is consistent with T max . One canestimate the hole concentration using p H = 1 / ( R H ec ), and the temperature dependenceof R H suggests that p H decreases on cooling, as shown in Fig. 9(a). There are no sharpchanges in the T dependence of p H near T ∗ (estimated as 410 K and 535 K for x = 0 . One of the first identifications of the pseudogap was based on Knight-shift measurements using Y nuclear mag-netic resonance (NMR) in YBCO [181]. 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 planes, after correction for a chaincontribution [182, 183]. In the original interpretation of the temperature dependence of the Knight shift data [181],the role of Cu moments (clearly detected by neutron scattering [184]) was ignored; instead, it was interpreted asthe 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 bythe Cu moments (at least for p (cid:46) . ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 and 0.14, respectively [187]], so that T ∗ should not be viewed as a phase transition. In-stead, Gor’kov and Teitel’baum [188] found that the temperature dependence of R H ( T )is well described by a thermally-activated carrier density. The excitation gap has a mag-nitude and doping dependence similar to that observed for the mid-infrared conductivity,shown in Fig. 6.In Fig. 9(a), p H has been normalized to the dopant concentration x . We see that p H ≈ x for small x , but ratio p H /x grows as x reaches optimal doping. This is consistentwith the doping dependence of the Drude weight shown in Fig. 7. Studies of opticalconductivity vs. temperature in underdoped LSCO show that the frequency-dependentconductivity becomes completely incoherent at high temperature [189, 190]), so that the T dependence of p H /x reflects a crossover from a larger density of incoherent carriers athigh T to a smaller density of carriers on cooling. In terms of χ , we have to rememberthat strong AF correlations result in a small χ . Hence, we see that, as AF correlationsgrow on cooling, the effective carrier concentration decreases. With doping, the strengthof AF correlations weakens, and p H /x grows. Pairing and Coherence
We know from experiment that the superconducting state involves pairing of holes [191],and there is broad acceptance that superconducting gap of optimally-doped cuprateshas d -wave symmetry. The amplitude of the gap, with zeros at the nodal wave vectorsand maxima in the antinodal regions, was first detected in ARPES studies [192, 193].The phase factor, corresponding to a sign change across each node, was determined withvarious ingenious Josephson junction devices [194, 195].The gap symmetry differs from the s -wave gap of BCS theory [6], the latter being aconsequence of an attractive interaction between quasiparticles associated with exchangeof a phonon. The possibility of replacing the phonon with an AF magnon leads to arepulsive interaction that can still lead to pairing if the gap has the d -wave form [196–198]. The challenges for applying this concept to cuprates are that 1) we do not have aFermi liquid with sharp quasiparticles all around the nominal Fermi surface [163], and2) we do not have sharp AF paramagnons to be exchanged [119].A simple picture of a d -wave gap is also challenged by ARPES measurements below T c in underdoped cuprates. As indicated in Fig. 10, the proper d -wave gap shape isonly observed over a finite arc of the nominal Fermi surface [200, 201], up to a coherentgap scale ∆ c , with the spectral function broadening and deviating at higher energies[199, 202, 203]. The scale ∆ c ∼ k B T has also been identified in Raman scatteringstudies [204, 205]. An interpretation of this behavior will be discussed later. An importantpiece of the story is the charge disorder already indicated by the variation among STMconductance curves on the same sample shown in Fig. 2.
3. Stripes
The most extreme form of intertwined orders occurs with stripe order. Charge and spinstripes provide an energetic compromise between the competing superexchange and ki-netic energies, as we will discuss here. In related oxides, stripe order can result in aninsulating ground state, as occurs with La − x Sr x NiO . While it is true that stripe or-der in the cuprates tends to frustrate bulk superconductivity, there is good evidencethat pairing, along with a distinct superconducting order, occurs. Furthermore, orderedstripes provide a simple way to connect pairing with a model system that is better under-17 ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3Figure 10. Schematic of electronic gaps around a quadrant of reciprocal space for a square CuO plane withlattice constant a = 1. Green line: below T c , a d -wave gap is found near the node, extrapolating to ∆ . Forunderdoped samples, the dispersion observed by ARPES tends to deviate upwards in the antinodal region [199],rising to the pseudogap energy ∆ pg . Blue line: above T c , the coherent gap (with maximum energy of ∆ c ) closesto form a Fermi arc. stood, both in terms of theory and experiment. Using a consistent interpretation of theincompatibility of the incommensurate AF spin correlations and spatially-uniform super-conducting coherence then leads to a systematic explanation of the correlation betweenthe AF spin gap and the coherent superconducting gap. Stripe order
Charge order turns out to be quite common among underdoped cuprates [206, 207];however, it frequently comes along with a substantial gap in the spin excitations [208].The focus here is on combined charge and spin stripe orders. The best examples occurin La − x Ba x CuO [166, 209], La . − x Nd . Sr x CuO [39, 210], and La . − x Eu . Sr x CuO [211, 212]. The maximum amplitude of stripe order occurs at x ≈ / Q AF ; in terms of reciprocal lattice units for asquare CuO plane, the spin-stripe peaks appear at (0 . ± δ, .
5) and (0 . , . ± δ ), due tothe presence of orthogonal stripe domains in nearest-neighbor layers. For p = x = 1 / δ ≈ /
8. The charge order peaks appear about fundamental Bragg peaks, split by 2 δ .They can be inferred from neutron scattering measurements, but the signal is due tomodulation of the lattice in response to the charge stripes, since neutrons do not coupleto charge. The modulation of the O 2 p hole density is directly detected by resonantsoft-x-ray scattering measurements at the O K edge [217]. It should be noted that thespin-stripe order develops at a temperature below that of the charge order.Based on the scattering measurements, we infer a picture of combined spin and chargestripe orders as shown in Fig. 11. The period in real space is inversely proportional to For an extended discussion of scattering measurements of stripe order, see [216] ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 Charge Modulation0 0 4 8Spin Modulation0
Figure 11. Bottom shows a cartoon of bond-centered charge and spin stripes. Arrows indicate the size of theordered magnetic moments on Cu atoms (circles), with color changing between antiphase domains. Blue intensityindicates hole density (with white = 0) on O sites (ellipses). Also shown are the sinusoidal modulations of thecharge (top) and magnetic domains (middle); the charge is always positive, in contrast to the spin modulation,which changes sign. (Only a single harmonic is detected experimentally for the charge and spin modulations, sothere is no evidence for modulations beyond simple sinusoidal forms.) the modulation wave vector, which means that the period of the charge order ( ∼ a for p = 1 /
8) is half that of the spin stripes. Experiments have not yet determined theregistry of the stripes with respect to the lattice; in the figure the charge stripes aresuggestively shown as bond-centered, for reasons that will be discussed shortly. Thelocally-antiferromagnetic spin stripes flip their phase on crossing a charge stripe, resultingin a doubled period of 8 a . (The phase difference between the charge and spin modulationsis based on theoretical predictions, discussed below.)The stripe order evolves with doping: for p < /
8, the incommensurability follows δ ≈ p , while δ saturates at higher doping [80, 209]. Stripe correlations of weaker amplitudeare also observed in LSCO over a broad range of doping [147, 219, 220]. The occurrencein LSCO has been rationalized by detection of a reduced crystallographic symmetry,favorable to stripe pinning [170]. Evidence of the stripe order is also provided by neutronscattering measurements of the low-energy inelastic magnetic excitations that rise fromthe wave vectors of the spin-stripe order [80, 221]; these excitations are gapless (in zeromagnetic field) for x (cid:46) .
13 [222]. At low doping, where LSCO becomes insulating( x < . δ measured in LSCO is plotted (graytriangles) for a large range of doping in Fig. 12. Static or dynamic spin stripes are alsoobserved over a significant doping range in Bi x Sr − x CuO y (blue circles) with thesame δ ≈ p relationship [223]. For YBCO, looking at the lowest-energy spin excitationsleads to a similar trend, even though a substantial spin gap develops for p (cid:38) .
09 [224–226].The existence of spin and charge stripes in hole-doped cuprates is well supported bytheory [230]. In fact, stripes were initially predicted in calculations on the Hubbard The sinusoidal modulation shown is consistent with the experimental observation of a single modulation wavevector. 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 NiO . [218]. As the diffractionintensity 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 harmoniccontent. ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 SCSG T ( K ) T SG T c LSCO :Bi2201 :YBCO :single-layerdouble-layer x =0.50.4 0.3 0.2 (cid:98) (r . l . u . ) Bi x Sr x CuO hole concentration ( p ) (cid:98) = p (a)(b) Figure 12. Hole concentration dependence of the incommensurability δ of low-energy spin fluctuations inBi x Sr − x CuO y (blue circles) compared with results for LSCO (gray triangles) [221, 227, 228] and YBCO(open squares) [224, 225] (with p estimated from T c via [229]). The dashed line represents δ = p . From [223]. model using the Hartree-Fock approximation [35–37, 231]; however, these mean-fieldcalculations yielded insulating charge stripes. An alternative approach, motivated byevidence for phase separation in the hole-doped t - J model [32], considered the role ofthe extended Coulomb interaction, which tends to frustrate the phase separation [33].Monte Carlo calculations on an effective model yielded both stripe and checkerboardphases [34]; however, such modeling was not able to address the electronic character ofsuch phases.A new approach of numerical variational techniques applied to correlated-electronHamiltonians allowing general inhomogeneous solutions began with the application ofthe density matrix renormalization group (DMRG) by White and Scalapino [232]. Eval-uating the t - J model on finite clusters at p = 1 /
8, they found evidence for metallic chargestripes and antiphase spin stripes with a form very similar to that shown in Fig. 11. Ex-tending the calculations to other hole concentrations provided theoretical evidence forvertical charge and spin stripes over a large range of p [233].A variety of numerical techniques appropriate to this problem have now been devel-oped. A comparison study applying 4 techniques to the single-band Hubbard model (withonly nearest-neighbor hopping) with U/t = 8 and p = 1 / d -wave order [234]. The key messages are thatthe various techniques yielded consistent results and a variety of inhomogeneous statesare close in energy. For a model that yields mean-field electronic structure that is closeto experimental results of ARPES studies [14], it is important to include longer-rangehoppings [104]. Recent DMRG studies of the Hubbard model with a broad range of t (cid:48) and p found that vertical stripes are the ground state for plausible values of t (cid:48) [235, 236].Monte-Carlo results for a three-band spin-fermion model yield half-filled, bond-centeredstripes [237]. 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-stripeperiod predicted by Hartree-Fock is inconsistent with experiment. If one starts with a single-band Hubbard model, calculations at large
U/t indicate that it is very unlikely thattwo electrons will sit on the same site, while electron spins on neighboring sites are coupled by J = 4 t /U . Anapproximate 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 [120]. To tune the calculated electronic structure nearthe chemical potential, variants may include next- or next-next-nearest-neighbor hoppings t (cid:48) and t (cid:48)(cid:48) , respectively. ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 Fluctuating stripes
Static stripes are the exception rather than the rule. Experimentally, they are associ-ated with a crystalline phase in which the rotational symmetry is reduced from 4-foldto 2-fold [39, 170]. Similarly, the calculations that yield stripes are typically done onclusters with open boundary conditions in one direction and periodic in the other, whichimpacts the results [232]. When rotational symmetry is restored, the stripe correlationsin experimental systems tend to be purely dynamical.A good example is LBCO with x = 1 /
8. Static stripe order is absent above thestructural transition at 56 K, but the magnetic excitation spectrum remains virtuallyunchanged across the transition [133], except at low energies ( (cid:46)
10 meV), where theimaginary part of the Q -integrated dynamical susceptibility changes from being roughlyconstant in the ordered state (as for antiferromagnetic spin waves) to varying linear with (cid:126) ω in the disordered state [166]. The low-energy incommensurability of the spin-stripeexcitations remains well-resolved at 100 K, but the incommensurability decreases withtemperature and evolves to a single broad peak at 200 K. (A more complete study of tem-perature and frequency dependence of the low-energy spin response has been reportedfor LSCO x = 0 .
16 [238]; for that composition, one obtains a spin gap, rather than spinorder, at low temperature [222, 239].) The charge stripe fluctuations in the disorderedphase have now been detected by resonant inelastic x-ray scattering at the Cu L edge[240]. The experimental evolution of the spin and charge correlations with temperature isqualitatively consistent with recent theoretical results. Calculations on the one-band Hub-bard model for p = 1 /
16 using the minimally-entangled typical thermal states (METTS)method find well-defined spin and (half-filled) charge stripe correlations at low temper-ature, but a disordered inhomogeneous state at a significantly higher temperature [245].Calculations of the spin dynamics have been performed for a 3-band Hubbard model us-ing the determinant quantum Monte Carlo (DQMC) technique [246]. Here the calculationis done at finite temperature, with the lowest temperature limited by the “sign problem”;the minimum temperature achieved for p = 1 / J/
4. One can see damped AFspin-wave-like excitations at high energy, which have a broad Q -width near the AF wavevector [246], consistent with the high-temperature neutron scattering results on LBCO p = 1 / Superconducting stripes
Before we discuss a model of superconductivity based on stripes, we need to considerthe evidence that charge stripes can be superconducting. Much of the evidence has beenreviewed previously [48, 49], but a few words are appropriate here to provide a completestory. The incommensurability of the charge-stripe fluctuations appears to increase with temperature, opposite to thespin correlations [240]. This might be a consequence of the apparent strong sensitivity of the RIXS measurementsto electron-phonon coupling [241–243], which, for the relevant Cu-O bond stretching response, increases greatlytoward the Brillouin-zone boundary [244]. ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 Pair-Density-Wave Modulation0 Spin Modulation0
Figure 13. Top: schematic of uniform d -wave pair wave function, emphasizing that the hole density is largely onthe oxygen sites and the pair wave function changes sign when rotated by 90 ◦ (blue = positive, orange = negative).Bottom: schematic pair wave function of the pair-density-wave order, which has the same period as the magneticorder but with a shifted phase. Middle panels show the modulations of the PDW state and the spin-stripe order. The charge carriers and interactions that drive the superconductivity live within theCuO planes. To achieve 3D superconducting order, it is necessary to lock together thesuperconducting phases of the planes through interlayer Josephson coupling [103]. Oncethe superconducting correlations within the planes become substantial, a very smallinterlayer coupling is enough for the order to become 3D.If the interlayer Josephson coupling is absent or frustrated, then one might observe2D superconductivity without 3D order. That turns out to happen in LBCO x = 1 / c -axis resistivity remains large (and continues to grow with cooling); the sus-ceptibility remains positive when a weak magnetic field is applied parallel to the planes,demonstrating the absence of Josephson currents between planes, which would be neededto shield the field [44, 167]. With further cooling, nonlinear voltage vs. current relationsprovide evidence [44] for a 2D ordering of the superconducting phase through a Kosterliz-Thouless transition [247]. Only at a temperature of (cid:46) ◦ from one layer to the next, theinterlayer Josephson coupling is frustrated. This frustration inhibits the development of22 ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 µ H ( T ) T (K) H D H D H U Q M
3D SC AB R s ( R Q ) R H ( x - c m / C ) T (K) T c3D T c2D UQM
Figure 14. Phase diagram of LBCO x = 0 .
125 in terms of sheet resistance. Interpolated color contour plot ofthe sheet resistance R s = ρ ab /d (where d is the interlayer spacing) as a function of temperature and magneticfield. Black vertical marks indicate measurement temperatures. The regimes of 3D and 2D superconductivitywith zero electrical resistance are labeled; the ultra-quantum metal phase occurs at fields above the dotted line.Characteristic fields H , H , and H UQM are over-plotted as solid, dashed, and dotted white lines, respectively.From [45].
3D superconducting order, but it puts no restriction on 2D order.To further test this picture, LBCO x = 1 / R s , where the unit is the quantum of resistance for pairs, R Q = h/ (2 e ) ; note that the temperature is on a logarithmic scale, while the magnetic fieldscale is linear. At zero field, one can see the drop in R s at ∼
40 K, corresponding to 2Dsuperconductivity with phase disorder, followed by the 2D phase ordering at ∼
16 K. Atbase temperature, increasing magnetic field causes the loss of 3D superconducting orderat H ≈
10 T, reentrant 2D superconductivity at H ≈
20 T, and then a rapid risefollowed by saturation at R s ≈ R Q in an ultra-quantum-metal (UQM) phase. The Hallcoefficient is essentially zero over the full range of fields for
T <
16 K, indicating particle-hole symmetry and suggesting that pair correlations may survive within the charge stripeseven after suppression of superconducting phase order. Allowing for disorder, this couldbe evidence of a Bose metal phase [249]. In any case, it suggests that the charge stripesdo not interfere with pairing, but do strongly impact the superconducting phase order.Related behavior is also found in LBCO x = 0 . T c = 32 K. Nevertheless,perturbing the system with either a c -axis magnetic field [250, 251] or substitution of 1%Zn [252, 253] enhances the stripe order and induces decoupling of the superconductinglayers. Other cuprate systems that exhibit stripe order also show evidence for layer de-coupling and PDW order [49, 254], such as Nd-doped LSCO [255], Eu-doped LSCO [256],and even LSCO at x = 0 .
125 [257]. In an alternative direction, the 2D superconductivityin LBCO x = 0 .
115 can be pushed toward 3D order by suitable application of strain[258]. Going beyond single-layer cuprates, decoupling of superconducting bilayers in a c -axis magnetic field has been observed in crystals of La − x Ca x Cu O with zero-field T c as high as 54 K [259]; stripe order has not been directly detected there, but the spinexcitations remain gapless below T c [260].Numerical calculations applying various techniques (variational Monte Carlo [47, 261,262], DMRG [263, 264], infinite projected-entangled pair states (iPEPS) [265–267]) to t - J and Hubbard models at 1/8 hole doping find that PDW order is one of severalcompetitive states that are close in energy, with the lowest-energy state varying with The name “ultra-quantum metal” is meant to convey the idea that we have a metallic phase that cannot beunderstood within a semi-classical model. ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 values of parameters such as next-nearest-neighbor hopping [47, 262, 265]. It is alsofound that stripe order, especially spin-stripe order, tends to compete with uniform d -wave order [266–268]. Given that stripe and PDW orders occur within restricted dopingranges within certain cuprates, it should not be surprising that PDW order does notstick out as a robust ground state in such calculations. It is important to keep in mindthat the models treated in these calculations are rough approximations that neglectsignificant factors such as the poorly-screened Coulomb interactions between neighboringsites. The fact that the PDW state is energetically competitive provides good supportfor the interpretations of experimental results discussed above.
4. Stripes and pairing
So far, we have seen that stripe order occurs in some cuprates; it tends to competewith spatially-uniform d wave superconductivity, but it is compatible with 2D super-conductivity that likely corresponds to PDW order. Unrestricted variational algorithmsapplied to simplified models of CuO layers containing only repulsive interactions appearto support both stripe and superconducting correlations, but they leave unclear therelationship between PDW and uniform d -wave orders. Furthermore, they provide noconnection with the dynamic correlations that should be relevant to the pairing scaleand transition temperature. In this section, the goal is to extract some deeper insightsfrom experiments.In 1997, Emery, Kivelson, and Zachar (EKZ) proposed a model for cuprate super-conductivity based on stripes [40]. They treated each charge stripe as a one-dimensionalelectron gas and noted that if a spin gap were present (as in a Luther-Emery liquid [271]),it would act as an amplitude for electron pair correlations. Superconducting phase orderwould require Josephson coupling between neighboring stripes.A challenge with this picture is that it was assumed that the spin gap on the chargestripes would be transferred from a spin gap in the neighboring spin stripes. That as-sumption fails in the case of LBCO x = 1 /
8, where spin-stripe order (and the absenceof a significant gap on spin stripes) coexists with 2D superconductivity [167]. In thissection, we will consider a variation on the EKZ model. To develop this approach, wewill start in a surprising place: we look at the spin excitations of insulating stripes inLa − x Sr x NiO . It turns out that at x = 1 /
3, the charge stripes exhibit 1D spin excita-tions that are decoupled from the 3D spin correlations of the neighboring spin stripes.Applying this lesson to cuprates, we reconsider the nature of the spin excitations inLBCO x = 1 / S = 1 / d -wave SC [267], the solution is to have antiphaseJosephson coupling between neighboring stripes, resulting in PDW order. To achieve the It is worth noting that it has been proven that one can obtain superconductivity from a model with only repulsiveinteractions [269]; however, the proof applies in the weak-coupling limit, where T c is quite small. The remainingchallenge is to show that one can get high transition temperatures from repulsive interactions. One can get dynamic correlations from quantum Monte Carlo (QMC) calculations [246], as we will considerbelow; however, it is challenging to extend QMC calculations down to temperatures comparable to T c because ofthe “sign problem” [270]. ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3Figure 15. Cartoon of the stripe order favored at low temperature in La − x Sr x NiO with x = 1 /
3. Circlesrepresent Ni sites; ovals are bridging O; gray indicates doped holes. Red arrows indicate S = 1 moments in spinstripes; gray arrows correspond to low-spin S = 1 / in-phase coupling necessary for uniform d -wave order, we need to gap the spin excitationsin the spin stripes (and avoid static order). As we will see, experiments indicate that thespin gap limits the coherent superconducting gap in cuprates with uniform d -wave SC[54]. Coupled and decoupled spin correlations La − x Sr x NiO has the same structure as La − x Sr x CuO , but it differs electronically.While it shares a tendency to develop stripe order [216], the stripes run diagonallywithin the NiO planes [273, 274] and there is a substantial charge excitation gap belowthe charge ordering temperature [98]. The ordering temperatures for charge and spinstripes are maximized at x = [275, 276]. Associated with the quasi-3D spin stripeorder are spin excitations with a strong 2D dispersion [277, 278]. In addition, Boothroydand coworkers [279] observed spin excitations with a 1D dispersion that appear below ∼
70 K, far below the ordering temperature for the spin stripes.In a reinvestigation of the 1D spin excitations [280], it became apparent that theyonly made sense in terms of Ni site-centered charge stripes, illustrated schematically inFig. 15, which become favored below a glass-like transition near 50 K. Within the spinstripes, the Ni ions have S = 1, with an anti-phase ordering across the charge stripes.Within the charge stripes, there is one doped hole per Ni site. Combining the Ni momentwith the hole spin on neighboring O sites, it seems reasonable to assume a low-spin S = 1 / ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 Q = (H,0.5,0) E ne r g y ( m e V ) New analysis of the cuprate magnetic spectrum
Emery, Kivelson, Zachar,
PRB , 6120 (1997) LBCO: JMT et al ., Nature , 163 (2004)
LSCO: Vignolle et al ., Nat. Phys. , 163 (2007) Figure 16. Left panel shows the distribution of magnetic spectral weight, χ (cid:48)(cid:48) ( Q , ω ), in LBCO [272] and relatedLSCO samples [281]. The excitations are sharp in Q at low energy, but they are very broad at energies wherecommensurate scattering appears. The white dashed line indicates the hourglass spectrum; the neck of the hour-glass is labelled E cross . The cartoons on the right indicate spin an charge correlations relevant to the observedspin dynamics, as discussed in the text. Ellipses indicate spin singlets, blue circles indicate holes; only Cu sitesare indicated. Making sense of the cuprate magnetic excitation spectrum
The left-hand side of Fig. 16 shows a schematic image of the magnetic spectral weightdetermined by neutron scattering experiments on LBCO x = 1 / E cross . There is anatural bias to look for explanations of such a spectrum based on a spin-only model. Fora stripe ordered system, an obvious choice is to ignore the charge stripes and considerthe excitations associated with ordered spin stripes. Calculations based on coupled spinladders can provide a spectrum comparable to the hourglass form [283–285]; however,the calculated excitations have comparable widths for energies above and below E cross . While the excitations at
E < E cross are relatively sharp in Q , those above E cross areextremely broad. In fact, integrating the excitations over all energies in LBCO x =1 / ∼ Q -integrated magnetic spectral weight as energy increases across E cross that is not capturedby considering the spin stripes alone.An alternative perspective is to apply the lesson from LSNO regarding the decoupledspin degrees of freedom in charge stripes. A difference here is that, if we take the chargestripes to be hole-doped 2-leg ladders as suggested by Fig. 11, then we expect the chargestripes to have a substantial spin gap. For an undoped spin ladder with superexchange J that is the same on legs and rungs, the gap is ∼ . J [51]. Various analyses find pairingcorrelations in doped t - J [50, 287] and Hubbard [288–292] 2-leg ladders. In the limit ofstrong superexchange on the rungs, hole-pairing occurs in order to minimize disruptionof the rung singlet correlations, and this tendency survives as one transforms towardsisotropic exchange. These analyses are backed up by experiment: neutron scatteringstudies of the undoped spin ladders in La Sr Cu O show a spin gap of 26 meV[293] that rises to 32 meV in Sr Cu O and, under pressure, the doped ladders inSr . Ca . Cu O . become superconducting [52].The cartoons on the right-hand side of Fig. 16 indicate how we can apply these ideas to An independent 2-component analysis of the spin excitations in LSCO, more in the spirit of the analysis below,is reported in [286]. ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 interpret the spectrum on the left. At low energy, the incommensurability of the excita-tions indicates that they are fluctuations of the antiphase spin stripes. The absence ofcommensurate response is consistent with a spin gap on the charge stripes. Within thispicture, it is natural to associate the onset of commensurate scattering at E cross with thesinglet-triplet excitation energy of the doped ladders. We can also associate this energywith the scale for pair-breaking within the charge stripes.For E (cid:38) E cross , spin excitations can occur at any site in a CuO plane. On thisenergy scale, the holes are no longer confined to pairs and the spins in charge stripesare no longer confined to singlets. Hence, when a neutron scatters and flips a Cu spin,that excitation can disperse through the AF correlations within a plane, but it will bedamped by the unbound holes. Indeed, we already saw in Fig. 6(a) that the 2-magnonresonance is renormalized downward by ∼ / p ∼ / . A fit to the peaks of theneutron spectra for LBCO x = 1 / J ≈
100 meV, down by ∼ / CuO [81, 82]. Applying this value of J in the formula for the singlet-triplet gap of an isotropic2-leg spin ladder gives 50 meV, consistent with the measured E cross . The observation thatall spins contribute to the excitations above E cross (but not below) is consistent with therise in magnetic spectral weight in that range [272, 281].In a spin only model, achieving order in an array of 2-leg spin ladders requires asignificant inter-ladder exchange coupling to overcome the large spin gap of the isolatedladders. In the actual system (including charge stripes), there is also the energetics ofthe charge stripes to take into account. Forming an antiphase order of the neighboringundoped ladders protects the charge stripe and enables a large spin gap on it. Of course,one should also take account of the effects of spin-orbit coupling [297], which can help tostabilize ordered spins and which has measurable effects in terms of the Dzyaloshinskii-Moriya interaction and spin canting in La CuO [100, 298]. Electron-phonon coupling isanother relevant factor that can contribute to stabilizing inhomogeneous phases [299].Now consider the facts that: 1) the onset of 2D superconductivity (and putative PDWorder) in LBCO x = 1 / d -wave andPDW orders. Both of these observations are consistent with the idea that local pairingcoherence is incompatible with overlapping spin order or slowly fluctuating spins. Asone can see from Fig. 13, the Josephson coupling between neighboring charge stripesmust be antiphase so that the pair wave function can go to zero where the spin stripeamplitude is maximum. It is not enough to have the charge order already established,with spin stripe correlations defined and spin fluctuations virtually gapless; the spindirections have to be frozen before the PDW order develops [167].In principle, uniform d -wave and PDW orders could locally coexist, in which case acharge modulation would be induced with the PDW period (a 1 q modulation comparedto the 2 q of the observed charge-stripe order) [248]. Such a 1 q modulation in the effectivedensity of states was measured by STM in magnetic vortex halos in near optimally-dopedBi Sr CaCu O δ [301]; however, that system has a large spin gap that is unlikely tobe closed near the vortex cores [302]. In a system such as LBCO with spin-stripe order,however, careful diffraction searches for 1 q order have been unsuccessful. Any patches Experimentally, the incommensurate spin excitations appear to disperse only inwards towards Q AF [272, 294,295], whereas calculations commonly indicate a more symmetric dispersion [296]. One may hope that the resolutionof this puzzle is connected with a proper understanding of the interstripe-couplings between spins and charges. Figure 6 shows results for YBCO; however, the Raman data show comparable results for other cuprates, includingLSCO [130]. Evidence for incompatibility of SC and AF orders is also provided by a study of La − x Sr x CuO /La CuO heterostructures [300]. ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 of uniform d -wave SC must occur where local deviations in charge density (to higherdoping) correspond to gapped spin correlations. As we will discuss below, the analysisof NMR measurements on LSCO by Imai and coworkers [303] provide evidence for asurprisingly broad distribution of local hole densities that is sufficient to support thisconcept. Hence, the 3D superconductivity that develops in LBCO x = 1 / d -wave order.
5. Spatially-uniform superconductivity
We have seen how stripe order stabilizes doped spin ladders in which pairing is inducedby the presence of a large singlet-triplet spin gap. I have proposed that this energycorresponds to the experimental scale E cross observed by neutron scattering, which shouldserve as an upper limit for the superconducting gap. In the case of ordered stripes, thestatic spins inhibit uniform d -wave superconductivity. But is this pairing mechanismunique to stripe-ordered samples? Here we consider how a version of this story can alsoexplain the behavior of cuprates with uniform d -wave superconductivity. Similar magnetic response in all cuprates
Past reviews have already made the case that the dispersion of magnetic excitationsis similar in all cuprate families where it has been studied [119, 304]. The parent in-sulator phases all show commensurate antiferromagnetic order with J in the range of100–150 meV. With hole doping, the spin waves of the antiferromagnetic phase evolveinto strongly damped excitations with upward dispersion above a commensurate gap of E cross . The systematic softening of high-energy spin excitations and gradual reductionof spectral weight was clearly pointed out by Stock et al. [136]. Below E cross , there aregenerally excitations that disperse downwards to varying degrees, the main exceptionbeing HgBa CuO δ , where there is simply a gap below E cross [308]. We have alreadynoted in Fig. 12 that the incommensurability of the lowest-energy spin excitations growswith p in underdoped samples, before saturating. Role of the spin gap
To see how the superconducting order can change while the pairing mechanism remainsthe same, let use consider LSCO. This system exhibits at least partial charge and spinorders for x (cid:46) .
13 [170, 222], and develops a gap in the incommensurate spin excitationsfor x (cid:38) .
13 [222, 239, 311]. Along with the loss of weight in χ (cid:48)(cid:48) at low energy, there is again in weight at higher energy [312]. There have been varying criteria used to define thespin gap, but it was argued in [54] that the appropriate criterion is to select the energy atwhich the weight transfer changes from negative to positive. For LSCO x = 0 .
17 and 0.21,it was observed that the spin gap is approximately equal to the coherent superconductinggap, ∆ c [313, 314]. This behavior has been observed in LSCO [281, 305], YBCO [306, 307], Bi Sr CaCu O δ [302], HgBa CuO δ [308, 309], and La − x Ca x Cu O [260]. The dispersion in HgBa CuO δ has been characterized as having the form of a “wine glass” rather than anhourglass [308]; however, given the large damping, the data appear roughly consistent with a parabolic upwarddispersion where the lower part of the dispersion is not resolved due to the damping. A downwardly-dispersingcomponent that was resolvable at T < T c has now been observed in a sample with T c = 88 K [309]. Such a changein damping across T c was previously observed in YBCO [306, 310]. ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 La Sr x CuO YBa Cu O δ HgBa CuO δ Bi Sr CaCu O δ Nd Ce x CuO δ ∆ c ( m e V ) ∆ spin (meV) E c r o ss ( m e V ) (a)(b) Figure 17. (a) Coherent superconducting gap, ∆ c , vs. the spin gap in the superconducting state, ∆ spin for arange of cuprates. (b) Shows the behavior of E cross , which is always larger than ∆ c . References to the originalsources for data shown here are given in [54]. Now, we have seen that stripe order locally frustrates coherent superconducting order,and spin fluctuations (in the absence of spin order) appear to inhibit the establishment ofantiphase Josephson coupling between charge stripes. It seems plausible that fluctuatingspin stripes will also frustrate spatially-uniform d -wave order, except at energies wherethose fluctuations are gapped; this concept is supported by a recent analysis of contribu-tions to pairing based on a phenomenological model for spin fluctuations in underdopedYBCO [53]. Indeed, experimental results are consistent with this picture. Figure 17(a)compares values of the coherent SC gap ∆ c , estimated from Raman spectroscopy, withthe gap ∆ spin determined by neutron scattering [54] for a number of cuprate families. Asone can see, ∆ c ≤ ∆ spin . Furthermore, ∆ c is always less than E cross [plotted in Fig. 17(b)],which I have argued is the upper limit for pairing.While optimally-doped LSCO has a similar normal-state carrier density to other opti-mally doped cuprates, it has a substantially reduced superfluid density, which correlateswith its modest T c [158]. We might expect all of the doped holes to be involved in pairingcorrelations; however, if the maximum energy scale for pairing correlations is E cross , andonly pairs at energies below ∆ spin can participate coherently in the long-range order, thenit is quite reasonable that the superfluid density is relatively small. Furthermore, reso-29 ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 nant soft x-ray scattering measurements on optimally-doped LSCO have demonstratedthat charge-stripe correlations grow as the temperature is reduced towards T c , but thendecrease below T c [147, 220]. This is consistent with the idea that the stripe correlationsare good for pairing, but stripe order is bad for uniform superconductivity.For cuprates with a higher T c , such as YBCO, Bi2212, and Hg1201, the spin gap nearoptimal doping tends to be much larger than in LSCO. Within the stripe-motivatedpicture, a bigger spin gap means that a larger fraction of the doped holes can participatein the superfluid, as observed experimentally [158]. Of course, with a bigger spin gap, oneis further from the limit of spin-stripe order, so that the nature of the pairing mechanismis obscured.It is also worth noting the impact that spin fluctuations have on the electronic selfenergy at energies above ∆ c . In considering the spin fluctuation spectrum in Fig. 16,we noted the impact of unpaired holes on the magnetic dispersion for E > E cross . Theconverse is also significant. Analysis of ARPES data on slightly underdoped Bi2212 showthat, while the electronic scattering rate is small for energies below the SC gap, theyjump to very large values for energies above the gap [315]. At antinodal wave vectors,the above-gap self energy is of order 2 J . The large jump in the scattering rate is alsoseen in optical reflectivity studies, where the jump occurs at ∼ E cross [316]. Charge disorder and granular superconductivity
To understand why there is a coherent-superconducting-gap scale ∆ c that is smallerthan the extrapolated d -wave gap maximum ∆ , we have to consider the role of chargedisorder resulting from random positioning of dopants and poor screening of the long-range Coulomb interaction [317]. This is distinct from the form of inhomogeneity that wehave already discussed: charge and spin stripes. The (“quenched”) charge disorder formsthe landscape within which stripe correlations develop, with a stripe period that dependson the local average hole density. For now, we will focus on underdoped cuprates, andthe overdoped case will be considered separately. Besides explaining ∆ c , charge disorderalso provides a way to make sense of experimental observations of pairing correlations at T (cid:38) T c .While it can be easy to overlook, charge disorder in cuprates should not come as asurprise. In the case of LSCO, the holes are introduced by partial substitution of ionsthat have a valence that differs by one from the host sublattice. The holes are introducedinto a correlated insulator state, which is inherently bad at screening Coulomb potentials.In fact, the hole doped into the CuO plane when La is replaced by Sr is the dominantchannel for screening, which limits the range over which the hole can spread [100].Electronic disorder is certainly not unique to cuprates. A new system that exhibitsregimes of strong electronic correlations and superconductivity is magic-angle bilayergraphene, where impressive phase diagrams can be mapped out on a single sample bytransport measurements as a function of carrier density, which is controlled by gatevoltage [318]. Despite the quality of the transport data, nanoscale mapping of Landaulevel with a scanning SQUID has imaged substantial disorder in the local twist angle[319].Returning to cuprates, early evidence of a spatial variation of the hole concentration inLa − x Sr x CuO was provided by an analysis of Cu nuclear quadrupole resonance [303]and O nuclear magnetic resonance [320] measurements as a function of x . For eachsample, there appears to be a spread of environments that can be described in terms of a SQUID = superconducting quantum interference device. ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 Q = (H,0.5,0) E ne r g y ( m e V ) YBCO UD61K c E cross (a) dI/dV (pS) Bi2212 UD75K c (b) |cos(k x a)-cos(k y a)|/2 Bi2212 UD75K c (c) Figure 18. (a) Magnetic spectral weight measured by neutron scattering in YBa Cu O . with T c = 61 K and p ≈ .
12 [306, 333]. (b) Bias voltage (binding energy) vs. conductance measured by STM for typical regions inBi2212 with T c = 75 K and p = 0 . T c = 75 K [200]; dashed line indicates the simple d -wave gap form. In each panel, the gray line indicates∆ c = 3 kT c based on Raman results [204, 205]. distribution of dopant concentations. Hence, for x ∼ .
15, the width of the distributionis ∆ x ∼ .
05 at low temperature, with a characteristic length scale of ∼
30 ˚A.In YBCO, the doping is achieved by adding O atoms to form Cu-O chains in layersintervening between CuO bilayers. X-ray scattering measurements with high spatial res-olution (300 ×
300 nm ) on YBa Cu O . has demonstrated that the patches of orderedchains have a typical domain diameter of ∼
75 ˚A [321], suggesting variations in the holedensity transferred to the planes on this length scale. The lack of effective charge screeningis also indicated by a study of the dynamic charge susceptibility with momentum-resolvedelectron-energy-loss spectroscopy in Bi2212, where the plasmon features, expected for aconventional metal, are missing [322].We have already seen the evidence of substantial charge disorder provided by STM ona doped-but-insulating sample of Bi2201 shown in Fig. 2. This is just the latest evidence,as from the earliest spectroscopic imaging with STM on Bi2212 at T (cid:28) T c , a commonpoint of emphasis has been the spatial variation and disorder in apparent superconductingcoherence peaks with energies in the range of 20–50 meV [323–325]; such behavior is alsoseen in superconducting Bi2201 [326]. The scale of the disorder in the coherence peaks ofBi2212 is tens of ˚A [327], but this is also the scale of the superconducting coherence length[328]. This has led to proposals that the inhomogeneity is associated with variations inboth the local hole concentration and the local pairing scale [327, 329].When measurements were eventually performed across T c , it was found that there isalso a correlation between the size of the local gap and the temperature at which itcloses, with larger gaps closing at higher temperatures that extend well above the bulk T c [330, 331]. The spread in gaps for T (cid:38) T c is also indicated in cuprates such as LSCO,YBCO, and Hg1201 by a study of nonlinear conductivity [332].Figure 18(b) shows typical low-temperature conductance curves (for positive bindingenergy) from STM measurements at various locations on the surface of underdopedBi2212 [334]. For a sample with T c = 75 K, the most frequently observed gap energyis 45 meV, with a broad spread of gaps observed around that value. In contrast tothe behavior of the coherence peaks, the conductance looks much more uniform at lowenergies [335–337]. The gray line indicates the magnitude of the coherent gap ∆ c obtainedfrom Raman scattering studies [205].Another measure of the superconducting gap is given by ARPES. Deviations from the d -wave gap dispersion are observed in underdoped cuprates [199, 338], as illustrated in31 ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 E c r o ss (a) LSCO YBCOHg1201Bi2212 E ne r g y ( m e V ) c AN (b) Bi2212 p T e m pe r a t u r e ( K ) SC(c) LSCO
Figure 19. (a) E cross from neutron scattering studies of La − x Sr x CuO [119], YBa Cu O x [307, 341–343],Bi Sr CaCu O δ [302, 344–347], and HgCa CuO δ [308, 309]. Gray line indicates the common trend amongthese families. (b) Antinodal gap (red filled circles) and coherent gap (dark red open circles) energies from Ramanscattering on Bi Sr CaCu O δ [205, 348]. The shading of the ∆ AN symbols reflects the change in peak area withdoping. (c) Violet contours indicate relative strength (logarithmic scale) of superconducting fluctuations abovethe superconducting phase (blue) obtained from Nernst effect measurements on La − x Sr x CuO [349]. Fig. 18(c) for Bi2212 with T c = 75 K [200]. The dashed line indicates the anticipated d -wave dispersion; significant deviations occur at energies above ∆ c . The ARPES mea-surements involve a photon beam that covers a large surface area compared to the regionprobed by STM, and hence should average over the the distribution of coherence peaksobtained by STM. I have argued that the energy E cross determined from the spin excitation spectrumprovides an upper limit for pairing, and hence a limit for superconducting coherencepeaks. To provide a comparison, Fig. 18(a) shows a schematic version of the imaginarypart of the dynamic susceptibility, χ (cid:48)(cid:48) ( q , ω ), in YBa Cu O . as determined by inelasticneutron scattering [226, 306]. While the doping levels are not quite the same, one cansee that E cross ∼
38 meV of the hourglass spectrum is comparable to the energies of thecoherence peaks measured with electronic spectroscopies.To appreciate why the local pairing scale may depend on hole concentration, Fig. 19(a)shows the doping dependence of E cross determined by neutron scattering in four familiesof cuprates. The trends are remarkably similar in all of these compounds. The generaldoping trend is approximated by the gray bars in the background. A key point here isthat, for rather underdoped cuprates, the limiting energy for pairing increases linearlywith hole concentration.Theoretically, it has been proposed that charge disorder can cause a cuprate to behavelike a granular superconductor [55, 350–354]. In particular, Imry et al. [55] considered amodel in which superconducting grains are effectively coupled together through Joseph- 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 [339]; it was used previously to simulateARPES results for LSCO [340]. No neutron scattering studies of underdoped Bi2212 crystals have been reported yet. ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 son junctions. Such a model is compatible with the early observation by Emery andKivelson [56] that the onset of superconducting order in underdoped cuprates is limitedby the development of superconducting phase coherence, rather than the onset of pair-ing. Both of these approaches provide an explanation for the observation of Uemura etal. [158] that T c grows in proportion to the superfluid density, within the underdopedregime.There has been quite a bit of work on systems that exhibit the characteristics ofgranular superconductors. A popular model system involves amorphous thin films ofInO x . STM measurements reveal a gap that is associated with coherence peaks below T c , but that does not close until the temperature is increased far above T c [355], asituation analyzed in [354]. Point-contact measurements reveal Andreev reflections thatoccur on an energy scale considerably smaller than the large energy gap obtained fromtunneling conditions [356]. (Related behavior is also seen in granular Al [357], wherean oxide surface layer limits coherent coupling.) Applying a magnetic field can destroythe Josephson coupling between grains, resulting in a superconductor-to-insulator (SIT)transition [358], and parallels have been drawn with the SIT behavior observed in veryunderdoped LSCO [359, 360].To the extent that cuprates show granular features, what limits the coherence betweengrains? One proposal is that there might be a mixture of antiferromagnetic and super-conducting domains, with the AF domains providing the barrier to coherence [350, 351].While there is no experimental evidence for commensurate AF order coexisting withsuperconductivity, we do have a varying degree of low-energy incommensurate spin fluc-tuations that can play a role. We already saw in Fig. 17 that ∆ c is limited by ∆ spin . Wecan now combine that with the fact that the ∆ spin tends to grow with doping, as does E cross , to interpret the observed behaviors of gap variation and appearance above T c .Imagine, for a moment, that a sample contains regions associated with just two differenthole dopings, one higher ( p + ) and one lower ( p − ), and that the size of these domains ismuch larger than ξ . The local pairing strength should be determined by a combination ofthe local E cross and ∆ spin , so that the local ∆ should be larger in the p + domains. The p + domains will start to develop local superconducting order at a temperature T + c > T c . Inorder to develop long-range order, it is necessary to achieve coherence in the p − domains,but the smaller (or possibly negligible) ∆ spin in these domains will limit this. EffectiveJosephson coupling through the p − domains will increase ∆ − spin to ∆ c , the gap scalebelow which coherent, spatially-uniform superconductivity occurs, and the resulting ∆ c will determine T c .A consequence of increasing ∆ − spin is that spin-fluctuation spectral weight must bepushed above ∆ c . This shift is associated with the appearance of a “resonance” peak atenergy E r > ∆ c . Past neutron scattering studies have demonstrated a correlation between E r and T c [363, 364]. In the present interpretation, this relationship is a consequence ofthe connections with ∆ spin and ∆ c .With that story in mind, consider now the results plotted in Fig. 19(b). Here the graybars summarizing the E cross trend are repeated to allow a comparison with measurements The concept of a spin resonance comes from the weak coupling perspective, in which the same electrons areresponsible for both the spin correlations and the superconductivity [361, 362]. In the superconducting state,where the antinodal electronic states should be gapped by 2∆ , a spin excitation at E r < requires a resonantenhancement relative to the bare Lindhard spin susceptibility. The “resonance” language is misleading whenapplied to the experimental results, because a weak-coupling description is not relevant. ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 of ∆ c and the AN gap ∆ AN determined by Raman scattering in Bi2212 [205, 348]. TheRaman result for ∆ AN is comparable to the mean of the coherence-peak energies fromSTM in Fig. 18(b) and the ARPES scale in Fig. 18(c). Besides indicating the energies bysymbol position, the shading of the symbols is indicative of the integrated intensity ofthe spectral feature, which extrapolates to zero at p ≈ .
10 [205]. Note that the absenceof the ∆ AN signal for p (cid:46) .
12 is found for other cuprate families, as well [204]. Also,ARPES measurements on Bi2212 with p (cid:46) . T c <
70 K) generally show an absenceof legitimate quasiparticle peaks in the AN region, where the large pseudogap tends todominate [202, 367–369]. Hence, it appears that ∆ AN (cid:46) E cross (with the exception ofLSCO, where ∆ AN (cid:28) E cross [54]).Figure 19b) only compares energy scales at low temperature; there are related obser-vations at T > T c . For example, ARPES studies of Bi2212 reveal weak intensities ofcoherent antinodal peaks that survive for a finite temperature range above T c [370–372].Measurements with high resolution in energy indicate that the gap in the near-nodalregion does not close uniformly at T c [373]. Again, these features occur within the dis-ordered landscape indicated by STM [330]. Note that the coherence length ξ is muchsmaller than the magnetic penetration depth λ , and as long as the grains that develop co-herence above T c are small compared to λ , they will have limited impact on bulk-sensitivemeasurements of superconductivity, such as magnetization.Considering the average behavior at T > T c , ARPES measurements on underdopedBi2212 and Bi2201 find that the superconducting gap closes only along a finite arc,centered on the nodal point, with a gap remaining in the AN region [163, 200, 338,373, 374]. The magnitude of the low-temperature gap that develops at the ends of thearcs provides another measure of ∆ c , as indicated in Fig. 10, one which is quantitativelyconsistent with the Raman results for ∆ c shown in Fig. 19(b).Figure 19(c) shows the temperature dependence of superconducting fluctuations as afunction of doping in LSCO, where the measure is the Nernst coefficient, plotted withroughly logarithmic intensity contours [349]. The maximum onset temperature occursfor p ∼ .
1, with the response falling rapidly at lower p . Confirmation that these fluctua-tions are associated with pairing correlations is provided by a recent study of tunneling-current noise involving LSCO films [375]. The trend of the fluctuations with doping isconsistent with the result in Fig. 19(b) that the maximum local pairing gap is optimizednear p ∼ .
12. In contrast, phase coherence and superfluid density are optimized at p ∼ p ∗ , as indicated by the plot of the superfluid density in Fig. 7. The overdoped regime
To discuss the changes on increasing p into the overdoped regime, it is convenient to startwith an interpretive summary, shown graphically in Fig. 20, and then consider the ex-perimental results that support this picture. In the underdoped region, as we have seen,the competition between superexchange-coupling of Cu moments and the kinetic energyof O holes leads to stripe correlations, but all good things must come to an end. In 214cuprates, the charge stripe period at p = 1 / p = 1 / Comparisons of 2∆ AN from a broad range of techniques have been presented elsewhere [365]; the other measuresare consistent with the Raman results. For the families YBCO, Bi2212, Tl Ba CuO δ , and HgBa CuO δ , ithas been observed that 2∆ c ≈ kT c [204, 205]. ∆ c is also detected by Andreev reflection in tunneling spectroscopy [366]. The Nernst effect is the transverse voltage measured in response to a longitudinal temperature gradient in thepresence of a perpendicular magnetic field; it is sensitive to vortex fluctuations for temperatures near T c [349]. ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 charge, spindistributionin real spacespinresponse (Q,E)FermisurfacecoherentSC gap p 1/8 p p* p > p* Q AF E ne r g y Q AF Q AF cos(k x a) - cos(k y a) Figure 20. Cartoon summary of changes in various properties with overdoping. Each column corresponds tothe hole concentration listed above the top row of panels. Top row: spatial distribution of charge (blue) and spin(red) densities in real space. Stripes indicate strong correlations, even in the absence of static order. Second row:spin response in the form of χ (cid:48)(cid:48) ( Q , E ) for Q = ( h, . , A ( k , E F )measured by ARPES. Bottom row: coherent SC gap along one quadrant of the Fermi surface, as a function ofcos( k x a ) − cos( k y a ). form uniformly-doped regions. Eventually, the strongly-correlated, striped regions willno longer percolate across the CuO planes. This crossover should correspond to thepoint, p = p ∗ ∼ .
19 [376], at which the large AN pseudogap due to Cu spin correlationseffectively disappears [368, 377], and a coherent SC gap is observed around the entireFermi surface, including the AN region [378]. The superfluid density reaches its maxi-mum at p ∗ , as seen in Fig. 7. For p > p ∗ , the superfluid density and T c decrease fairlyrapidly [139, 159, 379–382]. While the carrier density remains substantial, the couplingof electrons to the excitations that drive pairing drops to zero as T c →
0, as seen byboth ARPES [383] and optical spectroscopy [142]; in particular, the signature of thisinteraction is absent in the normal state for p > p ∗ [383].Inelastic neutron scattering measurements on LSCO indicate that the spin correlationscontinuously weaken with doping, remaining detectable at p > p ∗ , but disappearing asthe superconductivity disappears at large p [54, 384, 385]. In a related fashion, the2-magnon peak detected by Raman scattering in several cuprate families gradually shiftsto lower energy [as indicated in Fig. 6(a) for YBCO], until it becomes overdamped at p (cid:38) p ∗ [130]. Signatures (from neutron scattering, NMR, and ultrasound) of quasi-staticAF correlations induced by high magnetic fields disappear at p ∼ pc [222, 386]. Thecharacteristic temperature T ∗ determined from quantities such as the Hall coefficient In particular, there is no evidence of a collapsing of the excitation spectrum towards zero frequency, as onemight expect in the case of a quantum critical point. ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 and the bulk susceptibility (discussed in Sec. 2.4) trends toward zero near p ∗ [187, 188].These results are all consistent with a loss of percolation for the regions with strong localAF correlations at p ∼ p ∗ .The data suggest that cuprates with p > p ∗ have patches with local AF correlations,that can drive pairing, embedded within a metallic state. Spivak et al. [387] have analyzedhow such patches can drive bulk superconductivity through the proximity effect. Thedisorder can lead to granular behavior, with T c varying with the superfluid density,rather than the pairing strength [55, 57].
6. Discussion
While I have attempted to cover a large range of experimental work essential to under-standing the cuprates, there are some remaining significant topics that deserve comment.
Nematic order
When charge and spin stripe correlations are purely dynamic, they can still have in-teresting impacts on bulk properties. An analogy to liquid crystals led to a proposalof smectic and nematic electronic phases [43]. Even when static stripe correlations arepresent, quenched disorder may destroy long-range correlations; nevertheless, vestigialeffects can result in nematic order [390, 391].Evidence of nematic order [392, 393] and its vestigial character [394] has been providedby STM studies on Bi2212. Rotational anisotropy in the Nernst effect measured onYBCO indicates nematic order [396].Signatures of nematic responses have also been identified in Raman scattering exper-iments on very underdoped YBCO and LSCO [397]. A recent Raman study of Bi2212proposes a nematic quantum critical point at p ∗ ≈ .
22 [398]. While theory shows thatfluctuations at a nematic quantum critical point can, in principle, enhance superconduc-tivity [26, 27, 399], I would argue, on the basis of nematicity being a vestigial order, thatthis is not of fundamental importance. The underlying magnetic and charge correlationsprovide the dominant electron pairing, as discussed in Sec. 5.
Charge order
Charge-density-wave (CDW) order has now been observed in most cuprate families,especially by x-ray scattering techniques [206, 207]. While the modulation is alwaysparallel to the Cu-O bonds, as with the charge stripes in La − x Ba x CuO and other 214cuprates, a difference is that it tends to develop in the absence of spin order. Of course,this does not mean that there is an absence of AF correlations; to the contrary, CDWorder develops in YBCO (in zero magnetic field) over the range of doping 0 . (cid:46) p (cid:46) . as determined by Alternatively, it has been argued that some of the phenomenology observed in overdoped cuprates can beunderstood in terms of “dirty d -wave theory”, which is based on a conventional BCS picture [388, 389]. While thatmay be the case, it does not provide a specific understanding of the pairing mechanism, especially for p < p ∗ . There are experimental technique issues that may qualify the identification of nematic order by STM [395]. In YBCO, ordering of Cu-O chains provides a broken symmetry that enables the detection of a nematic effectthat turns on well below the chain-ordering temperature. Charge density waves and charge stripes are both names for a periodic modulation of the charge density. In bilayer cuprates, the AF coupling between neighboring CuO layers may enhance the tendency to developinga spin gap [364]. ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 the temperature dependence of the spin-lattice relaxation rate measured by Cu NMR[401, 402]. The short-range and static characters of the low-field CDW are confirmed byNMR [403].CDW order in Bi2212 was first detected by STM [79, 404, 405] and was later connectedwith measurements of resonant elastic x-ray scattering [406]. The study that discoveredPDW order in halos about magnetic vortices found the PDW modulation to be linkedto the CDW order that was previously seen in zero magnetic field [301]. Follow on work,combining STM spectroscopic imaging and theoretical mean-field analysis, made thecase that, even in zero field, the observed CDW order is consistent with calculationsfor PDW order that coexists with uniform d -wave order [407]. The observation ofperiodically-modulated Josephson tunneling demonstrates that the pairing correlationsare spatially-modulated and in-phase with the CDW order [408]. Complementary workon Bi2201 shows that the coherence peaks in the superconducting state are modulatedat the same period as the CDW in Bi2201 [409, 410]. At minimum, these results indicatethat the CDW order does not compete with pairing.Returning to YBCO, the zero-field CDW order involves equal and opposite atomicdisplacements of the CuO bilayers immediately above and below one chain layer [411],and the order decays on cooling below T c . In contrast, in c -axis magnetic field of sufficientstrength ( >
15 T for p = 0 .
12) there is a transition to a CDW phase with modulationsthat are in-phase in all layers, so that there is a finite correlation length along the c-axis T c .[412–414]. The development of this phase begins slightly below the onset of finiteresistivity [415]. Strikingly, it has been discovered that a similar CDW phase, with 3Dcorrelations, can be induced at zero field by application of strain along the a axis [416];it appears on cooling below the zero-strain T c and then disappears with the onset of bulksuperconductivity at a reduced T c [417].Given the relatively low onset field for the 3D CDW in YBCO and the associationbetween pair correlations and CDW order in Bi2212, it would be surprising if pair cor-relations were not present in the 3D CDW state of YBCO. Indeed, as discussed below,analysis of quantum oscillations in the high-field regime indicates that the “normal-state”response comes from only a fraction of the doped carriers [418, 419], and measures of theincreasing density of states with field indicate saturation in the high-field phase [420].A recent analysis suggests that the missing holes, presumably associated with antinodalstates, are likely gapped [421]. Also, weak diamagnetism above the nominal upper criticalfield, H c has been reported [422, 423]. These effects remain to be properly understood. Electron-phonon coupling
Analyses are often posed as a choice between electron interactions with phonons or withspin correlations. In reality, both are significant; however, they are not equally important[424]. In particular, the charge order in cuprates benefits from, but is not driven by,electron-phonon coupling.Given the low carrier density in cuprates, phonons make an important contributionto Coulomb screening. The phonon branch with the biggest impact is the Cu-O bond-stretching mode [425]. At q = ( π, π ), it forms a symmetric breathing mode of O − aboutCu ; at ( π,
0) and (0 , π ), we have half-breathing modes. In cuprates such as LSCOand YBCO, neutron scattering experiments show that the strongest phonon softeningis along the Cu-O bond direction, growing in strength as one moves from zone center While there is no static spin order to conflict with the uniform SC, it may be that defects locally make theCDW/PDW orders energetically competitive, as proposed in the case of Zn-doped LBCO [253]. ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 to zone boundary [426–428]. There is a strong dip near the charge-ordering wave vector[307, 429].If charge order were driven by electron-phonon coupling and the nesting of the noninter-acting Fermi surface, we would expect the charge order to be in the diagonal direction, asshown by many analyses [430, 431]. The inconsistency with experiment is direct evidencethat, while phonon softening lowers the electronic energy, this occurs to complement thestronger interactions resulting from superexchange.Resonant inelastic x-ray scattering (RIXS) is particularly sensitive to electron-phononcoupling (although with limited energy resolution, at present). A soft-mode behavior overa large energy range in connection with the CDW wave vector has been reported for Eu-doped LSCO [242] and Bi2212 [243, 432]. The results show an interesting anisotropy inintensity with momentum transfer q . The intensity rises from the CDW wave vector andconnects smoothly with the bond-stretching mode at large q . This appears to directlyillustrate the growing strength of electron-phonon coupling with q . The continuous vari-ation is a consequence of poor energy resolution. The weight must involve anticrossingtransfers among the many phonon modes between zero energy and the bond-stretchingbranch. For example, several non-resonant x-ray and neutron scattering studies, withmuch better energy resolution, have demonstrated in YBCO that the CDW is associatedwith phonon softening in acoustic [433, 434], bond-bending [435], and bond-stretchingbranches [427]. Quantum oscillations
The discovery of quantum oscillations in measurements of transverse and longitudingalresistance [436], magnetization [437], and specific heat [438] in underdoped YBCO as afunction of magnetic field created a sensation. In order for quantum oscillations to bedetectable, quasiparticles near the Fermi level must be sufficiently coherent that they cancomplete an orbit on the Fermi surface. Thus, their observation seemed to suggest thatthe state obtained by suppressing the superconducting order is a Fermi liquid, which isa challenge to relate to the zero-field normal state at
T > T c .From the oscillation frequency it is possible to estimate the size of the orbit in reciprocalspace, and that yields an associated density of carriers. It quickly became clear that onlya fraction of the doped carriers contributes to the oscillations [439]. A wide varietyof proposals was made for various mechanisms that could cause a modification of thenominal Fermi surface, including stripe order and biaxial CDW order [418, 440]. In thelatter case, a small pocket is effectively generated that includes the Fermi arcs, withscattering between arc ends at points connected by one of the CDW wave vectors. Thepresence of a spin gap [402, 412] helps to minimize the scattering rate for states on thearcs.The concept of a small pocket created from Fermi arcs due to biaxial CDW order iscertainly intriguing, but it leaves some open questions. For instance, what is the role ofthe uniaxial CDW order induced in high fields, and what happens with to the antinodalstates? Regarding the missing antinodal states, a recent study indicates that they arelikely gapped [421], as mentioned above.Besides considerations of reciprocal space, it is also necessary to consider real space.In particular, successful detection of quantum oscillations requires that the electronicstates be coherent over the cyclotron radius, which is largest at the lowest fields atwhich the oscillations first appear. Gannot et al. [441] pointed out that, in comparingthe cyclotron radius with the typical correlation lengths for the zero-field and high-field-38 ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 induced CDWs, only the high-field CDW has a correlation length consistent with theonset field. Such a connection is intriguing, but more details must be sorted out beforea complete understanding is reached. Strange and bad metals
Anderson noted that the in-plane resistivity is approximately linear in T down to low T near optimal doping [442]. This deviates from the T behavior expected in the T → ∼ (cid:126) /k B T , Zaanensuggested that the associated phenomenon be termed “Planckian dissipation” [444].Another unusual aspect of transport in cuprates occurs at high temperature. Resistivityin metals is normally observed to saturate when it rises to a level of ∼ .
15 mΩ-cm [445].This is presumed to be roughly the point at which the mean-free-path of an electronbecomes comparable to the interatomic spacing, which has come to be known as the Mott-Ioffe-Regel (MIR) limit [446, 447]. In contrast, the resistivity of underdoped cuprates risesthrough this limit without any change of behavior. Emery and Kivelson identified thisas “bad”-metal behavior [448].As we have seen from the temperature dependence of the Hall coefficient, the densityof charge carriers grows at high temperature. Hussey et al. [449, 450] have noted that theelectronic scattering rate saturates in a fashion roughly consistent with the MIR limit,and that the carriers at high temperature are incoherent, as indicated by the loss of theDrude peak in optical conductivity. From the perspective of the spin correlations, onecan imagine the completely incoherent state corresponding to soup of local singlet andtriplet correlations with rapid fluctuations due to the lack of segregation between chargesand moments.The conventional view of resistivity is in terms of diffusive motion of independent elec-trons. To describe the anomalous behavior in cuprates, a picture of hydrodynamic trans-port, involving a collective motion of a viscous electron fluid, has been proposed [451]. Inparticular, hydrodynamic effects associated with fluctuating charge density waves havebeen analyzed [452]. Results from related modelling have been applied to describe arange of experimental results for a Bi2201 sample [453]. There has also been progresson modelling Planckian dissipation by inclusion of random interactions among electronsthrough a variation of the Sachdev-Ye-Kitaev model [454].From the perspective of intertwined charge and spin correlations, the concept of col-lective, viscous transport is appealing. It also suggests that realistic models will needto take account of the AF correlations, which are a dominant source of scattering, aswe have seen. Recent quantum Monte Carlo calculations on the Hubbard model showpromising signs of this [455].
Tri-layer cuprates
Within a given family, T c varies with the number of consecutive CuO layers, showinga maximum for three layers [456]. Analysis of NMR Knight shift data on 3- and 4-layercuprates indicates that the hole concentrations are different on the inner and outer layers,with p being larger on the outer planes by ∼ .
04 [457]. The results shown in Figs. 7 and19 suggest why the distinct values of p could be beneficial for T c . The superfluid density39 ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 J S i S i + P a i r i ng S c a l e S upe r f l u i d D en s i t y p* p T c Figure 21. Schematic figure of hole-density dependence of (top to bottom) local AF correlations, pairing scale,superfluid density, and T c . reaches a maximum for p ∼ .
2, whereas pairing strength appears to be optimized for p ∼ .
12. The trilayer cuprates can simultaneously benefit from both of these features,consistent with a proposed mechanism for enhancing T c [458].
7. Conclusion
From the beginning, there have been attempts by theorists to extrapolate phenomenarealized in 1D to 2D. Anderson’s proposal of a 2D quantum spin liquid, in the formof an RVB state [13], was made in the light of the known quantum-spin-liquid state ofthe 1D Heisenberg spin- AF. When AF order 2D at T = 0 was confirmed by theory,considerable effort was devoted to models with frustration, such as second-neighbor AFexchange, to get a spin liquid or bond order prior to hole doping [459].A key feature of a quantum spin liquid is the absence of a spin gap. Nevertheless, earlymean-field analyses associated with the RVB model recognized that a spin gap (takento be spatially uniform) could be associated with pairing correlations [31]. The spin gapwas assumed to be large at low doping and to decrease linearly toward zero at large p .An alternative way to get a spin gap is to consider even-leg spin ladders; however, as oneextrapolates to 2D, the gap goes to zero [460].Experimentally, it is true that the average strength of local AF correlations decreaseswith p , eventually reaching a percolation transition at p ∗ , as indicated schematically inthe top panel of Fig. 21. However, the singlet gap that sets the pairing scale is absent inthe undoped system; it develops with doping due to the development of intertwined ordersas a consequence of competition between the hole kinetic energy with the superexchange40 ebruary 5, 2021 1:33 Advances in Physics striped˙lens˙v3 interactions that drive the AF correlations I have identified this gap with the energy E cross corresponding to the neck of the typical hourglass spin excitation spectrum foundin neutron scattering studies.It is the doping that causes the planes to segregate into quasi-1D stripes. Emery, Kivel-son, and Zachar [40] were the first to propose that stripes could be the source of pairingand superconductivity. I have proposed a slight variation on this, arguing that the chargestripes should be viewed as doped two-leg ladders (which are known to develop strongpairing [51]); they are isolated by that antiphase order of the neighboring spin stripes.Of course, those spin correlations obstruct spatially-uniform d -wave superconductivity,and they must be gapped to achieve it. The pairing scale on the ladders, as measuredby E cross , reaches its maximum at p ≈ , where stripe order is optimized. It decaysgradually with doping and gets diluted as the regions that support stripe correlationsare gradually replaced by uniformly-doped domains.The connection between pairing within charge stripes and spatially-uniform supercon-ductivity is clearest where the stripe fluctuations are slow compared to the pairing scale,as in optimally-doped LSCO. Obviously, the picture becomes fuzzier as the incommen-surate spin gap approaches the E cross . Nevertheless, the doping dependence of E cross issimilar in all cuprates studied so far, and the self-organization of doped holes and localAF correlations provides a consistent interpretation.Besides the stripe order/correlations, we also have variations in average charge densityover a larger length scale due to dopant disorder and poor screening. Because of thisgranular character, the development of superconducting phase order is dependent on thesuperfluid density [55, 56]. It follows that T c is determined by a balance between thepairing scale and the magnitude of the superfluid density. As indicated in Fig. 21, T c ismaximized midway between the peaks in the pairing scale and the superfluid density.Applications of advanced algorithms to simplified model Hamiltonians provide resultssupportive of pieces of my story, but they also demonstrate why reaching a conclusion hasbeen challenging. There are many competing states with similar energies, and establishingevidence for pairing order is a numerical challenge. Rapid progress is being made, and itwill be interesting to see to what extent future results may support and justify the storyI have presented here. Acknowledgments
I have benefited from discussions with and comments by I. Bozoviˇc, E. Dagotto, E.Fradkin, K. Fujita, C. C. Homes, M. H¨ucker, P. D. Johnson, S. A. Kivelson, 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. I am especially grateful to T. Egami for challenging me totell an interesting story. This work was supported at Brookhaven National Lab by theU.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciencesand Engineering Division under Contract No. DE-SC0012704.
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