Section 7: Emergent Behavior in the Cuprate Superconductors
We can best appreciate the remarkable properties of the cuprate superconductors by considering a candidate phase diagram (Figure 24) that has emerged following almost 25 years of experimental and theoretical study described in well over 100,000 papers. In it, we see how the introduction of holes in the CuO planes through chemical substitution in materials such corresponding to La1-xSrxCuO4 or YBa2Cu3O6+x, the low-temperature phase is one of antiferromagnetic order. The gateway to this emergent behavior is the very strong electrostatic repulsion between the planar quasiparticles. This causes the planar Cu d electron spins to localize (a process called "Mott localization" in honor of its inventor, Nevill Mott rather than be itinerant, while an effective antiferromagnetic coupling between these spins causes them to order antiferromagnetically. The magnetic behavior of these localized spins is remarkably well described by a simple model of their nearly two-dimensional behavior, called the "two-dimensional Heisenberg model;" it assumes that the only interaction of importance is a nearest neighbor coupling between spins of strength J.
The impact of adding holes
Figure 24: A candidate phase diagram based, in part, on magnetic measurements of normal state behavior, for the cuprate superconductors, in which are shown the changes in its emergent behavior and ordering temperatures as a function of the concentration of holes in the CuO planes.
Source: © David Pines. More info
When one adds holes to the plane, their presence has a number of interesting consequences for the localized Cu spins. Those, in turn, can markedly influence the behavior of the holes that coexist with them. The accompanying phase diagram (Figure 24) indicates some of the effects. Among them:
- Holes interfere with the long-range antiferromagnetic order of the localized spins, initially reducing its onset temperature, TN, and then eliminating it altogether for hole doping levels x > 0.03.
- At higher hole doping levels, 0.03 < x < 0.22, the local spins no longer exhibit long-range order. Instead they form a spin liquid (SL) that exhibits short-range spin order and scaling behavior controlled by their doping-dependent interaction. The measured scaling behavior of the SL can be probed in measurements using nuclear magnetic resonance to probe the temperature-dependent uniform magnetic susceptibility and measure the relaxation time of 63Cu probe nuclei. These show that for temperatures above T*(x), the SL can still be described by the 2-d Heisenberg model, with a doping-dependent interaction, Jeff(x), between nearest neighbor spins whose magnitude is close to the temperature, Tmax(x), at which the SL magnetic susceptibility reaches a maximum. As the density of holes increases, both quantities decrease linearly with x.
- x = 0.22 is a quantum critical point (QCP) in that, absent superconductivity, one would expect a quantum phase transition there from localized to itinerant behavior for the remaining Cu spins.
- Between Tmax and T*, the holes form an anomalous fermi liquid (AFL), whose anomalous transport properties are those expected for quantum critical matter in which the quasiparticles are scattered by the QC fluctuations emanating from the QCP at x ~ 0.22. Careful analysis of the nuclear spin-lattice relaxation rate shows that in this temperature range, the SL exhibits the dynamic quantum critical behavior expected in the vicinity of 2d AF order, hence its designation as a quantum critical spin lLiquid, QCSL.
- Below T*, a quite unexpected new state of quantum matter emerges, pseudogap matter, so called because in it some parts of the quasihole Fermi surface become localized and develop an energy gap; the SL, which is strongly coupled to the holes, ceases to follow the two-dimensional Heisenberg scaling behavior found at higher temperatures.
Figure 25: Schematic illustration of the temperature evolution of the Fermi surface in underdoped cuprates. The d-wave node below Tc (left panel) becomes a gapless arc in the pseudogap state above Tc (middle panel), which expands with increasing T to form the full Fermi surface at T* (right panel).
- For 0.05 < x < 0.17, the hole concentration that marks the intersection of the T* line with Tc, the superconducting state that emerges from the pseudogap state is "weak"; some of the available quasiparticles have chosen, at a temperature higher than Tc, to become localized by condensing into the pseudogap state, and are therefore not available for condensation into the superconducting state. Their absence from itinerant behavior, illustrated in Figure 25, is seen, for example, in an ARPES (angle-resolved photoemission spectroscopy) probe of quasiparticles at the Fermi surface. Pseudogap matter and superconductivity thus compete for the low-temperature ordered state of the hole Fermi liquid in much the same way as antiferromagnetism and superconductivity compete in heavy electron materials.
- For x > 0.17, superconductivity wins the competition and is "strong," in that all available quasiparticles condense into the superconducting state. At these dopings, the pseudogap state does not form unless a magnetic field strong enough to destroy superconductivity is applied; when it is, the pseudogap state continues to form until one reaches the QCP at x ~ 0.22, behavior analogous to that found for the AF state in CeRhIn5.
- Whether the superconductivity is weak or strong, the pairing state turns out to be the dx2-y2 state that, in the case of heavy electron materials, is the signature of a magnetic mechanism in which the magnetic quantum critical spin fluctuations provide the pairing glue. It is not unreasonable to conclude that the same physics is at work in the cuprates, with the nearly antiferromagnetic spin fluctuations playing a role for these unconventional superconductors that is analogous to that of phonons for conventional superconductors.
- The pseudogap state tends to form stripes. This tendency toward "inhomogeneous spatial ordering" reflects the competition between localization and itinerant behavior. It leads to the formation of fluctuating spatial domains that have somewhat fewer holes than the average expected for their doping level that are separated by hole-rich domain walls.
- Scanning tunneling microscope experiments (STM) (Figure 26) on the BSCCO members of the cuprate family at low temperatures show that, for doping levels less than x ~ 0.22, even the samples least contaminated by impurities exhibit a substantial degree of spatial inhomogeneity, reflected in a distribution of superconducting and pseudogap matter energy gaps.
- Just as in the case of heavy electrons, the maximum Tc is not far from the doping level at which the spatial order manifested in pseudogap behavior enters.
Ingredients of a theory
Figure 26: Left: A scanning tunneling microscope (STM) is a powerful instrument for imaging surfaces at the atomic level. Right: Inhomogeneous energy gaps measured in BSCCO; (a)-(d) correspond to doping levels that range from near optimal values of x = 0.19 seen in sample (a), for which Tc is 89 K, through levels of 0.15 (b), 0.13 (c) to the very underdoped material(d), for which x = 0.11 and Tc = 65 K; the color scales are identical.
Source: © Left: Wikimedia Commons, CC Share Alike 2.5 Generic License. Author: Royce Hunt, 5 March 2007. Right: K. McElroy, D.-H. Lee, J. E. Hoffman, K. M. Lang, J. Lee, E.W. Hudson, H. Eisaki, S. Uchida, and J. C. Davis. More info
We do not yet possess a full microscopic theory that explains these amazing emergent behaviors, but we see that the basic ingredients for developing such a theory are remarkably similar to those encountered in heavy electron materials. In both cuprates and heavy electron materials, local moments coexist with quasiparticles over a considerable portion of their generalized phase diagrams. Their mutual interaction and proximity to antiferromagnetism and a "delocalizing" quantum critical point lead to the emergence of quantum critical matter and d x2-y2 superconductivity, with the maximum Tc for the latter located not far from the QCP at which quasiparticle localization first becomes possible.
The principal differences are twofold: First, in the cuprates, the physical origin of the local moments is intrinsic, residing in the phenomenon of Mott localization brought about by strong electrostatic repulsion); second, in place of the AF order seen in heavy electron materials, one finds a novel ordered state, the pseudogap, emerging from the coupling of quasiparticles to one another and to the spin liquid formed by the Cu spins. It is the task of theory to explain this last result.
We can qualitatively understand the much higher values of Tc found in the cuprates as resulting from a mix of their much higher intrinsic magnetic energy scales as measured by the nearest neighbor LM interaction—J ~1000 K compared to the 50 K typically found in heavy electron materials—and their increased two-dimensionality.
Theories in competition
Our present understanding of emergent behaviors in the cuprates would not have been possible without the continued improvement in sample preparation that has led to materials of remarkable purity; the substantive advances in the use of probes such as nuclear magnetic resonance and inelastic neutron scattering, to study static and dynamic magnetic behavior in these materials; and the development of probes such as ARPES, STM, and the de Haas von Alphen effect that enable one to track their quasiparticle behavior in unprecedented detail. The brief summary presented here has scarcely done justice to the much more detailed information that has emerged from these and other experiments, while it is even more difficult to present at a level appropriate for this unit an overview of the continued efforts by theorists to develop a microscopic explanation of this remarkable range of observed emergent behaviors.
The theoretical effort devoted to understanding the cuprate superconductors is some orders of magnitude greater than that which went into the search for a microscopic theory of conventional superconductors. Yet, as of this writing, it has not been crowned by comparable success. Part of the reason is that the rich variety of emergent behaviors found in these materials by a variety of different experimental probes are highly sample-dependent; it has not yet proved possible to study a sample of known concentration and purity using all the different probes of its behavior. This has made it difficult to reconcile the results of different probes and arrive at candidate phenomenological pictures such as that presented above, much less to arrive at a fundamental theory.
Another aspect is the existence of a large number of competing theories, each of which can claim success in explaining some aspect of the phase diagram shown in Figure 24. The proponents of each have been reluctant to abandon their approach, much less accept the possibility that another approach has been successful. Since none of these approaches can presently explain the complete candidate phase diagram discussed above, developing a microscopic theory that can achieve this goal continues to be a major challenge in condensed matter theory.
Still another challenge is finding new families of superconductors. Theory has not notably guided that quest in the past. However, the striking similarities in the families of novel unconventional superconductors thus far discovered suggest one strategy to pursue in searching for new families of unconventional (and possibly higher Tc) superconductors: Follow the antiferromagnetism, search for layered materials with high values of J, and pay attention to the role that magnetic order can play in maximizing Tc. In so doing, we may argue that we have learned enough to speculate that, just as there was a "phonon" ceiling of some 30 K for Tc in conventional superconductors, there may be a "magnetic" ceiling for Tc in unconventional superconductors. Both may be regarded as reflecting a tendency for strong quasiparticle interactions to produce localization rather than superconductivity. The question, then, is whether we have reached this ceiling with a Tc of about 160 K or whether new materials will yield higher transition temperatures using magnetic glues, and whether there are nonmagnetic electronic routes to achieving still higher values of Tc.