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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

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.

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:

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).

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).

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Ingredients of a theory

 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.

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.