Section 6: New Superconductors
Figure 19: The first superconducting material was discovered in 1911 when mercury was cooled to 4 Kelvin (K). Seventy-five years later, thanks to the discovery of superconductivity in the family of cuprate materials by Bednorz and Mueller, scientists made a giant leap forward as they discovered many related materials that superconduct at temperatures well above 90 K.
The past three decades have seen an outpouring of serendipitous discoveries of new quantum phases of matter. The most spectacular was the 1986 discovery by IBM scientists J. Georg Bednorz and K. Alex Müller of superconductivity at high temperatures in an obscure corner of the periodic table: a family of ceramic materials of which LaxSr1-xCuO4 (containing the elements lanthanum, strontium, copper, and oxygen) was a first example. By the American Physical Society meeting in March 1987 (often referred to as the "Woodstock of physics"), it was becoming clear that this was just the first of a large new family of cuprate superconductors that possess two factors in common. They have planes of cupric oxide (CuO2) that can be doped with mobile electrons or holes. And the quasiparticles in the planes exhibit truly unusual behavior in their normal states while their superconducting behavior differs dramatically from that of the conventional superconductors in which phonons supply the pairing glue.
Over the past two decades, thanks to over 100,000 papers devoted to their study, we have begun to understand why the cuprate superconductors are so different. Moreover, it is now clear that they represent but one of an extended family of unconventional superconductors with three siblings: the heavy electron superconductors discovered in 1979; the organic superconducting materials discovered in 1981; and the iron-based superconductors discovered in 2006. Although there is a considerable range in their maximum values of Tc—about 160 K for a member of the cuprate family, HgBa2Ca2Cu3Ox, under pressure; roughly 56 K in the iron pnictides (LnFeAsO1-x); and 18.5 K for PuGaIn5, a member of the 115 (RMIn5) family of heavy electron materials—they show remarkable similarities in both their transport and magnetic properties in the normal and superconducting states.
In particular, for all four siblings:
- Superconductivity usually occurs on the border of antiferromagnetic order at which the magnetic moments of atoms align.
- The behavior of the quasiparticles, density fluctuations, and spin fluctuations in their normal state is anomalous, in that it is quite different from that of the Landau Fermi liquids found in the normal state of liquid 3He and conventional superconductors.
- The preferred superconducting pairing state is a singlet state formed by the condensation of pairs of quasiparticles of opposite spin in an orbital angular momentum, l, state, with l = 2; as a result, the superconducting order parameter and energy gap vary in configuration and momentum space.
Figure 20: Left: Conventional superconductors and right: heavy-electron superconductors.
Source: © Los Alamos National Laboratory. More info
In this section, we can explore only a small corner of this marvelous variety of materials whose unexpected emergent properties continue to surprise and offer considerable promise for commercial application. To understand these, we must go beyond the standard model in which phonons provide the glue that leads to attraction. We explore the very real possibility that the net effective attraction between quasiparticles responsible for their superconductivity occurs without phonons and is of purely magnetic origin. In so doing, we enter territory that is still being explored, and in which consensus does not always exist on the gateways to the emergent behavior we find there.
Heavy electron materials
We begin with the heavy electron materials for three reasons. First, and importantly, they can easily be made in remarkably pure form, so that in assessing an experiment on them, one is not bedeviled by “dirt” that can make it difficult to obtain reliable results from sample to sample. Second, the candidate organizing concepts introduced to explain their behavior provide valuable insight into the unexpected emergent behavior seen in the cuprates and other families of unconventional superconductors. Third, these materials display fascinating behavior in their own right.
Heavy electron materials contain a lattice, called the "Kondo lattice," of localized outer f-electrons of cerium or uranium atoms that act like local magnetic moments magnetically coupled through their spins to a background sea of conduction electrons. It is called a Kondo lattice because in isolation each individual magnetic moment would give rise to a dramatic effect, first identified by Jun Kondo, in which the magnetic coupling of the conduction electrons to the moment acts to screen out the magnetic field produced by it, while changing the character of their resistivity, specific heat, and spin susceptibility. When one has a lattice of such magnetic moments, the results are even more dramatic; at low temperatures, their coupling to the conduction electrons produces an exotic new form of quantum matter in which some of the background conduction electrons form a heavy electron Fermi liquid for which specific heat measurements show that the average effective mass can be as large as that of a muon, some 200 or more bare electron masses. The gateway to the emergence of this new state of matter, the heavy electron non-Landau Fermi liquid, is the collective entanglement (hybridization) of the local moments with the conduction electrons.
Remarkably, the growth of the heavy electron Fermi liquid, which we may call a "Kondo liquid (KL)" to reflect its origin in the collective Kondo lattice, displays scaling behavior, in that its emergent coherent behavior can be characterized by the temperature, T*, at which collective hybridization begins.
Below T*, the specific heat and spin susceptibility of the emergent Kondo liquid display a logarithmic dependence on temperature that reflects their coherent behavior and collective origin. Its emergent behavior can be described by a two-fluid model that has itself emerged only recently as a candidate standard phenomenological model for understanding Kondo lattice materials. In it, the strength of the emergent KL is measured by an order parameter, f (T/T*), while the loss in strength of the second component, the local moments (LMs) that are collectively creating the KL, is measured by 1-f(T/T*). KL scaling behavior, in which the scale for the temperature dependence of all physical quantities is set by T*, persists down to a temperature, T0 close to those at which the KL or LM components begin to become ordered.
Phase diagram of a work in progress
Figure 21: A candidate phase diagram for CeRhIn5 depicting the changes in its emergent behavior and ordering temperatures as a function of pressure.
Source: © David Pines. More info
The accompanying candidate phase diagram (Figure 21) for the heavy electron material CeRhIn5 (consisting of cerium, rhodium, and indium) gives a sense of the richness and complexity of the emergent phenomena encountered as a result of the magnetic couplings within and between the coexisting KL and LM components as the temperature and pressure are varied. It provides a snapshot of work in progress on these fascinating materials—work that will hopefully soon include developing a microscopic theory of emergent KL behavior.
Above T*, the local moments are found to be very weakly coupled to the conduction electrons. Below T*, as a result of an increasing collective entanglement of the local moments with the conduction electrons, a KL emerges from the latter that exhibits scaling behavior between T* and T0; as it grows, the LM component loses strength. T0 is the temperature below which the approach of antiferromagnetic or superconducting order influences the collective hybridization process and ends its scaling behavior.
At TN, the residual local moments begin to order antiferromagnetically, as do some, but not all, of the KL quasiparticles. The remaining KL quasiparticles become superconducting in a so-called dx2-y2 pairing state; as this state grows, the scale of antiferromagnetic order wanes, suggesting that superconductivity and antiferromagnetic order are competing to determine the low-temperature fate of the KL quasiparticles.
When the pressure, P, is greater than PAF, superconductivity wins the competition, making long-range antiferromagnetic LM order impossible. The dotted line continuing TN toward zero for P greater than PAF indicates that LM ordering is still possible if superconductivity is suppressed by application of a large enough external magnetic field. Experimentalists have not yet determined what the Kondo liquid is doing in this regime; one possibility is that it becomes a Landau Fermi liquid.
Starting from the high pressure side, PQC denotes the point in the pressure phase diagram at which local moments reappear and a localized (AF) state of the quasiparticles first becomes possible. It is called a "quantum critical point" because in the absence of superconductivity, one would have a T = 0 quantum phase transition in the Kondo liquid from itinerant quasiparticle behavior to localized AF behavior.
Since spatial order is the enemy of superconductivity, it should not be surprising to find that in the vicinity of PQC, the superconducting transition temperature reaches a maximum—a situation we will see replicated in the cuprates and one likely at work in all the unconventional superconducting materials. One explanation is that the disappearance of local moments at PQC is accompanied by a jump in the size of the conduction electron Fermi surface; conversely, as the pressure is reduced below PQC, a smaller Fermi surface means fewer electrons are capable of becoming superconducting, and both the superconducting transition temperature and the condensation energy, the overall gain in energy from becoming superconducting, are reduced.
In the vicinity of a quantum critical point, one expects to find fluctuations that can influence the behavior of quasiparticles for a considerable range of temperatures and pressures. Such quantum critical (QC) behavior provides yet another gateway for emergent behavior in this and other heavy electron materials. It reveals itself in transport measurements. For example, in CeRhIn5 at high pressures, one gets characteristic Landau Fermi liquid behavior (a resistivity varying as T2) at very low temperatures; but as the temperature increases, one finds a new state of matter, quantum critical matter, in which the resistivity in the normal state displays anomalous behavior brought about by the scattering of KL quasiparticles against the QC fluctuations.
What else happens when the pressure is less than PQC? We do not yet know whether, once superconductivity is suppressed, those KL quasiparticles that do not order antiferromagnetically exhibit Landau Fermi liquid behavior at low temperatures. But given their behavior for P less than Pcr, that seems a promising possibility. And their anomalous transport properties above TN and Tc suggest that as the temperature is lowered below T0, the heavy electron quasiparticles exhibit the anomalous transport behavior expected for quantum critical matter.
Superconductivity without phonons
We turn now to a promising candidate gateway for the unconventional superconducting behavior seen in this and other heavy electron materials—an enhanced magnetic interaction between quasiparticles brought about by their proximity to an antiferromagnetically ordered state. In so doing, we will continue to use BCS theory to describe the onset of superconductivity and the properties of the superconducting state. However, we will consider its generalization to superconducting states in which pairs of quasiparticles condense into states of higher relative angular momentum described by order parameters that vary in both configuration and momentum space.
Figure 22: The magnetic quasiparticle interaction between spins s and s' induced by their coupling to the spin fluctuations, , of the magnetic background material.
Source: © David Pines. More info
To see how the magnetic gateway operates, we turn to Figure 22 which illustrates how the magnetic interaction between two quasiparticles, whose spins are s and s', can be modified by their coupling to the spin fluctuations characteristic of the background magnetic behavior of the material in which they are located. Quite generally, spin, s, located, say, at the origin, acts to polarize the material by inducing a spin fluctuation; this induced magnetization, in turn, couples to the second spin, s', located a distance r away, producing an induced effective magnetic interaction which is analogous to the phonon-induced interaction responsible for superconductivity in ordinary BCS superconductors.
This induced effective magnetic interaction is highly sensitive to the magnetic properties of the background material. For an ordinary paramagnet exhibiting weakly magnetic behavior, the resulting magnetic interaction is quite weak and unremarkable. If, however, the background material is close to being antiferromagnetic, the spectrum of the spin fluctuations that provide the glue connecting the two spins becomes highly momentum dependent, exhibiting a significant peak for wave vectors that are close to those for which one finds a peak in the wave vector dependent magnetic susceptibility of the almost magnetically ordered material. As a result, the induced magnetic quasiparticle interaction will be strong and spatially varying.
Figure 23: Magnetic interaction potential in a lattice.
Source: © Reprinted by permission from Macmillan Publishers Ltd: Nature 450, 1177–1183 (18 December 2007). More info
Consider, for example, a magnetic material that is at a pressure near the point at which the material exhibits simple two-dimensional planar commensurate AF order (in which the nearest neighbor spins point in opposite directions). Its momentum dependent susceptibility will then have a peak at the commensurate wave vector Q =[/a, /a] where a is the lattice spacing, as will its spin fluctuation spectrum. The corresponding induced magnetic quasiparticle interaction in configuration space will then be repulsive at the origin, attractive at its nearest neighbor sites, repulsive at next nearest neighbor sites, etc., as shown in Figure 23. See the math
Such an interaction, with its mixture of repulsion and attraction, does not give rise to the net attraction required for superconductivity in the conventional BCS singlet s-wave pairing state with an order parameter and energy gap that do not vary in space. The interaction can, however, be remarkably effective in bringing about superconductivity in a pairing state that varies in momentum and configuration space in such a way as to take maximum advantage of the attraction while possessing nodes (zeros) that minimize the repulsion. A dx2-y2 pairing state, the singlet d-wave pairing state characterized by an order parameter and energy gap x2-y2 (k)= [cos (kxa)-cos (kya)], does just that, since it has nodes (zeroes) where the interaction is repulsive (at the origin or along the diagonals, for example) and is maximal where the interaction is maximally attractive (e.g., at the four nearest neighbor sites) as may also be seen in Figure 23.
We call such superconductors "gapless" because of the presence of these nodes in the gap function. Because it costs very little energy to excite quasiparticles whose position on the Fermi surface puts them at or near a node, it is the nodal quasiparticle excitations which play the lead role in determining the normal fluid density. Their presence is easily detected in experiments that measure it, such as the low-temperature specific heat and the temperature dependence of the London penetration depth. Nodal quasiparticle excitations are also easily detected in NMR measurements of the uniform susceptibility and spin-lattice relaxation rate, and the latter measurements have verified that the pairing state found in the "high Tc" heavy electron family of CeMIn5 materials, of which CeRhIn5 is a member, is indeed dx2-y2, the state expected from their proximity to antiferromagnetic order.
To summarize: In heavy electron materials, the coexistence of local moments and itinerant quasiparticles and their mutual interactions can lead to at least four distinct emergent states of matter: the Kondo heavy electron liquid, quantum critical matter, antiferromagnetic local moment order, and itinerant quasiparticle dx2-y2, superconductivity, while the maximum superconducting transition temperature is found close to the pressure at which one finds a QCP that reflects the onset of local moment behavior. In the next section, we will consider the extent to which comparable emergent behavior is observed in the cuprate superconductors.