Since their dramatic debut in 1986, cuprate superconductors have been some of the most well-studied materials in existence. Yet many mysteries about the materials have persisted, including perhaps the key question: What mechanism forces electrons to overcome their repulsion and pair up?
In conventional superconductors, Bardeen-Cooper-Schrieffer (BCS) theory describes how phonon vibrations bring electrons together into Cooper pairs. The material properties of these superconductors often obey the “Matthias rules”: no magnetism, no oxides, no insulators. Apart from sulfur hydrides, no BCS superconductors exceed temperatures of 40 K. None of this has prevented doped copper oxides, the main compounds of which are insulating antiferromagnets, from remaining superconductors at temperatures up to 135 K. As a further evidence against a BCS pairing mechanism, cuprate superconductors are mostly insensitive to changes in phonon frequency.
Cuprate superconductors vary in their chemical formulas, but they all contain the same essential building block: planes with a copper atom between two oxygen atoms. Hypotheses abound on the mechanism of the superconductivity of cuprates. Some theorists have suggested spin fluctuations; others believe that phonons are the answer. Less than a year after the discovery of cupric superconductivity by Georg Bednorz and Alex Müller, Philip Anderson proposed that the glue that holds electrons together comes from superexchange, in which the spins of copper atoms are coupled, creating a magnetic attraction between its electrons despite the non-magnetic oxygen atom. in between
Recently, several studies have begun to connect the key factors behind a potential superexchange mating mechanism. An important factor is the charge transfer gap (CTG), the energy required (usually a few eV) for an oxygen atom to take an electron from a copper atom. The larger the gap between the d-orbital of copper and the p-orbital of oxygen, the less likely it is that the oxygen will take an electron from the copper. Last year, theorists at the University of Sherbrooke in Quebec calculated the rate at which electron pairing varies with the CTG.
This prediction provided a key target for a team led by JC Séamus Davis, which has laboratories at the University of Oxford, University College Cork in Ireland and Cornell University. In a recent study in the Proceedings of the National Academy of Sciences (PNAS), Davis and colleagues report evidence that matches the predictions of the Canadian theorists, suggesting that the mechanism behind cuprate superconductivity is mediated superexchange by CTG.
Recovery of models
Although the BCS theory can be solved analytically (John Schrieffer solved the key equation for Cooper pairs in the subway), the theory behind high Tc superconductors is more complex. To simplify the picture, researchers have often resorted to a one-band Hubbard model in which the cuprate is approximated as a square lattice of spins. Anderson was able to use the model to show how superexchange might work; others have even used it to predict where cuprate phase transitions occur. But the Hubbard model on the one hand does not consider multiple electron orbitals between copper and oxygen because it essentially breaks oxygen and copper into one effective molecule.
As early as 1989, Vic Emery of Brookhaven National Laboratory introduced a more realistic three-band Hubbard model to address these dynamics. At the same time, other theorists were beginning to point out the importance of oxygen. Jeff Tallon, an experimenter at the Victoria University of Wellington, New Zealand, proposed that there was a correlation between the oxygen hole content (the amount of electron holes present in an oxygen atom) and the Tc maximum
Extracting responses from three-band Hubbard models has been out of reach until recently. Since the early 1990s, new algorithms and exponential increases in computing power have allowed theorists to capture the dynamics of many more atoms and previously intractable problems about magnetic impurities. With these tools, the theorists at the University of Sherbrooke returned to the problem.
Theorists began by trying to understand two experimental findings: that a large CTG correlates with a low Tc and that a low oxygen hole content correlates with a low Tc. By solving the three-band Hubbard model for lattice, the Sherbrooke researchers demonstrated the connection between these results. They found that increasing the CTG reduces the oxygen hole content by compressing the oxygen p orbitals, leaving less room for the holes. A larger CTG also limits the strength of the superexchange interaction because it presents a barrier to coupling. Taken together, the authors concluded that the electron pairing mechanism is a superexchange, which in turn depends on the CTG and oxygen hole content.
Sherbrooke’s theory paper, published last year in PNAS, “is a real milestone in the long journey to understand cuprates,” says Tallon. The authors also suggested an elegant explanation for why the cuprates are special: Among all the transition metals, the strongest covalent bond exists between copper and oxygen. Strong covalent bonds lead to more superexchange than weak or ionic bonds.
Critically, the Sherbrooke theorists also identified a quantifiable target for future experiments: they predicted how much a given change in the CTG would affect the Cooper pair density. “From the experimentalist’s point of view, now you have traction,” says Davis. “If you can measure the degree of control freedom, and if you can measure the response, then you can do real physics.”
Laboratory work
To verify Sherbrooke’s prediction, Davis and his colleagues chose the cuprate Bi2Sr2CaCu2O8+x (BSCCO, pronounced “bisco”) because of its unique periodic property. The height of the oxygen atom above the copper atom in BSCCO varies by up to 12%, a huge difference that appears as wavy lines in the topographic image of the sample. According to the Sherbrooke theorists, increasing the oxygen height would decrease the CTG, and a smaller CTG would lead to a larger superexchange interaction, which can be measured by the local density of Cooper pairs.
Top graph: Oxygen atoms (red dots) vary in height above copper atoms (blue dots). Lower graphs: measurements reveal that changes in the height of the out-of-plane oxygen atoms (grey) lead to a decrease in the charge transfer gap (green) and an increase in the density of Cooper pairs (orange) . Credits: Wangping Ren and Shane O’Mahony
Davis and colleagues used two very different scanning tunneling microscopy (STM) approaches to measure BSCCO at 15% hole doping. To measure the pair of electrons, the tip of the probe must reach within picometers of the surface of the flat, scaly material, where the electric field is of the order of 109 V/m. (The Josephson STM technique that Davis used for the measurement took a decade to develop, he says.) To measure the CTG, the probe must be 5,000 times farther away, like using a record player with a pencil to the other side of a room. , says Davis. He and his team had to split the experiment into two parts and perform the measurements with different STM tips.
Matching changes in CTG with differences in Cooper pair density allowed the researchers to demonstrate a strong and convincing correlation, perhaps the clearest evidence yet for a mechanism underlying cuprate superconductivity.
The Davis group’s paper is “an impressive tour de force,” says Tallon. But this does not mean that one of the most important questions in condensed matter physics has been answered. “Is this the final experiment to identify the long-sought microscopic origins of cuprate superconductivity?” he asks “With all due respect to the authors, my opinion is… not yet.”
Inna Vishik, a condensed matter experimentalist at the University of California, Davis, agrees. “It’s a correlation that proposes a mechanism, but ultimately it motivates more experimental work in terms of evaluating this in other compounds,” says Vishik, who was not involved in the recent studies.
The relationship between the charge transfer gap (left) and the density of Cooper pairs (right), visualized in the undulating ripples of the BSCCO layers. Where the CTG is larger (light), the Cooper pair density is lower (dark); where the CTG is smaller (dark), the Cooper pair density is higher (light). Taken together, the two measurements represent a visible link suggesting CTG-mediated superexchange as a mechanism for electron pairing. Credits: Wangping Ren and Shane O’Mahony
Another recent study, published in Nature Communications, points to superexchange as the pairing mechanism for mercury-based cuprates. “We were looking at these two systems with a 30% difference in Tc,” says lead author Yuan Li of Peking University. “The question we wanted to answer is very simple: Is the magnetic energy scale also 30% different between these two?” They found that the difference in magnetic energy corresponded exactly to the difference in Tc, suggesting a magnetic basis such as superexchange for the mechanism.
One problem with any cuprate study is doping. Unlike dopants in semiconductors, whose amounts are known to within one part per million, oxygen is complicated and difficult to determine better than one part percent. Differences in doping can have large effects on the electronic structure, even pushing the compound into the pseudogap region, making it neither an antiferromagnet nor a superconductor. If even a fraction of the BSCCO crystals were to slip from superconductivity to the pseudogap phase, it would seriously compromise the authors’ conclusions. Davis argues that his sample was far from the pseudogap region, but acknowledges that the pseudogap remains mysterious.
Also, there are exceptions: some cuprate superconductors, such as La2−xSrxCuO4, have a large superexchange…