In condensed matter physics, one of the greatest puzzles is to find the precise link between superconductivity and charge order in cuprate superconductors. In the case of superconductors, electrons move freely through the material—there is no resistance upon being cooled below its critical temperature.
However, the cuprates display charge order and superconductivity at the same time in patterns of alternating stripes. This is anomalous because charge order represents regions of confined electrons. How is it feasible for charge order and superconductivity to coexist?
Scientists from the University of Illinois at Urbana-Champaign have now collaborated with researchers at the SLAC National Accelerator Laboratory to throw light on how these distinct states can occur next to one another.
Illinois Physics post-doctoral researcher Matteo Mitrano, Professor Peter Abbamonte, and their colleagues used an innovative X-ray scattering method—time-resolved resonant soft X-ray scattering—which makes the most of the sophisticated equipment at SLAC.
This technique allowed the researchers to investigate the striped charge-order phase at an unparalleled energy resolution. For the first time, this has been achieved at an energy scale relevant to superconductivity.
The researchers evaluated the charge-order fluctuations in La2−xBaxCuO4 (LBCO), a prototypical copper-oxide superconductor. They discovered that the fluctuations had energy matched with the superconducting critical temperature of the material, suggesting that the material’s superconductivity—and, by projection, that of the cuprates—may be mediated by the fluctuations in the charge order.
Furthermore, the researchers showed that, if the charge order melts, the system’s electrons will reform the charge order’s striped areas within tens of picoseconds. It has been found that this process obeys a universal scaling law.
Mitrano and Abbamonte intended to perceive what they observed in their experiment and sought the help of Illinois Physics Professor Nigel Goldenfeld and his graduate student Minhui Zhu, who could apply theoretical techniques taken from soft condensed matter physics to explain the formation of the striped patterns. The study outcomes have been reported in the online journal Science Advances on August 16th, 2019.
Cuprates Have Stripes
The importance of this puzzle can be understood within the context of studies on high-temperature superconductors (HTS), particularly the cuprates—layered materials containing copper complexes. The cuprates are some of the first found HTS that have considerably higher critical temperatures when compared to “ordinary” superconductors (for example, lead and aluminum superconductors have a critical temperature below 10 K).
In the 1980s, it was found that the cuprate LBCO has a superconducting critical temperature of 35 K, or −396 °F, a discovery that won the Nobel Prize for Bednorz and Müller. The discovery led to a lot of research into the cuprates. Over the years, researchers have identified experimental evidence for inhomogeneities in LBCO and analogous materials: coexisting metallic and insulating phases.
In 1998, Illinois Physics Professor Eduardo Fradkin, Stanford Professor Steven Kivelson, and others hypothesized that Mott insulators—materials that must, by theory, conduct under traditional band theory but insulate because of repulsion between electrons—can host stripes of superconductivity and charge order.
An example of a Mott insulator is the parent compound of LBCO—La2CuO4. Upon adding Ba to that compound, some La atoms are replaced, thereby forming stripes owing to the spontaneous organization of holes—vacancies of electrons that function like positive charges.
Yet, there are other questions related to the behavior of the stripes. Are the charge order regions immobile? Do they fluctuate?
The conventional belief is that if you add these doped holes, they add a static phase which is bad for superconductivity—you freeze the holes, and the material cannot carry electricity. If they are dynamic—if they fluctuate—then there are ways in which the holes could aid high-temperature superconductivity.
Matteo Mitrano, Postdoctoral Researcher, Illinois Physics
Probing the Fluctuations in LBCO
In order to perceive the precise behavior of the stripes, Mitrano and Abbamonte devised an experiment to melt the charge order and look at the process of its reformation in LBCO. Mitrano and Abbamonte reconceptualized a measurement method known as resonant inelastic X-ray scattering, adding a time-dependent protocol to look at the way the charge order recovers over a duration of 40 ps.
The researchers shot a laser at the LBCO sample, lending additional energy to the electrons to melt the charge order and establish electronic homogeneity.
“We used a novel type of spectrometer developed for ultra-fast sources, because we are doing experiments in which our laser pulses are extremely short,” explained Mitrano. “We performed our measurements at the Linac Coherent Light Source at SLAC, a flagship in this field of investigation. Our measurements are two orders of magnitude more sensitive in energy than what can be done at any other conventional scattering facility.”
What is innovative here is using time-domain scattering to study collective excitations at the sub-meV energy scale. This technique was demonstrated previously for phonons. Here, we have shown the same approach can be applied to excitations in the valence band.
Peter Abbamonte, Professor, Illinois Physics
Hints of a Mechanism for Superconductivity
The first important outcome of this experiment is that the charge order fluctuates indeed, traveling with an energy that nearly matches the energy determined by LBCO’s critical temperature. This implies that Josephson coupling may be vital for superconductivity.
The concept behind the Josephson effect, found out by Brian Josephson in 1962, is that it is possible to connect two superconductors through a weak link, essentially an insulator or a normal metal. In a system of this type, there is a possibility that superconducting electrons will leak from the two superconductors into the weak link, producing within it a current of superconducting electrons.
Josephson coupling provides a possible explanation for the coupling between superconductivity and striped regions of charge order, wherein the stripes fluctuate such that superconductivity leaks into the areas of charge order, the weak links.
Obeying Universal Scaling Laws of Pattern Formation
Once the charge order was melted, the recovery of the stripes was evaluated by Mitrano and Abbamonte as they evolved in time. Upon approaching its full recovery, the charge order followed an unpredicted time dependence. This outcome was not similar to what the scientists had observed in the past. How can this possibly be explained?
The solution is taken from the area of soft condensed matter physics, and more particularly from a scaling law theory proposed by Goldenfeld 20 years before pattern formation in liquids and polymers was described. Goldenfeld and Zhu showed that the stripes in LBCO recover based on a dynamic, universal, self-similar scaling law.
According to Goldenfeld, “By the mid-1990s, scientists had an understanding of how uniform systems approach equilibrium, but how about stripe systems? I worked on this question about 20 years ago, looking at the patterns that emerge when a fluid is heated from below, such as the hexagonal spots of circulating, upwelling white flecks in hot miso soup.”
Goldenfeld continued, “Under some circumstances these systems form stripes of circulating fluid, not spots, analogous to the stripe patterns of electrons in the cuprate superconductors. And when the pattern is forming, it follows a universal scaling law. This is exactly what we see in LBCO as it reforms its stripes of charge order.”
Goldenfeld and Zhu used their calculations to explain the process of time-dependent pattern reformation in the experiment devised by Mitrano and Abbamonte. The stripes reform based on a logarithmic time dependence—a very gradual process. Moreover, compliance with the scaling law in LBCO suggests that it includes topological defects, or irregularities, in its lattice structure. This is the second important result from this experiment.
It was exciting to be a part of this collaborative research, working with solid-state physicists, but applying techniques from soft condensed matter to analyze a problem in a strongly correlated system, like high-temperature superconductivity.
Minhui Zhu, Graduate Student, Illinois Physics
Zhu added, “I not only contributed my calculations, but also picked up new knowledge from my colleagues with different backgrounds, and in this way gained new perspectives on physical problems, as well as new ways of scientific thinking.”
In future studies, Mitrano, Abbamonte, and Goldenfeld intend to further investigate the physics behind the fluctuations of charge order with the aim of fully melting LBCO’s charge order to look at the physics of stripe formation. They also intend to perform similar experiments with other cuprates, such as yttrium barium copper oxide compounds, also called YBCO.
Goldenfeld considers this and future experiments as having the potential to catalyze new research in HTS: “What we learned in the 20 years since Eduardo Fradkin and Steven Kivelson’s work on the periodic modulation of charge is that we should think about the HTS as electronic liquid crystals,” he said. “We’re now starting to apply the soft condensed matter physics of liquid crystals to HTS to understand why the superconducting phase exists in these materials.”
This study is funded by the U.S. Department of Energy. Support from the Alexander von Humboldt Foundation and the Gordon and Betty Moore Foundation is also acknowledged by M.M. and P.A., respectively.