Scientists from the US Department of Energy’s (DOE’s) Oak Ridge National Laboratory (ORNL) (CSA CSM) have identified an alternative route to superconducting for copper oxides, or cuprates, in the pseudogap phase, an in-between phase before superconductivity in which cuprates exhibit both insulating and conducting properties.
High-temperature superconductors are materials that can transport electricity with perfect efficiency at or near liquid nitrogen temperatures (-196°C). Researchers around the world have been working to develop a theory that explains the essential physics of high-temperature superconductors like cuprates, as a sound theory would not only explain why a material superconducts at high temperatures but also suggest other materials that could be created to superconduct at temperatures closer to room temperature.
Hyper-efficient electricity transmission could revolutionize power grids and electronic devices, enabling a wide range of new technologies. That future energy economy, however, is predicated on advancements in the understanding of how high-temperature superconductors work at the microscopic level.
The team at ORNL, led by Thomas Maier, used the Titan supercomputer at ORNL to simulate cuprates on the path to superconductivity. Titan, America’s fastest supercomputer for open science, is the flagship machine of the Oak Ridge Leadership Computing Facility (OLCF), a DOE Office of Science User Facility.
The conventional pathway to superconductivity has traditionally been considered blocked during the pseudogap phase examined in the experiment. Maier’s team, however, identified a possible alternative route mediated by the magnetic push-and-pull of cuprates’ electrons.
Simulating a 16-atom cluster, the team measured a strengthening fluctuation of electronic antiferromagnetism, a specific magnetic ordering in which the spins of neighboring electrons point in opposite directions (up and down) as the system cooled. The findings add context to scientific understanding of the pseudogap and how superconductivity emerges from the phase.
At extremely cold temperatures, electrons in certain materials do unexpected things. They pair up, overcoming a natural repulsion toward one another, and gain the ability to flow freely between atoms without resistance, like a school of fish in synchronized motion.
In conventional low-temperature superconductors such as mercury, aluminum and lead, the explanation for this phenomenon—called Cooper pairing—is well understood. In 1957, John Bardeen, Leon Cooper and John Robert Schrieffer proved that Cooper pairs arise from the interaction between electrons and a material’s vibrating crystal lattice (phonons). This theory, however, doesn’t seem to apply to cuprates and other high-temperature superconductors that are more complex in composition and electronic structure.
Cuprates consist of two-dimensional layers of copper and oxygen. The layers are stacked on top of each other with additional insulating elements in between. To set the stage for superconductivity, trace elements are substituted between the copper and oxygen layers to draw out electrons and create holes, impurities in the electrons’ magnetic ordering that act as charge carriers.
At sufficiently low temperatures, this hole doping process results in the emergence of a pseudogap, a transition marked by electronic stops and starts, like a traffic jam struggling to pick up speed.
“In a conventional superconductor, the probability of electrons forming Cooper pairs grows as the temperatures decreases,” Maier says. “In cuprates, the pseudogap’s insulating properties disrupt that mechanism. That begs the question, how can pairing arise?”
According to the team’s simulations, the antiferromagnetic fluctuations of electrons’ own spin is enough to form the glue.
“These spin fluctuations become much stronger as the material cools down,” Maier says. “The interaction is actually very similar to the lattice vibrations, or phonons, in conventional superconductors—except in high-temperature superconductors the normal state of electrons is not well-defined and the phonon interaction does not become stronger with cooling.”
Maier’s team approached the problem with an application called DCA++, calculating a cluster of atoms using a two-dimensional Hubbard model—a mathematical description of how electrons behave in solid materials. DCA++, which stands for “dynamical cluster approximation,” relies on a quantum Monte Carlo technique involving repeated random sampling to obtain its results.
“This model is very simple—it’s a very short equation—and yet it’s very hard to solve,” Maier says. “The problem is complex because it scales exponentially with the number of electrons in your system and you need a large number of electrons to describe thermodynamic transitions like superconductivity.”
With Titan, Maier’s team possessed the computing power necessary to solve the Hubbard model realistically and at low enough temperatures to observe pseudogap physics. The team gained access to Titan, a Cray XK7 with a peak performance of 27 petaflops (or 27 quadrillion calculations per second), through a 2015 Innovative and Novel Computational Impact on Theory and Experiment program allocation.
Designed by researchers at ORNL and ETH Zurich in Switzerland, DCA++ maximizes Titan’s hybrid architecture by making use of the GPUs on each of Titan’s 18,688 nodes. In past demonstrations on Titan, DCA++ has topped 15 petaflops. The DCA algorithm futhermore minimizes a common problem associated with calculating many-particle systems using the Monte Carlo method, the fermionic sign problem.
In physics, the quantum nature of electrons and other fermions is described by a wave function, which can switch from positive to negative, or vice versa, when two particles are interchanged. When the positive and negative values nearly cancel each other out, accurately calculating the many-particle states of electrons becomes tricky.
“The sign problem is affected by cluster size, temperature and the strength of the interactions between the electrons,” Maier says. “The problem increases exponentially, and there’s no computer big enough to solve it. What you can do to get around this is measure physical observables using many, many processors. That’s what Titan is good for.”
DCA++ works by measuring notable physical characteristics of the model as it walks randomly through the space of electronic configurations. Running on Titan, the code allows for larger clusters of atoms at lower temperatures, providing a more complete snapshot of the pseudogap phase than previously achieved.
Moving forward, Maier’s team is focused on simulating more complex and realistic cuprate systems to study the transition temperature at which they become superconducting, a point that can vary greatly within the copper-oxide family of materials. To take the next step, the team will need to use models with more degrees of freedom, or energy states, information that must be derived from first-principles calculations that take into account all the electrons and atoms in a system. “Once we get that, we can ask why the transition temperature is higher in one material and lower in another,” Maier says. “If you can answer that, you could do the same for any high-temperature superconductor or any material you want to simulate.”