New Study Suggests Charge Orderings in Superconducting Cuprates Share a Common Origin

Considered the biggest open problem in condensed matter physics, high-temperature superconductivity has posed a formidable challenge to scientists for over 35 years.

Eduardo Fradkin
Eduardo Fradkin
Peter Abbamonte
Peter Abbamonte
Greg MacDougall
Greg MacDougall
Edwin Huang
Edwin Huang
Sangjun Lee
Sangjun Lee
Thomas Johnson
Thomas Johnson

Daniel Inafuku
for Illinois Physics

Considered the biggest open problem in condensed matter physics, high-temperature superconductivity has posed a formidable challenge to scientists for over 35 years. Since the discovery of high-temperature superconductors in 1986 by Georg Bednorz and K. Alex Müller, physicists have faced the difficulty of developing an adequate theory that explains the host of different ordered phases of matter displayed by these materials and their mutual relationships.

Now, a collaborative team of condensed matter experimentalists and theorists at the University of Illinois Urbana-Champaign have begun to untangle a web of different effects found in a class of high-temperature superconductors known as the cuprates. The researchers discovered that differences in patterns of charge density among different cuprates may be manifestations of the same general behavior—behavior that is possibly universal across the cuprate family. These findings were published on April 4, 2022, in the Proceedings of the National Academy of Sciences.

Charge density waves behave differently in different materials—or do they?

The cuprates are a class of compounds made up of planes containing both copper and oxygen, as well as two (or more) elements between the copper-oxide planes. These materials have the highest transition temperatures among the high-temperature superconductors. Recent attention has focused on the behavior of the cuprates’ charge densities, which form ordered spatial patterns called charge density waves (CDWs). Since their discovery in lanthanum-based cuprates in the mid-1990s, the roles that CDWs play in cuprate superconductivity have been the subject of much debate and generated additional interest when about a decade later, CDWs were also found in other, non-lanthanum-based cuprates.

Today, CDWs are widely recognized as fundamental features of cuprates and respond to changes in temperature and doping—the addition of chemical elements that change the number of mobile charge carriers, which enables and enhances superconductivity. Studying CDWs, however, is far from straightforward, because they can interact with other effects present in a material, such as spin density waves (SDWs), leading to unexpected material-specific CDW behaviors.

This complex maze of interacting effects prompted condensed matter physicists, including Illinois Physics Professor Eduardo Fradkin and Stanford University Physics Professor Steven Kivelson, co-principal investigators of the current study, to describe the orderings of different effects as “intertwined.”

Fradkin explains, “These cuprates are strongly interacting materials, and as you change the amount of mobile charge—usually through doping—you end up in a regime where you observe superconducting states having the highest known transition temperatures, typically in the range of 40K to 150 K.

“As physicists have learned more about the cuprates, it has become apparent that there are significant differences between different cuprate families. Among lanthanum-based cuprates, for example, the CDW wave vector—the reciprocal of the CDW’s period—appears to first increase then flatten out as the degree of doping increases. By contrast, the wave vector decreases for yttrium- and bismuth-based cuprates. This apparent discrepancy seems to suggest that the origin of this phenomenon may be specific to each material and hence not the reflection of a general physical origin.”

A central task to understand the cuprates, therefore, is to separate out CDW-interacting effects from each other. Specifically, the authors wanted to learn if CDWs among different cuprate families share similar physics.

Co-principal investigator and Illinois Physics Research Professor Gregory MacDougall is an expert in growing crystals for condensed matter experimentation. He notes, “There are dozens of cuprate families, and they each have slightly different properties. One challenge is to sift through these properties to possibly find a generic one that contributes to superconductivity. Our goal was to determine whether these properties are family-specific or something more general.”

As a starting point, the team looked at the interaction between CDWs and spin density waves (SDWs). SDWs, which encode magnetic ordering, are prominent features of lanthanum-based cuprates, whereas SDWs are absent in yttrium-based and bismuth-based cuprates, suggesting the possibility that a CDW-SDW interaction is the source of the lanthanum-based cuprates’ unusual doping dependences.

“The source of the different cuprate behaviors has been a big mystery in condensed matter physics for years. We showed that the charge order, which we thought was unique to each cuprate family, was actually just different incidences of a larger, universal behavior.”

Gregory MacDougall
Research Professor, Illinois Physics

Pictured above: The four-mirror optical image furnace in the MacDougall lab at the Frederick Seitz Materials Research Laboratory in Urbana, used to grow the LESCO crystals for this study. Illinois Physics graduate student Thomas Johnson used the traveling solvent float zone method in an oxygen overpressure environment. The furnace is manufactured by Crystal Systems.

Scattering experiments reveal some surprises

To study the CDW-SDW interaction in detail, the researchers conducted resonant soft X-ray scattering (RSXS) and neutron scattering experiments to characterize CDWs and SDWs in the lanthanum-based cuprate La1.8-xEu0.2SrxCuO4 (LESCO) and developed the theory needed to interpret the experiments. Notably, the researchers studied LESCO crystals enriched with the isotope 153Eu, enabling neutron scattering studies of the SDW in this system for the first time.

In true “Urbana style,” this work was a collaborative effort: MacDougall and his group grew the unique samples and carried out the neutron scattering experiments; Illinois Physics Professor Peter Abbamonte and his group performed the RSXS studies of the CDW; and Fradkin and Kivelson, in collaboration with Illinois Physics postdoctoral scholar Edwin Huang, developed theory that accounted for the temperature and doping dependences of the CDW and SDW orders.

Lanthanum-based cuprates often possess a structural transition found near the onset temperature of CDWs as doping is adjusted.

This transition can interfere with the CDW and obscure its measurement. LESCO is unique in that its structural transition temperature is well separated from the CDW onset temperature, making LESCO a tractable yet typical representative of lanthanum-based cuprates.

Co-lead author and Illinois Physics graduate student Thomas Johnson, who grew the LESCO samples and performed the neutron scattering experiments, explains, “One advantage of LESCO is that the structural transition—from orthorhombic to tetragonal—is nearly doping independent. This feature makes it easier to identify which behaviors are attributable to which effects.”

Neutron scattering experiments, which probe magnetic ordering, showed that the CDW and SDW orders in LESCO strongly couple to each other at low temperatures. In particular, the SDW wave vector was found to be exactly half the value of the CDW wave vector. This relationship allowed the researchers to infer that the CDW wave vector first increased with doping before flattening out, as expected from a lanthanum-based cuprate. In addition, detectable SDWs died out at higher temperatures.

The CDW is an ordered state of electrons, which, like all waves, is described by an amplitude and a phase. These materials contain disorder, which tends to make the phase fluctuate from one location in the material to another while leaving the amplitude essentially unaffected. Consequently, there is no true long-range CDW order. The RSXS experiments showed that the CDW amplitude (not to be confused with the CDW wave vector) does not vary much with temperature. In fact, at all temperatures up to the highest recorded temperature of 270 Kelvin, the amplitude appeared to be nearly constant.

But the results also contained some surprises.

Remarkably, the CDW wave vector exhibited a non-monotonic dependence on temperature. At sufficiently high dopings, the CDW wave vector developed a V-shaped “kink” having a sharp minimum at a characteristic temperature, a new feature not seen before in lanthanum-based cuprates. The kink lay near the temperature corresponding to the emergence of spin order, as shown by the neutron scattering experiments, suggesting the presence of CDW-SDW interactions.

Untangling interacting effects

To explain the possible causes of these experimental surprises, the authors developed a theoretical model that incorporates an additional effect: because the samples are actually metals, electron interactions are screened, or damped, inside the material over long distances; when such screening occurs, charge carriers in the material form a compressible fluid. The compressibility of the charged fluid is the key to understanding why the CDW wave vector changes with temperature and doping.

By including compressibility as an adjustable parameter in their model—together with parameters encoding both charge and spin—the researchers obtained fits in good agreement with the RSXS data: the theory predicted kinks in the CDW wave vectors just as the experiments showed.

Critically, the model also enabled the researchers to simulate what would happen at temperatures beyond those used in the experiments—in particular, beyond temperatures where they suspected spin order is relevant. In other words, since detectable SDWs disappeared at higher temperatures, the researchers were able to numerically observe what they believe to be the intrinsic, “actual” CDW behavior in the absence of the suspected CDW-SDW interactive influence.

Charge density wave graph
Plot of the charge density wave (CDW) wave vector versus the degree of doping. The lower curve (solid blue line) depicts the low-temperature behavior of LESCO and is typical for lanthanum-based cuprates. The upper curve (solid red line) depicts the extrapolation of experimental data for LESCO to high temperatures, showing a decreasing trend similar to yttrium- and bismuth-based cuprates such as YBCO. Figure originally published in Lee et al., PNAS 119 (15) e2119429119.

Huang, a co-lead author and the lead theorist of the study, explains, “According to our model, we found that if one extrapolates the CDW wave vector from our experimental data to much higher temperatures, up through 400 K, then the wave vector follows a doping dependence that’s different from the one seen at lower temperatures.”

And that doping dependence wasn’t arbitrary: it obeys the same doping dependence seen in yttrium- and bismuth-based cuprates, where the wave vector decreases with increasing doping. This result indicates that lanthanum-based cuprates are more similar to non-lanthanum-based ones than previously thought, pointing to a universal mechanism shared by all the cuprates.

Huang continues, “We believe that at high temperatures, the CDW and the SDW are not interacting much. There’s some unknown mechanism that prefers this decreasing wave vector behavior in the yttrium-based cuprates and LESCO.

“Using the extrapolated results from our physically motivated model, we’re seeing behavior in LESCO, a lanthanum-based cuprate, that is quantitatively consistent with the behaviors in other, non-lanthanum-based cuprates. Our extrapolation showing an yttrium-like trend in LESCO was a real surprise.”

Fradkin elaborates, “The physics of this problem is that the CDW forms at high temperatures, whereas the SDW forms only at much lower temperatures. The change in the CDW wave vector is explained by separate effects. First, these materials studied are actually metallic, which leads to screening effects on charge fluctuations. This screening—this suppression of interaction among charges at long distances—is the origin of the temperature and doping dependences of the CDW wave vectors.

“Second, at lower temperatures where the SDW begins to form, a strong interaction arises between the nascent SDW and the already well-formed CDW. Because of this interaction, these modulated states tend to become commensurate with each other. In other words, below some temperature, the period of the modulations of the CDW and SDW wave vectors become locked to each other by a factor of 2. The process by which the CDW and SDW become locked together occurs within a very narrow temperature range. The interaction between the CDW and the SDW is the origin of the kink observed in the RSXS experiments.”

What’s next for the cuprates?

Having characterized the CDW-SDW interaction, the researchers have opened the door to a host of new experimental and modeling opportunities aimed at understanding the cuprates.

MacDougall sums up, “The source of the different cuprate behaviors has been a big mystery in condensed matter physics for years. We showed that the charge order, which we thought was unique to each cuprate family, was actually just different incidences of a larger, universal behavior.”

The researchers also acknowledge that their work says nothing yet about the exact microscopic physics of the cuprates’ CDWs.

Huang notes, “Regarding our model, we haven’t made a microscopic prediction for the value of the compressibility parameter, and it allows us to characterize only the behavior that we’re seeing.

“While we can’t say exactly what’s happening at high temperature from our data alone, our theoretical findings are still strong evidence for a universal mechanism of intertwined orders underlying the physics of the cuprates.”

Clearly, a next step is to study the root cause of intertwined orders in the cuprates and other strongly correlated systems.

Fradkin says, “Another open problem is to explain the microscopic physical mechanism behind the specific way that the charge and spin orders interact. Understanding this interaction is the key to understanding the origin of intertwined orders seen in these and many other materials.”

This work was funded by the U.S. Department of Energy, Office of Basic Energy Sciences under Grant Nos. DE-FG02-06ER4628, DE-SC0012368, and DEAC02-76SF00515; by the National Science Foundation under Grant No. DMR-1725401; and by the Gordon and Betty Moore Foundation’s EPiQS Initiative under Grant Nos. GBMF9452, GBMF4305, and GMBF8691. The findings presented are those of the researchers, and not necessarily those of the funding agencies.

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This story was published May 15, 2022.