A new paper observing the topologically non-trivial drumhead state emerging from nodal-line semimetals offers new insights into how these newly discovered states behave in real materials. The work, led by Sarah Burke and colleagues and highlighted as an Editor’s Choice paper published in the journal Physical Review B, gives researchers a new roadmap and techniques for studying the interesting phenomena emerging from this topological state in new materials.

Semimetals, like metals, have a continuous spectrum of states crossing the dividing line between where the electrons are and where they could be if they had more energy, what physicists call the “Fermi energy.” In semimetals, however, the electronic states that mostly make up occupied and unoccupied states cross over each other, usually leaving few states near the Fermi energy unlike regular metals.

In some semimetals, including nodal-line semimetals like those observed in this study, these Fermi energy crossings lead to a linear relationship between energy and the momentum of the electrons, making them behave more like photons (light particles) than electrons and leading to exciting analogies with high-energy particle physics similar to the playground that graphene offers.

These linear crossings are protected by the topology of the bands⁠—a property of the whole electronic structure⁠—which can also give rise to new states spanning these special crossings. While Weyl and Dirac semimetals have point crossings, these can also be extended into lines tracing across “momentum space”⁠—a way of looking at how the electrons are arranged in the material. In nodal-line semimetals that form a closed loop in momentum space, these crossings become 2-dimensional. Because the topology of the electrons inside and outside the loop are distinct, a new boundary mode must “stretch” across the loop: the drumhead state.

Pictured: an illustration of the drumhead state in nodal-line semi-metals as compared to related topological semi-metal counterparts. Image credit: Brandon Stuart.

Drumhead surface states are rare, and have mostly been studied using theoretical models. While those models lay important groundwork, they often neglect aspects that are unavoidable in real materials. Building on that theoretical work, these experimental findings offer a new look at an exotic phenomenon occurring in a real, accessible material that could be useful in electronic applications or magnetic sensors.

But as esoteric as the physics might be, this story is primarily about people.

“But does it cleave?”

In the hunt for real materials exhibiting this new physics, the so-called “square-net” materials have emerged as important candidates due to their symmetry-enforced topology and Dirac band-crossings. In 2016, at a Max Planck-UBC-UTokyo Center for Quantum Materials workshop, Burke, Associate Professor in the Department of Physics and the Department of Chemistry and Leslie Schoop, who at the time was a postdoctoral researcher with the Max Planck Institute for Solid State Research in Stuttgart, met at the right time to work together on an ideal candidate to see the drumhead state: ZrSiTe, or zirconium silicon telluride.

As Schoop presented a talk laying out a compelling case for studying ZrSiTe, Burke wondered if they might be able to probe the drumhead state by looking at how electrons scatter in it with the scanning tunnelling microscopes in her lab, a technique known as quasiparticle interference (QPI).

“The first thing I asked her was, ‘Does it cleave?’” said Burke, who excitedly approached Schoop during a coffee break to learn more. “And ‘can I have some?’”

“Cleaving” means breaking the material into pieces; many materials do not cleave cleanly, leaving erratic surfaces that are hard to study using surface-sensitive techniques such as scanning tunneling microscopy (STM), a technique Burke uses to observe surfaces at the atomic level.

“This material cleaves beautifully, and so it’s very amenable to investigation via ARPES and STM,” said Burke. “This material was also looked at with ARPES before we looked at it with STM, so while another group first resolved this topologically nontrivial state, we were able to map the full occupied and unoccupied states, while demonstrating that QPI could be used to probe drumhead states.”

QPI had been used to look at other square-net materials where a more restricted drumhead state should appear, but had only seen other trivial surface states and not the drumhead, leaving questions about whether it could be probed by STM.

ARPES, or angle-resolved spectroscopy, is a technique used to detect the movement and energy of electrons in a material. ARPES can only detect up to the level of the Fermi energy, which is where electrons sit at the lower limit of the thermodynamic temperature scale (zero degrees kelvin); STM can look above the Fermi energy and at the atomic scale.

“One challenge with drumhead states is that theoretical models⁠—toy models⁠—are missing a lot of the complexity that real materials have,” said Burke. “One of the things that the material that we looked at has that the toy models don’t have is spin-orbit coupling, so the question was whether these drumhead states would survive the presence of spin-orbit coupling.”

Burke credits Lukas Muechler (Assistant Professor, Pennsylvania State University) and Raquel Quieroz (Assistant Professor, Columbia University) for their contributions to the underlying theory describing both the material and scattering observed by STM.

Spin-orbit coupling tends to open up gaps in the electronic states⁠—as it does in ZrSiTe⁠—and can change the topological landscape of the electrons, so physicists were concerned it might preclude the presence of such states.

“What’s interesting is that we were able to show that the drumhead states survived across the whole range of energies that it was predicted to exist for,” said Burke. “The reason is that the energy scale of this topological state is much larger than the spin-orbit coupling, and so, at least under these conditions, the state survives the perturbation.”

Confirming that the drumhead state survives the presence of at least the modest spin-orbit coupling present in ZrSiTe shows a robustness of the topology and opens up a wider range of materials in which physicists can try to study the drumhead state.

Max Planck-UBC-UTokyo collaboration is a catalyst

None of what the team uncovered would have come to light if not for the CQM collaboration and a fortuitous coffee break six years ago.

At the time, a number of collaborators on the paper⁠—including Schoop⁠—were based at the Max Planck Institute for Solid State Research in Stuttgart, Germany. Since then, the project has continued as the collaborators moved on; Schoop is now an Assistant Professor in the Department of Chemistry at Princeton University.

“This is really a testament to the value of these meetings, and the connections you can make in person,” said Burke. “And, a lot of life has happened since that first interaction: one author finished a PhD and got married, three authors moved and started faculty positions, one got tenure, and four babies were born⁠—and, of course, we worked through a pandemic⁠—but these relationships we form when we are able to meet each other face to face and get excited about each other’s work⁠—that’s really irreplaceable.”

The Max Planck-UBC-UTokyo CQM was formed in 2017 as an expansion of the original Max Planck-UBC Centre for Quantum Materials that was established in 2012. In 2021, all three organizations committed to the extension of the partnership into 2027. A testament to the strength of the partnership and the collaborations that have emerged from it, this is the first time an international Max Planck centre has been extended beyond a ten-year period.

“This work couldn’t have happened without the CQM and the opportunity to get together in the first place,” said Burke. “Even though most of those people are no longer where they started, it was the CQM partnership that made it happen.”