Loop the loop, electron-style

HFML researchers have determined the shape of the low-energy electronic structure – or Fermi surface – of the Dirac nodal line semimetal ZrSiS with unprecedented accuracy. Due to the peculiar cage-like shape of the Fermi surface, the electron states are able to tunnel between adjacent disconnected parts of the Fermi surface, effectively causing the individual nodal lines to form a closed loop. These findings pave the way for future research that will focus on tuning the fascinating electronic properties in this class of materials.

Fermi surface – the electronic fingerprint of every metal

Electrons in a crystal occupy so-called energy bands up to a certain energy – the Fermi energy. These bands can be seen as the continuous analogue of the discrete energy states we know from atoms. A material with a non-fully filled band is defined as a metal and visualizing all bands at the Fermi energy results in an energy surface called the Fermi surface. For each metal this energy surface is unique and fully determines its electronic properties. Furthermore, the electron’s properties like mass and charge depend on the curvature of the energy band it belongs to. Due to that the state of an electron can either be electron-like or hole-like, with a hole being the positively charged counterpart of an electron. Metals with both electron- and hole-like states are often referred to as semimetals.

Loopy ZrSiS

ZrSiS belongs to a special class of semimetals called the topological semimetals. A key feature of many of these topological semimetals is that their energy bands have specific crossing points called Dirac points or ‘nodes’ which give rise to their topological character – topological in the sense that these point features in their energy band structure are robust against external stimuli, making them attractive materials for multiple quantum technologies. In certain rare cases, like in ZrSiS, these crossing points actually form lines – Dirac nodal lines – which almost connect to each other to form a cage-like ‘loop’ structure. The figure shows the predicted Fermi surface of ZrSiS (by calculation). The first thing to notice is the two-colour scheme – green for electron-like sections, blue for hole-like sections – characteristic of a semi-metal. The second thing to notice is the cage formed out of the individual and slightly chunky nodal lines. What is remarkable about ZrSiS is that its Fermi surface hosts only Dirac nodal lines, making it the ideal material system to study the unique properties of topological nodal line semimetals – at least in theory.  What we really need is a means to determine the shape of the Fermi surface as precisely as possible and to see if this cage-like structure is in fact realized.


Three-dimensional Fermi surface of ZrSiS. High-symmetry points in the center and the top surface of the Brillouin zone are indicated. All pockets observed in the dHvA oscillations are labeled by Greek letters. Electron and hole pockets are shown in green and blue, respectively

High fields to the rescue

Here the high magnetic fields come in. Weak magnetic fields cause electrons to deviate slightly from their original trajectory.  As the field strength increases, these deviations become bigger and as a result, the electrons are able to move over the contours of the Fermi surface. At large enough fields, however, the electrons are able to trace out orbits around sections of the Fermi surface. This in turn leads to an oscillatory signal in certain physical properties, like resistance or magnetization, the frequency of these oscillations being determined by the area of that particular section.

Magnetic cartography

The researchers measured the oscillatory signal in resistance and magnetization of ZrSiS crystals and used a setup that allows single crystals to be rotated in the applied magnetic field. In doing this, they could track the size and orientation of each orbit as the electrons follow the contours of the Fermi surface. The result was a map that captured the shape of the Fermi surface with unprecedented accuracy and that confirmed both the cage-like structure and the prediction that only Dirac nodal lines make up the Fermi surface of ZrSiS.

ZrSis magnetic cartography

Fast Fourier transform of the dHvA oscillations at T=0.34K

Trespassing forbidden regions

When the magnetic field was oriented along a certain crystallographic axis, the team noticed additional frequencies appearing in the oscillatory signal. These frequencies, indicated with red lines in the panel, are not due to some unknown orbits rather they are the product of electrons passing through forbidden regions between disconnected parts of the Fermi surface – the term ‘forbidden’ indicates the empty space in between the disconnected parts as there are no available states for the electrons. This effect is called magnetic breakdown and originates from the quantum mechanical principle of tunneling combined with the presence of a magnetic field. This essentially means that particles have a nonzero probability to pass through forbidden regions as long as the forbidden region is not too large. Due to its cage-like structure of the individual nodal lines, this condition was met for the hole- and electron-like sections labelled ‘a’ and ‘b’. The electrons living on these sections of the Fermi surface start tunneling to the other section as soon as the applied magnetic field reaches a certain value – the breakdown field value – and form so-called breakdown orbits that either enclose both sections up to multiple times or go around the cage in the plane of the ‘a’ and ‘b’ sections.

“They’re oscillations, Jim, but not as we know it…”

After having identified the shape of the Fermi surface of ZrSiS and being able to explain all observed frequencies, the team studied the orbits’ dependence on temperature. Normally, the oscillatory signal vanishes with increasing temperature as the oscillations are thermally suppressed. This typically happens just above liquid Helium temperatures of 4 Kelvin (roughly -269 degree Celsius). What surprised them, however, was that some of these oscillations were found to persist up to 100 Kelvin (roughly -173 degree Celsius) indicating that they had a different origin from the usual oscillations. To be able to explain this effect one, again, has to consider another property of quantum mechanics: electrons can behave as particles and as waves. These waves can then interfere with each other just lake ripples on water. The researchers then could explain this phenomenon by considering the interference between different breakdown orbits originating from the ‘a’ and ‘b’ sections. Furthermore, this interference effect can be linked to something called Stark interference which is known to occur in metals like magnesium but was never before observed in topological semimetals.

Correlating topology – towards new frontiers

One could think that everything is now known about ZrSiS and that one can move onto the next material. However, that’s not how our scientific curiosity works. The findings of this study barely mark the beginning of an even more intriguing scientific journey: engineering this material in a way to induce novel and interesting ‘correlation’ effects due to the (Coulomb) repulsion between the electrons. Combining this with the topological properties of this material could open up a completely unexplored field of physics. In order to plough this field, engineering techniques like electrical doping, chemical substitution, applied strain will be needed. PhD student Claudius Müller: “It was fascinating to unravel the properties of ZrSiS piece by piece. ZrSiS now belongs to the few topological semimetals whose low-energy electronic structure are fully known.  I am intrigued to see what other secrets lie hidden at the border where topology and electronic correlations meet.”

Related publication

Determination of the Fermi surface and field-induced quasiparticle tunneling around the Dirac nodal loop in ZrSiS, C.S.A. Müller, T. Khouri, M.R. van Delft, S. Pezzini, Y.-T. Hsu, J. Ayres, M. Breitkreiz, L.M. Schoop, A. Carrington, N.E. Hussey, and S. Wiedmann, Physical Review Research 2, 023217 (2020)

More information

Steffen Wiedmann
Nigel Hussey