The quantum Griffiths singularity (QGS) is a rare phenomenon in which, near a quantum phase transition, critical exponents diverge because of rare regions with anomalously slow dynamics. In two-dimensional superconductors, QGS has been observed for about a decade, while in three-dimensional systems it has been seen only in a handful of materials: MgTi2O4, MoTiN, and CaFe1−xNixAsF. Moreover, the behavior of these three systems differs: in MgTi2O4 and CaFe1−xNixAsF the critical exponent follows the activated law for 3D, whereas in MoTiN films it follows the 2D law, even though superconductivity there is bulk. The work on ZrNx stands apart: nearby in the archive are studies of magnetic vortices as qubits, thermal properties of superconductors, and the influence of vacuum on magnetism, but direct neighbors on the topic are almost absent.
ZrNx films about 200 nm thick were deposited by reactive magnetron sputtering on MgO(100) substrates at 740 K, with the nitrogen partial pressure varied from 5% to 20% to obtain compositions from x≈1.21 to 1.50. Structure was checked by X-ray diffraction, composition by energy-dispersive spectroscopy. Resistance was measured with a four-probe method in a cryostat with a 3He refrigerator down to 0.5 K in magnetic fields applied perpendicular and parallel to the film plane. For scaling analysis, the power-law dependence of resistance on field and temperature, accepted for two-dimensional systems, was used, even though the films are known to be three-dimensional in terms of superconductivity.
For films with x≳1.30, the magnetoresistance curves at different temperatures intersect not at a single point but over a broad field range, a signature of QGS. The effective critical exponent zν, extracted from power-law scaling, grows as temperature decreases and diverges as T→0. The dependence of zν on the critical field follows an activated law with exponent νψ≈0.7–0.8, close to the value 0.61 from simulations for 2D. Data near the transition collapse onto two branches in the coordinates of activated scaling for 2D. Meanwhile, the upper critical fields Bc2⊥ and Bc2∥ are anomalous: the parallel field is smaller than the perpendicular one, which is unusual for thin films.
This work is the first to demonstrate QGS with two-dimensional scaling in a three-dimensional superconductor with a cubic lattice, distinct from previously studied nitrides. Unlike neighboring works that investigated magnetic vortices, thermal conductivity, or the influence of vacuum, here the focus is on the quantum superconductor–insulator transition in a disordered system. The result supports the hypothesis that rare superconducting regions in a 3D material can behave as effectively two-dimensional because of a percolating network of superconducting sheets.
The topic may develop toward direct observation of the dynamics of rare superconducting regions using scanning tunneling microscopy or microwave spectroscopy. It would be interesting to check whether the two-dimensional scaling persists in other transition-metal nitrides with strong disorder, such as HfN or TiN. Theorists will need to explain why rare regions in a 3D medium acquire an effective dimension of 2.
The results will affect the understanding of quantum phase transitions in disordered superconductors and the development of superconducting devices where disorder is unavoidable. They are also important for generalizing QGS theory to three-dimensional systems.
The next step is to measure transport in ZrNx at even lower temperatures (millikelvin) and in tilted magnetic fields to separate the contribution of rare regions from bulk effects. Local spectroscopy to visualize superconducting islands would also be useful.
This work is directly connected to an unresolved problem in condensed matter physics: how disorder and rare regions govern quantum phase transitions in superconductors. It adds new data to the long-standing debate about the nature of the superconductor–insulator transition in three-dimensional systems and the role of effective dimensionality near the critical point.
This section is addressed to the author of the paper and their colleagues. Neighbouring works were found by meaning-based search across our archive, not by word overlap. We do not review: we show the field around the work.
What the work does
In this work, epitaxial ZrNx films about 200 nm thick were grown with controlled nitrogen content, and magnetoresistance was measured at ultralow temperatures in fields up to 9 T. It was found that the magnetoresistance curves at different temperatures intersect over a wide field range, and the effective critical exponent zν diverges as T→0, which is interpreted as a quantum Griffiths singularity with two-dimensional scaling in a three-dimensional superconductor.
Where its strength is
This work demonstrates a rare case of a quantum Griffiths singularity in a three-dimensional superconductor with a cubic lattice, where the scaling corresponds to a two-dimensional system. This expands the class of materials in which such behavior is observed and provides a new example for testing theoretical models.
Where it could go next
Worth trying
Why this may matter
Understanding quantum phase transitions in disordered superconductors is important for developing devices where superconductivity is suppressed by magnetic field or disorder, as well as for fundamental physics of quantum critical phenomena in three-dimensional systems.
Nearby works in our archive
The archive is sparse around this work: the closest review is only 0.52, against a usual 0.64. It looks like few have drilled nearby.
Zirconium nitride is one of the oldest known superconductors: its superconductivity was discovered back in 1952 at a temperature of 9.05 K. Since then, this material has been used in protective coatings and cutting tools, and now it turns out to be a testing ground for exotic quantum phenomena.