**Significance **

Superconductors are materials that carry electrical current with exactly zero electrical resistance. This means you can move electrons through it without losing any energy to heat. Superconductors are used to make incredibly strong magnets for magnetic levitation (maglev) trains where the train floats above the track using superconducting magnets; this eliminates friction and energy loss as heat, allowing the train to reach such high speeds. Another important application is for the magnetic resonance imaging (MRI) machines in hospitals. Indeed, the development of superconductors has improved the field of MRI as the superconducting magnet can be smaller and more efficient than an equivalent conventional magnet which allowed better clinical diagnosis of cancers. Different types of superconductors with different properties are suitable for different applications. Besides conventional superconductors, advances in technology have given rise to unconventional superconductors such as the heavy-fermion, iron-based, higher transition temperature and higher upper critical field superconductors. However, despite the extensive research progress, the physical origin and nature of the superconductivity of these emerging groups of materials are still poorly understood.

Most of these unconventional superconductors have strong anisotropic structures and upper critical fields, and they exhibit unique orientation dependencies. To this note, upper critical induction measurements in non-magnetic systems are one of the most useful tools for studying various superconductors. Being a thermodynamic measurement, it allows the extraction of anisotropy parameters, coherence lengths and pair-breaking mechanisms that are all vital in understanding superconductors. Moreover, the upper critical induction in unconventional superconductors also exhibits unique orientational dependencies.

While the orbital and spin-paramagnetic effects are the two main ways of inducing pair-breaking of singlet-pair superconductors, they also affect the upper critical induction of real materials and can be modified using the same effective mass anisotropy parameters. The occurrence can be affected by the Fermi surface anisotropy and the Pauli paramagnetic energy. Despite the important contributions of these factors, they are often ignored in most related studies. Additionally, there is no consensus on the orbital symmetry of the superconducting order parameters. Thus, there is a need to develop other experimental approaches for

Herein, Professor Richard Klemm from University of Central Florida together with Dr. Aiying Zhao and Professor Qiang Gu from the University of Science and Technology Beijing investigated the temperature and angular dependence of upper critical induction of clean s- and d_{x2-y2}-wave superconductors exhibiting Zeeman energy emanating from the ellipsoidal anisotropy of the Fermi surface with effective masses. In their approach, the Schrödinger-Dirac single particle Hamiltonian method generalized to an ellipsoidal effective mass anisotropy was employed to aid the self-consistent effects of the orbital and spin effective mass anisotropies upon the Pauli-limiting effects of the upper critical field. Their work is currently published in the peer-reviewed, *Journal of Physics: Condensed Matter.*

The research team showed that the upper critical induction was significantly larger in the lowest effective mass direction for anisotropic *s*-wave superconductors but proportional to the universal orientation factor. In contrast, the upper critical induction perpendicular to the highest effect mass direction exhibited a four-fold azimuthal pattern just below the transition temperature for d_{x2-y2}-wave pairing and vanishing planar effective mass anisotropy. Moreover, when the d_{x2-y2}-wave superconductor had a weak planar effective mass anisotropy; it exhibited a two-fold pattern. For both of these cases in a fixed field direction, the pattern rotated about the azimuthal direction as the temperature was lowered, providing new methods for distinguishing the s-wave and d-wave pairing symmetries in clean unconventional superconductors. Furthermore, the upper critical magnetic induction was calculated at arbitrary directions and temperatures T for both isotropic *s*-wave and anisotropic and d_{x2-y2}-wave superconducting order parameters with satisfactory accuracy.

In summary, the authors demonstrated that the temperature and angular dependence of the upper critical field in clean type-II superconductors can provide a new test of the orbital symmetry of the superconducting order parameter. Although the research work focused on s-wave and d-wave superconductors exhibiting effective anisotropic masses, it could be extended to an anisotropic pairing function. In a statement to *Advances in Engineering*, lead and corresponding author Professor Klemm explained that their study will advance our understanding of the upper critical induction in clean s-wave and unconventional d-wave superconductors.

**Reference**

Zhao, A., Gu, Q., & Klemm, R. A. (2022**). Angular dependence of the upper critical induction of** clean s- and -wave **superconductors with self-consistent ellipsoidal effective mass and Zeeman anisotropies**. Journal of Physics: Condensed Matter **34, **(35), 355601.