New method to test the orbital symmetry of the order parameter in unconventional superconductors


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 dx2-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 dx2-y2-wave pairing and vanishing planar effective mass anisotropy. Moreover, when the dx2-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 dx2-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.

About the author

Richard A. Klemm was an undergraduate at Stanford University working in organic chemistry with his father, Prof. LeRoy H. Klemm, and with H. Fritz Schaefer III and Prof. Frank E. Harris on atomic hyperfine structure calculations, receiving his B. S. in Physics in 1969. At Synvar Research Institute in Palo Alto, CA, he synthesized the first highly layered superconductor, 2H-TaS2(pyridine)1/2.  He received his M. S. in 1972 and his PhD in physics at Harvard University in 1974 with Asst. Prof. Alan H. Luther. His thesis was titled Layered Superconductors.  He was a postdoctoral fellow at Stanford University with Prof. Sebastian Doniach, and in 1976, an Asst. Professor in Physics at Iowa State University and a scientist at the Ames Laboratory.  After receiving tenure in 1980, he went to Exxon Research and Engineering Co. in Annadale, NJ, where he worked with Dr. J. Robert Schrieffer on the theory of charge-density wave conduction in quasi-one-dimensional metals, and he grew the crystals of o-TaS3 and NbSe3 for the experimental studies. While visiting Prof. M. Brian Maple at the University of California at San Diego, the new Exxon director fired half of the scientists. Dr. Klemm then went to UCSD, the Ames Laboratory, Oak Ridge National Laboratory working with Dr. Samuel H. Liu, and then Argonne National Laboratory.  In 1992, Argonne promoted him to Scientist in the group headed by Dr. Alexei A. Abrikosov.  After he and Prof. Kurt Scharnberg of Hamburg University published a theory of c-axis twist Josephson junctions of high-temperature superconductors, Dr. Klemm persuaded Dr. Qiang Li of Brookhaven National Laboratory to do the experiment with Bi2Sr2CaCu2O8+d, and they published the results in PRL showing strong evidence of s-wave superconductivity in 1999.  Therefore, Argonne fired him in 2000.  So, he went to the MPI-PKS Dresden, Germany, where he and Dr. Qiang Gu published a paper on cold atoms.  In 2002, he went to the University of North Dakota and in 2003 to Kansas State University, mostly unpaid. In 2007, he became a Research Professor at the University of Central Florida in Orlando.  After publishing Layered Superconductors Volume I (Oxford University Press, 2012), he became a tenured Associate Professor in 2015 and was promoted to Professor in 2017.  The 1999 PRL c-axis twist experiment was confirmed in 2022 by Tsinghua University, Beijing, but was disputed by Harvard and Pohang Universities.  Dr. Aiying Zhao’s second paper provides another test of this important topic.

About the author

Qiang Gu received his bachelor’s degree in 1993 from Lanzhou University and his PhD in 1999 from the Institute of Physics, Chinese Academy of Sciences. After postdoctoral research appointments at Tsinghua University, Max Planck Institute for the Physics of Complex Systems, and Hamburg University, he joined the University of Science and Technology Beijing (USTB) as a professor in 2005. He has served as the director of the Institute of Theoretical Physics at USTB since 2021.

He is especially interested in fundamental problems of quantum physics. His current research interests center on novel quantum phenomena in correlated electrons and quantum gases, including quantum magnetism, unconventional superconductivity and superfluidity.

About the author

Aiying Zhao received her bachelor’s degree in 2013 from Shangqiu Normal University and her Ph.D. in January 2023 from the Institute of Theoretical Physics of the University of Science and Technology Beijing (USTB). The title of her thesis is “Some consequences of special relativity in condensed matter physics”. She was a visiting scholar for two years at the University of Central Florida in Orlando, FL (UCF), supported by the China Scholarship Council (CSC) and worked with Prof. Richard A. Klemm during her doctoral studies. She mainly engages in theoretical research related to the upper critical field of superconductors and the novel quantum phenomena based on the anisotropic Dirac equation. She has published “The Zeeman, spin-orbit, and quantum spin Hall interactions in anisotropic and low-dimensional conductors”, “Angular dependence of the upper critical induction of clean s- and dx2−y2-wave superconductors with self-consistent ellipsoidal effective mass and Zeeman anisotropies” and “Type-II quantum spin Hall effect in two-dimensional metals” in refereed journals. Her research started with the Dirac equation in an orthorhombically anisotropic system and she found that the Zeeman interaction depends strongly upon the effective mass anisotropy, vanishing for one-dimensional propagation, and for two-dimensional conduction, only the normal component of its spin interacts with external electric and magnetic fields. Then she studied self-consistently the upper critical field of 𝑠 -wave and 𝑑𝑥2−𝑦2 -wave superconductors with self-consistent orthorhombically anisotropic electronic effective masses and Zeeman energy, based on the Schrödinger equation obtained from the anisotropic Dirac equation, and in the third paper, she proposed a new quantum spin Hall effect (Type- II QSH) generating a quantized angular spin current in a two-dimensional metallic Corbino disk, which is caused by a radial electric field and an azimuthal magnetic vector potential. This is very different from the conventional quantum spin Hall effect (Type-I QSH) present in either a topological insulator or a quantum well channel.


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.

Go To Journal of Physics: Condensed Matter

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