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  Eskildsen Group
Superconductivity and Vortices
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Research scope
Among the materials with remarkable electronic and magnetic properties, none are more extraordinary than the superconductors. Apart from the perfect loss of electrical resistance, they also possess intriguing magnetic properties: When a type-II superconductor is placed in a magnetic field, it is threaded by swirling whirlpools of electric current known as vortices or flux-lines. The vortices are truly fascinating creatures which behave like massive entities, and provide a unique probe into the nature of the superconducting state in the host material.

Some of the most significant advances in superconductivity will be precisely in the materials which are most complex to understand and control. This is largely owing to the fact that in many such materials the superconducting order parameter has unconventional character.

Furthermore, vortices in superconductors are of crucial relevance to the use of superconductors, and are often the limiting factor in practical applications.

In the absence of disorder, the vortices arrange themselves in a periodic vortex lattice (VL). Our research is centered around studies of vortices in unusual and/or unconventional superconductors as well as novel kinds of vortex lattice dynamics. The main areas of research within our group are outlined below.

Vortex lattice metastability
Metastable phases of matter are well-known, with famous examples including supercooling and superheating of liquids and diamond which is one of the many allotropes of carbon. Metastability is almost exclusively observed in connection with first-order transitions, and is often found in frustrated systems where the energy difference between the states is small.
The structure of the vortex lattice (VL) is known to be highly sensitive to changes in external parameters such as temperature and magnetic field and can therefore be expected to display metastability, for example, in connection with first order transitions such as the VL melting or the reorientation transition of the rhombic VL found in most superconductors with a four-fold in-plane anisotropy.
We have recently discovered an unprecedented degree of VL metastability in MgB2 in connection with a second-order rotation transition. This allows us, for the first time, to perform structural studies of a well-ordered, non-equilibrium VL. Presently the mechanism responsible for the longevity of the metastable states is not resolved, but is speculated to be due to a jamming of VL domains, preventing a rotation to the ground state orientation.

References:
P. Das et al., Phys. Rev. Lett. 108, 167001 (2012).
C. Rastovski et al., Phys. Rev. Lett. 111, 107002 (2013).

Superconductors with triplet pairing
The superconducting state emerges due to the formation and condensation of carrier Cooper pairs, although the exact microscopic mechanism responsible for the pairing in different materials varies and, in many cases, remains elusive.
An important step towards a microscopic understanding is detailed knowledge of the superconducting order parameter. This is especially important in materials such as Sr2RuO4 and UPt3 where the carrier spins in the Cooper pairs are believed to form a triplet, introducing an additional degree of freedom and complexity to the problem. As a result the exact nature of the order parameter in these compounds, that have become paradigms for unconventional superconductivity, is still unresolved.
We are studying the vortex lattice in both Sr2RuO4 and UPt3 to provide further information about the superconducting order parameter. Our results imposes significant constraints on the possible order parameter in in these materials.

References:
W. J. Gannon et al., Phys. Rev. B 86, 104510 (2012).
C. Rastovski et al., Phys. Rev. Lett. 111, 087003 (2013).
W. J. Gannon et al., New J. Phys. 17, 023041 (2015).

Pauli paramagnetic effects
The interaction of superconductivity with local moment magnetism is a field which still has many open questions. A prominent example is the prediction of an inhomogenous superconducting state; the so-called Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase.
Despite vigorous efforts clear experimental proof for the existence of the FFLO phase is still missing. However, we have recently observed strong effects of Pauli paramagnetism on the vortices in both TmNi2B2C and CeCoIn5.
In both cases the effect of Pauli paramagnetism on the superconducting state is enhanced by a polarization of the local moments, which are coupled to the conduction electrons through an exchange intercation. As a consequence the unpaired electrons in the vortex cores are spin-polarized, adding a substantial periodic magnetization to the field modulation.

References:
L. DeBeer-Schmitt et al., Phys. Rev. Lett. 99, 167001 (2007).
A. D. Bianchi et al., Science 319, 177 (2008).
J. S. White et al., New J. Phys. 12, 023026 (2010).
P. Das et al., Phys. Rev. Lett. 108, 087002 (2012).
P. Das et al., Phys. Rev. B 86, 144501 (2012).
S. J. Kuhn et al., Phys. Rev. B 93, 104527 (2016).

Direct space field reconstruction from SANS
One often seeks to parameterize the field modulation due to the vortex lattice in terms of two characteristic length scales: the penetration depth (λ) and the coherence length (ξ). While such an approach provides a simplified method of analyzing experimental results, it also requires the implicit acceptance of a particular theoretical model.
Recently we have begun a more complete, model independent analysis of small-angle neutron scattering (SANS) measurements of the VL in member of the nickelborocarbide superconductors, extended significantly beyond the first-order Bragg reflection, which is customarily the only one measured. This allows for a real-space reconstruction of the VL magnetic field profile, which among other things will provide information about the in-plane anisotropy of these materials.

References:
J. M. Densmore et al., Phys. Rev. B 79, 174522 (2009).
P. Das et al., Phys. Rev. B 86, 144501 (2012).

Pnictide superconductors
The discovery of superconductivity in the pnictide and chalcogenide superconductors with elevated critical temperatures increasing, has sparked a strong interest in these materials. Furthermore, the possibility of growing large high-quality single crystals allow small-angle neutron scattering (SANS) studies of the VL in this material.
We have studied the FL in Ba(Fe,Co)2As2. At all fields and temperatures a "powder" ring is observed, indicating a highly disordered vortex configuration, and with a rocking curve extending beyond the measurable range. The magnitude of the VL scattering vector indicates either small, randomly oriented domains of rhombic symmetry at all fields or a vortex glass with short-range hexagonal order. An analysis of the radial width provides an estimate of the radial in-plane correlation length, which is found to be only a few vortex spacings at all fields, indicating that a single-vortex pinning regime.

References:
M. R. Eskildsen et al., Phys. Rev. B 79, 100501(R) (2009).
M. R. Eskildsen et al., Physica C 469, 529 (2009).
P. Das et al., Supercond. Sci. Technol. 23, 054007 (2010).
M. R. Eskildsen, E. M. Forgan, and H. Kawano-Furukawa, Rep. Prog. Phys. 74, 124504 (2011).
S. J. Kuhn et al., Phys. Rev. B 93, 104527 (2016).

Research Funding
The U. S. Department of Energy, Office of Science, Basic Energy Sciences grant no. DE-FG02-10ER46783 entitled "Vortex Lattice Studies in Type-II Superconductors".
The U. S. National Science Foundation, Emerging Frontiers & Multidisciplinary Activities Division grant no. EFMA-1433490 entitled "EFRI 2-DARE: Monolayer Heterostructures: Epitaxy to Beyond-CMOS Devices".

Previous support includes:
The Alfred P. Sloan Foundation.
The U. S. National Science Foundation, Directorate for Mathematical & Physical Sciences, Materials Research grant no. DMR-0804887 entitled "Vortices and the Interplay between Superconductivity and Magnetism".

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Eskildsen Group, Department of Physics
225 Nieuwland Science Hall, Notre Dame, Indiana 46556
Phone: 574-631-4010 Email: eskildsen@nd.edu
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