The physics of grain boundaries
in strongly correlated electron systems has developed into a fascinating and
active field of experimental research in recent years. Because they are in many
cases the single factor most important in limiting electrical current flow, and
will almost certainly be present in large numbers in any commercial product,
understanding what happens at grain boundaries is extremely important for device
applications. Large angle grain boundary Josephson junctions are the primary
components of RSFQ circuits with operating temperatures of 50-60K, and
high-quality SQUIDs operating at 77K, which are already employed for a host of
applications such as fetal heart monitoring or the inspection of devices in
semiconductor fabrication lines. Texturing techniques to align grain boundaries
in high-Tc wires have achieved millions of A/cm^{2} critical currents in prototype cables, and improvements along these lines will certainly be facilitated by careful modelling of the grain boundaries themselves. In addition, a whole set of theoretical questions about what happens in correlated systems at interfaces is largely unexplored.

One big mystery peristing over many years was the measured
exponential dependence of the critical current on the grain boundary angle.
This could not be explained simply by assuming a *d*-wave order parameter.
We felt a host of different questions had to be posed, including:
What are the changes in electronic structure at the interfaces, and are they in fact generically underdoped as recent experiments suggest? Are such interfaces intrinsically magnetic due to disrupted magnetic correlations in the bulk materials?

After a great deal of effort (largely by Siggi Graser!),
we succeeded in understanding the
exponential decay of the critical current with misorientation angle. This
required an "end to end" style calculation which involved performing molecular
dynamics simulations to reconstruct the structure of the interface, microscopic
analysis of the resulting hopping matrix elements and local potentials due to
interface inhomogeneity, mapping onto an effective Cu-only Hamiltonian, and
calculation of the critical current in such a system. Our results
were presented in Graser et
al., Nat. Phys. 6, 609 (2010) . See also comment Freericks,
"Grain Boundaries: guilty as charged."

Left: calculated critical current vs. grain boundary angle compared
to experiment by Mannhart group. The missing factor of 10 was
explained as a strong correlation effect (see also Wolf et al., Phys. Rev.
Lett. 108, 117002 (2012)). Right: cover of Nature Physics with graphic
representing charging of CuO_{4} squares.

Most superconducting tapes or wires consist of tiny oriented crystallites or grains connected by boundaries, which limit the maximum critical current which the wire can carry.

Top view of a (410) YBCO grain boundary calculated with molecular dynamics.

Calculated currents flowing at interface of (410) grain boundary. Note existence of local loops and backward flow.

Nanoscale superconductivity is of interest due to 1) possible applications of ultrasmall superconducting devices; 2) new quantum effects which arise when the system dimensions become comparable to the coherence length; and 3) recent STM experiments indicate that at least some high-temperature superconductors are inhomogeneous on a length scale of 30 Angstrom or so. It may be possible to think of this system as a collection of weakly coupled nanoscale grains, although this picture is far from clear. An isolated nanoscale grain has a poorly defined superconducting phase, and exhibits interesting effects depending on whether the number of electrons is even or odd. If macroscopic grains are weakly coupled, one may describe them in terms of Josephson coupled superconducting islands, but at the nanoscale there is no complete description currently available. The Hirschfeld group has confined itself largely to simulating systems with weakly random local superconducting pairing interactions, treating phase degrees of freedom at the mean field level. While inadequate in general, such a description provides a remarkably good account of correlations among various STM observables in disordered cuprate materials. See for example Nunner et al., Phys. Rev. Lett. 177003 (2005).

STM "Gap map" (1/2 energy difference between two conductance peaks) of two samples of BSCCO-2212.