Is There a Lower Size Limit for Superconductivity?

The ultimate lower size limit for superconducting order to exist is set by the "Anderson criterion"-arising from quantum confinement-that appears to be remarkably accurate and universal. We show that carefully grown, phase-pure, nanocrystalline bcc-Ta remains superconducting (with ordering temperature, T_C ≈ 0.9 K) down to sizes 40% below the conventional estimate of the Anderson limit of 4.0 nm. Further, both the T_C and the critical magnetic field exhibit an unusual, nonmonotonic size dependence, which we explain in terms of a complex interplay of quantum size effects, surface phonon softening, and lattice expansion. A quantitative estimation of T_C within first-principles density functional theory shows that even a moderate lattice expansion allows superconductivity in Ta to persist down to sizes much lower than the conventional Anderson limit, which can be traced to anomalous softening of a phonon due to its coupling with electrons. This appears to indicate the possibility of bypassing the Anderson criterion by suitable crystal engineering and obtaining superconductivity at arbitrarily small sizes, an obviously exciting prospect for futuristic quantum technologies. We take a critical look at how the lattice expansion modifies the Anderson limit, an issue of fundamental interest to the study of nanoscale superconductivity.