  The interplay between unconventional superconductivity and quantum critical
ferromagnetism in the U-Ge compounds represents an open problem in strongly
correlated electron systems. Sample quality can have a strong influence on both
of these ordered states in the compound UCoGe, as is true for most
unconventional superconductors. We report results of a new approach at UCoGe
crystal growth using a floating-zone method with potential for improvements of
sample quality and size as compared with traditional means such as Czochralski
growth. Single crystals of the ferromagnetic superconductor UCoGe were produced
using an ultra-high vacuum electron-beam floating-zone refining technique.
Annealed single crystals show well-defined signatures of bulk ferromagnetism
and superconductivity at $T_c \sim$2.6 K and $T_s \sim$0.55 K, respectively, in
the resistivity and heat capacity. Scanning electron microscopy of samples with
different surface treatments shows evidence of an off-stoichiometric uranium
rich phase of UCoGe collected in cracks and voids that might be limiting sample
quality.
