  CaFe$_2$As$_2$ has been found to be exceptionally sensitive to the
application of hydrostatic pressure and superconductivity has been found to
exist in a narrow pressure region that appears to be at the interface between
two different phase transitions. The pressure - temperature ($P - T$) phase
diagram of CaFe$_2$As$_2$ reveals that this stoichiometric, highly ordered,
compound can be easily tuned to reveal all the salient features associated with
FeAs-based superconductivity without introducing any disorder. Whereas at
ambient pressure CaFe$_2$As$_2$ does not superconduct for $T > 1.8$ K and
manifests a first order structural phase transition near $T \approx 170$ K, the
application of $\sim 5$ kbar hydrostatic pressure fully suppresses the
resistive signature of the structural phase transition and instead
superconductivity is detected for $T < 12$ K. For $P \ge 5.5$ kbar a different
transition is detected, one associated with a clear reduction in resistivity
and for $P > 8.6$ kbar superconductivity is no longer detected. This higher
pressure transition temperature increases rapidly with increasing pressure,
exceeding 300 K by $P \sim 17$ kbar. The low temperature, superconducting dome
is centered around 5 kbar, extending down to 2.3 kbar and up to 8.6 kbar. This
superconducting phase appears to exist when the low pressure transition is
suppressed sufficiently, but before the high pressure transition has reduced
the resistivity, and possibly the associated fluctuations, too dramatically.
