  The idea of employing non-Abelian statistics for error-free quantum computing
ignited interest in recent reports of topological surface superconductivity and
Majorana zero modes (MZMs) in FeTe$_{0.55}$Se$_{0.45}$. An associated puzzle is
that the topological features and superconducting properties are not observed
uniformly across the sample surface. Understanding and practical control of
these electronic inhomogeneities present a prominent challenge for potential
applications. Here, we combine neutron scattering, scanning angle-resolved
photoemission spectroscopy (ARPES), and microprobe composition and resistivity
measurements to characterize the electronic state of
Fe$_{1+y}$Te$_{1-x}$Se$_{x}$. We establish a phase diagram in which the
superconductivity is observed only at sufficiently low Fe concentration, in
association with distinct antiferromagnetic correlations, while the coexisting
topological surface state occurs only at sufficiently high Te concentration. We
find that FeTe$_{0.55}$Se$_{0.45}$ is located very close to both phase
boundaries, which explains the inhomogeneity of superconducting and topological
states. Our results demonstrate the compositional control required for use of
topological MZMs in practical applications.
