  We investigate in detail antiferromagnetic (AF) and superconducting (SC)
phases as well as their coexistence in the two-dimensional Kondo lattice model
on a square lattice, which is a paradigmatic model for heavy fermion materials.
The results presented are mainly obtained using the variational cluster
approximation (VCA) and are complemented by analytical findings for the
equations of motion of pairing susceptibilities. A particularly interesting
aspect is the possibility to have s-wave SC near half filling as reported by
Bodensiek \textit{et al.} [Phys. Rev. Lett. \textbf{110}, 146406 (2013)]. When
doping the system, we identify three regions which correspond to an AF metallic
phase with small Fermi surface at weak coupling, an AF metal with a different
Fermi surface topology at intermediate coupling, and a paramagnetic metal with
a large Fermi surface at strong coupling. The transition between these two AF
phases is found to be discontinuous at lower fillings, but turns to a
continuous one when approaching half-filling. In the quest for s-wave
superconductivity, only solutions are found which possess mean-field character.
No true superconducting solutions caused by correlation effects are found in
the s-wave channel. In contrast, we clearly identify robust d-wave pairing away
from half-filling. However, we show that only by treating antiferromagnetism
and superconductivity on equal footing artificial superconducting solutions at
half-filling can be avoided. Our VCA findings support scenarios previously
identified by variational Monte Carlo approaches and are a starting point for
future investigations with VCA and further approaches such as cluster-embedding
methods.
