We investigate in detail antiferromagnetic (AF) and superconducting (SC)phases as well as their coexistence in the two-dimensional Kondo lattice modelon a square lattice, which is a paradigmatic model for heavy fermion materials.The results presented are mainly obtained using the variational clusterapproximation (VCA) and are complemented by analytical findings for theequations of motion of pairing susceptibilities. A particularly interestingaspect is the possibility to have s-wave SC near half filling as reported byBodensiek \textit{et al.} [Phys. Rev. Lett. \textbf{110}, 146406 (2013)]. Whendoping the system, we identify three regions which correspond to an AF metallicphase with small Fermi surface at weak coupling, an AF metal with a differentFermi surface topology at intermediate coupling, and a paramagnetic metal witha large Fermi surface at strong coupling. The transition between these two AFphases is found to be discontinuous at lower fillings, but turns to acontinuous one when approaching half-filling. In the quest for s-wavesuperconductivity, only solutions are found which possess mean-field character.No true superconducting solutions caused by correlation effects are found inthe s-wave channel. In contrast, we clearly identify robust d-wave pairing awayfrom half-filling. However, we show that only by treating antiferromagnetismand superconductivity on equal footing artificial superconducting solutions athalf-filling can be avoided. Our VCA findings support scenarios previouslyidentified by variational Monte Carlo approaches and are a starting point forfuture investigations with VCA and further approaches such as cluster-embeddingmethods.
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