  Geometric frustration adds a new competing energy scale to the
antiferromagnetic (AFM) Kondo lattice model (KLM). In this work, we
systematically study the doniach phase diagram on the square and triangular
lattices in the same theoretical framework and reveal unexpected responses of
it on the two lattices. The potential energy created by the geometric
frustration is comparable to the Ruderman-Kittel-Kasuya-Yosida (RKKY) coupling,
which completely suppresses the long-range antiferromagnetic (AFM) order on the
half-filled triangular lattice. While, on the square lattice, the long-range
AFM order successfully establishes and constitutes the conventional competition
between the RKKY and Kondo couplings. The geometrical frustration on the
triangular lattice is partially released when doped with holes, in which two
different magnetic orders emerge unexpectedly. The two orders closely relate to
the topology of the interacting Fermi surface. Our comprehensive comparison of
the KLM on the two lattices not only reveals a significant competition of
geometric frustration, RKKY, and Kondo couplings on low-dimensional systems but
also sheds light on experimentally finding new phases in related materials.
