  The competition between the indirect exchange interaction (IEC) of magnetic
impurities in metals and the Kondo effect gives rise to a rich quantum phase
diagram, the Doniach Diagram. In disordered metals, both the Kondo temperature
and the IEC are widely distributed due to the scattering of the conduction
electrons from the impurity potential. Therefore, it is a question of
fundamental importance, how this Doniach diagram is modified by the disorder,
and if one can still identify separate phases. Recently, it has been
investigated the effect of Ruderman-Kittel-Kasuya-Yosida (RKKY) correlations on
the Kondo effect of two magnetic impurities, renormalizing the Kondo
interaction based on the Bethe-Salpeter equation and performing the poor men's
renormalization group (RG) analysis with the RKKY-renormalized Kondo coupling.
In the present study, we extend this theoretical framework, allowing for
different Kondo temperatures of two RKKY-coupled magnetic impurities due to
different local exchange couplings and density of states. As a result, we find
that the smaller one of the two Kondo temperatures is suppressed more strongly
by the RKKY interaction, thereby enhancing their initial inequality. In order
to find out if this relevance of inequalities between Kondo temperatures
modifies the distribution of the Kondo temperature in a system of a finite
density of randomly distributed magnetic impurities, we present an extension of
the RKKY coupled Kondo RG equations. We discuss the implication of these
results for the interplay between Kondo coupling and RKKY interaction in
disordered electron systems and the Doniach diagram in disordered electron
systems.
