Magnetic quantum phase diagram of magnetic impurities in two-dimensional disordered electron systems

The quantum phase diagram of disordered electron systems as a function of the concentration of magnetic impurities nm and the local exchange coupling J is studied in the dilute limit. We take into account the Anderson localization of the electrons by a nonperturbative numerical treatment of the disorder potential. The competition between Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction JRKKY and the Kondo effect, as governed by the temperature scale T-K, is known to give rise to a rich magnetic quantum phase diagram, the Doniach diagram. Our numerical calculations show that in a disordered system both the Kondo temperature T-K and J(RKKY) as well as their ratio J(RKKY)/ T-K is widely distributed. However, we find a sharp cutoff of that distribution, which allows us to define a critical density of magnetic impurities n(c) below which Kondo screening wins at all sites of the system above a critical coupling J(c), forming the Kondo phase [see Fig. 3( b)]. As disorder is increased, J(c) increases and a spin coupled phase is found to grow at the expense of the Kondo phase. From these distribution functions we derive the magnetic susceptibility which show anomalous power-law behavior. In the Kondo phase that power is determined by the wide distribution of the Kondo temperature, while in the spin coupled phase it is governed by the distribution of J(RKKY). At low densities and small J < J(c) we identify a local-moment phase (LM). We also report results on a honeycomb lattice, graphene, where we find that the spin coupled phase is more stable against Kondo screening, but is more easily destroyed by disorder into a LM phase.