Criterion of applicability of coherent-potential approximation at low impurity concentrations

Dynamical diffusion and renormalization group equation for the Fermi velocity in doped graphene

“…In the other side, when impurity concentration is gradually increased, individual impurity states begin to overlap and the contribution from these states to the self-energy is becoming more pronounced in the vicinity of the impurity state energy and a spectrum rearrangement appears for a critical concentration (see [10]). This impose several restrictions to the possible values for the the concentration of impurities and the potential V 0 value (see [32] and [33]), which in turn impose several restrictions to the approximation used in this work, because it cannot be applied in a close vicinity of the Dirac point in the spectrum due to the increase in cluster scattering. Nevertheless, in [34] and [35], a E F → 0 limit is taken on the average Green function and by using the Ioffe-Regel criterion (see [36]), one of the solutions of this limit implies that the selfconsistent method is not applicable near the nodal point, which is equivalent to the conditions found in [32], but another low energy asymptotics solution exists, which impose more suitable conditions for the applicability of the Born approximation (see eq.…”