Determine the defect-density scaling exponent of Dirac-point stretching

Determine the scaling exponent governing the concentration dependence of the Dirac-point stretching \(\Delta E_{\mathrm{DP}}\) for substitutional Mn resonant defects in graphene, beyond the three-point concentration series measured for graphene/Cu(111).

Background

The paper compares the measured Dirac-point stretching with the self-consistent $T$-matrix prediction $$\Delta E_{\mathrm{DP}}\propto\sqrt{n_{\mathrm{def}}/|\ln(cn_{\mathrm{def}})|}$$ for resonant defects. The measurements show a monotonic increase in stretching from the nonimplanted reference to the two Mn concentrations, but the authors explicitly state that the available three-point series and the substantial graphene/Cu background do not permit determination of the scaling exponent. A denser concentration series would therefore be required to establish the quantitative concentration law.