## 3 Results and Discussion
### Free Charge Carrier Properties of MLG and QFS-MLG
All three free electron density values are below the saturation density of n-type doping of MLG of 10\({}^{13}\) cm\({}^{-2}\)[55], indicating successful efficient annealing of donors on and near the SiC surface. The observed differences with \(T_{\rm Ar}\), albeit small, are significantly below the error bar of 0.3 \(\times\)  
\(10^{12}\) cm\({}^{-2}\). Since the MLG with \(T_{\rm Ar}\) = 1300 \({}^{\circ}\)C was obtained for a considerably longer time (5 min as compared to 0 s) it is tempting to speculate that the longer annealing may have a positive effect on reducing the interface dangling bonds effectively reducing the density of the surface state and leading to a lower free electron density. We have previously shown that purging with N\({}_{2}\) (or inert gases) effectively removes the ambient acceptor dopant, which may require up to several days of purging [37; 40]. The free electron densities in the MLG samples with \(T_{\rm Ar}\) = 640 \({}^{\circ}\)C and \(T_{\rm Ar}\) = 800 \({}^{\circ}\)C after purging in dry N\({}_{2}\) for 9-10 days increased slightly to 5.1 \(\times\) 10\({}^{12}\) cm\({}^{-2}\) and 7.0 \(\times\) 10\({}^{12}\) cm\({}^{-2}\), respectively, remaining below the the saturation density of n-type doping. The electron mobility parameters in these two cases slightly decreased in comparison to the freshly annealed samples, most likely as a result of the slightly increased charge density. The MLG with \(T_{\rm Ar}\) = 1300 \({}^{\circ}\)C shows the opposite behavior with slightly decreased charge density and slightly increased mobility parameter. Overall the purging with dry N\({}_{2}\) led to very small changes in the MLG electron density and mobility, which can be considered as the intrinsic free-electron parameters of MLG. As expected after purging with moist air (RH of 45%) the electron density in the MLG samples decreased due to the acceptor redox reaction at the graphene surface. The samples with different \(T_{\rm Ar}\) show very similar electron density of \(\sim\)2 \(\times\) 10\({}^{12}\) cm\({}^{-2}\) after \(\sim\)20 h of purging. We have measured the in-situ variations of free charge carrier properties and found that approximately 45 h purging in moist air are needed to flip the conductivity of MLG from \(n\)-type to \(p\)-type with free hole density of 1.4 \(\times\) 10\({}^{12}\) cm\({}^{-2}\). Long-term exposure in ambient conditions (several months) leads to only a very small increase of free hole density to \(\sim\)2 \(\times\) 10\({}^{12}\) cm\({}^{-2}\) indicating saturation of p-type ambient doping in MLG. Again, very similar free hole densities are found for the samples with different \(T_{\rm Ar}\) = 1300\({}^{\circ}\). On the other hand, the free charge carrier mobility of the ambient doped MLG with \(T_{\rm Ar}\) = 1300 \({}^{\circ}\)C is more than 50% larger than the respective values of MLG with \(T_{\rm Ar}\) = 800 \({}^{\circ}\)C and \(T_{\rm Ar}\) = 640 \({}^{\circ}\)C. This is true for both the cases of free electrons and free holes (see Figure 7 right panel results for Air RH 45% and Ambient). This finding is very interesting considering that the samples with \(T_{\rm Ar}\) = 640 \({}^{\circ}\)C and \(T_{\rm Ar}\) = 800 \({}^{\circ}\)C have better MLG coverage of 99% and lower RMS \(\simeq\) 0.4 nm, compared with the \(T_{\rm Ar}\) = 1300 \({}^{\circ}\)C sample, which has 92% MLG coverage and RMS \(\simeq\) 0.75 nm. It was previously suggested that dominant scattering mechanisms at room temperature in graphene on SiC are the remote interface phonon scattering, as a result of coupling to the polar modes in the substrate, and scattering by impurities [56; 57; 58].