## 3 Results and Discussion
The dashed lines represent the probability evolution obtained in a small time interval for one configuration of dopants, the full lines represent the same curves averaged over nine independent configurations. Figure 3: (Color online) Probabilities of finding the electron before (\(P_{1}\)—black curve), inside (\(P_{2}\)—red curve), and after (\(P_{3}\)—blue curve) the scattering region, as a function of time, for a \(E=100\) meV wave packet propagating through the 40-nm wide nitrogen doped area with 0.4 at.% concentration. by about a factor of two when the N concentration ranges from 0.1 to 0.5 at.%. The overall increase of time with increasing impurity concentration is the result of the wave packet bouncing back and forth several times inside the perturbed region. By comparison, the transmission time of the wave packet in pure graphene is 150 fs, here defined as the time interval between the moment \(P_{2}\) starts to increase and the moment when \(P_{2}\) falls back to zero (Figure 2a). In the presence of scattering centers, the probability of finding the electron after the perturbed region decreases slightly, but significantly, as the concentration of dopant increases, as shown by the blue curve in Figures 2b, 3 and 4. The asymptotic value of \(P_{3}\) was extrapolated by an exponential law assumed to govern its large-\(t\) behavior:  
\[P_{3}(t)\approx T-\alpha\exp(-\gamma t). \tag{7}\]  
The parameters \(T\) (transmission coefficient), \(\alpha\) and \(\gamma\) were fitted to the numerical calculations of \(P_{3}(t)\) for \(t>200\) fs. The obtained transmission coefficient \(T\) together with the parameter \(\gamma^{-1}\), which is considered as an indicator of the dwell time, are listed in Table 1 vs. N concentration \(c\). From [PERSON] theory of quantum transport [62], \(T\) for a given dopant concentration is a measure of the electrical conductance \(G(c)\) of the 40 nm doped area at the energy of 100 meV. As mentioned above, the conductivity of N-doped graphene varies weakly with energy above the Dirac point [26]. This means that the arbitrary choice of 100 meV does not entail the trends of the results. The trends are clearly that the relative conductance \(G(c)/G(0)\), estimated by the ration \(T(c)/T(0)\), decreases with increasing concentration with an average slope 0.19 when \(c\) is in at.%. This behavior is in agreement with the results of DFT calculations of graphene conductivity performed for higher N concentrations with the Kubo formula [23]. These calculations, in addition, pinpoint the dominant role played by the \(\pi\) electrons in the decrease of conductance with increasing doping. This finding validates the use of \(\pi\)-electron Hamiltonian considered here. To finish this discussion, it is worth mentioning that a significant reduction of electrical conductance of graphene upon B or N doping has been reported experimentally [10]. The observed decrease of conductance \(G(c)\) reflects the decrease of mobility \(\mu=G(c)/n_{e}e\) with increasing doping [63]. In principle, the mobility should decrease with increasing the length of the doped region. However, Landauer-Buttiker calculations reported in ref. [26] show that the conductivity of N-doped graphene does not depend significantly on the channel length for electron energies above the Dirac point, as considered here. For that reason, it is expected that the data of Table 1 are not much sensitive to the 40 nm width of the perturbed region. Further comparisons of the data of Table 1 with the existing literature is not easy for several reasons. Landauer-Buttiker DFT calculations for doped graphene nanoribbons depend strongly on the edges (zigzag or armchair) and on the width of the ribbon [33; 34; 35]. Furthermore, the conductance is spin dependent in the case of narrow zigzag nanoribbons [34; 37], because of magnetic ordering of the edge spins.