## 3 Results and Discussion  
As previously mentioned, the choice of the pseudo-spinor components of the initial wave packet (Equation (1)) is such that in pure graphene, the wave packet conserves its spatial shape and travels at the constant velocity \(v_{F}\) set to \(8.7\times 10^{5}\) m/s = 0.87 nm/fs. This soliton-like propagation is clearly seen by the \(P_{2}\) probability displayed in Figure 2a in dashed line as a function of time, together with the probabilities of finding the electron in region 1 (\(P_{1}\)) and 3 (\(P_{3}\)). These curves are shown for comparison with the doped case. After 96 fs, the center of the wave packet has reached the coordinate \(y=0\) at the middle of region 2, \(P_{2}\) reaches its maximum value. The wave packet has almost completely left the region 2 after 175 fs, when the probability \(P_{3}\) saturates to one. Figures 2b, 3 and 4 represent the probabilities of finding the electron before (\(P_{1}\)), inside (\(P_{2}\)), and after (\(P_{3}\)) the scattering region, as a function of time, for wave packets propagating through the nitrogen-doped region, with concentrations 0.1 at.% and 0.4 at.%, respectively. The case 0.1 at.% is also illustrated in Figure 2a on a smaller time interval. The effects of doping show up after 100 fs. \(P_{1}\) drops to a very small value after 120 fs and then starts increasing slowly, as a consequence of reflection of the wave packet by the scatterers. Just like in pristine graphene, \(P_{2}\) reaches its maximum after 100 fs, decreases rapidly immediately after, but then much slower after 150 fs. This is because part of the wave packet remains trapped in the scattering region. Due to the finite extension of the perturbation region, the trapped fermions eventually escape from region 2 on both sides, giving rise to a non-zero reflection probability and to a transmission probability smaller than one. The wave function escapes as small packets, giving rise to amplitude oscillations on the probability density functions \(P_{i}\) (\(i=1,2\) and 3). However, since these oscillations come from the wave packet being bounced back and forth by the scatterers in the doped region, they are configuration dependent, so that averaging between several impurity distributions for the same dopant concentration leads to smooth curves, as illustrated in Figure 4. This figure reveals that the transmission probability fluctuates by less than 2% from one configuration to the other. The observed small configuration dependence of the results is partly due to the wave packet extending uniformly across the width of the graphene strip (see Figure 1), which produces an averaging process over the lateral positions of the dopant atoms. Thus, due to  
Figure 2: (Color online) (**a**) Probabilities of finding the electron before (\(P_{1}\)), inside (\(P_{2}\)), and after (\(P_{3}\)) the scattering region, as a function of time, for a \(E=100\) meV wave packet propagating in pristine graphene (dashed line) and through a 40 nm wide doped region with 0.1 at.% N (full line); (**b**) Same as in (a) over a longer time interval for the 0.1 at.% doping concentration. The shortest distance between impurities is 2 nm. The number of scattering centers for this concentration is 195.  
a prohibitively long time of computation, just one configuration was generated for the other dopant concentrations. It can be clearly observed that \(P_{3}\) saturates faster for the smallest values of N concentration, which suggests that the dwell time of the wave packet inside the scattering region becomes longer as the concentration of nitrogen increases. The dwell time, whose definition follows from Equation (7), is listed in Table 1 for different dopant concentrations. The data indicate that the dwell time increases  
Figure 4: (Color online) Effect of configurational averaging on the probabilities \(P_{1}\) (black curve), \(P_{2}\) (red curve), and \(P_{3}\) (blue curve) for the N concentration 0.3 at.%.