## 2 Theoretical Model  
We consider an infinite graphene sample, by setting periodic boundary conditions in both \(x\) and \(y\) directions, with a period 128 nm in the former and 8192 nm in the latter. The two ends of the ribbon in the \(y\) direction are then considered as leads, as a Gaussian shaped pulse of charge carriers is launched from one lead (at \(y=y_{0}=-80\) nm) and propagates through the system towards \(y\to+\infty\). Around a given region of the graphene plane, scattering potential wells mimic the effects of substitutional N dopants, while the dopants concentration is related to the number of scattering centers per area unit. The perturbed region covers the full width of the ribbon and has a spatial extension of 40 nm in the \(y\) direction. As initial wave packet, we consider a Gaussian wave front as sketched in Figure 1, namely, a wave packet that is invariant along the \(x\)-direction, but which has a width \(a_{y}\) in the \(y\)-direction, multiplied by a pseudo-spinor (\(1\)\(i\))\({}^{T}\) and by a plane wave  
\[\Psi(x,y,0)=N\left( \begin{array}{c}1\\ i\end{array} \right)e^{ik(y-y_{0})-\frac{(y-y_{0})^{2}}{2\sigma_{y}^{2}}}, \tag{1}\]  
where \(N\) is a normalization constant and \(k\) is the average wave vector, which, in graphene, is related to the wave-packet energy \(E\) by \(k=E/\hbar v_{F}\), where \(v_{F}\) is the Fermi velocity. Since this pseudo-spinor is such that \(\left\langle\sigma_{y}\right\rangle=1\) and \(\left\langle\sigma_{x}\right\rangle=\left\langle\sigma_{z}\right\rangle=0\), where \(\vec{\sigma}\) is the Pauli matrix vector, this choice of pseudo-spinor is convenient: according to Heisenberg equation, the velocity is given by  
\[\vec{v}=\frac{d\vec{x}}{dt}=\frac{1}{i\hbar}\left[\vec{x},H_{D}\right], \tag{2}\]  
where \(H_{D}=v_{F}\vec{\sigma}\cdot\vec{p}\) is the Weyl-Dirac Hamiltonian of low-energy electrons in graphene in the absence of external potentials. Since \(\left[\vec{x},H_{D}\right]=v_{F}\vec{\sigma}\cdot\left[\vec{p},\vec{x}\right]= i\hbar v_{F}\vec{\sigma}\), then \(\vec{v}=v_{F}\vec{\sigma}\), and the wave packet is therefore expected to propagate in the direction of its pseudo-spinor (in this case, the \(y\)-direction), with constant velocity \(v_{F}\), no matter how much energy it has [44; 45]. We chose to work with a wave front, instead of a circularly symmetric wave packet, in order to avoid _zitterbewegung_ (a trembling motion of the wave packet) [47; 54] along the \(x\)-direction. In the latter case, one would not have been able to define an exact value of \(k_{x}\) and, consequently, \(\vec{\sigma}\) --and, ultimately, the velocity--would not have been a constant of motion. All the results in the remainder of this paper were obtained for a wave packet of width \(a_{y}=20\) nm and energy \(E=100\) meV. In the following, the wave-packet energy will remains pinned at this 100 meV value for all N concentrations. Our primary objective is to investigate how the concentration of scattering centers affects the transport properties while keeping constant the Fermi energy. Kubo-Greenwood calcul  
Figure 1: (Color online) Sketch of the graphene ribbon, illustrating the geometry of the problem addressed in this work. The colors represent the potential energy \(V(x,y)\) (in eV) generated by a given configuration of \(N\) atoms located in the perturbed region. The graphene parts below and above the doped region serve as leads. The shaded area represents the initial Gaussian wave packet (Equation (1)) in the bottom lead and propagating upwards. conductance of graphene containing 4 at.% random N impurities does not vary with energy above the Dirac point [26]. This result makes it non-essential to adapt the Fermi energy to the doping level. In fact, the Fermi energy of the system depicted in Figure 1 is imposed by the graphene leads. The chosen value of 100 meV corresponds to graphene doped at a charge density \(n_{e}=10^{12}\) cm\({}^{-2}\), which could be the result of defects or be obtained by other means such as an external gate potential.