## 2 Bloch Function Wave Packet Construction, Time Evolution and Scattering in Graphene
This time evolution--i.e., that for the infinite lattice--can also be analytically calculated by inserting the \(exp\bigg{[}-iE\bigg{(}\overset{\rightarrow}{k}_{[PERSON]}\bigg{)}t\bigg{]}\) time propagator into the kernel of Equation (1), where \(E\bigg{(}\overset{\rightarrow}{k}_{[PERSON]}\bigg{)}\) is the graphene dispersion relation. The WP moves in the \(-y\) direction, because its spectral distribution is centered at the lowest \(K\) point of the graphene Brillouin zone. The spectral width of the WP is small, \(0.1\overline{\Gamma K}\), hence the dispersion relation is still in the linear regime. There is only a very small spreading in the time evolution resulting in that the width and the shape of the WP remain unchanged. Figure 2d-f show the time evolution of the same initial Bloch WP on a graphene lattice with a structural defect. The defect was modeled by adding a Gaussian potential to the pseudopotential. The Gaussian was centered on an atomic site, its height is 54 eV and its HWHM is 0.45 nm. This raises the \(-\)44 eV minimum and +21 eV maximum values of the pseudopotential to +11 eV and +75 eV, respectively. These parameters were chosen in order to demonstrate the anisotropic spreading effects, see below. In the future, we intend to calculate defect potentials that model real structural defects. We can see that the WP is indeed scattered on the defect and the angular distribution of the scattered WP has a hexagonal symmetry (Figure 2h and Video S1). Similar anisotropic property around localized defect was also proved by using analytical calculations of the Green's function [29] highlighting the accuracy of our WPD method. It is worth noting that anisotropy was also observed in the experiments around a point defect [30] originating from intervalley scattering processes. In order to directly reveal the intervalley scattering process in our  
WPD simulations we performed Fourier transformation on our calculated probability density images (Figure 2) and compared the FFT images with our STM measurements. Figure 3a is the 2D-FFT power spectrum of the probability density of the initial wave packet, \(F\Big{[}\left|\psi(x,y;t=0)\right|^{2}\Big{]}\). This Fourier image shows a thin vertical line section in the origin, repeated six times at the vortices of a hexagon. The line section has Gaussian density distribution along \(y\). It is the reciprocal space representation of the initial wave packet (Figure 2d), which is delocalized in the x direction and localized in the y direction. Figure 3b is the 2D-FFT power spectrum of the probability density of the scattered wave packet, \(F\Big{[}\left|\psi(x,y;t=7.2 fs)\right|^{2}\Big{]}\). The \(t=7.2 fs\) corresponds to a specific time value, when the WP has already scattered on the defect (Figure 2f), therefore, the incoming and scattered waves can lead to characteristic interference patterns, which can be compared with the FFT images of the STM measurements. In Figure 3b, we can still recognize the spot at the origin and its six copies at the vortices of the hexagon, but the spots are not narrow vertical lines anymore. They have indeed a finite size in both directions, because the scattering is caused by the localized defect: the incoming wave packet has only vertical momentum, but the outgoing wave packet has momentum components distributed along the whole polar angle. Six other spots (marked by green circles) are also visible on this Fourier image. These  
Figure 2: Time evolution of the probability density of a Bloch function wave packet in the graphene pseudopotential. (**a–c**) Without defect. (**d–f**) With defect. (**g**) Spectral distribution of the Bloch function wave packet in the Brillouin zone, only the real part of \(a\Big{(}\overset{\rightarrow}{k}\Big{)}\) is shown. The red bar shows the reciprocal space width \(\Delta k\). (**h**) Scattering pattern, absolute value of the difference of the WPs shown on (**c**,f**). The size of the calculation window is 14.48 nm. (see Video S1 for the time development of the Bloch WP).