## 2 Bloch Function Wave Packet Construction, Time Evolution and Scattering in Graphene  
For any \(V\left(\overset{\rightarrow}{r}\right)\) potential which is periodic in space, the solutions of the stationary Schrodinger equation have the form of \(\varphi\left(\overset{\rightarrow}{r},\overset{\rightarrow}{k}_{[PERSON]}\right)=u \left(\overset{\rightarrow}{r},\overset{\rightarrow}{k}_{Bloch}\right)e^{i \overset{\rightarrow}{k}_{Bloch}^{\rightarrow}},\)where \(u\left(\overset{\rightarrow}{r},\overset{\rightarrow}{k}_{[PERSON]}\right)\) is a periodic function and \(\overset{\rightarrow}{k}_{[PERSON]}\) is the Bloch wave vector. These \(\varphi\left(\overset{\rightarrow}{r},\overset{\rightarrow}{k}_{[PERSON]}\right)\) functions are called Bloch functions. Bloch functions and the corresponding \(E\left(\overset{\rightarrow}{k}_{[PERSON]}\right)\) energies can be easily computed numerically for any periodic potential, with the Fourier transformed form of the Schrodinger equation. Figure 1 shows some characteristic Bloch functions for the graphene surface, computed by using the graphene pseudopotential [21]. The symmetry properties of Bloch functions depend both on the symmetry of the lattice and the \(\overset{\rightarrow}{k}_{[PERSON]}\) Bloch wave vector. The effect of \(\overset{\rightarrow}{k}_{[PERSON]}\) on the Bloch function is twofold: (i) the \(e^{i\overset{\rightarrow}{k}_{Bloch}\overset{\rightarrow}{\tau}}\) plane wave part introduces a \(\overset{\rightarrow}{k}_{Bloch}\) dependent phase factor and (ii) the shape of the \(u\bigg{(}\overset{\rightarrow}{\tau},\overset{\rightarrow}{k}_{Bloch}\bigg{)}\) function depends on \(\overset{\rightarrow}{k}_{Bloch}\). The \(\overset{\rightarrow}{k}_{Bloch}\) dependent phase factors determine the interference patterns of two or more Bloch functions. Such interference patterns are visible in FFT-LDOS images obtained by STM. In the case of the \(\Gamma\) point (Figure 1a-c), the symmetry of the wave function is identical to the potential itself, a hexagonal symmetry. However, increasing the Bloch wavelength along the vertical direction (Figure 1d-l) introduces a \(y\) dependent modulation into the Bloch function. Next, we constructed localized wave packets as a superposition of Bloch states. It is possible to build maximally localized Wannier functions [24; 25] by a carefully chosen superposition, but those superpositions span in the whole Brillouin zone. If we want to have a fairly narrow spectral distribution together with a sufficiently narrow spatial distribution, we can use a superposition with a simple Gaussian amplitude function \(a\bigg{(}\overset{\rightarrow}{k}\bigg{)}\):  
\[\phi\bigg{(}\overset{\rightarrow}{r}\bigg{)}=\int e^{-\frac{|\overset{ \rightarrow}{k}-\overset{\rightarrow}{k}_{Bloch}|^{2}}{4\Delta k^{2}}}e^{i \overset{\rightarrow}{r}_{0}\overset{\rightarrow}{k}^{2}}\ \varphi\bigg{(}\overset{ \rightarrow}{r},\overset{\rightarrow}{k}\bigg{)}d^{3}\overset{ \rightarrow}{k}, \tag{1}\]  
where we wrote \(\overset{\rightarrow}{k}\) for \(\overset{\rightarrow}{k}_{Bloch}\) in this formula for brevity (see Appendix A), \(\overset{\rightarrow}{k}_{0}\) is the momentum space vector, \(\overset{\rightarrow}{r}_{0}\) is the direct space initial position of the Gaussian and \(\Delta k\) is its momentum width. Figure 2a shows the constructed WP localized in its \(y\) coordinate in direct space. Its spectrum is concentrated on one of the graphene \(K\) points and Figure 2g shows the plot of the real part of the \(a\bigg{(}\overset{\rightarrow}{k}\bigg{)}\) amplitude function, which is a Gaussian multiplied with a plane wave. Figure 2a-c show the time evolution of this Bloch WP for an unperturbed graphene lattice calculated by numerically solving the time-dependent Schrodinger equation with the split-operator FFT method [26; 27; 28]. We used the \(\hat{H}=\hat{K}_{free}+\hat{V}\) Hamilton operator, where \(\hat{K}_{free}\) is the free space kinetic energy operator and \(\hat{V}=V\bigg{(}\overset{\rightarrow}{r}\bigg{)}\) is the graphene pseudopotential [21].