#
Wave Packet Dynamical Simulation of Quasiparticle Interferences in 2D Materials^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

^{2}carbon nanosystems by using a variationally calculated local carbon one-electron pseudopotential [21]. This pseudopotential has two major advantages: (i) it brings the specific electronic dynamics of the bands (linear dispersion near the K points for electrons near the Fermi energy (E

_{F}), trigonal warping for hot electrons, etc.) into the WPD calculation and (ii) it allows us to handle localized defects. We were able to exploit this feature of the pseudopotential in calculating the transport properties of different 0D and 1D graphene defects [22,23].

_{2}single layers [20] and now we extend it for graphene structural defects opening the way for the WPD investigations of defect scattering processes in other 2D materials.

## 2. Bloch Function Wave Packet Construction, Time Evolution and Scattering in Graphene

## 3. Band Structure Governed Wave Packet Dynamics

_{2}single sheets in Reference [20]. Those calculations successfully reproduced the trigonal warping effect and the anisotropic WP spreading characteristic of these 2D materials and showed different symmetries of the WP spreading, depending on the band structure and the spectral distribution of the initial WP. As we emphasized in Reference [20], similar calculations can be easily performed for any crystalline material, where the dispersion relation is known.

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

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**Figure 1.**Construction of the graphene Bloch wave functions for different points along the ΓK line in the extended Brillouin zone. Left column: ${e}^{i{\overrightarrow{k}}_{Bloch}\overrightarrow{r}}$. Middle column: $u\left(\overrightarrow{r},{\overrightarrow{k}}_{Bloch}\right)$. Right column: $\phi \left(\overrightarrow{r},{\overrightarrow{k}}_{Bloch}\right)$. See Section 2 for details. Real parts of the wave functions are shown, green is positive, red is negative. The graphene lattice is shown by blue lines. The insets in a, d, g, j show the position of ${\overrightarrow{k}}_{Bloch}$ (yellow dot) relative to the Brillouin zone.

**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 $a\left(\overrightarrow{k}\right)$ is shown. The red bar shows the reciprocal space width $\mathsf{\Delta}k$. (

**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).

**Figure 4.**Time evolution of the probability density of a wave packet on the graphene surface with- and without defect. The band structure of the perfect crystal is incorporated into the kinetic energy operator, hence the potential is everywhere zero, except in the defect region. (

**a**–

**c**) Without defect. (

**d**–

**f**) With defect. The defect is shown by a blue spot. (

**g**) Spectral distribution of the initial wave packet in the Brillouin zone (absolute value) is shown by red circles in the K and K’ points, superimposed on the graphene band structure. The radius of the red circles is $\mathsf{\Delta}k/2$, the intensity of the red color is proportional to the spectral weight at each $\left({\overrightarrow{k}}_{x},{\overrightarrow{k}}_{y}\right)$ points. (

**h**) Scattering pattern, absolute value of the difference of the WPs shown on (

**c**,

**f**). The size of the calculation window is 23.04 nm.

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**MDPI and ACS Style**

Vancsó, P.; Mayer, A.; Nemes-Incze, P.; Márk, G.I.
Wave Packet Dynamical Simulation of Quasiparticle Interferences in 2D Materials. *Appl. Sci.* **2021**, *11*, 4730.
https://doi.org/10.3390/app11114730

**AMA Style**

Vancsó P, Mayer A, Nemes-Incze P, Márk GI.
Wave Packet Dynamical Simulation of Quasiparticle Interferences in 2D Materials. *Applied Sciences*. 2021; 11(11):4730.
https://doi.org/10.3390/app11114730

**Chicago/Turabian Style**

Vancsó, Péter, Alexandre Mayer, Péter Nemes-Incze, and Géza István Márk.
2021. "Wave Packet Dynamical Simulation of Quasiparticle Interferences in 2D Materials" *Applied Sciences* 11, no. 11: 4730.
https://doi.org/10.3390/app11114730