# Projects for students

## Enabling GW calculations for molecules on surfaces (Master project)

In this project, we focus on molecules adsorbed on surfaces, e.g. a pentacene molecule on a NaCl surface [1]. In various single-molecule electron transport experiments [2], it is crucial to describe the relative position of electronic levels of the molecule with respect to electronic bands of the surface.

The GW method is the state-of-the-art method to compute electronic levels in molecules and band structures in solids. We propose in this project to apply a recently developed efficient GW implementation [3] to molecules on surfaces to explore and understand the fundamental gap of the molecule as function of adsorption site and surface geometry. (fundamental gap = difference between the energy of the highest occupied molecular orbital and the lowest unoccupied molecular orbital)

A fundamental first part of this project is to accelerate the GW implementation [3] further: Gaussian basis functions are used in the algorithm [3], that are centered on atomic positions. The efficiency of the GW implementation crucially relies on vanishing overlap between basis functions. We propose to expand the Gaussian basis functions with small exponents (wide-spread) by Gaussians with high exponents (localized function). Consequently, the overlap is much smaller, can be removed from the calculation and the GW algorithm [3] is accelerated substantially. A detailed description of the idea can be found in a
PDF document.

[1]  J. Repp, G. Meyer, S. M. Stojković, A. Gourdon, C. Joachim, Phys. Rev. Lett. 94, 026803 (2005).
[2]  F. Evers, R. Korytár, S. Tewari, J. M. van Ruitenbeek, arxiv:1906.10449 (2019).
[3]  J. Wilhelm, D. Golze, L. Talirz, J. Hutter, C. A. Pignedoli, J. Phys. Chem. Lett. 9, 306-312 (2018).

## GW bandgap calculations on phosphorene sheets (Bachelor project)

The rise of two-dimensional materials began with the synthesis of graphene in 2004. In the following years, there have been many efforts to find new two-dimensional materials. Scientists succeeded in 2014 to synthesize two-dimensional phosphorene that is composed of a single layer of phosphorus atoms. [1] Phosphorene can be viewed as a single layer of the material black phosphorus, much in the same way that graphene is a single layer of graphite. Phosphorene is predicted to be a strong competitor to graphene because, in contrast to graphene, phosphorene has a non-zero bandgap and is therefore promising for semiconductor applications.

There is a debate ongoing how large the bandgap of free-standing 2d phosphorene is and how the bandgap evolves from small rectangular sheets of phosphorene to the 2d periodic phosphorene. [2] In this project, bandgaps of finite (non-periodic) 2d sheets of phosphorene will be calculated using a highly efficient GW implementation [3] to better understand the bandgap evolution from small to large phosphorene ribbons.

[1]  L. Li, Y. Yu, G. J. Ye, Q. Ge, X. Ou, H. Wu, D. Feng, X. H. Chen, Y. Zhang, Nat. Nanotechnol. 9, 372-377 (2014).
[2]  T. Frank, R. Derian, K. Tokár, L. Mitas, J. Fabian, I. Štich, Phys. Rev. X 9, 011018 (2019).
[3]  J. Wilhelm, D. Golze, L. Talirz, J. Hutter, C. A. Pignedoli, J. Phys. Chem. Lett. 9, 306-312 (2018).

## High-harmonic generation in topological insulators (2x Bachelor project)

When irradiating a solid with ultra-short laser pulses (~ 200 fs), electrons are excited from valence bands to conduction bands. The electric field of the laser pulse $E(t)=E_0 \cos (\omega_0t)\exp(-t^2/\sigma^2)$ features a Gaussian envelope and is typically modulated with a cosine of frequency $\omega_0$ ~ 30 THz ~ 1/(30 fs) [1]. The Fourier transform $E(\omega)\sim \exp(-\sigma^2(\omega-\omega_0)^2)$ of the laser pulse is centered around $\omega_0$.

From a driven harmonic oscillator, we expect the charge density to also oscillate with a frequency $\omega_0$ as response to the electric field. However, due to multiple excitation of electrons between bands and due to acceleration of already excited electrons by the electric field $E(t)$, the electron charge density oscillates not only with frequency $\omega_0$, but also with higher frequencies $n\omega_0, n\in \mathbb{N}$. Consequently, the emitted radiation by the charge density movement features high-harmonic frequencies $n\omega_0$, where $n>20$ can be observed [1]. We employ semiconductor Bloch equations [2] to model high-harmonic generation as function of the material-specific bandstructure depending on pulse amplitude and shape.

In the project "Effects of interactions on high harmonic generation in topological insulators", we propose to extend our semiconductor Bloch equations code to capture Coulomb interactions of the involved charge carriers. Once having the coding part finalized, we propose to investigate the effects of Coulomb interactions on the high-harmonic spectrum using the developed code.

In a second project "Manifestation of a topological phase transition in high harmonic generation", we propose to investigate the effect of the bandstructure (topological versus non-topological) on the high-harmonics spectrum using our semiconductor Bloch equations code.

If you are interested in a project concerning high-harmonic generation, please contact Prof. F. Evers or Dr. Jan Wilhelm.

[1]  O. Schubert et al., Nat. Photonics 8, 119-123 (2014).
[2]  J. Li et al., Phys. Rev. A 100, 043404 (2019).

## Simulating the time evolution of excitations in superconducting phases (Bachelor project)

High-harmonic generation (see also project above) is mainly investigated in non-superconducting phases. We propose this analytical project to investigate effects of Bogoliubov-de Gennes pairing terms describing superconductivity on the charge carrier dynamics.

If you are interested in a project concerning high-harmonic generation in superconductors, please contact Prof. F. Evers.