The formation of polarons, i.e. phonon-”dressed” charge carriers (holes or electrons), plays an important role in electronic structure, optical spectra and electrical conductivity of materials. Depending on the electron-phonon coupling strength, polarons can be either delocalized over many unit cells of the underlying lattice (large polarons), or localized in one or several unit cells (small polarons). The type of polaron determines, for example, the mechanism and temperature dependence of conductivity.

To accurately describe charge localization and, consequently, polarons in materials is a challenging task for density-functional theory. First, depending on the polaronic nature, the required supercell size to host a polaron may be rather large. More importantly, standard xc functionals of density-functional theory (like LDA or GGAs) suffer from the self-interaction error, leading to underestimation of localization. The self-interaction error can be reduced by replacing a part of local or semi-local exchange by the exact exchange. However, the resulting hybrid functionals contain the fraction of exact exchange as a parameter, which needs to be determined in some way, for example by comparing the DFT results to higher-level methods such as many-body perturbation theory. On the other hand, both hybrid functionals and the higher-level methods are computationally much more expensive than semi-local functionals in particular for geometry optimization. Moreover, for some values of the fraction of exact exchange small polarons may not be found.

Here, we propose a way to address the above challenges. Using constraints from exact DFT, namely the IP-theorem, the binding energy of polarons can be reformulated only in terms of energies obtained from the neutral system, which results in self-interaction error cancellation. The above reformulation also allows for calculating forces. In this way, the dependence of polaron binding energy on the underlying exchange-correlation functional can be drastically reduced, so that the search for polaronic distortions using the more computationally expensive hybrid functionals can be avoided. An alternative way, but in the same spirit, is to use the selfconsistent SCAN functional, the newest development by John Perdew.

Once we have corrected binding energies obtained by the above approach, the optimal fraction of exact exchange in the hybrid functional that yields an accurate binding energy can be determined as the one for which the hybrid functional satisfies the IP theorem. The above scheme will be tested for MgO, TiO_{2}, and ZnO and then applied to group-III oxides. For the latter, we plan to start our investigation with Ga_{2}O_{3} which shows the highest localization energy among the oxides. We plan to address the following questions:

How does the degree of polaron localization depend on bonding, local environment, and alloying? To this extent, we will explore the polaronic nature in various polymorphs and in strained systems. In addition, we will explore the impact of alloying elements and vacancies. This will give us a first idea about the possible behaviour at surfaces and interfaces.

Another issue that we may address concern differences between charged an neutral polarons. In fact, it has been argued that formation of (charged) polarons may hinder p-type doping in oxides. As the usage of periodic boundary conditions hamper the treatment of charged systems, this poses the question of how to compensate the access charge effectively. Possible mechanisms that need to be evaluated are (i) adding a constant background charge, (ii) population of the conduction band, or (iii) the virtual crystal approximation.

An distinctively different method to investigate charge-neutral polarons is provided by Green function approaches, as pioneered by Engelsberg and Schrieffer as well as Heine, Allen, and Cardona. This procedure is based on the solution of the Dyson equation with the self-energy containing the electron-phonon interaction, and it involves the evaluation of the phonon spectrum, the corresponding electron-phonon coupling parameters and the Eliashberg function. While on the one hand, the computation of these quantities is rather involved, the method does on the other hand not rely on supercells. Another advantage is that it also gives access to polaron lifetimes. However, like in the case of constrained DFT, the role of the xc functional in the computation of phonons must be explored.

**Major accomplishments expected:**

- Development of an accurate approach for calculating polarons in oxide materials
- Accurate results for lattice distortions and polaron binding energies
- Insight into the impact of bonding, strain, and alloying on the polaronic nature
- Alternative concepts of exploring polarons

**Collaboration with partners in the project:**

The work is extremely challenging and thus requires two PhD students, one supported through GraFOx, the other one by FHI. Experimental work is performed at the IKZ (M. Albrecht) and the PDI.

If you have queries about the project, please contact the PI:

Claudia Draxl, Humboldt Universität zu Berlin

Matthias Scheffler, Fritz-Haber-Institut

The Leibniz ScienceCampus GraFOx is a network of two Leibniz Institutes, two universities and one institute of the Max Planck Society. It is based in Berlin, Germany.

**Coordinator:**

Kai Hablizel

Paul-Drude Institut für Festkörperelektronik (PDI)

Leibniz-Institut im Forschungsverbund Berlin e.V.

Tel.: +49 30 20377-342

Email: grafox@pdi-berlin.de

**Speaker: **

Prof. Dr. Henning Riechert, PDI

**Scientific Coordinator: **

Dr. Oliver Bierwagen, PDI

Dr. Martin Albrecht, IKZ