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## Calculation of Optical Properties of Nanostructures from First Principles

#### CECAM-HQ-EPFL, Lausanne, Switzerland

#### Organisers

The optical properties of nanostructures is the centerpiece of many recently established research fields. Here we highlight three promising techniques/goals of optical spectroscopies. First, the study of optical properties is the ubiquitous characterization technique to learn about the electronic structure, internal electric fields, magnetic moments, morphology, composition, defect properties etc... of nanostructures. In general, despite the extraordinary accuracy of the optical spectroscopies which is not reached by other techniques, the understanding and the interpretation of measured spectra require detailed theoretical support. Moreover, for complex nanostructure-based systems the predictive power of theoretical-numerical studies is needed. On the one hand, it is nowadays possible to measure the photoluminescence of single semiconductor nanostructures (quantum dots, QDs) with a spectral resolution of few micro-eV. On the other hand, spectra of nanocrystalline systems are measured in a wide range of photon energies, e.g. by means of spectroscopic ellipsometry. Moreover, nanostructures in form of superlattices and even embedded or colloidal QDs are used to enhance the efficiency or lower the cost of photovoltaic devices. Nanostructures are used to adopt the fundamental absorption edge. These QDs can be used to absorb the sun's light directly, or to accept the charge carriers photogenerated by dye molecules (Grätzel cell). In both cases, a fundamental knowledge of the optical properties, that remains preliminary today, would be required to engineer materials with targeted properties. Third, nanostructures have emerged as interesting candidates for the study of quantum optical phenomena. Some of the effects well studied in quantum optics are presently investigated in the solid state counterpart of the atom, namely the semiconductor nanostructure quantum dot. Observations of coherent decay rates limited mainly by spontaneous emission and photon antibunching are strong indications that indeed these structures have many similarities with atoms. However, with the significant advantage to be ``artificially" engineered in terms of their sizes, compositions, shapes, which leads to a control of their quantum optical light emission properties. The possibility to engineer artificial structures that behave as few level systems known from atoms in quantum optics opens up a large field of research related to quantum information processing.

Third, nanostructures have emerged as interesting candidates for the study of quantum optical phenomena. Some of the effects well studied in quantum optics are presently investigated in the solid state counterpart of the atom, namely the semiconductor nanostructure quantum dot.

Observations of coherent decay rates limited mainly by spontaneous emission and photon antibunching are strong indications that indeed these structures have many similarities with atoms.

However, with the significant advantage to be ``artificially" engineered in terms of their sizes, compositions, shapes, which leads to a control of their quantum optical light emission properties.

The possibility to engineer artificial structures that behave as few level systems known from atoms in quantum optics opens up a large field of research related to quantum information processing.

The theoretical challenges to connect to these three fields of research are fueled by two facts. First, the main topic deals with optical properties and hence two-particle excitations such as excitons, which is at the frontier of todays ab-initio approaches. Then, the structures are non- periodic along at least one of the dimensions calling for the treatment of large simulation cells with often many-atoms.

The present theoretical approaches can be cast into two groups of basic approaches to describe the optical properties of nanosystems. In the early days of the development the computations of optical properties were based on rough approximations for the electronic structure, especially to describe the emission properties near the absorption edge, such as effective-mass and k · p treatments. At this mesoscopic level, the limit is towards small systems (in contrast to atomistic approaches) in a top-down approach. The question being: “how far can the size of the structures be reduced and still be described accurately by continuum models?”. The continuum approaches are not anymore applicable for nanostructures whose characteristic extents are not large compared to the characteristic atomic distances. Therefore, the majority of current developments is based on an atomistic description of the structures and follow a bottom-up approach, limited towards large structure sizes. These are the type of approaches we indent to discuss in our workshop.

These atomistic approaches can be cast into three different groups that we wish to bring together. The hope is thereby to learn from each other in a way that synergy effects will ensue. The three groups in question are the following:

DFT -> GW -> BSE In this approach [1–10], the Kohn-Sham orbitals, calculated at different levels of density functional theory (LDA, GGA, Hybrid,...), yield the Green’s function needed in the GW-approximation. Several variants of the GW approximation exist, ranging from non-self-consistent “single-shot” calculations using pseudopotentials and the plasmon pole approximation to a fully-self-consistent all-electron calculation including vertex corrections. The increased level of accuracy goes along with a substantial increase in the computational demand. The GW-stage is appropriate for single charged excitations (ionization energies, electron affinity) but not for the two-particle excitations (excitons) required in the calculation of optical properties. The Bethe-Salpeter equation (BSE) has to be solved for complex systems with hundreds or better thousands of atoms starting from the quasiparticle electronic structures of their arrangements. This is especially true for nanostructures, where the confinement leads to very large (compared to the cases of extended systems) excitonic effects (exciton binding energies of 1 eV can be involved). The number of atoms in a nanostructure determines the rank of the two-particle electron-hole Hamiltonian matrix. Novel numerical methods are needed for ranks above 100000.

TDDFT Following the idea to stay with a density based formalism, avoiding the use of wave functions, but moving to a time dependent formulation [11–21] yields informations about the excited states properties, hence the optical properties. The method requires a time dependent exchange correlation potential, as well as its density variation, which calls for (as in DFT) appropriate approximations. The method is rather direct (no intermediate step) and has the potential advantage to require only a two-point equation which, compared to the four-point equation of BSE, may represent a key advantage. While it’s applicability to extended structures has been shown to be satisfactory, nanostructures with confined states are still challenging.

AEP -> CI The atomic effective pseudopotentials (AEPs) are generated from the Kohn-Sham self-consistent effective potential of bulk materials and are used subsequently for nanostructures [22–24]. The ensuing wave functions can be used to construct excited state Slater determinants that are used in a configuration interaction treatment [22, 25]. The higher level of approximation underlying the AEPs and the model-screening function used in the calculation of Coulomb and exchange integrals in the CI treatment allow to address a large number of atoms [25].

## References

**Germany**

Friedhelm Bechstedt ( University of Jena ) - Organiser & speaker

Gabriel Bester ( Max Planck Institute for Solid State Research, Stuttgart ) - Organiser & speaker

Claudia Draxl ( Humboldt-Universität zu Berlin ) - Organiser