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## Auger scattering and carrier multiplication

Auger scattering and carrier multiplication are opposite phenomena (figure 1.3), and belong to the inelastic-type scattering as the number of particles is not conserved. The latter involves the recombination of an electron with a hole, whose resulting photon is absorbed by another conduction electron, while the former refers to an intraband transition whose released energy is captured by a valence electron then promoted to the conduction band. These processes have a characteristic time of 102 ps [21], and are dependent upon the carrier density and the band gap. Those are not strictly speaking « cooling processes », but interfere on a long time scale in the establishment of the average carrier temperature.

### The role of phonons in carrier cooling

LO-phonons emitted from intraband scattering are not stable (figure 1.3), but decay into lower energy phonons within few picoseconds [43]. A detailed description of this process will be given in chapter II and III. Previous to this decay, there is a certain probability, related precisely to the phonon lifetime, that the electrons would reabsorb LO-phonons. It was proven by several independent theoretical Monte Carlo studies [44, 19, 45, 46] and Boltzmann equation fitting of experimental data from room temperature femtosecond spectroscopy [47] that the hot carriers energy relaxation time is strongly dependent on the LO-phonon lifetime, and that the longer it is, the slower the cooling for both electrons and holes. As an example, table 1.3 displays the average energy and temperature of a photoexcited hot-electron population, as a function of the LO-phonon lifetime, calculated within a full-band Monte Carlo solving of the Boltzmann Transport Equation in In0.53Ga0.47As (details about the implementation can be found in [44]). 2.5 ps corresponds to the LO-phonon lifetime at room temperature, and is representative of most of III-V semiconductors in the FCC crystal structure [49]. A gain of one order of magnitude on this parameter induces a significant enhancement of the conduction electrons average energy: tuning the LO-phonon lifetime appears, then, as a key leverage on hotcarrier cooling dynamics, which is of fundamental significance for the hot-carrier solar cell concept.

### Engineering phonon band structure for hot-carrier solar cells

From a quantum mechanical point of view, the fewer the final states are, the less probable the decay is, and thus the greater is the LO-phonon lifetime. Hence, the first basic idea in phonon engineering for hot-carrier solar cells is to find or design an absorbing material that exhibits few allowed two-phonon final states. The idea of using the gap formation in the density of states may come from an indirect reading of a paper from B. K. Ridley [88], in which he noticed that the gap between the optic and acoustic branches in bulk GaP is too large to allow LO-phonon decay though Klemens channels, and that other channels have to be considered (as it happened, the Ridley channels). In a first approach, with a one-dimensional ball-and-spring model [102], it appears that the mass ratio between the atoms is a key parameter. As an example, the phonon band structure was calculated for a one-dimensional (GaSb)NL−(AlSb)NL superlattice, together with bulk GaSb and AlSb cases, within the approximation of nearest neighbour-only interaction, where NL is the number of layers of each binary, the whole structure having periodic boundary conditions (figure 2.3). In the bulk cases to start with, the gap opening between the acoustic and the optic region is solely determined by the atomic mass ratio: at the zone boundary, the acoustic and optic frequencies are respectively given by ωO = � 2C m and ωA = � 2C M , m and M being the mass of the lightest and heaviest atom, C the stiffness of the interatomic springs, that are assumed atomic species-independent. Hence, the opening is simply ωO ωA = � M m . Considering the superlattice, forbidden energy ranges.

**Quantum dot: how gaps vanish because dots are not round**

To expand the gap as observed in superlattices in the direction perpendicular to the interface, one would need, at the atomic scale, a flat interface in every direction around a centre, i.e. a perfectly round dot. In the limit of an infinitely small dot, the atomic feature of matter forbid to consider an isotropically smooth interface. In the limit of a large dot, it is tantalizing to treat the system in an elastic continuum model. Yet, this would not include the enormous number of internal degrees of freedom whose corresponding modes vibrational energy is likely to span a large range below the highest optic frequency. In addition, when depopulating a dense frequency region, one has to be careful not to populate a previously empty range that may create new two-phonon final states. This is intrinsically in contradiction with criteria (iii) (see section (1.4.2)) since a wide frequency range would be required to avoid density peaks overlaps.

**How it turns out that gaps are sufficient but not necessary**

We showed that engineering phonon band structure with, as a goal, the creation of well energetically positioned gaps in the density of states is a very complicated task. It is even more true considering the fact that we have restricted ourselves to cubic systems. For crystal with lower symmetry, the increasing number of degree of freedom decreases the chance to find reasonably large forbidden phonon energy range. However, it is important at this point to recall that the experimental pieces of evidence of the hot-carrier effect have not been highlighted in such materials. In the numerous GaAs/AlxGa1−xAs superlattices systems in which it has not been reported [67, 68, 54], it is manifest, referring to figure 2.6, 2.10, and to the previous discussion, that there is no chance to witness an appropriate phonon gap. This later feature is thus a sufficient but not necessary condition for a material to be a good candidate for hot-carrier solar cells from a phonon point of view. From the results presented in this chapter, it appears that such gaps are more likely to be found in bulk materials, such as in SiSn.

**Table of contents :**

Introduction

**1 The Hot-Carrier Solar Cell **

1.1 Third generation Photovoltaics: beyond the Shockley-Queisser limit .

1.1.1 The last decade objective

1.1.2 « There’s plenty of room above »

1.1.3 Promising already existing technologies

1.2 Carrier cooling, the hot topic

1.2.1 Carrier-carrier scattering

1.2.2 Intravalley versus intervalley scattering

1.2.3 Auger scattering and carrier multiplication

1.2.4 The role of phonons in carrier cooling

1.3 Hot Carrier Solar Cell: the ultimate PV device

1.3.1 Principle and comparison with a single p-n junction solar cell

1.3.2 Maximum efficiencies expected

1.3.3 Recent experimental achievements on hot-carrier solar cell issues .

1.4 Materials for hot-carrier solar cell absorber

1.4.1 Carrier cooling and atomic scale engineering

1.4.2 Formal requirements

1.4.3 Physical challenges for materials science

1.5 Open questions

**2 Phonons: density of final states **

2.1 The Physics of phonons

2.1.1 Interatomic force constants

2.1.2 Equation of motion, dynamical matrix and secular equation

2.1.3 Phonon dispersion and one-phonon density of states

2.2 Phonon decay

2.2.1 Conservation rules

2.2.2 Decay channels nomenclature

2.2.3 Engineering phonon band structure for hot-carrier solar cells

2.2.4 Need for a complete picture

2.3 Phonons within Density Functional Perturbation Theory

2.3.1 Principle and formalism

2.3.2 Successes and limitations

2.4 Two-phonon final states in bulk semiconductors

2.4.1 Practical implementation

2.4.2 Detailed two-phonon states analysis

2.4.3 Discussion

2.5 Gaps in the density of states of nanostructured materials

2.5.1 Practical implementation of PhDOS calculation on large systems

2.5.2 Application to superlattices

2.5.3 Application to quantum dots

2.6 Gaps in the density of states, a lost cause ?

2.6.1 Superlattice: how gaps are filled

2.6.2 Quantum dot: how gaps vanish because dots are not round

2.6.3 How it turns out that gaps are sufficient but not necessary

2.7 Conclusion on the two-phonon final states investigation

**3 Phonons: decay rate **

3.1 LO-phonon lifetime

3.1.1 Phonon decay from experiment

3.1.2 Lifetime in bulk materials

3.2 Formalism – Beyond the harmonic approximation

3.2.1 Derivation

3.2.2 State of the art of phonon lifetime calculation

3.2.3 Drawbacks of tantalizing approximations

3.2.4 Misleading success in fitting experimental data

3.3 Third order anharmonic tensor within DFPT

3.4 Practical implementation

3.4.1 Phonons eigenvectors and eigenfrequencies

3.4.2 q-space mesh and energy smearing

3.4.3 The two-phonon density of final states

3.4.4 Validation and commentary on DFPT for HCSC

3.5 Application to SiSn

3.5.1 Two-phonon density of states

3.5.2 Lifetime of the LO-phonon: beyond the zone centre approximation .

3.5.3 Lifetime dependence on the LO-phonon reciprocal position

3.5.4 Eventual photovoltaic efficiency

3.6 Towards higher orders-phonon processes

3.6.1 Previous discussions on four-phonon processes

3.6.2 Estimation of the four-phonon processes contribution

3.6.3 Consecutive three-phonon processes

3.6.4 Crystal symmetry and higher-order anharmonic tensors

3.7 Conclusions on LO phonon lifetime calculations for HCSC

3.7.1 SiSn features as references

3.7.2 Limitations

3.7.3 Towards other ways to hinder carrier cooling

**4 Electron-phonon interaction **

4.1 Dimensionality issue

4.1.1 Electron cooling in superlattices

4.1.2 Self-induced electric field and intraband scattering

4.1.3 [InAs]n-[GaAs]n

4.2 Directionally dependent electron-phonon interaction model

4.2.1 Electron-phonon coupling strength

4.2.2 Electron wave function

4.2.3 Electron evolution equations

4.3 Full-band cascade: practical implementation

4.3.1 Electronic band

4.3.2 Phonon polar field

4.3.3 Electronic cascade

4.3.4 Validation: bulk case

4.4 Effects of the superlattice size

4.4.1 Effect on the electric field

4.4.2 Effect on the electron-phonon interaction dimensionality

4.5 Conclusions on the electron-phonon interaction in superlattices .

4.5.1 Approach and results

4.5.2 Approximations and subsequent limitations

4.5.3 Perspective on the hot-carrier effect in superlattices

**A Elements of Density Functional Theory **

**B Elements of Density Functional Perturbation Theory **

**C SiSn electronic band structure **

**D DFT calculation using hybrid functional **

**E Transferability of the interatomic force constants **

**F Conservation rules Gaussian width **

**G Average fourth-order phonon-phonon coupling constant **

Résumé en français

**Bibliography **