In this thesis we use global structural prediction methods (Particle Swarm Optimization and Minima Hopping Method) and high-throughput techniques to predict crystal structures of different systems under different conditions. We performed structural prediction by using the Crystal structure Analysis by Particle Swarm Optimization(CALYPSO) combined with Density Functional Theory (DFT) that made possible to unveil several stable compounds, so far unknown, on the phase diagrams of Ba-Si and N-H-O. Afterwards, we performed a high-throughput investigation on ternary compounds of composition ABX 2 , where A and B are elements of the periodic table up to Bi, and X is a chalcogen (O, S, Se, and Te) by using DFT and combining calculations of crystal prototypes with structural prediction (Minima Hopping Method). The following paragraphs summarize the content by chapter of this document.

Chapter 1 is a short introduction of this thesis.

Chapter 2 includes the basic theory used in this thesis. Firstly, a short introduction of DFT is presented. Then, we describe some approximate exchange-correlation functions that make DFT practical. Next, we introduce different structural prediction algorithms, especially Particle Swarm Optimization and Minima Hopping Method which we used in this thesis. Finally, we discuss how to determine if a new structure is thermodynamically stable.

In Chapter 3, we first consider the Ba-Si system. Using an unbiased structural search based on a particle-swarm optimization algorithm combined with DFT calculations, we investigate systematically the ground-state phase stability and structural diversity of Ba-Si binaries at high pressure. The phase diagram turns out to be quite intricate, with several compositions stabilizing/destabilizing as a function of pressure. In particular, we identify novel phases of BaSi, BaSi 2 , BaSi 3 , and BaSi 5 that might be synthesizable experimentally over a wide range of pressures. Our results not only clarify and complete the previously known structural phase diagram, but also provide new insights for understanding the Ba-Si binary system.

Chapter 4 contains the investigation of the phases diagram of the N-H-O system. By using an ab initio evolutionary structural search, we report the prediction of two novel stable crystalline phases in the N-H-O ternary phase diagram at high pressure, namely NOH 4 and HNO 3 (nitric acid). Our calculations reveal that the new C2/m phase of NOH 4 becomes stable at 71 GPa, while the P2 1 /m phase of HNO 3 stabilizes at 39 GPa. Both phases remain thermodynamically stable at least up to 150 GPa, the maximum pressure we considered. The C2/m phase of NOH 4 contains two O-H layers and one "dumbbell cluster" layer formed by two NH 3 molecules linked by a N-N covalent bond. The P2 1 /m phase of HNO 3 contains a surprising "quasi clover" layer formed N-O-H covalent bonds. Further calculations show that both phases are semiconductors, with a band gap of 6.0 eV and 2.6 eV for NOH 4 and NHO 3 , respectively. We also confirmed that the composition NOH 5 (NH 3 H 2 O) becomes unstable at pressures above 122 GPa, decomposing into NH 3 and H 2 O.

Chapter 5 focuses on p-type transparent electrodes of ternary chalcogenides. We use a high-throughput approach based on DFT to find delafossite and related layered phases of composition ABX 2 , where A and B are elements of the periodic table, and X is a chalcogen (O, S, Se, and Te). From the 15 624 compounds studied in the trigonal delafossite prototype structure, 285 are within 50 meV/atom from the convex hull of stability. These compounds are further investigated using global structural prediction methods to obtain their lowestenergy crystal structure. We find 79 systems not present in the "materials project database" that are thermodynamically stable and crystallize in the delafossite or in closely related structures. These novel phases are then characterized by calculating their band gaps and hole effective masses. This characterization unveils a large diversity of properties, ranging from normal metals, magnetic metals, and some candidate compounds for p-type transparent electrodes.

At the end of the thesis, we give our general conclusions and an outlook.

Résumé

Dans cette thèse nous utilisons des méthodes globaux de prédiction des structures cristallographiques avec les algorithmes d'optimisation par essaims particuliers (« Particle Swarm Optimization » en anglais) et de « Minima Hopping Method » combinés à des techniques de grande capacité de traitement de données afin de prédire la structure cristalline de différents systèmes et dans des conditions thermodynamiques variées. Nous avons réalisé des prédictions structurales utilisant l'analyse cristalline par optimisation par essaims particuliers (CALYPSO) combinés avec la Théorie Fonctionnelle de la Densité (DFT) ce qui a permis de mettre en évidence la stabilité de plusieurs composés jusqu'à la inconnus dans le digramme de phases du système Ba-Si et dans le système N-H-O. Nous avons également réalisé une étude à haute capacité de traitement de données sur un système ternaire de composition ABX 2 , avec A et B des éléments du tableau périodique allant jusqu'au bismuth et X étant un chalcogénure (O, S, Se ou Te) . Nous avons utilisé la Théorie Fonctionnelle de la Densité combinant calculs de prototypes structuraux à partir des prédictions structurelles avec la méthode de « Minima Hopping ». Dans les paragraphes suivants nous résumons le contenu de différents chapitres de cette thèse.

Le premier chapitre qui constitue une brève introduction au travail de cette thèse est suivi du chapitre 2 présentant les aspects théoriques utilisés dans ce travail. D'abord il est fait une brève introduction à la Théorie Fonctionnelle de la Densité. A continuation nous décrivons quelques fonctions d'échange-corrélation choisies qui constituent des approximations rendant l'utilisation de la DFT efficace. Ensuite nous présentons différents procédés de prédiction structurale, et en particulier les algorithmes d'optimisation par essaims particuliers et de « Minima Hopping » eu nous avons utilisés dans cette thèse. Finalement il est discuté comment doit-on se prendre pour évaluer la stabilité thermodynamique des nouvelles phases identifiées.

Dans le chapitre 3, nous considérons le système Ba-Si. A travers l'utilisation d'une recherche structurale non-biaisée basée sur l'algorithme d'optimisation par essaims particuliers combinée avec des calculs DFT, nous faisons une étude systématique de la stabilité des phases et de la diversité structurale du système binaire Ba-Si sous haute pression. Le diagramme de phases résultant est assez complexe avec plusieurs compositions se stabilisant et se déstabilisant en fonction de la pression. En particulier, nous avons identifié des nouvelles vi phases de stoechiométrie BaSi, BaSi 2 , BaSi 3 et BaSi 5 qui devraient pouvoir être synthétisées expérimentalement dans un domaine de pressions étendu. Nos résultants permettent non seulement de compléter le diagramme de phase de Ba-Si, mais ils donnent plus largement des clefs pour la compréhension du système binaire Ba-Si.

Dans le chapitre 4 est présentée notre étude du diagramme de phases du système N-H-O. S'appuyant sur une recherche structural «évolutive » de type ab initio, nous prédisons deux nouvelles phases du système ternaire N-H-O qui sont NOH4 et HNO 3 (acide nitrique). La nouvelle phase de NOH 4 a une symétrie C2/m et devient stable à partir de 71 GPa tandis que la phase P2 1 /m de HNO 3 est stabilisée à 39 GPa. Les deux phases restent thermodynamiquement stables jusqu'à 150 GPa, la pression limite de notre étude. La phase C2/m de NOH 4 présente deux couches O-H et une couche de type agrégat d'haltères formée de deux molécules NH 3 connectées par des liaisons covalentes N-N. La phase P2 1 /m de HNO 3 (acide nitrique) contient une couche surprenante de type quasi-trèfle formée des liaisons covalentes N-O-H. Le calcul de la structure électronique montre que les deux phases sont semiconductrices. Les gaps électroniques pour les phases de NOH 4 et NHO 3 sont de 6.0 et 2.6 eV respectivement. Nous confirmons également que la composition NOH 5 (NH 3 H 2 O) perd son stabilité pour des pressions supérieures à 122 GPa se décomposant en NH 3 et H 2 O à cette pression.

Le chapitre 5 se focalise sur les électrodes transparentes de type-p à base des chalcogénures ternaires. Nous utilisons une approche à grande capacité de traitement de données basée sur la DFT pour obtenir la delafossite et d'autres phases voisines de composition ABX 2 , avec A et B des éléments du tableau périodique et X des chalcogénures (O, S, Se et Te). A partir d'an total de 15624 composés étudiés dans la structure prototypique de la delafossite, 285 structures se trouvent dans la limite de stabilité d'enveloppe convexe avec une marge de dispersion de 50 meV/atome. Ces composés sont étudiés par les méthodes de prédiction structurale afin d'obtenir leur structure cristalline de plus basse énergie. Nous trouvons 79 systèmes qui sont absents de la base de données « Materials project database », qui sont stables du point de vue thermodynamique et qui cristallisent soit dans la structure delafossite, soit dans des structures très proches. Ces nouvelles phases sont ensuite caractérisées déterminant leur gap électronique et la masse effective des trous. Cette caractérisation révèle une grande diversité de propriétés allant depuis les métaux ordinaires aux métaux magnétiques et permettant d'identifier quelques candidats pour des électrodes transparents de type-p.

Nous présentons enfin à la fin du manuscrit nos conclusions générales et les perspectives de ce travail.

through high-throughput computational materials design. Finally, in Chapter 6 we present a summary and relevant conclusions of this thesis.

c

), and optimized parameters a = 0.19, b = 0.65, and μ = 0.334 bohr -1 . Another example in this class of approximations is the ωB97X exchange-

Chapter 1 Introduction

The discovery of new materials often lies at the base of important innovations for industrial applications. Moreover, it provides insight into fundamental scientific issues. We can obtain new materials by tailoring intrinsic materials parameters that can be controlled during synthesis and processing, such as dimensionality, composition and geometric size of the sample features. On the other hand, we can also use external conditions to induce changes in the material, such as temperature, pressure, electric, magnetic fields or others. In particular, pressure is a fundamental thermodynamic variable that can be used to control the properties of materials, because it can reduce interatomic distances, change bond patterns and profoundly modify electronic orbitals. It is thus a powerful tool for finding novel materials that are inaccessible at ambient conditions. Pressure can enable the creation of new materials mainly through four routes. Firstly, pressure can produce new materials through structural phase transitions. High pressure can change the energetics of the various possible structures by altering the interatomic distances and bonding patterns thus stabilizing new materials. A well known example is synthetic diamond which is made by the compression of graphite. The sp 2 C-C bonding in graphite changes into sp 3 C-C bonding in diamond due to the applied pressure. Secondly, pressure can stabilize new stoichiometries. Structural phase transitions originate from qualitative changes in the potential energy landscape. Pressure can also effectively modify the compositional landscape, leading to stabilization of the exotic stoichiometries that would not be expected from the conventional wisdom based on chemical rules at ambient conditions. Examples include Li x Al 1-x [START_REF] Sarmiento-Pérez | Novel phases of lithium-aluminum binaries from first-principles structural search[END_REF], Na x Au 1-x [START_REF] Sarmiento-Pérez | Sodium-gold binaries: novel structures for ionic compounds from an ab initio structural search[END_REF] and unusual stoichiometries of hydrogen sulfide H 3 S [START_REF] Bernstein | What superconducts in sulfur hydrides under pressure and why[END_REF]. Thirdly, pressure can enable new chemical reactions. Pressure can affect the chemical reactivity of elements by influencing the electronic orbitals and their occupancy. The outer atomic orbitals (such as the s, p and d orbitals) will hybridize as a result of the changing of the energetic ordering of the orbitals induced by pressure. Finally, pressure can change the electronic properties of materials.

Even in the case when the structure and stoichiometry remain unchanged under pressure, the electronic properties can be fundamentally modified, because pressure can reduce the interatomic distances and strengthen interatomic interactions. This can broaden energy bands thus enabling insulator-to-metal, semiconductor-to-metal or metal-to-superconductor transitions. For example, superconductivity of BaSi 2 changes remarkably under pressure [START_REF] Flores-Livas | Enhancing the Superconducting Transition Temperature of BaSi 2 by Structural Tuning[END_REF]. The main goals of this thesis are to discover new stoichiometries and new phases of the Ba-Si and N-H-O systems under high pressure and to explore their properties and the underlying physical mechanisms. Our study can give us a fundamental understanding and potential applications and prospects of these two systems.

As the demand for the materials is endlessly growing, high-throughput computational materials design also became a burgeoning area in recent years, in particular, because experimental discovery is bound by high costs and is very time-consuming. The high-throughput computational materials design is based on the marriage between computational quantummechanical-thermodynamic approaches and a multitude of techniques rooted in database construction and intelligent data mining. The concept is simple yet powerful: create a large database containing the calculated thermodynamic and electronic properties of existing and hypothetical materials, and then intelligently interrogate the database in the search of materials with the desired properties. Clearly, the entire construction should be validated by reality, namely the existing materials must be predicted correctly and the hypothetical ones should be eventually synthesized. Such reality check feeds back to the theory to construct better databases and increased predictive power. With the advent of efficient and accurate theoretical tools and inexpensive computers, high-throughput computational materials design has become an important tool for finding new materials. This method has successfully screened many different kind of materials and applications, such as solar energy harvesting [START_REF] Yu | Identification of potential photovoltaic absorbers based on firstprinciples spectroscopic screening of materials[END_REF], water photospliting [START_REF] Lin | In silico screening of carbon-capture materials[END_REF], carbon capture and gas storage [START_REF] Krishna | In silico screening of metal-organic frameworks in separation applications[END_REF], topological insulators [START_REF] Yang | A search model for topological insulators with high-throughput robustness descriptors[END_REF] and thermoelectric materials [START_REF] Yang | Evaluation of Half-Heusler Compounds as Thermoelectric Materials Based on the Calculated Electrical Transport Properties[END_REF]. In the domaing solar energy harvesting, another goal of this thesis is to find delafossite and related layered phases of composition ABX 2 which possess p-type transparent conduction, where A and B are elements of the periodic table, and X is a chalcogen (O, S, Se, and Te) by using high-throughput computational materials design. This study can provide precious indications for further experimental synthesis and opens the way for more specific theoretical and experimental studies of dopability and transport properties, which can be restricted only to the most interesting candidates. This thesis is organized as follows: Chapter 2 is an introduction to the theoretical framework. Then Chapter 3 and Chapter 4 are about novel phases of the Ba-Si system under high pressure and the phases diagram of the N-H-O system under high pressure. Chapter 5 is concerned with the investigation of ternary chalcogenides for p-type transparent electrodes Chapter 2

Theoretical background

Ab initio or first-principles methods have emerged in the last two decades as a powerful tool to probe the properties of matter at the microscopic scale. These approaches are used to derive macroscopic observables under the controlled condition of a "computational experiment," and with a predictive power rooted in the quantum-mechanical description of interacting atoms and electrons. Density functional theory (DFT) has become the default method of choice for most applications, due to its combination of reasonable scaling with system size and good accuracy in reproducing most ground state properties. Such an electronic structure approach can then be combined with molecular dynamics to provide an accurate description of thermodynamic properties and phase stability, atomic dynamics, and chemical reactions, or as a tool to sample the features of a potential energy surface. Due to the development of computer technology, large and complex systems are now accessible. For example, the largest system that simulated with DFT contains over 2 million atoms [START_REF] Bowler | Calculations for millions of atoms with density functional theory: linear scaling shows its potential[END_REF].

In this chapter, I will introduce the basic concepts and methods that were used in this thesis. As DFT is one of the most basic theoretical tools, I give a short introduction of this theory. Then, in the following sections, I describe different structural prediction algorithms. Finally, I give a discussion on the stability of crystal phases.

Density functional theory

We consider a N-electron system (atom, molecule, or solid) in the Born-Oppenheimer approximation [START_REF] Born | Zur quantentheorie der molekeln[END_REF] and non-relativistic approximation. The Schrödinger equation [START_REF] Dreizler | Density Functional Theory[END_REF] can be written as ĤΨ(x 1 , x 2 , . . . , x N ) = EΨ(x 1 , x 2 , . . . , x N ) , (2.1) where x i comprehends the spatial and spin coordinates of electron i (r i , σ i ). The Hamiltonian operator is formally written as

Ĥ = T + Vext + Vee , ( 2.2)
where

T = - N ∑ i=1 1 2 ∇ 2 i , ( 2.3)
is the kinetic enery operator,

Vext = N ∑ i=1 v ext (r i ) , (2.4) v ext (r i ) = -∑ I Z I r iI , ( 2.5)
is the sum of the external potential operators. Here, capital indices stand for the nuclear coordinates and Z I is the atomic number of the nuclei I. Hartree atomic units are used throughout. Furthermore Vee = N ∑ i< j 1 r i j , (2.6)

r i j = |r i -r j | , ( 2.7)
is the electron-electron interaction operator. To obtain the total energy of the system we have also to add the nuclei-nuclei repulsive energy contribution,

V nn = ∑ I<J Z I Z J r IJ .
(2.8)

The Hohenberg-Kohn theorem

Consider an electronic system with an arbitrary external local potential v ext . The corresponding ground-state wave function Ψ can be obtained by solving the Schrödinger equation (2.1), from which the associated ground-state density n 0 (r) can be deduced. Therefore, one has a mapping from the potential v ext to the ground-state density n 0 (r). In 1964, Hohenberg and Kohn [START_REF] Hohenberg | Inhomogeneous electron gas[END_REF] showed that this mapping can be inverted, i.e, the ground-state density n 0 (r) determines the potential v ext up to an arbitrary addictive constant. This can be easily proved by reductio ad absurdum. The ground-state density n 0 (r) determines the potential v ext , which in turn determines the Hamiltonian, and everything about the many-body problem. In other words, the potential v is a unique (up to an additive constant) functional of the ground-state density n, and for instance the ground-state wave function Ψ is also a unique functional of n (which can be denoted by Ψ[n]). Consequently, Hohenberg and Kohn defined the universal (i.e., independent from the external potential) density functional

F[n] = Ψ[n]| T + Vee |Ψ[n] , (2.9)
which can be used to define the total electronic energy under an external potential v ext as a functional of the density:

E v ext [n] = Ψ| T + Vext + Vee |Ψ = F[n] + dr n(r)v ext (r) .
(2.10)

Hohenberg and Kohn further showed that the density functional E v ext [n] satisfies a variational property: The ground-state energy E 0 of the system considered is obtained by minimizing this functional with respect to N-electron densities n that are ground-state densities associated with some local potential (referred to as v-representable densities) .11) the minimum being reached for the ground-state density n 0 (r) corresponding to the potential v ext . This can also be written as

E 0 = min {n} E v ext [n] . ( 2
δ δ n(r) E v ext [n] -μ n(r) dr = δ F[n] δ n(r) + v ext (r) -μ = 0 . (2.12)
where μ is a Lagrange multiplier that ensures the normalization of the density to the total number of electrons. Therefore, solving Eq. (2.12) would give us the exact ground-state of any electronic system.

The existence of a mapping from a ground-state density to a local potential, the existence of the universal density functional, and the variational property with respect to the density constitute the Hohenberg-Kohn theorem. Even though the original Hohenberg-Kohn theorem assumes a nondegenerate ground-state, it was showed later that the theorem is in fact also valid for a degenerate ground states (see, e.g., Ref. 12).

Kohn-Sham equations

As we discussed above, Hohenberg and Kohn in 1964 presented two theorems that are the core of the modern DFT [START_REF] Hohenberg | Inhomogeneous electron gas[END_REF]. They are known as the first and second Hohenberg-Kohn theorems. The Hohenberg-Kohn theorems tell us that the ground-state energy is a functional of the electronic density, however, the form of this functional remains unknown. The energy include kinetic energy and exchange-correlation energy parts which can not be written as an explicit functional of the density. To deal with this problem, Kohn and Sham considered an explicit system of non-interacting electrons ( Vee = 0) with the same density of the real system [START_REF] Kohn | Density Function Theory[END_REF]. In this case, the many-body ground state wave function can be written as a Slater determinant of single particle wave functions ψ i ,

Ψ s (x 1 , x 2 , . . . , x N ) = 1 √ N! ψ 1 (r 1 ) ψ 2 (r 1 ) • • • ψ N (r 1 ) ψ 1 (r 2 ) ψ 2 (r 2 ) • • • ψ N (r 2 ) . . . . . . . . . . . . ψ 1 (r N ) ψ 2 (r N ) • • • ψ N (r N ) , ( 2.13)
that satisfy:

- 1 2 ∇ 2 + v KS (r) ψ i (r) = ε i ψ i (r) . (2.14)
Here, v KS is a fictitious external potential. The ground-state density is obtained from the N lowest occupied orbitals and can be written as:

n(r) = N ∑ i=1 |ψ i (r)| 2 . (2.15)
For the interacting system, the universal functional can be written as

F HK [n] = T s [n] + E H [n] + E xc [n] , (2.16)
where E H [n] is the Hartree energy functional

E H [n] = 1 2 dr dr n(r) n(r ) |r -r | , ( 2.17)
that represents the classical electrostatic repulsion energy for the charge distribution. While T s is the independent-particle kinetic energy that can be given by

T s = - 1 2 N ∑ i=1 ψ i | ∇ 2 |ψ i = 1 2 N ∑ i=1 |∇ψ i | 2 , (2.18) E xc [n]
is the exchange-correlation energy functional that remains to be approximate. This function can be written as

E xc [n] = F HK [n] -(T s [n] + E H [n]) , (2.19)
or in the more revealing form

E xc [n] = T [n] +V ee [n] -(T s [n] + E H [n]) . (2.20)
which contains a kinetic contribution and a potential contribution. With these modified potentials the Euler equation of the Hohenberg-Kohn equation (2.12) can be written as

δ T s [n] δ n(r) + v ext (r) + 1 2 dr n(r ) |r -r | + δ E xc [n] δ n(r) -μ = 0 . (2.21)
We then define the Kohn-Sham potential

v KS [n](r) = v ext (r) + v H [n](r) + v xc [n](r) , (2.22)
where

v H [n](r) = 1 2 dr n(r ) |r -r | , (2.23) v xc [n](r) = δ E xc [n] δ n(r) . (2.24)
The Euler equation of the Hohenberg-Kohn equation (2.21) can be rewritten as

δ T s [n] δ n(r) + v KS [n](r) -μ = 0 . (2.25)
Solving this equation is equivalent to solving the single-particle Schrödinger equation:

- 1 2 ∇ 2 + v KS [n](r) ψ i (r) = ε i ψ i (r) . (2.26)
Equations (2.25) and (2.26) are known as the Kohn-Sham equations [START_REF] Kohn | Self-consistent equations including exchange and correlation effects[END_REF]. The orbitals satisfying (2.26) are called the Kohn-Sham orbitals. The Kohn-Sham Hamiltonian can be written as:

ĤKS = - 1 2 ∇ 2 + v KS [n](r) , (2.27)
v KS is called Kohn-Sham potential. The Kohn-Sham equations are similar to the Hartree-Fock equations, with the difference that they involve a local exchange potential instead of a nonlocal one, and an additional local correlation potential.

Making DFT practical: approximations

We review here the main classes of approximate exchange-correlation functionals. These classes are roughly ordered from the simplest to the most sophisticated ones. The discussion is focused on the contents of the approximations, not their performance in practical calculations. On average, more sophisticated approximations are usually more accurate than simpler ones, even though many exceptions can be found.

The local density approximation

In the local-density approximation (LDA), introduced by Kohn and Sham [START_REF] Kohn | Self-consistent equations including exchange and correlation effects[END_REF], the exchangecorrelation functional is written as

E LDA xc [n] = n(r)ε unif xc (n(r))dr , ( 2.28)
where ε unif xc (n) is the exchange-correlation energy per particle of the infinite uniform electron gas with the density n. The uniform electron gas represents a family of systems of interacting electrons with an arbitrary spatially constant density n that acts as a parameter. Thus, in the LDA, the exchange-correlation energy per particle of an inhomogeneous system at a spatial point of density n(r) is approximated as the exchange-correlation energy per particle of the uniform electron gas of the same density.

The function ε unif xc (n) is a sum of exchange and correlation contributions,

ε unif xc (n) = ε unif x (n) + ε unif c (n) . (2.29)
The exchange energy per particle of the uniform electron gas can be calculated analytically

ε unif x (n) = c x n 1/3 , ( 2.30)
where

c x = -(3/4)(3/π) 1/3 . (2.31)
The LDA exchange functional is associated with the names of Dirac [START_REF] Dirac | Note on exchange phenomena in the Thomas atom[END_REF] and Slater [START_REF] Slater | A simplification of the Hartree-Fock method[END_REF].

The correlation energy per particle ε unif c (n) of the uniform electron gas can not be calculated analytically. This quantity has been obtained numerically for a number of densities n using accurate quantum Monte Carlo calculations [START_REF] Ceperley | Ground state of the electron gas by a stochastic method[END_REF], and fitted to a parametrized function of n satisfying the known high-and low-density expansions. Expressed in terms of the Wigner-Seitz radius r s = (3/(4πn)) 1/3 , the first terms of the high-density expansion (r s → 0) have the form

ε unif c (n) = Alnr s + B +Cr s lnr s + O(r s ) , (2.32)
and the first terms of the low-density expansion (r s → +∞) have the form

ε unif c (n) = a r s + b r 3/2 s + O( 1 r 2 s ) , ( 2.33)
where A, B, C, a, and b are constants. The two most used parameterizations are the one of Vosko, Wilk, and Nusair (VWN) [START_REF] Vosko | Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis[END_REF] and the one of Perdew and Wang (PW92) [START_REF] Perdew | Accurate and simple analytic representation of the electron-gas correlation energy[END_REF].

Their forms are too complicated to be given here. These parameterizations also include the generalization to spin densities ε unif c (n ↑ , n ↓ ) differing from ε unif c (n) for spin-polarized systems (n ↑ = n ↓ ) (sometimes specifically referred to as the local-spin-density (LSD) approximation).

Generalized gradient approximations

The next logical step beyond the LDA is the gradient-expansion approximation (GEA). One way of deriving the GEA is to start from the uniform electron gas, introduce a weak and slowly-varying external potential v r , and expand the exchange-correlation energy in terms of the gradients of the density. At second order, one obtains a functional of the form

E GEA xc (n) = E LDA xc (n) + C xc n(r) 4/3 ∇n(r) n(r) 4/3 2 dr , ( 2.34)
where

C xc = C x +C c , (2.35)
is the sum of the exchange and correlation coefficients of the second order gradient expansion. Note that the gradient expansion is most naturally written in terms of the reduced density gradient ∇n(r)/n(r) 4/3 which is a dimensionless quantity. The GEA should improve over the LDA provided that the reduced density gradient is small. Unfortunately, for real systems, the reduced density gradient can be large in some regions of space, and the GEA turns out to be a worse approximation than the LDA.

The failure of the GEA lead to the development of generalized-gradient approximations (GGA), which really started in the 1980s, of the generic form

E GGA xc (n) = n(r) f (n(r), ∇n(r))dr , (2.36)
where f is some function. The GGA are semilocal approximations in the sense that f does not only use the local value of the density n(r) but also its gradient ∇n(r). For simplicity, we consider here only the spin-independent form, but in practice GGA functionals are more generally formulated in terms of spin densities (n ↑ , n ↓ ) and their gradients (∇n ↑ , ∇n ↓ ).

Many GGA functionals have been proposed. We very briefly review here some of the most widely used ones.

B88 exchange functional

The Becke 88 (B88 or B) [START_REF] Becke | Density-functional exchange-energy approximation with correct asymptotic behavior[END_REF] exchange functional is formulated as an additive correction to LDA. It consists in a compact function of n(r) and |∇n(r)|/n(r) 4/3 chosen to satisfy the exact asymptotic behavior of the exchange energy per particle, and with an empirical parameter fitted to Hartree-Fock exchange energies of rare-gas atoms.

LYP correlation functional

The Lee-Yang-Parr (LYP) [START_REF] Lee | Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density[END_REF] correlation functional is one of the rare functionals which have not been constructed starting from the LDA. It originates from the Colle-Salvetti [START_REF] Colle | Approximate calculation of the correlation energy for the closed shells[END_REF] correlation-energy approximation depending on the Hartree-Fock pair density and containing four parameters fitted to helium data. By making a further reasonable approximation, LYP turned the Colle-Salvetti expression into a density functional depending on the density n(r), the square of the density gradient |∇n(r)| 2 , and the Laplacian of the density ∇ 2 n(r). The dependence on the Laplacian can be exactly eliminated by an integration by parts [START_REF] Miehlich | Results obtained with the correlation energy density functionals of[END_REF].

PBE exchange-correlation functional

For the Perdew-Burke-Ernzerhof functional (PBE) [START_REF] Perdew | Generalized gradient approximation made simple[END_REF], the exchange and correlation energies per particle are expressed as simpler functions of n(r) and |∇n(r)| enforcing less exact conditions and with no fitted parameters. Specifically, the function used for correlation enforces the second-order small-gradient expansion in the high-density limit, the vanishing of correlation in the large-gradient limit, and the removal of the logarithm divergence of the LDA in the high-density limit (see Equation 2.32). The function used for exchange is chosen to cancel out the second-order small-gradient expansion of correlation and to enforce the Lieb-Oxford bound in the large-gradient limit.

Meta-generalized gradient approximations

The meta-generalized-gradient approximations (meta-GGA or mGGA) are of the generic form,

E mGGA xc = n(r) f (n(r), ∇n(r), ∇ 2 n(r), τ(r))dr .
(2.37)

They use more ingredients than the GGAs, namely the Laplacian of the density ∇ 2 n(r) and the non-interacting kinetic energy density τ(r):

τ(r) = 1 2 N ∑ i=1 |ψ i (r)| 2 , ( 2.38)
where ψ i (r) are the KS orbitals. Note that meta-GGAs are only implicit functionals of the density since τ(r) is itself an implicit functional of the density through the orbitals. This tacitly implies an extension of the usual KS method. The meta-GGAs are considered part of the family of semilocal approximations, in the sense that τ(r) depends only the gradient of the orbitals. Again, we consider here only the spin-independent form, but meta-GGA are more generally formulated in terms of spin-resolved quantities

(n ↑ , n ↓ , ∇n ↑ , ∇n ↓ , ∇ 2 n ↑ , ∇ 2 n ↓ , τ ↑ , τ ↓ ).
One motivation for the introduction of the variable τ(r) is that it appears in the short-range expansion of the exchange hole [START_REF] Becke | Hartree-Fock exchange energy of an inhomogeneous electron gas[END_REF], which for the case of a closed-shell system is

n x (r, r ) = - n(r) 2 - 1 3 ∇ 2 n(r) -4τ(r) + |∇n(r)| 2 2n(r) |r -r | 2 + . . . . (2.39)
Another important motivation is that τ(r) can be use to make the correlation energy per particle correctly vanish in spatial regions where the density has a one-electron character. This is done by comparing τ(r) with the von Weizsäcker kinetic energy density

τ W (r) = |∇n(r)| 2 8n(r) . (2.40)
Which is exact for one-electron systems, e.g. by introducing the ratio τ(r)/τ W (r).

One the most used meta-GGA approximation is the Tao-Perdew-Staroverov-Scuseria (TPSS) [START_REF] Tao | Climbing the density functional ladder: Nonempirical meta-generalized gradient approximation designed for molecules and solids[END_REF] exchange-correlation functional.

Hybrid approximations

Based on arguments relying on the adiabatic-connection formalism, in 1993 Becke [START_REF] Becke | A new mixing of Hartree-Fock and local density-functional theories[END_REF] proposed to mix a fraction of the exact or Hartree-Fock (HF) exchange energy with GGA functionals. In particular, he proposed a three-parameter hybrid (3H) approximation [START_REF] Becke | Density-functional thermochemistry. III. The role of exact exchange[END_REF] of the form

E 3H xc = aE HF x + bE GGA x + (1 -a -b)E LDA x + cE GGA c + (1 -c)E LDA c , ( 2.41)
where the three parameters a, b, and c are determined by fitting to experimental data. Note that E HF x depends on the occupied orbitals and has the exact-exchange expression:

E HF x = - 1 2 N ∑ i=1 N ∑ j=1 ψ * i (r 1 )ψ j (r 1 )ψ * j (r 2 )ψ i (r 2 ) |r 1 -r 2 | dr 1 dr 2 . (2.42)
This constitutes an extension of the usual KS method, sometimes referred to as generalized Kohn-Sham [START_REF] Seidl | Generalized Kohn-Sham schemes and the band-gap problem[END_REF]. The main benefit of adding a fraction of HF exchange is to decrease the self-interaction error in the exchange functional which tends to favor delocalized electron densities over localized electron densities. The most famous and widely used three-parameter hybrid approximation is B3LYP [START_REF] Stephens | Ab initio calculation of vibrational absorption and circular dichroism spectra using density functional force fields[END_REF], which uses the B88 exchange functional and the LYP correlation functional, and the parameters a = 0.20, b = 0.72, and c = 0.81.

In 1996, Becke proposed a simpler one-parameter hybrid (1H) approximation [32]

E 1H xc = aE HF x + (1 -a)E DFA x + E DFA c , (2.43)
where E DFA

x and E DFA c can be any (semilocal) density-functional approximation (DFA), and the fraction a of HF exchange has to be determined. The fraction of HF exchange should be small enough to keep the compensation of errors usually occurring between the approximate semilocal exchange functional E DFA x and the approximate semilocal correlation functional E DFA c . Fits to experimental data often give an optimal parameter a around 0.25. A rationale has also been proposed in favor of this value [START_REF] Perdew | Rationale for mixing exact exchange with density functional approximations[END_REF]. For example, PBE0 [START_REF] Adamo | Toward reliable density functional methods without adjustable parameters: The PBE0 model[END_REF][START_REF] Ernzerhof | Assessment of the Perdew-Burke-Ernzerhof exchangecorrelation functional[END_REF] is a popular one-parameter hybrid approximation which uses a = 0.25 and the PBE exchange and correlation functionals.

A strategy that has been sometimes used to construct approximations of the form of Equation 2.43 is to employ parameterized flexible functions for E DFA x and E DFA c and systematically optimize all the parameters (including the fraction a of HF exchange) on large sets of physicochemical properties of molecular systems. For example, the Becke 97 (B97) exchange-correlation functional [START_REF] Becke | Density-functional thermochemistry. V. Systematic optimization of exchange-correlation functionals[END_REF] is a hybrid GGA approximation containing 13 parameters optimized on atomic exchange and correlation energies, atomization energies, ionization potentials, and proton affinities. Another example is the so-called family of "Minnesota" functionals, and in particular the M06 exchange-correlation functional [START_REF] Zhao | The M06 suite of density functionals for main group thermochemistry, thermochemical kinetics, noncovalent interactions, excited states, and transition elements: two new functionals and systematic testing of four M06-class functionals and 12 other functionals[END_REF] which is a hybrid meta-GGA approximation containing 36 parameters optimized on a very large set of diverse physicochemical properties concerning main-group thermochemistry, reaction barrier heights, noncovalent interactions, electronic spectroscopy, and transition metal bonding.

Double-hybrid approximations

In 2006, Grimme [START_REF] Grimme | Semiempirical hybrid density functional with perturbative second-order correlation[END_REF] introduced a two-parameter double-hybrid (2DH) approximation

E 2DH xc = a x E HF x + (1 -a x )E DFA x + (1 -a c )E DFA c + a c E MP2 c , ( 2
E MP2 c = - 1 4 N ∑ i=1 N ∑ j=1 ∑ a≥N+1 ∑ b≥N+1 |(ia| jb) -(ib| ja)| 2 ε a -ε b -ε i -ε j , ( 2.45)
where (ia| jb) are two-electron integrals, ε are orbital energies, and i, j and a, b run over occupied and virtual spin orbitals, respectively. The presence of nonlocal MP2 correlation allows one to use a larger fraction of nonlocal HF exchange. For example, Grimme proposed the B2-PLYP approximation which uses the B88 exchange and LYP correlation functionals, and the parameters a x = 0.53 and a c = 0.27 optimized on experimental heats of formation of a set of molecules.

In 2011, Sharkas, Toulouse, and Savin [START_REF] Sharkas | Double-hybrid density-functional theory made rigorous[END_REF] showed how the double-hybrid approximations can be rigorously reformulated using the adiabatic-connection formalism, which leads to a one-parameter double-hybrid (1DH) approximation

E 1DH xc = λ E HF x + (1 -λ )E DFA x + (1 -λ 2 )E DFA c + λ 2 E MP2 c , ( 2.46)
where the fraction HF exchange a x = λ is now connected to the fraction of MP2 correlation a c = λ 2 . It turns out that using a x = λ = 0.53 nearly reproduces the parameter a c = λ 2 = 0.28 0.27 independently optimized in B2-PLYP.

The double-hybrid approximations are examples of correlation functionals depending on virtual orbitals. Another example of a correlation functional depending on virtual orbitals is the random-phase approximation (RPA), which goes beyond second order and has been the subject of intensive developments since the 2000s.

Range-separated hybrid approximations

Based on earlier ideas of Savin [START_REF] Seminario | Recent developments and applications of modern density functional theory[END_REF] in 2001, Iikura, Tsuneda, Yanai, and Hirao [START_REF] Iikura | A long-range correction scheme for generalized-gradient-approximation exchange functionals[END_REF] proposed a long-range correction (LC) scheme

E LC xc = E lr,HF x + E sr,DFA x + E DFA c , ( 2.47)
where E lr,HF

x is the HF exchange energy for a long-range electron-electron interaction

E lr,HF x = - 1 2 N ∑ i=1 N ∑ j=1 φ * i (r 1 )φ j (r 2 )φ * j (r 2 )φ i (r 1 )ω lr ee dr 1 dr 2 , ( 2.48)
where ω lr ee = er f (μr 12 )/r 12 where r 12 = |r 1r 2 | is the inter-electronic distance and er f is the error function

er f (x) = 2 √ π x 0 e -t 2 dt , ( 2.49)
and E sr,DFA x is the DFA exchange energy for the complementary short-range interaction. In addition, the short-range exchange part is easier to approximate with a semilocal DFA. The parameter μ in the error function controls the range of the separation and must be chosen, e.g. by fitting to experimental data. In practice, a value around μ 0.3 -0.5 bohr -1 is often found to be optimal. A popular example of such LC approximations is LC-ω PBE [START_REF] Vydrov | Assessment of a long-range corrected hybrid functional[END_REF] which uses a short-range version of the PBE exchange functional, as well as the standard PBE correlation functional. Note that the LC scheme has also been referred to as the range-separated hybrid exchange (RSHX) scheme [START_REF] Gerber | Hybrid functional with separated range[END_REF].

In 2004, Yanai, Tew, and Handy [START_REF] Yanai | A new hybrid exchange-correlation functional using the Coulomb-attenuating method (CAM-B3LYP)[END_REF], introduced a more flexible scheme called the Coulomb-attenuating method (CAM) [START_REF] Yanai | A new hybrid exchange-correlation functional using the Coulomb-attenuating method (CAM-B3LYP)[END_REF] in which fractions of HF exchange are added at both short and long range

E CAM xc = aE sr,HF x + bE lr,HF x + (1 -a)E sr,DFA x + (1 -b)E lr,DFA x + E DFA c , ( 2.50)
where E sr,HF x is the HF exchange energy for the short-range interaction and E lr,HF x is the DFA exchange energy for the long-range interaction. The reintroduction of HF exchange at short range improves thermodynamic properties such as atomization energies. According to this scheme, the authors proposed the CAM-B3LYP approximation which uses short-and long-range versions of the B88 exchange functional, the same correlation functional used in B3LYP (i.e., 0.81 E LYP c + 0.19E LDA correlation functional [START_REF] Chai | Systematic optimization of long-range corrected hybrid density functionals[END_REF] which is based on the B97 exchange-correlation functional with reoptimized parameters, and uses a = 0.16, b = 1, and μ = 0.3 bohr -1 . Another functional that can be considered as part of this class of approximations is the Heyd-Scuseria-Ernzerhof (HSE) exchange-correlation functional [START_REF] Heyd | Hybrid functionals based on a screened Coulomb potential[END_REF], which uses the parameters a = 0.25, b = 0 (i.e. no long-range HF exchange), and μ = 0.15 bohr -1 , with a long-range version of the PBE exchange correlation functional and the standard PBE correlation functional. The absence of HF exchange at long range makes this approximation particularly useful for solids.

Structural prediction algorithms

The crystal structure occupies a central and often critical role in material science, particularly when establishing a correspondence between material performance and its basic composition, since the properties of a material are intimately tied to its crystal structure. Experimentally, the structure is determined by x-ray diffraction. It has solved numerous problems of crystal structure. However, only a small faction of systems have been studied experimentally, especially at high pressure. Therefore, it is often useful to be able to predict the crystal structure with theoretical techniques. There are two main different ways to predict the structure of a material. They are high-throughput methods and global crystal structure prediction methods. It is well known that nature often chooses the same solution to similar problems. Therefore, one can take a known, experimentally characterized structure, change its chemical composition, and hope that this will be also a stable phase for a new composition. This is the main idea of high-throughput methods. This method was already used successfully for many systems, such as delafossites, perovskites and clathrates. Higher success rates can be achieved by using sophisticated machine learning algorithms, which can help to choose the most likely substitutions. This approach is currently widely used with very good results, and applications to several areas have already been published [START_REF] Fujimura | Accelerated Materials Design of Lithium Superionic Conductors Based on First-Principles Calculations and Machine Learning Algorithms[END_REF][START_REF] Ceder | Opportunities and challenges for first-principles materials design and applications to Li battery materials[END_REF][START_REF] Curtarolo | The high-throughput highway to computational materials design[END_REF][START_REF] Carrete | Finding unprecedentedly low-thermal-conductivity half-Heusler semiconductors via high-throughput materials modeling[END_REF]. However, as it relies heavily on experimental data, it fails to find compounds that are very different from what is already known. In this case, the global structure prediction methods that only use the chemical composition are possible. However, these methods are rather complex as they basically involve classifying a huge number of energy minima on the potential energy surface. About 20 years ago John Maddox even published an article in Nature to question the predictive power provided with only the knowledge of chemical composition [START_REF] Maddox | Waves caused by extreme dilution[END_REF]. His words still remain largely true, as evidenced by poor results of the latest blind test for crystal structure prediction [START_REF] Day | A third blind test of crystal structure prediction[END_REF]. Owing to significant progress in both computational power and basic material theory, it is now possible to predict the crystal structure using standard methods based on DFT and beyond. Some of these are simulated annealing [START_REF] Kirkpatrick | Optimization by simmulated annealing[END_REF], genetic algorithms [START_REF] Woodley | The prediction of inorganic crystal structures using a genetic algorithm and energy minimisation[END_REF][START_REF] Abraham | A periodic genetic algorithm with real-space representation for crystal structure and polymorph prediction[END_REF][START_REF] Oganov | Crystal structure prediction using ab initio evolutionary techniques: Principles and applications[END_REF][START_REF] Glass | USPEX-evolutionary crystal structure prediction[END_REF][START_REF] Trimarchi | Global space-group optimization problem: Finding the stablest crystal structure without constraints[END_REF], basin hopping [START_REF] Wales | Global optimization by basin-hopping and the lowest energy structures of Lennard-Jones clusters containing up to 110 atoms[END_REF][START_REF] Nayeem | A comparative study of the simulated-annealing and Monte Carlo-with-minimization approaches to the minimum-energy structures of polypeptides:[Met]-enkephalin[END_REF], metadynamics [START_REF] Martoňák | Predicting crystal structures: The Parrinello-Rahman method revisited[END_REF][START_REF] Martoňák | Simulation of structural phase transitions by metadynamics[END_REF], random sampling [START_REF] Pickard | Structure of phase III of solid hydrogen[END_REF][START_REF] Pickard | Highly compressed ammonia forms an ionic crystal[END_REF][START_REF] Pickard | High-pressure phases of silane[END_REF], data mining [START_REF] Mujica | Theoretical study of the high-pressure phase stability of GaP, InP, and InAs[END_REF], particle-swarm optimization [START_REF] Wang | Crystal structure prediction via particle-swarm optimization[END_REF][START_REF] Wang | A method for crystal structure prediction[END_REF] and minima hoping [START_REF] Goedecker | Minima hopping: An efficient search method for the global minimum of the potential energy surface of complex molecular systems[END_REF][START_REF] Amsler | Crystal structure prediction using the minima hopping method[END_REF]. In the next paragraph we will discuss the two main algorithms that were used in this thesis: particle-swarm optimization (CALYPSO) and minima hoping method (MHM).

Particle swarm optimization

Particle-swarm optimization was first proposed by Kennedy and Eberhart in the mid 1900s [START_REF] Kennedy | Particle swarm optimization {C}[END_REF][START_REF] Lu | Dynamic population strategy assisted particle swarm optimization in multiobjective evolutionary algorithm design[END_REF]. As a stochastic global optimization method, particle-swarm optimization is inspired by the choreography of a bird flock and can be seen as a distributed behavior algorithm that performs multidimensional search. According to the particle-swarm optimization, the behavior of each individual is affected by either the best local or the best global individual to help it fly through a hyperspace. Moreover, an individual can learn from its part experiences to adjust its flying speed and direction. Therefore, all the individuals in the swarm can quickly converge to the global position and near-optimal geographical position by the behavior of the flock and their flying histories. Particle-swarm optimization (PSO) has been verified to perform well on many optimization problems. Y. Ma and Y. Wang implemented PSO algorithm for crystal structure prediction in the crystal structure analysis by particle swarm optimization CALYPSO code [START_REF] Wang | Crystal structure prediction via particle-swarm optimization[END_REF][START_REF] Wang | A method for crystal structure prediction[END_REF]. In this algorithm, a structure (an individual) in the searching phase space is viewed as a particle x t i, j . A set of individual particles is called a population or a generation. During the generation of particles, the new particle, x t i, j , depends on its previous position, x t i, j , and its velocity, v t i, j , as x t i, j = x t i, j + v t i, j . It is necessary to note that the velocity plays an important role on the determination of the speed and direction of the particle movement. The new velocity of each individual i at the jth dimension is calculated based on its previous location (x t i, j ) before optimization, previous velocity v t i, j , current location pbest t i, j with an achieved best fitness, i.e., lowest enthalpy, of this individual, and the population global location gbest t i, j with the best fitness value for the entire population according to equation 2.51

v t i, j = ωv t i, j + c 1 r 1 (pbest t i, j -x t i, j ) + c 2 r 2 (gbest t i, j -x t i, j ) , ( 2.51)
where j ∈ (1, 2, 3), ω (in the range of 0.9-0.4) denotes the inertia weight and controls the momentum of the particle. c 1 is a self-confidence factor and expresses how much the particle trusts its own past experience while c 2 is a swarm confidence factor and expresses how much it trusts the swarm. r 1 and r 2 are two separately generated random numbers and uniformly distributed in the range [0, 1]. The velocity update formula includes random parameters r 1 and r 2 to ensure good coverage of the searching space and avoid entrapment in local optima. As shown in equation 2.51, it is quite obvious that the movement of particles in the search space is dynamically influenced by their individual past experience (pbest t i, j , v t i, j ) and successful experiences attained by the whole swarm gbest t i, j . Thus the velocity makes the particles to move toward to global minimum and accelerates the convergence speed. The possible potential energy face of each particle can be shown in Fig. 2.1.

Minima hopping method

The Minima Hopping Method, developed by S. Gödecker and M. Amsler [START_REF] Goedecker | Minima hopping: An efficient search method for the global minimum of the potential energy surface of complex molecular systems[END_REF][START_REF] Amsler | Crystal structure prediction using the minima hopping method[END_REF], is a structural prediction algorithm that tries to find the global minimum of a potential energy surface by jumping between local minima using short molecular dynamics (MD) runs (solid arrows in Fig. 2.2), followed by local geometry optimizations at each minima (dashed arrows). An energy based acceptance criteria assures that each new minimum has a 50% chance of being accepted (δ E in the figure). This means that every time a new minimum is found, the acceptance criterion is adjusted to guarantee that, on average, half of the local minima are accepted. In order to avoid revisiting already explored minima, the temperature of the MD runs is adjusted dynamically during the search. This is done to allow escaping local minima surrounded by high energy barriers. A softening process is used to accelerate the minimization problem as it is expected that (on average) crossing low energy barriers will lead to lower energy minima [START_REF] Roy | Bell-Evans-Polanyi principle for molecular dynamics trajectories and its implications for global optimization[END_REF]. Hence, the MD velocities should be aligned along directions of low curvature -soft modes. In practice, as a perfect alignment would eliminate the random character of the escape step, these directions are chosen only to avoid hard mode directions.

Figure 2.2 Hypothetical potential energy surface for minima hopping method. The x axis represents some internal degree of freedom, while y is the energy for a given configuration.

This method is able to optimize both cell vectors and atomic positions, being the only constraints the stoichiometry and the number of unit cells. In our runs, we start always from a random structure, assuring only that the minimum distance between atoms is at least equal to the sum of their covalent radii.

The minima hopping method has been used for structural prediction in a wide range of materials [START_REF] Amsler | Crystal Structure of Cold Compressed Graphite[END_REF][START_REF] Huan | Low-Energy Polymeric Phases of Alanates[END_REF][START_REF] Amsler | Conducting Boron Sheets Formed by the Reconstruction of the α-Boron (111) Surface[END_REF][START_REF] Cerqueira | The crystal structure of p-type transparent conductive oxide CuBO 2[END_REF], including the dependence on pressure [START_REF] Flores-Livas | High-Pressure Structures of Disilane and Their Superconducting Properties[END_REF] and the exploration of binary phase diagrams [START_REF] Sarmiento-Pérez | Novel phases of lithium-aluminum binaries from first-principles structural search[END_REF][START_REF] Valencia-Jaime | Novel crystal structures for lithium-silicon alloy predicted by minima hopping method[END_REF] with remarkable results.

Structural stability

How can we determine a new structure is thermodynamically stable? Here we discuss this common problem. As we know, even for a simple system there are also lots of possible compositions. However most of them are not thermodynamically stable. In order to determine if a structure is thermodynamically stable, let us consider a system at zero temperature. We should compare the formation enthalpy of the new compound with all the possible decomposition channels through Eq. 2.52,

ΔH| P ≡ H(A y B 1-y ) -yH(A) -(1 -y)H(B) .
(2.52)

Figure 2.3 An example of convexhull

The distance to the convex hull is the measure of the energy released by decomposing into the stable phases. The structures that are above the convex hull are thermodynamic unstable, while the structures that on the convex hull are thermodynamic stable and may be synthesized experimentally. Fig. 5.2 shows a convex hull (in green) for a A-B binary system. Blue points indicate phases that are not on the hull and therefore unstable and red points indicate stable phases. For instance, the construction shows directly that the phase γ at AB will decompose into α 1 and β 2 .

Another important aspect to take in account is the mechanically stability. When we predict a new structure that is thermodynamically stable, we should calculate the phonon spectrum of this new structure to insure that it is mechanically stable. If there are no imaginary phonon frequencies we can say this structure is stable, otherwise this structure is not stable even though it is thermodynamically stable. In this case it is means we did not find the ground state structure for this system.

Chapter 3

New phases of Ba-Si system under high pressure

Introduction

Alkali-earth silicides are extremely versatile materials, capable of crystallizing in a large variety of structures. For example, BaSi 2 , perhaps the most studied member of this family, exhibits four different phases at ambient conditions, [START_REF] Evers | Electrical properties of CrB type alkaline earth monosilicides and monogermanides[END_REF] namely an orthorhombic (Pnma), cubic (P4 3 32), trigonal (P 3m1), and hexagonal (P6/mmm) phase [START_REF] Schäfer | BaSi2, a phase with discrete Si4 tetrahedra[END_REF][START_REF] Evers | Crystal Structure of Barium Disilicide at High Pressures[END_REF][START_REF] Evers | A New High-Pressure Phase of Barium Disilicide[END_REF][START_REF] Evers | Transformation of three-connected silicon in BaSi 2[END_REF]. This rich structural diversity naturally leads to a wealth of interesting physical properties. The orthorhombic and cubic phases are semiconductors [START_REF] Kumar | BaSi2 as a promising low-cost, earth-abundant material with large optical activity for thin-film solar cells: A hybrid density functional study[END_REF][START_REF] Imai | Electrical resistivity of metastable phases of BaSi 2 synthesized under high pressure and high temperature[END_REF] with a suitable band-gap for photovoltaic applications [START_REF] Pokhrel | Synthesis and characterization of barium silicide (BaSi 2 ) nanowire arrays for potential solar applications[END_REF] and a large absorption coefficient [START_REF] Morita | Optical and electrical properties of semiconducting BaSi 2 thin films on Si substrates grown by molecular beam epitaxy[END_REF]. On the other hand, the trigonal phase is metallic, and becomes superconducting at low temperatures [START_REF] Flores-Livas | Enhancing the Superconducting Transition Temperature of BaSi 2 by Structural Tuning[END_REF][START_REF] Imai | Superconductivity of trigonal BaSi 2[END_REF][START_REF] Flores-Livas | Superconductivity in intercalated group-IV honeycomb structures[END_REF][START_REF] Flores-Livas | Superconductivity in layered binary silicides: A density functional theory study[END_REF]. It is interesting to note that this trigonal phase is composed of corrugated planes of silicon intercalated by barium atoms, and can be viewed as the silicene counterpart of intercalated graphite. It is also worth mentioning that alkali-earth disilicides are also used as percursors for the synthesis of silicon clathrates [START_REF] Toulemonde | High pressure synthesis and properties of intercalated silicon clathrates[END_REF][START_REF] Yamanaka | High-pressure synthesis of a new silicon clathrate superconductor[END_REF]. At ambient pressure there is still another stable composition in the Ba-Si phase diagram, namely BaSi [START_REF] Evers | Electrical properties of CrB type alkaline earth monosilicides and monogermanides[END_REF][START_REF] Rieger | Alkaline earth silicides, germanides and stannides with CrB structure type[END_REF], that crystallizes in a metallic CrB-type phase (Cmcm).

Unfortunately, at high-pressures there is relatively little experimental and theoretical work performed on this phase diagram. The composition BaSi 6 , has been synthesized at around 15 GPa [START_REF] Yamanaka | Structural evolution of the binary system Ba-Si under high-pressure and high-temperature conditions[END_REF], but nothing else is known about this system. Moreover, it is by now well known that high pressure can change dramatically the chemistry of even the simplest elements [START_REF] Dong | How do chemical properties of the atoms change under pressure[END_REF], leading to the stabilization of diverse, and often unexpected compositions [START_REF] Zhang | Unexpected Stable Stoichiometries of Sodium Chlorides[END_REF][START_REF] Zhu | Reactions of xenon with iron and nickel are predicted in the Earth's inner core[END_REF].

The objective of this work is therefore to perform a systematic theoretical study of the high-pressure Ba-Si phase diagram, especially in the silicon rich region. We will concentrate on the pressure range from 0 to 150 GPa as these pressures can nowadays be easily reached in diamond anvil cells. This will be done through an unbiased structural search based on a particle-swarm optimization Algorithm [START_REF] Wang | Crystal structure prediction via particle-swarm optimization[END_REF][START_REF] Wang | A method for crystal structure prediction[END_REF] (see Chapter 2) and a first principles density-functional approach to evaluate total energies and forces. The effectiveness of this method has already been demonstrated by several predictions under high pressure [START_REF] Cui | Novel high-pressure crystal structures of boron trifluoride[END_REF][START_REF] Wang | Superconductive sodalite-like clathrate calcium hydride at high pressures[END_REF][START_REF] Zhu | Spiral chain O4 form of dense oxygen[END_REF][START_REF] Li | Metallic Icosahedron Phase of Sodium at Terapascal Pressures[END_REF][START_REF] Liu | Proton or deuteron transfer in phase IV of solid hydrogen and deuterium[END_REF][START_REF] Peng | Predicted lithium-boron compounds under high pressure[END_REF][START_REF] Lu | Structural evolution of carbon dioxide under high pressure[END_REF][START_REF] Lv | Predicted novel high-pressure phases of lithium[END_REF][START_REF] Cui | Hydrogen segregation and its roles in structural stability and metallization: silane under pressure[END_REF]. We remark that a surprising structural richness has recently been revealed using the same structural search algorithm in the Ca-Si phase diagram [START_REF] Gao | Novel Si Networks in the Ca/Si Phase Diagram under Pressure[END_REF].

Methods

We investigated the stoichiometries BaSi x (x=1, 2, 3, 4, 5, 6), using cells containing 2, 4, 6, and 8 formula units in a pressure range from 0 to 150 GPa. Ground-state structures were obtained using the CALYPSO (Crystal structure AnaLYsis by Particle Swarm Optimization) global structure prediction code [START_REF] Wang | Crystal structure prediction via particle-swarm optimization[END_REF][START_REF] Wang | A method for crystal structure prediction[END_REF]. Total energies and forces were calculated with an ab initio density functional approach [START_REF] Kohn | Density Function Theory[END_REF] within the Perdew-Burke-Ernzenhof [START_REF] Perdew | Generalized gradient approximation made simple[END_REF] generalized gradient approximation as implemented in the VASP code [START_REF] Kresse | Ab initio molecular dynamics for liquid metals[END_REF]. We adopted the all-electron projector-augmented wave method [START_REF] Kresse | From ultrasoft pseudopotentials to the projector augmentedwave method[END_REF] using the valence configurations 5s 2 5p 6 6s 2 (for Ba) and 3s 2 3p 2 (for Si). We used a cutoff energy of 600 eV and uniform Monkhorst-Pack k-meshes [START_REF] Monkhorst | Special points for Brillouin-zone integrations[END_REF]. The convergence of all ab initio calculations was ensured to better than 1 meV/atom. The phonon density of states was calculated using a finite displacement method [START_REF] Parlinski | First-principles determination of the soft mode in cubic ZrO 2[END_REF] as implemented in the PHONOPY package [START_REF] Togo | First-principles calculations of the ferroelastic transition between rutile-type and CaCl 2 -type SiO 2 at high pressures[END_REF]. Electron-phonon coupling (EPC) calculations were carried out with density function perturbation theory using the program Quantum ESPRESSO [START_REF] Espresso | a modular and open-source software project for quantum simulations of materials[END_REF] with a cutoff of 80 Ry, A 12 × 12 × 12 q-point mesh, and a Gaussian smearing of 0.05 Ry. We note that all calculations were performed at 0 K neglecting the zero-point correction to the energy, that in any case is expected to be small for our systems.

Results and discussion

We investigated the energetic stability of the Ba-Si system by studying the enthalpy of formation for different BaSi x compounds with pressures from 0 to 150 GPa. The formation enthalpy H of Ba y Si 1-y (0 < y < 1) is defined as

ΔH| P ≡ H(Ba y Si 1-y ) -yH(Ba) -(1 -y)H(Si) , ( 3.1)
where H(Ba) and H(Si) represent the enthalpy of formation of bulk Ba and Si. For Ba, We used the Im 3m [START_REF] Probst | Superconductivity of bcc barium under pressure[END_REF] and P6 3 /mmc [START_REF] Kenichi | High-pressure structural study of barium to 90 GPa[END_REF] structures under ambient pressure and high pressure, respectively, while for pure Si we used the Fd 3m phase at ambient pressure, the Si-VI (Cmca) [START_REF] Hanfland | Crystal structure of the highpressure phase silicon VI[END_REF] phase at 20 GPa, the Si-VII (P6 3 /mmc) [START_REF] Olijnyk | Structural phase transitions in Si and Ge under pressures up to 50 GPa[END_REF] and Si-X (Fm 3m) [START_REF] Duclos | Experimental study of the crystal stability and equation of state of Si to 248 GPa[END_REF] phases at high pressure. The enthalpy of formation ΔH, and the respective convex hull of thermodynamic stability, is depicted in Fig. 3.1 for pressures of 0, 5, 20, 50, 100, and 150 GPa. From the convex hulls in Fig. 3.1 we can see that all Ba-Si compounds were found to have a negative formation enthalpy at ambient pressure. Both BaSi and BaSi 2 are thermodynamically stable, in good agreement with experimental results [START_REF] Evers | Electrical properties of CrB type alkaline earth monosilicides and monogermanides[END_REF][START_REF] Schäfer | BaSi2, a phase with discrete Si4 tetrahedra[END_REF][START_REF] Evers | Crystal Structure of Barium Disilicide at High Pressures[END_REF][START_REF] Evers | A New High-Pressure Phase of Barium Disilicide[END_REF][START_REF] Evers | Transformation of three-connected silicon in BaSi 2[END_REF], with the former being the most stable composition. The situation, however, becomes quite complex when we increase the pressure. At 5 GPa, also BaSi 6 becomes thermodynamically stable, again in agreement with experiment [START_REF] Yamanaka | Structural evolution of the binary system Ba-Si under high-pressure and high-temperature conditions[END_REF]. At 20 GPa, however, the formation enthalpy of BaSi 5 decreases, and it becomes thermodynamically stable, thereby destabilizing BaSi 6 . At this pressure, also BaSi 3 has a phase that is extremely close to the convex hull. As the pressure is further increased to 50 GPa, all these compositions with high silicon content become thermodynamically unstable. At 100 GPa BaSi also becomes unstable, and finally, at 150 GPa we do not find any stable binary compound in the Ba-Si system. The same behavior was observed for Ca-Si binaries at pressure above 250 GPa [START_REF] Gao | Novel Si Networks in the Ca/Si Phase Diagram under Pressure[END_REF]. The explanation given in Ref. [START_REF] Gao | Novel Si Networks in the Ca/Si Phase Diagram under Pressure[END_REF] is that while both energy and PV contributions to the enthalpy favor compound formation at moderate pressures, the ΔPV term quickly grows negative and dominates enthalpy variations at high pressure. The higher compressibility of Ca with respect to Si and Ca-Si binaries is at the origin of the phase separation at higher pressure. A similar explanation applies also to our Ba-Si phase diagram, as a consequence of the higher compressibility of Ba. It can be easily found from Fig. 3.2. The structural parameters of our predicted stable structures for Ba-Si compounds at their corresponding pressures are shown in Tab. 3.1.

The phase transitions of BaSi under pressure are further illustrated in Fig. 3.3, where we plot the relative enthalpies per formula unit with respect to the Cmcm structure (that is the ground-state at ambient conditions), including the enthalpy of the most likely decomposition channel.

In the following we give a more detailed analysis of the new stable structures found during our structural prediction runs. The structures are shown in Fig. 3.4. We calculated phonon spectra, which confirmed the dynamical stability of all the new phases. Finally, we determined the electronic character (semiconducting or metallic) by inspection of their electronic density of states.

BaSi

We now focus on Fig. 3.1 and Fig. 3.3 to discuss the stability of BaSi under pressure. We can see that the BaSi composition is stable in a large range of pressures, from ambient pressure to more than 65 GPa, after which the decomposition into Ba and BaSi 2 becomes energetically favorable. At ambient pressure, our structural prediction algorithm obtains a ground-state structure with Cmcm symmetry. This is the phase already known experimentally, in which Ba atoms form distorted hexagonal structures with one-dimensional channels that are filled with zigzag chains of Si atoms [START_REF] Evers | Electrical properties of CrB type alkaline earth monosilicides and monogermanides[END_REF][START_REF] Rieger | Alkaline earth silicides, germanides and stannides with CrB structure type[END_REF]. However, this phase is quickly destabilized by pressure, and a novel structure with space group Imma becomes the ground-state at pressures of around 6 GPa.

The new predicted Imma phase of BaSi is shown in Fig. 3.4 (a). Its primitive cell has two Ba atoms and two Si atoms. There are striking similarities between this structure and the low-pressure Cmcm phase. Both phases contain one-dimensional hexagonal channels formed by Ba atoms, but now the hexagons are regular due to a rearrangement of the Ba-atoms. The Si chains have the same arrangement in both cases. The distance between the two Figure 3.3 Relative enthalpy per formula unit of the different phases of BaSi with respect to the Cmcm phase [START_REF] Evers | Electrical properties of CrB type alkaline earth monosilicides and monogermanides[END_REF][START_REF] Rieger | Alkaline earth silicides, germanides and stannides with CrB structure type[END_REF] as a function of pressure. The Imma phase is a new structure predicted in this work. We also plot the enthalpy of the most likely decomposition channel into Ba and BaSi 2 . Note that the green line with diamonds refers to a different, meta-stable Cmcm phase and not to the ground-state Cmcm phase at low-pressures. silicon atoms in the Imma phase is 2.31 Å, while the angle of the chain is about 105.34 • at 20 GPa. To demonstrate the dynamical stability of this structure, we calculated the phonon dispersion for the Imma phase at 20 GPa, as shown in Fig. 3.5a. There are no imaginary phonon frequencies, which indicates that the Imma structure is mechanically stable. The electronic band structure and projected density of states (DOS) for the Imma structure at 20 GPa are also shown in Fig. 3.5b, showing that this is a metallic phase.

BaSi 2

From Fig. 3.1 we can see that BaSi 2 is a stable composition from ambient pressure to more than 100 GPa. At ambient pressure our structural prediction runs find many possible lowenergy structures, in good agreement with previous studies [81-84, 95, 120-123]. The lowest energy structure is the orthorhombic Pnma phase, with several other phases competing in the narrow energy range of a few tenths of an meV/atom. The phase transition from the Pnma to the P4 3 32 phase is predicted to occur at 1 GPa, followed by a phase transition to a P3m1 structure at about 3.5 GPa, and then to a P6/mmm phase at around 18 GPa. All these results are in good agreement with previous works [START_REF] Schäfer | BaSi2, a phase with discrete Si4 tetrahedra[END_REF][START_REF] Evers | Crystal Structure of Barium Disilicide at High Pressures[END_REF][START_REF] Evers | A New High-Pressure Phase of Barium Disilicide[END_REF][START_REF] Evers | Transformation of three-connected silicon in BaSi 2[END_REF]. However, at pressures above 20.8 GPa we found that the ground-state structure is a new cubic phase with Fd 3m symmetry. Moreover, this is the only low-enthalpy phase that we found in our simulations at high pressure. From Fig. 3.6 we can see that the Fd 3m phase is stable with respect to decomposition in Ba and Si in a large range of pressures from 21 GPa to more than 100 GPa.

The new Fd 3m phase can seen in Fig. 3.4 (b). In our opinion this is a remarkable structure for several reasons. Four neighboring silicon atoms form a regular tetrahedron with a side of 2.54 Å at 50 GPa. These tetrahedral then combine into larger tetrahedral, as shown on Fig. 3.4 (b). 6-fold coordination is observed in the first high pressure phase of pure silicon (β -tin structure), even if it is not a regular octahedron. Due to this unique structural arrangement, the Fd 3m phase is the ground-state in a large range of pressure. In order to ensure the thermal stability of Fd 3m, we calculated the phonon dispersion of this phase at 50 GPa, as shown in Fig. 3.7a. As expected, The Fd 3m phase is mechanically stable. We also calculated the electronic band structure and projected density of states, as shown in Fig. 3.7b. It is evident that the Fd 3m phase is also a metallic phase. We obtained the superconducting transition temperature of the Fd 3m phase by calculating the electronphonon coupling parameter λ and the Eliashberg phonon spectral function α 2 F(ω). The results are shown in Fig. 3.8. The overall integrated λ is 0.4, while the logarithmic average of the phonon frequency ω log is 320 K. The superconducting critical temperature T c estimated by using the McMillan equation [START_REF] Mcmahon | The properties of hydrogen and helium under extreme conditions[END_REF] modified by Allen and Dynes. is 1.24 K at 50 GPa with the screened Coulomb interaction μ * =0.12. All these results are in good agreement with a previous work [START_REF] Flores-Livas | Computational and experimental studies of sp 3 -materials at high pressure[END_REF]. X-ray diffraction at ambient conditions shows the pressure induced amorphization of BaSi 2 beyond 10 GPa [START_REF] Nishii | X-ray diffraction study of amorphous phase of BaSi 2 under high pressure[END_REF]. Consequently the experimental identification of the new Fd 3m phase will most probably need to perform a combined high pressure and high temperature experiment.

There are strong similarities between the tetrahedral arrangement of the Si atoms and the so-called T-carbon, a recently proposed allotrope of diamond [START_REF] Sheng | A Novel Carbon Allotrope[END_REF]. Both phases are constructed from tetrahedra, however T-carbon is obtained by occupying a diamond lattice site with a tetrahedra. In order to better compare both geometrical arrangements we studied the Si framework of our Fd 3m structure by removing the Ba atoms, and compared it to T-silicon i.e., the phase obtained by replacing C by Si in the T-carbon structure. T-silicon is considerably less dense (with a volume of 52.21 Å 3 /atom) than our structure (21.61 Å 3 /atom). For comparison, silicon in the diamond structure has a volume of 20.35 Å 3 /atom within our numerical approximations. It turns out that our structure is considerably lower in energy than T-silicon (by 0.38 eV per atom), but still considerably higher than silicon in the diamond structure (by 0.66 eV/atom). Unfortunately, such large energy differences probably mean that it is impossible to stabilize these silicon phases at ambient pressure. The composition BaSi 3 is very close to the convex hull for pressures of around 20 GPa. Our predicted structure [see Fig. 3.4 (c)] has I4/mmm symmetry and 4 formula units in its primitive cell. This is a very complicated and unusual structure, with corrugated Ba planes separated by Si layers. In these layers we find silicon cubes connected between themselves through bridging Si atoms in a two-dimensional lattice. The smallest distance between two silicon atoms at 20 GPa is 2.34 Å, while the longest Si-Si bond is 2.47 Å.

On the other hand, we found that BaSi 5 crystallizes in a fascinating Cmmm structure [see Fig. 3.4 (d)] that is on the convex hull in a small pressure range at around 20 GPa. This phase consists of two superposed flat hexagonal Si layers intercalated by Ba atoms that are located at the center of the Si hexagons. Between the two Si planes one finds further Si atoms also at the center of the hexagons. The Si-Si bond distance is in this case 2.43 Å.

The composition BaSi 6 is stable at pressures from 5 to 20 GPa. Our structural prediction runs find an interesting structure with Cmcm symmetry, where silicon atoms form a beer barrel type polyhedron, with the polyhedra linked by sharing faces to form channels containing the Ba atoms. This is the same structure found in previous works [START_REF] Yamanaka | Structural evolution of the binary system Ba-Si under high-pressure and high-temperature conditions[END_REF]. From the phonon dispersion of the I4/mmm phase of BaSi 3 and the Cmmm phase of BaSi 5 , as shown in Fig. 3.9a and Fig. 3.10a, we can ensure that both structures are mechanically stable. We calculated the band structure and the electronic density of states for the I4/mmm phase of BaSi 3 and the Cmmm phase of BaSi 5 . As shown in Fig. 3.9b and Fig. 3.10b, both phases are metallic.

Conclusions

In summary, using a global structural prediction method we studied systematically the phase diagram and the electronic properties of Ba-Si compounds. We identify five novel stoichiometric BaSi x phases with unexpected structures that might be experimentally synthesizable over a wide range of pressure. These include BaSi, BaSi 2 , BaSi 3 , BaSi 5 , and BaSi 6 . For the BaSi system, we find a new metallic Imma phase that is stable under pressure from 6 GPa to 65 GPa. For BaSi 2 , there is a new metallic phase Fd 3m that is stable from 21 GPa to 100 GPa. This phase, in which each Si forms six bonds with other neighboring Si atoms, becomes superconducting below 1.2 K at 50 GPa. Finally, at around 20 GPa two unexpected compositions become stable, namely BaSi 3 (I4/mmm) and BaSi 5 (Cmmm), both displaying metallic behavior. These results show the remarkable variety of Ba-Si compounds, and can serve as an important guide for further experimental studies of this binary system.

Chapter 4

Phase diagram of N-H-O system under high pressure

Introduction

Nitrogen, oxygen and hydrogen are some of the simplest and better studied elements of the periodic table. Moreover, they are some of the most abundant elements in the solar system and form several textbook compounds (such as water, nitrogen oxides, ammonia, nitric acid, etc.) that are fundamental not only to life, but also to modern industry [START_REF] Wigner | On the possibility of a metallic modification of hydrogen[END_REF][START_REF] Gregoryanz | infrared, and x-ray evidence for new phases of nitrogen at high pressures and temperatures[END_REF][START_REF] Streib | Single-Crystal X-Ray Diffraction Study of β Nitrogen[END_REF][START_REF] Hanfland | Structures of molecular nitrogen at high pressures[END_REF][START_REF] Hörl | Structure and structure imperfections of solid β -oxygen[END_REF]. The ternary phase diagram H-N-O is extremely well characterized as a function of temperature at ambient pressure. There are also a large amount of studies, both experimental and theoretical, focused on the binary phase diagrams H-N, H-O, and N-O up to very high pressures [START_REF] Datchi | Solid ammonia at high pressure: A single-crystal x-ray diffraction study to 123 GPa[END_REF][START_REF] Gauthier | Raman scattering study of ammonia up to 75 GPa: Evidence for bond symmetrization at 60 GPa[END_REF][START_REF] Gauthier | Brillouin study of liquid and solid ammonia up to 20 GPa[END_REF]. However, much less is known about the ternary diagram at high pressure.

Our aim in this chapter is to perform a systematically study of the ternary H-N-O phase diagram as a function of pressure. Our main tool are global structural prediction methods. These methods have proved extremely important in exploring phase-diagrams under pressure in the last years, and have yielded a remarkable number of important results. For example, they were used to show that the xenon apparently missing from the Earth is in fact bound with iron at the interior of our planet [START_REF] Zhu | Reactions of xenon with iron and nickel are predicted in the Earth's inner core[END_REF], or that simple alkali metals become semiconducting and transparent under pressure [START_REF] Ma | Transparent dense sodium[END_REF]. More recently, structural prediction was able to find superconductivity with record transition temperature in hydrogen sulfide [START_REF] Li | The metallization and superconductivity of dense hydrogen sulfide[END_REF][START_REF] Duan | Pressure-induced metallization of dense (H2S) 2H2 with high-Tc superconductivity[END_REF], a prediction that was quickly confirmed experimentally one year later [START_REF] Drozdov | Conventional superconductivity at 203 kelvin at high pressures in the sulfur hydride system[END_REF].

It is simple to understand the importance and attractiveness of structural prediction in high pressure science. High-pressure changes substantially the chemistry, leading to novel and unexpected bonding patterns, making predictions based on our (ambient pressure) intuition often to fail spectacularly. Furthermore, high-pressure investigations, requiring delicate diamond anvil cell experiments, are difficult and expensive. Computers, on the other hand, are by now ubiquitous and have reached unparalleled performances. Computer experiments appear therefore as a fast and cost-efficient method to explore the phase diagram of solid substances as a function of pressure.

There are several powerful global structural prediction algorithms developed over the past decade (such as genetic algorithms [START_REF] Woodley | The prediction of inorganic crystal structures using a genetic algorithm and energy minimisation[END_REF][START_REF] Abraham | A periodic genetic algorithm with real-space representation for crystal structure and polymorph prediction[END_REF][START_REF] Oganov | Crystal structure prediction using ab initio evolutionary techniques: Principles and applications[END_REF], particle swarm methods [START_REF] Wang | Crystal structure prediction via particle-swarm optimization[END_REF][START_REF] Wang | A method for crystal structure prediction[END_REF], random search [START_REF] Pickard | Structure of phase III of solid hydrogen[END_REF], minima hopping method [START_REF] Goedecker | Minima hopping: An efficient search method for the global minimum of the potential energy surface of complex molecular systems[END_REF][START_REF] Amsler | Crystal structure prediction using the minima hopping method[END_REF], etc.), and these are now available in a number of freely available codes. It is therefore not surprising that global structural prediction studies of bulk solids under pressure have mushroomed over the past years. A large majority of these works concentrate on single stoichiometries or binary phase diagrams. Studies of ternary phase diagrams are of course rarer, due to the much larger number of possible compositions, and the consequent increase of the required computational effort.

Elementary and binary phase diagrams

The H-N-O phase diagram in its elementary and binary versions has been extensively studied, both experimentally and theoretically, in a wide range of pressures and temperatures. In the following, we will try to summarize briefly the current knowledge of this phase diagram, starting with the elemental phases.

Hydrogen as the first and arguably the simplest element of the periodic table, has attracted much attention since Wigner and Huntington proposed in 1935 that solid hydrogen might dissociate into an atomic metal at around 25 GPa [START_REF] Wigner | On the possibility of a metallic modification of hydrogen[END_REF]. Unfortunately, the extremely weak X-ray scattering of hydrogen and the lack of convenient experimental methods hindered for a long time experimental studies of the low-temperature and high-pressure structures. Phase I of hydrogen consists of a disordered hcp structure [START_REF] Hazen | Single-crystal x-ray diffraction of n-H 2 at high pressure[END_REF], while the low temperature and high pressure phases II and III of hydrogen were extensively studied both experimentally and theoretically [START_REF] Pickard | Structure of phase III of solid hydrogen[END_REF][START_REF] Liu | Proton or deuteron transfer in phase IV of solid hydrogen and deuterium[END_REF][START_REF] Hemley | Phase transition in solid molecular hydrogen at ultrahigh pressures[END_REF][START_REF] Mazin | Quantum and classical orientational ordering in solid hydrogen[END_REF][START_REF] Goncharov | Vibron frequencies of solid H2 and D2 to 200 GPa and implications for the P-T phase diagram[END_REF][START_REF] John | Evidence from molecular dynamics simulations for nonmetallic behaviour of solid hydrogen above 160 GPa[END_REF][START_REF] Kitamura | Quantum distribution of protons in solid molecular hydrogen at megabar pressures[END_REF][START_REF] Tolédano | Symmetry breaking in dense solid hydrogen: Mechanisms for the transitions to phase II and phase III[END_REF][START_REF] Liu | Room-temperature structures of solid hydrogen at high pressures[END_REF]. More recently, experimental results suggest a phase transition to a metallic, atomic phase at around 447 GPa [START_REF] Mcminis | Molecular to atomic phase transition in hydrogen under high pressure[END_REF], but these results are still under intense debate.

Nitrogen has a remarkably rich phase-diagram, with no less than 13 solid molecule phases [START_REF] Gregoryanz | infrared, and x-ray evidence for new phases of nitrogen at high pressures and temperatures[END_REF][START_REF] Bini | High-pressure phases of solid nitrogen by Raman and infrared spectroscopy[END_REF][START_REF] Gregoryanz | High-pressure amorphous nitrogen[END_REF][START_REF] Gregoryanz | High P-T transformations of nitrogen to 170 GPa[END_REF][START_REF] Lipp | Transformation of molecular nitrogen to nonmolecular phases at megabar pressures by direct laser heating[END_REF][START_REF] Eremets | Semiconducting nonmolecular nitrogen up to 240 GPa and its low-pressure stability[END_REF][START_REF] Ma | Novel high pressure structures of polymeric nitrogen[END_REF][START_REF] Wang | Cagelike diamondoid nitrogen at high pressures[END_REF], an amorphous state [START_REF] Gregoryanz | High-pressure amorphous nitrogen[END_REF] and two non-molecular phases [START_REF] Eremets | Singlebonded cubic form of nitrogen[END_REF][START_REF] Tomasino | Pressure-induced symmetry-lowering transition in dense nitrogen to layered polymeric nitrogen (LP-N) with colossal Raman intensity[END_REF]. At 300 K, nitrogen freezes at 3 GPa to β -N 2 , and further compression results in the δ , δ loc , ε, ζ and κ phases at 5, 11, 16, 60, and 115 GPa, respectively [START_REF] Streib | Single-Crystal X-Ray Diffraction Study of β Nitrogen[END_REF][START_REF] Hanfland | Structures of molecular nitrogen at high pressures[END_REF][START_REF] Gregoryanz | High P-T transformations of nitrogen to 170 GPa[END_REF][START_REF] Eremets | Structural transformation of molecular nitrogen to a singlebonded atomic state at high pressures[END_REF][START_REF] Gregoryanz | On the -ζ transition of nitrogen[END_REF]. Finally, at about 150 GPa, there is a transition to the amorphous η which exhibits substantial pressure hysteresis [START_REF] Gregoryanz | High-pressure amorphous nitrogen[END_REF][START_REF] Goncharov | Optical evidence for a nonmolecular phase of nitrogen above 150 GPa[END_REF]. Recently, Frost and co-workers synthesized λ -N 2 through low temperature compression and observed it to have a remarkably wide stability range [START_REF] Frost | Novel high-pressure nitrogen phase formed by compression at low temperature[END_REF].

Oxygen also exhibits a rich polymorphism with seven unambiguously established crystalline phases. At ambient pressure and as a function of temperature, oxygen solidifies to the paramagnetic γ-phase, then to the magnetically disordered β -phase [START_REF] Hörl | Structure and structure imperfections of solid β -oxygen[END_REF][START_REF] Lesar | Character of the α-β phase transition in solid oxygen[END_REF] and ultimately to the anti-ferromagnetic α-phase [START_REF] Meier | Neutron-diffraction study of α-and β -oxygen[END_REF]. At the pressure of 6 GPa, the α-phase transforms into the anti-ferromagnetic δ -phase [START_REF] Schiferl | Structure of'orange'18O2 at 9.6 GPa and 297 K[END_REF][START_REF] Gorelli | Crystal structure of solid oxygen at high pressure and low temperature[END_REF][START_REF] Goncharenko | Direct determination of the magnetic structure of the delta phase of oxygen[END_REF]. As the pressure increases to about 8 GPa, the δ -phase of oxygen is destroyed, leading to the ε-O 8 phase consisting of O 8 clusters [START_REF] Fujihisa | O 8 cluster structure of the epsilon phase of solid oxygen[END_REF][START_REF] Lundegaard | Observation of an O8 molecular lattice in the phase of solid oxygen[END_REF]. Above 96 GPa, ε-O 8 was observed to transform into a metallic ζ -phase [START_REF] Desgreniers | Optical response of very high density solid oxygen to 132 GPa[END_REF][START_REF] Akahama | New high-pressure structural transition of oxygen at 96 GPa associated with metallization in a molecular solid[END_REF] (that intriguingly exhibits superconductivity with a transition temperature of 0.6 K [START_REF] Shimizu | Superconductivity in oxygen[END_REF]). Zhu and co-workers predicted another phase of oxygen at 1.92 TPa exhibiting polymeric spiral chains of O 4 [START_REF] Zhu | Spiral chain O4 form of dense oxygen[END_REF]. These studies explored the high-pressure phase diagrams of solid oxygen up to the maximum pressure of 2 TPa.

Like the elementary substances, also the binary N-H, N-O, and O-H systems have been extensively studied. The low temperature and pressure phase-I (P2 1 3) of NH 3 undergoes first-order phase transitions into phase-IV (P2 1 2 1 2 1 ) at around 3-4 GPa and then into phase-V at about 14 GPa [START_REF] Datchi | Solid ammonia at high pressure: A single-crystal x-ray diffraction study to 123 GPa[END_REF][START_REF] Gauthier | Raman scattering study of ammonia up to 75 GPa: Evidence for bond symmetrization at 60 GPa[END_REF][START_REF] Gauthier | Brillouin study of liquid and solid ammonia up to 20 GPa[END_REF]. In 2008, Pickard et al. found that the phase-V of ammonia also belongs to the space group of P2 1 2 1 2 1 . Furthermore, they predicted two new ionic phases (Pma2 and P2 1 /m) under high pressure that then re-transform into a molecular phase (Pnma) at about 440 GPa [START_REF] Pickard | Highly compressed ammonia forms an ionic crystal[END_REF].

As one of the most important binary compounds, H 2 O ice has a very rich phase diagram. Till now, at least 15 solid phases have been identified experimentally [START_REF] Petrenko | Physics of ice[END_REF][START_REF] Salzmann | Ice XV: a new thermodynamically stable phase of ice[END_REF] and another 7 high-pressure phases have been predicted through theoretical studies [START_REF] Benoit | New high-pressure phase of ice[END_REF][START_REF] Militzer | New phases of water ice predicted at megabar pressures[END_REF][START_REF] Wang | High pressure partially ionic phase of water ice[END_REF][START_REF] Pickard | Decomposition and terapascal phases of water ice[END_REF]. Experimentally, all crystalline phases were found to consist of water molecules connected by hydrogen bonds except ice-X. This is an atomic crystal that is stable above 70 GPa at low temperature.

Nitrogen oxides like NO, N 2 O 5 , NO 2 and N 2 O have also been studied both experimentally and theoretically. NO is a metastable compound at low pressure, while a polymeric NO structure becomes stable at 198 GPa [START_REF] Li | Nitrogen oxides under pressure: stability, ionization, polymerization, and superconductivity[END_REF]. Molecular N 2 O 5 phases are stable in a wide pressure range from 9 to 446 GPa, when they decompose into NO and O [START_REF] Li | Nitrogen oxides under pressure: stability, ionization, polymerization, and superconductivity[END_REF]. The cubic phase (Im -3) of NO 2 forgoes two phase transitions at 7 GPa (to a P2 1 /c crystal) and then at 64 GPa (to an another P2 1 /c phase), before decomposing at 91 GPa [START_REF] Li | Nitrogen oxides under pressure: stability, ionization, polymerization, and superconductivity[END_REF]. N 2 O follows a series of phase transitions from Pa 3 [START_REF] Hamilton | DISORDER IN SOLID NITROUS OXIDE BY NEUTRON DIFFRACTION1[END_REF] → Cmca [START_REF] Mills | Crystal structures of N2O to 12 GPa by x-ray diffraction[END_REF] → C2m [START_REF] Zhu | Exploring the metallic phase of N 2 O under high pressure[END_REF] → P2 1 /m [START_REF] An | Stability of NNO and NPO Nanotube Crystals[END_REF] at the pressures of 5, 57, 177 and 194 GPa, respectively.

Finally, the ternary phase AMH I [START_REF] Olovsson | The crystal structure of ammonia monohydrate[END_REF] of NH 3 H 2 O transforms into AMH II [START_REF] Fortes | Crystal structure of ammonia monohydrate phase II[END_REF] at about 2.8 GPa and then transforms into a P4/nmm and then to a Ima2 phase at 3.3 GPa and 42 GPa [START_REF] Mafety | Étude ab initio des glaces d , ammoniac fluoré et hydraté sous conditions thermodynamiques extrêmes[END_REF] respectively. On the other hand, a new high-pressure phase (P2 1 /c) of

N 2 O 2 H 2 NH 3 N 2 O N 2 O 5 H 2 O HNO 3 NOH 4

Methods

For our study we chose to perform structural prediction with a method based on a particleswarm optimization algorithm as implemented in the CALYPSO code [START_REF] Wang | Crystal structure prediction via particle-swarm optimization[END_REF][START_REF] Wang | A method for crystal structure prediction[END_REF]. This method is unbiased by any known structural information, and has been successfully used for the prediction of various systems under high pressure [START_REF] Lv | Predicted novel high-pressure phases of lithium[END_REF][START_REF] Cui | Hydrogen segregation and its roles in structural stability and metallization: silane under pressure[END_REF][START_REF] Shi | Investigation of new phases in the Ba-Si phase diagram under high pressure using ab initio structural search[END_REF]. The underlying ab initio structural relaxations and the electronic band structure calculations were performed within the framework of density functional theory (DFT) as implemented by the VASP code [START_REF] Kresse | Ab initio molecular dynamics for liquid metals[END_REF]. The calculations were carried out at the level of the generalized gradient approximation using the Perdew-Burke-Ernzerhof [START_REF] Perdew | Generalized gradient approximation made simple[END_REF] (PBE) approximation for the exchange correlation functional. The electronic wave functions were expanded in a plane wave basis set with a cutoff energy of 600 eV. The electron-ion interaction was described by means of projector augmented wave (PAW) [START_REF] Kresse | From ultrasoft pseudopotentials to the projector augmentedwave method[END_REF] pseudopotentials with 1s 1 , 2s 2 2p 3 and 2s 2 2p 4 electrons in the valence for the H, N and O atoms, respectively. Monkhorst-Pack k-point [START_REF] Monkhorst | Special points for Brillouin-zone integrations[END_REF] meshes with a grid density of 0.03 Å -1 were chosen to achieve a total energy convergence of better than 1 meV per atom. The phonon dispersion curves were computed by the direct supercell calculation method as implemented in the Phonopy program [START_REF] Togo | First-principles calculations of the ferroelastic transition between rutile-type and CaCl 2 -type SiO 2 at high pressures[END_REF].

Results and discussion

We investigated systematically the phase stability of stoichiometric NOH x (x = 1, 2, 3, 4, 5), HON x (x = 1, 2, 3, 4, 5), HNO x (x = 1, 2, 3, 4, 5) and H 3 NO 4 compounds at the pressure of 50, 100 and 150 GPa. The structural predictions were performed with simulation cell sizes of 2 and 4 formula units (f.u.). We calculated the convex hull with respect to the decomposition into all known phases of the H-N-O ternary system in their corresponding stable pressure range. They are: H 2 (C2/c), N 2 (I2 1 3), O 2 (C2/m), H 2 O (phase X for the pressure above 70 GPa and phase VIII below 70 GPa), N 2 O (P2 1 /c) and N 2 O 5 (C2/c).

Our main result is the ternary phase diagram of H-N-O, that is shown in Fig. 4.1 for a pressure of 150 GPa (the largest pressure considered). Meta-stable compositions that appear A complete summary of the results is shown in Tab. 4.1, which includes the structural parameters of our predicted stable structures for the N-H-O ternary system. In the following we analyze systematically these two new phases.

We calculated the dependence of the enthalpy on the applied pressure for the decomposition of NOH 5 and NOH 4 into water, ammonia, and nitrogen, as shown in Fig. 4.2. We find that NH 3 H 2 O is stable in a large range of pressures, in good agreement with previous works [START_REF] Mafety | Étude ab initio des glaces d , ammoniac fluoré et hydraté sous conditions thermodynamiques extrêmes[END_REF], but that it becomes thermodynamically unstable at around 122 GPa when it decomposes into NH 3 and H 2 O. We find the opposite behavior for NOH 4 that becomes stable at around 71 GPa, as shown in Fig. 4 quantity is a valuable tool developed in quantum chemistry to visualize bonds, lone pairs, etc. It assumes values in the range from zero to one, with one meaning perfect localization of electrons (such as in a bond). We find that the ELF has large values between two NH 3 molecules, indicating a covalent bond between the two N atoms. We thus conclude that the NH 3 layer is in fact built from N 2 H 6 "dumbbell" clusters.

The H-O distance is 1.117 Å and the H-O-H angle is 112.87 • at 150 GPa. In the chains, the electrons are mainly distributed in the center of the O-H bonds. The covalent character of these bonds can also be confirmed by the charge density plot in Fig. 4.3 (b). On the other hand, the N-N distance is 1.355 Å at 150 GPa. The N-N bond is a single bond, which is confirmed by the shape of the ELF.

The C2/m phase of NOH 4 at 150 GPa is dynamically stable as it does not exhibit any imaginary phonon frequencies, as shown in Fig. 4.6. The electronic band structure and projected density of states (DOS) are shown in Fig. 4.5. We see that the C2/m phase is an insulator with a large band gap of 6.0 eV at 150 GPa. Note that this value was calculated with the PBE functional, that systematically underestimate band-gaps. HNO 3 (nitric acid) at 150 GPa crystallizes in a novel phase of P2 1 /m symmetry and with an interesting structural configuration containing two parallel layers forming a "quasi clover" The relative enthalpy per formula unit of the new P2 1 /m phase of HNO 3 , with respect to the phase II (P2 1 /c) [START_REF] Allan | The crystal structures of the low-temperature and high-pressure polymorphs of nitric acid[END_REF], is shown in Fig. 4.7 as a function of pressure. Our calculations show that the new phase of becomes energetically favorable in a large range of pressures from about 39 GPa to more than 150 GPa.

Finally, we calculated the phonon dispersion of this phase at 150 GPa, as depicted in Fig. 4.8, that proves that the P2 1 /m phase is mechanically stable. We also calculated the electronic band structure and density of states, as shown in Fig. 4.9. It reveals that the P2 1 /m phase of HNO 3 is a semiconductor with a PBE band gap of 2.6 eV.

Conclusions

In summary, we explored the ternary phase diagram of the N-H-O system at high-pressure using a global structural prediction method combined with first-principle calculations. We found only two stable ternary phases up to 150 GPa, namely NOH 4 and HNO 3 . The novel phase of NOH 4 possesses C2/m symmetry while the new phase of HNO 3 has P2 1 /m symmetry. While the former composition becomes thermodynamically stable at 71 GPa, the P2 1 /m phase of HNO 3 is stabilized with respect to the phase II at pressures higher than 39 GPa. Both are layered systems, with the C2/m phase of NOH 4 containing HO chains separated by NH 3 -NH 3 "dumbbell clusters", and the P2 1 /m phase of HNO 3 composed of "quasi-clovers" with a H tail. These remarkable and unexpected structures are a consequence of the high-pressure affected bonding behavior of even simple elements. Our results can be viewed as a guide for further experimental synthesis of the ternary N-H-O system.

Chapter 5

Novel p-type transparent electrodes of ternary chalcogenides

Introduction

The delafossite mineral group is composed of ternary compounds with a general formula ABX 2 . This crystal structure, named after the French crystallographer Gabriel Delafosse, has been known since the 19th century, but it started to be intensively studied after the discovery in 1997 of p-type electrical conductivity in a transparent thin-film of CuAlO 2 delafossite [START_REF] Kawazoe | P-type electrical conduction in transparent thin films of CuAlO 2[END_REF]. Transparent conductive materials (TCMs) possess two usually mutually exclusive characteristics which are transparency and conductivity. TCMs are now essential components of many modern technologies as they are used as transparent electrodes for optoelectronic device applications [START_REF] Minami | Transparent conducting oxide semiconductors for transparent electrodes[END_REF] (e.g., flat-panel displays and solar cells).

The atomic structure of delafossite crystals consists of planes of linearly coordinated A cations forming a triangular lattice, stacked between layers of flattened edge-sharing octahedra BO 6 . The cation A is usually a noble metal like Cu, Ag, Pd or Pt; the cation B is typically a transition or group 13 element or a rare earth species; finally, the anion X is usually oxygen, although few delafossites with other anions can be found in material databases [START_REF] Jain | The Materials Project: A materials genome approach to accelerating materials innovation[END_REF][START_REF] Saal | Materials design and discovery with high-throughput density functional theory: the open quantum materials database (OQMD)[END_REF][START_REF] Curtarolo | ORG: A distributed materials properties repository from high-throughput ab initio calculations[END_REF]. The trigonal delafossite structure (space group 166) is shown in the left panel of Fig. From this perspective, there is no difference between this phase of AgBiS 2 and the trigonal (space group 166) structure of TlScSe 2 . The A cations of ABX 2 form a triangular lattice that alternates with layers composed of distorted edge-sharing BX 6 octahedra. The similarity between the left and right panels is evident: only the alignment of A and X atoms differentiates the two structures (the dotted lines are a guide to the eye meant to emphasize this difference).

Due to the rather large range of possible stable configurations, it is not surprising that delafossites exhibit a significant richness in properties and potential applications. Several important technological applications have been proposed for members of this family, such as electrodes for hydrogen production by photo-electrochemical water splitting [START_REF] Younsi | Synthesis and characterization of oxygen-rich delafossite CuYO 2+x -Application to H 2 -photo production[END_REF][START_REF] Saadi | Hydrogen photoproduction over new catalyst CuLaO 2[END_REF], as luminescent materials [START_REF] Liu | Luminescence of delafossitetype CuAlO 2 fibers with Eu substitution for Al cations[END_REF], in solar cells [START_REF] Daniel | Photovoltaic Performance of (Al, Mg)-Doped CuCrO 2 for p-type Dye-sensitized Solar Cells Application[END_REF][START_REF] Zakutayev | Design of nitride semiconductors for solar energy conversion[END_REF][START_REF] Jiang | Copper borate as a photocathode in p-type dye-sensitized solar cells[END_REF], for thermoelectricity [START_REF] Ono | Structural, magnetic and thermoelectric properties of Delafossite-type oxide[END_REF][START_REF] Yasuhiro Ono | Magnetic and Thermoelectric Properties of Delafossite-type Oxide, CuCr 1-x Mg x O 2 (0 ≤ x ≤ 0.05)[END_REF][START_REF] Hayashi | Structure and High Temperature Thermoelectric Properties of Delafossite-Type Oxide CuFe 1-x Ni x O 2 (0 ≤ x ≤ 0.05)[END_REF][START_REF] Ohkubo | Anisotropic Anomalies of Thermoelectric Transport Properties and Electronic Structures in Layered Complex Nitrides AMN 2 (A = Na, Cu; M = Ta, Nb)[END_REF], as possible transparent superconductors [START_REF] Chutirat Taddee | Characterization of transparent superconductivity Fe-doped CuCrO 2 delafossite oxide[END_REF], etc. Recently, it was also found that delafossite PtCoO 2 behaves like a nearly free-electron system, turning this surprising 5d transition-metal system in the most conductive oxide known [START_REF] Kushwaha | Nearly free electrons in a 5d delafossite oxide metal[END_REF].

The discovery of p-type conductivity in CuAlO 2 was particularly significant, as all technologically relevant transparent conducting oxides (TCOs), such as indium-tin-oxide (ITO) or Al-and Ga-doped ZnO (AZO and GZO), are n-type semiconductors. Unfortunately, neither CuAlO 2 nor other recently proposed p-type TCOs possess adequate conductivity and transparency for technological applications.

The reasons for the difficulty of finding good p-type TCOs are well understood and they are related to the strong localization of holes at the valence band edge of most oxides. In fact, the O-2p orbitals, which form hole transport paths at the top of the valence band of many oxides, are rather localized and generally much lower in energy than the valence orbitals of the metal species. These two facts imply difficulties in hole doping and large hole effective masses (and therefore low hole mobilities).

In the race for obtaining such elusive material, delafossites have been widely explored for the past years. Getting inspiration from the structure/property relations in the delafossite family, Hosono and co-workers [START_REF] Yanagi | Chemical Design and Thin Film Preparation of p-Type Conductive Transparent Oxides[END_REF][START_REF] Kawazoe | Transparent p-Type Conducting Oxides: Design and Fabrication of p-n Heterojunctions[END_REF] Following Hosono's rules, the discovery of p-type conductivity in CuAlO 2 triggered the experimental and theoretical characterization of other delafossites [START_REF] Marquardt | Crystal chemistry and electrical properties of the delafossite structure[END_REF][START_REF] Scanlon | Understanding conductivity anomalies in CuI-based delafossite transparent conducting oxides: Theoretical insights[END_REF][START_REF] Scanlon | Understanding the p-Type Conduction Properties of the Transparent Conducting Oxide CuBO 2 : A Density Functional Theory Analysis[END_REF][START_REF] Scanlon | Understanding the p-type defect chemistry of CuCrO 2[END_REF][START_REF] Ginley | Handbook of transparent conductors[END_REF] and of transparent p-n junctions. CuCrO 2 and CuGaO 2 are also p-type TCOs [START_REF] Han | Structural, electronic band transition and optoelectronic properties of delafossite CuGa 1-x Cr x O 2 (0 < x < 1) solid solution films grown by the sol-gel method[END_REF][START_REF] Ueda | Epitaxial growth of transparent p-type conducting CuGaO 2 thin films on sapphire (001) substrates by pulsed laser deposition[END_REF]. CuInO 2 [START_REF] Yanagi | Bipolarity in electrical conduction of transparent oxide semiconductor CuInO 2 with delafossite structure[END_REF] is particularly interesting because it is amphoteric, i.e. it can be alternatively acceptor-doped or donor-doped. This ability is explained by the unique presence of two cationic environments in the delafossite structure: an octahedral indium environment that favors strong hybridization and n-type dopability, together with a linear copper environment that favors hybridization at the top valence and p-type doping [START_REF] Ginley | Handbook of transparent conductors[END_REF]. Among delafossite compounds, CuCr 1-x Mg x O 2 [START_REF] Nagarajan | P-type conductivity in CuCr 1-x Mg x O 2 films and powders[END_REF] possesses the highest conductivity of 220 S cm -1 , but transparency in the visible range is only about 30%, much smaller than for the traditional n-type TCOs. The recently synthesized Cu deficient CuCrO 2 has a conductivity in the range of 1-12 S cm -1 [START_REF] Farrell | Synthesis of nanocrystalline Cu deficient CuCrO 2 -a high figure of merit p-type transparent semiconductor[END_REF]. It was also claimed that the transparency can be improved by increasing the temperature, at the cost of higher resistivity [START_REF] Nagarajan | P-type conductivity in CuCr 1-x Mg x O 2 films and powders[END_REF][START_REF] Nagarajan | P-type conductivity in the delafossite structure[END_REF][START_REF] Wang | Structural, optical and electrical properties of Mg-doped CuCrO 2 thin films by sol-gel processing[END_REF]. Recently, A. Barnabé and co-workers studied Mg-doped CuCrO 2 thin films by using RF-sputtering [START_REF] Barnabé | p-Type conducting transparent characteristics of delafossite Mg-doped CuCrO 2 thin films prepared by RF-sputtering[END_REF]. They found that, at 600 • C, the Mg-doped CuCrO 2 structure displayed an optical transmittance of only 63% in the visible range with, however, a conductivity of 1.6 S cm -1 .

Another idea, again put forward by Hosono and co-workers [START_REF] Hosono | Recent progress in transparent oxide semiconductors: Materials and device application[END_REF], is to replace oxygen with a chalcogen (S, Se, or Te) with more delocalized p orbitals. This has led to the synthesis of LaCuOS and LaCuOSe p-type oxides [START_REF] Ueda | Transparent p-type semiconductor: LaCuOS layered oxysulfide[END_REF][START_REF] Hiramatsu | Degenerate p-type conductivity in wide-gap LaCuOS 1-x Se x (x = 0 -1) epitaxial films[END_REF]. Unfortunately, these latter compounds have either a too low mobility (LaCuOS) or a too small band gap (LaCuOSe). However, this strategy looks appealing as the p states of the chalcogen atoms are usually higher and therefore make the compound more prone to be p-doped. Recent calculations suggest that this is the case for Ba(Cu,Ag) 2 (S,Se) 2 compounds [START_REF] Krishnapriyan | First-principles study of band alignments in the p-type hosts BaM 2 X 2 (M = Cu, Ag; X = S, Se)[END_REF].

Going beyond simple empirical rules, experimental databases have been recently explored by performing high-throughput ab initio calculations [START_REF] Hautier | Identification and design principles of low hole effective mass p-type transparent conducting oxides[END_REF][START_REF] Sarmadian | Easily doped p-type, low hole effective mass, transparent oxides[END_REF] to single out compounds with large band gaps (> 3 eV) and low effective masses. These previous investigations focused on oxides and showed that small hole effective masses are much rarer that small electron effective masses. Hautier et al. [START_REF] Hautier | Identification and design principles of low hole effective mass p-type transparent conducting oxides[END_REF] found that hole effective masses of 0.5-1.0 electron masses (m e ) in transparent oxides like PbTiO 3 , PbZrO 3 , PbHfO 3 , K 2 Sn 2 O 3 , and Tl 4 V 2 O 7 are due to the hybridization of the oxygen 2p orbitals with s states of Sn 2+ , Pb 2+ , Tl 1+ . Similarly, hole effective masses smaller than 1.0 m e and band gaps larger than 3.0 eV are reported by Sarmadian et al. [START_REF] Sarmadian | Easily doped p-type, low hole effective mass, transparent oxides[END_REF] for X 2 SeO 2 , with X = La, Pr, Nd, and Gd. Recently, Cerqueira et al. [START_REF] Cerqueira | Identification of novel Cu, Ag, and Au ternary oxides from global structural prediction[END_REF] performed a search of new compounds of the form (Cu,Ag,Au)XO 2 by combining global structural prediction methods and high-throughput calculations. They showed that there are 45 new (quasi) stable phases in this system, some of which are candidate p-type transparent conductors. However, these new stable phases contain Ag and Au, which are not ideal elements for commercial applications.

We remind that finding a large band gap and low hole effective masses is not a sufficient condition for a good p-type TCO. In fact, another essential condition, much harder to translate to the minimization/maximization of a simple material property, is the p-type dopability of the system. As an example, we can think about ZnO: the hole effective mass values for ZnO range from 0.31 m e to 0.59 m e and its band gap is about 3.5 eV. Despite these ideal parameters, numerous attempts to produce p-type doped ZnO have failed [START_REF] Ginley | Handbook of transparent conductors[END_REF].

It appears by now clear that if we work only with known materials it will be very hard to improve both transparency and conductivity to technologically acceptable limits. However, the search for improved p-type TCMs remains an important open question. Fortunately, we only know experimentally a fraction of all possible stable compounds. Delafossite structures, which combine the advantages of linear and octahedral environments for the two cations remain particularly attractive for p-type TCMs and ambipolar doping. We therefore asked ourselves: Are there more delafossite compounds that wait to be synthesized and characterized? And more precisely: can good p-type TCMs be found in the delafossite family or should we resign ourselves and direct future search towards completely different crystal families?

Here we provide an answer to the questions above by performing a comprehensive firstprinciples investigation of ABX 2 systems that may crystallize in the delafossite structure. We extend in fact our study to all ABX 2 compositions where A and B cover the periodic table from H to Bi (excluding the rare gases and the lanthanides) and X is a chalcogen (O, S, Se, and Te). This set contains 63 × 62 × 4 = 15 624 compositions and includes the already known delafossites crystals. At this point, we take all A and B elements and do not make any bias towards non-toxic or abundant elements. Such filtering can however be easily performed a posteriori.

We note that most research on delafossites is restricted to oxide materials. Also most of the systems known have X=O. For example, the materials project database [START_REF] Jain | The Materials Project: A materials genome approach to accelerating materials innovation[END_REF], contains 90 trigonal ABX 2 oxides (with space group 166) while only 23 sulfides, 10 selenides and 7 tellurides. It is unclear if this is because the delafossite structure prefers the highly electronegative oxygen, or if it simply reflects a bias against sulfides, selenides, and tellurides.

To screen the large amount of possible compositions of ABX 2 compounds, we use a combination of high-throughput density-functional theory [START_REF] Morgan | High-throughput and data mining with ab initio methods[END_REF] and global structural prediction methods [START_REF] Oganov | Modern Methods of Crystal Structure Prediction[END_REF]. This two-step process is essential to ensure that our predicted structures are (i) thermodynamically stable, i.e. that do not decompose to other more stable species, and (ii) are indeed delafossites, i.e. that there are no other lower energy phases for the given composition. The resulting stable delafossite phases are then characterized theoretically using state-of-the-art ab initio methods. The final results provide precious indications for further experimental synthesis and open the way for more specific theoretical and experimental studies of dopability and transport properties, which can be restricted only to the most interesting candidates.

Methods

As starting prototype unit cell we used the rhombohedral (space group 166) delafossite structure, and not the hexagonal (space group 194) one. In fact, these two layered phases only differ in the stacking of the layers, and we have verified [START_REF] Cerqueira | Identification of novel Cu, Ag, and Au ternary oxides from global structural prediction[END_REF] that they exhibit very similar total energies (with differences of the order of few meV/atom) and electronic properties. In that sense these two crystal structures are equivalent in our calculations. However, the former contains 4 atoms in the primitive unit cell, while the latter contains 8, which would lead to a considerable increase of the computational burden of our high-throughput search.

As the A and B atoms are in the inequivalent crystallographic positions 1a and 1b of the delafossite structure, there are 15 624 possible ABX 2 compositions that combine the 64 chemical elements considered here. It is true that this number could be quite reduced by eliminating compositions on the basis of empirical rules based on oxidation states or the radius of the A, B, and X elements. However, we decided not to follow this path in order to allow for delafossites that do not conform to our intuition.

For each one of these structures we optimized the geometry and calculated the total energy. This was done within ab initio density functional theory as implemented in the computer code VASP [START_REF] Kresse | Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set[END_REF][START_REF] Kresse | Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set[END_REF]. All parameters were set to guarantee compatibility with the data available in the materials project database [START_REF] Jain | The Materials Project: A materials genome approach to accelerating materials innovation[END_REF] and open quantum materials database [START_REF] Saal | Materials design and discovery with high-throughput density functional theory: the open quantum materials database (OQMD)[END_REF]. We used the PAW [START_REF] Blöchl | Projector augmented-wave method[END_REF] datasets of version 5.2 with a cutoff of 520 eV and Γ-centered k-point grids, as dense as required to ensure an accuracy of 2 meV/atom in the total energy. All forces were converged to better than 0.005 eV/Å. We followed the same protocol as in Ref. [START_REF] Cerqueira | Identification of novel Cu, Ag, and Au ternary oxides from global structural prediction[END_REF]: spin-polarized calculation using the Perdew-Burke-Ernzerhof [START_REF] Perdew | Generalized gradient approximation made simple[END_REF] (PBE) exchange-correlation functional, with the exception of oxides and fluorides containing Co, Cr, Fe, Mn, Mo, Ni, V, W, where an on-site Coulomb repulsive interaction U with a value of 3.32, 3.7, 5.3, 3.9, 4,38, 6.2, 3.25, and 6.2 eV, respectively, was added to correct the d-states. We are of course aware that the PBE (+U) functional is not always able to determine the correct electronic and spin ground-state. Moreover, it is well-known that the PBE, as well as many other exchange correlation functionals, including hybrid functionals [START_REF] Tran | Rungs 1 to 4 of DFT Jacob's ladder: Extensive test on the lattice constant, bulk modulus, and cohesive energy of solids[END_REF], yield an average error for the energies of formation substantially larger than the so-called chemical accuracy (around 1 kcal/mol 43 meV) [START_REF] Sarmiento-Pérez | Optimized Exchange and Correlation Semilocal Functional for the Calculation of Energies of Formation[END_REF]. Nevertheless, the optimized crystal structure and the ordering of low-energy phases, resulting from differences of formation energies, is very often qualitatively and quantitatively good within PBE (+U), even in many cases in which the electronic states are not correctly described.

The second step of our search is to identify which compositions are thermodynamically stable (see Chapter 2 for more details). We worked always at zero temperature and pressure, and we neglected the effects of the zero point motion of the phonons (that are expected to be negligible for these materials). In this case, the thermodynamic quantity of interest is the total energy. Fortunately, it is no longer necessary to calculate the total energy of all possible reservoir compounds, as this information is already available in excellent public databases, such as the materials project [START_REF] Jain | The Materials Project: A materials genome approach to accelerating materials innovation[END_REF], open quantum materials database, [START_REF] Saal | Materials design and discovery with high-throughput density functional theory: the open quantum materials database (OQMD)[END_REF] and the ab-initio electronic structure library AFLOWLIB [START_REF] Curtarolo | ORG: A distributed materials properties repository from high-throughput ab initio calculations[END_REF]. We chose to use the materials project database for our reference energies, and to determine the distances to the convex hull of stability with PYMATGEN [START_REF] Ong | Python Materials Genomics (pymatgen): A robust, open-source python library for materials analysis[END_REF]. The materials project database includes most of the experimentally known inorganic crystals that are present in the ICSD database [START_REF] Bergerhoff | Inorganic Crystal Structure Database[END_REF][START_REF] Belsky | New developments in the Inorganic Crystal Structure Database (ICSD): accessibility in support of materials research and design[END_REF] and an increasing number of theoretically predicted phases, including those proposed in Ref. [START_REF] Cerqueira | Identification of novel Cu, Ag, and Au ternary oxides from global structural prediction[END_REF]. Note that, as we only use the compounds included in the database to build the convex hull, new materials will appear with negative distances to the hull. The next step is to perform global structural prediction runs for the most stable structures stemming from the high-throughput runs. There are several good algorithms available in the market for this task. Our method of choice is the minima hopping method (see Chapter 2).

Having in mind the possible final application as TCM, for a first characterization of the structures we determined hole effective masses and band gaps. For the calculation of effective masses we used the same framework as for the calculation of the total energies, that relies on the PBE functional. Effective masses were calculated with the program BOLTZTRAP [START_REF] Madsen | A code for calculating bandstructure dependent quantities[END_REF] for a concentration of 10 18 cm -3 and a temperature of 300 K. We remark that while it is clearly desirable to have dispersive valence bands close to the band gap to ensure high hole mobilities, this is not the only possibility as other transport mechanisms, e.g., based on small polarons, have been proposed for delafossite p-type TCOs [START_REF] Ginley | Handbook of transparent conductors[END_REF].

Although the PBE functional is a quite reasonable approximation for the curvature of the bands, it is well known that this functional underestimates considerably the band gaps. Therefore we decided to adopt the hybrid exchange-correlation functional of Heyd, Scuseria, and Ernzerhof (HSE06) [START_REF] Heyd | Hybrid functionals based on a screened Coulomb potential[END_REF] to obtain reliable band gaps.

Finally, we give a first estimate of the p-type dopability by testing if acceptor-like defects can be easily formed in the system and whether one can expect or not the generation of compensating native defects (e.g., anion vacancies). To this purpose, a simple criterion with significant predictive power is the determination of the branch point energy. The branch point energy [START_REF] Sarmadian | Easily doped p-type, low hole effective mass, transparent oxides[END_REF][START_REF] Schleife | Branch-point energies and band discontinuities of III-nitrides and III-II-oxides from quasiparticle bandstructure calculation[END_REF][START_REF] Robertson | Limits to doping in oxides[END_REF], or charge neutrality level, is defined as the energy at which the defect states induced in the gap change their character from predominantly donor-like to acceptor-like. The position of this energy with respect to the band edges is particularly important as the formation of acceptor (donor) defect states in the gap becomes favorable above (below) it. Hence, in a material with a branch point energy lying in the conduction band (such as ZnO) or high in the band gap, donor impurity defects tend to be shallow, while acceptor impurities, if they can exist, will be necessarily deep. The condition to favor p-type doping is therefore to have the branch point energy at least in the middle of the gap. In general, the lower the value of this energy the easier p-type doping can be expected. To calculate this quantity we follow Schleife et al. [START_REF] Schleife | Branch-point energies and band discontinuities of III-nitrides and III-II-oxides from quasiparticle bandstructure calculation[END_REF] and we define it as a weighted average of the mid-gap energies (calculated with a hybrid HSE06 functional) over the Brillouin zone. The number of valence and conduction states included in the calculation follows the prescription of Ref. [START_REF] Schleife | Branch-point energies and band discontinuities of III-nitrides and III-II-oxides from quasiparticle bandstructure calculation[END_REF], which has already been adopted in other works [START_REF] Sarmadian | Easily doped p-type, low hole effective mass, transparent oxides[END_REF][START_REF] Robertson | Limits to doping in oxides[END_REF]. This method is a particularly convenient criterion to use in high-throughput calculations as it only requires the energy levels of the perfect crystals. We note, however, that reliable conclusions on the p-type dopability of the system can be obtained only by performing extensive defect calculations.

Crystal structures were visualized with VESTA [START_REF] Momma | VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data[END_REF].

Results and discussion

Overview

The main results of our high-throughput search are summarized in the stability maps of Fig. 5.2, where we plot the distance to the convex hull of the delafossite phase for all the compositions studied here. In order to put into evidence the chemical similarity between the chemical elements, we order them using an optimized [START_REF] Glawe | The optimal one dimensional periodic table: a modified Pettifor chemical scale from data mining[END_REF] Pettifor scale [START_REF] Pettifor | A chemical scale for crystal-structure maps[END_REF]. Each point in the plots corresponds to a different compound, and the color code is used to indicate the distance to the convex hull of stability (light green -stable, red -unstable). It is quite evident that the large majority of the compositions are highly unstable in the delafossite phase, with distances to the hull larger than 400 meV/atom. However, there are still many systems that are either thermodynamically stable or close to stability.

We then filter this list using a threshold of 50 meV/atom above the convex hull. This is a rather conservative threshold, that tries to minimize the number of false positives, but at the same time takes into account the errors in evaluating differences of PBE energies [START_REF] Sarmiento-Pérez | Optimized Exchange and Correlation Semilocal Functional for the Calculation of Energies of Formation[END_REF] and the possible reduction of the energy of formation related to small distortions of the lattice, native defects, temperature and disorder, etc. This also allows us to include in our list some experimentally known delafossite structures that exhibit (small) positive distances to the hull in our formalism, such as CuInO 2 (7 meV/atom).

Within this threshold we find 285 compositions (109 oxides, 66 sulfides, 67 selenides, and 43 tellurides). We can find entries in the materials project database for 101 of the 285 compositions, namely for 72 oxides (including our predictions of Ref. [START_REF] Cerqueira | Identification of novel Cu, Ag, and Au ternary oxides from global structural prediction[END_REF], 11 sulfides, 11 selenides, and 7 tellurides. Obviously, there are many more entries ABX 2 with X=O, S, Se, Te in the materials project database (463 in total at the time of writing), but most of these, such as, e.g., the chalcopyrites, have atomic arrangements very different from the delafossites, and do not appear therefore as a result of our search. We could also find in literature some (unfortunately incomplete) information on other compounds that are not part of the materials project database: MgNiO 2 , PdCrO 2 [START_REF] Takatsu | Magnetic structure of the conductive triangular-lattice antiferromagnet PdCrO 2[END_REF][START_REF] Shannon | Chemistry of noble metal oxides. I. Syntheses and properties of ABO 2 delafossite compounds[END_REF], PdRhO 2 [START_REF] Shannon | Chemistry of noble metal oxides. I. Syntheses and properties of ABO 2 delafossite compounds[END_REF], KBiS 2 [START_REF] Boon | The crystal structure of NaBiS 2 and KBiS 2[END_REF], KScS 2 [START_REF] Havlak | Structure determination of KScS 2 , RbScS 2 and KLnS 2[END_REF], KYS 2 [START_REF] Havlak | Structure determination of KScS 2 , RbScS 2 and KLnS 2[END_REF], RbScS 2 [START_REF] Havlak | Structure determination of KScS 2 , RbScS 2 and KLnS 2[END_REF], AgRhS 2 [START_REF] Pachoud | Robustness of Antiferromagnetism and Pyroelectricity in AgCr 1-x Rh x S 2[END_REF], AgYS 2 [START_REF] Ballestracci | Une classe de nouveaux composés sulfurés de terres rares et d'argent de type AgTS 2[END_REF], HgTiS 2 [START_REF] Sidorov | Structural Investigation of Mercury-Intercalated Titanium Disulfide. 2. HRTEM of Hg x TiS 2[END_REF], TlMnS 2 [START_REF] Veliev | Magnetization, paramagnetic susceptibility, and electrical conductivity of layered TIMnS 2 and TIMnSe 2 antiferromagnets[END_REF], TlMnSe 2 [START_REF] Veliev | Magnetization, paramagnetic susceptibility, and electrical conductivity of layered TIMnS 2 and TIMnSe 2 antiferromagnets[END_REF], KCrSe 2 [START_REF] Dijkstra | Electronic structure of the half-metallic ferromagnet KCrSe 2[END_REF], CsBiTe 2 [START_REF] Trippel | Synthesis and properties of some compounds ABiTe 2[END_REF], TlYTe 2 [START_REF] Kabré | Julien Pouzol Sur une nouvelle famille de tellurures doubles de thallium(I) et de terres rares[END_REF]. The fact that we also identify these compounds as thermodynamically stable is another confirmation of the validity of our approach.

It is quite interesting to compare these numbers at this stage. First, they indicate that there are probably many more stable ternary compounds than the ones known experimentally (or predicted theoretically and already present in the databases). Second, we can see the strong experimental bias towards oxides in detriment of other chalcogenides.

Up to now, we only know that the delafossite phase is stable (or almost) regarding decomposition into other compounds. The following step in our investigation is to evaluate if the delafossite phase is indeed the ground-state structure, or if there exist other phases lower in energy. We combined two different approaches for all stoichiometries that did not already appear in the materials project database: (i) We tested a set of common prototype structures for the composition ABX 2 (see Table 5.1). This amounts to 28 further structural optimizations per composition. (ii) For each composition, we performed two independent structural prediction runs including 8 atoms (2 formula units) in the unit cell. The length of each run was tuned in order to obtain ∼50 minima.

Our final step consists in filtering again the results and characterizing the remaining materials. We retain only the stoichiometries that are thermodynamically stable or slightly unstable (up to 25 meV/atom above the convex hull, which corresponds approximately to room temperature), and whose ground state structure is a delafossite or a closely related structure. We define as closely related structure a crystal structure that maintains the essential characteristics of the delafossite: the presence of both a linear and an octahedral environment for the two cations. Note however, that the linear and octahedral coordinations can be significantly distorted in some quasi-delafossites. For these structures we finally calculate indirect and direct band gaps, magnetic moments per formula unit and hole effective masses whenever the system is non-metallic. Our results concerning all new structures, i.e. the ones not already present in the materials project database, are summarized in Tables 5.2, 5.3, 5.4, 5.5, 5.6, 5.7 and 5.8. For completeness, we decided to also include the information concerning the phases reported in the literature but not present in the materials project database. These can be found in bold in the tables. All crystallographic information files are given as Appendix A.

We can observe that there are 79 delafossite systems that are stable (and 123 close to stability). The tables include 16 [START_REF] Bowler | Calculations for millions of atoms with density functional theory: linear scaling shows its potential[END_REF] new oxides, 40 [START_REF] Tao | Climbing the density functional ladder: Nonempirical meta-generalized gradient approximation designed for molecules and solids[END_REF] sulfides, 38 [START_REF] Perdew | Generalized gradient approximation made simple[END_REF] selenides, and 29 [START_REF] Slater | A simplification of the Hartree-Fock method[END_REF] tellurides. (The numbers in parentheses are the number of phases strictly below the hull of stability). The list includes a series of rather exotic systems that depart from the usual A + B +3 X -2 2 oxidation states of the delafossites. Actually, we do find often an element of the IA group (Li, Na, K, Rb, or Cs) or IB group (Cu, Ag, Au) in position A. However many other possibilities appear such as Hg, In, Tl, Bi, or even halogens such as Br or I.

Most of the new stable compositions crystallize either in the trigonal delafossite structure of, e.g., CuAlO 2 , or in the closely related trigonal structures (see Fig. 5.1) of, e.g., AgBiS 2 (space group 164) or TlScSe 2 (space group 166). However, we also find a variety of structures showing small distortions or different stacking patterns in comparison to the standard delafossite arrangement. We will now discuss in detail our results separately for oxides, sulfides, selenides, and tellurides.

ABO 2

Among the 3906 possible delafossite oxides (compatible with our selection of chemical elements) there are 93 phases that within appear at least 25 meV/atom below the convex hull of stability. Of these, 77 can already be found in the materials project database [START_REF] Jain | The Materials Project: A materials genome approach to accelerating materials innovation[END_REF] or in Ref. [START_REF] Cerqueira | Identification of novel Cu, Ag, and Au ternary oxides from global structural prediction[END_REF]. The remaining 16 compositions are reported in Table 5.2.

All compounds with A=Pd, Pt, Hg, Tl, and Br crystallize in the delafossite structure with space group 166. The only exception is BrLaO 2 where the stacking is slightly changed, leading to a monoclinic ground-state (prototype of NaCuO 2 ) which is 14 meV/atom lower in energy than the trigonal delafossite phase. On the other hand, when A is an alkali or an alkali earth metal, the ground-state phase has the trigonal structure of TlScSe 2 (space group 166), that is very similar to the trigonal structure of AgBiS 2 (space group 164) shown in the right panel of Fig. 5.1.

The crystals KInO 2 , RbInO 2 , RbRhO 2 , CsLaO 2 , HgMgO 2 , and BrLaO 2 are semiconductors with gaps between 3 and 4 eV, and not too large effective masses of around 2 m e (with the notable exception of RbRhO 2 ). We note that hole effective masses of 1∼2 m e are not small in absolute terms, but they are comparable with those of CuAlO 2 and CuCrO 2 . Among the metals, the compounds containing Ni and Cr are magnetic, with a total magnetization that can reach 3.0 bohr magnetons for the ferromagnetic compound PdCrO 2 .

ABS 2

Among all the stable sulfides that we found in this work, only 10 compositions could be found in the materials project database [START_REF] Jain | The Materials Project: A materials genome approach to accelerating materials innovation[END_REF]. These are NaScS 2 , NaInS 2 , KCrS 2 , KLaS 2 , RbYS 2 , CsLaS 2 , AgBiS 2 , AuCrS 2 , TlBS 2 , and TlInS 2 . All the other novel 40 compositions can be found in Tables 5.3 and 5.4. There are 6 systems with A=Ag and 9 with A=Au, however we did not find any new system with A=Cu. Most of these have the standard delafossite structure with the exception of AgScS 2 and AgYS 2 that exhibit a different stacking of the layers (space group 160 and 156 respectively). We also find 5 systems with A=Hg, 4 with A=Bi, 4 with A=Tl, and 3 with A=Pb. Interestingly, we also find a couple of delafossites with trivalent In and tetravalent Sn in the Wyckoff 1a position. Finally, there are a series of systems with an alkali in the 1a position, all assuming the trigonal lattice depicted in Fig. 5.1. The only systems that do not crystallize in one of the structures of Fig. 5.1 are HIrS 2 and HgTiS 2 . While the former just exhibits a different stacking, in the latter Hg forms layers with square and not hexagonal symmetry.

Many of the novel systems are semiconducting with indirect band gaps varying in the rather large range of 0.7-3.4 eV. Hole effective masses, as expected considering the more delocalized nature of S p states, are significantly lower than in oxides. We would like to point out materials like (K,Rb)YS 2 , composed of relatively abundant elements, with rather large band gaps of the same order of magnitude as the usual Cu delafossite TCOs, and with rather small hole effective masses (of the order of 1.2-1.3 m e ), if compared to CuAlO 2 . AgYS 2 , with an HSE band gap of 3.16 eV and a hole effective mass of 0.71 m e , is the best system identified in this study. Again, stable metals with Cr or Mn in the 1b position are magnetic with relatively large magnetization.

ABSe 2

For the selenides ABSe 2 we find 48 phases below 25 meV/atom, of which can be found in the materials project database. The results for the remaining 38 phases can be found in Tables 5.5 and 5.6. For the compounds with a IA element in the 1a Wyckoff position we find that stable compounds tend to have larger alkalis (Rb, Cs) in detriment of the lighter ones (Na, K). The same does not seem to happen with the IB group, and we can observe several compounds containing S and Se together with both Ag and Au. We also witness the appearance of the rather exotic INiSe 2 at the limit of our threshold of 25 meV/atom. Again most of the compounds crystallize in one of the structures of Fig. 5.1, with the following exceptions: HScSe 2 has a monoclinic ground-state structure where the layers of H are buckled; HgTiSe 2 , like other Hg compounds, has Hg layers with square symmetry; In HgVSe 2 we find a different stacking of the layers leading to a monoclinic ground-state; HgNbSe 2 and TlVSe 2 present BSe 2 layers that are distorted with respect to the usual delafossite structure; Finally, SbMnSe 2 has buckled Sb planes.

We again obtain the usual mixture of metals and semiconductors, although, and as expected, with gaps generally smaller than for the sulfides. Compounds including Cr and Mn are magnetic, but AuCoSe 2 , BrNiSe 2 , and INiSe 2 are non-magnetic.

ABTe 2

From the systems found in our search, there are only 7 ABTe 2 phases in the material project database. These structures are AgSbTe 2 , AgBiTe 2 , KLaTe 2 , KYTe 2 , TlSbTe 2 , TlScTe 2 and TlBiTe 2 . Besides these known tellurides, we identify 29 new phases that are stable or very close to stability (see Tables 5.7 and 5.8). As expected by the chemical similarity between Se and Te, the situation here is analogous to what was described for the Se systems. Perhaps the biggest difference regards the chalcogen in the Wyckoff position 1a. We find a series of Br and I metallic compounds with the 1b position occupied by a transition metal. Some of these systems, like BrPtTe 2 , are quite below the convex hull of stability, which, together with the overall consistency of the results, makes us believe that these exotic systems can be experimentally synthesized.

From the compositions that do not crystallize in the structures of Fig. 5 The two compounds with Mn in the 1b position are ferromagnetic, but, more surprisingly, we also find that HgVTe 2 are TlTiTe 2 have a relatively small magnetic moment. For the semiconducting compounds there is also a further decrease of the magnitude of the gap with respect to the selenides.

Gaps and hole effective masses

Finally, Fig. 5.3 is a scatter plot of the direct band gap and the average hole effective mass for all delafossite-like compounds within our window of stability. We include both the new systems found in this work, as well as the other compounds from the materials project database (such as CuAlO 2 , CuGaO 2 , RbYO 2 , AgYO 2 , etc.), and the Cu, Ag, and Au oxides from Ref. [START_REF] Cerqueira | Identification of novel Cu, Ag, and Au ternary oxides from global structural prediction[END_REF]. Such a plot allows us to quickly recognize potential outliers for p-type transparent conductors. In particular, we are interested in compounds in the lower central part of the plot, with a band-gap larger than the visible range and a small hole mass. We chose to plot the direct band gap, as this is the one most directly related to optical absorption. The numbers were calculated with the HSE06 approximation to the xc potential of DFT, that usually gives accurate gaps. We should keep in mind, however, that HSE underestimates systematically the band gap of medium-large gap semiconductors. Moreover, some of these materials can have substantial quasi-particle corrections and large excitonic bindingenergies [START_REF] Vidal | Effects of Electronic and Lattice Polarization on the Band Structure of Delafossite Transparent Conductive Oxides[END_REF][START_REF] Trani | Band structures of delafossite transparent conductive oxides from a self-consistent GW approach[END_REF]. The hole masses discussed here, on the other hand, are averaged values, and may not include all the physics in cases of valence bands with multiple valleys.

We also label a few relevant points in Fig 5 .3, such as CuAlO 2 and CuGaO 2 . It is evident, for example, that CuAlO 2 , synthesized for the first time in 1997 [START_REF] Kawazoe | P-type electrical conduction in transparent thin films of CuAlO 2[END_REF], possesses a large band gap of around 3.37 eV (in good agreement with the experimental value of 3.53 eV [START_REF] Pellicer-Porres | On the band gap of CuAlO 2 delafossite[END_REF]), but has a relatively high hole effective mass (≈2.5). We can observe several new phases that possess a sufficiently large band gap and a smaller hole effective mass. These phases are not only oxides, but also sulfides, and few selenides and tellurides. For example, some candidate p-type TCO systems are BrLaO 2 (gap 3.92 eV, hole mass 2.23), CsLaO 2 (gap 4.11 eV, hole mass 2.16), KYS 2 (gap 3.37 eV, hole mass 1.23), AgYS 2 (gap 3.16 eV, hole mass 0.71).

Looking at the branch point energies of these candidates, we found a similar behavior for all of them, with values that lie in the middle of the HSE band gap, at distances of 1.4-1.9 eV from the lowest conduction band. The fact that the branch point energy is not close to or within the conduction band is good news, as this would imply that p-type doping is very difficult (as in the case of ZnO). However, this result does not allow to conclude that it is possible to p-dope these materials. When the branch point energy is in the middle of the gap all scenarios are in fact open: the compound may be p-or n-type dopable, or even ambipolar. Accurate defect calculations are then the only safe way to draw reliable conclusions.

Conclusions

We performed a high-throughput investigation using DFT and combining calculations of crystal prototypes with structural prediction. We focused on ternary compounds of composition ABX 2 , where A and B are elements of the periodic table up to Bi, and X is a chalcogen (O, S, Se, and Te) in search of thermodynamically stable delafossite or closely related structures. The starting set of compositions was made of 15 624 compounds, for which we found the lowest-energy structure under the constraint to preserve the symmetries of the trigonal delafossite structure. After this step 285 compounds turn out to be within 50 meV/atom from the convex hull of stability. These compounds were further studied using the minima hopping method for global structural prediction, removing any symmetry constraint to obtain the ground-state crystal structure. At the end of this procedure, we could count 79 ternary systems not present in the materials project database that are thermodynamically stable (and therefore potentially synthesizable) and crystallize in the delafossite or in closely related structures. Among these new delafossites, 10 are oxides, 27 sulfides, 25 selenides and 17 tellurides. If we include also the new structures that are above the convex hull of stability within a threshold of 25 meV per atom, the number of new systems increases to [START_REF] Pani | The phase diagram of the Ba-Si system[END_REF].

By calculating the band gaps and hole effective masses of these structures we identified few systems that could be better p-type transparent conducting materials than those already known. In particular, AgYS 2 , with an HSE band gap of 3.16 eV and a hole effective mass of 0.71 m e , is probably the best system that emerged from this study. KYS 2 and RbYS 2 have even larger band gaps (more than 4 eV) and effective masses of only about 1 m e . We also found several systems, mainly containing Cr and Mn, that have relatively large magnetization densities.

These results provide us with a much more complete picture of the delafossite structure and can be used as a guide for further experimental and theoretical exploration of this fascinating family of compounds. The crystal structures of these new compounds are now available for more accurate theoretical characterization, hoping that some other interesting properties that have not been screened in this first study can come on the scene and motivate experimentalists to try to synthesize some of these compounds. of NOH 4 , there is a layer formed by a NH 3 -NH 3 "dumbbell cluster" which is between two layers formed by O-H chains. On the other hand, for the P2 1 /m phase of HNO 3 , the layer is like a "quasi clover" with a H atom tail formed by three O atom, one N atom and one H atom. Surprisingly we find N-H-O covalent bonds in the P2 1 /m phase of HNO 3 , revealing that pressure plays a major role for forming N-H-O covalent bonds in the N-H-O system. Electronic property calculations revealed that both the C2/m phase of NOH 4 and the P2 1 /m phase of HNO 3 are insulators with a band gap of 6.0 eV and 2.6 eV, respectively.

Novel p-type transparent electrodes of ternary chalcogenides We performed a highthroughput investigation using density functional theory of possible thermodynamically stable phases of delafossite oxides, sulfides, selenides, and tellurides. The stable compositions were then studied using global structural prediction methods that allowed us to obtain their ground-state crystal structures. We found a total of 79 new delafossite (or related) phases that are stable, and therefore potentially possible to synthesize. By calculating the band gaps and hole effective masses of these structures we identified few systems that could be better p-type transparent conducting materials than those already known. In particular, AgYS 2 , with an HSE band gap of 3.16 eV and a hole effective mass of 0.71 m e , is probably the best system identified in this study. KYS 2 and RbYS 2 have even larger band gaps (more than 4 eV) and effective masses of about 1 m e . We also found several systems, mainly containing Cr and Mn, that have relatively large magnetization densities. These results provide us with a much more complete picture of the interesting delafossite family and can be used as a guide for further experimental and theoretical exploration of this fascinating family of compounds.

In conclusion, the ab-initio prediction of new compounds is by now a mature subject. We showed that combining the power of DFT with high-throughput techniques and standard prediction methods is an efficient way to discover new materials with improved properties. This path is orders of magnitude faster than experimental synthesis and may allow us to have a rather complete view on more synthesizable compounds in a foreseeable future. a (R -3 2/m) b=3.61618 Å Ir 0.00000 0.00000 0.50000 c=20.86395 Å S 0.00000 0.00000 -0.11166 α=β =90 • ,γ=120 • a=3.33532 Å Au 0.00000 0.00000 0.00000 AuMnS 2 a (R -3 2/m) b=3.33532 Å Mn 0.00000 0.00000 0.50000 c=21.81510 Å S 0.00000 0.00000 -0.10786 α=β =90 • ,γ=120 • a=3.58538 Å Au 0.00000 0.00000 0.00000 AuRhS 2 a (R -3 2/m) b=3.58538 Å Rh 0.00000 0.00000 0.50000 c=20.98328 Å S 0.00000 0.00000 -0.11063 α=β =90 • ,γ=120 • a=3.82016 Å Au 0.00000 0.00000 0.00000 AuScS 2 a (R -3 2/m) b=3.82016 Å Sc 0.00000 0.00000 0.50000 c=22.01038 Å S 0.00000 0.00000 -0.10493 α=β =90 • ,γ=120 • a=4.13622 Å Au 0.00000 0.00000 0.00000 AuYS 2 a (R -3 2/m) b=4.13622 Å Y 0.00000 0.00000 0.50000 c=22.10901 Å S 0.00000 0.00000 -0.10430 α=β =90 • ,γ=120 • a=3.63137 Å Hg 0.00000 0.00000 0.00000 HgHfS 2 b (R -3 2/m) b=3.63137 Å Hf 0.00000 0.00000 0.50000 c=28.69144 Å S 0.00000 0.00000 0.21733 α=β =90 • ,γ=120 a=b=3.59750 Å Tl 0.00000 0.00000 0.00000 TlHfS 2 164 (P -3 2/m 1) c=8.54200 Å Hf 0.00000 0.00000 0.50000 α=β =90 • ,γ=120 • S 0.33333 0.66667 -0.32445 a=3.37108 Å Tl 0.00000 0.00000 0.00000 TlMnS 2 166 a (R -3 2/m) b=3.37108 Å Mn 0.00000 0.00000 0.50000 c=26.36703 Å S 0.00000 0.00000 -0.11764 α=β =90 • ,γ=120 • a=b=3.47225 Å Tl 0.00000 0.00000 0.00000 TlTiS 2 164 (P -3 2/m 1) c=8.17414 Å Ti 0.00000 0.00000 0.50000 α=β =90 • ,γ=120 • S 0.33333 0.66667 -0.32576 a=b=3.64723 Å Tl 0.00000 0.00000 0.00000 TlZrS 2 164 (P -3 2/m 1) c=8.25608 Å Zr 0.00000 0.00000 0.50000 α=β =90 • ,γ=120 • S 0.33333 0.66667 0.31475 a=3.46684 Å Sn 0.00000 0.00000 0.00000 SnCrS 2 166 a (R -3 2/m) b=3.46684 Å Cr 0.00000 0.00000 0.50000 c=24.44877 Å S 0.00000 0.00000 -0.11396 α=β =90 • ,γ=120 • a=3.39805 Å Sn 0.00000 0.00000 0.00000 SnTiS 2 166 a (R -3 2/m) b=3.39805 Å Ti 0.00000 0.00000 0.50000 c=25.67830 Å S 0.00000 0.00000 -0.11020 α=β =90 • ,γ=120 • a=3.46683 Å Pb 0.00000 0.00000 0.00000 PbCrS 2 166 a (R -3 2/m) b=3.46683 Å Cr 0.00000 0.00000 0.50000 c=25.11917 Å S 0.00000 0.00000 -0.11541 α=β =90 • ,γ=120 • a=3.43824 Å Pb 0.00000 0.00000 0.00000 PbTiS 2 166 a (R -3 2/m) b=3.43824 Å Ti 0.00000 0.00000 0.50000 c=26.24558 Å S 0.00000 0.00000 -0.11203 α=β =90 • ,γ=120 • a=3.60049 Å Pb 0.00000 0.00000 0.00000 PbZrS 2 166 a (R -3 2/m) b=3.60049 Å Zr 0.00000 0.00000 0.50000 c=26.99142 Å S 0.00000 0.00000 -0.10959 α=β =90 • ,γ=120

Figure 2 . 1

 21 Figure 2.1 Depiction of the velocity and position updates in PSO. The black solid line approximates a typical energy landscape. Arrows represent either the velocity at the relative positions of particle (here represent a structure). An initial velocity v t develops to v t after one PSO step through Eq. 2.51.

Figure 3 . 1

 31 Figure 3.1 Calculated enthalpy of formation (ΔH in eV/atom) of various Ba-Si compounds with respect to the elemental decomposition into bulk Ba and Si at 0, 5, 20, 50, 100, and 150 GPa. The solid lines represent the convex hulls, while the dotted lines are just a guide for the eyes.

Figure 3

 3 Figure 3.2 (a) Compressibility, V0/V, for Ba and Si. (b) Ground-state energy and PV components of the enthalpy of the elements relative to BaSi.

Figure 3 . 4

 34 Figure 3.4 Structures predicted in Ba-Si system: (a) Imma phase of BaSi; (b) Fd 3m phase of BaSi 2 ; (c) I4/mmm phase of BaSi 3 ; and (d) Cmmm phase of BaSi 5 . The green and blue balls represent Ba and Si atoms, respectively.

Figure 3 . 6

 36 Figure 3.6 Enthalpy per formula unit of the different phases of BaSi 2 relative to the P4 3 32 phase as a function of pressure.

Figure 3

 3 Figure 3.7 (a) Phonon dispersions of the Fd 3m phase of BaSi 2 at 50 GPa. (b) Electronic band structures and densities of state of the Fd 3m phase of BaSi 2 at 50 GPa.

Figure 3 . 8

 38 Figure 3.8 Eliashberg phonon spectral function α 2 F(ω) and integrated EPC parameter λ as a function of frequency for the Fd 3m structure at 50 GPa.

3. 3 . 3

 33 BaSi 3 , BaSi 5 and BaSi 6

Figure 3

 3 Figure 3.9 (a) Phonon dispersions of the I4/mmm phase of BaSi 3 at 20 GPa. (b) Electronic band structures and densities of state of the I4/mmm phase of BaSi 3 at 20 GPa.

Figure 3 .

 3 Figure 3.10 (a) Phonon dispersions of the Cmmm phase of BaSi 5 at 20 GPa. (b) Electronic band structures and densities of state of the Cmmm phase of BaSi 5 at 20 GPa.

Figure 4 . 1

 41 Figure 4.1 Ternary phases diagram of the N-H-O system at 150 GPa. The novel phases are marked in red.

NOH 4 (

 4 C2/m) NOH 5 (NH 3 H 2 O) (Ima2)

Figure 4 . 2

 42 Figure 4.2 Enthalpy per formula unit of the Ima2 phase of NOH 5 (NH 3 H 2 O) and of the C2/m phase of NOH 4 as a function of pressure. The zero line corresponds to the decomposition into NH 3 and H 2 O (for NOH 5 ) and into NH 3 , H 2 O, and N 2 (for NOH 4 ).

Figure 4

 4 Figure 4.3 (a) Crystal structure of NOH 4 with space group of C2/m. The white, blue and red atoms represent H, N and O, respectively. (b) Charge density plot. (c) Electron-localization function on a plane perpendicular to the (010) crystallographic direction.

. 2 .

 2 The new phase of NOH 4 has C2/m symmetry (space group #12) with 24 atoms per unit cell as shown in Fig. 4.3 (a), and is composed by O-H chains separated by a layer of NH 3 molecules. To understand the bonding character of this structure, we plot in Fig. 4.3 (b) the charge density and in Fig. 4.3 (c) the electron localization function (ELF). This latter

Figure 4 .

 4 Figure 4.4 (a) and (b) The P2 1 /m phase of HNO 3 . The red, light blue, and white atoms represent respectively O, N and H. (c) Charge density plot of the P2 1 /m phase of HNO 3 . (d) Slabs cut along the (010) direction of the ELF. For clarity, we used here a (2×1×2) supercell.

Figure 4 . 5 Frequencies (cm - 1 )Figure 4 . 6

 45146 Figure 4.5 Band structure and density of states of the C2/m phase of NOH 4 at 150 GPa.

Figure 4 . 7 Frequencies (cm - 1 )Figure 4 . 8

 47148 Figure 4.7 Relative enthalpy per formula unit of the new phase P2 1 /m of HNO 3 with respect to phase II (P2 1 /c) of nitric acid as a function of pressure.

  with a H atom tail. There are 10 atoms in the unit cell, as shown in Fig.4.4 (a) and (b). The "quasi clover" is formed by three nitrogen atoms, one oxygen atom and one hydrogen atom. Every N atom in the middle of the "quasi clover" has three nearest O atoms while one of the three O atoms is bonded with one H atom.The charge density plot of the P2 1 /m phase of HNO 3 is shown in Fig.4.4 (c). We can clearly find that the electron cloud is almost entirely distributed around the region between the N and O atoms. Obviously, there are two different kinds of N-O covalent bonds with bond lengths of 1.195 Å and 1.246 Å at 150 GPa. From the plot of the ELF [see Fig.4.4 (d)] one can distinguish the three N-O covalent bonds, together with the lone pairs around the oxygen atoms. The O-H bond distance is 1.042 Å at 150 GPa.

Figure 4 . 9

 49 Figure 4.9 Band structure and density of states of the P2 1 /m phase of HNO 3 at 150 GPa.

  5.1. A polytype of this structure with hexagonal symmetry (space group 194) can be formed when two neighboring A layers are stacked with each layer rotated by 180 • in relation to one another. If A belongs to the IA group, one often finds another closely related trigonal structure (space group 164) with a slightly different stacking and where the layers are closer (see right panel of Fig. 5.1).

Figure 5 . 1

 51 Figure 5.1 Left: Trigonal delafossite structure (space group 166) of CuAlO 2 (Cu dark blue, Al cyan, O red); Right: Trigonal structure (space group 164) of AgBiS 2 (Ag gray, Bi magenta, S yellow).From this perspective, there is no difference between this phase of AgBiS 2 and the trigonal (space group 166) structure of TlScSe 2 . The A cations of ABX 2 form a triangular lattice that alternates with layers composed of distorted edge-sharing BX 6 octahedra. The similarity between the left and right panels is evident: only the alignment of A and X atoms differentiates the two structures (the dotted lines are a guide to the eye meant to emphasize this difference).

  proposed few empirical rules for p-type TCOs, aiming at favoring at the same time large band gaps and dispersive top valence bands: (i) The A cations should possess closed d shells to prevent from absorbing light in the visible range. Optical transparency has also been associated with dipole forbidden transitions related to the linear O-A-O coordination motif. (ii) The energy levels of the A cations should overlap with the anion p levels (O 2p) to allow for strong hybridization at the top of the valence band. (iii) Hole paths should be favored, for example providing B states that can mix strongly with O 2p states into the valence band to build a A-O-B-O-A pathway.

Figure 5 . 2

 52 Figure 5.2 Stability maps for ABO 2 , ABS 2 , ABSe 2 and ABTe 2 . The colors indicate the distance to the convex hull of stability (in meV per atom), with green meaning that the composition is thermodynamically stable. The order of the elements along both axes follows an optimized Pettifor scale.

Figure 5 . 3

 53 Figure 5.3 Hole effective masses as a function of the HSE06 band gap for all compounds that are thermodynamically stable.

Table 3 .

 3 1 Calculated structural parameters of our predicted stable structures for Ba-Si compounds at their corresponding pressures

	Space group	Lattice	Atomic coordinates (fractional)
	Pressure	parameters
	Imma-BaSi	a=3.67280 Å	Ba(4e) 0.50000 0.25000 0.39804
	20 GPa	b=5.08260 Å	Si(4e) 0.50000 -0.25000 0.17580
		c=9.43680 Å
		α=β =γ=90 •
	Fd 3m-BaSi 2	a=7.18670 Å	Ba(8a) 0.25000 0.25000 0.25000
	50 GPa	b=7.18670 Å	Si(16d) 0.62500 0.12500 0.12500
		c=7.18670 Å
		α=β =γ=90 •
	I4/mmm-BaSi 3	a=6.84750 Å	Ba1(4e) 0.50000 0.50000 0.31654
	20 GPa	b=6.84750 Å	Ba5(4d) 0.00000 0.50000 0.25000
		c=12.17490 Å	Si1(16m) 0.68117 0.31883 0.09624
		α=β =γ=90 •	Si17(8i) 0.66915 0.00000 0.00000
	Cmmm-BaSi 5	a=4.2121 Å	Ba1(2c) 0.00000 0.50000 0.50000
	20 GPa	b=7.2510 Å	Si1(8n) -0.00000 0.83323 0.81577
		c=6.3456 Å	Si9(2b) 0.00000 0.50000 -0.00000
		α=β =γ=90 •

Table 4 .

 4 1 Calculated structural parameters of our predicted stable structures of the N-H-O compounds at their corresponding pressures.

	Space group	Lattice	Atomic coordinates (fractional)
	Pressure	parameters
	C2/m-NOH 4	a=4.80513 Å	H (8j) 0.73862 0.27769 0.18187
	150 GPa	b=3.94998 Å	H (2a) 0.00000 0.00000 0.00000
		c=3.72274 Å	H (2b) 0.00000 0.50000 0.00000
		α=γ=90 •	H (4i) 0.47412 0.00000 0.83563
		β =101.9 •	O (4i) 0.86866 0.00000 0.21498
			N (4i) 0.60060 0.00000 0.65717
	P2 1 /m-HNO 3	a=3.76343 Å	H (2e) 0.09859 0.25000 0.06669
	150 GPa	b=3.71923 Å	O (2e) 0.34502 0.25000 0.01337
		c=3.79534 Å	O (2e) 0.67110 0.25000 0.65678
		α=γ=90 •	O (2e) -0.01431 0.25000 0.33338
		β =119.7 •	N (2e) 0.67195 0.25000 0.34246
	Ima2-NOH 5	a=3.82308 Å	H (8h) 0.00000 0.05335 0.26087
	(NH 3 H 2 O)	b=4.11076 Å	H (8i) 0.27414 0.25000 0.02918
	100 GPa	c=5.62304 Å	H (4d) 0.25000 0.25000 0.75000
		α=β =γ=90 •	O (4e) 0.00000 0.25000 0.86373
			N (4e) 0.00000 0.25000 0.37254

Table 5 .

 5 [START_REF] Sarmiento-Pérez | Novel phases of lithium-aluminum binaries from first-principles structural search[END_REF] Common prototype structures for the composition ABX 2 used in this study. We list the space group (spg) number, the number of atoms in the unit cell, and an example of a composition that crystallizes in the phase.

	spg	# atoms	example
	166	4	CuAlO 2
	122	8	CuInSe 2
	62	16	CaBaO 2
	14	16	CuBrTe 2
	15	8	KFeSe 2
	141	8	CuGaSe 2
	12	4	NaCuO 2
	63	8	CuSrO 2
	15	32	KAlTe 2
	140	8	KAlTe 2
	160	4	CuAsSe 2
	33	16	LiAlSe 2
	123	4	AgSbTe 2
	9	8	AgInSe 2
	156	4	AgAlS 2

Table 5 .

 5 2 Distance to the convex hull (E hull in meV/atom), HSE06 gaps (indirect and direct in eV), average hole effective mass for the semiconducting phases (m * h in m e ), magnetic moment per formula unit (μ in bohr magnetons, and space group (spg) of the new (quasi) stable delafossite and closely related phases with composition ABO 2 . The compounds that are not in the materials project database, but that are reported in literature are indicated in bold. a See left panel of Fig. 5.1. b See right panel of Fig. 5.1.

	Structure	E hull	E ind Gap	E dir Gap	m * h	μ	spg
	KInO 2	-106	3.49	3.63	2.71	0	166 b
	RbInO 2	-27	3.38	3.57	2.68	0	166 b
	RbRhO 2	-185	3.06	3.49	34.84	0	166 b
	CsLaO 2	-44	4.10	4.15	2.16	0	166 b
	MgNiO 2	-4	0	0	-	2.0	166 b
	PdAlO 2	-7	0	0	-	0	166 a
	PdCrO 2	-12	0	0	-	3.0	166 a
	PdRhO 2	22	0	0	-	0	166 a
	PtAlO 2	12	0	0	-	0	166 a
	PtNiO 2	17	0	0	-	1.0	166 a
	HgMgO 2	0	3.25	3.77	2.39	0	166 a
	TlRhO 2	23	0	0	-	0	166 a
	BrCdO 2	-29	0	0	-	0	166 a
	BrLaO 2	-13	3.92	4.59	2.23	0	12 a
	BrNiO 2	14	0	0	-	1.4	166 a
	BrTlO 2	24	0	0	-	0	166 a

Table 5 .

 5 [START_REF] Bernstein | What superconducts in sulfur hydrides under pressure and why[END_REF] Properties of delafossite sulfides ABS 2 . For an explanation see caption of Table5.2.

	Structure E hull	E ind Gap	E dir Gap	m * h	μ	spg
	HIrS 2	0	2.49	2.81	1.13 0	160
	KBiS 2	-184 2.14	2.53	5.09 0	166 b
	KScS 2	-158 2.89	3.72	1.20 0	166 b
	KYS 2	-194 3.37	4.14	1.23 0	166 b
	RbScS 2	-137 2.88	3.69	1.29 0	166 b
	AgCoS 2	-22	1.29	2.50	1.77 0	166 a
	AgMnS 2	-53	0	0	-	2.1 166 a
	AgRhS 2	-26	1.69	2.19	1.35 0	166 a
	AgScS 2	-3	2.29	3.06	0.66 0	160
	AgIrS 2	11	1.87	2.16	0.87 0	166 a
	AgYS 2	-15	3.16	3.24	0.71 0	156
	AuAlS 2	-29	2.72	2.97	0.65 0	166 a
	AuBiS 2	20	1.35	2.28	0.93 0	166 a
	AuCoS 2	-48	1.36	2.68	1.19 0	166 a
	AuInS 2	-13	1.70	1.95	0.68 0	166 a
	AuIrS 2	-23	2.13	2.52	0.83 0	166 a
	AuMnS 2	-41	0	0	-	2.0 166 a
	AuRhS 2	-54	1.72	2.39	1.19 0	166 a
	AuScS 2	24	2.28	3.14	0.52 0	166 a
	AuYS 2	23	2.95	3.68	0.62 0	166 a

Table 5 .

 5 4 Properties of delafossite sulfides ABS 2 . Continuation of Table. 5.3. For an explanation see caption of Table 5.2. The ground-state of TlMnS 2 has space group number 15. The delafossite structure is only 10 meV/atom above this phase.

	Structure E hull E ind Gap	E dir Gap	m * h	μ	spg
	HgHfS 2	14	0.79	1.34	-	2.4 b
	HgMnS 2	-36	0	0	-	3.0 a
	HgPtS 2	16	1.24	1.42	0.71 0	b
	HgTiS 2	-14	0	0	-	0
	HgZrS 2	11	0.66	1.12	0.19 0	b
	BiAlS 2	23	0	0	-	0	a
	BiCrS 2	-7	0	0	-	3.0 a
	BiTiS 2	6	0	0	-	0	a
	BiIrS 2	-5	0	0	-	0	a
	InTiS 2	-25	0	0	-	0	a
	InZrS 2	9	0	0	-	0	a
	TlHfS 2	-5	0	0	-	0	a
	TlMnS 2	-33	0	0	-	2.6 a *
	TlTiS 2	-21	0	0	-	0	a
	TlZrS 2	-43	0	0	-	0	a
	SnCrS 2	24	0	0	-	2.8 a
	SnTiS 2	-4	0	0	-	0	a
	PbCrS 2	25	0	0	-	2.8 a
	PbTiS 2	-16	0	0	-	0	a
	PbZrS 2	5	0	0	-	0	a
	*

Table 5 .

 5 [START_REF] Yu | Identification of potential photovoltaic absorbers based on firstprinciples spectroscopic screening of materials[END_REF] Properties of the delafossite selenides ABSe 2 . For an explanation see caption of Table5.2. The ground-state of HMnSe 2 has space group number 8. The delafossite structure is 8 meV/atom above this phase.

	Structure	E hull	E ind Gap	E dir Gap	m * h	μ	ΔE GS
	HMnSe 2	-15	0	0	-	4.0	166 a *
	HScSe 2	-15	2.28	2.51	0.97	0	11
	KCrSe 2	-83	0	0	-	3.0	166 b
	KScSe 2	-157	2.37	3.11	0.98	0	166 b
	KRhSe 2	-94	1.82	1.87	1.38	0	166 b
	KYSe 2	-203	2.85	3.72	1.00	0	166 b
	RbScSe 2	-141	2.40	3.13	1.03	0	166 b
	RbRhSe 2	-80	1.78	1.92	1.26	0	166 b
	CsScSe 2	-142	2.30	2.97	1.10	0	166 b
	CsLaSe 2	-159	3.00	3.32	1.45	0	166 b
	CsRhSe 2	-72	1.76	1.84	1.13	0	166 b
	CsYSe 2	-206	2.74	3.45	1.10	0	166 b
	AgMnSe 2	-54	0.64	0.84	-	4.0	166 b
	AgRhSe 2	-1	0.97	1.73	0.30	0	166 a
	AuCoSe 2	15	0.60	1.86	-	0	166 a
	AuCrSe 2	2	0	0	-	3.0	166 a
	AuMnSe 2	-44	0	0	-	4.0	166 a
	AuRhSe 2	-13	1.01	1.99	0.08	0	166 a
	HgHfSe 2	12	1.06	1.40	3.51	0	166 b
	HgMnSe 2	-67	0	0	-	2.9	166 b
	*

Table 5 .

 5 [START_REF] Lin | In silico screening of carbon-capture materials[END_REF] Properties of the delafossite selenides ABSe 2 . Continuation of Table5.5. For an explanation see caption of Table5.2.

	Structure	E hull	E ind Gap	E dir Gap	m * h	μ	ΔE GS
	HgNbSe 2	0	0	0	-	0	38
	HgPtSe 2	14	1.28	1.66	0.90	0	166 b
	HgRhSe 2	13	0	0	-	0	166 a
	HgTiSe 2	1	0	0	-	0	1
	HgVSe 2	-7	0	0	-	0	6
	HgZrSe 2	11	0	0	1.46	0	166 b
	InZrSe 2	17	0	0	-	0	164 b
	TlHfSe 2	1	0	0	-	0	164 b
	TlMnSe 2	-98	0	0	-	4.0	166 b
	TlRhSe 2	-63	0	0	-	0	164 b
	TlTiSe 2	23	0	0	-	0	164 b
	TlVSe 2	-7	0	0	-	0	187
	TlZrSe 2	-40	0	0	-	0	164 b
	SbMnSe 2	-4	0	0	-	4.0	11
	BiCrSe 2	17	0	0	-	3.0	166 a
	BiMnSe 2	12	0	0	-	3.7	166 a
	BrNiSe 2	-2	0	0	-	0	166 a †
	INiSe 2	25	0	0	-	0	166 a
	† The ground-state of BrNiSe 2 has space group number 2. The delafossite structure is 22
	meV/atom above this phase.

  .1 we have: HgVTe 2 , with displays distorted planes; BrIrTe 2 with has a different stacking; BrPdTe 2 with distorted Br layers; finally INiTe 2 and IPdTe 2 have a different stacking.

Table 5 .

 5 [START_REF] Krishna | In silico screening of metal-organic frameworks in separation applications[END_REF] Properties of the delafossite tellurides ABTe 2 . For an explanation see caption of Table5.2. The ground-state of AgMnTe 2 has space group number 72. The trigonal structure is 8 meV/atom above this phase.

	Structure	E hull	E ind Gap	E dir Gap	m * h	μ	ΔE GS
	LiYTe 2	-265	1.56	2.48	0.61	0	164 b
	NaYTe 2	-314	1.89	2.94	0.84	0	166 b
	RbLaTe 2	-108	2.30	2.87	1.10	0	166 b
	CsBiTe 2	-68	1.57	1.75	1.18	0	166 b
	CsHfTe 2	12	0	0	-	0	166 b
	CsScTe 2	-70	1.64	2.20	0.77	0	166 b
	CsYTe 2	-268	2.03	2.76	0.80	0	166 b
	BaCaTe 2	25	2.26	2.94	0.58	0	166 b
	AgMnTe 2	-4	0.01	0.28	-	4.0	166 b *
	HgHfTe 2	10	0	0	-	0	166 b
	HgTiTe 2	7	0	0	-	0	166 b
	HgVTe 2	9	0	0	-	1.2	11
	HgZrTe 2	-65	0	0	0	0	166 b
	InYTe 2	-169	0.50	1.44	-	0	166 b
	TlHfTe 2	-3	0	0	-	0	164 b
	*

Table 5 .

 5 [START_REF] Yang | A search model for topological insulators with high-throughput robustness descriptors[END_REF] Properties of the delafossite tellurides ABTe 2 . Continuation ofTable 5.7. For an explanation see caption of Table5.2.

	Structure	E hull	E ind Gap	E dir Gap	m * h	μ	ΔE GS
	TlMnTe 2	-47	0	0	-	4.0	164 b
	TlTiTe 2	4	0	0	-	0.4	164 b
	TlYTe 2	-230	1.20	2.05	0.24	0	166 b
	TlZrTe 2	-59	0	0	-	0	164 b
	BrCoTe 2	2	0	0	-	0	166 a
	BrIrTe 2	-4	0.92	1.72	1.48	0	160
	BrNiTe 2	-30	0	0	-	0	166 b
	BrPdTe 2	18	0	0	-	0	12
	BrPtTe 2	-68	0	0	-	0	166 b
	BrRhTe 2	16	0	0	-	0	166 a
	INiTe 2	8	0	0	-	0	13
	IPdTe 2	14	0	0	-	0	12
	IPtTe 2	-8	0	0	-	0	166 b
	IRhTe 2	25	0	0	-	0	166 a

Table A .

 A 1 Structual parameters of the new stable delafossite and closely related phases with composition ABO 2 . Table A.2 Structual parameters of the new stable delafossite and closely related phases with composition ABO 2 . Continuation of Table A.1. Table A.3 Structual parameters of the new stable delafossite and closely related phases with composition ABS 2 .

	Compounds Spg nunmber (name) Lattice parameters Atom Coordinates parameters
			a=3.65516 Å	Ir	0.00000 0.00000 0.19100
	HIrS 2	160 (R 3 m)	b=3.65516 Å	S	0.00000 0.00000 -0.21000
	Compounds Spg nunmber (name) Lattice parameters Atom Coordinates parameters a=3.35949 Å In 0.00000 0.00000 0.50000 KInO 2 166 b (R -3 2/m) b=3.35949 Å K 0.00000 0.00000 0.00000 c=18.54410 Å O 0.00000 0.00000 -0.22894 α=β =90 • ,γ=120 • a=3.39981 Å In 0.00000 0.00000 0.00000 RbInO 2 166 b (R -3 2/m) b=3.39981 Å Rb 0.00000 0.00000 0.50000 c=19.36914 Å O 0.00000 0.00000 -0.27463 α=β =90 • ,γ=120 • a=3.29582 Å Rh 0.00000 0.00000 0.00000 RbRhO 2 166 b (R -3 2/m) b=3.29582 Å Rb 0.00000 0.00000 0.50000 c=17.953 Å O 0.00000 0.00000 -0.28092 α=β =90 • ,γ=120 • a=3.78465 Å La 0.00000 0.00000 0.00000 CsLaO 2 166 b (R -3 2/m) b=3.78465 Å Cs 0.00000 0.00000 0.50000 c=20.84 Å O 0.00000 0.00000 -0.27684 α=β =90 • ,γ=120 • a=3.00143 Å Ni 0.00000 0.00000 0.50000 MgNiO 2 166 b (R -3 2/m) b=3.00143 Å Mg 0.00000 0.00000 0.00000 c=14.558 Å O 0.00000 0.00000 0.25082 α=β =90 • ,γ=120 • a=2.86607 Å Pd a=2.85681 Å Pt c=17.47866 Å S 0.00000 0.00000 -0.41600 0.00000 0.00000 0.00000 PtAlO 2 α=β =90 • ,γ=120 • H 0.00000 0.00000 -0.33500 166 a (R -3 2/m) b=2.85681 Å Al 0.00000 0.00000 0.50000 c=18.15331 Å O a=4.15325 Å Bi 0.00000 0.00000 0.00000 0.00000 0.00000 0.11357 KBiS 2 166 b (R -3 2/m) b=4.15325 Å K 0.00000 0.00000 0.50000 α=β =90 • ,γ=120 • a=2.92246 Å Pt c=22.36348 Å S 0.00000 0.00000 -0.26288 0.00000 0.00000 0.50000 PtNiO 2 α=β =90 • ,γ=120 • 166 a (R -3 2/m) b=2.92246 Å Ni 0.00000 0.00000 0.00000 c=18.21565 Å O a=3.84107 Å Sc 0.00000 0.00000 0.00000 0.00000 0.00000 -0.39247 KScS 2 166 b (R -3 2/m) b=3.84107 Å K 0.00000 0.00000 0.50000 α=β =90 • ,γ=120 • a=3.25902 Å c=21.87898 Å S 0.00000 0.00000 -0.26922 Hg 0.00000 0.00000 0.00000 α=β =90 • ,γ=120 • HgMgO 2 166 a (R -3 2/m) b=3.25902 Å Mg 0.00000 0.00000 0.50000 c=18.25154 Å O a=4.05568 Å Y 0.00000 0.00000 0.00000 0.00000 0.00000 0.11034 KYS 2 166 b (R -3 2/m) b=4.05568 Å K 0.00000 0.00000 0.50000 α=β =90 • ,γ=120 • a=3.16242 Å c=22.12079 Å S 0.00000 0.00000 -0.26596 Tl 0.00000 0.00000 0.00000 TlRhO 2 α=β =90 • ,γ=120 • 166 a (R -3 2/m) b=3.16242 Å Rh 0.00000 0.00000 0.50000 c=20.77487 Å O a=3.87064 Å Rb 0.00000 0.00000 0.00000 0.00000 0.00000 -0.12102 RbScS 2 166 b (R -3 2/m) b=3.87064 Å Sc 0.00000 0.00000 0.50000 α=β =90 • ,γ=120 • a=3.51303 Å c=22.80286 Å S 0.00000 0.00000 0.22769 Cd 0.00000 0.00000 0.00000 α=β =90 • ,γ=120 • BrCdO 2 166 a (R -3 2/m) b=3.51303 Å Br 0.00000 0.00000 0.50000 c=18.64577 Å O a=3.35973 Å Ag 0.00000 0.00000 0.00000 0.00000 0.00000 0.39411 AgCoS 2 166 a (R -3 2/m) b=3.35973 Å Co 0.00000 0.00000 0.50000 α=β =90 • ,γ=120 • a=8.25488 Å c=21.26877 Å S 0.00000 0.00000 -0.11332 La 0.00000 0.00000 0.00000 α=β = • ,γ=120 • 0.00000 0.00000 0.50000 PdAlO 2 166 a (R -3 2/m) b=2.86607 Å Al 0.00000 0.00000 0.00000 c=18.09580 Å O 0.00000 0.00000 -0.38662 α=β =90 • ,γ=120 • a=3.00903 Å Pd 0.00000 0.00000 0.50000 PdCrO 2 166 a (R -3 2/m) b=3.00903 Å Cr 0.00000 0.00000 0.00000 c=18.23102 Å O 0.00000 0.00000 -0.38882 α=β =90 • ,γ=120 • a=3.08303 Å Pd 0.00000 0.00000 0.00000 PdRhO 2 166 a (R -3 2/m) b=3.08303 Å Rh 0.00000 0.00000 0.50000 c=18.30676 Å O 0.00000 0.00000 0.11048 BrLaO 2 12 (A 1 2/m 1) b=3.94162 Å Br 0.50000 0.50000 0.50000 c=6.85559 Å O a=3.32591 Å Ag 0.00000 0.00000 0.00000 -0.16000 0.00000 1.00000 AgMnS 2 166 a (R -3 2/m) b=3.32591 Å Mn 0.00000 0.00000 0.50000 α=γ=90 • ,β =134.99 • a=3.11474 Å c=22.33769 Å S 0.00000 0.00000 -0.10862 Br 0.00000 0.00000 0.00000 BrNiO 2 α=β =90 • ,γ=120 • 166 a (R -3 2/m) b=3.11474 Å Ni 0.00000 0.00000 0.50000 c=17.78526 Å O a=3.57217 Å Ag 0.00000 0.00000 0.00000 0.00000 0.00000 -0.11013 AgRhS 2 166 a (R -3 2/m) b=3.57217 Å Rh 0.00000 0.00000 0.50000 α=β =90 • ,γ=120 • a=3.63606 Å c=21.49311 Å S 0.00000 0.00000 -0.11138 Tl 0.00000 0.00000 0.50000 BrTlO 2 α=β =90 • ,γ=120 • 166 a (R -3 2/m) b=3.63606 Å Br 0.00000 0.00000 0.00000 c=18.94165 Å O a=b=3.76128 Å Ag 0.00000 0.00000 0.00658 0.00000 0.00000 -0.11016 α=β =90 • ,γ=120 • 160 (R 3 m) c=20.68221 Å Sc 0.00000 0.00000 0.15680 AgScS 2 α=β =90 • S 0.00000 0.00000 0.41700 α=β =90 • ,γ=120 • γ=120 • S 0.0000 0.00000 -0.11144
			a=3.60138 Å	Ir	0.00000 0.00000 0.50000
	AgIrS 2	166 a (R -3 2/m)	b=3.60138 Å	Ag	0.00000 0.00000 0.00000
			c=21.37937 Å	S	0.00000 0.00000 -0.11235
			α=β =90 • ,γ=120

Table A .

 A 4 Structual parameters of the new stable delafossite and closely related phases with composition ABS 2 . Continuation of Table A.3.

			a=b=4.05641 Å	Ag 0.33300 0.66700 -0.42100
	AgYS 2	(P 3 m 1)	c=6.63987 Å α=β =90 •	Y S	0.00000 0.00000 -0.00178 0.33300 0.66700 0.20456
			γ=120 •	S	0.66700 0.33300 -0.23684
			a=3.56258 Å	Au 0.00000 0.00000 0.00000
	AuAlS 2 a (R -3 2/m) b=3.56258 Å	S	0.00000 0.00000 -0.10762
			c=21.56252 Å	Al 0.00000 0.00000 0.50000
			α=β =90 • ,γ=120 •
			a=4.28266 Å	Bi 0.00000 0.00000 0.50000
	AuBiS 2 a (R -3 2/m) b=4.28266 Å	Au 0.00000 0.00000 0.0000
			c=22.26884 Å	S	0.00000 0.00000 -0.10296
			α=β =90 • ,γ=120 •
			a=3.36726 Å	Au 0.00000 0.00000 0.00000
	AuCoS 2 a (R -3 2/m) b=3.36726 Å	Co 0.00000 0.00000 0.50000
			c=20.78827 Å	S	0.00000 0.00000 -0.11258
			α=β =90 • ,γ=120 •
			a=3.94181 Å	Au 0.00000 0.00000 0.00000
	AuInS 2	a (R -3 2/m) b=3.94181 Å	In	0.00000 0.00000 0.50000
			c=22.00386 Å	S	0.00000 0.00000 -0.10468
			α=β =90 • ,γ=120 •
			a=3.61618 Å	Au 0.00000 0.00000 0.00000
	AuIrS 2

Table A .

 A 5 Structual parameters of the new stable delafossite and closely related phases with composition ABS 2 . Continuation of Table A.4.

			a=3.34606 Å	Hg 0.00000 0.00000 0.00000
	HgMnS 2 166 a (R -3 2/m) b=3.34606 Å	Mn 0.00000 0.00000 0.50000
			c=27.51030 Å	S	0.00000 0.00000 -0.11876
			α=β =90 • ,γ=120 •
			a=3.57244 Å	Hg 0.00000 0.00000 0.50000
	HgPtS 2	166 b (R -3 2/m) b=3.57244 Å	Pt	0.00000 0.00000 0.00000
			c=26.37048 Å	S	0.00000 0.00000 -0.28647
			α=β =90 • ,γ=120 •
			a=5.90492 Å	Hg -0.44390 -0.49609 1.00000
			b=3.39530 Å	Hg 0.48651 -0.42345 -0.49379
			c=9.60591 Å	Ti	-0.01571 -0.22243 0.00483
	HgTiS 2	1 (P 1)	α=87.44096 • β =105.41234 •	Ti S	0.48495 0.28364 0.00500 -0.11507 0.26144 0.16022
			γ=89.99510 •	S	0.08440 0.30054 -0.14989
				S	0.38464 -0.23908 0.15946
				S	-0.41558 -0.19991 -0.15015
			a=3.67081 Å	Hg 0.00000 0.00000 0.50000
	HgZrS 2 166 b (R -3 2/m) b=3.67081 Å	Zr 0.00000 0.00000 0.00000
			c=28.05292 Å	S	0.00000 0.00000 -0.28090
			α=β =90 • ,γ=120 •
			a=3.53328 Å	Bi 0.00000 0.00000 0.00000
	BiAlS 2	166 a (R -3 2/m) b=3.53328 Å	S	0.00000 0.00000 -0.11385
			c=24.70228 Å	Al 0.00000 0.00000 0.50000
			α=β =90 • ,γ=120 •
			a=3.52727 Å	Bi 0.00000 0.00000 0.00000
	BiCrS 2	166 a (R -3 2/m) b=3.52727 Å	Cr 0.00000 0.00000 0.50000
			c=24.57201 Å	S	0.00000 0.00000 -0.11492
			α=β =90 • ,γ=120 •
			a=3.44094 Å	Bi 0.00000 0.00000 0.50000
	BiTiS 2	166 a (R -3 2/m) b=3.44094 Å	Ti	0.00000 0.00000 0.00000
			c=25.83420 Å	S	0.00000 0.00000 -0.38896
			α=β =90 • ,γ=120 •
			a=3.59387 Å	Bi 0.00000 0.00000 0.00000
	BiIrS 2	166 a (R -3 2/m) b=3.59387 Å	Ir	0.00000 0.00000 0.50000
			c=24.25913 Å	S	0.00000 0.00000 -0.11851
			α=β =90 • ,γ=120 •
			a=3.39205 Å	In	0.00000 0.00000 0.00000
	InTiS 2	166 a (R -3 2/m) b=3.39205 Å	Ti	0.00000 0.00000 0.50000
			c=26.02497 Å	S	0.00000 0.00000 -0.11113
			α=β =90 • ,γ=120 •
			a=b=3.61485 Å	In	0.00000 0.00000 0.50000
	InZrS 2	164 (P -3 2/m 1) c=8.18948 Å α=β =90 • ,γ=120 • S Zr 0.00000 0.00000 0.00000 0.33333 0.66667 0.18891

Table A .

 A 6 Structual parameters of the new stable delafossite and closely related phases with composition ABS 2 . Continuation of Table A.5.

Table A .

 A 9 Structual parameters of the new stable delafossite and closely related phases with composition ABSe 2 . Continuation of Table A.8.

		a=3.80643 Å	Hg 0.00000 0.00000 0.00000
	HgRhSe 2 166 a (R -3 2/m)	b=3.80643 Å	Rh 0.00000 0.00000 0.50000
		c=23.93630 Å	Se 0.00000 0.00000 -0.11563
		α=β =90 • ,γ=120 •
		a=3.52743 Å	Hg 0.48447 0.00794 -0.02072
		b=10.10198 Å	Hg 0.47672 -0.00430 0.47278
		c=6.11632 Å	Se -0.23259 -0.33594 0.16527
	HgTiSe 2 1 (P 1)	α=102.59452 • β =90 •	Se 0.11387 0.34262 0.21573 Se 0.26668 -0.33700 -0.33558
		γ=99.61212 •	Se -0.38583 0.34373 -0.28350

Table A .

 A 11 Structual parameters of the new stable delafossite and closely related phases with composition ABTe 2 .

	Compounds Spg nunmber (name) Lattice parameters Atom Coordinates parameters
			a=b=4.30381 Å	Te	0.33333 0.66667 -0.25725
	LiYTe 2	164 (P -3 2/m 1)	c=7.15747 Å α=β =90 • ,γ=120 • Li Y	0.00000 0.00000 0.00000 0.00000 0.00000 0.50000
			a=4.43028 Å	Te	0.00000 0.00000 0.24412
	NaYTe 2	166 b (R -3 2/m)	b=4.43028 Å	Y	0.00000 0.00000 0.50000
			c=22.70594 Å	Na	0.00000 0.00000 0.00000
			α=β =90 • ,γ=120 •
			a=4.74251 Å	La	0.00000 0.00000 0.50000
	RbLaTe 2	166 b (R -3 2/m)	b=4.74251 Å	Te	0.00000 0.00000 -0.23545
			c=25.61822 Å	Rb	0.00000 0.00000 0.00000
			α=β =90 • ,γ=120 •
			a=4.65619 Å	Bi	0.00000 0.00000 0.00000
	CsBiTe 2	166 b (R -3 2/m)	b=4.65619 Å	Cs	0.00000 0.00000 0.50000
			c=26.88465 Å	Te	0.00000 0.00000 0.26897
			α=β =90 • ,γ=120 •
			a=4.20620 Å	Hf	0.00000 0.00000 0.00000
	CsHfTe 2	166 b (R -3 2/m)	b=4.20620 Å	Cs	0.00000 0.00000 0.50000
			c=27.05162 Å	Te	0.00000 0.00000 0.27177
			α=β =90 • ,γ=120 •
			a=4.37131 Å	Cs	0.00000 0.00000 0.00000
	CsScTe 2	166 b (R -3 2/m)	b=4.37131 Å	Te	0.00000 0.00000 0.22681
			c=26.50526 Å	Sc	0.00000 0.00000 0.50000
			α=β =90 • ,γ=120 •
			a=4.55740 Å	Cs	0.00000 0.00000 0.50000
	CsYTe 2	166 b (R -3 2/m)	b=4.55740 Å	Te	0.00000 0.00000 0.26999
			c=26.74329 Å	Y	0.00000 0.00000 0.00000
			α=β =90 • ,γ=120 •
			a=4.76440 Å	Ba	0.00000 0.00000 0.00000
	BaCaTe 2	166 b (R -3 2/m)	b=4.76440 Å	Te	0.00000 0.00000 -0.24129
			c=23.39713 Å	Ca	0.00000 0.00000 0.50000
			α=β =90 • ,γ=120 •
			a=4.03881 Å	Te	0.00000 0.00000 0.24288
	AgMnTe 2	166 b (R -3 2/m)	b=4.03881 Å	Ag	0.00000 0.00000 0.00000
			c=20.81605 Å	Mn	0.00000 0.00000 0.50000
			α=β =90 • ,γ=120 •
			a=3.95145 Å	Hg	0.00000 0.00000 0.00000
	HgHfTe 2	166 b (R -3 2/m)	b=3.95145 Å	Hf	0.00000 0.00000 0.50000
			c=29.71098 Å	Te	0.00000 0.00000 0.22664
			α=β =90 • ,γ=120 •
			a=3.73742 Å	Hg	0.00000 0.00000 0.50000
	HgTiTe 2	166 b (R -3 2/m)	b=3.73742 Å	Te	0.00000 0.00000 0.27456
			c=29.77145 Å	Ti	0.00000 0.00000 0.00000
			α=β =90

• ,γ=120 •

Acknowledgements

First, I tender my deepest thanks to my dedicated supervisors, Prof. Alfonso San Miguel, Prof. Miguel Marques and Prof. Silvana Botti for giving me the opportunity to work on this project, for your guidance and encouragement throughout my thesis work, for the discussions and corrections of my manuscript, and for your care of my living in Lyon and Jena as well. I also want to thank Dr. Stéphane Pailhès, Dr. Georges Boulon and Dr. Tristan Albaret, for the fruitful discussions. I would like to thank the rest of my thesis committee, for their encouragement, insightful comments, and hard questions.

I thank my fellow labmates in our group: Dr. Tiago Cerqueira, Dr. Sabine Korbel, Dr. Abraao Torres Dias, Pedro Borlido, Sun Lin, Dr. Claudia Rödl, Ivan Guilhon Mitoso Rocha, Valerio Armuzza, Christoph Otzen, Thomas Bischoff, Dr. Rafael A. Sarmiento Pérez , Silvio Domingos Silva Santos, Dr. Aude Stolz, Amani Tlili, and Helainne Thomeny Girao for the stimulating discussions and for all the fun we have had in the last three years.

To my friends, Zonglong Yang, Ye Tian, Xingjie Ma, Tianwen Huang, Kefeng Zhang, Zeya Jia, Ruihan Guo, Quanyi Yin, Biwen Zhu, Tingting Niu, Dan Yi, Mingchao Ji, Yang Wang, Yunpeng Luo, Jianbei Huang and other friends whose names are not listed here, thank you all for your company and help during my stay in Lyon and Jena.

I gratefully acknowledge Prof. Yanming Ma for your advice and help for applying the scholarship from China Scholarship Council (CSC). I also express my acknowledgement to CSC for funding this thesis.

I owe a great thanks to my girlfriend Wenwen Cui, thank you so much for your understanding and all your support on my work. Last but not the least, I would like to thank my family, for supporting me spiritually throughout my life.

Chapter 6

Summary and conclusions

In this thesis, we used a combination of high-throughput and ab initio global structure prediction methods to study a wide range of different systems under different conditions. In the following, a summary of the most notable results is presented.

New phases of Ba-Si system under high pressure Using a global structural prediction method we studied systematically the phase diagram and the electronic properties of Ba-Si compounds. We identified five novel stoichiometric BaSi x phases with unexpected structures that might be experimentally synthesizable over a wide range of pressure. These include BaSi, BaSi 2 , BaSi 3 , BaSi 5 , and BaSi 6 . For the BaSi system, we found a new metallic Imma phase that is stable under pressure from 6 GPa to 65 GPa. For BaSi 2 , there is a new metallic phase Fd 3m that is stable from 21 GPa to 100 GPa. This phase, in which each Si forms six bonds with other neighboring Si atoms, becomes superconducting below 1.2 K at 50 GPa. Finally, at around 20 GPa two unexpected compositions become stable, namely BaSi 3 (I4/mmm) and BaSi 5 (Cmmm), both displaying metallic behavior. These results show the remarkable variety of Ba-Si compounds, and can serve as an important guide for further experimental studies of this binary system.

Phases diagram of N-H-O system under high pressure We explored the structures of ternary N-H-O system at 150 GPa using structural prediction (CALYPSO) combined with first-principle calculations. Two novel phases of NOH 4 and HNO 3 were predicted to exist at 150 GPa while NH 3 H 2 O will decompose into NH 3 and H 2 O when the pressure is above 122 GPa. The novel phase of NOH 4 possesses C2/m symmetry while the new phase of HNO 3 has a P2 1 /m symmetry. The C2/m of NOH 4 becomes stable at 71 GPa, while the P2 1 /m phase of HNO 3 will be more energetically favorable than the phase II (P2 1 /c) [START_REF] Allan | The crystal structures of the low-temperature and high-pressure polymorphs of nitric acid[END_REF] from 39 GPa to 150 GPa, the maximum pressure that we have studied. For the C2/m phase

List of Publications

Appendix A Structual parameters of the novel phases with composition ABX 2 . (Chapter 5)

We summarize structual parameters of the new stable delafossite and closedly related phases with composition ABO 2 , ABS 2 , ABSe 2 and ABTe 2 .