These De L'universite De Lyon Computational And

N◦ d’ordre 127-2012 Ann´ee 2012

THESE DE L’UNIVERSITE DE LYON

D´elivr´ee par

L’UNIVERSITE CLAUDE BERNARD LYON 1

ECOLE DOCTORALE DE PHYSIQUE ET ASTROPHYSIQUE

DIPLOME DE DOCTORAT (arrˆet´edu 7 aout 2006)

soutenue publiquement le 18 septembre 2012

par

M. Jos´eA. Flores Livas

COMPUTATIONAL AND EXPERIMENTAL STUDIES OF sp3-MATERIALS AT HIGH PRESSURE

Directeur de Thèse: M. Miguel A. L. Marques Co-directeur de Thèse: M. Stéphane Pailhès

JURY

Mme. Isabelle Daniel Président du Jury M. Xavier Gonze Rapporteur M. Aldo H. Romero Rapporteur M. Stefan Goedecker Examinateur M. Yann Gallais Examinateur M. Romain Viennois Examinateur M. Stéphane Pailhès Co-directeur de Thèse M. Miguel A. L. Marques Directeur de Thèse ii ABSTRACT

COMPUTATIONAL AND EXPERIMENTAL STUDIES OF sp3-MATERIALS AT HIGH PRESSURE

M. Jos´eA. Flores Livas Universit´ede Lyon 1, 2012

We present experimental and theoretical studies of sp3 materials at high pressure, alkaline-earth-metal (AEM) disilicides, disilane (Si2H6) and carbon. First, we review the thermodynamic phase diagram of pressure and temperature of AEM disilicides. In particular, we study the case of a layered phase of BaSi2 which has an hexagonal structure with sp3 bonding of the silicon atoms. This electronic environment leads to a natural corrugated Si-sheets. We demonstrate experimentally and theoretically an enhancement of superconducting transition temperatures from 6 to 8.9 K when silicon planes flatten out in this structure. Our extensive ab initio calculations based on density-functional theory guided the experimental research and permit explain how electronic and phonon properties are strongly affected by changes in the buckling of silicon plans in AEM disilicides. Second, we investigated the crystal phases of disilane at the megabar range of pressure. A novel metallic phase of disilane is proposed by using crystal structure prediction methods. En- thalpically this structure is more favourable than previously proposed structures of disilane up to 280 GPa. The calculated transition temperatures yielding a superconducting Tc of around 20 K at 100 GPa and decreasing to 13 K at 220 GPa. These values are significantly smaller than previously predicted Tc’s and put serious drawbacks in the possibility of high- 3 Tc superconductivity based on silicon-hydrogen systems. Third, we studied the sp -carbon structures at high pressure through a systematic structure search. We found a new allotrope of carbon with Cmmm symmetry which we refer to as Z-carbon. This phase is predicted to be more stable than graphite for pressures above 10 GPa and is formed by sp3-bonds. It is a wide band-gap semiconductor and has a hardness comparable to diamond. Experimental and simulated XRD, Raman spectra suggest the existence of Z-carbon in micro-domains of graphite under pressure.

iii iv PUBLICATIONS DURING THIS WORK

1. Enhancing the superconducting transition temperature of BaSi2 by structural tuning. J. A. Flores-Livas, R. Debord, S. Botti, A. San Miguel, M. A. L. Marques, and S. Pailh`es. Phys. Rev. Lett. 106, 087002 (2011).

2. High-pressure structures of disilane and their superconducting properties. J. A. Flores- Livas, M. Amsler, T. J. Lenosky, L. Lehtovaara, S. Botti, M. A. L. Marques, S. Goedecker. Phys. Rev. Lett. 108 117004 (2012).

3. Raman activity of sp3 carbon allotropes under pressure: A density functional theory study. J. A. Flores-Livas, L. Lehtovaara, M. Amsler, S. Goedecker, S. Botti, S. Pailh`es, A. San Miguel and M. A. L. Marques. Phys. Rev. B 85, 155428 (2012).

4. Superconductivity in layered binary silicides: A density functional theory study. J. A. Flores-Livas, R. Debord, S. Botti, A. San Miguel, S. Pailh`es, and M. A. L. Marques. Phys. Rev. B 84, 184503 (2011).

5. Crystal structure of cold compressed graphite. M. Amsler, J. A. Flores-Livas, L. Lehto- vaara, F. Balima, S. A. Ghasemi, D. Machon, S. Pailh`es, A. Willand, D. Caliste, S. Botti, A. San Miguel, S. Goedecker and M.A.L. Marques. Phys. Rev. Lett. 108, 065501 (2012).

6. Novel structural motifs in low energy phases of LiAlH4. M. Amsler, J. A. Flores-Livas, Huang Zhou, S. Botti, M. A. L. Marques, S. Goedecker. Phys. Rev. Lett. 108 205505 (2012).

7. p-doping in expanded phase of ZnO: An ab anitio study. D. Hapiuk, M. A. L. Marques, P. Melinon, J. A. Flores-Livas, S. Botti, and B. Masenelli. Phys. Rev. Lett. 108 115903 (2012).

8. Low-energy silicon allotropes with strong absorption in the visible for photovoltaic applications. S. Botti, J. A. Flores-Livas, M. Amsler, S. Goedecker and M. A. L. Marques. Submitted (2012).

9. Prediction of a novel monoclinic carbon allotrope. M. Amsler, J. A. Flores-Livas, S. Botti, M. A. L. Marques, S. Goedecker. http://arxiv.org/abs/1202.6030 (2012).

v vi TABLE OF CONTENTS

Page

CHAPTERS

I TheoreticalBackground ...... 3 I.1 Latticevibrations: Phonons ...... 3 I.1.1 Harmonic Potential Approximation ...... 4 I.1.2 SemiclassicalApproach ...... 6 I.2 IntroductiontoSuperconductivity ...... 8 I.2.1 EnergyGapandBCStheory ...... 9 I.2.2 Eliashberg and McMillan equations ...... 12 I.3 Ab-initiotechniques ...... 14 I.3.1 Density-FunctionalTheory ...... 15 I.3.2 Density-Functional Perturbation Theory ...... 18 I.4 CrystalandPhononSymmetry ...... 22 II Background of Experimental Techniques ...... 24 II.1 HighPressurePhysics ...... 24 II.1.1 BeltApparatusPress ...... 25 II.1.2 DiamondAnvilCell ...... 26 II.2 Characterization techniques ...... 27 II.2.1 X-ray Diﬀraction Technique ...... 27 II.2.2 Low Temperature Electrical Resistivity ...... 29 II.2.3 RamanScattering ...... 31 III Studies on Alkaline Earth Metal Disilicides ...... 36 III.1 Introduction: Properties of AEM-disilicides ...... 36 III.1.1 CrystalsofAEMDisilicides ...... 37 III.1.2 Ba-Disilicide: Electronic Properties ...... 39 III.1.3 Superconductivity in AEM Disilicides ...... 41 III.2TheCaseofBa-Disilicide ...... 42 III.2.1 Structure: XRD and DFT Calculations ...... 44 III.2.2 Raman Activity of Ba-Disilicide ...... 46 III.2.3 Electronic Transport in Ba-Disilicide ...... 48 III.3 Superconductivity in Ba-Disilicide ...... 52 III.3.1 Layered Structure as a Function of Pressure and Temperature . . . . 52 III.3.2 Lattice Dynamics as a Function of Pressure and Temperature . . . . 54 III.3.3 Parameter z and Its Electronic Repercussions ...... 57 III.3.4 Superconductivity in Ba-Disilicide ...... 58 III.4Conclusions ...... 62 IV Superconductivity in Intercalated AEM Disilicides ...... 64 IV.1 Intercalation of Layered Crystals ...... 64 IV.2CrystalStructures ...... 65 IV.3 Electronic Structure and Fermi Surface...... 67 IV.4 Phonons in Intercalated Disilicides ...... 70 IV.5 Transition Temperature in AEM-Disilicides ...... 71 IV.6Conclusions ...... 76

vii V SuperconductivityinDisilane ...... 77 V.1 Metallic Structures of Disilane ...... 80 V.2 Superconducting Disilane at High Pressure ...... 82 V.3 Conclusions ...... 84 VI Z-Carbon:Anewallotrope ...... 85 VI.1 Carbon allotropes and Crystal Prediction ...... 86 VI.2Z-carbon...... 86 VI.3 Raman SpectraofCarbonat HighPressure ...... 89 VI.4Conclusions ...... 94 CurriculumVitæ ...... 119

viii 1

INTRODUCTION

Due to an increasing orientation towards environment, the design of recyclable energy devices and energy-saving research is attracting considerable attention. In order to realize energy efficient materials it is crucial to improve their characteristics, and to realize new devices using new materials such as high-temperature superconductors and highly efficient thermoelectric materials. Additionally, these materials have to fulfil imperative needs such as being environmentally friendly, abundant and technologically applicable from the indus- trial point of view. The study of superconductors and thermoelectric materials has always been an active research field. The ultimate goal is obviously the enhancement of transition temperatures

Tc (in the case of superconductors) and of the figure of merit (in the case of thermoelectric materials). Superconductivity represents one of the most interesting and enigmatic phenomenon in science, and the potential applications of superconductors are numerous. If one would find a material with a Tc close to ambient temperature, industry and technology could enter in a revolution changing our life style. Over the past 100 years, the effort of thousands of researches steadily increased our understanding of superconductors and increased Tc. However, room temperature superconductivity still remains a dream. The current state-of-the-art in computational materials science allows to calculate material properties using ab initio quantum-mechanical techniques, whose only input information is the chemical composition of the material. Therefore, the impact in understanding superconducting phenomena, and modelling superconducting behaviour, has expanded greatly over the past two decades. Moreover, since the discovery of the surprising Tc of 39 K in 1 a chemically simple compound as MgB2, studies of superconductivity entered an era of enhanced research efforts. The basic electronic, phononic and magnetic structures are an essential input for theoretical models of superconduction to calculate the transition temperature. These approaches are accurate enough to predict the superconducting state in many materials. Indeed, once the material is synthesized and Tc is measured very often a good agreement with theoretical predictions is found. The central issue in designing superconducting materials concerns the maximization of

Tc. In a phonon-mediated superconductor in order to get a high Tc one generally needs a substantial interaction between electrons and phonons. In layered materials, the softening of phonons and the strong anisotropy of the electronic Fermi surface lead to higher temperatures of superconductivity. This is of course an incentive to continue searching 2D-like systems. Moreover, pressure can also maximize Tc. Early predictions of superconductivity 2 in simple elements at high pressures opened new roads towards high temperature superconductors. Pressure is a simple tool to alter the chemistry and the volume of materials. In fact, high pressure studies become powerful tools to help physicists to discover new states of matter, and understand the underlying physics of high pressure phenomena. Another fundamental problem remaining in condensed matter physics is the theoretical description of crystals and their prediction for a desirable thermodynamic condition. Recently, thanks to the use of advanced search algorithms in combination with ab initio methods, permits to explore low-lying energetic structures not yet synthesized or found in nature. This methodology has been proved as efficient strategy to screen possible atomic configurations. And thanks to the increasing development of codes and supercomputers, more realistic situations can be simulated. Consequently, systems ranging from few atoms to nanostructures, and even more real-complex materials can be explored.2,3 This dissertation is focused on the understanding of specific material properties, namely electronic transport and superconductivity, of alkaline earth metal (AEM) disilicides, disilane compounds and carbon-based materials. The outline of Chapters in this dissertation is as follows. Chapter I introduces the theoretical background and general concepts of phonons and superconductivity. Then, the theoretical approaches such as density-functional theory (DFT) and density-functional perturbation theory (DFPT) ab initio techniques will be briefly explained. Chapter II describes the underlying theory behind the experimental techniques used. Chapters III and IV are devoted to study experimentally and computationally the alkaline earth metal (AEM) disilicides. In particular Seebeck coefficients of semi-metallic structures and superconductivity of layered compounds. Chapter V reviews the high pressure phases of disilane and the correct description of their superconducting properties at the megabar range of pressure. In Chapter VI we will discuss about the carbon allotropes at high pressure. Finally in the last Chapter, general conclusions of the dissertation are exposed. CHAPTER I. THEORETICAL BACKGROUND 3

CHAPTER I

THEORETICAL BACKGROUND

In this Chapter we review the theoretical methods used in this dissertation. First, we look at lattice dynamics of crystals in a simple model of harmonic oscillators based in the adiabatic or Born-Oppenheimer approximation.4 We also included a short description of phonon-drive superconductivity, where we introduced the key concepts of BCS theory,5 Ginzburg-Landau theory,6 Eliashberg theory,7 and McMillian-Allen-Dynes formulation. The third section presents an introduction of the Kohn-Sham formalism of density- functional theory (DFT).8 At this point we introduce density-functional perturbation theory (DFPT) a perturbative approach for the calculation of response properties.9,10 The last section describes concepts in crystal and phonon symmetry.

I.1 LATTICE VIBRATIONS: PHONONS

Phonons represent the intrinsic vibrations of the lattice. The simplest approach that can be used to understand these vibrations is the classical harmonic oscillator. First, we consider a classical Hamiltonian for an harmonic oscillator of mass m and force constant k

p2 kx2 = + (I.1.1) H 2m 2 which yields,

mx¨ = kx , (I.1.2) − where x and p denote the amplitude and the linear momentum, respectively. The solution of this equation yields a continuous and smooth dependence of the energy on the mean square amplitude of the oscillation, x2 ,

E = mω2 x2 , where ω is the frequency of oscillation, ω = k/m. Making this energy equal to the 1 ~ value derived from quantum-mechanics, E = (n+ 2 ) ω, with n = 0, 1, 2, ..., we obtain the quantization for the mean square amplitude, that gives for the zero point vibration, at T = 0, x2 = ~/2mω. If we consider now a crystal with two atoms per unit cell with reduced −1 −1 −1 mass , = (m1 + m2 ) , and N unit cells, we can derive the corresponding expression for the vibrational amplitude of a phonon by just summing the contribution of all cells and substituting m by 2.9,11 4 CHAPTER I. THEORETICAL BACKGROUND

~ x2 = ∝ −1/2 , (I.1.3) 4Nω which yields the mass dependence of the average amplitude of the atomic displacements. This example shows that is possible to extract general trends from the harmonic oscillator by introducing simple quantization arguments in the classical formalism. Another application of the harmonic oscillator is in the derivation of the Bose-Einstein distribution from the quantization of the energy and the Boltzmann thermal occupation of levels. If En is the energy of the n-th level, the average energy is given by

∞ En ∞ ∞ − ∂ −βE ∂ −~ωβ(n+ 1 ) E e kB T e n e 2 n − ∂β − ∂β E = n=0 = n=0 = n=0 , (I.1.4) ∞ En ∞ ∞ − −~ωβ(n+ 1 ) e kBT e−βEn e 2 n=0 n=0 n=0 where β = 1 and k is the Boltzmann constant. The sums can be easily evaluated since kBT B ∞ they correspond to geometric progressions of ratio less than the unity, yielding e−β~/2(1+ n=0 e−β~ω)−1. The average number of states occupying the n th excited level, n can be − obtained from the average energy E as follows:

−∂ −β~ω/2 −β~ω −1 E 1 e (1 e ) 1 n = = ∂β − , ~ω − 2 ~ωe−β~ω/2(1 e−β~ω)−1 − 2 − (I.1.5) 1 T : T ∝ →∞ = −β~ω , with following limits e 1 = 0 : T 0 −  → here, n is the so-called Bose-Einstein distribution for bosons, and yields the temperature  dependence of the Bose-Einstein distribution.

I.1.1 Harmonic Potential Approximation

Let us start considering a perfect crystal lattice. The Hamiltonian of such a crystal in absence of external ﬁelds is

= + + H He Hi He−i where and are the purely electronic and ionic contributions to the Hamiltonian and He Hi is the potential energy representing the interaction between electrons and ions, He−i Ve−i

= T + , He e Ve = T + , Hi i Vi = . He−i Ve−i CHAPTER I. THEORETICAL BACKGROUND 5

T and represent the kinetic and the potential energy for electrons and ions, respec- e,i Ve,i tively. This separation of electronic and nuclear motion is known as the adiabatic or Born-Oppenheimer approximation.4,12 This assumes that the nuclei are so much more massive than the electrons, that they must accordingly have much smaller velocities. Thus it is plausible that, on the typical time-scale of the nuclear motion, the electrons relax very rapidly to the instantaneous ground-state configuration and are able to follow instanta- neously the movements of the ions. The Born-Oppenheimer ansatz allows us to decouple the electronic and the ionic parts of the Hamiltonian at the cost of neglecting some terms. Within this approximation, we write the solution of the Schrödinger equation factorized in the form Ψ(r, R)= ψ(r, R)φ(R), being r and R the electronic and the ionic positions. The terms Ψ(r, R) and φ(R) represent the ionic and the electronic parts of the wavefunctions, respectively.12 The Schrödinger equation can be expressed as

= ( + + ) ψφ H He Ve−i Hi ~2 (I.1.6) = ( + ) ψφ + ψ φ φ 2ψ + 2 ψ φ , He Ve−i Hi − 2m ∇ ∇ ∇ i i where mi represents the mass of the i-th ion and where we have substituted in the last term the kinetic energy of the ions by its corresponding diﬀerential operator. The adiabatic approximation tells us that we can neglect the last term in parenthesis in Eq. I.1.6, leading to the following equation for the ions

[T + + E (R)]φ [T + V (R)]φ = Eφ , (I.1.7) i Vi e ≡ i where Ee(R) is the instantaneous solution to the electronic part of the Hamiltonian for ﬁxed ionic positions R,

( + )ψ = E (R)ψ , He Hei e and V (R) is an eﬀective potential energy. Ee(R) represents the adiabatic contribution of the electrons to the lattice energy.12 For metals the term neglected in this approximation 13 contributes to the phonon frequency by less than ω(~ω/εF ), where εF is the Fermi energy. For semiconductors or insulators, this term is also negligible since phonon energies are much smaller than the electronic gap. In general, this contribution amounts to less than 1% of the frequencies. In the case of phenomena where the dynamics of electrons are directly involved, the adiabatic approximation may not hold and the term neglected in I.1.7 should be properly considered. Relevant physical properties as electric resistance due to scattering of electrons by phonons: the temperature dependence of energy band gaps in semiconductors, and formation of the Cooper-pairs in superconductors require to go beyond 6 CHAPTER I. THEORETICAL BACKGROUND this approximation. As crystals are stable below its melting point, one can further simplify Eq. I.1.7 by performing a Taylor expansion of the eﬀective potential energy around the equilibrium positions of the ions

1 ∂2V (R) V (R)= V (R )+ u(l, κ)u(l′, κ′) 0 u u ′ ′ 2 ′ ′ ∂ (l, κ)∂ (l , κ ) lκ,l κ 0 1 ∂3V (R) + u(l, κ)u(l′, κ′)u(l′′, κ′′)+ ... (I.1.8) u u ′ ′ u ′′ ′′ 3! ′ ′ ′′ ′′ ∂ (l, κ)∂ (l , κ )∂ (l , κ ) lκ,l κ ,l κ 0 V (R)= + + + ... V0 V2 V3 In the previous Eq. I.1.8 the linear terms are zero, as they are proportional to the Hellmann-Feynman forces that are zero at equilibrium. We can also expand the electron- ion potential energy

= V [R(l, κ), r] , Ve−i κ lκ in a Taylor series

2 ∂Vκ[R(l, κ), r] ∂ Vκ[R(l, κ), r] 2 e−i = Vκ[R0(l, κ), r]+ u(l, κ)+ u (l, κ)+ ... V ∂u(l, κ) ∂u2(l, κ) lκ lκ 0 lκ 0 where Vκ(R) is the electron-ion potential of the κ-th atomic species, and r denotes the electrons positions. The ﬁrst term is actually the lattice periodic potential, and is usually included in the electronic Hamiltonian, . He

I.1.2 Semiclassical Approach

Lets us consider the expansion of the eﬀective potential energy given by Eq. I.1.8. The term corresponds to the binding energy of the crystal, which can be set to zero when V0 solving the lattice dynamical problem. The second term contains all the information about the lattice dynamics within the harmonic approximation, and can be re-written as

1 V = Φ l, κ; l′κ′ u (l, κ) u l′κ′ , 2 α,β α β lκ,l′κ′ α,β where Φα,β is the second derivative of the potential energy V with respect to the components of u. It is also called the force constants matrix (the three-dimensional equivalent of the spring constant in the harmonic oscillator). In the following we will use α, β and γ to CHAPTER I. THEORETICAL BACKGROUND 7 denote Cartesian coordinates of vectors. The equations of motion become a generalization of the Eq. I.1.2,

m u¨ (l, κ)= Φ l, κ; l′κ′ u l′κ′ (I.1.9) κ α − α,β β l′,κ′,α′ The force constant matrix Φ has two important properties: lattice translational symmetry, which allows us to reduce the problem of solving Eq. I.1.9 to a problem with 3n variables (where n is the number of atoms in the unit cell). And the second, translational invariance of the crystal, which means that there is no force acting on any atom when all atoms are equally displaced:

′ ′ Φαβ l, κ; l , κ = 0 l′,κ′ In parallel to these properties, the force constant matrix is symmetric with respect to the exchange of atomic displacements due to the symmetry of the Hessian matrix of Eq. I.1.8. Using translational symmetry, Bloch’s theorem allows us to deﬁne solutions to the equations of motion which are linear combinations of Bloch waves depending on a wavevector q :

1 i[qr(l)−ωt] uα(l, κ)= u˜α (q κ) e , (I.1.10) √mκ q | where r(l) is the equilibrium position vector of the l-th unit cell andu ˜ (q κ) are the coef- α | ﬁcients of the Fourier expansion of uα(l, κ), that are independent of l. The substitution of this trial solution in Eq. I.1.9 yields 3n equations of motion,

ω2u˜ (q, κ)= D κκ′ q u˜ q κ′ , (I.1.11) α αβ | β | κ′,β′ where Dαβ represents the so-called dynamical matrix of the crystal, and is obtained from

Φαβ by making a Fourier transformation

′ 1 ′ ′ iqr(l′) Dαβ(κκ q)= Φαβ(0, κ; l κ )e . (I.1.12) ′ | √mκmκ ′ l Non-trivial solutions of Eq.I.1.11 can be obtained by solving the determinantal equation,

′ 2 D (κκ q) ω δ δ ′ = 0 . | αβ | − αβ κκ | This equation has 3n eigenvalues for a given q, denoted by ω2(q), which represent the normal modes of vibration. Due to the translational invariance of Φαβ, three of the modes are degenerate for q = 0 and have zero frequency ωq=0 = 0, since they correspond to rigid 8 CHAPTER I. THEORETICAL BACKGROUND translations of the crystal along the three independent axis. The relation between ω and q is called the dispersion relation of the frequencies. There are therefore 3n branches of these dispersions relations, j = 1, 2, 3, ...3n. Among them we can distinguish 3 acoustical branches and 3n 3 optical branches, the acoustical being the three branches with the limit − ω = 0 for q = 0. Moreover, for each of the 3n eigenvalues ωj for a given q, there exists a normalized eigenvector e(qj κ) deﬁned by |

ω2(q)e (qj κ)= D (κκ′ )e (qj κ) . j α | αβ | β | αβ These eigenvectors fulﬁl the orthonormality and completeness relations given by

∗ ′ e (qj κ)e (qj κ)= δ ′ , α | α | jj κα ∗ e (qj κ)e (qj κ)= δ δ ′ . α | β | αβ κκ j From an experimental point of view, to fully determine the force constants of the system one needs to know not only the phonon dispersion relations, but also the eigenvectors.

Furthermore, as the dynamical matrix (Dαβ) is Hermitian, it is assured that the eigenvalues ω2 are real. However, under certain conditions (with pressure, temperature or under strong electronic interactions) some phonon dispersion branches might decrease in value (frequency) until they become zero. Eventually, these so-called soft modes can formally lead to negative values of ω2 and therefore to an imaginary frequency ω. This can be understood from Eq. I.1.10 as an instability of the crystal structure, caused by anharmonic or strong electron-phonon interaction eﬀects. The existence of soft modes is normally associated with phase transitions or strong electron-phonon interaction.11,12 Thus, one can determine the dynamical-structural-stability for a given crystal by calculating the phonon dispersion relation over the entire Brillouin zone and assuring real frequency values.

I.2 INTRODUCTION TO SUPERCONDUCTIVITY

Superconductivity is the phenomenon of zero electrical resistivity below a certain critical † temperature Tc. This characteristic of some metals was discovery a century ago by H. Kamerlingh Onnes in Leiden.14 Another characteristic of superconductors discovered by W. Meissner and R. Ochsenfeld is that, below the critical temperature, superconductors show perfect diamagnetism. In other words, they repeal magnetic ﬁelds, a phenomenon called the Meissner eﬀect.15 These two properties of superconductors are illustrated in Fig. 1.1. In (a)

†A recommended text book in superconductivity is “Introduction to Superconductivity” of Michael Tin- kham. 17 CHAPTER I. THEORETICAL BACKGROUND 9

B=0

FIG. 1.1: (a) Electrical conductivity of a metal superconductor as function of temperature. Below Tc the electrical resistivity drops to zero. (b) Temperature dependence of the critical fields. Below Tc and Hc one finds the superconducting state. (c) Schematic diagram depicting the exclusion of magnetic flux from the interior of a massive superconductor. Here, λ represent the penetration depth, normally of the order of a few hundred A.˚

The electrical resistivity in metals drops to zero below the transition temperature Tc and

(b) the Meissner effect is destroyed by a critical magnetic field Hc, and this can be related thermodynamically to the free-energy difference between the normal and superconducting state at zero field. (c) The exclusion of the magnetic field is screened exponentially from the interior of superconductors with a depth penetration λ. This quantity, the penetration depth, is empirically found to be approximately described by λ(T ) λ(0)[1 (T/T )4]1/2.16 ≈ − c

I.2.1 Energy Gap and BCS theory

One of the first steps in the understanding of superconductors was the establishment of the existence of an energy gap ∆, of order kTc, between the ground state and the quasiparticle excitation of the system. This concept had been suggested by Daunt and Mendelsshon in 1946 to explain the observed absence of thermoelectric effects in materials at low temperatures.18 Later on in 1956, Bardeen, Cooper and Schrieffer published the BCS5 theory of superconductivity, in which they proposed that a weak attractive interaction between electrons, such as that one created in second order by the electron-phonon interaction, can cause an instability of the ordinary Fermi-sea ground state of the electron gas. This leads to the formation of bound pairs of electrons occupying states with equal and opposite momentum and spin.5 The bound pairs are the so-called Copper pairs, and one of the key BCS predictions of BCS theory was that a minimum energy Eg = 2∆(T ) should be required to break a Cooper pair. This ∆(T ) was predicted to increase from zero at Tc to a limiting value 10 CHAPTER I. THEORETICAL BACKGROUND

BCS Eg (0) = 2∆(0) = 3.528kTc for T <

The origin of the negative potential attraction Vk,k′ needed for superconductivity, appears only when one takes the motion of the ions cores into account.17 The physical idea is that the first electron polarizes the medium by attracting positive ions; these positive ions in turn attract the second electron, usually in an effective attractive interaction between the electrons. If this attraction is strong enough to override the repulsive screened Coulomb interaction, it gives rise to a net attractive interaction, and superconductivity arises. His- torically the importance of the electron-lattice interaction in explaining superconductivity was first suggested by Fröhlich in 1950.19 This suggestion was confirmed experimentally 1/2 by the discovery of the isotope effect, i.e., the proportionality of Tc and Hc to M for isotopes of the same element. Since these lattice deformations are also responsible for what makes a solid elastic, it is clear that the characteristic phonon frequencies play a role. For the electronically screened Coulomb interaction, the characteristic frequency is the plasma frequency, which is so high, that can be concluded instantaneous.12,17 From momentum conservation, we can expected that if an electron is scattered from k to k’, the relevant phonon must carry the momentum q = k k′ and the characteristic frequency must then − be the phonon frequency ωq. As a result, it is plausible that the phonon contribution to the screening function be proportional to (ω2 ω2)−1. Evidently, this resonance denominator − q gives a negative sign if ω<ωq, corresponding to the physical argument above; for higher frequencies, i.e., electron energy differences larger than ~ωq, the interaction becomes repulsive. Thus, the cutoff energy ~ω of Cooper’s attractive matrix element V is expected to c − be of the order of the Debye energy ~ωD = kΘ, which characterizes the cut-off of phonon frequencies.20,21

Ginzburg-Landau Theory The Ginzburg-Landau (GL) theory concentrates entirely on the superconducting electrons rather than on excitations (as BCS teory). It is based on the work of London and the analysis of non-local electrodynamics proposed by Pippard.16 Ginzburg and Landau introduced a complex pseudo-wavefunction ψ that describes the superconducting electrons CHAPTER I. THEORETICAL BACKGROUND 11 and the local density of superconducting electrons, n = ψ(x) 2. This theory handled s | | successfully the so-called intermediate state of superconductors, in which superconducting and normal domains coexist in the presence of a near critical magnetic ﬁeld H H . (see ≈ c Fig. 1.1.) When ﬁrst proposed, the theory appeared rather phenomenological and its importance was not generally appreciated.17 However, in 1959 Gor’kov demonstrated that the GL theory was, in fact, a limiting form of the microscopic BCS theory. Physically, ψ can be thought of as the wavefunction of the centred of mass motion of the Cooper pairs. The GL theory re-introduces a characteristic length, or coherence length as,

~ ξ(T )= , 2m α(T ) 1/2 | ∗ | which characterizes the distance over which ψ(r) can vary without undue energy increase. The ratio of the two characteristic lengths deﬁnes the Ginzburg-Landau parameter κ, as

λ κ = . ξ This is an important measurable parameter in superconductors. Here, λ is the penetration depth (Meisnner effect), the screened length of the magnetic field in the interior of bulk superconductor [see panel (c) of Fig. 1.1] and ξ is the coherence length of GL theory. In bulk superconductors λ diverges as (T T )1/2 near T , and this dimensionless ratio is approx- c − c imately independent of temperature. For typical classic superconductors, λ 500 A˚ and ≈ ξ 3,000 A,˚ so κ 1. In this case there is a positive surface energy associated with a ≈ ≪ domain wall between normal and superconducting state. This positive energy surface sta- bilizes domains in the intermediate state, with a scale of subdivision intermediate between the microscopic length ξ and the macroscopic sample size. Moreover, in 1957 Abrikosov published a remarkable paper in which he investigated what would happen if in GL-theory κ was large instead of small.22 Figure 1.2 illustrates the flux penetration behaviour in type I superconductors (centre panel) and the decay of ψ of the GL theory (left panel). If ξ < κ, then we have the reverse situation. First this leads to a negative surface energy, which is radically different from the classic intermediate-state or type I superconductors. Abrikosov called these type II superconductors and at the same time showed that the threshold between two regimes was at κ = 1/√2. For materials with κ > 1/√2, he found that instead of a discontinuous breakdown of superconductivity in a first-order transition at Hc (as type I, see central panel of Fig. 1.2), there was a continuous increase in flux penetration starting at lower critical field Hc1 and reaching B = H at an upper field Hc2. Figure 1.2 shows a typical flux penetration of type II superconductors (central panel) with two characteristic critical fields. 12 CHAPTER I. THEORETICAL BACKGROUND

Type I Type II

Type II

Type I

FIG. 1.2: Central panel: comparison of ﬂux penetration behaviour of type I and type II superconductors with the same thermodynamic critical ﬁeld. Left and right panels: schematic diagram of variation of h and ψ in a domain wall. Type I (positive energy wall) for κ 1 and type II superconductivity for negative energy wall. 17,22 ≪

I.2.2 Eliashberg Equations and McMillan’s Formula

Since 1970s, Eliashberg theory is the state-of-the-art in phonon-drive superconductivity.7 Contrary to BCS, the strong coupling theory of Eliashberg takes into account the retarded nature of the phonon-induced interaction and treats properly the damping of the excitations. At the core of the Eliashberg theory is the electron-phonon interaction described by the spectral function α2F (ω).7 This quantity can be derived from the underlying electronic structure (the electron bands and density of states), the phonon dispersion curves and phonon density of states and the electron-phonon coupling constants.17 The general equation for the electron-phonon spectral density α2F (ω) that enters the Eliashberg equations is,

2 2 Ωa 1 3 3 2 α F ( )= d kδ(ǫk) d k’ δ(ǫk’) g δ[ ω (k’-k)] , (I.2.1) V (2π)3 N(0)~ | kk’i| V − i i where Ωa is the atomic volume, ǫk is the energy of the Bloch state ψk with band index implied in the momentum label k, i is a phonon branch index with ωi(k’-k) and gkk’i is the phonon-electron matrix element. The two Dirac delta functions depending on the energy limit the integrations over momentum to the Fermi surface, and the Fermi energy has been taken to be zero. Finally the single-spin density of state is,

Ω Ω dS a 3k a k N(0) = 3 d δ(ǫk)= 3 . (2π) (2π) ~ k FS | V |

This integration is over the Fermi surface only, where dSk is an element of the Fermi surface area and k is the Fermi velocity. The electron-phonon matrix element is related to the V CHAPTER I. THEORETICAL BACKGROUND 13 matrix element of the gradient of the crystal potential and the phonons through

~ 1/2 i g = ψk ǫ (k k’) V ψk’ , kk’i | − ∇ | 2Mω (k-k’) i where M is the ion mass, V is the crystal potential, and ǫi(k) is the polarization vector for the (ik)th phonon mode. Under a number of assumptions and approximations the

Eliashberg equations are written as a set of two non-linear equations for the gap ∆(iωn) † and renormalization factors Z(iωn) on the imaginary Matsubara frequency axis,

∆ iω )Z(iω )= πT [λ(iω iω ) ∗(ω )θ(ω ω )] (I.2.2) ( n n m − n − c c − | m| m ∆(iωm) 2 2 × ωm +∆ (iωm) and

πT ωm Z(iωn)=1+ λ(iωm iωn) (I.2.3) ω − 2 2 n m ωm +∆ (iωm) In order to solve Eqs. I.2.2 and I.2.3 and to get an estimate for Tc, one needs to know the inputs parameters, α2F (Ω) and ∗. McMillan and Rowell where the ﬁrst who investigated the form of this electron-phonon spectral function in tunnelling data experiments, and derived a formula to approximate T within an accuracy of 1% based in strong coupling c ∼ theory. They proposed the formula as,

Θ 1.04(1 + λ) T = D exp , (I.2.4) c 1.45 −λ ∗(1 + 0.62λ) − where ΘD is the Debye temperature, which was initially used for the characteristic phonon frequency on simple metals, and λ is deﬁned as,

λ = 2 α2F (ω)dω . (I.2.5) Allen and Dynes introduced further corrections to the formula proposed by McMillan, in particular ΘD was replaced by Ωlog that is deﬁned as,

∞ 2 α2F (ω) Ω = exp ln(ω) dω . (I.2.6) log λ ω  0   The formula to calculate the transition temperature obtained by McMillan and Rowell is much more precise that the one obtained from BCS theory,

†For details, see derivations in the review of Carbotte. 23 14 CHAPTER I. THEORETICAL BACKGROUND

1 T BCS = 1.14 < ω > exp − . c | | N(0)V

The McMillan’s formula has been extremely popular for the prediction of Tc in many compounds and using these equations one can compute from ﬁrst principles the transition temperature for a given material. In fact, we can use them even for the prediction of superconductivity in systems not yet synthesized or under extreme conditions not yet reached experimentally.

I.3 AB-INITIO TECHNIQUES

In 1964, Walter Kohn and Pierre Hohenberg introduced density-functional theory (DFT).8 In this pioneer work they proposed the electron density n(r) as the fundamental variable, and at the same time they proved the existence of an exact ground state energy functional, E[n(r)], and a variational principle based on it. For fixed nuclear positions, the ground-state energy is the lowest possible energy that can be obtained from a search over all valid densities. Thus, the density that yields the ground state energy is the ground state density.8,24 Nowadays, the so-called Kohn-Sham DFT represents one of the most advanced tool in Computational Physics, and proved to be a very efficient method to calculate energies, forces, etc. This theory is widely used both in Quantum Chemistry and Condensed Matter Physics.25–27 Currently, Kohn-Sham DFT is routinely performed on systems of hundred of atoms where one can explore the potential energy surface, compute the response of the systems to small perturbations, etc. An accurate calculation of the total energy allows us to obtain electronic band structures of crystals and get insights in electronic and magnetic properties. The failures and limitations remaining in DFT are associated principally with deficiencies in the exchange and correlation functional. However, the major drawback of DFT is that it is a theory for non-degenerate ground-states, and therefore many interesting properties are beyond its reach. There are several extensions of basic DFT that were proposed over the past years to solve this problem. For example, the time-dependent DFT of Runge and Gross, 28–30 or even DFT for superconductors (proposed by Oliveira, Kohn and Gross).31 Alternatively, one can have access to those properties using many-body methods such as GW theory.32 CHAPTER I. THEORETICAL BACKGROUND 15

Initial guess Calculate electron density Self-Consistent ? No!

Yes

Calculate effective potential Solve KS equation Output Energy, Forces, etc.

FIG. 1.3: Iterative ﬂow chart of the self-consistent loop for the solution of Kohn-Sham equations. 25

I.3.1 Density-Functional Theory

Here, we give a brief description of the equations that lie at the heart of DFT. We will only introduce the main equations and highlight the principal concepts.† KS-DFT ab initio codes usually solve the Kohn-Sham equations,

1 2 + υ [n](r) ψ (r)= ǫ ψ (r) , (I.3.1) −2∇ KS i i i where υKS[n] represents the Kohn-Sham potential which has a functional dependence on the electronic density n, deﬁned in terms of the Kohn-Sham wavefunctions:

occ n (r)= ψ (r) 2 . (I.3.2) 0 | i | 1 The potential υKS is deﬁned as the sum of the external potential (normally the potential generated by the nuclei), the Hartree term and the exchange and correlation (xc) potential

υKS[n](r)= υext(r)+ υHartree[n](r)+ υxc[n](r) . (I.3.3)

The standard procedure to solve this equation is iterating until self-consistency is achieved. The ﬂow chart of the self-consistent steps is depicted in Fig. 1.3. Usually one starts the iterative procedure with a well behaved guess for the density, n0(r), but in principle, any positive function normalized to the total number of electrons should work. Other options include to construct n (r) from a sum of atomic densities, as n (r)= n (r R ), 0 0 α α − α where Rα and nα represent the position and atomic density of the nucleus α. For an atom,

† Specialized books of ab initio-electronic methods and DFT include those of R. Martin, F. Nogueira et al., and E. Economou. 25–27 16 CHAPTER I. THEORETICAL BACKGROUND a convenient choice could be the Thomas-Fermi density. We can then evaluate the Kohn-

Sham potential (see I.3.3) with this density. Each of the components of υKS is calculated separately and each one of them poses a diﬀerent numerical problem. The external potential is typically a sum of nuclear potentials centred at the atomic positions

υ (r)= υ (r R ) (I.3.4) ext α − α α In some applications, υext is simply the Coulomb attraction between the bare nucleus and the electrons, υ (r) = Z /r, where Z is the nuclear charge. Often it is a sum of ext − α α pseudopotentials. The next term in υKS is the Hartree potential,

n(r′) υ (r)= d3r′ (I.3.5) Hartree r r′ | − | There are several diﬀerent techniques to evaluate this integral, either by direct integration or by solving the equivalent diﬀerential (Poisson) equation,

2υ (r)= 4πn(r) . ∇ Hartree − Finally, we have the xc potential, which is formally deﬁned through the functional derivative of the xc energy:

δE υ (r)= xc . (I.3.6) xc δn(r) There are hundreds of approximate xc functionals in the literature. But the ﬁrst to be proposed and, in fact, the simplest is the local-density approximation (LDA).33

LDA d HEG LDA 3 HEG υxc (r)= ε (n) , Exc = d rε (n) , dn n=n(r) n=n(r) HEG where ε (n) stands for the xc energy per unit volume of the homogeneous electron gas (HEG) of constant density n. Note that εHEG(n) is a simple function of n, which was tabulated for several densities using Monte Carlo methods by Ceperley and Alder and later parametrized by a series of authors.33 In the case of the generalized gradient approximations (GGA) the functional has a similar form, but now ε does not depend solely on the density n, but also on its gradient n. The evaluation of the GGA xc potential is also straightforward. ∇ More complicate families of xc functionals will involve dependences on other quantities like the kinetic energy density (the meta-GGA’s) or the Kohn-Sham orbitals (the orbital functionals like the exact exchange EXX).†

†Further details in Chapter 2 of Ref. 27 CHAPTER I. THEORETICAL BACKGROUND 17

At this point we have the Kohn-Sham potential, and we can solve the Kohn-Sham equation I.3.1. The goal is to obtain the p lowest eigenstates of the Hamiltonian HKS, where p is half the number of electrons (for a spin-unpolarized calculation). For an atom, or for any other case where the Kohn-Sham equations can be reduced to a one-dimensional differential equation, a very efficient integration method exists. In other cases, when using basis sets, real-space methods or plane-waves, one has to diagonalize the Hamiltonian matrix, HKS. One should keep in mind that fully diagonalizing a matrix is a N 3 problem, where N is the dimension of the matrix (which is roughly proportional to the number of atoms in the calculation). For practical purpose, one does not need to write-down this Hamiltonian matrix HˆKS, and one usually resorts to iterative methods. The knowledge of how HˆKS applies to a test wavefunction is then sufficient. These methods also scale much better with the dimension of the matrix. Nonetheless, diagonalizing the Kohn-Sham Hamiltonian is usually the most time-consuming part of an ordinary Kohn-Sham calculation. Once the electronic density from I.3.2 is obtained, the self-consistency cycle is stopped when some convergence criterion is reached. The two most common criteria are based on the difference of total energies or densities from iteration i and i 1. − To accelerate the convergence one uses density mixing schemes, the simplest being linear mixing. In this method, the density supplied to start the new iteration, n(i+1) is a linear combination of the density obtained from I.3.2, n′, and the density of the previous iteration, n(i), as ni+1 = βn′ + (1 β)n(i), where β is of the order of 0.3. Other methods are − more complex, such as Anderson or Broyden mixing,34 in which n(i+1) is an educated extrapolation of the densities of several previous iterations. Finally, at the end of the calculation, we can evaluate several observables, the most important of which the total energy. From this observable, one can obtain equilibrium geometries, phonon dispersion curves, or even ionization potentials. In Kohn-Sham theory, the total energy is given by a contribution of the non-interacting (Kohn-Sham) kinetic energy, the external potential, the Hartree and the xc energies, respectively.

occ 2 1 n(r)n(r′) E = d3rψ∗(r)∇ ψ (r)+ υ (r)n(r)d3r + d3r d3r′ + E , − i 2 i ext 2 r r′ xc i | − | or, using the Kohn-Sham equations, occ 1 E = ε d3r + υ (r)+ υ (r) n(r)+ E . (I.3.7) i − 2 Hartree xc xc i This is the formula implemented in most DFT codes. For geometry optimization or nuclear dynamics, one needs to add to the total energy a repulsive Coulomb term that accounts for the nucleus-nucleus interactions. 18 CHAPTER I. THEORETICAL BACKGROUND

I.3.2 Density-Functional Perturbation Theory

It is nowadays possible to calculate the lattice-dynamical properties relatively easily for a vast number of systems. The interest of phonon calculations in metals and semiconductors arises from the possibility of access valuable information concerning transport properties, such as thermal conductivity, electrical conductivity and especially superconductivity. Besides phonon dispersion curves, we can calculate electron-phonon coupling matrix elements, the Eliashberg spectral function, and superconducting transition temperatures for materials. Thanks to the development of density-functional perturbation theory (DFPT) these calculations are routinely performed using ab initio quantum-mechanical techniques based on the linear-response theory of lattice vibrations.9,10,35,36 In this section we give an overview of linear-response phonon calculations. We used in section I.1.2 the adiabatic approximation of Born and Oppenheimer4 which allows to decouple the vibrational from the electronic degrees of freedom in a solid. Within this approximation, the lattice-dynamical properties of a system are determined by the eigenvalues ε and eigenfunctions Φ of the Schr¨odinger equation,

~2 ∂2 2 + E(R) Φ(R)= εΦ(R) , (I.3.8) − 2MI ∂RI I therefore, the calculation of phonon dispersions relation requires to solution of the equation,

1 ∂2E(R) det ω2 = 0 , (I.3.9) √MI MJ ∂RI ∂RJ − where RI is the coordinate of the Ith nucleus, MI its mass, R RI the set of all nuclear ≡ { } coordinates, and E(R) the clamped-ion energy of the system, which is often referred to as the Born-Oppenheimer energy surface. In practice E(R) is the ground state energy of a system of interacting electrons moving in the field of fixed nuclei. Calculation of the equilibrium geometry and of the vibrational properties of a system is accomplish by computing the first and second derivatives of its Born-Oppenheimer energy surface. Then, using the Hellmann-Feynman theorem,37,38 which states that the first derivative of the eigenvalues of a Hamiltonian, Hλ, that depends on a parameter λ is given by the expectation value of the derivative of the Hamiltonian,

∂Eλ ∂Hλ = Ψλ Ψλ , (I.3.10) ∂λ ∂λ where Ψλ is the eigenfunction of the Hamiltonian, corresponding to the Eλ eigenvalue: HλΨλ = EλΨλ. In the Born-Oppenheimer approximation, nuclear coordinates act as parameters in the electronic Hamiltonian, and the force acting on the Ith nucleus in the CHAPTER I. THEORETICAL BACKGROUND 19 electronic ground state is thus

∂E(R) ∂HBO(R) FI = = Ψ(R) Ψ(R) − ∂RI − ∂RI where Ψ(r, R) is the electronic ground-state wavefunction of the Born-Oppenheimer Hamil- tonian. This Hamiltonian depends on R via the electron-ion interaction that couples to the electronic degrees of freedom only through the electron charge density. The Hellmann- Feynman theorem states in this case,

∂VR(R) ∂EN(R) F = n(r) dr , I − ∂R − ∂R I I 2 ZI e where VR = is the electron nucleus interaction, and nR is the ground-state iI |ri−RI | electron charge density corresponding to the nuclear conﬁguration R. The Hessian matrix of the Born-Oppenheimer energy surface appearing in Eq. I.3.9 is obtained by diﬀerentiating the Hellmann-Feynman37,38 forces with respect to nuclear coordinates,

2 2 2 ∂E (R) ∂FI ∂nR(r) ∂VR(r) ∂ VR(r) ∂ EN(R) = dr + nR(r)dr + . (I.3.11) ∂R ∂R ≡−∂R ∂R ∂R ∂R ∂R ∂R ∂R I J J J I I J I J The calculation of the Hessian matrix of the Born-Oppenheimer energy surfaces (or matrix of the interatomic force constants) requires the calculation of the ground-state electron charge density nR(r) as well as of its linear response to a distortion of the nuclear 36 geometry, ∂nR(r)/∂RI . Of course we assumed that the external potential acting on the electrons is a diﬀerentiable function of a set of parameters, λ λ , λ R (in the case of ≡ { i} ≡ I lattice dynamics). In crystalline solids the nuclear positions appearing in the interatomic force constants Eq. I.3.9 are labelled by an index I, which indicates the unit cell l to which a given atom belongs and the positions of the atom within that unit cell I l,s . The ≡ { } position of the Ith atom is,

R R + τ + u (l) . l ≡ l s s

Where Rl is the position of the lth unit cell in the Bravais lattice, τs is the equilibrium position of the atom in the unit cell, and us(l) indicates the deviation from equilibrium of the nuclear position. Because of translational invariance, the matrix of the interatomic force constants (Eq. I.3.11) depends on l and m only through the diﬀerence R R R , ≡ l − m

2 αβ ∂ E αβ C (l,m) = C (Rl Rm) , st ≡ α β st − ∂us (l)∂ut (m) 20 CHAPTER I. THEORETICAL BACKGROUND where the Greek superscripts indicate Cartesian components. The Fourier transform of αβ ˜αβ Cst (R) with respect to R, Cst (q) can be seen as the second derivative of the Born- Oppenheimer energy surface (Eq. I.3.11) with respect to the amplitude of a lattice distortion of deﬁnite wave vector,

1 ∂2E C˜αβ(q) e−iqRCαβ(R)= , st ≡ st N ∗α β R c ∂us (q)∂ut (q) where Nc is the number of unit cells in the crystal, and the vector us(q) is deﬁned by −iqR the distortion pattern, RI [us(q)] = Rl + τs + us(q)e . The phonon frequencies ω(q) are the solutions of the equation,

1 ˜αβ 2 det Cst (q) ω (q) = 0 (I.3.12) √MsMt − Translational invariance can be alternatively stated in this context by saying that a lattice distortion of wave vector q does not induce a force response in the crystal at wave vector q’ = q. Due to this property, interatomic force constants are most easily calculated in reciprocal space and, when they are needed in direct space, can be readily obtained by Fourier transform.

Third-order derivatives of the total energy can be obtained from the ﬁrst-order derivatives of the wavefunctions. This is due to the 2n + 1 theorem that is consequence of the variational principle in Quantum Mechanics, and that was proved within DFT by Gonze and Vigneron in 1989.39 Observable quantities that can be obtained from third- order derivatives are, for example, third-order anharmonicites, phonon-phonon scattering, phonon lifetimes, infrared and (oﬀ-resonance) Raman cross sections, thermal properties, etc. One such response property of particular interest for the study of materials are Raman intensities. Born and Huang40 derived an expression that reduces the computation of the integral Raman intensities to the evaluation of the imaginary part of the linear Raman-susceptibility. The Kramers–Kronig transformation of the Huang and Born expression yields,41

~ 4 2π (ωL ωi) 2 Ii,γβ = 4 − [n(ωi) 1](αi,γβ) (I.3.13) c ωi − In this expression the polarization is along γ, and ﬁeld along the direction β, for the i-mode and ω is the laser frequency of the excitation source, n(ω) 1 represent the Bose L − occupation number,

−~ω −1 ( i ) n(ω) 1= 1 e κβT − − CHAPTER I. THEORETICAL BACKGROUND 21

where ωi the frequency of mode i, kB the Boltzmann constant and αi,γβ, the Raman tensor susceptibility is deﬁned as

√Ω −1/2 ∂ǫγβ(ωL) αi,γβ = Riαβ,nγeinγ Mn with, Riγβ,nν = , (I.3.14) 4π ∂uinν nγi where Mn is the mass of atoms, einγ is the eigenvector of the dynamical matrix, and Ω is the unit cell volume.41,42 The matrix Raman tensor susceptibilities can be calculated from ﬁrst principles using DFPT.36,43–45 Two cases have to be distinguish, Eq. I.3.13 is applicable to an oriented single crystal; in the case of a powder, one needs to average the intensity over the possible directions of the crystallites,41,42 then, the ﬁnal expressions are,

1 G(0) = (α + α + α )2 , i 3 ixx iyy izz

1 G(1) = (α α )2 + (α α )2 + (α α )2 , i 2 ixy − iyx ixz − izx izy − iyz 1 G(2) = (α + α )2 + (α + α )2 + (α + α )2 + i 2 ixy iyx ixz izx izy iyz 1 (α α )2 + (α α )2 + (α α )2 . 3 ixx − iyy ixx − izz iyy − izz The reduced intensity for a polarized and depolarized light beam in a back scattering ⊥ geometry is,

powder 4 1+ n (ωi) (0) (2) Ii (ωL ωi) 10Gi + 4Gi , ∼ − 30ωi

powder 4 1+ n (ωi) (1) (2) Ii⊥ (ωL ωi) 5Gi + 3Gi , ∼ − 30ωi

powder powder powder IiT OT = Ii + Ii⊥ .

Applying this expression for each of the Raman active i (of the 3N 3) optical modes − for a solid at q = 0, one can build a spectra by convolution with Lorentzian or Gaussian functions. We should keep in mind that the full width at half mean (FWHM, Γi) is a parameter independent of the intensity and that is related directly with sample quality, experimental conditions and anharmonic eﬀects in materials. We used this expression to construct the Raman spectra of carbon allotropes under pressure. Further details can be found in Chapter VI. 22 CHAPTER I. THEORETICAL BACKGROUND

I.4 CRYSTAL AND PHONON SYMMETRY

Symmetry is a fundamental property often used in the mathematical description of nature and is one of the most complex and elegant fields in physics. In fact, symmetry is found everywhere: from electronic transitions inside atoms, to the arrangement of atoms, to the vibration modes of arranged atoms, molecules, cells, up to large scale symmetry in planets, galaxies and the universe. Thanks to it, many laborious calculations in Solid State Physics can be greatly simplified We review in this section some concepts of crystal symmetry and elements of phonon symmetry; fundamental for the development of this Thesis. In crystallography there are only 230 ways to describe how identical objects can be arranged in orderly arrays in an infinite three-dimensional lattice network. We refer to this object as a crystal, an ordered state in which positions of the nuclei are repeated periodically in space. The description of the entire crystal is therefore based on the lattice and the motif. If one applies translations to the lattice, the result is the characteristic and well-known 14 Bravais cells. Including a proper set of operations inside the motif, 32 point group can be discriminated.46,47 The point group is then, the description of an ensemble of symmetry operations for a given lattice, and as group must fulfil certain mathematical requirements. Therefore, the combination of all available symmetry operations in the point groups (mir- rors, fractional translations, glides and screws) with the Bravais translations leads to exactly 230 combinations, the 230 space groups in crystallography.† The crystal symmetry is then the classification of certain motif (atom) arranged in space. There are two common nomen- clatures to describe space groups in crystallography, the Schönflies notation (very often used in Chemistry) and the international notation known as Hermann-Mauguin notation (frequently used in Solid State Physics). Point groups in both notations are tabulated in crystallographic actas together with the 14 Bravais cells and the resulting 230 space groups. In this work we used International Table-A (ITA) to describe crystal systems and symmetry operations.47 For phonons, the normal modes of vibration of a crystal are labelled by a wave vector k and by a branch index i. The symmetry description of phonons in crystals is given Sfrom the analysis of the eigenvectors e(κ ki) which are obtained from the Hermitian matrix { | } D (κκ′ k) , the Fourier-transformed dynamical matrix. The eigenvectors e(κ ki) of { αβ | } { | } this matrix describe the displacement pattern of the atoms comprising the crystal when they are vibrating in the mode (ki).48 Knowledge of the forms of these eigenvectors and of their transformation properties under the symmetry operations is often useful for the

†No co–representations of magnetic space groups are included. CHAPTER I. THEORETICAL BACKGROUND 23

A B E T

1D 1D 2D 3D On-phase Anti-phase Double degenerate Triple degenerate

FIG. 1.4: Mulliken symbols used for describe phonon vibrations in solids. A represents vibration on-phase, B anti-symmetric vibrations, E doubly degenerate mode and T triply degenerate mode.

solution of certain types of lattice dynamical problems. Among these, for example, is the establishment of selection rules for processes such as two-phonon lattice absorption and the second-order Raman effect, or phonon-assisted electronic transitions.48–50 The use of group- theoretical arguments greatly simplifies the determination of the form of the dynamical matrix for a crystal and of its eigenvector. Furthermore, the Mulliken symbols are used to label the discriminated vibrations and identify the irreducible representation of the modes. The symbols for describing phonon vibrations are: A, B, E, and T. Fig. 1.4 illustrates examples of eigenvectors for different modes. A is a 1-D representation for degenerate states that is symmetric (vibration on-phase) with respect to the rotation around the principal axis; B is a 1-D representation of a degenerate state which is anti-symmetric with respect to rotation around the principal axis; E indicates a 2-D representation of doubly degenerate modes with two sets of atom vector displacements and T is a 3-D representation of triply degenerate modes with three sets of atom vector displacements. Subscripts in the symbols provide further information about vibration with respect to certain symmetry operations.47,50 During the development of this work, the use of crystal and phonon symmetry was key to understand our experimental results and guide further experiments. 24 CHAPTER II. BACKGROUND OF EXPERIMENTAL TECHNIQUES

CHAPTER II

BACKGROUND OF EXPERIMENTAL TECHNIQUES

This Chapter comprises an introduction to high pressure and spectroscopy techniques to characterize materials. The Chapter is divided in two parts: the first part describes high pressure techniques to synthesize materials and the second part gives an overview of the physics behind characterization techniques. The ensemble of experimental work includes the use of techniques such as: belt apparatus, diamond anvil cell (DAC), Raman scattering, X-ray diffraction, squid measurements and electrical conductivity. Although the experimental techniques used were mostly available in the Laboratoire de Physique de la Matière Condensée et Nanostructures, we carried collaborations with other groups at the Ecole´ Nor- mal Superior de Lyon, the Universitéde Montpelier II, and the UniversitéDiderot-Paris 7, that gave us access to experimental measurements to complement our work. Further- more, during the development of this thesis several proposals were approved to perform experiments in large facilities, such as neutron diffraction experiments at laboratoire Léon Brillouin (reactor orphée) CEA Saclay and measures of angle resolved X-ray photo-electron spectroscopy (ARPES) at synchrotron soleil, in beamline cassiopée.

II.1 HIGH PRESSURE PHYSICS

High-pressure research is now leading to the identiﬁcation of families of materials with new technological applications. The experimental progress in this ﬁeld is remarkable; the technology developed during the past 40 years permits us to reach pressures of the order 300 GPa in quasi-hydrostatic conditions and to measure physical properties at such pressures. Moreover, it is now relatively easy to reach pressures of tens of giga-pascals without the use of expensive or complex installations. Finding ways to achieve higher values of transition temperatures in superconductors remains a great challenge. High-pressure physics can have a deep impact in the research of high-Tc superconductors. An interesting example is the case of HgBa2Ca2Cu3O8+δ (known as the Hg-1234 compound), which is the record holder of the highest Tc at ambient pressure at around 133 K.51 However, if one applies pressure, the critical temperature can be further increased to the record-breaking T of 160 K at 30 GPa.52 In fact, this value is the highest c ∼ † Tc measured so far. This sharp increase of Tc is a clear evidence of the strong dependence of Tc on the lattice strains. Therefore, the use of pressure to alter or modify the physical

† There is published work that claims higher values of Tc above 200 K. However, the evidence is weak and not reproducible. http://www.superconductors.org/200K.htm CHAPTER II. BACKGROUND OF EXPERIMENTAL TECHNIQUES 25 and chemical states of materials is a primary research tool. In fact, simply by reducing the volume of a substance, one can vary and explore new properties of materials. Ideally, high-pressure physics aims not only to ﬁnd new structures with interesting properties at high pressures (as Tc) but also stabilize them (structures) at ambient conditions. Moreover, the determination of the existence of new high pressure compounds and the study of their pressure-temperature phase relations leads to the establishment of new synthetic routes to materials under metastable conditions. This section describes two diﬀerent high- pressure devices used to synthesized materials: the belt apparatus, a press which we used to acquired metastable phases of alkaline earth metal disilicides at high pressure and high temperature treatments, and a second device, probably the most known in high-pressure research, the diamond anvil cell (DAC).

II.1.1 Belt Apparatus Press

The belt apparatus press is one of the most effective high pressure and high temperature device used in high pressure physics. This apparatus was conceived in 1953 by H. Tracy Hall at the General Electric Research Laboratory,53,54 and consists in two conical pistons and a conical shaped chamber (forming a toroidal belt around the sample, where its name) together with a compressible, sandwich gasket to generate pressure. A belt is capable of generate pressures of 2 GPa up to 10 GPa, simultaneously with steady state temperatures of 500 K to 2000 K. In left panel of Fig 2.5 we illustrate the core elements of a belt apparatus. Two conical pistons of tungsten-carbide WC (coloured in light-blue) are thrust into each other at the end of a conically shaped chamber (coloured in light-grey ) by an hydraulic press. In our experiments the samples were placed inside a crucible (chemically no reactive made of h-BN) of a few mm diameter. Then, the crucible itself was placed in an outer tube of graphite, and this tube was, as well, inserted in an outer large tube made of pyrophyllite. Finally, it was closed with electrical contacts. In order to avoid metal-metal contacts we used Teflon gaskets. In this apparatus the pressure is generated by the advancing piston and is transmitted to the sample, the pyrophyllite plays the crucial role of pressure transmitting medium. Once the pressure is stabilized, the sample is heated by the current passing trough the graphite cylinder ( 2 ampères). In addition to transmitting pressure, the pyrophyllite also serves ∼ as thermal and electrical insulation, and teflon rings provide thermal insulation at the end of the heater-sample tube preventing electrical chunks. Pressures and temperatures were previously calibrated.† The belt used in our studies covers the range of 2.5 to 5 GPa and

†Further details can be found in the work of San Miguel and Toulemonde. 55,56 26 CHAPTER II. BACKGROUND OF EXPERIMENTAL TECHNIQUES

DAC (30 GPa, < 300 K)

FIG. 2.5: Left figure: core-scheme of belt-apparatus. Right figure: standard configuration of opposed diamond anvils.

temperatures from 500 K to 1400 K. The main advantage of this apparatus is the combination of pressure, temperature and of large volume, which is important for experimental techniques where large volume of sample is required. This apparatus was used previously to synthesized clathrates with diﬀerent magnetic metals as host atoms. However, we oriented our research to explore high pressure phases of disilicides, principally BaSi2 that is used as starting material to synthesized clathrates.

II.1.2 Diamond Anvil Cell

The diamond anvil cell (DAC) is probably the smallest apparatus used in the field of modern high pressure research (it can fit into the palm of the hand). Despite its size and small sample volume, DAC remains an essential tool for the rapid exploration of pressure-temperature diagrams and the identification of new phases. This device is capable of generating megabar pressures on materials of microscopic dimensions and combined with spectroscopy techniques, in-situ investigations of optical absorption, reflectivity measurements, Raman, Brillouin scattering, XRD, and IXS are possible. Recently, electrical and magnetic susceptibility measurements were performed in DAC under pressure and low temperatures.57 DAC had considerable success identifying new elemental superconductors and new phase transitions in a series of elements and compounds which are not superconducting at ambient conditions of pressure.58 The principle of the DAC is simple: a sample (few microns) is placed between two opposite diamond flat facets. Figure 2.5 illustrates CHAPTER II. BACKGROUND OF EXPERIMENTAL TECHNIQUES 27 the standard configuration of a DAC. An important element present in all configurations is the metallic gasket. This element serves two basic functions, it avoids the direct contact of the culet† in diamonds, and it restricts the volume of the chamber; restricting the pressure reached in the experiment.59,60 In our experimental set-up we used iconel gaskets with a 150-120 hole drilled. The determination of the pressure in a DAC’s experiment is normally verified by ruby luminescence, placing 2 or 3 ruby chips in the compression chamber and measuring the luminescence at each stage of pressure. This method can be used up to pressures of 90 GPa. Beyond this limit, in order to determined the pressure one requires an extrapolation of the equation of state (EOS) of simple fcc metals like Pt, Au, Ag or Re measured by XRD.

II.2 CHARACTERIZATION TECHNIQUES

Characterization techniques are of capital importance in condensed matter physics. An accurate determination of the properties of matter is acquired by perturbing the system with probes (as one does with theoretical methods) and by measuring its response. The widely used probes in experiments are: photons (lasers, X-rays) electrons, positrons, muons, neutrons, atoms and ions, etc. In some cases the perturbation can also be performed by varying the pressure and temperature. In that case the information obtained is of major interest but largely complex. The crystal structure of a system is experimentally determined from their characteristic diffraction pattern. We used X-ray diffraction (XRD) to determined the crystal structure of our materials and to confirm theoretical predictions. Moreover, we performed electrical and magnetic susceptibility measurements in order to obtain information about the electronic states. The lattice vibrations were measured with spectroscopy techniques such as Raman scattering. Furthermore, Raman spectroscopy combined with diamond anvil cells allow us to measure the vibrational properties of solids under pressure.

II.2.1 X-ray Diﬀraction

In this section we review basic concepts of X-ray diﬀraction in inﬁnite crystals. The lattice structure of a crystalline solid can be determined experimentally by elastic scattering of X-rays or neutrons. Photons (X-rays), being neutral, interact weakly with solids and, having a wavelength comparable with the spacing between atoms, are ideal for penetrating in solids and construct interference patterns. In this case, the Bragg relation is used to describe the constructive scattering pattern of the crystal lattice.

†In jewellery, the point on the bottom of a diamond’s pavilion is called a culet. From French: –la partie inf´erieure d’une pierre pr´ecieuse.– 28 CHAPTER II. BACKGROUND OF EXPERIMENTAL TECHNIQUES

Lattice d Atoms

dsin

Bragg Diffraction Debye-Scherrer rings

FIG. 2.6: Left: incident ki and diﬀracted kf beam, construct an interference pattern for nλ = 2dsin(θ). Right: obtained pattern of LaB6 sample used to calibrate experimental machine-parameters for Rietveld reﬁnement.

Consider an incident beam of X-rays (electromagnetic radiation having typical wave- lengths of 0.3-1.8 A)˚ that is reflected by a family of atomic planes. The path difference between neighbouring planes causes a phase difference between the diffracted beams. Thus, the reflected radiation will show constructive and destructive interference, the Bragg relation is

nλ = 2dsin(θ) .

This relation predicts that constructive interference only occurs if the ratio between the phase difference and the wavelength is an integer value of n (see in Fig. 2.6). In this formula θ is an experimental measured variable and is the angle between the incident or the reflected beam and atomic planes. Assuming elastic scattering, incident and reflected beams must keep the same wavelength λ. Finally, d is the spacing between planes. † In order to characterized the structure of materials studied in this work. Experimental X- ray diffraction were performed in θ 2θ using 2 different geometries of the spectrometer, − Bragg-Brentanno and Debye-Scheer configuration. In the first case we used a Bruker D8

Advance powder diffractometer with the Kα1,2 of Cu as wavelength and the second machine used a Xcalibur-Oxford model X with an excitation source of Mo,Kα1. The measurements were performed at the Centre de Diffractomètrie Henri Longchambon at the University of Lyon. As an example, we show in right panel of Fig. 2.6 a characteristic diffraction pattern acquired in Debye-Scheer geometry for LaB6 at ambient conditions. This sample was used to set parameters required for the Rietveld refinement of the XRD spectra.

†For a detailed description of scattering phenomena in solids, consult Ashcroft-Mermin’s book. 61 CHAPTER II. BACKGROUND OF EXPERIMENTAL TECHNIQUES 29

II.2.2 Low Temperature Electrical Resistivity

The temperature dependence of the electrical resistivity provides fundamental information on the physical properties of materials. A characteristic property of metals is the electrical conductivity (σe) or resistivity (ρe = 1/σe) and the evolution of this quantity with temperature reﬂects the intrinsic scattering mechanism in the system. For metals, information regarding conduction bands can be obtained and for semiconductors, information concerning valence bands and energy gap (Egap) is acquired. This section describes common scattering mechanism present in materials.

In the framework of a single conduction band and above the temperature of Debye ΘD, the electron-electron scattering dominates (electrons scatter oﬀ each other via a screened Coulomb interaction). This scattering has been predicted to give a resistivity proportional to T . However, below the Debye temperature, the quantized lattice vibrations interact with electrons giving arise to elementary process of type k + q k′. Then, at low temperatures the electron-electron scattering is reduced because of the conservation of crystal momentum and the smallness of q (on average q is the energy of scattered phonons at low temperature). Thus, electrons can only be scattered through a small angle and this reduces drastically resistivity in metals as temperature goes down. Electrical resistance varies approximately according to the law

m ρ ∝ T , for T < ΘDx, and 0

The scenario is much more complicated at low temperatures than the simple regime at high temperatures. At low temperatures electrical conductivity in metals depends on many factors that adjust the power m of T in Eq. II.2.1. For instance, factors as the shape and area of the Fermi surface, the effective mass of conduction bands, impurities (sample quality) and the interaction of electrons and phonons (among other factors) play a crucial role.12 Furthermore, the comparison of experimental measured curves with a more formal theoretical framework is far to be trivial. There are several factors that alter and contribute to the scattering process such as, impurity scattering, lattice scattering, surface scattering, piezoelectric scattering, and alloy scattering. This latter is of capital importance specially in the case of binary compounds, because different effective masses change the distribution of electron velocities at the Fermi surface. In order to account correctly for electron mobility, and therefore for the scattering mechanism in materials, we can use the Matthiensen rule to build a sum of scattering process as,

ρs(T )= ρ0(T )+ ρe−p(T )+ ρm(T ) , 30 CHAPTER II. BACKGROUND OF EXPERIMENTAL TECHNIQUES

where ρ0 represents the residual (ﬁnite) scattering (defects) of a system in the normal state, ρe−p accounts for scattering of conduction electrons by quantized lattice vibrations

(phonons), and ρm is introduced for magnetics impurities. However, in our case we neglect this term since non magnetic elements are studied. In the same spirit, resistivity coming from the scattering of conduction electrons by phonons as function of temperature can be accounted by the Bloch-Gr¨uneisen equation,

T m ΘD/T xmex dx ρ (T )= ρ + (m 1)ρ′Θ . (II.2.2) s 0 − D Θ (ex 1)2 D 0 − This equation is used to ﬁt experimental curves of resistivity as a function of temperature. The residual resistivity ρ0 accounts for lattice defects (in the normal state), ΘD is the Debye temperature and ρ′ is a temperature coeﬃcient of resistivity that can be expressed as

6πk λ ρ′ = B tr , 2N(0)~e2 υ2 F where N(0) is the electron density of states (eDOS) at the Fermi level, υ2 is the mean- F square electron velocity at the Fermi surface, λtr is a dimensionless constant for electrical transport and kB is the Boltzmann constant. From this relation one obtains experimental information of quantities that can be calculated by ab initio approaches. Experimentally we carried out electrical measurements in diﬀerent samples (semiconducting and metallic) in order to elucidate the scattering mechanism dominating in our materials. Samples were prepared by high pressure and high temperature synthesis. Parallelepiped shaped blocks were prepared and metallic contacts pasted with silver epoxy and dried on argon atmosphere. We used a conventional four-probe method to measure the electrical conductivity. The sample holder was placed in a vacuum jacket Dewar with an He compressor that cooled down to 15 K. Calibrated resistances in good thermal contact were used to measure the temperature. The experimental set-up allows to apply pressure in samples using an He compressor of three stages. The pressure reached is of the order of 400 kbar.†

†Further details of the experimental set-up can be found in the work of Caillier, San Miguel, et. al. 62 CHAPTER II. BACKGROUND OF EXPERIMENTAL TECHNIQUES 31

II.2.3 Raman Scattering

Raman scattering is a powerful spectroscopy used to study the intrinsic vibrational states of a crystal. This technique is based on the analysis of inelastically scattered light. The source of the Raman process is the scattering from optical modes of quasiparticles: normally optical phonons, but also optical magnons, plasmons and eventually localized electronic excitations can contribute to this process.49,50 Raman scattering is originated by the change in polarizability of molecules or the change in susceptibility of crystals by excited quasiparticles. In the quantum-mechanical model, the Raman process is the simple excitation of an electron by a photon. In this picture, the scattering geometry is determined by momentum conservation as shown in panels (a) and (b) of Fig. 2.7. The corresponding mathematical relationships are ~ω = ~ω ~Ω i s ± and ~κ = ~κ ~q, where the subindex i and s refer to scattered and incident light. The i s ± “+” sign is for phonon generation and “-” is for absorption. An important remark is that incoming scattering can only occur for quasiparticles with small values of q, as compared to the size of the Brillouin zone. For a backscattering configuration of 180◦ the maximum allowed value of q is obtained. It is related to k and k by q = κ + κ 2κ . Since κ is i s max i s ≈ i i of the order of 105 cm−1, in the visible spectral range only scattering with phonons from the centre of the Brillouin zone with q 0 are allowed. Moreover, depending on whether the ≈ quasiparticle is absorbed or emitted the process can be classified as anti-Stokes or Stokes scattering. To calculate the probability of two optical transitions (absorption followed by emission) second-order perturbation is required. † Here, we treat only the vibrational part explicitly by evaluating the matrix element for transitions between the initial state i and the final state f. For a scattering process the transition is driven by the polarization induced by light P = χε0 E. The matrix element has therefore the form,

P = f P i = f ε χE i . (II.2.3) fi | | | 0 | Since f and i are generalized wevefunctions the integration runs over all electronic and | | nuclear coordinates. If the wavelength of light is much larger that the interatomic distances, the electric ﬁeld can be considered to be constant in II.2.3, so that we can extract from the equation a generalized form of the susceptibility known as the transition susceptibility,

[χ ] = f χ i . (II.2.4) mn fi | mn| The left part of this equation is a material-speciﬁc quantity determined by electronic orbitals

†Advanced textbooks of spectroscopy as Hollas or Kuzmany are recommended for further details. 50 32 CHAPTER II. BACKGROUND OF EXPERIMENTAL TECHNIQUES

Intermediate state Fano lines

Final state f

I. state i GrSt

FIG. 2.7: (a) Momentum conservation for the light-scattering process with generation of a phonon. ki, ks and q: wavevector for incident, scattered, and phonon, respectively. (b) Energy diagram level of a Raman process, where the ﬁnal state is reached from the initial state via an intermediate state. This state diﬀers from the initial state by the generation or absorption of a quasiparticle with energy ~Ω. (c) Energy diagram of Fano interference between the excitation from the ground state (GrSt) to a discrete level s and to a continuum ǫ′.

in the crystal. If the ﬁnal and the initial states are both a ground state and if one applies the adiabatic approximation, Eq. II.2.4 yields the susceptibility as4,50

∗ ∗ [χmn]fi = ̺f (X)ρf (x, X)χmnρi(x, X)̺i(X)dxdX , (II.2.5) First we consider the integration over electronic coordinates x. The wavefunctions are factorized for electrons ρ and for atoms ̺. Assuming that the ﬁnal and initial electronic state are the same renders for this integral the electronic part χmn(X) of the transition susceptibility. Without using the Condon approximation in Eq. II.2.5, the susceptibility still depends on the nuclear positions X. If we introduce normal coordinates Qk, we can expand the electronic part of the susceptibility with respect to the normal coordinates. Considering only the linear term of the expansion and extracting the expansion coeﬃcient from the integral we obtain,

∂χmn [χmn]fi = (χmn)0 ...υfk ...... υik . . . + ...υfk . . . Qk ...υik . . . . (II.2.6) | ∂Qk 0 | | k The bra and ket symbols represent the total vibrational wavefunctions from the integral II.2.5. They are expressed as the product of harmonic-oscillator wavefunctions with occupation numbers υfk or υik. CHAPTER II. BACKGROUND OF EXPERIMENTAL TECHNIQUES 33

υ ,...υ ,...υ = υ f1 fk f,n| fk| k (II.2.7) υ ,...υ ,...υ = υ , i1 ik i,n| ik| k where υ and υ are the harmonic-oscillator wavefunctions for occupation numbers υ fk| ik| fk and υik, respectively. Since we do not used the Franck-Condon principle the oscillators are non-shifted and the expectation values in Eq. II.2.5 are

0 for υfk = υik υ υ = (II.2.8) fk| ik  1 for υfk = υik and, 

0 for υfk = υik  1/2 υfk Qk υik = (υik + 1) ~/2Ωk for υfk = υik + 1 (II.2.9) | |   (υ )1/2 ~/2Ω for υ = υ 1 . ik k fk ik −  Because of orthogonality of the wavefunctions all expectation values from II.2.6 can be factorized into relationships like II.2.8 and II.2.9. The ﬁrst term in II.2.6 is only diﬀerent from zero if υfk = υik for all k. This means that the quantum state of the system has not changed. If (χmn)0 is properly calculated, it describes the process of absorption or Rayleigh scattering. The second term is responsible for the Raman process as is evident from the appearance of the susceptibility. According to II.2.8 and II.2.9 it is only non-zero if for all ′ k = k then υ ′ = υ ′ and for the mode k υ = υ 1 holds. In this case the transition fk ik fk ik ± susceptibility II.2.6 has the form,

1/2 ∂χmn [χmn] = (υik + 1) ~/2Ω (II.2.10) υik+1,υik ∂Q k 0 and 1/2 ∂χmn [χmn] = (υik) ~/2Ω , (II.2.11) υik−1,υik ∂Q k 0 The two equations above describe the Stokes and anti-Stokes Raman process. For comparison with experimental results attention must be paid to the dependence of the intensities on the vibronic occupation number υk. Since the latter is determined by a Boltzmann factor,

exp( εk/kBT ) exp( ~Ωk(υk + 1/2)/kB T ) W (εk)= − = − , (II.2.12) Z exp [ ~Ωk(υk + 1/2)/kbT ] υk − A thermal averaging of the form, 34 CHAPTER II. BACKGROUND OF EXPERIMENTAL TECHNIQUES

(υk + 1)W (εk) υ k is required to obtain the eﬀective square of the Raman tensor from II.2.10 and II.2.11. This is similar to the thermal averaging used for the evaluation of the optical absorption from localized wavefunctions. In the case of Stokes scattering the average is nk + 1 where nk is given by the Bose-Einstein distribution for the mode k,

1 nk = fE(Ωk)= ~ . (II.2.13) e(− Ωk/KB T ) 1 − For anti-Stokes scattering the average yields nk. Taking the convenient description of the orientation of incident and scattered light with respect to the scattering plane using the symbols and ,† the scattering intensity per steradian for an incident intensity I is ⊥ i

~ 4 2 d Φ (ω Ωk) νuχ (nk + 1)Iiν ⊥ − yx,k = 2 4 , dΩ 32π c0Ωk where (nk + 1) is the thermal averaging. It is important to add that the temperature dependence of Raman intensities is very often a convenient quantity to measure, simply because it is the ratio between the anti-Stokes and the Stokes intensities of one mode. This relation actually provides a good check of temperature eﬀects due to the incident laser when it is focused down on the crystal. The width of Raman lines can also change with temperature, pressure, and the nature of the scattering mechanism of phonons. Thus, the shape of the Raman response measured experimentally is a measure of the intrinsic decay mechanism of phonons and is normally modelled either by Lorentzian or Gaussian contributions with a width at half mean (FWHM = Γ).∗ Strictly, this quantity is also related to the sample quality and quantity of defects. However, besides Gaussian, Lorentzian or a combination of both, the response of certain optical modes may require an elaborated model of the lineshape to be correctly interpreted. For example, if there is a strong interaction of the electronic background with the lattice dynamics in a crystal, one has to use the so-called Fano line. The physics behind this eﬀect is illustrated in panel (c) of Fig. 2.7, and can be consulted in original work of U. Fano in he studied which the energy loss of electrons scattered in He.63 From the schema of panel (c), we can imagine a transition of the system from the ground state to either a single discrete ′ state s with energy ǫs and described by a wavefunction ψs, or a continuum of states ǫ described by a wavefunction ψ′. The interaction between discrete and continuum states is described with the Hamiltonian H with matrix elements V ′ = ψ ′ H ϕ . From this υ sǫ ǫ | υ| s †Details of experimental geometries and polarizations can be consulted in Ref. 50 ∗In the following we use Γ to refer to the FHMW. CHAPTER II. BACKGROUND OF EXPERIMENTAL TECHNIQUES 35

interaction the discrete state is renormalized to a new state described by a wavefunction Φs and energy ǫ =ǫ ¯ F (ǫ) given by, s s − 2 ′ ′ Vsǫ′ Φ = ϕ + P dǫ V ′ ψ ′ and F (ǫ)= P dǫ | | , s s sǫ ǫ ǫ ǫ′ − where P indicates the principal value of the integral. The total interacting system is described by wavefunctions Ψǫ. The lineshape for any transition in the total system originating from a perturbation T can be described by the renormalization and a reduced energy ǫ¯,

ǫ ǫ F (ǫ) ǫ ǫ F (ǫ) ǫ¯ = − s − = − s − . π V ′ 2 Γ/2 | sǫ |

2 The spectral width of the interacting discrete state is Γ = 2φ V ′ and Q is the lineshape | sǫ | f parameter (a phenomenological shape parameter) of the form,

Φs T GrSt Qf = | | . φV ′ ψ ′ T GrSt sǫ ǫ | | Expressing the lineshape as the ratio between the transition probability in the system to the transition probability for the unperturbed continuum, yields

2 2 Ψ T GrSt (Qf +ǫ ¯) | | | | 2 = 2 , (II.2.14) ψ ′ T GrSt 1 +ǫ ¯ | ǫ | | | which is the scattering cross section of the so-called Fano profile. The normalized value of 2 the Fano profile is defined by dividing by (1 + Qf ). The three different cases of resonance profiles are: for Q = 0 antiresonance, for 0 < Q < asymmetric resonance, and for f f ∞ Q = the characteristic Lorentzian shape.50 The Fano profile is widely used in condensed f ∞ matter physics, in non-linear optics, photonics and nuclear physics. This profile is used in Chapter III to fit our experimental measures. 36 CHAPTER III. STUDIES ON ALKALINE EARTH METAL DISILICIDES

CHAPTER III

STUDIES ON ALKALINE EARTH METAL DISILICIDES

In this Chapter we study diﬀerent alkaline earth metal (AEM) disilicides, namely, CaSi2,

SrSi2 and BaSi2. Based on the experimental work of Evers and Imai we review the pressure- temperature phase diagram and crystal structures adopted by AEM disilicides. Experimen- tally several techniques are used to characterized diﬀerent properties of disilicides such as XRD, Raman, XPS and SQUID measurements. We complement our studies with theoretical density-functional theory (DFT) calculations and using the Boltzmann transport formalism we obtain Seebeck coeﬃcients for the case of cubic structures of AEM disilicides.

Furthermore, we present a detailed study of BaSi2 that includes the calculation of structural, vibrational and electrical properties. Experimentally, we found evidence of a natural buckling of the Si plans that tends to enhanced superconductivity. Our results from DFT calculations clarify the origin of the increase of Tc in BaSi2. The last part of this Chapter is dedicated to the evolution of superconductivity in BaSi2 under pressure.

III.1 INTRODUCTION: PROPERTIES OF AEM-DISILICIDES

Metal disilicides have been recently the subject of extensive studies due to their possible technological applications in many fields. Among this vast family of compounds, the so- called Kankyo semiconductors have attracted special attention because they are composed of naturally abundant and non toxic elements.64 In particular, we are interested in the family of Zintl-silicides which are considered as eco-friendly semiconductors and potentially useful in thermoelectric materials. Binary Zintl silicides are intermetallic compounds formed between a strongly electro-positive metal (alkali metals, alkaline earth metals and lanthanides) and a less electro-positive metal (normally d-block or p-block elements). This family was named after E. Zintl who studied these compounds in early 1930s.65–67 This family was proposed as a promising candidate for applications ranging from photovoltaics, thermoelectrics, hard materials and superconductivity.66 The first systematic investigation of the family of AEM disilicides was carried out by Evers and Weiss in 1974.68 In these works, the authors explored different stoichiometric compositions mixing Si (silicon, column-IV A) and metal atoms (alkaline earth metals, column-II A) following the 8-N rule proposed by Zintl.66 (In Zintl silicides, silicon is the anionic part of the structure and fulfils the octet shell).

The research presented in this Chapter was published in: J.A. Flores-Livas, et al., Phys. Rev. Lett. 106, 087002 (2011) and J.A. Flores-Livas, et al., Phys. Rev. B 84, 184503 (2011) CHAPTER III. STUDIES ON ALKALINE EARTH METAL DISILICIDES 37

Si Si Ba Si

Rhombohedral Cubic Orthorhombic TR6

FIG. 3.1: Ground state structures found for diﬀerent alkaline earth metal disilicides. CaSi2 has a 69 rhombohedral structure, SrSi2 crystallizes in a cubic phase and BaSi2 in an orthorhombic system.

III.1.1 Crystal Structures at Ambient Condition and High Pressure

The crystal structures in which AEM disilicides crystallize at ambient conditions are depicted in Fig. 3.1. These structures represent the ground state for the chemical formula of the form MSi2, where M is a divalent metal, namely Ca, Sr, and Ba. CaSi2 crystallizes in a rhombohedral phase formed by intercalated layers of Ca and Si. SrSi2 presents the characteristic cubic phase, build from a Si-network with Sr atoms in the semi-cages. And for BaSi2, an orthorhombic crystal constructed by tetrahedral-units of Si and isolated Ba atoms, is the ground-state structure. The structural transition in AEM disilicides was studied by Imai and Kikegawa in 2003 using synchrotron radiation.70 In these studies the experimental pressure-temperature phase diagram was investigated in a diamond anvil cell and in situ heating with a high power laser. Their results revealed that the structural sequence of AEM disilicides is diﬀerent from that 70 already known in AX2-type compounds. Figure 3.2 shows the experimental pressure- temperature phase diagram reported by Imai and Kikegawa. We can distinguish that AEM disilicides adopt a series of polymorphs† along diﬀerent thermodynamic conditions, and that these structures can be metastabilized at ambient conditions.

CaSi2 can adopt four diﬀerent crystallographic forms along diﬀerent conditions of pressure and temperature (P-T). Three structures belong to the trigonal crystal system and one to the tetragonal system. The ground state structure of CaSi2 is a rhombohedral crystal,

†Polymorphism is the crystallization of a compound in at least two distinct forms 38 CHAPTER III. STUDIES ON ALKALINE EARTH METAL DISILICIDES

FIG. 3.2: Experimental pressure-temperature phase diagram for alkaline earth metal disilicides 70 68,71 reported by Imai et al. including the previous studies of Evers. For the case of CaSi2, also the experimental work of Bordet, Sanﬁlippo, et al. is included. 72,73

the trigonal CaSi -type structure (space group R 3m, see Fig. 3.1), in which corrugated 2 − Si hexagonal layers alternate with hexagonal Ca layers along the [001] direction and both layers stack in AABBCC sequence. This phase can obtained by quick cooling of molten

CaSi2. Also a diﬀerent sequence of type ABC can be synthesized. The second trigonal

CaSi2 phase has the EuGe2-type structure. This phase appears at high pressures ranging from 9 to 16 GPa and can be quenched to ambient conditions. The third trigonal CaSi2 phase has the AlB2-like structure (space group P 6/mmm) in which ﬂat Si honeycomb layers alternate with Ca layers in AA sequence. The IV phase (AlB2) appears in high-pressure experiments and was reported not to be quenchable to ambient conditions.72 Another polymorph reported for CaSi2 is the α-SrSi2-like structure, consisting of a tetragonal cell (space group I41/amd, see Fig. 3.1). Note that the rhombohedral phase of CaSi2 is the same as that of EuGe2-phase (trigonal-CaSi2) except for the diﬀerence in the stacking sequence.

SrSi2 presents two crystallographic forms, a cubic ground state of SrSi2-type (space group P 4332) and a tetragonal α-ThSi2-type system (space group I41/amd). The tetragonal phase is stable at high pressures and can be quenched to ambient conditions. This tetragonal phase is formed by a three-dimensional Si-network of hexagons not completely closed and intercalated by the Sr-metal. This phase exhibit sp2 bonding and bonds angles < 110 ◦.

In Fig. 3.2 we can distinguish that BaSi2 has three phases at pressures up to 7 GPa and temperatures from 300 to 1300 K. First, there is an orthorhombic crystal (Pnma symmetry) formed by isolated Si-tetrahedra (unit coloured in blue) and Ba atoms separated from each other (see Fig. 3.3). In this structure the Si-Si distance lies between 2.3 to 2.45 A,˚ ∼ and bond angles are close to 60 ◦.68 The structural transformation into a cubic system is illustrated in Fig. 3.3. This cubic structure is build by a Si net formed by 43 helices where CHAPTER III. STUDIES ON ALKALINE EARTH METAL DISILICIDES 39

Orthorhombic Cubic

Trigonal

P = 0 GPa or T > 800 K P > 3 GPa and T > 700 K P > 4.5 GPa and T > 550 K

FIG. 3.3: Crystal structures of BaSi2. The orthorhombic system (24 atoms in cell, 8 Ba) is a semiconductor with an energy gap of 1.3 eV. The cubic crystal (12 atoms in cell, 4 Ba) is a semimetal, and the trigonal structure (3 in cell atoms, 1 Ba) is a metallic compound. 69

every Si atom in such a helix uses one bond for connection with other helices. The angle between Si-Si atoms increases to values around 115 ◦.71 In a well deﬁned domain of P-T, as shown in the right panel of Fig. 3.2, the next structural transformation is into a trigonal phase which is classiﬁed as the EuGe2-like crystal. This structure can be viewed as an hexagonal Si-honeycomb with metal atoms (Ba) between Si planes having an AA stacking. The remaining tetrahedral (sp3) environment of Si atoms deform the network rendering the

Si sheets corrugated. Note that this phase is equivalent to the III-phase of CaSi2.

III.1.2 Electronic and Transport Properties of AEM Disilicides

Alkaline earth metals were recently suggested as promising candidates for thermoelectric and solar cell applications. The advantage of those systems consists in an efficient thermal conductivity as well as good electrical conductivity (with possibility to dope n or p). Thanks to these features, AEM disilicides have been extensively studied in their electrical and transport properties. In Fig. 3.4 we show the work of Affronte and Imai that performed electrical conductivity measurements as a function of temperature for AEM disilicides. The 74 case of CaSi2 has also been reported by Affronte et al., where they explored the resistivity of the body-centred tetragonal phase of CaSi2. This compound exhibits a metallic behaviour and a superconducting state below 1.58 K. For SrSi2 the electrical resistivity was measured by Imai et al.,75 in the cubic phase. The electrical resistivity as a function of temperature 40 CHAPTER III. STUDIES ON ALKALINE EARTH METAL DISILICIDES

FIG. 3.4: Electrical resistivity of alkaline earth metal disilicides as function of temperature. The tetragonal phase of CaSi2 (left panel) exhibits a metallic behaviour with a superconducting transition 74 temperature of 1.58 K. The resistivity curve of SrSi2 (centre panel) in its cubic phase presents a 75 semimetallic behaviour. BaSi2 (right panel), orthorhombic and cubic phases show a negative temperature dependence characteristic of semiconductors. On the contrary, the trigonal phase exhibits a metallic character where the dominant carriers are electrons. Image from Imai, et al. 76

for this compound shows the characteristic behaviour of a semimetal compound. Further studies of the Hall coefficient showed that the dominant carriers are holes in this material. There are not experimental reports of the electrical properties as a function of temperature for the tetragonal phase of SrSi2. The electrical measurement in phases of BaSi2 was reported by Imai et al.,76 and is shown in right panel of Fig. 3.4. For this compound it should be noted that the resistivity and the temperature dependence is strongly dependent on the crystal structure. The resistivity decreases remarkably as the structure changes from the orthorhombic phase into the trigonal phase through the cubic phase. Therefore, the orthorhombic phase shows the highest resistivity among the three phases and a large negative temperature dependence. This phase is a semiconducting material with an indirect band gap energy of 1.3 eV.68,77–79 This system has been studied for its applications in solar cell materials due to its large absorption coefficient (α) and energy band gap close to the desirable value of 1.4 eV.80,81 The cubic phase also shows a negative temperature ∼ dependence but the resistivity is considerably lower than that of the orthorhombic phase. This system presents a characteristic activation energy of a semimetal, contrary to the two first structures. The trigonal phase shows the lowest resistivity and a positive temperature dependence. In spite of the large residual resistivity this result indicates that the layered phase of BaSi2 is a metal. Additionally, Imai, et al. reported a superconducting transition temperature of 6.8 K.82 CHAPTER III. STUDIES ON ALKALINE EARTH METAL DISILICIDES 41

CaC6 MgB2 BaSi2

Ca Mg Ba

C B Si

FIG. 3.5: Superconducting layered crystals (hexagonal crystal systems): CaC6, MgB2 and BaSi2. 69 The ﬁrst two materials present completely ﬂat planes, contrary to BaSi2 that has buckled plans.

III.1.3 Superconductivity in AEM Disilicides

In fact, one of the main characteristics of high temperature superconductors (cuprates oxides, iron pnictides or MgB2) is the reduced dimensionality related to their layered na- 83–86 ture. The surprising discovery of the record-breaking superconductivity in MgB2 at 39 K1 has served as a guide for much of the experimental and theoretical research on conventional superconductivity for the past years. In fact, it is not surprising that other layered 2 87–89 materials composed of ﬂat plans with sp bonding, such as CaC6 and YbC6 have been extensively studied in the years following the discovery of MgB2. In practice, MgB2 can be considered as an iso-electronic structure of graphite and the structures of the two materials can be understood in terms of the honeycomb graphene plane. While in graphite the carbon atoms form stacked hexagonal planes, in MgB2, the boron atoms form an hexagonal net and magnesium atoms are placed between hexagonal planes. Each Mg atom provides two valence electrons, so that the total electron valence count per cell is the same as for graphite and MgB2. Thus we can expect the band structures to be closely related and the bands near the Fermi level to be similar.

In the case of layered disilicides, like BaSi2 or CaSi2 (trigonal phases) one can also think that the structure is iso-structural to MgB2. The layered phase of BaSi2 crystallizes in an hexagonal structure close to that of graphite, CaC6 or MgB2. This is illustrated in Fig. 3.5, where we show an atomistic and geometric model of the structures. We can see in the top 42 CHAPTER III. STUDIES ON ALKALINE EARTH METAL DISILICIDES panels the similar honeycomb networks viewed along the z-axis. However, observed along y-axis (bottom panels) there are major differences. First, in MgB2 and CaC6 completely flat sheets are present, contrary to the case of BaSi2, in which, due to the electronic tendency of Si atoms to form sp3-bonding, there is a buckling of Si-plans (corrugated Si-hexagonal planes). Buckled plans are nevertheless reported to play a fundamental role in high temperature superconductors. For example, in cuprates, the CuO2 plane can alter drastically the superconducting transition temperature.85,90 Disilicides are therefore intriguing as they share the 2D-character of MgB2 (the highest Tc reported for a conventional superconductor) and buckled plans as in the case of cuprates. Therefore it is not surprising that early studies of superconductivity in CaSi2 by Sanfilippo, et al., showed an enhancement of transition temperatures as the layered structure was modified by pressure. In CaSi2 the transition 73 temperature was enhanced from a mK to a maximal Tc of 14 K. In the next section we present experimental and theoretical studies of BaSi2 that indicate a similar trend of the superconducting transition temperature as the layered structure is modified.

III.2 THE CASE OF BaSi2

Synthesis of High Pressure Phases of BaSi2

BaSi2 in its simple chemical formula tends to form an orthorhombic crystal (the ground state structure) but also can form metallic compounds, as we have seen in the previous section. Experimentally, our goal was to synthesized the diﬀerent structures following the phase diagram studied by Evers and Imai. According to the thermodynamic phase diagram of BaSi2 presented in right panel of Fig. 3.2, the metallic phase is synthesized above pressures of 4.5 GPa and temperatures greater than 500 K. In order to synthesized the trigonal phase, our starting point was a powder of BaSi2 (orthorhombic structure) with a purity of 98.5% (Sr < 0.6%) from CERACr. As barium disilicide is water-reactive and sensible to moisture of hydrogen, the samples were stored and fully handled in a glove-box with an inert atmosphere of Ar. The loading of powder (BaSi2) into the cartridge of the belt or the chamber of the diamond anvil cell was cautiously performed (see Chapter II). Using the belt-type apparatus a series of diﬀerent samples of trigonal and cubic phases were synthesized by changing the † temperature and the pressure conditions according to the phase diagram of BaSi2. Details of First-Principle Calculations We carried out extensive ab initio calculations in disilicides in order to confront our experimental stuidies. Ab initio calculations were performed within DFT as implemented in the abinit code.43 We used the Perdew, Burke, and Erzernhof (PBE) generalized gradient approximation93 for the exchange-correlation functional. The electron-ion interaction

†The kinetic time for the phase transformation was studied by Evers. 71,91 CHAPTER III. STUDIES ON ALKALINE EARTH METAL DISILICIDES 43

BaSi

Experiment

4000 (101)

Model

3000

= Cu-K

2000 (100) C ounts (a.u.) Trigonal

1000 (001)

* * *

10 20 30 40 50 60 70

20000

Experiment

16000

= Cu-K

12000

8000 Cubic C ounts (a.u.)

4000

10 20 30 40 50 60 70

1.50

Experiment

1.25

= Mo-K

1.00

0.75

* * C ounts (a.u.)

0.50 Orthorhombic

0.25

10 15 20 25 30 35

2- º

FIG. 3.6: Experimental X-ray diffraction patterns taken at ambient conditions for different allotropes of BaSi2. (Top) trigonal phase: experimental (black dots) and calculated structural model (red line) using Rietveld refinement. 92 (Centre) cubic phase, and (bottom) orthorhombic phase. Dots-stars depict the principal impurity peaks, the phases can be pure as 98%.

was described by norm-conserving Troullier-Martins pseudopotentials94 generated with the same functional. In addition to those calculations, we used the projector augmented-wave approach95 in a plane-wave pseudopotential formalism as implemented in the code vasp.96 The phonon spectrum and the electron-phonon matrix elements were obtained employing density-functional perturbation theory.36,97 Proper convergence was ensured using a cut- oﬀ energy of 30 Ha and converged k-meshes with the Monkhorst-Pack98 sampling of the Brillouin zone.99 44 CHAPTER III. STUDIES ON ALKALINE EARTH METAL DISILICIDES

III.2.1 Crystal Structures of BaSi2, XRD and Energy Calculations

We performed XRD measurements on synthesized samples of BaSi2 to determine the crystal structure. The structural characterizations were performed by X-ray diffraction in θ 2θ (see details of diffraction in Chapter II). Figure 3.6 shows the experimental X-ray − diffraction patterns taken at ambient conditions for different meta-stabilized polytypes of

BaSi2. We confirmed the XRD patters of the phases found by Evers and Imai.71,91 According to our experiments, the thermodynamic P-T phase diagram of BaSi2 shows a well defined zone where the different crystal structures can be meta-stabilized (quenched) at ambient conditions. Further studies of experimental X-ray diffraction of BaSi2 showed important differences when samples were prepared in the thermodynamic conditions of the trigonal phase. Among the different thermodynamic conditions, two principal zones in the trigonal boundaries were of major interest because they presented the most pronounced structural differences. These were the samples synthesized at (4.5 GPa, 800 K) and (4.5 GPa, 1300 K). In these two extreme conditions of temperature, the lattice coordinates exhibits sizeable variations depending on the pressure and the temperature of synthesis. For the trigonal phase there are three parameters to model using Rietveld refinement92 of the measured diffraction pattern, namely, the two lattice parameters a and c and the atomic coordinates of Si, especially the ’z’ component related to the motif in the crystal represented by the 2d-Wyckoff position of Si atoms, where values of z=0.5 corresponds to completely flat Si plans. The measured cell parameters modelled were a=4.061(3) A,˚ c=5.293(3) A,˚ and z= Exp.A ˚3 ˚ ˚ 0.565(1) for 800 K (V0 = 75.59 A ) and a=4.065(3) A, c=5.347(4) A, and z= 0.546(3) Exp.B ˚3 99 for 1300 K (V0 = 76.51 A ).

DFT Calculations on AEM Crystal Structures We explored the ensemble of structures of AEM disilicides using density-functional theory (DFT). Figure 3.7 shows the calculated enthalpy as function of pressure for the crystal phases of CaSi2, SrSi2 and BaSi2. In our calculations, we also analysed the high pressure regime up to 40 GPa. As expected, the volume of the MSi2 compounds (Ca,Sr,Ba) per chemical formula unit becomes shorter when the atomic number of the alkaline earth metal becomes smaller. For the case of CaSi2 we compared three structures (tetragonal, trigonal and AlB2). The phase transitions are in excellent agreement with the experimental P-T phase diagram, Fig. 3.7 (marked by arrows). For BaSi2, the ground state (orthorhombic structure) changes to the cubic phase as pressure is applied, as can be seen in top-right panel of Fig. 3.7, roughly above 1 GPa. Further increase of pressure suggests that the trigonal phase is enthalpically more favourable above pressures of 3 GPa. We should note that this CHAPTER III. STUDIES ON ALKALINE EARTH METAL DISILICIDES 45 / / / /

Volume shrink

FIG. 3.7: Density-functional theory calculations of the enthalpy as a function of pressure for different crystal phases of CaSi2, SrSi2 and BaSi2. In BaSi2 (top panels) two phase transitions are distinguishable; from the orthorhombic to the cubic crystal around 1 GPa, and from the cubic to the trigonal phase above 3 GPa. Our calculations in SrSi2 do not show any crossing in the enthalpy from the ground state (cubic) and the tetragonal phase. In CaSi2, comparing to the tetragonal phase two phase transitions can occur at 10 GPa and 16 GPa. The cases of BaSi2 and CaSi2 show an excellent agreement with the experimental P-T phase diagram.

trend is in good agreement with the experimental phase diagram, even if temperature is not included in our DFT ground state calculations.

As we had noticed in the experimental XRD patterns of BaSi2 (trigonal), sizeable variations exist on the lattice constant and the internal parameter z. In order to have insights on the origin of these eﬀects in BaSi2, we further investigated these variations within DFT. For the trigonal phase, we found that lowest energy structure has lattice parameters (a = 4.08 A,˚ c = 5.42 A,˚ and z = 0.56 A).˚ In fact these geometry optimizations carried out for CaSi2,

SrSi2 and BaSi2 are in good agreement with the experimental lattice constant measured in XRD patterns, and these values are consistent with the typical error of PBE calculations. DF T Figure 3.8 shows the calculated total energy (for ﬁxed V0 ) as a function of the internal 46 CHAPTER III. STUDIES ON ALKALINE EARTH METAL DISILICIDES

350

300

High-buckled

250

200

150

Planar structure

Low-buckled 100 T o tal E n erg y (m eV)

< 50 meV

0.50 0.52 0.54 0.56 0.58 0.60

Crystal Coordinate Z

FIG. 3.8: Variation of the total energy of trigonal BaSi2 per unit cell as a function of the buckling parameter z. The low-buckled regime 0.545 z 0.575 (yellow area) corresponds to values of z that can be explored experimentally.

coordinate z. One can observe that the total energy curve can be divided in intervals of z, depending on the buckling degree. The intervals of values of z are: a planar region for z 0.545, a high-buckled region for z 0.575, and a low-buckled (experimentally found) between 0.545 z 0.575. In the low-buckled interval, the total energy changes by less than 50 meV (580 K) per unit cell. The yellow-area corresponds to values of z that can be explored experimentally. This fact could explain why it is easy to deform the structure, therefore making it possible to obtain experimental samples with diﬀerent z.

III.2.2 Lattice Dynamics of BaSi2: Raman and DFPT Calculations

We performed Raman spectroscopy measurements to study the dynamics of the lattice in the different phases of BaSi2 and the effects of different z in the trigonal phase. On right panels of Fig. 3.9 we show the characteristic Raman spectra acquired for the three crystal structures of BaSi2 measured with an excitation laser of 514.5 nm at ambient conditions. The orthorhombic sample has high frequency optical modes that are generated by vibrations of the Si-tetrahedra at frequencies around 485 cm−1, and this isolated Si-tetrahedral splits the phonon dispersion in narrow bands of frequency in the whole Brillouin zone (this is seen in our calculations of the phonon dispersion from Z-Γ.) The acoustic modes in this phase are predominantly dominated by Ba vibrations. In the cubic phase (right panel), four Raman actives modes are labelled with their corresponding symmetry. All these optical modes are CHAPTER III. STUDIES ON ALKALINE EARTH METAL DISILICIDES 47

q= 0

Calculations Experiments

- 1 ) 500

485 c m -1

- 1 cm

374 c m 400 (

- 1

353 c m

- 1

290 c m 300

- 1

275 c m

200

- 1

150 c m - 1

137 c m

- 1

- 1 121 c m

112 c m Orthorhombic

100 Wave number

Acoustic modes

Z )

400 -1

-1 T +E

338 c m 2

-1

316 c m cm

( T

2 300

-1

220 c m T +A

2 1

C ubic 200 -1

207 c m

100

optic

AET34

1 2 48ac Wave number

Acoustic modes

- 1 )

403 cm E 400

g -1 cm (

300

- 1

227 cm

200

Trigonal

100

Acoustic modes Wave number

Intensity (a. u.)

FIG. 3.9: Left panels: calculated linear-response phonon dispersion for the three phases of BaSi2. Right panels: the corresponding experimental -Raman spectra taken at ambient conditions. Our calculations describe well the phonon energies and the corresponding symmetry for q =0.

vibrations of the Si-helix that disperse more than in the orthorhombic crystal. The phonon calculations for the cubic phase are down-shifted (underestimated), when one compares to the measured Raman frequencies roughly about 30 cm−1. This disagreement is related to an overestimation of the volume in the cubic phase. However, the phonon frequency at q=0 (calculated and measured peaks) are in perfect agreement. For the trigonal phase, the phonon dispersion from K to Γ was calculated for z=0.56 (low energy z) and two Raman active modes were measured and identiﬁed with their corresponding symmetry. These two Raman modes correspond to the vibrations of Si plans, the out-of-plane movements with

A1g symmetry and the double degenerate Eg mode that involves in-plane vibrations. As even small changes in z lead to a shift of the frequencies, for this phase we marked in green 48 CHAPTER III. STUDIES ON ALKALINE EARTH METAL DISILICIDES

350

0 GPa

300

= 520 K

250 0.3 GPa

T = 1.95 )

200 (

150

100

0 50 100 150 200 250 300

Temperature (K)

FIG. 3.10: Temperature dependence of electrical resistivity in the trigonal phase of BaSi2 at ambient pressure at 0.3 GPa. The red and blue lines are the respective ﬁt using the Bloch-Gr¨uuneisen equation. The electrical resistivity decreases linearly with an exponent of T = 1.95 and Debye temperature ΘD = 520 K.

(for q = 0) the variations of the frequency that one obtains if z is adjusted within the sizeable experimental range (yellow-area in Fig. 3.8). A remarkable agreement with the Raman spectra of the trigonal phase is thus obtained.

III.2.3 Electronic and Transport Properties of structures of BaSi2

The electrical resistivity of the different phases of BaSi2 was measured using a conventional four-probe method (see details in Chapter II). The measured trigonal samples showed a positive dependence of the resistivity (indicating a metal) where the predominant carriers are holes. Figure 3.10 shows the measured dependence of the resistivity as a function of temperature and in solid lines the fit performed with the Bloch-Grüneisen equation. From the fit we can obtain qualitative information concerning the scattering mechanism. For this structure (trigonal) the picture of weak electron-phonon scattering is the most suitable. This comes from the T 2 dependence that holds independently of the measured samples (different z). By applying pressure (few kbar) the residual resistivity is shifted to lower values. However the same parameters for the fit hold for the case of applied pressure. From this analysis we can assume that the decrease of resistivity is due to a reduction of the distance between grain boundaries of the polycrystalline sample, and not the effect of pressure in the electronic or phononic properties. The Debye temperature is in good agreement with CHAPTER III. STUDIES ON ALKALINE EARTH METAL DISILICIDES 49

E eDOS-HSE

F Ba:5d

~ 2 eV

eDOS-PBE States/cell

XPS

Si:3s

Si:3(p +p )

x y

Si:3p

Ba:5p C onduction B and 3/2 Intensity (a.u.)

Valence Band

-20 -15 -10 -5 0

Energy (eV)

FIG. 3.11: Bottom curve: X-ray photoelectron spectroscopy (XPS) measured in BaSi2. Calculated electronic density of states (eDOS) with DFT: (centred) using GGA-PBE as exchange correlation functional and (top) hybrid functional HSE06.

previously reported values for disilicides structures.73,82

We also performed the analysis of chemical composition: X-ray photoelectron measurements (XPS) were carried for trigonal samples using a GAMMADATA-SCIENTA SES2002 electron analyzer and high-ﬂux laser with Mg-anode (monochromatic). The energy resolution was better than 2 meV and the base-pressure of the spectrometer was 5x10−11 Torr. Samples were fractured in situ to obtain clean surfaces at room temperature. The ﬁrst − curve of Fig. 3.11 shows the valence band photoelectron spectra of polycrystalline trigonal

BaSi2. From this curve we can distinguish three peaks: two prominent peaks that can be assigned to Ba:5p1/2-p3/2 states and Si:3px-py states, and a third coming from Si:3pz state dominating below the Fermi level. These peaks suggest the strong interaction of Ba atoms and Si layers to form a stable two dimensional system. In Fig. 3.11, we also show the total electron density of states (eDOS) integrated over the Brillouin zone obtained from ﬁrst principle calculations using the GGA-PBE as exchange-correlation functional (centre curve). We can see in this plot a discrepancy in the position of the principal peaks; shifted to low energies. It is known that standard DFT functionals fails to describe very localized states in the valence band, but the use of hybrid functionals usually allows to circumvent these problems. Therefore, we calculated the electronic density of states using the hybrid functional HSE06 (see top curve of Fig. 3.11). We ﬁnd a good agreement with the experimental density of states, even for the deep p-states of Ba around -15 eV. 50 CHAPTER III. STUDIES ON ALKALINE EARTH METAL DISILICIDES

FIG. 3.12: Left panels: electronic dispersion in the Brillouin zone along high symmetry points. Centred panels: Fermi surfaces. Right panels: Seebeck coeﬃcient (thermopower S) as a function of chemical potential ( ) of cubic CaSi2, SrSi2 and BaSi2 at 300 K.

AEM disilicides as thermoelectric materials: Seebeck coefficients We now turn to the discussion of conductivity proprieties such as the Seebeck coefficient. Zintl silicides have been proposed as excellent candidates to serve as thermoelectric materials in devices. Thermoelectric devices can be used for cooling or heating and re- cover energy from waste heat. The efficiency of a thermoelectric material (TM) can be measured by a dimensionless quantity called figure of merit (ZT), and this quantity is de- 2 fined as ZT = (S σT )/κ(e−p), where σ is the electrical conductivity, S is the thermopower

(Seebeck coefficient), T is the absolute temperature, and κe−p is the thermal conductivity which has electronic and phononic contributions. Improving this quantity (ZT) is not ob- vious because the different parameters entering it are linked and compete with each other. Other aspects to be considered are the cost of materials, their toxicity and availability. For that reason, the so called “Kankyo semiconductors” were recently proposed as thermoelectric materials because they fulfil the cited points. Additionally, larger values of ZT = 1.1 CHAPTER III. STUDIES ON ALKALINE EARTH METAL DISILICIDES 51

100 † were reported for n-doped Mg2(Si,Sn) and FeSi2. Therefore, several disilicides such as 103 LaSi, SrSi2 and BaSi2 (orthorhombic) were extensively explored experimentally. In this section we present calculations of the electronic structure and the transport properties of

MSi2 (M = Ca, Sr, Ba) in the SrSi2-cubic type structure (P 4322). The electronic structure was obtained from standard DFT calculations with the GGA-PBE as exchange-correlation functional. Electronic transport calculations were performed using the Boltzmann transport formalism with the constant relaxation time approximation using BoltzTraP.104 First we consider the electronic properties, left panels of Fig. 3.12 show the electronic band structure plotted along high symmetry points in k-space for CaSi2, SrSi2 and BaSi2 in the cubic structure. In this plot one can see a band degeneracy at Γ for Ca, Sr and Ba. Decreasing the atomic number of the substituted metal in the structures, one can distinguish that the degenerated states shift to lower values of energy. The central panels illustrate the Fermi surface in the ﬁrst Brillouin zone. From this ﬁgure we can distinguish that the metallic character decreases as the metal atom changes. With Ca, two bands occupy a large proportion of the Fermi surface, and the conduction bands in CaSi2 have a sharp dispersion between M - Γ - R, a characteristic not desired for thermoelectric applications.

Changing to Sr, the Fermi surface is only occupied for a top valence band (mostly Si:pz states) but also a sharp dispersion is visible. In the case of Ba, a small gap is opened and a few occupation is visible fairly dispersive in the Fermi surface. We should point out the strong inﬂuence of 4d-electrons in Ca, that make that this compound does not have a

SrSi2-cubic structure as a stable polymorph, contrary to SrSi2 and BaSi2 which both share this structure as a metastable polymorph. Right panels of Figure. 3.12 show the Seebeck coeﬃcient (S) and its variation with the chemical potential (), for AEM disilicides calculated within the constant relaxation time approximation. The highest Seebeck number (S) found is for BaSi2 at 300 K (non- doped case) that reaches a maximum value of -25 V/K, then, SrSi -10 V/K and for ∼ 2 ∼ CaSi 10 V/K. Note that in the hypothetical case of p doped CaSi , eventually using 2 ∼ 2 Al or B the maximum attained values are of the order of 20 V/K. For substitution with Sr or Ba and exploring doping values (p-n) the maximum now is limited to 120 V/K, a relatively low value. Comparing with the experimental values of S reported by Hashimoto et al. for cubic SrSi2, our calculations are an order of magnitude lower to those reported.

Moreover, the sign of S is reported to be positive in the case of SrSi2, indicating that the majority of carriers are holes. The origin of this disagreement comes from the absence of a narrow gap as reported experimentally in cubic-SrSi2. In our calculations this structure is a metal with ﬁnite occupation at the Fermi level (seen in centred panel of Fig. 3.12). Our

† 101,102 The highest value of ZT reported so far is 2.4 for a superlattice of p-type Bi2Te3/Sb2Te3. 52 CHAPTER III. STUDIES ON ALKALINE EARTH METAL DISILICIDES

1.00 Volume

High-buckled 0.58

0.95 )

0.56 0

0.90

Low-buckled

0.54

0.85

0.80 Volume (V/V

0.52 Crystal Coordinate Z

0.75

P lanar

0.50

0.70

0 5 10 15 20 25 30

P ressure (GPa)

FIG. 3.13: Left axis: dependence of the volume with pressure for layered structure of BaSi2 calculated with DFT (solid line). Right axis: z parameter as a function of pressure (dashed line). Above 20 GPa the structure is completely planar.

calculations reveal the importance of the semimetallic character present (experimentally) in cubic structures of AEM disilicides. The presence of narrow gaps and its variation in temperature is the principal drawback for the correct description of transport properties.

III.3 SUPERCONDUCTIVITY IN BaSi2

III.3.1 Layered Structure as a Function of Pressure and Temperature

In the previous section we studied the trigonal phase of BaSi2 and the sizeable variations of the internal coordinate z. In this section we explore the evolution of volume and z under pressure and its effects on the electronic and lattice properties. One of the difficulties found in our XRD experiments was the accurate determination of z (buckling degree) under pressure. The difficulty arises from the quality of the samples (as our samples are composed of polycrystalline domains), thus the diffraction pattern of XRD includes a sum of z’s. Although we can obtain a value of z at ambient conditions, the measurements and precise determination of z at high pressures is rather complicated. In this situation, DFT calculations are extremely helpful to interpret experiments. In Fig. 3.13 we show the volume as a function of pressure calculated for the trigonal phase of BaSi2. DF T ˚3 The left axis represents the normalized volume with V0 = 77.16 A . For comparison the CHAPTER III. STUDIES ON ALKALINE EARTH METAL DISILICIDES 53

FIG. 3.14: Panel (a): experimental Raman spectra of the A1g mode for trigonal BaSi2 as a function of temperature from RT to 2 K. The calculated eigenvector of the mode is depicted in the top of the right panel. Panels (c) and (d) show the corresponding evolution of phonon energies (peaks positions) and width FWHM (Γ) of the measured response as a function of temperature.

Exp.A ˚3 Exp.B ˚3 experimental volumes are: V0 = 75.59 A and V0 = 76.51 A . In addition, the right axis shows the optimized z as a function of pressure. The P-V curve is characteristic of a layered material. In these materials the c-component of the lattice decreases substantially faster than in-plane directions, a and b. Moreover, from the ﬁgure (see in right axis of Fig. 3.13) we can see that the starting value of z = 0.560 at 0 GPa, decreases rapidly as pressure increases and the structure evolves to a low buckled-zone (the yellow area indicates the low-buckled zone found at ambient conditions). For pressures smaller than 20 GPa it becomes completely planar. 54 CHAPTER III. STUDIES ON ALKALINE EARTH METAL DISILICIDES

III.3.2 Lattice Dynamics as a Function of Pressure and Temperature

This section is dedicated to the study of the lattice dynamics and their evolution with temperature and pressure in trigonal samples of BaSi2. To reach that goal, several experiments and calculations were carried out to complement those presented in the previous section. Raman scattering measurements were performed for different samples of BaSi2 (trigonal) as function of temperature, from 300 K down to 2 K. In Fig. 3.14 and Fig. 3.15, we show the temperature dependence of the two Raman actives modes present in the trigonal structure and the respective eigenvectors of each mode. The evolution in temperature of the phonon frequency of the two Raman active modes increases linearly by cooling down, as can be seen in panel (c) of the respective figures. The hardening of the phonon energy is associated with a shortening of the bonding distance and the reduction in the linewidth. This effect is significantly higher for the A1g mode visible in panel (d) of Fig. 3.14. The spectra taken at 2 K presents a linewidth reduction of 10 cm−1 compared to that at 300 K, and the phonon energy drastically increases by around 30 cm−1. Comparing this value with Ra- 105 man measures of other layered superconductors like MgB2, in BaSi2 it is much stronger.

Furthermore, the doubly degenerate Eg mode shows a normal evolution with an increase of energy of around 8 cm−1 from 300 K to 2 K, We should note that in the Raman spectra showed above, there is no sign of structural transformation when one cools down the layered phase of BaSi2. Our measures of the Raman modes for the trigonal system yield valuable information about the coupling mechanism. In particular, one can obtain fruitful information from Raman spectra taken below the superconducting transition temperature. In fact, for the degenerate Eg mode, we found an asymmetric line shape at 2 K (blue line of Fig. 3.16 shows the Eg mode fitted by a Lorentzian response). We can distinguish that the fit is not fully satisfactory for this mode. In order to take correctly into account the shape of the response 2 2 † we recurred to the Fano model. The red-line is the Fano fit with IF [(Qf + ε) /1+ ε ]. The characteristic Fano-shape comes from the interference between a discrete phonon excitation and a continuum electronic excitation, thus, revealing that the electronic-background has a strong interaction with the quantization of the lattice in BaSi2. Additionally, the asymmetric parameter q, which is proportional to the inverse of electron-phonon interaction, could give us insights on the coupling mechanisms. However, the parameter q depends strongly on the shape of the background.

To study experimentally the layered BaSi2 under pressure, we used a diamond-anvil cell (DAC) to apply pressure in trigonal samples (see details of high pressure devices in Chapter

II). The two Raman actives modes (A1g and Eg) were measured under pressures up to

† Further details of Fano model are presented in Chapter II. CHAPTER III. STUDIES ON ALKALINE EARTH METAL DISILICIDES 55

FIG. 3.15: Panel (a): experimental Raman spectra of the Eg mode of trigonal BaSi2 as a function of temperature from room temperature to 2 K. The calculated eigenvector of the mode is depicted in the top of right panel. Panel (c) and (d) show the evolution of phonon (peaks) position and width FWHM (Γ) of the measured response.

13 GPa. In order to avoid possible noise in the spectra, no pressure transmitting medium was used. In Fig. 3.17 we summarize the measured frequencies versus applied pressure for the optical modes of BaSi2. For the optical Eg mode one can see that the frequency increases as pressure is induced in the structure. On the other hand, as pressure is applied, −1 the frequency of the A1g immediately drops to lower values, changing by around 8-9 cm . A further increase of pressure results in minor oscillations or modifications of the frequency, as we can see in the right panel. For this optical mode we obtained a negative pressure dependence. Moreover, in both modes we can distinguish abrupt changes immediately when pressure is applied: in Eg a sudden increase of frequency, and in A1g a drop the frequency below the value 2.5 GPa (this effect is visible in the change of slopes). We can even ∼ identify two regimes, the first is related to the softness of the c-component of the lattice parameter, which decreases faster than the in-plane components a and b. This is intrinsic 56 CHAPTER III. STUDIES ON ALKALINE EARTH METAL DISILICIDES

2 K Intensity (a.u.) Intensity

Wave number (cm-1)

FIG. 3.16: Raman spectra of Eg mode taken at 2 K with a wavelength excitation of 514.5 nm. −1 −1 Lorentzian ﬁt parameters: ωi = 421.5 cm and Γi = 5.79 cm . Fano ﬁt parameters: ωi = −1 −1 422.6 cm , Γi = 5.86 cm and Qf = 10.07. The Fano model reproduce satisfactory the experimental spectra.

to the hexagonal symmetry. The second regime arises when the crystal reaches the minimal energy conﬁguration between the three components. This is clearly visible experimentally and in our calculations for several modes with Raman and IR-activity in BaSi2. This can be also observed for other hexagonal crystals.72

To corroborate our experimental ﬁndings, we calculated the pressure dependence of the optical phonon modes in the trigonal phase of BaSi2. In Fig. 3.17 we included for both modes (blue-dashed line) the values of phonon frequencies calculated. The frequencies of the Eg modes are in good agreement with the measured values. However, the A1g modes are severely underestimated in our DFPT calculations probably due to strong anharmonic eﬀects. Nevertheless, the trend of the two frequency regimes discussed above for the hexagonal lattice is in fair agreement. In what concerns the rest of phonon modes, the IR-active modes show a similar softening in pressure, while the acoustical lines gain in energy as the structure decreases its volume. Furthermore, we found that the reminiscent anharmonicity in the A1g mode leads to a dynamical instability in the structure for pressures above 14 GPa and for z values close to 0.525, as the modes exhibit imaginary frequencies indicating a structural instability. CHAPTER III. STUDIES ON ALKALINE EARTH METAL DISILICIDES 57

E A

g 1g

240

520 220

200

500 ) )

180 -1 -1

480 cm cm

( ( 160

460

140

440

120 F req u en cy F req u en cy

100

420

400

0 5 10 15 0 5 10 15

Pressure (GPa) Pressure (GPa)

FIG. 3.17: Raman active modes of BaSi2 (trigonal structure) as a function of pressure. Left panel: Eg mode, right panel: A1g mode. Black squares represent the experimental data and red lines were added as a visual guide. Blue-dots are the calculated phonon frequencies with DFPT.

III.3.3 Parameter z And Its Repercussions In The Electronic Structure

We now turn to the discussion of the electronic density of states at the Fermi level

N(EF ), an important quantity as it can provide insights about the structure stability and other electronic properties such as transport or superconductivity. First we turn to the eﬀects of varying z (buckling of Si plans) in the electronic properties of the system. To respond to this question, we plotted in Fig. 3.18 the total electronic density of states at the

Fermi level as a function of the parameter z. The red curve shows the calculated N(EF ), DF T in the optimized volume in DFT (V0 ) for a wide range of z. This curve (red-line dots) shows a linear increase of electrons at the Fermi level as one decreases z, specially in the range of values depicted in yellow on Fig. 3.8 (those experimentally explored). Therefore, an increase of N(EF ) is directly related to the decrease of z. A maximum occupation number DF T 99 of N(EF )=0.63 is reached for V0 before the structure instability occurring for z=0.545.

For the calculations of N(EF ) optimizing the volume and z for each pressure, one can see in Fig. 3.18 (blue-line squares) that also N(EF ) tends to increase as the structure becomes planar and the maximum occupation attained before the dynamical instability (z=0.525) is N(EF )=0.63. The shaded-green area serves as a guide for this limit in Fig. 3.18. In 58 CHAPTER III. STUDIES ON ALKALINE EARTH METAL DISILICIDES

Limit of stability

0.7

0.6

14.5 GPa

0.5

DFT

V 10GPa

0.4 States/cell (eV/spin) ) 0 GPa F E

0.3 ( N

P lanar Low-buckled HB

0.2

0.52 0.53 0.54 0.55 0.56 0.57 0.58

Crystal Coordinate Z

FIG. 3.18: Electronic density of states at the Fermi level N(EF ) and its dependence on the DF T parameter z. Red dots (linear curve) are the numbers calculated for a ﬁx volume V0 and blue squares show the values of N(EF ) calculated for optimized volume and z under pressure.

addition our results show the possibility of, to some extent, being able to tune the electronic properties by changing the volume of the system.

III.3.4 Superconductivity in BaSi2 as a Function of Pressure

As we know, high pressure has been employed extensively in the study of superconductivity to enhance transition temperatures; the simple reason is that normally one ﬁnds higher transition temperatures in the vicinity of dynamical instabilities, or close to a phase transition. In our investigations, having the possibility to study experimentally under pressure the layered compound of BaSi2, we measured the superconducting transition temperatures

(Tc) of different samples (presenting different z). The measures were performed in a SQUID (QD MPMS 5XL). The principal results are shown in Fig. 3.19, where in the left panel we can see the temperature dependence of the zero-field-cooled (ZFC) magnetic susceptibility, revealing a Tc onset of 6 K for the phase with z=0.565 (high buckling). This value is in agreement with the previously reported by Imai.82 The sample with z=0.546 (low buckled) exhibited a higher Tc of 8.7 K. For this sample, the M-H loop was measured at 2 K (z=0.546). For convenience, we plot the differences χ(T ) χ(10K), normalized by the value at 2 K, − as shown in the right panel of Fig. 3.19. In the trigonal sample with z=0.546, the lower CHAPTER III. STUDIES ON ALKALINE EARTH METAL DISILICIDES 59

0.2

z=0.563

0.0

z=0.546

0.0 K

-0.3

-0.2

6 K

-0.4

-0.6 ) /

K 8.9 K

-0.6

-0.9 M (e.m.u. per mol f.u.)

( -0.8

-1.0

-1.2

2 3 4 5 6 7 8 9 10 0 100 200 300 400

Temperature (K) Applied Magnetic Field (Oe)

FIG. 3.19: Left panel: temperature dependence of the ZFC dc-magnetic susceptibility measured at 20 Oe of two trigonal samples of BaSi2 marked by their z coordinate. Right panel: initial mag- netization curve as a function of the applied magnetic ﬁeld measured at 2 K. Evidencing a type-II superconducting state (see Chapter II).

critical field H 75 Oe is determined from the departure from linearity (red line) at low c1 ∼ field. The upper critical field Hc2 has been estimated at about 7 kOe by the magnetic field at which the M-H reverse legs merge at high magnetic field using the criteria of ∆M < | | −5 10 emu. The values of Hc1 and Hc2 include the demagnetization effect assumed to be 1/3 in cgs units. This measure evidences a type-II superconducting state with H 75 Oe c1 ∼ and H 7 kOe. Furthermore, using the Ginzburg-Landau† equations6 we estimate the c2 ∼ penetration depth λ 3300 A˚ and the coherence length ξ = 104 A.˚ ∼ The calculations presented in the previous sections have shown how easy it is to modify the electronic structure in the trigonal phase by changing the z parameter. Fact that was confirmed by thorough experimental evidence. Moreover, our calculations showed the possibility to tune the internal parameter z, suggesting the possibility of increasing the transition temperature in samples with low-buckled structures. This is clear, as the flatter the structure is, the larger of density of states at the Fermi level is (see Fig. 3.18). In order to verify this point, the superconducting state was studied with the strong coupling theory of superconductivity (Eliashberg equations).7 In this context, we need to

†For details of Ginzburg-Landau equations, consult Chapter II. 60 CHAPTER III. STUDIES ON ALKALINE EARTH METAL DISILICIDES

* = 0.1

Exp. T

4 T (K)

0.55 0.56 0.57 0.58

1.0

0.8

0.545

0.8

0.6 ) 0.55

0.6

0.4

0.56 F(

0.4

0.2

0.54 0.56 0.58

0.2

0.0

0 10 20 30 40 50

(meV)

2 DFT FIG. 3.20: Bottom panel: Eliashberg spectral function α F (ω) calculated for V0 (in the inset: electron-phonon coupling constant λ is show as a function of z). Top panel: calculated Tc with the ∗ McMillan formula. The shaded area shows the interval of Tc values when varies from 0.06 to 0.125. Squares dashed-line represent the experimental values of Tc.

calculate several quantities such as λ and Ωlog, which measure the average electron-phonon interaction in the system. Using density-functional perturbation theory one can have access to the spectral function α2F (ω).7 Once the spectral function is converged, the calculation of quantities as λ and Ωlog was performed from the integrals of Allen-Dynes and its modification of the McMillan equation.106,107 We assumed a value for the Coulomb pseudopotential ∗ of 0.1, which is a standard choice for this kind of material. The calculated Eliashberg spectral function α2F (ω) is shown in bottom panel of Fig. 3.20. From these spectra (for different z we can distinguish two main peaks contributing to λ: the first, due to the acoustic modes, is fairly insensitive to changes in z; the second peak, due to the strong electron-phonon coupling of the A1g optical modes, moves to lower frequency as z becomes flatter, increasing its area and consequently its contribution to λ. In the inset of the bottom panel of Fig. 3.20 (λ as function of z) one can see how this leads to a dramatic increase of the electron-phonon coupling constant. The value of Ωlog, the weighted average of the phonon frequencies, remains basically unchanged in the optimal range 0.55 0.58, and therefore can not be responsible for the variations of T . At z = 0.56 − c we obtain the maximal value Ωlog = 182 K; at z = 0.55 it is slightly smaller, Ωlog=170 K, for z = 0.545 it drops to 156 K and for z = 0.57 and z = 0.58 we found Ωlog = 179 K and CHAPTER III. STUDIES ON ALKALINE EARTH METAL DISILICIDES 61

TABLE I: Superconducting transition temperatures calculated for BaSi2-trigonal phase. The transition temperatures were calculated using Allen-Dynes modiﬁcation of McMillan’s formula.

Pressure Tc ∗ (GPa) λ Ωlog z =0.1 0 0.498 181.9 0.560 2.20 0.5 0.503 183.9 0.559 2.30 1.0 0.510 185.3 0.558 2.44 1.5 0.516 186.0 0.557 2.57 2.0 0.524 186.4 0.556 2.72 3.5 0.540 190.3 0.553 3.08 5.0 0.569 190.7 0.550 3.71 8.0 0.615 187.6 0.543 4.62 10.0 0.678 183.8 0.537 5.90 12.5 0.735 174.0 0.529 6.76 14.0 0.873 136.7 0.524 7.58

171 K, respectively. Therefore, the maximum theoretical value of Tc calculated is around

6 K using the McMillan formulation (top panel of Fig. 3.20). The experimental values of Tc are plotted (red-square), and for comparison the shaded area shows the interval of Tc calculated with McMillan formula when ∗ varies from 0.06 to 0.125. The slight underestimation of the calculated Tc is due to the neglect of multi-band eﬀects that are known to enhance 108 superconductivity in similar systems, like MgB2. Note that the experimental trend for the dependence of Tc of BaSi2 on the buckling is perfectly reproduced by our calculations.

The calculated transition temperatures for BaSi2 (trigonal phase) under pressure are summarized in table I. In this compound, as pressure leads to the contraction of z yielding a planar structure, and plays an important role in the superconducting properties of BaSi2 (see table I and left panel of Fig. 3.21). The z parameter has a minimum value of 0.525 at 14 GPa, and beyond this pressure a dynamical instability suggests a phase transition. In this compound a maximum theoretical value of Tc = 7.58 K is attained. We can see how λ increases steadily as pressure is applied, and Ωlog oscillates between 190 and 137 K. Our results also offer an explanation to previous experimental findings in other similar 72 compound as CaSi2. Right panel of Fig. 3.21 shows the transition temperatures as a function of pressure, measured for different starting crystal of CaSi2 reported by Sanfilippo et al.73 In this compound pressure leads for the trigonal structure an enhancement of transition temperature, from less than 1 K up to 14 K. For BaSi2 taking into account the experimental distribution of measured values at ambient pressure, and comparing to our values found within the McMillan formula, we can expect a maximal Tc for the trigonal structure of 62 CHAPTER III. STUDIES ON ALKALINE EARTH METAL DISILICIDES

Calculated Experimental Transition Temperature (K) Temperature Transition

FIG. 3.21: Left chart: calculated transition temperature for the trigonal phase of BaSi2 as a function of pressure. In this structure the parameter z down to a minimum value of 0.524 at 14 GPa, beyond this pressure a dynamical instability suggest a phase transition, therefore; theoretically, the maximum value of Tc attained is 7.58 K. Right chart: experimental evolution of Tc in pressure for 73 CaSi2, reported by Sanﬁlippo, et al., In this compound the highest Tc attained is 14 K at 16 GPa.

Basi of 14 K for pressures between 10 and 15 GPa (just below the structural modiﬁcation 2 ∼ predicted theoretically at 15 GPa).

III.4 CONCLUSIONS

In this Chapter we presented an experimental and theoretical study of alkaline earth metal (AEM) disilicides. In the first part we reviewed their thermodynamic pressure- temperature phase diagram, structural properties and synthesis of high pressure phases. Then, we investigated the electronic and lattice dynamics in the semiconducting phases. Using the Boltzmann transport formalism, the Seebeck coefficient was calculated for cubic phases of AEM disilicides (CaSi2, SrSi2 and BaSi2). The electronic band structure revealed the metallic nature of the systems. Additionally we obtained low Seebeck coefficients, independently of the metal substitution and over a wide range of doping.

Furthermore, we studied the layered phase of BaSi2, and we show that the buckling of the Si sheets can be modified experimentally by using different pressure and temperature synthesis conditions. The reason for such a behaviour can be found in our DFT calculations, which evidences a broad low-buckling interval where the total energy changes by less than 50 meV (580 K) per unit cell as a function of z. The electronic band structure calculations demonstrate that the density of states at the Fermi level significantly increases by reducing the buckling of the Si planes. However, the flattening of the Si layers is limited by a CHAPTER III. STUDIES ON ALKALINE EARTH METAL DISILICIDES 63 structural instability concomitant with the softening of the Si optical buckling phonon mode

(A1g). The coupling of this mode with electrons is also dramatically enhanced by flattening the Si planes, leading to an increase from 6 K to 8.7 K of the superconducting transition temperature. Such competition between superconductivity and structural distortion can be found in a wide variety of conventional superconductors, like the Chevrel-phases, the transition metal carbides, the cubic Ba1−xKxBiO3, etc. This scenario is in full agreement with our experimental and theoretical findings of an increase of Tc when Si planes flatten 72,73 out, and it is compatible with previous measurements on the disilicide CaSi2. Moreover, the mechanism of tuning Tc by controlling the buckling of the layers is likely to be present in many other layered superconductors, and therefore can provide a new path for optimizing

Tc. 64 CHAPTER IV. SUPERCONDUCTIVITY IN INTERCALATED AEM DISILICIDES

CHAPTER IV

SUPERCONDUCTIVITY IN INTERCALATED AEM DISILICIDES

In the previous Chapter we showed the importance of the parameter z in BaSi2, also we demonstrated how the structure can be tuned via high pressure and high temperature treatment. Variations of parameter z can alter the electronic and the phononic structure resulting in strong effects in the superconducting properties of BaSi2. In particular, the flattening of Si planes tends to increase the transition temperature. In fact, to reach this flattening of Si planes one requires of high pressure and high temperature treatments. However, this situation is particularly difficult from the practical and experimental point of view. To circumvent this problem, in CaSi2, a flat metastable structure at ambient pressure can be in principle stabilized through hole doping by filling the π-bands and increasing the sp2 character of silicon.109,110 Therefore, via doping or with chemical compression, one can expect avoid the use of high-pressure and high-temperature treatments and have stable structures with a relatively high Tc. In this Chapter, we give an overview on intercalation in superconducting layered structures of alkaline earth metal (AEM) disilicides. Our aim is to provide a better understanding of the intercalation process and the mechanisms behind the variation of Tc in AEM disilicides. First, we performed DFT calculations for different intercalating atoms in the MSi2 form, where M takes a divalent metal of the alkaline-earth column; Mg, Ca, Sr, Ba and Ra. Next, we analysed the electronic structure, phonon spectrum, and the electron-phonon coupling. A the end, we correlate these properties to the transition temperatures obtained from McMillan’s formula in order to give hints on how superconductivity can be enhanced in those layered systems.

IV.1 INTERCALATION OF LAYERED CRYSTALS

In this section we discuss the possibility of controlling z in two diﬀerent ways: either by external pressure (as we have seen in previous Chapter), or chemically (i.e., by changing the intercalate metal). Based on the available Zintl-silicides (according to the 8-N rule),66 a large number of elements can serve as intercalating atoms, namely alkaline-earth metals (Ca, Sr, Ba) and lanthanides (Eu, La, Ce, Pr, Th, Nd, Gd, Sm, Y, Dy, Yb, Er, Tm, and Lu). Most of these Zintl binary silicides synthesized experimentally share the same layered

EuGe2 structure ( a trigonal system depicted in Fig. 3.3). In view of the large variety of

The results presented in this Chapter were published in: J.A. Flores-Livas, et. al. Phys. Rev. B 84, 184503 (2011). CHAPTER IV. SUPERCONDUCTIVITY IN INTERCALATED AEM DISILICIDES 65 atoms that can be used to intercalate and the fact that those systems always present a metal phase, makes disilicides an ideal testbed to study superconductivity in layered systems. In order to study the eﬀect of chemical pressure, we mainly focused on intercalation with atoms of the column II-A (Mg, Ca, Sr, Ba, Ra). In addition, the eﬀect of mechanical pressure

(external) could be explored for CaSi2 thanks to the availability of experimental data. Note that in our set of calculations, we incorporated the hypothetical trigonal structures of MgSi2 † and RaSi2 in order to better investigate the variation of Tc along the II-A column. Calculations were performed within DFT as implemented in the abinit code43. We used the Perdew, Burke, and Erzernhof (PBE) generalized gradient approximation93 for the exchange-correlation functional. The electron-ion interaction was described by norm- conserving Troullier-Martins pseudopotentials94 generated with the same functional. We also used the projector augmented-wave approach95 in a plane-wave pseudopotential formalism using the vasp code96. The phonon spectrum and the electron-phonon matrix elements were obtained trough density-functional perturbation theory36,97. The spectral function α2F (ω), that appears in the Eliashberg equations7, was calculated applying the tetrahedron technique for the integration over the Fermi surface. Proper convergence of these quantities was ensured by setting a cut-oﬀ energy of 30 Ha, a 16 16 16 Monkhorst- × × Pack sampling of the Brillouin zone98, and a 4 4 4 q-grid for the phonon wave-vectors. × × Superconducting transition temperatures were calculated through the McMillan-Dynes formula.106,107

IV.2 CRYSTAL STRUCTURES

In Table IV.2 we present the lattice parameters a, c, and zopt that resulted from the ab initio geometry optimization. As expected, going down the II-A column the unit cell expands (with the exception of RaSi2). Moreover, for heavy atoms we obtain low buckled structures (close to the AlB2 phase with a z=0.5) while for light atoms the distortion increases leading to a highly buckled structure (larger z). In Fig. 4.1, we plot the total energy as a function of the internal z parameter at fixed volume. Besides the variation of values of z corresponding to the minimum of the energy curves, we can also notice that the concavity of the curves increase with the increasing atomic number. This concavity is related to the phonon mode that describes the off-plane deformation of the Si layers. One can deduce that, especially for heavy metal atoms, it costs very little energy to change significantly the corrugation of the layers. This was also verified experimentally, as it was possible to produce samples of BaSi2 with different values of z by changing the synthesis

† Detailed information about the synthesis and properties of these compounds can be found in excellent reviews on Silicon polymorphs and 2D Si-puckered layers by Yamanaka, 66 Imai 70 and Demchyna. 67 66 CHAPTER IV. SUPERCONDUCTIVITY IN INTERCALATED AEM DISILICIDES

TABLE I: Optimized lattice parameters and z values obtained trough DFT structural optimization for the trigonal phase. Note that z tends to decrease as the mass of the intercalated metal atom increases. We remind that z =0.5 corresponds to ﬂat layers.

System a c Volume zopt (A)˚ (A)˚ (A˚3) MgSi2 3.70 4.86 57.6 0.602 CaSi2 3.89 5.01 65.5 0.585 SrSi2 3.96 5.13 69.6 0.573 BaSi2 4.07 5.40 77.4 0.560 RaSi2 4.07 5.09 73.1 0.556

71,91,99 procedure. In the case of BaSi2 it turned out that the structures with values of z 99 closer to 1/2 were more interesting as they exhibited larger values of Tc. In order to explore the superconducting properties of disilicides we need to investigate a reasonable range of values of z. To have a working deﬁnition for this range, we kept the cell parameters ﬁxed at their theoretical minimum (energy), while allowing z to vary in a region that maximizes N(EF), λ and Tc, but at the same time assuring the stability of the structures by verifying that phonon frequencies are real. In our opinion, this recipe gener- ates a reasonable range of z that we can use to discuss the maximum possible transition temperature for each one of the elements of the MSi2 family studied in this work. We will refer as “tuned z” to the value of z = ztuned that maximizes the density of states at the Fermi level and the superconducting transition temperature, while yielding real phonons. Those values of z are marked by empty squares in Fig. 4.2 and listed in Table II. Figure 4.2 summarizes the ranges of z we used in this work (left panel), together with the corresponding variations in the Si–Si distances (right panel). We emphasize that all these intervals correspond to variations of the total energy of only a few tenths of meV per MSi2 unit (for example for BaSi2 it correspond to less than 50 meV.) Small atomic numbers (Mg, Ca) favor highly buckled structures, while the SrSi2 system exhibits an intermediate behaviour. Finally, the heavier and larger atoms (Ba and Ra) tends to decrease the corrugation due to the increasing chemical pressure that they exert on the Si planes. It is clear that by changing the intercalating atom, i.e. by applying chemical pressure, we can gain control on the corrugation of the hexagonal Si planes. Another way to modify the buckling of the Si planes is by applying external pressure. As a function of increasing external pressure, it is intuitive that the buckling of the Si-Si planes should decrease, while the Si–Si distance shortens (as the volume decreases). In the case of CaSi2, for example, the stable z values go from 0.585 at zero pressure to z = 0.555 at CHAPTER IV. SUPERCONDUCTIVITY IN INTERCALATED AEM DISILICIDES 67

500

Planar LB High-Buckled

MgSi 400

CaSi

300 SrSi

BaSi

200

RaSi

100 T o tal E n erg y (m eV )

0.50 0.52 0.54 0.56 0.58 0.60 0.62

z (Crystal coordinate)

FIG. 4.1: Total energy per MSi2 unit calculated at 0 GPa and the its dependence on the z parameter for the diﬀerent disilicides. The concavity of the curve increases with increasing atomic number. This concavity is related to the phonon mode that describes the oﬀ-plane deformation of the Si layers.

16 GPa, while the Si–Si distance reduces from 2.40 A˚ at zero pressure to 2.21 A˚ at 16 GPa. It is important to remark that the typical Si–Si bonding distance in a layered structure is around 2.3–2.7 A,˚ 66 a range that is respected for all structures studied here at zero pressure. The shortening of the Si–Si distance can be explained from the experimental compressibility along the c-axis, larger by a factor of two than the one along a or b axes.70 Note that under pressure, as the buckling decreases and the atoms approach, the hybridization of the states increases leading to a strong modiﬁcation of the electronic properties. In the case of the silicides this eﬀect leads to an enhancement of Tc.

IV.3 ELECTRONIC STRUCTURE AND FERMI SURFACE

The calculated Fermi surfaces for the diﬀerent metal intercalation are illustrated in

Fig. 4.3. For elements of the column II-A, at the tuned ztuned values (see Table II and empty squares in Fig. 4.2), the Fermi surface is composed of two parts: a large surface centred at the A points and smaller pockets around the M-points. For these compounds the bands around the A-point have mostly Si 3pxy character and disperse much more in the x-y plane than along vertical directions. For the BaSi2 compound we found that at the limit of structural stability (smallest z), a small third sheet of the Fermi surface composed purely of Si 3s states appears in the Brillouin zone. This third sheet can already been seen 68 CHAPTER IV. SUPERCONDUCTIVITY IN INTERCALATED AEM DISILICIDES

0.61

2.45

High-buckled

0.60 0 GPa

2.40

0.59 0 GPa 16 GPa

0.58

2.35

0.57

2.30 16 GPa Si-Si distance (A)

0.56 z (Crystal coordinate)

2.25

Low-buckled

0.55

0.54 2.20

Mg Ca Ca Sr Ba Ra Mg Ca Ca Sr Ba Ra

Intercalated atom

FIG. 4.2: Range of z values (left panel) and Si–Si distances (right panel) for which phonon- frequencies are real for the diﬀerent systems studied. The black dots represent the fully relaxed zopt/Si–Si distance, while the empty square shows the “tuned z” positions ztuned and the bar indicates the range of stable structures. In the case of CaSi2 we consider two diﬀerent pressures of 0 GPa and 16 Gpa.

at zero pressure in RaSi2, due to the negative chemical pressure exerted by this large atom. A surprising fact is the inﬂuence of d-states of the metal on the electronic properties, that probably leads to the stability of the layered phase. In fact, MgSi2 [see panel (b)] has a quite diﬀerent Fermi surface than the other compounds; composed of small pockets around the H point. This is very likely one of the reasons why neither Mg nor Be form spontaneously a disilicide system. Finally in panels (c) and (d) of Fig. 4.3 one can distinguish the Fermi surface of CaSi2 at zero and 16 GPa. (Our calculations agree with the electronic structure reported previously by Fahy, et. al.111) The increase of the surface with pressure is evident, both around A-points, and in the pockets around M-points; at 16 GPa the whole M–L region is covered. Note that these latter states are composed predominantly by the hybridization of Si 3pz and Ca 4d orbitals. We now turn to the discussion of the electronic density of states at the Fermi level

N(EF ), an important quantity for superconductivity. From Fig. 4.4, we can see that CHAPTER IV. SUPERCONDUCTIVITY IN INTERCALATED AEM DISILICIDES 69

FIG. 4.3: (a) Brillouin zone illustrating high symmetry points. Fermi surfaces of (b) MgSi2, (c) CaSi2 at 0 GPa, (d) CaSi2 at external pressure of 16 GPa, (e) SrSi2 and (f) RaSi2. For RaSi2, note a band appearing at Γ composed purely of 3s-states coming from silicon.

for relaxed zopt the value of N(EF ) with respect to the intercalating atom does not vary monotonously across the II-A group (green dashed bars), it decreases from Mg to Ca, and then increases again until Ra. The situation is different if one uses the tuned ztuned (grey solid bars), N(EF ) now increases as a function of the atomic number of the metal. For light metals (like Mg), the out-of-plane Si phonon mode is stiff, and therefore there is not much freedom in flatting the planes. This leads to a small difference between N(EF ) at the relaxed and tuned z. On the other hand, for Sr and Ca one can increase the density of states by 40% and 60% respectively by tuning z. This effect is even larger under pressure. For example, at 16 GPa and at the low buckling state, one can increase the density of states at the Fermi level of CaSi2, relatively to its zero pressure and relaxed zopt by as much as 80%. This effect is explained by the large compressibility of along c-axis, which leads to an important decrease of volume, and by the softness of the off-plane Si phonon mode that easily allows for large variations of z. 70 CHAPTER IV. SUPERCONDUCTIVITY IN INTERCALATED AEM DISILICIDES

1.0

Tuned N(E )

Normal N(E ) 0.8

0.6 16 GPa

0.4 0 GPa States/cell (ev/spin)

) f

E 0.2 ( N

0.0

Mg Ca Ca Sr Ba Ra

Element intercalated

FIG. 4.4: Occupation of the electronic density of states at the Fermi level N(EF) for the systems studied in this work. Light-green bars represent the calculated with relaxed geometry z = zopt, while black bars are obtained for the tuned values z = ztuned.

IV.4 PHONONS IN INTERCALATED DISILICIDES

Since there are 3 atoms per unit cell in the trigonal phase of metal-disilicides, the phonon band structures are composed by three acoustic and six (3 3-3) optical branches. The me- × chanical representation for motif in the 1a and 2d (crystallographic sites) in the space group

P -3m1 is M = A1g + 2A2u + 2Eu + Eg, where A1g and Eg (double degenerate) are Raman active modes, and A2u and Eu (double degenerate) are IR active modes (infrared). The three higher frequency optical modes involve vibrations of Si atoms composing the hexagonal planes, while the other three lower energy optical modes are out-of-phase vibrations of Si and metal atoms. These six optical modes relative to the P -3m1 crystal symmetry, are depicted in Fig. 4.5. The Eg modes describe the in-plane oscillations of Si atoms, while the

Eu mode corresponds to the combined out-of-phase movement of Si and metal atoms in the xy plane. On the other hand, the A1g and A2u modes are characterized by the out-of-plane oscillations of Si atoms, and out-of-phase Si and metal atoms. CHAPTER IV. SUPERCONDUCTIVITY IN INTERCALATED AEM DISILICIDES 71

FIG. 4.5: 6 optical phonon modes present in the trigonal phase (P -3m1) of AEM disilicides. Eg modes involve the movement of Si atoms in the xy plane and A1g modes represent the out-of-plane motion of Si atoms. Double degenerate Eu modes are displacements of individual metal and Si atoms in the xy plane and A2u modes are the corresponding out-of-plane displacements.

IV.5 TRANSITION TEMPERATURE IN AEM-DISILICIDES

Figure 4.6 shows the phonon band dispersions for diﬀerent metal disilicides studied in this work calculated at tuned values (ztuned) of the internal parameter z. In view of the 99 previous experimental and theoretical ﬁndings for BaSi2, these results give the highest

Tc that is possible to attain for each system. There are two important consequences of decreasing the values of z: (i) There is a softening of the phonon modes. In the case of

BaSi2, for example, the major eﬀect is on the A1g mode. (ii) As we approach an unstable phase, we generally witness an increase of the electron-phonon interaction. For each phonon band structure plotted in the left panel of Fig. 4.6 we also plotted the corresponding spectral function α2F (ω), in order to elucidate the origin of the superconducting properties. We remind that, in conventional phonon-driven superconductivity, the basic recipe suggests that we should have high-frequency phonon modes strongly coupled to electrons in order to attain large transition temperatures.

Quantities important for superconductivity like Ωlog, and λ were calculated from the 2 electron-phonon spectral-function α F (ω) [see Table II]. Tc was calculated solving the McMillan-Allen-Dynes’s equation.106,107 There are two principal issues concerning the use 72 CHAPTER IV. SUPERCONDUCTIVITY IN INTERCALATED AEM DISILICIDES

F( )

0.0 0.2 0.4

600 i S g M

500 ) -1

400

(cm 300 q

200 2

100

A H K M L

0.0 0.2 0.4

600

500 Si aS C )

400 -1

300 (cm

200 2

100

A H K M L

0.0 0.2 0.4

600

500 Si rS S )

400 -1

300 (cm q 2 200

100

A H K M L

0.0 0.3 0.6 0.9

600

500 Si aS B )

400 -1

300 (cm q

200 2

100

A H K M L

0.0 0.2 0.4 0.6

600

500 Si aS R )

400 -1

300 (cm q

200 2

100

H K M L A

F( )

FIG. 4.6: Left panels: phonon band structures of disilicides studied in this work. Right panels: 2 Eliashberg spectral function α F (ω) for each structure, calculated for the values of ztuned reported in Table II. CHAPTER IV. SUPERCONDUCTIVITY IN INTERCALATED AEM DISILICIDES 73 of this formula: (i) this formula requires a value for the parameter ∗, that describes the decrease of the transition temperature due to the breaking of the Cooper pairs interacting through the repulsive Coulomb interaction. In the overwhelming majority of calculations of

Tc this value is semi-empirical, or speciﬁed through a more or less educated guess. Typical ∗ ∗ values of are between 0.1 and 0.2. For Tc values show in Table II, we used = 0.1. (ii) McMillan’s formula neglects multi-band eﬀects that are known to increase considerably

Tc for materials like MgB2. As the structures studied here have several bands crossing the

Fermi level, one can expect that our calculated values for Tc are slightly underestimated. This eventually could explains small diﬀerences between our calculated transition temperatures and experimental values for BaSi2 and CaSi2. In spite of these two limitations, trends should, however, be correctly described.

As expected, we ﬁnd in general a softening of the phonon modes as the mass of the intercalated metal atom increases. The acoustic modes are also found to disperse less for heavier metals. From the α2F (ω) plots we see that all phonon modes couple to electrons, but that the relative weights of each phonon vary with the intercalating metal. For example, for Mg, acoustic modes are by far the most important, while for Ca we also see a big peak coming from the high-energy Eg phonon branch, and for Sr, Ba, and Ra the mixed Si-metal lower energy optical modes play an important role. For the lightest II-A metal studied, Mg, decreasing the buckling does not have a major inﬂuence on the A1g out-of-plane vibration −1 that remains (at Γ) close to 340 cm , while the Eg frequency at Γ remains at around 470 cm−1. In this case, the phonon modes that cause the structural instability with decreasing z are the acoustic modes that become imaginary at the points M and H of the Brillouin zone. Regarding the spectral function α2F (ω) we see that the strongest electron-phonon coupling involves acoustic phonons. Consequently ωlog is relatively low, which, combined with the moderate λ = 0.79, leads to a maximum Tc for this structure of around 6 K.

73 Superconductivity in CaSi2 has been studied experimentally by Sanﬁlippo et al. as a function of pressure, and it was found that this compound can achieve transition temperatures of the order of 14 K (see right panel of Fig. 3.21). The samples with highest Tc turned out to be close to the phase transition (at around 16 GPa) between the EuGe2 phase and 112 the AlB2 structure of CaSi2 predicted by Kusakabe , which consist in completely ﬂat Si planes (z = 1/2). However, linear-response calculations of the AlB2 phase lead to imaginary phonons, and geometry optimization starting from the AlB2 structure with a slightly broken symmetry yields again a highly buckled structure.113 These are clear indications that the AlB2 structure is not stable in the relevant pressure range. On the other hand, the superconducting transition temperatures of EuGe2 phase had been previously studied 114 113 by Nakanishi and Loison. Both works, found a theoretical Tc much smaller than the 74 CHAPTER IV. SUPERCONDUCTIVITY IN INTERCALATED AEM DISILICIDES maximum experimental value of 14 K at 16 GPa.73 To resolve this issue we note (as we know from BaSi2) that it is possible to modify z experimentally by applying external pressure, and consequently to enhance superconductivity in CaSi2.

In Fig. 4.6 we show the phonon dispersion for CaSi2 calculated at the smallest z for which the lattice is theoretically stable. The highest phonon mode (that is strongly coupled to the −1 electronic states) is the Eg in-plane vibration of the Si atoms at around 580 cm . Then, at −1 −1 Γ we ﬁnd the A2u mode at 240 cm , and the A1g around 310 cm . Note that these values are lower than the ones calculated by Nakanishi114 and Loison113 for the trigonal phase of

CaSi2. This can be easily explained by the diﬀerent z used in the calculations, as these previous works used the larger value obtained by geometry minimization within standard DFT. Moreover, it is interesting to see how the superconducting transition temperature increases with decreasing value of z. At 16 GPa, and for the highly buckled sample z = 114 0.5876, λ = 0.3 and Tc = 0.1 K in agreement with the previously calculations. Decreasing z to 0.57 leads to Ωlog = 220 K and to a slight increase of λ = 0.44 and Tc = 1.5 K. As one reaches z = 0.56, there is a sudden increase of the electron-phonon coupling constant to λ

= 0.90 with a Tc = 5.85 K. Finally, just before the structure instability indicated by the presence of imaginary phonon frequencies, we obtain for z = 0.557 the values of λ = 1.1,

Ωlog=100 K, and Tc = 8 K. This number is of the same order of magnitude of the maximum experimental Tc = 14K found for CaSi2. The slight underestimation can, moreover, be attributed to the neglect of multi-band eﬀects in the McMillan’s formula used. We can now see that the large experimental values of Tc for CaSi2 can be explained without resorting to exotic superconducting mechanisms. Furthermore, the presence of samples with diﬀerent 99 values of z, can justify, in the same way as for BaSi2, the dispersion of values of Tc found experimentally. From panel (b) of Fig. 4.6 we can see that for CaSi2 the phonon modes 2 that contribute more to α F (ω) and therefore to λ, are the two Eg modes and the acoustic modes. We should note that, in this case, the acoustic Ca-modes become imaginary at low values of z, and therefore are responsible for driving the system to instability.

In the case of heavier intercalated atoms (Sr, Ba, and Ra), we find that the phonon dispersions show very similar structures, the main difference being the energy of the eigen- modes. The spectral function reveal two main peaks contributing to superconductivity: the first, due to the acoustic modes, less sensitive to changes in z, and the second peak, due to the coupling of electrons with the A1g mode, that moves to lower frequencies for smaller values of z. Descending the II-A column we can also see how the decrease of z induced by the chemical pressure (i.e., by increasing the size of the intercalating atom) leads to an increase of the electron-phonon coupling constants and of the superconducting transition temperature. In fact, the Sr compound has the lowest value of λ with 0.50, leading to Tc = CHAPTER IV. SUPERCONDUCTIVITY IN INTERCALATED AEM DISILICIDES 75

TABLE II: Superconducting transition temperatures calculated for different MSi2 silicides. The transition temperatures were calculated using Allen-Dynes modification of McMillan’s formula. 106,107 108 The neglect of multi-band effects in McMillan’s formula can underestimated Tc.

System P (GPa) ztuned λ Ωlog (K) Tc (K) MgSi2 0 0.593 0.79 132 6.0 CaSi2 0 0.565 0.42 190 1.0 CaSi2 16 0.558 1.1 100 8.0 SrSi2 0 0.565 0.50 187 2.2 BaSi2 0 0.545 0.77 156 6.8 RaSi2 0 0.544 0.93 134 8.3

2.2 K, followed by Ba with λ = 0.77 and Tc = 6.8 K. Finally, Ra leads to the highest values

λ = 0.93 and Tc = 8.3 K.

In fact, the order of compounds from the highest Tc to the lowest is as follows: SrSi2,

MgSi2, BaSi2, CaSi2, and RaSi2. For the heavy intercalating metals, the lattice instability related to the flattening of the hexagonal Si layers comes through the acoustic metal-branch that becomes imaginary for small values of z. This is a further evidence of flattening as a mechanism to enhance Tc. Moreover, it is interesting to consider the relationship between superconductivity trends in the MSi2 family and silicon clathrates having the same guest atoms (Ca, Sr, Ba). Both families are intercalated silicon structures, but they exhibit different structural constraints. Actually, there is no analogous to the buckling tuning parameter of MSi2 in silicon clathrates, which on the other hand possess various intercalation sites. If we consider the simplest and more studied case of the type-I clathrate structure M8Si46 with only two intercalation sites, it has been suggested that the McMillan approximation holds 56,115 with Tc essentially controlled by the density of states at the Fermi level. This leads to 116 a stronger dependence of Tc on the nature of the guest atom as there is no additional structural tuning parameter. 76 CHAPTER IV. SUPERCONDUCTIVITY IN INTERCALATED AEM DISILICIDES

IV.6 CONCLUSIONS

In the current Chapter, we studied several compounds belonging to the family of AEM disilicides, namely MgSi2, CaSi2, SrSi2, BaSi2, and RaSi2. We calculated using ab initio density-functional theory the electronic properties, phonon dispersions, electron-phonon coupling and superconducting transition temperatures. All these quantities were obtained as a function of the internal coordinate z, an important parameter that measures the buckling of the hexagonal Si-planes in the structure. Our results show that it is possible to increase the density of electronic states at the Fermi surface in two diﬀerent ways, either by reducing the buckling of the Si planes by means of external pressure, or by intercalating heavier atoms between the Si sheets. Both paths are equally eﬀective. However from the practical side, the intercalation is easier to achieve than the high pressure and high temperature treatment. The phonon modes show, as expected, a softening with increasing atomic number of the metal, and the largest electron-phonon coupling comes normally from the acoustic and lowest-lying optical modes.

The calculated electron-phonon coupling constants explain the superconductivity in CaSi2 and BaSi2 without the need of invoking exotic superconducting mechanisms. Furthermore, the experimental dispersion of values of Tc can be easily explained by the presence of samples with diﬀerent values of z. CHAPTER V. SUPERCONDUCTIVITY IN DISILANE 77

CHAPTER V

SUPERCONDUCTIVITY IN DISILANE

In this Chapter we study the crystal structures of disilane and its superconducting state in the megabar range of pressure. In 1968, Ashcroft predicted the possibility of metallization of hydrogen and the superconducting state at high pressures. These predictions were based on the BCS-theory, and motivated by the simple idea of the existence of high phonon frequencies due to the light H mass, and strong covalent bonding related to the lack of core electrons.117 Since that work, the possibility of pressure induced superconductivity in simple elements has intrigued many researchers. In the following years, superconductivity under pressure was discovered in several elements and compounds.58 The most notable examples showing an increase of the transition temperature at high pressures are the family of: alkali and alkaline-metals Li, Ca, Sr, and Ba,118–121 lanthanides such as La, Ce, Y, Lu and Sc,122,123 transition metals, V and Zr,124,125 on semimetals as Si and B,126,127 non- metals such as P and S,128,129 and plutonium-based compounds,130 even diamond131 and in oxygen was reported the possibility of attain metallization and superconductivity.132 In fact, 128 Sulphur (S) is currently the elemental high-Tc record holder, achieving 17 K at 160 GPa. Nevertheless, we should note that the transition temperatures found in all the examples listed above are less than 20 K in the experimentally explored range of pressures. More recently, the early predictions of Aschcroft on superconductivity in hydrogen were reinforced with the use of novel theoretical techniques, such as the density-functional theory 31,133,134 for superconductors. The calculated Tc were recently estimated to be as high as 240 K at pressures of around 450 GPa.135,136 However, the synthesis of metallic hydrogen has been found to be experimentally challenging, and even at extremely high pressures ( 340 GPa) metallization has not been observed.137,138 Indeed, this is in agreement with ≈ theoretical calculations, that predicted the metallic transition above 400 GPa (a pressure beyond current experimental capabilities). To circumvent this problem, it was recently suggested that metallization could be achieved at relatively lower pressures in hydrogen rich materials, where hydrogen is chemically “pre-compressed.”139 Several investigations of such compounds have appeared in the literature, primarily focusing on group IV-hydrides, such as: highly compressed silane, 140–145 germane,146 stannane,147,148 and inclusively platinum-hydride.149 These works showed the possibility of metallic phases with relatively high Tc and indeed, at moderate pressures. Experimentally, Eremets et al. studied the case of silane SiH4, for it was

The study presented in this Chapter was published in: J.A. Flores-Livas, M. Amsler, et. al. Phys. Rev. Lett. 108, 117004 (2012) 78 CHAPTER V. SUPERCONDUCTIVITY IN DISILANE

1 2Si+3H2 SiH4+Si+H2 Cmcma 0.5 P-1a C2/2b P-1b Pm-3mb 0 H (eV) ∆

-0.5

Cmc21 -1 50 100 150 200 250 300 350 Pressure GPa

FIG. 5.1: Enthalpy per formula unit of disilane as a function of pressure with respect to elements in their solid form 2Si(s)+3H2(s). The decomposition enthalpies were computed from the predicted structures of hydrogen 154 and high pressure phases of silicon. 155,156 Disilane structures with superscript a are proposed in this work and superscript b from Ref. 157 The dynamical instability towards the Cmc21 phase is indicated by the arrow.

reported crystallization and metallicity above 50–130 GPa150–153 and even a superconducting behaviour found above 190 GPa.150 Following this idea of “pre-compressed” hydrogen rich materials, we investigated the high pressure phases of disilane (Si2H6) and its superconducting behaviour. This Chapter is the result of a fruitful collaboration with the theoretical group of Professor Goedecker at the University of Basel. In the ﬁrst part of this Chapter, we present results of the phase diagram of disilane and the evolution of the crystal structures under pressure. The rest of the Chapter concerns the analysis of the electronic structure, phonons, and superconductivity at the megabar range of pressure. At the end, we discuss our conclusions.

Details of the Minima Hopping calculations We used the recently developed minima hopping method (MHM) of Amsler† and Goedecker159,160 for the prediction of low-enthalpy structures. The only input was the chemical composition of a system. The MHM aims at ﬁnding the global minimum on the enthalpy surface while gradually exploring low-lying structures. Moves on the enthalpy

†For details see the PhD Thesis of Amsler. 158 CHAPTER V. SUPERCONDUCTIVITY IN DISILANE 79

Cmcm P-1

Si Si

H H

FIG. 5.2: Crystal structures of the Cmcm phase at 200 GPa and the P –1 phase at 300 GPa. Si and H are pictured in yellow and light-grey, respectively. 163

surface are performed by using variable cell shape molecular dynamics with initial velocities approximately chosen along soft mode directions. The relaxations to local minima are performed using the fast inertia relaxation engine161 by taking into account both atomic and cell degrees of freedom. We performed simulations for cells containing 1, 2, and 3 formula units of disilane (Si2H6) covering a wide range of pressure between 40 and 400 GPa. The initial screening of the enthalpy surface was carried out using the MHM together with Lenosky’s tight-binding scheme extended to include hydrogen.162

The most promising structures found during the initial sampling were further studied at DFT level using the abinit code43,164 within the Perdew-Burke-Ernzerhof exchange- correlation functional93 and norm-conserving Hartwigsen-Goedecker-Hutter (HGH) pseudopotentials.165 The plane-wave cutoﬀ energy was set to 1400 eV, and Monkhorst-Pack k-point meshes98 with a grid spacing denser than 2π 0.025 A˚ were used, resulting in a × total energy convergence better than 1 meV/atom. Finally, in order to conﬁrm that the tight-binding scheme was able to correctly sample the enthalpy surfaces, we performed MHM simulations for the selected pressures of 100 GPa, 200 GPa, 280 GPa, and 320 GPa at the DFT level. We further characterized the structures found by calculating the phonon spectrum, the electron-phonon coupling, and the superconducting transition temperature

Tc. The phonon spectrum and the electron-phonon matrix elements were obtained from density-functional perturbation theory.36 The spectral function α2F (ω) was integrated over 80 CHAPTER V. SUPERCONDUCTIVITY IN DISILANE

F( )

2500

2000

H )

1500 -1 (cm

1000

500

Z S R 0.0 0.2 0.4 0.6

FIG. 5.3: Left panel: phonon band dispersion of the Cmcm structure at 200 GPa. Right panel: the corresponding calculated Eliashberg spectral function α2F (ω) (solid line) and decomposition of phonon partial density of states (yellow: silicon, light-blue: hydrogen).

the Fermi surface by applying the tetrahedron technique. Convergence of the these quantities was ensured by a 16 16 16 Monkhorst-Pack k-point sampling,98 and a 4 4 4 × × × × q-point sampling for the phonon wave-vectors. The above settings result in a Tc converged within less than 1 K.

V.1 METALLIC STRUCTURES OF DISILANE

In Fig. 5.1 the enthalpy of the diﬀerent phases found in our MHM simulations is shown with respect to decomposition towards elemental silicon and hydrogen. During our simulations we found a new low-enthalpy metallic phase of disilane (see Fig. 5.2). This structure belongs to the Cmcm space group, and is the lowest enthalpy structure from 100 GPa up to 280 GPa. At 200 GPa, its conventional cell parameters are a = 7.965 A,˚ b = 2.705 A,˚ and c = 4.728 A,˚ with one silicon atom occupying the 8e crystallographic site at (0.141, 0, 0) and three hydrogen atoms occupying 8g, 8g and 8f sites at coordinates (0.293, 0.173, 0.250), (0.086, 0.302, 0.250) and (0.000, 0.311, 0.895), respectively. A sketch of the proposed Cmcm structure is shown in panel (a) of Fig. 5.2. The crystal arrangement is composed of hydrogen atoms embedded into a framework of ﬁve-fold coordinated silicon atoms. The silicon-silicon bond length is around 2.28 A,˚ and each silicon atom is surrounded by six hydrogen atoms with an average bond length of 1.52 A.˚ Furthermore, as seen in Fig. 5.1, crystalline disilane CHAPTER V. SUPERCONDUCTIVITY IN DISILANE 81

90 GPa 230 GPa

2500 2500

2000 2000 ) )

1500 -1

1500 -1 cm cm ( (

1000

500

T Z S R T T Z S R Y Y T

(a) (b)

260 GPa 300 GPa

2500 2500

2000 2000 ) )

1500 1500 -1 -1 cm cm ( (

1000 1000

500 500

0 0

-500 -500

S X Z U R T V Y T Z R

FIG. 5.4: Phonon band dispersions of Cmcm structure at 90 GPa, 230 GPa and at 260 GPa. The phonon band dispersion of P –1 structure is shown at 300 GPa (d).

is enthalpically unstable toward decomposition to elemental silicon and hydrogen below

95 GPa. Note that the decomposition to silane SiH4 together with elemental silicon and hydrogen is enthalpically possible up to pressures of 190 GPa. This compositional instability represent a challenge for the synthesis of crystalline disilane, whose outcome depends on bar- rier heights and on the dynamics of the decomposition. Also from Fig. 5.1 we can note that at pressures above 280 GPa, the Pm–3m phase is favoured, competing with several other structures reported by Jin et al.157 In addition to these previously known structures, our simulations revealed another low-lying phase with P –1 symmetry beyond 300 GPa, whose structure is shown in Fig. 5.2 panel (b). However, in the high pressure limit, all these structures lie in a very small enthalpy range which is within our numerical precision. In fact, taking into account the zero-point vibrational energies might easily change the energy or- dering, and in general one can expect that at ﬁnite temperature the competing low-enthalpy 82 CHAPTER V. SUPERCONDUCTIVITY IN DISILANE phases are present as an admixture.

V.2 SUPERCONDUCTING DISILANE AT HIGH PRESSURE

In order to calculate the transition temperatures (Tc) of the proposed structures of disilane, we evaluated the dynamical stability of the system at several pressures. The phonon band dispersion of Cmcm phase at 200 GPa can be seen in left panel of Fig. 5.3, while the partial phonon density of states is shown in the right panel. As expected, the low frequency domain (<700 cm−1) is dominated by the vibrations of the silicon framework, whereas the high end of the spectrum extending up to 2300 cm−1 is solely due to the light hydrogen atoms. The Cmcm structure is dynamically stable up to 220 GPa. However, if the pressure is increased beyond 225 GPa a dynamical instability arises. This is clearly visible in panel (b) of Fig. 5.4, the phonon band dispersion at 230 GPa, where arrows indicate the imaginary (plotted as negative) frequencies.

(a) (b) (c) (d)

FIG. 5.5: Fermi surface of the Cmcm phase at several pressures: (a) 100 GPa, (b) 140 GPa, (c) 160 GPa, and (d) 220 GPa. 166 The diminution of the Fermi surface is visible in the (blue-light) spherical region near the centre of the Brillouin zone.

Following the eigendisplacements of this mode (along the soft mode related with the dynamical instability) and performing a full relaxation of the structure with DFT, we found a new structure (dynamically stable) with Cmc21 symmetry. Compared to the Cmcm phase, the silicon framework remains essentially intact while the hydrogen atoms are slightly displaced, partially breaking the symmetry. Due to the strong similarities between the Cmcm and the Cmc21 structures we do not expect large diﬀerences in their phonons or superconducting properties. A similar analysis as above has been carried out following a further imaginary frequency arising at the S-point when the pressure is increased above 260 GPa [panel (c)]. The resulting structure found by following the corresponding eigendisplacements resulted in a structure with P 1 symmetry. In order to investigate the superconducting properties of the Cmcm phase, we use CHAPTER V. SUPERCONDUCTIVITY IN DISILANE 83

TABLE I: Superconducting properties of the disilane’s Cmcm phase. The transition temperatures were calculated using McMillan’s formula.

Pressure (GPa) λ Ωlog (K) Tc (K) Tc (K) ∗ = 0.1 ∗ = 0.13 100 0.84 478 24.5 20.1 140 0.68 553 18.0 13.6 160 0.66 556 16.9 12.8 200 0.68 501 16.4 12.4 220 0.76 384 16.4 13.0

106,107 McMillan’s approximate formula for the superconducting transition temperature Tc. As we discussed already in the previous Chapters (IV and V) McMillan’s formula requires the weighted-average of the phonon frequencies Ωlog, the average of the electron-phonon ∗ interaction λ and the dimensionless Coulomb pseudopotential .Ωlog and λ were calculated from the Eliashberg spectral function α2F (ω). In right panel of Fig. 5.3, solid lines represent the Eliashberg spectral function of Cmcm phase at 200 GPa. One can distinguish three main features from the spectra: (i) the contribution of low-optical modes of the silicon framework vibrating below 700 cm−1 that couple efficiently with the electrons, (ii) two intense contributions from hydrogen-electrons around 1450 cm−1 and 1600 cm−1, and (iii) high frequency modes of hydrogen above 2000 cm−1 contributing significantly to the spectra. The superconducting properties of Cmcm structure calculated for different pressures are summarized in Table I, using two different values for the Coulomb pseudopotential, ∗ = 0.1 and ∗ = 0.13. Assuming the larger of those values, the superconducting transition temperature Tc is 20.1 K at 100 GPa, decreasing to 13.0 K at 220 GPa. We should emphasize that the Tc of the Cmcm phase is approximately a factor of 6.5 smaller than the reported for the P –1 structure157 at 200 GPa, and that the Cmcm phase is the lowest enthalpy phase. This raises serious doubts on whether high-Tc superconductivity will ever be achieved in silane materials under reasonable pressure. Furthermore, a decreasing Tc with respect to increasing pressure167 has also been reported in other hydrogen rich materials, experimentally and theoretically in silane and in PtH.149,150,168

The superconducting properties of Cmcm structure are strongly related to its electronic structure. We show in Fig. 5.5 the evolution of the Fermi surface as a function of pressure. Three electronic states compose the Fermi surface. The ﬁrst (magenta) and the second (yellow) states cover an important portion of the Brillouin zone and remain nearly unaltered as 84 CHAPTER V. SUPERCONDUCTIVITY IN DISILANE the pressure increases, whereas the third (cyan) state changes substantially. The contribution of this third state to the Fermi surface consists on spherical regions near Γ-points. This band connects two main portions of the Fermi surface. Therefore, a high density of state at the Fermi level (Nf ) will be reﬂected in higher values of Tc at low pressures: λ = 0.84 and Ωlog = 480 with a Tc of 20 K at 100 GPa. However, as the volume of the structure decreases with increasing pressure, this sphere-like feature of the Fermi surface is abruptly reduced. This is illustrated in Fig. 5.5. Consequently, at 160 GPa the superconductivity Tc and (λ) clearly decrease, while Ωlog only slightly increases; λ = 0.66, Ωlog = 556, yielding

Tc = 12.4 K.

V.3 CONCLUSIONS

In this Chapter we presented a thorough investigation of the high-pressure phases of disilane by using first-principles calculations. Applying the minima hopping method to explore the potential energy surface of disilane, we found a metallic structure which is enthalpically favourable compared to the previously proposed structures of disilane. In fact, in cases for a given stoichiometry the actual arrangement of the atoms (or energetic landscape) differs just few kbT , and this difference actually counts for the correct description of a ground state. This is especially true when one wants to predict accurately the physical properties of the ground-state of any new material. Additionally, the systematic study of the superconducting properties as a function of pressure shows that the Cmcm phase possesses a moderate electron-phonon coupling, leading to a superconducting transition temperature in the 10–20 K range. The transition temperatures calculated by Jin et al. for several structures of disilane yield values of Tc roughly an order of magnitude higher than to those 169 presented in this work. This result stands in sharp contrast with the Tc found in the structures previously proposed for disilane under pressure. Moreover, we observed that the transition temperature of the Cmcm structure has the tendency to decrease monotonically with applied pressure, which can be understood by the shrinking of a part of the Fermi surface. This decrease of Tc is in agreement with most theoretical and experimental results of hydrogen rich materials, including silane.149,150,168 Certainly, this does not imply that superconductivity in hydrogen rich materials is limited to relatively low values of Tc for a reasonably high pressure, but our results do impose strong constraints on the possibility of high-Tc superconductors based in silicon-hydrogen systems. We conclude this Chapter leaving a clear message, the necessity of perform accurately global geometry optimizations to truly predict the physical properties of the ground state of any new material. CHAPTER VI. A NEW CARBON ALLOTROPE AT HIGH PRESSURE 85

CHAPTER VI

A NEW CARBON ALLOTROPE AT HIGH PRESSURE

Carbon is a special element thanks to the extreme feasibility to form sp, sp2, and sp3 bonds. At ambient pressure, it is usually found as graphite (the most stable structure) or as diamond, but the richness of its phase diagram does not end there, and many other structures can coexist at ambient conditions, such as fullerenes, carbon nanotubes (different size) and amorphous carbon, etc. Recently, forceful experimental data indicated a structural transformation of graphite into a new superhard phase with full sp3 bonding occurring in the range of pressure of 10–20 GPa.170 Many structures were proposed during the past years to explain this structural transformation. Specially we proposed a new sp3-carbon allotrope called Z-carbon. This structure was found by ab initio calculations by exploring the energy landscape of carbon at high pressures using the minima hopping method.159,160 A series of competitive structures have been also proposed from other theoretical groups.140,171–174 It is surprising the fact that most of these proposed sp3-structures are more stable than graphite exactly in the experimental range of pressure of 10–20 GPa (where the structural transformation occurs) and also most of the structures proposed could eventually explain the experimental observations. Despite the potential applications of this new superhard phase of carbon, still highly debated which is the crystal structure behind the transformation of graphite. Therefore, the aim of this Chapter is to shed-light on the phase transformation of graphite. First we present different sp3-carbon structures and the experimental evidence for the existence of Z-carbon. Then, we will explore the fingerprint signal characteristic of Raman-active modes in graphite under pressure. At the end of the Chapter, we analyse the simulated first-order Raman spectra of sp3-carbon allotropes under pressure and compare with our experimental measurements. This Chapter is the outcome of a successful collaboration of three different groups, the theoretical group at the Univer- sität Basel, our theoretical group and the experimental group of high-pressure physics at the Universitéde Lyon 1.

The research presented in this Chapter was published in: M. Amsler, J. A. Flores-Livas et al., Phys. Rev. Lett. 108, 065501 (2012) and J. A. Flores-Livas, L. Lehtovaara, et. al., Phys. Rev. B 85, 155428 (2012) 86 CHAPTER VI. A NEW CARBON ALLOTROPE AT HIGH PRESSURE

VI.1 CARBON ALLOTROPES AND CRYSTAL PREDICTION

The possibility of a new phase of carbon was set forth in pioneering works early in the 1960s by Bundy et. al.175 In the following years, evidences for a structural phase transition in compressed graphite were reported in numerous experiments using diﬀerent techniques, such as XRD, IXS, Raman scattering, resistivity and optical measurements, etc.170,175–179 This phase transition was suggested to appear above 10 GPa at ambient temperatures∗ (< 300 K) and this phase seems to be metastable when cooled down below 77 K.180 In addition, from the theoretical side the pressure-induced phase transition of graphite has been intensively studied.181 Crystal structure prediction methods, such as evolutionary algorithm171,172,182 (US- PEX), random sampling (AIRSS),140,173 particle-swarm optimization (PSO) algorithm,174 and minima hopping method (MHM),159,160 have been employed to reveal the crystal structures of carbon at high pressure. Indeed, this vast systematic search predicted a new series of carbon allotropes, such as 3D-carbon nanotubes,183 superdense-superhard carbon allotropes,184,185 and several carbon crystals under pressure: hybrid sp2–sp3 186,187 188 189,190 191 diamond-graphite structures, M-carbon, Bct-C4, W-carbon, and recently Z-carbon,192 to name a few.

VI.2 Z-CARBON

The common procedure to search for new crystal structures is to perform a systematic survey of the enthalpy surface using some structure prediction method. Crystal prediction is a complex and challenging task. Therefore, one needs to use an efficient methodology to explore the potential energy landscape in order to achieve the global minimum of the system. In this work, we used the minima hopping algorithm,158,182 which is based on two main principles: (i) first, a feedback mechanism to avoid trapping that recognizes which regions were already visited previously during the run, (ii) second, it exploits the Bell-Evans- Polanyi principle, which gives a higher probability that the catchment basin into which one hops belongs to a low-energy local minimum. We employed for the first time the minima hopping method159 generalized to periodic systems160,193 to explore the possible low-enthalpy phases of carbon. By using simulation cells with 4 and 8 carbon atoms at constant pressure of 15 GPa, we found, besides all the previously proposed structures of compressed graphite, a novel carbon phase that we called Z-carbon. This structure has Cmmm symmetry (see Fig. 6.1) and, like diamond, is composed

∗This is contrary to the synthesis of synthetic-diamond which require chemical precursors and temperatures above 1500 K. CHAPTER VI. A NEW CARBON ALLOTROPE AT HIGH PRESSURE 87

Z-Carbon Cub-Dia

Hex-Dia

M-Carbon W-Carbon BcT-C4

FIG. 6.1: Several carbon structures with sp3 chemical bonding studied in this work. 163 The structure of Z-carbon (orthorhombic crystal) has four-membered and non-planar eight-membered rings. Cubic-diamond, hexagonal-diamond, M-carbon (monoclinic), Bct-C4, and W-carbon (orthorhombic).

of sp3 bonds. The conventional unit cell has 16 atoms with cell parameters at 0 GPa of 3 a = 8.668 A,˚ b = 4.207 A,˚ and c = 2.486 A,˚ yielding a cell volume of V0 = 90.7 A˚ . The structure has two inequivalent carbon atoms occupying the 8p and 8q crystallographic sites with coordinates (1/3, y, 0) and (0.089, y, 1/2), where y = 0.315. The structure contains four-, six- and eight-membered rings, where planar four-membered rings and non-planar eight-membered rings join together buckled graphene sheets. The most plausible transition pathway from graphite to Z-carbon is a combination of sliding and buckling of the graphene sheets. The naturally staggered, i.e. AB stacked, graphene sheets slide along the [210] direction to an aligned AA stacking while the inter-layer distance decreases, and the aligned graphene sheets deform to create an alternating armchair-zigzag buckling. In order to investigate the relative stability of Z-carbon, the calculated enthalpy diﬀerence with respect to graphite of several allotropes are compared in Fig. 6.2 as a function of pressure. Z-carbon has the lowest enthalpy among all proposed cold-compressed graphite phases, becoming more stable than graphite at 9.9 GPa (around 2.5 GPa below W -carbon).† To complement our studies, we used the many-body (GW) technique to calculate the electronic structure, revealing that Z-carbon is an indirect band-gap material with a gap of around 4.7 eV.

†Recently also other groups in Germany and China reported Z-carbon as the most stable structure, see references. 194,195 88 CHAPTER VI. A NEW CARBON ALLOTROPE AT HIGH PRESSURE

Graphite 200 bct-C4 M-carbon W-carbon 100 Hex-diamond Cub-diamond Z-carbon 0 H (meV) ∆ 9.87 GPa -100

-200 0 5 10 15 20 25 Pressure (GPa)

FIG. 6.2: Calculated enthalpy diﬀerence per atom with respect to graphite of several carbon allotropes as a function of pressure. Graphite is the horizontal line at zero. Z-carbon becomes more stable than graphite at around 10 GPa.

As consequence, this material is expected to be optically transparent, in agreement with experiments.176,196 Furthermore, from a ﬁt of the Murnaghan equation of state we obtained 197 a bulk modulus B0 = 441.5 GPa, and a calculated Vicker’s hardness Hv = 95.4 GPa. Both bulk modulus and hardness are extremely high and very close to the values of diamond diamond diamond (B0 = 463.0 GPa and Hv = 97.8 GPa), which is compatible with the observed ring cracks in diamond anvil cells.170

We simulate the XRD spectra of Z-carbon and compare with the spectrum taken from graphite under pressure.† In Fig. 6.3, we can see that the broadening of the peaks present in the XRD spectra at high pressure can be explained by the coexistence of graphite and Z- carbon. The (002) and the (101) reﬂections are the characteristic of 2D-staggered graphite. As pressure increases, the intensity of those peaks is drastically reduced. Thence, at angles of 9 10 ◦ in 2-Θ space, a considerable broadening is visible at pressure of 23 GPa. The − broadening and the increased intensity in this zone match perfectly the principal reﬂection peak of Z-carbon, seen in the top panel of Fig. 6.3.

†Lorentzian and Gaussian contributions were used to build the simulated spectrum, parameters such as GU, GV and GW were carefully tuned in order obtain a more realistic spectra. CHAPTER VI. A NEW CARBON ALLOTROPE AT HIGH PRESSURE 89

23.9 GPa

Z-Carbon

Graphite Simulated

13.7 GPa

23.9 GPa Intensity a.u.

13.7 GPa

(002) (101) Experimental (110) (112) (100)

5 7 9 11 13 15 17

FIG. 6.3: Experimental XRD spectrum of compressed graphite (Bottom panel) at two diﬀerent pressures, taken from Mao, et al. 170 and our simulation of XRD spectrum for Z-Carbon at 23.9 GPa and graphite at 13.7 GPa (top panel). The main peaks of Z-carbon match the observed changes.

VI.3 RAMAN SPECTRA OF CARBON AT HIGH PRESSURE

One of the most significant features of Raman spectroscopy is that it provides a unique fingerprint of the intrinsic vibrational properties of a crystal. Therefore, Raman spectroscopy is an adequate technique to explore the phase transition of compressed graphite. We performed extensive Raman scattering experiments carried out at room temperature (300 K) in order to find signatures of the structure transformation. We used a diamond anvil cell, (see details in Chapter II) to apply pressure on different samples (single crystals of graphite and highly oriented pyrolitic graphite), were placed inside a 120-m hole drilled in an iconel gasket. We employed a Jobin-Yvon HR-800 Labram spectrometer with double-notch filtering with resolution better than 2 cm−1 and the excitation wavelenght of 514.5 nm line of an Ar+ laser. The pressure was determined by the ruby luminescence of a small chip (< 30 microns) and the laser was focused down to 3 microns with a power of less than 20 mW on the sample. We started by investigating the evolution of the principal Raman active mode of graphite, the so-called G-band vibrating at 1579 cm−1 (at 0 GPa), characteristic of sp2- 196 carbon atoms vibrating in-plane with E2g symmetry. In previous works, Hanfland et al. showed a sub-linear behaviour of this E2g mode under pressure, and an anomalous pressure dependence of the Raman linewidhts at pressures above 10 GPa. In our experiments, 90 CHAPTER VI. A NEW CARBON ALLOTROPE AT HIGH PRESSURE

Hanfland et al.

Argon-HOP G

P araffin-HOP G )

60 -1

40 Linewidth(cm

0 5 10 15 20 25

P ressure (GPa)

FIG. 6.4: Evolution of the measured linewidht (ΓLor) of graphite under pressure, black-circles show the reported values of Hanﬂand et al. 196 and the yellow area shows the threshold pressure for the structural transformation; independent of the pressure transmitting medium.

we found a similar effect on the phonon energy and the linewidth of the G-band under hydrostatic pressure, the results are summarized in left panel of Fig. 6.4. From the chart one can observe the linewidth remaining nearly constant until around 9–10 GPa. Above this value, the linewidth begins to broaden rapidly. This behaviour was interpreted as important changes coming from the Raman cross section, indeed, caused by the interlayer coupling and the formation of sp3 bonds. In other words, in graphite it is a sign of a structural transformation at this pressure. As the pressure transmitting medium can eventually play a role, two different pressure transmitting medium were tested in this study, namely argon (Ar) and liquid paraffin. Our conclusion is that the influence of the pressure transmitting can be neglected at relatively low pressures (P < 30 GPa). Additionally, previous works of Hanfland, et al. and Mao, et al. explored different transmitting medium, such as KCl (solid salt) and liquid He. In fact, all transmitting medium giving the same threshold pressure above 10 GPa (see left panel of Fig. 6.4). We may note that in Fig. 6.2, Z-carbon becomes enthalpically favoured with respect to graphite at around 10 GPa, whereas all other proposed structures cross the graphite line at significantly higher pressures. Of course this evidence can not be considered as a direct proof of the existence of Z-carbon. But moti- vates further investigations; now we will present a deep analysis on the vibrational modes of carbon structures at high pressure.

In fact, the Raman scattering measurements carried-out at high pressure in diamond CHAPTER VI. A NEW CARBON ALLOTROPE AT HIGH PRESSURE 91 anvil cells (DAC), present a principal drawback to experimentalists; the DAC used in all −1 high-pressure experiments lead to an intense peak at 1332 cm coming from T2g optical mode of cubic-diamond, and this extends to higher frequencies as pressure is increased. Unfortunately, is expected for sp3-carbon that their principal Raman-active modes will lies in the same spectral range of cubic-diamond. In consequence, the identification of possible peaks coming from new structures could be hidden around the diamond zone, and as the Raman scattering efficiency of the longitudinal mode (T2g) of cubic-diamond is large, it is very hard to distinguish a possible signal coming from the sample behind the background signal of the cell. Nevertheless, the recently proposed allotropes belong to different crystal families (e.g, monoclinic, orthorhombic), therefore; one could expected that the crystals exhibit other Raman-active modes in a different range of frequency. In addition, the evolution of these modes with respect to increasing pressure, can also provide information which allows us to compare with experiments. Therefore, we examined the first-order Raman activity for several carbon allotropes and their pressure dependence. The carbon structures studied during this Thesis are show in Fig. 6.1. We included as reference systems cubic-diamond (F d 3m) and hexagonal-diamond (P 6 /mmc) and the − 3 recently proposed sp3-carbon allotropes: (i) M-carbon,188 a monoclinic system (C2/m) with 189,190 8 atoms per primitive cell, (ii) Bct-C4, a tetragonal system (I4/mm) with 4 atoms per cell, (iii) W-carbon,191 an orthorhombic system (Pnma) with 16 atoms per cell, and (iv) Z-carbon,192 an orthorhombic system (Cmmm) with 8 atoms per primitive cell. We optimized the crystal volume and relaxed the atomic positions for the presented carbon structures using a Troullier-Martins pseudopotential and the local density approximation (LDA) as exchange-correlation functional. The planewave cut-off energy for all runs was 30 Hartree, and the k-point meshes were constructed with the Monkhorst-Pack scheme using: 8 8 8 for 16-atom cells, 12 12 12 for 8-atom cells, and 16 16 16 for × × × × × × 4-atom cells. Phonon calculations were performed within density functional perturbation theory as implemented in abinit.43,164 The Raman activity of crystals was determined by group-theoretical selection rules examining the space group and atomic sites. In addition, from the high-order energy derivatives of the total energy, the Raman tensor was calculated using DFPT36,41,44,45 (described in Chapter I) and the intensities were averaged isotropically to mimic powder spectra,42 that were broadened by a Lorentzian with full width at half −1 198 maximum, ΓLor=10 cm .

In Fig. 6.5 we show the non-resonant Raman spectra at 10 GPa for sp3-carbon structures presented in Fig. 6.1. For clarity and for comparison with the experimental Raman frequency of diamond under pressure, we added the shaded (green) area that limits the experimental evolution in pressure measured for the diamond anvil cell up to 10 GPa.199,200 92 CHAPTER VI. A NEW CARBON ALLOTROPE AT HIGH PRESSURE

Diamond zone

x 1000 Exp.

(9.8 GPa)

Cub-Dia.

E 2g

Hex-Dia.

A A

g A g B B g Z-Carbon

1g 3g

B A

Bct-C

A A

g g A B A

g g A

Relativ e In ten sity (a.u .) M-Carbon g

B A B

A 3g g 3g

g W-Carbon

400 600 800 1000 1200 1400

-1

Wave number (cm )

FIG. 6.5: Calculated Raman spectra intensities for sp3-carbon allotropes at 10 GPa (only principal modes are labelled for clarity). The yellow area shows the experimental spectra at high pressure; focus in graphite (x1000) and near to the interface with the diamond anvil cell. Marked in green the diamond zone.

We remark that the diamond peak and its pressure evolution have been extensively studied both experimentally and theoretically.199–205 The calculated optical mode of cubic-diamond,

T2g, and the characteristic modes of hexagonal-diamond, whose mechanical representation Hex 205–207 is Γ = A1g +E1g +E2g are in good agreement with previously reported values. For Bct Bct-C4 the Raman active modes are Γ = A1g + B1g + B2g + Eg, and the two principal

(with high intensity) are labelled in Fig. 6.5. The A1g mode is a longitudinal optical mode −1 vibrating at 1350 cm at 0 GPa, whose value is above the T2g mode of cubic-diamond and with a much larger pressure coeﬃcient. Other features present in Bct-C4 are the B1g mode and the Eg mode, this last one originated by the double degenerate vibration in plane of the four membered carbon rings (squares) present in the structure. These modes are below the diamond area and they could appear as a small shoulder in Raman measurements above a pressure of 18.4 GPa – the pressure at which the structure becomes (enthalpically) 189,190 M more stable than graphite. For M-carbon (Γ = 2Ag + Bg) two modes can be used to identify the structure: the family of the Ag’s mode which are in-phase vibrations (this mode is activated independently of geometry and laser polarization) and the Bg’s mode, an anti- W phase vibration. The Raman modes of W-carbon: Γ = 2Ag + B1g + 2B2g + B3g, and for CHAPTER VI. A NEW CARBON ALLOTROPE AT HIGH PRESSURE 93

Z Z-carbon: Γ = 4Ag + 4B1g + 2B2g + 2B3g and are presented in Fig. 6.5 respectively. These two crystals belong to the same point group, and in consequence the same labels describe the vibrations present in the crystal. The Ag mode is the usual in-phase mode and the

Bg is a non-degenerate anti-phase mode, where the sub-index refers to the symmetry with respect to diﬀerent mirror planes. A signiﬁcant feature in W- and Z-carbon is that these two structures posses lower frequency Raman modes than the rest of the sp3-structures.

These lowest vibrations (with Raman activity) of W-carbon are the Ag and the B3g modes −1 at around 427 cm , and the lowest modes on Z-carbon are the B3g and the B1g modes at around 525 cm−1 and 570 cm−1, respectively. These modes have a sub-linear pressure dependence, therefore; could be distinguishable in experiments.

From the calculated Raman spectra intensities showed in Fig. 6.5, one can distinguish for Bct-C4 two modes with relatively large intensity which are outside the diamond zone −1 −1 at 10 GPa: the A1g (1385 cm ) and the B1g (1107 cm ). For M-carbon most of the −1 −1 modes with a large scattering eﬃciency are those with Ag symmetry at 897 cm , 1250 cm and 1375 cm−1. For W-carbon the mode with the highest scattering eﬃciency is vibrating at 1295 cm−1 just below the diamond zone at 10 GPa. The vibration corresponds to an −1 Ag symmetry. Z-carbon shows three principal peaks (at 10 GPa) vibrating at 1093 cm , 1221 cm−1 and 1360 cm−1.

Experimentally, we extensively explored the zone of frequencies described above in order to detect any fingerprint that could come from a new allotrope of carbon. In Fig. 6.4 (yellow area) we present the experimental Raman spectra of two range of frequency: the interface sample-DAC and the range below the diamond zone. In the first, the Raman peak coming from the diamond anvil cell starting at 1332 cm−1 and the measured peak around 1357 cm−1 at 9.8 GPa. In the second zone, we can observe that a clear peak appears at 1082 cm−1 for pressures higher than 9.8 GPa (x1000). This peak cannot be explained by neither graphite, hexagonal-diamond, cubic-diamond nor M-carbon, or by the solidification of the pressure medium (in this case argon). Other possible explanation for this systematic peak comes from experiments of Raman scattering at ambient pressure, where a sideband Raman peak at 1090 cm−1 can be observed in samples of “nanocrystalline” diamond.208,209 Although, the presence of “nanodiamonds” in our sample might be enthalpically possible. The formation “nanodiamonds” has been shown not to be stable in high pressure synthesis. Actually, the observed G-band broadening is fully reversible under pressure unload, thus this possibility of “nanodiamonds” is less suitable in our experiments.210 Nevertheless, in this frequency range of vibrations the possible candidates that could explain the measured peak in the

Raman spectra are Bct-C4-carbon, Z-carbon and W-carbon. However, our experiments are far to be conclusive and further investigations is needed. In summary, in order to 94 CHAPTER VI. A NEW CARBON ALLOTROPE AT HIGH PRESSURE identify the several sp3 structures using Raman scattering experiments we propose the use of the following fingerprints: (i) For Bct-C4 the most important mode is the B1g mode at −1 1124 cm , that should appear above 18.0 GPa; (ii) M-carbon can be identified by the Ag −1 −1 mode at 900 cm at around 13.5 GPa. (iii) For W-carbon the Ag mode at 1300 cm , i.e., below the diamond zone, above 12.3 GPa; (iv) Finally, the transition to Z-carbon at 10 GPa −1 −1 can be identified by the Ag modes at around 1220 cm and 1090 cm . However, we should emphasize that the calculated frequencies and transition pressures contain an error caused by the theoretical framework.

VI.4 CONCLUSIONS

In this Chapter we used the minima hopping method to predict low-enthalpy crystal structures under pressure and we compared the structural and the vibrational properties against respective experimental evidence. The proposed allotropic structure of carbon, Z-carbon, becomes more stable than graphite above 10 GPa. From all known carbon allotropes, only cubic and hexagonal diamond have lower enthalpy at high pressures.† Our calculations demonstrated that the new Z-carbon phase is as hard as diamond, and is transparent in the optical region. Overall, a wide range of experimental data can only be explained by the presence of Z-carbon in samples of compressed graphite: first, the features of the X-ray diffraction spectra of graphite under pressure exhibit a broadening that matches the main peaks of Z-carbon. Second, the principal Raman signal of graphite, the G-band mode, suffers an abrupt increase of the linewidth above 9-10 GPa — the pressure range where Z-carbon becomes more stable than graphite. Third, an emerging peak at 1082 cm−1 appears in the Raman spectrum of graphite at around 10 GPa, at the frequency of a Raman active mode of Z-carbon. The most appropriate candidate among all carbon allotropes proposed so far that can explain all above features simultaneously is Z-carbon. In conclusion, we gathered a considerable amount of experimental evidence that points to the plausible existence of Z-carbon. In addition, in order to provide insights to the researchers for the identification of the structure, we calculated the Raman spectra of different carbon structures under pressure. A positive and incontestable determination of the structure of compressed graphite will require the identification of several peaks in different frequency ranges. Our results, however, can serve as useful guide for experimentalists performing

†At the moment of ﬁnishing this thesis new hypothetical structures of carbon have been reported, see the work of Niu, et al. 211–213 CHAPTER VI. A NEW CARBON ALLOTROPE AT HIGH PRESSURE 95

GENERAL CONCLUSIONS

In this thesis we have consolidated both theoretical predictions and experimental findings, which was its first objective. Indeed, theoretical predictions were corroborated by experiments in some cases, while in some others where the agreement between the two was not so immediate they stipulated at least new discussion and re-analysis of experimental data. Along this dissertation, we went beyond prediction of properties of existing materials, and advanced to predict even the structures themselves by making use of the minima hopping method. Such prediction schemes are very promising and powerful tools, and we expect them to lead to significant advances in material design, a domain still in fledgling stages.

The core of this work was presented in Chapter III, where alkaline-earth-metal (AEM) disilicides were studied. First we reviewed their thermodynamic pressure-temperature phase diagram, structure properties and synthesis of high pressure phases. We investigated the electronic and the lattice dynamics, and for cubic structures of AEM disilicides, the electronic transport properties were calculated. We report low Seebeck coeﬃcients; independently of metal substitution and over a wide range of doping values. Moreover, the superconductivity on layered structures of BaSi2 was measured and we found that the buckling of Si-planes in this disilicide has a noteworthy impact on the electronic and the phononic structure. In addition to the thorough of experimental investigations, extensive DFT-DFPT calculations were carried out, and the necessary ingredients of phonon-driven superconductivity were not only evaluated as a functions of pressure, but also as a function of the buckling degree for this structure. Most of the theoretical results were satisfactorily connected to the experimental evidences, in some cases even, our results were used for predict properties, and conﬁrmed a posteriori along the development of the Thesis.

Chapter IV was devoted to studied intercalation in layered disilicides by metal atoms of column II-A of the periodic table. We studied for different AEM disilicides, energetic issues behind the intercalation of layered materials. In fact, our results show that it is possible to increase the density of electronic states at the Fermi level in two different ways, either by reducing the buckling of the Si planes by means of external pressure, or by intercalating heavier atoms between the Si sheets. Both paths are equally effective to enhance superconductivity, however from the practical side, the intercalation by heavy species is easier to achieve than the high pressure and high temperature treatments.

In Chapter V we treated superconductivity in disilane under a wide range of pressure by calculating the transition temperature. Contrary to expectations, and already reported 96 CHAPTER VI. A NEW CARBON ALLOTROPE AT HIGH PRESSURE work, and despite its typical metal behaviour under pressure; disilane exhibits a relatively small transition temperature of the order of 25 K. In order to draw this non-trivial conclusion, it was crucial to search for diﬀerent pressures the corresponding disilane ground state, i.e. the most stable state in terms of enthalpy. Indeed, several metal phases were predicted to be more suitable than previously reported. This once again stresses the power and the usefulness of crystal structure prediction methods such as, the minima hopping.

In Chapter VI the minima hopping method allows us to propose a new allotrope of carbon, termed Z-carbon, which is enthalpically more suitable for the structural transformation of graphite above pressures of 10 GPa. This structure is hard as diamond and transparent in the optical region. In order to characterize this and compare with other new carbon structures under pressure, we calculated their Raman spectra. On one hand, this suggested the possible identiﬁcation of Z-carbon at high pressures, and on the other hand, provides a useful guide for further experiments, which is a subject of on-going research. BIBLIOGRAPHY 97

BIBLIOGRAPHY

[1] J. Nagamatsu, T. Muranaka, Y. Zenitani, and J. Akimitsu, “Superconductivity at 39 K in Magnesium Diboride,” Nature (London), vol. 410, Feb 2001. 1, 41

[2] B. Kang and G. Ceder, “Battery materials for ultrafast charging and discharging,” Nature, vol. 458, pp. 190–193, 2009. 2

[3] G. Ceder, G. Hautier, A. Jain, and S. Ong, “Recharging lithium battery research with ﬁrst-principle methods,” MRS Bulletin, vol. 36, no. 3, pp. 185–190, 2011. 2

[4] M. Born and R. Oppenheimer., “Zur quantentheorie der molekeln,” Ann. Phys., vol. 84, pp. 457–484, 1927. 3, 5, 18, 32

[5] J. Bardeen, L. N. Cooper, and J. R. Schrieﬀer, “Theory of superconductivity,” Phys. Rev., vol. 108, pp. 1175–1204, Dec 1957. 3, 9, 10

[6] V. Ginzburg and L. Landau, “English translation: Type II Superconductors and The Vortex Lattice,” Zh. Eksp. Teor. Fiz., vol. 1064, p. 20, 1950. 3, 59

[7] G. Eliashberg, “Eliashberg Equations of Strong Coupling Theory,” Teor. Fiz 38, 966 (1960)[Sov. Phys. FETP, vol. 38, 1960. 3, 12, 59, 60, 65

[8] P. Hohenberg and W. Kohn, “Inhomogeneous electron gas,” Phys. Rev., vol. 136, pp. B864–B871, Nov 1964. 3, 14

[9] E. N. Zein, “Fiz. Tverd. Tela (Leningrad),” Sov.Phys. Solid State 26, 1825, vol. 26, p. 3024, 1984. 3, 18

[10] S. Baroni, P. Giannozzi, and A. Testa, “Green’s-function approach to linear response in solids,” Phys. Rev. Lett., vol. 58, pp. 1861–1864, May 1987. 3, 18

[11] J. Serrano Gutiérrez, Phonon- and electron-phonon interaction in semiconductors: Isotopic effects. Ph.D. Thesis, Fakultät Physik der Universität Stuttgart, Jul 2003. 3, 8

[12] Z. J. M., Principles of the theory of solids, second edition. Cambridge, University Press., Sept 1972. 5, 8, 10, 29

[13] P. Br¨uesch, Phonons: Theory and Experiments I. Springer-Verlag, Berlin, 1982. 5

[14] H. Kamerlingh Onnes, “Further experiments with liquid helium. a,b,c. on the electrical resistance of pure metals, the disappearance of the resistance of mercury.,” Comm. Phys. Lab. Univ. Leiden, vol. 122b, 1911. 8 98 BIBLIOGRAPHY

[15] W. Meisnner and R. Ochsenfeld, “Ein neuer eﬀekt bei eintritt der supraleitf¨ahigkeit,” Naturwissenschaften, vol. 21, no. 44, pp. 787–788, 1933. 8

[16] F. London and H. London, “The electromagnetic equations of the supraconductor,” Proc. Roy. Soc. (London), vol. A149, no. 866, 1935. 9, 10

[17] M. Tinkham, Introduction to Superconductivity. McGraw-Hill Book Co., 1996. 8, 10, 11, 12

[18] W. S. Corak, B. B. Goodman, C. B. Satterthwaite, and A. Wexler, “Exponential temperature dependence of the electronic speciﬁc heat of superconducting vanadium,” Phys. Rev., vol. 96, pp. 1442–1444, Dec 1954. 9

[19] H. Fr¨ohlich, “Theory of the superconducting state. I. the ground state at the absolute zero of temperature,” Phys. Rev., vol. 79, pp. 845–856, Sep 1950. 10

[20] J. Bardeen and D. Pines, “Electron-phonon interaction in metals,” Phys. Rev., vol. 99, pp. 1140–1150, Aug 1955. 10

[21] D. Pines, “Superconductivity in the periodic system,” Phys. Rev., vol. 109, pp. 280– 287, Jan 1958. 10

[22] A. A. Abrikosov, “Zh. eksperim,,” i Teor. Fiz 32, 1442 (1957)[Sov. Phys. FETP, vol. 32, 1957. 11, 12

[23] J. P. Carbotte, “Properties of boson-exchange superconductors,” Rev. Mod. Phys., vol. 62, pp. 1027–1157, Oct 1990. 13

[24] W. Kohn and L. J. Sham, “Self-consistent equations including exchange and correlation eﬀects,” Phys. Rev., vol. 140, pp. A1133–A1138, Nov 1965. 14

[25] R. M. Martin, Electronic Structure, Basic Theory and Practical Methods. Cambridge University Press, April 2004. 14, 15

[26] E. N. Economou, The Physics of Solids. Springer, Berlin, 2010.

[27] F. Nogueira, A. Castro, and M. Marques, A Primer in Density-Functional Theory, vol. 620. Lecture Notes in Physics Springer Berlin., 2003. 14, 15, 16

[28] E. Runge and E. K. U. Gross, “Density-functional theory for time-dependent systems,” Phys. Rev. Lett., vol. 52, pp. 997–1000, Mar 1984. 14

[29] E. K. U. Gross and W. Kohn, “Local density-functional theory of frequency-dependent linear response,” Phys. Rev. Lett., vol. 55, pp. 2850–2852, Dec 1985. BIBLIOGRAPHY 99

[30] M. Marques, N. Maitra, F. Nogueira, E. Gross, and A. Rubio, Fundamentals of Time- Dependent Functional Theory, vol. 837. Springer-Verlag, Berlin, 2012. 14

[31] L. N. Oliveira, E. K. U. Gross, and W. Kohn, “Density-functional theory for superconductors,” Phys. Rev. Lett., vol. 60, pp. 2430–2433, Jun 1988. 14, 77

[32] L. Hedin, “New method for calculating the one-particle green’s function with application to the electron-gas problem,” Phys. Rev., vol. 139, pp. A796–A823, Aug 1965. 14

[33] D. M. Ceperley and B. J. Alder, “Ground State of the Electron Gas by a Stochastic Method,” Phys. Rev. Lett., vol. 45, pp. 566–569, Aug 1980. 16

[34] D. G. Anderson, “Iterative procedures for nonlinear integral equations,” J. Assoc. Comput. Mach., vol. 12, p. 547, 1965. 17

[35] X. Gonze, “Adiabatic density-functional perturbation theory,” Phys. Rev. A, vol. 52, pp. 1096–1114, Aug 1995. 18

[36] S. Baroni, S. de Gironcoli, A. Dal Corso, and P. Giannozzi, “Phonons and related crystal properties from density-functional perturbation theory,” Rev. Mod. Phys., vol. 73, pp. 515–562, Jul 2001. 18, 19, 21, 43, 65, 79, 91

[37] H. Hellmann, “Einf¨uhrung in die Quantenchemie, (Introduction to Quantum Chem- istry),” Leipzig: Franz Deuticke. p. 285., 1937. 18, 19

[38] R. P. Feynman, “Forces in molecules,” Phys. Rev., vol. 56, pp. 340–343, Aug 1939. 18, 19

[39] X. Gonze and J.-P. Vigneron, “Density-functional approach to nonlinear-response coeﬃcients of solids,” Phys. Rev. B, vol. 39, pp. 13120–13128, Jun 1989. 20

[40] M. Born and K. Huang, Dynamical Theory of Crystal Lattices. Oxford University Press, Oxford, 1996. 20

[41] G. Placzek, “Rayleigh-Streuung und Ramaneﬀekt,” In Handbuch der Radiologie, vol. 6, p. 208, 1934. 20, 21, 91

[42] S. A. Prosandeev, U. Waghmare, I. Levin, and J. Maslar, “First-order Raman spectra

of AB12B12O3 double perovskites,” Phys. Rev. B, vol. 71, p. 214307, Jun 2005. 21, 91 100 BIBLIOGRAPHY

[43] X. Gonze, G. M. Rignanese, M. J. Verstraete, J. M. Beuken, Y. Pouillon, R. Caracas, F. Jollet, M. Torrent, G. Zerah, M. Mikami, P. Ghosez, M. Veithen, J. Y. Raty, V. Olevano, F. Bruneval, L. Reining, R. W. Godby, G. Onida, D. R. Hamann, and D. C. Allan, “A brief introduction to the ABINIT software package,” Zeitschrift f¨ur Kristallographie, vol. 220, pp. 558–562, 2005. 21, 42, 65, 79, 91

[44] M. Veithen, X. Gonze, and P. Ghosez, “First-principles study of the electro-optic eﬀect in ferroelectric oxides,” Phys. Rev. Lett., vol. 93, p. 187401, Oct 2004. 91

[45] M. Veithen, X. Gonze, and P. Ghosez, “Nonlinear optical susceptibilities, Raman eﬃ- ciencies, and electro-optic tensors from ﬁrst-principles density functional perturbation theory,” Phys. Rev. B, vol. 71, p. 125107, Mar 2005. 21, 91

[46] S. Miller and W. Love, Tables of Irreducible Representations of Space Groups an CoRepresentations of Magnetic Space Groups. Pruett Press, Boulder, CO, 1967, 1967. 22

[47] “Bilbao Crystallographic Server, a project of Universidad del Pais Vasco/Euskal Her- riko Unibertsitatea.” http: www.cryst.ehu.es. 22, 23

[48] A. A. Maradudin and S. H. Vosko, “Symmetry properties of the normal vibrations of a crystal,” Rev. Mod. Phys., vol. 40, pp. 1–37, Jan 1968. 22, 23

[49] M. Cardona and G. G¨untherodt, Light Scattering in Solids vol. VIII. Topics in Applied Physics (Springer, Berlin), 2000. 31

[50] H. Kuzmany, Solid-State Spectroscopy. Springer, Berlin, 1998. 23, 31, 32, 34, 35

[51] C. W. Chu, L. Gao, F. Chen, Z. J. Huang, R. L. Meng, and Y. Y. Xue, “Super-

conductivity above 150K in HgBa2Ca2Cu3O8 at high pressures,” Nature, vol. 365, pp. 323–325, Sept. 1993. 24

[52] L. Gao, Y. Y. Xue, F. Chen, Q. Xiong, R. L. Meng, D. Ramirez, C. W. Chu, J. H.

Eggert, and H. K. Mao, “Superconductivity up to 164 k in HgBa2Ca2Cu3O8 under quasihydrostatic pressures,” Phys. Rev. B, vol. 50, pp. 4260–4263, Aug 1994. 24

[53] H. T. Hall, “Ultra-high-pressure, high-temperature apparatus: the ”Belt”,” The Re- view of Scientiﬁc Instruments, vol. 31, pp. 125–131, Feb. 1959. 25

[54] K. Gschneidner, M. Hepworth, and N. Parlee, Metallurgy at high pressures and high temperature, high pressure-temperature apparatus. Gordon and Breach Science Pub- lishers, New York, 1964. 25 BIBLIOGRAPHY 101

[55] A. San Miguel and P. Toulemonde, “High-pressure properties of group IV clathrates,” High Pressure Research, vol. 25, pp. 159–185, 2005. 25

[56] P. Toulemonde, C. Adessi, X. Blase, A. San Miguel, and J. L. Tholence, “Supercon-

ductivity in the Ba8Si46 clathrates: Experimental and ab initio investigation,” Phys. Rev. B, vol. 71, p. 094504, Mar 2005. 25, 75

[57] X.-J. Chen, V. V. Struzhkin, Y. Yu, A. F. Goncharov, C.-T. Lin, H. K. Mao, and R. J. Hemmey, “Enhancement of superconductivity by pressure-driven competition in electronic order,” Nature, vol. 466, no. 19, pp. 950–953, 2010. 26

[58] C. Buzea and K. Robbie, “Assembling the puzzle of superconducting elements: A review,” Supercond. Sci. Technol., vol. 18, pp. R1–R8, 2005. 26, 77

[59] H.-k. Mao and R. J. Hemley, “Ultrahigh-pressure transitions in solid hydrogen,” Rev. Mod. Phys., vol. 66, pp. 671–692, Apr 1994. 27

[60] A. Jayaraman, “Diamond anvil cell and high-pressure physical investigations,” Rev. Mod. Phys., vol. 55, pp. 65–108, Jan 1983. 27

[61] N. W. Ashcroft and N. D. Mermin, Solid State Physics. Saunders College, 1976. 28

[62] C. Caillier, A. Ayari, V. Gouttenoire, A. San Miguel, V. Jourdain, M. Picher, and J.-L. Sauvajol, “Gold contact to individual metallic carbon nanotubes: A sensitive nanosensor for high-pressure,” Appl. Phys. Lett., vol. 173111, 2010. 30

[63] U. Fano, “Eﬀects of conﬁguration interaction on intensities and phase shifts,” Phys. Rev., vol. 124, pp. 1866–1878, Dec 1961. 34

[64] Y. Makita, “Proceding of the Japan-UK Joint Workshop on Kankyo-Semiconductors (Ecologically friendly Semiconductors),” Society of Kankyo Semiconductors in Japan, Japan, pp. 1– , 1999. 36 ∞ [65] E. Zintl and H. Kaiser, “Uber¨ die F¨ahigkeit der Elemente zur Bildung negativer Ionen,” Z. Anorg. Allg. Chem., vol. 211, no. 113–131, 1933. 36

[66] S. Yamanaka, “Silicon clathrates and carbon analogs: high pressure synthesis, structure, and superconductivity,” Dalton Trans., vol. 39, pp. 1901–1915, Dec. 2009. 36, 64, 65, 67

[67] R. Demchyna, S. Leoni, H. Rosner, and U. Schwarz, “High-pressure crystal chemistry of binary intermetallic compounds,” Z. Kristallogr., vol. 221, pp. 420–434, 2006. 36, 65 102 BIBLIOGRAPHY

[68] J. Evers and A. Weiss, “Electrical properties of alkaline earth disilicides and diger- manides,” Mater. Res. Bull., vol. 9, p. 549, 1974. 36, 38, 40

[69] K. Momma and F. Izumi, “VESTA-3 for three-dimensional visualization of crystal, volumetric and morphology data,” Journal of Applied Crystallography, vol. 44, pp. 1272–1276, Dec 2011. 37, 39, 41

[70] M. Imai and T. Kikegawa, “Phase transitions of Alkaline-Earth-Metal Disilicides

MAESi2 (MAE: Ca, Sr, and Ba at high pressures and high temperatures,” Chem. Mater., vol. 15, pp. 2543–2551, 2003. 37, 38, 65, 67

[71] J. Evers, “Transformation of three-connected silicon in BaSi2,” J. Solid State Chem., vol. 32, pp. 77–86, March 1980. 38, 39, 42, 44, 66

[72] P. Bordet, M. Affronte, S. Sanfilippo, M. Núñez Regueiro, O. Laborde, G. L. Olcese, A. Palenzona, S. LeFloch, D. Levy, and M. Hanfland, “Structural phase transitions

in CaSi2 under high pressure,” Phys. Rev. B, vol. 62, pp. 11392–11397, Nov 2000. 38, 56, 61, 63

[73] S. Sanfilippo, H. Elsinger, M. Núñez Regueiro, O. Laborde, S. LeFloch, M. Affronte,

G. L. Olcese, and A. Palenzona, “Superconducting high pressure CaSi2 phase with Tc up to 14 K,” Phys. Rev. B, vol. 61, pp. R3800–R3803, Feb 2000. 38, 42, 49, 61, 62, 63, 73, 74

[74] M. Aﬀronte, O. Laborde, G. Olcese, and A. Palenzona, “Low temperature properties of calcium disilicides,” J. Alloys Compd., vol. 274, pp. 68–73, 1998. 39, 40

[75] M. Imai, T. Naka, T. Furubayashi, and K. Yagasaki, “Electrical properties of poly-

crystalline SrSi2,” Applied Physics Letters, vol. 86, no. 032102, 2005. 39, 40

[76] T. H. M. Imai, “Elec. Resistivity on phases of BaSi2 synthesized under high pressure and high temperature,” J. Alloys Compd., vol. 224, pp. 111–116, June 1995. 40

[77] M. Imai, T. Hirano, T. Kikegawa, and O. Shimomura, “In situ measurements

of the orthorhombic-to-trigonal transition in BaSi2 under high-pressure and high- temperature conditions,” Phys. Rev. B, vol. 55, pp. 132–135, Jan 1997. 40

[78] T. Nakamura, T. Suemasu, and K. Tanaka, “Investigation of the energy band struc-

ture of orthorhombic BaSi2 by optical and electrical measurements and theoretical calculations,” Mater. Res. Bull., vol. 81, pp. 1031–1034, 2002. BIBLIOGRAPHY 103

[79] D. B. Migas, V. L. Shaposhnikov, and V. E. Borisenko, “Isostructural BaSi2, BaGe2

and SrGe2: electronic and optical properties,” Phys. stat. sol. (b), vol. 244, pp. 2611– 2618, 2007. 40

[80] T. Suemasu, K. Morita, and M. Kobayashi, “Molecular beam epitaxy of band gap tun-

able ternary semiconducting silicides Ba1xSrxSi2 for photovoltaic application,” Jour- nal of Crystal Growth, vol. 301302, pp. 680–683, april 2007. 40

[81] Y. Imai and A. Watanabe, “Consideration of the band-gap tunability of BaSi2 by alloying with ca or sr based on the electronic structure calculations,” Thin Solid Films, vol. 515, pp. 8219–8225, Feb 2007. 40

[82] K. H. M. Imai and T. Hirano, “Superconductivity of trigonal BaSi2,” Physica C, vol. 245C, pp. 12–14, 1995. 40, 49, 58

[83] J. G. Bednorz and K. A. M¨uller, “Possible high Tc superconductivity in the BaLaCuO system,” Zeitschrift f¨ur Physik B Condensed Matter, vol. 64, no. 189-193, 1986. 41

[84] Z. Z. Sheng and A. M. Hermann, “Superconductivity in the rare-earth-free TlBaCuO system above liquid-nitrogen temperature,” Nature, vol. 332, Feb 1988.

[85] O. Chmaissem, J. D. Jorgensen, S. Short, A. Knizhnik, Y. Eckstein, and H. Shaked,

“Scaling of transition temperature and CuO2 plane buckling in a high-temperature superconductor,” Nature (London), vol. 397, no. 45, 1999. 42

[86] W. E. Pickett, J. M. An, H. Rosner, and S. Y. Savrasov, “Role of two dimensionality

in MgB2,” Physica C, vol. 387, no. 117121, 2003. 41

[87] N. Emery, C. Hérold, M. d’Astuto, V. Garcia, C. Bellin, J. F. Marêché, P. La-

grange, and G. Loupias, “Superconductivity of bulk CaC6,” Phys. Rev. Lett., vol. 95, p. 087003, Aug 2005. 41

[88] M. Calandra and F. Mauri, “Theoretical explanation of superconductivity in C6Ca,” Phys. Rev. Lett., vol. 95, p. 237002, Nov 2005.

[89] L. Boeri, G. B. Bachelet, M. Giantomassi, and O. K. Andersen, “Electron-phonon interaction in graphite intercalation compounds,” Phys. Rev. B, vol. 76, p. 064510, Aug 2007. 41

[90] M. K. Wu, J. R. Ashburn, C. J. Torng, P. H. Hor, R. L. Meng, L. Gao, Z. J. Huang, Y. Q. Wang, and C. W. Chu, “Superconductivity at 93 k in a new mixed-phase YBaCuO compound system at ambient pressure,” Phys. Rev. Lett., vol. 58, pp. 908– 910, Mar 1987. 42 104 BIBLIOGRAPHY

[91] M. Imai, T. Hirano, T. Kikegawa, and O. Shimomura, “Phase transitions of BaSi2 at high pressures and high temperatures,” Phys. Rev. B, vol. 58, pp. 11922–11926, Nov 1998. 42, 44, 66

[92] H. M. Rietveld, “A proﬁle reﬁnement method for nuclear and magnetic structures,” Journal of Applied Crystallography, vol. 2, pp. 65–71, Jun 1969. 43, 44

[93] J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized Gradient Approximation Made Simple,” Phys. Rev. Lett., vol. 77, pp. 3865–3868, Oct 1996. 42, 65, 79

[94] M. Fuchs and M. Scheﬄer, “Ab initio pseudopotentials for electronic structure calculations of poly-atomic systems using density-functional theory,” Comput. Phys. Com- mun., vol. 119, pp. 67–98, 1999. 43, 65

[95] P. E. Bl¨ochl, “Projector augmented-wave method,” Phys. Rev. B, vol. 50, pp. 17953– 17979, Dec 1994. 43, 65

[96] G. Kresse and J. Furthm¨uller, “Eﬃciency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set,” Comput. Mat. Sci., vol. 6, pp. 15–50, 1996. 43, 65

[97] S. Y. Savrasov and D. Y. Savrasov, “Electron-phonon interactions and related physical properties of metals from linear-response theory,” Phys. Rev. B, vol. 54, pp. 16487– 16501, Dec 1996. 43, 65

[98] H. J. Monkhorst and J. D. Pack, “Special points for Brillouin-zone integrations,” Phys. Rev. B, vol. 13, pp. 5188–5192, Jun 1976. 43, 65, 79, 80

[99] J. A. Flores-Livas, R. Debord, S. Botti, A. San Miguel, M. A. L. Marques, and

S. Pailh`es, “Enhancing the superconducting transition temperature of basi2 by structural tuning,” Phys. Rev. Lett., vol. 106, p. 087002, Feb 2011. 43, 44, 57, 66, 71, 74

[100] V. K. Zaitsev, M. I. Fedorov, E. A. Gurieva, I. S. Eremin, P. P. Konstantinov, A. Y.

Samunin, and M. V. Vedernikov, “Highly eﬀective Mg2Si1−xSnx thermoelectrics,” Phys. Rev. B, vol. 74, p. 045207, Jul 2006. 51

[101] R. Venkatasubramanian, E. Siivola, T. Colpitts, and B. O’Quinn, “Thin-ﬁlm thermoelectric devices with high room-temperature ﬁgures of merit,” Nature, vol. 413, no. 0028-0836, 2001. 51 BIBLIOGRAPHY 105

[102] C. B. Vining, “An inconvenient truth about thermoelectrics,” Nature, vol. 8, no. 1476- 1122, 2009. 51

[103] K. Hashimoto, K. Kurosaki, Y. Imamura, H. Muta, and S. Yamanaka, “Thermoelec-

tric properties of BaSi2, SrSi2, and LaSi,” J. Appl. Phys, vol. 102, p. 063703, 2007. 51

[104] G. Madsen and D. Singh, “Boltztrap. a code for calculating bandstructure dependent quantities,” Computer Physics Communications, vol. 175, 2006. 51

[105] H. Shi, L. Zhang, L. Chen, and Y. Feng, “The Raman spectrum and lattice parameters

of MgB2 as a function of temperature,” J. Phys.: Condens. Matter, vol. 16, no. 6541, 2004. 54

[106] W. L. McMillan, “Transition Temperature of Strong-Coupled Superconductors,” Phys. Rev., vol. 167, pp. 331–344, Mar 1968. 60, 65, 71, 75, 83

[107] P. B. Allen and R. C. Dynes, “Transition temperature of strong-coupled superconductors reanalyzed,” Phys. Rev. B, vol. 12, pp. 905–922, Aug 1975. 60, 65, 71, 75, 83

[108] A. Y. Liu, I. I. Mazin, and J. Kortus, “Beyond Eliashberg Superconductivity in

MgB2: Anharmonicity, Two-Phonon Scattering, and Multiple Gaps,” Phys. Rev. Lett., vol. 87, p. 087005, Aug 2001. 61, 75

[109] G. Satta, G. Profeta, F. Bernardini, A. Continenza, and S. Massidda, “Electronic

and structural properties of superconducting MgB2, CaSi2, and related compounds,” Phys. Rev. B, vol. 64, p. 104507, Aug 2001. 64

[110] C. Bersier, A. Floris, A. Sanna, G. Profeta, A. Continenza, E. K. U. Gross, and S. Massidda, “Electronic, vibrational, and superconducting properties of CaBeSi: First-principles calculations,” Phys. Rev. B, vol. 79, Mar 2009. 64

[111] S. Fahy and D. R. Hamann, “Electronic and structural properties of CaSi2,” Phys. Rev. B, vol. 41, pp. 7587–7592, Apr 1990. 68

[112] J. Kusakabe, T. Ogitsu, and S. Tsuneyuki, “First-principles study of calcium disilicide under high pressure,” J. Phys.: Condens. Matter, vol. 10, p. 11561, 1998. 73

[113] C. Loison and H. Rosner, “CaSi2, the antibonding sister of MgB2,” Solid State Com- mun., vol. 150, p. 2076, 2010. 73, 74 106 BIBLIOGRAPHY

[114] H. N. A. Nakanishi, T. Ishikawa and K. Kusakabe, “Origin of enhanced supercon-

ducting transition temperature through structural transformation in CaSi2,” J. Phys. Conf. Ser., vol. 121, p. 052010, 2008. 73, 74

[115] Y. Noat, T. Cren, P. Toulemonde, A. San Miguel, F. m. c. Debontridder, V. Dubost, and D. Roditchev, “Two energy gaps in the tunneling-conductance spectra of the

superconducting clathrate Ba8Si46,” Phys. Rev. B, vol. 81, p. 104522, Mar 2010. 75

[116] P. Toulemonde, A. San Miguel, A. Merlen, R. Viennois, S. LeFloch, X. Adessi, Ch. Blase, and J. Tholence, “High pressure synthesis and properties of intercalated silicon clathrates,” J. Phys. Chem. Solids, vol. 67, p. 1117, 2006. 75

[117] N. W. Ashcroft, “Metallic hydrogen: A high-temperature superconductor?,” Phys. Rev. Lett., vol. 21, p. 1748, Dec. 1968. 77

[118] B. Vasvari, A. O. E. Animalu, and V. Heine, “Electronic Structure of Ca, Sr, and Ba under Pressure,” Phys. Rev., vol. 154, pp. 535–539, Feb 1967. 77

[119] C. Probst and J. Wittig, “Superconductivity of bcc barium under pressure,” Phys. Rev. Lett., vol. 39, pp. 1161–1163, Oct 1977.

[120] K. Shimizu, H. Ishikawa, D. Takao, T. Yagi, and K. Amaya, “Superconductivity in compressed lithium at 20 K,” Nature, vol. 419, pp. 597–599, 2002.

[121] G. Profeta, C. Franchini, N. N. Lathiotakis, A. Floris, A. Sanna, M. A. L. Marques, M. L¨uders, S. Massidda, E. K. U. Gross, and A. Continenza, “Superconductivity in Lithium, Potassium, and Aluminum under Extreme Pressure: A First-Principles Study,” Phys. Rev. Lett., vol. 96, p. 047003, Feb 2006. 77

[122] M. B. Maple, J. Wittig, and K. S. Kim, “Pressure-Induced Magnetic-Nonmagnetic Transition of Ce impurities in La,” Phys. Rev. Lett., vol. 23, pp. 1375–1377, Dec 1969. 77

[123] J. Wittig, “Pressure-Induced Superconductivity in Cesium and Yttrium,” Phys. Rev. Lett., vol. 24, pp. 812–815, Apr 1970. 77

[124] M. Ishizuka, M. Iketani, and S. Endo, “Pressure eﬀect on superconductivity of vanadium at megabar pressures,” Phys. Rev. B, vol. 61, pp. R3823–R3825, Feb 2000. 77

[125] Y. Akahama, M. Kobayashi, and H. Kawamura, “Superconductivity and phase transition of Zirconium under high pressure up to 50 GPa,” J. Phys. Soc. Japan, vol. 59, pp. 3843–5, 1990. 77 BIBLIOGRAPHY 107

[126] K. J. Chang, M. M. Dacorogna, M. L. Cohen, J. M. Mignot, G. Chouteau, and G. Martinez, “Superconductivity in High-Pressure Metallic Phases of Si,” Phys. Rev. Lett., vol. 54, pp. 2375–2378, May 1985. 77

[127] M. Eremets, V. V. Struzhkin, H. K. Mao, and H. R. J., “Superconductivity in boron,” Science, vol. 293, pp. 272–274, 2001. 77

[128] V. V. Struzhkin, R. J. Hemley, H. K. Mao, and Y. A. Timofeev, “Superconductivity at 10–17 K in compressed sulphur,” Nature, vol. 390, pp. 382–384, 1997. 77

[129] I. Shirotani, J. Mikami, T. Adachi, Y. Katayama, K. Tsuji, H. Kawamura, O. Shimo- mura, and T. Nakajima, “Phase transitions and superconductivity of black phosphorus and phosphorus-arsenic alloys at low temperatures and high pressures,” Phys. Rev. B, vol. 50, pp. 16274–16278, Dec 1994. 77

[130] J. L. Sarrao, L. A. Morales, J. D. Thompson, B. L. Scott, G. R. Stewart, F. Wastin, J. Rebizant, P. Boulet, P. Boulet, and G. H. Lander, “Plutonium-based superconductivity with a transition temperature above 18K,” Nature, vol. 420, pp. 297–299, 2002. 77

[131] E. A. Ekimov, V. A. Sidorov, E. D. Bauer, N. N. Mel’nik, N. J. Curro, J. D. Thompson, and S. M. Stishov, “Superconductivity in diamond,” Nature, vol. 428, pp. 542–545, 2004. 77

[132] K. Shimizu, K. Suhara, M. kumo, M. I. Eremets, and K. Amaya, “Superconductivity in oxygen,” Nature, vol. 393, pp. 767–769, 1998. 77

[133] M. L¨uders, M. A. L. Marques, N. N. Lathiotakis, A. Floris, G. Profeta, L. Fast, A. Continenza, S. Massidda, and E. K. U. Gross, “Ab initio Theory of Superconduc- tivity. I. Density functional formalism and approximate functionals,” Phys. Rev. B, vol. 72, p. 024545, Jul 2005. 77

[134] M. A. L. Marques, M. L¨uders, N. N. Lathiotakis, G. Profeta, A. Floris, L. Fast, A. Continenza, E. K. U. Gross, and S. Massidda, “Ab initio Theory of superconductivity. II. Application to Elemental Metals,” Phys. Rev. B, vol. 72, p. 024546, Jul 2005. 77

[135] T. Barbee, A. Garc´a, and M. L. Cohen, “First-principles prediction of high- temperature superconductivity in metallic hydrogen,” Nature, vol. 340, pp. 369–371, 1989. 77 108 BIBLIOGRAPHY

[136] P. Cudazzo, G. Profeta, A. Sanna, A. Floris, A. Continenza, S. Massidda, and E. K. U. Gross, “Ab initio description of high-temperature superconductivity in dense molecular hydrogen,” Phys. Rev. Lett., vol. 100, p. 257001, June 2008. 77

[137] C. Narayana, H. Luo, J. Orloﬀ, and A. L. Ruoﬀ, “Solid Hydrogen at 342 GPa: no evidence for an Alkali Metal,” Nature, vol. 393, pp. 46–49, May 1998. 77

[138] P. Loubeyre, F. Occelli, and R. LeToullec, “Optical studies of solid hydrogen to 320 GPa and evidence for black hydrogen,” Nature, vol. 416, no. 6681, pp. 13–17, 2002. 77

[139] N. W. Ashcroft, “Hydrogen dominant metallic alloys: High temperature superconductors?,” Phys. Rev. Lett., vol. 92, p. 187002, May 2004. 77

[140] C. J. Pickard and R. J. Needs, “High-Pressure phases of silane,” Phys. Rev. Lett., vol. 97, p. 045504, July 2006. 77, 85, 86

[141] J. Feng, W. Grochala, T. Jaro, R. Hoﬀmann, A. Bergara, and N. W. Ashcroft, “Struc-

tures and potential superconductivity in SiH4 at high pressure: En route to Metallic hydrogen,” Phys. Rev. Lett., vol. 96, p. 017006, Jan. 2006.

[142] Y. Yao, J. S. Tse, Y. Ma, and K. Tanaka, “Superconductivity in high-pressure SiH4,” Europhysics Letters (EPL), vol. 78, p. 37003, May 2007.

[143] X. Chen, J. Wang, V. V. Struzhkin, H.-k. Mao, R. J. Hemley, and H. Lin, “Super- conducting behavior in compressed solid SiH 4 with a layered structure,” Phys. Rev. Lett., vol. 101, no. 7, p. 077002, 2008.

[144] D. Y. Kim, R. H. Scheicher, S. Lebgue, J. Prasongkit, B. Arnaud, M. Alouani, and R. Ahuja, “Crystal structure of the pressure-induced metallic phase of SiH4 from ab initio theory,” Proceedings of the National Academy of Sciences, vol. 105, pp. 16454 –16459, Oct. 2008.

[145] M. Martinez-Canales, A. R. Oganov, Y. Ma, Y. Yan, A. O. Lyakhov, and A. Bergara, “Novel Structures and Superconductivity of Silane under Pressure,” Phy. Rev. Lett., vol. 102, p. 087005, Feb. 2009. 77

[146] G. Gao, A. R. Oganov, A. Bergara, M. Martinez-Canales, T. Cui, T. Iitaka, Y. Ma, and G. Zou, “Superconducting high pressure phase of germane,” Phy. Rev. Lett., vol. 101, no. 10, p. 107002, 2008. 77 BIBLIOGRAPHY 109

[147] J. S. Tse, Y. Yao, and K. Tanaka, “Novel Superconductivity in Metallic SnH4 under High Pressure,” Phy. Rev. Lett., vol. 98, p. 117004, Mar. 2007. 77

[148] G. Gao, A. R. Oganov, P. Li, Z. Li, H. Wang, T. Cui, Y. Ma, A. Bergara, A. O. Lyakhov, T. Iitaka, and G. Zou, “High-pressure crystal structures and superconduc-

tivity of Stannane (SnH4),” Proceedings of the National Academy of Sciences, vol. 107, pp. 1317–1320, Jan. 2010. 77

[149] D. Y. Kim, R. H. Scheicher, C. J. Pickard, R. J. Needs, and R. Ahuja, “Predicted formation of superconducting platinum-hydride crystals under pressure in the presence of molecular hydrogen,” Phys. Rev. Lett., vol. 107, p. 117002, Sep 2011. 77, 83, 84

[150] M. I. Eremets, I. A. Trojan, S. A. Medvedev, J. S. Tse, and Y. Yao, “Superconductivity in hydrogen dominant materials: Silane,” Science, vol. 319, pp. 1506–1509, Mar. 2008. 78, 83, 84

[151] X. Chen, V. V. Struzhkin, Y. Song, A. F. Goncharov, M. Ahart, Z. Liu, H.-k. Mao, and R. J. Hemley, “Pressure-induced metallization of silane,” Proceedings of the National Academy of Sciences, vol. 105, pp. 20–23, Jan. 2008.

[152] O. Degtyareva, J. E. Proctor, C. L. Guillaume, E. Gregoryanz, and M. Hanﬂand, “For- mation of transition metal hydrides at high pressures,” Solid State Communications, vol. 149, pp. 1583–1586, Oct. 2009.

[153] M. Hanﬂand, J. E. Proctor, C. L. Guillaume, O. Degtyareva, and E. Gregoryanz, “High-Pressure Synthesis, Amorphization, and Decomposition of Silane,” Phy. Rev. Lett., vol. 106, p. 095503, Mar. 2011. 78

[154] C. J. Pickard and R. J. Needs, “Structure of phase III of solid hydrogen,” Nat Phys, vol. 3, pp. 473–476, July 2007. 78

[155] Olijnyk, H. and Sikka, S. K. and Holzapfel, W. B., “Structural phase transitions in Si and Ge under pressures up to 50 GPa,” Phys. Lett. A, vol. 103, pp. 137–140, June 1984. 78

[156] S. J. Duclos, Y. K. Vohra, and A. L. Ruoﬀ, “Hcp to CC transition in silicon at 78 GPa and studies to 100 GPa,” Phys. Rev. Lett., vol. 58, p. 775, Feb. 1987. 78

[157] X. Jin, X. Meng, Z. He, Y. Ma, B. Liu, T. Cui, G. Zou, and H.-k. Mao, “Super- conducting high-pressure phases of disilane,” Proceedings of the National Academy of Sciences, 2010. 78, 81, 83 110 BIBLIOGRAPHY

[158] M. Amsler, Crystal Structure Prediction Based on Density Functional Theory. Ph.D. Thesis, Philosophich-Naturwissenschaftlichen Fakult¨at der Universit¨at Basel, Basel, June 2012. 78, 86

[159] S. Goedecker, “Minima hopping: An eﬃcient search method for the global minimum of the potential energy surface of complex molecular systems,” The Journal of Chemical Physics, vol. 120, no. 21, p. 9911, 2004. 78, 85, 86

[160] M. Amsler and S. Goedecker, “Crystal structure prediction using the minima hopping method,” The Journal of Chemical Physics, vol. 133, no. 22, p. 224104, 2010. 78, 85, 86

[161] E. Bitzek, P. Koskinen, F. G¨ahler, M. Moseler, and P. Gumbsch, “Structural relaxation made simple,” Phys. Rev. Lett., vol. 97, p. 170201, Oct 2006. 79

[162] T. J. Lenosky, J. D. Kress, I. Kwon, A. F. Voter, B. Edwards, D. F. Richards, S. Yang, and J. B. Adams, “Highly optimized tight-binding model of silicon,” Phys. Rev. B, vol. 55, p. 1528, Jan. 1997. 79

[163] We used Pov-Ray to build the ﬁgures, “Persistence of Vision Raytracer (Version 3.6),” 2004. http://www.povray.org/download/. 79, 87

[164] X. Gonze, B. Amadon, P. Anglade, J. Beuken, F. Bottin, P. Boulanger, F. Bruneval, D. Caliste, R. Caracas, M. Côté, T. Deutsch, L. Genovese, P. Ghosez, M. Giantomassi, S. Goedecker, D. Hamann, P. Hermet, F. Jollet, G. Jomard, S. Leroux, M. Mancini, S. Mazevet, M. Oliveira, G. Onida, Y. Pouillon, T. Rangel, G. Rignanese, D. San- galli, R. Shaltaf, M. Torrent, M. Verstraete, G. Zerah, and J. Zwanziger, “ABINIT: first-principles approach to material and nanosystem properties,” Computer Physics Communications, vol. 180, pp. 2582–2615, Dec. 2009. 79, 91

[165] C. Hartwigsen, S. Goedecker, and J. Hutter, “Relativistic separable dual-space Gaus- sian pseudopotentials from H to Rn,” Phys. Rev. B, vol. 58, pp. 3641–3662, Aug 1998. 79

[166] Anton and Kokalj, “Computer graphics and graphical user interfaces as tools in simulations of matter at the atomic scale,” Computational Materials Science, vol. 28, no. 2, pp. 155–168, 2003. XCRYSDEN was used to plot the Fermi surfaces. 82

[167] T. F. Smith and C. W. Chu, “Will Pressure Destroy Superconductivity?,” Phys. Rev., vol. 159, pp. 353–358, Jul 1967. 83 BIBLIOGRAPHY 111

[168] X.-F. Zhou, A. R. Oganov, X. Dong, L. Zhang, Y. Tian, and H.-T. Wang, “Supercon- ducting high-pressure phase of platinum hydride from ﬁrst principles,” Phys. Rev. B, vol. 84, p. 054543, Aug 2011. 83, 84

[169] M. Amsler, J. A. Flores-Livas, L. Lehtovaara, F. Balima, S. A. Ghasemi, D. Machon, S. Pailh`es, A. Willand, D. Caliste, S. Botti, A. San Miguel, S. Goedecker, and M. A. L. Marques, “Crystal structure of cold compressed graphite,” Phys. Rev. Lett., vol. 108, p. 065501, Feb 2012. 84

[170] W. L. Mao, H. kwang Mao, P. J. Eng, T. P. Trainor, M. Newville, C. chang Kao, D. L. Heinz, J. Shu, Y. Meng, and R. J. Hemley, “Bonding changes in compressed superhard graphite,” Science, vol. 302, pp. 425–427, Oct. 2003. 85, 86, 88, 89

[171] A. Oganov and C. W. Glass, “Crystal structure prediction using ab initio evolutionary techniques: Principles and applications,” J. Chem. Phys., vol. 124, no. 24, p. 244704, 2006. 85, 86

[172] C. W. Glass, A. Oganov, and N. Hansen, “USPEX – evolutionary crystal structure prediction,” Comput. Phys. Commun., vol. 175, no. 11-12, pp. 713–720, 2006. 86

[173] R. T. Strong, C. J. Pickard, V. Milman, G. Thimm, and B. Winkler, “Systematic prediction of crystal structures: An application to sp3-hybridized carbon polymorphs,” Phys. Rev. B, vol. 70, p. 045101, Jul 2004. 86

[174] Y. Wang, J. Lv, L. Zhu, and Y. Ma, “Crystal structure prediction via particle-swarm optimization,” Phys. Rev. B, vol. 82, p. 094116, Sep 2010. 85, 86

[175] F. P. Bundy, “Hexagonal DiamondA new form of carbon,” J. Chem. Phys., vol. 46, no. 9, p. 3437, 1967. 86

[176] A. F. Goncharov, I. N. Makarenko, and S. M. Stishov, “Graphite at pressures up to 55GPa: optical properties and Raman scattering-amorphous carbon?,” Sov. Phys. JETP, vol. 69, p. 380, Aug. 1989. 88

[177] Y. X. Zhao and I. L. Spain, “X-ray diﬀraction data for graphite to 20 GPa,” Phys. Rev. B, vol. 40, p. 993, July 1989.

[178] W. Utsumi and T. Yagi, “Light-Transparent phase formed by room-temperature compression of graphite,” Science, vol. 252, pp. 1542–1544, June 1991.

[179] T. Yagi, W. Utsumi, M.-a. Yamakata, T. Kikegawa, and O. Shimomura, “High- pressure in situ x-ray-diﬀraction study of the phase transformation from graphite 112 BIBLIOGRAPHY

to hexagonal diamond at room temperature,” Phys. Rev. B, vol. 46, no. 10, p. 6031, 1992. 86

[180] E. D. Miller, D. C. Nesting, and J. V. Badding, “Quenchable transparent phase of carbon,” Chemistry of Materials, vol. 9, pp. 18–22, Jan. 1997. 86

[181] R. Khaliullin, H. Eshet, J. K¨uhne, T. D. Behler, and M. Parrinello, “Nucleation mechanism for the direct graphite-to-diamond phase transition,” Nature Materials, vol. 10, pp. 693–697, 2011. 86

[182] A. R. Oganov, Modern Methods of Crystal Structure Prediction. Wiley-VCH Verlag GmbH Berlin, Nov 2010. 86

[183] Z. Zhao, B. Xu, L.-M. Wang, X.-F. Zhou, J. He, Z. Liu, H. Wang, and Y. Tian, “Three Dimensional Carbon-Nanotube Polymers,” ACS NANO, vol. 5, no. 9, pp. 7226–7234, 2011. 86

[184] Q. Zhu, A. R. Oganov, M. A. Salvad´o, P. Pertierra, and A. O. Lyakhov, “Denser than diamond: Ab initio search for superdense carbon allotropes,” Phys. Rev. B, vol. 83, p. 193410, May 2011. 86

[185] A. O. Lyakhov and A. R. Oganov, “Evolutionary search for superhard materials:

Methodology and applications to forms of carbon and TiO2,” Phys. Rev. B, vol. 84, p. 092103, Sep 2011. 86

[186] R. Baughmana, A. Liub, C. Cuia, and P. Schieldsc, “A carbon phase that graphitizes at room temperature,” Synth. Met., vol. 86, p. 2371, 1997. 86

[187] F. J. Ribeiro, P. Tangney, S. G. Louie, and M. L. Cohen, “Structural and electronic properties of carbon in hybrid diamond-graphite structures,” Phys. Rev. B, vol. 72, p. 214109, Dec. 2005. 86

[188] Q. Li, Y. Ma, A. R. Oganov, H. Wang, H. Wang, Y. Xu, T. Cui, H. Mao, and G. Zou, “Superhard monoclinic polymorph of carbon,” Phys. Rev. Lett., vol. 102, p. 175506, Apr. 2009. 86, 91

[189] K. Umemoto, R. M. Wentzcovitch, S. Saito, and T. Miyake, “Body-centered tetragonal C4: A viable sp3 carbon allotrope,” Phys. Rev. Lett., vol. 104, p. 125504, Mar. 2010. 86, 91, 92

[190] X.-F. Zhou, G.-R. Qian, X. Dong, L. Zhang, Y. Tian, and H.-T. Wang, “Ab initio study of the formation of transparent carbon under pressure,” Phys. Rev. B, vol. 82, p. 134126, Oct 2010. 86, 91, 92 BIBLIOGRAPHY 113

[191] J. Wang, C. Chen, and Y. Kawazoe, “Low-Temperature Phase Transformation from Graphite to sp3 Orthorhombic Carbon,” Phys. Rev. Lett., vol. 106, p. 075501, Feb. 2011. 86, 91

[192] M. Amsler, J. A. Flores-Livas, L. Lehtovaara, F. Balima, S. A. Ghasemi, D. Machon, S. Pailh`es, A. Willand, D. Caliste, S. Botti, A. San Miguel, S. Goedecker, and M. A. L. Marques, “Crystal structure of cold compressed graphite,” Phys. Rev. Lett., vol. 108, p. 065501, Feb 2012. 86, 91

[193] M. Amsler, J. A. Flores-Livas, T. D. Huan, S. Botti, M. A. L. Marques, and

S. Goedecker, “Novel Structural Motifs in Low Energy Phases of LiAlH4,” Phys. Rev. Lett., vol. 108, p. 205505, May 2012. 86

[194] Z. Zhao, B. Xu, X.-F. Zhou, L.-M. Wang, B. Wen, J. He, Z. Liu, H.-T. Wang, and

Y. Tian, “Novel Superhard Carbon: C-Centered Orthorhombic C8,” Phys. Rev. Lett., vol. 107, p. 215502, Nov 2011. 87

[195] D. Selli, I. A. Baburin, R. Martoˇn´ak, and S. Leoni, “Superhard sp3 carbon allotropes with odd and even ring topologies,” Phys. Rev. B, vol. 84, p. 161411, Oct 2011. 87

[196] M. Hanﬂand, H. Beister, and K. Syassen, “Graphite under pressure: Equation of state and ﬁrst-order Raman modes,” Phys. Rev. B, vol. 39, p. 12598, June 1989. 88, 89, 90

[197] F. Gao, J. He, E. Wu, S. Liu, D. Yu, D. Li, S. Zhang, and Y. Tian, “Hardness of Covalent Crystals,” Phys. Rev. Lett., vol. 91, p. 015502, July 2003. 88

[198] J. A. Flores-Livas, L. Lehtovaara, M. Amsler, S. Goedecker, S. Pailh`es, S. Botti, A. San Miguel, and M. A. L. Marques, “Raman activity of sp3 carbon allotropes under pressure: A density functional theory study,” Phys. Rev. B, vol. 85, p. 155428, Apr 2012. 91

[199] F. Occelli, P. Loubeyre, and R. LeToullec, “Properties of diamond under hydrostatic pressures up to 140 GPa,” Nature Materials, vol. 2, pp. 151–154, Mar. 2003. 91, 92

[200] Y. Akahama and H. Kawamura, “High-pressure Raman spectroscopy of diamond anvils to 250GPa: method for pressure determination in the multimegabar pressure range,” J. Appl. Phys., vol. 96, no. 7, p. 3748, 2004. 91

[201] M. Hanﬂand, K. Syassen, S. Fahy, S. G. Louie, and M. L. Cohen, “Pressure dependence of the ﬁrst-order Raman mode in diamond,” Phys. Rev. B, vol. 31, pp. 6896–6899, May 1985. 114 BIBLIOGRAPHY

[202] H. Boppart, J. van Straaten, and I. F. Silvera, “Raman spectra of diamond at high pressures,” Phys. Rev. B, vol. 32, pp. 1423–1425, Jul 1985.

[203] M. Eremets, “Megabar high-pressure cells for Raman measurements,” J. Raman Spec- trosc., vol. 34, pp. 515–518, 2003.

[204] M. Popov, “Pressure measurements from Raman spectra of stressed diamond anvils,” J. Appl. Phys., vol. 95, no. 7, p. 5509, 2004.

[205] N. Dubrovinskaia, L. Dubrovinsky, and M. Caracas, R. Hanﬂand, “Diamond as a high pressure gauge up to 2.7 Mbar,” Appl. Phys. Lett., vol. 97, p. 5509, 2010. 92

[206] B. R. Wu and J.-a. Xu, “Zone-center modes of cubic and hexagonal diamond under high pressure: A ﬁrst-principles study,” Phys. Rev. B, vol. 60, pp. 2964–2967, Aug 1999.

[207] K. Kunc, I. Loa, and K. Syassen, “Equation of state and phonon frequency calculations of diamond at high pressures,” Phys. Rev. B, vol. 68, p. 094107, Sep 2003. 92

[208] S. S. Prawer, K. Nugent, D. Jamieson, L. Bursill, and J. Peng, “The Raman spectrum of nanocrystalline diamond,” J. Chem. Phys. Lett., vol. 332, pp. 93–97, 2000. 93

[209] A. C. Ferrari and J. Robertson, “Origin of the 1150 cm−1 Raman mode in nanocrystalline diamond,” Phys. Rev. B, vol. 63, p. 121405, Mar 2001. 93

[210] T. T. Irifune, A. Kurio, S. Sakamoto, T. Inoue, and H. Sumiya, “Materials: Ultrahard polycrystalline diamond from graphite,” Nature (London), vol. 421, p. 599, 2003. 93

[211] H. Niu, X.-Q. Chen, S. Wang, D. Li, W. L. Mao, and Y. Li, “Families of superhard crystalline carbon allotropes constructed via cold compression of graphite and nanotubes,” Phys. Rev. Lett., vol. 108, p. 135501, Mar 2012. 94

[212] Z. Li, F. Gao, and Z. Xu, “Strength, hardness, and lattice vibrations of z-carbon and w-carbon: First-principles calculations,” Phys. Rev. B, vol. 85, p. 144115, Apr 2012.

[213] Q. Zhu, Q. Zeng, and A. R. Oganov, “Systematic search for low-enthalpy sp3 carbon allotropes using evolutionary metadynamics,” Phys. Rev. B, vol. 85, p. 201407, May 2012. 94 115

NOMENCLATURE AND ABBREVIATIONS

kB Boltzmann constant AEM Alkaline earth metal m Mass BCS Bardeen, Cooper, Schrieﬀer P Pressure GL Ginzburg-Landau

EF Fermi energy DFT Density-functional theory

Eg Band gap energy DFPT Density-functional perturbation theory Chemical potential LDA local-density approximation

νk Fermi velocity GGA Gneralized-gradient density approximation n(r) Electron density B Magnetic ﬁled

Ωa Atomic volume Hc Magnetic Critical ﬁeld D Dynamical matrix T Superconducting transition temperature { αβ} c ωi Phonon energy eDOS Electronic density of state

ΘD Debye temperature PDOS Phonon density of state

V ′ BCS attractive potential − k,k BCS Eg BCS energy gap λ Electron-phonon strength

Ωlog Phonon average ∗ Screen Coulomb interaction κ GL-parameter ξ Coherence length χ Magnetic susceptibility ZT Figure of merit

ρe Electrical resitivity

τs Relaxation time 116 117

ACKNOWLEDGMENTS

First of all, I would like to acknowledge CONACyT-México (The National Council of Science and Technology in México) for providing me the scholarship of excellence that I received during the past years. Also I want to thank the economical support received from the ANR(08-CEXC8-008-01) French project of Dr. Marques. The thorough of our calculations were ran in the local cluster and also in the national computer centres of France and Switzerland. I thank all members of the jury, Prof. Isabelle Daniel, Prof. Xavier Gonze, Prof. Aldo H. Romero, Prof. Stefan Goedecker, Dr. Yann Gallais, Dr. Romain Viennois, Dr. Stéphane Pailhès and Dr. Miguel A. L. Marques for attending the defence and providing valuable criticisms to my work.

Foremost, I thank my advisor Dr. Miguel A. L. Marques for his guidance, support and encouragement throughout my Ph.D. studies. He has always been willing to share his knowledge and discuss ideas and new projects as well as sharing his experience of how to communicate Science, the strong criticism received were for me a constant challenging to move and improve at each step. I acknowledge him for his patience and willingness to help me in any regard. I would also like to thank Dr. Stéphane Pailhès for his effort and for giving me access to perform complex experiments on large facilities. I am indebted with professor Alfonso San Miguel, he opened the door for starting my Ph.D. research. When I arrived in Lyon I was warmly welcomed by the high pressure group. Also I appreciate all fundamental knowledge transmitted concerning high pressure physics. This thesis could not have been finished without the help and support of many people who are gratefully acknowledged: Dr. Silvana Botti, Prof. Stefan Goedecker, Dr. Regis De- bord, Dr. Franck Legrand, Dr. Vittoria Pischedda, Dr. Silvie Le Floch, Dr. Denis Machon, Dr. Valentina Giordano and postdocs of the theoretical group, Dr. David Kammerlander and Dr. Lauri Lehtovaara. I would like to thank personal from the secretary and administration office for helping me in many laborious tasks, Mrs. Cristelle Macheboeuf and Mrs. Delphine Kervella. Moreover, I am deeply grateful to Dr. Silvana Botti, Dr. David Kammerlander, Dr. Lauri Lehtovaara, Dr. Stéphane Pailhès and Dr. Miguel A. L. Marques for their thorough reading of the manuscript and for their comments and suggestions. Thanks for all the help and good company Ive had in general in the LPMCN (in both, theoretical and high pressure group). It has been such a valuable experience to work with excellent physicists with many different backgrounds. In particular, I want to thank Maximilian Amsler who has contributed so much to the projects in this thesis: Thank you, Max, for always asking relevant questions and share all you know about science. The good atmosphere of collaboration between us has been wonderful. Furthermore, I also want to thank Mirjam and Max for hosting my visits in Basel, I really enjoyed my stays there. During the past years, I interacted a lot with Max, Felix, Dimitri (Ph.D. students at the time), Lauri and David; all of them with different character, and personality, but sharing motivation to work and huge-flourished capacities, sincerely I learned tons from you guys. Finally, my days in Lyon have been fantastic, playing and doing: petanque, darts, video games, pallet, paint-ball, snowboarding, football, rock-climbing and hanging out with: (alphabetically ordered) Abraao, Acrisio, Arnaud D., Arnaud (Maquina), Camille S., Claire, Clément (Alejandro oúBin-bin), David C. David K., Delphine, Dimitri, Dragos, Felix, Guilherme, Jean Christhope, Julien, Lauri, Lucas, Marion, Massimo, Maxime, Romain, Romain (Pastis), Sam, Simon, Thomas, Yoan. With all these guys I did unforgettable and crazy trips all over the pubs and bars in Lyon and the surroundings. I would like to mention all interesting people that marked a part of my life and I meet in Lyon and along my thesis, Julie, Maryline, Elise, Laura, Céline, Lucy, Camille P., Cécile, Eilin, Deniz, Anna, Raphaelle, Marie, Estelle, Ryan, Florian and Stephan. Pequetishina, thank you for patiently be beside me the last couple of years, sharing the fun times and challenges alike. My work is dedicated to my Family. Thank you for putting up with me living so far away, for your love and support as I explore the world.

Todo mi trabajo esta dedicado a mi familia: A mi mama, a mi papa, a mis hermanos, a mis abuelos, a mis t´os y a mis primos. Por soportar todas las ausencias, cuando estoy lejos de casa y sin saber nada mi, por soportar este perro callejero que busca solo ser feliz.

“Hay hombres que luchan un da y son buenos, hay quiénes luchan un año y son mejores, hay quiénes luchan muchos años y son muy buenos, pero hay quiénes luchan toda la vida, esos son los imprescindibles”

Eugen Berthold Friedrich Brecht. CURRICULUM VITÆ

JOSE´ A. FLORES LIVAS

December 27th, 1985. San Pedro de las Colonias, Coahuila, M´exico.

Schools 1991 - 1997 Primary school, Centenario (San Pedro, M´exico). 1997 - 2000 High school, Secundaria 11 (Monterrey, M´exico). 2000 - 2002 College school, UANL-15 Madero (Monterrey).

Undergraduate Studies 2002 - 2007 Studies of Physics at the Universidad Autonoma de Nuevo Le´on. 2007 - 2008 Thesis at FCFM, “Structure determination of AuPd icosahedral nanoparticles: experimental and computational studies.”

Graduate Studies 2009 - Present Ph.D. student at the Universit´ede Lyon 1, France.

Based on British English Typeset using LATEX.