Density Functional Theory in Computational Materials Science164957/FULLTEXT01.pdf · it turns out that the solution of these equations are impossible to ﬁnd. In the last four decades,

Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 1001

Density Functional Theory inComputational Materials Science

BY

JORGE MARIO OSORIO GUILLÉN

ACTA UNIVERSITATIS UPSALIENSISUPPSALA 2004

Dissertation at Uppsala University to be publicly examined in Polhemssalen, Angstrom Lab-oratory, Friday, September 24, 2004 at 10:15 for the Degree of Doctor of Philosophy. Theexamination will be conducted in English

AbstractOsorio Guillen, J. 2004. Density Functional Theory in Computational Materials Science. ActaUniversitatis Upsaliensis. Comprehensive Summaries of Uppsala Dissertations from the Facultyof Science and Technology 1001. viii + 59 pp. Uppsala. ISBN 91-554-6016-X

The present thesis deals with the application of first-principles self-consistent total-energycalculations within the density functional theory on different topics in materials science.

Crystallographic phase-transitions under high-pressure have been study for TiO2, FeI2,Fe3O4, Ti,the heavy alkali metals Cs and Rb, and C3N4. A new high-pressure polymorphof TiO2 has been discovered. Also, the crystal structures of Cs and Rb metals have beenstudied under high compressions. Our results confirm the recent high-pressure experimentalobservations of new complex crystal structures for the Cs-III and Rb-III phases. Thus, it is nowcertain that the famous isostructural phase transition in Cs is rather a new crystallographic phasetransition.

The elastic properties of the new superconductor MgB2 and Al-doped MgB2 have beeninvestigated. The elastic constants as well as the bulk moduli in the a and c directions arepredicted. Our analysis suggests that the high anisotropy of the calculated elastic moduli is astrong indication that MgB2 should be a rather brittle material. Al doping decreases the elasticanisotropy of MgB2 in the a and c directions but it will not change the brittle behaviour of thematerial considerably.

The three most relevant battery properties, namely average voltage, energy density andspecific energy, as well as the electronic structure of the Li/LixMPO4 systems, where M iseither Fe, Mn, or Co have been calculated. The mixing between Fe and Mn in these materialsis also examined. Our calculated values for these properties are in good agreement with recentexperimental values. Further insight is gained from the electronic structure of these materials,through which conclusions about the physical properties of the various phases are made.

The electronic and magnetic properties of the dilute magnetic semiconductor Mn-dopedZnO has been calculated. We have found that for a Mn concentration of 5.6 %, the ferromagneticconfiguration is energetically stable in comparison to the antiferromagnetic one. This materialis a future candidate for spintronics applications.

Keywords: Density Functional Theory, High Pressure, Phase Transitions, Elastic Properties,Lithium Batteries, Dilute Magnetic Semiconductors

Jorge Mario Osorio Guillen, Department of Physics. Uppsala University. Box530, SE - 751 21Uppsala, Sweden

c© Jorge Mario Osorio Guillen 2004

ISBN 91-554-6016-XISSN 1104-232Xurn:nbn:se:uu:diva-3556 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-3556)

To Gloria

Cover illustrationCs-II → Cs-III crystallographic phasetransition. Cs-II fcc transforms intoa complex orthorhombic crystal struc-ture with 84 atoms in the unit cellwhich ruled out the classical isostruc-tural phase transformation of Cs underpressure.

List of papers

This thesis is based on the collection of papers given below. Each article willbe referred to by its Roman numeral.

I Experimental and theoretical identification of a new high-pressureTiO2 polymorphN. A. Dubrovinskaia, L. S. Dubrovinsky, R. Ahuja, V. B. Prokopenko,V. Dmitriev, H. P. Weber, J. M. Osorio-Guillen and B. JohanssonPhys. Rev. Lett. 87, 275501-1 (2001).

II Pressure-induced structural transformations in the Mott insulatorFeI2G. Kh. Rozenberg, M. P. Pasternak, W. M. Xu, L. S. Dubrovinsky, J.M. Osorio-Guillen, R. Ahuja, B. Johansson and T. Le BihanPhys. Rev. B 68, 064105 (2003).

III The structure of the metallic high-pressure Fe3O4 polymorph: ex-perimental and theoretical studyL. S. Dubrovinsky, N. A. Dubrovinskaia, C. McCammon, G. Kh.Rozenberg, R. Ahuja, J. M. Osorio-Guillen, V. Dmitriev, H-P Weber,T. Le Bihan and B. JohanssonJ. Phys.: Condens. Matter 15, 7697 (2003).

IV Titanium metal at high pressure: Synchrotron experiments andab-initio calculationsR. Ahuja, L. S. Dubrovinsky, N. A. Dubrovinskaia, J. M. Osorio-Guillen, M. Mattesini, B. Johansson and T. Le BihanPhys. Rev. B 69, 184102 (2004).

V Structural phase transitions in heavy alkali metal under pressureJ. M. Osorio-Guillen, R. Ahuja and B. JohanssonChemPhysChem (accepted).

iii

VI Pressure effects on the structure and vibrations of C3N4

M. Marques, J. M. Osorio-Guillen, R. Ahuja, M. Florez and J. M. Re-cioPhys. Rev. B (accepted).

VII Bonding and elastic properties of superconducting MgB2

J. M. Osorio-Guillen, S. I. Simak, Y. Wang, B. Johansson and R. AhujaSol. Stat. Comm. 123, 257 (2002).

VIII Elastic properties of Mg1−xAlxB2 from first-principles theoryP. Souvatzis, J. M. Osorio-Guillen, R. Ahuja, A. Grechnev and O.ErikssonJ. Phys.: Condens. Matter 16, 5241 (2004).

IX Calculation of surface stress for fcc transition metalsJ. Kollar, L. Vitos, J. M. Osorio-Guillen and R. AhujaPhys. Rev. B 68, 245417 (2003).

X A theoretical study of olivine LiMPO4 cathodesJ. M. Osorio-Guillen, B. Holm, R. Ahuja and B. JohanssonSol. Stat. Ionics 167, 221 (2004).

XI Electronic structure of MgxMo3S4 and related materials for green-rechargeable magnesium batteriesJ. M. Osorio-Guillen, B. Holm and R. AhujaIn manuscript.

XII Role of titanium in hydrogen desorption in crystalline sodiumalanateR .Ahuja, P. Jena, J. M. Osorio-Guillen and C. M. AraujoSubmitted to Angewandte Chemie.

XIII Ferromagnetism above room temperature in bulk and transparentthin films of Mn-doped ZnOP. Sharma, A. Gupta, K. V. Rao, F. J. Owens, R. Sharma, R. Ahuja, J.M. Osorio-Guillen, B. Johansson and G. A. GehringNature Materials 2, 673 (2003).

XIV Electronic and magnetic properties of Mn-doped ZnO from first-principles theoryJ. M. Osorio-Guillen and R. AhujaIn manuscript.

iv

XV The electronic structure of Mn doped ZnO film from X-ray spec-troscopy and DFT calculationsJ.-H. Guo, A. Gupta, P. Sharma, K. V. Rao, C. I. Dong, J. M. Osorio-Guillen, S. M. Butorin, M. Mattesini, C. L. Chang and R. AhujaSubmitted to Phys. Rev. Lett. .

XVI Optical properties of 4H-SiCR. Ahuja, A. Ferreira da Silva, C. Persson, J. M. Osorio-Guillen, I.Pepe, K. Jarrendahl, O. P. A. Lindquist, N. V. Edwards, Q. Wahab andB. JohanssonJ. Appl. Phys. 91, 2099 (2002).

XVII Electronic and optical properties of lead iodideR. Ahuja, H. Arwin, A. Ferreira da Silva, C. Persson, J. M. Osorio-Guillen, J. Souza de Almeida, C. Moyses Araujo, E. Veje, N. Veissid,C. Y. An, I. Pepe and B. JohanssonJ. Appl. Phys. 92, 7219 (2002).

XVIII Spectroscopy studies of 4H-SiCA. C. de Oliveira, J. A. Freitas Jr., W. J. Moore, A. Ferreira da Silva, I.Pepe, J. Souza de Almeida, J. M. Osorio-Guillen, R. Ahuja, C. Persson,K. Jarrendahl, O. P. A. Lindquist, N. V. Edwards and Q. WahabMaterials Research 6, 43 (2002).

XIX Electronic and optical properties of γ-Al2O3 from ab-initio theoryR. Ahuja, J. M. Osorio-Guillen, J. Souza de Almeida, B. Holm, W. Y.Ching and B. JohanssonJ. Phys.: Condens. Matter 16, 2891 (2004).

XX Electronic structure of a thermoelectric material: CsBi4Te6

R. Ahuja, J. Souza de Almeida, J.M. Osorio-Guillen and B. JohanssonSubmitted to Journal of Physics and Chemistry of Solids.

Reprints were made with permission from the publishers.

The following paper is not included in the thesis:

• Photoacoustic and transmission studies of SiC polytypesA. C. de Oliveira, J. A. Freitas Jr., W. J. Moore, A. Ferreira da Silva, I. Pepe,J. Souza de Almeida, G. C. B. Braga, J. M. Osorio-Guillen, C. Persson andR. AhujaMaterials Research 6, 47 (2002).

v

Comments on my contributionIn the papers where I am the first author I am responsible for the main part ofthe work. Concerning the other papers I have contributed in different ways,such as ideas, either the whole or part of the calculations and participation inthe analysis.

vi

Contents

List of papers iii

1 Introduction 1

2 Density Functional Theory 52.1 The Many-Body Problem . . . . . . . . . . . . . . . . . . . . 52.2 The Hohenberg-Kohn Theorem . . . . . . . . . . . . . . . . . 82.3 The Self-Consistent Kohn-Sham Equations . . . . . . . . . . 92.4 The Local Density Approximation . . . . . . . . . . . . . . . 112.5 The Generalised Gradient Approximation . . . . . . . . . . . 12

3 Electronic Structure Calculation 133.1 Bloch Electrons . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Full-Potential Linear-Muffin-Tin-Orbital Method . . . . . . . 143.3 Projector Augmented-Wave Method . . . . . . . . . . . . . . 15

4 Crystallographic Phase transitions Under High Pressure 194.1 The Equation Of State For Crystals . . . . . . . . . . . . . . . 204.2 Phase Transitions In Crystalline Matter . . . . . . . . . . . . . 21

5 The Elastic Properties Of Crystals 255.1 The Strain And Stress Tensors . . . . . . . . . . . . . . . . . 255.2 The Elastic Constants . . . . . . . . . . . . . . . . . . . . . . 26

5.2.1 The Elastic Constants For A Hexagonal Crystal . . . . 27

6 Lithium Insertion Materials For Rechargeable Batteries 316.1 Thermodynamics Of The Electrochemical Cell . . . . . . . . 326.2 Electronic Structure . . . . . . . . . . . . . . . . . . . . . . . 36

vii

7 Dilute Magnetic Semiconductors 397.1 Theory Of Ferromagnetism In DMS . . . . . . . . . . . . . . 407.2 Results For Mn-doped ZnO . . . . . . . . . . . . . . . . . . . 41

8 Optical Properties 458.1 Theoretical Justification Of DFT Calculations For The Optical

Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458.2 Calculation Of The Dielectric Function . . . . . . . . . . . . . 46

9 Sammanfattning Pa Svenska 51

Acknowledgements 53

viii

Chapter 1Introduction

During the last century quantum mechanics has been one of the important keyto understand the properties of matter and their evolution from macroscopicobjects such as stars to microscopic objects such as electrons, atomic nuclei,etc.. One of the most clear example of the knowledge and understandingreached is by its manifested growing technology applications which withoutquantum mechanics would not have been possible. The Schrodinger equationin the non-relativistic case and the Dirac equation in the relativistic case arethe fundamental equations which allow us, in principle, to know the structureof matter and its properties. When we want to use these fundamental equationson macroscopic systems, where the number of particles is of the order of 1023,it turns out that the solution of these equations are impossible to find. In thelast four decades, an alternative approach , the so-called Density FunctionalTheory (DFT), has became the standard calculation tool in solid state physics.The main variable in DFT is the density distribution, n(r), which only de-pends on three coordinates, rather than the many-particle wave function whichdepends on 3N coordinates, where N is the number of particles. In solid statephysics, DFT allows parameter free calculations of densities, spin densities,ground state energies, as well as related quantities such as lattice structuresand constants, bulk moduli, phases transitions as a function of pressure, elasticconstants, phonon dispersion relations, magnetic moments, etc.. In this thesis,by means of DFT, I have investigated some of the above quantities for differentmaterials covering various topics of materials science.

We have studied crystallographic phase-transitions of several materials suchas TiO2, the heavy alkali metals Cs and Rb, etc.. Outstanding properties ofsome of the titania (TiO2) polymorphs have not only make those phases ex-tremely useful in many applications, but they have also identified them asprototype materials for experimental and theoretical studies. TiO2 is of par-ticular interest because of its use in a wide variety of commercial applications

1

Chapter 1. Introduction

including pigment, catalysis, electronics, electrochemical and ceramic indus-tries. High-pressure transformations of TiO2 have attracted special attentionbecause this material is regarded as a low-pressure analog of SiO2, the mostabundant component of the Earth’s mantle.

Among phase transformations in solids, there are two special cases whichhave caught a lot of attention both experimentally and theoretically. Theseare the isostructural phase transitions in the cerium and cesium metals underpressure. In accordance with the lack of a broken crystal symmetry, thesetransformations have been referred to as electronic phase changes involving the4f -electrons in the case of Ce and a drastic change of the number of d-electronsin the case of Cs. However, most recently, high pressure experiments on Cshave suggested that the isostructural phase change in fact is a crystallographictransition. Since this will involve a drastic change in our previous generalunderstanding of this phase transition. It becomes highly interesting to performa new theoretical investigation of compressed Cs.

We have also investigated the mechanical properties of the recently dis-covered superconductor, MgB2. Our motivation was the lack of theoreticalas well as experimental information about mechanical properties of this su-perconductor. Knowledge of the mechanical properties is important becausethese properties are often the cause of limitation in practical applications ofsuperconducting materials. The first step for a theoretical consideration of me-chanical properties should be a detailed investigation of the elastic propertiesof the system. In particular, values of the elastic constants provide an impor-tant information about the degree of anisotropy which is known to correlatewith a tendency to either ductility or brittleness.

Also, the promising lithium insertion battery cathode LiMPO4, where M iseither Mn, Fe, or Co has been investigated. The typical properties of interest ina battery such as the open-circuit voltage (OCV), the energy density (EΩ), andthe specific energy (Em) has been calculated. As experiments always needa large amount of time and is often costly, first-principles calculations canbe used for a systematic study of the parameters mentioned above and otherfactors, such as the determination of the structures of the phases, instabilities,conductivities, etc., of the novel materials. The promising cathode materialMgxMo6S8 for magnesium rechargeable batteries has also been studied in thisthesis.

Recently, there has been intense searching for ferromagnetic ordering indoped, dilute magnetic semiconductors focusing on possible spin-transportproperties, which has many potentially interesting device applications. Wehave studied Mn-doped ZnO, which is a future candidate for spintronics appli-cations.

The above applications of DFT only represents a very small segment ofthis field within the solid state physics. The reader is invited to go through

2

the published literature for further applications of this theory in other fields ofmaterials science.

3

Chapter 1. Introduction

4

Chapter 2Density Functional Theory

The foundation of the electronic structure of matter is the non-relativisticSchrodinger equation for the the many-electron wave function Ψ(r1, r2, ..., rn),where ri denotes the position and the spin of the particle i. In a solid, we haveto solve the Schrodinger equation for a number of interacting electrons of theorder of 1023 on the macroscopic scale. Even, in solids with a crystalline struc-ture, if we take in account translational symmetry and further symmetries suchas rotations, invertions, etc., we still face a pharaonic task.

In Density Functional Theory (DFT), which is an alternative approach to thetheory of electronic structure of matter, the electron density distribution, n(r),which only depends on three spatial coordinates and the spin, rather than themany-electron wave function plays the central role. DFT in the Hohenberg-Kohn formulation1 is deduced from the n-particle Schrodinger equation, butit is finally expressed entirely in terms of the electron density n(r), whereasin the Kohn-Sham2 formulation it is expressed in terms of n(r) and single-particle wave functions ϕi(r). In solid state physics, DFT allows parameterfree calculations of densities, ground state energies, as well as related quanti-ties such as lattice structures and constants, elastic constants, phonon disper-sion relations, magnetic moments, etc. It also leads to nominal energy bands,which are usually a very useful approximation to the physical bands. In thischapter a very short sketch of DFT will be presented, and a complete review issomewhere else.3

2.1 The Many-Body Problem

The non-relativistic time-independent Hamiltonian characterising a multicom-ponent system composed by A components in the presence of external poten-tials v1(r),v2(r),...,vA(r) which coupled to the densities n1(r),n2(r),...,nA(r)

5

Chapter 2. Density Functional Theory

of the different components is∗

H = T + W + V

=A∑

α=1

∫d3r ψ†

α(r)

(− 1

2mα∇2

)ψα(r) +

+12

A∑α,β=1

∫ ∫d3r d3r′ ψ†

α(r)ψ†β(r′)wα,β(r, r′)ψβ(r′)ψα(r) +

+A∑

α=1

∫d3r ψ†

α(r)vα(r)ψα(r), (2.1)

where ψα(r) =∑

kλ φkλ(r)bkλα is a field operator for the αth component,ψ†

α(r) is its Hermitian conjugate; bkλα is an annihilation operator for a particleof the component α and φkλ(r) is some complete set of orthonormalised wavefunctions. wα,β(r, r′) is the interaction potential between a particle of thecomponent α at point r and a particle of component β at point r′.

The multicomponent system most often used in condensed matter physics iscomposed by nuclei and electrons†. If we consider this 2-component system inthe absence of external potentials and the interaction potentials wen, wnn, wee

being the electron-nuclei, nuclei-nuclei and electron-electron Coulomb inter-action respectively; the non-relativistic time-independent Hamiltonian for thissystem consisting of M nuclei and N electrons is

∗atomic units are used, where e = me = = 1†other multicomponent systems used in condensed matter physics are the electron-hole liq-

uid, the electron-positron liquid, etc.

6

2.1. The Many-Body Problem

Htot = Tn + Te + Wnn + Wee + Wne

=∑α

∑I

∫d3R φ†

αI(R)

(− 1

2Mα∇2

)φαI(R) +

+∑

δ

∫d3r ψ†

δ(r)(−1

2∇2)ψδ(r) +

+12

∑α,β

∑IJ

∫ ∫d3R d3R′ φ†

αI(R)φ†βJ(R′) ×

× ZαZβ

|R − R′| φβJ(R′)ψαI(R) +

+12

∑δ,γ

∫ ∫d3r d3r′ ψ†

δ(r)ψ†γ(r′)

1

|r − r′| ψγ(r′)ψδ(r) +

−∑α

∑I

∑δ

∫ ∫d3r d3R ψ†

δ(r)φ†αI(R) ×

× Zα

|r− R| φαI(R)ψδ(r), (2.2)

where Zα and Mα are charge and mass of the nucleus α; ψδ and φαI arefield operators for an electron with spin δ and a nucleus α with spin I , re-spectively. These field operators satisfy the ordinary commutation or anti-commutation relations depending upon their statistics.

In solid state physics calculations, the Born-Oppenheimer approximationis used. Under this approximation, the motion of nuclei and electrons can beseparated because the mass ratio of an electron compare to a nucleus is verysmall. The electronic wave function and the electronic energy obtained fromthis approximation depend only on the nuclear coordinates parametrically. Theparametric dependence of the electronic energy on nuclear coordinates givesthe concept of the Born-Oppenheimer energy surface, from which the study ofthe nuclear dynamics is developed. Once the Born-Oppenheimer approxima-tion is made, the problem is reduced to solving the many-electron problem fora fixed nuclear geometry. Thus, the kinetic energy term, Tn, in Eq. (2.2) canbe dropped and the third term, Wnn, is a constant which is not consider it untilthe calculation of the total energy. Finally, if we regard the Coulomb interac-tion between nuclei and electrons to be the external potential, vne(r), actingon the electrons, the non-relativistic time-independent Hamiltonian describingthe many-electron system is

7


He = Te + Vne + Wee

=∑

δ

∫d3r ψ†

δ(r)(−1

2∇2)ψδ(r) +

−∑

δ

∑α

∫d3r ψ†

δ(r)Zα

|r − Rα| ψδ(r)

+12

∑δ,γ

∫ ∫d3r d3r′ ψ†

δ(r)ψ†γ(r′)

1

|r − r′| ψγ(r′)ψδ(r). (2.3)

This Hamiltonian will be in the following the centre of our studies, and thephysical properties that we are going to investigate are driven by it.

2.2 The Hohenberg-Kohn TheoremThe Hohenberg-Kohn theorem for a bound system of interacting electrons,described by the Hamiltonian (2.3), states that: The ground state expectationvalue of any observable O is a unique functional of the exact ground stateelectron density ⟨

Ψ[n(r)]∣∣O∣∣Ψ[n(r)]

⟩= O[n(r)]. (2.4)

The proof of this theorem is quite easy and is given somewhere else.3 A sec-ond statement of the Hohenberg-Kohn theorem shows the variational characterof the energy functional

Ev0[n(r)] =

⟨Ψ[n(r)]

∣∣T + W + V0

∣∣Ψ[n(r)]⟩

= FHK [n(r)] +

∫d3 rv0(r)n(r), (2.5)

with

FHK [n(r)] =⟨Ψ[n(r)]

∣∣T + W∣∣Ψ[n(r)]

⟩, (2.6)

where V0 is the external potential of the system with ground state densityn0(r) and ground state energy E0. By the Rayleigh-Ritz principle the func-tional Ev0

has the property

E0 = Ev0[n0(r)] < Ev0

[n(r)]. (2.7)

Then, we can find the the exact ground state energy by minimisation of thefunctional Ev0

[n(r)]

8

2.3. The Self-Consistent Kohn-Sham Equations

E0 = minn∈N

Ev0[n(r)], (2.8)

where N is the set of ground states densities.Thus, the pharaonic problem of finding the minimum of the energy with

respect the wave function has been replaced into the most simple of findingthe minimum of the functional (2.5) with respect to the three-dimensional trialfunction n(r). Although, it remains the question how to construct the exactfunctional FHK or a least how to approximate it.

2.3 The Self-Consistent Kohn-Sham EquationsIf we consider a system of N non-interacting electrons, by the Hohenberg-Kohn theorem, the energy functional of this system is

Es[n(r)] = Ts[n(r)] +

∫d3r vs(r)n(r), (2.9)

where Ts[n(r)] denotes the kinetic energy functional of the non-interactingelectrons. For such system the variational equation δEs[n(r)] = 0 yields theexact ground state density ns(r) of the non-interacting system.

The main statement used to establish the Kohn-Sham equations is that forany interacting system, there exist a local single-particle potential vs(r) suchthat the exact ground state density n(r) of the interacting system equals theground state density of the non-interacting system (n(r) = ns(r)).

Returning to the problem of interacting electrons with external potentialv0(r) and ground state density n0(r), we rewrite the total energy functional ofthe interacting electrons (Eq. (2.5)) in terms of the kinetic energy functionalof the non-interacting electrons as

Ev0[n(r)] = Ts[n(r)] +

∫d3r v0(r)n(r) +

+12

∫ ∫d3r d3r′

n(r)n(r′)

|r − r′| + Exc[n(r)], (2.10)

where the exchange-correlation functional Exc[n(r)] is defined as

Exc[n(r)] = FHK [n(r)] − 12

∫ ∫d3r d3r′

n(r)n(r′)

|r − r′| − Ts[n(r)]. (2.11)

The corresponding Euler-Lagrange equations of the energy functional ofthe interacting electrons is

9


δEv0[n(r)] =

∫δn(r)

δTs[n(r)]

δn(r)

∣∣∣∣n(r)=n0(r)

+ veff (r) − ε

= 0, (2.12)

where

veff (r) = v0(r) +

∫d3r′

n0(r′)

|r − r′| + vxc[n0; r] (2.13)

and

vxc[n0; r] =δExc[n(r)]

δn(r)

∣∣∣∣n(r)=n0(r)

. (2.14)

Here ε is a Lagrange multiplier to assure particle conservation. Then, theproblem of minimising density n(r) is giving by solving the single particleequation

(−12∇2 + veff (r) − εi

)ϕi = 0, (2.15)

with

n0(r) =N∑

i=1

|ϕi(r)|2, (2.16)

and

veff (r) = v0(r) +

∫d3r′

n0(r′)

|r − r′| + vxc[n0; r]. (2.17)

The set of equations (2.15)-(2.17) represent the Kohn-Sham equations. Sincethe single-particle potential depends on the density, the whole set of equationshas to be solved in self-consistent fashion.

The ground state energy is giving by

E0 =∑

i

εi − 12

∫ ∫d3r d3r′

n(r)n(r′)

|r − r′| + Exc[n] −∫

d3r vxc(r). (2.18)

If we know th exact Exc and vxc all the many-body effects are in principleincluded. Practical development of DFT depends entirely on whether approx-imations for the functional Exc could be found and if they are sufficientlysimple and sufficiently accurate.

10

2.4. The Local Density Approximation

2.4 The Local Density ApproximationThe functional Exc are not known exactly and different approximations havebeen used in atomic, molecular and solid state physics.4 The most widely usedapproximation is the local density approximation (LDA) defined by

ELDAxc =

∫d3r ehom

xc (n0)|n0→n(r), (2.19)

where ehomxc (n0)|n0→n(r) is the exchange-correlation energy per particle of

a homogeneous electron gas with the replacement of the constant density n0

by the local density n(r) of the actual inhomogeneous system, i.e., the LDAapproximates the exchange-correlation energy density of a real, spatially inho-mogeneous system at each point r by that of a homogeneous electron gas withdensity equal to the local n0. For practical calculations, we must then deter-mine what exc is. Although no general form is known exactly, the low-densityand high-density limits can be calculated analytically. Usually, the density isexpressed in terms of the dimensionless parameter rs which is the radius ofthe sphere that can be assigned to each electron, measured in units of the Bohrradius a0. In the low-density limit (rs 1), the electrostatic potential energydominates, and the electrons condense into what is known as a Wigner crys-tal, the energy of which can be calculated. While the density of the Wignercrystal is not strictly uniform, we can still use the energy per electron of thissystem to develop an estimate for exc at low densities. In the high-density limit(rs 1), the kinetic energy dominates, and the random-phase approximationbecomes exact. Unfortunately, real materials have rs of the order unity. Usu-ally, one then uses one of several interpolation schemes than join the low andhigh-density limits of exc. In the local spin density approximation (LSDA), theexchange-correlation energy is frequently separated into exchange and corre-lation parts. These two quantities are then calculated a zero and unit polari-sations. Different parametrisations of the exchange-correlation are used suchas Hedin and Lundqvist,5 von Barth and Hedin,6 and a parametrisation of theMonte-Carlo results of Ceperley and Alder7 by Perdew and Zunger.8 In spiteof its formal domain of validity, is for very slowly varying densities, the LDAapproximation has been successful in many practical applications. This is at-tributed to the fact that LDA satisfies the sum rule for the exchange and corre-lation hole.

For atoms and simple molecules, the LDA gives good results for geomet-rical quantities, such as bond lengths, and for electron densities, vibrationalfrequencies, and energy differences such as ionisation potentials. These resultsare often an improvement over results obtained using the Hartree-Fock approx-imation. For open shell atoms, the LDA tends to overestimate the ground stateenergy in comparison with the experimental values, and with the Hartree-Fock

11


approximation. Another important quantity for molecules is the bond dissocia-tion energy, or atomisation energy. The LDA typically does a rather poorly forthis quantity, overestimating it with errors around 20%, and sometimes over100%.

The band structure of solids are usually calculated by interpreting the Kohn-Sham eigenvalues εk,n for Bloch states of wave vector k in a band n as beingband energies. Although there is no formal justification for this interpreta-tion, it usually works remarkably accurately. However, if one tries calcu-late the band gap of insulators and semiconductors by taking the differencebetween the highest occupied LDA Kohn-Sham eigenvalue and the lowestunoccupied one, the result typically underestimates the real band gap by asmuch as 50%, or even in some cases of strongly correlated insulators theLDA gives no band gap at all. The error has two distinct sources. One isthe LDA exchange-correlation potential. The other is related to the fact thatthe exchange-correlation potential must have discontinuities in its functionalderivative at integer particle numbers. In order to evaluate the band gap of anN electron system, one must include this discontinuity at N .

2.5 The Generalised Gradient ApproximationAn improvement of LDA is the generalised gradient approximation (GGA),which express the exchange-correlation energy in terms of the densities and itslocal gradients

EGGAxc =

∫d3r eGGA

xc (n,∇n). (2.20)

The function eGGAxc (n,∇n) is not uniquely defined and many forms have

been proposed. Their construction make use of sum rules, general scalingproperties, asymptotic behaviour of effective potentials, etc. We have used inour calculations the GGAs proposed by Perdew, Burke and Ernzerhof9 andPerdew and Wang 91.10, 11

The great strength of the GGA lies in the dramatic improvement it givesover the LDA in calculating such properties as bond dissociation energies,which the LDA may overestimate by a much as 100%, while the GGA gives er-rors typically of the order of 10% or less. The GGA also gives usually a goodimprovement over the LDA bulk modulus of solids, with an error of around10%, compared with around 20% for the LDA.

12

Chapter 3Electronic Structure Calculation

3.1 Bloch ElectronsIt has been established that the ground state energy can be determined uniquelyby the electron density, and the self-consistent Kohn-Sham equations (2.15-2.17) provide a path to obtain the ground state electron density and groundstate energy for the interacting electron system. For a crystal, it is obvious thatthe effective potential Eq. (2.17) will have the symmetry of the crystal lattice,and our goal is to solve the Kohn-Sham equations for this effective potential.

An infinite crystal has the property that is invariant under lattice translations,so the effective crystal potential is invariant under lattice translations as well

veff (r + R) = veff (r). (3.1)

The translation vector R is defined in terms of a set of vectors a1,a2,a3 by

R = m1a1 + m2a2 + m3a3, (3.2)

where m1, m2, m3 are integers. The set of vectors a1,a2,a3 are known asthe Bravais lattice vectors of the crystal.

The effect of translational symmetry with periodic boundary conditions(where the value of any wave function is the same at equivalent points onopposite sides of the crystal) on the Kohn-Sham one-particle wave functions isexpressed by the Bloch’s theorem⟨

r + R∣∣kn

⟩= eik·R

⟨r∣∣kn

⟩, (3.3)

the phase factor eik·R is an eigenvalue of the translation lattice operator

TR

∣∣kn⟩

= eik·R∣∣kn

⟩, (3.4)

such that its value will be unit if k correspond exactly to a reciprocal latticevector defined by k = 2π(n1b1 +n2b2 +n3b3), where n1, n2, n3 are integers

13

Chapter 3. Electronic Structure Calculation

and b1,b2,b3 are the basis vectors of the reciprocal lattice. Since k·R = 2πj,where j is an integer, the wave function (3.3) is not uniquely defined. Anywave function Eq. (3.3) with reciprocal lattice vector K = k + k′ defineequally the wave function. But, we only need to specify the wave functionslying within the Brillouin zone (BZ). Apart from its translational symmetry,the Bravais lattice will be invariant under rotations. This leads us to consideronly the irreducible part of the BZ to solve the band structure problem. Finally,the index, n, called the band index, is used to label the Bloch wave functionswhich are solutions within the same vector k.

At this point, we introduce an ansatz for the wave function in order to solvethe Kohn-Sham equations HKS

∣∣kn⟩

= Ekn

∣∣kn⟩

∣∣kn⟩

=∑

j

cknj

∣∣kj⟩, (3.5)

where∣∣kj

⟩are also Bloch wave functions and form a complete basis set.

However, it must be possible to represent the complete basis set using only asmall subset of the original infinite set (truncated basis set). Written in termsof the truncated basis set, the Kohn-Sham equations is reduced to a generalisedmatrix eigenvalue problem∑

j

cknj

⟨ki∣∣HKS

∣∣kj⟩

= Ekn

∑j

cknj

⟨ki∣∣kj

⟩. (3.6)

We can write the Kohn-Sham equations in matrix notation as

H · cn = EnO · cn. (3.7)

The matrix O is called the overlap matrix. The generalised matrix eigen-value problem could be transformed into the standard symmetric eigenvalueproblem via the Cholesky decomposition, and then it can be solved using astandard eigenvalue subroutine package. The remainder of this chapter will beabout the choice of basis set.

3.2 Full-Potential Linear-Muffin-Tin-Orbital MethodThe full-potential linear-muffin-tin-orbital (FP-LMTO) method has its originsin the LMTO method.12, 13 The implementation used in this thesis is fullydescribed somewhere else14 and a very brief sketch is presented here.

The FP-LMTO method begins dividing up the crystal into non-overlappingmuffin-tin spheres around the ions of the crystal at positions R + t, where R

is a lattice vector and t is a basis vector of the unit cell. The remainder of thecrystal is called the interstitial. Once we have divided the crystal, we introducethe FP-LMTO ansatz for the wave function

14

3.3. Projector Augmented-Wave Method

∣∣kn⟩

=∣∣kn

⟩MT+

∣∣kn⟩INT

, (3.8)

where∣∣kn

⟩MTand

∣∣kn⟩INT

are the wave functions inside the muffin-tinspheres and in the interstitial regions, respectively. This wave functions areconstructed as linear combinations of Bloch sums∣∣kn

⟩MT=

∑Rlm

χknlmte

k·R∣∣Rt lm

⟩ ∣∣Rt lm⟩

(3.9)

and ∣∣kn⟩INT

=∑R

ek·RK

knκtlm

∣∣κRtlm⟩, (3.10)

where∣∣Rtlm

⟩is the radial function, which is calculated numerically by

solving the Kohn-Sham equations in the spherically average potential insidethe muffin-tin sphere centred at R + t,

∣∣Rtlm⟩

has the angular dependencepart of the wave function centred at the muffin-tin sphere and

∣∣κRtlm⟩

issometimes called the envelope function. In the the position representation isdefined by

⟨r∣∣κRtlm

⟩= −κl+1ilYlm(r − R − t)

−h+

l (κr) if κ2 ≥ 0

nl(κr) if κ2 ≤ 0,, (3.11)

where h+l (κr) is a spherical Hankel function of the first kind and nl(κr) is

a spherical Neumann function. The parameter κ2 is called the kinetic tail en-ergy. Very often a basis will contain functions with the same quantum numbersRtlm but with different κ2 and ε, being the latter a fixed energy eigenvalue ofthe radial function.

The FP-LMTO basis set is used to rewrite the Kohn-Sham equation as

(H0 + H1 − E(k)O)∣∣kκRtlm

⟩= 0, (3.12)

where H0 is the spherical part of the Hamiltonian, O is the overlap op-erator. H1 is the part of the Hamiltonian which include all the non-sphericalterms coming from the non-spherical parts of the potential inside the muffin-tinspheres, the kinetic energy, and the interstitial potential.

3.3 Projector Augmented-Wave MethodThe projector augmented-wave (PAW) method15, 16 is conceived as a map ofthe original Hamiltonian HKS , via a linear transformation T, into a more easilyto solve pseudo-Hamiltonian Hps

KS

15


T†HKST = Hps

KS . (3.13)

The PAW method starts dividing up the crystal in the augmentation regionwhich are made up of non-overlapping spheres enclosing each ion of the crys-tal and the interstitial region which consists of the remainder of the crystal.Once we have divided the crystal we introduce the PAW ansatz in two steps.The first step is to choose a complete basis set of atomic wave functions,∣∣ηRtlm

⟩atcentred on an ion at R + t, to expand the complete wave func-

tion∣∣kn

⟩within the augmentation region; and in the interstitial region we can

expand∣∣kn

⟩in terms of plane waves. The second step of the PAW ansatz is to

introduce a two new classes of functions: The pseudo-atomic wave functions∣∣ηRtlm⟩ps−at

and the projector functions∣∣ηRtlm

⟩in order to have some

practical scheme to insure that the complete wave function is continuous anddifferentiable across the augmentation-interstitial boundary, and to make theplane wave part of the complete wave function to be identically zero inside theaugmentation spheres.

The pseudo-atomic wave functions∣∣ηRtlm

⟩ps−atmust satisfy certain con-

ditions. First, they must be eigenfunctions of the Khom-Sham equations foran isolated pseudo-atom. Second, the eigenvalue Eη must be the same for∣∣ηRtlm

⟩atand

∣∣ηRtlm⟩ps−at

. Third, outside the augmentation sphere∣∣ηRtlm⟩at

=∣∣ηRtlm

⟩ps−at. Fourth, inside the augmentation spheres, it

must be possible to represent the plane wave expansion of the complete wavefunction,

∣∣kn⟩pw

, in terms of the pseudo-atomic wave functions∣∣kn⟩pw

=∑

ηRtlm

aknηRtlm

∣∣ηRtlm⟩ps−at

, (3.14)

the expansion coefficients aknηRtlm are found by evaluation of the overlap

integral between the interstitial wave function∣∣kn

⟩pwand the projector func-

tions

aknηRtlm =

⟨ηRtlm

∣∣kn⟩pw

. (3.15)

The projector functions are mathematical constructs which form a connec-tion between the augmentation and the interstitial regions. They are definedsuch that for each pair (

∣∣ηRtlm⟩at

,∣∣ηRtlm

⟩ps−at) there must be a corre-

sponding projector function∣∣ηRtlm

⟩. Furthermore, they must fulfil the fol-

lowing completeness and orthogonality relations∑ηlm

∣∣ηRtlm⟩ps−at⟨

ηRtlm∣∣ = 1, (3.16)

⟨ηlm

∣∣η′l′m′⟩ps−at

= δηlm,η′l′m′ . (3.17)

16

3.3. Projector Augmented-Wave Method

Now, with these relations, we can write the PAW ansatz as follows

∣∣kn⟩

=∣∣kn

⟩pw+

+∑

ηRtlm

⟨ηRtlm

∣∣kn⟩pw∣∣ηRtlm

⟩at+

−∑

ηRtlm

⟨ηRtlm

∣∣kn⟩pw∣∣ηRtlm

⟩ps−at, (3.18)

or in operator form as

∣∣kn⟩

=

1 +∑

ηRtlm

(∣∣ηRtlm⟩at − ∣∣ηRtlm

⟩ps−at) ⟨

ηRtlm∣∣ ∣∣kn

⟩pw

= T∣∣kn

⟩pw. (3.19)

The operator T makes the connection between the Hilbert space spanned bythe wave function

∣∣kn⟩

and the PAW Hilbert space spanned by the plane wavepart of the PAW wave function

∣∣kn⟩pw

.

By means of the transformation T, the Kohn-Sham equations is transformedto (

Hps − EOps) ∣∣ηRtlm

⟩ps−at= 0, (3.20)

where Hps = T†HT is the pseudo-Hamiltonian and the overlap operator is

Ops = T†OT.

17


18

Chapter 4Crystallographic PhaseTransitions Under High Pressure

In the last decade, high-pressure chemistry and physics have experienced agreat expansion in many basic and applied scientific areas due to the increasingpower of the experimental and computational techniques. The improvementin diamond anvil cell methods combined with third generation synchrotronsources, which improve the performance of novel structural and spectroscopicexperiments, have became an essential tool to understand the nature of theconstituent materials at the Earth’s mantle such as SiO2,17–20 iron oxides,21–23

etc.. From the technological point of view high-pressure studies are a veryuseful and powerful tool to define synthesis routes of new promising materi-als such as TiO2

24 and binary and ternary nitrides25–29 because some of theirpressure-induced phases are qualified as potential super-hard materials.

From the fundamental aspect, high-pressure research is one of the most im-portant probe to understand the correlation between the electronic and struc-tural properties of matter. Among phase transformations in solids there aretwo special cases which have caught a lot of attention both experimentally andtheoretically.30 These are the isostructural volume collapses in the cerium andcesium metals under pressure. In accordance with the lack of a broken crys-tal symmetry, these transformations have been referred to as electronic phasechanges involving the 4f -electrons in the case of Ce and a drastic change ofthe number of d-electrons in the case of Cs. However, most recently, highpressure experiments on Cs have suggested that the isostructural phase changein fact is a crystallographic transition.31 This will involve a drastic changeof the previous general understanding of this phase transition. On the otherhand, the titanium group of elements and their alloys are situated in the begin-ning of the transition metal series and their electronic structure is dominatedby a comparatively narrow d-band hybridising with a broad s-band. Under

19

Chapter 4. Crystallographic Phase transitions Under High Pressure

compression the titanium group elements are expected to transform from theirhcp (α) structure to the bcc (β) structure by applying pressure because of thes → d electronic charge transfer.32 Also, a considerable progress in the stud-ies Mott insulators phenomena in transition-metal compounds in which theprocess of pressure-induced Mott-transition occurs, namely, the strongly cor-related d-electrons systems transform into normal metallic state.33

4.1 The Equation Of State For CrystalsThe equation of state concept means the complete set of thermodynamic func-tions on the equilibrium surface of a material. The Helmholtz free energyserves as a generating function for the equation of state. In practical appli-cations, it is important to evaluate all thermodynamic functions from a singlefree energy, so as to achieve thermodynamic consistency among all the derivedfunctions. For a crystal, the most useful thermodynamic functions are the inter-nal energy, the entropy and the stresses, with independent variables being thecrystal structure and temperature. When the applied stress is isotropic pres-sure, the independent variables are volume and temperature.

Following the universal formulation developed by Wallace,34 the Helmholtzfree energy (F ) of a crystal which is derived from the exact many-particleHamiltonian (Eq. (2.1)) with approximations appropriate to condensed matter(see Chapter 2), could be arranged in order of generally decreasing importanceof terms for applications to equations of state:

F = E0 + Fph + Fel + Fanh + Fep, (4.1)

where E0 is the electronic ground state energy when the nuclei are fixedat the crystal lattice sites (Eq. (2.18)). Fph describes the energy of quasi-harmonic phonons, Fel account electronic excitation energies from the refer-ence ground state energy, Fanh is the energy coming from anharmonic effects(phonon-phonon interaction), and Fep is the energy of the coupling of elec-tronic excitations and phonons (electron-phonon interaction).

To construct an equation of state, one needs to calibrate the parametric func-tions contained in Eq.(4.1). In practice all the experimental and theoreticalreliable information is used to perform this calibration.

As I perform all calculations in this thesis at zero temperature (T = 0K), the first term in Eq. (4.1), the ground state energy, is the energy whichI calculated in this work. The nearest experimental quantity is the isother-mal compression curve. For densities not far from normal density, the ex-perimental compression curve is prescribed by the isothermal bulk modulus(BT ), normally available from experiments at various temperatures and pres-sures (P ). The experimental data always contain all the contributions implied

20

4.2. Phase Transitions In Crystalline Matter

by Eq. (4.1), but the major contribution is from E0 and Fph. Then, one ap-proximate Eq. (2.18) at T = 0 K to F = E0 + Fph. Hence, we are able tocalculate P and BT

P = −(

∂F

∂Ω

)T

, (4.2)

and

BT = −Ω

(∂P

∂Ω

)T

, (4.3)

where Ω is the volume of the crystal.

4.2 Phase Transitions In Crystalline MatterExperiments are usually done a fixed pressure and temperature. At a givenP and T , phase stability is determined by the Gibbs free energy G(P, T ):The phase with lowest G is absolutely stable, and any other phase, if present,is metastable. Two phases α and β are in equilibrium when Gα(P, T ) =Gβ(P, T ). In theoretical work, it is most convenient to choose Ω and T asindependent variables. Then the phase boundary is located by the Helmholtzconstruction: At a given T , F (Ω, T ) is drawn for each phase, and the commontangent locates the equilibrium condition Gα(P, T ) = Gβ(P, T ), since P =− (∂F/∂Ω)T .

DFT is currently able to reproduce experimental observations of crystal-crystal transitions as function of pressure at temperatures far below the meltingpoint. Theory is done at T = 0 K and F (Ω, T ) is replaced by E0. Curves of E0

are calculated for a number of volumes for each crystal structure, and phasestransitions are located from Helmholtz constructions. The level of agreementbetween theory and experiment is illustrated in this thesis for several materialsas is shown in papers I to VI.

Among phase transformations in solids there are two special cases whichhave caught a lot of attention both experimentally and theoretically.30 Theseare the isostructural volume collapses in the cerium and cesium metals underpressure. In accordance with the lack of a broken crystal symmetry, thesetransformations have been referred to as electronic phase changes involvingthe 4f -electrons in the case of Ce and a drastic change of the number of d-electrons in the case of Cs. The classical isostructural interpretation in Cs hasdeveloped as follows: In 1941 Bridgman35 reported on two phase transitionsin Cs at room temperature under pressure (Cs-I → Cs-II and Cs-II → Cs-III).Hall et al.36 confirmed by in situ high pressure X-ray diffraction studies thefcc structure assigned to Cs-II, stable in the pressure range 2.3-4.25 GPa. They

21


also showed that Cs-III attains the fcc phase as well, but with a considerablysmaller lattice constant. From resistance and X-ray diffraction studies theyobserved still another phase transition located only 0.05 GPa higher than theCs-II → Cs-III transition. The X-ray pattern of this new phase ( (Cs-IV) couldnot be satisfactorily indexed. In spite of this complication, the discovery ofan isostructural transition in Cs at 4.25 GPa suggests that this is an electronictransition. Later experiments by Jayaraman et al.37 and McWhan et al.38 alsoconfirmed this picture.

Recently, McMahon at al.31 performed new single-crystal diffraction stud-ies of Cs-III using a synchrotron X-ray source. In contrast to the previouslyreported fcc structure, they identified a very complex type of structure withorthorhombic symmetry belonging to the space group C2221 having 84 atomsin the unit cell as it is shown in Fig. 4.1.

Figure 4.1: Crystal structure of Cs-III along different crystal axes.

In paper V, we have studied the phase stabilities of these Cs phases. InFig. 4.2, we show the calculated total energy difference between the pertinentcrystallographic structures for Cs near a volume of about V/V0 ≈ 0.43. Herewe compare the fcc, Cs-III and Cs-IV phases, where the energy of the Cs-IIIstructure (orthorhombic C2221) is taken as reference level. It is most satisfyingto notice that our calculations exactly reproduce the observed crystal structure

22

4.2. Phase Transitions In Crystalline Matter

sequence; fcc → Cs-III → Cs-IV, with increasing pressure.In order to study the stabilisation of the different structures in Cs, we have

plotted the density of states (DOS) for three different volumes; a) a volumewhere Cs-II is stable, b) at a volume where Cs-III is stable and c) at a volumewhere Cs-IV is stable. Inspection of the density of states in Fig. 4.3a showsthat the band width is a little bit larger for the Cs-II phase than for Cs-III.The DOS around the Fermi level, say ±0.5 eV, is very flat for the Cs-II phasewhereas for the Cs-III phase the DOS shows more structure and pseudo gaps.In fact, at the volume where the Cs-II phase is more stable, the Fermi levelfor the Cs-III phase is located near a sharp peak, marked as A, in the DOSsuggesting an instability of the Cs-III phase. When the pressure is increasedand we reach a volume where Cs-III is the stable phase, one can see that thesharp peak A has now become fully occupied and the Fermi level is situatedexactly between two peaks in the pseudo gap, making the Cs-III phase morestable. The Fermi level for the Cs-II phase is still situated in a very flat regionof the density of states. After a further increase of the pressure, the DOS (Fig.4.3c) for the Cs-III phase shows that the Fermi level has now left the gap andlies on a peak of the DOS. This makes the Cs-III structure unstable. Howeverfor the Cs-IV phase, the Fermi level is situated on an almost flat region. Inaddition Cs-IV has a pronounced fully occupied peak well below the Fermilevel, a circumstance which stabilises this phase relative to the Cs-III phase.

0.4 0.41 0.42 0.43

V/V0

-0.5

0

0.5

1

1.5

E-E

Cs-

III /

(m

Ry

per

ion)

Cs-IICs-IV

0.4 0.41 0.42 0.43-7.4

-7.3

-7.2

Cs-IIICs-II

Cs-IV

Figure 4.2: Energy difference between the fcc, Cs-III and Cs-IV crystal structures forCs as a function of volume (V/V0, V0= experimental equilibrium volume). A spd-basis set has been used in these calculations. In the inset sp-basis set has been used.The Cs-III structure is used as the zero energy reference level.

23


-3 -2 -1 0 1 2

Energy / eV

0

1

2 Cs-IICs-III

0

1

2

Den

sity

of

stat

es /

num

ber

of s

tate

s pe

r at

om

Cs-IICs-III

0

1

2

3

Cs-IVCs-III

(a)

(b)

(c)V/V

0=0.3547

V/V0=0.4259

V/V0=0.4341

A

A

Figure 4.3: Calculated total density of states (DOS) for Cs in fcc and Cs-III structure(a) at a volume where Cs-II (fcc) phase is stable (b) at a volume where Cs-III phaseis stable and (c) where Cs-III is unstable. The Fermi level is set at zero energy andmarked by a vertical line.

24

Chapter 5The Elastic Properties Of Crystals

5.1 The Strain And Stress TensorsAny solid body undergoes deformations when it is under the action of appliedforces. In order to describe its change of shape, we define the vector field u(r),which is called the displacement field, such that

u(r) = r′ − r, (5.1)

where r is the position of some particular point in the body before deforma-tion and r′ is its position after deformation.

When a solid body is deformed, the distances between its points change aswell. If we consider only small deformations, the distances between pointsbefore and after deformation dl and dl′ are related by

dl′2 = dl2 + 2∑ik

uikdxi dxk, (5.2)

where the tensor uik is given by

uik = 12

(∂ui

∂xk+

∂uk

∂xi+

∂ul

∂xi

∂ul

∂xk

). (5.3)

The tensor uik is called the strain tensor. By definition the strain tensor issymmetric, i.e. uik = uki. In cases where ui and their derivatives are small forsmall deformations, we can neglect the last term in (5.3), thus the strain tensoris reduced to

uik = 12

(∂ui

∂xk+

∂uk

∂xi

). (5.4)

Also, when a deformation occurs, the arrangement of the molecules is changedand the solid body leaves its original equilibrium state. Therefore, forces arise

25

Chapter 5. The Elastic Properties Of Crystals

which tends to return the body to equilibrium. These internal forces are calledinternal stresses. If no deformation occurs, there are not internal stresses. Ifwe consider the total force on some portion of the body, this can be written as∫

d3r Fi =∑

k

∫d3r

∂σik

∂xk=

∑k

∮dfk σik, (5.5)

where the tensor σik is called the stress tensor and the term∑

k dfk σik isthe ith component of the force on the surface element df .

The work done by the internal stresses when the solid body undergoes asmall deformation is the product of the force Fi =

∑k

∂σik

∂xkby the displace-

ment δui integrated over the volume of the body∑ik

∫d3r

∂σik

∂xkδui =

∫d3r δR, (5.6)

where δ R denotes the work done by the internal stresses per unit volume.Integrating by parts, we obtain the work in terms of the change in the straintensor ∫

d3r δR = −∑ik

∫d3r σikδuik. (5.7)

With the above relations, we can obtain the fundamental thermodynamicrelation for deformed bodies. An infinitesimal change dE in the internal en-ergy is equal to the difference between the heat acquired by the unit volumeconsidered and the work dR done by the internal stresses. If we consider areversible process this difference is

dE = TdS +∑ik

σikduik. (5.8)

5.2 The Elastic ConstantsIn order to make the above thermodynamic relation practical, we must knowthe internal energy E of the body as a function of the strain tensor. This maybe done for small deformations expanding the internal energy in powers of uik.We only consider crystals a zero temperature and a formulation for isotropicbodies is given somewhere else.39 This power expansion with respect to theinitial internal energy E0 of the undeformed crystal is

E(Ω, u) = E(Ω0, 0) + Ω0

(∑ik

τikuik + 12

∑iklm

ciklmuikulm

), (5.9)

26

5.2. The Elastic Constants

where Ω0 is the volume of the undeformed crystal and E(Ω0, 0) being thecorresponding energy. The tensor of rank four ciklm is called the elastic modu-lus tensor. Since the strain tensor is symmetric implies that the tensor ciklm canbe defined with the same symmetry properties ciklm = ckilm = cikml = clmik.A tensor of rank four having this symmetry properties has a number of dif-ferent components equal to 21. If the crystal presents symmetry, relationsbetween the various components of the tensor ciklm arise and the number ofindependent components is less than 21. An analysis of symmetry for thecrystal classes and their corresponding number of independent components isfound somewhere else.39

5.2.1 The Elastic Constants For A Hexagonal CrystalIn papers VII and VIII, we show the results for the recently discovered su-perconductor MgB2

40 and Al-doped MgB2. The crystal symmetry of MgB2

is hexagonal (Fig. 5.1), i.e., there are two lattice parameters a and c and fiveindependent elastic moduli.

Figure 5.1: Crystal Structure of MgB2. Space group P6/mmm (no. 191).

First, we rewrite Eq. (5.9) in Voigt’s notation which uses for ciklm the newnotation cjn with j and n taking values from 1 to 6 in correspondence withxx, yy, zz, yz, zx, xy; and the strain tensor uik is written as ej , so

E(Ω, e) = E(Ω0, 0) + Ω0

∑j

τjej + 12

∑jn

cjnejξjenξn

. (5.10)

The factor ξj introduced in Eq. (5.10) takes the value 1 if j is 1,2 or 3 andtakes the value 2 if j is 4, 5 or 6. The five elastic moduli for a hexagonal crystal

27


are called c11, c12, c13, c33 and c55. Since we have five independent elasticmoduli, we need five different strains to determine these. The five distortionsused to deform the hexagonal lattice of MgB2 are described as follows:

The first distortion is written as 1 + e 0 0

0 1 + e 0

0 0 1 + e

, (5.11)

and it gives compression or expansion to the system. This preserves thesymmetry but changes the volume. The strain energy associated with this dis-tortion is

E(Ω, e) = E(Ω0, 0) + Ω0[(τ1 + τ2 + τ3)e +

+12(2c11 + 2c12 + 4c13 + c33)e

2]. (5.12)

The second distortion (1 + e)−1/3 0 0

0 (1 + e)−1/3 0

0 0 (1 + e)2/3

, (5.13)

gives the volume and symmetry-conserving variation of c/a. The energyassociated with this distortion is

E(Ω, e) = E(Ω0, 0) + Ω0[(τ1 + τ2 + τ3)e +

+1

9(c11 + c12 − 4c13 + 2c33)e

2]. (5.14)

The third distortion 1 + e 0 0

0 1 − e 0

0 0 1

, (5.15)

distorts the basal plane by elongation along a and compression along b andkeeps the volume constant. The energy associated with this distortion is

E(Ω, e) = E(Ω0, 0) + Ω0[(τ1 − τ2)e + (c11 − c12)e2]. (5.16)

The fourth distortion

28

5.2. The Elastic Constants

1 0 e

0 1 0

e 0 1

, (5.17)

distorts the hexagonal crystal to a triclinic crystal keeping the volume con-stant. The energy associated with this distortion is

E(Ω, e) = E(Ω0, 0) + Ω0[τ5e + 2c55e2]. (5.18)

The fifth distortion 1 0 0

0 1 0

0 0 1 + e

, (5.19)

gives compression or expansion of the c-axis while keeping the other axesunchanged. This preserves the symmetry but changes the volume. The energyassociated with this distortion is

E(Ω, e) = E(Ω0, 0) + Ω0

[τ5e +

c33

2e2]. (5.20)

By solving the linear equations given above, we obtain all the five elasticmoduli. Also an important quantity in crystals is the elastic anisotropy whichis known to correlate with a tendency to either ductility or brittleness.41 Inorder to calculate the elastic anisotropy, we compute the bulk modulus Ba andBc along the axes a and c respectively

Ba = adP

da=

Λ

2 + β, (5.21)

and

Bc = cdP

dc=

Ba

β, (5.22)

where Λ = c11 + 2c12 + c22 + 4c13β + c33β2 and β = c11+c12−2c13

c33−c13.

The FP-LMTO method14 has been used to solve the Kohn-Sham equationsand to calculate the energies associated with these distortions. Elastic constantsand the ratio Ba/Bc for MgB2 are found in paper VII.

Also an analysis of the bonding picture of MgB2 by means of the so-calledElectron Localisation Function (ELF), which is a useful tool for the character-isation of bonds in molecules and solids, has been done. All kind of channels,ionic (between the magnesium atoms), metallic (between the boron layers) andcovalent (within the boron network) can be well distinguished in Fig. 5.2. Ac-cording to the ELF analysis, the nature of the bonding situation in MgB2 is

29


Below 0.00

0.00 - 0.05

0.05 - 0.10

0.10 - 0.15

0.15 - 0.20

0.20 - 0.25

0.25 - 0.30

0.30 - 0.35

0.35 - 0.40

0.40 - 0.45

0.45 - 0.50

0.50 - 0.55

0.55 - 0.60

0.60 - 0.65

0.65 - 0.70

0.70 - 0.75

0.75 - 0.80

0.80 - 0.85

0.85 - 0.90

0.90 - 0.95

0.95 - 1.00

Above 1.00

ELF

Figure 5.2: Electron localisation function for the section through both, magnesiumand boron atoms, in MgB2.

very diverse. The important finding is the highly covalent and therefore verystrong bonding revealed within the boron layers, whereas interlayer bondingis found to be essentially weaker. This raises an expectation of rather highanisotropy in elastic moduli. Our calculated elastic properties also suggestthat MgB2 is highly anisotropic. Using the analogy with other known highlyanisotropic materials we expect MgB2 to be rather brittle and therefore not astechnologically attractive as one would like it to see.

In paper IX, calculation of the surface stress for fcc transition metals ispresented.

30

Chapter 6Lithium Insertion Materials ForRechargeable Batteries

The use of portable electronic devices for purposes of aerospace applications,communication, data processing and transmission, entertainment etc. is rapidlyincreasing. Furthermore, the vision of massive commercial use of electricallypowered vehicles is becoming more and more realistic. This development hasled to a demand for continued pursue of more efficient batteries with a num-ber of crucial properties. These include a relatively flat open-circuit voltage(OCV), low cost, environmental benefit, easy to fabricate, and safety in han-dling and operation.

The reversible insertion/extraction of lithium into host materials have provento be one of the most successful solutions to achieve these goals for low loadapplications as rechargeable battery electrodes.42–44 In some materials the ra-tio of lithium to metal ions can be varied between 0 and 1 without substantialchanges in the structure of the host material. This Li insertion/extraction mech-anism is the basis for its application as an electrode in a rechargeable battery.When the Li-ions are inserted into the host material not only the structure butalso the electronic structure is changed. It is believed that Li is fully ionisedin some host materials and donates its electron to the host bands without muchaffecting them. This makes it possible to control the band filling of the hostmaterial by varying the Li content electrochemically. In electrochromic appli-cations, band filling is used to adjust the electronic and optical properties.

A crucial key towards a good battery design is concerned with the choiceof cathode material. The cathode should be both an ionic and electronic con-ductor in order to allow for the insertion/extraction of lithium and electronicconduction during the discharge/charge of the battery. Further, it is desirablethat the volume change of the cathode during its operation is as small as pos-sible. This ensures good reversibility and cycle life.

31

Chapter 6. Lithium Insertion Materials For Rechargeable Batteries

Intensive research on transition metal oxides has lead to the utilisation ofthe layered rock salt systems45, 46 and the manganese- spinel framework sys-tems.47–49 These systems operate in a voltage regime between 3 and 4 V.However, the applications of these materials in the cases of electric vehicles orload-levelling systems demand very large batteries. Hence, the cost of transi-tion metals, the low voltage and environmental problems make the materialsmentioned unfavourable for these purposes.

Another line of research has been devoted to compounds containing polyan-ions, (XO4)−3 where X=S, P or As, mainly Li3Fe2(PO4)350 and LixFe2(SO4)351

and the olivine structure LiFePO4,52–54 all within the NASICOM framework.The LiFePO4 compound in principle satisfies the main requirements for a cath-ode material, It also shows an electronic conduction problem when lithium iscompletely extracted caused by a metal-insulator transition. This in turn couldgive rise to a poor performance on the discharge cycle of the battery. As a rem-edy, the doping of the material with manganese has been investigated52, 55, 56 inorder to obtain an optimum performance for this cathode candidate. We herepursue this line of investigation through our theoretical approach.

First-principles calculations have by now become a useful tool for batterydesign,57–59 since they can be used to calculate their most important proper-ties without any requirement of experimentally measured input data. Typicalproperties of interest include the open-circuit voltage (OCV), the energy den-sity (EΩ), and the specific energy (Em). As experiments always need a largeamount of time and is often costly, first-principles calculations can be used fora systematic study of the parameters mentioned above and other factors, suchas the determination of the structures of the phases, instabilities, conductivi-ties, etc., of novel materials. Thereby the experimental investigations can befocused on those materials which appear to be the most promising.

6.1 Thermodynamics Of The Electrochemical CellMaterials made by insertion at room temperature are often metastable, if heated,they can change their structure or decompose into other materials. At roomtemperature, the ratio of Fe, P and O in a host like FePO4 is fixed. From thepoint of view or thermodynamics, the constrain that the host remain FePO4

means that we can regard an insertion material like LixFePO4 as a ‘pseudo-binary’ material instead of a quaternary one.

An electrochemical cell produces electric energy on discharge and consumeenergy on charge. The discharge process can be simulated by describing thesystem in thermodynamic equilibrium when the electrochemical cell is in opencircuit. In open circuit the resistance is infinity, the current is zero and the workdone by the system is the reversible electric work. Consider an electrochemicalcell with two host materials (the electrodes IMA and IMB) where a guest Ion

32

6.1. Thermodynamics Of The Electrochemical Cell

(Li, Mg) can be insert and extracted (Fig. 6.1). If the guest ion has chargene in the solution of the cell, one ion is inserted for every n electrons passedthrough the external circuit.

Figure 6.1: Electrochemical cell on discharge and charge. IMA and IMB are theinsertion materials (electrodes).

On discharge the extraction reaction on the negative current collector is

IonxAIMA −→ IonxA−dxIMA + dxIonn+ + ne−dx, (6.1)

and on the positive current collector the insertion reaction is

IonxBIMB + dxIonn+ + ne−dx −→ IonxB+dxIMB, (6.2)

overall

IonxAIMA + IonxB

IMB −→ IonxA−dxIMA + IonxB+dxIMB. (6.3)

On the other hand, on charge the insertion reaction on the negative currentcollector is

IonxAIMA + dxIonn+ + ne−dx −→ IonxA+dxIMA, (6.4)

and on the positive current collector the extraction reaction is

33


IonxBIMB −→ IonxB−dxIMB + dxIonn+ + ne−dx, (6.5)

overall

IonxAIMA + IonxB

IMB −→ IonxA+dxIMA + IonxB−dxIMB, (6.6)

where xA and xB are the ionic guest concentration in IMA and IMB, re-spectively and dx is the number of inserted/extracted guest ions.

The reversible electric work (W ) done on the cell is equal to the variationof the Gibbs free energy (G)

dW (x) = dG(x) = −V (x)dq = −V (x)nedx, (6.7)

where V is the potential difference between IMA and IMB.By applying the general definition of the G: dG(x) =

∑i µi(x)dxi, where

µi is the chemical potential for each species, we can obtain the following equa-tion (on discharge)

dG(x) =(µIMB

ION (x)dx + µIMAION (x)(−dx)

)T,P

=(µIMB

ION (x) − µIMAION (x)

)T,P

dx. (6.8)

From Eqs. (6.7) and (6.8) we obtain the OCV of the electrochemical cell asa function of the chemical potential of the guest ion on the electrodes

V (x) = −µIMBION (x) − µIMA

ION (x)

ne(6.9)

Thermodynamics implies that guest ion insertion into the cathode (on dis-charge is IMB) increases its chemical potential and consequently the OCV,V (x), of the cell decreases.

It is difficult to compute the change in the Gibbs free energy as a functionof x, due to the limited knowledge about the nature of the local ordering ofguest ions on the host structures, as for example lithium ions in LixFePO4 for0 < x < 1. Instead of the calculation of the V (x) curve, we can compute theaverage OCV, V , between two insertion limits x1 and x2.

V =1

x2 − x1

∫ x2

x1

dq V (x)

= − 1

ne(x2 − x1)

∫ x2

x1

dq(µIMB

ION (x) − µIMAION (x)

). (6.10)

Using the definition of the the chemical potential µ(x) = (∂G(x)/∂x)T,P

we can rewrite Eg.(6.10) as

34

6.1. Thermodynamics Of The Electrochemical Cell

V = − 1

x2 − x1

∫ x2

x1

dx

[(∂GIMB(x)

∂x

)−(

∂GIMA(x)

∂x

)]= − 1

x2 − x1

[(GIMB(x2) − GIMB(x1)) − (GIMA(x2) − GIMA(x1))

]= −∆Greac

x2 − x1, (6.11)

when one of the electrodes is either metallic Li or Mg, the chemical poten-tial of the anode (on discharge) is constant and Eq.(6.11) becomes

V = − 1

x2 − x1

[(GIMB(x2) − GIMB(x1)) − ((x2 − x1)G

Li/Mg)]. (6.12)

Finally, one more approximation is done. The difference of Gibbs free en-ergy, ∆Greac = ∆Ereac + P∆Ωreac − T∆Sreac, can be approximate by thedifference of internal energy, ∆Ereac , at 0 K. Since P∆Ωreac is of the orderof 10−5 eV, T∆Sreac is of the order of the thermal energy which is also verysmall, and ∆Ereac is of the order of 3-5 eV; this approximation is quite goodto calculate the average OCV.

V = − ∆Ereac

(x2 − x1). (6.13)

Improvements in the calculation of the OCV taking in account the localordering of guest ions and the effects of the temperature could be done. Usingthe real-space cluster expansion technique, it is possible to parametrise thefree energy on site disorder and to obtain the V (x) curve as it was done forLixCoO2.60–62 Temperatures effects can be treated using canonical or grandcanonical Monte Carlo simulations on the cluster expansion60, 62 or at roomtemperature including the vibrational energy term calculated by the harmonicapproximation of the vibrations of the ions in the solid.57

The energy density EΩ and specific energy Em of the battery are given by

EΩ = −∆Greac

Ω, (6.14)

Em = −∆Greac

m, (6.15)

where Ω denotes the average volume of one formula unit of the reactant andone formula unit of the product of the electrochemical reaction. The quantitym denotes the mass of one molecular formula unit of either the reactant or theproduct of the electrochemical reaction (the two are equal).

35


Results obtained here for the battery system Li/Lix(MI1−yMIIy)PO4 (Fig.6.2) are discussed in more detail in paper X. The FP-LMTO method14 has beenused to solve the Kohn-Sham equations and to calculate the internal energiesof the compounds. By applying Eq. (6.11). we obtain the average OCV,the energy density EΩ and the specific energy Em of this system. Also, inpaper VII we have studied the battery system Mg/MgxMo6S8.

Figure 6.2: Crystal structure of LiFePO4 (space group Pnma).

6.2 Electronic StructureIn order to design potential electrodes and electrolytes for a rechargeable lithium-insertion battery, one has to consider some fundamentals requirements. The

36

6.2. Electronic Structure

electrodes should be both an ionic and electronic conductor in order to allowfor the insertion/extraction of lithium and electronic conduction during the dis-charge/charge of the battery, and the electrolyte must be an ionic conductor andan electronic insulator. Also, to avoid reduction or oxidation of the electrolyte,the Fermi level of the anode should be lie below the conduction band of theelectrolyte and the Fermi level of the cathode should be lie above the valenceband of the electrolyte. Of course, maximisation of the cell potential requiresa good matching of the Fermi levels of the anode and the cathode with theenergy band gap of the electrolyte. The electronic band structure can provideus all the information required above in order to choose the components for agood lithium-insertion battery design. Furthermore, if the electronic structureis calculated, the electronic charge density or other functions as the ELF canbe deduced to understand the nature of the bonding in the solid.

In paper X, the electronic structure (Fig. 7.2) of the olivine type compoundsLix(MI1−yMIIy)PO4 for x = 0, 1 and MI, MII = Mn, Fe, Co is shown andtheir implications on the performance of the rechargeable lithium battery isanalysed.

-12 -10 -8 -6 -4 -2 0 2 4

Energy (eV)

0

10

20

30

40LiFePO

4

FePO4

0

10

20

30

40

Den

sity

of

stat

es (

num

ber

of s

tate

s/un

it ce

ll an

d eV

)

LiFe0.8

Mn0.2

PO4

Fe0.8

Mn.2PO4

0

10

20

30

40LiFe

0.5Mn

0.5PO

4

Fe0.5

Mn0.5

PO4

0

10

20

30

40

50LiMnPO

4

MnPO4

(a)

(b)

(c)

(d)

Figure 6.3: Total DOS for LiFe1−yMnyPO4 and Fe1−yMnyPO4 for y = 1, 0.5,0.2and 0. The Fermi level is indicated by a vertical doted line.

37


38

Chapter 7Dilute Magnetic Semiconductors

Recently, there has been intense search for ferromagnetic ordering in dilutemagnetic semiconductors (DMS)63–65 focusing on possible spin-transport prop-erties (spintronics), which has many potentially interesting device applica-tions. Spintronics devices such as spin-valve transistors, spin light-emittingdiodes, non-volatile memory, logic devices, optical isolators and ultrasoft opti-cal switches are some of the areas of interest for introducing the ferromagneticproperties at room temperature in a semiconductor. Recent reviews on DMSand spintronics are found in Ref.66 and Ref.67, respectively.

Most of the past work on DMS has focused on (Ga,Mn)As and (In,Mn)As.But the highest reported Curie Temperature (Tc) in the single phase samplesgrown by MBE range from ≈ 35 to 172 K.68 This quest for a room tempera-ture ferromagnetic semiconductor, gained momentum, following a theoreticalprediction by Dietl et al., that ZnO and GaN would exhibit ferromagnetismabove room temperature on doping with Mn, provided that the hole density issufficiently high.64 Several researchers have since then reported observation ofroom temperature ferromagnetism in doped semiconductors. A considerableattention has been paid to semiconductors doped with ferromagnetic metals(Co, Fe, and Ni). In these types of systems, the fundamental issue of muchconcern is that the ferromagnetic ordering could be a result of metal precipi-tates e.g., Co in Co-doped TiO2 and Co-doped ZnO.69–71 Moreover, a definitepicture regarding the actual mechanism of ferromagnetic ordering in these sys-tems has not been established.

A ZnO based DMS would be very promising because of its widespread ap-plications in electronic devices, such as transparent conductors, gas sensors,varistors, surface acoustic wave devices, optical wave guides, acoustoopticmodulators/deflectors, ultra violet laser source, and detectors.72 Out of all thetransition metals (TM), the doping of Mn in ZnO is most favourable becauseMn has the highest possible magnetic moment73 and also the first half of the

39

Chapter 7. Dilute Magnetic Semiconductors

d-band is full, creating a stable fully polarised state. The theoretical studieson Mn-doped ZnO also proved its novelties in the fabrication of room temper-ature spintronics devices.64, 74, 75 A Tc higher than 300 K for p-type ZnO hasbeen predicted theoretically, but this has not been realized experimentally.64

Despite uncertainty in the mechanism of ferromagnetism in doped semicon-ductors, and the fact that the obtained magnetisation is lower than the theoret-ically predicted value in most of the reports appearing in literature, the resultsreported thus far, provide a pathway for exploring the Mn doped DMS. It ishowever, imperative to understand the phenomenon and the factors affectingthe magnetisation value in order to realize commercially applicable devices.

7.1 Theory Of Ferromagnetism In DMSThere are several attempts to understand the ferromagnetism in these systems.It is believe that the basic mechanism of magnetism in these materials is theinteraction of the magnetic moment of the Mn ion and the itinerant-hole sys-tem of the semiconductor. The theoretical model commonly used for carrier-induced ferromagnetism (double exchange)76 in this system so far has beenbased on a mean field scheme of treating the double exchange process whereinthe local spin of the impurity interacts with the electrons or holes of the semi-conductor.64, 77 However, this model has not been successful in explaining theexperimental observations.

The basic model78 used in almost all the theoretical discussions is describedby the Hamiltonian of the semiconductor hole gas treated in the effective-massapproximation, (sometimes using the Hubbard model) doped with Mn ions.The hole gas is assumed to be of sufficiently low density that we could firsttreat them as if they are non-interacting. The effective Hamiltonian is

H = H0 + Himp + Hext, (7.1)

where

H0 =

∫d3r

∑σ

ψ†σ(r)

(− 1

2m∗∇2 − µ

)ψσ(r), (7.2)

Himp =∑

t

∫d3r J(r − Rt)S(Rt) · s(r). (7.3)

Here m∗ is the effective mass of the holes, µ is the chemical potential of theholes, S(Rt) is the S = 5/2 spin operator representing the Mn ion located atsome position Rt, and the itinerant hole-spin density is expressed in terms ofthe hole field operators as

40

7.2. Results For Mn-doped ZnO

s(r) =1

2

∑σ,σ′

ψ†σ(r)τσ,σ′ψσ′(r), (7.4)

where τ is the vector of spin Pauli matrices.In this model, the basic parameters are the bandwidth W (or, equivalently,

the density of holes p) and the exchange splitting ∆ = NMnJS of the itiner-ant carriers due to an average effective magnetic field induced by the Mn ions,where NMn is the Mn concentration. In the indirect exchange mechanism viacharge carriers, the so-called Ruderman-Kittel-Kasuya-Yosida (RKKY) inter-action, is valid when JS W (or, equivalently, ∆ EF , where EF isthe Fermi energy), that is high charge density. This condition is invalid whenthe charge density limit is low where one has the opposite limit, JS W(∆ EF ), where the double exchange process dominates. In the frameworkof band structure,79 double exchange essentially means the following: Nearthe half filling, and when the exchange splitting is bigger than the bandwidth,the band energy of the ferromagnetic state is lower than the antiferromagneticstate if a sufficient (usually rather small) number of holes (or electrons) ex-its. This type of theory results in only a limited qualitative agreement withexperimental results.

7.2 Results For Mn-doped ZnOAccording to the theoretical prediction, ferromagnetic ordering in transitionmetal (TM) doped ZnO can be achieved without using any additional carrierswith V, Cr, Fe, Co and Ni74, 75 whereas with Mn doping additional p-type dop-ing is needed.64, 74, 75 Despite this Mn remains the primary dopant of interestbecause solubility of Mn in ZnO is larger than 10 mol% and the electron massis as large as ≈ 0.3me, where me is the free electron mass.80 In fact, Fuku-mura et al. showed that Mn atoms could be doped into ZnO up to 35% by PLDtechnique without phase segregation.81 Therefore, the amount of injected spinin the host material can be very large with Mn doping.

The model by Dietl et al. predicts that the transition temperature in di-lute magnetic semiconductors (DMS) will scale with a reduction in the atomicmass of the constituent elements due to an increase in p − d hybridisation anda reduction in spin-orbit coupling. The theory predicts a TC greater than 300K for p-type ZnO doped with Mn mediated by heavy p-type doping, with TC

dependent on the concentration of magnetic ions and holes.64 This predictionby Dietl initiated pursuit of a Mn-doped ZnO based DMS. However, no reportsfor confirmation of the theoretical work appeared until 2003(see paper XIII).Fukumura et al. found a spin-glass behaviour.81 Whereas Tiwari et al.didnot find any ferromagnetism in their MnxZn1−xO films deposited on sapphiresubstrate using PLD.82 Jung et al. reported that ferromagnetism appears at

41


low temperature in MnxZn1−xO films grown on Al2O3 (001) substrates usinglaser MBE.83 The transition temperature obtained from magnetisation mea-surements was 30 K for x = 0.1 film and 45 K for x = 0.3 one.

On the theoretical side, there are very few calculations for Mn-doped ZnO.Sato et al.using the CPA approximation whitin the LDA, have shown that ferro-magnetism can be achieved in doped ZnO without using any additional carrierswith V, Cr, Fe, Co and Ni whereas with Mn doping additional p-type doping isneeded.74, 75 However, the energy difference between the ferromagnetic (FM)and antiferromagnetic (AFM) states in Mn-doped ZnO shows a clear trend tostabilise the FM states when decreasing the Mn concentration, and at 5% Mnconcentration is almost negligible. Spladin has done LDA calculation usingpseudopotentials with localised atomic-orbital basis sets.84 She has found inthis work that for pure Mn-doped ZnO, the energy difference between FM andAFM states for a Mn concentration of 12.5%, is small (of the order of fewmeV) with the AFM state being more favourable for separated Mn configura-tion, and FM state being more favourable for close Mn configuration. Further-more, the energy difference between the FM and AFM states, with FM statestable was found to increase with the additional doping of Cu in Mn-dopedZnO. The additional doping of Cu in Mn-doped ZnO is used to provide holesto mediate the ferromagnetic coupling between Mn-Mn in Mn-doped ZnO.

0 0.05 0.1 0.15 0.2 0.25x

-20

0

20

40

60

80

100

SE

= E

FM -

EA

FM (

meV

)

short configurationslong configurations

Figure 7.1: Concentration x in MnxZn1−xO vs. total energy difference in meV/Mnatom (∆E = EFM − EAFM).

In paper XIV we present our results for this promising material for applica-tions on spintronics. In Fig. 7.1 we show total energy difference between FMand AFM configurations (∆E = EFM −EAFM) vs. Mn concentration (x). Aswe can see for the 5.6% Mn concentration a crossover from the AFM to FMconfigurations has occurred. From the extrapolation of this results to lower

42

7.2. Results For Mn-doped ZnO

Mn concentrations in ZnO, we might concluded that the FM phase is stabilisedeven further. This results are in agreement with previous LDA CPA calcula-tions74, 75 and LDA pseudo-potential calculations84 for the 25 and 12.5% Mnconcentrations but not for the 5.6% Mn concentration.

Our calculated density of states (DOS) for one the FM arrangement studiedhere of Mn2Zn34O36 is shown in Fig. 7.2. The DOS of ZnO is also shown.There is a very deep valence band located around -20 eV and -18 eV relativeto the Fermi level of Mn2Zn34O36, which is composed of O s-states. The Znd-states and O p-states of the host material are hybridised in a band locatedaround -7 eV and -1.4 eV and when the Mn impurity is present, Mn d-stateshybridise with those Zn d-states and O p-states. Between the energies around-1.3 eV and 0.05 eV, there are only majority spin (spin up) Mn d-states, wherethe Fermi level is located, leading to the so-called half-metallic behaviour. Ourcalculated magnetic moment is 4.97 µB per Mn atom. The next conductionband is located around 0.2 eV and 8.5 eV relative to the Fermi level, this bandis mainly built of O p-states and spin down Mn d-states. We can also see thatthe conduction band where the Fermi level is located is not fully occupied.In these cases, the the 3d-electron in the partially occupied 3d-states of theMn impurity is allowed to hop to the 3d-states of neighbouring Mn ions if theneighbouring Mn ions have parallel magnetic moments. This might explainthe stabilisation of ferromagnetism at the 5.6% Mn concentration.

-15

-10

-5

0

5

10

15

20

-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10Energy (eV)

-15

-10

-5

0

5

10

15

Den

sity

of

stat

es (

arbi

trar

y un

its)

(a)

(b)

Spin up

Spin down

Spin up

Spin down

Figure 7.2: Density of states (a) Solid line total density of states for Mn2Zn34O36

and dashed line total density of states for ZnO. (b) Mn 3d states. The Fermi level ofMn2Zn34O36 is denoted by a vertical doted line.

43


44

Chapter 8Optical Properties

From last few years spectroscopic techniques are becoming the standard toolfor measuring excited states of materials. In particular optical, magneto-optical,magnetic X-ray dichoism are now becoming routine procedure to probe thestructural and magnetic properties of materials.

Theoretically, the above properties can be calculated by means of DFT cal-culations, where the band structure of solids are usually calculated by inter-preting the Kohn-Sham eigenvalues εk,n for Bloch states of wave vector k ina band n as being band energies. Although, there is no formal justification forthis interpretation.

8.1 Theoretical Justification Of DFT Calculations ForThe Optical Spectra

DFT is a theory for the ground state energy of an interacting electron systemand there are also extensions to thermodynamic equilibria and excited states.While DFT is formally exact and can in principle be used to determine theexpectation value of any observable of the system. Practical calculations asLDA and GGA focus on the exact formulation for the ground state energyand electron density, and they are constructed to give good approximations forthese quantities. It is therefore not surprising that quantities extracted directlyfrom the ground state energy and electron density tend to be more accurate thanquantities found for which there exists no formal justification. Extensions ofDFT to excited electronic states were developed by Mermin85 and others;86, 87

but they are extremely complicated for practical calculations.In quasi-particle (QP) theory, the electronic structure of an interacting many-

body system is described by the single-particle eigenstates resulting from theinteraction of this single particle with the many-body electron gas of the sys-tem. In order to obtain the single-particle eigenstates, one solves a Schrodinger

45

Chapter 8. Optical Properties

like equation containing the non-local and energy-dependent self-energy in-stead of the exchange-correlation potential in the Kohn-Sham equations:

(−1

2∇2 + v0(r) +

∫d3r′

n0(r′)

|r − r′|)

Ψ(r) +

+

∫d3r′ Σ(r, r′, E)Ψ(r′) = EΨ(r), (8.1)

where Σ is the self-energy which contains all the many-body effects. Al-most, all first-principles QP studies were performed within the so-called GWapproximation, where Σ is calculated by the Hedin’s GW approximation.88

This method approximate Σ as the convolution of the LDA self-consistentGreen function and the screened Coulomb interaction within the random-phaseapproximation. The QP eigenvalues are often obtained using first-order per-turbation theory starting from LDA eigenvalues and eigenvectors.89 The GWpredicts optical excitations energies of semiconductors around 0.1 eV fromthe experimental results and the amazing fact is that the QP wave functionsare almost identical to the ones produced by the LDA.89 Then, the Kohn-Shamequations could be thought as derived from a quasi-particle theory where theself-energy is local and time average, i.e., Σ(r, r′, t) ≈ vxc(r)δ(t), where vxc

is the exchange-correlation potential. Thought in this way, the Kohn-Shameigenvalues are then approximate QP energies and could be compared to ex-perimental data. This argument is supported by QP calculations within the GWapproximation showing that the valence QP energies of semiconductors are ingood agreement with the obtained using LDA, and the conduction QP energiesdiffer by approximately a rigid energy shift.89–91

This finding is important and shows that the LDA eigenvalues have somemeaning and could be used to calculate excited states.

8.2 Calculation Of The Dielectric FunctionThe (q = 0) dielectric function is calculated in the momentum representation,which requires matrix elements of the momentum operator, p, between oc-cupied and unoccupied eigenstates. To be specific the imaginary part of thedielectric function, ε2(ω) ≡ Im ε(q = 0, ω), was calculated from92

εij2 (ω) =

4π2e2

Ωm2ω2

∑knn′σ

⟨knσ

∣∣pi

∣∣kn′σ⟩⟨

kn′σ∣∣pj

∣∣knσ⟩×

×fkn

(1 − fkn′

)δ(ekn′ − ekn − ω

). (8.2)

46

8.2. Calculation of the dielectric function

In Eq. (8.2), e is the electron charge, m its mass, Ω is the crystal volumeand fkn is the Fermi distribution. Moreover,

∣∣knσ⟩

is the crystal wave functioncorresponding to the nth eigenvalue with crystal momentum k and spin σ.

The determination of the momentum matrix elements have been calculatedin this thesis by means of the basis set of the FP-LMTO.14 With our sphericalwave basis functions, the matrix elements of the momentum operator are con-veniently calculated in spherical coordinates and for this reason the momentumis written93

p =∑

µ

e∗µpµ, (8.3)

where µ is -1, 0, or 1, and

p−1 =1√2(px − ipy), (8.4)

p0 = pz, (8.5)

and

p1 = − 1√2(px + ipy), (8.6)

e∗−1 =1√2(px − ipy), (8.7)

e∗0 = pz, (8.8)

and

e∗1 =−1√

2(px + ipy). (8.9)

In practice we calculate matrix elements of the symmetrised momentumoperator

〈i∣∣↔pµ

∣∣j〉 ≡ (〈i|pµj〉 + (−)µ〈p−µi|j〉)2

. (8.10)

The evaluation of the matrix elements in Eq. (8.2) is done over the muffin-tin region and the interstitial separately. The integration over the muffin-tinspheres is done in a way similar to what Oppener94 and Gasche92 did in theircalculations using the atomic sphere approximation (ASA). A full detaileddescription of the calculation of the matrix elements was presented some-where else.95 In our theoretical method the wave function inside the muffin-tinspheres is atomic-like in the sense that it is expressed as a radial component

47


times spherical harmonic functions (also involving the so called structure con-stants), i.e.,

Ψkmt(r) =

∑t

ctχkt , (8.11)

where inside the sphere R′ the basis function is

χkt =

Φt(r − R)

Φ(SRmt)

δR,R′ −∑t′

Φt′(r − R′)

Φ(SR′

mt)Sk

t,t′ , (8.12)

and

Φt(r) = ilY ml (r)(ϕ(r) + ω(D)ϕ(r)). (8.13)

In the Eqs. (8.12-8.13) SRmt is the muffin-tin radius for atom R, S k

t,t′ isthe structure constant, D is the logarithmic derivative, ϕ(r) is the numericalsolution to the spherical component of the muffin-tin potential and ϕ(r) is theenergy derivative of ϕ(r).12, 13 Therefore, this part of the problem is quiteanalogous to the ASA calculations92, 94 and we calculate the matrix elementsin Eq. (8.2) as ⟨

knσ∣∣pµ

∣∣kn′σ⟩

=∑t,t′

ctc∗t′⟨χt′

∣∣pµ

∣∣χt

⟩. (8.14)

Since, χt involves a radial function multiplied with a spherical harmonicfunction, i.e., f(r)Y m

l (we will label this product∣∣l, m⟩

), we can calculate thematrix elements in Eq. (8.14) using the relations

⟨l + 1, 0

∣∣p0

∣∣l, 0⟩ =l + 1√

(2l + 1)(2l + 3)

⟨f∗(r)(

δ

δr− l

r)f(r)

⟩, (8.15)

and

⟨l − 1, 0

∣∣p0

∣∣l, 0⟩ =l√

(2l − 1)(2l + 1)

⟨f∗(r)(

δ

δr+

l + 1

r)f(r)

⟩. (8.16)

A general matrix element,⟨l′, m′

∣∣pi

∣∣l, m⟩is then calculated using the Wigner-

Eckart theorem.93 Matrix elements of the momentum over the interstitial areobtained from the relation∫

Ωint

d3r(ψ∗i ∇ψj) =

1

κ2j

∫Ωint

d3r(ψ∗i ∇2∇ψj). (8.17)

By use of Green’s second theorem the expression above can be expressedas

48

8.2. Calculation of the dielectric function

∫Ωint

d3rψ∗i ∇2∇ψj = −

∫SMT

dS(ψ∗i ∇

d

drψj). (8.18)

In the expression above the surface integral is taken over the muffin-tinspheres and since the interstitial wave function by construction is the same asthe muffin-tin wave function we can use for ψ the numerical wave functiondefined inside (and on the boundary of) the muffin-tin spheres. In this way theevaluation of the integral above is done in a quite similar way as done for themuffin-tin contribution (Eqs. (8.15-8.17)) to the gradient matrix elements.

The summation over the Brillouin zone in Eq. (8.2) is calculated using lin-ear interpolation on a mesh of uniformly distributed points, i.e., the tetrahedronmethod. Matrix elements, eigenvalues, and eigenvectors are calculated in theirreducible part of the Brillouin-zone. The correct symmetry for the dielectricconstant was obtained by averaging the calculated dielectric function. Finally,the real part of the dielectric function, ε1(ω), is obtained from ε2(ω) using theKramers-Kronig transformation,

ε1(ω) ≡ Re(ε(q = 0, ω)

)= 1 +

1

π

∫ ∞

0dω′ ε2(ω

′)( 1

ω′ − ω+

1

ω′ + ω

).

(8.19)In papers XVI-XX we have calculated the optical properties of important

technological materials such as 4H-SiC, PbI2, γ-Al2O3 and the thermoelectricmaterial CsBi4Te6.

49


50

Chapter 9Sammanfattning Pa Svenska

Under forra arhundradet vaxte kvantmekniken fram mot den viktiga roll denspelar idag for var forstaelse av materien och dess tillkomst, bade pa denmakroskopiska skalan med stjarnor och andra astronomiska objekt och denmikroskopiska skalan med atomer, elektroner o.s.v. Ett av de mest tydligaexemplen pa kvantmekanikens betydels ar alla de teknologiska tillampningarsom inte hade varit mojliga att utfora och forsta utan denna. Schrodinger ekva-tionen i det icke relativistiska fallet samt Dirac ekvationen i det relativistiska, arde fundamentala ekvationerna som i princip later oss forstamateriens strukturoch dess egenskaper. Om man onskar tillampa dessa fundamentala ekvationerpa mikroskopiska system, som typiskt innehaller storleksordningen 1023 par-tiklar ar en exakt losning givetvis omojlig. Under de senaste 40 aren har densa kallade tathetsfunktional teorin (eng. density functional theory DFT) tagitform och ar nu ett kraftfullt verktyg for beraknandet och beskrivningen avmanga partiklars samverkan i ett system. Teorin har blivit ett standardverktygi dagens materialfysik. Den huvudsakliga variabeln i DFT ar tathetsdistribu-tionen n(r), som endast peror pa tre lageskoordinater, snarare an mangpartikelvagfunktion som beror pa 3N koordinater, dar N ar antal partiklar i systemet.

Inom fasta tillstandets fysik mojliggor DFT parameterfria berakningar avdensiteter, spindensiteter, grundtillstandsenergier samt relaterade storhetersasom kristall-stuktur, bulkmoduler, fasovergangar vid tryckforandringar,fonondispersioner, magnetiska moment m.m. I denna avhandling undersoksdessa egenskaper for ett antal olika material med hjalp av tathetsfunktionalteorin allteftersom olika problemstallningar inom material vetenskapen gasigenom.

Vi har studerat kristallina fasovergangar i ett antal olika material sasomTiO2, de tunga alkalimetallerna Cs och Rb o.s.v. Vi har upptackt att titanoxi-den uppvisar faser med exceptionellt intressanta egenskaper for manga tillamp-ningar saval som for prototyper for experimentella och teoretiska

51

Chapter 9. Sammanfattning Pa Svenska

undersokningar. Materialet ifraga ar speciellt intressant pa grund av de mangakomersiella tillampningarna som inkluderar pigment, katalys, elektronik, elek-trokemisk och keramisk industri. Hog-trycks transformationer av titanoxid haraven vackt uppmarksamhet ur ett geologiskt perspektiv pa grund av analoginmed kvarts (SiO2), den mest forekommande foreningen i jordens mantel.

Bland fasovergangar i fasta material har nyligen tva speciella fall dragit tillsig mycket uppmarksamhet bade experimentellt och teoretiskt. Det ar fraganom isostrukturella fasovergangar i cerium respektive cesium under tryck. Efter-som kristall symmetrin ej andras kallas dessa elektroniska fasovergangar sominvolverar 4f elektronerna i Ce samt en drastisk forandring av antalet d elek-troner i Cs. Nyligen har dock hogtrycksexperiment indikerat att det faktisktror sig om en kristallin fasovergang i fallet for Cs, varfor det finns anledningatt omprova den teoretiska forstaelsen for denna overgang.

Aven mekaniska egenskaper hos den nyupptackta supraledaren MgB2 un-dersoks, motiverat av avsaknaden av teoretisk och experimentell informationsom ar viktig vid en eventuell teknisk anvandning av materialet. De mekaniskaegenskaperna kan namligen utgora en begransning av praktiska tillampningar.En forsta undersokning borde vara en detaljerad studie av elastiska egenskaper,som ar nara forknippade med formbarhet och bracklighet.

Vidare undersoks ett antal material som kanlampa sig som katoder i diversebatteri tillampningar. Ett lovande material for forbattringen av Li batteriersprestanda utgors av LiMPO4, dar M kan vara Mn, Fe eller Co. Det finns avenett okat intresse av Mg batterier, varfor vi aven undersokt den potentiella ka-toden MgxMo6S8. Vi har teoretiskt undersokt olika relevanta egenskaper somspanning, energi densitet och specifik energi. Har lampar sig teoretiska simu-leringar synnerligen lampligt da man kan utfora en systematisk studie av olikastrukturer, ledningsformagor, instabiliteter o.s.v.

Pa sistone har man intensivt sokt efter ferromagnetisk ordning i dopademagnetiska halvledare med tanke pa mojlig spin transport och mojliggorandetav tillverkningen avintressanta elektroniska komponenter. Vi har i samman-hanget studerat Mn dopad ZnO.

Dessa tillampningar av tathetsfunktonal teorin utgor en liten del av fastatillstandsfysiken, och den intresserade lasaren uppmuntras att ta del av denpublicerade litteraturen for tillampningar i andra omraden av materialveten-skapen.

52

Acknowledgements

I am most grateful to my supervisor Rajeev Ahuja who has encouraged mystudies all these years. He has lead me to work in different projects where Ihave learnt a lot of solid state physics and materials science. And I am evenmuch more thankful for his friendship, jokes and advises.I am very thankful to Borje Johansson. He has provided all the necessarythings which has made possible this work. It is very nice when he comes latein the evening and says: “hello, hello”, cause people here cares about the groupand we just love to work at CMT.I would like to thank Olle Eriksson, he is always open to teach and share hisknowledge. I also want to express my gratitude to Biplad Sanyal to make mylife a little more complicated with his thoughts about physics.Thanks to Clas Persson, Maurizio Mattesini,Olivier Bengone, Sergei Simak,Eyvaz Isaev and Peter Oppeneer. They always listen to my questions and cameup with good ideas when some of my research turned out to be difficult. Bythe way Clas, the next time we go to watch football we must to choose a bettermatch, and GSE sucks! Big thanks to Bengt Holm, without him it couldn’t bepossible for me have been at CMT, and for his great support during the writingof this thesis.I thank Alexei Grechnev, Weine Olovsson, Velimir Meded, Leonid Pourovskii,and Adrian Taga for all the nights of drinking and chatting. Guys, out of Weineyou need to improve a little be more :-)Thanks to Sa Li, Jailton Souza de Almeida and Moyses Araujo for all thesupport during these years. I hope to go on the near future to enjoy the Salvadorbeaches with you guys.I want to thank to our former and actual system administrators Lars Bergqvist,Weine Olovsson, Thomas Dziekan, Peter Larsson, Carlos Ortiz and Bjorn Sku-bic for all their patient with me.I wish to thank Hakan Hugosson for the AC/DC tickets, we’ve rock in Goter-

53

borg; Massimiliano Colarieti-Tosti for all his help in the beginning and I don’tknow whether say thanks or not to Elisabeth Sjoestedt, you almost got me withthe cooking course ;-)I thank Erik Holmstrom and Raquel Lizarraga, Mauro Torres and Paula Col-ima, Raul Valdivia and Lennart Harsegrstad for their friendship during all theseyears in Uppsala.Last, but no least, I want to say thanks to my parents, my brother and especiallyGloria “mi munequita” for their help, support and love.

54

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59

Acta Universitatis UpsaliensisComprehensive Summaries of Uppsala Dissertations

from the Faculty of Science and TechnologyEditor: The Dean of the Faculty of Science and Technology

Distribution:Uppsala University Library

Box 510, SE-751 20 Uppsala, Swedenwww.uu.se, [email protected]

ISSN 1104-232XISBN 91-554-6016-X

A doctoral dissertation from the Faculty of Science and Technology, UppsalaUniversity, is usually a summary of a number of papers. A few copies of thecomplete dissertation are kept at major Swedish research libraries, while thesummary alone is distributed internationally through the series ComprehensiveSummaries of Uppsala Dissertations from the Faculty of Science and Technology.(Prior to October, 1993, the series was published under the title “ComprehensiveSummaries of Uppsala Dissertations from the Faculty of Science”.)

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