Materials Design from ab initio Calculations - DiVA portal164658/FULLTEXT01.pdf · Materials Design...

Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 982

Materials Design from ab initioCalculations

BY

SA LI

ACTA UNIVERSITATIS UPSALIENSISUPPSALA 2004

Printed in Sweden by Universitetstryckeriet, Uppsala 2004

To my family

Front page illustration

The electronic charge density distribution for

the -AlOOH phase in the (001) plane.

List of Publications

I S. Li, R. Ahuja, and B. JohanssonPressure-induced phase transitions of KNbO3

J. Phys. C 14, 10873 (2002)

II Sa Li, R. Ahuja, and B. JohanssonHigh pressure theoretical studies of actinide dioxidesHigh Pressure Research 22, 471 (2002)

III J. K. Dewhurst, R. Ahuja, S. Li, and B. JohanssonLattice Dynamics of Solid Xenon under PressurePhys. Rev. Lett. 88, 075504 (2002)

IV B. Holm, R. Ahuja, S. Li, and B. JohanssonTheory of the ternary layered system Ti-Al-NJ. of Appl. Phys. 91, 9874 (2002)

V H.W.Hugosson, G.E.Grechnev, R.Ahuja, U.Helmersson, L.Sa,and O.ErikssonStabilization of potential superhard RuO2 phases: Atheoretical studyPhys. Rev. B 66, 174111 (2002)

VI Z. M. Sun, R. Ahuja, S. Li, and J. M. SchneiderStructure and bulk modulus of M2AlCAppl. Phys. Lett. 83, 899 (2003)

VII S. Li, R. Ahuja and Y. Wang, and B. JohanssonCrystallographic structures of PbWO4

High Pressure Research 23, 343 (2003)

VIII Z. M. Sun, S. Li, R. Ahuja, and J. M. SchneiderCalculated elastic properties of M2AlCSolid State Commun. 129, 589 (2004)

IX A. B. Belonoshko, S. Li, R. Ahuja, and B. JohanssonHigh-pressure crystal structure studies of Fe, Ru andOsJ. Phys. Chem. Sol. (In press)

X S. Li, R. Ahuja, and B. JohanssonWolframite: the post-fergusonite phase in YLiF4

J. Phys. C 16, 983 (2004)

XI S. Li, R. Ahuja, and B. JohanssonThe Elastic and Optical Properties of the High-PressureHydrous Phase δ-AlOOHSubmitted to Phys. Rev. B

XII J. -P. Palmquist, S. Li, P. O. A. Persson, J. Emmerlich, O.Wilhelmsson, H. Hogberg, M. I. Katsnelson, B. Johansson,R. Ahuja, O. Eriksson, L. Hultman, and U. JanssonNew MAX Phases in the Ti-Si-C System Studied byThin Film Synthesis and ab initio CalculationsSubmitted to Phys. Rev. B

XIII P. Finkel, J. D. Hettinger, S. E. Lofland, K. Harrell, A. Gan-guly, M. W. Barsoum, Z. sun, S. Li, and R. AhujaLow Temperature Elastic, Electronic and TransportProperties of Ti3SixGe1−xC2 Solid SoutionsSubmitted to Phys. Rev. B

XIV M. Magnuson, J. -P. Palmquist, M. Mattesini, S. Li, R.Ahuja, O. Eriksson, J.Emmerlich, O. Wilhelmsson, P. Ek-lund, H. Hogberg, L. Hultman, and U. JanssonElectronic structure of the MAX-phases Ti3AC2 (A=Al,Si, Ge) investigated by soft X-ray absorption andemission spectroscopiesSubmitted to Phys. Rev. B

XV Z. M. Sun, D. Music, R. Ahuja, S. Li, and J. M. SchneiderBonding and classification of nanolayered ternary car-bidesSubmitted to Phys. Rev. B

XVI R. Ahuja, H. W. Hugosson, S. Li, B. Johansson, and O.ErikssonElectronic structure and optical properties of C60

Submitted to Phys. Rev. B

XVII R. Ahuja, L. M. Huang, S. Li, and Y. WangHigh pressure structural phase transition in Zircon

(ZrSiO4)Submitted to Phys. Rev. B

XVIII A. Grechnev, S. Li, R. Ahuja, O. Eriksson, and O. JanssonA possible MAX new phase, Nb3SiC2, predicted fromFirst Principles TheorySubmitted to Appl. Phys. Lett.

XIX S. Li, H. Pettersson, C. G. Ribbing, B. Johansson, M. W.Barsoum, and R. AhujaOptical properties of Ti3SiC2 and Ti4AlN3

In manuscript

Comments on my contribution to the papers

For those papers I, II, VII, X, XI and XIX, I’m the first author of, Itook the major responsibilities in calculation design and their execution,as well as paper drafting. For papers IX and XIV, I have performed cal-culations and contributed to the paper writings. In the experimentalpaper XII, I have carried out calculations and written up the theoreti-cal parts of the publication. As regards paper III, IV, V, VI, VIII, IX,XIII, XV, XVI, XVII and XVIII, I have contributed either with ideasor calculations and result analysis.

Contents

1 Introduction 3

2 Many body problem 52.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 The Hartree approximation . . . . . . . . . . . . . . . . . 62.3 Hartree-Fock approximation . . . . . . . . . . . . . . . . . 72.4 Density functional theory . . . . . . . . . . . . . . . . . . 7

3 Computational methods 93.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Electronic structure methods . . . . . . . . . . . . . . . . 113.3 The LMTO method . . . . . . . . . . . . . . . . . . . . . 12

3.3.1 Muffin-tin orbitals . . . . . . . . . . . . . . . . . . 123.3.2 The LMTO-ASA method . . . . . . . . . . . . . . 14

3.4 Full potential LMTO method . . . . . . . . . . . . . . . . 143.4.1 The basis set . . . . . . . . . . . . . . . . . . . . . 153.4.2 The LMTO matrix . . . . . . . . . . . . . . . . . . 163.4.3 Total energy . . . . . . . . . . . . . . . . . . . . . 17

3.5 Projector Augmented Wave Method . . . . . . . . . . . . 183.5.1 Wave function . . . . . . . . . . . . . . . . . . . . 183.5.2 Charge density . . . . . . . . . . . . . . . . . . . . 203.5.3 Total energy . . . . . . . . . . . . . . . . . . . . . 21

3.6 Ultrasoft pseudopotential . . . . . . . . . . . . . . . . . . 223.7 PAW and US-PP . . . . . . . . . . . . . . . . . . . . . . . 23

4 Phase transitions 254.1 Static total energy calculation . . . . . . . . . . . . . . . . 254.2 Elastic stability criteria . . . . . . . . . . . . . . . . . . . 264.3 Bain path . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.4 Dynamical stability (phonon calculation) . . . . . . . . . . 274.5 Equation of state . . . . . . . . . . . . . . . . . . . . . . . 29

4.5.1 Murnaghan equation of state . . . . . . . . . . . . 294.5.2 Birch-Murnaghan equation of state . . . . . . . . . 30

4.5.3 Universal equation of state . . . . . . . . . . . . . 314.5.4 Comparison of different EOS for Fe . . . . . . . . . 31

5 Semiconductor optics 335.1 Dielectric function . . . . . . . . . . . . . . . . . . . . . . 335.2 Dielectric function of PbWO4 . . . . . . . . . . . . . . . . 345.3 Solar energy materials . . . . . . . . . . . . . . . . . . . . 36

6 MAX phases 396.1 MN+1AXN phase . . . . . . . . . . . . . . . . . . . . . . . 396.2 Phase stability in Ti-Si-C system . . . . . . . . . . . . . . 406.3 Chemical bonding in 312 phases . . . . . . . . . . . . . . 426.4 XAS and XES calculation . . . . . . . . . . . . . . . . . . 436.5 DOS with electrical conductivity . . . . . . . . . . . . . . 436.6 Optical properties . . . . . . . . . . . . . . . . . . . . . . 466.7 Surface energy . . . . . . . . . . . . . . . . . . . . . . . . 486.8 Ductility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

7 Sammanfattning pa svenska 53

Acknowledgments 57

Bibliography 59

Chapter 1

Introduction

In 1998, Walter Kohn was awarded Nobel Prize in chemistry. Hiswork has enabled physicists and chemists to calculate the properties ofmolecules and solids using computers, without performing experimentsin the laboratory. Already around 1930 physicists were fully aware ofthe quantum mechanical equations governing the behavior of systems ofmany electrons, but were incapable of exactly solving them in all but thevery simplest cases. They developed several approximation schemes, butnone of them was very successful. In 1964 Walter Kohn [1] and PierreHohenberg and somewhat later Walter Kohn and Lu Jeu Sham [2] hadproved an idea that was essential to their solution scheme, which isnow called density functional theory (DFT). DFT differs from quantumchemical methods and does not yield a correlated N-body wavefunction.Many of the chemical and electronic properties of molecules and solidsare determined by electrons interacting with each other and with atomicnuclei. In DFT, knowing the average density of electrons at all pointsin space is enough to uniquely determine the total energy, hence alsoa number of other properties of the system. DFT theory is based onone-electron theory and shares many similarities with the Hartree-Fockmethod. DFT has come to prominence over the last decade as a methodpotentially capable of generating very accurate results at relatively lowcost.

By means of state-of-the-art DFT, great achievements in calculatingmechanical and electronic properties of solids have been made. Differentapproximations related to the construction of exchange-correlation func-tionals used in DFT for ground and excited states have been introduced,for example, local density approximation (LDA) [3] and generalized gra-dient approximations (GGA) [4], LDA+U [5] and GW [6] approximation.Applications to materials modeling in general as well as to nanotubes,quantum dots, and artificial molecules are incorporated. The calcula-

3

4 CHAPTER 1. INTRODUCTION

tions have developed so rapidly and reached such an advanced level thatit now becomes the right time for a theory-based approach, to supportas well as supplement experiment. This development has also been fa-cilitated by the continued upgrading of powerful computers, which havemade it possible to calculate materials properties with an impressive ac-curacy. The main thrust of the present thesis is to use highly accuratetheoretical ab initio methods to predict materials properties and searchfor the new engineering materials.

Recently, the nanolayered ternary compounds MN+1AXN (MAX)[7], where N =1, 2 or 3, M is an early transition metal, A is an A-group (mostly IIIA and IVA) element, and X is either C and/or N, haveattracted increasing interests owing to their unique properties. Theseternary carbides and nitrides combine the unique properties of bothmetals and ceramics. Like metals, they are good thermal and electri-cal conductors with electrical and thermal conductivities ranging from0.5 to 14×106 Ω−1m−1 [8], and from 10 to 40 W/m·K [9], respectively.They are relatively soft with Vickers hardness of about 2-5 GPa. Likeceramics, they are elastically stiff, some of them, like Ti3SiC2, Ti3AlC2

and Ti4AlN3 also exhibit excellent mechanical properties at high tem-peratures. They are resistant to thermal shock and unusually damagetolerant, and exhibit excellent corrosion resistance. Above all, unlikeconventional carbides or nitrides, they can be machined by conventionaltools without lubricant. All these excellent properties make MAX phasesa new family of technologically important materials. Systematic studieson MAX phases regarding their electronic, bonding, elastic and opticalproperties have been carried out and described in this thesis. The rela-tion between mechanical and bonding properties of these solids elabo-rated by the present electronic structure studies provides robust supportto the exploration of the behavior of these ternary compounds.

The rest of the thesis is organized as the following. The secondchapter describes the main idea of density functional theory. The thirdchapter deals with the computational methods. My research results canbe classified in three parts, 1) Phase transition and its related properties,such as phase stability and equations of states, which are elaborated inChapter 4. 2) The calculated results for the linear optical properties ofcertain semiconductors, such as the scintillating crystal PbWO4 and thesolar energy materials CuIn1−xGaxSe2 are described in Chapter 5. 3)In Chapter 6, applications of DFT in MAX phases are presented, whichinclude their electronic, bonding, mechanical and optical properties.

Chapter 2

Many body problem

5

2.1 Introduction

In this work our focus is restricted to bulk materials. This means thatpossible surface effects are excluded and that we consider the bulk tobe an infinite crystal. To study the properties of atoms, molecules andsolids, the so-called Schrodinger Equation has become the basic tool thatthe solid state theorists work with. The time-independent SchrodingerEquation has the form

HΨ = EΨ, (2.1)

which has been proven to quite exactly solve the problem with onenucleus and one electron, such as the hydrogen atom. However fora solid the system is described by the many electron wave functionΨ(r1, r2, ..., rN ), where ri gives the position and spin of particle i. Ina solid we are typically dealing with 1023 particles and this makes theproblem very complex. Let us have a closer look at the Hamiltonianthat describes the whole bulk system:

H = − h2

2

∑

k

2

2Mk− h2

2m

∑

i

2i +

1

2

∑

k =l

Z2e2

|Rk − Rl|

+1

2

∑

i=j

e2

|ri − rj | −∑

k

∑

i

Ze2

|ri − Rk|, (2.2)

where h is Planck constant, Rk is the nuclear coordinate for the k’thnucleus, ri the electronic coordinate for the i’th electron and Mk andm are the corresponding masses. Z is the nuclear charge. The first twoterms in Eq. (2.2) are the kinetic energy operators for the nuclei andelectrons, respectively and the third term describes the nuclei-nucleiinteraction, VNN. The next term in Eq. (2.2) is the electron-electroninteraction, Vee. The last term is the interaction between the electrons

6 CHAPTER 2. MANY BODY PROBLEM

and nuclei and could be regarded as an external potential, Vext, actingupon the electrons.

Since the Hamiltonian describes a strongly coupled system involvingboth electrons and nuclei, it is quite difficult to solve. The first approx-imation we introduce is Born-Oppenheimer approximation. Within thisapproximation the nuclei are taken to be stationary, so that the nuclearkinetic energy will be zero. Now the total energy Hamiltonian can beexpressed as

H = Te + VNN + Vee + Vext = − h2

2m

∑i

2i +

1

2

∑k =l

Z2e2

|Rk − Rl|

+1

2

∑i=j

e2

|ri − rj | −∑k

∑i

Ze2

|ri − Rk| (2.3)

2.2 The Hartree approximation

The Hartree approximation provides one way to reduce Eq. ( 2.3) toa problem which we can solve easily. In Eq. (2.3), the potential whicha certain electron feels depends upon all the other electrons’ positions.However this potential can be approximated by an average single-particlepotential

Vd(ri) = e2∑j =i

nj|ψj(rj)|2|ri − rj | , (2.4)

where nj are the orbital occupation numbers and ψj(rj) is a single-particle wave-equation, i.e. a solution to the one-particle wave-equation,[

− h2

2m2 +Vext + Vd(ri)

]ψi(ri) = εiψi(ri) (2.5)

With this simplification the set of equations now become separable.However the equations are still non-linear and have to be solved self-consistently by iteration.

According to the Pauli exclusion principle, two electrons can notbe in the same quantum state. However the wave function in Hartreetheory

Ψ(r1σ1, r2σ2 ..., rNσN ) =N∏i

ψi(ri, σi) (2.6)

is not antisymmetric under the interchange of electron coordinates andaccordingly does not follow the Pauli principle. Furthermore, the Hartreeapproximation fails to represent how the configuration of the N −1 elec-trons affects the remaining electrons. This problem has been rectifiedby Hartree-Fock theory.

2.3. HARTREE-FOCK APPROXIMATION 7

2.3 Hartree-Fock approximation

We assert that a solution to HΨ = EΨ is given by any state Ψ thatmakes the following quantity stationary:

E =(Ψ,HΨ)

(Ψ,Ψ). (2.7)

According to the variational principle [10], the normalized expectationvalue of energy is minimized by the ground-state wave function Ψ.

A better description is to replace wave function Eq. (2.6) by a Slater-determinant of one-electron wave functions

Ψ(r1σ1, r2σ2 ..., rNσN ) =1√N !

∣∣∣∣∣∣∣∣∣∣ψ1(r1σ1) ψ1(r2σ2) · · · ψ1(rNσN )ψ2(r1σ1) ψ2(r2σ2) · · · ψ2(rNσN )

......

. . ....

ψN (r1σ1) ψN (r2σ2) · · · ψN (rNσN )

∣∣∣∣∣∣∣∣∣∣.

(2.8)This is a linear combination of products of the form given by of Eq.(2.6) and all other products obtainable from the permutation of the riσi

among themselves. The Hartree-Fock equation which follows from anenergy-minimization is given by:[

− h2

2m 2 +Vext(ri) + Vd(ri)]ψi(ri)

−∑j

∫dr′ e2

|r−r′|ψ∗j (r

′)ψi(r′)ψj(r)δsisj = εiψi(ri). (2.9)

The last term on the left side due to exchange originates from the wavefunction (Slater determinant). This term only operates between elec-trons having the same spin, this is called the exchange term. In additionto this, there should also be a correlation interaction between electrons,which is not included here. Consequently, the correlation energy can bedescribed as the difference between the exact energy and the Hartree-Fock energy. Another more effective approach to treat the electrons ina solid will be introduced in the following sections.

2.4 Density functional theory

The density functional theory is based on two fundamental theoremsintroduced by Hohenberg and Kohn [1], and later extended by Kohnand Sham [2].

First, the ground-state energy E of a many electron system is shownto be a unique functional of the electron density n(r),

E[n] =

∫drVext(r)n(r) + F [n]. (2.10)

8 CHAPTER 2. MANY BODY PROBLEM

According to the Hohenberg and Kohn theory, we can separate the func-tional F [n] into two terms

F [n] =

∫ ∫n(r)n(r′)|r − r′| drdr′ + G[n]. (2.11)

The first term on the right is the usual electron-electron Coulomb contri-bution, and the second term G[n] is a universal functional of the electrondensity. Kohn and Sham proposed the following approximation for thefunctional G[n],

G[n] = T [n] + Exc[n], (2.12)

where T [n(r)] is the kinetic energy of a system of non-interacting elec-trons with electron density n(r). However, it is impossible to find anexact expression for the exchange-correlation energy Exc.

To deal with the problem of the exchange-correlation energy Exc, amost useful approximation has been introduced, namely the local densityapproximation (LDA). This approximation is exact in the limit of slowlyvarying densities. In LDA the exchange-correlation energy is replacedby

Exc[n] =

∫n(r)εxc[n]dr, (2.13)

where the εxc is the exchange and correlation energy per particle of ahomogeneous electron gas.

Now we write the electron density in terms of one-electron wavefunc-tions, ψ(r), as

n(r) =N∑

i=1

ψ∗i (r)ψi(r), (2.14)

where N is the total number of electrons.The one-particle Schrodinger equation now becomes

[−2 +Veff(r)]ψi(r) = εiψi(r), (2.15)

where the atomic unit h = 2me = e2/2 = 1 has been used. The effectiveone-electron potential, Veff , is given by

Veff(r) = Vext(r) +

∫2n(r′)|r − r′|dr + Vxc(r), (2.16)

where

Vxc(r) =δ(n(r)εxc[n(r)])

δ(n(r)). (2.17)

The set of equations, (2.15)− (2.17) are known as the Kohn-Sham equa-tions, which have to be solved in a self-consistent way, just like theHartree and Hartree-Fock approximations.

Chapter 3

Computational methods

3.1 Introduction

In the previous chapter we have obtained an effective one-electron equa-tion which can be solved in a self-consistent way,

[−2 +Veff(r)]ψi(r) = εiψi(r). (3.1)

The method of solving this eigenvalue equation makes use of the sym-metry of the crystal structure.

For an infinite crystal the potential is periodic, i.e. invariant underlattice translations R. For a monoatomic solid we have

V (r + R) = V (R), (3.2)

where R is defined by

R = n1a1 + n2a2 + n3a3, (3.3)

in which ni are integers and the set of vectors ai are the real space Bra-vais lattice vectors that span the crystal cell. According to the Bloch’stheorem, the eigenstates can be chosen to take the form of a plane wavetimes a function with the periodicity of the Bravais lattice;

ψk(r + R) = eik·Rψk(r). (3.4)

where the k is the so called Bloch wave vector. Now, the one-electronfunction can be characterized by the Bloch vector k. As a consequence,Eq. (3.1) can be written as

Heff(r)ψn(k; r) = εn(k)ψn(k; r), (3.5)

where the index i in Eq. (3.1) has been substituted by the quantumnumber n, the band index. The one-electron wave function ψn and the

9

10 CHAPTER 3. COMPUTATIONAL METHODS

corresponding eigenvalues, εn are now be characterized by the Blochwave vector k. The Bloch vector k used to label the one-electron statesis conveniently viewed as a vector in the reciprocal space. A latticevector G in the reciprocal space is constructed as

G = x1b1 + x2b2 + x3b3, (3.6)

where the xi are the integers and the bi are the basic vectors of thereciprocal lattice:

ai · bj = δij. (3.7)

Regarded as functions of the wave vector k, the energy eigenvalues andwave functions have the translational symmetry of the reciprocal lattice,

ε(k) = ε(k + G) (3.8)

ψ(k, r) = ψ(k + G, r). (3.9)

As the Wigner-Seitz cell is the smallest unit that characterizes the crys-tal structure, the Brillouin Zone (BZ) is the smallest unit that builds upthe whole reciprocal lattice by repeating itself periodically. By meansof translational symmetry as well as other point group symmetries (ro-tations, mirror, inversion operations), we can reduce the problem to theirreducible part of the BZ, the smallest zone which defines a completeset of wave vectors. For example, in a lattice with full cubic symmetry,the irreducible part of the BZ is only 1/48 of the full BZ. It is only inthis part we need to solve the electronic structure problem.

According to the Pauli exclusion principle the eigenstates with eigen-value εi(k) are occupied from the lowest eigenvalue up to the Fermienergy, εF . The Fermi energy is defined by

N =

∫ εF

−∞D(ε)dε, (3.10)

where N is the number of valence electrons and D(ε) is the density ofstates (DOS),

D(ε) =2

8π3

∫S(ε)

dS

| ε(k)| . (3.11)

The integration is carried out over a surface of constant energy, S(ε), inthe first BZ. The one electron states most relevant for most of the phys-ical properties are those with energies around the Fermi level. Thesestates are closely related to crystal structure stability, transport prop-erties, susceptibility, etc..

3.2. ELECTRONIC STRUCTURE METHODS 11

3.2 Electronic structure methods

In practice the solution to equation (3.5), ψn(k; r), can be expanded insome basis sets. To solve the problem we need to resort to one of manyavailable electronic structure methods.

For the different selection of the basis set, electronic structure meth-ods can be divided into two parts [11]:

Fixed basis set Variable Basis Set

Plane Wave Augmented Plane WaveTight Binding Korringa Kohn Rostoker

Pseudopotential Linear Augmented Plane WaveOrthogonalized Plane Wave Linearized Muffin Tin Orbitals

Linear Combination of Atomic Orbitals Augmented Spherical Wave

The first set of methods obey the Bloch condition explicitly. Thatis, in the expansion

ψ(r) =∑n

cnφn(r) (3.12)

the basis functions are fixed and the coefficients cn are chosen to mini-mize the energy. One disadvantage of these methods is that the wave-functions are fixed. This often leads to great difficulty in obtaining asufficiently converged basis set.

In the second set of methods, the wavefunctions are varied. Thisis performed by introducing energy dependent wavefunctions φn(ε, r).The wavefunctions are energy dependent and have the form of

ψ(ε, r) =∑n

cnφn(ε, r), (3.13)

However, the Bloch condition is not automatically fulfilled. The solu-tions in one unit cell are chosen to fit smoothly to those of the neighborcells, thus fulfilling the Bloch condition “indirectly”. As the wavefunc-tion can be modified to the problem at hand, these techniques convergevery fast in the number of required basis function. In APW and KKR,the price for doing so is the additional parameter ε. At every k-pointof the band structure, equation (3.5) must be solved for a large numberof ε. Solutions only exist for those ε that are actual eigenvalues. Whilethese methods are accurate, they are also time consuming. The solutionto this problem is to linearize the energy dependent orbitals as is donein LAPW, LMTO, and ASW. They are expanded as a Taylor expansionin ε so that the orbitals themselves are energy independent, although


Figure 3.1: Muffin-tin part of the crystal potential V(r) and radial wavefunction. SMT is the radius of the muffin-tin sphere, SE is the radius ofthe escribed sphere and VMTZ is the potential of the interstitial region.

the expansion retains the energy dependence. The variational equation(3.5) thus has to be solved only once for each k-point. These methodsare extremely rapid and only slightly less accurate than other non-linearmethods.

3.3 The LMTO method

During the last decades, the linear-muffin-tin-orbital (LMTO) [12] me-thod has become very popular for the calculation of the electronic struc-ture of crystalline systems. The LMTO method combines the followingadvantages: (1) it uses a minimal basis, which leads to high efficiencyand makes calculations possible for large unit cell; (2) it treats all ele-ments in the same way, so that d and f metals as well as atoms witha large number of core states can be considered; (3) it is very accurate,due to the augmentation procedure which gives the wave function thecorrect shape near the nuclei; (4) it uses atom-centered basis functions ofwell-defined angular momentum, which makes the calculated propertiestransparent [13].

3.3.1 Muffin-tin orbitals

The crystal is divided into non-overlapping muffin-tin spheres surround-ing the atomic sites and an interstitial region outside the spheres. Insidethe muffin-tin sphere the potential is assumed to be spherically symmet-ric while in the interstitial region the potential, VMTZ, is assumed to be

3.3. THE LMTO METHOD 13

constant or slowly varying. Because the potential in the interstitial isconstant we can shift the energy scale so as to set it to zero. In thefollowing, we consider a crystal with only one atom per primitive cell.Within a single muffin-tin well we define the potential

VMT(r) =

V (r) − VMTZ , |r| < SMT

0 , |r| > SMT(3.14)

Here V (r) is the spherically symmetric part of the crystal potential. Theradii of the muffin-tin spheres are chosen so that they do not touch eachother. In the following, SMT is expressed by S.

Now we try to solve the Schrodinger equation for muffin-tin potential,

[−2 +VMT]ψ(ε, r) = (ε − VMTZ)ψ( r). (3.15)

We define the kinetic energy κ2 in the interstitial region by

κ2 = ε − VMTZ (3.16)

For an electron moving in the potential from an isolated muffin-tin wellembedded in the flat potential VMTZ, the spherical symmetry can extendthroughout all space and the wave functions are

ψL(ε, r) = ilY ml (r)ψl(ε, r) (3.17)

where we use the convention that r = |r| and r is the direction of r. Aphase factor il is included.

To obtain basis functions which are approximately independent ofenergy, reasonably localized, and normalizable for all values of κ2, An-derson [14] accomplished these by Muffin-tin orbitals. A spherical Besselfunction that cancels the divergent part of ψl(ε, κ, r) and simultaneouslyreduces the energy and potential dependence of the tails, we have themuffin-tin orbitals in form of

χlm(ε, r) = ilY ml (r)

ψl(ε, r) + Pl(ε)

(r/S)l

2(2l+1) , |r| < S

(r/S)−l−1 , |r| > S(3.18)

where ψl(ε, r) is a solution of the radial Schrodinger equation inside theatomic sphere. The potential function

Pl(ε) = 2(2l + 1)Dl(ε) + l + 1

Dl(ε) − l(3.19)

and the normalization of ψl(ε, r) are determined by satisfying differen-tiability and continuity of the basis function on the sphere boundary.Here the Dl(ε) is the logarithmic derivative of the wave function. Thetail of the basis function, i.e. the part outside the muffin-tin spherecan in general be written as Neumann function. But in Eq. (3.16) thekinetic energy of this tail, known as κ2, is chosen to be zero. Thereforethe Neumann function has a simple form like this.


3.3.2 The LMTO-ASA method

In the atomic sphere approximation, LMTO-ASA, the muffin-tin spheresare overlapping in such a way that the total volume of muffin-tin sphereis the same as the atomic volume. This means that the muffin-tin radiusS is equal to the Wigner- Seitz radius SWS where the total volume peratom is given by V = (4π/3)S3

WS. In the ASA, the potential is alsoassumed to be spherically symmetric inside each muffin-tin sphere andthe kinetic energy of the basis functions defined in the interstitial isrestricted to be constant, actually zero in the calculation.

In order to construct a linear method, the energy dependent termsin the muffin-tin spheres of the Eq. (3.18) are replaced by the energyindependent function Φ. The function is defined as a combination ofradial functions and their energy derivative

Φ(D, r) = φl(r) + ω(D)φl(r), (3.20)

where w(D) is a function of the logarithmic derivative and w(D) shouldmake the energy dependent orbitals χlm(ε, r) defined in the Eq. (3.18)continuous and differentiable at the sphere boundary S. The boundarycondition determines D = −l − 1, The so obtained energy independentorbital can now be written as

χlm(ε, r) = ilY ml (r)

Φl(D, r) , |r| < S(r/S)−l−1 . |r| > S

(3.21)

3.4 Full potential LMTO method

The FP-LMTO calculations are all electron, fully relativistic, withoutshape approximation to the charge density or potential. The crystal isdivided into non-overlapping muffin-tin sphere and an interstitial regionoutside the spheres. The wave function is then represented differentlyin the two types of regions. Inside a muffin-tin spheres, the basis func-tions are as in the LMTO-ASA method. They are Bloch sum of linearmuffin-tin orbitals and are expanded by structure constant, φν(r) andφν(r). However the kinetic energy is not, as in the ASA approximation,restricted to the zero in the interstitial region. For simplicity, here weonly consider a monoatomic solid, and suppress the atomic site index.The κ dependent linear muffin-tin orbitals can now be written as

ψκlm(k, r) = χκlm(r) +∑lm

Jκlm(r)Sκlm,l′m′(k), (3.22)

where

χlm(r) = ilY ml (r)

−iκhl(κS) Φ(Dh,r)

Φ(Dh,S) , |r| < S

−iκhl(κr) , |r| > S(3.23)

3.4. FULL POTENTIAL LMTO METHOD 15

and

Jκlm(r) = ilY ml (r)

Jl(κS)(κS) Φ(DJ ,r)

Φ(DJ ,S) , |r| < S

Jl(κr) . |r| > S(3.24)

Inside the muffin-tin at τ , we can also expand the electron densitiesand potential in spherical harmonics times a radial function,

nτ (r)|τ =∑h

nτ (h; rτ )Dh(rτ ), (3.25)

Vτ (r)|τ =∑h

Vτ (h; rτ )Dh(rτ ), (3.26)

where Dh are linear combinations of spherical harmonics, Y ml (r). Dh

are chosen here because we need an invariant representation of the localpoint group of the atomic site contained in the muffin-tin. The expansioncoefficients nτ (h; rτ ) and Vτ (h; rτ ) are numerical functions given on aradial mesh.

In the interstitial region, the basis function, charge densities andpotential are expressed as Fourier series,

ψ(k; r)|I =∑G

ei(k+G)·rψ(k + G), (3.27)

nI(r)|I =∑G

nGei(k+G)·r, (3.28)

VI(r)|I =∑G

VGei(k+G)·r, (3.29)

where G are reciprocal lattice vectors spanning the Fourier space.

3.4.1 The basis set

Envelope function is the basis function in the interstitial region. Bychoosing appropriate envelope functions, such as plane waves, Gaus-sians, and spherical waves (Hankel functions), we can generate variouselectronic structure methods (LAPW, LCGO, LMTO, etc.). The LMTOenvelope function is represented as below,

Klm(κ; r) = −κl+1ilY ml (r)

−h+

l (κr) , κ2 ≤ 0nl(κr) , κ2 > 0

(3.30)

where nl is a spherical Neumann function and h+l is a spherical Hankel

function of the first kind. The envelope function is a singular Hankel orNeumann functions with regards to the sign of the kinetic energy. This


introduces a κ dependence for the basis functions inside the muffin-tinsphere through the matching conditions at the sphere boundary. Thisis not a problem. Using a variational method, the ground state still hasseveral basis functions with the same quantum numbers, n, l and m, butdifferent κ2. This is called a double basis.

The basis set can always contain different bases corresponding to theatomic quantum number l but with different principle quantum numbersn. A basis constructed in this way forms a fully hybridizing basis set,not a set of separate energy panels.

To illustrate the way the basis set is constructed, we take fcc Ce [15]as an example. The ground state configuration is 4f 15d16s2. Thus weinclude the 6s, 6p, 5d, 4f as valence states. To reduce the core leakageat the sphere boundary, we also treat the core states 5s and 5p as semi-core states. By this kind of construction, the basis set becomes morecomplete.

3.4.2 The LMTO matrix

We now introduce a convenient notation for the basis functions:

|χi(k)〉 = |φi(k)〉 + |ψi(k)〉, (3.31)

where |φ〉 is the basis function inside the muffin-tin spheres and |ψi(k)〉represents the basis functions, tails, outside the spheres.

We can construct a wave function Ψkn(r) by a linear combination ofLMTO basis functions, χi. Hence the linear combination can be writtenas

|Ψ〉 =∑

i

Ai|χi〉 (3.32)

The Hamiltonian operator is

H = H0 + Vnmt + VI (3.33)

where H0 is the Hamiltonian operator containing the kinetic operatorand the spherical part of the muffin-tin potential, Vnmt represents thenon-spherical part of the muffin-tin potential, and VI is the interstitialpotential. Then by using the variational principle for the one-electronHamiltonian the LMTO secular matrix follow as∑

j

[〈χi(k)|H0 + Vnmt + VI |χj(k)〉 − ε(k)〈χi(k)|χj(k)〉]Aj = 0 (3.34)

We can reduce it to ∑j

[H0ij + H1

ij − ε(k)Oij ]Aj = 0 (3.35)

3.4. FULL POTENTIAL LMTO METHOD 17

where

H0ij = 〈φi(k)|H0|φj(k)〉 (3.36)

Oij = 〈φi(k)|φj(k)〉 + 〈ψi(k)|ψj(k)〉 (3.37)

H1ij = 〈φi(k)|Vnmt|φj(k)〉+ 1

2(κi

2 +κj2)〈ψi(k)|ψj(k)〉+ 〈ψi(k)|VI |ψj(k)〉

(3.38)where |ψj(k)〉 is an eigenfunction to 2 with eigenvalue κ2

j . H0ij is

the spherical muffin-tin part of Hamiltonian matrix. Oij is the overlapbetween the orbitals inside the sphere as well as in the interstitial. H 1

ij

contains the corrections to the Hamiltonian matrix coming from themuffin-tin and interstitial region. The first term in Eq. (3.38) is thenon-spherical potential matrix. The next term is the expectation valueof the kinetic energy operator in the interstitial region. The last term isthe interstitial potential matrix.

3.4.3 Total energy

The total energy for whole crystal can be expressed as [16]

Etot = Tval + Tcor + Ec + Exc, (3.39)

where Tval and Tcor are the kinetic energy for the valence and core elec-trons, Ec is electrostatic energy including electron-electron, electron-nucleus and nucleus-nucleus energy, and Exc is our familiar term whichhas been studied in LDA. The kinetic energy is usually expressed as theexpectation value of the kinetic operator −2. By using the eigenvalueequation the expectation value can be expressed as the sum over oneelectron energies minus the effective potential energy. The core eigen-values εiτ are obtain as exact solution to the Dirac equation with thespherical part of the muffin-tin potential.

The total energy can be written as

Etot =occ∑kn

wnkεkn +∑µτ

fiτεiτ +

∫Vc

n(r)[1

2Vc(r) − Vin(r)]dr

− 1

2

∑j

ZτjVc(τj; 0) +

∫Vc

n(r)εxc(n(r))dr, (3.40)

where the integral is over the unit cell [14]. The sum j is over the corestates. The density n(r) is the total charge density, valence as well ascore electrons. Vin is the input potential obtained from LDA. Madelungterm Vc(τ ; 0) is the Coulomb potential at the nucleus less the Z/r selfcontribution and εxc is the exchange-correlation energy.


3.5 Projector Augmented Wave Method

Blochl [17] developed the projector augumented wave method (PAW)by combining the ideas from pseudopotentials and linear augmented-plane-wave (LAPW) methods. PAW method is an all-electron electronicstructure method. It describes the wave function by a superposition ofdifferent terms: the plane wave part, the so-called pseudo wave function,and expansions into atomic and pseudo atomic orbitals at each atom.

On one hand, the plane wave part has the flexibility to describe thebonding and tail region of the wave functions, but if it is used alone itwould require prohibitive large basis sets to describe correctly all the os-cillations of the wave function near the nuclei. On the other hand, the ex-pansions into atomic orbitals can describe correctly the nodal structureof the wave function near the nucleus, but lack the variational degrees offreedom for the bonding and tail regions. The PAW method combinesthe virtues of both numerical representations in one well-defined basisset.

To avoid the dual efforts by performing two electronic structure cal-culations, both plane waves and atomic orbitals, the PAW method doesnot determine the coefficients of the atomic orbitals variationally. In-stead, they are unique functions of the plane wave coefficients. The totalenergy, and most other observable quantities can be broken into threealmost independent contributions: one from the plane wave part and apair of expansions into atomic orbitals on each atom. The contributionsfrom the atomic orbitals can be broken down furthermore into contribu-tions from each atom, so that strictly no overlap between atomic orbitalson different sites need to be computed.

In principle, the PAW method is able to recover rigorously the den-sity functional total energy, if plane wave and atomic orbital expansionsare complete. This provides us with a systematic way to improve thebasis set errors. The present implementation uses the frozen core ap-proximation, it provides the correct densities and wave functions, andthus allows us to calculate other parameters of the system. By makingthe unit cell sufficiently large and decoupling the long-range interactions,limitations of plane wave basis sets to periodic systems (crystals) caneasily be overcome. Thus this method can be used to study molecules,surfaces, and solids within the same approach.

3.5.1 Wave function

Firstly, we will introduce a transformation matrix τ . There are twoHilbert spaces, one called all electron (AE) Hilbert, and the other called

3.5. PROJECTOR AUGMENTED WAVE METHOD 19

pseudo (PS) Hilbert. We need to map the AE valence wave functionsonto to the fictitious PS wave functions.

Every PS wave function can be expanded into PS partial waves

|Ψ〉 =∑

i

|φi〉ci (3.41)

The corresponding AE wave function is of the form

|Ψ〉 = τ |Ψ〉 =∑

i

|φi〉ci (3.42)

From the two equations above, we can derive

|Ψ〉 = |Ψ〉 −∑

i

|φi〉ci +∑

i

|φi〉ci (3.43)

because we need the transformation τ to be linear, the coefficients mustbe linear functions of the PS wave functions. Therefore the coefficientsare scalar products of PS wave function with projector functions 〈pi|,〈pi|Ψ〉. The projector functions must fulfill the condition∑

i

|φi〉〈pi| = 1 (3.44)

within the augmentation region ΩR, which implies that

〈pi|φj〉 = δij . (3.45)

Finally, the transformation matrix can be deduced from Eq. (3.42) andEq. (3.43) with the definition ci = 〈pi|Ψ〉

τ = 1 + (∑

i

|φi〉 − |φi〉)〈pi|. (3.46)

Using this transformation matrix, the AE valence wave function can beobtained from PS wave function by

|Ψ〉 = |Ψ〉 +∑

i

(|φi〉 − |φi〉)〈pi|Ψ〉 (3.47)

The core states wave functions |Ψ〉c are decomposed in a way similarto the valence wave functions. They are decomposed into three contri-butions:

|Ψ〉c = |Ψ〉c + |φ〉c − |φ〉c. (3.48)

Here |Ψ〉c is a PS core wave function, |φ〉c is AE core partial wave andlastly |φ〉c is the PS core partial wave. Comparing to the valence wavefunctions no projector functions are needed to be defined for the corestates, and the coefficients of the one-center expansion are always unity.


Figure 3.2: PAW method illustration

3.5.2 Charge density

The charge density at point r in space is composed of three terms:

n(r) = n(r) + n1(r) − n1(r) (3.49)

The soft pseudo charge density n(r) is the expectation value of real-spaceprojection operator |r〉〈r| on the pseudo-wave-functions.

n(r) =∑n

fn〈Ψn|r〉〈r|Ψn〉 (3.50)

The onsite charge densities n1 and n1 are treated on a radial supportgrid. They are given as:

n1(r) =∑n

fn〈Ψn|pi〉〈φi|r〉〈r|φj〉〈pj |Ψn〉 = ρij〈φi|r〉〈r|φj〉 (3.51)

here ρij is the occupancies of each augmentation channel (i, j) and theyare calculated from the pseudo-wave-functions applying the projectorfunctions: ρij =

∑n fn〈Ψn |pi〉〈pj | Ψn〉, and

n1(r) =∑n

fn〈Ψn|pi〉〈φi|r〉〈r|φj〉〈pj |Ψn〉 = ρij〈φi|r〉〈r|φj〉 (3.52)

We will focus on the frozen core case, n, n1, n1 are restricted to thevalence quantities. Besides that, we introduce four quantities that will beused to describe the core charge density: nc, nc, nZc, nZc. nc denote thecharge density of frozen core all-electron wave function in the referenceatom. The partial core density n is introduced to calculate nonlinearcore corrections. nZc is defined as the sum of the point charge of nucleinZ and frozen core AE charge density nc: nZc = nZ + nc,

Lastly, the pseudized core density is a charge distribution that isequivalent to nZc outside the core radius and have the same moment asthe nZc inside the core region.

∫Ωr

nZc(r)d3r =

∫Ωr

nZc(r)d3r (3.53)

3.5. PROJECTOR AUGMENTED WAVE METHOD 21

The total charge density nT [18] is decomposed into three terms:

nT = n + nZc

= (n + n + nZc) + (n1 + nZc) − (n1 + n + nZc)

= nT + n1T − n1

T (3.54)

A compensation charge n is added to the soft charge densities n+nZc

and n1 + nZc to reproduce the correct multipole moments of the AEcharge density n1 + nZc that is located in each augmentation region.Because nZc and nZc have exactly the same monopole −Zion(chargeof an electron is +1), the compensation charge must be chosen so thatn1+n has the same moments as the AE valence charge density n1 withineach augmentation sphere.

3.5.3 Total energy

The final expression for the total energy can also be split into threeterms:

E(r) = E(r) + E1(r) − E1(r). (3.55)

Where E(r), E1(r), E1(r) are given by

E(r) =∑n

fn〈Ψn| − 1

2∆|Ψn〉 + Exc[n + n + nc] + EH [n + n]

+

∫vH [nZc][n(r) + n(r)]dr + U(R,Zion) (3.56)

U(R,Zion) is the electrostatic energy of point charges Zion in an uniformelectrostatic background,

E1(r) =∑i,j

ρij〈φi| − 1

2∆|φj〉 + Exc[n1 + nc] + EH [n1]

+

∫vH [nZc]n

1(r)dr (3.57)

Here∫

vH [nZc]n1(r)dr is the electrostatic interaction between core and

valence electrons and EH is electrostatic energy

EH [n] =1

2(n)(n) =

1

2

∫dr

∫dr′

n(r)n(r′)|r − r′| (3.58)

E1(r) =∑i,j

ρij〈φi| − 1

2∆|φj〉 + Exc[n1 + n + nc] + EH [n1 + n]

+

∫vH [nZc][n

1(r) + n(r)]dr (3.59)

The overline means that the corresponding terms must be evaluated onthe radial grid within each augmentation region.


3.6 Ultrasoft pseudopotential

It is unaffordable to treat first-row elements, transition metals, and rare-earth elements by standard Norm-conserving Pseudopotentials (NC-PP). Therefore, various attempts have been made to generate the socalled soft potentials, and Vanderbilt [19] ultrasoft pseudopotentials(US-PP) has been proved to be the most successful one among them.There are number of improvements in US-PP method: 1) nonlinear corecorrections were included in the US-PP. 2) Lower cutoff energy, namelyreduced number of plane waves, was required in US-PP than NC-PP.This enables us to perform molecular dynamics simulations for systemscontaining first-row elements and transition metals.

Because E is exactly the same in the PAW method and US-PP me-thod, we only need to consider the linearization of E1 and E1. We obtainE1 to the first order by linearization of the E1 in the PAW total energyfunctional around atomic reference occupancies ρij

E1 ≈ C +∑ij

ρij〈φi| − 1

2∆ + υa

eff |φj〉 (3.60)

with υaeff = υH [n1

a + nZc] + υxc[n1a + nc] and C is a constant.

A similar linearization can also be done for E1

E1 ≈ C +∑ij

[ρij〈φi| − 1

2∆ + υa

eff |φj〉 +

∫QL

ij(r)υaeff (r)dr] (3.61)

with

υaeff = υH [n1

a + na + nZc] + υxc[n1a + na + nZc] (3.62)

QLij(r) is a pseudized augmentation charge in the US-PP approaches.

Given QLij(r) = Qij(r) = φ∗

i (r)φj(r) − φ∗i (r)φj(r),

E1 − E1 =∑ij

ρij(〈φi| − 1

2∆|φj〉 − 〈φi| − 1

2∆|φj〉). (3.63)

Now, we can compare the PAW functional with the US-PP func-tional. In the PAW method, if the sum of compensation charge andpseudocharge density, n1 + n, is equivalent to the onsite AE charge den-sity n1, and nZc = nZc, nc = nc,we can derive the same E1 − E1 fromEq. (3.57) and Eq. (3.59). In this limiting case, The PAW method isequivalent to the US-PP method.

3.7. PAW AND US-PP 23

3.7 PAW and US-PP

The general rule in Vienna ab initio simulation package (VASP) is touse PAW potential wherever possible, the PAW potentials are especiallygenerated for improving the accuracy for magnetic materials, alkai andalkai earth elements, 3d transition metals, lanthanides and actinides.For these materials, the treatment of semicores states as valence statesare desirable. The PAW method is as efficient as the FLAPW method,it is easy to unfreeze of low lying core states, only one partialwave (andproject) for the semicore states is included.

Differences between PAW and US-PP are only related to the pseudi-zation of the augmentation charges. By choosing very accurate pseudizedaugmentation function, discrepancies of both methods can be removed.However, augmentation charges must be represented on a regular gridwith the US-PP approach. Therefore, hard and accurate pseudized aug-mentation charges are expensive in terms of computer time and memory.The PAW method avoids these drawbacks by introducing radial supportgrids. The rapidly varying functions can be elegantly and efficientlytreated on radial support grids.

The PAW potentials are generally slightly harder than US-PP andthey retains similar hardness across the periodic table. Vice versa, theUS-PP Potential become progressively softer when moving down in theperiodic table. For multi species compounds with very different covalentradii mixed, the PAW potentials are clearly superior, except for onecomponent system, the US -PP might be slightly faster at the price ofreduced precision. Most PAW potential were optimised to work at acutoff of 250-300 eV, which is only slightly higher than in the US-PP.

Chapter 4

Phase transitions

4.1 Static total energy calculation

Static total energy is still the main method of ab initio simulation. Thesecalculations evaluate the energy and stability of an ‘ideal’, zero temper-ature crystal in which all atoms are located on their lattice positions.Pressure induced phase transitions can be reliably predicted by evaluat-ing the enthalpy (total energy plus PV) for each phase as a function ofpressure.

At a given pressure, the stable structure is the one which has thelowest minimum enthalpy. The phase transition pressure can also bededuced from the common tangent between curves on a total energy vsvolume graph corresponding to the two phases. The transition pressureis given by PT = (F2−F1)/(V1−V2) where F1 and F2 are the Helmholtzfree energy for phase 1 and 2 respectively (this is identical to the totalenergy for T = 0). The free energy is minimised with respect to theinternal coordinates and unit cell parameters in each phase. One cannotevaluate PT directly from the above equation. So one has to calculateequations of states for the two phases separately, and then compared.

The hydrous phases δ-AlOOH (paper XI) have recently been sub-jected to various studies at high pressure and high temperature [20, 21].One of the most important hydrous phase is the so called Egg phase,AlSiO3OH [22, 23]. Recent x-ray diffraction studies at high pressureand high temperature have shown that this Egg phase decomposes intoδ-AlOOH and stishovite SiO2 at 23 GPa and 1000C. This decomposi-tion reaction suggests that water stored in the phase Egg can be carriedfurther by δ-AlOOH into the deep lower mantle. The stability field ofδ-AlOOH was reported to be from 17 GPa up to at least 25 GPa ataround 1000C to 1200C [24]. In our calculation, a phase transitionfrom α-AlOOH to δ-AlOOH was calculated to take place at 17.9 GPa

25

26 CHAPTER 4. PHASE TRANSITIONS

0 10 20 30 40 50

Pressure (GPa)

-3

-2

-1

0

1

2

3

Dif

fere

nce

in e

ntha

lpy

(mR

y/at

om) δ-AlOOH

α-AlOOH

Figure 4.1: The 0 K enthalpy as a function of pressure for two differentcrystal structures: α-AlOOH and δ-AlOOH. The enthalpy of the α-AlOOH phase is taken as the energy zero. The transition from α-AlOOHto δ-AlOOH takes place at 17.9 GPa.

and with a volume collapse of 3%, as shown in Fig. 4.1.

4.2 Elastic stability criteria

For a cubic crystal with elastic constants C11, C12 and C44, the gen-eralized elastic stability criteria are C11 + 2C12 > 0, C44 > 0 andC11 − C12 > 0 [25, 26]. The transition metals, Ti, Zr and Hf stabi-lized in the hcp structure at the ambient conditions. The bcc phase iscalculated to be unstable and with a negative C′ (C ′ = (C11 − C12)/2).With the increasing pressure, C′ becomes positive at the V/V0 = 0.73. Ahigh pressure bcc phase is predicted to be stable, this can be understoodas an effect of the s → d transfer under compression. This means thatTi behave more like its nearest neighbor V, which has the bcc crystalstructure [27] at ambient conditions.

4.3 Bain path

The structural path for going from bcc phase to fcc phase within thetetragonal structure is well known as Bain Path [28]. Body-centeredand face-centered cubic crystal can be considered as special cases of abody-centered tetragonal crystal with c/a = 1 and

√2, respectively.

Starting with a normally fcc element Ce [29] (Fig. 4.2), and calculatingenergy as a function of c/a, we obtain two local minimums at c/a = 1.41

4.4. DYNAMICAL STABILITY (PHONON CALCULATION) 27

1.4 1.5 1.6

c/a ratio

-0.706

Tot

al E

nerg

y (R

y)

-0.7025

V/V0=0.57

V/V0=0.58

Figure 4.2: The comparison between the c/a ratio of V/V0 = 0.57 andV/V0 = 0.58 for Ce. When V/V0 = 0.58, the fcc (c/a =

√2) phase is

more stable. When V/V0 = 0.57, the bct phase is more stable.

and c/a = 1.66. The curvature of E(c/a) around the maximum and theminimum directly correspond to C′ we mentioned above. Before thetransition, the fcc phase (c/a =

√2) is more stable compared to the bct

phase and after the transition, the bct phase starts to win in the totalenergy as the volume is changing from V/V0 = 0.58 to V/V0 = 0.57.

4.4 Dynamical stability (phonon calculation)

The soft phonon phase transition is one of the best established mech-anisms by which a crystal structure can change [30]. In the pressure-induced case, the frequency of a given vibration in the lattice goes to zeroas the transition is approached: zero frequency implies that the latticestructure has become unstable, and will transform to a new phase.

Considering a system consists of N atoms, the Hamiltonian of thesystem can then be expressed as [31]

H =1

2

∑i

Mi[u(i)]2 +1

2

∑ij

∑αβ

φαβ(i, j)uα(i)uβ(j) (4.1)

Here mi is the mass of atom i and u(i) is its displacement away fromits equilibrium position, while α and β subscripts denote one of theCartesian components of a vector. φαβ(i, j) is the so called force tensor,which is simply the second derivative of the potential.

φαβ(i, j) =∂2U

∂uα(i)∂uβ(j)(4.2)


The substitution e(i) =√

Miu(i) yields

H =1

2

∑i

e(i)2 +∑ij

e(i)φ(ij)√MiMj

e(j) (4.3)

We define the dynamical matrix for the system,

Dαβ(i, j) =1√

MiMjφαβ(i, j) (4.4)

which can be constructed from force constant tensor. The size of thedynamical matrix is 3N × 3N . Diagonalizing the dynamical matrix weobtain all of its eigenvalues λm,m = 1 · · · 3N . In the harmonic approx-imation, the knowledge of these frequencies is sufficient to determineother thermodynamic quantities of the system. For example, the freeenergy of the system can now be calculated through [32]

F =kBT

N

3N∑m=1

ln[2sinh(hωm

2KBT)]. (4.5)

In a crystal, the determination of the normal modes is somewhatsimplified by the translational symmetry of the system. Say n denotes

the number of atoms per unit cell, u

(li

)is the displacements of atom

i in cell l away from its equilibrium position, and Φ

(l l′

i j

)is the force

constant relative to atom i in cell l and atom j in cell l ′ and let

e

(li

)= exp [ι2π(k · l)] e

(0i

), (4.6)

where ι =√−1, l denotes the Cartesian coordinates of one corner of

cell l, and k is a point in the first Brillouin zone. This fact reducesthe problem of diagonalizing the 3N × 3N matrix D to the problem ofdiagonalizing a 3n × 3n matrix D(k) for various values of k. This canbe shown by a simple substitution of Eq. (4.6) into Eq. (4.3). Thedynamical matrix D(k) to be diagonalized is given by

D(k) =∑

l

exp[ι2π(k · l)]

Φ

(o l1 1

)√

M1Mn· · ·

Φ

(o l1 n

)√

M1Mn

.... . .

...

Φ

(o ln 1

)√

M1Mn· · ·

Φ

(o ln n

)√

M1Mn

(4.7)

4.5. EQUATION OF STATE 29

As before, the resulting eigenvalues λ(k) for i = 1, · · ·, n give the fre-quencies of the normal modes ω(k) = 1

2π

√λ(k). The function ω(k) for

a given i is called a phonon branch, while the plot of the k dependenceof all branches along a given direction in k space is called the phonondispersion curve. In periodic systems, the phonon DOS, which gives thenumber of modes of oscillation having a frequency lying in the energyinterval [ω, ω + dω] is defined as

g(ω) =3n∑i=1

∫BZ

δ[ω − ωi(k)]dk (4.8)

where the integral is taken over the first Brillouin zone. Phonon calcu-lation in Ti was performed by means of PWSCF package (Plane-WaveSelf-Consistent Field). PWSCF is a first-principles energy code that usespseudopotentials (PP) and ultrasoft pseudopoentials (US-PP) withinDFT. In contrast to the frozen-phonon method, it includes linear re-sponse method, which allows the treatment of arbitrary phonon wavevectors q. The phonon dispersion curves along several of high symmetryk points for the Ti high pressure γ phase is presented. Vohra et al. [33]observed a transformation from an ω phase to an orthorhombic phase(γ phase) at a pressure of 116 ± 4 GPa, and this phase is stable up to146 GPa. In this pressure range (see Fig. 4.3), our calculated phonondispersion curves give an imaginary phonon frequency indicating thatthe γ phase is unstable in the pressure range we calculated, from 119GPa to 173 GPa.

4.5 Equation of state

The equation of state (EOS) of solids is of great importance in basicand applied science. The measurable properties of solids, such as theequilibrium volume (V0), the bulk modulus (B0) and its first pressurederivative (B′

0) are directly related to the EOS. The high pressure EOShas been represented in various functional forms, for example, the Mur-naghan equation, Birch-Murnaghan (BM) equation, universal equationand recently a new equation of state which is appropriate at strongcompressions has been put forward by Holzapfel [34, 35].

4.5.1 Murnaghan equation of state

Murnaghan EOS was introduced by Murnaghan et al. in the forties [36].The E-V form Murnaghan EOS can be represented as,

E(V ) = B0V0

B′

0

[1

B′

0−1

(V0

V

)B′

0−1

+ VV0

− B′

0

B′

0−1

]+ Ecoh (4.9)


0

4

8

12

16

0

4

8

12

16Fr

eque

ncy

(TH

z)

Γ Z T Y Γ S R

0

4

8

12

16

Figure 4.3: The phonon dispersion curves for the γ phase of Ti: fromup to down, P=119, 140 and 173 GPa respectively.

Ecoh is the cohesive energy and is treated as an adjustable parameter.Since the pressure can be obtained from P (V ) = −∂E(V )/∂V , the Mur-naghan equation can be expressed in its usual form

P (V ) = B0

B′

0

((V0

V

)B′

0 − 1

). (4.10)

The bulk modulus is derived through the volume derivative of the equa-tion above, B = −V (∂P/∂V ),

B(V ) = B0

(V0

V

)B′

0

. (4.11)

4.5.2 Birch-Murnaghan equation of state

Birch et al. [37, 38] expanded the Gibb’s free energy F in terms ofEulerian strain ε, with V0/V = (1 − 2ε)3/2. The integrated energy-volume form of the third order BM-EOS,

E(V ) = − 916B0[(4 − B′

0)V 3

0

V 2 − (14 − 3B′0)

V7/3

0

V 4/3

+(16 − 3B′0)

V5/3

0

V 2/3] + E0 (4.12)

Using the obtained B0, B′0 and V0 from a least-square fit of the calculated

V-E curves to the EOS above, the hydrostatic pressure P was determined

4.5. EQUATION OF STATE 31

from the P-V form of the BM EOS, which is the volume derivative ofthe former equation. The second order BM-EOS can be written as

P (V ) = 1.5B0

[(V0

V

)7/3 −(

V0

V

)5/3]

(4.13)

While the third order BM-EOS [39] has the analytical form;

P (V ) = 1.5B0

[(V0

V

)7/3 −(

V0

V

)5/3]

·

1 + 34(B′

0 − 4)

[(V0

V

)2/3 − 1

](4.14)

The bulk modulus corresponding to Eq. (4.14) is

B(V ) = 1.5B0

[73

(V0

V

)7/3 − 53

(V0

V

)5/3]

·

1 + 34 (B′

0 − 4)

[(V0

V

)2/3 − 1

]+1.5B0

[(V0

V

)7/3 −(

V0

V

)5/3] [

12 (B′

0 − 4)(

V0

V

)2/3]

(4.15)

4.5.3 Universal equation of state

Vinet et al. [40] have reported a universal form the EOS for all classesof solids, such as ionic, metallic, covalent and rare-gas solid, under com-pression. Their P-V relation can be represented as [41]

P (V ) =

[3B0

(1 − x)

x2

]exp[η(1 − x)] (4.16)

Here η is fixed in terms of B ′0

η = 3/2(B′0 − 1) (4.17)

and x = ( VV0

)1/3. Poirier [42, 43] has pointed out that the UniversalEOS can be obtained by the same derivation as the Birch-MurnaghanEOS, using a strain parameter ε = (V0/V )1/3 − 1 and the free energyF = F0(1 + Aε)exp(−Aε) (F0 and A are constants).

4.5.4 Comparison of different EOS for Fe

Iron has attracted a lot of attention from geophysicists because it isconsidered to be the major constituent of the earth core. At low tem-perature, a Fe bcc (α) - hcp (ε) transition takes place around 13 GPa[44], and the hcp phase is believed to be stable up to the pressure of


the inner core. In paper IX, we show that the c/a ratio of Fe increaseswith increasing pressure and eventually approaches the ideal ratio of√

83 = 1.633 at extreme pressures. Three different equations of states

(as shown in Fig. 4.4) have been used to fit the same set of GGA cal-culated E-V data set for the hcp phase. Because B0 and B′

0 are highlycorrelated parameters in the Birch-Murnaghan equation of state (EOS)[45], we show the comparison when B ′

0 is taken to be the experimen-tal value 5.8 [46]. These three EOS have similar behavior in the lowpressure range, but they start to diverge at a pressure around 150 GPa.The Vinet EOS shows a better agreement with the experimental dataof Mao et al. and Dubrovinsky et al. under pressure. All these EOSfitted equilibrium volume (6.14 cm3/mol) is lower than that obtainedfrom the experimental data (6.73 cm3/mol [46]). The treatment of anti-ferromagnetically ordered structure gives a better equilibrium V0=6.35cm3/mol [47]. However, none of them can successfully reproduce theexperimental equilibrium volume.

4 4.5 5 5.5 6 6.5

Volume (cm3/mol)

0

50

100

150

200

250

300

350

400

450

500

Pres

sure

(G

Pa)

Jephcoat et al.Mao et al.Dubrovinsky et al.Vinet EOSMurnaghan EOSBM EOS

Fe

Figure 4.4: Comparison between the experimental and theoretical EOSusing three different EOS for hcp Fe. The filled circle data points areexperiments from Mao et al. [48] up to 304 GPa without medium. Thesquare data points correspond to 300 K x-ray diffraction measurementswith Ar and Ne medium up to to 78 GPa [49]. The dot-dashed line isthe latest data from Dubrovinsky et al. [50]. The Murnaghan, Birch-murnaghan and Vinet EOS are denoted by dashed, dotted and solid linerespectively.

Chapter 5

Semiconductor optics

5.1 Dielectric function

Materials having an energy band gap, Eg, in the range 0 < Eg ≤ 4eV are called semiconductors and those where having a gap Eg ≥ 4eV are insulators [51]. Semiconductors with a gap approximately belowor near 0.5 eV are named as narrow-gap semiconductors; on the otherhand, materials with a gap between 2 eV and 4 eV are called wide-gapsemiconductors. If Eg is close to zero, they are called semimetals, suchas TiC.

It is known that the measurement of optical properties is help-ful to explain the electronic structure of materials. The knowledgeof the refractive indice and absorption coefficient of semiconductorsis especially important in the design and analysis of heterostructurelasers and other semiconductor devices [52]. The dielectric function,ε(w) = ε1(w) + iε2(w), fully describes the optical properties of mediumat all photon energies, hω. The (q = 0) dielectric function can be cal-culated in the momentum representation, which requires matrix ele-ments of the momentum, p, between occupied and unoccupied eigen-states. To be specific, the imaginary part of the dielectric function,ε2(w) = Imε(q = 0, w), can be calculated from

εij2 (w) = 4π2e2

Ωm2w2

∑knn′σ < knσ|pi|kn′σ >< kn′σ|pj|knσ >

·fkn(1 − fkn′)δ(ekn′ − ekn − hw). (5.1)

In Eq. (5.1), e is the electron charge, m is its mass, Ω is the crys-tal volume, and fkn is the Fermi distribution. Moreover |knσ〉 is thecrystal wave function corresponding to the nth eigenvalue with crystalmomentum k and spin σ. With our spherical wave basis functions, thematrix elements of the momentum operator are conveniently calculatedin spherical coordinates and for this reason the momentum is written in

33

34 CHAPTER 5. SEMICONDUCTOR OPTICS

p =∑

µ e∗µpµ, where µ is -1, 0, or 1, and p−1 = 1/√

2(px − ipy), p0 = pz,

and p1 = 1/√

2(px + ipy).

The evaluation of the matrix elements in Eq. (5.1) is done over themuffin-tin region and the interstitial separately. The integration over theprimitive cell is done in way similar to what Oppeneer [53] and Gasche[11] did in their calculations.

The summation over the Brillouin zone in Eq. (5.1) is calculated us-ing linear interpolation on a mesh of uniformly distributed points, i.e.,the tetrahedron method. Matrix element, eigenvalues, and eigenvectorsare calculated in the irreducible part of the Brillouin-zone. The cor-rect symmetry for the dielectric constant is obtained by averaging thecalculated dielectric function.

Through the Kramers-Kronig relations, we have derived the real partof the dielectric function ε1.

ε1(w) = 1 +1

π

∫ ∞

0dω′ε2(ω

′)(1

ω′ − ω+

1

ω′ + ω) (5.2)

The relation between the dielectric function and the complex refrac-tive index N = n + ik is given by

ε1 = n2 − k2 (5.3)

and

ε2 = 2nk (5.4)

Given n and k, we can derive the normal-incidence reflectivity R

R =(n − 1)2 + k2

(n + 1)2 + k2(5.5)

and the absorption coefficient α

α =4πk

λ(5.6)

5.2 Dielectric function of PbWO4

Under the ambient conditions, lead tungstate has two stable crystallo-graphic structures, raspite and scheelite. For raspite, with a monoclinicstructure, the total dielectric function is composed of three different di-electric functions along the a, b and and c axis. But for scheelite, havinga tetragonal structure, we only need to average over two components toget the total dielectric function, namely the components corresponding

5.2. DIELECTRIC FUNCTION OF PBWO4 35

to light polarized parallel and perpendicular to the c axis. In this casethe total, orientation averaged ε2 is given by

εtot2 (w) =

ε‖2(w) + 2ε⊥2 (w)

3(5.7)

In the discussion that follows we will refer to εij2 (w) as ε⊥2 (w) when

i=j=x and as ε‖2(w) when i=j=z.

0 5 10 15 20 25

PHOTON ENERGY (eV)

0

5

10

15

0

5

10

15

20

ε 2

experimentcalculation

3.3 3.4 3.5

3456

2.6 2.8 30

1

2

3 (b) raspite PbWO4

(a) scheelite PbWO4

↑↑

A

B

↓C

Figure 5.1: The imaginary part of the dielectric function for both scheel-ite (a) and raspite structure (b) of PbWO4. The experimental data arefrom Itoh et al. [54]. The inserts show an expanded part of ε2.

In Fig. 5.1, we show the averaged ε2 for both scheelite and raspitestructure. Our calculated dielectric function agrees very well with theexperimental data even without any broadening. From the experimentaldielectric function, we can see the energy gap, around 3.5 eV acts as athreshold for interband excitations for both structures. In the scheelitestructure, the doublet peaks observed by Shpinkov et al. [55] and Itoh et

al. can be clearly seen at 3.30 eV and 3.44 eV from our calculation. Thesharp intensive experimental peak at 4.18 eV which these two doubletpeaks contribute to was reproduced by a rather prominent calculatedpeak at 3.35 eV. According to the selection rule, l = ±1, only suchtransitions that change of the angular momentum quantum number of1 are allowed. These two peaks in the calculated spectrum can be as-signed to Pb s → p interband transitions. In the raspite structure, mostfeatures of the measured data are well reproduced by our calculations.


The weak peak at 3.75 eV was reproduced by a calculated peak at 2.95eV in Fig. 5.1 (b). Since the energy gap was underestimated by around1 eV in the LDA approximation, and if we shift the calculated curve by1 eV, the agreement will be even better.

5.3 Solar energy materials

Solar cells are devices which convert solar energy into electricity, eitherdirectly via the photovoltaic effect, or via the intermediate of heat orchemical energy. The ideal solar energy cell is required to have a bandgap in the visible light region as well as to have high absorption in thevisible region. CuInSe2 (CIS) has a band gap of only 1 eV, with Gaaddition to CIS, forming the CuIn1−xGaxSe2 (CIGS) alloy, the gap israised, and thus increases the open circuit voltage [56]. At present, theputatively best CIGS solar cells are made with 30% doping of Ga [57].

The effect of the Ga doping to CuInSe2 can be observed in the changeof the band gap [58]

Eg(x) = (1 − x)Eg(CIS) + xEg(CGS) − bx(1 − x), (5.8)

b=0.21 was given by Wei et al. [56] CIS exhibits a direct band gap atΓ point, which indicates that the photon energy is directly convertedinto the creation of electron/hole pairs in the semiconductor material.This is preferable to the material that exhibits an indirect band gap,where there is some energy loss due to the creation of phonons. Theband gap in CIS is 1.04 eV [59], and 1.68 eV [59] in CGS, accordingto Eq. (5.8) CuIn0.75Ga0.25Se2, CuIn0.5Ga0.5Se2 should have the bandgap around 1.16 eV and 1.31 eV respectively. Although the chalcopy-rite structure resembles to the zinc-blende structure, the band gaps ofthe ternary (I-III-VI2) semiconductors are only less than half of theirbinary analogs (II-VI). This band gap anomaly makes CIS one of themost known absorbers in the solar spectrum. The p-d hybridization andstructure anomaly attribute to the anomaly in the band gap [59]. Fromupper panel of Fig. 5.2, we can easily observe the opening of the bandgap with Ga addition to CuInSe2. The band gap is directly obtainedfrom calculation without any shift.

CIS has a high optical absorption coefficient, which provides informa-tion about optimum solar energy conversion efficiency. The absorptioncoefficient of a material indicates how far light of a specific wavelength(or energy) can penetrate into the material before being absorbed.

5.3. SOLAR ENERGY MATERIALS 37

0

2

4

6

8

10

12CuInSe2CuIn

0.75Ga

0.25Se

2

CuIn0.5

Ga0.5

Se2

CuGaSe2

2 4 6 8PHOTON ENERGY (eV)

0

2

4

6

8

10

12

ε 2

E ⊥ c

E || c

Figure 5.2: The imaginary part of the dielectric function of CuInSe2,CuIn0.75Ga0.25Se2, CuIn0.5Ga0.5Se2 and CuGaSe2 for both E⊥c and E‖c.

0

2

4

6

8E⊥ c

0 1 2 3 4 5Energy (eV)

0

2

4

6

8

Abs

orpt

ion

coei

ffic

ient

(×

105 )

E || c

CuInSe2

Figure 5.3: The absorption coefficient of CuInSe2. The experimentaldata and theoretical results are denoted by dashed and solid line respec-tively. The experimental data are taken from Kawashima et al. [60].


A small absorption coefficient means that light is not readily absorbed bythe material. The depth of penetration (1/α = d in this case) is definedby the distance at which the radiant power decreases to 1/e of its incidentvalue. The fundamental absorption corresponds to a strong absorptionregion which is in order of 105 cm−1 to 106 cm−1. The fundamentalabsorption area is manifested by the rapid rise in absorption and is usedto determine the energy gap of the semiconductor. In Fig. 5.3, theinterband transitions start at around 1 eV, and CuInSe2 shows a highabsorption coefficient up to 8 × 105 cm−1 in the energy range where weare interested in.

Chapter 6

MAX phases

6.1 MN+1AXN phase

MN+1AXN (MAX) (N=1 to 3) phases are a series of ceramics but with acombination of ductility, conductivity and machinability comparable tometals. M is an early transition metal, A is an A-group element (mostlyIIIA and IVA) and X is either C or N. These phases have hexagonallayered structures and belong to the space group P63/mmc, whereinMN+1XN layers have the rock salt structure interleaved with pure layersof the A-group elements (Fig. 6.1). The structures of the vast majorityof these compounds were determined by Nowotny [61] and co-workersin the sixties. Up tp now, there are roughly fifty M2AX phases, threeM3AX2 and one M4AX3 phases known [7] (Table 6.1).

Figure 6.1: Crystal structures of 211, 312 and 413.

39

40 CHAPTER 6. MAX PHASES

Table 6.1: Known MAX phases

211 Ti2AlC Nb2AlC Ti2GeC Zr2SnC Hf2SnC Ti2SnCNb2SnC Zr2PbC Hf2PbC Ti2AIN0.5C0.5 Zr2SC Ti2SCNb2SC Hf2SC Ti2GaC V2GaC V2AsC Nb2AsCTi2CdC Sc2InC Ti2InC Zr2InC Nb2InC Hf2InCTi2AlN (Nb,Ti)2AlC Cr2AlC Ta2AlC V2AlC V2PCNb2PC Ti2PbC Cr2GaC Nb2GaC Mo2GaC Ta2GaCTi2GaN Cr2GaN V2GaN V2GeC Ti2InN Zr2InNHf2InN Hf2SnN Ti2TlC Zr2TlC Hf2TlC Zr2TlC

312 Ti3SiC2 Ti3AlC2 Ti3GeC2

413 Ti4AlN3

6.2 Phase stability in Ti-Si-C system

In the paper XII, we presented the formation energy calculation forTi-Si-C system. The phase stability has been predicted by comparingthe cohesive energy of the MAX phase with the cohesive energy of thecompeting equilibrium phases at corresponding composition as given bythe phase diagram in Fig. 6.2. All the calculations have been carried outon stoichiometric phases without consideration to homogeneity ranges(e. g. TiCx; x=1 and Ti5Si3Cx; x=0). Ti3SiC2 is the only ternary phasereported to exist in this system.

Figure 6.2: Phase diagram at 1250 C for the Ti-Si-C system [62]

6.2. PHASE STABILITY IN TI-SI-C SYSTEM 41

The phase diagram shows that the competing phases for Ti4SiC3,Ti5SiC4 and Ti7Si2C5 are Ti3SiC2 and TiC, while the competing phasesfor Ti2SiC and Ti5Si2C3 are Ti3SiC2, TiSi2 and Ti5Si3 (in reality Ti5Si3Cx).Table 6.2 lists the cohesive energy per atom (Ecoh) for each of the cal-culated MAX phases and for the competing equilibrium phases. Thestability of a given MAX phase is determined by the energy difference(∆E) between the MAX phase and its competing phases. A negative∆E indicates a stable phase. A positive ∆E suggests that it will de-compose or not formed at all in favor for the competing phases.

From Table 6.2, it can be seen that there is a very small, but neg-ative, energy difference of -0.008 eV/atom, between the 211 phase andits competing phases. This suggests that the 211 could be stable, butit should be noted that the calculated energy difference is so small thatit is approaching the accuracy of the calculations. This phase may bemetastable and can be synthesized as thin films. A more significant neg-ative energy difference is calculated for the 413 compound with −0.029eV/atom. This suggests that the 413 phase actually is stable. Forthe 514 compound, the calculated energy difference is clearly positive,+0.037 eV/atom and this phase should therefore not be stable. Thecalculations of the new MAX phase structures, 523 and 725, show thatthe energy difference is positive compared with the competing phases;+0.007 eV/atom for the 523 compound (again, this energy differenceapproaches the accuracy of the calculations) and +0.030 eV/atom forthe 725 compound. This suggests that both the observed inter-grownphases should not be stable. However, it should be noted that eventhough ∆E > 0 the calculated energy differences are very small andthat the compound still may be formed under metastable conditions.

Table 6.2: Formation energy in Ti-Si-C system

Si/Ti decompositions ∆E

Ti2SiC 0.5 0.75Ti3SiC2+0.107TiSi2+0.1429Ti5Si3 -0.0076Ti3SiC2 0.33Ti4SiC3 0.25 0.75Ti3SiC2 + 0.25 TiC -0.0288Ti5SiC4 0.2 0.6Ti3SiC2 + 0.4 TiC 0.0373

Ti5Si2C3 0.4 0.9Ti3SiC2 + 0.0429 TiSi2 + 0.057 Ti5Si3 0.0359Ti7Si2C5 0.286 0.8571Ti3SiC2 + 0.1429 TiC 0.0278


6.3 Chemical bonding in 312 phases

Three types of bonding, namely, metallic, covalent and ionic contributeto the bonding in MAX phases. The high electrical conductivity isattributed to the metallic bonding parallel to the basal plane (TiI andTiII) and high bulk modulus is attributed to the strong Ti-C covalentbond. The calculated balanced crystal orbital overlap (BCOOP) [63]is used as a tool to study the chemical bonding in the Ti-Si-C system.BCOOP can easily indicate the covalent bonding between two types oforbitals. In Ti3SiC2, we are interested in the bonding between α1: Ti3d states and α2: C 2p states

BCOOPα1,α2=

∑n,k

δ(ε − εn(k))〈α1 |α2〉∑

α 〈α |α〉 (6.1)

Here εn(k) is the Kohn-Sham eigenvalue, each eigenvector can be de-composed into non-orthogonal contributions |n, k〉 =

∑α |α〉, and |α〉 =∑

i ci(n, k) |i〉. The denominator in Eq. (6.1),∑

α 〈α1 |α2〉, is introducedto balance the bonding and antibonding states. BCOOP has positivevalue for bonding states and negative value for the antibonding states,the intensity of peaks and areas gives an indication of the strength ofthe bonding.

-0.4

-0.2

0.0

0.2

0.4

-6 -4 -2 0 2 4 6E-EF (eV)

-0.4

-0.2

0.0

0.2

0.4-0.4

-0.2

0.0

0.2

0.4

BC

OO

P (1

/eV

)

-0.4

-0.2

0.0

0.2

0.4

TiC

Ti3SiC2

Ti3AlC2

Ti3GeC2

Ti I-C Ti II-C Ti II-A

Figure 6.3: BCOOP for Ti3SiC2, Ti3AlC2, Ti3GeC2 and TiC.

6.4. XAS AND XES CALCULATION 43

In Fig. 6.3, the area under the TiII-C peaks is larger than the area un-der TiI-C peaks indicates that TiII-C bond is stronger than TiI-C bond.While TiII-Al BCOOP located closer to the fermi level compared withTiII-Ge and TiII-Si suggests that TiII-Al bond is weaker than the othertwo types of bonds. This can easily be understood from the electronicconfiguration. Al has three valence electrons instead of four valence elec-trons in Si and Ge, which should form less strong covalent bond becauseof less saturated bonds.

6.4 XAS and XES calculation

The X-ray absorption (XAS) and X-ray emission (XES) in paper XIVhas been calculated using WIEN2K [64] code. The electric-dipole ap-proximation has been employed, which means that only the transitionsbetween the core states with angular momentum l to the l ± 1 compo-nents of the conduction bands have been considered.

In Paper XIV, we show the calculated XES spectra of Ti3SiC2,Ti3AlC2 and Ti3GeC2 and TiC. The calculated 3d DOS was projectedby 3d, 4s → 2p dipole transition matrix elements. We may notice thatTi3AlC2 shows a much more pronounced double peak structure com-pared with Ti3SiC2 and Ti3GeC2 (Fig. 6.4).

There is an extra peak located at -1 eV below the fermi level forTi3AlC2. To understand the orgin of this special double-structure inTi3AlC2, we plot the band structure for these three phases (as shownin Fig. 6.5). A flat band between L to M, and K to H around -1 eVcorresponds to a high density of state in DOS, and this originates fromSi p state. The extra peak at -1 eV in Fig. 6.4 could due to the Ti-Sihybridization. The other flat band between K to H symmetry pointsattributes to the higher DOS of Ti at EF in Ti3AlC2.

6.5 DOS with electrical conductivity

The density of states at Fermi level N(EF) in the simplest approximationis directly related to the electrical conductivity. Our calculated N(EF)for 211, 312, 413 and 514 phases are 0.36, 0.33, 0.29 and 0.25 states/eVper atom, respectively. This can be compared to TiC, which has apseudo gap at the Fermi level with N(EF) close to 0.1 states/eV peratom, showing only weak metallic behavior. A trend of decreasing N(EF)in the sequence of 211, 312, 413 and 514 compounds can be seen asdecreasing metallicity of MAX phases with increasing number of n. Acalculated N(EF) = 3.96 states/eV per unit cell in 312 is in very good


-20 -15 -10 -5 0 5 10E-EF (eV)

Inte

nsity

(arb

. uni

ts)

-20 -15 -10 -5 0 5 10E-EF (eV)

-20 -15 -10 -5 0 5 10E-EF (eV)

Ti3SiC2 Ti3AlC2 Ti3GeC2 TiC

Ti I

Ti II

Ti I + Ti II L3

L2

Figure 6.4: Calculated Ti L edge XES spectra of Ti3SiC2, Ti3AlC2 andTi3GeC2 and TiC for TiI, TiII and the sum.

-2

-1

0

1

2

-2

-1

0

1

2

Ene

rgy

(eV

)

ΓA LM Γ KH A-2

-1

0

1

2

Figure 6.5: Band structures of Ti3SiC2, Ti3AlC2 and Ti3GeC2.

6.5. DOS WITH ELECTRICAL CONDUCTIVITY 45

-15 -10 -5 0 5 10 15

E-EF (eV)

0

1

2

3

4

DO

S (e

V/s

tate

s un

it ce

ll)

TiI

TiII

Ti3SiC

2

Figure 6.6: TiI and TiII type of DOS in Ti3SiC2

agreement with experimental data of 4.42 states/eV per unit cell [66]derived from heat capacity data.

All Tin+1SiCn (n=1 to 4) MAX-phase films show excellent conduct-ing properties (see Table 6.3). This is in good agreement with the DFTcalculations, which show a relatively high DOS at Fermi level for allthese compounds. The electrical conductivity properties of Tin+1SiCn

should approach TiC with increasing number of Ti layers. Zhou et al

[65] suggest that the difference in charge density distribution betweenTiI and TiII layers indicate that the TiII layers contribute more to theelectrical conductivity than TiI layers. We also observe this from differ-ent contribution to DOS at EF from TiI and TiII, as shown in Fig. 6.6for Ti3SiC2.

Therefore, the number of TiII layers per unit cell plays an importantrole in the electrical conductivity. The highest calculated N(EF) in theTin+1SiCn system suggests that 211 has higher conductivity than theother MAX phases, which is also consistent with that it only containsTiII type. As shown in Table 6.3, the ratio of TiII/Ti in one unit cell are

Table 6.3: Density of state at fermi level (NEF) and electrical resistivity.

TiII/Ti N(EF) Resistivity(µΩcm)

TiC 0.1 200-260Ti2SiC 1 0.36 -

Ti3SiC2 0.667 0.33 25-30Ti4SiC3 0.5 0.29 50Ti5SiC4 0.4 0.25 -


1 (4/4), 0.667 (4/6), 0.5 (4/8), 0.4 (4/10) in the sequence of 211, 312,413 and 514, suggesting a decrease of the electrical conductivity withincreasing number of n.

6.6 Optical properties

In metals, optical absorption occurs through two processes. One is theintraband transition, the excitation of the electrons at the fermi level.The Drude term represents a phenomenological way to describe the in-traband transition, which contributes at low energies [67]. The secondprocess is interband transitions, i.e. the excitation of electrons from anoccupied band to an empty band. These two contributions combined togive the total optical response.

Drude term is composed of two parts: relaxation time and plasmafrequency.

1) Relaxation time. This plays a fundamental role in the theory ofmetallic conduction. We pick an electron randomly in the free electrongas, on the average, it travels for a time τ before its next collision. Therelaxation time can be estimated via the resistivities ρ and electron den-sity n, τ = m

ρne2 . At room temperature, τ is typically 10−14 to 10−15 sec

[68]. Although there is no formally correlation between the relaxationtimes characterizes the (longitudinal) electrical conductivity and relax-ation obtained in the Drude expression (describing transverse response),we believe that these two relaxation time are strongly correlated to eachother. The inverse of τ , i.e. the collision frequency, often given thesymbol Γ is a measure of the broadening of the the phenomenologicalDrude feature.

2) Plasma frequency. For the free electron gas of density n, the os-

cillations of the electron plasma have frequency, ωp =√

4πne2

m . The ωp isthe frequency where the real part of the dielectric function goes throughzero from below, and the imaginary part approaches zero from above.A sharp reflectance drop at ωp is a characteristic for high conductancemetal.

It is known TiN has a high reflectivity in the infrared, and a lowreflectivity for shorter wavelengths. It is this property makes TiN apromising material for solar energy applications. The low reflectancein the region of blue and violet light (2.8-3.5 eV) for TiN renders itgoldlike color [70]. With Al addition to TiN, the reflectivity decreasesdramatically in the infrared region and increase in the visible light region(as shown in Fig. 6.7). The decrease of the reflectivity in the infraredregion is because of lower plasma frequency. The dip shifts from 2.8

6.6. OPTICAL PROPERTIES 47

1.5 2 2.5 3 3.5 4 4.5

ENERGY (eV)

0

0.2

0.4

0.6

0.8

RE

FLE

CT

IVIT

Y

TiCTiNTi

3SiC

2

Ti4AlN

3

Figure 6.7: Reflectivity for Ti3SiC2 and Ti4AlN3. The experimentaldata for TiC and TiN are taken from Fuentes et al. [69].

eV to 3.7 eV, i.e. outside the region of visible light. This shift is notgood for thermal solar collector applications, which require high solarabsorption, i.e. low reflectance in the range 1.5-3.5 eV. To minimize ra-dioactive losses the collector surface should simultaneously exhibit lowthermal emittance at the working temperature, i.e. high reflectance inthe infrared region. As just mentioned the alloying of aluminum intoTiN annihilates both of these optical advantages, but with maintainedmechanical strength as well as thermal and chemical inertness. Thispoint makes another technical application for Ti4AlN3. The reflectivityminimum 0.3 (30%) for Ti4AlN3 is higher than 0.2 (20%) for pure TiN.It can therefore be used to avoid solar heating on space-crafts and alsoincrease the radioactive cooling due to the increased thermal emittancecompared to TiN [71]. The MAX-phases are therefore candidate mate-rials for coatings in future space-missions to Mercury. The reflectanceof Ti3SiC2 shows similar behavior with TiC, with almost constant valuein the region we have studied. The constant reflectance in the visiblelight region makes Ti3SiC2 look metallic grey.

It is known that in TiC the interband band transition occurs atlow energies even less than 0.1 eV [72], there is almost no free-electronlike region. In contrast, TiN exhibits a free-electron like region, below2.5 eV, with high and constant reflectance [73]. Early band structurecalculations by Neckel et al. [74] explained different onset of transitionenergy for TiC and TiN. The position where EF situated is the keyto the onset of interband transition energy or plasma energy. In TiN,EF is located at the metal Ti-d dominated states, the free-electron likebehavior comes from the d electron-gas. While in TiC, C has one electronless than N, EF moves down to the minimum of DOS, it becomes semi-


0

5

10

15

20

25

30

C-pTi-d

-20 -15 -10 -5 0 5 10 15Energy (eV)

0

5

10

15

20

25

30D

OS

(Sta

tes/

eV u

nit c

ell)

N-pTi-d

Ti3SiC

2

Ti4AlN

3

Figure 6.8: Density of states for Ti3SiC2 and Ti4AlN3 .

metal. The plasma energy is proportional to the position of EF abovethe DOS minimum. In Fig. 6.8, we plot the density of states for Ti3SiC2

and Ti4AlN3, We can observe that as Si addition to TiC, EF moves upto higher energy [75]; while as Al addition to TiN, EF moves down tothe bottom of DOS valley [76]. This indicates that Ti3SiC2 should havehigher plasma energy also higher conductivity than Ti4AlN3.

6.7 Surface energy

Surface energy is one of the fundamental properties in the surface sci-ence. In the Ti-Si-C system, surface energy calculations for the (0001)surface have been made. The surface energy for TiC (111) was used asa reference to compare the the Ti-C bond strength between TiC andthe Ti-Si-C system. Because MAX phases have the layered hcp struc-ture along the [111] direction of the TiC, if we stack fcc TiC along [111]direction, and replace a single layer of Si with C, we will get the hcpTi3SiC2 structure. Due to different bond strength between Ti-C andTi-Si bond, the atom positions are slightly rearranged along the z direc-tion in Ti3SiC2 compared with TiC. In order to study the (0001) surfaceenergy of the Ti-Si-C system, it would be very important to look at thesurface energy in TiC.

Firstly, we compared the calculated surface energy for the unpolar(001) surface of TiC with other existing results. The basis vector for fccwas represented using a primitive tetragonal basis (1/2, -1/2, 0) (1/2,1/2, 0) (0 0 1). The resulting surface energy 2.26 J/m2 agrees wellwith 2.254 J/m2 reported by Arya et al. [77], which shows that ourcalculations should be reliable.

The Ti terminated surface has the lowest energy for the polar TiC

6.7. SURFACE ENERGY 49

(111) surface, while the C surface terminations yield much high surfaceenergies. A calculated Ti terminated (111) surface energy Ef = 3.122J/m2 was reported by Arya et al. [77]. The anisotropy in fcc TiC can bedescribed by the ratio γ100/γ111 [78], a ratio of 1 gives a perfect isotropicstructure. A ratio 1.875 for TiC indicates that it is strongly anisotropic.In order to study the Ti-C bond strength in TiC, we take the Ti and Cterminated surface. The calculated average surface energy 5.4 J/m2 ismuch higher than the result of Arya et al., which indicates that the Cterminated surface might have even higher surface energy.

The (0001) surface energy in the Ti-Si-C system was calculated byinserting vacuum layers with the thickness 12 atomic layers (1 lattice pa-rameter along z direction). As we know, there are three types of bondingTiI-C, TiII-C and Ti-Si, we insert the vacuum at different position tobreak the different types of bonds. The average of all these three typesof surface energy is the average surface energy of the (0001) plane,

E =Evac − Ebulk

2. (6.2)

where Evac is the total energy of surface slab plus vacuum layers andEbulk is the bulk energy. To get reliable surface energies, we have toconverge our surface energies with respect to k-points and vacuum layers.In this calculation, we have checked convergence up to 24 vacuum layers.The average surface energies are shown in Table 6.4. Our calculatedaverage surface energy is strongly correlated to the bond strength. Therelation between bulk modulus and the bond strength is illustrated, onecan see from Table 6.4, the bulk modulus increases from 205 to 245 withincreasing bond strength. But all these MAX phases is softer than TiCbecause of the much weaker Ti-Si bond. The Ti-C (both TiII-C andTiI-C) bond is calculated to be much stronger than the Ti-Si bond. TheTiII-C type bond is slightly stronger than the TiI-C type bond, and theTiI-C bond strength is similar to the Ti-C bond in TiC.

Table 6.4: The surface energy (eV) and bulk modulus (GPa) in theTi-Si-C system.

Ti-Si TiI-C TiII-C γs B (GPa)

TiC 3.052 3.052 289211 1.593 3.330 2.462 205312 1.585 3.083 3.583 2.750 233413 1.583 3.317 3.586 2.951 245


6.8 Ductility

Crystalline materials divide into two categories depending on their num-ber of slip planes. Materials have a large multiplicity of easy glide slipdemonstrate significant ductility, and the dislocation activity in this sys-tem is irreversible. Meanwhile, materials that exhibit limited amountsof slips are generally brittle (e.g. ceramics) [79], and constrain their usein a number of technological applications.

One criterion proposed by Paugh for isotropic materials is the ra-tio between B (bulk) and G (shear modulus). There B represents theresistance to fracture, while G represents the resistance to plastic defor-mation. The critical value around 1.75 differentiates ductile from brittlematerials [80].

An approximation to calculate the elastic properties for cubic struc-ture from Gruneisen constant was introduced by Korzhavyi et al. [81].

We take the simple expression for the determination:

γ = −1 − V

2

∂3E/∂V 3

∂2E/∂V 2(6.3)

E is the total energy, which can take the integrated form of the thirdorder BM-EOS,

E(V ) = − 916B0[(4 − B′

0)V 3

0

V 2 − (14 − 3B′0)

V7/3

0

V 4/3

+(16 − 3B′0)

V5/3

0

V 2/3] + E0 (6.4)

Then we can derived the γ at V = V0

γ =1

2B′ − 1

2, (6.5)

This γ is exactly the Dugdale-MacDonald Gruneisen constant, whichis proved to lie closest to the that determined from ab initio calculation[82].

The relation between the Gruneisen constant and Poisson’s ratio is

σ =4γ − 3

6γ + 3(6.6)

The ratio between B and G can be determined through Poisson’sratio

B/G =2(1 + σ)

3(1 − 2σ). (6.7)

Our calculated Poisson’s ratio is in very good agreement with theab initio value 0.25 calculated by Holm et al. [83] and in reasonable

6.8. DUCTILITY 51

Table 6.5: Comparison between TiC, Ti2SiC, Ti3SiC2 and Ti4SiC3. forbulk modulus (B0), first pressure derivative of bulk modulus (B ′

0), shearmodulus (G), Young’s modulus (E) and Poisson’s ratio (σ).

TiC 211 312 413

B0 289 205 233 245B′

0 3.892 4.043 4.045 4.069G 183 120 137 142E 454 302 343 357σ 0.238 0.254 0.255 0.257

B/G 1.578 1.703 1.704 1.725

agreement with the experimental value 0.20 [84]. In Table 6.5, we listthe B/G ratio in the sequence of TiC, Ti2SiC, Ti3SiC2 and Ti4SiC3.

It is well known that TiC is quite brittle. We can observe the pro-nounced change of B/G with Si addition to TiC and this makes it be-come more ductile. However, in our calculation, the B/G is very sen-sitive to the fitting parameter B ′ and the slight increase or decrease ofB′ can affect the ratio. Therefore to get reliable values for B/G, elasticconstants calculations have to be carried out for this system.

Chapter 7

Sammanfattning pa svenska

Walter Kohn fick nobelpris i kemi 1998 for sina insatser inom elektron-strukturteorin. Kohns vetenskapliga arbeten har gjort det mojligt forkemister och fysiker att berakna fundamentala egenskaper hos molekyleroch fasta amnen med hjalp av datorer, utan att behova gora nagra somhelst experiment i laboratoriet.

Redan under 1930-talet var fysiker i full fart med att utarbeta dekvantmekaniska ekvationer som bestammer uppforandet hos ett systemmed manga elektroner. Tyvarr kunde de inte losa dessa ekvationer utomfor de allra enklaste fallen. De teoretiska fysikerna utvecklade en mangdapproximationsmetoder, men ingen av dem var tillrackligt slagkraftigeller tillrackligt enkel att genomfora. 1964 bevisade Kohn tillsammansmed Pierre Hohenberg en hypotes som sedan blev central for deraslosning till elektronproblemet, nagot som idag kallas for densitetsfunk-tionalteori (DFT). DFT skiljer sig fran de traditionella kvantkemiskametoderna och ger inte en korrelerad N-kroppars vagfunktion. Mangaav de kemiska egenskaperna hos molekyler och de elektroniska egen-skaperna hos fasta amnen bestams av elektroner som vaxelverkar mellanvarandra och med atomkarnorna. Enligt en tidig version av DFT somKohn utarbetade tillsammans med Sham var det tillrackligt att kannatill medeltatheten for elektronerna i rummets alla punkter for att enty-digt bestamma den totala energin, och darigenom en mangd andra egen-skaper hos systemet. DFT-teorin ar i grunden en s.k. en-elektronteorioch uppvisar manga likheter med Hartree-Fock-teorin. DFT har kom-mit att bli den dominerande metoden under det sista artiondet, varandeden som ar potentiellt mest kapabel till att leverera noggranna resultattill laga kostnader.

Med hjalp av den allra senaste formuleringen av densitetsfunktion-alteorin har stora framsteg gjorts vad betraffar berakning av mekaniskaoch elektroniska egenskaper hos fasta amnen. Olika approximationer

53

54 CHAPTER 7. SAMMANFATTNING PA SVENSKA

vad betraffar behandlingen av de utbytes-korrelationfunktionaler, somanvands inom densitetsfunktionalteorin for grundtillstand och exciter-ade tillstand har lagts fram, exempelvis lokaldensitetsapproximationen(LDA), generaliserande gradient approximationen (GGA), LDA+U ochGW-approximationen. Mangder av tillampningar inom materialmod-ellering i allmanhet liksom for nanotuber, kvantprickar och molekyler(aven artificiella) har genomforts. Eftersom teorin har utvecklats my-cket snabbt och natt en sadan avancerad niva har tiden nu har blivitmogen for ett teoribaserat angrepp vad betraffar support och supple-ment till experiment. Denna utveckling har ocksa fatt stor hjalp genomden fortsatta forbattring av kraftfulla datorer som har gjort det mojligtatt berakna materialegenskaper med en imponerande noggrannhet. Hu-vudtemat i denna avhandling ar att anvanda mycket noggranna teo-retiska ab initio-metoder for att forutsaga materialegenskaper och sokaefter nya teknologiskt anvandbara material.

Nyligen har monolagrade ternara foreningar MN+1AXN, dar N=1,2och 3, M en tidig overgangsmetall, A ar ett element fran A-gruppen(huvudsakligen III A och IV A), och X ar antingen kol och/eller kvave,dragit till sig ett alltmer okat intresse pa grund av deras unika egen-skaper. De temara karbiderna och nitriderna kombinerar egenskaper hosbade metaller och keramer. Liksom metallerna ar de goda termiska ochelektriska ledare med elektriska och termiska konduktiviteter varierandefran 0.5 till 14×106 Ω−1m−1, och fran 10 till 40W/m·K, respektive. Dear relativt mjuka med en Vickers hardhet pa omkring 2-5 GPa. Pasamma satt som keramer ar de elastiskt styva. Nagra av dem sasomTi3SiC2, Ti3AlC2 och Ti4AlN3 uppvisar aven excellenta mekaniska egen-skaper vid hoga temperaturer. De motstar termisk chock och ar ovanligttaliga gentemot skador, och uppvisar utmarkt korrosionsbestandighet.Framfor allt, till skillnad fran konventionella karbider och nitrider, kande formbehandlas med konventionella verktyg utan smorjmedel, vilkethar stor praktisk betydelse for anvandandet av MAX-faserna. De ovannamnda egenskaperna gor MAX-faserna till en ny familj av teknologisktviktiga material. Systematiska studier av MAX-faserna vad betraffarelektronstruktur, bindning, elastiska och optiska egenskaper har utfortsi denna avhandling. Elektronstrukturen och bindningsegenskaper hosdessa fasta amnen ar nyckeln till att exploatera egenskaperna hos MAX-faserna.

Avhandlingen ar uppdelad enligt foljande. Sektion 2 beskriver hu-vuddragen hos densitetsfunktionalteorin. Sektion 3 beskriver de berakningsmetoder som jag har anvant for mina berakningar. Mina forskn-ingsresultat ar indelade i tre delar:

1) Fasovergangar och relaterade egenskaper sasom fasstabilitet och

55

tillstandsekvationer, vilket behandlas i kapitel 3.2) Berakning av linjara optiska egenskaper hos vissa halvledare, sasom

solcellsmaterialen CuIn(Ga)Se2 och den scintillerande kristallen PbWO4.3) I kapitel 5 behandlas tillampningar av densitetsfunktionalteorin pa

MAX - faserna, inkluderande deras elektroniska, bindnings - , mekaniskaoch optiska egenskaper.

Acknowledgments

The work presented above was made possible by support of the Con-densed Matter Theory Group (Fysik IV) at the Uppsala University (UU)under the supervision of Docent Rajeev Ahuja. Many other colleaguesin the group have greatly contributed to my work and enriched my life.Words are not enough to express my cherishment of all the memorablemoments of my stay in Fysik IV. I would like to extend my sincere grati-tude to all those that had given me a helpful hand in the past four years.Among them, I am especially grateful to:

Docent Rajeev Ahuja, my supervisor, for your indispensable hand-to-hand education and visionary supervision; for all of the encouragementsand supports you gave me, on my research work, on my publicationand thesis write-up. I am especially thankful to you for being the onerecruited me to the group.

Prof. Borje Johansson, for your patience in revising my papers andthesis. Thank you especially for treating me as a friend, enlighteningme with your wisdom in academics as well as in life.

Prof. Olle Eriksson, for your interesting lectures, kind jokes, and forthe pleasant working experience.

My great thanks will also be directed to my co-workers J. M. Wills,Anna Delin and Kay Dewhurst who implemented the code, which fun-damentally helped my work.

Prof. Ulf Jansson, Prof. C-G Ribbing, Prof. Michel Barsoum, Prof.Jochen M. Schneider, Dr. Zhimei Sun, Dr. Martin Magnuson and Dr.Jens -P. Palmquist, I appreciate your collaborations, which significantlyenriched my knowledge on MAX phases.

Dr. Yi Wang, for your kind helps in solving the problems I encoun-tered during the work. Ms. Lunmei Huang and all my Chinese friendsin Uppsala for your sharing the feelings of living in foreign country inour mother language.

Jorge Osorio, Alexei Grechnev, Weine Olovsson, Jailton Souza deAlmeida, Erik Holmstrom and all other colleagues in the CondensedMatter Theory Group in Uppsala University, for your generous assis-tance in my study, my computer problems and everything else. Most

57


importantly, for your care-free fellowships!Sincere gratitude is to my dear parents, and Enyin, Lingli, Siyun as

well as all of other family members, for your love, your deep-touchinglove!

Lastly, Yupeng, for your dearest love! Life would not be the sameas meaningful with you out.

Bibliography

[1] P. Hohenberg and W. Kohn, Phys. Rev. B 136, 864 (1964).

[2] W. Kohn and L. J. Sham, Phys. Rev. A 140, 1133 (1965).

[3] L. Hedin and B. I. Lundqvist, J. Phys. C 4, 2064 (1971).

[4] J. P. Perdew, K.Burke, and M.Ernzerhof, Phys. Rev. Lett. 77, 3865(1996).

[5] V. I. Anisimov, J. Zaanen, and O. K. Andersen, Phys. Rev. B 44,943 (1991).

[6] F. Gygi and A. Baldereschi, Phys. Rev. Lett. 62, 2160 (1989).

[7] M. W. Barsoum, Solids. Prog. Solid. St. Chem. 28, 201 (2000).

[8] M. W. Barsoum, H. I. Yoo, I. K. Polushina, V. Y. Rud, Y. V. Rud,and T. El-Raghy Phys. Rev. B 62, 10194 (2000).

[9] H. I. Yoo, M. W. Barsoum, and T. El-Raghy, Nature 407, 581-582(2000).

[10] N. W. Ashcroft and N. D. Mermin Solid State Physics, P. 332.

[11] T. Gasche Ground State, Optical and Magneto-Optical Proper-

ties from First Principles Theory PhD thesis, Uppsala University(1993).

[12] H. L. Skriver The LMTO Method (Springer-Verlag, Berlin, 1984).

[13] M. Methfessel, Phys. Rev. B 38, 1537 (1988).

[14] O. K. Andersen Computational Methods in Band Theory P. 178

[15] B. Johansson, Philos. Mag. 30, 469 (1974).

[16] Joakim Trygg First Principles Studies of Magnetic and Structural

Properties of Metallic Systems PhD thesis, Uppsala University(1995).

59

60 BIBLIOGRAPHY

[17] P. E. Blochl, Phys. Rev. B 50, 17953 (1994).

[18] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).

[19] K. Laasonen, A. Pasquarello, R. Car, C. Lee, and D. Vanderbilt,Phys. Rev. B 47, 10142 (1993).

[20] M. W. Schmidt and S. Poli, Earth Planet. Sci. Lett. 163, 361 (1998).

[21] S. Ono, J. Geophys. Res. 103, 18253 (1998).

[22] S. Ono, Contrib. Mineral. Petrol. 137, 83 (1999).

[23] M. W. Schmidt, L. W. Finger, R. J. Angel, and R. E. Dinnebier,Amer. Mineral. 83, 881 (1998).

[24] E. Ohtani, K. Litasov, A. Suzuki, and T. Kondo, Geophys. Res.Lett. 28, 3991 (2001).

[25] G. Grimval, Thermophysical properties of materials (North-HollandPhysics Publishing, Amsterdam 1986).

[26] B. B. Karki, G. J. Ackland, and J. Crain, J. Phys. C 9, 8579 (1997).

[27] R. Ahuja, J. M. Wills, B. Johansson and O. Eriksson, Phys. Rev.B 48, 16269 (1993).

[28] P. Alippi, P. M. Marcus, and M. Scheffler, Phys. Rev. Lett. 78,3892 (1997).

[29] P. Soderlind, O. Eriksson, B. Johansson and J. M. Wills Phys. Rev.B 52, 13169 (1995).

[30] G. Ackland RIKEN Review 29, 34 (2000).

[31] A. van de Walle and G. Ceder, Rev. Modern Phys. 74, 11 (2002).

[32] Yi Wang, Theoretical Studies of Thermodynamic Properties of Con-

densed Matter under High Temperature and High Pressure PhD the-sis, KTH (2004).

[33] Y. K. Vohra and P. T. Spencer, Phys. Rev. Lett. 86, 3068 (2001).

[34] W. B. Holzapfel, High Press. Res. 7, 290 (1991).

[35] W. B. Holzapfel, Europhys. Lett. 16, 67 (1991).

[36] F. D. Murnaghan, Proc. Natl. Acad. Sci. 30, 244 (1944).

BIBLIOGRAPHY 61

[37] F. Birch, Phys. Rev. B 71, 809 (1947).

[38] F. Birch, J. Geophys. Res. 83, 1257 (1978).

[39] T. Tsuchiya and K. Kawamura, J. chem. Phys, 114, 10086 (2001).

[40] P. Vinet, J. Ferrante, J. R. Smith and J. H. Rose, J. Phys. C 19,467 (1986).

[41] P. Vinet, J. H. Rose, J. Ferrante. and J. R. Smith, J. Phys. C 1,1941-1963 (1989).

[42] J. P. Poirier, Introduction to the Physics of the Earth’s Interior

(Cambridge University Press, 2000, 2nd ed.)

[43] N. Sata, G. Shen, M. L. Rivers and S. Sutton, Phys. Rev. B 65104114 (2002).

[44] J. P. Rueff, M. Krisch, Y. Q. Cai, A. Kaprolat, M. Hanfland, M.Lorenzen, C. Masciovecchio, R. Verbeni and F. Sette, Phys. Rev. B60, 14510 (1999).

[45] A.B. Belonoshko and L.S. Dubrovinsky, Phys. Earth and Planet.Inter. 98, 47 (1996).

[46] R. J. Hemley and H.-K. Mao, Int. Geol. Rev. 43, 1 (2001).

[47] G. Steinle-Neumann, L. Stixrude, and R. E. Cohen, Phys. Rev. B60, 791 (1999).

[48] H. K. Mao, Y. Wu, L. C. Chen, and J. F. Shu, J. Geophys. Res.95, 21737 (1990).

[49] A. P. Jephcoat, H. K. Mao, and P. M. Bell, J. Geophys. Res. 91,4677 (1986).

[50] L. S. Dubrovinski, S. K. Saxena, F. Tutti and S. Rekhi, Phys. Rev.Lett. 84, 1720 (2000).

[51] C. F. Klingshirn, Semiconductor Optics (Springer-Verlag, Berlin,1997).

[52] C. C. Kim, J. W. Garland, H. Abad, and P. M. Raccah, Phys. Rev.B 45, 11749 (1992).

[53] P. M. Oppeneer, T. Maurer, J. Sticht, and J. Kubler, Phys. Rev. B45, 10924 (1992).

62 BIBLIOGRAPHY

[54] M. Itoh and M. Fujita, Phys. Rev. B 62, 12825 (2000).

[55] I. N. Shpinkov, I. A. Kamenskikh, M. Kirm, V. N. Kolobanov, V.V. Mikhailin, A. N.Vasilev and G. Zimmerer, Phys. Status Solidi A170, 167 (1998).

[56] S. H. Wei, S. B. Zhang and A. Zunger, Appl. Phys. Lett. 72, 3199(1998).

[57] A. M. Gabor, J. R. Tuttle, D. S. Albin, M. A. Contreras, R. Noufiand A. M. Hermann, Appl. Phys. Lett. 65, 198(1994).

[58] D. S. Albin, J. J. Carapella, J. R. Tuttle, and R. Noufi, Mater. Res.Soc. Symp. Proc. 228, 267 (1991).

[59] J. E. Jaffe and Alex Zunger, Phys. Rev. B 27, 5176 (1983).

[60] T. Kawashima, S. Adachia, H. Miyake, and K. SuSugiyama, J.Appl. Phys. 84, 5202 (1998).

[61] H. Nowotny, Prog. Solid State Chem., H. Reiss, Ed. 2, 27 (1970).

[62] W. J. J. Wakelkamp, F. J. J. van Loo, and R. Metselaar, J. Eur.Ceram. Soc. 8, 135 (1991).

[63] A. Grechnev, R. Ahuja, and O. Eriksson, J. Phys. C 15, 7751(2003).

[64] P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka, and J. Luitz,WIEN2k, An Augmented Plane Wave + Local Orbitals Program for

Calculating Crystal Properties (Karlheinz Schwarz, Techn. Univer-sity Wien, Austria), 2001, ISBN 3-9501031-1-2.

[65] Y. Zhou and Z. Sun, J. Phys. C 12, 457 (2000).

[66] J. C. Ho, H. H. Hamdeh, M. W. Barsoum and T. El-Raghy, J. Appl.Phys. 85, 7970 (1999).

[67] R. Ahuja, S. Auluck, J. M. Wills, M. Alouani, B. Johansson, andO. Eriksson. Phys. Rev. B 55, 4999 (1997).

[68] C. Kittel, Introduction to solid state physics, P. 272.

[69] G. G. Fuentes, E. Elizalde, and J. M. Sanza, J. Appl. Phys. 90,2737 (2001).

[70] A. Delin, O. Eriksson, R. Ahuja, B. Johansson, M. S. S. Brooks, T.Gasche, S. Auluck, and J. M. Wills, Phys. Rev. B 54, 1673 (1996).

BIBLIOGRAPHY 63

[71] M. Brogren, G. L. Harding, R. Karmhag, C-G. Ribbing, G.A.Niklasson, L. Stenmark, Thin Solid Films 370, 268-277 (2000).

[72] J. F. Alward, C. Y. Fong, M. E. Batanouny, and F. Wooten, Phys.Rev. B 12, 1105 (1975).

[73] B. Karlsson, Optical Properties of Solids for Solar Energy Conver-

sion PhD thesis, Uppsala University (1981).

[74] A. Neckel, P. Rastl, R. Eibler, P. Weinberger, and K. Schwarz J.phys. C 9, 579 (1976).

[75] R. Ahuja, O. Eriksson, J. M. Wills, B. Johansson, Appl. Phys. Lett.76, 2226 (2000).

[76] B. Holm, R. Ahuja, S. Li, and B. Johansson, J. Appl. Phys. 91,9874 (2002).

[77] A. Arya, and E. A. Carter, J. Chem. Phys. 118, 8982 (2003).

[78] J. Kollar, L. Vitos, B. Johannson, and H. L. Skriver, phys. stat. sol.217, 405 (2000).

[79] M. W. Barsoum, T. Zhen, S. R. Kalidindi, M. Radovic, and A.Murugaiah, Nature 2, 107 (2003).

[80] P. Ravindran, L. Fast, P. A. Korzhavyi, B. Johansson, J. M. Wills,and O. Eriksson, J. Appl. Phys. 84, 4891 (1998).

[81] P. A. Korzhavyi, A. V. Ruban, S. I. Simak, and Yu. Kh. Vekilov,Phys. Rev. B 49, 14229 (1994).

[82] L. Vocadlo, J. P. Poirer, and G. D. Price, Ame. Miner. 85, 390(2000).

[83] B. Holm, R. Ahuja, Y. Yourdshahyan, B. Johansson, and B. I.Lundqvist, Phys. Rev. B 59, 12777 (1999).

[84] P. Finkel, M. W. Barsoum, and T. El-Rahy, J. Appl. Phys. 84, 5817(1998).

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 91-554-5976-5

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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