densityfunctionaltheorystudy ofbulkpropertiesofmetallic...

Density Functional Theory Study

of Bulk Properties of MetallicAlloys and Compounds

Liyun Tian

Doctoral ThesisSchool of Industrial Engineering and Management, Department of

Materials Science and Engineering, KTH, Sweden, 2017

MaterialvetenskapKTH

SE-100 44 StockholmISBN 978-91-7729-350-7 Sweden

Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolanframlägges till offentlig granskning för avläggande av doktorsexamen fredagenden 2 Juni 2017 kl 10:00 i F3, Kungliga Tekniska Högskolan, Lindstedtsvägen26, Stockholm.

© Liyun Tian, 2017

Tryck: Universitetsservice US AB

Abstract

First-principles methods based on Density functional theory (DFT) are nowadopted routinely to calculate the properties of materials. However, one of thebiggest challenges of DFT is to describe the electronic behaviors of randomalloys. One of the aims of this thesis is to study binary alloys, e.g. Ti-Al,Cu-Au, and multi-component alloys by using two models for chemically randomstructures: the special quasi-random structure (SQS) and coherent potentialapproximation (CPA).I investigate these approaches by focusing on the local lattice distortion (LLD)

and the crystal symmetry effects. Within the SQS approach, the LLD effect canbe modeled in a straightforward manner by relaxing the positions of atoms inthe supercell. However, within this approach it is difficult to model the randommulti-components alloys due to the large size of the supercells. On the otherhand, the CPA approach uses single-site approximation and thus it is not limitedby the number of alloy components. But CPA suffers from the neglect of thelocal lattice relaxation effect, which in some systems and for some propertiescould be of significant importance.In my studies, the SQS and CPA approaches are combined with the pseu-

dopotential method as implemented in the Vienna Ab-initio Simulation Package(VASP) and the Exact Muffin-Tin Orbitals (EMTO) methods, respectively. Themixing energies or formation enthalpies and elastic parameters of fcc Ti1−xAlxand Cu1−xAux (0 ≤ x ≤ 1) random solid solutions and high-entropy multi-component TiZrVNb, TiZrNbMo and TiZrVNbMo alloys are calculated as afunction of concentration. By comparing the results with and without locallattice relaxations, we find that the LLD effect is negligible for the elastic con-stants C11, C12 and C44. In general, the uncertainties in the elastic parametersassociated with the symmetry lowering in supercell studies turn out to be supe-rior to the differences between the two alloy techniques including the effect ofLLD. However, the LLD effect on the mixing energies or formation enthalpiesis significant and depends on the degree of size mismatch between alloy con-stituents. In the cases of random Cu-Au and high-entropy alloys, the formationenthalpies and mixing energies are significantly decreased when the LLD effectis considered. This finding sets the limitations of CPA for the mixing energiesor formation enthalpies of alloys with large atomic size differences.The other goal of the thesis is to study the effect of exchange-correlation

functionals on the formation energies of ordered alloys. For this investigation,we select the Cu-Au binary system which has for many years been in the fo-cus of DFT and beyond DFT schemes. The Perdew-Burke-Ernzerhof (PBE)approximation to the exchange-correlation term in DFT is a mature approachand have been adopted routinely to investigate the properties of metallic alloys.In most cases, PBE provides theoretical results in good agreement with exper-iments. However, the ordered Cu-Au system turned out to be a special casewhere large deviations between the PBE predictions and observations occur. Inthis work, we make use of a recently developed exchange-correlation functional,the so-called quasi-nonuniform exchange-correlation approximation (QNA), tocalculate the lattice constants and formation energies for ordered Cu-Au alloys

iv

as a function of composition. The calculations are performed using the EMTOmethod and verified by a full-potential method. We find that the QNA func-tional leads to an excellent agreement between theory and experiment. ThePBE strongly overestimates the lattice constants for ordered Cu3Au, CuAu,CuAu3 compounds and also for the pure metals which are nicely corrected bythe QNA approach. The errors in the formation energies of Cu3Au, CuAu,CuAu3 relative to the experimental data decrease from 38-45% obtained withPBE to 5-9% calculated for QNA. This excellent result demonstrates that onecan reach superior accuracy within DFT for the formation energies and thereis no need to go beyond DFT. Furthermore, it shows that error cancellationcan be very effective for the formation energies as well and that the main DFTerrors obtained at PBE or LDA levels originate from the core-valence overlapregion, which is correctly captured by QNA due to its particular construction.Our findings are now extended to disordered alloys, which is briefly discussedalready in one of my published papers.

v

PrefaceList of included publications:

I Elastic constants of random solid solutions by SQS and CPA approaches:the case of fcc Ti-AlLi-Yun Tian, Qing-Miao Hu, Rui Yang, Jijun Zhao, Börje Johanssonand Levente Vitos, J. Phys.: Condens. Matter. 27, 316702 (2015).

II CPA descriptions of random Cu-Au alloys in comparison with SQS ap-proachLi-Yun Tian, Li-Hua Ye, Qing-Miao Hu, Jijun Zhao, Börje Johanssonand Levente Vitos, Computational Materials Science 128, 302-309 (2017).

III Alloying effect on the elastic properties of refractory high-entropy alloysLi-Yun Tian, Guisheng Wang, Joshua S. Harris, Douglas Irving, JijunZhao, Levente Vitos, Materials and Design 114, 243-252 (2017).

IV Exchange-Correlation Catastrophe in Cu-Au: A Challenge for SemilocalDensity Functional ApproximationsLi-Yun Tian, Henrik Levämäki, Matti Ropo, Kalevi Kokko, Ágnes Nagy,and Levente Vitos, Physical Review Letters 117, 066401 (2016).

Comment on my own contribution:Paper I: 50% calculations and the data analysis was jointly, literature survey;the manuscript was written jointly.Paper II: This work was done jointly including the calculations, the data anal-ysis, literature survey and the manuscript.Paper III: All calculations, the data analysis was done jointly, literature survey;the manuscript was written jointly.Paper IV: This work was done jointly including the calculations, the data analy-sis, literature survey and the manuscript. However, the idea was established byme, I made the first calculations, demonstrated the excellent results and madesubstantial contribution to writing the paper and including relevant literature.

List of papers not included in the thesis:

I A First Principles Study of the Stacking Fault Energies for fcc Co-basedBinary AlloysLi-Yun Tian, Raquel Lizárraga, Henrik Larsson, Erik Holmström, LeventeVitos, submitted to Acta Materialia.

II Challenges in automated high-throughput ab inito thermodynamics ofmagnetic high-entropy alloysHenrik Levämäki, Matti Ropo, Li-Yun Tian, in manuscript.

Contents

List of Figures x

List of Tables xi

1 Introduction 1

2 Theoretical Methodology 5

2.1 Hamiltonian of Electronic Structure . . . . . . . . . . . . . . . . 5

2.2 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Hohenberg-Kohn Theorems . . . . . . . . . . . . . . . . . 6

2.2.2 Kohn-Sham Theory . . . . . . . . . . . . . . . . . . . . . 7

2.3 Solution of Kohn-Sham equations . . . . . . . . . . . . . . . . . . 9

2.4 Exchange-Correlation functional . . . . . . . . . . . . . . . . . . 9

2.4.1 Local Density Approximation - LDA . . . . . . . . . . . . 9

2.4.2 Generalized Gradient Approximation - GGA . . . . . . . 10

2.5 Exact Muffin-Tin Orbitals . . . . . . . . . . . . . . . . . . . . . . 13

2.6 Random structural models . . . . . . . . . . . . . . . . . . . . . . 14

2.6.1 SQS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.6.2 CPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Elastic Properties of Materials 17

3.1 Single-Crystal Elastic Constants . . . . . . . . . . . . . . . . . . 17

3.1.1 Stress-strain relations . . . . . . . . . . . . . . . . . . . . 18

3.1.2 Energy-strain relations . . . . . . . . . . . . . . . . . . . . 19

3.2 Elastic Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

viii CONTENTS

4 CPA and SQS descriptions of binary alloys 234.1 Lattice constants of random alloys . . . . . . . . . . . . . . . . . 23

4.1.1 Lattice constants . . . . . . . . . . . . . . . . . . . . . . . 234.1.2 Electron numbers . . . . . . . . . . . . . . . . . . . . . . . 24

4.2 Mixing energy of Ti-Al and formation enthalpy of Cu-Au . . . . 274.2.1 Mixing energy and formation enthalpy . . . . . . . . . . . 274.2.2 Estimated enthalpy for Cu-Au alloys . . . . . . . . . . . . 294.2.3 Local lattice distortion effect . . . . . . . . . . . . . . . . 30

4.3 Elastic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.3.1 Elastic constants . . . . . . . . . . . . . . . . . . . . . . . 314.3.2 Elastic parameters versus chemical composition . . . . . . 344.3.3 Strain effect on the relaxation energy . . . . . . . . . . . . 35

5 High-entropy random solid solutions 375.1 LLD effect on lattice constants . . . . . . . . . . . . . . . . . . . 375.2 LLD effect on elastic constants . . . . . . . . . . . . . . . . . . . 39

5.2.1 Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . 395.2.2 Elastic parameters . . . . . . . . . . . . . . . . . . . . . . 405.2.3 Rule of mixtures . . . . . . . . . . . . . . . . . . . . . . . 42

5.3 Mixing energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.4 Local lattice distortion . . . . . . . . . . . . . . . . . . . . . . . . 45

6 QNA functional study of Cu-Au system 496.1 Performance of GGA-level . . . . . . . . . . . . . . . . . . . . . . 496.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.2.1 Exchange-correlation energy . . . . . . . . . . . . . . . . . 516.2.2 Alloying effect . . . . . . . . . . . . . . . . . . . . . . . . 52

6.3 QNA study of the other alloys . . . . . . . . . . . . . . . . . . . . 54

7 Conclusions and Future Work 577.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

8 Sammanfattning 59

A Appendix 63

Bibliography 71

List of figures

1.1 Schematic illustration of the relaxation in high-entropy alloys. . . 3

4.1 Theoretical lattice parameters for fcc Ti1−xAlx and Cu1−xAuxalloys plotted as a function of composition. . . . . . . . . . . . . 25

4.2 s, p and d electron numbers n of Ti1−xAlx plotted as a functionof composition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.3 Theoretical mixing energies for fcc Ti1−xAlx (a) and formationenthalpies for fcc Cu1−xAux alloys (b) plotted as a function ofcomposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.4 Local lattice distortion ∆d and relaxation energy ∆Er for fccTi1−xAlx and Cu1−xAux alloys plotted as a function of composition. 31

4.5 Single crystal elastic constants of fcc Ti1−xAlx and Cu1−xAuxalloys plotted as a function of compositions. . . . . . . . . . . . . 32

4.6 Tetragonal elastic constant C ′ and bulk modulusB of fcc-Ti1−xAlxas a function of composition. . . . . . . . . . . . . . . . . . . . . 34

4.7 The relaxation energies ∆Er for Cu0.5Au0.5 alloy as a functionof the deformation strains ε. . . . . . . . . . . . . . . . . . . . . . 35

5.1 Theoretical lattice constants of high-entropy TiZrVNb, TiZrNbMoand TiZrVNbMo alloys are shown for both SQS and CPA calcu-lations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.2 Comparison between the theoretical (VASP and EMTO) and ex-perimental elastic constants for the refractory elements. . . . . . 39

5.3 Single-crystal elastic constants of bcc TiZrVNb, TiZrNbMo andTiZrVNbMo alloys obtained by unrelaxed SQS (SQSu), relaxedSQS (SQSr) and CPA calculations. . . . . . . . . . . . . . . . . . 40

5.4 Theoretical and estimated lattice parameters and elastic con-stants plotted for the present TiZrNbMo, TiZrVNb and TiZrNbMoValloys. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

x LIST OF FIGURES

5.5 Mixing energies of high-entropy TiZrVNb, TiZrNbMo and TiZrVNbMoalloys obtained for both SQS (unrelaxed and relaxed) and CPAapproaches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.6 Histogram plot of LLD for the high-entropy TiZrNbMo (top),TiZrVNb (middle) and TiZrVNbMo (bottom) alloys. . . . . . . . 46

5.7 Average distances and the standard deviations for the first nearestneighbor (NN) pairs. . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.1 The present lattice constants (upper panel) and formation ener-gies (lower panel) for the Cu-Au system calculated at PBE, QNAand SCAN levels. . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.2 Enhancement function maps for PBE, QNA(Cu) and QNA(Au).The s vs rs curves are drawn for pure elements and for elementsin the Cu3Au alloy. . . . . . . . . . . . . . . . . . . . . . . . . . . 54

List of tables

2.1 Optimal QNA and PBE parameters µ and β for Cu and Au. . . 12

4.1 Contributions for the formation enthalpies of random Cu0.75Au0.25,Cu0.5Au0.5 and Cu0.25Au0.75 alloys come from local lattice dis-tortion, DFT (XC), short range order (SRO). . . . . . . . . . . . 30

5.1 The theoretical lattice constants for the bcc underlying lattice,elastic constants, the Zener anisotropy and Cauchy pressure, poly-crystalline elastic moduli, Poisson’s ratio and the Pugh ratio ofthe TiZrVNb, TiZrNbMo and TiZrVNbMo alloys calculated us-ing the EMTO-CPA and VASP-SQS methods . . . . . . . . . . . 41

6.1 Alloy component resolved exchange-correlation formation energies. 526.2 Formation energies of selected binary alloys at LDA, PBE, QNA,

SCAN and HSE [26]. . . . . . . . . . . . . . . . . . . . . . . . . . 55

Chapter 1

Introduction

First-principles methods based on density functional theory (DFT) [1–3] arepowerful tools in condensed mater theory and materials science to predict theproperties of materials. While these methods still have some problems to de-scribe directly the properties of random alloys. Further approximations have tobe made in order to describe the randomness. Three random structural models,special quasi-random structure (SQS) [4, 5], coherent potential approximation(CPA) [6, 7] and virtual-crystal approximation (VCA) [8, 9], are the mostly usedapproaches to describe the behavior of metallic alloys. The features of these ap-proaches are different. SQS was designed by randomly occupying N sites by thealloy components in a periodic supercell, while CPA and VCA were developedas single-site approximations. Since the accuracy of VCA is limited to systemswith nearest-neighber elements, here we do not discuss this approach.In the thesis, I calculate the electronic properties of disordered solid solu-

tions, such as lattice parameters, formation enthalpies and elastic constants,using SQS and CPA approaches. The local lattice distortion (LLD) effect in-duced by local chemical environment is properly taken into account within theSQS. The limitation of SQS approach is the size of the supercells especially whenapplied to multi-component alloys. In the CPA method, the LLD effect can notbe considered due to the single-site approximation, e.g. the A-A, A-B and B-Bbonds are equal in binary A1−xBx alloy. Nevertheless, CPA is a very successfulmean-field approach for random alloys. Due to the single-site nature, the com-putational load is mitigated. Furthermore, in contrast to the SQS technique,the coherent Green’s function retains the symmetry of the alloy matrix and thusCPA provides properties which reflect the symmetry of the underlaying lattice.The first aim of the thesis is to compare these two approaches and determinethe effect of LLD in the properties of materials.By using the two random Ti-Al and Cu-Au binary alloys as examples, we

demonstrate the performances of SQS and CPA approaches by comparing thetrends of calculated structural and elastic properties against composition withand without the influence of local lattice relaxation. Both of these systems areconsidered in the face centered cubic (fcc) crystallographic lattice.Ti-Al and Cu-Au alloys are two representative examples due to the atomic size

mismatch with 1.6% for Ti-Al alloys and 12.2% for Cu-Au alloys. Some distinctcharacteristics are shown for the two cases. Previous studies of Cu-Au alloys

2 CHAPTER 1. INTRODUCTION

mainly focused on the order-disorder transitions [10–12] and the properties ofordered phase [13]. For disordered Cu-Au alloys and the hypothetical randomTi-Al alloys, the elastic properties are scarce. Here we adopt the SQS andCPA methods to describe these alloys on ab initio level. Furthermore, we givequantitative values for the effect of LLD and lattice symmetry on the mixingenergies or formation enthalpies and elastic constants of random Ti-Al and Cu-Au alloys.The high-entropy alloys (HEAs), which are equiatomic or nearly-equiatomic

multicomponent alloys, have attracted attention due to their unique mechanicalproperties [14–20]. In addition to the first fcc HEAs, alloys with body centeredcubic (bcc) and hexagonal close packed (hcp) structures have also been synthe-sized. In particular, the TiZrNbMoVx alloys have been verified to adopt thebody centered cubic (bcc) phase and possess excellent mechanical properties [21,22]. In contrast to the intensive experimental investigations, so far there is arelatively small number of theoretical efforts focusing on HEAs.For the multicomponent HEAs, the CPA technique should be a well suited

approach to study the mechanical properties awning to the single-site approxi-mation which the calculations are low computational cost. Recently, an increas-ing number of ab initio calculations on HEAs have been performed by the CPAapproach [22–24]. On the other hand, SQS approximation requires large super-cells, consisting of different elements with random occupation. Nevertheless, incontrast to CPA, SQS can account for LLD. In Fig. 1.1, the schematic relaxationin the HEAs is shown. It is clear that when LLD is taken into account, LLDconsideration one can provide more accurate information [25].In the present case, three bcc TiZrVNb, TiZrNbMo and TiZrNbMoV alloys

are taken as examples to study the elastic properties of multicomponent alloyswith and without relaxation. CPA and SQS techniques in combination withfirst-principles methods were adopted to obtain the total energy. Energy-strainrelation was used to calculate the elastic constants of multicomponent alloys.Sufficiently large supercells were considered in the calculations to mitigate theperiodic boundary error and ensure closeness to the fully random structures.DFT is one important quantum-mechanical theory used for the calculations

of electronic properties. The core point of DFT is to solve the noninteractingelectron Kohn-Sham equations as I will describe in the following sections. Theapplication of DFT involves different approximations of the exchange-correlation(XC) functional needed to obtain more accurate electronic total energy. There-fore, the main aspect of DFT calculations is the choice of exchange-correlationfunctionals. The local density approximation (LDA) and the family of gener-alized gradient approximations (GGAs) are two most widely used functionalsfor solid states. However, there are still several problems associated with thelimited features of these approximations.In a recent investigation, it was found that the theoretical formation en-

ergies of ordered Cu-Au system are far smaller (in absolute value) than theexperimental values [26]. This finding has created some doubts concerning thescope of DFT and in particular of Perdew-Burke-Ernzerhof (PBE) [27] exchange-correlation functional for the Cu-Au system. In that work, the Heyd-Scuseria-Ernzerhof (HSE) hybrid functional [28, 29] was used and the computed resultsare in excellent agreement with the experimental data. However, the HSE ex-

3

(a)

(b)

Figure 1.1: Schematic illustration of the relaxation in high-entropy alloys [25]. (a)The ideal bcc structure of high-entropy alloys. (b) The relaxed structure ofhigh-entropy alloys.

change functional is mixed by Hartree-Fork and semilocal exchange functional,which undoubtedly requires a high computational cost.Quasi-nonuniform exchange-correlation approximation (QNA) [30, 31], which

is a new and improved XC approximation, produces excellent volumes and bulkmoduli for metallic solids. In our present work, QNA gives the perfect formationenergies for the Cu-Au system which is consistent with the experimental andHSE results. This study shows that QNA removes most shortcomings of PBEand is a more effective approximation than HSE functional due to the fact thatdepends only on density and density gradient.

Chapter 2

Theoretical Methodology

The mechanical properties of solid materials are investigated in this thesis us-ing first-principles methods based on DFT. In a number of aspects of materialproperties, DFT can give accurate and reasonable results relative to experimen-tal values. However, the strength of DFT lies on its ability to predict trendsin materials properties. As such DFT is a powerful aid in the design of newmaterials with improved properties. Today, DFT as the basic theory is widelyused in computational material science. In this chapter, we briefly introduceand discuss DFT.

The many-body Schrödinger equation contains the physical knowledge we areafter, but its usefulness is limited by the ability to find a good description ofthe many-body effects contained in the exchange-correlation functional. If wehave full knowledge of this functional DFT would be an exact theory. In recentyears, the different exchange-correlation functionals are developed and success-fully applied in practical calculations. More details will be given in followingdiscussions.

2.1 Hamiltonian of Electronic Structure

First-principles theory for a system of nuclei and electrons is based on solvingthe many body Schrödinger equation. The many-body Schrödinger equationcan be described as

HΨ(RI , ri) = EΨ(RI , ri) , (2.1)

where H is the Hamiltonian of the system, and E is the total energy. Ψ(RI , ri)is the many-body wavefunction describing the state of the system; RI and riare the positions of ion I and electron i.

In general, the Hamiltonian H for a system of electrons and nuclei can beexpressed as

6 CHAPTER 2. THEORETICAL METHODOLOGY

H = − ~2

2MI

∑I∇2

I −~2

2mi

∑i

∇2i

+ 12

I 6=J∑IJ

ZI ZJe2

|RI −RJ |−∑Ii

ZIe2

|RI − ri|+ 1

2

i6=j∑ij

e2

|ri − rj |,

(2.2)

where ~ is the Planck’s constant divided by 2π, e is the electron charge, ZI isthe charge of the nuclei, MI is the mass of Ith nuclei and mi is the mass ofith electron. Therefore, the contributions to the Hamiltonian come from thekinetic energy T of nuclei and electrons (the first and second terms) and thepotential energy due to the Coulomb interactions of nuclei-nuclei (namely VIJ),nuclei-electron (namely external potential Vext) and electron-electron (namelyinternal potential Vint), respectively.The Hamiltonian in the equation 2.2, including all interactions within the

Schrödinger equation [32, 33], expresses the full theory without any approxi-mation. In real materials, however, this question is not solvable (with very fewexceptions). Within Born-Oppenheimer approximation [34, 35], explicitly thekinetic contribution from nuclei is treated as a parameter due to the relativelylarge mass of the nuclei compared to the electrons. In addition, the term ofnuclei-nuclei interaction potential VIJ is neglected, because it is a constant [36].Therefore, the Hamiltonian of the simplified (only electronic) system becomes

H = − ~2

2mi

∑i

∇2i −

∑Ii

ZIe2

|RI − ri|+ 1

2

i 6=j∑ij

e2

|ri − rj |

= T + Vext + Vint ,

(2.3)

Unfortunately, since there is a large number of electrons in a solid the compli-cation of solving the electronic problem still remains, which leads us to furthersimplifications and approximations.

2.2 Density Functional TheoryIn the following we will solve the problems of many-electron Hamilton by thedensity functional approximations. It should be noted that we will discuss theapproximations based on the ground state of a system.

2.2.1 Hohenberg-Kohn Theorems

In 1964, Hohenberg and Kohn introduced the fundamentals of density functionaltheory [37] based on two theorems. The wave-function of the density functionaltheory is described by the ground states density n(r). The density n(r) is definedas the number of electrons per volume at the point r in space

N =∫n(r)dr. (2.4)

2.2 Density Functional Theory 7

Theoretically the corresponding ground state energy and density can be de-termined. In the following the two theorems are stated.

Theorem 1 : For a system of interacting electrons in an external potential,the potential is uniquely determined by the ground state electron density n(r).

EHK[n] = 〈Ψ|H|Ψ〉

= 〈Ψ|T + Vint + Vext|Ψ〉

= T [n] + Eint[n] + Eext[n]

= FHK[n] + Eext[n] ,

(2.5)

where FHK[n] is a universal functional of the electron density n(r) and thesecond term Eext[n] is the interaction energy with the external potential.

Corollary : The Hamiltonian and the ground state energy are also determineduniquely in terms of the electron density.

Theorem 2 : The ground state density minimizes the total energy functionalof the system EHK[n].

EHK[n] = minΨ→n(r)

〈Ψ|T + Vint + Vext|Ψ〉 (2.6)

The Hohenberg-Kohn equation as the exact density functional formulate ofa many-electron system is reasonable and have been proved. The Hohenberg-Kohn theorems are the basis of DFT.It has been established that a functional can be defined using the ground

state electron density and the exact ground state energy and density can befound by minimizing the energy functional. However one is still left with theproblem of writing the energy functional. Kohn and Sham tackled this problemby assuming the independent particle wavefunctions, which have been provedthat this is very successful for the development of density functional theory.

2.2.2 Kohn-Sham Theory

The total densities of an N-electron system were defined by Kohn and Sham [2]as

n(r) =N∑i

ni(r) =N∑i

|ψi(r)|2 , (2.7)

where ni(r) = |ψi(r)|2 is the density of single-particle.According to Eq.2.7, the total energy functional can be expressed by Hohen-

berg-Kohn theorem in terms of the independent particle wave functional:

EKS = 〈Ψ|T |Ψ〉+ 12

∫ ∫n(r)n(r′)|r− r′| drdr′ +

∫Vext(r)n(r)dr + Exc[n]

= Ts[n] + EH[n] + Eext[n] + Exc[n] ,(2.8)


where Ts[n] is the independent particle kinetic energy, EH[n] is the Hartreeenergy due to the interaction between the electrons and Exc(n) is the exchange-correlation energy which includes all the quantum mechanical effects.Comparing the Hohenberg-Kohn equation 2.5, the exchange-correlation en-

ergy can be written as

Exc[n] = FHK[n]− (Ts[n] + EH[n])

= (T [n]− Ts[n]) + (Eint[n]− EH[n]).(2.9)

Unfortunately, so far the minimum of the total energy functional could notbe established directly because the form of the kinetic energy functional is stillunknown. Furthermore, the true form of the exchange-correlation energy func-tional is not known either. A practical way to overcome this problem is to usethe Kohn and Sham scheme. The Kohn-Sham scheme introduces non-interactingkinetic energy operator and all the other terms (including the remaining kineticenergy) are incorporated in the so called effective or Kohn-Sham one-electronpotential. Therefore, the independent single-electron Kohn-Sham equations canbe written as [

− 12∇

2i + Veff

]ψi(r) = εiψi(r) , (2.10)

where εi are the eigenvalues of the hypothetical KS particles. The highestlaying eigenvalue defines the chemical potential for electrons (Fermi level) andit is the only quantity which has a well defined physical interpretation. Theindependent-particle wave function only depends on the effective potential Veffand ψi(r).The DFT solution for one independent-particle system can be viewed as the

question of minimization of the ground state electronic energy. Actually, inequation 2.8, the kinetic energy Ts can be expressed as a functional of theorbitals ψi(r) and the other terms (EH, Eext, Exc) are functionals of the densityn. The minimization of the ground state electronic energy can be expressed as

δEKS

δψ∗i (r) = δ

δψ∗i (r)

(Ts[n] + Eext[n] + EH[n] + Exc[n]

)

= δTs[n]δψ∗i (r) +

(δEext[n]δn

+ δEH[n]δn

+ δExc[n]δn

)δn

δψ∗i (r)

= δTs[n]δψ∗i (r) +

(Vext[n] + VH[n] + Vxc[n]

)δn

δψ∗i (r) = 0,

(2.11)

where VH is Hartree potential describing the Coulomb repulsion that can bedescribed by

VH[n] = 12

∫ ∫n(r)n(r′)|r− r′| drdr′ . (2.12)

Vxc is exchange-correlation potential defined by

Vxc[n] = δExc(n(r))δn(r) . (2.13)

The Kohn-Sham effective potential Veff [n] is given by

Veff [n] = Vext[n] + VH[n] + Vxc[n] . (2.14)

2.3 Solution of Kohn-Sham equations 9

Thus the Eq. 2.11 can be rewritten as

δEKS

δψ∗i (r) = δTs

δψ∗i (r) + δVeff

δn

δn

δψ∗i (r) = 0 . (2.15)

This equation is equivalent with that written for non-interacting electronssitting in an effective potential. Hence, within the Kohn-Sham scheme oneeffectively moves all interactions into an effective potential term and solves non-interacting equations self-consistently. Based on the discussion above, the solu-tion of Kohn-Sham equations (Eq. 2.10) is to find the exact electronic densityn(r) and the kinetic energy of an independent-particle system. The success ofthe Kohn-Sham equation is the description of effective potential Veff (Eq. 2.14),including the external, Hartree and exchange-correlation potentials. Next, weshould solve the Kohn-Sham equations.

2.3 Solution of Kohn-Sham equationsSo far, the Kohn-Sham equation of an independent-particle system, in practice,is the problem of the kinetic energy and the effective potential Veff . In thefollowing, the solution of Kohn-Sham equation is given briefly.Kinetic energy Ts[n] can be described as a function of orbitals ψi(ri). The

independent-particle wave-function can be expressed using a Slater determinantSDψ1(r1)ψ2(r2) · · ·ψN (rN ). According to Eq. 2.7, the kinetic energy is also afunction of the ground state density. As in the Hartree-Fock theory, the kineticenergy can be easily obtained in terms of the one-electron orbitals.Effective potential Veff is the sum of Hartree potential, external potential and

exchange-correlation potential. The Hartree potential can be calculated exactlyaccording the Eq. 2.12, due to the unknown part of electron-electron interac-tion is put in the exchange-correlation term. According to the Eq. 2.9, theexchange-correlation energy Exc[n] is the difference between exact energy andthe other contributions. Therefore, the main problem of Kohn-Sham equationis the solution of the exchange-correlation potential. It is difficult to calcu-late it exactly, and approximations for the exchange-correlation functional aretherefore important.

2.4 Exchange-Correlation functionalIn this section, we review several exchange-correlation approximations, includ-ing Local Density Approximation (LDA), two Generalized Gradient Approxima-tions (GGA), such as PBE and one most new Quasi-non-uniform gradient-levelapproximation (QNA). They are in the first and second rungs in the Jacob’sladder. In the following discussions, we mainly consider the spin-unpolarizedcases.

2.4.1 Local Density Approximation - LDA

Several approximations of the exchange-correlation functionals have been de-veloped during the last decades. The simplest one is the homogeneous electron


gas approximation (HEG) [1] or the local density approximation (LDA) [2]. Inthis case, the electron density is assumed to be uniform around one point in thespace. The exchange-correlation potential at each position is known

Vxc(r) = V HEGxc [n(r)]. (2.16)

The exchange-correlation energy of LDA can be expressed as

ELDAxc (n) =

∫n(r)εLDA

xc (n(r))dr , (2.17)

where εLDAxc is the exchange-correlation energy density per volume unit, and it

can be divided into two parts

εLDAxc = εLDA

x + εLDAc . (2.18)

LDA is successful for many systems, such as bulk metals, which the electronicdensity is close to uniform. However, the limitation of LDA is obvious, whichmay be failed to the systems with inhomogeneous electronic density. Therefore,the better approximations should be constructed.

2.4.2 Generalized Gradient Approximation - GGATo address the issue of inhomogeneities in the electronic density, it is crucial todevelop new functionals that more accurately represent the exact functional andimplement it in a mathematical form that can be efficiently solved for systemswith a large number of electrons. One approach is carried out by the gradientexpansion of the density [38, 39]. This has led to the development of functionalsof the so-called GGA family, such as, PBE [27], PBEsol [40], AM05 [41, 42]and QNA [30, 31]. In most cases, GGA is more accurate than LDA to describethe properties of solid solutions, because GGA includes more physical informa-tion about the local electron density n(r) and the local gradient in the electrondensity ∇n(r) or higher order derivatives. These GGA functionals were con-structed to satisfy different conditions. PBE functional is the most commonlyused functional for solid-state calculations. PBEsol functional is more accurateto calculate the solid and surface due to the β parameter was fitted to TPSSexchange-correlation energies of jellium surface. AM05 functional was designedto use different approximations in different subsystem regions, and it performswell for systems with surfaces. QNA was constructed by fitting parameters toexperimental data and very good performance for the solids, especial for metals.The exchange-correlation energy of GGA is the function of density and sec-

ond-order density gradient,

Exc[n] =∫n(r)εGGA

xc (n(r),∇n(r))dr

=∫n(r)εLDA

x (n(r))Fxc(rs(r), s(r))dr ,(2.19)

where the enhancement factor Fxc as a function of Wigner-Seitz radius rs andthe reduced density gradient s. The form of Wigner-Seitz radius is

rs =(

34πn

)1/3=(

9π4

)1/3 1kF

2.4 Exchange-Correlation functional 11

and the density gradient can be written as

s = |∇n|2 3√

3π2n4/3.

The GGA-level XC functional is based on the development of the enhance-ment factor, by deriving appropriate mathematical expressions (such as PBE)or fitting the parameters of the functional to experimental data (such as QNA).Here, the exchange-correlation functionals PBE and QNA will be given in moredetail.

PBE

The PBE exchange-correlation functional can be expressed as

EPBExc =

∫nεLDA

x (n)FPBExc (rs, s)dr , (2.20)

where FPBExc (rs, s) = FPBE

x (s) + FPBEc (rs, s) denotes the PBE enhancement

factor.PBE functional satisfies some conditions, such as the inhomogeneous electron

gas. Theoretically, GGA PBE provides more accurate features than LDA. ForPBE exchange functional, the enhancement factor is given by

FPBEx (s) = 1 + κ− κ

1 + µκs

2 , (2.21)

where κ = 0.804 and µ = 0.219 515.Perdew and Wang [43] assumed the following for PBE correlation energy

EPBEc =

∫n

(εLDAc (rs, ζ) +H(rs, ζ, t)

)dr, (2.22)

where ζ is the relative spin polarization and a dimensionless density gradient t= |∇n|/[2φksn] = |∇n|/[2φ 2

√4kFπa0

n]. More details can be found in the Ref. [27].The function H can be expressed as

H = e2

a0γφ3 × In

[1 + β

γt2(

1 +At2

1 +At2 +A2t4

)](2.23)

with

A = β

γ

(e−ε

LDAc /(γφ3e2/a0) − 1

)−1,

where β = 0.066 725 and γ = 0.031 091. The behavior of PBE correlationenergy is determined by the parameter β with the relation µ = βπ2/3.

QNA

QNA is one of the newest exchange-correlation functional and was developedby Levämäki et al. [30, 31]. This functional has the same analytical formas the exchange-correlation PBE/PBEsol functional, while it is different from


PBE/PBEsol, because the so-called optimal parameters µopt, βopt were usedin QNA approach for different alloy components. The approximation is an em-pirical approach by minimizing the absolute relative errors of the equilibriumvolumes and the bulk moduli (relative to the experimental values) to find thebest set µopt, βopt. It has been proved that this functional has excellent per-formance in the volumes, bulk moduli and formation energy of metallic alloysand pure metals [31].

Similar with PBE/PBEsol functional, the exchange-correlation functional ofQNA can be expressed as

EQNAxc =

∫nεLDA

x (n)FQNAxc (rs, s)dr , (2.24)

where FQNAxc is the QNA enhancement parameter. The expression of FQNA

xc is

FQNAxc (rs, s) =

∑i

F optixc (rs, s) , (2.25)

where F optixc is the optimal enhancement function based on the optimal param-eters µ, βopti

for the alloy component i.

In the Ref. [31], some optimal parameters µoptiand βopti

of pure metals werelisted. According to the conception of QNA F opt

xc , one component has one uniqueoptimal µ, β. It should be noted that FQNA

xc (rs, s) reduces to FLDAxc (rs) for s

approaching to 0.

QNA exchange-correlation functional can also be considered as part of thesubsystem functional approximations (SFA) which is the another gradient-levelapproximation. The core conception of subsystem functional approach [44]is the individual subsystems using different functionals. Therefore, in quasi-nonuniform exchange-correlation approximation, each element in the periodictable can be viewed as one subsystem. Each functional with the unique optimalparameters µ and β is used in the individual subsystem.

Table 2.1: Optimal QNA and PBE parameters µ and β for Cu and Au.

Element QNA PBE

µopt βopt µ β

Cu 0.079 5 0.005 0 0.219 515 0.066 725Au 0.125 0 0.100 0

In Table 2.1, the QNA parameters µopt and βopt for Cu and Au are listedwhich were used in the calculations of ordered Cu-Au alloys.

Based on the discussion above, the QNA enhancement function Foptixc as a

function of rs and s can be plotted the different maps for different alloy com-ponents. From the maps, it is easy to analyze the physical behaviors [see paperIV].

2.5 Exact Muffin-Tin Orbitals 13

2.5 Exact Muffin-Tin OrbitalsDFT has been implemented in several computed programs. The Exact Muffin-Tin Orbitals (EMTO) [45–48] method is one of them. EMTO is an improvedscreened Koringa-Kohn-Rostoker method (KKR) [49], which offers a very effi-cient scheme for solving the Kohn-Sham equations. The main idea of the EMTOapproach is to use optimized overlapping Muffin-Tin (OOMT) potential spheresinstead of non-overlapping ones.Within the EMTO method, the Muffin-Tin potential VMT of one single-elec-

tron is approximated as

VMT(r) = V0 +∑Ri

[V (r −Ri)− V0] , (2.26)

where V (r−Ri) is the spherical potential centered at one site Ri in the Muffin-Tin sphere. Outside the sphere, the Muffin-Tin potential is a constant V0.

For fixed potential spheres, the OOMT potential FΩ are determined by opti-mizing the mean of the squared deviation between VMT(r) and V (r), which canbe expressed as

FΩ[V (r −Ri), V0] =∫

Ω

[V (r)− V0 −

∑Ri

(V (r −Ri)− V0

)]2dr , (2.27)

where Ω is the region of the potential optimization. The minimum of the spher-ical potentials FΩ are∫

ΩδV (r −Ri)

δFΩ[V (r −Ri), V0]δV (r −Ri)

dr = 0 for any R , (2.28)

andδFΩ[V (r −Ri), V0]

δV0dr = 0 . (2.29)

From Eq. 2.28 and 2.29, we can obtain the optimal VR(rR) and V0. Thesolution for the above equations leads to the OOMT potential.To solve the single-electron Schrödinger equation, we could construct the KS

orbital ψi(r) in terms of exact MT orbitals ψaRL(εi, rR). The form of KS orbitalψi(r) is

ψi(r) =∑RL

ψa

RL(εi, r −Ri)υaRL,i , (2.30)

where υaRL,i are the expansion coefficients, determined by demanding that thesolutions are smooth functions in the entire space. L = (l,m) represents a multi-index, denoting the set of the orbital (l) and magnetic (m) quantum numbers,respectively.The EMTO consists of two parts with different basis functions. Inside the

potential spheres at sR, the partial waves φRL(ε, r−Ri) are defined for any realor complex energy ε and inside the OOMT potential sphere. In the interstitialregion, the screened spherical waves ψRL(ε− v0, rR) are used as basis functions,where the potential is approximated by v0. Combining the screened spheri-cal waves and the partial waves, it can accurately describe the single-electronoverlapping potential.


In fact, accurate potential spheres should overlap, however, a screened spheri-cal wave behaves only on its own a-sphere. Therefore, an additional free-electronfunction ϕaRL(ε, r−Ri) has to be introduced, which connects the screened spher-ical wave and the partial wave.Finally, the EMTO is constructed using different basis functions in the dif-

ferent regions:

ψa

RL(ε, r−Ri) = φaRL(ε, r−Ri) +ψaRL(ε− υ0, r−Ri)−ϕaRL(ε, r−Ri) . (2.31)

By solving the so-called kink-cancellation equation related to the boundary con-dition in the region aR ≤ r − Ri ≤ sR, we can find the solution of the KSequation.

2.6 Random structural models

The so-called random structural models are SQS and CPA, which are widelyemployed in the first-principles calculations of random alloys. They have theiradvantages and disadvantages in the DFT applications. In this section, themathematical framework is introduced.

2.6.1 SQS

SQS was developed by Zunger et al. [4] to model the behavior of random alloysby one small ordered supercells. In the random supercell, such as A1−xBx, Aand B atoms are randomly occupying N sites. The special quasi-random theorywas introduced based on the cluster expansion formalism. More details can befound in the Ref. [4, 50]. The multisite correlation functions Πk,m are used tocharacterize the structure of one random alloy. For one fully random A1−xBxstructure, the correlation functions can be described as

〈Πk,m〉R = (2x− 1)k . (2.32)

with k vertices separated by an mth-neighbor distance.The central ideal of SQS is to find the best correlation functions between

neighboring site occupations matching the real 〈Πk,m〉R for a given supercellwith N atoms. The advantages of the special quasi-random structure model is:(i) the short- or long-range order between the atomic distances can be included;(ii) the relaxation effect is easy to consider. However, comparing with thehomogeneous models of random alloys, the disadvantages of SQS are obvious:(i) the computational cost will be increased due to the big supercells; (ii) therandom structures with more elements are not easy to design, such as highentropy alloys.The performances of two random structure models, such as SQS and CPA,

are shown in Chapter 4 and 5 and papers I, II and III. Overall, SQS is aneffective approach to construct random structures of metallic alloys.

2.6 Random structural models 15

2.6.2 CPA

In this subsection, I will introduce the coherent potential approximation (CPA)[46, 48, 51, 52], a very powerful technique that describes the properties of ran-dom alloys. The main idea of CPA is using one component effective mediumto describe the average of alloy components. This approximation is accurateand effective. Unlike SQS with big supercells, CPA treats the impurity problemwithin the uniform effective medium in one atomic site. Since the local envi-ronment around one atom is same, the effects of atomic relaxation and short-or long-range order are neglected.Two main approximations in the CPA are summarized:(1) Within an alloy the local potentials around a certain type of atom are the

same;(2) The site independent coherent potential P of a monoatomic sub-system

describes the whole system.Within the CPA effective medium, one average (coherent) Green’s function g

has been used to describe one alloy. The effective medium is determined by theindependent coherent potential P and the coherent Green’s function g can bewritten as

g =[S − P

](2.33)

where S is the structure constant matrix.For A1−xBx alloy, the single-site coherent Green’s function g can be written

asg = (1− x)gA + xgB , (2.34)

where gA and gB denote the single-site Green’s functions for the atoms A andB which embedded in the effective medium.

The Green’s functions of the alloy compositions, gA and gB, is determinedby the real atomic potential PA and PB by solving the Dyson equation in realspace,

gA(B) = g + g(PA(B) − P

). (2.35)

The three equations above are solved iteratively, and the output g and gA(B)are used to determine the electronic structure, charge density and total energyof the random alloy.

Chapter 3

Elastic Properties ofMaterials

Elastic properties are important to describe various fundamental characteristicsof solid materials. The elastic constants are essential parameters that provide adetailed information on the mechanical properties of materials, such as Poisson’sratio, specific heat, sound wave velocities, Debye temperature, etc.

3.1 Single-Crystal Elastic Constants

When solid materials are subject to small stresses, the crystals deform almost ina linear elastic manner and the deformation is homogeneous and elasticated. Ingeneral, the stress is described by the stress tensor σ. Similarly, the deformationsof crystals caused by the small stress are described by the elastic strain tensorε. The linear relationship between the stress tensor σ and the strain tensor ε isprovided by the Hooke’s law

σi =6∑j=1

Cijεj . (3.1)

Additionally, the number of independent parameters required to specify thetensor Cij depends on the symmetry of the system, such as, three indepen-dent elastic constants for the cubic crystals, C11, C12 and C44, whereas fiveindependent elastic constants for hexagonal crystals, C11, C12, C13, C33, andC44.

The strain matrix is defined as:

D(ε) =

ε1

12ε6

12ε5

12ε6 ε2

12ε4

12ε5

12ε4 ε3

. (3.2)

18 CHAPTER 3. ELASTIC PROPERTIES OF MATERIALS

The stress tensor σi is obtained by the first derivative of total energy withrespect to the strain

σi = 1V

∂E

∂εi. (3.3)

The elastic constant Cij is given by the second derivative

Cij = 1V

∂2E

∂εi∂εj. (3.4)

For a small deformation, the total energy with small strains can be expressedby the Taylor polynomial in the following way

E(V, εi) = E(V0, 0) + V0

6∑i=1

σiεi + V0

2

6∑i,j=1

Cijεiεj +O(εi3) (3.5)

where V0 is the equilibrium volume and O(εi3) stands for the neglected highorder terms.In addition, the relation between the total energy and the tress tensor can be

expressed asE = V0

2∑i

σiεi . (3.6)

Two approaches are usually used to calculate the elastic constants, one is thestress-stain relationship and the other one is the energy-stain relationship. Thecomparison of the two methods can be found in Ref. [53]. Usually, we usedthe latter approach in EMTO-CPA calculations. The total energies are usuallycomputed for six distortions ε running from 0 to 0.05 with step of 0.01. Next,more details for the two methods are given.

3.1.1 Stress-strain relationsThe stress-strain method can be directly used to calculate the stress tensor. Theelastic constants matrix also can be derived according to Eq. 3.1. Because wehave three independent elastic constants C11, C12 and C44, the three distortionsare used in the Stress-strain calculations and every strain matrix is symmetrical[54].In the thesis, the stress-strain (σ-ε) relation is used to fit the elastic constants

in the SQS calculations of binary alloys. The symmetry of the SQS supercellis generally very low due to the quasi-random distribution of the atoms. Thismakes the calculations of some symmetry dependent properties inconvenient.In particular, the number of independent elastic constants Cij increases greatlydue to the lowering of the lattice symmetry.For one given strain tensor σ, the strain matrix (Eq. 3.2) of cubic systems

along one direction deformation can be written asε 0 0

0 0 12ε

0 12ε 0

. (3.7)

3.1 Single-Crystal Elastic Constants 19

According to the σ - ε relationship (Eq. 3.1), the independent elastic constantsC11, C12 and C44 are obtained

σ1 = C11ε,

σ2 = σ3 = C12ε,

σ4 = C44ε.

(3.8)

However, since the cubic symmetry is broken in the SQS supercells, the straintensors with different orthogonal directions, i. e.

0 0 12ε

0 ε 0

12ε 0 0

, (3.9)

and 0 1

2ε 0

12ε 0 0

0 0 ε

, (3.10)

must be used, where the strains ε varied from -0.004 to 0.004 in steps of 0.002 inthe SQS calculations. To obtain the elastic constants of such random system, onegenerally averages the relevant elastic parameters [55–57]. For a cubic system,the three independent elastic constants are obtained according to the followingaverage relations:

C11 = 13(C11 + C22 + C33);

C12 = 13(C12 + C23 + C13);

C44 = 13(C44 + C55 + C66).

(3.11)

3.1.2 Energy-strain relations

In this thesis, the elastic constants are fitted by the energy-strain relation forall CPA calculations. The elastic constants are obtained from the total ener-gies computed as a functional of small strains. We use volume-conserving or-thorhombic and monoclinic deformations on the fcc or bcc unit cell as describedby

εo 0 0

0 −εo 0

0 0 ε2o/(1− ε2

o)

, (3.12)

20 CHAPTER 3. ELASTIC PROPERTIES OF MATERIALS

and 0 εm 0

εm 0 0

0 0 ε2m/(1− ε2

m)

. (3.13)

The strains εo and εm vary between 0 and 0.05 with step of 0.01. The changesin total energy are obtained by applying a series of small strains at the equi-librium lattice. The elastic constants are deduced by the deformations. Theelastic constants are identified as proportional to the second order coefficient ina polynomial fit of the total energy as a function of the distortion parameterε. The cubic shear elastic constants C ′ and C44 are evaluated by fitting theassociated total energies against the distortions, namely

E(εo) = E(0) + 2V0C′ε2o +O(ε4

o) , (3.14)

andE(εm) = E(0) + 2V0C44ε

2m +O(ε4

m) , (3.15)

respectively. Here O(ε4m) stands for the neglected terms, which are of order of

ε4. For cubic systems, the relation of bulk modulus with elastic constants is

B = 13(C11 + 2C12) (3.16)

and the tetragonal shear modulus can be expressed as

C ′ = 12(C11 − C12) . (3.17)

Combining Eq. 3.14, 3.15, 3.16 and 3.17, we can obtain C11, C12 and C44.The theoretical equilibrium volume and the bulk modulus are determined

from an exponential Morse-type function [58] fitted to the ab initio total ener-gies of cubic structures calculated for nine different atomic volumes. The bulkmodulus is calculated from the volume derivative of the pressure as

B(V ) = −V ∂P∂V

= V∂2E(V )∂V 2 . (3.18)

3.2 Elastic ModulusThe polycrystalline elastic modulus for metallic alloys can be determined fromthe three independent elastic constants for cubic materials. There are two ap-proximation methods to calculate the elastic modulus: a uniform strain forVoigt averaging method [59] and a uniform stress for Reuss method [60]. Forcubic crystals, within the Voigt approach, the bulk and shear moduli can beexpressed as

BV = C11 + 2C12

3 , (3.19)

andGV = (C11 − C12 + 3C44)

5 . (3.20)

3.2 Elastic Modulus 21

While within the Reuss approximation, the expressions are

BR = BV , (3.21)

andGR = 5(C11 − C12C44)

4C44 + 3(C11 − C12) . (3.22)

According to the Hill averaging method [61], Voigt-Reuss-Hill (VRH) average,the arithmetic average for the bulk and shear moduli can be estimated fromVoigt and Reuss bounds

B = BV +BR

2 , G = GV +GR

2 . (3.23)

The Young’s modulus (E) and the Poisson ratio(v) are connected to B andG by the following relationships

E = 9BG3B +G

, (3.24)

andv = 3B − 2G

6B + 2G . (3.25)

The elastic anisotropy is expressed by AVR [62] and AZ [63] characterized thepolycrystalline solids. It is useful to measure

AVR = GV −GR

GV +GR. (3.26)

and the Zener ratioAZ = 2C44

C11 − C12. (3.27)

For isotropic crystals, AVR = 0 and AZ = 1.

Chapter 4

CPA and SQS descriptionsof binary alloys

In this chapter, we consider two random alloys as examples to demonstrate ofthe capabilities of SQS and CPA. We compare both methods and monitor thetrends of structural and elastic parameters as a function of composition. Wealso investigate the influence of local lattice relaxation. The two examples withdistinct atomic mismatch, approximately 1.6% for Ti-Al and 12.2% for Cu-Au,show different behaviors with and without considering the effect of LLD.The single-site CPA is frequently criticized for not being able to take into

account the LLD induced by solutes. In general, CPA should work well for therandom alloys with small atomic mismatch due to the negligible effect of LLD.While for the random alloys with large atomic mismatch, the influence of locallattice distortion may be significant. Therefore, the influence of LLD inducedby local chemical environment should be properly assessed. SQS approach ismore suited to be used in the DFT calculations with big supercells which allowto take LLD into account.

4.1 Lattice constants of random alloys

4.1.1 Lattice constants

Volumes of random alloys are one important parameter which depend on thestructures and decide on other mechanical properties of alloys. In this section,the equilibrium lattice parameters of random Ti1−xAlx and Cu1−xAux (0 ≤ x ≤1) alloys are shown as a function of composition. Two cases are considered inSQS calculations, with and without LLD. In Fig. 4.1, we compare the CPA andSQS lattice constants. The dash lines denote the trend according to Vegard’slaw.As shown in Fig. 4.1, the CPA and SQS results are in good agreement

with each other for random Ti1−xAlx and Cu1−xAux alloys, respectively. ForTi1−xAlx alloys, the lattice parameters decrease linearly with the compositionup to x = 0.375, reach a minimum in between 0.75 and 0.875, and then increasewith further increasing the composition x. The largest error being of the order

24 CHAPTER 4. CPA AND SQS DESCRIPTIONS OF BINARY ALLOYS

of 0.2%. In fact, this difference appears already for pure fcc Ti, and representsthe characteristic deviation between EMTO and VASP results for monoatomicsystems. In subsection 4.1.2, one attempt to give the explanation about thevariation of lattice parameters at the Al-rich side which results in a pronouncednegative deviation relative to Vegard’s rule.In Fig. 4.1 (b), the CPA and SQS equilibrium lattice parameters of random

Cu1−xAux alloys are shown. It is found that the lattice parameters increasealmost linearly with Au concentrations. The CPA and SQS lattice constantswithout LLD are in excellent agreement with each other, with the largest errorbeing 0.1% at both ends.Next we address the question of the atomic relaxation effect on the lattice

parameters in the two solid solutions cases. Comparing the lattice parame-ters of random Ti1−xAlx alloys from SQS calculations with and without atomicrelaxations, we find that the equilibrium lattice parameters remain almost un-changed at Ti-rich side upon LLD. At Al-rich side, the lattice parameters withatomic relaxation are only slightly larger than those with fixed atomic positions.Therefore, the influence of LLD on the lattice parameters of random Ti1−xAlxalloys is negligible because of the similar size of chemical species.While the lattice parameters of random Cu1−xAux alloys are slightly de-

creased when considering the influence of LLD, as shown in Fig. 4.1 (b). Whenatomic relaxations are performed in the SQS calculations, the lattice parame-ters are slightly decreased between 0.125 and 0.875 upon allowing the SQS cellsto relax. The maximum deviation is about 0.56% at x = 0.375. This may beregarded as a small effect and in most of the case can be neglected.For the two cases of Ti1−xAlx and Cu1−xAux alloys, the trends of lattice pa-

rameters show very different behaviors. Unlike Cu1−xAux alloys, the Ti1−xAlxlattice parameters exhibit an opposite trend relative to Vegard’s rule in the Al-rich side. The Cu1−xAux CPA and SQS calculated trends are consistent withVegard’s law (shown by dashed lines), considering that the atomic radius of Auis bigger than that of Cu. Furthermore, the formation energies for the two caseswith distinct mismatch show different behaviors with and without relaxations.

4.1.2 Electron numbers

As seen in Fig. 4.1 (a), for Ti1−xAlx alloys, the lattice parameter of Al decreasesupon Ti addition up to about 0.25. It shows a negative deviation relative toVegard’s rule. To estimate the alloying effects in the anomalous lattice constantsof Ti-Al alloys, we introduce the pressure expressions [64, 65]. Here we recallthat the s partial pressure depends on the average s electron density, whereasthe other partial pressures depend on the actual number of the electrons andvolume.The total pressure is decomposed into a homogeneous positive sp pressure

and a negative d pressure, viz.,

P (V ) = Psp(nsp/V ) + Pd(nd, V ), (4.1)

where nsp and nd denote the sp electron number and d electron number, re-spectively, and V is the atomic volume. The bulk modulus is defined as thevolume derivative of the pressure: B = -V0∂P/∂V which connects the changes

4.1 Lattice constants of random alloys 25

0.00 0.25 0.50 0.75 1.004.02

4.04

4.06

4.08

4.10

4.12

0.00 0.25 0.50 0.75 1.00

3.6

3.7

3.8

3.9

4.0

4.1

4.2 (b) Cu-Au

(a) Ti-Al

SQS Unrelaxed SQS Relaxed CPA

Latti

ce p

aram

eter

s (Å

)

x

Figure 4.1: Theoretical lattice constants for fcc Ti1−xAlx (a) and Cu1−xAux alloys (b)(0 ≤ x ≤ 1) plotted as a function of composition. Results are shown for bothSQS (with and without local lattice distortion) and CPA calculations. Thedash lines denote the trend according to Vegard’s law.

of volume with the occupation numbers. According to the Refs. [66, 67], theexpression of volume change can be expressed as ∆V/V0 ≈ K∆nnp/nsp with K= −V0/Bnd(10− nd)∂W/∂V and W is the potential parameter describing thed bandwidth.

However, in the present system we have metallic bonds ranging from puretransition-metal-type (Ti) to p-metal-type (Al) and hence the alloying effectsare expected to manifest in complex bonding properties. Ti is an early transi-tion metal with two s electrons and two d electrons. In general, s electrons giverise to a broad band and stabilize a large lattice parameter dictated mainly bythe balance between the repulsive kinetic energy of the average electron densityand the attractive exchange-correlation hole. In contrast, the d states are moreshort-ranged than the delocalized s states and prefer small lattice constants. Asa result, the equilibrium volume of Ti is obtained as an interplay between the ho-


mogeneous positive s pressure ps(ns/V ) and the negative d pressure pd(nd, V ).Here ns and nd denote the number of s and d electrons, respectively. The pres-ence of an attractive d pressure in Ti is also reflected by the ∼ 35% decrease ofthe Wigner-Seitz radius as we go from Ca to Ti. Similarly to the d electrons inTi, the p electron in Al also gives rise to an attractive pressure pp(np, V ) whichshrinks the Wigner-Seitz radius of Al by ∼ 12% compared to that of Mg.

0.00 0.25 0.50 0.75 1.00-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

Elec

tron

num

bers

x

s p d

Figure 4.2: s, p and d electron numbers n of fcc Ti1−xAlx (0 ≤ x ≤ 1) alloys plottedas a function of composition. The electron numbers are plotted relative to theconcentration weighted average of the values calculated for pure Ti and Al.

In Ti1−xAlx alloys, the partial s, p and d pressures vary continuously withcomposition from their values in pure Ti to those in pure Al. Since the changein the equilibrium volume with alloying is relatively small, one can assume thatps is proportional to the number of s electrons. In addition, since both p andd states are bonding (less than half-filled bands), one may consider that pp andpd also scale with the number of p and d electrons, respectively. In Fig. 4.2, weplot the changes in ns, np and nd relative to the linear trend corresponding to(1− x)no(Ti) + xno(Al) (o stands for s, p or d). It is found that nd shows onlya weak deviation relative to the linear mixing, meaning that the attractive pdpressure is removed from the system nearly linearly upon Al addition. However,both ns and np show strong deviations from the linear trend. The positivedeviation for np suggests additional attractive pp pressure compared to thatcorresponding to the linear increase of the Al content. The negative deviationfor ns, on the other hand, signals a lowering of the repulsive ps pressure. Both ofthese mechanisms lead to a lattice contraction relative to Vegard’s rule, in linewith Fig. 4.1. The change in the s− p hybridization (increasing s→ p electrontransfer with Ti addition) turns out to be more pronounced in the Al-rich end,which explains why the equilibrium volume has the negative slope as a functionof Ti-content in the Al-rich alloys.

4.2 Mixing energy of Ti-Al and formation enthalpy of Cu-Au 27

4.2 Mixing energy of Ti-Al and formation en-thalpy of Cu-Au

4.2.1 Mixing energy and formation enthalpy

In this section, the mixing energies of random Ti1−xAlx and formation en-thalpies of random Cu1−xAux alloys (0 ≤ x ≤ 1) are discussed. The aim ofthis section is to show the influence of LLD on the these parameters of randomalloys with different atomic mismatches, although small or negligible effect onthe equilibrium volumes.The mixing energy or formation enthalpy of random binary alloys is defined

as∆E = E(Veq)− (1− x)E(VA)− xE(VB) (4.2)

with E(Veq) being the total energy per atom of binary random A1−xBx alloys atequilibrium volumes and E(VA) and E(VB) are the energies for per atom of Aand B in fcc equilibrium volumes. The calculated results for random Ti1−xAlxand Cu1−xAux alloys are presented in Fig. 4.3 against the composition.

First, discussions are presented for the CPA and SQS calculations with fixedatomic positions. It is very helpful for the CPA users in the future calcula-tions. Without taking LLD into account, the CPA and SQS results are ingood agreement with each other for Ti1−xAlx (Fig. 4.3 (a)) and Cu1−xAux(Fig. 4.3 (b)) alloys, respectively. The random Ti1−xAlx alloys are ideal reg-ular fcc solid solutions with the chemical species of similar size, and the mix-ing energies show a nearly perfect parabolic behavior with a minimum at x= 0.5. For random Cu1−xAux alloys, as the Fig. 4.3 (b) shown, one maxi-mum of formation enthalpies is approximatively 52.72 meV/atom at x = 0.375.Furthermore, the present results for Cu-Au are in agreement with the previ-ous theoretical calculations by different methods. For example, the forma-tion enthalpies of random fcc Cu0.75Au0.25, Cu0.5Au0.5 and Cu0.25Au0.75 al-loys are 54.64, 44.32 and 19.82 meV/atom by linear muffi-tin orbitals (LMTO)method [68], and 46.40, 37.73 and 17.78 meV/atom by using the augmented-spherical-wave (ASW) method [69], respectively. Ozolin, š et al. [12] obtained56.5 and 37.8 meV/atom for Cu0.75Au0.25 and Cu0.25Au0.75 alloys using SQSapproach, respectively, when the atomic relaxation is not taken into account in14 atoms SQS supercells.The all VASP-SQS results presented here are obtained at PBE level. In all

calculations, the ideal random Ti1−xAlx and Cu1−xAux alloys are presentedin the thesis without considering the effect of short-range order or long-rangeorder.In Fig. 4.3 (a), we find that SQS calculations with atomic relaxations slightly

reduce the mixing energies of random Ti1−xAlx alloys with Al composition0.375 ≤ x ≤ 0.75. Outside of the interval, the effect on the mixing energies isnegligible. While the formation enthalpies of random Cu1−xAux alloys are lowersubstantially in the whole composition range when taking LLD into account.The largest reduction in SQS calculations is about 52.79 meV/atom at x =0.375. The formation enthalpy shows the minimum value with approximately-7.1 meV/atom at x = 0.5, meaning that the relaxed random Cu0.5Au0.5 alloyis stable with respect to the end members. Our calculated relaxed enthalpies


0.00 0.25 0.50 0.75 1.00-300

-250

-200

-150

-100

-50

0

0.00 0.25 0.50 0.75 1.00

-60

-40

-20

0

20

40

60

(b) Cu-Au

(a) Ti-Al

Mix

ing

ener

gy (m

eV/a

tom

)

Expt.1 (720 K) Expt.2 (800 K)

SQS Unrelaxed SQS Relaxed CPAPBE

CPAPBE Relaxed CPAQNA

CPAQNA RelaxedFo

rmat

ion

enth

alpy

(meV

/ato

m)

x

Figure 4.3: Theoretical mixing energies for fcc Ti1−xAlx (a) and formation enthalpiesfor fcc Cu1−xAux alloys (0 ≤ x ≤ 1) plotted as a function of composition. Re-sults are shown for both SQS (with and without LLD) and CPA calculations.For Cu1−xAux alloys, the CPA-PBE and CPA-QNA formation enthalpies arepresented and the relaxed results are estimated using the effective tetrahedronmodel [70]. The experimental data are shown at 720 K [71] and 800 K [72].

for Cu0.75Au0.25 and Cu0.25Au0.75 are 0.38 and -3.83 meV/atom, respectively,which are consistent with previous theoretical results (5.5 and -5.2 meV/atom)obtained by Ozolin, š et al [12].

One conclusion is presented here based on the discussions above. The largedifference between relaxed and unrelaxed formation enthalpies is connected tothe large elastic strain energy that develops in rigid fcc lattice due to the largevolume mismatch between Cu and Au atoms. The influence of LLD on themixing energies is negligible in the systems with small atomic size mismatch,e.g., random fcc-Ti-Al alloys.

4.2 Mixing energy of Ti-Al and formation enthalpy of Cu-Au 29

4.2.2 Estimated enthalpy for Cu-Au alloysIn Fig. 4.3 (b), the experimental and theoretical formation enthalpies of randomCu1−xAux alloys are presented. PBE and QNA functionals are considered inthe present CPA calculations. While the experimental data are much lower thanthose calculated results at PBE level. The important influencing factors are notclear. In paper IV, we know the excellent performance of QNA functional onthe ordered Cu-Au system. Therefore, we try to understand the deviation fromdifferent XC functionals.The formation enthalpies can be not obtained directly by the single-site CPA

approach. Here we use the effective tetrahedron approach [70] which considersthe relaxation enthalpy by relaxing the volumes of smallest tetrahedral clusters,fcc Cu4, L12 Cu3Au, L10 Cu2Au2, L12 CuAu3 and fcc Au4. The approachcan effectively estimate the atomic relaxation effects on the random formationenthalpies within the EMTO-CPA scheme.The CPA-LLD formation enthalpy of random Cu1−xAux alloys can be esti-

mated as

∆ELLDCPA = ∆ECPA + (1− x)4ECu4 + 4(1− x)3xECu3Au

+6(1− x)2x2ECu2Au2 + 4(1− x)3xECuAu3 + x4EAu4 , (4.3)

where ∆ECPA is the formation enthalpy without LLD, and ECu4 , ECu3Au,ECu2Au2 , ECuAu3 and EAu4 are the tetrahedral energies with considering lo-cal lattice relaxations. The estimated formation enthalpies of Cu1−xAux alloysare shown at PBE and QNA levels in Fig. 4.3 (b), respectively. The estimatedvolumes of five tetrahedral clusters are used in the method according to theharmonic spring model as described in Ref. [70].As shown in Fig. 4.3 (b), the significant difference between PBE and QNA

formation enthalpies demonstrates that the influence from XC functionals canbe not ignored in the present system. Here, we define the difference betweenPBE and QNA functionals ∆Exc = EQNA - EPBE. In Table 4.1, ∆Exc forCu0.75Au0.25, Cu0.5Au0.5 and Cu0.25Au0.75 alloys are listed. As expected, theenthalpies are strongly decreased in the whole composition range as going fromPBE to QNA. However, compared with the experimental values, the QNA for-mation enthalpies including the SQS relaxation energy ELLDSQS are still far toolarge.The short-range order (SRO) is described as the second factor to induce the

discrepancy. Significant SRO effect in Cu-Au system above the ordering tem-perature (about 200K) was reported in Refs. [71, 73, 74]. Here the formationenthalpies of random Cu0.75Au0.25, Cu0.5Au0.5 and Cu0.25Au0.75 alloys includ-ing SRO effect are estimated according to the previous work studied by Ozolin, šet al. [12]. Therefore, the formation enthalpy of random Cu1−xAux alloys shouldbe described as

∆E = ELLDSQS + ∆Exc + ∆ESRO, (4.4)

where ELLDSQS SQS formation enthalpy with LLD, and ∆Exc is the difference of

enthalpies induced by DFT error and ∆ESRO = E(T = 800K) - E(T =∞) thereduction of formation enthalpy when considering the SRO effect. ELLD

SQS , ∆Excand ∆ESRO are listed in Table 4.1.


These theoretical formation enthalpies are surprisingly close to the experi-mental values [71, 72]. Based on the simple estimation, we conclude that thedeviation seen in Fig. 4.3 (b) between the PBE-level theoretical values (relaxed)and the experimental data is due to two major effects: one is the SRO effect andthe other is the QNA approximation used for the exchange-correlation term. Asshown in Table 4.1, the latter effect turns out to be larger than the SRO effectand should be properly addressed when aiming for an accurate description ofthe formation enthalpies in random Cu-Au system.

Table 4.1: Contributions for the formation enthalpies of random Cu0.75Au0.25,Cu0.5Au0.5 and Cu0.25Au0.75 alloys come from local lattice distortion, DFT(XC), short range order (SRO). ∆E denotes the formation enthalpy consider-ing all contributions. The experimental results are shown at 700 K [71].

Enthaly (meV/atom) Cu0.75Au0.25 Cu0.5Au0.5 Cu0.25Au0.75

ELLDSQS 0.38 -7.1 -3.83

∆Exc -23.8 -25.8 -20.3∆ESRO -19.9 -20.9 -6.6∆E -43.3 -53.8 -30.7Expt. [71] -45.6 -53.2 -30.8

4.2.3 Local lattice distortion effect

For a quantitative assessment of the LLD for different alloys in SQS calculations,the local lattice distortion ∆d and relaxation energy ∆E are defined to measurethe average size of the lattice distortion. The lattice distortion ∆d can beexpressed as

∆d = 1N

32∑i=1

√(xi − x′i)2 + (yi − y′i)2 + (zi − z′i)2 (4.5)

where (xi, yi, zi) and (x′i, y′i, z′i) are the SQS atomic positions with and withoutLLD, and N is the atomic number in the supercells. In the present calculations,32-atom SQS supercells are used and the structures can be found in Appendix.The relaxation energy ∆E are expressed similarly with ∆d by the energies withand without LLD.In Fig. 4.4, the local lattice distortion ∆d and relaxation energy ∆E of random

Ti1−xAlx and Cu1−xAux alloys are shown as a function of composition. Forboth examples, the ∆d and ∆E increase with the compositions in the Ti/Curich side, respectively. At the Al/Au rich side, decreased trends occur withthe compositions. For the Ti1−xAlx alloys, the maximum of ∆d occurs at thecomposition of about 0.75, while the curve of ∆d shows a nearly parabolicbehavior for Cu1−xAux alloys. Both ∆E and ∆d exhibit similar trends for bothsystems.It is seen from Fig. 4.4, due to the different atomic mismatches, the local

lattice distortions ∆d and relaxation energies ∆E of random Cu1−xAux alloys

4.3 Elastic properties 31

0.00 0.25 0.50 0.75 1.000

3

6

9

12

15

18

0.00 0.25 0.50 0.75 1.00

0

10

20

30

40

50

60(b)(a)

Ti-Al

Loca

l lat

tice

dist

ortio

nd

(10

3 )

x

Cu-Au

Rel

axat

ion

ener

gyE

(meV

/ato

m)

Ti-Al

Cu-Au

x

Figure 4.4: Local lattice distortion ∆d (a) and relaxation energy ∆Er (b) for fccTi1−xAlx and Cu1−xAux alloys (0 ≤ x ≤ 1) plotted as a function of com-position.

are larger than those of Ti1−xAlx alloys. That means the LLD effect is moreimportant for the alloys of sizable lattice mismatch.

4.3 Elastic properties

4.3.1 Elastic constants

In the section, I show the LLD effect on the elastic constants of random Ti1−xAlxand Cu1−xAux alloys. Both alloys are shown to compare the behaviors of elasticconstants based on two different kinds of sizable lattice.The symmetry of the supercell is broken by the randomness of the alloys,

therefore the number of elastic constants for each composition is increased andthe values are quite scattered. Here, the average method is used to obtain theelastic parameters for both examples. The individual elastic constants and someof the strain tensors from SQS calculations with and without atomic relaxationare calculated. It is noted that, the trends of the individual elastic constantsdeviate significantly from those of the average elastic constants. That meansthe one should not rely on one particular strain tensor when determining thecomposition dependence of the elastic constants of random alloys. Furthermore,in general, the individual SQS elastic constants with local lattice distortion isa little more scattered than those without local lattice distortion. More detailscan be found in papers I and II.In Fig. 4.5 (a) and (b), the average elastic constants Cij from SQS calculations

with and without LLD are shown as a function of composition for Ti1−xAlx andCu1−xAux alloys, respectively. Here, I split the discussions for the two cases.

Ti-Al alloys

In Fig. 4.5 (a), we compare the single crystal elastic constants of randomTi1−xAlx alloys from SQS and CPA calculations without relaxing the atomicpositions. As seen in the figure, the CPA trends of C11, C12 and C44 are almostin agreement with those of SQS without LLD. The error in C11 between CPA


SQS Unrelaxed SQS Relaxed CPA EAM (Pezold et al., 2010)80

100

120

140

160

180

40

60

80

100

120

0.00 0.25 0.50 0.75 1.0020

40

60

80

100

C11

(GPa

)

C12

(GPa

)

(a) Ti1-xAlx

SQS Unrelaxed SQS Relaxed CPA EAM (Pezold et al., 2010)

SQS Unrelaxed SQS Relaxed CPA EAM (Pezold et al., 2010)

C44

(GPa

)

x

120

140

160

180

100

110

120

130

140

0.00 0.25 0.50 0.75 1.000

20

40

60

80

100



C11

(GPa

)

C12

(GPa

)

(b) Cu1-xAux


C

44 (G

Pa)

x

Figure 4.5: Single crystal elastic constants of fcc Ti1−xAlx and Cu1−xAux (0 ≤ x ≤1) alloys plotted as a function of compositions. Results are shown from SQScalculations with and without atomic relaxations. The dashed lines are theEAM predictions by Pezold et al. [55].

and SQS is approximately 15 GPa at the Ti-rich side. For C12, the CPA valueis lower about 10 GPa than the SQS one at x = 0.75. However, the C44 fromSQS calculations are lower almost by a constant over the entire concentrationrange. Moreover, the above differences are to a large extent present also for thepure end members. This suggests that the deviations between CPA and SQSC44 come from the methods rather than the corresponding alloy formalism.

Nevertheless, CPA and SQS elastic constants show very similar compositiondependence. C11 decreases with x at the Ti-rich side up to the concentrationof about 0.5 and a bump occurs at the Al-rich side. C12 and C44 from bothCPA and SQS calculations show a maximum at x ≈ 0.375. The consistenttrends of the elastic constants against the composition from CPA and SQScalculations indicate that both techniques catch reliably the chemistry behindthe composition dependence of the elastic constants.

Next the effect of LLD should be considered in the SQS calculations. One


can see from Fig. 4.5 (a), C12 remains almost unchanged with and withoutLLD. While LLD reduces C11 and C44 in the range between 0.25 and 0.75. Thelargest deviation about 12 GPa is found for C11 at x = 0.625 and about 7 GPafor C44 at x = 0.75, which the corresponding errors are about 11% and 12%,respectively.In Fig. 4.5 (a), the previous values of fcc-Ti1−xAlx [55] are plotted by dashed

lines which were calculated using molecular dynamic (MD) simulation with semi-empirical EAM potential. As shown in the figure, C11 deviates substantiallyfrom the CPA and SQS calculations: we get a minimum C11 at x ∼ 0.625whereas EAM calculations generated a maximum at x = 0.5. The maximumC12 and C44 from the EAM calculations occur also at x = 0.5, however, theyare at x ∼ 0.375 from the first-principles CPA and SQS calculations. Sincethose authors employed a semi-empirical method and our results are based onab initio theory, we tend to believe that the true trends are more accuratelyrepresented by the present data. SQS and CPA Cij values follow very similarvariations as a function of concentration (Fig. 4.5 (a)). The discussion in thenext subsection 4.3.2 gives support to the conclusions, where we identify themain electronic structure mechanism behind the calculated trends for the elasticconstants.

Cu-Au alloys

As discussed above in the case of Ti1−xAlx, the effect of LLD on the elasticconstants is negligible due to the similar atomic sizes of Ti and Al. However, forCu1−xAux alloys with larger size mismatch, the properties of elastic constantsshould be very interesting when the LLD effect has been taken into accountexplicitly. Similar with Ti1−xAlx alloys, two approaches CPA and SQS withsame size supecells (N = 32) are used in the calculations of Cu1−xAux alloys.

In Fig. 4.5 (b), the CPA and SQS average elastic constants of Cu1−xAux alloysare shown. The data of individual elastic constants for three crystallographicdirections can be found in the paper II. In general, the trends of C11, C12 andC44 for CPA calculations are in good agreement with the SQS results withoutLLD. Similar with Ti1−xAlx alloys, CPA C44 of Cu1−xAux are larger than SQSvalues in the whole composition range induced by the calculated methods. Whenconsidering the effect of LLD, it is interesting that the elastic constants C11, C12and C44 are very close to the corresponding values without atomic relaxation,respectively. The effect of atomic relaxation on the C12 is more significant thanthe other two which has one largest deviation about 10 GPa at x = 0.125. Theerror is much smaller than the scattering constants of the individual elasticconstants induced by the symmetry breaking effect (20 GPa at most). Theeffect of the reduced symmetry of the SQS supercells on the elastic constants isat least as significant as the effect of atomic relaxation. The observation is inline with that reported for random Ti-Al alloys.Based on the discussions above of the two random alloys, Ti1−xAlx alloys

consisting of chemical special of similar size and Cu1−xAux alloys with sizablelattice mismatch, one conclusion is obtained: the local lattice distortion doesnot influence significantly the elastic constants. To understand the negligibleeffect of local lattice distortion on the elastic constants, one explanation is givenin the next subsection 4.3.3.


4.3.2 Elastic parameters versus chemical compositionAs the discussion in subsection 4.1.2, the composition dependence of the s, p andd occupation numbers has implications on the trends obtained for the elasticparameters as well. The negative deviation of the equilibrium lattice parameterrelative to Vegard’s rule (Fig. 4.1) is consistent with the positive deviation of thebulk modulus plotted in Fig. 4.6. We recall that lower volume usually resultsin increased bulk modulus and vice versa. Combining the bulk modulus andthe tetragonal shear modulus, we get C11 = B + 4/3C ′ and C12 = B − 2/3C ′.Since for most of the alloys considered here C ′ << B, the trend obtained for Bshould be dominated in both C11 and C12. This is indeed the case for C12 in thewhole concentration range and very roughly also for C11 (Fig. 4.5). However,in Al-rich alloys, we find a clear local maximum in C11 around x = 0.80 and alocal minimum near x = 0.50. In addition, C11 has negative slope as a functionof composition in Ti-rich alloys as compared to the positive slope found forthe bulk moduli. We point out that this non-monotonous behavior of C11 asa function of Al content is to a large degree missing from the previous EAMdata but it is present in both SQS and CPA results (although the minima areat somewhat shifted position).

0.00 0.25 0.50 0.75 1.000

5

10

15

20

25

30

C' (

GPa

)

x

70

80

90

100

110

Bulk

mod

ulus

(GPa

)

Figure 4.6: Tetragonal elastic constant C′ and bulk modulus B of fcc-Ti1−xAlx as afunction of composition. ⊕ represents C′ obtained at volumes correspondingto Vegard’s law. The dashed lines show the linear rule of mixing.

The non-monotonous trend of C11 is due to the strongly nonlinear behaviorof C ′ plotted in Fig. 4.6. In Al-rich alloys, C ′ increases with Ti addition andshows a local maximum near 0.80 Al. At larger Ti content, C ′ decreases anddrops below the linear mixing value. We notice that the effect of C ′ is somewhatweaker in C12 than in C11 manifesting in the nearly linear trend of C12 in Al-richalloys. In Fig. 4.6, a few C ′ values obtained for volumes estimated based onVegard’s rule are shown. The small deviation between the two sets of C ′ valuesindicates a more profound electronic structure origin behind the non-linear trend


than the simple volume effect. A similar effect on C ′ was found in the case ofAl-Mg alloys as well, where a very strong positive deviation of C ′ relative to thetrend corresponding to the linear mixing was predicted [75]. In the Ti-rich side,gradually removing the bonding d electrons results in a substantial drop in C ′.The s→ p charge transfer in Al-rich alloys turns the system more Al-like. Thisexplains why C ′ is relatively large in these alloys and thus C11 exhibits a localmaximum near 0.20 Ti.

4.3.3 Strain effect on the relaxation energy

The relaxation energies ∆Er as a function of the deformation strains were cal-culated for the Cu0.5Au0.5. The values are shown in Fig. 4.7. The strains werecalculated within the range from -0.04 to 0.04, which is much larger than thoseused in the VASP-SQS calculations of the elastic constants. As shown in Fig. 4.7,∆Er ∼ ε shows a nearly parabolic behavior in the considered strain range. Weobserve that even within the large strain interval, the relaxation energy changesrelatively little relative to the largest change around 0.05 meV/atom.

-0.04 -0.02 0.00 0.02 0.04

-1.70

-1.69

-1.68

-1.67

-1.66

-1.65

-1.64

Rela

xatio

n En

ergy

Er

(meV

/ato

m)

Strain

Er=-1.65+0.34 -26.32 2

Figure 4.7: The relaxation energies ∆Er for Cu0.5Au0.5 alloy as a function of thedeformation strains ε. The relaxation energy is an average of the individualenergies correspond to different strain tensors. The curve is a parabolic fit ofthe ∆Er ∼ ε dataset.

The unrelaxed elastic constant C may be expressed as C = V0d2E(ε)dε2 and

relaxed one Cr = V0d2Er(ε)dε2 . Thus, the difference between the unrelaxed and

relaxed elastic constant ∆Cr = C - Cr = V0d2∆Er(ε)

dε2 with ∆Er(ε) = E(ε) -Er(ε). With the parabolic behavior Er ∼ ε, one would expect the relaxedelastic constants be systematically lower than the unrelaxed one. However, thestrain tensors are not volume-conserving, which leads to change the curvatureof the ∆Er ∼ ε2 relationship. Furthermore, with much smaller strain range,e.g. from -0.004 to 0.004, ∆Er approaches a linear strain dependence meaningthat d2∆Er(ε)

dε2 in small strain limit is close to zero. Therefore, ∆Cr is expectedto be very small. This is why local lattice distortion does not affect the elastic


constant significantly and validates the CPA calculations of elastic constants ofalloys with either large or small atomic size difference between atomic species.

Chapter 5

High-entropy random solidsolutions

The lattice distortion induced by the size of different elements can affect thestrength of HEAs [76]. Therefore, it is essential to consider the influence of LLDon the mechanical properties of HEAs. According to the experiments, the HEAalloys TiZrVNb, TiZrNbMo and TiZrVNbMo, which are composed of refrac-tory elements (Ti, Zr, V, Nb and Mo), display single-phase bcc structures [77–79]. These alloys exhibit superior mechanical properties at high temperature,however, the ab initio calculations for the three alloys are few and the physicalbehaviors are not clear, not like the famous CrCoFeNiMn alloys.In this chapter, the elastic properties of three alloys TiZrVNb, TiZrNbMo and

TiZrVNbMo are investigated by carrying out ab initio SQS and CPA calcula-tions. We place special attention on the impact of LLD on the bulk propertieswithin SQS approach and parallel establish the accuracy of single-site CPAmethod for the three alloys.

5.1 LLD effect on lattice constantsIn this section, first, the accuracy of the theoretical methods (EMTO and VASP)is assessed. The lattice parameters for pure Ti, Zr, V, Nb and Mo metals aretested using VASP and EMTO approaches. All refractory elements are adopteda bcc crystal structure below their melting temperatures. The inset of Fig. 5.1shows the present lattice parameters for five elemental metals and are comparedwith the experimental values [31]. The ab initio lattice parameters using thetwo approaches are in excellent agreement with each other and also with theexperimental values. It is noted that, at low temperature, the crystal structureis hcp for Ti and Zr and bcc for V, Nb, Mo and the three HEAs consideredhere [14, 19]. With increasing temperature, both Ti and Zr undergo an hcp-bccstructural transition. The bcc experimental lattice parameters are used to asa comparison which extrapolated to 0 K from high temperatures and correctedfor the zero point phonon effect [31].In Fig. 5.1, the present theoretical lattice constants of bcc TiZrVNb, TiZrNbMo

and TiZrVNbMo alloys are displayed. The disordered SQS crystal structures

38 CHAPTER 5. HIGH-ENTROPY RANDOM SOLID SOLUTIONS

were generated with 128 atoms (4 × 4 × 4) for TiZrVNb and TiZrNbMo alloysand 250 atoms (5 × 5 × 5) for TiZrVNbMo alloy. The atomic positions of theSQS supercells can be found in Ref. [80].

3.22

3.24

3.26

3.28

3.30

3.32

3.34

La

ttice

con

stan

t (Å

)

TiZrVNb TiZrNbMo TiZrVNbMo

SQS Unrelaxed SQS Relaxed CPA Expt. Expt.

Ti Zr V Nb Mo1.01.52.02.53.03.54.0

a (Å

)

VASP EMTO Expt.

Figure 5.1: Theoretical lattice constants obtained for bcc high-entropy TiZrVNb,TiZrNbMo and TiZrVNbMo alloys using EMTO-CPA and VAPS-SQS ap-proaches. The experimental values for TiZrVNb and TiZrVNbMo alloys arelisted [77, 81]. The inset shows the lattice constants for the refractory elementswith bcc structure calculated from EMTO and VASP. The present results forthe refractory elements are compared with the experimental values [31].

As shown in Fig. 5.1, the CPA and SQS lattice parameters exhibit a goodagreement when the LLD effect is not considered. The differences of the theo-retical predictions between the SQS and CPA are 0.12%, 0.15% and 0.12% forTiZrVNb, TiZrNbMo and TiZrVNbMo, respectively. Similar deviations are alsoobserved for the pure elements and thus they may be considered as the charac-teristic differences between EMTO and VASP results for the lattice parameters.Compared the two SQS calculations, we find that the lattice parameters are

increased slightly for the three alloys when considering the LLD effect. The dif-ferences between the relaxed and unrelaxed SQS calculations are 0.40%, 0.48%and 0.43% for TiZrVNb, TiZrNbMo and TiZrVNbMo alloys, respectively. Thesedeviations should still be considered very small indicating that the present the-oretical equilibrium lattice parameters are robust and they represent the correctDFT values obtained at PBE level.We conclude that the two ab initio approaches, CPA and SQS (with and

without LLD), predict very similar equation of states for the present HEAs.Furthermore, the LLD effect is negligible on the equilibrium volumes of thethree alloys.

5.2 LLD effect on elastic constants 39

5.2 LLD effect on elastic constants

5.2.1 Assessment

In this section, the ab initio elastic parameters of the present three TiZrVNb,TiZrNbMo and TiZrVNbMo alloys are discussed. First, the assessment of elas-tic parameters (C11, C12 and C44) are displayed for the five refractory elementsin Fig. 5.2. We find that the computed C11 values for elemental metals inEMTO or CPA are in good agreement with the experimental data at room-temperature [82], except for Ti and Zr. Because there is not available experi-mental values for bcc Ti and Zr with unstable structures at room temperature.For C12 and C44, the agreement is less perfect. In particular, EMTO giveslower C12 for Nb and Mo relative to the VASP and experimental values. Theunderestimated C12 by EMTO is reflected in positive C ′ = (C11 − C12)/2 forTi (5 GPa) and Zr (7 GPa), which is in contrast to the small but negative C ′values predicted by VASP for these two metals (-8 and -3 GPa, respectively).EMTO predicts that Ti and Zr are barely stable according the mechanical sta-bility requires. Nevertheless, the above absolute deviations between VASP andEMTO C ′ values are below those found for V, Nb and Mo. For C44, the EMTOvalues for the five refractory elements are higher than those by VASP, however,eventually closer to the experimental values. The experimental data are as areference, the C44 errors for V, Nb and Mo are 46, 39 and 23% in VASP and22, 12 and 6% found in EMTO.

Mo

Nb

V

Zr

Ti

0 100 200 300 400 500

C11 (GPa)

VASP EMTO Expt.

Mo

Nb

V

Zr

Ti

0 50 100 150 200

C12 (GPa)

Mo

Nb

V

Zr

Ti

0 30 60 90 120

C44 (GPa)

Figure 5.2: Comparison between the theoretical (VASP and EMTO) and experimentalelastic constants for the refractory elements in bcc structures. The experimen-tal single-crystal elastic constants are from the Ref. [82].

Based on the discussions above, we conclude that in general the two ab initiomethods provide consistent results for elemental metals. Somewhat larger de-viations between VASP and EMTO predictions are observed in the case of C12and C44 elastic constants. Since none of the methods performs clearly better forall parameters when compared to the experimental data, we continue to presentresults obtained by both approaches and discuss the alloying effects and rule ofmixtures separately for the two sets of theoretical data.


5.2.2 Elastic parameters

In this section, the single-crystal elastic constants (Cij) for the present alloysdenote the average values in three main crystallographic directions. The resultsfor the C11, C12 and C44 are summarized in Fig. 5.3. The individual SQS elasticconstants due to the low symmetry induced by the chemical quasi-randomnessin SQS calculations can be found in the paper III. By comparing the individualvalues one can learn about the effect of reduced-symmetry on the elastic pa-rameters. For instance, the deviation is about 12.8 GPa for C44 of TiZrNbMo,representing about 65% of the average value. One expects that increasing thesize of the SQS cell should reduce the differences between the individual elasticconstants and eventually restore the cubic symmetry for very large cells.

Figure 5.3: Single-crystal elastic constants of bcc TiZrVNb, TiZrNbMo andTiZrVNbMo alloys obtained by unrelaxed SQS (SQSu), relaxed SQS (SQSr)and CPA calculations. For SQS, shown are the average values.

In Table 5.1, the CPA and average SQS values (both SQSu and SQSr) arealso listed. As shown in the Table, the mechanical stability criteria (C11 + 2C12> 0, C44 > 0, C11 − C12 > 0) for cubic phase structures [83] are fulfilled byall three present alloys. However, when compared the CPA and SQS results,we find that some elastic constants are sensitive to the employed methods. Forexample, C ′ and C44 obtained in CPA calculations are by ∼ 15 and ∼ 30 GPa,respectively, larger than those calculated using the SQS method. We shouldrecall that similar differences between VASP and EMTO single-crystal elastic

5.2 LLD effect on elastic constants 41

constants are found for the pure metals as well (Fig. 5.2). This observationsuggests that the main deviations between the CPA and SQS values in Table5.1 in fact originate from the errors associated with the underlying DFT methodsrather than from the way how the chemical randomness is treated by CPA orSQS.

Table 5.1: The theoretical lattice constants for the bcc underlying lattice (unit in Å),elastic constants C11, C12, C44 and C′ (unit in GPa), the Zener anisotropy(AZ = C44/C

′) and Cauchy pressure (C12-C44) (unit in GPa), polycrystallineelastic moduli (unit in GPa), Poisson’s ratio (ν) and the Pugh ratio (B/G) ofthe TiZrVNb, TiZrNbMo and TiZrVNbMo alloys calculated using the EMTO-CPA and VASP-SQS methods. For SQS both unrelaxed (SQSu) and relaxed(SQSr) results are shown. For reference, we also list the previous CPA results(CPAa) obtained by Tian et al. [22].

Method a C11 C12 C44 C ′ AZ C12-C44 B G E ν B/G

TiZrVNbCPA 3.290 165.8 92.8 50.5 36.5 1.385 42.2 117.1 44.4 118.2 0.332 2.639CPAa 3.281 166.4 94.7 53.8 35.9 1.500 41.0 118.6 45.7 121.1 0.330 2.604SQSu 3.294 156.9 111.7 21.6 22.6 0.956 90.0 126.7 22.0 62.4 0.418 5.756SQSr 3.303 159.8 114.3 18.5 22.8 0.810 95.8 129.5 20.1 57.3 0.426 6.447

TiZrNbMoCPA 3.306 209.6 98.8 49.9 55.4 0.901 48.9 135.8 52.0 138.4 0.330 2.608CPAa 3.304 209.9 101.0 52.6 54.4 0.966 48.4 137.3 53.3 141.7 0.328 2.575SQSu 3.311 198.9 122.1 26.9 38.4 0.702 95.1 146.9 31.1 87.1 0.401 4.729SQSr 3.322 203.4 121.5 29.6 41.0 0.723 91.8 148.2 33.8 94.1 0.394 4.408

TiZrVNbMoCPA 3.252 216.3 100.2 47.8 58.1 0.824 52.4 138.9 51.7 137.9 0.335 2.688CPAa 3.248 213.7 100.7 50.9 56.5 0.900 49.8 138.5 53.2 141.1 0.330 2.608SQSu 3.256 203.7 121.1 24.1 41.3 0.583 97.1 148.7 29.9 84.1 0.406 4.969SQSr 3.266 209.3 123.0 26.9 43.1 0.623 96.2 151.8 32.5 91.0 0.400 4.669

As shown in Fig. 5.3 and Table 5.1, the theoretical elastic constants of thepresent three alloys remain almost unchanged with and without LLD. Therelaxed C11 increases by 1.8%, 2.3% and 2.7% for TiZrVNb, TiZrNbMo andTiZrVNbMo, respectively. The relative LLD induced changes in C44 are some-what larger due to the small absolute value.

In Table 5.1, the Zener anisotropy ratio AZ = C44/C′, the Young’s modulus

E, the shear modulus G, the Cauchy pressure (C12−C44), the Pugh ratio B/Gand the Poisson’s ratio ν are also listed. The SQS and CPA values for elasticmoduli are consistent with each other. The large differences obtained for theshear and Young’s moduli are primarily due to the strong underestimation ofthe C44 elastic constants of pure metals and the present alloys by VASP ascompared to the EMTO and experimental data (Figure 5.2). Note that allHEAs considered here are relatively isotropic (AZ is close to 1), which placesG close to both single-crystal shear elastic constants (C ′ and C44). The abovephenomenological ductility indicators are weakly affected by LLD.


0 50 100 150 200 250

0

50

100

150

200

250 (a) EMTO-CPA

C11 C12 C44 G E a TiZrVNb TiZrNbMo TiZrVNbMo

Cij (G

Pa)

Cijest (GPa)

3.24 3.26 3.28 3.30 3.32 3.34

3.24

3.26

3.28

3.30

3.32

3.34

aest (Å)

a (Å

)

0 50 100 150 200 250

0

50

100

150

200

250 (b) VASP-SQS C11 C12 C44 G E a (SQS

u )


Cij (G

Pa)

Cijest (GPa)

C11 C12 C44 G E a (SQSr )


3.24 3.26 3.28 3.30 3.32 3.34

3.24

3.26

3.28

3.30

3.32

3.34

a (Å

)

aest (Å)

Figure 5.4: Theoretical and estimated lattice parameters and elastic constants plottedfor the present TiZrNbMo, TiZrVNb and TiZrNbMoV alloys using CPA andSQS approaches.

5.2.3 Rule of mixtures

The rule of mixtures has often been employed to make an initial screening of theproperties of alloys and compounds. It gives a simple estimation of the selectedand often unknown property of the alloy based on the accessible properties ofthe end members. In the section, we make a more robust assessment of therule of mixtures for the present HEAs and investigate if at least the generaltrends could be reproduced by such simple estimate. A widely used exampleis Vegard’s rule where the lattice constant of a binary alloy is estimated forma linear interpolation between the lattice parameters of the constituents. Herewe possess sufficient amount of reliable ab initio data for the alloys and theircomponents to be able to test such rule in the case of refractory HEAs.

5.3 Mixing energy 43

Based on the data obtained for the elemental metals, we estimate the latticeparameters and elastic constants of HEAs according

pest = 1N

N∑i=1

pi (5.1)

where pi stands for elastic constants or lattice parameters of metal i in bccstructure and N is the number of components in the alloy. We compare theestimated physical parameter with the one computed in fully self-consistent abinitio calculation. The results are shown in Fig. 5.4. Upper and lower panelcontain the data obtained using the EMTO and VASP methods, respectively.Since bcc Zr and bcc Ti are dynamically unstable at static conditions one cannotdefine a physically meaningful shear modulus for these two hypothetical metals.Hence, G (same applies to E) cannot be obtained directly from the rule ofmixtures. But these polycrystalline elastic constants can be computed from theaveraged single-crystal data. Note that the Poisson’s ratio can easily be derivedfrom G and E.

We observe that in general the trends of the lattice parameters and elasticconstants are well reproduced by the rule of mixtures. The largest deviationsbetween calculated and estimated data are obtained for the lattice parameters.The differences are not systematic: Eq. (5.1) overestimates the CPA latticeparameters of TiZrNbMo and TiZrVNbMo and slightly underestimates that ofTiZrVNb. Very similar deviations are found for the SQS lattice parametersas well. On the other hand, the estimated elastic constants are surprisinglyclose to the theoretical values in both CPA and SQS calculations. One canconclude that the linear rule of mixtures provides a rather useful estimate forthe elastic constants of unknown HEAs assuming that the elastic constants ofthe alloy constituents are known for the same crystal structure. Nevertheless,one should notice that the good agreement between CPA and SQS calculationsstrictly holds only on the scale of the present elastic constants (0-250 GPa), butfails when one looks at the individual elastic parameters on their own scales.

5.3 Mixing energyThe formation enthalpy is one of the key parameters for high-entropy alloyscontrolling the solid solution phase formation [84]. While the formation enthalpyis defined with respect to the ground state structure of the alloy constituents,i.e. hcp for Ti and Zr, and bcc for V, Nb and Mo. The mixing energy is definedwith respect to the total energies of the alloy components in the bcc structure.In this way, when comparing different theoretical results we can exclude effectscoming from the hcp-bcc structural energy differences between various methods.The mixing energy can be expressed as

∆E = Ealloy −∑i

ciEbcci , (5.2)

where Ealloy is the total energy per atom of the HEA and Ebcci is the energy

of the ith alloy component calculated for the bcc structure. ci stands for theconcentration.


In the previous work, the formation enthalpies of the present three alloyswere estimated as -0.03 (TiZrVNb), -2.50 (TiZrNbMo) and -2.72 (TiZrVNbMo)kJ/mol [85]. These values are all located inside the empirical limits from -15 to+5 kJ/mol [86], needed to form a solid solution.The present calculated mixing energies for the three alloys are shown in

Fig. 5.5. One can see that ∆E calculated from CPA and SQSu are in reasonableagreement with each other. The largest deviation of ∼ 4 kJ/mol (correspond-ing approximatively to 25% difference) is obtained for TiZrVNb. Here we usethe screened impurity model [87] with a fixed screening parameter to describethe electrostatic potential within CPA. Adjusting the screening parameter, onecould in principle reproduce with very high accuracy the SQSu results by CPA.On the other hand, the ∆E values obtained by the rigid-lattice SQSu approxi-mation turn out to be far too large when compared to the relaxed SQS results,which makes such tuning of the screening parameter rather useless. As seen inFig. 5.5, the theoretical ∆E values are significantly decreased when we take intoaccount the local lattice relaxations. Actually, the LLD effect on the mixing en-ergies is substantially larger than the CPA-SQSu differences, which makes theCPA results for the mixing energy less reliable.

-4

0

4

8

12

16

20

24

Mix

ing

ener

gy (k

J/m

ol)

CPA SQSu

SQSr


Figure 5.5: Mixing energies of high-entropy TiZrVNb, TiZrNbMo and TiZrVNbMoalloys obtained for both SQS (unrelaxed and relaxed) and CPA approaches.

In our present calculations, we observe that the fully relaxed mixing energiesfollow the trend ∆E (TiZrNbMo) < ∆E(TiZrVNbMo) < ∆E(TiZrVNb), whichis different from the above quoted trend of the formation enthalpies [85]. Thepresent VASP (EMTO) total energy differences between bcc and hcp Ti and Zrare 10.4 (8.1) and 7.7 (5.5) kJ/mol, respectively. These values are in line withthe LDA results reported previously [88]. Using the above VASP bcc-hcp struc-tural energy differences in combination with the SQSu mixing energies, for theformation enthalpies of TiZrVNb, TiZrNbMo and TiZrVNbMo we get 11.1, 0.15and 6.3 kJ/mol, respectively. The rather high formation enthalpy of TiZrVNbmay indicate that the solid solution formation is at least questionable for thisHEA in the light of the empirical upper limit of 5 kJ/mol by Yang et al. [86].We notice that the CPA and SQSu mixing energies yield formation enthalpies

5.4 Local lattice distortion 45

far above the Yang’s upper limit. Hence the rigid lattice results completely ruleout the possibility of solid solution formation for all HEAs considered here. Itis clear that without proper account for the LLD effect, no meaningful theoret-ical prediction can be made on the thermodynamics of HEAs. Before endingthis discussion, we should recall that the above-quoted empirical solid solutionformation limits have to be regarded with precautions in view of the recent de-velopments by Laurent-Brocq et al. [89] that probed the present systems againstphase separation, segregation or intermetallic formation.

5.4 Local lattice distortion

In the previous sections, we have discussed at large the effect of LLD. Here wetry to quantity the magnitudes of the LLD which can be directly evaluated fromthe atomic coordinates obtained through the VASP-SQS geometry optimization.For each relaxed HEA system, a histogram of the radial distributions for everyatom in the cell is presented in Fig. 5.6. Significant distortion may be observedin the relaxed cells, which results in an overlap of the first and second nearestneighbor (NN) shells represented in the histograms. This overlap is more likelyin the present alloys due to the close proximity of the first two NN shells inthe bcc lattice (

√3ao/2 and ao). Before the LLD is analyzed, we will briefly

describe the procedure by which the results were obtained.The histograms in Fig. 5.6 were generated by calculating the distance to

each atom around every site and retaining only those within a radial cut off ofapproximately 4 Å. The histograms count atoms in bins of width 0.03 Å between2.49 and 3.69 Å. Because this prescription double counts every pair, the numberof atoms in each bin was then reduced by half. Due to the overlap of NN shellsin the relaxed systems, the distances determined from the relaxed supercellsare by themselves insufficient to determine whether a given pair belongs in thefirst or second NN shell. Here, we use the ideal unrelaxed SQS to determinethe indices of first and second NNs. These indices are then used to identifythe pair distances plotted in Fig. 5.6. Pairs of atoms that were first NNs in theunrelaxed system are binned in one histogram, while those that were second NNsin the unrelaxed system are binned in a separate histogram, both of which arerepresented on the same plot. It should be noted that the standard deviationsdetermined for the radial distribution of all sites for the first and second NNsin each system are identical to 4 decimal places, which should be the case fora periodic bulk supercell and indicates the degree of relaxation from the ideallattice.From inspection of Fig. 5.6, it is clear that the most significant LLD is found

in the TiZrVNb. Smaller, yet still significant, LLD is found in TiZrNbMo andTiZrVNbMo alloys. It is evident in Fig. 5.6 that the radial distribution his-tograms for first and second NN pairs in TiZrVNb are more spread out thanthe corresponding histograms for the other HEAs. Moreover, the histogramsfor TiZrVNb seem to be superpositions of small subpeaks, whereas the his-tograms for the other HEAs are more centered around a single peak. In orderto understand the more significant distortion found in TiZrVNb as comparedto TiZrNbMo and TiZrVNbMo, we have calculated the average pair distanceand standard deviation for first NN pairs at every site and also at each site of


Figure 5.6: Histogram plot of LLD for the high-entropy TiZrNbMo (top), TiZrVNb(middle) and TiZrVNbMo (bottom) alloys. The 1st and 2nd nearest neighbors(NNs) are colored red and gray, respectively. The ideal (unrelaxed) positionsof the first and second NNs based on the lattice parameters are indicated withdashed lines.

a given element in each of these alloys. The results are presented in Fig. 5.7.Each four-component alloy has 32 unique chemical sites, whereas there are

50 unique chemical sites in the five-component alloy. With 8 first NNs in thebcc unit cell, this leads to sample sizes of 256 and 400 for the four- and five-component systems, respectively. Looking at the results in Fig. 5.7, it is appar-ent that within each HEA there is little variation of the standard deviations fordifferent chemical sites. The average radial distance around each site, however,varies more significantly from site to site. Generally, the average radial distanceis largest around the largest atom, Zr, and smaller around the smaller atomsMo and V. Ti and Nb are intermediate in average radial distance as comparedto the other elements, and are generally closer in average radial distance to theoverall average of the bulk.The more significant standard deviations in TiZrVNb originate from the vari-

ation in average radial distances around each atomic site. This is a product ofboth atomic size and the resultant electronic structure of the alloy. TiZrVNb

5.4 Local lattice distortion 47

Ti Zr V Nb2.4

2.6

2.8

3.0

3.2

Ave

rage

dist

ance

s rav

g 2.832 2.880 2.866

Ti Zr Nb Mo2.55

2.70

2.85

3.00

3.15 (c)(b)(a) TiZrVNb TiZrNbMo

Ti Zr V Nb Mo

2.55

2.70

2.85

3.00

3.15 TiZrVNbMo

Figure 5.7: Average distances and the standard deviations for the first nearest neigh-bor (NN) pairs around each site in high-entropy TiZrNbMo (left), TiZrVNb(middle) and TiZrVNbMo (right) alloys.

and TiZrNbMo are very close in lattice spacing, which is clear from inspection ofthe ideal first and second NN distances in Fig. 5.6 or the average radial distancevalues of all sites in Fig. 5.7. The average radial distance for first NN pairsaround V sites is smaller as compared to Mo sites. Comparison of the four-component alloys shows that Ti and Nb have smaller average radial distancesin the alloy containing V, while Zr has a slightly larger average radial distancein the alloy containing V.If we envision the histograms in Fig. 5.6 as the superposition of a set of

element-specific histograms for radial distributions, we can readily explain theirshapes. In such a picture, an average value in Fig. 5.7 would represent the loca-tion of the peak of the distribution around a given element, while the standarddeviation would represent the width of that peak. We see from Fig. 5.7 thatthe average values, and therefore the peak locations, are more spread out forTiZrVNb than for the other two HEAs; and the standard deviations, that is, thewidth of each peak, are higher. Therefore, we expect the overall distribution ofradial pair distances in TiZrVNb to feature multiple broad peaks, whereas theoverall distributions for the other two HEAs should feature relatively narrowpeaks at roughly the same location. This is exactly what we observe in Fig. 5.6:multiple flat peaks for the first NN distribution in TiZrVNb, and a single narrowpeak for the first NN distributions in TiZrNbMo and TiZrVNbMo. A similaranalysis applies to the shape observed for the histograms of the second NNdistributions.In summary, significant LLD is found in the three high-entropy TiZrVNb,

TiZrNbMo, and TiZrVNbMo alloys. These have been illustrated by the presen-tation of radial distribution histograms as well as quantitative analysis. How theLLD influence other physical properties of the alloys will be analyzed throughthe use of both the EMTO-CPA and the VASP-SQS approaches.

Chapter 6

QNA functional study ofCu-Au system

DFT with its various practical approximate forms has come to have a deep im-pact on many different fields of science. The local and semilocal XC schemes,such as LDA and GGA, are two of the most important levels [90] on the DFTXC approximation gamut. Recently, there have also been important develop-ments in meta-GGAs, such as nonempirical MGGA-MS2 [91] and SCAN [92],where the functional also includes a dependence on the kinetic energy density.These functionals can lead to significant improvements in both structure andenergetics [93].In recent work, Zhang et al. found that the conventional semilocal DFT sim-

ply falls short of being able to accurately predict key properties in alloy theory,such as the formation energy [26]. One of the most spectacular failures happenswith the well-known Cu-Au system showing a series of intermetallic compounds.The experimental formation energies of these intermetallics are far smaller inmagnitude than their (semi) local DFT counterparts and concluded that non-local exchange interaction schemes, such as the Heyd-Scuseria-Ernzerhof (HSE)hybrid functional [28, 29], are necessary in order to mitigate the delocalizationerror of standard approximations and to increase the accuracy of the theoreticalpredictions.In this work, we adopt an energy functional perspective and investigate wheth-

er an appropriate GGA functional better suited for the particular alloy system,would be able to eliminate the observed discrepancies in the formation energies.QNA functional is one semilocal exchange-correlation approach utilized SFA.The details have been given in Chapter 2. The challenge of this work is to usethe QNA approach to calculate the lattice constants and formation energies ofthe ordered Cu-Au system. Furthermore, the standard PBE and SCAN meta-GGA are used as comparisons.

6.1 Performance of GGA-levelThe Cu-Au system shows three intermetallic compounds at the compositions of25 at.% (Cu3Au), 50 at.% (CuAu) and 75 at.% (CuAu3) possessing the L12,

50 CHAPTER 6. QNA FUNCTIONAL STUDY OF CU-AU SYSTEM

L10 and L12 crystal structures [71, 94, 95], respectively. For L10-CuAu alloy,the c/a is 0.915 consistent with the theoretical values (0.915) [96] and the exper-imental data (0.926) [97]. In the section, I evaluate the relative merits of PBEand QNA approximations within the framework of DFT in the case of orderedCu3Au, CuAu and CuAu3 alloys and the pure fcc Cu and Au by comparingtheir performances on the equilibrium lattice constants and formation energies.All PBE and QNA calculations are performed within the EMTO scheme.

First, I compare the present lattice constants of Cu, Cu3Au, CuAu, CuAu3and Au with the previous theoretical values and the available experimental data.In Fig. 6.1 (upper panel), we find that the PBE lattice constants are in goodagreement with the former Projector Augmented Wave (PAW) [98, 99] andthe Linear Muffin-Tin Orbials (LMTO) [68] results, however, they are stronglyoverestimated relative to the experiments [97, 100], especially for Au-rich al-loys. Additionally, the lattice constants are also consistent with the theoreticalresults obtained by the Augmented Spherical Wave (ASW) method [69]. QNAand SCAN can give excellent results: lattice constants are extremely closedto the experimental values over the whole concentration range. The errors ofthe present PBE, QNA ans SCAN lattice parameters are shown in the insetof Fig. 6.1 (upper panel). The findings are also consistent with the previousresearch [31, 101].

Another interesting point is the formation energies of Cu-Au system. InFig. 6.1 (lower panel), the formation energies at PBE, QNA and SCAN levels,the experimental data and previous HSE results and the errors relative to theexperimental data [26] are shown. QNA gives the formation energies that agreevery well with the experimental results. The errors are 4.73%, 6.67%, 5.38%for ordered Cu3Au, CuAu and CuAu3, respectively. However, the PBE resultshave large deviations compared to the experimental values: namely the errorsare 39.32%, 38.49% and 37.95%, respectively. The PBE formation energies arestrongly overestimated by nearly a factor of 2 in magnitude. This failure ofPBE for the Cu-Au system has previously been noticed by Zhang et al [26]and Ozolin, š [12]. For meta-GGA functional, SCAN gives a noticeable improve-ment over the PBE GGA, but its average performance remains slightly inferiorcompared to the QNA.

As we known, QNA is designed to describe lattice constants, or volumes, veryaccurately and high accuracy in volume which is expected to improve the de-scription of other quantities as well. Here we show the so called “volume effect”on the formation energies, such as the bad PBE formation energies induced bythe overestimated volumes. Furthermore, we calculate the formation energiesat the PBE equilibrium volumes using QNA energies. Similarly, the PBE for-mation energies are calculated at the QNA equilibrium volumes. The resultsare shown in Fig. 6.1 (lower panel). It is seen that the “volume effect” cannotexplain the large improvement of the QNA results with respect to the PBEresults. Further studies are needed to understand the real reason behind thisinteresting effect.

6.2 Analysis 51

Figure 6.1: The present lattice constants (upper panel) and formation energies (lowerpanel) for the ordered Cu-Au system calculated at PBE, QNA and SCANlevels. Previous theoretical values [26, 68, 69] and experimental data [26] areshown as a function of composition. The insets show the errors of latticeconstants and formation energies relative to the experimental data.

6.2 AnalysisWhy does QNA show excellent performance on the Cu-Au system, while PBEdoes not? In this section, I will probe the origin of this question from two parts:(1) the exchange-correlation energies at LDA, PBE and QNA levels; (2) thechanges of enhancement function induced by alloying.

6.2.1 Exchange-correlation energy

According to Kohn-Sham equation 2.8, the term exchange-correlation (XC) en-ergy is expressed by the relation


∆Exc = ∆Etotal − (∆T + ∆Eext + ∆EH)︸︷︷︸≡∆Eother

, (6.1)

where ∆T is the kinetic energy, and ∆Eext is the interaction energy with theexternal potential and ∆EH is the Hartree energy coming from Hartree poten-tial, respectively. Here we can use ∆Eother to express the contributions of ∆T ,∆Eext and ∆EH.The XC formation energy can be further broken down to the contributions of

each alloy constituent,

∆Exc(CumAun) = m∆Exc(Cu) + n∆Exc(Au)m+ n

, (6.2)

where ∆Exc(Cu) is the XC energy of Cu and ∆Exc(Au) is the XC energy of Au,m and n denote the constituents of Cu and Au, respectively.

Table 6.1: Alloy component resolved exchange-correlation formation energies (unit:meV/atom) for Cu3Au (L12), CuAu (L10) and CuAu3 (L12) calculated at LDA,PBE and QNA levels.

Cu3Au ∆Exc (Cu) ∆Exc (Au) ∆Exc ∆Eother ∆Etotal

LDA -1574 4468 -64 18 -46PBE -1623 4617 -63 18 -46QNA -1612 4490 -87 18 -69

In Table 6.1, the component-resolved exchange-correlation formation energiesof Cu3Au, are listed for LDA, PBE and QNA functionals. It should be notedthat the QNA equilibrium volume (VQNA) was used in the calculations. ∆Eotherare same for the three XC functionals due to the all three total energies beingevaluated from LDA charge density. Drastic differences are found in the XCformation energies between PBE/LDA and QNA. However, the inconsistentPBE and LDA ∆Exc(Cu) and ∆Exc(Au) are existed that means the fail reasonsare different. As seen from Table 6.1, the PBE and QNA ∆Exc(Cu) are verysimilar, and LDA greatly overestimates the ∆Exc(Cu). While it is conversefor the ∆Exc(Au) which is overestimated by PBE. The variations of exchange-correlation formation energies are consistent with XC equilibrium volumes. LDAvolume for Au is accurate, while PBE strongly overestimates it. Additionally,the PBE volume for Cu is acceptable, while the LDA greatly underestimatesit. This observation suggests that the accurate formation energies for the Cu-Au system require XC functionals that are suitable for both alloy constituents.Within the semilocal GGA functionals, the QNA is designed to meet this strictdemand.

6.2.2 Alloying effect

Additionally, we can analyze the variations of formation energies from the alloy-ing effect, due to the changes of electronic structure. The form of XC formation

6.2 Analysis 53

energies can be expressed by the formation energy density exc(r),

∆Exc = ∆∫exc(r)dr. (6.3)

Furthermore, the formation energy density as a function of electronic Wigner-Seitz radius rs and reduced density gradient s

exc = εLDAx (rs)Fxc(rs, s), (6.4)

with rs = 3√

3/4πn and s = | ∇n |/(2 3√

3π2n4/3). εLDAx is the LDA exchange

energy density, Fxc(rs, s) is the enhancement function and n is the electrondensity.Hence, the change in the alloy component resolved XC formation energy den-

sity is

∆exc = dεLDAx (rs)drs

Fxc(rs, s)∆rs

+ εLDAx (rs)

(∂Fxc(rs, s)

∂rs∆rs + ∂Fxc(rs, s)

∂s∆s),

(6.5)

where ∆rs and ∆s stand for the changes in rs and s upon alloying.Since these changes are computed from LDA densities, they remain same

for all semilocal functionals, and thus the differences between the three ∆Exc(Cu/Au) values in Table 6.1 should be associated with the enhancement factors.Actually, the first two terms in Eq. 6.5 are very similar for the three function-als, which places the ∂Fxc/∂s term in the spotlight. The maps of ∂FPBE

xc /∂s,∂F

QNA[Au]xc /∂s and ∂FQNA[Cu]

xc /∂s are shown in Fig. 6.2 as a function of rs ands. In the figure, we also plot the actual spherically averaged (rs, s) values forCu and Au in the Cu3Au alloy and in pure Cu and Au. The shape of the PBEenhancement function map is the same for all elements, while QNA maps arevaried with the elements due to the different “optimal” parameters (Table 2.1).In Fig. 6.2, the core-valence overlap region (CVOR) are shown with circle

and diamond symbols. The region is important for the equilibrium values. Itis important to notice that the changes in ∆rs and ∆s and thus the physicallyrelevant area for ∆Exc are located in the highlighted CVOR.

As one can see in Fig. 6.2, some interesting conclusions are obtained from theregion of CVOR,(i) Within the CVOR[Cu], ∆s is positive and ∂FQNA[Cu]

xc /∂s is positive andclose to ∂FPBE

xc /∂s. Therefore, the QNA XC energies ∆EQNAxc [Cu] is close to

∆EPBExc [Cu], however, ∆EQNA

xc [Cu] is smaller than ∆ELDAxc [Cu] ;

(ii) Within the CVOR[Au], ∆s is negative, and ∂FQNA[Au]xc /∂s is close to zero.

Therefore, ∆EQNAxc [Au] is close to ∆ELDA

xc [Au] and lower than ∆EPBExc [Au] .

Therefore, the largely positive ∂FPBExc /∂s around the CVOR[Au] leads to

a heavily overestimated ∆EPBExc [Au]. The above discussion can fully explain

the numbers in the Table 6.1 and the good performance of QNA-SFA on theformation energies of Cu-Au system.


0.0 0.2 0.4 0.6 0.8 1.0

s

0.0

0.5

1.0

1.5

2.0

rs

PBE dFxc(rs, s)/ds

Cu(pure)

Cu(CuAu)

Au(pure)

Au(CuAu)

0.2 0.4 0.6 0.8 1.0

s

QNA[Cu] dFxc(rs, s)/ds

Cu(pure)

Cu(CuAu)

0.2 0.4 0.6 0.8 1.0

s

QNA[Au] dFxc(rs, s)/ds

Au(pure)

Au(CuAu)

−0.12

−0.10

−0.08

−0.06

−0.04

−0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

0.24

0.26

0.28

0.30

Figure 6.2: Enhancement function maps for PBE, QNA[Cu] and QNA[Au]. The s vs rs

curves are drawn for pure elements and for elements in the Cu3Au alloy. Theparts of the curves drawn using symbols represent the core-valence overlapregion and the size of the symbols indicates the magnitude of the changebetween the pure element curve and the alloy component curve. Regionsoutside the relevant core-valence overlap region are drawn using dashed lines.

6.3 QNA study of the other alloysIn order to verify that the ordered Cu-Au system is not a special case, we alsocalculate the formation energies of some selected binary alloys at LDA, PBE,QNA and SCAN levels. We demonstrate the robustness and universality ofour QNA-SFA scheme with various other binary alloy systems in Table 6.2.The QNA, which handles both alloy components correctly, turns out to givea substantial improvement over the classical approximations. SCAN, whichhas been shown to uniquely yield accurate formation energies across all MnO2polymorphs [93], performs well for the Cu-Au system and CuPt, but seems togive overestimates for CuAg, AgPd, AgAu and NiAl.In the case of the QNA exchange-correlation functional, we identify the main

cause for this to be connected with the fact that this functional was designedto describe lattice constants accurately simultaneously for all alloy components.The increased accuracy in volume ensures a better description of other quantitiesas well, such as the formation energy. Nevertheless, merely the volume effect(i.e., using the experimental volumes) by itself cannot account for the improvedtheoretical trends.

6.3 QNA study of the other alloys 55

Table6.2:

Form

ation

energies

(unit:

meV

/atom)of

selected

bina

ryalloys

calculated

atLD

A,PBE,QNA,SC

AN

and

HSE

[26].

Availablelow

tempe

rature

expe

rimentald

ataarelistedforr

eferen

ce.The

disordered

AgP

dwas

mod

eled

bya32-atom

fccspecialq

uasirand

omstructure.

The

best

semilo

calresults

arein

boldface

(for

CuA

gwith

respectt

otheHSE

hybrid

functio

nal).

The

last

columnrepo

rtst

hemeanab

solute

relativ

eerror(M

ARE).

The

Fritz

Hab

erInstitu

teabinitio

molecular

simulations

(FHI-aims)

package[102–104]w

asused

tocalculateSC

AN(aim

s)an

dPBE(aim

s)results

.

XC

Cu 3

Au

CuA

uCuA

u 3CuP

dCuA

gCuP

tAgP

dAgA

uNiA

lMARE

(L1 2

)(L

1 0)

(L1 2)

(B2)

(L1 0)

(L1 1)

(fcc)

(L1 0)

(B2)

(in%)

LDA

-39

-54

-18

-138

103

-119

-39

-64

-731

36PB

E-45

-57

-24

-143

93-145

-42

-59

−67

830

PBE(

aims)

-38

-47

-20

-120

103

−15

4-46

-60

-655

38QNA

−70

−87

−41

−14

176

−15

4−

27−

46-689

7SC

AN(aim

s)-64

-78

−37

-126

124

−19

3-56

-83

-783

36HSE

-71

-91

-53

-170

74...

...

-52

...

...

Expt.

-74[26]

-93[26]

-39[26]

-140±

21[26]

...

-174

[105]

-23±

3[106]

-48[12]

-680

[107]

...

Chapter 7

Conclusions and FutureWork

7.1 ConclusionsIn this thesis, first-principles calculations based on DFT have been performedusing LDA and GGA-level XC functionals, such as PBE and QNA.One of the aims is to investigate the effect of LLD on the lattice parameters,

elastic constants and mixing energies or formation enthalpies for the randombinary and multicomponent alloys. The corresponding electronic structures andbonding characteristic with and without atomic relaxation are discussed.Coherent potential approximation (CPA) and special quasi-random structure

(SQS) have been adopted to investigate the elastic properties of disorderedmetallic alloys. Ab initio total energies are obtained within the framework ofExact Muffin-Tin Orbitals (EMTO) and Vienna Ab initio Simulation Package(VASP).Based on the results presented in the thesis, we can conclude the following:

(a) the calculated results for CPA and SQS are consistent with each other; (b)the influence of local lattice relaxation on the lattice constants is negligible. Forrandom Ti-Al alloys, the lattice constants increase slightly between 0.375 and0.875 Al concentration when considering the effect of LLD, however, for randomCu-Au, the relaxed lattice constants have a small reduction and the differencesfor the present HEAs are 0.40%, 0.48% and 0.43% between the unrelaxed andrelaxed structures, respectively; (c) compared to the results without LLD, therelaxed mixing energies and formation enthalpies are decreased for all presentalloys. Especially, the deviation between relaxed and unrelaxed energies is largefor the random Cu-Au and high-entropy alloys. That means the LLD effect onthe mixing energies and formation enthalpies is significant for the large atomicmismatch systems; (d) the present calculations show that the effect of LLD onthe elastic constants is negligible;The other conclusion which we get from the calculations of ordered Cu-Au

system, is that QNA gives the excellent lattice constants and formation energieswhen compared to the experimental results, whereas a large deviation arises forPBE. This means that semilocal DFT within the SFA framework is capable

58 CHAPTER 7. CONCLUSIONS AND FUTURE WORK

of accurately describing the formation energies of metallic binary alloys. Thepresent functional shows that it is important for the development of DFT.

7.2 Future workFurthermore, in subsection 4.2.2 of Chapter 4, the formation enthalpies forrandom Cu1−xAux (0 ≤ x ≤ 1) alloys are shown in Fig. 4.3. However, theexperimental data are much lower than the ab initio results. The effects ofLLD and XC functionals and short-range order (SRO) are considered as threemain factors to induce the large difference between the present results and theexperimental data. The idea has also been confirmed and the calculated andestimated results are shown in Table 4.1.For the XC effects, one effective tetrahedron model is used to estimate the

relaxed formation enthalpies of random Cu-Au alloys in EMTO-CPA method.It is interesting and important if we can give more information about the XCand SRO effects.In my future work, I will focus on the formation enthalpies of random Cu1−xAux

(0 ≤ x ≤ 1) alloys. The aim is to do the calculations of formation enthalpieswithin one GPAW code [108] including the effects of LLD, QNA functionaland SRO. Therefore, a more comprehensive assessment of the QNA for randommetallic alloys will also be part of my future efforts. It is my goal to extendthe present and planned studies to multicomponent systems and see how theemployed XC approximation, LLD and SRO affect their mechanical and ther-modynamical properties.

Chapter 8

Sammanfattning

Täthetsfunktionalteori (density functional theory, DFT) är en effektiv och pre-cis metod för att beskriva elektronstrukturen i material. I praktiken så behövsapproximationer för korrelation och elektronutbyte i denna teori. Utvecklan-det utav funktionaler för korrelation och utbyte (exchange-correlation, XC) harvarit en viktig uppgift under de senaste åren. PBE-funktionalen är den idagmest använda funktionalen för beräkningar i det fasta tillståndet. I den häravhandlingen så har XC-funktionaler baserade på LDA och GGA använts förDFT-beräkningar.DFT är en framgångsrik metod för att beräkna materialegenskaper, men

för legeringar blir ekvationerna svårlösta. SQS och CPA, två ofta användaslumpmodeller för strukturell oordning, används här för att modellera oord-nade strukturer inom DFT. En av skillnaderna mellan modellerna är om manskall ta hänsyn till lokal kristalldistorsion inom DFT-beräkningarna eller ej.CPA- och SQS-beräkningar har utförts för att undersöka elastiska egenskaperhos legeringar inom ramverket av EMTO och VASP. Elektronstrukturen ochbindningen med och utan atomär avslappning diskuteras.En annan begränsning hos DFT-metoder är att XC-funktionalerna har utveck-

lats för att tillfredsställa vissa förhållanden. Detta gör att vissa funktionalerinte kan prestera väl för vissa system, som till exempel för Cu-Au-system. Pågrund av detta så består avhandlingens andra del i utforskandet utav QNA-funktionalens prestanda, som visar sig ge utmärkta resultat. Detta visar attsemilokal DFT inom ramverket av SFA kan beskriva formationsenergin hos met-alliska legeringar.

Acknowledgement

These words that appear at the end of my PhD are hardly to describe the deepgratitude I fell. I can only try and express my gratefulness to so many people.First, No one deserve my thanks more than my wonderful supervisor Prof.

Levente Vitos. Levente has not only been an excellent thesis supervisor, but hehas also offered me plenty of opportunities to learn density functional theoryand has shown me the practical side of calculated work in EMTO and in sodoing he has been extremely patient. While working with him, I have learned alot from his advice and understanding about the subject. I really admire of hiscalmness and deep insight on the subject.Grateful acknowledgment also goes to Prof. Qing-Miao Hu. In the LLD

project, especially for random Ti-Al and Cu-Au alloys, Prof. Qing-Miao Hugiven me a lot of suggestions and comments. The project of Ti-Al alloys is myfirst project during my PhD in KTH and the first time I did the EMTO-CPAand SQS calculations. It is very important in my research.I would like to thank my office roommate Dr. Raquel Lizárraga for her ongoing

encouragement and professional guidance in my research. In the 22 months,you give me many happiness and let my study become more interesting. Mythesis can be successfully finished because of your great help and suggestionsand comments. I want to thank Dr. Wei Li for helping on the computer andlearning EMTO. He is funny and likes talking some funny stories during thedinner at the ITM kitchen and making the happy time. I also would like tothank Henrik Levämäki. I learned a lot of DFT knowledge form him includingQNA functional. We have at least two co-worked projects. In the process, I amconstantly making progress. You are the first reader of my thesis and correctsome English grammar problems.Furthermore, I would like to thank my co-supervisor Prof. Jijun Zhao in

China. Although I have made a decision giving up my Chinese PhD degree, itwas happy time during my master in your group.I am very grateful to our group members: Song, Dongyoo, Stephan, Xiaoqing,

He, Zhihua, Shuo, Zongwei, Xiaojie and Ruiwen for their assistance and forcreating a pleasant working atmosphere.Last but not the least, I would like to thank my family for their understanding,

continuous support and encouragement.The Swedish Research Council, the Swedish Steel Producers’ Association, the

European Research Council, the Swedish Foundation for Strategic Research,the China Scholarship Council, the Hungarian Scientific Research Fund, are

62 CHAPTER 8. SAMMANFATTNING

acknowledged for financial support. The computations simulations were per-formed on resources provided by the Swedish National Infrastructure for Com-puting (SNIC) at the National Supercomputer Centre in Linköping.

Appendix A

Appendix

The SQS structures used in the calculations of random Ti-Al and Cu-Au alloysof the thesis are given in the Tables.

64 APPENDIX A. APPENDIX

A28B41.0000 0.0000 0.00000.0000 1.0000 0.00000.0000 0.0000 1.0000A B28 4Direct0.00 0.00 0.00 Ti/Cu0.00 0.50 0.00 Ti/Cu0.75 0.75 0.00 Ti/Cu0.50 0.00 0.00 Ti/Cu0.50 0.50 0.00 Ti/Cu0.25 0.75 0.00 Ti/Cu0.25 0.25 0.00 Ti/Cu0.00 0.75 0.75 Ti/Cu0.00 0.25 0.75 Ti/Cu0.75 0.50 0.75 Ti/Cu0.50 0.75 0.75 Ti/Cu0.50 0.25 0.75 Ti/Cu0.25 0.00 0.75 Ti/Cu0.25 0.50 0.75 Ti/Cu0.00 0.00 0.50 Ti/Cu0.00 0.50 0.50 Ti/Cu0.75 0.75 0.50 Ti/Cu0.75 0.25 0.50 Ti/Cu0.50 0.00 0.50 Ti/Cu0.50 0.50 0.50 Ti/Cu0.25 0.75 0.50 Ti/Cu0.25 0.25 0.50 Ti/Cu0.00 0.75 0.25 Ti/Cu0.00 0.25 0.25 Ti/Cu0.75 0.50 0.25 Ti/Cu0.50 0.25 0.25 Ti/Cu0.25 0.00 0.25 Ti/Cu0.25 0.50 0.25 Ti/Cu0.75 0.25 0.00 Al/Au0.75 0.00 0.75 Al/Au0.75 0.00 0.25 Al/Au0.50 0.75 0.25 Al/Au

65

A24B81.0000 0.0000 0.00000.0000 1.0000 0.00000.0000 0.0000 1.0000A B24 8Direct0.50 0.00 0.00 Ti/Cu0.25 0.75 0.00 Ti/Cu0.25 0.25 0.00 Ti/Cu0.75 0.50 0.75 Ti/Cu0.50 0.75 0.75 Ti/Cu0.25 0.50 0.75 Ti/Cu0.75 0.75 0.50 Ti/Cu0.50 0.00 0.50 Ti/Cu0.50 0.50 0.50 Ti/Cu0.25 0.75 0.50 Ti/Cu0.00 0.75 0.25 Ti/Cu0.75 0.00 0.25 Ti/Cu0.75 0.50 0.25 Ti/Cu0.50 0.75 0.25 Ti/Cu0.00 0.50 0.00 Ti/Cu0.75 0.75 0.00 Ti/Cu0.75 0.25 0.00 Ti/Cu0.00 0.75 0.75 Ti/Cu0.00 0.00 0.50 Ti/Cu0.00 0.50 0.50 Ti/Cu0.25 0.25 0.50 Ti/Cu0.00 0.25 0.25 Ti/Cu0.50 0.25 0.25 Ti/Cu0.25 0.50 0.25 Ti/Cu0.00 0.00 0.00 Al/Au0.50 0.50 0.00 Al/Au0.00 0.25 0.75 Al/Au0.75 0.00 0.75 Al/Au0.50 0.25 0.75 Al/Au0.25 0.00 0.75 Al/Au0.75 0.25 0.50 Al/Au0.25 0.00 0.25 Al/Au


A20B121.0000 0.0000 0.00000.0000 1.0000 0.00000.0000 0.0000 1.0000A B20 12Direct0.00 0.00 0.00 Ti/Cu0.00 0.50 0.00 Ti/Cu0.75 0.75 0.00 Ti/Cu0.50 0.50 0.00 Ti/Cu0.00 0.75 0.75 Ti/Cu0.75 0.50 0.75 Ti/Cu0.50 0.25 0.75 Ti/Cu0.00 0.50 0.50 Ti/Cu0.75 0.75 0.50 Ti/Cu0.75 0.25 0.50 Ti/Cu0.50 0.50 0.50 Ti/Cu0.25 0.75 0.50 Ti/Cu0.25 0.25 0.50 Ti/Cu0.00 0.25 0.25 Ti/Cu0.75 0.00 0.25 Ti/Cu0.75 0.50 0.25 Ti/Cu0.50 0.75 0.25 Ti/Cu0.50 0.25 0.25 Ti/Cu0.25 0.00 0.25 Ti/Cu0.25 0.50 0.25 Ti/Cu0.75 0.25 0.00 Al/Au0.50 0.00 0.00 Al/Au0.25 0.75 0.00 Al/Au0.25 0.25 0.00 Al/Au0.00 0.25 0.75 Al/Au0.75 0.00 0.75 Al/Au0.50 0.75 0.75 Al/Au0.25 0.00 0.75 Al/Au0.25 0.50 0.75 Al/Au0.00 0.00 0.50 Al/Au0.50 0.00 0.50 Al/Au0.00 0.75 0.25 Al/Au

67

A16B161.0000 0.0000 0.00000.0000 1.0000 0.00000.0000 0.0000 1.0000A B16 16Direct0.00 0.00 0.00 Ti/Cu0.50 0.00 0.00 Ti/Cu0.25 0.75 0.00 Ti/Cu0.25 0.25 0.00 Ti/Cu0.75 0.50 0.75 Ti/Cu0.50 0.75 0.75 Ti/Cu0.50 0.25 0.75 Ti/Cu0.25 0.50 0.75 Ti/Cu0.75 0.75 0.50 Ti/Cu0.50 0.00 0.50 Ti/Cu0.50 0.50 0.50 Ti/Cu0.25 0.75 0.50 Ti/Cu0.00 0.75 0.25 Ti/Cu0.75 0.00 0.25 Ti/Cu0.75 0.50 0.25 Ti/Cu0.50 0.75 0.25 Ti/Cu0.00 0.50 0.00 Al/Au0.75 0.75 0.00 Al/Au0.75 0.25 0.00 Al/Au0.50 0.50 0.00 Al/Au0.00 0.75 0.75 Al/Au0.00 0.25 0.75 Al/Au0.75 0.00 0.75 Al/Au0.25 0.00 0.75 Al/Au0.00 0.00 0.50 Al/Au0.00 0.50 0.50 Al/Au0.75 0.25 0.50 Al/Au0.25 0.25 0.50 Al/Au0.00 0.25 0.25 Al/Au0.50 0.25 0.25 Al/Au0.25 0.00 0.25 Al/Au0.25 0.50 0.25 Al/Au


A12B201.0000 0.0000 0.00000.0000 1.0000 0.00000.0000 0.0000 1.0000A B12 20Direct0.75 0.25 0.00 Ti/Cu0.50 0.00 0.00 Ti/Cu0.25 0.75 0.00 Ti/Cu0.25 0.25 0.00 Ti/Cu0.00 0.25 0.75 Ti/Cu0.75 0.00 0.75 Ti/Cu0.50 0.75 0.75 Ti/Cu0.25 0.00 0.75 Ti/Cu0.25 0.50 0.75 Ti/Cu0.00 0.00 0.50 Ti/Cu0.50 0.00 0.50 Ti/Cu0.00 0.75 0.25 Ti/Cu0.00 0.00 0.00 Al/Au0.00 0.50 0.00 Al/Au0.75 0.75 0.00 Al/Au0.50 0.50 0.00 Al/Au0.00 0.75 0.75 Al/Au0.75 0.50 0.75 Al/Au0.50 0.25 0.75 Al/Au0.00 0.50 0.50 Al/Au0.75 0.75 0.50 Al/Au0.75 0.25 0.50 Al/Au0.50 0.50 0.50 Al/Au0.25 0.75 0.50 Al/Au0.25 0.25 0.50 Al/Au0.00 0.25 0.25 Al/Au0.75 0.00 0.25 Al/Au0.75 0.50 0.25 Al/Au0.50 0.75 0.25 Al/Au0.50 0.25 0.25 Al/Au0.25 0.00 0.25 Al/Au0.25 0.50 0.25 Al/Au

69

A8B241.0000 0.0000 0.00000.0000 1.0000 0.00000.0000 0.0000 1.0000A B8 24Direct0.00 0.00 0.00 Ti/Cu0.50 0.50 0.00 Ti/Cu0.00 0.25 0.75 Ti/Cu0.75 0.00 0.75 Ti/Cu0.50 0.25 0.75 Ti/Cu0.25 0.00 0.75 Ti/Cu0.75 0.25 0.50 Ti/Cu0.25 0.00 0.25 Ti/Cu0.50 0.00 0.00 Al/Au0.25 0.75 0.00 Al/Au0.25 0.25 0.00 Al/Au0.75 0.50 0.75 Al/Au0.50 0.75 0.75 Al/Au0.25 0.50 0.75 Al/Au0.75 0.75 0.50 Al/Au0.50 0.00 0.50 Al/Au0.50 0.50 0.50 Al/Au0.25 0.75 0.50 Al/Au0.00 0.75 0.25 Al/Au0.75 0.00 0.25 Al/Au0.75 0.50 0.25 Al/Au0.50 0.75 0.25 Al/Au0.00 0.50 0.00 Al/Au0.75 0.75 0.00 Al/Au0.75 0.25 0.00 Al/Au0.00 0.75 0.75 Al/Au0.00 0.00 0.50 Al/Au0.00 0.50 0.50 Al/Au0.25 0.25 0.50 Al/Au0.00 0.25 0.25 Al/Au0.50 0.25 0.25 Al/Au0.25 0.50 0.25 Al/Au


A4B281.0000 0.0000 0.00000.0000 1.0000 0.00000.0000 0.0000 1.0000A B4 28Direct0.75 0.25 0.00 Ti/Cu0.75 0.00 0.75 Ti/Cu0.75 0.00 0.25 Ti/Cu0.50 0.75 0.25 Ti/Cu0.00 0.00 0.00 Al/Au0.00 0.50 0.00 Al/Au0.75 0.75 0.00 Al/Au0.50 0.00 0.00 Al/Au0.50 0.50 0.00 Al/Au0.25 0.75 0.00 Al/Au0.25 0.25 0.00 Al/Au0.00 0.75 0.75 Al/Au0.00 0.25 0.75 Al/Au0.75 0.50 0.75 Al/Au0.50 0.75 0.75 Al/Au0.50 0.25 0.75 Al/Au0.25 0.00 0.75 Al/Au0.25 0.50 0.75 Al/Au0.00 0.00 0.50 Al/Au0.00 0.50 0.50 Al/Au0.75 0.75 0.50 Al/Au0.75 0.25 0.50 Al/Au0.50 0.00 0.50 Al/Au0.50 0.50 0.50 Al/Au0.25 0.75 0.50 Al/Au0.25 0.25 0.50 Al/Au0.00 0.75 0.25 Al/Au0.00 0.25 0.25 Al/Au0.75 0.50 0.25 Al/Au0.50 0.25 0.25 Al/Au0.25 0.00 0.25 Al/Au0.25 0.50 0.25 Al/Au

Bibliography

[1] W. Kohn P. Hohenberg. In: Phys. Rev. B 136.3B (1964), p. 864 (cit. onpp. 1, 10).

[2] W. Kohn and L. J. Sham. In: Phys. Rev. 140A.1951 (1965), p. 1133 (cit.on pp. 1, 7, 10).

[3] R. M. Dreizler and E. K. U. Gross. In: Springer-Ver lag Berlin Heidelberg.New York London (1990) (cit. on p. 1).

[4] A. Zunger, S. H. Wei, L. G. Ferreira, and J.s E. Bernard. In: Phys. Rev.Lett. 65 (1990), p. 353 (cit. on pp. 1, 14).

[5] S. H. Wei, L. G. Ferreira, James E. Bernard, and Alex Zunger. In: Phys.Rev. B 42.15 (1990), p. 9622 (cit. on p. 1).

[6] P. Soven. In: Phys. Rev. 156.3 (1967), p. 809 (cit. on p. 1).[7] B. L. Gyorffy. In: Phys. Rev. B 5.6 (1972), p. 2382 (cit. on p. 1).[8] L Bellaiche and David Vanderbilt. In: Phys. Rev. B 61.12 (2000), p. 7877

(cit. on p. 1).[9] Nicholas J Ramer and Andrew M Rappe. In: Phys. Rev. B 62.2 (2000),

R743 (cit. on p. 1).[10] V. Kumar and K. H. Bennemann. In: Phys. Rev. lett. 53.3 (1984), p. 278

(cit. on p. 2).[11] T. Hashimoto, T. Miyoshi, and H. Ohtsuka. In: Phys. Rev. B 13.3 (1976),

p. 1119 (cit. on p. 2).[12] V. Ozolin, š, C. Wolverton, and A. Zunger. In: Phys. Rev. B 57.11 (1998),

p. 6427 (cit. on pp. 2, 27–29, 50, 55).[13] Z. W. Lu, S-H Wei, and A. Zunger. In: Phys. Rev. B 45.18 (1992),

p. 10314 (cit. on p. 2).[14] J-W Yeh, S-K Chen, S-J Lin, J-Y Gan, T-S Chin, T-T Shun, C-H Tsau,

and S-Y Chang. In: Adv. Eng. Mater. 6.5 (2004), p. 299 (cit. on pp. 2,37).

[15] Yeh J-W. In: Ann. Chim. Sci. Mat 31.6 (2006), p. 633 (cit. on p. 2).[16] Y. Y. Chen, T. Duval, U. D. Hung, J. W. Yeh, and H. C. Shih. In: Corros.

Sci. 47.9 (2005), p. 2257 (cit. on p. 2).[17] H. P. Chou, Y. S. Chang, S. K. Chen, and J-W Yeh. In: Mater. Sci. Eng.

B Solid-State Mater. Adv. Technol. 163.3 (2009), p. 184 (cit. on p. 2).

72 BIBLIOGRAPHY

[18] O. N. Senkov, J. M. Scott, S. V. Senkova, D. B. Miracle, and C. F.Woodward. In: J. Alloys Compd. 509.20 (2011), p. 6043 (cit. on p. 2).

[19] V. Dolique, A-L Thomann, P. Brault, Y. Tessier, and P.e Gillon. In: Surf.Coat. Technol. 204.12 (2010), p. 1989 (cit. on pp. 2, 37).

[20] O. N. Senkov and C. F. Woodward. In: Mater. Sci. Eng. A 529 (2011),p. 311 (cit. on p. 2).

[21] Y. Zhang, X. Yang, and P. K. Liaw. In: JOM 64.7 (2012), p. 830 (cit. onp. 2).

[22] F. Tian, L. K. Varga, N. Chen, J. Shen, and L. Vitos. In: J. AlloysCompd. 599 (2014), p. 19 (cit. on pp. 2, 41).

[23] F. Tian, L. K. Varga, N. Chen, L. Delczeg, and L. Vitos. In: Phys. Rev.B 87.7 (2013), p. 075144 (cit. on p. 2).

[24] A. J. Zaddach, C. Niu, C. C. Koch, and D. L. Irving. In: JOM 65.12(2013), p. 1780 (cit. on p. 2).

[25] I. Toda-Caraballo, J. S. Wróbel, S. L. Dudarev, D. Nguyen-Manh, andP. E. J. Rivera-Díaz-del-Castillo. In: Acta Mater. 97 (2015), p. 156 (cit.on pp. 2, 3).

[26] Y. Zhang, G. Kresse, and C. Wolverton. In: Phys. Rev. Lett. 112.7 (2014),p. 1 (cit. on pp. 2, 49–51, 55).

[27] J. P. Perdew, K. Burke, and M. Ernzerhof. In: Phys. Rev. Lett. 77.18(1996), p. 3865 (cit. on pp. 2, 10, 11).

[28] Jochen Heyd, Gustavo E Scuseria, and Matthias Ernzerhof. In: J. Chem.Phys. 118.18 (2003), p. 8207 (cit. on pp. 2, 49).

[29] Jochen Heyd, Gustavo E Scuseria, and Matthias Ernzerhof. In: J. Chem.Phys. 124.21 (2006), p. 219906 (cit. on pp. 2, 49).

[30] H. Levämäki, M. P. J. Punkkinen, K. Kokko, and L. Vitos. In: Phys.Rev. B 86 (2012), p. 201104 (cit. on pp. 3, 10, 11).

[31] H. Levämäki, M. P. J. Punkkinen, K. Kokko, and L. Vitos. In: Phys.Rev. B 89 (2014), p. 115107 (cit. on pp. 3, 10–12, 37, 38, 50).

[32] Walter Heitler and Fritz London. In: Z. Phys 44.6-7 (1927), p. 455 (cit.on p. 6).

[33] F. Bloch. In: Z. Phys 52 (1928), p. 555 (cit. on p. 6).[34] M. Born and R. Oppenheimer. In: Ann. d. Physik 84 (1927), p. 457 (cit.

on p. 6).[35] M. Born and K. Huang. Dynamical Theoty of Crystal Lattices. Vol. 134.

Oxford university, 1954 (cit. on p. 6).[36] Efthimios Kaxiras. Atomic and electronic structure of solids. Cambridge

University Press, 2003 (cit. on p. 6).[37] P. Hohenberg and W. Kohn. In: Phys. Rev. 136 (1964), B864 (cit. on

p. 6).[38] E. K. U. Gross and R. M. Dreizler. In: Z. Phys. A: Hadrons Nucl. 302.2

(1981), p. 103 (cit. on p. 10).[39] J. .P Perdew. In: Phys. Rev. Lett. 55.16 (1985), p. 1665 (cit. on p. 10).

BIBLIOGRAPHY 73

[40] J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E. Scuseria,X. Constantin L. A .and Zhou, and K. Burke. In: Phys. Rev. Lett. 100.13(2008), p. 136406 (cit. on p. 10).

[41] R. Armiento and A. E. Mattsson. In: Phys. Rev. B 72 (2005), p. 085108(cit. on p. 10).

[42] A. E. Mattsson and R. Armiento. In: Phys. Rev. B 79.15 (2009), p. 155101(cit. on p. 10).

[43] J. P. Perdew and Y. Wang. In: Phys. Rev. B 45.23 (1992), p. 13244 (cit.on p. 11).

[44] R. Armiento and A. E. Mattsson. In: Phys. Rev. B 66.16 (2002), p. 165117(cit. on p. 12).

[45] L. Vitos, H. L. Skriver, B. Johansson, and J. Kollár. In: Comput. Mater.Sci. 18 (2000), p. 24 (cit. on p. 13).

[46] L. Vitos, I. A. Abrikosov, and B. Johansson. In: Phys. Rev. Lett. 87.15(2001), p. 156401 (cit. on pp. 13, 15).

[47] L. Vitos. In: Phys. Rev. B 64 (2001), p. 14107 (cit. on p. 13).[48] L. Vitos. Computational Quantum Mechanics for Materials Engineers.

London: Springer-Verlag, 2007 (cit. on pp. 13, 15).[49] O. K. Andersen, O. Jepsen, G. Krier, et al. In: World Science, Singapore

(1994), p. 63 (cit. on p. 13).[50] Axel van de Walle. In: Calphad 33.2 (2009), p. 266 (cit. on p. 14).[51] B. L. Gyorffy. In: Phys. Rev. B 5 (1972), p. 2382 (cit. on p. 15).[52] A. Taga, L. Vitos, B. Johansson, and G. Grimvall. In: Phys. Rev. B 71

(2005), p. 014201 (cit. on p. 15).[53] M. A. Caro, S.n Schulz, and E. P. OReilly. In: J. Phys.: Condens. Matter

25.2 (2012), p. 025803 (cit. on p. 18).[54] P. Ravindran, L. Fast, P. A. Korzhavyi, B. Johansson, J. Wills, and O.

Eriksson. In: J. Appl. Phys. 84.9 (1998), p. 4891 (cit. on p. 18).[55] Johann von Pezold, Alexey Dick, Martin Friák, and Jörg Neugebauer.

In: Phys. Rev. B 81.9 (2010), p. 094203 (cit. on pp. 19, 32, 33).[56] F. Tasnádi, M. Odén, and I. A Abrikosov. In: Phys. Rev. B 85.14 (2012),

p. 144112 (cit. on p. 19).[57] D. Holec, F. Tasnadi, P. Wagner, M. Friák, J. Neugebauer, P. H. Mayrhofer,

and J. Keckes. In: Phys. Rev. B 90.18 (2014), p. 184106 (cit. on p. 19).[58] V. L. Moruzzi, J. F. Janak, and K. Schwarz. In: Phys. Rev. B 37.2 (1988),

p. 790 (cit. on p. 20).[59] G. Grimvall. Thermophysical properties of materials. Elsevier, 1999 (cit.

on p. 20).[60] A Reuss. In: ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift

für Angewandte Mathematik und Mechanik 9.1 (1929), p. 49 (cit. onp. 20).

[61] J. Kollár, L. Vitos, and H. L. Skriver. In: Electronic Structure and Physi-cal Properties of Solids: the uses of the LMTO method. Ed. by H. Dreyssé.Berlin: Springer-Verlag, 2000, p. 85 (cit. on p. 21).

74 BIBLIOGRAPHY

[62] A. G. Every. In: Phys. Rev. B 22.4 (1980), p. 1746 (cit. on p. 21).[63] Clarence Zener. Elasticity and anelasticity of metals. University of Chicago

press, 1948 (cit. on p. 21).[64] O. K. Andersen. In: Solid State Commun. 13.2 (1973), p. 133 (cit. on

p. 24).[65] D.G. Pettifor. In: J. Phys. F: Metal Phys. 8.2 (1978), p. 219 (cit. on

p. 24).[66] K. Kádas, Z. Nabi, S. K. Kwon, L. Vitos, R. Ahuja, B. Johansson, and

J. Kollár. In: Surf. Sci. 600.2 (2006), p. 395 (cit. on p. 25).[67] E. K. Delczeg-Czirjak, L. Delczeg, M. Ropo, K. Kokko, M. P. J. Punkki-

nen, B. Johansson, and L. Vitos. In: Phys. Rev. B 79.8 (2009), p. 085107(cit. on p. 25).

[68] A. V. Ruban, I. A. Abrikosov, and H. L. Skriver. In: Phys. Rev. B 51.19(1995), p. 12958 (cit. on pp. 27, 50, 51).

[69] K. Terakura, T. Oguchi, T. Mohri, and K. Watanabe. In: Phys. Rev. B35.5 (1987), p. 2169 (cit. on pp. 27, 50, 51).

[70] A. V. Ruban, S. I. Simak, S. Shallcross, and H. L. Skriver. In: Phys. Rev.B 67.21 (2003), p. 214302 (cit. on pp. 28, 29).

[71] R. L. Orr. In: Acta Metall. 8.7 (1960), p. 489 (cit. on pp. 28–30, 50).[72] R. Hultgren, P. D. Desai, D. T. Hawkins, M. Gleiser, and K. K. Kelley.

Selected values of the thermodynamic properties of binary alloys. Tech.rep. DTIC Document, 1973 (cit. on pp. 28, 30).

[73] B. W. Roberts. In: Acta metall. 2.4 (1954), p. 597 (cit. on p. 29).[74] B. Borie and B. E. Warren. In: J. Appl. Phys. 27.12 (1956), p. 1562 (cit.

on p. 29).[75] J. Zander, R. Sandström, and L. Vitos. In: Comput. Mater. Sci. 41.1

(2007), p. 86 (cit. on p. 35).[76] J-W Yeh, S-J Lin, T-S Chin, J-Y Gan, S-K Chen, T-T Shun, C-H Tsau,

and S-Y Chou. In: Metall. Mater. Trans. A 35.8 (2004), p. 2533 (cit. onp. 37).

[77] O. N. Senkov, S. V. Senkova, C. Woodward, and D. B. Miracle. In: ActaMater. 61.5 (2013), p. 1545 (cit. on pp. 37, 38).

[78] J. P. Couzinié, G. Dirras, L. Perrière, T. Chauveau, E. Leroy, Y. Cham-pion, and I. Guillot. In: Mater. Lett. 126 (2014), p. 285 (cit. on p. 37).

[79] O. N. Senkov, G.B. Wilks, D.B. Miracle, C.P. Chuang, and P.K. Liaw.In: Intermetallics 18.9 (2010), p. 1758 (cit. on p. 37).

[80] M. C. Gao, C. Niu, C. Jiang, and D. L. Irving. “Applications of specialquasi-random structures to high-entropy alloys”. In: High Entropy Alloys.Springer, 2016, p. 333 (cit. on p. 38).

[81] I. Kunce, M. Polanski, and J. Bystrzycki. In: Int. J. Hydrogen Energy39.18 (2014), p. 9904 (cit. on p. 38).

[82] Per Söderlind, Olle Eriksson, J M Wills, and A M Boring. In: Phys. Rev.B 48.9 (1993), p. 5844 (cit. on p. 39).

BIBLIOGRAPHY 75

[83] Jinghan Wang, Sidney Yip, SR Phillpot, and Dieter Wolf. In: Phys. Rev.Lett. 71.25 (1993), p. 4182 (cit. on p. 40).

[84] Y. Zhang, Y. J. Zhou, J. P. Lin, G. L. Chen, and P. K. Liaw. In: Adv.Eng. Mater. 10.6 (2008), p. 534 (cit. on p. 43).

[85] F. Tian, L. K. Varga, N. Chen, J. Shen, and L. Vitos. In: Intermetallics58 (2015), p. 1 (cit. on p. 44).

[86] X. Yang and Y. Zhang. In: Mater. Chem. Phys. 132.2 (2012), p. 233 (cit.on p. 44).

[87] P. A. Korzhavyi, A. V. Ruban, I. A. Abrikosov, and H. L. Skriver. In:Phys. Rev. B 51.9 (1995), p. 5773 (cit. on p. 44).

[88] R. Ahuja, J. M. Wills, B. Johansson, and O. Eriksson. In: Phys. Rev. B48.22 (1993), p. 16269 (cit. on p. 44).

[89] Mathilde Laurent-Brocq, Loïc Perrière, Rémy Pirès, and Yannick Cham-pion. In: Mater. Des. 103 (2016), p. 84 (cit. on p. 45).

[90] J. P. Perdew, K. Schmidt, V. Van Doren, C. Van Alsenoy, and P. Geer-lings. “Jacob′s ladder of density functional approximations for the exchange-correlation energy”. In: AIP Conference Proceedings. Vol. 577. 1. AIP.2001, p. 1 (cit. on p. 49).

[91] J. Sun, R. Haunschild, B. Xiao, I. W. Bulik, G. E. Scuseria, and J. P.Perdew. In: J. Chem. Phys. 138.4 (2013), p. 044113 (cit. on p. 49).

[92] J. Sun, A. Ruzsinszky, and J. P. Perdew. In: Phys. Rev. Lett. 115.3 (2015),p. 036402 (cit. on p. 49).

[93] D. A. Kitchaev, H. Peng, Y. Liu, J. Sun, J. P. Perdew, and G. Ceder. In:Phys. Rev. B 93.4 (2016), p. 045132 (cit. on pp. 49, 54).

[94] H. Okamoto, D. J. Chakrabarti, D. E. Laughlin, and T. B. Massalski.In: J. Phase Equilib. 8.5 (1987), p. 454 (cit. on p. 50).

[95] P. Villars. In: ASM International Materials Park, OH (2006) (cit. onp. 50).

[96] M. Sanati, L. G. Wang, and A. Zunger. In: Phys. Rev. lett. 90.4 (2003),p. 045502 (cit. on p. 50).

[97] Gene Simmons, Herbert Wang, et al. In: (1971) (cit. on p. 50).[98] V. N. Staroverov, G. E. Scuseria, J. Tao, and J. P. Perdew. In: Phys.

Rev. B 69.7 (2004), p. 075102 (cit. on p. 50).[99] S. L. Shang, A. Saengdeejing, Z. G. Mei, D. E. Kim, H. Zhang, S Gane-

shan, Yi Wang, and Z. K. Liu. In: Comput. Mater. Sci. 48.4 (2010), p. 813(cit. on p. 50).

[100] H. H. Kart, M. Tomak, and T. Çağin. In: Model. Simul. Mater. Sci. Eng.13.5 (2005), p. 657 (cit. on p. 50).

[101] M. Ropo, K. Kokko, and L. Vitos. In: Phys. Rev. B 77.19 (2008), p. 195445(cit. on p. 50).

[102] V. Blum, R. Gehrke, F. Hanke, P. Havu, V. Havu, X. Ren, K. Reuter,and M. Scheffler. In: Comput. Phys. Commun. 180.11 (2009), p. 2175(cit. on p. 55).

76 BIBLIOGRAPHY

[103] V. Havu, V. Blum, P. Havu, and M. Scheffler. In: J. Comput. Phys.228.22 (2009), p. 8367 (cit. on p. 55).

[104] FHI-aims employs numerical atomic-centered orbital basis sets. The tier2 level basis set with a “tight” integration grid and which introduces neg-ligible errors compared to fully self-consistent SCAN calculations. 1688(1099 for the AgPd sqs system) k points in the irreducible Brillouin zonewas used. SCAN energies were evaluated non-self-consistently using con-verged PBE densities. In: () (cit. on p. 55).

[105] Z. W. Lu, S-H Wei, A. Zunger, S. Frota-Pessoa, and L. G. Ferreira. In:Phys. Rev. B 44.2 (1991), p. 512 (cit. on p. 55).

[106] D. Feng and P. Taskinen. In: J. Mater. Sci. 49.16 (2014), p. 5790 (cit. onp. 55).

[107] R. E. Watson and M. Weinert. In: Phys. Rev. B 58.10 (1998), p. 5981(cit. on p. 55).

[108] J E Enkovaara, C. Rostgaard, Jens J. Mortensen, J. Chen, M Dułak, L.Ferrighi, J.e Gavnholt, Christian Glinsvad, V Haikola, and H A Hansen.In: J. Phys.: Condens. Matter 22.25 (2010), p. 253202 (cit. on p. 58).

densityfunctionaltheorystudy ofbulkpropertiesofmetallic...

Documents