Quantum-Mechanical Modeling

of High-Entropy Alloys

SHUO HUANG

Doctoral Thesis

School of Industrial Engineering and Management, Department of

Materials Science and Engineering, KTH, Sweden, 2018

Materialvetenskap

KTH

SE-100 44 Stockholm

ISBN 978-91-7729-765-9 Sweden

Akademisk avhandling som med tillstånd av Kungliga Tekniska högskolan framlägges

till offentlig granskning för avläggande av doktorsexamen tisdag den 12 juni 2018 kl

10:00 i B2, Kungliga Tekniska högskolan, Brinellvägen 23, Stockholm.

© Shuo Huang , 2018

Tryck: Universitetsservice US AB

iii

Abstract

High-entropy alloys (HEAs) consisting of multi-principal elements open up a

near-infinite compositional space for materials design. Extensive attention has

been put on HEAs, and interesting structural, physical and chemical properties

are being continuously revealed. Based on first-principle theory, here we focus

on the fundamental characteristics of HEAs, as well as on the optimization and

prediction of alternative alloy with promising technological applications.

The relative phase stability of different-types of HEAs is investigated from

the minimum of structural energy, and the composition-, temperature-, and

ordering-induced phase transformations are presented. The elastic properties

are discussed through the single-crystal and polycrystalline elastic moduli by

making use of a series of phenomenological models. The competition between

full slip, twinning, and stacking fault in face-centered cubic HEAs is analyzed

by studying the generalized stacking fault energy. The magnetic characteristics

are provided through the Heisenberg Hamiltonian model in connection with

Monte-Carlo simulation, and the Curie temperature of a large number of cubic

HEAs is mapped out with the help of mean-filed approximation. The thermal

expansion behavior is estimated by using the Debye-Grüneisen model.

This work provides some fundamental and pioneering theoretical points of

view to understand the intrinsic physical mechanisms in HEAs, and reveals

alternative opportunities for optimizing and designing properties of materials.

The challenge of comprehending the observed complex behavior behind the

multi-component nature of HEAs is great, on the other hand, the potential to

enhance the underlying theoretical understanding is remarkable.

iv

Sammanfattning

Högentropilegeringar, som består av flera viktiga grundämnen, är nuförtiden

ett av de mest spännande forskningsområdena inom materialvetenskap.

Baserat på första princips teori, fokuserar vi på de fundamentala egenskaperna

hos högentropilegeringar samt optimering och förutsägelse av alternativ

legeringar med lovande tekniska applikationer.

Det elastiska beteendet tillhandahålls från enkelkristall och polykristallin

elasticitetsmoduler genom användning av en rad fenomenologiska modeller.

Tävlingen mellan full glidning, tvillinggränser och staplingsfel under plastisk

deformation i ytcentrerat kubiskt högentropilegeringar analyseras från den

generella staplingsfelenergin. Magnetiska egenskaperna undersöks genom en

modell för den Hamiltonoperatoren i samband med Monte-Carlo simulering.

Curietemperaturen kartläggs för ett stort antal kubiska ekvimolära

högentropilegeringar med hjälp av medelfältapproximation. Temperatur-

variationen av termisk utvidgningskoefficient beräknas baserat på Debye-

Grüneisen modellen.

Det här arbetet ger några grundläggande och banbrytande teoretiska

synvinklar för att förstå de inneboende fysiska mekanismerna bakom den

multifunktionella naturen hos högentropilegeringar, och avslöjar alternativa

möjligheter att utforma och optimera materialets egenskaper.

v

Preface

List of included publications:

I. Temperature dependent stacking fault energy of FeCrCoNiMn high entropy alloy

Shuo Huang, Wei Li, Song Lu, Fuyang Tian, Jiang Shen, Erik Holmström, and

Levente Vitos, Scripta Materialia, 108, 44-47 (2015).

II. Phase stability and magnetic behavior of FeCrCoNiGe high-entropy alloy

Shuo Huang, Ádám Vida, Dávid Molnár, Krisztina Kádas, Lajos Károly Varga, Erik

Holmström, and Levente Vitos, Applied Physics Letters, 107, 251906 (2015).

III. Mechanism of magnetic transition in FeCrCoNi-based high entropy alloys

Shuo Huang, Wei Li, Xiaoqing Li, Stephan Schönecker, Lars Bergqvist, Erik

Holmström, Lajos Károly Varga, and Levente Vitos, Materials and Design, 103, 71-74

(2016).

IV. Thermal expansion in FeCrCoNiGa high-entropy alloy from theory and experiment

Shuo Huang, Ádám Vida, Wei Li, Dávid Molnár, Se Kyun Kwon, Erik Holmström,

Béla Varga, Lajos Károly Varga, and Levente Vitos, Applied Physics Letters, 110,

241902 (2017).

V. Thermal expansion, elastic and magnetic properties of FeCoNiCu-based high-entropy

alloys using first-principle theory

Shuo Huang, Ádám Vida, Anita Heczel, Erik Holmström, and Levente Vitos, JOM,

69, 2107-2112 (2017).

VI. Mechanical performance of FeCrCoMnAlx high-entropy alloys from first-principle

Shuo Huang, Xiaoqing Li, He Huang, Erik Holmström, and Levente Vitos, Materials

Chemistry and Physics, 210, 37-42 (2018).

VII. Mapping the magnetic transition temperatures for medium- and high-entropy alloys

Shuo Huang, Erik Holmström, Olle Eriksson, and Levente Vitos, Intermetallics, 95,

80-84 (2018).

VIII. Strengthening induced by magneto-chemical transition in Al-doped Fe-Cr-Co-Ni high-

entropy alloys

Shuo Huang, Wei Li, Erik Holmström, Levente Vitos, submitted to Phys. Rev. Appl.

IX. Phase-transition assisted mechanical behavior of TiZrHfTax high-entropy alloys

Shuo Huang, Wei Li, Erik Holmström, Levente Vitos, submitted to Sci. Rep.

vi

X. Elasticity of high-entropy alloys from ab initio theory

Shuo Huang, Fuyang Tian, Levente Vitos, submitted to J. Mater. Res.

Comment on my own contribution:

Literature survey, research planning, data analysis, and manuscript preparation

done jointly. My personal contributions to the above research steps are on the

average 75%, 80%, 70%, and 75%, respectively. Calculations were done

almost entirely by me (papers I, II, IV, V, VII, VIII, IX 90%, papers III and VI

70%). My contribution to Paper X is about 80 %.

List of publications not included in the thesis:

XI. C.H. Zhang, S. Huang, J. Shen, N.X. Chen, Chin. Phys. B, 21 (2012) 113401.

XII. S. Huang, C.H. Zhang, J. Sun, J. Shen, Chin. Phys. B, 22 (2013) 083401.

XIII. C.H. Zhang, S. Huang, J. Shen, N.X. Chen, J. Mater. Res., 28 (2013) 2720-2727.

XIV. S. Huang, C.H. Zhang, J. Sun, J. Shen, Mod. Phys. Lett. B, 27 (2013) 1350195.

XV. J. Sun, P. Qian, J. Shen, J.C. Li, S. Huang, J. Alloys Compd., 580 (2013) 522-526.

XVI. J. Sun, J. Shen, P. Qian, S. Huang, Physica B, 427 (2013) 110-117.

XVII. Z.F. Zhang, P. Qian, J.C. Li, Y.P. Li, S. Huang, J. Sun, J. Shen, Y. Liu, N.X. Chen,

Chin. J. Phys., 51 (2013) 606-618.

XVIII. C.H. Zhang, S. Huang, R.Z. Li, J. Shen, N.X. Chen, Int. J. Mod. Phys. B, 27 (2013)

1350147.

XIX. S. Huang, R.Z. Li, C.H. Zhang, J. Shen, Chin. J. Phys., 52 (2014) 891-902.

XX. S. Huang, R.Z. Li, S.T. Qi, B. Chen, J. Shen, Phys. Scr., 89 (2014) 065702.

XXI. S. Huang, R.Z. Li, S.T. Qi, B. Chen, J. Shen, Int. J. Mod. Phys. B, 28 (2014) 1450087.

XXII. S. Huang, R.Z. Li, S.T. Qi, B. Chen, J. Shen, Solid State Commun., 184 (2014) 52-55.

XXIII. S. Huang, C.H. Zhang, R.Z. Li, J. Shen, N.X. Chen, Intermetallics, 51 (2014) 24-29.

XXIV. C.H. Zhang, S. Huang, J. Shen, N.X. Chen, Intermetallics, 52 (2014) 86-91.

XXV. Á. Vida, L.K. Varga, N.Q. Chinh, D. Molnár, S. Huang, L. Vitos, Mater. Sci. Eng. A,

669 (2016) 14-19.

XXVI. Á. Vida, Z. Maksa, D. Molnár, S. Huang, J. Kovac, L.K. Varga, L. Vitos, N.Q. Chinh,

J. Alloys Compd., 743 (2018) 234-239.

XXVII. H. Huang, X. Li, Z. Dong, W. Li, S. Huang, D. Meng, X. Lai, T. Liu, S. Zhu, L. Vitos,

Acta Mater., 149 (2018) 388-396.

XXVIII. S. Huang, H. Huang, W. Li, D. Kim, S. Lu, X. Li, E. Holmström, S.K. Kwon, L. Vitos,

Twinning in metastable high-entropy alloys, submitted to Nat. Commun.

XXIX. S. Huang, W. Li, E. Holmström, S.K. Kwon, L. Vitos, Plastic deformation transition

in FeCrCoNiAlx high-entropy alloys, in manuscript.

vii

Contents

Abstract ........................................................................................................ iii

Sammanfattning ........................................................................................... iv

Preface ........................................................................................................... v

Contents ....................................................................................................... vii

1 Introduction .............................................................................................. 1

2 Theoretical methodology .......................................................................... 3

2.1 Schrödinger equation ............................................................................ 3

2.2 Density functional theory ..................................................................... 4

2.3 Exact muffin-tin orbital method ........................................................... 5

2.4 Coherent potential approximation ........................................................ 6

3 Structural properties ................................................................................ 8

3.1 Phase stability ....................................................................................... 8

3.1.1 Composition dependence .............................................................. 8

3.1.2 Temperature effect .......................................................................10

3.1.3 Ordering behavior ........................................................................11

3.2 Lattice parameter .................................................................................13

4 Deformation performance .......................................................................15

4.1 Elasticity ..............................................................................................15

4.1.1 Single-crystal elastic constants ....................................................15

4.1.2 Polycrystalline elastic moduli ......................................................16

4.2 Plasticity ..............................................................................................17

4.2.1 Generalized stacking fault energy ................................................17

5 Magnetic behavior ...................................................................................19

5.1 Magnetic moment ................................................................................19

viii

5.2 Curie temperature ................................................................................ 20

5.2.1 Monte-Carlo simulation ............................................................... 20

5.2.2 Mean-field approximation ........................................................... 23

6 Thermo-physical characteristics ............................................................ 26

6.1 Debye temperature .............................................................................. 26

6.2 Thermal expansion .............................................................................. 27

7 Summary and outlook ............................................................................. 30

Acknowledgement ....................................................................................... 31

Bibliography ................................................................................................ 32

1

Chapter 1

Introduction

High-entropy alloys (HEAs), first brought to general attention in 2004 [1, 2],

represent an exciting research area in materials science, since it is characterized

by many new findings, unexplained results, and vigorous controversies.

Being different from the conventional alloys, Yeh [3] defined HEAs based

on composition and configurational entropy: i, containing at least five principle

elements with the concentration of each element being between 5 and 35 at. %;

ii, having configurational entropy at a random state larger than 1.5 𝑅 (𝑅 is the

gas constant, ~ 8.314 Jmol-1K-1), despite showing single- or multi-phase at

room temperature. Notice that the definitions of HEAs are just guidelines, not

laws, that the basic principle is having high mixing entropy to enhance the

formation of solid solution phases, and avoid complicated structures [4].

In this scenario, a large number of different-types of HEAs has been reported

[4]. Interestingly, HEAs consisting of late 3d transition metals tend to form a

face-centered cubic (fcc) phase [5], those composed of refractory metals often

have a body-centered cubic (bcc) phase [6], while HEAs based on rare-earth

metals usually crystalize in a hexagonal closed-packed (hcp) phase [7]. In an

attempt to understand the phase selection, several parametric approaches have

been proposed by using physiochemical parameters, such as mixing enthalpy,

mixing entropy, melting point, atomic size mismatch, and valence electron

concentration [8-11]. On the other hand, the potential to enhance the empirical

rules to scientific theories is remarkable to tackle challenges in the family of

multi-component alloys.

Along with the appearance of the special microstructure, interesting physical

and chemical properties are being continuously revealed over the recent years.

In the case of the equiatomic fcc CrMnFeCoNi, for example, the strength and

ductility increase simultaneously at cryogenic temperatures [12, 13], which run

2

counter to many other materials where an inverse dependence of strength and

ductility is invariably seen [14].

The physical metallurgy principles might be different for HEAs as compared

to the conventional alloys. Meanwhile, although a large body of work has been

carried out since 2004, there are still many open questions that concern people

in the academic and metallurgic research communities [4]. For instance, are

HEAs thermodynamically stable or unstable in nature? How about the phase

diagrams of HEAs? What features are crucial for producing HEAs with

exceptional properties? How to accelerate the exploration of high-performance

single- or multi-phase HEAs? All these facts making HEAs attractive.

In this thesis, we present a theoretical study of fundamental characteristics

for a variety of HEAs, including the phase stability, elastic/plastic deformation,

magnetic behavior, and thermo-physical properties. This work provides some

theoretical points of view to study the intrinsic physical mechanisms behind

the complex behavior as observed in HEAs, and reveals some opportunities for

optimizing and designing properties of materials.

3

Chapter 2

Theoretical methodology

Many physical and chemical properties of materials depend on the interactions

of electrons and nuclei, which require a quantum-mechanical treatment. In this

scenario, one should find the solution of Schrödinger wave equation for multi-

particle system. However, it is incredible complicate to solve such equation in

practice. Thus, reasonable approximations and simplifications are necessary.

This chapter briefly presents some key approaches used in the present work.

2.1 Schrödinger equation

In quantum mechanics, the state of system is described by the multi-particle

Schrödinger equation

ℋΨ = 𝐸Ψ, (2.1)

where 𝐸 is the ground state energy, Ψ = Ψ(𝐫1, … , 𝐫N, 𝐑1, … , 𝐑M) is the wave-

function for N electrons and M nuclei system with positions 𝐫𝑖 (i = 1, …, N)

and 𝐑𝑗 (j = 1, …, M), respectively. The many-body Hamiltonian

ℋ = −ℏ2

2𝑚𝑒

∑ ∇𝐫𝑖

2𝑖 −

ℏ2

2∑

∇𝐑𝑗2

𝑀𝑗𝑗 − ∑ ∑

𝑒2𝑍𝑗

|𝐫𝑖−𝐑𝑗|𝑗𝑖 +1

2∑

𝑒2

|𝐫𝑖−𝐫𝑗|𝑖≠𝑗 +1

2∑

𝑒2𝑍𝑖𝑍𝑗

|𝐑𝑖−𝐑𝑗|𝑖≠𝑗 , (2.2)

consisting of electrons with mass 𝑚𝑒 and charge 𝑒, and nuclei with mass 𝑀𝑗

and charge −𝑍𝑗𝑒. Here ℏ = ℎ 2𝜋⁄ is the reduced Planck constant. The first two

terms denote the kinetic energy operators for electrons and nuclei, respectively.

The last three terms represent the corresponding Coulomb interactions of

electron-nuclei, electron-electron, and nuclei-nuclei, respectively. The above

Hamiltonian expresses the full theory. However, in real materials, solving Eq.

2.1 within Eq. 2.2 is a Herculean task.

4

Nuclei are much more massive than electrons, which implies that electrons can

respond to the nuclei moving instantaneously. Hence, one may neglect the

nuclei kinetic energy term in Eq. 2.2, i.e., Born-Oppenheimer approximation.

The nuclei-nuclei interactions are neglected, which shift the eigenvalues only.

Then the Hamiltonian is simplified as

ℋ = −ℏ2

2𝑚𝑒

∑ ∇𝐫𝑖

2𝑖 − ∑ ∑

𝑒2𝑍𝑗

|𝐫𝑖−𝐑𝑗|𝑗𝑖 +1

2∑

𝑒2

|𝐫𝑖−𝐫𝑗|𝑖≠𝑗 = 𝑇 + 𝑉 + 𝑈, (2.3)

where 𝑇 is the kinetic energy operator, 𝑉 represents the external potential from

the static nuclei background, and 𝑈 represents the internal potential from the

electron-electron interaction.

2.2 Density functional theory

Density functional theory (DFT) is one of the most popular approaches to the

theory of electronic structure, in which the electronic density plays the central

role, rather than the many-electron wave function [15].

In the framework of DFT, there are two important theorems [16]: i, the

external potential 𝑉(𝒓) is a unique functional of the electronic density 𝑛(𝒓),

apart from a trivial additive constant; ii, there exists a universal functional,

𝐹[𝑛(𝒓)], independent of 𝑉(𝒓), such that the expression

𝐸[𝑛(𝒓)] ≡ ∫ 𝑉(𝒓)𝑛(𝒓)𝑑𝒓 + 𝐹[𝑛(𝒓)], (2.4)

has its minimum value for the correct 𝑛0(𝒓). Based on the variational principle,

the minimum of 𝐸 equals the total energy of the electronic system. The 𝐹[𝑛] is usually represented as

𝐹[𝑛] =1

2∬

𝑛(𝒓)𝑛(𝒓′)

|𝒓−𝒓′|𝑑𝒓𝑑𝒓′ + 𝐺[𝑛], (2.5)

where 𝐺[𝑛] ≡ 𝑇𝑠[𝑛] + 𝐸xc[𝑛] is a universal functional like 𝐹[𝑛]. Here, 𝑇𝑠[𝑛]

is the kinetic energy of a system of non-interacting electrons, and 𝐸xc[𝑛] is the

exchange-correlation energy representing all electron-electron interactions

omitted in the other terms.

Based on the Kohn-Sham (KS) scheme [17], 𝑛(𝒓) is expressed as 𝑛(𝒓) =

∑ |𝜓𝑖(𝒓)|2𝑁𝑖=1 , where 𝜓𝑖(𝒓) is the KS orbitals describing electrons moving in

an effective potential, which contains all types of interactions. Therefore, one

can obtain 𝑛(𝒓) by solving the single-particle Schrödinger equation

[−1

2∇𝑖

2 + 𝑉(𝒓) + ∫𝑛(𝒓′)

|𝒓−𝒓′|𝑑𝒓′ + 𝑉xc] 𝜓𝑖(𝒓) = ϵ𝑖𝜓𝑖(𝒓), (2.6)

5

where ϵ𝑖 denotes the eigenvalues of the hypothetical KS particles, and 𝑉xc =

𝛿𝐸xc[𝑛] 𝛿𝑛⁄ represents the corresponding exchange-correlation potential.

As the exact form of 𝐸xc is not known, although the existence of exchange-

correlation is guaranteed by the basic theorem, one needs to resort to various

approximations. Nowadays, the local density approximation (LDA) [18] and

the generalized gradient approximation (GGA) [19] are two most widely used

approximations for the exchange-correlation term in computational materials

science based on DFT.

2.3 Exact muffin-tin orbital method

Developing accurate and efficient numerical approaches for solving the KS

equation challenges the computational materials science community. Several

techniques were introduced: i, full-potential methods, in which the wave-

function is taken into account with high accuracy, while at the price of very

expensive computational effort; ii, pseudopotential methods, in which a full-

potential description is kept in the interstitial region, and the true Coulomb-

like potential is replaced with a weak pseudopotential in the region near the

nuclei; iii, muffin-tin potential methods, in which the Kohn-Sham potential is

substituted by spherically symmetric potentials centered on atoms plus a

constant potential in the interstitial region.

In the present work, the exact muffin-tin orbitals (EMTO) method is applied.

The details about this method and its self-consistent implementation can be

found in previous work [20]. This section briefly introduces some key points.

According to the overlapping muffin-tin approximation, the effective single-

electron potential is approximated by the spherical potential wells 𝑉𝑅(𝑟𝑅) − 𝑉0

centered on lattice sites 𝑅 with notation 𝒓𝑅 ≡ 𝑟𝑅�̂�𝑅 = 𝒓 − 𝑹, plus a constant

potential 𝑉0, viz.,

𝑉eff = 𝑉(𝒓) + ∫𝑛(𝒓′)

|𝒓−𝒓′|𝑑𝒓′ + 𝑉xc ≈ 𝑉mt(𝒓) = 𝑉0 + ∑ [𝑉𝑅(𝑟𝑅) − 𝑉0]𝑅 , (2.7)

where 𝑉𝑅(𝑟𝑅) becomes equal to 𝑉0 outside the potential sphere of radius 𝑠𝑅.

The Eq. 2.6 within Eq. 2.7 are solved by expanding the KS orbital 𝜓𝑖(𝒓) in

terms of exact muffin-tin orbitals �̅�𝑅𝐿𝛼 (𝜖𝑖 , 𝒓𝑅), viz.,

𝜓𝑖(𝒓) = ∑ �̅�𝑅𝐿𝛼 (𝜖𝑖 , 𝒓𝑅)𝑣𝑅𝐿,𝑖

𝛼𝑅𝐿 , (2.8)

6

where the expansion coefficients 𝑣𝑅𝐿,𝑖𝛼 are determined from the condition that

the above expansion should be a solution for Eq. 2.6 in the entire space, and

the multi-index 𝐿 = (𝑙, 𝑚) represents the set of the orbital (𝑙) and magnetic

(𝑚) quantum numbers, respectively. The �̅�𝑅𝐿𝛼 (𝜖𝑖 , 𝒓𝑅) contains different basis

functions in different regions

�̅�𝑅𝐿𝛼 (𝜖𝑖 , 𝒓𝑅) = 𝜙𝑅𝐿

𝛼 (𝜖𝑖 , 𝑟𝑅) + 𝜓𝑅𝐿𝛼 (𝜖𝑖 − 𝑣0, 𝒓𝑅) − 𝜑𝑅𝐿

𝛼 (𝜖𝑖, 𝑟𝑅)𝑌𝐿(𝑟𝑅), (2.9)

where 𝜙𝑅𝐿𝛼 (𝜖𝑖 , 𝑟𝑅) is the partial wave inside the potential sphere (𝑟𝑅 ≤ 𝑠𝑅),

𝜓𝑅𝐿𝛼 (𝜖𝑖 − 𝑣0, 𝒓𝑅) is the screened spherical wave in the interstitial region, i.e.,

outside of the non-overlapping sphere 𝑎𝑅 (𝑎𝑅 ≤ 𝑠𝑅), and 𝜑𝑅𝐿𝛼 (𝜖𝑖 , 𝑟𝑅)𝑌𝐿(𝑟𝑅) is

the backward extrapolated free-electron wave function, matched continuously

and differentiable to the partial waves at 𝑠𝑅 and continuously to the screened

spherical waves at 𝑎𝑅. From the so-called kink-cancellation equation, which is

related to the boundary condition in the region 𝑎𝑅 ≤ 𝑟𝑅 ≤ 𝑠𝑅, one can find the

solution of the KS equation.

2.4 Coherent potential approximation

The main difficulty in the application of DFT to real system is the presence of

various kinds of disorder. One powerful technique is the coherent potential

approximation (CPA) [21-23]. It is based on the assumption that the alloy may

be replaced by an ordered effective medium, the parameters of which are

determined self-consistently. The single-site approximation is applied to the

impurity problem, i.e., one single impurity is embedded in an effective medium

and no information is provided about the individual potential and charge

density beyond the sphere or polyhedron around the impurity.

Basically, two main approximations are involved within the CPA: i, assume

that the local potentials around a certain type of atom from the alloy are the

same, i.e., the effect of local environment is neglected (all atoms belonging to

one particularly type are the same); ii, the system is replaced by a monoatomic

set-up described by the site independent coherent potential �̃� . In terms of

Green functions, one approximates the real Green function 𝑔 by a coherent

Green function �̃�. A single-site Green function 𝑔𝑖 is introduced for each alloy

component i = A, B, C, … in substitutional alloy AaBbCc…, where a, b, c, …

stand for the atomic fractions of the A, B, C, … atoms, respectively.

First, the coherent Green function �̃� is derived from the coherent potential

�̃� using an electronic structure method

�̃� = [𝑆 − �̃�]−1, (2.10)

7

where 𝑆 denotes the structure constant (slope) matrix corresponding to the

underlying lattice. Next, the Green functions of the alloy components, 𝑔𝑖, are

calculated by substituting the coherent potential of the CPA medium �̃� by the

real atomic potentials 𝑃𝑖 , viz.,

𝑔𝑖 = �̃� + �̃�(𝑃𝑖 − �̃�)𝑔𝑖 , 𝑖 = A, B, C …, (2.11)

where the condition is expressed via real-space Dyson equation. Finally, the

average of the individual Green functions should reproduce the single-site part

of the coherent Green function

�̃� = 𝑎𝑔A + 𝑏𝑔B + 𝑐𝑔C + ⋯. (2.12)

The above three equations are solved iteratively, and the output �̃� and 𝑔𝑖 are

used to determine the electronic structure, charge density and total energy of

the random alloy.

8

Chapter 3

Structural properties

Structural properties often hold the key to understand the material properties

from a microscopic point of view. For many reported HEAs, despite containing

multiple components with different crystal structures in their ground states,

there exists a preference for the formation of simple yet chemically disordered

solid solution phase rather than complex intermetallic structures. This chapter

briefly presents some fundamental structural characteristics of HEAs.

3.1 Phase stability

Experiments indicated that the equiatomic FeCrCoMnAl adopts a single bcc

solid solution phase [24]. Figure 3.1 (left panel) shows the total energy of the

FeCrCoMnAl as a function of Wigner-Seitz radius for the bcc, fcc, and hcp

phases at the ferromagnetic (FM) and paramagnetic (PM) states, respectively.

Here, all energies are plotted with respect to the equilibrium total energy of

FM bcc. As can be seen, the bcc structure is energetically favorable over the

fcc and hcp structures irrespective of the magnetic state. Extending the above

study to other compositions, as shown in Fig. 3.1 (right panel), we find that the

FeCrCoMnAlx (0.6 ≤ x ≤ 1.5) HEAs prefer the bcc structure rather than the

other two close-packed structures.

3.1.1 Composition dependence

In the widely studied FeCrCoNiAlx HEAs, the as-cast structure evolves from

the initial single fcc phase to a mixture of fcc and bcc duplex phases, and then

a single bcc phase with the increase of Al concentration [25, 26].

9

2.58 2.64 2.70 2.76

0.0

4.5

9.0

13.5

Ener

gy

dif

fere

nce

(m

Ry/a

tom

)

Wigner-Seitz radius (Bohr)

bccFM

bccPM

fccFM

fccPM

hcpFM

hcpPM

0.5 1.0 1.5

0.0

4.5

9.0

13.5

Ener

gy

dif

fere

nce

(m

Ry/a

tom

)

Al content (x)

fccFM

hcpFM

Figure 3.1 Left panel: Theoretical total energy of the FeCrCoMnAlx (x = 1.0) as a function of

Wigner-Seitz radius for the bcc, fcc and hcp phases at the FM and PM states, respectively. Right

panel: Comparison of the equilibrium total energies for the fcc and hcp phases to that of the bcc

phase at the FM state for x = 0.6, 1.0 and 1.5, respectively.

Motived by such interesting phenomenon, the effect of sp element Ge addition

on the microstructure properties of the FeCrCoNi were investigated [27, 28].

The results indicated that this alloy is decomposed into a mixture of fcc and

bcc duplex phases. Following experimental information, here we consider the

FeCrCo(NiGe)x (0.167 x 3.5) HEAs as a pseudobinary (FeCrCo)1-y(NiGe)y

alloy with y = 2x/(3+2x) and 0.1 y 0.7.

Then, the relative formation energy can be expressed as ∆𝐺𝛼(𝑦) = 𝐺𝛼(𝑦) −

[1 − (10𝑦 − 1) 6⁄ ]𝐺fcc(0.1) − [(10𝑦 − 1) 6⁄ ]𝐺fcc(0.7), where 𝛼 stands for

fcc or bcc, and 𝐺𝛼(𝑦) is the Gibbs free energy per atom for (FeCrCo)1-y(NiGe)y

in the 𝛼 phase. Here we estimate the 𝐺𝛼(𝑦) by the following approximation

relation 𝐺𝛼(𝑦) ≈ 𝐸𝛼(𝑦) − 𝑇𝑆conf(𝑦), where 𝑇 is the temperature, and 𝐸𝛼(𝑦)

is the total energy per atom for (FeCrCo)1-y(NiGe)y in the 𝛼 phase. The mixing

configurational entropy 𝑆conf is calculated by using the mean-field expression

𝑆conf = −𝑘B ∑ 𝑐𝑖 ln 𝑐𝑖𝑖 , where 𝑐𝑖 is the concentration of the ith element and 𝑘B

is the Boltzmann constant. Accordingly, the additional contributions from the

vibrational, magnetic, and electronic parts are neglected.

Figure 3.2 shows the composition dependent ∆𝐺 of the (FeCrCo)1-y(NiGe)y

(0.1 y 0.7) HEAs at room temperature for the fcc and bcc phases at the FM

and PM states, respectively. As can be seen, the fcc and bcc phases at the FM

state arrive at equilibrium around y = 0.34 (x = 0.773) as the Gibbs free energy

difference vanishes. According to the rule of common tangent line, we find

10

that at room temperature the (FeCrCo)1-y(NiGe)y is likely to form bcc phase for

y 0.25 (x 0.5), fcc phase for y 0.42 (x 1.1), and a mixture of fcc and bcc

duplex phases between the above limits.

0.1 0.2 0.3 0.4 0.5 0.6 0.7

-3.0

-1.5

0.0

1.5

3.0

-3.0

-1.5

0.0

1.5

3.0 T = 300 K

x=1.1x=0.5

Gib

bs

free

ener

gy

G (

mR

y)

Gib

bs

free

ener

gy

G (

mR

y)

NiGe content (y)

fccPM

bccPM

fccFM

bccFM

Figure 3.2 Comparison of the Gibbs free energies of the (FeCrCo)1-y(NiGe)y (0.1 y 0.7) HEAs

as a function of NiGe content at room temperature for the fcc and bcc phases at the FM and PM

states, respectively. Here y = 2x/(3+2x), where x is the atomic fraction of NiGe in FeCrCo(NiGe)x.

3.1.2 Temperature effect

Generally, the relative phase stability at ambient pressure and as a function of

temperature can be investigated from the free energies calculated for various

structures. Here, we decompose the free energy as 𝐹 = 𝐸 + 𝐹conf + 𝐹mag +

𝐹vib + 𝐹el, where 𝐸 is the internal energy, and 𝐹conf, 𝐹mag, 𝐹vib and 𝐹el are

the additional temperature-dependent contributions for the configurational,

magnetic, vibrational and electronic free energies, respectively.

The configurational free energy for an ideal solid solution can be estimated

by 𝐹conf = −𝑇𝑆conf. For a disordered paramagnetic state, the magnetic free

energy is expressed as 𝐹mag = −𝑇𝑆mag = −𝑘B𝑇 ∑ 𝑐𝑖 ln(1 + 𝜇𝑖)𝑖 , where 𝜇𝑖 is

the local magnetic moment of the 𝑖th element. The vibrational free energy can

be derived from the Debye-Grüneisen model 𝐹vib = 9𝑘B𝜃D 8⁄ + 3𝑘B𝑇 ln(1 −

𝑒−𝜃D 𝑇⁄ ) − 𝑘B𝑇𝐷(𝜃D 𝑇⁄ ), where 𝜃D is the Debye temperature, and 𝐷 is the

Debye integral [29]. The electronic free energy is defined as 𝐹el = 𝐸el − 𝑇𝑆el,

where 𝐸el and 𝑆el are the electronic energy and entropy, respectively, which

are obtained directly from the EMTO calculations with the finite-temperature

Fermi distribution [30].

11

Figure 3.3 shows the temperature-dependent free energies of the FeCrCoNiGa

for the fcc and bcc phases at the FM and PM states, respectively. For clarity,

all free energies are plotted with respect to the free energy of FM bcc. As can

be see, the fcc and bcc phases at the FM state arrive in equilibrium around

room temperature as the free energy difference vanishes. When comparing all

four free energies, we find that the bcc phase at the FM state is energetically

stable at low temperatures (cryogenic conditions), and the fcc phase at the PM

state becomes favorable at high temperatures.

0 600 1200-5.0

-2.5

0.0

2.5

5.0

0 600 1200-5.0

-2.5

0.0

2.5

5.0

Ener

gy

dif

fere

nce

(m

Ry)

Temperature (K)

fcc

FM

PM

643 K

691 K

Ener

gy

dif

fere

nce

(m

Ry)

bcc

FM

PM

Figure 3.3 Temperature-dependent free energies of the FeCrCoNiGa for the fcc and bcc phases

at the FM and PM states, respectively. All energies are plotted with respect to the corresponding

energy of FM bcc.

3.1.3 Ordering behavior

Previous experimental work indicated that in the Al-doped FeCrCoNi HEAs,

disordered bcc precipitates embedded in ordered B2 matrix appear at high Al

concentrations [31-33].

To investigate this ordering behavior, here we consider two alloy categories

with bcc underlying lattice: (i) the completely disordered phase (type I), which

stands for the reference structure where the alloy components randomly

occupy all sites, and (ii) the B2 ordered phase (type II), where the corner sites

(denoted by A) and the body-center sites (denoted by B) are occupied by

different types of atoms. We treat the equiatomic FeCrCoNiAl as a pseudo-

binary 𝐴0.8𝐵0.2 alloy, where 𝐴 represents the sub-system containing four

selected elements with equal concentration, and 𝐵 denotes the rest of the alloy

components. Then, the degree of chemical order can be controlled by changing

12

the composition at the corner positions as 𝐴0.8+𝑐𝐵0.2−𝑐 and at the body-center

positions as 𝐴0.8−𝑐𝐵0.2+𝑐, represented by (𝐴0.8+𝑐𝐵0.2−𝑐)(𝐴0.8−𝑐𝐵0.2+𝑐).

Accordingly, we change 𝑐 from 0 (i.e., 𝐵 atoms occupy both sites with equal

probability) to 0.2 (i.e., the probability of finding 𝐵 atoms on the corner site is

zero). The degree of chemical disorder can be expressed as 𝜂 = 5𝑐. We should

notice that although 𝑐 = 0 (𝜂 = 0) describes chemically equivalent A and B

sites, the magnetic difference is still allowed and thus this set up accounts for

a possible magnetic transition from B2-type to bcc-type magnetism.

Figure 3.4 (left panel) plots the equilibrium total energy difference between

type I and type II as a function of order parameter 𝜂 for situations when one

particular element is gradually moved from random distribution to sublattice

B. The obtained results reveal that Cr ordering is energetically unfavorable,

whereas Al ordering is always preferred, independent of 𝜂. Namely, Al is more

likely to occupy a particular sublattice rather than being randomly distributed.

Since there are still four alloy components which are randomly distributed,

one can go further and investigate the simultaneous ordering of two or more

elements. Below we consider only the case of Al co-ordering with Al order

parameter equal 1. For example, we move all available Cr atoms from the Al-

sublattice to the other one, while keeping the others randomly distributed. This

case is represented as Cr/Al-FeCoNi.

By studying a series of different kinds of Al co-ordering configurations in

the present work, as shown in Figure 3.4 (right panel), we find that Ni and Co

are prefer to be located on the Al-free site.

0.0 0.5 1.0-4.2

-2.1

0.0

2.1

4.2

6.3

A

B

A

B

En

erg

y d

iffe

ren

ce (

mR

y)

Order parameter ()

Fe

Cr Ni

Co AlII

chemical

transition

0 -10 -20

NiCo/FeAl-Cr

NiCo/CrAl-Fe

Ni/CoAl-FeCr

Ni/CrAl-FeCo

Ni/FeAl-CrCo

NiCo/Al-FeCr

NiCr/Al-FeCo

NiFe/Al-CrCo

Ni/Al-FeCrCo

Co/Al-FeCrNi

Cr/Al-FeCoNi

Fe/Al-CrCoNi

Al

co-o

rder

ing

co

nfi

gu

rati

on

Energy difference (mRy)

Figure 3.4 Equilibrium total energy of the FeCrCoNiAl for type II phase (with respect to type I)

as a function of order parameter for each alloy component in one-atom partition configurations

(left panel), and for a series of different kinds of Al co-ordering configurations (right panel).

13

3.2 Lattice parameter

Table 3.1 shows the calculated equilibrium Wigner-Seitz radius of a large

number of equiatomic medium- and high-entropy alloys with solid solution

phases, along with the corresponding values converted from the experimental

lattice parameters. The majority of selected alloys are based on 3d transition

metals. It is found that the theoretical values are in reasonable agreement with

the available experimental data.

For instance, the calculated equilibrium Wigner-Seitz radius of the FM fcc

FeCoNiCu is 2.641 bohr, which compares well with the experimental value of

2.648 bohr [34]. Alloying with Cr decreases the volume of the FeCoNiCu host,

in line with the experimental observation [1, 34].

Table 3.1. List of equiatomic medium- and high-entropy alloys crystallizing in the fcc and/or bcc

phases, respectively. The calculated equilibrium Wigner-Seitz radius for the fcc and bcc phases at

the FM (PM) state are presented, respectively, along with the available experimental data.

Alloy Phase fcc bcc

Cal. Expt. Cal. Expt. Ref.

AlNiCu fcc + bcc 2.711 2.690 2.710 2.767 [35]

CrFeNi fcc 2.626 (2.619) 2.636 (2.632) [5]

CrCoNi fcc 2.609 (2.604) 2.636 2.622 (2.618) [36]

MnFeNi fcc 2.617 (2.624) 2.646 (2.652) [5]

FeCoNi fcc 2.629 (2.606) 2.658 2.635 (2.624) [37]

AlFeCoNi bcc 2.668 (2.654) 2.669 (2.663) 2.682 [37] AlCoNiCu fcc + bcc 2.674 (2.670) 2.661 2.676 (2.674) 2.737 [35]

VCrFeMo bcc 2.790 (2.789) 2.771 (2.769) [38]

CrFeCoNi fcc 2.619 (2.604) 2.644 2.630 (2.624) [26] MnFeCoNi fcc 2.616 (2.606) 2.657 2.655 (2.640) [39]

FeCoNiCu fcc 2.641 (2.632) 2.648 2.647 (2.642) [34]

AlTiFeNiCu fcc + bcc 2.767 (2.765) 2.762 (2.760) [40] AlCrMnFeCo bcc 2.667 (2.660) 2.670 (2.658) 2.680 [24]

AlCrFeCoNi fcc + bcc 2.661 (2.653) 2.667 2.664 (2.659) 2.683 [26]

AlCrFeNiCu fcc + bcc 2.689 (2.688) 2.686 (2.683) [41] AlCrCoNiCu fcc + bcc 2.671 (2.671) 2.650 2.673 (2.671) 2.726 [35]

AlFeCoNiCu fcc + bcc 2.672 (2.665) 2.670 2.674 (2.670) 2.668 [42]

TiVCrFeMo bcc 2.836 (2.835) 2.799 (2.794) [38] TiCrFeCoNi fcc 2.690 (2.682) 2.691 (2.684) [43]

VMnFeCoNi fcc 2.638 (2.636) 2.664 (2.652) [44]

VFeCoNiCu fcc 2.662 (2.652) 2.666 (2.661) [8] CrMnFeCoNi fcc 2.607 (2.604) 2.659 2.640 (2.630) [45]

CrMnFeCoCu fcc 2.647 (2.630) 2.656 (2.647) [44]

CrMnFeNiCu fcc + bcc 2.664 (2.647) 2.664 (2.655) [41] CrFeCoNiCu fcc 2.638 (2.626) 2.642 2.643 (2.638) [1]

CrFeCoNiPd fcc 2.716 (2.706) 2.694 2.717 (2.723) [46]

CrFeNiCuMo fcc 2.743 (2.740) 2.740 (2.739) [41] MnFeCoNiCu fcc 2.646 (2.643) 2.671 (2.657) [47]

MnFeCoNiGa fcc + bcc 2.676 (2.673) 2.715 2.695 (2.685) 2.654 [39]

MnFeCoNiMo fcc 2.721 (2.717) 2.735 (2.731) [44]

14

FeCoNiCuMo fcc 2.731 (2.726) 2.736 (2.733) [47]

FeCoNiCuAg fcc 2.752 (2.750) 2.757 (2.756) [47] FeCoNiCuPt fcc 2.748 (2.745) 2.757 (2.755) [47]

AlSiFeCoNiCu fcc + bcc 2.673 (2.666) 2.693 2.677 (2.672) 2.638 [42]

AlTiCrFeCoNi fcc + bcc 2.721 (2.716) 2.718 (2.715) [43] AlTiFeCoNiCu fcc + bcc 2.730 (2.724) 2.681 2.729 (2.723) 2.709 [42]

AlCrFeCoNiCu fcc + bcc 2.669 (2.661) 2.669 2.670 (2.665) 2.671 [42]

TiCrMnFeCoNi fcc + bcc 2.686 (2.677) 2.688 2.687 (2.678) [2] VCrMnFeCoNi fcc 2.637 (2.637) 2.644 2.654 (2.645) [2]

CrMnFeCoNiCu fcc 2.652 (2.630) 2.651 2.656 (2.646) [2]

CrMnFeCoNiGe fcc + bcc 2.679 (2.663) 2.644 2.682 (2.676) [2] AlSiCrFeCoNiCu fcc + bcc 2.670 (2.664) 2.647 2.672 (2.668) 2.723 [35]

Table 3.2. Prediction of Wigner-Seitz radius of some equiatomic medium- and high-entropy alloys

for the bcc phase at the FM (PM) state. The “average” values obtained according to Vegard’s rule

by using the corresponding experimental data of the alloy components.

Alloy EMTO-CPA average Alloy EMTO-CPA average

AlVCr 2.775 2.829 AlVCrMn 2.717 2.797 AlVMn 2.742 2.835 AlVCrFe 2.723 (2.716) 2.789

AlVFe 2.756 (2.747) 2.824 AlVMnFe 2.708 (2.701) 2.793

AlCrMn 2.690 2.792 AlVNbMo 2.947 2.951 AlCrFe 2.704 (2.699) 2.781 AlVNbW 2.959 2.953

TiVNb 2.972 2.979 AlCrMnFe 2.672 (2.667) 2.760

TiVMo 2.909 2.931 TiVCrMn 2.741 2.812 TiCrMo 2.866 2.888 TiVCrMo 2.848 2.870

TiNbMo 3.003 3.017 TiVNbMo 2.956 2.966

VCrMn 2.668 2.732 VCrMnFe 2.660 (2.656) 2.716 VCrFe 2.678 (2.672) 2.721 VCrNbMo 2.881 2.874

VMnFe 2.666 (2.658) 2.727 AlTiVCrMn 2.776 2.848

VNbMo 2.943 2.937 AlVCrMnFe 2.692 (2.688) 2.771 AlTiVMn 2.813 2.889 AlVCrNbW 2.905 2.899

AlTiVFe 2.821 (2.811) 2.881 AlVCrNbMoW 2.911 2.904

AlTiCrMn 2.773 2.857

Table 3.2 lists some promising candidate alloy compositions, which are

suggested to adopt a single bcc solid solution phase at 873 K [48]. The Wigner-

Seitz radius of the AlVCrNbMoW, for example, is 2.904 bohr as determined

from Vegard’s rule by using the corresponding experimental data of the alloy

components [20, 49]. This estimated value is close to the EMTO-CPA result

of 2.911 bohr. From the detailed comparison of the two sets of data, we find

that there is a general good agreement between these theoretical predictions.

15

Chapter 4

Deformation performance

There exists a shape change in material when a sufficient load is applied, which

is called deformation. This chapter briefly presents two types of deformation:

i, elastic deformation, a temporary shape change that is self-reversing after the

load is removed; ii, plastic deformation, a permanent shape change that without

fracture under a sufficient load.

4.1 Elasticity

The elastic properties govern the stress-strain relation before yielding, and they

are sensitive to various internal and external conditions, e.g., ordering, pressure,

temperature. On an atomic scale, macroscopic elastic strain is manifested as

subtle changes in the interatomic spacing.

4.1.1 Single-crystal elastic constants

For a cubic lattice, the elastic properties can be fully characterized by three

independent elastic constants: 𝐶11, 𝐶12, and 𝐶44. Here, the bulk modulus 𝐵 =(𝐶11 + 2𝐶12) 3⁄ is derived from the equation of state fitted to the ab initio total

energies for a series of different volumes. The tetragonal shear modulus 𝐶′ =(𝐶11 − 𝐶12) 2⁄ and the cubic elastic modulus 𝐶44 are derived from the volume-

conserving orthorhombic and monoclinic deformations, respectively [20].

Experiments indicated that the FeCoNiCu and FeCoNiCuX (X = V, Cr, Mn)

HEAs adopt a single fcc solid solution phase [1, 8, 34, 50-53]. The present ab

initio data confirm the stability of the fcc structure relative to the bcc and hcp

16

structures [54]. Table 4.1 lists the calculated single-crystal elastic constants of

the FeCoNiCu and FeCoNiCuX (X = V, Cr, Mn) HEAs at the FM and PM

states, respectively. Usually, the mechanical stability requirement in a cubic

lattice leads to the following restrictions [55]: 𝐶11 > 0, 𝐶44 > 0, 𝐶11 − 𝐶12 >

0 and 𝐶11 + 2𝐶12 > 0. As shown in Table 4.1, the considered alloys all satisfy

the above criteria.

Table 4.1. Theoretical single-crystal elastic constants (𝐶11, 𝐶12, 𝐶44, and 𝐶′ = (𝐶11 − 𝐶12) 2⁄ , in

units of GPa) of the FeCoNiCu and FeCoNiCuX (X = V, Cr, Mn) HEAs for the fcc phase at the

FM and PM states, respectively.

Alloy 𝑪𝟏𝟏 𝑪𝟏𝟐 𝑪𝟒𝟒 𝑪′

FeCoNiCu FM 211.9 157.8 126.9 27.0

PM 213.1 147.7 138.2 32.7

FeCoNiCuV FM 196.3 158.2 124.1 19.1 PM 207.2 158.0 132.8 24.6

FeCoNiCuCr FM 209.2 149.6 142.2 29.8

PM 223.7 155.3 151.3 34.2 FeCoNiCuMn FM 189.8 123.8 139.2 33.0

PM 187.1 122.8 139.3 32.2

4.1.2 Polycrystalline elastic moduli

In general, the bulk modulus 𝐵 represents the resistance to volume change by

applied pressure, and the shear modulus 𝐺 indicates the resistance to reversible

deformations upon shear stress. The Young’s modulus 𝑌 is defined as the ratio

of the tensile stress to the corresponding tensile strain [56]. The 𝐵/𝐺 ratio has

often been used to describe the ductile/brittle behavior of materials. According

to the Pugh criterion [57], a high 𝐵/𝐺 ratio indicates a tendency for ductility,

while a small one for brittleness with the critical value around 1.75.

Here, 𝐺 is estimated from the arithmetic Hill average 𝐺 = (𝐺V + 𝐺R) 2⁄ ,

where 𝐺V and 𝐺R are the Voigt and Reuss bounds, respectively [58]. The

Young’s modulus 𝑌 and Poisson ratio 𝜐 are connected to 𝐵 and 𝐺 by the

relations 𝑌 = 9𝐵𝐺 (3𝐵 + 𝐺)⁄ and 𝜐 = (3𝐵 − 2𝐺) (6𝐵 + 2𝐺)⁄ , respectively.

Figure 4.1 shows the theoretical polycrystalline elastic moduli (𝐵, 𝐺, and 𝑌)

of the FeCoNiCu and FeCoNiCuX (X = V, Cr, Mn) HEAs at the FM and PM

states, respectively. The elastic moduli characterized by using the respective

properties of the pure elements are plotted for comparison [49]. As can be seen,

the simple rule of mixture of the selected property of HEAs can provide useful

first-level insight of the general trend as compared to the present ab initio data.

17

0 100 2000

50

100

150

200

250

0 100 200 0 100 2000

50

100

150

200

250

B

Ela

stic

mod

uli

(G

Pa)

Y

FM PM

FeCoNiCu

X = V

X = Cr

X = Mn

G

Average elastic moduli (GPa)

Ela

stic

mod

uli

(G

Pa)

Figure 4.1 Theoretical polycrystalline elastic moduli (𝐵 , 𝐺 , and 𝑌 ) of the FeCoNiCu and

FeCoNiCuX (X = V, Cr, Mn) HEAs for the fcc phase at the FM and PM states, respectively. The

“average” values obtained according to Vegard’s rule by using the corresponding experimental

data of the alloy components.

4.2 Plasticity

The activations of plastic deformation mechanism play an important role in

determining the mechanical performance of crystalline materials.

According to phenomenological model, the competition between the leading

deformation mechanisms in fcc metals, i.e., stacking fault (SF), twinning (TW),

and full-slip (SL), is related to the size of stacking fault energy (SFE) [59]. For

example, austenitic steels with SFE higher than 10-16 mJ/m2 are suggested to

show twinning [60-62], whereas alloys with SFE below this critical value

should exhibit deformation-induced phase transformation.

In the following, we briefly introduce an alternative approach [63] based on

the effective energy barriers (EEBs) in combination with the intrinsic material

parameters that obtained from the generalized stacking fault energy (𝛾-surface).

4.2.1 Generalized stacking fault energy

In fcc metals, the active deformation mode is decided by the competitions

between corresponding EEBs [63]: 𝛾SF

(𝜃) = 𝛾usf/ cos 𝜃 , 𝛾TW

(𝜃) = (𝛾utf −

18

𝛾isf)/ cos 𝜃, and 𝛾SL

(𝜃) = (𝛾usf − 𝛾isf)/ cos(60° − 𝜃) for SF, TW, and SL,

respectively. Here, 𝜃 reflects the effect of grain orientation relative to the

resolved shear direction, 𝛾isf is the intrinsic staking fault energy, 𝛾usf is the

unstable stacking fault energy, and 𝛾utf is the unstable twin fault energy.

Twinning is possible for 𝛾TW

≤ 𝛾SF

, otherwise martensitic transformation

occurs. Particularly, SF and SL or TW and SL can co-exist in material, whereas

SF and TW are exclusive to each other independently of 𝜃.

Figure 4.2 shows the room-temperature 𝛾-surfaces of the Fe40Mn40Co10Cr10,

CrMnFeCoNi, and CrCoNi alloys. As can be seen, all three alloys have 𝛾usf ≈

𝛾utf with small (or negative) 𝛾isf . This particular shape of 𝛾-surface differs

from that of the conventional fcc alloys [63, 64], and has important implication

on the deformation mechanisms.

Here, the 𝛾-surface is obtained from alias shear, i.e., by shifting the upper

part of the fcc {111} layers along the ⟨112̅⟩ direction. In practice, this process

requires a critical resolved shear stress, which raises a small but finite affine

shear, i.e., all atomic planes are actually elastically strained along the shear

direction [65]. Obviously, increasing affine shear strain increases (𝛾SF

− 𝛾TW

),

and thus favors twin nucleation for all considered alloys.

0.0 0.5 1.0 1.5 2.0

0

200

400

esfisffcc

isf

utf

usf

(

mJ/

m2)

Fault pathway (b)

Fe40

Mn40

Co10

Cr10

CrMnFeCoNi CrCoNi

0.0 0.8 1.6

alias

Strain (%)

affine

0

20

40

SF -

T

W (m

J/m2)

Figure 4.2 Generalized stacking fault energy curves of the Fe40Mn40Co10Cr10, CrMnFeCoNi, and

CrCoNi alloys at room-temperature. b =1

6⟨112⟩ is the Burgers vector of the partial dislocation

(in units of fcc lattice parameter). 𝛾isf, intrinsic stacking fault energy; 𝛾usf, unstable stacking fault

energy; 𝛾utf, unstable twining fault energy. Right panel shows (𝛾SF

− 𝛾TW

) as a function of strain.

19

Chapter 5

Magnetic behavior

Magnetism is one of the fundamental features of matter in solid state physics.

In general, the magnetization decreases with increasing temperature until the

long-range magnetic order disappears at Curie temperature (𝑇C). The bcc Fe,

for example, is characterized by the long-range ferromagnetic order below 𝑇C

1043 K. Notice that HEAs could be exciting candidate materials for magnetic

refrigeration applications based on the recent experimental work. This chapter

briefly presents some fundamental magnetic characteristics of HEAs.

5.1 Magnetic moment

Table 5.1 summarizes the calculated total and partial magnetic moments of the

FeCrCoNiAlx (0 x 2) HEAs at corresponding equilibrium volumes. Here,

the fcc phase is considered for x 1 and the bcc phase for x 0.5 based on the

microstructure observations [25, 26] and previous theoretical predictions [66].

Table 5.1. Total and partial magnetic moments (units of B per atom) of the FeCrCoNiAlx HEAs

as a function of Al fraction x for the fcc and bcc phases at the FM state, respectively.

fcc bcc

x = 0 x = 0.5 x = 1 x = 0.5 x = 1 x = 1.5 x = 2

Fe 1.96 1.97 1.98 2.23 2.17 2.13 2.08

Cr -0.72 -0.64 -0.60 -0.09 -0.07 -0.05 -0.03

Co 1.12 1.04 0.97 1.44 1.34 1.26 1.19 Ni 0.30 0.23 0.19 0.31 0.27 0.24 0.21

Al - -0.05 -0.04 -0.03 -0.03 -0.03 -0.03

Total 0.66 0.57 0.50 0.86 0.74 0.64 0.57

20

The changes of magnetic moments depend on the Al concentration and crystal

structure. For instance, the magnetic moment per Co atom decreases from 1.12

B at x = 0 to 0.97 B at x = 1 in the fcc phase, and in the bcc phase it varies

from 1.44 B to 1.19 B when x changes from 0.5 to 2. The total magnetic

moment per atom of the FeCrCoNiAl in the bcc phase is 0.74 B, which is 48 %

larger than the one in the fcc phase. Irrespective of the structure, Al addition is

found to decrease the total magnetic moments, which is consistent with the fact

that the Al is a non-magnetic metal.

5.2 Curie temperature

The Heisenberg model is a statistical mechanical model, which is widely used

to study the critical point and phase transition. Here, the effective Heisenberg-

like Hamiltonian is expressed as 𝐻 = − ∑ 𝐽𝑖𝑗𝐦𝑖 ∙ 𝐦𝑗𝑖,𝑗 , where Jij denotes the

strength of magnetic exchange interaction between atomic sites i and j with

magnetic moments mi and mj. Several approaches, such as Monte-Carlo (MC)

simulation and mean-field approximation (MFA), can be applied to evaluate

𝑇C of magnetic system within the Heisenberg Hamiltonian.

5.2.1 Monte-Carlo simulation

Figure 5.1 plots the magnetic exchange interactions between alloy components

of the equiatomic FeCrCoNiAl for the fcc and bcc phases, respectively. Here,

a reduced exchange interaction parameter 𝐽𝑖𝑗′ = 𝑧𝑝𝐽𝑖𝑗𝒎𝑖𝒎𝑗 , where zp is the

coordination number of the pth coordination shell, is employed to illustrate the

crystal structural effect.

As can be seen, the interactions show long-range oscillatory behavior, e.g.,

the Fe-Fe interaction is predominantly ferromagnetic but may have anti-

ferromagnetic contribution depending on the distance between the Fe atoms.

The ferromagnetic interactions in the fcc phase are mainly from the nearest-

neighbor Fe-Fe, Fe-Co, and Co-Co pairs, and anti-ferromagnetic interactions

from the nearest-neighbor Fe-Cr and Cr-Cr pairs. Notice that Cr is anti-parallel

with Fe/Co/Ni as shown in Table 5.1, hence, the positive interactions between

Cr and Fe/Co/Ni indicate an anti-ferromagnetic coupling. In the case of the bcc

phase, the dominating ferromagnetic interactions (Fe-Fe, Fe-Co and Co-Co

pairs at the nearest-neighbor shell) are much stronger than those in the fcc

phase. These interesting features demonstrate that the crystal structure has a

strong impact on the ferromagnetic behavior of the FeCrCoNiAl.

21

0 5 10 15 20-5.5

0.0

5.5

11.0

16.5

0 5 10 15 20-5.5

0.0

5.5

11.0

16.5

bcc

Fe-Fe

Fe-Cr

Fe-Co

Fe-Ni

Fe-Al

Cr-Cr

Cr-Co

Mag

net

ic e

xch

ange

inte

ract

ion

(m

Ry

)

pth coordination shell

fcc

Mag

enti

c ex

chan

ge

inte

ract

ion

(m

Ry

)

Cr-Ni

Cr-Al

Co-Co

Co-Ni

Co-Al

Ni-Ni

Ni-Al

Al-Al

Figure 5.1 Magnetic exchange interactions of the FeCrCoNiAl as a function of coordination shell

for the fcc and bcc phases, respectively.

Starting from the computed magnetic exchange interactions, here we perform

MC simulation implemented in the UppASD program [67] to estimate 𝑇C of

the FeCrCoNiAl for the fcc and bcc phases, respectively.

The MC simulation boxes contained up to 108000 atoms (subject to periodic

boundary conditions) for the fcc underlying lattice and up to 128000 atoms for

the bcc one. Random distributions of alloy components in the supercells were

generated, and 20000 MC steps have been used for equilibration, then followed

by 20000 steps for obtaining thermodynamic averages.

Figure 5.2 (a) shows the normalized magnetization (M/M0, with M and M0

being the magnetization at T and 0 K, respectively) of the FeCrCoNiAl as a

function of temperature for the fcc and bcc phases, respectively. The crystal

structure effect is clearly observed, i.e., the magnetic transition temperature in

the bcc phase is much higher than that in the fcc phase.

Figure 5.2 (b) shows the magnetic susceptibility (𝜒 = [⟨𝑀2⟩ − ⟨𝑀⟩2]/𝑘𝐵𝑇)

of the FeCrCoNiAl as a function of temperature for the fcc and bcc phases,

respectively. Results obtained from MC simulation with varying system size

are presented for comparison. Notice that 𝜒 diverges at the critical temperature

in the thermodynamic limit at the absence of external magnetic field. From the

robust susceptibility peak, we find that 𝑇C of the FeCrCoNiAl is 205 5 K in

the fcc phase, and 355 5 K in the bcc phase.

22

0 100 200 300 400 500 600

0.00

0.25

0.50

0.75

1.00

0.00

0.25

0.50

0.75

1.00

(a)

No

rmal

ized

mag

net

izat

ion

Norm

aliz

ed m

agnet

izat

ion

Temperature (K)

fcc

bcc

120 160 200 240 280

0.00

0.05

0.10

0.15

0.20

280 320 360 400 440

0.00

0.05

0.10

0.15

0.20

bcc

N = 32000

N = 62500

N = 108000

Mag

net

ic s

usc

epti

bil

ity

(a.

u.)

Temperature (K)

fcc(b)

Mag

net

ic s

usc

epti

bil

ity

(a.

u.)

N = 54000

N = 85750

N = 128000

Figure 5.2 Temperature dependent (a) normalized magnetization and (b) magnetic susceptibility

of the FeCrCoNiAl for the fcc and bcc phases, respectively. The magnetic susceptibilities are

shown for different simulation boxes (number of atoms N is indicated in the legend).

By applying the above procedure, in Fig. 5.3 we present the concentration

dependent 𝑇C for the fcc and bcc phases, respectively, along with the available

experimental values [68]. Clearly, 𝑇C of the considered HEAs depends on the

chemical composition, and especially on the crystal structure. For the alloys

with a single fcc or bcc phase, as well as a mixture of fcc and bcc duplex phases

(assuming equal phase fractions), 𝑇C decreases with increasing Al content,

which is in line with the experimental observation.

23

0.0 0.5 1.0 1.5 2.0

0

100

200

300

400

500

0

100

200

300

400

500

0 200 400 600

0.0

0.5

1.0

bccfcc + bccfcc

Curi

e te

mper

ature

(K

)

Curi

e te

mper

ature

(K

)

Al content (x)

fcc

bcc

Ref. a

Temperature (K)

Normalized

Magnetization

Figure 5.3 Theoretical Curie temperature of the FeCrCoNiAlx HEAs as a function of Al content

for the fcc and bcc phases, respectively. The values from Ref. a [68] are presented for comparison.

Experiments indicated that the equiatomic FeCrCoNi adopts a single fcc solid

solution phase, and has 𝑇C far below the room temperature. Combining the

previous results on the relative phase stability, here we may conclude that at

room temperature, the appearance of the bcc phase (as Al is gradually added)

causes the magnetic state change from paramagnetic to ferromagnetic.

5.2.2 Mean-field approximation

Alternatively, with the help of MFA, one can evaluate 𝑇C from the ab initio

energies, i.e., 3𝑘B𝑇C = 2(𝐸PM − 𝐸FM) (1 − 𝑐)⁄ , where 𝑐 is the concentration

of nonmagnetic alloy components, and 𝐸PM and 𝐸FM are the equilibrium total

energies per atom at the PM and FM states, respectively.

Considering the documented success and high efficiency, here we carry out

the 𝑇C mapping of a large number of HEAs by using the above approach. Table

5.2 summarizes the calculated 𝑇C of the selected equiatomic medium- and

high-entropy alloys. Results for the fcc and bcc phases are given to bring to

light the related properties as a function of crystal structure. The detailed phase

information can be found in Table 3.1. Here, the available experimental and

theoretical data are listed for comparison.

24

Table 5.2. List of equiatomic medium- and high-entropy alloys crystallizing in the fcc and/or bcc

phases, respectively. The calculated Curie temperature (K) for the fcc and bcc phases are presented,

respectively, along with the available experimental and other theoretical data.

Alloy fcc bcc Expt./cal. Ref. Phase

AlNiCu - - fcc + bcc

CrFeNi 80 225 fcc CrCoNi 25 120 fcc

MnFeNi 43 71 fcc

FeCoNi 804 1147 995 [69] fcc AlFeCoNi 497 763 bcc

AlCoNiCu 197 245 fcc + bcc

VCrFeMo 21 251 bcc CrFeCoNi 155 414 120-130 [68, 70] fcc

MnFeCoNi 166 580 fcc

FeCoNiCu 796 987 826 [71] fcc AlTiFeNiCu 213 286 fcc + bcc

AlCrMnFeCo 32 463 bcc

AlCrFeCoNi 136 334 fcc + bcc AlCrFeNiCu 124 159 fcc + bcc

AlCrCoNiCu 25 70 fcc + bcc AlFeCoNiCu 493 663 fcc + bcc

TiVCrFeMo 31 97 bcc

TiCrFeCoNi 95 339 fcc VMnFeCoNi 27 470 fcc

VFeCoNiCu 246 404 fcc

CrMnFeCoNi 27 397 20 [72] fcc CrMnFeCoCu 59 477 fcc

CrMnFeNiCu 60 191 fcc + bcc

CrFeCoNiCu 251 382 172 [51] fcc

CrFeCoNiPd 440 414 440 [46] fcc

CrFeNiCuMo 102 144 fcc

MnFeCoNiCu 146 540 400 [47] fcc MnFeCoNiGa 77 567 fcc + bcc

MnFeCoNiMo 68 540 fcc

FeCoNiCuMo 328 508 657 [47] fcc FeCoNiCuAg 805 881 fcc

FeCoNiCuPt 837 879 864 [47] fcc

AlSiFeCoNiCu 281 438 fcc + bcc AlTiCrFeCoNi 93 264 fcc + bcc

AlTiFeCoNiCu 274 438 fcc + bcc

AlCrFeCoNiCu 176 300 fcc + bcc TiCrMnFeCoNi 37 380 fcc + bcc

VCrMnFeCoNi 10 288 fcc

CrMnFeCoNiCu 124 371 fcc CrMnFeCoNiGe 97 346 fcc + bcc

AlSiCrFeCoNiCu 104 213 fcc + bcc

The calculated 𝑇C of the FeCoNiCu is 796 K, compares well with the previous

theoretical value of 826 K [71]. A significant decrease in 𝑇C is revealed when

adding equimolar V, Cr, and Mn to the FeCoNiCu host. One possible reason

for this decrease is the appearance of antiferromagnetic coupling between the

25

alloying elements (V, Cr, and Mn) and the Fe-Co-Ni matrix, as demonstrated

in previous work focused on the FeCrCoNi-based HEAs [27, 73-75].

Furthermore, according to MC simulation, 𝑇C of the CrMnFeCoNi is 10

10 K in the fcc phase, and 340 10 K in the bcc phase. We recall that 𝑇C

obtained with the help of MFA in the fcc and bcc phases are 27 K and 397 K,

respectively. The good agreement between these two approaches gives support

for the 𝑇C maps as shown in Table 5.2.

For most of the alloys considered here, 𝑇C is predicted to be larger in the bcc

phase than in the fcc phase, as shown in Figure 5.4. Here, we would like to

highlight a particularly experimental fact which gains theoretical support in the

mirror of the present ab initio data.

Experiments indicated that the equiatomic CrMnFeCoNi adopts a single fcc

solid solution phase, and in a paramagnetic state down to a temperature of 93

K (i.e., the magnetic transition temperature should be below 93 K) [76]. During

mechanical alloying and spark plasma sintering process, a minority bcc phase

formed in the fcc solid solution matrix. Measuring the magnetic hysteresis

curves at room temperature indicated that the as-milled powder possesses a

characteristic feature of soft magnet [77], which is fully supported by the

presently predicted TC in the bcc phase.

0 600 1200

0

300

600

900

1200

0

100

200

300

400

Curie temperature (K)

bcc

Curi

e te

mper

atu

re (

K)

Crystal structure

MFA

MC

CrMnFeCoNi

fcc

In t

he

bcc

ph

ase

In the fcc phase

Figure 5.4 Left panel: Comparison of Curie temperature obtained via MFA and MC approaches

of the CrMnFeCoNi for the fcc and bcc phases, respectively. Right panel: Curie temperature

obtained via MFA approach of the selected systems listed in Table 5.2 for the fcc and bcc phases.

26

Chapter 6

Thermo-physical characteristics

Thermo-physical performance can be simply viewed as material properties that

vary with the state variables, e.g., pressure and temperature, without altering

the chemical identity. Notice that some relevant parameters which today are

directly accessible from the ab initio calculations. This chapter briefly presents

some fundamental thermo-physical characteristics of HEAs.

6.1 Debye temperature

The Debye temperature 𝜃𝐷 is an important parameter that correlates with many

thermal characteristics of materials. It can be derived from elastic modulus,

electrical resistivity, thermal expansion, specific heat measurement, melting

point, X-ray and neutron diffraction intensity data, and so on. Since 𝜃𝐷 vary

with temperature, one should be careful when comparing them that obtained

by different methods.

Here, we calculate 𝜃𝐷 based on the average sound velocity 𝑣𝑚, viz., 𝜃D =

(ℏ 𝑘B⁄ )(6𝜋2 𝑉⁄ )1/3𝑣m with 𝑣m = [(1 𝑣L3⁄ + 2 𝑣T

3⁄ ) 3⁄ ]−1/3 , where 𝑉 is the

atomic volume, 𝑣L and 𝑣T are the longitudinal and transverse sound velocities,

respectively. The sound velocities are related to the bulk modulus 𝐵, shear

modulus 𝐺, and density 𝜌, viz., 𝑣L = √(𝐵 + 4𝐺 3⁄ ) 𝜌⁄ and 𝑣T = √𝐺 𝜌⁄ .

Figure 6.1 shows the calculated 𝑣L, 𝑣T, 𝑣m, and 𝜃D of the FeCrCoNiGa for

the fcc and bcc phases at the FM and PM states, respectively. As can be seen,

𝑣L, 𝑣T, and 𝑣m are about 5.37-5.71 km/s, 2.69-2.84 km/s, and 3.02-3.19 km/s,

respectively, which compare well with the corresponding experimental values

of 5.22 km/s, 2.82 km/s, and 3.15 km/s. In the case of 𝜃D, we get the values of

27

394 K (417 K) and 416 K (396 K) for the fcc and bcc phases at the FM (PM)

state, respectively. These values are close to 410 K as estimated in experiment.

0.0

2.5

5.0

7.5

So

un

d v

elo

city

(km

/s)

vL

vT

vm

0

200

400

600

fccFM

fccPM

bccFM

bccPM

Expt.

Deb

ye

tem

per

atu

re (

K)

Figure 6.1 Sound velocity (left panel) and Debye temperature (right panel) of the FeCrCoNiGa

for the fcc and bcc phases at the FM and PM states, respectively. The experimental results are

shown by the shaded bars. The experimental error bars for the sound velocities are ~2.5%.

Note: the present experimental data are from my co-authors as list in preface.

6.2 Thermal expansion

The Debye-Grüneisen model [29] is applied in the present work to estimate the

thermal expansion behavior of HEAs. The Grüneisen parameter 𝛾 describes

the anharmonic effects and gives the volume dependent Debye temperature by

𝜃D(𝑉) = 𝜃D(𝑉0)(𝑉0 𝑉⁄ )𝛾. Notice that 𝛾 can be expressed as 𝛾 = −𝑓 + 𝐵′ 2⁄ ,

where 𝐵′ is the pressure derivative of the bulk modulus. For pure metals, the

factor 𝑓 = 1/2 provides a good agreement with the experimental data [20, 29].

The obtained total energy as a function of volume in the static approximation

is carried out to determine the structural parameters at ground state, and then

to derive the macroscopic properties as a function of temperature.

Figure 6.2 presents the calculated lattice parameter 𝐿 and linear thermal

expansion coefficient 𝛼 of the FeCrCoNiGa in the temperature range of 0 −

900 K. The values of 𝐿 at 300 K are 3.637 (3.627) Å and 2.885 (2.884) Å for

the fcc and bcc phases at the FM (PM) state, respectively, and they increase by

1.11 (1.33) % and 0.68 (1.08) % when the temperature reaches 900 K. It is

evident that the temperature gives a larger influence in the fcc phase than in

28

the bcc phase. In addition, the calculated 𝛼 for all considered phases increase

rapidly at low temperatures and gradually turn towards a linear trend at high

temperatures. For the considered highest temperature region, the propensity of

increment becomes moderate, especially for the FM bcc.

0 300 600 9003.60

3.63

3.66

3.69

0 300 600 9002.85

2.88

2.91

2.94

Temperature (K)

L i

n t

he

fcc

ph

ase

(Å)

fcc

FM

PM

(a)

L i

n t

he

bcc

ph

ase

(Å)

bcc

FM

PM

0 300 600 9000.0

6.5

13.0

19.5

26.0

0.0

6.5

13.0

19.5

26.0(b)

Ther

mal

expan

sio

n

(

10

-6 K

-1)

Ther

mal

expan

sio

n

(

10

-6 K

-1)

Temperature (K)

fccFM

bccFM

fccPM

bccPM

Expt.

Figure 6.2 Temperature dependent (a) lattice parameter 𝐿 and (b) linear thermal expansion

coefficient 𝛼 of the FeCrCoNiGa for the fcc and bcc phases at the FM and PM states, respectively.

The experimental results are depicted by circles with error bars.

Note: the present experimental data are from my co-authors as list in preface.

We recall that according to experiments, this alloy possesses a mixture of fcc

and bcc duplex phases [28]. Hence, the actual mean 𝛼 should be estimated by

averaging over the individual phases. For example, assuming equal fractions

for the two phases around room temperature, as indicated from Fig. 3.3, we get

29

the thermal expansion coefficient 𝛼300K ≈ 13.3×10-6 K-1, which compares well

with the experimental data shown in Fig. 6.2. On the other hand, this value is

close to that of several different HEAs, e.g., ~ 14 ×10-6 K-1 for the FeCoNiCr

and ~ 15 ×10-6 K-1 for the FeCoNiCrMn [45, 69].

Interestingly, the measured 𝛼 increases sharply around Curie temperature.

Theory predicts 𝛼 values which increase in the order of FM bcc, PM bcc, FM

fcc and PM fcc in the entire temperature range. As a consequence, a mixture

of fcc and bcc duplex phases of the FeCrCoNiGa at the FM state results in a

small 𝛼 derived as the mean value below the critical temperature. Furthermore,

the measured 𝛼 is close to the theoretical value obtained for the FM bcc phase

at cryogenic conditions and to that for the PM fcc phase at high temperatures.

Hence, the sizable difference of the calculated 𝛼 values between the FM and

PM states can explain the observed anomalous thermal expansion behavior.

30

Chapter 7

Summary and outlook

The structural, deformation, magnetic, and thermos-physical characteristics of

different-types of HEAs are investigated in the present work by using first-

principle theory in combination with a series of phenomenological models.

The present calculations confirm and predict the relative phase stability of

several HEAs, e.g., a mixture of fcc and bcc duplex phases of the FeCrCoNiGa

at room temperature. The single-crystal elastic constants obtained from the

equation of state and the volume-conserving orthorhombic and monoclinic

deformations are presented and discussed. The elastic moduli characterized by

using the respective properties of the pure elements are shown for comparison.

An atomic-level transparent approach based on the effective energy barriers

are adopted to shed light on the origin of twinning in metastable fcc HEAs.

Analysis of magnetic characteristics of the FeCrCoNiAlx through Monte-Carlo

simulation indicates that the alloy’s magnetic state is controlled by the stability

of the Al-induced single- and/or dual-phase. The fact that certain HEAs have

drastically different ordering temperature for the fcc and bcc phases offers a

way to identify alloys with a very sharp change in the magnetic properties close

to a structural transition. The combined experimental and theoretical results

indicate that the FeCrCoNiGa has an anomalous thermal expansion behavior.

The results show that engineering the structural and/or magnetic transition,

offers rich opportunities for optimizing and designing HEAs with interesting

properties. The revealed strong coupling between the two degrees of freedom

brings the reported/predicted HEAs into the focus of technological including

magneto-caloric applications.

I plan to continue the research by involving multi-scale materials modeling

tools, e.g., hybrid Monte-Carlo/molecular dynamics simulations.

31

Acknowledgement

Thanks to my supervisor Prof. Levente Vitos for his continuous and invaluable

support, and professional guidance during my PhD work.

Thanks to my co-supervisor Dr. Erik Holmström, and Prof. Nanxian Chen, for

their ongoing encouragement and guidance.

Thanks to our group members: Andreas, Dávid, Dongyoo, Fuyang, Guijiang,

Guisheng, He, Henrik, Liyun, Raquel, Ruihuan, Ruiwen, Song, Stephan, Wei,

Wenyue, Xiaojie, Xiaoqing, Xun, Zhihua, and Zongwei for helpful discussion

and collaboration.

Thanks to Ádám Vida, Anita Heczel, Béla Varga, Chuanhui Zhang, Jiang Shen,

Krisztina Kádas, Lajos Károly Varga, Lars Bergqvist, Olle Eriksson, and Se

Kyun Kwon for helpful discussion and collaboration.

The China Scholarship Council, the Swedish Research Council, the Swedish

Foundation for Strategic Research, the Swedish Foundation for International

Cooperation in Research and Higher Education, the Carl Tryggers Foundation,

the Sweden’s Innovation Agency, the Swedish Energy Agency, the National

973 Project of China, the National Natural Science Foundation of China, and

the Hungarian Scientific Research Fund are acknowledged for financial

support. The Swedish National Infrastructure for Computing at the National

Supercomputer Centers in Linköping and Stockholm are acknowledged for

computational facilities.

Last but not the least, I would like to thank my family, relatives, and friends

for their continuous support and encouragement.

32

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