Ab-initio statistical mechanics for ordering and segregation at the (110) surface of Ni90%-Al [Elektronische Ressource] / vorgelegt von Ralf Drautz
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Ab-initio statistical mechanics for ordering and segregation at the (110) surface of Ni90%-Al [Elektronische Ressource] / vorgelegt von Ralf Drautz

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191 Pages
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Ab-initio Statistical Mechanicsfor Ordering and Segregationat the (110) Surface of Ni90%-AlRalf Drautz– Typeset by FoilT X – 1EAb-initio Statistical Mechanicsfor Ordering and Segregationat the (110) Surface of Ni90%-AlVon der Fakultat Physik der Universitat Stuttgart zur Erlangung der Wurde eines Doktors der Naturwissenschaften(Dr. rer. nat.) genehmigte Abhandlungvorgelegt vonRalf Drautzgeboren in Bad CannstattHauptberichter: Prof. Dr. M. FahnleMitberichter: Prof. Dr. G. WunnerTag der Einreichung: 27. Februar 2003Tag der mundlichen Prufung: 2. Mai 2003 Max-Planck-Institut fur MetallforschungStuttgart2003Contents1 Introduction 12 Experimental ndings 32.1 Introduction to x-ray di raction . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Surface x-ray di raction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.1 Geometrical considerations . . . . . . . . . . . . . . . . . . . . . . . 62.2.2 Information on the rods . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Results for Ni-10%Al(110) . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.1 Bulk phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.2 Behavior of surfaces of the Ni-rich NiAl-alloy . . . . . . . . . . . . . 103 Electron theory 153.1 Adiabatic approximation for the nuclei . . . . . . . . . . . . . . . . . . . . 153.1.1 Born-Oppenheimer approximation . . . . . . . . . . . . . . . . . . 163.2 Density functional theory . . . . .

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Ab-initio Statistical Mechanics
for Ordering and Segregation
at the (110) Surface of Ni90%-Al
Ralf Drautz
– Typeset by FoilT X – 1EAb-initio Statistical Mechanics
for Ordering and Segregation
at the (110) Surface of Ni90%-Al
Von der Fakultat Physik der Universitat Stuttgart zur
Erlangung der Wurde eines Doktors der Naturwissenschaften
(Dr. rer. nat.) genehmigte Abhandlung
vorgelegt von
Ralf Drautz
geboren in Bad Cannstatt
Hauptberichter: Prof. Dr. M. Fahnle
Mitberichter: Prof. Dr. G. Wunner
Tag der Einreichung: 27. Februar 2003
Tag der mundlichen Prufung: 2. Mai 2003
Max-Planck-Institut fur Metallforschung
Stuttgart
2003Contents
1 Introduction 1
2 Experimental ndings 3
2.1 Introduction to x-ray di raction . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Surface x-ray di raction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1 Geometrical considerations . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.2 Information on the rods . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Results for Ni-10%Al(110) . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.1 Bulk phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.2 Behavior of surfaces of the Ni-rich NiAl-alloy . . . . . . . . . . . . . 10
3 Electron theory 15
3.1 Adiabatic approximation for the nuclei . . . . . . . . . . . . . . . . . . . . 15
3.1.1 Born-Oppenheimer approximation . . . . . . . . . . . . . . . . . . 16
3.2 Density functional theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 Kohn-Sham equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.4 Local density approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.5 Implementation of the Kohn-Sham equations . . . . . . . . . . . . . . . . . 19
4 Many-body potentials 21
4.1 Energetics in an alloy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Many-body potential expansion . . . . . . . . . . . . . . . . . . . . . . . . 22
(N)4.2.1 De nition of W . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
(N) (N)4.2.2 Expansion of W in many-body potentials V . . . . . . . . . . 23
(N)4.2.3 Symmetry of the many-body potentials V . . . . . . . . . . . . . 26
(L)
4.2.4 Determination of the coe cien ts a . . . . . . . . . . . . . . . . 27NN
4.3 Limitations for the form of the many-body potential . . . . . . . . . . . . . 29
4.4 Expansion of e ectiv e pair potentials . . . . . . . . . . . . . . . . . . . . . 30
5 Cluster Expansion 33
5.1 Separation of con gurational and spatial variables . . . . . . . . . . . . . . 35
5.2 Introduction of clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.3 From many-body potentials to expansion coe cien ts . . . . . . . . . . . . 39
iii
5.3.1 Example: Expansion coe cien ts from pair potentials . . . . . . . . 40
5.3.2 Screening of the pair potentials in NiAl -
importance of higher order interactions . . . . . . . . . . . . . . . . 40
5.3.3 Independence of cluster expansion coe cien ts . . . . . . . . . . . . 40
5.3.4 Local quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.4 Introduction of a crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.5 Generalization for multicomponent systems, orthogonality and completeness 44
5.6 Energy of formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.7 Taylor expansion of the cluster expansion coe cien ts . . . . . . . . . . . . 50
5.8 Elastic interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.9 The Russian school . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6 Cluster Expansion in con ned geometry 53
6.1 Breaking the symmetry at a surface . . . . . . . . . . . . . . . . . . . . . 53
6.1.1 Broken bond model . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6.1.2 Surface modi cations . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.1.3 energy of the elements . . . . . . . . . . . . . . . . . . . . . 55
6.1.4 Energy of formation for a surface . . . . . . . . . . . . . . . . . . . 56
6.1.5 The case of a thin lm . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.1.6 A potentials point of view . . . . . . . . . . . . . . . . . . . . . . . 57
6.1.7 Surface description using pair potentials . . . . . . . . . . . . . . . 59
6.2 Alloying on top of a substrate . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.2.1 Substrate induced interactions . . . . . . . . . . . . . . . . . . . . . 64
6.2.2 Crystal lattice sites . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.3 Partial coverage of a substrate by a single atomic species . . . . . . . . . . 65
6.3.1 Substrate induced interactions . . . . . . . . . . . . . . . . . . . . . 66
6.3.2 Relation between expansion coe cien ts . . . . . . . . . . . . . . . . 66
7 Practical Cluster Expansion 67
7.1 Extracting expansion coe cien ts from ab-initio data . . . . . . . . . . . . . 70
7.1.1 Selection of clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
7.1.2 Inversion procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
7.2 Procedures used in this work . . . . . . . . . . . . . . . . . . . . . . . . . . 72
7.2.1 Cluster expansion coe cien ts for bulk material . . . . . . . . . . . 72
7.2.2 coe cien ts for surface . . . . . . . . . . 74
7.2.3 Statistical averaging . . . . . . . . . . . . . . . . . . . . . . . . . . 75
7.2.4 Flowchart of the cluster expansion procedure . . . . . . . . . . . . . 77
8 Temperature e ects 79
8.1 Characterizing the state of order . . . . . . . . . . . . . . . . . . . . . . . . 79
8.2 The Cluster Variation method . . . . . . . . . . . . . . . . . . . . . . . . 80
8.2.1 The internal energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
8.2.2 The con gurational entropy . . . . . . . . . . . . . . . . . . . . . . 82iii
8.2.3 Cluster algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
8.2.4 The Bragg-Williams approximation (BW) . . . . . . . . . . . . . . 83
8.2.5 Minimization of the free energy . . . . . . . . . . . . . . . . . . . . 84
8.2.6 Calculation of phase diagrams . . . . . . . . . . . . . . . . . . . . 86
8.2.7 Extensions for phonons . . . . . . . . . . . . . . . . . . . . . . . . . 87
8.3 The Monte Carlo method . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
8.4 Excitations not included in the lattice gas model . . . . . . . . . . . . . . . 88
8.5 Inversion of the CVM equations . . . . . . . . . . . . . . . . . . . . . . . 88
8.5.1 Information from x-ray experiments . . . . . . . . . . . . . . . . . 89
9 The (110) surface of Ni90%-Al 91
9.1 NiAl bulk material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
9.1.1 Relaxation: No simple atomic size e ect . . . . . . . . . . . . . . . 93
9.2 The (110) surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
9.2.1 Characterization from ab-initio calculations . . . . . . . . . . . . . 99
9.2.2 Note on nite size e ects, limited accuracy and possible conclusions 107
9.3 Cluster Expansion of the surface energetics . . . . . . . . . . . . . . . . . 110
9.4 Can we understand what is happening at
the surface? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
9.4.1 Charge density modi cations at the surface . . . . . . . . . . . . . . 114
9.4.2 The covalent bond energy at the surface . . . . . . . . . . . . . . . 116
10 Finite temperature considerations for the surface 117
10.1 Model calculations 1: nearest-neighbor interactions . . . . . . . . . . . . . 118
10.2 Model 2: the 2d Ising model . . . . . . . . . . . . . . . . . . . 121
10.3 The NiAl surface at nite temperature . . . . . . . . . . . . . . . . . . . . 125
10.3.1 What could we learn? . . . . . . . . . . . . . . . . . . . . . . . . . 125
10.3.2 Correlations and segregation at the disordered surface . . . . . . . . 127
11 Summary and conclusions 131
A Ordering the summation in many-body potentials 135
A.1 Regrouping for di eren t limits . . . . . . . . . . . . . . . . . . . . . . . . . 135
A.2 Summation over distinct sets . . . . . . . . . . . . . . . . . . . . . . . . . . 135
B Computational details 139
C The covalent bond energy 141
C.1 From ab-initio DFT to semi-empirical tight-binding . . . . . . . . . . . . . 142
C.1.1 No gauge invariance of E . . . . . . . . . . . . . . . . . . . . . 144band
C.2 Algebra in a non-orthogonal basis . . . . . . . . . . . . . . . . . . . . . . . 144
C.3 The reformulated covalent bond energy . . . . . . . . . . . . . . . . . . . . 145
C.3.1 E and basis transformations . . . . . . . . . . . . . . . . . . . . . 146cov
C.4 A new covalent bond energy . . . . . . . . . . . . . . . . . . . . . . . . . . 148iv
C.4.1 A modi ed band energy . . . . . . . . . . . . . . . . . . . . . . . . 148
C.4.2 A modi ed covalent bond energy . . . . . . . . . . . . . . . . . . . 148
D CVM approximations used 151
D.1 Low temperature behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
D.1.1 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . 152
D.1.2 The free energy at low temperatures . . . . . . . . . . . . . . . . . 154
D.1.3 Adding a hard core potential to the free energy . . . . . . . . . . . 155
E Determination of the parameters for the 2d Ising model 159
F More on surface CVM calculations 161
F.1 Note on the calculation of surface phase diagrams with the CVM . . . . . 161
F.2 Di eren t CVM approximations for the disordered surface . . . . . . . . . . 162
G Zusammenfassung 167
G.1 der einzelnen Kapitel . . . . . . . . . . . . . . . . . . . 168
G.1.1 Kapitel 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
G.1.2 3: Dichtefunktionaltheorie . . . . . . . . . . . . . . . . . . . 168
G.1.3 Kapitel 4: Mehrkorp er-Potentiale . . . . . . . . . . . . . . . . . . . 168
G.1.4 5: Clusterentwicklung . . . . . . . . . . . . . . . . . . . . . 169
G.1.5 Kapitel 6:twicklung in eingeschrankter Geometrie . . . . 169
G.1.6 7: Praktische Clusterentwicklung . . . . . . . . . . . . . . . 169
G.1.7 Kapitel 8: Modellierung des Ein u es der Temperatur . . . . . . . . 170
G.1.8 9: Die (110) Ober ache von Ni90%-Al . . . . . . . . . . . . 170
G.1.9 Kapitel 10: Die (110) Ober ache von bei endlicher Tem-
peratur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
G.2 Ab-initio statistische Mechanik fur Ni90%-Al(110) . . . . . . . . . . . . . . 172Chapter 1
Introduction
Unlike one century ago, when quantum mechanics and statistical mechanics were devel-
oped, nowadays materials science has nothing to do with nding laws of nature. Laws of
nature in materials science are known and accepted. At the beginning of the 21st century
materials science is devoted to understanding, predicting and engineering the consequences
of the laws found one hundred years ago. The advent of powerful computer systems in the
past years allowed rst-principles theories to be developed. First-principles theories that
can be built on the laws of nature aiming to describe real materials as opposed to model
systems. In order to implement rst-principles theories, it is crucial to nd transforma-
tions, enumerations and coarse graining algorithms that allow to systematically deal with
the \most relevant states" from the vast amount of possibilities that result if only few par-
ticles are brought together without having to handle all possibilities explicitly. This purely
numerical di cult y requires a \physical understanding" of the system investigated. In
this context \physical understanding" means that the general behavior of a system can be
understood by simple quantities. Hence, phenomenological models and their terminology
are also necessary.
In this work the (110) surface of Ni90%-Al is approached using a rst-principles the-
ory. Chapters 3 to 8 introduce the building blocks of the theory for the examination of
the Ni90%-Al surface. Only few surface calculations are necessary to show that simple
phenomenological \broken bond" models completely fail to describe experimental facts.
The (110) surface indeed behaves di eren tly from the bulk. Although the energetics of
the surface turns out to be not completely di eren t from the bulk, surface modi cations
together with frustration e ects present in the bulk but not in the surface, provide the
understanding of ordering and segregation at the surface. A deeper insight to the surface
energetics from further calculations then raises the question whether the (110) surface of
Ni90%-Al behaves more like a 1, 2 or 3-dimensional system.
12
ObservationExperiment System
Laws of
ModelNature
1st PRINCIPLES FIT
Figure 1.1: \Material scientists in Platos Cave". An experiment is looked at as a pro-
jection. The experimental observations are compared to a shadow on the wall. Hitherto
materials scientists had to explain and understand the results of their experiments within
self-contained models, parameters had to be tted to the experimental result so that the
model reproduced the experimental result. First-principles theories directly relate the pa-
rameters required in a model to the laws of nature.
Figure 1.2: Segregation of Al to the (110) surface of Ni (chapter 9).
The fact that the surface behaves di eren t from the bulk is of interest also from a
materials science and engineering point of view: The surface behavior is of great importance
because any bulk material makes contact to the surroundings via its surface, such as
chemical reactions, adsorption phenomena, heat transfer, etc. Thus a rst-principles theory
provides valuable information for the understanding and engineering of materials and their
surface properties.