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- Herausgeber: Wiley-VCH
- Kategorie: Fachliteratur
- Sprache: Englisch
An overview of the latest computational materials science methods on an atomic scale. The authors present the physical and mathematical background in sufficient detail for this highly current and important topic, but without unnecessary complications. They focus on approaches with industrial relevance, covering real-life applications taken from concrete projects that range from tribology modeling to performance optimization of integrated circuits. Following an introduction to the fundamentals, the book describes the most relevant approaches, covering such classical simulation methods as simple and reactive force field methods, as well as highly accurate quantum-mechanical methods ranging from density-functional theory to Hartree-Fock and beyond. A review of the increasingly important multiscale approaches rounds off this section. The last section demonstrates and illustrates the capabilities of the methods previously described using recent real-life examples of industrial applications. As a result, readers gain a heightened user awareness, since the authors clearly state the conditions of applicability for the respective modeling methods so as to avoid fatal mistakes.
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Veröffentlichungsjahr: 2015
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Title Page
Copyright
Preface
Part I : Basic Physical and Mathematical Principles
Chapter 1: Introduction
Chapter 2: Newtonian Mechanics and Thermodynamics
2.1 Equation of Motion
2.2 Energy Conservation
2.3 Many Body Systems
2.4 Thermodynamics
Chapter 3: Operators and Fourier Transformations
3.1 Complex Numbers
3.2 Operators
3.3 Fourier Transformation
Chapter 4: Quantum Mechanical Concepts
4.1 Heuristic Derivation
4.2 Stationary Schrödinger Equation
4.3 Expectation Value and Uncertainty Principle
Chapter 5: Chemical Properties and Quantum Theory
5.1 Atomic Model
5.2 Molecular Orbital Theory
Chapter 6: Crystal Symmetry and Bravais Lattice
6.1 Symmetry in Nature
6.2 Symmetry in Molecules
6.3 Symmetry in Crystals
6.4 Bloch Theorem and Band Structure
Part II : Computational Methods
Chapter 7: Introduction
Chapter 8: Classical Simulation Methods
8.1 Molecular Mechanics
8.2 Simple Force-Field Approach
8.3 Reactive Force-Field Approach
Chapter 9: Quantum Mechanical Simulation Methods
9.1 Born–Oppenheimer Approximation and Pseudopotentials
9.2 Hartree–Fock Method
9.3 Density Functional Theory
9.4 Meaning of the Single-Electron Energies within DFT and HF
9.5 Approximations for the Exchange–Correlation Functional
9.6 Wave Function Representations
9.7 Concepts Beyond HF and DFT
Chapter 10: Multiscale Approaches
10.1 Coarse-Grained Approaches
10.2 QM/MM Approaches
Chapter 11: Chemical Reactions
11.1 Transition State Theory
11.2 Nudged Elastic Band Method
Part III : Industrial Applications
Chapter 12: Introduction
Chapter 13: Microelectronic CMOS Technology
13.1 Introduction
13.2 Work Function Tunability in High-
k
Gate Stacks
13.3 Influence of Defect States in High-
k
Gate Stacks
13.4 Ultra-Low-
k
Materials in the Back-End-of-Line
Chapter 14: Modeling of Chemical Processes
14.1 Introduction
14.2 GaN Crystal Growth
14.3 Intercalation of Ions into Cathode Materials
Chapter 15: Properties of Nanostructured Materials
15.1 Introduction
15.2 Embedded PbTe Quantum Dots
15.3 Nanomagnetism
References
Index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Part I : Basic Physical and Mathematical Principles
Begin Reading
List of Illustrations
Chapter 2: Newtonian Mechanics and Thermodynamics
Figure 2.1 Trajectory of a point particle.
Figure 2.2 Potential energy landscape of a conservative force field with two different paths from point to point .
Figure 2.3 Illustration of the first (a) and second (b) thermodynamic law
Figure 2.4 Illustration of energy and entropy changes for the example of an oxyhydrogen reaction.
Figure 2.5 Illustration of the microcanonical (
N, V, E
), canonical (
N, V, T
), and isothermal-isobaric (
N, p, T
) ensemble.
Chapter 3: Operators and Fourier Transformations
Figure 3.1 Illustration of the complex plane and two different possible representations of a complex number.
Figure 3.2 Illustration of the transformation of three example functions.
Chapter 4: Quantum Mechanical Concepts
Figure 4.1 One-dimensional wave function (gray dashed line) and its absolute square (black line).
Figure 4.2 Arbitrary one-dimensional probability density (gray line) and the probability to find the particle within the region between and (shaded region).
Figure 4.3 Comparison between Newtonian and quantum mechanic particles.
Chapter 5: Chemical Properties and Quantum Theory
Figure 5.1 Illustration of spherical coordinates.
Figure 5.2 Discrete energy levels (a) and radial distribution functions (b) of an electron in a Coulomb potential.
Figure 5.3 Illustration of the s-, p-, d-, and f-orbitals of a hydrogen atom.
Figure 5.4 Energetic order of the atomic orbitals. The maximal occupation of the orbitals is indicated (each ball may be occupied by a couple of spin-paired electrons). There are some exceptions to this ordering among the transition metals and heavier elements.
Figure 5.5 Tetrahedron of structure, bonding, and material type [20, 21].
Figure 5.6 Schematic illustration of a hydrogen molecule ion .
Figure 5.7 Energy eigenvalues of the hydrogen molecule ion .
Figure 5.8 Molecular orbital diagram of a hydrogen and a molecule. The occupation of the orbitals is illustrated by black balls.
Figure 5.9 Schematic illustration of some bonding molecular orbitals.
Figure 5.10 Illustration of the sp, , and hybrid orbitals.
Figure 5.11 Examples of molecules with different hybridization degrees.
Chapter 6: Crystal Symmetry and Bravais Lattice
Figure 6.1 Schematic illustration of a water molecule and its symmetry operations. The water molecule belongs to the point group.
Figure 6.2 Conventional unit cells of the cubic crystal structures and their Wigner–Seitz cells in real and reciprocal space.
Figure 6.3 Illustration of the periodicity of a lattice and its consequences for the Fourier components.
Figure 6.4 Illustration of the differences in the electronic properties of metals, semiconductors, and insulators: left panel - electronic band structure; middle panel - schematic picture of the valence and conduction bands; right panel - electronic density of states (DOS). The band gap region is indicated by a white background color.
Chapter 7: Introduction
Figure 7.1 Classification of the different atomistic simulation methods according to their underlying theory.
Figure 7.2 Classification of the different atomistic simulation methods according to their possible application scenario. * Chemical reactions can be described with a limited accuracy using, for example, the NEB method (see 11.2). ** The hybrid functionals are a semiempirical method (see Section 9.5). *** Classical FF methods can usually not be used to describe chemical reactions.
Figure 7.3 Overview over different higher-level methods and their fields of application.
Chapter 8: Classical Simulation Methods
Figure 8.1 Illustration of the steepest descent algorithm. The black lines indicate contours of constant potential energy, and the white arrows indicate the steepest descent steps.
Figure 8.2 Illustration of the principal work flow of a MM simulation.
Figure 8.3 Schematic illustration of different simple FF-potentials.
Figure 8.4 Relevant regions of different contributions of the van-der-Waals forces.
Figure 8.5 Total bond order BO
ij
and its components.
Figure 8.6 Shape of the and potential terms within the ReaxFF approach.
Figure 8.7 Simulation steps of an ab-initio-based ReaxFF approach for the example of a DNA helix.
Chapter 9: Quantum Mechanical Simulation Methods
Figure 9.1 External potential (black line) generated by a superposition of several atomic Coulomb potentials. The horizontal gray lines indicate possible electronic niveaus.
Figure 9.2 Flow diagram for the optimization of atomic coordinates.
Figure 9.3 Flow diagram of the HF self-consistency loop.
Figure 9.4 Flow diagram of the KS self-consistency loop
Figure 9.5 Three different ways to calculate the quasiparticle band gap.
Figure 9.6 Real-space and -space unit cells of two different (red and green) periodic systems. The
G
vector grids are illustrated by red and green points (within the -space), respectively.
Figure 9.7 Possible application scenarios of different methods to calculate the electronic properties of a material.
Figure 9.8 Different self-consistency loops for different levels of GW approximation.
Chapter 10: Multiscale Approaches
Figure 10.1 Sizes and time scales of different physical systems. Furthermore, applicable simulation methods are indicated.
Figure 10.2 Illustration of a polymer chain and possible supermolecules of a coarse-grained model system.
Figure 10.3 Schematic illustration of the QM/MM approach.
Chapter 11: Chemical Reactions
Figure 11.1 Energy landscape and a transition path from the reactants to the products of a chemical reaction.
Figure 11.2 Illustration of the NEB method. The dashed line reflects a first guess for the transition path. Along the optimized transition path, the virtual springs are indicated.
Chapter 13: Microelectronic CMOS Technology
Figure 13.1 General topology of a CMOS structure. The different fabrication steps, FEoL and BEoL, are indicated.
Figure 13.2 General structure of a NMOS (a) and a PMOS (b) transistor. The Figure have been produced by E. Nadimi (AQcomputare GmbH).
Figure 13.3 Characteristics of the valence and conduction band edges at different applied gate voltages. The channel region (perpendicular to the image plane) is indicated by a light green color, while the band gap region of Si and is colored in gray.
Figure 13.4 Schematic illustration of the SCE.
Figure 13.5 Atomistic structure of a NMOS gate stack with a high-
k
material layer (green background color). O, Si, Hf, Ti, and N atoms are represented by red, beige, dark green, gray, and blue balls, respectively. The Figure has been produced by E. Nadimi (AQcomputare GmbH).
Figure 13.6 Mechanism of the DT of channel electrons through the layer at gate voltages .
Figure 13.7 Used crystal structures and their conventional unit cells: (a) m-, (b) -cristobalite , (c) fcc TiN, and (d) fcc Si. O, Si, Hf, Ti, and N atoms are represented by red, beige, dark green, gray, and blue balls, respectively.
Figure 13.8 Influence of the supercell size on the impurity concentration.
Figure 13.10 Construction of the used HKMG stack supercell. Gray regions indicate the surface unit cells of the different materials and surface orientations.
Figure 13.9 Schematic illustration of the used HKMG stack supercell. H, O, Si, Hf, Ti, and N atoms are represented by white, red, oliv, dark green, gray, and blue balls, respectively.
Figure 13.11 Applied calculation scheme for band edge offsets.
Figure 13.12 Averaged electrostatic potential within the constructed HKMG stack supercell without (a) and with (b) La impurities (light blue balls). H, O, Si, Hf, Ti, and N atoms are represented by white, red, oliv, dark green, gray, and blue balls, respectively. The Figure have been produced by E. Nadimi (AQcomputare GmbH).
Figure 13.13 Shift of valence band offset and the relative total energies versus the
z
position of the La impurities ( impurities–also see Ref. [183]). The Figure has been produced by E. Nadimi (AQcomputare GmbH).
Figure 13.14 Mechanism of the TAT of channel electrons through the layer for . The rough energetic positions of the DDS and SDS are indicated.
Figure 13.15 Determination of charge transition levels.
Figure 13.16 Formation energies (obtained with the PBE0 functional) of the considered oxygen vacancies in m- versus the Fermi energy. The Figure has been taken from Ref. [206] with permission of Wiley.
Figure 13.17 Energetic position of the two types of charge transition levels with respect to the conduction band edges of the HKMG stack for different applied gate voltages.
Figure 13.18 Degradation of the ULK material during the reactive ion beam etching of trenches or vias.
Figure 13.19 Chemical structure of the considered repair chemicals.
Figure 13.20 Schematic illustration of a silylation process. The proton transfer is indicated by dashed arrows: (a) silylation of a hydroxyl group by a repair chemical with one reactive group. (b) and (c) stepwise silylation of two hydroxyl groups by a repair chemical with two reactive groups.
Figure 13.21 Reaction path of a simple silylation process. The corresponding reaction energy and the activation energy are indicated.
Figure 13.22 Schematic illustration of the pore-filling effect.
Figure 13.23 Possible reactions of DMADMS with silanol. (a) desired silylation reaction, (b) possible side reaction, (c) consecutive reaction of the product dimethylamine from reaction (b). The Figure has been taken from Ref. [206] with permission of Elsevier.
Chapter 14: Modeling of Chemical Processes
Figure 14.1 Chemical processes.
Figure 14.2 HVPE process for the example of GaN growth.
Figure 14.3 Schematic illustration of the trenches within the substrate surface and the occurrence of island coalescence.
Figure 14.4 The applied general simulation approach.
Figure 14.5 The automated ReaxFF parameter training scheme.
Figure 14.6 (a) Some small GaN and GaN:H structures included in ; (b) Ga–N binding energy of several GaN training structures. Simple structures, ring structures, and cage structures are indicated by green, blue, and red background colors, respectively.
Figure 14.7 (a) The geometry of the HVPE reactor of [268] by means of the isosurfaces of the particle velocity field [m/s] (reprinted from Ref. [269]); (b) model geometry for our ReaxFF simulations. The Figure has been produced by O. Böhm (AQcomputare GmbH).
Figure 14.8 Top view of a GaN(0001) surface with two reconfigurated 4-core edge dislocations indicated by red circles (see text). The Ga and N atoms are illustrated by beige and blue balls, respectively. The Figure have been produced by O. Böhm (AQcomputare GmbH).
Figure 14.9 Top view of a GaN(0001) surface with two 4-core edge dislocations. Snapshot of the MD simulation after 0.141 ns (a) and 0.742 ns (b); top (with substrate) and bottom (without substrate).
Figure 14.10 General scheme of the working principle of a Li ion battery.
Figure 14.11 (a) Schematic illustration of the unit cell; the occurring pyramidal structures are indicated. (b) The supercell used for the calculation of the Li intercalation.
Figure 14.12 Cell parameters of the structures: (a) lattice vector
a
, (b) lattice vector
c
, (c) ratio
c
/
a
, (d) volume of the unit cell. The experimental values are taken from Ref. [288]. The Figure have been produced by O. Böhm (AQcomputare GmbH).
Figure 14.13 (a) The monoclinic -phase of for . The red, gray, and violet balls represent O, V, and Li atoms, respectively. (b) Comparison between the stacking order within the , and -phases. The corresponding unit cells are indicated by a gray background color.
Figure 14.14 Calculated voltage discharge curve (black dots) of a single Li-ion battery cell with the cathode material and bulk Li as anode. The experimental data (green) are taken from Ref. [272], while the theoretical data (red) are taken from Braithwaite
et al.
[291]. The dashed lines are just guides to the eye.
Chapter 15: Properties of Nanostructured Materials
Figure 15.1 Nano-objects of different dimensionality: top row - 0d-objects, middle row - 1d-objects, bottom row - 2d-objects. The pictures are (a) produced by H. Dorn (Virginia Tech), (b) provided by Evident Technologies Inc., (c) reprinted from Ref. [294], (d) reprinted from Ref. [295, 296] with permission of Wiley and Elsevier, (e) reprinted from Ref. [297] with permission of the Royal Society of Chemistry, (f) reprinted from Ref. [298] with permission of Wiley, (g) reprinted from Ref. [299] with permission of A.R. Barron, (h) produced by K. Hermann (FHI Berlin) using the Balsac software, and (i) produced by AQcomputare (www.matcalc.de)
Figure 15.2 Photoluminescence of indium phosphide QDs. The material emits light with different colors by tuning the QD size and material composition. The picture was provided by J. Yurek (Nanosys Inc.).
Figure 15.3 Interplay between theory and experiment.
Figure 15.4 Wulff construction to obtain the ECS of an arbitrary object.
Figure 15.6 The upper row shows ECSs of PbTe QDs embedded in CdTe matrix (not shown). The green facets represent {110}, the red facets {100}, and the blue facets {111} faces. The ECS at the left is constructed using the values of Table 15.2; for the ECS in the middle, we have changed the (111) interface energy to 0.22 J (within the estimated error bar), and the ECS at the right is constructed using equal interface energies of 0.2 J. In the lower row, a projection along the [110] axis, with the abscissa along [] and the ordinate along [001], is shown. The Figure is reprinted from Ref. [182].
Figure 15.5 (a) Cross-sectional TEM image of a PbTe/CdTe heterostructure after annealing at C showing QDs with the shape of small rhombo-cubo-octahedrons. (b) Centrosymmetric PbTe dots with different height/length aspect ratios. (c) Size distribution of the QDs. The solid line represents an aspect ratio of 1. The two gray areas indicate highly symmetric dots with aspect ratios between 0.8 and 1.2 and elongated dots with heights smaller than 27 nm. (d) Centrosymmetric PbTe dot with an aspect ratio of 1. The occuring interface orientations are indicated. The pictures (a)–(c) are reprinted from Ref. [315] with permission from AIP Publishing LLC; picture (d) is reprinted in a modified version from Ref. [325] with permission from IOP.
Figure 15.7 Theoretically predicted ECS (b) and the constructed atomistic model structure (a) of an embedded PbTe QD. The different PbTe/CdTe interface terminations are indicated. Reprinted from Ref. [326].
Figure 15.8 Bond lengths of neighboring Pb and Te atoms within embedded PbTe QDs with different diameters: 0.64 nm (a), 1.28 nm (b), 1.92 nm (c). The dashed horizontal lines represent the average interbilayer bond length, while the solid horizontal lines represent the average intrabilayer bond length. The increased bond lengths at the QD-matrix interfaces are indicated by dashed ellipses. Reprinted from Ref. [294].
Figure 15.9 (a) Fourier-filtered electrostatic potential (arbitrary units) shown in the (10) and (01) planes. Blue colors correspond to negative values and red colors to positive values. The atomic positions of PbTe QD (inside the white line) and CdTe matrix are indicated by a stick and ball model. (b) The red solid line shows the plane average of the Fourier-filtered electrostatic potential along the [111] direction. The estimated slope of the electrostatic potential inside the dot region is indicated by a black dashed line. The illustration is reprinted from Ref. [294].
Figure 15.10 Spatial (upper panel) and energetic (lower panel) separation of the highest occupied (red) and lowest empty (green) energy level within an embedded PbTe QD. The black solid line represents the valence and conduction band edges of the QD and matrix region, respectively.
Figure 15.11 Current achievements and open questions regarding the integration of magnetic Si QDs into spintronic devices. The questions considered in the presented example are indicated by the dark-gray box.
Figure 15.12 The applied construction principle of the Si QDs.
Figure 15.13 Five-shell Si QD. The positions of the selected dopant sites are indicated by red (substitutional) and blue (interstitial) dots. The Figure is reprinted from Ref. [348].
Figure 15.14 (a) Relative formation energy of a singly Mn-doped Si QD with a diameter of 2.18 nm versus the variation in the Si chemical potential with respect to its bulk value. The corresponding doping sites are indicated. (b) Schematic illustration of the stability of the different doping configurations “int”, “sub”, and “mix” in doubly doped Si QDs.
Figure 15.15 Atomic geometries and energies of the , , and configurations for the example of a Si QD with .
Figure 15.16 Relative formation energy for the most stable substitutional (red) and interstitial (blue) doping sites. Solid lines illustrate the incorporation of Mn and dashed lines of Fe atoms. The Figure is reprinted from Ref. [349].
Figure 15.17 Calculated magnetic coupling constant
J
(with error bars) as a function of the Mn–Mn distance. Results for the free-standing clusters of Ref. [358] are illustrated by a red line. The different magnetic coupling regimes are indicated.
Figure 15.18 Vector-field representation of the magnetization density of an int (3.5-3.5) configuration within a Si QD with a diameter of 1.82 nm (). The color of the drawn vectors indicates the absolute value of the magnetization (red = large, blue = small). The angle of about between both magnetic moments is clearly visible. The Figure is reprinted from Ref. [348].
List of Tables
Chapter 3: Operators and Fourier Transformations
Table 3.1 List of some of the most commonly used operators in physics.
Chapter 5: Chemical Properties and Quantum Theory
Table 5.1 Denotation of the atomic orbital functions in dependence on
n
and
l
.
Chapter 6: Crystal Symmetry and Bravais Lattice
Table 6.1 Definition of the seven lattice systems and the corresponding Bravais lattices.
Chapter 9: Quantum Mechanical Simulation Methods
Table 9.1 Comparison of the different wave function representation schemes.
Chapter 13: Microelectronic CMOS Technology
Table 13.1 Comparison of the obtained bulk lattice constants and the corresponding experimental values. Additionally, the used
k
-point densities for the Brillouin zone sampling are given.
Table 13.2 Summary of the obtained band gap values within the DFT-GGA approach, the empirical scissor parameters
S
, and the experimental band gap values [174].
Table 13.3 Obtained offsets of the electrostatic potential, the valence, and the conduction band edges at different interfaces of the HKMG stack.
Table 13.4 Band gap values of m- calculated at different levels of approximation. The required computer time in CPUh has been estimated from the average over several electronic optimization runs (with fixed atomic geometry).
Table 13.5 Formation energies (obtained with the PBE0 functional) of the considered oxygen vacancies in m- at a Fermi energy of eV [194, 195].
Table 13.6 Charge transition levels (obtained with the PBE0 functional) of and with respect to the valence band edge of m-.
Table 13.7 Thermochemical properties of the considered repair chemicals for silylation reactions with the prototypical ULK fragment . The size of the molecules has been determined as their largest interatomic distance. The data are taken from Ref. [206].
Chapter 14: Modeling of Chemical Processes
Table 14.1 Structural and energetic properties of some representative structures included in .
Table 14.2 Comparison of the obtained lattice parameters of three different phases with .
Chapter 15: Properties of Nanostructured Materials
Table 15.1 Theoretical and experimental lattice constants () and band gaps (eV) of PbTe and CdTe as well as the effective longitudinal electron and hole masses at the
L
point of PbTe in units of .
Table 15.2 Calculated average interface energies in [J/] of low energy PbTe/CdTe interfaces taken from Ref. [182].
Table 15.3 Characteristic properties of the considered TM-doped Si QDs.
Table 15.4 Total magnetic moment in units of of a TM-doped Si QD with a diameter of 1.82 nm (). Only results for typical dopant configurations are listed. The superscript “” denotes interstitial sites beneath a {001} QD facet.
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Roman Leitsmann, Philipp Plänitz, andMichael Schreiber
In-vitro Materials Design
Modern Atomistic Simulation Methods for Engineers
Authors
Dr. Roman Leitsmann
AQcomputare GmbH
Annaberger Straße 240
09125 Chemnitz
Germany
Dr. Philipp Plänitz
AQcomputare GmbH
Annaberger Straße 240
09125 Chemnitz
Germany
Michael Schreiber
Technische Universität Chemnitz
Institute of Physics
Reichenhainer Str. 70
09126 Chemnitz
Germany
Cover picture courtesy of
Sang-Woo Kim, Ph.D., Professor
School of Advanced Materials Science & Engineering
SKKU Advanced Institute of Nanotechnology (SAINT)
Sungkyunkwan University (SKKU)
Cheoncheon 300
Suwon 440-746
South Korea
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A catalogue record for this book is available from the British Library.
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Preface
In many academic and industrial R&D projects, physicists, chemists and engineers are working together. In particular, the development of advanced functionalized materials requires an interdisciplinary approach. In the last decades, the size of common devices and used material structures has become smaller and smaller. This has led to the emergence of the so-called nanotechnology, that is, a technology that uses material systems with an extent of less than several hundred nanometers. The enormous technical advances in this field are subject to two mutually amplifying effects. On the one hand, modern experimental techniques have been developed that allow the observation, manipulation, and manufacturing of materials at an atomic length scale with an industrially relevant production rate. On the other hand, the enhancements in the computer technology have led to a tremendous growth of the scientific field of computational material sciences. Nowadays, modern simulation methods are indispensable for the design of new and functionalized nanomaterials. They are essential to understand the chemical and physical processes beyond many macroscopic effects.
However, the basic concepts of modern atomistic simulation methods are not very well established in common engineering courses. Furthermore, the existing literature either deals with very specific problems or is at a very deep physical or mathematical level of theory. Therefore, the intention of this book is to give a comprehensive introduction to atomic scale simulation methods at a basic level of theory and to present some recent examples of applications of these methods in industrial R&D projects. Thereby, the reader will be provided with many practical advices for the execution of proper simulation runs and the correct interpretations of the obtained results.
For those readers who are not familiar with basic modern mathematical and physical concepts, Part I will give a rough introduction to Newtonian and quantum mechanics, thermodynamics, and symmetry-related properties. Furthermore, necessary mathematical concepts will be introduced and the reader will be provided with the denotation and terminology that will be used later on. Readers with a fundamental physical and mathematical knowledge may skip this part and look up certain aspects later, if it is necessary.
Part II gives a brief introduction to important aspects of state-of-the-art atomic scale simulation techniques. In particular, the basics of classical and reactive force field methods, the density functional and Hartree–Fock theory, as well as multiscale approaches will be discussed. Possible fields of application will be depicted, and limitations of the methods are illustrated. Furthermore, several more advanced methods, which are able to overcome some of these limitations, will be shortly mentioned. The intention of this part is to enable the reader to decide which simulation method (with which limitations) would be optimal to investigate a certain problem of interest.
The last part illustrates possible application scenarios of atomic scale simulation techniques for industrially relevant problems. It is divided into three chapters that consider three different industrial fields: microelectronics, chemical processes, and nanotechnology. Real industrial problems and the corresponding contributions of atomic scale simulations will be presented to the reader. Thereby, the set up, the execution, and the analysis of the results will be discussed in detail, and many practical hints for potential users of atomic scale simulations are provided.
Roman Leitsmann
Chemnitz
April 2015
Part I
Basic Physical and Mathematical Principles
Chapter 1Introduction
The scope of this part is to provide the reader with basic physical and mathematical principles that are necessary to understand the discussions in the following chapters. Furthermore, a notation is introduced, which will be utilized throughout the remaining book. No special previous knowledge is required from the readership. Nevertheless, a basic scientific knowledge is advantageous. Part I makes no claim to provide a complete overview. Many things can be discussed only very briefly. For a more detailed description of special topics and background information, the readers are provided with suitable references.
Those readers who are already familiar with the physical and mathematical concepts can skip this part and look up certain points later if necessary.
Chapter 2Newtonian Mechanics and Thermodynamics
Classical or Newtonian mechanics describes the motion of objects, from small particles to astronomical objects. Newtonian mechanics provides extremely accurate results as long as the domain of study is restricted to macroscopic objects and velocities far below the speed of light. When the objects being dealt with become sufficiently small, it becomes necessary to include quantum mechanical effects (see Chapter 4). In the case of velocities close to the speed of light, classical mechanics has to be extended by special or general relativity.
The following section introduces the basic concepts of classical Newtonian mechanics and its application to atomistic objects. At the end of this section, a critical discussion about the restrictions of this approach is given.
2.1 Equation of Motion
Quite often, objects are treated as point particles, that is, objects with negligible size. The motion of a point particle is characterized by a small number of parameters: its position, its mass, and its momentum.
Note:
In reality, all objects have a nonzero size. However, often, they can be treated as point particles, because effects related to the finite size are either not of interest or have to be described by more sophisticated theories such as quantum mechanics.
The position of a point particle can be defined with respect to an arbitrary fixed reference point in space.1 In general, the point particle does not need not be stationary relative to , so is a function of the time t
Without loss of generality, the reference point can always be assumed to be at the origin of the used coordinate system, that is,
Note:
The position of the point particle and all similar quantities are three-dimensional vectors. They must be dealt with using vector analysis. They will be denoted by
where x, y, and z are the Cartesian coordinates of the point particle.
The velocity , or the rate of change of position with time, is defined as the derivative of the position with respect to the time
The acceleration, or rate of change of velocity, is the derivative of the velocity with respect to time (the second derivative of the position with respect to time)
The acceleration can arise from a change with time of the magnitude of the velocity or of the direction of the velocity or both.
Note:
If only the magnitude of the velocity decreases, this is sometimes referred to as deceleration, but generally, any change in the velocity with time, including deceleration, is simply referred to as acceleration.
As we all know from our everyday life, an acceleration of an object requires the action of a force on it. Sir Isaac Newton was the first who mathematically described this relationship, which is known today as Newton's second law2
The quantity introduced in this equation is called (canonical) momentum. The force acting on a particle is thus equal to the rate of change of the momentum of the particle with time.
As long as the forces acting on a particle are known, Newton's second law is sufficient to completely describe the motion of the particle. Hence, written in a slightly different form, it is also called equation of motion
where the sum of all forces acting on the particle yields the total net force .3
If at a time , the position and the velocity of a point particle are known and all forces acting on that particle are given, then the motion of the particle can be determined for its whole future and past by solving the equation of motion yielding the particle trajectory (see Figure 2.1). This illustrates the deterministic character of Newtonian mechanics.
Figure 2.1Trajectory of a point particle.
Example: Free particle:
In the case of a free particle, no forces are acting on it. Hence, the equation of motion becomes quite simple
Using Eq. (2.4) and carrying out two integrations over the time t, the trajectory of the particle becomes
with the integration constants (initial velocity) and (initial position). This is the textbook formula well known from basic physics courses.
2.2 Energy Conservation
Imagine a constant force is applied to a point particle and causes a finite displacement . The work done by the force is defined as the scalar product of the force and the displacement vector
In a more general case, the force may vary as a function of position as the particle moves from to along a path C. The work done on the particle is then given by the path integral
In the special case that the work done in moving the particle from to is the same no matter which path is taken, the force is said to be conservative. For example, gravity is a conservative force, as well as the force of an idealized spring (Hooke's law). On the other hand, the force due to friction is nonconservative. All conservative forces can be expressed as the gradient of a scalar function
Except for an arbitrary constant shift c, this function is equal to the potential energy
of the point particle.
Example: Potential energy landscape:
In Figure 2.2, a potential energy landscape is illustrated. The thin solid lines correspond to lines along which the value of the scalar function is constant—the so—called equipotential lines. The force acting on a particle is equal to the gradient of (Eq. (2.11)). The denser the equipotential lines are, the larger the force acting on the particle is.
Two different paths connecting point and point are illustrated.
The first one runs through a valley, an area with small changes in
. Hence, only small forces are acting on a particle along this path.
The second path crosses a mountain, an area with strong changes in
. Hence, large forces are acting on the particle. However, when the particle first climbs up the mountain, but then moves down again, the forces are directed in opposite directions.
Altogether, the work done by moving a particle from point to point is the same for both paths.
Figure 2.2 Potential energy landscape of a conservative force field with two different paths from point to point .
The kinetic4 energy of a point particle5 of mass m and speed v (i.e., the magnitude of the velocity) is given by
The work-energy theorem states that for a point particle of constant mass m, the total work W done on the particle is equal to the change in kinetic energy of the point particle:
If all the forces acting on a particle are conservative and is the total potential energy, the following equalities are satisfied
This result is known as the conservation of energy and states that the total energy
is constant in time. This result is a general (maybe the most general) concept in physics. It holds not only in conservative systems, but also in all physical systems; only the types of energy to be considered must be adapted. In nonconservative open systems, besides the kinetic and potential energy, also the energy exchange with the environment, the change of the internal energy (see Section 2.4), the friction energy, and other energy types have to be taken into account.
2.3 Many Body Systems
Up to now, we have considered only one single point particle and external forces acting on it. In the current subsection we will expand the discussion to a system of N point particles, which may interact with each other. Hereby, interacting particles are those particles that induce forces acting on other particles. As these forces have their source within the considered system of N particles, they are called internal forces (in contrast to external forces that may be applied to the system from outside). The most prominent examples of internal forces are electrostatic forces acting between charged particles or the gravitational force acting between massive particles. However, other types of forces such as van-der-Waals forces or bending and torsion forces also belong to this category.
Many forces are acting pairwise between two different particles i and j. For this type of forces, Newton's third law () holds:
It means that the force induced by particle i on particle j has the same magnitude as force induced by particle j on particle i but acts in opposite direction. Hence, in the case of pairwise acting forces, the sum over all internal forces must vanish:
The total force acting on particle i induced by the remaining particles
is obviously an internal force. With Eq. (2.18), this adds up to
The last equation holds not only for forces acting pairwise, but also more generally for all kinds of internal forces. If the forces are moreover conservative forces, that is, they can be expressed by a scalar potential function V as a generalization of Eq. (2.11)
the total energy of the N-particle system is conserved. A typical example for such a system is an infinite one-dimensional chain of particles coupled by ideal springs.
According to Eq. (2.6), a system in which only internal forces are acting between the particles and no external forces are applied can be described by the following set of equations of motion:
For real systems, this set of differential equations may get quite complicated and its solution can be obtained only approximately in most cases. Nevertheless, the deterministic character of the theory remains. If one knows at a certain time
