Table of Contents

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Title Page

Preface

Part I : Basic Physical and Mathematical Principles

Chapter 1: Introduction
Chapter 2: Newtonian Mechanics and Thermodynamics
1. 2.1 Equation of Motion
2. 2.2 Energy Conservation
3. 2.3 Many Body Systems
4. 2.4 Thermodynamics
Chapter 3: Operators and Fourier Transformations
1. 3.1 Complex Numbers
2. 3.2 Operators
3. 3.3 Fourier Transformation
Chapter 4: Quantum Mechanical Concepts
1. 4.1 Heuristic Derivation
2. 4.2 Stationary Schrödinger Equation
3. 4.3 Expectation Value and Uncertainty Principle
Chapter 5: Chemical Properties and Quantum Theory
1. 5.1 Atomic Model
2. 5.2 Molecular Orbital Theory
Chapter 6: Crystal Symmetry and Bravais Lattice
1. 6.1 Symmetry in Nature
2. 6.2 Symmetry in Molecules
3. 6.3 Symmetry in Crystals
4. 6.4 Bloch Theorem and Band Structure

Part II : Computational Methods

Chapter 7: Introduction
Chapter 8: Classical Simulation Methods
1. 8.1 Molecular Mechanics
2. 8.2 Simple Force-Field Approach
3. 8.3 Reactive Force-Field Approach
Chapter 9: Quantum Mechanical Simulation Methods
1. 9.1 Born–Oppenheimer Approximation and Pseudopotentials
2. 9.2 Hartree–Fock Method
3. 9.3 Density Functional Theory
4. 9.4 Meaning of the Single-Electron Energies within DFT and HF
5. 9.5 Approximations for the Exchange–Correlation Functional $c09-math-0106$
6. 9.6 Wave Function Representations
7. 9.7 Concepts Beyond HF and DFT
Chapter 10: Multiscale Approaches
1. 10.1 Coarse-Grained Approaches
2. 10.2 QM/MM Approaches
Chapter 11: Chemical Reactions
1. 11.1 Transition State Theory
2. 11.2 Nudged Elastic Band Method

Part III : Industrial Applications

Chapter 12: Introduction
Chapter 13: Microelectronic CMOS Technology
1. 13.1 Introduction
2. 13.2 Work Function Tunability in High-k Gate Stacks
3. 13.3 Influence of Defect States in High-k Gate Stacks
4. 13.4 Ultra-Low-k Materials in the Back-End-of-Line
Chapter 14: Modeling of Chemical Processes
1. 14.1 Introduction
2. 14.2 GaN Crystal Growth
3. 14.3 Intercalation of Ions into Cathode Materials
Chapter 15: Properties of Nanostructured Materials
1. 15.1 Introduction
2. 15.2 Embedded PbTe Quantum Dots
3. 15.3 Nanomagnetism

References

Index

End User License Agreement

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 $c02-math-043$ to point $c02-math-044$ .
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 $c04-math-042$ between $c04-math-043$ and $c04-math-044$ (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 $c05-math-087$ .
Figure 5.7 Energy eigenvalues of the hydrogen molecule ion $c05-math-088$ .
Figure 5.8 Molecular orbital diagram of a hydrogen $c05-math-095$ and a $c05-math-096$ 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, $c05-math-118$ , and $c05-math-119$ 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 $c06-math-0009$ 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 $c08-math-0085$ and $c08-math-0086$ 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 $c09-math-0147$ -space unit cells of two different (red and green) periodic systems. The G vector grids are illustrated by red and green points (within the $c09-math-0148$ -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 $c13-math-0003$ 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 $c13-math-0032$ layer at gate voltages $c13-math-0033$ .
Figure 13.7 Used crystal structures and their conventional unit cells: (a) m- $c13-math-0046$ , (b) $c13-math-0047$ -cristobalite $c13-math-0048$ , (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 $c13-math-0093$ 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 $c13-math-0174$ and the relative total energies $c13-math-0175$ versus the z position of the La impurities ( $c13-math-0176$ $c13-math-0177$ 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 $c13-math-0200$ layer for $c13-math-0201$ . 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- $c13-math-0272$ 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 $c13-math-0311$ and the activation energy $c13-math-0312$ 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 $c14-math-0026$ ; (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 $c14-math-0075$ unit cell; the occurring $c14-math-0076$ pyramidal structures are indicated. (b) The $c14-math-0077$ supercell used for the calculation of the Li intercalation.
Figure 14.12 Cell parameters of the $c14-math-0107$ 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 $c14-math-0131$ -phase of $c14-math-0132$ for $c14-math-0133$ . The red, gray, and violet balls represent O, V, and Li atoms, respectively. (b) Comparison between the stacking order within the $c14-math-0134$ , and $c14-math-0135$ -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 $c14-math-0155$ 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 $c15-math-0019$ (within the estimated error bar), and the ECS at the right is constructed using equal interface energies of 0.2 J $c15-math-0020$ . In the lower row, a projection along the [110] axis, with the abscissa along [ $c15-math-0021$ ] 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 $c15-math-0022$ 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 ( $c15-math-0058$ 10) and (0 $c15-math-0059$ 1) 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 $c15-math-0092$ of a singly Mn-doped Si QD with a diameter of 2.18 nm versus the variation in the Si chemical potential $c15-math-0093$ 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 $c15-math-0099$ , $c15-math-0100$ , and $c15-math-0101$ configurations for the example of a Si QD with $c15-math-0102$ .
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 $c15-math-0121$ 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 ( $c15-math-0122$ ). The color of the drawn vectors indicates the absolute value of the magnetization (red = large, blue = small). The angle of about $c15-math-0123$ 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- $c13-math-0231$ 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- $c13-math-0261$ at a Fermi energy of $c13-math-0262$ eV [194, 195].
Table 13.6 Charge transition levels (obtained with the PBE0 functional) of $c13-math-0291$ and $c13-math-0292$ with respect to the valence band edge of m- $c13-math-0293$ .
Table 13.7 Thermochemical properties of the considered repair chemicals for silylation reactions with the prototypical ULK fragment $c13-math-0322$ . The size of the molecules $c13-math-0323$ 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 $c14-math-0030$ .
Table 14.2 Comparison of the obtained lattice parameters of three different $c14-math-0136$ phases with $c14-math-0137$ .

Chapter 15: Properties of Nanostructured Materials

Table 15.1 Theoretical and experimental lattice constants $c15-math-0005$ ( $c15-math-0006$ ) and band gaps (eV) of PbTe and CdTe as well as the effective longitudinal electron and hole masses $c15-math-0007$ at the L point of PbTe in units of $c15-math-0008$ .
Table 15.2 Calculated average interface energies in [J/ $c15-math-0017$ ] 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 $c15-math-0103$ in units of $c15-math-0104$ of a TM-doped Si QD with a diameter of 1.82 nm ( $c15-math-0105$ ). Only results for typical dopant configurations are listed. The superscript “ $c15-math-0106$ ” denotes interstitial sites beneath a {001} QD facet.

Related Titles

Frenking, G., Shaik, S. (eds.)

The Chemical Bond

2 Volume Set

2014

ISBN: 978-3-527-33318-9;