materials interaction with femtosecond lasers

Materials Interaction with Femtosecond Lasers

Bernd Bauerhenne

Materials Interactionwith Femtosecond LasersTheory and Ultra-Large-Scale Simulationsof Thermal and Nonthermal Pheomena

Bernd BauerhenneTheoretical Physics IIUniversity of KasselKassel, Germany

ISBN 978-3-030-85134-7 ISBN 978-3-030-85135-4 (eBook)https://doi.org/10.1007/978-3-030-85135-4

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https://doi.org/10.1007/978-3-030-85135-4

Foreword

Femtosecond laser pulses are extremely short and extremely intense light pulsesthat were first generated four decades ago and have been further developed andimproved since then. Nowadays, femtosecond laser pulses are used in research to,among other things, observe chemical reactions in real time, create nanostructures ormanipulate material surfaces. Femtosecond laser pulses are currently being appliedin different industries, in archaeology and in medicine. Themajor advantage of usingsuch light pulses is the possibility to manipulate matter far beyond the thermody-namic limits. Femtosecond laser pulses create an extreme non-equilibrium state inmaterials, in which the electrons can reach temperatures of tens of thousands ofdegrees Celsius, while the ions initially remain at near room temperature. In thisstate, very exciting and novel non-thermal phenomena can be observed, in whichcollective ionic motions take place that cannot occur under normal conditions, andwhich are theoretically described bymeans of precise density functional simulations.However, and due to their complexity, such calculations are only feasible for smallsystems up to a maximum of around thousand atoms. The above-mentioned laser-induced non-equilibrium state has a very short lifetime, which mainly depends onthe so-called electron-phonon interaction, i.e., on the collisions between electronsand ions. During this process, energy is transferred from the electrons to the ions,which leads to the buildup of a common temperature of the ionic lattice and the elec-trons. This happens on a time scale of a few picoseconds. From then on, the motionof the ions is thermal and can be determined using large-scale classical moleculardynamics simulations based on analytical interatomic potentials. Such moleculardynamics simulations can account for hundreds of millions of atoms. Laser-inducedstructural effects only become experimentally noticeable on a nano- to microsecondstime scale.

The understanding of the transition from non-thermal to thermal thermal ionicmotion is of fundamental importance for the description of laser laser processingof materials. Until Dr. Bauerhenne’s works, there was no theory that accuratelydescribes this transition. Dr. Bauerhenne was aiming at developing a unified theoryfor non-thermal and thermal laser-induced structural phenomena with atomic preci-sion able solve thismajor open problem of Solid State Physics. And I can confirm that

v

vi Foreword

he has perfectly succeeded. His work enables the description of ultrafast structuralphase transitions in materials with millions of atoms without having to sacrifice theaccuracy of density functional theory. Dr. Bauerhenne has not only developed thedesired theory but also invented amethod for constructing complex interatomic inter-actions that depend on the electron temperature, in order to describe laser-excitedsystems. On the basis of these interatomic potentials and the unified theory describedbefore, Bernd Bauerhenne performed large-scale atomistic simulations on variousmaterials.His results reproduce one-to-one the experimental observations based uponsophisticated pump-probe techniques. The present book describes the achievementsof Bernd Bauerhenne and provides a solid basis for the understanding of mechanicalproperties of materials under extreme conditions. The results shown in this book andobtained by B. Bauerhenne during his PhD work has resulted in numerous publi-cations in renowned journals. Many other novel results of his doctoral thesis, alsopresented here, are the subject of publications that are still in preparation.

I got to know Mr. Bauerhenne as a student in the 3rd semester when he attendedmy lecture “Theoretical Mechanics”. He immediately caught my attention due to hisinterest as well as his very original solutions to the exercises. Later on he joined mygroup for performing his Diploma work. Dr. Bauerhenne completed his studies ofphysics and mathematics simultaneously and with honors. His outstanding perfor-mance during his studies also qualified him to participate in the 66th Lindau NobelLaureate Meeting in 2016, which was dedicated to physics. With the method forgeneration of high-precision laser-excited potentials, he won the doctoral studentprize at European Materials Research Society (EMRS) meeting in Strasbourg in2017. Dr. Bauerhenne is both an extremely talented and a very hardworking youngscientist with a great deal of perseverance. He certainly belongs to the top 10% scien-tists at his current career stage. His Ph.D. thesis is one of the best ones I have eversupervised or reviewed.

The present book is scientifically rigorous and, at the same time, easy to under-stand. It is very well written, perfectly structured and its chapters look like anextremely successful lecture notes. In addition, many basic formulas that are inthe work are explained in more detail than in most textbooks. The notation is wellthought out. Each variable and quantity has its own designation. There are no repeti-tions of letters, which is remarkable with respect to the large amount of quantities thatare defined throughout the book. I am therefore very pleased that Mr. Bauerhenne ismaking his work available to a wide readership in the form of this extremely valuablebook.

Prof. Dr. Martin E. GarciaTheoretical solid-state and ultrafast

physicsUniversity of Kassel

Kassel, Germany

Acknowledgements

Atfirst, I amgrateful toZacharyEvenson and his editorial teamof SpringerNature fortheir support and endeavors, which has enabled the publication of this academicworkas a monograph. I also thank very much Prof. Dr. Martin E. Garcia from the Univer-sity of Kassel for supporting me during writing this comprehensive book project,which was originally a dissertation with the title “Unified theory for non-thermaland thermal effects occurring in matter following a femtosecond laser-excitation”at the University of Kassel in the department of natural sciences with the date ofthe disputation 10.12.2020. I would also like to thank very much Prof. Dr. FelipeValencia Hernandez from the National University of Columbia for giving me veryuseful comments and suggestions on this book and for the calculation of the electron-phonon coupling constant in antimony, which was essential for my simulations oflaser-excited antimony.

I acknowledge the wonderful cooperation with Dr. Sascha Epp from the MaxPlanck Institute for Structure and Dynamics of Matter. He performed ultrafast X-ray diffraction experiments on femtosecond laser-excited thin antimony films, withwhich I compared my theoretical calculations, and he provided me useful commentson this book. I am also grateful to Dr. Dmitry Ivanov for giving me very usefulhints and tips regarding ultra large scale MD simulations with classical interatomicpotentials to simulate laser-excited matter. I would like to acknowledge Dr. VladimirLipp for the useful scientific discussions.

I acknowledge gratefully the financial support by the “Promotionsstipendium desOtto-Braun Fonds” and by the “Abschlussstipendium der Universtät Kassel”. It wasessential for my research that I was able to perform my large scale calculations onthe IT Servicecenter (ITS) University of Kassel, on the Lichtenberg High Perfor-mance Computer (HHLR) TU Darmstadt, and on the computing cluster FUCHS inFrankfurt.

vii

About This Book

This book presents a unified view of the response of materials as a result offemtosecond laser excitation, introducing a general theory that captures bothultrashort-time non-thermal and long-time thermal phenomena. It includes a novelmethod for performing ultra-large-scale molecular dynamics simulations extendinginto experimental and technological spatial dimensions with ab-initio precision. Forthis, it introduces a new class of interatomic potentials, constructed from ab-initiodata with the help of a self-learning algorithm, and verified by direct comparisonwithexperiments in two differentmaterials—the semiconductor silicon and the semimetalantimony.

In addition to a detailed description of the new concepts introduced, as well asgiving a timely review of ultrafast phenomena, the book provides a rigorous intro-duction to the field of laser–matter interaction and ab-initio description of solids,delivering a complete and self-contained examination of the topic from the very firstprinciples. It explains, step by step from the basic physical principles, the under-lying concepts in quantum mechanics, solid-state physics, thermodynamics, statis-tical mechanics, and electrodynamics, introducing all necessary mathematical theo-rems as well as their proofs. A collection of appendices provide the reader with anappropriate review ofmany fundamentalmathematical concepts, aswell as importantanalytical and numerical parameters used in the simulations.

ix

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Ab-initio Description of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 Quantum Mechanical Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Born-Oppenheimer Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 Nuclei Motion in the Harmonic Approximationin Crystalline Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.3.1 Hohenberg-Kohn Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.3.2 Kohn–Sham Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.3.3 Approximations to the Exchange Correlation

Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.3.4 Bloch Waves in Crystalline Systems . . . . . . . . . . . . . . . . . . . . 512.3.5 Using a Set of Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . 532.3.6 Solving the Kohn–Sham Equations Self Consistently . . . . . . 552.3.7 Density Mixing to Speed up the Solution of the Kohn–

Sham Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.3.8 Pseudopotentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602.3.9 Electronic Band Structure of Solids . . . . . . . . . . . . . . . . . . . . . 63

2.4 Te-dependent Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . 642.4.1 Basic Considerations of Thermodynamics . . . . . . . . . . . . . . . 642.4.2 Basic Considerations of Statistical Mechanics . . . . . . . . . . . . 662.4.3 Mermin’s Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822.4.4 Te-dependent Kohn–Sham Equations . . . . . . . . . . . . . . . . . . . 86

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3 Ab-initio Description of a Fs-laser Excitation . . . . . . . . . . . . . . . . . . . . . 1033.1 Basic Considerations of Electrodynamics . . . . . . . . . . . . . . . . . . . . . . 103

3.1.1 Maxwell Equations in Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . 1043.1.2 Radiation of Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . 118

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xii Contents

3.1.3 Energy in Electromagnetic Fields . . . . . . . . . . . . . . . . . . . . . . . 1203.1.4 Interaction of a Charged Particle

with an Electromagnetic Wave . . . . . . . . . . . . . . . . . . . . . . . . . 1233.2 Basic Considerations of Second Quantization . . . . . . . . . . . . . . . . . . . 126

3.2.1 Second Quantization for Electrons . . . . . . . . . . . . . . . . . . . . . . 1263.2.2 Second Quantization for Phonons . . . . . . . . . . . . . . . . . . . . . . 132

3.3 Reduced Electron Density Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 1453.4 Effects of a Fs-laser Interaction on Matter . . . . . . . . . . . . . . . . . . . . . . 151

3.4.1 Effects of the Fs-Laser Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 1523.4.2 Electron Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1553.4.3 Electron-Phonon Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1563.4.4 Electron-Phonon Coupling Strength . . . . . . . . . . . . . . . . . . . . 161

3.5 Physical Picture of the Fs-laser Excitation . . . . . . . . . . . . . . . . . . . . . . 1713.6 Code for Highly Excited Valence Electron Systems (CHIVES) . . . . 1733.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

4 Ab-Initio MD Simulations of the Excited Potential EnergySurface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1794.1 Molecular Dynamics Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . 180

4.1.1 Velocity Verlet Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1814.1.2 Preparation of Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . 184

4.2 Calculation of the Diffraction Peak Intensities . . . . . . . . . . . . . . . . . . 1944.3 Fs-Laser Induced Thermal Phonon Squeezing

and Antisqueezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2004.4 DFT Calculations and MD Simulations of Si at Various Te’s . . . . . . 205

4.4.1 Equilibrium Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2064.4.2 Cohesive Energies at Various Te’s . . . . . . . . . . . . . . . . . . . . . . 2074.4.3 Phonon Band Structure at Various Te’s . . . . . . . . . . . . . . . . . . 2084.4.4 MD Simulations of Thermal Phonon Antisqueezing

at Moderate Te’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2104.4.5 MD Simulations of Non-thermal Melting at High Te’s . . . . . 2134.4.6 Behavior of the Electronic Indirect Band Gap . . . . . . . . . . . . 2164.4.7 MD Simulations of a Thin-Film at Various Te’s . . . . . . . . . . . 2204.4.8 Summary of the Effects Induced by an Increased Te . . . . . . . 224

4.5 DFT Calculations and MD Simulations of Sb at Various Te’s . . . . . . 2264.5.1 Equilibrium Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2264.5.2 Cohesive Energies at Various Te’s . . . . . . . . . . . . . . . . . . . . . . 2304.5.3 Potential Energy Surface and Displacive Excitation

of the A1g Phonon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2314.5.4 Phonon Band Structure at Various Te’s . . . . . . . . . . . . . . . . . . 2354.5.5 MD Simulations of the A1g-Phonon Excitation

at Various Te’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2364.5.6 MD Simulations of Thermal Phonon Antisqueezing

at Moderate Te’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

Contents xiii

4.5.7 MD Simulations of Non-thermal Melting at High Te’s . . . . . 2444.5.8 MD Simulations of a Thin-Film at Various Te’s . . . . . . . . . . . 2474.5.9 Summary of the Effects Induced by an Increased Te . . . . . . . 253

4.6 THz Emission from Coherent Phonon Oscillations in BNNTs . . . . . 2554.6.1 Equilibrium Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2554.6.2 Displacive Excitation of Coherent Phonons in BNNTs . . . . . 2594.6.3 THz Radiation from Coherent Phonon Oscillations

in the (5, 0) Zigzag BNNT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2644.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

5 Empirical MD Simulations of Laser-Excited Matter . . . . . . . . . . . . . . . 2755.1 Interatomic Potentials for Ground State Electrons in Solid

State Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2765.1.1 Classical Analytical Interatomic Potential Models . . . . . . . . 2815.1.2 Determining of Interatomic Potential Parameters . . . . . . . . . 2895.1.3 Machine Learning Interatomic Potentials . . . . . . . . . . . . . . . . 2915.1.4 Performing Large Scale MD Simulations . . . . . . . . . . . . . . . . 292

5.2 Simulation of Laser Excitation via Two Temperaturesand Velocity Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

5.3 Simulation of Laser Excitation via Bond-Softeningin the Tersoff Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

5.4 Te-Dependent Interatomic Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . 2985.4.1 Si Potential of Shokeen and Schelling . . . . . . . . . . . . . . . . . . . 2995.4.2 Si Potential of Darkins et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3045.4.3 MD Simulations with a Te-Dependent Interatomic

Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3075.5 Universal Interatomic Potential Parameter Fitting Program . . . . . . . 310

5.5.1 Construction of Fit Error Function . . . . . . . . . . . . . . . . . . . . . . 3115.5.2 General Definition of the Analytical Form

of the Interatomic Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3135.5.3 Analytical Expressions for the Interatomic Potential

Parameter Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3145.5.4 Efficient and Parallelized Implementation in Fortran . . . . . . 318


6 Ab-Initio Theory Considering Excited Potential EnergySurface and e−-Phonon Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3236.1 Usage of Global Temperatures in the Simulation Cell . . . . . . . . . . . . 324

6.1.1 Implementation in the Velocity Verlet Algorithm . . . . . . . . . 3326.1.2 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

6.2 Usage of Local Temperatures in the Simulation Cell . . . . . . . . . . . . . 3396.2.1 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3476.2.2 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

6.3 Polynomial Te-Dependent Interatomic Potential Model . . . . . . . . . . 352

xiv Contents

6.3.1 Polynomial Functional Form . . . . . . . . . . . . . . . . . . . . . . . . . . . 3536.3.2 Fitting of Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3576.3.3 Optimal Polynomial-Degree Combination Selection

Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3616.3.4 Easy Evaluation via Power Lists . . . . . . . . . . . . . . . . . . . . . . . . 3626.3.5 Efficient Evaluation of the Three-Body Term . . . . . . . . . . . . . 3626.3.6 Efficient Evaluation of the Four-Body Term . . . . . . . . . . . . . . 369


7 Study of Femtosecond-Laser Excited Si . . . . . . . . . . . . . . . . . . . . . . . . . . . 3797.1 Te-Dependent Interatomic Potential for Si . . . . . . . . . . . . . . . . . . . . . . 380

7.1.1 Ab-Initio Reference Simulations Used for Fitting . . . . . . . . . 3807.1.2 Parameter Fitting of Classical Interatomic Potentials . . . . . . 3817.1.3 Polynomial Interatomic Potential �(Si)(Te) . . . . . . . . . . . . . . 3837.1.4 Physical Properties of Polynomial �(Si)(Te) . . . . . . . . . . . . . . 3867.1.5 Thermophysical Properties of Polynomial �(Si)(Te) . . . . . . . 391

7.2 MD Simulations of Excited PES and EPC with Polynomial�(Si)(Te) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3977.2.1 Direct Comparison of the Bragg Peak Intensities

with Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3987.2.2 MD Simulations of a Femtosecond-Laser Excited Si

Film . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4057.2.3 MD Simulations of Femtosecond-Laser Excited Bulk

Si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4107.3 Correction of the Melting Temperature . . . . . . . . . . . . . . . . . . . . . . . . 415

7.3.1 Correction of the 3-Body Potential Coefficients . . . . . . . . . . . 4167.3.2 Melting Temperature and Slope Study on Test

Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4227.3.3 Correction of the 2-Body and 3-Body Potential

Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4277.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434

8 Study of Femtosecond-Laser Excited Sb . . . . . . . . . . . . . . . . . . . . . . . . . . 4378.1 Te-dependent Interatomic Potential for Sb . . . . . . . . . . . . . . . . . . . . . . 438

8.1.1 Ab-initio Reference Simulations Used for Fitting . . . . . . . . . 4388.1.2 Optimization of the Functional Formof the Polynomial

Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4398.1.3 Physical Properties of Polynomial �(Sb)(Te) . . . . . . . . . . . . . 441

8.2 Optical Properties of Sb as a Function of the Peierls Parameter . . . . 4468.3 MD Simulations of Excited PES and EPC with Polynomial

�(Sb)(Te) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4498.3.1 Direct Comparison of the Bragg Peak Intensities

with Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4518.3.2 Laser-Induced A7 to Sc Transition . . . . . . . . . . . . . . . . . . . . . . 459

Contents xv


9 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4739.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

9.1.1 THz Emission from Coherent Phonon Oscillations . . . . . . . . 4749.1.2 Universal Behavior of the Indirect Electronic Band

Gap in Laser-Excited Si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4749.1.3 Theory Allowing MD Simulations Considering

Excited Potential EnergySurface andElectron-PhononCoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474

9.1.4 Construction of Efficient and Highly AccurateTe-Dependent Interatomic Potentials . . . . . . . . . . . . . . . . . . . . 475

9.1.5 Te-Dependent Interatomic Potential �(Si)(Te) for Si . . . . . . . 4769.1.6 Correction of the Melting Temperature of �(Si)(Te)

to the Experimental Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4769.1.7 MD Simulations of Femtosecond Laser-Pulse Excited

Si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4779.1.8 Te-Dependent Interatomic Potential �(Sb)(Te) for Sb . . . . . . 4779.1.9 MD Simulations of Femtosecond Laser-Pulse Excited

Sb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4789.2 Future Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

Appendix A: Additional Information and Tables . . . . . . . . . . . . . . . . . . . . . 481

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

About the Author

BerndBauerhenne conducts research in the Solid State andUltrafast Physics Groupat the Institute of Theoretical Physics of the University of Kassel. One focus of hisresearch is the theory of ultrafast phenomena in solids and nanostructures, in partic-ular, the description of ultrafast structural changes induced by an intense femtosecondlaser. Among other things, he develops highly accurate interatomic potentials usingself-learning algorithms, performs ultra large scale molecular dynamics simulations,and applies electronic temperature-dependent density functional theory. He studiedmathematics and physics at the University of Kassel and at the University of Luxem-burg with a focus on numerics and dynamical systems inmathematics and theoreticalmodeling of solids in physics. He received a diploma in mathematics and a diplomain physics and he completed his Ph.D. in theoretical physics at the University ofKassel. He obtained the Otto Braun Fund doctoral fellowship and the University ofKassel final fellowship, won the Ph.D. student award of the SymposionX at the Euro-pean Materials Research Society meeting in Strasbourg 2017, and had the honor ofattending the 66th Lindau Nobel Laureate Meeting, which was dedicated to physics.

xvii

Acronyms

Au Goldbcc Body centered cubicBNNT Boron nitride nanotubeCHIVES Code for Highly excIted Valence Electron SystemsDFT Density functional theorydia Diamond-like structuree− Electrone.g. Exempli gratia =∧ for exampleEPC Electron-phonon couplingfcc Face centered cubicfs FemtosecondFWHM Full width at half maximumGGA Generalized gradient approximationi.e. Id est =∧ that isLDA Local density approximationMD Molecular dynamicsMEAM Modified Embedded Atom MethodMo MolybdenumMPI Message passing interfaceOpenMP Open multi-processingPES Potential energy surfaceps PicosecondRPA Random-phase approximationsc Simple cubicSb AntimonySi SiliconTTM Two-temperature modelTTM-MD Two-temperature model combined with molecular dynamics simula-

tionsW Tungsten

xix

List of Symbols

1 Unity matrix1 Unity operatora Lattice parameterαs Super diffusion exponentαf Fractional diffusion exponentαabs Absorption coefficientℵ2 Parameter to manipulate the two-body potential in order to

correct the melting temperature of the interatomic potentialℵ3 Parameter to manipulate the three-body potential in order to

correct the melting temperature of the interatomic potentialarg(r) Argument functiona1, a2, a3 Lattice vectors in R3 of the Bravais gridA Vector potential in R3 used in Electrodynamics

b∧†

jq Phonon creation operator that creates a phonon in state jq

b∧

jq Phonon annihilation operator that creates a phonon in state jqb1,b2,b3 Lattice vectors in R3 of the reciprocal gridB Magnetic field vector in R3

Cip Specific heat of the ions at constant pressureCe Specific heat of the electronsC (i)e Local specific heat of the electrons in sub cell i

c Speed of light in vacuumC Set of all complex numbersC11,C12,C44 Independent elastic constants of Si in the bulk diamond-like

crystal structurecx , cy, cz Cell lengths of the simulation cell in the x , y, and z directionc(q)

2 Coefficients of the two-body potentialc(q1q2q3)3 Coefficients of the three-body potentialc(q1q2q3q4q5q6)4 Coefficients of the four-body potentialc(q1q2)ρ Coefficients of the embedded atom potential

xxi

xxii List of Symbols

C Orthonormal matrix in R3Nat×3Nat that diagonalizes the dynam-ical matrix D

C(qn) Unitary matrix in C3Nb×3Nb that diagonalizes the Fourier trans-formed dynamical matrix D(qn)

C jk(q) Matrix element in row j and column k of matrix C(qn)c∧†�k Electron creation operator that creates an electron in state �kc∧

�k Electron annihilation operator that destroys the electron in state�k

dfilm Thickness of the filmD Diffusion coefficient�x,�y,�z Cell lengths of the sub cells in the x , y, and z directiond Electrical dipole moment vector in R3

d(I) Electrical dipole moment vector in R3 caused by the atoms ofset I

d(I)e Electrical dipole moment vector in R3 caused by the electrons

of the atoms of set Id(I)ions Electrical dipole moment vector in R3 caused by the ions of set

Idυ Relative position vector in R3 of atoms belonging to basis υ

δi j Kronecker deltaδ(r) Delta functionD(E) Electron density of states at energy ED Dynamical matrix in R3Nat×3Nat

DTnυα,Tn′ υ ′α′ Dynamical matrix elementD(qn) Fourier transformed dynamical matrix in C3Nb×3Nb at wave

vector qnDυα,υ ′α′(qn) Matrix element of the Fourier transformed dynamical matrix at

wave vector qne Electronic chargee Euler’s numbererf(x) Gauss error functionE (s�)err (Te) Normalized root-mean square deviation in cohesive energies of

reference simulation s� at electronic temperature TeE (s�)sum(Te) Sum of the cohesive energies of the ab-initio forces of reference

simulation s� at electronic temperature TeE (Internal) EnergyEtotal Total energyEphot Photon energyEkin Kinetic energy of the nucleiE (T)kin Thermal kinetic energy of the nuclei of a link cell

E (i)kin Local kinetic energy of the nuclei in sub cell i

EkinMkKinetic energy in the phonon modes of setMk

E (i)kinM(i)

k

Local kinetic energy in the phonon modes of setM(i)k in sub cell

i

List of Symbols xxiii

ELtot Total energy that is absorbed from the laserEL(t�) Total laser energy that is penetrating into the total simulation

volume up to time t��E (i)

L (t�) Total laser energy that is penetrating into the sub cell i duringtime step t�

�E(itop)L (t�) Laser energy that is penetrating into the top sub cells itop during

time step t�ELabs(t) Energy that is deposited by the laser in the electronic system up

to time tE (i)Labs

(t) Energy that is locally deposited by the laser in the electronicsystem of sub cell i up to time t

�ELabs(t�) Energy that is deposited by the laser in the electronic systemduring time step �t�

�E (i)Labs

(t�) Energy that is locally deposited by the laser in the electronicsystem in sub cell i during time step �t�

Ec Cohesive energyE (s�)c (Te, tk) Cohesive energy in reference simulation s� at time step tk and

electronic temperature TeEF Fermi energyEfusion Enthalpy of fusionEi Total energy of the ionsE (i)D (t�) Energy that is transferred from sub cell i to the neighboring sub

cells up to time t� due to heat conductivity caused by differentelectronic temperatures

ED(t�) Total energy that was transferred in or out of the completesimulation cell due to heat conductivity up to time t�

�E (i)D (t�) Total electronic heat diffusion of sub cell i during time step t�

�E (i,i′)D (t�) Electronic heat diffusion from sub cell i′ to sub cell i during time

step t�Ee Total energy of the electronsE (i)e Local internal energy of the electrons of sub cell i

�Ee(t�) Total change of the internal energy of the electrons at time stept�

�E (i)e (t�) Total change of the local internal energy of the electrons of sub

cell i at time step t�Eep Energy transfer rate between electrons and nuclei due to

electron-phonon couplingE (i)ep Local energy transfer rate between electrons and nuclei due to

electron-phonon coupling in sub cell i�Eep(t�) Energy that is transferred from the nuclei to the electrons due to

electron-phonon coupling at time step t��E (i)

ep (t�) Energy that is locally transferred in sub cell i from the nuclei tothe electrons due to electron-phonon coupling at time step t�

Egap Electronic band gap

xxiv List of Symbols

Enn Interaction energy of the nucleiEHK Hohenberg-Kohn energy functionalEHa Hartree energy functionalEKS Kohn-Sham energy functionalExc Exchange correlation functionalE (LDA)xc Exchange correlation functional in the local density approxima-

tionE (GGA)xc Exchange correlation functional in the generalized gradient

approximationEα(r1, . . . , rNat ) Potential energy surface of the electrons in state α for nuclei at

positions r1, . . . , rNat

E Electrical field vector in R3

e∧

x , e∧

y, e∧

z Unit vectors in Cartesian coordinates x, y, ze∧

r , e∧

φ, e∧

ϑ Unit vectors in spherical coordinates r, φ, ϑ

e(i) i-th Column vector in R3Nat of Ct describing the nuclei motionsof the i-th eigenmode

e(i)(qn) i-th Column vector in R3Nb of C†(qn) describing the nucleimotions in the basis cell of the i-th eigenmode at wave vector qn

ε0 Vacuum permittivityε(hom)xc Exchange-correlation energy density of the homogeneous elec-

tron gasfβ Equilibrium occupation number of orbital βFnγ Force on atom n in direction γ obtained from the interatomic

potentialF Helmholtz free energyFHK Hohenberg-Kohn functional describing the kinetic and interac-

tion energy of the electronsFM Mermin’s Helmholtz free energy functionalFKS Kohn-Sham Helmholtz free energy functionalfc(r) Cutoff functionfG(x) Gaußian probability distribution functionfF(E, Te) Fermi distribution functionfepsf(E, E ′, ω) Electron-phonon spectral functionfEl(ω) Eliashberg functionf (s�)err (Te) Normalized root-mean square deviation in atomic forces of

reference simulation s� at electronic temperature Tef (s�)sum (Te) Sumof the squares of the ab-initio forces of reference simulation

s� at electronic temperature Tef Force vector in R3

fi Force vector in R3 on atomf (s�)i ((Te, tk)) Force vector in R3 on atom i in reference simulation s� at time

step tk and electronic temperature Tef (s)i Stochastic force vector in R3 on atom iF Vector inR3Nat of all atomic forces obtained from the interatomic

potential

List of Symbols xxv

F (i) Vector in R3N (i)at of forces of all atoms located in the sub cell i

obtained from the interatomic potentialFtot Vector in R3Nat of all total atomic forcesF (i)tot Vector in R3N (i)

at of total forces of all atoms located in the sub cellgi Random number following a normal Gaußian distributionGn Vector in R3 belonging to the reciprocal gridGep Electron-phonon coupling constantG(i)

ep Local electron-phonon coupling constant in sub cell iGepMk

Electron-phonon coupling constant for the phonon modes of setMk

G(i)epM(i)

k

Local electron-phonon coupling constant for the phonon modes

of setM(i)k in sub cell i

G(x) Integral of the normal Gaußian distributionG−1(x) Inverse function of G(x)� Gamma function�(i → j) Transition rate from state i to state jH Hamilton functionH (harm) Hamilton function in the harmonic approximationHL Additional term in the Hamilton function describing the electro-

magnetic field interaction on a charged particleHL Operator describing the electromagnetic field interaction on the

electronsH Hamilton operatorH (harm) Hamilton operator in the harmonic approximationHe Electronic Hamilton operatorH (1)i One-particle Hamilton operator for electron i

H (L)i Operator describing the laser field interaction on electron i

HKS Kohn-Sham Hamilton operatorH Set of quantum numbers of the Hamilton operatorHMk (t�+1) Kinetic energy of the phonon mode set Mk at time step t�+1, if

the influence of the electron-phonon coupling is neglected at thetime step t�+1

� Reduced Planck’s constantIinc(x, y, t) Spatial incident laser intensity at position x, y and time tIinctot (x, y) Total incident laser fluence at position x, yILtot (x, y) Total absorbed laser fluence at the surface at position x, yI (damage)Ltot

Damage threshold for the incident laser fluenceIq(t) Diffraction peak intensity for reciprocal vector q at time tIhkl(t) Diffraction peak intensity for the Miller indices (hkl) at time tIm(x) Imaginary part of xi Imaginary uniti Integer vector in N3

0 labeling the sub cells in the simulation cell

xxvi List of Symbols

I Set of all integer vectors i labeling the sub cells that are locatedin the simulation cell

J Charge current density vector in R3

kB Boltzmann constantKe Electronic heat conductivityK (i)

e Local electronic heat conductivity of sub cell iκ(4i)q2q4q7q6q9 Term used to calculate the four-body potential like a two-body

potentialk Electron wave vector in R3

L Lorenz constantLmin Minimal periodical length of a nanotubeLo Optical penetration depthLb Ballistic rangeλ Wave lengthMSD(t) Mean-square displacements of the atoms at time tMSDM(t) Mean-square displacements of the atoms at time t in direction

of the phonon modes of setMMSDx (t) Mean-square displacements of the atoms at time t in the

x-directionMSDy(t) Mean-square displacements of the atoms at time t in the

y-directionMSDz(t) Mean-square displacements of the atoms at time t in the

z-directionm Massmk Mass of atom kmυ Mass of atoms belonging to basis υ

me Electron massM ( jq)

�k+q+G,�′k Electron-phonon coupling matrix elementMs(Te) Set of ab-initio reference simulations s� used for the interatomic

potential fit at electronic temperature TeMk k-th subset containing the indices of the corresponding phonon

modesM(i)

k k k-th subset containing the indices of the corresponding phononmodes in sub cell i

MN (θ)4

(q1, q2, q3) Set that contains the allowed powers (q4, q5, q6) of the bond

angles as a function of N (θ)4 and the distance powers (q1, q2, q3)

μ Chemical potentialμe Chemical potential of the electronsμ0 Vacuum permeabilityn Index of refractionNb Number of atoms in the basis cellNe Total number of electronsNat Total number of atomsN (i)at Number of atoms in sub cell i

List of Symbols xxvii

N (s�)at Number of atoms used in reference simulation s� for fitting

N (s�)t Number of (time) steps used in reference simulation s� for fitting

ND Number of time steps �tD for solving the diffusion equationsthat are performed during one time step �t for solving the ionicequations of motion

Nx , Ny, Nz Number of sub cells used in the simulation cell in the x , y, andz direction

NM Total number of different subsets Mk of phonon modesN (r)2 Degree of the two-body potential

N (r)3 Radial degree of the three-body potential

N (θ)3 Angular degree of the three-body potential

N (r)4 Radial degree of the four-body potential

N (θ)4 Angular degree of the four-body potential

N (r)ρ Number of measures ρ

(q1)i of the atomic density used in the

embedded atom potentialN (ρ)

ρ Density degree of the embedded atom potentialNC Total number of coefficients used in the polynomial potentialNd Total number of data points used for fitting of the interatomic

potentialN (q3, q4) Function to numerate the coefficients of the spherical harmonicsn�k Electron occupation number operator that counts the number of

electrons in state �kn�k Number of electrons in state �kn jq Phonon occupation number operator that counts the number of

phonons in state jqn jq Number of phonons in state jqne(r) Electron density operatorne(r) Electron densityne0(r) Ground state electron densityn(out,k)e (r) Output electron density in mixing step k

n(in,k)e (r) Input electron density in mixing step k

N Set of all positive integer numbersNi Set containing the indices of the neighboring atoms of atom in Vector in Z3 characterizing the position in the Bravais or

reciprocal gridηi j Measure for the bond order of bond between atom i and jηloc Function determining how a bond contributes to the bond order

of a given bond� Number of statesρ Charge densityρat Atomic density at ambient conditionsρi Atomic density surrounding atom iρq1i Measure for the atomic density surrounding atom i

xxviii List of Symbols

ρloc Function determining how an atom contributes to the atomicdensity ρi surrounding atom i

ρ Density matrixρ(mc) Density matrix of the microcanonical ensembleρ(c) Density matrix of the canonical ensembleρe Electronic density matrixρ(1)e One-electron reduced density matrix

ρ(2)e Two-electron reduced density matrixp Momentump Pressurep ProbabilityP(t) Total emitted power at time t℘�(x)� �-th Legendre polynomialp

(�)i (r) Radial projector in the pseudopotentialp Momentum vector operatorpnυ Momentum vector in R3 of nucleus at position nυ

pnυα Operator describing the momentum in direction α of nucleus atposition nυ

pnυα Momentum in direction α of nucleus at position nυ

P Vector in R3Nat of all mass-normalized nuclei momentaPυα(Tn) Mass-normalized momentum in direction α of nucleus at posi-

tion nυ

P Vector of all mass-normalized nuclei momenta in direction ofthe eigenmodes

P(qn) Vector of all mass-normalized nuclei momenta in reciprocalspace at wave vector qn

Pυα(qn) Mass-normalized nuclei momentum in direction α of nuclei ofbasis υ in reciprocal space at wave vector qn

Pυα(qn) Operator describing the mass-normalized nuclei momentum indirection α of nuclei of basis υ in reciprocal space at wave vectorqn

P(qn) Vector of all mass-normalized nuclei momenta in direction ofthe eigenmodes in reciprocal space at wave vector qn

P j (qn) Momentum operator of j-th phonon eigenmode in reciprocalspace at wave vector qn

PM Projection matrix that projects onto the directions of the phononmodes in setM

P(i)M(i)

k

Local projection matrix of sub cell i that projects onto the

directions of the phonon modes in set M(i)k

π Circle number� Interatomic potential�loc Local energy contribution�0 Potential energy of an isolated atom�2 Pair or two-body potential

List of Symbols xxix

�3 Three-body potential�

(tot)3 Total three-body term

�4 Four-body potential�

(tot)4 Total four-body term

�2b Bond-order potential�ρ Embedded atom potentialφs Scalar potential used in Electrodynamics� Wave functionQ Heatq Electrical chargeqi i-th generalized coordinateqn Phonon wave vector in R3

R Reflectivityr (c) (Global) Cutoff radiusr (c)ρ Cutoff radius of the embedded atom potential

r (c)2 Cutoff radius of the two-body potentialr (c)3 Cutoff radius of the three-body potentialr (c)4 Cutoff radius of the four-body potentialr (c)2b Cutoff radius of the bond-order potentialri j Distance between atom i and jri j Distance vector in R3 between atom i and jri j Normalized distance vector in R3 between atom i and jrs Wigner-Seitz sphere radiusrk Randomnumber following anuniformdistribution in the interval

[0, 1]RMSDz(t) Root mean-square displacement of the atoms at time t in the

z-directionRMSDy(t) Root mean-square displacement of the atoms at time t in the

y-directionRMSDxy(t) Root mean-square displacement of the atoms at time t in the

x, y-directionsRMSDxz(t) Root mean-square displacement of the atoms at time t in the

x, z-directionsR Set of all real numbersR Vector in R3Nat of all atomic coordinatesR(i) Vector in R3N (i)

at of the coordinates of all atoms located in subcell i

ri Coordinate vector in R3 of atom ir(s�)i (Te, tk) Coordinate vector in R3 of atom i in reference simulation s� at

time step tk and electronic temperature Ter(opt)i Coordinate vector in R3 of the equilibrium position of atom iRi Coordinate vector in R3 of electron iS EntropySe Electronic entropy

xxx List of Symbols

S(i)e Local electronic entropy of sub cell i

Sbest Subset of efficient polynomial-degree combinationsSFq Structure factor for wave vector qS(V) Surface of volume (V)

S Poynting vector in R3

σ Spin of the electronσel Electrical conductivitys PermutationSNe Set of all permutations s of the integer set {1, . . . , Ne}t Time�t Time increment�tD Time increment to solve the heat diffusion equationtr Retarded timeτ Full width at half maximum (FWHM) time widthT TemperatureTm Melting temperatureTe Electronic temperatureT (i)e Local electronic temperature of sub cell i

�Te(t�) Change of the electronic temperature at time step t��T (i)

e (t�) Change of the local electronic temperature in sub cell i at timestep t�

Te Kinetic energy operator of the electronsTi Ionic temperatureT (i)i Local ionic temperature in sub cell i

TiMkIonic temperature of the phonon modes of set Mk

T (i)iM(i)

k

Local ionic temperature of the phonon modes of setM(i)k in sub

cell iTi Kinetic energy operator of the nucleiTaux Kinetic energy functional of the auxiliary system of non-

interacting electronsTr( A) Trace of the operator A

Translation operator related to the vector Tn

tn Eigenvalue of the translation operatorTn Vector in R3 belonging to the Bravais gridθi jk Angle between the bond from atom i to j and i to ku jk(r) Periodical function within the Bloch functionuem Energy density of the electromagnetic fieldsumech Mechanical energy densityunυ Displacement vector in R3 of nucleus at position nυ

unυα Displacement in direction α of nucleus at position nυ

unυα Operator describing the displacement in direction α of nucleusat position nυ

U Vector in R3Nat of all mass-normalized nuclei displacementsu Vector in R3Nat of all nuclei displacements

List of Symbols xxxi

Uυα(Tn) Mass-normalized displacement in direction α of nucleus atposition nυ

Uυα(Tn) Operator describing themass-normalized displacement in direc-tion α of nucleus at position nυ

U Vector in R3Nat of all mass-normalized nuclei displacements indirection of the eigenmodes

U(qn) Vector in R3Nb of all mass-normalized nuclei displacements inreciprocal space at wave vector qn

Uυα(qn) Mass-normalized nuclei displacement in direction α of nucleiof basis υ in reciprocal space at wave vector qn

Uυα(qn) Operator describing the mass-normalized nuclei displacementin direction α of nuclei of basis υ in reciprocal space at wavevector qn

U(qn) Vector in inR3Nb of all mass-normalized nuclei displacements indirection of the eigenmodes in reciprocal space at wave vectorqn

U j (qn) Displacement operator of j-th phonon eigenmode in reciprocalspace at wave vector qn

v j Velocity vector in R3 of nuclei jv(c) Velocity vector in R3 of the collective motion in the link cellv(T)i Velocity vector in R3 of the thermal motion of atom iVext External potential operator of the electronsVext(R) External potential of the electronsVxc(R) Exchange and correlation potentialV (hom)xc (R) Exchange and correlation potential of the homogeneous electron

gasVeff(R) Effective potential of the electrons in the Kohn-Sham equationsVint Operator describing the interaction of the electrons with each

otherV (int)i j Operator that describes the interaction of electrons i and j

V (ep)i Operator describing the interaction of electron i with the

phononsV (ep) Operator describing the interaction of the electrons with the

phononsV (ep)

�k,�′k′ Matrix element of the electron-phonon coupling operatorV (υ)(r) Single nuclei potentialV (υ)k (r) Fourier coefficient at wave vector k of the single nuclei potential

Vloc(r) Local part of the pseudopotentialVnl(r, r′) Non-local part of the pseudopotentialV VolumeVs Volume of the simulation cellVb Volume of the basis cellV Vector in R3Nat of all atomic velocitiesV(i) Vector in R3N (i)

at of velocities of all atoms located in sub cell i

xxxii List of Symbols

W WorkWerr Interatomic potential fit error functionW(t�+1) Vector in R3Nat that contains the velocities of all nuclei at time

step t� + 1, if the electron-phonon coupling is neglected at timestep t� + 1

ωi Eigenfrequency of the i-th eigenmodew Wrapping vector in R3 to form the BNNTW Diagonal matrix in R3Nat×3Nat that contains ω2

i on the diagonalW(qn) Diagonal matrix in R3Nb×3Nb that contains ω2

i (qn) on thediagonal at wave vector qn

w(s�)f (Te) Fit weight for the forces of reference simulation s� at electronic

temperature Tew(s�)E (Te) Fit weight for the cohesive energies of reference simulation s�

at electronic temperature Teξ Constant describing the acceleration strength of the ions due to

the electron-phonon couplingξMk Constant describing the acceleration strength of the ions in direc-

tion of the phonon modes of setMk due to the electron-phononcoupling

ξ(i)M(i)

k

Local constant describing the acceleration strength of the ions

in direction of the phonon modes of setM(i)k due to the electron-

phonon coupling in sub cell iξi j Measure for the bond order of bond between atom i and jξloc Function determining how a bond contributes to the bond order

of a given bondχi j Measure for the bond order of bond between atom i and jχloc Function determining how a bond contributes to the bond order

of a given bondχ(3i)q1q3q4 Term used to calculate the three-body potential like a two-body

potentialχ(4i)q1q4q7q5q8 Term used to calculate the four-body potential like a two-body

potentialY�m(r) Spherical harmonic functionZk Proton number of atom kZ (c) Quantum mechanical partition function of the canonical

ensembleZc Classical partition function of the canonical ensembleZ Set of all integer numbersz Peierls parameterζ Constant occurring in the classical partition functionZ(c) of the

canonical ensemble∞ Infinity

materials interaction with femtosecond lasers

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