Ab initio Investigations of the Intrinsic Optical Properties of Germanium and Silicon Nanocrystals D i s s e r t a t i o n zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) vorgelegt dem Rat der Physikalisch-Astronomischen Fakultät der Friedrich-Schiller-Universität Jena von Dipl.-Phys. Hans-Christian Weißker geboren am 12. Juli 1971 in Greiz Gutachter: 1. Prof. Dr. Friedhelm Bechstedt, Jena. 2. Dr. Lucia Reining, Palaiseau, Frankreich. 3. Prof. Dr. Victor Borisenko, Minsk, Weißrußland. Tag der letzten Rigorosumsprüfung: 12. August 2004. Tag der öffentlichen Verteidigung: 19. Oktober 2004. Why is warrant important to knowledge? In part because true opinion might be reached by arbitrary, unreliable means. Peter Railton1 1Explanation and Metaphysical Controversy, in P. Kitcher and W.C. Salmon (eds.), Scientific Explanation, Vol. 13, Minnesota Studies in the Philosophy of Science, Minnesota, 1989. Contents 1 Introduction 7 2 Theoretical Foundations 13 2.1 Density-Functional Theory . 13 2.1.1 The Hohenberg-Kohn Theorem . 13 2.1.2 The Kohn-Sham scheme . 15 2.1.3 Transition to system without spin-polarization . 17 2.1.4 Physical interpretation by comparison to Hartree-Fock . 17 2.1.5 LDA and LSDA . 19 2.1.6 Forces in DFT . 19 2.2 Excitation Energies . 20 2.2.1 Quasiparticles . 20 2.2.2 Self-energy corrections . 22 2.2.3 Excitation energies from total energies . 25 QP 2.2.3.1 Conventional ∆SCF method: Eg = I − A . 25 QP 2.2.3.2 Discussion of the ∆SCF method Eg = I − A . 25 2.2.3.3 ∆SCF with occupation constraint . 28 2.2.4 Other methods to calculate excitation energies of nanostructures . 29 2.2.5 Spin: singlet vs. triplet excitons . 31 2.2.6 Stokes shifts . 32 2.3 Optical Properties . 33 2.3.1 Dielectric function . 33 2.3.2 Radiative recombination . 35 2.4 Projector-Augmented-Wave Method and Matrix Elements . 35 3 Model, Method, and Numerical Implementation 39 3.1 Model . 39 3.1.1 Free crystallites . 39 3.1.2 Crystallites embedded in a crystalline matrix . 40 3.2 Electronic-Structure Calculations . 41 3.2.1 Supercell method and description of NCs . 41 3.2.2 Algorithms and potentials . 41 3.2.3 k points and cell size . 42 3.2.4 Cut-off energies . 44 3.3 Ionic Relaxations . 44 3.4 Calculation of the Dielectric Function . 45 3.4.1 Number of conduction bands . 46 2 CONTENTS 3 3.4.2 BZ integration: Tetrahedron method . 46 3.4.3 Extrapolation . 49 3.4.4 Resampling – tetrahedron mesh . 50 3.4.5 Band kissing / Anticrossing correction . 53 3.4.6 Spurious transitions . 55 3.4.7 Matrix element extrapolation . 56 3.5 Results for Constituents . 57 4 Results 61 4.1 Free, H-terminated Nanocrystals . 61 4.1.1 Structure and importance of relaxation . 61 4.1.2 Excitation energies . 65 4.1.3 Exchange splitting . 70 4.1.4 Transition probabilities . 71 4.1.5 Radiative lifetimes . 72 4.1.6 Spectra . 74 4.2 Alloying: Germanium and Silicon . 77 4.3 Embedment of the NCs . 80 4.3.1 Electronic properties . 80 4.3.2 Optical spectra . 82 4.3.3 Hexagonal matrix and NCs . 83 4.3.4 Influence of relaxation . 84 4.4 Beyond the Ground-State Equilibrium . 86 4.4.1 Pressure / Strain . 86 4.4.2 Stokes shifts . 88 5 Conclusion and Prospectives 91 Acknowledgments 97 References 99 Deutsche Zusammenfassung i Ehrenwörtliche Erklärung vii Lebenslauf ix Publikationsliste x Vortragsliste xii List of Figures 2.1 Schematic of Vxc discontinuity . 22 2.2 Cancellation of Coulomb and self-energy effects . 24 2.3 Stoke shift: Schematic . 32 3.1 Cubic GeSi NC in hexagonal SiC . 40 3.2 Model structure: 83-atom NC free and embedded. 40 3.3 Number of Ge valence electrons: 4 vs. 22 . 42 3.4 Comparison of PAW and norm-conserving pseudopotential: Spectra . 42 3.5 Convergence of electronic structure: k points. 43 3.6 Convergence with cell size: Bands . 43 3.7 Convergence with cell size: Spectra . 43 3.8 Convergence: Energy cut-off for PAW data sets . 44 3.9 Convergence: Energy cut-off and spectra . 44 3.10 Relaxation algorithms . 45 3.11 Convergence of ionic relaxation: Si41H60 excited state . 45 3.12 Convergence of ionic relaxation: Si41H60 ground state . 45 3.13 Contribution of band pairs to spectra. 46 3.14 Schematic of tetrahedron . 48 3.15 Quality of matrix elements . 49 3.16 Resampling: Tetrahedron meshes . 49 3.17 Linear vs. quadratic extrapolation . 49 3.18 Weight problem vs. symmetry-equivalent tetrahedron corners . 52 3.19 Band integration parameter . 52 3.20 Extrapolation: Number of bands in k·p expression . 53 3.21 Anticrossing / “Kissing” correction . 53 3.22 Applicability tetrahedron method: cell size . 53 3.23 Supercells: Recovering the joint density of states . 55 3.24 Density of states . 55 3.25 Embedded structures: Applicability of extrapolative tetrahedron method . 55 3.26 Bulk bands Ge: Theoretical lattice constant vs. SiC lattice constant . 58 3.27 Bulk bands Si: Theoretical lattice constant vs. SiC lattice constant . 58 3.28 Bulk bands SiC: Theoretical lattice constant . 58 3.29 Spectra of the constituent materials of the NCs . 58 4 LIST OF FIGURES 5 4.1 Averaged interatomic distances . 62 4.2 Bond-lengths distribution . 62 4.3 Relaxation pattern . 63 4.4 Wave functions near HOMO-LUMO gap . 63 4.5 Effect of relaxation on excitation energies . 64 4.6 Effect of relaxation on radiative lifetimes . 64 4.7 Effect of relaxation on oscillator strengths of lowest transitions . 64 4.8 Si: Excitation energies . 66 4.9 Fits and hypotheses: Si . 66 4.10 ∆SCF excitation energies vs. LDA HOMO-LUMO gap . 67 4.11 Excitation energies Ge . 68 4.12 Excitation energies Ge and Si: Unrelaxed . 68 4.13 Excitation energies Ge and Si: Relaxed . 68 4.14 Exchange splitting: Si and Ge . 70 4.15 Oscillator strengths and dielectric function . 71 4.16 Level scheme in Si and Ge NCs . 72 4.17 Radiative lifetimes Si . 73 4.18 Radiative lifetimes Si . 73 4.19 Radiative lifetimes Ge . 73 4.20 Optical spectrum: Si . 75 4.21 Optical spectrum: Ge . 75 4.22 Absorption Ge: Comparison with experiment . 75 4.23 Spectra and transition probabilities: nonsymmetric Ge and Si NCs . 76 4.24 GeSi alloy NCs: interatomic distances . 77 4.25 Excitation energies: Alloying . 77 4.26 Lifetimes GeSi NCs . 78 4.27 Spectra GeSi NCs . 78 4.28 Band structure of Ge NC embedded in SiC . 80 4.29 Band structure of Si NC embedded in SiC . ..