Quasiparticle scattering, lifetimes, and spectra using the GW approximation by Derek Wayne Vigil-Fowler A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Steven G. Louie, Chair Professor Feng Wang Professor Mark D. Asta Summer 2015 Quasiparticle scattering, lifetimes, and spectra using the GW approximation c 2015 by Derek Wayne Vigil-Fowler 1 Abstract Quasiparticle scattering, lifetimes, and spectra using the GW approximation by Derek Wayne Vigil-Fowler Doctor of Philosophy in Physics University of California, Berkeley Professor Steven G. Louie, Chair Computer simulations are an increasingly important pillar of science, along with exper- iment and traditional pencil and paper theoretical work. Indeed, the development of the needed approximations and methods needed to accurately calculate the properties of the range of materials from molecules to nanostructures to bulk materials has been a great tri- umph of the last 50 years and has led to an increased role for computation in science. The need for quantitatively accurate predictions of material properties has never been greater, as technology such as computer chips and photovoltaics require rapid advancement in the control and understanding of the materials that underly these devices. As more accuracy is needed to adequately characterize, e.g. the energy conversion processes, in these materials, improvements on old approximations continually need to be made. Additionally, in order to be able to perform calculations on bigger and more complex systems, algorithmic devel- opment needs to be carried out so that newer, bigger computers can be maximally utilized to move science forward. 2 In this work we discuss our endeavors to improve approximations and algorithms to an- swer the challenge of better describing material properties. After an introduction to define and discuss all the important concepts that appear later, we first discuss the calculation of so-called satellite properties in the photoemission spectra (PES) of doped graphene. While the GW approximation accurately produces the quasiparticle energies across a range of materials from nanostructures and molecules to bulk metals and semiconductors, it does not accurately produce the satellite properties seen in PES experiments. We find that a more advanced treatment of the Green’s function, the cumulant expansion, is needed to adequately describe the satellite properties of doped graphene on SiC. In addition to this more advanced Green’s function treatment, a novel technique is devised for including the screening due to the SiC substrate on which the doped graphene is placed. This more ad- vanced treatment of the substrate is also crucial for obtaining agreement with experiment. Next, we show how the cumulant expansion can be used to accurately predict the ARPES spectra of bulk Si and the time-domain capacitance spectra of two-dimensional electron gases (2DEGs) in semiconductor quantum wells, with both the quasiparticle and satellite features given correctly (unlike in GW theory, in which only the quasiparticle properties are predicted accurately). We then discuss carrier lifetimes from the GW approximation in bulk Si and GaAs, showing how theory can provide access to detailed microscopic in- formation that could be of use in designing more efficient photovoltaics. In chapter 6, we discuss the effect of the pseudopotential approximation on excited-state GW calculations. Finding a small amount of error due to the use of nodeless pseudowavefunctions when us- ing pseudopotentials, we are able to understand the tendency of GW calculations that use pseudowavefunctions to overestimate the band gap in many common semiconductors. We quantify this error and suggest improved techniques for applications where this error is too large. In the last section on research, we discuss the effect of self-consistency in GW calculations. Chapter 7 is on computational algorithm development, and there we discuss some algorithmic advances made in improving the BerkeleyGW code. A technique for bet- ter distributing the data during the calculation of the inverse dielectric matrix is discussed and shown to give very good performance improvements, especially for the large systems that are becoming increasingly common. Other small improvements that allow for a more accurate calculation of quasiparticle lifetimes are also discussed. Finally, a few appendices are included for completeness. i To Rafael, I cannot wait to meet you. ii Contents List of Figures v List of Abbreviations vi Acknowledgments vii I Background 1 1 Introduction 2 1.1 What is many-body perturbation theory (MBPT)? . ...... 2 1.2 Themany-bodyhamiltonian . .. 7 1.2.1 TheBorn-OppenheimerApproximation . .. 8 1.2.2 Symmetries : translational and point-group . ...... 9 1.2.3 PlanewavesandPseudopotentials . ... 10 1.3 Mean-fieldtheories ............................... 12 1.3.1 Hartreetheory .............................. 13 1.3.2 Hartree-Focktheory ........................... 13 1.3.3 Densityfunctionaltheory . 14 1.3.4 Single Slater determinants and particle interactions .......... 16 1.4 TheGWapproximation ............................. 17 1.5 Quasiparticlelifetimes . .... 19 1.6 BeyondGW:thecumulantexpansion . ... 20 CONTENTS iii II Research 27 2 PlasmonSatellitesinDopedGrapheneARPES 28 2.1 Summary ..................................... 28 2.2 Background .................................... 28 2.3 Methods...................................... 29 2.3.1 Mean-fieldcalculation . 30 2.3.2 GWcalculation .............................. 30 2.3.3 GW+Ccalculation ............................ 32 2.3.4 Linear-bandsmodelcalculation . ... 33 2.4 Results....................................... 34 2.4.1 Suspendedgraphene ........................... 35 2.4.2 Grapheneonsiliconcarbide . 40 2.5 Conclusions .................................... 44 3 OtherStudiesofPlasmonSatellites: BulkSiand2DEGs 47 3.1 Summary ..................................... 47 3.2 Introduction.................................... 47 3.3 Results....................................... 48 3.3.1 BulkSi................................... 48 3.3.2 2DEGsinsemiconductorquantumwells . .. 48 3.4 Discussion..................................... 52 4 Carrierlifetimesduetoelectron-electronscattering 53 4.1 Summary ..................................... 53 4.2 Introduction.................................... 53 4.3 Results....................................... 54 4.3.1 BulkSi................................... 54 4.3.2 BulkGaAs ................................ 54 4.4 Discussion..................................... 55 5 Pseudowavefunctions in the GW approximation 58 5.1 Summary ..................................... 58 5.2 Background .................................... 58 5.3 Methods...................................... 60 5.4 Results....................................... 61 CONTENTS iv 6 Self consistency in GW calculations 69 6.1 Summary ..................................... 69 6.2 Introduction.................................... 69 6.3 ComputationalDetails . .. 72 6.4 Results....................................... 73 III Computational Methods 75 7 Improved Accuracy and Scaling of Full-Frequency GW calculations 76 7.1 Parallelfrequencies . ... 78 7.1.1 BerkeleyGW’s calculation of ǫ−1 ..................... 78 7.1.2 Why you should calculate ǫ−1(ω) at multiple frequencies in parallel . 86 7.2 X+CORvs.COH+SEXdivisionsofΣ . 89 7.3 PrincipalvalueintegralevaluationofΣ . ........ 91 IV Appendix 93 8 COHSEX Derivation 94 8.1 DerivationofCOHSEXSelfEnergy . .. 94 9 Doped Graphene Refinement 101 Bibliography 104 v List of Figures 2.1 Contour plot: ab initio/linear bands GW/GW+C for isolated graphene . 36 2.2 Spectral functions: ab initio/linear bands GW/GW+C for isolated graphene 38 2.3 ImΣ for linear bands GW/GW+C for isolated graphene . 39 2.4 Scaling plots: ab initio/linear bands GW/GW+C for isolated graphene . 41 2.5 Contour plot: ab initio/linear bands GW/GW+C for graphene on SiC . 42 2.6 Spectral functions: ab initio/linear bands GW/GW+C for graphene on SiC 43 2.7 ab initio GW+C contour plot and experiment for graphene on SiC . 45 3.1 ab initio GW, GW+C contour plots and experiment for bulk Si . 49 3.2 ab initio GW,GW+Cspectralfunctionplotsfor2DEG . 50 3.3 ab initio GW, GW+C quasiparticle, satellite peak positions vs experiment. 51 4.1 Carrier lifetimes in Si with electron-electron, electron-phonon contributions. 55 4.2 Brillouin-zone resolved carrier lifetimes in bulk Si. ............. 56 4.3 ImpactionizationrateinbulkGaAs. ... 57 5.1 ExchangechargedensityinatomicSi . .... 62 5.2 ExchangepotentialinatomicSi . ... 63 5.3 Product of exchange charge density and exchange potentialinatomicSi . 64 7.1 Depiction of calculation of polarizability as matrix multiplication.. 80 7.2 Distribution of dielectric matrix and bands. ......... 81 7.3 Demonstration of matrix communication scheme. ........ 82 7.4 Demonstration of parallel frequencies scheme. .......... 88 vi List of Abbreviations ARPES Angle Resolved Photoemission Spectroscopy DFT Density Functional Theory DOS Electronic Density of States PDOS Projected Density of States or Partial Density of States PES Photoemission Spectroscopy UPS Ultraviolet Photoemission Spectroscopy XPS X-ray Photoemission Spectroscopy vii Acknowledgments The process of obtaining my Ph.D has been a long, challenging and beautiful experience, and I could never have gotten through