Quasiparticle scattering, lifetimes, and spectra using the GW approximation Derek Wayne Vigil ...

126 Pages·2015·1.28 MB·English

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
(cid:13)
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 The many-body hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.1 The Born-Oppenheimer Approximation . . . . . . . . . . . . . . . . . 8
1.2.2 Symmetries : translational and point-group . . . . . . . . . . . . . . 9
1.2.3 Planewaves and Pseudopotentials . . . . . . . . . . . . . . . . . . . . 10
1.3 Mean-field theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.1 Hartree theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.2 Hartree-Fock theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.3 Density functional theory . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.4 Single Slater determinants and particle interactions . . . . . . . . . . 16
1.4 The GW approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5 Quasiparticle lifetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.6 Beyond GW: the cumulant expansion . . . . . . . . . . . . . . . . . . . . . . 20
iii
CONTENTS
II Research 27
2 Plasmon Satellites in Doped Graphene ARPES 28
2.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.1 Mean-field calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.2 GW calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.3 GW+C calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3.4 Linear-bands model calculation . . . . . . . . . . . . . . . . . . . . . 33
2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4.1 Suspended graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4.2 Graphene on silicon carbide . . . . . . . . . . . . . . . . . . . . . . . 40
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3 Other Studies of Plasmon Satellites : Bulk Si and 2DEGs 47
3.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3.1 Bulk Si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3.2 2DEGs in semiconductor quantum wells . . . . . . . . . . . . . . . . 48
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4 Carrier lifetimes due to electron-electron scattering 53
4.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3.1 Bulk Si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3.2 Bulk GaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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
iv
CONTENTS
6 Self consistency in GW calculations 69
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.3 Computational Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
III Computational Methods 75
7 Improved Accuracy and Scaling of Full-Frequency GW calculations 76
7.1 Parallel frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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+COR vs. COH+SEX divisions of Σ . . . . . . . . . . . . . . . . . . . . . 89
7.3 Principal value integral evaluation of Σ . . . . . . . . . . . . . . . . . . . . . 91
IV Appendix 93
8 COHSEX Derivation 94
8.1 Derivation of COHSEX Self Energy . . . . . . . . . . . . . . . . . . . . . . . 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+C spectral function plots for 2DEG . . . . . . . . . . . 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 Impact ionization rate in bulk GaAs. . . . . . . . . . . . . . . . . . . . . . 57
5.1 Exchange charge density in atomic Si . . . . . . . . . . . . . . . . . . . . . 62
5.2 Exchange potential in atomic Si . . . . . . . . . . . . . . . . . . . . . . . . 63
5.3 Product of exchange charge density and exchange potential in atomic Si . . 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