Ab initio theory for ultrafast electron dynamics in metallic nanoparticles [Elektronische Ressource] / von Yaroslav Pavlyukh

Sujets

• Physik

Informations

• Publié par: martin-luther-universitat_halle-wittenberg
• Publié le: 01 janvier 2003
• Nombre de lectures: 27
• Langue: English
• Poids de l'ouvrage: 6 Mo

Extrait

Ab initio theory for ultrafast
electron dynamics in metallic nanoparticles
Dissertation
zur Erlangung des akademischen Grades
doctor rerum naturalium (Dr. rer. nat.)
vorgelegt der
Mathematisch-Naturwissenschaftlich-Technischen Fakult at
(mathematisch-naturwissenschaftlicher Bereich)
der Martin-Luther-Universit at Halle-Wittenberg
von Herrn Yaroslav Pavlyukh
geb. am 26.02.1976 in Drogobych
Gutachter:
1.
2.
3.
Halle (Saale),
urn:nbn:de:gbv:3-000005208
[ http://nbn-resolving.de/urn/resolver.pl?urn=nbn%3Ade%3Agbv%3A3-000005208 ]Contents
0.1 Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
0.2 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1 Introduction 5
2 Concepts of Many-Particle Theory 11
2.1 Electronic states in different systems . . . . . . . . . . . . . . . . . . . . . 11
2.2 excitations in clusters . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.1 Many-body Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.2 Quasiparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.3 Collective excitations . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Dynamics in many-body systems . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.1 Four approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 Methods 25
3.1 TDHF equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.2 Details of the numerical implementation . . . . . . . . . . . . . . . 29
3.1.3 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.4 Alternative implementations . . . . . . . . . . . . . . . . . . . . . . 31
3.2 GW approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.2 Justification of the GW approximation . . . . . . . . . . . . . . . . 41
3.2.3 Numerical implementation . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 Differences and similarities between HF, LDA, and GW . . . . . . . . . . . 47
3.4 SHG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4 Results I: Electron dynamics from TDHF theory 52
4.1 Deviation from adiabaticity . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 SHG response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3 Finite life-time from TDHF theory . . . . . . . . . . . . . . . . . . . . . . 58
+4.4 Power spectra of Na and Pt metal clusters . . . . . . . . . . . . . . . . . 609 3
12 Contents
5 Results II: Numerical results of GW calculations 66
0 05.1 Comparison of G W and GW approaches . . . . . . . . . . . . . . . . . . 67
+5.1.1 Na cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679
5.1.2 Two-body random interaction model . . . . . . . . . . . . . . . . . 71
+5.2 Sodium clusters Na , N from 15 to 25 . . . . . . . . . . . . . . . . . . . . 75N
5.3 Pt cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823
Conclusions 85
A Computation of some integrals over Gaussian basis functions 86
B Optimization of basis functions 90
C Simplified derivation of Hedin’s equations 92
Bibliography 940.1. Abbreviations 3
0.1 Abbreviations
BBGKY Bogolyubov, Born, Green, Kirkwood, Yvon (hierarchy)
CI configuration interaction (method)
DFT density functional theory
DMFT dynamical mean-field theory
ECP effective core potential
ERI electron repulsion integral
FFT fast Fourier transform
+GW approximation for the self-energy Σ(12)=iG(12)W(1 2)
0 0G W simplified GW approximation, G and W are computed from the LDA
or HF calculations.
0GW simplified GW approximation, G is involved in self-consistency loop,
while W is computed from the LDA or HF calculations.
HF Hartree-Fock approximation
HOMO highest occupied molecular orbital
HRR horizontal recurrence relation
lanl2dz Los Alamos National Laboratories double zeta (basis set)
LCAO linear combination of atomic orbitals
LDA local density approximation
LTH linearized time-dependent Hartree (approximation)
LUMO lowest unoccupied molecular orbital
MBPT many-body perturbation theory
ODE ordinary differential equation
ph particle-hole (excitation)
RPA random phase approximation
RHF restricted Hartree-Fock (method, approximation)
RHS right-hand side
RWA rotating wave approximation
SCF self-consistent field
SHG second harmonic generation
TBRIM two-body random interaction model
TDDFT time-dependent density functional theory
TDHFt Hartree-Fock (approximation)
TDLDA time-dependent local density approximation
TR-2PPE time-resolved two photon photoemission
VRR vertical recurrence relation
0.2 Units
In this thesis we adopted the following conventions:
† Formulae for the observables are shown in SI units in order to make them comparable with
experimental results.
† For the abstract quantities that cannot be measured in experiments like operators, Green’s
functions, Hamiltonians etc. we use a system of units in which they look most naturally.
In the case of cluster physics this is atomic units.4 Contents
¡10˚ ˚† Interatomic distances in clusters are shown in A (1 A=1¢10 m).
¡19† Energy levels, photon energies are shown in eV. (1 eV= 1:602188¢10 J).
Throughout the text we use atomic units of length, energy, time etc. The atomic unit of length
is the so-called Bohr radius
2 2 ¡10 ˚1a =¯h =me =0:529¢10 m=0:529A:B
The atomic unit of energy (so called Hartree) is
4 21Hr=me =h¯ =27:21eV:Chapter 1
Introduction
Thesuccessofphysicsasasciencecanbeexplainedtoalargeextendbyitsrefusaltobuild
a complete picture of the whole world and by its general method to reduce complicated
phenomenatosimplemodels. Themostrepresentativeexampleclosesttoourdiscussionof
thisis,probably,scatteringtheory. Wehaveinitiallyatarget(elementaryparticle,atomor
clusterinourcase)andaparticleinteractingwiththis(anotherelementaryparticle,
electron or photon). Based on the initial information about the position and velocity of
the particles the scattering theory predicts the final state of the system after the inter-
action process has been completed. The processes, that happened during the interaction
normally are not considered to be important and are assumed to be instantaneous. In the
application to the interaction of an atom with light, normally, one uses the terminology
that the atom absorbs the photon and goes to the excited state. The energy of the final
excited state, as well as a ground state can be computed on different levels of theory, for
example configuration interaction (CI), that takes into account the internal properties of
the system such as the number of electrons, spin-multiplicity etc., but does not care about
the excitation process itself.
Althoughthemany-bodyproblemoftheelectronsinanatom,molecule,orclusterisnot
solvableingeneral,manyapproximatemethodshavebeendevelopedtotreatthesesystems
approximately. The oldest, but still in many cases reliable Hartree-Fock approximation,
treats electrons on the mean-field level. Attempts to go beyond that, in the many-particle
theory terminology, to take account of the correlations — the part of the electron energy,
not taken into account in the mean-field approach – have lead to the development of
density functional theory (DFT). This approach owes its origin to the Hohenberg-Kohn
theorem, publishedin1964, whichdemonstratestheexistenceofauniquefunctionalwhich
determines the ground state energy and density exactly. The theorem does not provide
the form of this functional, however. One has to use some approximation to derive it for
simplesystemssuchasthehomogeneouselectrongas,andthentotransferthisdependence
on the real systems. The diagram technique of many-body perturbation theory provides
necessary tools for that.
There is, however, one question, that has only recently gained sufficient attention,
namely, what happens to the system between the initial and final state. The answer
requires the extension of the model, that is exhausted by describing only static properties.
56 Chapter 1. Introduction
Thewordtransition mustnowacquireadeepermeaning,revealedinlife-timesofthestates,
typical switching speeds, no longer being something, that happens instantly and traceless.
There are two cases that can be described relatively easy in the framework of perturbation
theory: thelimitofinstantaneousdisturbanceandthelimitofslowlyvaryingperturbation.
The intermediate situation, when the time scale of the excitation is comparable with the
speed of internal processes in the system, is the most interesting, but at the same time,
the most difficult one.
The systems under investigation in this work are metallic clusters. Our interest in
metallic clusters is raised by the recent advances in nanotechnology, fabrication and inves-
tigation of quantum dots, improvement of quantum chemical ab initio methods as well as
ofcomputationalfacilities, whichenablethe modellingofhundredsofatoms. On the other
hands progress in technology with its steady tendency to the miniaturization is constantly
demanding for novel materials. Clusters, that may be cons