Non-linear dynamics of metal clusters on insulating substrates [Elektronische Ressource] / vorgelegt von Matthias Bär

Non-Linear Dynamicsof Metal Clusters on InsulatingSubstratesDer Naturwissenschaftlichen Fakult¨atder Friedrich-Alexander-Universit¨atErlangen-Nu¨rnbergzurErlangung des Doktorgradesvorgelegt vonMatthias B¨araus CoburgAls Dissertation genehmigt von der Naturwissenschaftlichen Fakulta¨t derUniversita¨t Erlangen-Nu¨rnbergTag der mu¨ndlichen Pru¨fung: 16. April 2008Vorsitzender der Promotionskommission: Prof. Dr. Eberhard B¨anschErstberichterstatter: Prof. Dr. Dr. h.c. Paul-Gerhard ReinhardZweitberichterstatter: Prof. Dr. Eric SuraudZusammenfassungDie vorliegende Arbeit widmet sich der Untersuchung kleiner Natrium-Cluster, die inKontaktmitisolierendenSubstraten,insbesondereMgO(001)stehen. ImVordergrundsteht dabei die Frage, inwieweit die (geringe) Wechselwirkung zwischen Cluster undOberfla¨che die Merkmale des Adsorbats beeinflusst.Wa¨hrend Grundzustandseigenschaften deponierter Cluster durch bereits existie-rende quantenmechanische Modelle zuga¨nglich sind, gilt das nicht fu¨r den Bereichstarker Anregungen des Systems. In dieser Arbeit wird nun ein hierarchisches Mo-dell vorgestellt, das diese Lu¨cke schließen soll. Im Rahmen dieses Modells werdendie elektronisch aktivsten Komponenten des Systems, die Valenzelektronen des Clus-ters, quantenmechanisch mit Hilfe der zeitabh¨angigen Dichtefunktionaltheorie, dieelektronisch passiveren Komponenten hingegen durch klassische Molekulardynamikbeschrieben.
Publié le : mardi 1 janvier 2008
Lecture(s) : 26
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Source : WWW.OPUS.UB.UNI-ERLANGEN.DE/OPUS/VOLLTEXTE/2008/911/PDF/MATTHIASBAERDISSERTATION.PDF
Nombre de pages : 127
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Non-Linear Dynamics
of Metal Clusters on Insulating
Substrates
Der Naturwissenschaftlichen Fakult¨at
der Friedrich-Alexander-Universit¨atErlangen-Nu¨rnberg
zur
Erlangung des Doktorgrades
vorgelegt von
Matthias B¨ar
aus CoburgAls Dissertation genehmigt von der Naturwissenschaftlichen Fakulta¨t der
Universita¨t Erlangen-Nu¨rnberg
Tag der mu¨ndlichen Pru¨fung: 16. April 2008
Vorsitzender der Promotionskommission: Prof. Dr. Eberhard B¨ansch
Erstberichterstatter: Prof. Dr. Dr. h.c. Paul-Gerhard Reinhard
Zweitberichterstatter: Prof. Dr. Eric SuraudZusammenfassung
Die vorliegende Arbeit widmet sich der Untersuchung kleiner Natrium-Cluster, die in
KontaktmitisolierendenSubstraten,insbesondereMgO(001)stehen. ImVordergrund
steht dabei die Frage, inwieweit die (geringe) Wechselwirkung zwischen Cluster und
Oberfla¨che die Merkmale des Adsorbats beeinflusst.
Wa¨hrend Grundzustandseigenschaften deponierter Cluster durch bereits existie-
rende quantenmechanische Modelle zuga¨nglich sind, gilt das nicht fu¨r den Bereich
starker Anregungen des Systems. In dieser Arbeit wird nun ein hierarchisches Mo-
dell vorgestellt, das diese Lu¨cke schließen soll. Im Rahmen dieses Modells werden
die elektronisch aktivsten Komponenten des Systems, die Valenzelektronen des Clus-
ters, quantenmechanisch mit Hilfe der zeitabh¨angigen Dichtefunktionaltheorie, die
elektronisch passiveren Komponenten hingegen durch klassische Molekulardynamik
beschrieben. DieKopplung zwischen Klassik und Quantenmechanik gelingt durch den
Einsatz von Pseudopotentialen und Atom-Atom-Potentialen. Sie erfolgt nichtadia-
batisch, so dass durch zeitabh¨angige Rechnungen im Prinzip beliebige Anregungen
realisiert werden ko¨nnen. Die Teilchen des Substrats erhalten als zusa¨tzlichen inneren
Freiheitsgrad ein Dipolmoment, durch dessen Propagation dynamische Polarisations-
effekte beru¨cksichtigt werden k¨onnen.
Das Modell wird zuna¨chst dazu verwendet, um Eigenschaften des Grundzustands,
insbesondere die geometrische Struktur der adsorbierten Cluster und deren Anre-
gungsspektrum zu berechnen. Die gefundenen Strukturen dienen als Ausgangspunkt
fu¨rdieweiteren Simulationen. Imna¨chsten SchrittwirddieDynamikdes Depositions-
prozesses von Na und Na auf MgO(001) bei verschiedenen Einschlagenergien unter-6 8
sucht. Die Ergebnisse werden mit analogen Rechnungen mit einem Argonsubstrat
verglichen. Desweiteren wird das Modell zur Photoelektronenspektroskopie am adsor-
bierten Cluster Na eingesetzt. Neben der Untersuchung von globalen Gro¨ßen wie der8
mittleren Gesamtzahl der emittierten Elektronen, werden auch deren 4π-aufgel¨oste
Winkelverteilungen, sowie die kinetischen Energiespektren berechnet.
Schließlich wird in einem abschließenden Kapitel der extrem nichtlineare Prozess
von Coulomb-Explosionen eines Na Clusters auf der Oberfla¨che betrachtet. Die8
Wechselwirkung mit dem Substrat fu¨hrt zu einer deutlichen Erho¨hung des kritischen
Ladungszustands fu¨r instantane Explosion.
iiiAbstract
The present work is dedicated to the investigation of small sodium clusters which are
in contact with an insulating substrate, in most cases MgO(001). The influence of the
small interaction between cluster and surface on the properties of the adsorbate is of
primary concern.
While ground state features of deposited clusters are accessible by existing quan-
tum mechanical models, the regime of strong excitations of the system has so far been
out of reach. The aim of this work therefore is to introduce a model which fulfills this
task. Within this model the electronically most active components of the system (the
valence electrons of the cluster) are treated quantum mechanically in the framework
of time-dependent density functional theory, whereas the electronically inert compo-
nents are described by classical molecular dynamics. The classical part is coupled to
the quantum mechanical part by pseudo-potentials and atom-atom potentials. The
coupling is done non-adiabatically, so that even strong excitations can be realized by
time-resolved calculations. As additional internal degree of freedom each particle of
the substrate may acquire a dipole moment. The propagation of the dipole moment
allows to incorporate dynamical polarization effects.
First of all the model is applied to determine cluster properties close to the ground
state, namely the ion geometry of the adsorbed cluster and the optical response. The
structures which are determined, are used as starting point for the further calcula-
tions. As a next step, the deposition process for Na and Na on MgO(001) depending6 8
on their impact energy is investigated in detail. The results are compared with corre-
spondingcalculationsusinganargonsubstrateinstead. Furthermorethemodelisused
for performing photoelectron spectroscopy of deposited Na . Aside of global quanti-8
ties as the average total number of the emitted electrons, their 4π-resolved angular
distributions as well as their kinetic energy spectra are determined.
Finally the last chapter investigates the extremely non-linear process of Coulomb
explosions of a Na cluster on the surface. The result is a significant increase of8
the critical charge state for instantaneous explosion due to the interaction with the
substrate.
ivContents
1 Introduction 1
2 Theoretical Description 5
2.1 The Cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.1 Electronic Degrees of Freedom: DFT . . . . . . . . . . . . . . . . 7
2.1.2 Ionic Cores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.3 Internal Cluster Energy . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 A Model for the Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.1 Madelung Potential . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.2 The Gaussian Shell Model for MgO . . . . . . . . . . . . . . . . . 15
2.2.3 Dividing up the Surface . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.4 Cluster-Surface Coupling . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Calibration of the Surface-Electron Pseudopotentials . . . . . . . . . . . 20
3 Numerical Realization 24
3.1 Solution and Propagation of the Kohn-Sham Equations . . . . . . . . . 24
3.1.1 Numerical Representation . . . . . . . . . . . . . . . . . . . . . . 24
3.1.2 Static KS Equations: Accelerated Gradient Step . . . . . . . . . 25
3.1.3 Dynamic KS Equations: T-V-Splitting . . . . . . . . . . . . . . . 26
3.2 Pseudo-Densities and Subgrids . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Ionic Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4 Electron Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4.1 Absorbing Boundary Conditions . . . . . . . . . . . . . . . . . . 28
3.4.2 Angular Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4.3 Kinetic Energy Distribution . . . . . . . . . . . . . . . . . . . . . 30
4 Structure of Deposited Clusters 32
4.1 Cooling Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2 Ionic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.3 Global Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.4 Static Polarizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5 Optical Response 41
5.1 Some Basic Features of Free Metal Clusters . . . . . . . . . . . . . . . . 41
5.2 Determination of the Electronic Excitation Spectrum . . . . . . . . . . 42
v5.3 Spectra of Deposited Clusters . . . . . . . . . . . . . . . . . . . . . . . . 43
6 Cluster Deposition 48
6.1 Preparation of the System . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.2 Site Properties: Na Monomer on MgO . . . . . . . . . . . . . . . . . . . 50
6.3 General Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.4 Energy Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.5 Electron-Ion Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.6 MgO versus Argon Substrate . . . . . . . . . . . . . . . . . . . . . . . . . 66
7 Photoelectron Spectroscopy 68
7.1 Total Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7.1.1 Influence of Laser Polarisation . . . . . . . . . . . . . . . . . . . . 69
7.1.2 Separating Linear From Non-Linear Regimes . . . . . . . . . . . 73
7.2 Angular Distribution of Emitted Electrons . . . . . . . . . . . . . . . . . 74
7.2.1 Azimuthal Dependence . . . . . . . . . . . . . . . . . . . . . . . . 74
7.2.2 Azimuthally Integrated Angular Distributions . . . . . . . . . . . 80
7.2.3 Comparison With Argon Substrate . . . . . . . . . . . . . . . . . 85
7.3 Kinetic Energy Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
8 Coulomb Explosion on a Surface 90
8.1 Suppressed Explosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
8.2 Trends With Increasing Charge State . . . . . . . . . . . . . . . . . . . . 95
9 Conclusion and Outlook 98
A Model Parameters 102
B Madelung Potential 105
C Single Particle Spectrum of Na 1068
D Abbreviations 107Chapter 1
Introduction
For the past 30 years cluster physics has been a rapidly evolving field of science with
stronginterdisciplinaryaspects,bothtowardschemistryandmaterialsscience. Atomic
clusters – nano particles – are bound systems consisting of N equal atoms, where N
5rangesfromperhaps3toabout10 . Thusclusterphysicsaimstocoverthegapbetween
well-established branches like atomic and molecular physics on the one side and solid
state physics on the other side. Although cluster physics is a very young discipline,
applications of clusters can, in fact, be traced back to ancient times. The Roman
Lycurus cup, for instance, made of glass with traces of iron, appears green in reflected
light (as during daytime), but it looks red, when watching it in transmitted light,
i.e. when the light source is inside the cup [Bam06]. The producers – without knowing
abouttheexistenceof atomsor clusters, of course– madeuse ofa prominentquantum
mechanical feature of metal clusters, the Mie plasmon. The same technique has been
used in medieval times for the construction of coloured windows in cathedrals. In
moderntimestheapplicationsofclustersarewidespread. Apopularexample,although
alreadyoutdatedbydigitaltechnology,isthefilm-basedphotography. MolecularAgBr
clusters are converted into Ag clusters by exposing the film to light [Rei03]. Today’s
and future industrial applications range from systematic control of surface properties
to nanocatalysis.
The various kinds of atomic clusters can roughly be categorized by their dominant
type of binding [Hab92]. For instance, rare gas clusters are held together mainly by
van-der-Waals interaction, whereas atoms like carbon assemble to clusters predomi-
nantly by covalent binding. An example for this latter type is the famous fullerene
C . The present work focuses on the third important type, the class of metal clus-60
ters. Their common feature is the metallic binding, i.e. the valence electrons of the
cluster are heavily delocalized. The simplest, but also purest representatives of metal
clusters are those composed of alkaline atoms like sodium, potassium or cesium. Each
atom possesses one loosely bound valence electron, whereas all the other electrons are
deeply bound, localized and usually do not participate in chemical reactions. The
simple metallic behaviour of alkaline atoms is one of the main reasons, why sodium is
an ideal candidate for theoretical investigations of metal clusters.
The early phase of cluster physics focussed on the discovery of shell effects and the
opticalpropertiesofsinglemetalclustersinvacuum, bothexperimentallyandtheoret-
12 Chapter 1. Introduction
ically. In particular collective phenomena like the so-called Mie plasmon, were subject
to intensive research. By today the optical properties are among the best understood
features of metal clusters. For a review on this topic, see [Kre93]. These early dis-
coveries, namely shell effects and plasmon resonance revealed intriguing parallels to
nuclear physics. Considering the different types of interaction (long ranged Coulomb
and short ranged nuclear forces), these discoveries are all the more surprising.
Today’sresearchonclusterscoversawiderangeoftopics. Thediscoveryofcatalytic
properties of noble metal clusters [San99,Lim05] initiated intensive efforts to under-
stand the synthesis and reactivity of such nanocatalysts (see [Ha¨k03,Hub06,Vaj06,
Win06a]). The investigation of magnetic properties of clusters is another quickly
developing branch. Current research in this field concentrates for instance on sur-
face and interface effects on magnetic properties, when the cluster is supported by a
substrate [Zit02,Fav06,Ban05,Mav06]. The list of interesting physical problems and
applications in the literature has assumed vast proportions and could easily be ex-
tended, real-timespectroscopy of photochemical reactions [Koy07] or studies on phase
transitions like melting of clusters [Sch01,Hab05] are only a few to mention.
Until recently most theoretical treatments in cluster physics concentrated on free
clusters in vacuum or gas phase. The majority of experiments, however, in particular
those mentioned above, are performed for clusters which are in contact with some
environment. Examples for such a environments are rare gas droplets to cool down
the cluster after its creation, or a substrate the cluster is attached to. The theoret-
ical, quantum mechanical description of clusters in contact with a macroscopically
large environment is much more complicated than for free clusters and requires spe-
cial models. In the last few years, sophisticated models have been developed for the
purpose of simulating supported clusters. In most cases the models are designed for
thedeterminationofgroundstatepropertieslikebindinggeometry, adsorptionenergy,
etc. Usually, ordinary density functional theory (DFT) is used to describe the cluster
electrons and a small part of the substrate, typically 10-20 atoms. The rest of the
surface is treated classically and may be allowed to relax. In more advanced schemes
like [Nas01], the particles of the environment are even allowed to polarize. While
the schemes involving static DFT are numerous, there are only a handful of mod-
els which are adapted to perform time-dependent calculations [Mos02,Bu¨r06]. The
typical approach in these papers is to employ time-dependent DFT in local density
approximation (TD-LDA, see section 2.1.1) for the cluster and a few single atoms of
the substrate. The TD-LDA is then coupled to molecular dynamics. Aside of the
cluster and a few atoms of the substrate, the rest of the surface is frozen and un-
able to relax, because performing TD-LDA is extremely time consuming. Hence it is
practically impossible to treat a large scale surface plus adsorbate by TD-LDA. The
scheme is usually applied for surface chemistry problems like cluster induced catalysis
on substrates [Hub06]. By construction – fixed environment and in most cases Born-
Oppenheimer dynamics – these models are restricted to dynamical processes close to
the ground state. Whenever the physical situation is such that the surface is signifi-
cantly distorted or polarized, or whenever the cluster electrons are strongly excited,3
the models fail.
TheErlangen-Toulousecollaborationhasbeenworkingforyearstodevelopamodel
which isadaptedfor fullynon-lineardynamics ofclusters onsurfaces. Firstexperience
with supported clusters was gained in [Koh98,Koh98a,Cal98]. This first and simple
approach treated the substrate as a static interface potential. Using this scheme it
was possible to calculate ground state properties like adsorption geometry and the
excitation spectrum of the deposited clusters. In 2004 a more advanced, hierarchical
model was introduced in [Feh04] to describe sodium clusters in argon environment,
treating the cluster by TD-LDA and the environment by classical molecular dynam-
ics, see for instance [Feh06a,Feh06b,Din07]. The coupling between classical particles
and quantum mechanical wave functions is realized via atom-atom potentials and
pseudopotentials. Dynamical polarization of the atoms in the environment is taken
into account. As the calculations are numerically extremely expensive, the quantum
mechanical problem has been effectively reduced to two-dimensional wave functions
(CAPS, [Mon94] and appendix D) by confining oneself to axially symmetricproblems.
Heterogeneous materials like NaCl or MgO, which are commonly used in experiments
and which consist of two components in contrast to solid Ar, break this symmetry and
were out of reach so far.
The aim of the present work is twofold. The first goal is to improve the previ-
ous hierarchical model such that it is capable of describing time-resolved and fully
non-linear processes of sodium clusters which are in contact with an insulating two
component system like the ionic crystal surface MgO(001). The model shall incor-
porate the successful features of the earlier approach and overcome its drawbacks;
namely, the substrateis to be treated microscopically and classically, dynamically and
polarizably, and the quantum mechanical problem is to be treated without any sym-
metry restrictions. Hence, fully triaxial calculations are necessary. The model for the
MgO surface adds new steps to the hierarchy. The second goal is to exploit the flex-
ibility and power of the new model. For this purpose, it will be applied to a variety
of physical problems ranging from ground state properties to energetic impacts and
interaction with strong laser pulses.
The thesis is organized as follows. Chapter 2 presents the hierarchical model that
wasdevelopedinthiswork. ItgivesashortintroductiontoDFTandTD-LDAwhichis
usedforthequantummechanicaldescriptionofthevalenceelectronsoftheclusterand
discusses in detail the model for the surface as well as its coupling to the cluster. The
free parameters of the particle-based pseudopotentials for the cluster-surface coupling
are fixed by reference data from static ab initio calculations [Win06]. Chapter 3 gives
a summaryof thenumerical schemes used to realizethemodel. Aside ofthenumerical
solutionofthequantummechanicalequations,itconcentratesonhowelectronemission
is described in TD-LDA andhow differentialionization cross sections can beobtained.
Chapter 4 presents first results obtained from the model developed here, namely
the ground state structure of a set of small sodium clusters on the MgO surface. The
geometries are compared with those found for clusters in gas phase. The electronic
excitation spectrum of the deposited clusters is subject of chapter 5. It will be shown4 Chapter 1. Introduction
that the presence of the substratehas a severe influence on the spectrum. The various
forms of interaction due to the surface are disentangled and discussed.
The first truly non-linear time-resolved processes are presented in chapter 6, which
is dedicated to the deposition dynamics of small sodium clusters on MgO. It focuses
on basic questions regarding the efficiency of deposition, i.e. under what conditions
adsorption (or possibly reflexion from the surface) takes place, to what extent the
cluster structure is affected and the surface reacts. In this connection, the question
regarding energy and momentum transfer between cluster and surface is of particular
importance. Previous calculations for the deposition of sodium clusters on solid argon
allow to compare the deposition dynamics on materials of most different kinds.
Thenextapplicationofthehierarchicalmodelisphotoelectronspectroscopy(PES)
offreeanddepositedclusters. Forthispurpose,theclustersareexposedtofemtosecond
laser pulses. The primary aim here is to investigate the influence of the substrate in
order to estimate how far the tools of PES are applicable to analyze properties of
supported clusters. The first observable of interest here is the total number of emitted
electrons as a function of laser frequency. This quantity is closely related to the total
ionizationcrosssection. Asanextstep,the4π-resolvedangulardistributionofemitted
electronsisevaluatedanddiscussed. 4π-resolveddistributionshavenotbeenpublished
before, not even for free clusters. And finally, the kinetic energy distribution of the
emitted electrons, both for free and deposited clusters, is presented.
Chapter 8 goes over to the extremely non-linear case of Coulomb explosions on
the MgO surface. The laser pulses employed here are so strong that they leave the
cluster in a highly charged state. The resulting Coulomb pressure forces the cluster
to explode. In previous calculations treating sodium clusters packed all around in
an argon droplet [Feh07c], it was shown that the Coulomb explosion is hindered by
the environment. In fact, the critical charge state, above which the cluster undergoes
instantaneous Coulomb explosion, is increased. Chapter 8 investigates the influence
of a hard, solid substrate on the Coulomb explosion and will come to comparable
conclusions as in [Feh07c] for the argon droplet.

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