Switching and memory effects in electron-vibron systems [Elektronische Ressource] : from single-site junctions to chains and networks / vorgelegt von Pino D'Amico

universitat_regensburg

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Publié par	universitat_regensburg
Publié le	01 janvier 2010
Nombre de lectures	10
Langue	English
Poids de l'ouvrage	9 Mo

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Switching and memory eﬀects in electron-vibron systems:
from single-site junctions to chains and networks
D I S S E R T A T I O N
zur Erlangung des
DOKTORGRADES DER NATURWISSENSCHAFTEN (DR. RER. NAT.)
der Naturwissenschaftlichen Fakult¨at II - Physik
der Universit¨at Regensburg
vorgelegt von
PINO D’AMICO
ausQuadri (Italien)
im Mai 2010Switching and memory eﬀects in electron-vibron systems:
from single-site junctions to chains and networks
DISSERTATION
Die Arbeit wurde angeleitet von:
Prof. Dr. Klaus Richter
Promotionsgesuch eingereicht am:
13. 01. 2010
Pru¨fungsausschuß:
Vorsitz: Prof. Dr. Franz J. Gießibl
Erstgutachten: Prof. Dr. Klaus Richter
Zweitgutachten: Prof. Dr. Milena Grifoni
Weiterer Pru¨fer: Prof. Dr. Andreas Scha¨fer
2Contents
1 Introduction 5
1.1 Molecular Electronics . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.1 Experiments on single molecule junctions . . . . . . . . 7
1.1.2 Theoretical approaches for transport . . . . . . . . . . 8
1.2 Switching and bistability in nature . . . . . . . . . . . . . . . 9
1.3 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Model and Methods 13
2.1 Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Green functions . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1 Equation of Motion method . . . . . . . . . . . . . . . 18
2.3 Density Matrix approach . . . . . . . . . . . . . . . . . . . . . 20
2.3.1 Generalized Master Equation . . . . . . . . . . . . . . 21
2.3.2 Master Equation for the populations . . . . . . . . . . 22
3 Single-site junctions: charge memory eﬀects 27
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Intermediate coupling to the leads . . . . . . . . . . . . . . . . 29
3.2.1 Spectral function, average charge and current . . . . . 30
3.2.2 EquationofMotionmethodforthesingle-levelelectron-
vibron Hamiltonian . . . . . . . . . . . . . . . . . . . . 31
3.2.3 Self-consistent Hartree approximation . . . . . . . . . . 32
3.2.4 Second approximation . . . . . . . . . . . . . . . . . . 33
3.2.5 Results and discussion for the intermediate case . . . . 34
3.3 Weak system-to-leads coupling . . . . . . . . . . . . . . . . . . 38
3.3.1 Eigenstates and Master Equation . . . . . . . . . . . . 38
3.3.2 Charge, current and life-times . . . . . . . . . . . . . . 41
3.3.3 Conclusion for weak coupling case . . . . . . . . . . . . 44
34 CONTENTS
4 Single-site junctions: spin memory eﬀects 47
4.1 Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Weak lead-to-molecule coupling . . . . . . . . . . . . . . . . . 49
4.2.1 Model, states and energies . . . . . . . . . . . . . . . . 50
4.2.2 Rates and Master Equation . . . . . . . . . . . . . . . 52
4.2.3 Charge, spin polarization and lifetimes . . . . . . . . . 52
4.3 Intermediate lead-to-molecule coupling . . . . . . . . . . . . . 56
5 Chains of electron-vibron systems 65
5.1 Two-level-systems: intermediate coupling . . . . . . . . . . . . 67
5.1.1 Bias-independent and bias-dependent energy levels . . 69
5.1.2 Extended analysis for the bias-dependent case . . . . . 72
5.1.3 Memory eﬀects . . . . . . . . . . . . . . . . . . . . . . 75
5.2 Two-level-system: analytical considerations for weak coupling 77
5.3 Three-levelsystem intheintermediatelead-to-moleculecoupling 82
5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6 Networks of electron-vibron elements 89
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.2 Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 90
6.3 2 by 2 square lattice . . . . . . . . . . . . . . . . . . . . . . . 91
6.4 3 by 3 square lattice . . . . . . . . . . . . . . . . . . . . . . . 94
7 Conclusions and perspectives 101
Bibliography 104Chapter 1
Introduction
1.1 Molecular Electronics
The general framework in which this thesis is embedded is called Molecular
Electronics[1]. Inthisﬁeldthedreamistobeabletoproducestablejunctions
in which a given molecule is in contact with a certain number of electrodes.
Those allow to apply voltages and to perform speciﬁc tasks, exploiting the
functionality of the molecule itself.
Diﬀerent kinds ofmolecules have speciﬁc electronic, structural andvibra-
tional properties, but there is something that can be thought as a general
property: the typical dimension of a molecule is in general very small (of the
order of nanometers or smaller). Molecules can undergo structural changes
when additional charges are inserted through electron-tunneling in transport
setups. Because of that, the electronic and the vibrational degrees of free-
dom are strongly related in molecules and their mutual interaction plays a
fundamental role in the investigation of a molecular junction and in view of
possible applications.
In generalwe can consider a molecule as a very tiny object that is ﬂexible
and has localized vibrations. This property is peculiar of molecules and is
absentinsemiconductor devices like quantum-dots, two dimensionalelectron
gases and bulk materials. In those systems the vibrational properties are
associatedtothephononstructure,i.e. tothelatticestructureofthematerial
one considers. The ﬂexibility of the molecules make them interesting and
diﬀerentfromsemiconductorsdevices,openingnewperspectivesandbringing
new eﬀects into the game.
The idea of using single molecule junctions in order to obtain functional
devices like switches, rectiﬁers and memory elements, dates back to 1974. In
[2]AviramandRatnerproposedtouseasingleorganicmoleculeasarectiﬁer.
56 CHAPTER 1. INTRODUCTION
Figure 1.1: (Top) Original schematic representation of a molecular junction
from [2]. The scheme represents the energetics of a molecular junction made
oftwometallicleadsandamoleculeplacedbetween them. (Bottom)Current
rectiﬁcation calculated with the original proposed model [2] (left) and the
measured current from [3] (right).
Only recently the corresponding experimental realization has been achieved
by employing two weakly coupled π-systems with mutually shifted energy
levels [3]. A scheme of the molecular rectiﬁer from the original proposal is
showninFig. 1.1,togetherwiththecalculatedcurrentandtheexperimental
measurement from [3]. The Aviram-Ratner rectiﬁer is based on a acceptor-
donor sites system . If the acceptor and the donor are well isolated among
eachother, acurrent canﬂowonlyinonedirection resulting inarectiﬁcation
eﬀect.
Though the Aviram-Ratner theoretical proposal has been experimentally
observed, it has to be mentioned that the ﬁrst measurement on a single
molecule junction has been achieved by Reed et al. [4]. The molecule was a
benzene-1,4-dithiol.
In the following we will review the experimental and theoretical methods
used to investigate molecular junctions.1.1. MOLECULAR ELECTRONICS 7
1.1.1 Experiments on single molecule junctions
The most challenging part of the experiments with single molecule junctions
is to have a controllable method to contact the molecule and the reservoirs.
Making a stable junction with a single molecule as a bridge and active part
of the system is a very delicate task to accomplish for experimentalists. In
generalitispossibletorealizesinglemoleculejunctionsintwodiﬀerentways:
• attachingasinglemoleculetotheexternalleadsthroughasingleatomic
contact using the molecule as a bridge;
• using a Scanning Tunneling Microscopy (STM) technique where the
molecule is deposited on a given substrate and investigated through
the tip of the apparatus.
Theﬁrstsetup(we callitbridge setup)canberealized withtheMechanically
Controllable Break-Junction (MCBJ) technique (see for example chapter 9
in [1] and references therein) and also with the electromigration technique
[5]. In both methods the ﬁnal task is to obtain a gap into a metallic wire.
The two metallic segments are then used as electrodes. The desired molecule
is placed between the two electrodes and probed through a bias voltage (and
possibly a gate voltage).
The STM setup is realized placing an ultrathin insulating layer on a
metallic surface. The molecule (or the atom) of interest is deposited on top
of the insulating layer. The tip of the STM is placed on top of the molecule
in order to probe it with a voltage between the tip and the metallic surface
[6, 7, 8].
InFig.1.2weshowthebreakingresulting fromtheelectromigrationtech-
nique and an image taken with the STM setup.
In the bridge setup one has to be very careful in order to attribute the
measurement to a very single molecule, because more than one molecule
could be attached between the electrodes. In order to avoid this diﬃculty
statistical approaches can be used to analyze the data, as for example in [9].
The STM setup gives instead the possibility to manipulate single molecules
in a very precise way and to image molecular orbitals with high resolution
producing beautiful images: it is really possible to look at the molecular
orbitals. Thinking about possible realistic electronic applications, the bridge
setup is more suitable because a single molecule clamped between electrodes
can be a very small system and a chip-integration can be envisaged. On the
other hand, an STM setup is usually a big experi