Shot noise control in coherent nanoscale conductors [Elektronische Ressource] / von Michael Straß

Sujets

• Physik

Informations

• Publié par: universitat_augsburg
• Publié le: 01 janvier 2006
• Nombre de lectures: 16
• Langue: Deutsch
• Poids de l'ouvrage: 1 Mo

Extrait

Shot noise control in coherent
nanoscale conductors
Zur Erlangung des akademischen Grades eines
Doktors der Naturwissenschaften
der Mathematisch–Naturwissenschaftlichen Fakultät
der Universität Augsburg vorgelegte
Dissertation
von
Dipl. Phys. Michael Straß
aus
Donauwörth
Augsburg, im Januar 2006Erstberichterstatter: Priv. Doz. Dr. Sigmund Kohler
Zweitberichterstatter: Prof. Dr. Ulrich Eckern
Tag der mündlichen Prüfung: 15. März 2006
iiContents
Important acronyms and symbols v
1. Introduction 1
2. Time dependent scattering formalism 7
2.1. The model system . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2. Electrical current . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1. Landauer scattering approach . . . . . . . . . . . . . . 11
2.2.2. Heisenberg equations of motion . . . . . . . . . . . . . 12
2.2.3. Retarded Green function . . . . . . . . . . . . . . . . . 14
2.2.4. Average current . . . . . . . . . . . . . . . . . . . . . . . 17
2.3. Current fluctuations . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.1. Noise power . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.2. Shot noise . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3. Dissipative Floquet theory 23
3.1. Solution in composite Hilbert space . . . . . . . . . . . . . . . 23
3.1.1. Tight binding system driven by a dipole field . . . . . 23
3.1.2. Decomposition into Floquet basis . . . . . . . . . . . . 25
3.1.3. Numerical methods . . . . . . . . . . . . . . . . . . . . 29
3.2. Fundamental symmetries . . . . . . . . . . . . . . . . . . . . . 30
3.2.1. Time reversal symmetry . . . . . . . . . . . . . . . . . . 32
3.2.2. Time reversal parity . . . . . . . . . . . . . . . . . . . . 33
3.2.3. Generalized parity . . . . . . . . . . . . . . . . . . . . . 35
4. Rotating wave approximation 37
4.1. Coherent destruction of tunneling . . . . . . . . . . . . . . . . 38
4.2. Rotating wave approximation for driven transport . . . . . . . 39
iiiContents
5. Coherent shot noise control 45
5.1. Unbiased two level system . . . . . . . . . . . . . . . . . . . . . 46
5.2. Suppression of shot noise . . . . . . . . . . . . . . . . . . . . . 48
5.2.1. High frequency approximation . . . . . . . . . . . . . . 48
5.2.2. Comparison with exact results . . . . . . . . . . . . . . 51
5.3. Current suppression in heterostructures . . . . . . . . . . . . . 54
6. Noise in a nonadiabatic electron pump 59
6.1. The double dot model . . . . . . . . . . . . . . . . . . . . . . . 60
6.2. Resonant electron pumping . . . . . . . . . . . . . . . . . . . . 61
6.2.1. Symmetry considerations . . . . . . . . . . . . . . . . . 61
6.2.2. High frequency driving . . . . . . . . . . . . . . . . . . 63
6.2.3. Comparison with exact result . . . . . . . . . . . . . . . 64
6.2.4. Adiabatic vs. nonadiabatic pump . . . . . . . . . . . . . 66
6.2.5. Tuning the pump . . . . . . . . . . . . . . . . . . . . . . 67
6.3. Current–voltage characteristics . . . . . . . . . . . . . . . . . . 69
7. Summary and outlook 75
A. Static conductor 79
A.1. Scattering formalism . . . . . . . . . . . . . . . . . . . . . . . . 79
A.2. Two level system . . . . . . . . . . . . . . . . . . . . . . . . . . 80
A.3. Three level . . . . . . . . . . . . . . . . . . . . . . . . . 82
References 83
Acknowledgement 95
ivImportant acronyms and symbols
CDT coherent destruction of tunneling (see chapter 4)
I–V current–voltage
PAT photon assisted tunneling (see chapter 6)
RWA rotating wave approximation (see chapter 4)
TLS two level system (see chapter 5)
n index of wire site (n = 1,...,N)
‘ lead index (‘ = L: left,‘ = R: right)
n wire site connected to lead‘ (n = 1, n = N)‘ L R
α,β indices of Floquet state
k sideband/Fourier index
Ω angular frequency of driving field
T driving period (= 2π/Ω)
Δ tunneling matrix element of adjacent wire sites
†c ,c creator and annihilator of wire electron in site nnn
†c ,c creator and of lead electronq‘q‘
|ni wire site, orbital (n = 1,...,N)
|χ (t)i,|φ (t)i Floquet states, Floquet modesα α
e −ih¯γ complex quasienergyα α
vImportant acronyms and symbols
0G(t,t ) retarded Green function
(k)G (e) Fourier coefficient of retarded Green function
Σ imaginary part of self energy
Γ (e) spectral density of lead‘‘
ξ (e) noise operator of lead‘‘
−1f(e) Fermi function (= [1+exp(e/k T)] )B
vi1. Introduction
The development in chip technology over the past 50 years has been truly
breathtaking. As a consequence of the ongoing miniaturization, we have
reached a situation where the fabrication of integrated circuits based on
complementarymetaloxidesemiconductors(CMOS)encountersseverelim
itations. The structure size of the next chip generation expected in 2007 will
be 45nm with a gate oxide that is only three atoms thick. Pursuing the top
downapproachfurtherbymanufacturingwiththehelpoflithographyeven
smaller structures, undesirable quantum mechanical effects like tunneling
start to play a decisive role. The tunneling of electrons results in consider-
able leakage currents which is one of the main issues in microelectronics
nowadays (Narendra and Chandrakasan, 2005).
A conceptually different idea is the bottom up method approaching from
the other side: Atoms or molecules constitute the functional units of inte
grated circuits at the nanoscale. This idea initiated the field of molecular
electronics. A milestone of molecular electronics is the paper by Aviram
and Ratner (1974) in which they suggested electrical rectification by a sin
gle molecule with suitable asymmetry. One of the first experiments in the
field has been performed by Mann and Kuhn (1971) who studied the trans
port through alkane chains in ordered Langmuir Blodgett monolayers. The
Langmuir Blodgett technique and self assembly are by now a standard way
to form a monolayer of molecules on a surface (Ulman, 1991). By sandwich
ing such a film between metal electrodes, a rectification effect characterized
by an asymmetric current–voltage curve has been observed (Geddes et al.,
1992; Metzger et al., 1997). Meanwhile, there exists a rich variety of molecu
lar rectifiers (for an exhaustive survey see Metzger, 2003).
All the measurements discussed so far have in common that presumably
manymoleculesareinvolvedintheelectrontransfer.Averyelegantwayfor
probing conductance through single molecules is the technique of mechan
11. Introduction
ically controllable break junctions which has been developed in the context
of atomic point contact experiments (for a recent review see Agraït et al.,
2003). The first experiment of this kind by Reed et al. (1997) used molecules
bonding via thiol groups to the gold electrodes of an open break junction.
They concluded from conductance measurements that the number of active
molecules could be as few as one. A similar but more systematic and clear-
cut experiment has been performed by Reichert et al. (2002, 2003) using
bothasymmetricandanasymmetricmolecule.Thesymmetrypropertiesof
the sample are reflected in the current–voltage characteristics. This as well
as the sample to sample fluctuations in the conductance clearly pointed at
transport mediated by an individual molecule. A large number of review
articles, special issues and books on the topic of molecular electronics have
been published recently(Joachim et al., 2000; Hänggi et al., 2002; Heath and
Ratner, 2003; Nitzan and Ratner, 2003; Cuniberti et al., 2005).
Theinvestigationoftransportphenomenainsuchnanoscaleconductorsis
afascinatingfield.Inordertogainamoreprofoundinsightintothephysics
at work, the examination of the noise characteristics in small electric con
ductors proves to be a powerful tool (Blanter and Büttiker, 2000; Beenakker
and Schönenberger, 2003; Kohler et al., 2005). This is best summarized by
the saying of Rolf Landauer: “The noise is the signal.” From an experimen
tal point of view, an instructive noise signal in nanoconductors is extremely
small and it is a challenging task to detect fluctuations which can be solely
attributed to features of the conductor itself. The mostly undesired back
ground noise inherent to the measuring apparatus might be of the same
order of magnitude and might even exhibit similar characteristics. Noise in
electrical currents was first discussed by Schottky (1918) for vacuum tubes,
where the current in the device fluctuates due to the stochastic nature of
the electron emission process. This so called shot noise possesses a spec
tral density which is proportional to the time averaged current. However,
if quantum coherence is important for the electrical conduction, then noise
properties different from shot noise are to be expected. Exploring theoreti
callythenoisebehaviorofconductorshelpstointerpretexperimentalresults
and might yield suggestions for improving the setup.
In order to construct useful devices, however, it is not sufficient to have a
2currentflowingthroughamolecule,butonealsoneedstheabilitytocontrol
this current. This can in principle be achieved by the so called single elec
trontransistorsetupinwhichagateelectrodeisplacedclosetothemolecule.
Applying a gate voltage thus allows influence upon the transport across the
molecule. In more complex circuits, the need for a large numb