Dynamics of Shuttle Devices

CHAPTER 1. INTRODUCTION
1.2 A new transport regime
The shuttle devices are a particular kind of NEMS. The characteristic compo-
nent that gives the name to these devices is an oscillating object of nanometer
size that transfers electrons one-by-one between a source and a drain lead.
The device represents the nano-scale analog of an electromechanical bell in
which a metallic ball placed between the plates of a capacitor starts to os-
cillate when a high voltage is applied to the plates. The oscillations are
sustained by the external bias that pumps energy into the mechanical sys-
tem: when the ball is in contact with the negatively biased plate it gets
charged and the electrostatic field drives it towards the other capacitor plate
where the ball releases the electrons and returns back due to the oscillator
restoring forces1 and the cycle starts again.
In the first proposal [1] of a shuttle device the movable carrier is a metallic
grain confined into a harmonic potential by elastically deformable organic
molecular links attached to the leads. The transfer of charge is governed by
tunneling events, the tunneling amplitude being modulated by the position of
the oscillating grain. The exponential dependence of the tunneling amplitude
of the grain position leads to an alternating opening and closing of the left
and right tunneling channels that resembles the charging and discharging
dynamics of the macroscopic analog.
Different models for shuttle devices have been proposed in the literature
since this first seminal work by Gorelik et al. [1]. The mechanical degree
of freedom has been treated classically (using harmonic [2, 3, 4, 5] or more
realistic potentials [6]) and quantum mechanically [7, 8, 9]. Armour and
MacKinnon proposed a model with the oscillating grain flanked by two static
quantum dots [7, 10, 11, 12]. More generally the shuttling mechanism has
been applied to Cooper pair transport [13, 14] and pumping of superconduct-
ing phase [15] or magnetic polarization [16].
The essential feature of the nano-scale realization is the quantity trans-
ferred per cycle (a charge up to 1010 electrons for a macroscopic bell) that
is scaled down to 1 quantum unit (electron, spin, Cooper pair in the differ-
ent realizations). We can already guess the basic properties of the shuttle
transport:
1. Charge-position correlation: the shuttling dot loads the charge on one
side and transfers it on the other side, it releases it and returns back
1Due to the large amount of electrons in this macroscopic realization the ball gets
positively charged at the second plate by loosing some extra electrons and the restoring
force contains also an electrostatic component. The system is perfectly symmetric under
commutation of charge sign.
12
1.3. EXPERIMENTAL IMPLEMENTATIONS
to the starting point;
2. Matching between electronic and mechanical characteristic times (non-
adiabaticity);
3. Quantized current determined by the mechanical frequency;
4. Low current fluctuations: the stochasticity of the tunneling event is
suppressed due to an interplay between mechanical and electrical prop-
erties. The tunneling event is only probable at some particular short
time periods fixed by the mechanical dynamics (i.e. when the oscillat-
ing dot is close to a specific lead).
1.3 Experimental implementations
An experimental realization of the shuttle device has been produced by Erbe
et al. [17]. The structure consists of a cantilever with a quantum island at
the top placed between source and drain leads. Two lateral gates can set
the cantilever into motion via a capacitive coupling. An ac voltage applied
at these gates makes the cantilever vibrating and brings the tip alternatively
closer to the source or drain lead and thus allows the shuttling of electrons.
The device (shown in Fig. 1.1) is built out of silicon-on-insulator (SOI)
materials (using Au for the metallic parts) using etch mask techniques and
optical and electron beam lithography. The cantilever is 1µm long and the
corresponding resonant frequency is of the order of 100MHz. The source elec-
trode and the cantilever are at an average distance of approximately 0.1µm.
Shuttling experiments have been performed by Erbe et al. at different tem-
peratures. For experiments at 4.2K and 12K they measured a pronounced
peak in the current through the cantilever for a driving frequency of approx-
imately 120MHz corresponding to the natural frequency of the first mode
of the oscillator. The peak in the current corresponds to a rate of shuttle
success of about one electron per 9 mechanical cycles. The Erbe experiment
is very close to the original proposal by Gorelik. The only difference is in the
external driving of the mechanical oscillations. In the original proposal the
bias was time independent and the driving induced by the electrostatic force
on the charged oscillating island. The initially stochastic tunneling events
would eventually cause the shuttling instability and drive the system into
a self-sustained mechanical oscillation combined with periodic charging and
discharging events.
Another experiment often mentioned in the context of quantum shuttles
is the C60-experiment by Park et al. [18] In this experiment a C60 molecule is
13
CHAPTER 1. INTRODUCTION
Figure 1.1: Electron micrograph of the nano-mechanical resonator. The cantilever
(C) can be set into motion by applying an ac-voltage between the two gates G1 and
G2, and by applying a bias across the metallic tip of the cantilever (the island)
electrons are shuttled from the source (S) to the drain (D). The picture is taken
from [17].
Figure 1.2: The C60 experiment by Park et al. The C60 molecule (of mass M) can
be considered as being attached to a spring with spring constant k and corresponding
quantized excitation energy hf of the order of 5 meV. The figure is taken from [18].
14
1.4. THIS THESIS
deposited in the gap between two gold electrodes. The gap, produced with
break junction technique, has a width of 1 nm. The molecule (of diameter 0.7
nm) is bound to the electrodes due to van der Waals interaction. Around the
equilibrium position the potential can be approximated by a harmonic po-
tential and the molecule can be considered as attached by springs. We show
a schematic representation of this idea in Fig. 1.2. In the experiment Park
et al. sweep the voltage across the junction and register sudden increases in
the current. The steps are separated by 5 meV. Since the lowest internal ex-
citation energy of the C60 molecule is 35 meV one concludes that the slower
center of mass motion could be involved in the process. This hypothesis is
confirmed by the fact that the separation between the steps is independent
of the charge on the C60 molecule and the theoretical estimate for the energy
corresponding to the center of mass oscillation in the van der Waals potential
is exactly 5 meV. The IV-curve measured in these experiments can be inter-
preted in terms of shuttling [19], but also alternative explanations have been
promoted [20, 21, 22]. Whether we are in presence of coherent or incoherent
shuttling transport and to what extent the shuttling mechanism is involved
in this set-up cannot be completely clarified by current measurement. The
current-current correlation (also called current noise) was proposed by Fe-
dorets et al. [19] for a better understanding of the underlying dynamics of
the current jumps. For this reason the calculation of noise in shuttle devices
has been performed by many different groups. We proposed a completely
quantum formulation [23] that is also explained in detail in this thesis.
1.4 This thesis
This thesis contains the description and analysis of the dynamics of two
realizations of quantum shuttle devices. The models we consider describe
both the mechanical and electrical degrees of freedom quantum mechanically.
For the single dot quantum shuttle we extended an existing classical model
proposed by Gorelik et al. [1]. For the triple dot case we adopted the model
invented by Armour and MacKinnon [7]. In the following we outline the
contents of the thesis:
In Chapter 2 we introduce the two models called Single Dot Quantum
Shuttle and Triple Dot Quantum Shuttle, the first being the quantum ex-
tension also for the mechanical degree of freedom of the model originally
proposed by Gorelik et al. [1] while the second is the model invented by
Armour and MacKinnon [7]. Also in this model the oscillator is treated
quantum mechanically and the central moving dot is flanked by two static
dots.
15
CHAPTER 1. INTRODUCTION
We dedicate Chapter 3 to the derivation of the Generalized Master Equa-
tion (GME) that describes the shuttle device dynamics. Due to the different
coupling strengths we treat the mechanical and electrical baths with two dif-
ferent approaches. The Gurvitz approach for the electrical and the standard
Born-Markov approximation for the mechanical bath. The derivation a` la
Gurvitz represents a large part of this chapter. In order to facilitate the
understanding of this non-standard method and appreciate our extension to
the shuttle device we have given a long introduction in which we analyze in
great detail simpler models with increasing physical complexity. This shows
on one hand the essence of the derivation in simpler cases and also under-
lines the potentiality of this approach. An important aspect of this method
is also to be a natural prelude to full counting statistics since it naturally
produces a GME that counts the number of electron which tunneled through
the device at a certain time. We close the chapter with the description of
the numerical iterative method that we adopted for the calculation of the
stationary solution of the GME.
In Chapter 4 we introduce the concept of Wigner distribution function
and derive the corresponding Klein-Kramers equation for the SDQS start-
ing from the GME that we obtained in the previous chapter. The Wigner
function description is motivated by the effort to keep the complete quantum
treatment we achieved with the GME without losing as much as possible the
intuitive classical picture and with the possibility to handle the quantum-
classical correspondence.
Chapter 5 is dedicated to the definition and application of the three inves-
tigation tools we have chosen to analyze the properties of shuttle devices: the
charge resolved phase-space distribution, the current and the current-noise.
The phase space analysis reveals the shuttling transition and the charge po-
sition correlation typical of this operating regime. It also gives a very clean
way to appreciate “geometrically” the quantum to classical transition of the
shuttling behaviour for different device realizations. The second investigation
tool that we consider is the current. From the current calculation we obtain
also in the quantum treatment the quantized value of one electron per cycle
found in the semiclassical treatments of similar devices. We then present
current-noise calculations based on the MacDonald formula. The derivation
is strongly dependent on the derivation a` la Gurvitz of the n-resolved GME.
The low noise quasi-deterministic behaviour of the shuttling transport is clear
from the extremely low Fano factors found for this regime. In general we are
able with all the three investigation tools to identify three operating regimes
of shuttle devices: the tunneling, shuttling and coexistence regimes.
Chapter 6 is dedicated to a qualitative description of the these regimes
and also to the identification of the relation between different times and
16
1.4. THIS THESIS
length scales that define the three regimes in terms of the model parameters.
In Chapter 7 we consider separately the tunneling, shuttling and coexis-
tence regime. The specific separation of time scales allows us to identify the
relevant variables and describe each regime by a specific simplified model.
Models for the tunneling, shuttling and coexistence regime are analyzed in
this chapter. We also give a comparison with the full description in terms of
Wigner distributions, current and current-noise to prove that the models, at
least in the limits set by the chosen investigation tools, capture the relevant
dynamics.
A summary of the arguments treated in this thesis opens Chapter 8. We
conclude with a list of some of the open questions that could encourage a
continuation of the present work.
17
Chapter 2
The models
We describe in this chapter two models of quantum shuttles: the Single-
Dot Quantum Shuttle and the Triple-Dot Quantum Shuttle. Due to the
nanometer size of these devices we decide to treat quantum mechanically
not only the electrical but also the mechanical dynamics. This approach was
suggested by the work of Armour and MacKinnon [7] for the triple dot device
and implemented for the first time by us in the single dot device.
2.1 Single-Dot Quantum Shuttle
The Single-Dot Quantum Shuttle (SDQS) consists of a movable quantum
dot (QD) suspended between source and drain leads. One can imagine the
dot attached to the tip of a cantilever or connected to the leads by some
soft legands or embedded into an elastic matrix. In the model the center of
mass of the nanoparticle is confined to a potential that, at least for small
diplacements from its equilibrium position, can be considered harmonic. We
give a schematic visualization of the device in figure 2.1.
Due to its small diameter, the QD has a very small capacitance and thus
a charging energy that exceeds (in the most recent realizations almost at
room temperature [24]) the thermal energy kBT 1. For this reason we assume
that only one excess electron can occupy the device (Coulomb blockade)
and we describe the electronic state of the central dot as a two-level system
(empty/charged). Electrons can tunnel between leads and dot with tunneling
amplitudes which are exponentially dependent on the position of the central
island. This is due to the exponentially decreasing/increasing overlapping of
1A quick estimate of the charging energy can be obtained for an isolated 2D metallic
disk: e2/C = e2/(8²r²0R) where R is the disk radius and ²r = 13 in GaAs. For a dot of
radius 10 nm this yields e2/C = 20meV = kB230K
19
CHAPTER 2. THE MODELS
t exp(-X /    )L λ R λt  exp(X /    )
1-10 nm
QDSource Drain
µ L
µ R
Figure 2.1: Schematic representation of the Single-dot Shuttle: electrons tunnel
from the left lead at chemical potential (µL) to the quantum dot and eventually to
the right lead at lower chemical potential µR. The position dependent tunneling
amplitudes are indicated. X is the displacement from the equilibrium position. The
springs represent the harmonic potential in which the central dot can move.
the electronic wave functions.
The Hamiltonian of the model reads:
H = Hsys +Hleads +Hbath +Htun +Hint (2.1)
where
Hsys =
pˆ2
2m +
1
2
mω2xˆ2 + (ε1 − eE xˆ)c†1c1
Hleads =
∑
k
(εlkc
†
lk
c
lk
+ εrkc†rkcrk)
Htun =
∑
k
[Tl(xˆ)c
†
lk
c1 + Tr(xˆ)c†rkc1] + h.c.
Hbath +Hint = generic heat bath
(2.2)
Using the language of quantum optics we call the movable grain alone the
system. This is then coupled to two electric baths (the leads) and a generic
heat bath. The system is described by a single electronic level of energy
ε1 and a harmonic oscillator of mass m and frequency ω. When the dot is
charged the electrostatic force (eE) acts on the grain and gives the electrical
influence on the mechanical dynamics. The electric field E is generated by
the voltage drop between left and right lead. In our model, though, it is
kept as an external parameter, also in view of the fact that we will always
assume the potential drop to be much larger than any other energy scale
of the system (with the only exception of the charging energy of the dot).
The operator form xˆ, pˆ for the mechanical variables is due to the quantum
treatment of the harmonic oscillator. In terms of creation and annihilation
operators for oscillator excitations we would write:
20
2.1. SINGLE-DOT QUANTUM SHUTTLE
xˆ =
√
~
2mω (d
† + d)
pˆ = i
√
~mω
2
(d† − d)
pˆ2
2m +
1
2
mω2xˆ2 = ~ω
(
d†d+ 1
2
)
(2.3)
The leads are Fermi seas kept at two different chemical potentials (µL
and µR) by the external applied voltage (∆V = (µL−µR)/e ). The oscillator
is immersed into a dissipative environment that we model as a collection of
bosons and is coupled to that by a weak bilinear interaction:
Hbath =
∑
q
~ωqdq†dq
Hint =
∑
q
~g(dq + dq†)(d+ d†)
(2.4)
where the bosons have been labelled by their wave number q. The damping
rate is given by:
γ(ω) = 2pig2D(ω) (2.5)
where D(ω) is the density of states for the bosonic bath at the frequency of
the system oscillator. A bath that generates a frequency independent γ is
called Ohmic.
The coupling to the electric baths is introduced by the tunneling Hamilto-
nian Htun. The tunneling amplitudes Tl(xˆ) and Tr(xˆ) depend exponentially
on the position operator xˆ and represent the mechanical feedback on the
electrical dynamics:
Tl,r(xˆ) = tl,r exp(∓xˆ/λ) (2.6)
where λ is the tunneling length. The tunneling rates from and to the leads
(Γ¯L,R) can be expressed in terms of the amplitudes:
Γ¯L,R = 〈ΓL,R(xˆ)〉 =
⟨
2pi
~
DL,R exp
(
∓2xˆ
λ
)
|tl,r|
2
⟩
(2.7)
where DL,R are the densities of states of the left and right lead respectively
and the average is taken with respect to the quantum state of the oscillator.
The model presents three relevant time scales: the period of the oscillator
2pi/ω, the inverse of the damping rate 1/γ and the average injection/ejection
21
CHAPTER 2. THE MODELS
time 1/Γ¯L,R. It is possible also to identify three important length scales:
the zero point uncertainty ∆xz =
√
~
2mω , the tunneling length λ and the
displaced oscillator equilibrium position d = eEmω2 . Different relations between
time and length scales distinguish different operating regimes of the SDQS.
2.2 Triple-Dot Quantum Shuttle
The Triple-Dot Quantum Shuttle (TDQS) was proposed by Armour and
MacKinnon [7]. The system consists of an array of three QD’s: a movable
dot, that we assume confined to a harmonic potential, flanked by two static
ones. Relying on low temperature and on the low capacitance of the system
with respect to the leads, we again assume strong Coulomb blockade: only
one electron at a time can occupy the three-dot device. The Hamiltonian for
the model reads:
H = Hsys +Hleads +Hbath +Htun +Hint (2.8)
Only the system and tunneling part of the Hamiltonian differ from the
one dot case:
Hsys =²0|0〉〈0|+
∆V
2
|L〉〈L| − ∆V
2x0
xˆ|C〉〈C| − ∆V
2
|R〉〈R|+ ~ω
(
d†d+ 1
2
)
+ tR(xˆ)(|C〉〈R|+ |R〉〈C|) + tL(xˆ)(|C〉〈L|+ |L〉〈C|)
Htun =
∑
k
Tl(c
†
lk
|0〉〈L|+ c
lk
|L〉〈0|) + Tr(c†rk |0〉〈R|+ crk |R〉〈0|)
(2.9)
where Hosc is the harmonic-oscillator Hamiltonian, |α〉, α = 0, L, C,R are the
vectors that span the electronic part of the system Hilbert space . The tun-
able injection and ejection energies (the energy levels of the outer dots, that
we can assume fixed by external gates) simulate a controlled bias through the
device (∆V ) and the position dependent tunneling amplitudes are now be-
tween elements within the system. These amplitude are assumed to be expo-
nentially dependent on the position of the central dot tL(xˆ) = −V0e−(x0+xˆ)/λ
and tR(xˆ) = −V0e−(x0−xˆ)/λ. Tunneling from the leads is allowed only to the
nearest dot and the corresponding tunneling amplitude is independent of the
position of the oscillator. The “device bias” ∆V also gives rise to an electro-
static force on the central dot, when charged. A schematic representation of
the Triple-dot Shuttle is given in figure 2.2.
For reasons that will become clearer in the following, we assume that all
the energy levels of the system (except the Coulomb charging energy that
22
2.2. TRIPLE-DOT QUANTUM SHUTTLE
µ L
µ R
t (x)L t  (x)R
1-10 nm
Source Drain
ΓL ΓR
L C R
Figure 2.2: Schematic representation of the Triple-dot Shuttle: the leads and the
three-dot array are represented. The arrows mimic the electrical dynamics. Single
and double arrows indicate that the tunneling from and to the lead is always in a
given direction and incoherent while the internal dynamics of the system is subject
to coherent oscillations. The mechanical motion of the central dot confined to a
harmonic potential is represented by the springs.
ensures the strong Coulomb blockade regime) lie well inside the bias window.
In practice we will take the limit µL → ∞ and µR → −∞. This is reflected
in the directional flow of electrons from the source and to the drain.
23
Chapter 3
Generalized Master Equation
The state of a physical system is determined by the measurement of a certain
number of observables. Repeated measurements of a given observable always
return the same expectation value when the system is in an eigenstate for that
particular observable. The uncertainty principle ensures us that, for quantum
systems, there are incompatible observables that can not be measured at the
same time with indefinite precision.
Given a generic quantum system S and a complete set of compatible ob-
servables Ai [25], an eigenstate of the system for all observables is defined by
the set of the corresponding expectation values, i.e. the quantum numbers
ai. Each of the possible sets of expectation values is associated with an eigen-
vector in the Hilbert space of the system. More precisely the Hilbert space
of the system is spanned by the eigenvectors of a complete set of compatible
observables.
A pure state of the system is represented by a radius (class of equivalence
of normalized vectors with arbitrary phase) of this Hilbert space. We call |ψ〉
a representative vector of a radius. Observables are associated to Hermitian
operators on the Hilbert space of the system. The dynamics of the quantum
system is governed by the Hamiltonian operator, i.e. the operator associated
to the observable energy. Given an initial vector, the Schro¨dinger equation
prescribes the evolution of this vector at all times:
i~
d
dt
|ψ(t)〉 = H|ψ(t)〉 (3.1)
with the initial condition |ψ(0)〉 = |ψ0〉. An equivalent formulation of the
dynamics can be given in terms of projector operators |ψ〉〈ψ|. A projector is
independent from the arbitrary phase of the vector |ψ〉, it is then equivalent
to a radius of the Hilbert space and represents a pure state of the system.
Using the Leibnitz theorem for derivatives and the Schro¨dinger equation we
25
CHAPTER 3. GENERALIZED MASTER EQUATION
derive the equation of Liouville-von Neumann:
dρ
dt
= − i
~
[H, ρ] (3.2)
where ρ ≡ |ψ〉〈ψ|, [A,B] ≡ AB − BA is the commutator of the operators A
and B. The operator ρ is usually called density operator. For each basis of
the Hilbert space all the operators have a matrix representation. The matrix
that corresponds to the density operator is called density matrix. Each vector
of the basis of the Hilbert space corresponds to a particular eigenstate of
the system defined by a set of quantum numbers. The diagonal elements
of the density matrix are called populations. Each population represents
the probability that the system in the pure state |ψ〉〈ψ| is in the eigenstate
defined by the corresponding set of quantum numbers. The trace of the
density matrix is one and supports this probabilistic interpretation. The
off-diagonal terms of the density matrix are the coherencies of the system.
They reflect the linear structure of the Hilbert space. A linear combination
of eigenvectors gives rise to a pure state with non-zero coherencies.
Not all density operators correspond to pure states. A convex linear
combination of pure states |ψn〉〈ψn|, n = 1, ..., N is called statistical mixture:
ρ =
N∑
n=1
Pn|ψn〉〈ψn| (3.3)
where Pn ∈ [0, 1),
∑
n Pn = 1. This is an incoherent superposition of pure
states. Also statistical mixtures obey the Liuoville-von Neumann equation
of motion (3.2).
The master equation is an equation of motion for the populations. It is
a coarse grained1 equation that neglects coherencies. It was derived the first
time by Pauli under the assumption that coherencies have random phases in
time due to fast molecular dynamics. It reads:
dPn(t)
dt
=
∑
m
[ΓnmPm(t)− ΓmnPn(t)] (3.4)
where Pn is the population2 of the eigenstate n and Γnm is the rate of prob-
ability flow from eigenstate n to m [26].
1
In the sense that it describes the effective dynamics on a time scale long compared to
the typical times of the fastest processes in the physical system.
2Since a density matrix without coherencies is a statistical mixture of eigenstates we
have adopted the notation Pn ≡ ρnn
26
3.1. COHERENT DYNAMICS OF SMALL OPEN SYSTEMS
3.1 Coherent dynamics of small open systems
The master equation is usually derived for models in which a“small” system
with few degrees of freedom is in interaction with a “large” bath with effec-
tively an infinite number of degrees of freedom. The Liouville von-Neumann
equation of motion for the total density matrix is very complicated to solve
and actually contains too much information since it also takes into account
coherencies of the bath. It is useful to average it over bath variables and
obtain an equation of motion for the density matrix of the system (the re-
duced density matrix ). With no further simplification this equation is called
Generalized Master Equation (GME) since it involves not only the pop-
ulations but also the coherencies of the small subsystem. The derivation
of the GME from the equation of Liouville-von Neumann is far from trivial
and also non-universal: it involves a series of approximations justified by
the physical properties of the model at hand. Despite the apparent similar-
ities, the two equations are deeply different: the equation of Liouville-von
Neumann describes the reversible dynamics of a closed system; the GME,
instead, describes the irreversible dynamics of an open system that continu-
ously exchange energy with the bath3.
Shuttle devices are small systems coupled to different baths (leads, ther-
mal bath) but they maintain a high degree of correlation between electrical
and mechanical degrees of freedom captured by the coherencies of the reduced
density matrix. The GME seems to be a good candidate for the description
of their dynamics.
In the next two sections we will derive two GMEs using two different
approaches. They are both necessary for the description of the shuttling
devices since they correspond to the different coupling of the system to the
mechanical and electrical baths.
3.2 Quantum optical derivation
The harmonic oscillator weakly coupled to a bosonic bath is a typical problem
analyzed in quantum optics. This model well describes in shuttling devices
the interaction of the mechanical degree of freedom of the NEMS with its
environment. Following section 5.1 of the book “Quantum Noise” by C. W.
Gardiner and P. Zoller [27] we start considering a small system S coupled to
a large bath B described by the generic Hamiltonian:
3How can irreversibility be derived from reversibility? The solution of this dilemma lies
in time scales: system+bath recurrence time is “infinite” on the time scale of the system.
The GME holds on the time scales of the system.
27