THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
Electromechanical Phenomena inSuperconducting and Normal
Nanostructures
MILTON EDUARDO PEÑA AZA
Department of Physics
University of Gothenburg
SE-412 96 Göteborg, Sweden 2013
Electromechanical Phenomena in Superconducting and Normal Nanostructures
MILTON EDUARDO PEÑA AZA
ISBN: 978-91-628-8699-8
Electronic version available at:
http://hdl.handle.net/2077/32677
Doktorsavhandling vid Göteborgs Universitet
c©Milton Eduardo Peña Aza, 2013
Condensed Matter Theory Group
Department of Physics
University of Gothenburg
SE-412 96 Göteborg
Sweden
Telephone +46 (0)31 786 0000
Typeset in LATEX
Figures created using MATLAB, GIMP, POV-Ray, MATHEMATICA and Paint.
The suspended carbon nanotubes illustrations were created by Yury A. Tarakanov,
Gustav Sonne and Tomasz Antosiewicz. The time evolution of voltage-driven An-
dreev levels image was done by Gustav Sonne.
Printed by Kompendiet
Göteborg, Sweden 2013
Electromechanical Phenomena in Superconducting and Normal Nanostructures
MILTON EDUARDO PEÑA AZA
Condensed Matter Theory
Department of Physics
University of Gothenburg
ABSTRACT
This thesis summarizes a series of theoretical studies on the electromechanical prop-
erties of nanostructures made of superconducting and/or metallic elements. The first
part of the work is devoted to the analysis of the interactions between the electronic
and mechanical degrees of freedom in suspended nanowires. In particular, a metallic
carbon nanotube fixed between two superconducting leads and acting as a supercon-
ducting weak link is considered. This system is denoted as a nanoelectromechanical
Josephson junction. If biased by a dc voltage, such a nanodevice possesses the ability to
self-cool through the transfer of energy from the flexural vibrations of the suspended
nanowire to voltage-driven Andreev states and then to quasiparticle electronic states
in the superconducting leads. The electromechanical coupling required to accomplish
the energy transfer process can be attained by applying an external magnetic field. It
gives rise to a Lorentz force that couples displacements of the carbon nanotube to the
electrical current that is carried by Andreev states.
Further investigations of the nanoelectromechanical Josephson junction extend
the analysis of the first study to a case in which the system is subjected to a nonuni-
form magnetic field. In this case, inhomogeneity of the field causes the conducting
nanoresonator to execute a whirling movement. The analysis of the time evolution
of the amplitude and relative phase of the nanowire motion shows that the coupled
amplitude-phase dynamics presents different regimes depending on the degree of
inhomogeneity of the magnetic field: time independent, periodic, and chaotic.
The second part of the thesis describes the dynamics of a spatially symmetric
shuttle-system subjected to an ac gate voltage. In this system, parametric excitation
gives rise to mechanical vibrations at the resonant frequency, i.e., when the frequency
of the ac signal is close to the eigenfrequency of the mechanical subsystem. The para-
metrically excited mechanical oscillations result in a dc shuttle current in a certain
direction due to spontaneous symmetry breaking, where the direction of the current
is determined by the phase shift between the ac voltage and the induced mechanical
oscillations.
Keywords: Nanoelectromechanical systems, nanoelectromechanical Josephson junc-
tion, metallic carbon nanotubes, ground-state cooling, chaos, parametric excitation,
dc shuttle current.
Research publications
This thesis is an introduction to and summary of the work contained in the following
research articles printed as appendices and referred to, by capital Roman numerals in
the text, as Papers I-IV.
PAPER I
Cooling of a suspended nanowire by an ac Josephson current flow
G. Sonne, M.E. Peña-Aza, L.Y. Gorelik, R.I. Shekhter and M. Jonson
Physical Review Letters 104, 226802 (2010).
PAPER II
Voltage-driven superconducting weak link as a refrigerator for cooling of nanomechanical vi-
brations
G. Sonne, M.E. Peña-Aza, R.I. Shekhter, L.Y. Gorelik and M. Jonson
Fizika Nizkikh Temperatur 36, Nos. 10/11, 1128 (2010).
PAPER III
Dynamics of a suspended nanowire driven by an ac Josephson current in an inhomogeneous
magnetic field
M.E. Peña-Aza
Physical Review E 86, 046208 (2012).
PAPER IV
Parametric excitation of a dc shuttle current via spontaneous symmetry breaking
M.E. Peña-Aza, Alessandro Scorrano and L.Y. Gorelik
arXiv:1212.1035 (2012)
Submitted to Physical Review B.
These articles are appended at the end of the thesis.
I
The specific contribution by the author, M.E. Peña-Aza, to the appended papers was
the following,
PAPERS I - II
I performed preliminary model calculations for the problem, optimized the numer-
ical code in order to obtain the dc current through the system, and took part in the
analysis and discussion. I also participated in the writing process of the articles.
PAPER III
I am the sole author.
PAPER IV
I performed the model calculations and numerical simulations. I took part in dis-
cussions and analysis of the results. I also participated in the writing process of the
article.
II
TABLE OF CONTENTS
Research publications I
Table of Contents III
1 Introduction 1
1.1 Mechanical properties of suspended
nanowires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Mechanical systems in the quantum regime . . . . . . . . . . . . . . . . 4
1.4 Superconducting weak links . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4.1 Andreev levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4.2 DC Josephson current . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4.3 AC Josephson effect . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.4 Crisis and chaotic attractors . . . . . . . . . . . . . . . . . . . . . 7
1.4.5 Shuttle mechanism of charge transport . . . . . . . . . . . . . . 8
1.4.6 Parametric resonance . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Cooling of a suspended nanowire 12
2.1 System and electromechanical coupling . . . . . . . . . . . . . . . . . . 12
2.2 Mechanical Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Electronic Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Interaction Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5 Total system Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.6 Transition between Andreev levels . . . . . . . . . . . . . . . . . . . . . 18
2.7 Density Matrix Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.8 Evolution of the density matrix . . . . . . . . . . . . . . . . . . . . . . . 20
2.9 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.10 DC current through the junction . . . . . . . . . . . . . . . . . . . . . . 23
2.11 Quasiparticle spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.12 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 Nonlinear dynamics of a suspended nanowire 26
3.1 System and equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 Numerical results and discussion . . . . . . . . . . . . . . . . . . . . . . 29
3.3 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
III
Table of Contents
4 Parametric effects in a shuttle system 38
4.1 System and equations of motion . . . . . . . . . . . . . . . . . . . . . . 38
4.2 DC shutle current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.3 Analysis and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.4 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5 Summary 45
6 Acknowledgements 47
Appendix 1: Adiabatic condition for steady level population 48
Appendix 2: Rotating Wave Approximation 49
Bibliography 52
IV
CHAPTER 1
Introduction
Recent developments in nanotechnology have made possible the design and fabrica-
tion of nanomechanical resonators with high resonance frequencies (106-109 Hz), high
quality factors Q (103-105 ), and small masses (10−15-10−17 kg) [1–5]. These attributes
make nanoelectromechanical systems (NEMS) excellent devices for investigations in
basic science and engineering. Indeed, nanoelectromechanical systems have been
widely used for the exploration of the quantum world and for the development of
new technological applications [6–9]. In basic research NEMS are considered promis-
ing candidates for studying the quantum limit of mechanical motion [?,10–13]. It is ex-
pected that the quest for the quantum regime in such devices will elucidate questions
of fundamental nature in physics, for instance, the quantum-mechanical description
of macroscopic objects [14–18]. In addition, mechanical systems at the quantum limit
may become useful for applications in high precision measurements [19–21].
This thesis presents theoretical studies of two different nanoelectromechanical
systems. The first part of the work is mainly based on the effects of electromechanical
coupling in a nanodevice where an oscillating suspended nanowire forms a weak-link
between two superconductors, a nanoelectromechanical Josephson junction. The material
covered in the second part of the thesis deals with parametric effects in a nanoelec-
tromechanical shuttle system.
Three important outcomes from our research are the possibility to cool down the
flexural vibrations of the oscillating suspended nanowire, to devise a setup for study-
ing nonlinear dynamics and chaos at the nanoscopic level and, to generate a dc cur-
rent in a completely symmetric shuttle system.
Before continuing, we take a brief moment to introduce some concepts and termi-
nology that will be used in this thesis. This is done in the following sections.
1.1 Mechanical properties of suspended
nanowires
One of the main objectives of this thesis is to study how the oscillatory motion (me-
chanical degrees of freedom) of a suspended nanowire affects its charge transport
properties (electronic degrees of freedom). Thus, it is important to understand the
mechanical properties of suspended nanowires. The modal analysis of a nanome-
chanical resonator can be obtained from the Euler-Bernoulli formalism [22]. The ge-
ometry under consideration consists of a nanobeam clamped at the two ends as de-
picted in Fig. 1.1
1
Chapter 1. Introduction
Figure 1.1: Schematic diagram of a nanomechanical oscillator. A nanowire of length
L is suspended between two pillars.
The physical parameters of the nanobeam are: length L, elasticity modulus E, mo-
ment of inertia I , density ρ, and cross-sectional area Ac. For an isotropic nanobeam
with a uniform cross-section, in the limit of small amplitudes, the position dependent
displacement in the x-direction, Z(x, t), satisfies the following differential equation
ρAc∂
2Z(x, t)
∂t2 + EI
∂4Z(x, t)
∂x4 = f(x, t) . (1.1)
In Eq. (1.1) the first term on the left hand side comes from the inertial effect of the
motion and the second one represents the stress in the beam due to deformations. On
the right hand side, it is assumed that the nanobeam is under a time-varying external
force f(x, t). To evaluate further, the displacement of the nanotube, (in the z-direction
as shown in Fig. 1.1), Z(x, t) can be expanded in terms of a complete set of orthogonal
functions ψn(x)
Z(x, t) =
∑
n
ψn(x)θn(t) , (1.2)
where θn(t) is the time-dependent amplitude of motion and the set functions ψn(x)
are the normal modes of oscillation for the doubly-clamped beam. the index n =
0, 1, 2, 3, · · · , accounts for the oscillation mode. The form of the normal modes ψn(x)
and their corresponding frequencies ωn can be obtained by substituting Eq. (1.2) in
Eq. (1.1)
EI
∑
n
θn(t)∂
4ψn(x)
∂x4 + ρAc
∑
n
ψn(x)∂
2θn(t))
∂t2 = f(x, t) . (1.3)
Here on, the modal analysis is restricted to the homogeneous case f(x, t) = 0. As a
consequence, Eq. (1.3) leads to the following equations
∂4ψn(x)
∂x4 − β
4
nψn(x) = 0 , (1.4)
∂2θn(t)
∂t2 = −β
4
n
EI
ρAc θn(t) = −ω
2
nθn(t) . (1.5)
2
1.2. The harmonic oscillator
Here, βn = (ρAc/EI)1/4ω1/2n is a constant for each mode n that depends on the geom-
etry and the material. A general solution of Eq. (1.4) is
ψn(x) = an cos(βnx) + bn sin(βnx) + cn cosh(βnx) + dn cosh(βnx) . (1.6)
The boundary conditions for a doubly-clamped nanobeam are
ψn(0) = ψn(L) = ∂ψn(x)∂x
∣∣
x=0 =
∂ψn(x)
∂x
∣∣
x=L = 0 . (1.7)
When these boundary conditions are applied to Eq. (1.6), they imply that an = −cn,
bn = −dn
ψn(x) = an
[cos(βnx)− cosh(βnx)
]+ bn
[sin(βnx)− sinh(βnx)
] , (1.8)
an
bn =
cosh(βnL)− cos(βnL)
sin(βnL) + sinh(βnL) , (1.9)
cos(βnL) cosh(βnL)− 1 = 0 . (1.10)
The zeroes of Eq. (1.10) can be found numerically, with
βnL ∼ 4.73 , 7.85 , 10.99 , 14.14 . . . (1.11)
These values can be used for the estimation of the corresponding resonance frequency
of each mode,
ωn = β2n
√
EI
ρAc . (1.12)
This expression provides only an order of magnitude estimate for the frequency in
real nanomechanical resonators. Vibrating systems at the nanometer scale are sensi-
tive to mechanical stress resulting from coupling with the external environment.
1.2 The harmonic oscillator
This thesis is about nanoelectromechanical systems. Our frameworks to study NEMS
are the languages of classical and quantum mechanics. It is the objective of this Sec-
tion to give a short introduction to the physics of classical and quantum harmonic
oscillators. Several systems can be modeled as a harmonic oscillator, e.g., a mass on
a spring, a pendulum or the relative motion of atoms in molecules. In Papers I - III
we will consider oscillations of suspended nanowires which can also be described as
harmonic oscillators. From the mathematical point of view, the classical harmonic
oscillator is described by the equation
mv2
2 +
kx2
2 = E. (1.13)
Here, on the left-hand side of the equation, the first term corresponds to the energy
that the oscillator possesses due to its motion, i.e., kinetic energy. Similarly, the second
term is the energy associated with the position of the object in space, i.e., potential
energy. On the right-hand side of the equation, E is the total energy of the system
3
Chapter 1. Introduction
which is a conserved quantity and can take any value. At the equilibrium position
(oscillator at rest position) the energy of the oscillator is zero. In Eq. (1.13), m is the
mass of the oscillator, v is its velocity, x is its deflection coordinate and k the spring
constant.
The description of the quantum harmonic oscillator is different. The dynamics is
described through the Schrödinger equation,
Hˆ|n〉 =
( pˆ2
2m +
mω2xˆ2
2
)
|n〉 = En|n〉, (1.14)
for the energy eigenstate |n〉. In the Schrödinger equation ω is the frequency at which
the object oscillates, pˆ and xˆ are the momentum and position operators, respectively. It
turns out that the energy of the quantum harmonic oscillator can not take any value
as in the classical case. Rather, the energy can take discrete values En = ~ω(n +
1/2) where n = 0, 1, 2.... Therefore, the energy of the quantum harmonic oscillator is
quantized in multiples of the energy scale of the oscillator ~ω.
In accordance with the above energy equation at n = 0 the energy of the oscillator
is ~ω/2, and we say that the oscillator is in its ground state. To attain the ground state
the system must be cooled down to temperatures close to the absolute zero tempera-
ture. Even at zero temperature the oscillator can vibrate with an amplitude which is
known as the zero-point amplitude.
1.3 Mechanical systems in the quantum regime
The position measurement of any oscillator is limited by quantum mechanics, i.e.,
the harmonic oscillator is no longer considered a classical object when the position
amplitude fluctuations of the resonator become comparable to the width of its wave-
function [23]:
∆x =
√
~
2mω , (1.15)
This quantity, which is the root mean square amplitude of quantum fluctuations or the
zero-point amplitude, defines the standard quantum limit [16]. To enter the quan-
tum regime the thermal energy, kBT , is required to be much less than the associated
mechanical energy quantum, ~ω, i.e.
kBT ¿ ~ω . (1.16)
In the last equation kB is the Boltzmann constant, ~ is the reduced Planck constant,
and T is the temperature of the environment.
1.4 Superconducting weak links
A superconducting weak link is an insulating or conducting element that connects
two superconductors [24]. Considering a normal metal weak link, the resulting struc-
ture is an S-N-S junction. Before describing the electronic properties of such a super-
conducting device a few words about superconductivity and the Josephson effect will
be mentioned.
4
1.4. Superconducting weak links
• Superconductivity is a phenomenon in which some materials expulse the mag-
netic field and have no electrical resistence below a critical temperature TC [25].
Superconductivity was discovered by H. Kamerlingh Onnes in 1911 [26] and
the microscopic description of the phenomenon was formulated by Bardeen,
Cooper and Schrieffer (BCS-theory) in 1957 [27]. The main idea behind the the-
ory is that an effective attractive interaction between electrons will lead the for-
mation of Cooper pairs. These are correlated and form a quantum-mechanical
condensate characterized by the order parameter ∆0 exp(iφ). The magnitude
of the order parameter is ∆0 and φ is the superconducting phase. The ground
state is separated from the excited states by an energy gap of size 2∆0.
• The Josephson effect is a phenomenon in which a non-dissipative current can
flow between two weakly coupled superconductors separated by a non super-
conducting thin barrier or a narrow channel [28]. The current is driven by the
phase difference between the two superconductors. The Josephson effect can be
classified depending on whether an external voltage V is applied to the junc-
tion. In the first case, in the absence of voltage, a dc current can flow between
the superconductors (dc Josephson effect). In the second case, under the effect
of a constant voltage V , the current across the junction will oscillate in time at
a frequency proportional to the applied voltage (ac Josephson effect).
Proceeding with the initial discussion, the electronic properties of the S-N-S Joseph-
son junction can be understood by introducing a scattering process called Andreev
Reflection [29, 30]. At the interface between a superconductor and a normal metal
element, an electron incident from the normal part with an energy E < ∆ may be re-
flected as a hole. In this process the incoming electron combines with a time-reversed
electron below the Fermi energy and both enter the superconductor material as a
Cooper pair. A hole is created in order to conserve the charge of the system and it
travels in the opposite direction. Similarly, when this hole reaches the opposite S-
N interface, it will be reflected as an electron and multiple electron/hole reflections
will occur. Andreev reflection is an elastic process, thus all the particle reflections
can constructively interfere and create two bound states: Andreev states. One of the
most important characteristics of the Andreev bound states is the possibility to carry
current [31–33]. This can be realized by noting that the Andreev scattering process
is an effective electronic transport mechanism as it moves Cooper pairs between the
superconductors. In the following sections the spectrum of the Andreev levels and
the Josephson effect are studied in detail.
1.4.1 Andreev levels
The quantum-mechanical properties of the S-N-S junction can be determined by using
the Bogoliubov-de Gennes equations for the two-component wave function Ψ [34]:
HΨ = EΨ, Ψ =
(ν
ϕ
)
H =
( T ∆
∆∗ −T
)
, (1.17)
where T = −~2∇2/2m− µ is the kinetic energy operator and µ is the chemical poten-
tial.
5
Chapter 1. Introduction
In order to find the energy dependance of the Andreev states, it is assumed that
the length of the superconducting weak link L is shorter than the coherence lenght in
the superconductors ξ0, i.e., L ¿ ξ0 = ~vF /∆0 where vf is the Fermi velocity. This
regime corresponds to the short-junction limit where the spatial dependance of the
order parameter can be described avoiding the problem of self-consistency.
Considering that the region |x| < L/2 corresponds to the normal metal weak link
and the space |x| > L/2 to the superconductors, which for simplicity are assumed
to be identical, i.e., ∆1 = ∆2 = ∆0, TC1 = TC2 = TC . It is assumed that the order
parameter ∆(x) changes as |∆(x)| = ∆0 when |x| > L/2 and |∆(x)| = 0 when |x| <
L/2 on the interface between the normal metal and the superconductor1. However,
the phases of the order parameters in the superconductors may differ as arg∆(x) = φ1
for the superconductor placed in the region x < −L/2 and arg∆(x) = φ2 when x >
L/2. Therefore, the spacial dependence of the order parameter can be approximated
as [36]
∆(x) =



∆0 exp(iφ1), if x < −L/2,
0, if |x| < L/2,
∆0 exp(iφ2), if x > L/2.
(1.18)
The solution of Eq. (1.17) can be found by matching the wave functions in the
different regions [36–39]. The energy spectrum depends on the scattering processes
taken into consideration.
• By considering only Andreev reflections in the system, the energy spectrum as
a function of the phase difference φ = φ2 − φ1 between the superconductors
is [36]:
E±(φ) = ±∆0 cos(φ/2) . (1.19)
This mathematical expression for the energy spectrum implies that there are
two bound Andreev levels, i.e., a single bound state at positive energies E > 0
and its mirror image at negative energies E < 0.
• By including Andreev reflections and electronic scattering due to impurities in
the normal part, the energy espectrum is given by [37, 40, 41]:
E±(φ) = ±∆0
√
1−D sin2(φ/2) . (1.20)
In the last equation, D = 1−R, is the normal transmission coefficient. It can be
noticed that the case, D = 1, corresponds to the clean junction, Eq.(1.19).
1.4.2 DC Josephson current
The Josephson current is calculated from the Andreev level spectrum as [40]:
I(φ) = 2e~
∂E(φ)
∂φ . (1.21)
1This model is known as rigid boundary condition model, and it is discussed in Refs [35]
and [24]
6
1.4. Superconducting weak links
For a clean junction, the bound state current can be found by direct substitution of
Eq. (1.19) into Eq. (1.21) and it is,
I(φ) = ±2e~
∂E(φ)
∂φ
= ∓∆0e~ sin(φ/2) . (1.22)
The current in Eq. (1.22) is proportional to the sine of the phase difference between
the superconductors. For junctions sustaining both normal and Andreev reflections,
the current is given by
I(φ) = ±2e~
∂E(φ)
∂φ
= ∓∆0e2~
D sin(φ)√
1−D sin2(φ/2)
. (1.23)
This current is obtained by substituting Eq. (1.20) into Eq. (1.21).
1.4.3 AC Josephson effect
By applying an external voltage V between the superconductors, the superconducting
phase difference φ is related to the bias voltage through the expression [42]:
dφ
dt =
2eV
~ . (1.24)
A solution of Eq. (1.24) is:
φ(t) = 2eV t~ + φ0 . (1.25)
Hence, the phase difference evolves linearly in time. In Papers I - II, it is assumed that
the adiabatic condition for the phase evolution is fulfilled, i.e,
~φ˙ ¿ ∆0. (1.26)
Therefore, one can assume that the Andreev levels move adiabatically within the su-
perconductor energy gap.
E±(φ) → E±(φ(t)) . (1.27)
This adiabatic motion will play an important role on the research topic covered in
Papers I-II.
1.4.4 Crisis and chaotic attractors
One of the research topics covered in this thesis is NEMS as dynamical systems ex-
hibiting complex behavior. We will focus our attention on the occurrences of sudden
qualitative changes of chaotic attractors as a parameter is varied. These phenomena
are well known in bifurcation theory and they are denoted crises.
Our research in this thesis concerns chaotic attractors and their dynamics. We
would like to define an attractor as a compact set with a neighborhood such that,
7
Chapter 1. Introduction
for almost every initial condition in this neighborhood, the limit set of the orbit as
time tends to +∞ is the atractor. In other words, an attractor is a set towards which
a variable for a given dynamical system evolves over time in accordance with the
differential equation that determines its dynamics. Thus, points that get close enough
to the attractor keep close even if they are slightly disturbed.
A chaotic attractor can be thought of as a surface in the phase space of the dynam-
ics variables, to which the system orbit is asymptotic in time and on which it wanders
in a chaotic fashion. In a chaotic attractor there is a sensitive dependence on initial
conditions. From the mathematical point of view a chaotic attractor is one for which
typical orbits on the attractor have a positive Lyapunov exponent [43].
In this thesis we study sudden qualitative changes of chaotic dynamics (crises)
and it is possible to identify different types of crises [44],
• Boundary or exterior crisis. A boundary crisis occurs when a chaotic attractor
collides with an unstable periodic orbit on the basin boundary, converting the
attractor into a nonattracting chaotic set and generating transient chaos. In this
case, for parameter values just past the crisis point, the attractor no longer ex-
ists. Nonetheless, typical trajectories initialized in the region formerly occupied
by the destroyed attractor appear to move about in this region chaotically, as be-
fore the crisis occurred, but only for a finite time after which the orbit rapidly
leaves the region [45].
• Interior crisis. In the case of an interior crisis, there is a sudden increase in the
size of a chaotic attractor as a parameter passes through the crisis point.
• Attractor merging crisis. In this case, for parameter values just before the crisis
point, two chaotic attractors coexist, each having its own basin of attraction.
As the parameter is increased, the two attractors enlarge, and at the crisis point
they collide with the basin boundary separating their basins. As a consequence,
the two chaotic attractors merge together to increase in size. Merging crisis can
happen in systems possessing some symmetry whereby the precrisis attractors,
as well as their basins, are symmetric images of each other in the phase space.
In Chapter 4 we will introduce a nanoelectromechanical Josephson junction in an in-
homogeneous magnetic field and analyze how the dynamical bahavior of the system
leads to an example of an attractor merging crisis. For a review on dynamical system,
nonlinear phenomena and crises the reader is referred to Refs. [43–48].
1.4.5 Shuttle mechanism of charge transport
Some years ago, a novel form of electron transport (shuttle transport of electrons)
based on the mechanical vibrations of a metallic nanoparticle coupled to two elec-
trodes via elastic molecular links was proposed in Ref. [49]. Since then, the shut-
tle phenomena has been a subject of intensive experimental and theoretical research
[50–55].
The basic idea of the mechanism for shuttle transport is that the electrostatic en-
ergy, due to the tunneling of electrons from the leads to the mobile metallic nanopar-
ticle, can be large enough to deform the system. When the leads are voltage biased,
8
1.4. Superconducting weak links
above the critical voltage VC , the charged nanoparticle is pushed by the bias voltage
towards the opposite electrode where reverse charging takes place, and consequently
the nanoparticle experiences reverse motion. In this sense the metallic nanostructure
acts as a shuttle of electrons. In the proposal of Gorelik and co-workers oscillations
of the shuttle are driven via self-excitations, originating from the work done which is
determined by correlations between the charge of the nanoparticle and its position.
The shuttle nanostructure is assymmetric in the sense that a bias voltage is needed to
drive the system, thus the electrical field serves as a breaking symmetry agent.
In the shuttle mechanism of charge transport, Coulomb blockade phenomena play
an important role as they limit the number of electrons inside the nanoparticle due to
the high energy required to add an electron. For systems with continuous energy
spectra, the charging energy is Ec = Q2/2C, where Q is the extra charge and C the
capacitance of the island [56]. The main feature of the shuttle phenomenon is that
a constant potential difference, applied between two fixed electrodes, leads to a dy-
namical instability that causes the metal nanoparticle to oscillate. In the limit of low
dissipation, a dc current through the system, induced by the voltage drop between the
electrodes, becomes proportional to the frequency of the mechanical oscillations [49],
I = 2eNf where N =
[CV
e +
1
2
]
, (1.28)
e is the elementary charge, V is the voltage and f is the frequency of the nanoparticle.
The sequential transfer of electrons in shuttling phenomena was described in the
framework of classical mechanics and stochastic processes, in this case the electron
dephasing time τ0 is much shorter than the tunneling charge relaxation time RTC,
where RT is the tunneling resistance of the double junction [56]. The idea of shuttle
phenomena was also extended to the quantum realm [57–60].
Nanoelectromechanical shuttle systems have been also studied in the regime of ac
excitation and several interesting effects on the transport properties and the dynamics
of the shuttle system have been found [61–65]. In particular, a shuttle structure driven
by a time-dependent bias voltage has been considered in Refs. [66, 67]. It was shown
that in case of assymetric configuration such a setup can act as a rectifier, where the
intensity of the dc current depends on the ratio between the frequency of the external
oscillating voltage and the eigenfrequency of the mechanical subsystem. In Paper IV
we investigate the dynamics of a spatially symmetric shuttle-system subjected to an
ac gate voltage. We demonstrate that, despite the lack of a bias voltage, a shuttle dc
current can be generated. This mechanism of electron transport is an extension of the
shuttle transport proposed by Gorelik et al. [49] in which the direction of the shuttle
transport does not rely on the presence of any bias voltage.
1.4.6 Parametric resonance
In Paper IV we introduce a new form of shuttle transport of electrons. In this novel
idea, the phenomenon of parametric resonance is a crucial point. In this section we
discuss in some detail the main features of parametric resonance.
According to the conventional classification of oscillations by their method of ex-
citation, oscillations are denominated forced when the dynamical system is subjected
9
Chapter 1. Introduction
to an external periodic input. If the frequency of the external force ω is close to the
frequency oscillator ω0, the amplitude of the steady-state forced oscillations can reach
a large value. This phenomenon is called resonance [68].
Another way to excite oscillations in a system consists of a periodic variation of
some parameters of the system; oscillations are called parametric when the amplitude
of oscillation caused by the periodic modulation of some parameters, to which the
motion of the system is sensitive, increases steadily.
To discuss the parametric resonance quantitatively we consider an undamped free
oscillator in which the spring constant experiences a periodic modulation, i.e., the fre-
quency of the oscillator is time-dependent, ω(t), and differs slightly from the natural
frequency of the resonator ω0. It can be described as
ω2(t) = ω2(1 + h cos(Ωt)), (1.29)
where the constant h ¿ 1 is referred to as the parametric modulation amplitude and Ω
is the modulation frequency. Parametric resonance is strongest when the modulation
frequency of ω(t) is nearly twice ω0. Therefore we set Ω = 2ω0 + ², where ² ¿ ω0.
The solution of the equation motion for the free oscillator,
x¨+ ω2[1 + h cos(2ω0 + ²)t]x = 0, (1.30)
can be cast in the form
x = a(t) cos((ω0 + ²/2)t) + b(t) sin((ω0 + ²/2)t), (1.31)
where the functions a(t) and b(t) are time-dependent functions which vary slowly in
comparison with the trigonometrical factors.
After substitution of Eq. (1.31) into Eq. (1.30), a perturbative analysis indicates
that the instability frequency window in which the parametric resonance occurs is
(see details in Ref. [69])
−hω0/2 < ² < hω0/2 (1.32)
At the parametric resonance the equilibrium becomes unstable and the system is
excited2, in this regime the amplitude of oscillations increases. The growth of both
amplitude and energy of oscillations during parametric excitation is provided by the
work of forces that periodically change the parameter. The energy transfer process
to the dynamical system during parametric excitation can also take place when the
frequency of the periodic modulated parameter fulfill the following condition,
ω = 2ω0/n, (1.33)
where n = 1, 2, .... In general, the amount of energy given to the oscillating system
decreases with the order n of the parametric resonance.
An important difference between parametric and forced oscillations is the way
in which the growth of energy depends on the energy already stored in the system.
In the regime of parametric excitation the increment of energy is proportional to the
square of the amplitude, i.e., to the energy stored in the system. In case of forced
2Any small disturbance from equilibrium leads to a parametric instability
10
1.5. Thesis overview
oscillations, the increment of energy is proportional to the amplitude of oscillations,
i.e., to the square root of the energy.
Energy losses due to dissipative forces are also proportional to the energy stored
in the system. Therefore, in the regime of parametric oscillations the increment of
energy caused by a periodic variation of some parameters and the dissipative losses
are proportional to the square of the amplitude, thus their ratio is independent of
the amplitude of oscillation. As a consequence, parametric resonance takes place
when the increment of energy exceeds the amount of energy dissipated. Once the
this threshold value has been overcome, the frictional losses of energy cannot affect
the growth of the amplitude. In linear systems, the amplitude of the parametrically
excited oscillations grows without any limit. However, in real systems, the growth
amplitude is restricted by nonlinearities. For a review about parametric resonance,
the reader is referred to Refs. [68, 69].
1.5 Thesis overview
The scientific results presented in this thesis are based on Papers I-IV. This thesis is or-
ganized in the following manner. In Chapter 2, we summarize the material presented
in Papers I-II where we study a mechanism to cool the quantized vibrations of a sus-
pended nanowire in a voltage-biased superconducting junction. We focus our atten-
tion in the electromechanical coupling generated by a homogeneous magnetic field.
In Chapter 3, we consider the same voltage-biased nanoelectromechanical systems as
in Chapter 2, but now extend the analysis to a case in which the nanoelectromechan-
ical Josephson junction is subjected to a nonuniform magnetic field. In Chapter 4, we
study parametric excitation of a dc current in a symmetric shuttle-system. Finally, in
Chapter 5, we provide the summary of the thesis.
11
CHAPTER 2
Cooling of a suspended nanowire
It is the objective of this Chapter to discuss the conditions necessary for ground state
cooling of an oscillating nanowire suspended between two voltage-biased supercon-
ducting leads. In our description we consider the high transparency limit of the na-
noelectromechanical Josephson junction and use the language of quantum mechanics
in order to treat the oscillating nanowire. Below we describe the proposed cooling
scheme presented in Papers I-II, where the possibility to cool the vibrations of the
nanoresonator relies on the transfer of energy from the mechanical vibrations of the
nanowire to the electronic quasiparticle bath in the superconducting leads. This en-
ergy transfer process is achieved by inducing transitions between the bound Andreev
levels. In Papers I-II we analyze the situation in which these transitions can be ac-
complished by applying a uniform magnetic field perpendicular to the long axis of
the nanobeam.
2.1 System and electromechanical coupling
The diagram in Fig. 2.1 is a schematic illustration of a superconducting hybrid nan-
odevice, a superconductor-normal-superconductor (S-N-S) nanoelectromechanical Joseph-
son junction driven by a dc voltage bias V in an homogeneous magnetic field. The
junction consists of a metallic carbon nanotube suspended between two voltage-biased
superconducting leads. In such a geometry, the nanotube is simultaneously serving as
a mechanical resonator and as a weak link between the superconducting electrodes.
This double functionality of the metallic nanotube can be explained as follows:
• By considering the system as a doubly-clamped beam, the nanowire forms a
mechanically compliant element.
• By considering the system as an electronic device, the metallic nanowire acts as
a weak link between two superconductors.
It is worth mentioning that without any external influence (e.g. electromagnetic radi-
ation, magnetic field, electric field, electrostatic gates, etc.) there is no electromechan-
ical coupling between the mechanical and electronic subsystems and they remain in-
dependent each other. The properties of each subsystem has already been discussed
(see Chapter 1). However, by applying a uniform magnetic field perpendicular to the
12
2.1. System and electromechanical coupling
Figure 2.1: S-N-S nanoelectromechanical Josephson junction in a magnetic field.
direction of the current carried by the Andreev states, the Lorentz force couples vibra-
tions in the carbon nanotube to the current flowing through it, i.e., the mechanical and
electronic subsystems are interconnected. This electromechanical coupling opens an
energy transfer channel through which vibrational energy from the harmonic oscilla-
tor can be transferred into the electronic Andreev levels. The corresponding energy
uptake of the electronic subsystem is later released into the continuum states, leading
to an effective cooling of the resonator.
In order to describe in more detail the energy transfer mechanism, the dynamics
of the voltage-biased Andreev levels coupled to the mechanical subsystem is pre-
sented in Fig. 2.2. This plot is a schematic diagram of the evolution of the coupled
electromechanical system. Here, the applied dc voltage bias V causes the supercon-
ducting phase difference to evolve in time in accordance with equation (1.24), and
drives the adiabatic motion of the Andreev levels according to Eqs. (1.26) and (1.27).
In Fig. 2.2, the evolution in time of the Andreev levels is indicated by solid lines and
correspond to the periodic trajectories definied by E±(φ(t)) = ±
√
1−D sin2(φ(t)/2),
the period of the energy spectrum is TV = pi~/(eV ). In our analysis, we assume that
the thermal energy is much smaller than the initial separation between the levels,
kBT ¿ 2∆0. This assumption implies that at the start of the period (t = 0) the lower
Andreev level (•) is populated while the upper one is empty (◦). In Papers I-II we
analyze the case in which a transverse magnetic field couples the vibrations of the
nanowire to the Andreev levels, and transitions from the lower to the upper Andreev
level might occur by absorption of a quantum of mechanical energy ~ω. The prob-
abilily of absorption of a quantum of mechanical energy is increased at time t ≈ t0
when the electromechanical coupling in the system is maximum and the energy gap
between the Andreev levels attains its minimum value
Egap = 2∆0
√
R . (2.1)
On condition that the quantum of mechanical energy of the nanoresonator,
Emech = ~ω (2.2)
matches the minimum gap value between Andreev levels,
2∆0
√
R = ~ω . (2.3)
13
Chapter 2. Cooling of a suspended nanowire
Figure 2.2: Dynamics of the Andreev levels coupled to the mechanical subsystem.
Here, |±, n〉 = |±〉 ⊗ |n〉 denote the states of the junction; |±〉 labels the upper and
lower Andreev level, and |n〉 is the quantum state of the oscillator. At the start of the
period, a pair of Andreev states are created and their occupation depends on the dis-
tribution of quasiparticle excitations in the leads. As kBT ¿ 2∆0, the upper Andreev
level is empty (◦) while the lower one is the populated (•). These bound states carry
current through the nanodevice. Under conditions of adiabatic motion eV 6 4R∆0,
they evolve in time with period TV . Transitions from |−, n〉 to |+, n − 1〉 occur when
the energy gap between the levels matches the mechanical energy quantum ~ω and,
they are described through the scattering matrix Sˆ. After one period, the Andreev
states merge with the comtinuum states (arrows) and new states, orthogonal to the
old states are created. Here, the electronic subsystem is reset at the start of each pe-
riod. See discussion in the text.
14
2.2. Mechanical Hamiltonian
Then, transitions from the lower branch to the upper branch can occur and the lev-
els would become mixed. Afterwards, at the end of one period, t ≈ tV , the bound
Andreev levels merge with the continuum states in the superconducting leads and
new states are formed. Here we assume that the Andreev levels are reset at the start
of each period (see discussion in Section 2.11) and the energy transfer process is re-
peated again over many periods leading to an effective cooling of the nanowire. Note
that the mechanical subsystem is not affected by the dissolution of the Andreev levels
into the continuum spectrum in the leads, thus, they can be overcooled in the subse-
quent periods.
Finally, in order to complete our cooling scheme, a few words about adiabatic mo-
tion of the Andreev levels under weak bias must be mentioned. As discussed earlier
(see Section 1.4.3), the voltage-biased Andreev levels will move adiabatically within
the superconductor energy gap on condition that 2eV ¿ ∆0 (cf. Eq. (1.26)). In addi-
tion, our proposal for cooling implies that without any electromechanical coupling,
the population of the Andreev levels will remain constant at any time. As a conse-
quence, the applied voltage should be restricted so that it can not induce Landau-
Zener transitions at time t ≈ t0. It turns out that the applied bias must be less than or
equal to a certain critical voltage in order to fulfill the conditions for adiabatic motion
and steady population of the Andreev levels without any electromechanical coupling.
It can be estimated (as discussed further in Appendix 1) as
V 5 VC = 4∆0R/e . (2.4)
In Papers I-II it is shown that through the energy mechanism described above the
nanowire may be cooled to its ground state. In the following sections we give a more
in depth description of the applied model and mathematical methods used in our
research.
2.2 Mechanical Hamiltonian
In our analysis, the vibrating nanowire is modeled as a quantum harmonic oscilla-
tor, taking only the fundamental bending mode into account. In the framework of
quantum mechanics, the Hamiltonian for the harmonic oscillator is [70]
Hˆmech = ~ωbˆ†bˆ . (2.5)
Here, ~ is the Planck constant and ω is the fundamental frequency. The operator bˆ†(bˆ)
creates (annihilates) one quantum of vibrational energy ~ω of vibronic energy. These
operators satisfy the commutation relation [bˆ, bˆ†] = 1.
2.3 Electronic Hamiltonian
As discussed earlier, from the electronic point of view, the nanowire constitutes a
weak link for a short S-N-S Josephson junction. This type of junctions can experience
Andreev reflections at the metal and superconductors interfaces. Therefore, the elec-
tronic degrees of freedom can be cast in the form of a pair of bound Andreev states.
15
Chapter 2. Cooling of a suspended nanowire
The spatial extent of these states will be of the order of the superconducting coher-
ence length ξ. In our analysis we assume that the coherence length is much larger
than the length of the nanowire L. Assuming that the adiabatic conditions are ful-
filled Eqs. (1.26) and (2.4), the Andreev levels describe a two-level system and the
electronic Hamiltonian can be written in terms of temporally evolving energy levels.
This two-level Hamiltonian reads
Hˆele(t) = ±E(φ(t))σˆz +∆0
√
R sin(φ(t)/2)σˆx . (2.6)
Here, the first term E(φ(t)) = ∆0 cos(φ(t)/2) is the Andreev level for a single state
in a transparent junction. The second term is the energy contribution of an impurity
in the carbon nanotube with reflection coefficient R and σˆi (i = x, y, z) are the Pauli
matrices. The physical interpretation of these two terms is the following: For the com-
pletely transparent junction, the upper level of backward travelling electrons and the
lower level of forward moving electrons are independent of each other when there is
no normal scattering. As soon as the impurity is present, it reflects electrons and the
Andreev levels become coupled, therefore the energy degenerancy at φ(t) = pi is lifted
and an energy gap of size Egap = 2∆0
√R appears in the quasiparticle energy spec-
trum. This qualitative description is in agreement with solution of the Bogoliubov-de
Gennes equations given by Eq. (1.20).
2.4 Interaction Hamiltonian
In our proposal the externally applied magnetic field is used for coupling the Andreev
levels to deflections of the nanowwire. In this case, the Lorentz force that couples the
mechanical and electronic degrees of freedom in the system is given by
F = HIL, (2.7)
where H is the applied magnetic field, I the current across the junction and L the
length of the suspended part of the mechanical nanoresonator. Consequently, the
time-dependent interaction term reads
Hˆint(t) = HIˆ(φ(t))Lyˆ , (2.8)
where yˆ is the resonator position operator and Iˆ(φ(t)) the phase-dependent current
operator given by Eq. (1.22). The displacement of the nanoresonator in terms of the
creation and annihilation operators is
yˆ = y0[bˆ† + bˆ] . (2.9)
In Eq. (2.9), y0 is the amplitude of the zero-point oscillations in the nanoresonator and
it is defined as
y0 =
√
~/(2mω) . (2.10)
By substituting Eq. (2.9) and Eq. (1.22) into Eq. (2.8), the interaction Hamiltonian be-
comes
Hˆint(t) = H 2e~
∂E(φ(t))
∂φ(t) Ly0[bˆ
† + bˆ]σˆz
= e∆0~ LHy0 sin(φ(t)/2)[bˆ
† + bˆ]σˆz .
(2.11)
16
2.5. Total system Hamiltonian
In order to simplify the notation in Eq. (2.11), a dimensionless magnetic flux can be
introduced
Φ = 2LHpiy0/Φ0 . (2.12)
Where
Φ0 = h/2e , (2.13)
is the magnetic flux quantum. After the introduction of these quantities, the interac-
tion Hamiltonian in Eq. (2.11) reads
Hˆint(t) = ∆0Φsin(φ(t)/2)︸ ︷︷ ︸
g(φ(t))
[bˆ† + bˆ]σˆz . (2.14)
From the last equation, it is concluded that the coupling term g(φ(t)) attains its max-
imum at φ(t) = pi, which coincides with a minimal energy gap between Andreev
levels. Since the energy scales of the superconductive order parameter and the me-
chanical vibrations are very different, 2∆0 À ~ω, transitions between the Andreev
levels are most probable when the electromechanical coupling is strongest and the
energy gap between the Andreev levels equals the energy scale set by the harmonic
oscillator, i.e., when the resonance condition1,
~ω = 2∆0
√
R , (2.15)
is satisfied. Consequently, transitions between Andreev levels are likely to occur
when φ(t) = pi (at t = t0) as it can be seen in Fig. ??.
2.5 Total system Hamiltonian
The Hamiltonian describing the system is
Hˆsys(t) = Hˆmech + Hˆele(t) + Hˆint(t) ,
= ~ωbˆ†bˆ+ E(φ(t))σˆz +∆0
√
R sin(φ(t)/2)σˆx +∆0Φsin(φ(t)/2)[bˆ† + bˆ]σˆz .
To proceed further, the system Hamiltonian given in Eq. (2.16) is expressed in the
basis where the energy of the two electronic states takes the form
E(t) = ±∆0
√
1−D sin2(φ(t)/2) , (2.16)
and the space is spanned by the states Ψ±(φ(t)). In this space the Hamiltonian of the
system reads,
Hˆeff (t) = ~ωbˆ†bˆ+∆0
√
1−D sin2(φ(t)/2)τˆz +∆0Φsin(φ(t)/2)[bˆ† + bˆ]τˆx . (2.17)
In the last equation τi (i = x, y, z) are the Pauli matrices in the space of the wave func-
tions Ψ±(φ(t)). At the resonant phase φ(t) = pi, the Andreev states ψ±(φ(t) = pi) with
energies E±(φ(t) = pi) = ±∆0
√R are superpositions of symmetric and antisymmetric
states carrying current in opposite directions.
1This condition requires the weak link to have a high transparency coefficient D [71].
17
Chapter 2. Cooling of a suspended nanowire
2.6 Transition between Andreev levels
The description of the process continues by estimating the transition probabilities
between Andreev levels through the absorption of one quantum of mechanical energy
over one period. In order to calculate them, the rotating wave approximation (RWA)
is applied to Eq. (2.17) (see discussion in Appendix 2). In the RWA framework the
Hamiltonian becomes
Hˆeff (t) =
( E(t)− ~ω/2 ∆0Φsin(φ(t)/2)bˆ
∆0Φsin(φ(t)/2)bˆ† −E(t) + ~ω/2
)
. (2.18)
By using the Josephson relation, Eq. (1.24), it may be concluded that the transitions
between Andreev levels are more likely to occur at t0 = pi~/2eV (φ(t0 = pi)) when
the electromechanical coupling (off-diagonal elements of the matrix) is maximum.
Therefore, the analysis of Eq.(2.18) can be restricted to the vicinity of this time. By
performing a series analysis, the second order Taylor expansion of the energy E(t)
around t0, where |E(t0)| = ~ω/2 and E˙(t0) = 0, generates the following dimension-
less differential equation
i∂τ
(c+,n−1(τ)
c−,n(τ)
)
=
( τ2 Γ√n
Γ√n −τ2
)(c+,n−1(τ)
c−,n(τ)
)
, (2.19)
for the probability amplitudes c±,n(τ) of finding the state of the system in the upper(+)/lower(-
) Andreev level with the oscillator in the state n. The dimensionless variables intro-
duced in Eq. (2.19) are
τ = (t− t0)(ξ/~)1/3 , (2.20)
Γ = Φ∆0~ω
(Vc
V
)2/3
, (2.21)
ξ = ∂
2E(t)
∂t2
∣∣∣∣
t0
= D(~ω)
3
~2
( V
Vc
)2
. (2.22)
The coupling terms responsible for the interlevel transitions correspond to the off-
diagonal elements in Eq. (2.19). They are proportional to Γ and can be estimated by
considering the following parameters:
1. Quantum of mechanical energy ~ω =1µ eV
2. Superconducting gap ∆0 = 10~ω =10µ eV
3. Amplitude of the zero-point oscillations y0 =20 pm
4. Length of the metallic nanotube L =100 nm
5. Applied magnetic field2 H =1 T
6. Critical voltage Vc =10−7 V
2Such a magnetic field is not going to destroy the superconducting properties of thin film
leads. In these materials the critical magnetic field Hc can be greater than 1 T [72].
18
2.7. Density Matrix Analysis
By substituting these parameters into Eq. (2.21) and taking into account that in the
adiabatic regime V . Vc, the numerical analysis yields Γ ¿ 1. As a consequence,
by focusing on the infinitesimal interval (−δτ, δτ ) around the transition point τ0 = 0,
the parameter Γ may be considered as a perturbation of |c−,n(−δτ)|2 = 1 and the
probability to find the system in the upper Andreev level at δτ after the resonant
point is,
|c+,n−1(δτ)|2 ≈ piΓ2n . (2.23)
Therefore, the probability of a transition between the Andreev levels is linear with the
initial vibronic population of the mechanical resonator after crossing the resonance.
From Eq. (2.20) and Eq. (2.22), the characteristic time scale of the electromechanical
interactions can be estimated as
δt ≈ (~ωΦ/∆0)1/2~/(eV ) ¿ 20ns . (2.24)
The efficiency of the energy transfer process (cooling mechanism) depends on the
competition between the probability of transitions between the Andreev levels and
the thermal damping. This analysis will be done by studying the density matrix of
the system.
2.7 Density Matrix Analysis
In this section, the dynamics of the coupled electromechanical system is described
in the framework of the density matrix formalism. Here we investigate the form in
which the nanoresonator is affected by interactions with the external heat bath. In
our description, the interaction of the mechanical subsystem with the thermal envi-
ronment is modeled through the following integral collision,
Lˆ(ρˆ) = −(1 + nB)
(
bˆ†bˆρˆ+ ρˆbˆ†bˆ− 2bˆρˆbˆ†
)
− nB
(
bˆbˆ†ρˆ+ ρˆbˆbˆ† − 2bˆ†ρˆbˆ
)
, (2.25)
where nB = (exp[~ω/kBT ]−1)−1 is the corresponding occupation number of the oscil-
lator at temperature T . The evolution of the density matrix over one period depends
on which regime is studied:
1. In the adiabatic regime, where the Andreev levels evolve independently and
the system only interacts with the environment, the evolution of the density
matrix is determined by
∂ρˆ(t)
∂t = −
i
~
[
Hˆeff (t), ρˆ(t)
]
+ γ2 Lˆ(ρˆ(t)) , (2.26)
where γ = ω/Q is the thermal damping rate of the vibrational modes and Q
denoting the quality factor.
2. In the transition regime, where the levels interact for a short interval of time
(−δt, δt), the dynamics of the density matrix is described through the expression
ρˆ(t0 + δt) = Sˆρˆ(t0 − δt)Sˆ† . (2.27)
19
Chapter 2. Cooling of a suspended nanowire
Here, the unitary scattering matrix Sˆ is introduced and it has the following
structure
Sˆ =
(
κ1(nˆ) i ν1(nˆ)√nˆ+1 bˆ
ibˆ† ν2(nˆ+1)√nˆ+1 κ2(nˆ)
)
. (2.28)
In the scattering Sˆ-matrix the term
nˆ = bˆ†bˆ , (2.29)
corresponds to the number operator of mechanical vibrons. The subscripts 1, 2
refer to the upper(+)/lower(-) Andreev level respectively. The elements of the
S-matrix are:
• κi=1,2(nˆ) is the probability amplitude for the electronic subsystem to re-
main in the top/bottom electronic branch.
• νi=1(nˆ) is the probability amplitude for the electronic subsystem to be
found in lower level after the interaction.
• νi=2(nˆ) is the probability amplitude for the electronic subsystem to be
found in upper level after crossing the resonant point.
These coefficients satisfy the relationship
|κi=1,2(n)|2 + |νi=1,2(n)|2 = 1 . (2.30)
From this analysis it is possible to conclude that
|c+,n−1(δτ)|2 = |ν2(n)|2 ' pinΓ2 . (2.31)
Additionally, |ν2(n)|2 = |ν1(n − 1)|2 which follows from the symmetry of Eq. (2.19).
As outlined before the probability of Andreev level transitions is proportional to the
magnetic field and the quantum state of the oscillator.
2.8 Evolution of the density matrix
In order to evaluate the evolution of the density matrix during a single period, it is
assumed that the thermal energy is much smaller than the initial gap between the
Andreev levels, kBT ¿ 2∆0 , and the following boundary conditions:
ρˆ(t = nTV + ε) =
(0 0
0 1
)
⊗ ρˆp(nTV ) . (2.32)
ρˆp(nTV ) = Trelρˆ(nTV − ε) . (2.33)
Here
TV = pi~/eV . (2.34)
In the previous expressions n = 0, 1, 2, · · · , is an integer which labels the number of
periods and ε is infinitesimal time around nT . With these considerations at the start
of the period, the initial density matrix
ρˆin ≡ ρˆ(t = nTV + ε) , (2.35)
20
2.9. Results
will evolve during one period into
ρˆf ≡ ρˆ(t = [n+ 1]TV − ε) . (2.36)
In the limit of high mechanical quality factor (Q ∼ 105 [73]) the mechanical sub-
system is weakly coupled to the external thermal environment and the interaction
between both systems can be treated as a perturbative term in Eq. (2.26). From stan-
dard perturbation theory, the time evolution of the density matrix over one period
is
ρˆf = SˆρˆinSˆ† + γ2
TV
2 SˆLˆ(ρˆ
in)Sˆ† + γ2
TV
2 Lˆ(Sˆρˆ
inSˆ†) . (2.37)
Now, in light of the above boundary conditions (Eq. (2.32) and Eq. (2.33)), the expres-
sion for the evolution of the density matrix after one period at time t = [n+ 1]TV + ε
is
ρˆ =
(0 0
0 1
)
Trel
(
SˆρˆinSˆ† + γ2
TV
2 SˆLˆ(ρˆ
in)Sˆ† + γ2
TV
2 Lˆ(Sˆρˆ
inSˆ†)
)
. (2.38)
The stationary regime is achieved when
ρˆ(nTV + ε) = ρˆ([n+ 1]TV + ε). (2.39)
It turns out that the stationary density matrix at time t = nTV + ε has the form
ρˆst =
∞∑
n=0
P stn
(0 0
0 1
)
|n〉〈n| . (2.40)
Hence, in the stationary regime
∞∑
n=0
P stn
(0 0
0 1
)
|n〉〈n| =
(0 0
0 1
)
Trel
(
SˆρˆstSˆ† + γ2
TV
2 SˆLˆ(ρˆ
st)Sˆ† + γ2
TV
2 Lˆ(Sˆρˆ
stSˆ†)
)
.
(2.41)
In the last equations, P stn , is the stationary distribution of mechanical excitations.
2.9 Results
In order to find the stationary probability distribution of mechanical excitations a
numerical analysis of Eq. (2.41) must be done. This can be achieved by projecting
Eq. (2.38) on the mechanical space and expressing the result in matrix form. Through
this procedure, a matrix equation for the probability distribution of vibronic states is
obtained,
APn = ζPn. (2.42)
Here, P is the probability distribution of vibronic modes in the nanowire, A is the
probability matrix generated from Eq. (2.38) and ζ is an eigenvalue. Hence, finding
the stationary vibronic distribution, P stn , is an eigenvalue problem.
We can evaluate the average population of vibrational modes 〈n〉 when the popu-
lation of the vibrational modes does not change over one period, i.e., in the stationary
21
Chapter 2. Cooling of a suspended nanowire
0 5 10 15 20 250
0.05
0.1
0.15
0.2
0.25
0.3
n
P n
 
 
0 5 10 15 20 25
0
0.1
0.2
0.3
0.4
n
|ν 2
(n
)|2
 
 
N = 1
N = 1 × 101
N = 2 × 102
N = 3 × 102
N = 3 × 103
Figure 2.3: Probability distribution of the population of the vibrational modes Pn
in the system. In the plot, N labels the number of periods and the inset displays
the transition probability as a function of the occupation number of the quantum
state of the oscillator n. The curve N = 1 (blue) corresponds to the initial thermal
distribution, P thermaln ∝ e−~ωn/kBT . The oscillating carbon nanotube is cooled to its
motional ground state 〈n〉 . 1 after 3000 periods (curve in magenta). The system
parameters are: kBT = 5~ω, Vc =10−7 V, ~ω =10−6 eV, ∆0 = 10~ω, y0 =20 pm,
L =100 nm, H =1 T, Q = 105 and TV = pi~/eV ∼ 20 ns.
22
2.10. DC current through the junction
regime. From the mathematical point of view, it corresponds to finding the proba-
bility distribution with eigevalue ζ = 1. By calculating the expectation value of the
vibronic occupation number,
〈n〉 =
∑
n
Pnn . (2.43)
The corresponding evolution of the distribution of the vibrational modes is shown
in Fig. 2.3. As can be seen clearly, the energy transfer process discussed above has
lowered the vibronic excitations from the originally thermal distributed (blue) to the
stationary one (magenta). The metallic carbon nanotube is effectively cooled as
〈nthermal〉 ∼ 4.5 À 〈nst〉 ∼ 0.1 . (2.44)
This result is confirmed by the perturbative solution,
〈nst〉 ' nBQΦ2
~ω
∆0
( V
Vc
)1/3
∼ 0.1 . (2.45)
The stationary distribution in Fig. 2.3 corresponds to Teff < T , where Teff is the
effective temperature of the metallic carbon nanotube in the stationary regime.
2.10 DC current through the junction
In Paper II the possibility to probe the stationary distribution of the mechanical sub-
system is considered. There we suggest studying the dc current through the nanome-
chanical weak link as this directly measures the population of the vibrational modes.
To show this we evaluate the dc current (on resonance) induced by the transition be-
tween Andreev levels over one period when the mechanical subsystem has reached
the stationary regime. This current arises due to the scattering process which pro-
motes transition from the lower to the upper Andreev level. This process will result
in a net charge transfer through the nanostructure. We can evaluate the total current
(over a period) as
Idc = 2e~
2
TV
[ ∫ TV /2
0
∂E−(φ(t))
∂φ(t) dt+ (1− Pstat)
∫ TV
TV /2
∂E−(φ(t))
∂φ(t) dt+
+ Pstat
∫ TV
TV /2
∂E+(φ(t))
∂φ(t) dt
]
, (2.46)
where, Pstat is the total probability for the system to be scattered from the lower en-
ergy level to the upper one. Since n quanta of mechanical energy are excited with
probability P (n)stat, and since in this case the scattering probability is |ν2(n)|2 ≈
pinΓ2, the total probability is straightforwardly obtained. As a result, the expression
for Idc reads
Idc = 4e~ ∆0Γ
2(1−
√
R)〈n〉 , (2.47)
which is plotted in Fig. 2.4 as a function of the quality factor.
23
Chapter 2. Cooling of a suspended nanowire
10−6 10−4 10−2 100 102
0
0.05
0.1
0.15
Q−1
I D
C[n
A]
Figure 2.4: DC current as a function of the inverse quality factor in the stationary
regime. In the limit of very high quality factor the dc current goes zero. In the oppo-
site limit the dc current approaches a constant value which depends on the external
temperature, Idc ∝ nB . Here, the system parameters are the same as in Fig. 2.3.
Equation (2.47) shows that the total dc current over the junction is a linear func-
tion of the average vibron population in the stationary regime. From Fig. 2.4 we
notice that in the limit of small quality factor, i.e., high damping, the dc current satu-
rates to a constant value (0.15 nA). It can be understood by the fact that in this limit
the stationary distribution of the vibrational modes is given by the thermal distribu-
tion P statn ∝ exp(−n~ω/kBT ), i.e. the interactions with the environment are so strong
that they always drive the mechanical system into thermal equilibrium, no matter
how the inter-Andreev level scattering changes this distribution. As such, Idc ∝ nB
in this regime. In the opposite regime, Q → ∞, the dc current goes to zero (within
the rotating wave approximation) as expected as this regime corresponds to complete
ground state cooling of the mechanical subsystem. If this can be achieved, the prob-
ability of Andreev level scattering also goes to zero, |ν2(0)|2 = 0, hence the electronic
subsystem stays in the lower Andreev branch throughout and the Idc = 0.
2.11 Quasiparticle spectrum
At the end of each period, when φ(t) = 2pi, the bound Andreev levels reach the con-
tinuum spectrum and the absorbed vibronic energy is transfered to the quasiparticle
states in the leads. This description is based on the following arguments: Firstly, at
the edge of each period, the adiabatic condition is no longer valid and the Andreev
levels decay into the continuum state. As was highlighted in Ref. [74], this process
corresponds to a delocalization in real space and it is the reason for the energy trans-
24
2.12. Final remarks
fer process. Secondly, the orthogonality between Andreev states guarantees that at
the beginning of each period the Andreev levels will start again at the lower elec-
tronic level [75]. Finally, the analysis showed that the possibility of heating due to
the energy transfer process is negligible. Indeed, for a quasiparticle excitation created
at the point φ(t) = 2pi with characteristic energy δE = ∆0(eV/∆0)1/3 and velocity
vqp = vF (eV/∆0)1/3, where vF is the Fermi velocity. The traveled distance during
one period TV = pi~/eV is δx ∼ ξ0(∆0/(eV ))2/3 where ξ0 = ~vF /∆0 is the coher-
ence length of the superconductors. Considering that the size of the junction corre-
sponds to the extension of the Adiabatic Andreev level ξ = ξ0(∆0/(eV ))2/3 ¿ δx.
Consequently, the quasiparticle has moved away from the junction when the next
quasiparticle is created at the boundary edge and the quasiparticle excitation releases
energy in the leads which will increase the energy of the thermal bath at the rate
Q˙ ¿ ∆20/~(eV/∆0)4/3. As the energy relaxation process is proportional to the volume
of the leads, the above rate is small for heating the leads.
2.12 Final remarks
This chapter summarizes the content of Papers I-II where we have shown that quan-
tum mechanical cooling of a metallic carbon nanotube acting as a nanoelectrome-
chanical Josephson junction is feasible. In the proposed cooling scheme the externally
applied magnetic field couples the deflection of the nanowire to the current carried
by the Andreev levels. As a consequence, the electronic system can change from one
level to the other through the absorption or emission of a quantum of mechanical
energy (vibron) given the right experimental parameters. It is also shown that the
energy released into the superconducting leads should be experimentally observable
through the associated dc current over the junction as a function of the quality factor.
25
CHAPTER 3
Nonlinear dynamics of a suspended nanowire
In this Chapter we analyze nanoelectromechanical systems (NEMS) as tools for the-
oretical and experimental studies of nonlinear dynamical systems. As mentioned
earlier, NEMS systems are currently considered promising candidates to study non-
linear and complex phenomena due to the inherent properties of these devices, which
includes low damping, desired resonant operation, and the presence of nonlinear
potential fields [76]. Examples of complex dynamical phenomena in NEMS are nu-
merous and include chaotic behavior [77–80], bifurcation-topology amplification [81],
nonlinear switching dynamics [82], and nonlinear frequency pulling [83] to name but
a few. For a review on the fundamentals and applications of nonlinearities in NEMS
resonators the reader is referred to Refs. [76, 84, 85].
By making use of the potential offered by NEMS resonators, Sonne et al. [86] stud-
ied the nonlinear dynamics of a suspended carbon nanotube coupled to two voltage-
biased superconducting electrodes. In presenting their work, Sonne and collabora-
tors assumed that the nanoelectromechanical junction was subjected to a homogeneous
magnetic field perpendicular to the axes of the nanowire. For such a system, the au-
thors demonstrated the possibility to pump energy from the electronic subsystem into
the mechanical vibrations; they also demonstrated that the system had more than one
regime of finite-amplitude stationary nonlinear oscillations. In particular, a region of
bistability was found and the authors showed that it should be detected in the corre-
sponding dc Josephson current (see discussion in Ref. [86]).
In this Chapter of the thesis, we consider the same voltage-biased nanoelectrome-
chanical system studied by Sonne and co-workers [86], but now extend the analysis
to a case in which the NEMS resonator is subjected to a nonuniform magnetic field. As
will be discussed in the following sections, inhomogeneity of the field causes the con-
ducting nanoresonator to execute a whirling movement resembling a jump-rope-like
motion. The purpose of the subsequent sections is to analyze the time evolution of the
amplitude and relative phase of the nanotube whirling motion. We will demonstrate
that the coupled amplitude-phase dynamics exhibits different stationary regimes de-
pending on the degree of the magnetic field inhomogeneity: time independent, peri-
odic, and chaotic. This Chapter presents a brief statement of the main points of Paper
III.
3.1 System and equations
The diagram in Fig. 3.1 is a schematic illustration of a superconducting hybrid nanos-
tructure, a superconducting-normal-superconducting (S-N-S) nanoelectromechanical
26
3.1. System and equations
Josephson junction driven by a dc voltage bias V . The junction consists of a metal-
lic carbon nanotube of length L suspended between two voltage-biased supercon-
ducting leads. In such a geometry, the nanotube is simultaneously serving as a me-
chanical resonator and as a weak link between the superconducting electrodes. The
phase difference between the weakly coupled leads will be denoted by φ and, in ac-
cordance with the second Josephson relation, its time evolution is φ(t) = ωJ t, with
ωJ = 2eV/~ the Josephson frequency. In the layout of the system, the NEMS junc-
tion is under the influence of an external inhomogeneous magnetic field H, gener-
ated by a magnetic force microscope (MFM) cantilever tip in the form of a wedge,
parallel to the axis of the nanotube at a distance d. Considering a magnetic field of
the form H = (0,Hy,Hz), a first order Taylor expansion of the magnetic field yields
H = (0, y∂yHy(x, 0, 0),Hz(x, 0, 0) + z∂zHz(x, 0, 0)), where Hz(x, 0, 0) and ∂iHi(x, 0, 0)
represent the magnitudes of, respectively, the magnetic field z-component and the
magnetic field gradients, both evaluated at the axis of the nanotube. A straightfor-
ward calculation from the Maxwell equation ∇ ·H = 0, indicates that ∂yHy(x, 0, 0) =
−∂zHz(x, 0, 0) ≡ −H ′z . Setting Hz(x, 0, 0) ≡ Hz , the magnetic field reads
H = (0,−H ′zy,Hz +H ′zz
) . (3.1)
This field is also obtained from the equation H = ∇×A, with the vector potential A
is given by
A = −([Hz +H ′zz
]y, 0, 0) . (3.2)
In this scenario, the coupling between the Josephson current and the inhomogeneous
magnetic field gives rise to a non-planar whirling displacement of the nanotube due
to the Laplace force. As a consequence of the motion in the magnetic field, an elec-
tromotive force is induced along the nanomechanical weak-link and its magnitude
depends on the rate of change of the nanowire profile in the z-y plane, i.e., the rate
of change of the magnetic flux through the circuit. In this description, now, the su-
perconducting phase difference is not only a function of the bias voltage, but also of
the nanowire deflection in the z-y plane. We decompose the nanotube motion in this
plane into two independent deflections y(x, t) = u0(x)a(t) and z(x, t) = u0(x)b(t),
where u0(x) is the normalized and dimensionless profile of the fundamental mode in
both directions. Then, the expression for the superconducting phase difference has
the form
ϕ(t) = φ(t)2 +
eL
4~
[hz + h′zb(t)
]a(t) . (3.3)
The parameters hz = αHz and h′z = βH ′z are the renormalized magnetic field and
magnetic field gradient in the z-direction calculated at the axis of the tube, respec-
tively, with α, β ∼ 1 correctional factors originating from geometrical considerations.
The nanotube mechanics is thus described through the projection amplitudes a(t) and
b(t) and for the conjugate variables {a(t), pa(t)} and {b(t), pb(t)} (pj(t) denoting the
generalized momenta) one can formulate the following Hamiltonian function
H(pa, pb, a, b, t) = 12m
(p2a + p2b
) + mω
2
2
(a2 + b2) − 2D∆0 cos(ϕ(a, b, t)) , (3.4)
where m and ω are the mass and the mechanical eigenfrequency of the nanores-
onator, respectively. The last term in Eq. (3.4) corresponds to the Josephson energy
27
Chapter 3. Nonlinear dynamics of a suspended nanowire
Figure 3.1: (a) Schematic diagram of the voltage-biased S-N-S nanoelectromechanical
Josephson junction considered in Paper III. A doubly clamped metallic carbon nan-
otube suspended over a trench of length L, forms a weak link between two voltage-
biased superconducting electrodes. The junction is influenced by an inhomogeneous
magnetic field H, generated by a wedge-shaped MFM cantilever tip at a distance d
from the nanotube at rest (blue circle). (b) Nanotube displacement (green circles) in
the z-y plane induced by a Laplace force. RT is the curvature radius of the magnetic
tip.
28
3.2. Numerical results and discussion
EJ(ϕ(a, b, t)), with D the transmission coefficient of the junction and ∆0 the super-
conducting order parameter. The equations of motion for a(t) and b(t) are then ob-
tained from the Hamilton equations. Written in terms of the dimensionless deflection
coordinates
Y (t) = [eLhz
/4~]a(t),
Z(t) = [eLhz
/4~]b(t), (3.5)
the resulting set of differential equations for the nanotube amplitudes is:
Y¨ (τ) + γ˜Y˙ (τ) + Y (τ) = −ε [1 + κZ(τ)] sin
(
V˜ τ + [1 + κZ(τ)]Y (τ)
)
, (3.6)
Z¨(τ) + γ˜Z˙(τ) + Z(τ) = −εκY (τ) sin
(
V˜ τ + [1 + κZ(τ)]Y (τ)
)
. (3.7)
Here, we have added a dimensionless phenomenological damping coefficient γ˜ =
[γ/mω]. In these equations ε = [2eL2h2zjc/m~ω2] with jc = [D∆0e/2~], the critical
current through the junction. We also set the timescale to τ = ωt and, consequently,
V˜ = [eV/~ω]. The parameter
κ = 4~eLhzRT , (3.8)
where RT =
[hz/h′z
] denotes the curvature radius of the magnetic cantilever tip,
characterizes the degree of inhomogeneity of the magnetic field and will be referred
to as the control parameter. It turns out that κ determines the dynamical behavior
of the system. One can then realize the significance of κ by considering fixed system
parameters (L, hz) and by letting RT vary. In the limit RT →∞, the control parameter
vanishes and the equations of motion given by Eqs. (4.13) reduce to the case discused
by Sonne et al. [86] where the magnetic field is uniform and, therefore, the amplitude
of the driving force acting in the Z-direction becomes zero.
3.2 Numerical results and discussion
For a qualitative and quantitative discussion of the dynamic behavior of the nanowire
amplitudes, we consider the following system parameters: a carbon nanotube of ra-
dius r = 1 nm and length L = 1µm [87], superconducting order parameter ∆0 ∼ 1 meV,
Josephson critical current jc ∼ 100 nA [88], and quality factor Q ∼ 103 [89]. We also
assume that γ˜ = 1/Q, hz ∼ 40 mT, ε = 0.012, and RT is varied from 53 nm to 544 nm.
From Eq. (3.8), the control parameter will take values in the interval 0.12 ≤ κ ≤ 1.25
Numerical simulations of Eqs. (4.13a) and (4.13b) allow us to study the time aver-
age of the nanotube deflection coordinates Y (τ) and Z(τ) as functions of the driving
voltage V˜ . In doing so, one can notice that the system response presents a series of
resonance peaks at integer values of the driving voltage, i.e., the amplitude of the
nanoresonator is not damped provided the resonant condition is fulfilled: the Joseph-
son frequency matches the mechanical frequency (see Fig. 3.2). This resonant phe-
nomenon was first studied by Sonne et al. [86], who attributed a direct resonance at
V˜ = 1 and a parametric resonance at V˜ = 2. Accordingly, the same conclusion can be
drawn from the results presented in Fig. 3.2. In the remainder of the chapter we will
29
Chapter 3. Nonlinear dynamics of a suspended nanowire
Figure 3.2: Numerical simulations of Eqs. (4.13a) and (4.13b) for the time averaged
coordinates Y and Z as function of the driving voltage V˜ . The system presents a
series of resonance peaks at integer values of the driving voltage and the amplitude
of the nanotube is not damped when the Josephson frequency matches the mechanical
frequency. Further analysis is performed at parametric resonance V˜ = 2. The plots
are calculated for γ˜ = 0.001, ε = 0.012, κ = 1 (L = 1µm, hz = 40 mT, RT = 66 nm).
30
3.2. Numerical results and discussion
Figure 3.3: Numerical simulations for the stability analysis of Eqs. (3.10). Stable and
unstable stationary points are indicated by colored and numbered diamonds (♦) and
black crosses (×) respectively. Diamonds of identical color and number indicate the
four envelopes of the stable solution Xs, crosses in the same region (I or II) of both
phase planes belong to the same unstable solution Xu. The zero solution, X ≡ 0, is
unstable. Here, γ˜ = 0.001, ε = 0.012, κ = 0.12 (L = 1µm, hz = 40 mT, RT = 544 nm).
be mainly focusing on the parametric regime and take V˜ = 2. In this case, the dy-
namic behavior of the amplitudes can be analized by postulating a solution for both
deflection coordinates in the form
Y (τ) = Ar(τ) cos(τ) +Ai(τ) sin(τ) , (3.9a)
Z(τ) = Br(τ) cos(τ) +Bi(τ) sin(τ) . (3.9b)
Here, Ai;Bi and Ar;Br are the in-phase and quadrature amplitude components of the
proposed ansatz for Y ;Z given by Eqs. 3.9, respectively. On condition that γ˜, ε, εκ ¿
1, the four envelopes in the vector of amplitudes X = (Ar(τ), Ai(τ), Br(τ), Bi(τ))
vary slowly in time, i.e., dX/dτ ¿ 1 and, an averaging method [90] can be employed
in order to derive the equation of motion for X. By substituting the ansatz provided
in Eqs. (3.9) into the system of equations in Eqs. (4.13) and integrating over the fast
oscillations one gets
dAr
dτ +
γ˜Ar
2 =
∂G
∂Ai ,
dAi
dτ +
γ˜Ai
2 = −
∂G
∂Ar ,
dBr
dτ +
γ˜Br
2 =
∂G
∂Bi ,
dBi
dτ +
γ˜Bi
2 = −
∂G
∂Br ,
(3.10)
where
G(κ) = − ε2pi
∫ pi
−pi
cos([1 + κBr cos(Θ) + κBi sin(Θ)]
× [Ar cos(Θ) +Ai sin(Θ) + 2Θ]
)dΘ . (3.11)
31
Chapter 3. Nonlinear dynamics of a suspended nanowire
Here, G(κ) is the generating Hamiltonian function for Eqs. (3.10).
The study proceeds by performing a stability analysis based on Eqs. (3.10). In
general, we shall study solutions of a system of coupled ODEs, dX/dτ = f(X,κ),
and solutions of a system of algebraic equations 0 = f(X,κ). The discussion com-
mences by highlighting the symmetric nature of the stationary solutions for the al-
gebraic system in Eqs. (3.10). For the above considered system parameters with
κ = 0.12 (RT = 544 nm), numerical simulations for the slowly-varying envelopes
show that the system presents several stationary points. Such qualitative behavior
is visualized in the phase space diagrams for the nanotube amplitudes, Y and Z, in
Fig. 3.3. There, solutions pictured by colored and numbered diamonds (♦) and black
crosses (×) correspond to stable and unstable stationary points, respectively. Simi-
larly, the phase portraits are divided in two regions, denoted I and II, by diagonal
symmetry axes. The symmetry axes in the Y and Z coordinates satisfy the relations
Ai − Ar = 0 and Bi + Br = 0, respectively. From Fig. 3.3 it follows that the pre-
sented pattern of solutions has a mirror-image symmetry with respect to the line of
symmetry that divides the phase diagrams into regions I and II, where diamonds
of the same color and number represent the four components of a stable computed
solution Xs and crosses in the same region (I or II) of both phase spaces account for
the envelopes of an unstable solution Xu. From a symmetric point of view, the system
is characterized by two stable and two unstable (including X ≡ 0) stationary points
in each region. However, the dimensionless angular momentum of the nanowire,
Lx = Y dZ/dτ − ZdY/dτ , in both regions is equal in magnitude but opposite in sign.
Table 3.1: Bifurcation pattern in the symmetric regions of solutions I and II as a
function of the control parameter κ. Solution X ≡ 0 is included.
Number of Number of
κ-interval stable solutions unstable solutions
κ ≤ 0.120 2 2
0.120 < κ ≤ 0.125 3 3
0.125 < κ ≤ 0.140 1 3
Continuing our exploration, the computed results indicate that the bistable regime
is only attained for κ ≤ 0.120. By letting the control parameter increase further, bista-
bility is abandoned and the number of stationary solutions of Eqs. (3.10) is modified
as well as their stability. Indeed, Table 3.1 presents the number of stable and unsta-
ble stationary solutions in the interval 0.120 ≤ κ ≤ 0.140 in both regions for these
equations. As can be seen from the table, the system displays a phenomenon called
branching or bifurcation, which is a distinctive fingerprint of nonlinear dynamical sys-
tems [91]. The exhibited branching pattern formation can be described as follows.
For κ ≤ 0.120, bistability is accompanied by two unstable stationary points, one of
them corresponds to X ≡ 0 (see Fig. 3.3). Once the control parameter is slightly in-
creased over the threshold value κ = 0.120, the nonzero unstable stationary point
becomes stable, and simultaneously, two new unstable solutions appear and the sys-
tem has now three stable and three unstable (including the trivial solution) stationary
points. Next, by varying the control parameter in the interval 0.120 ≤ κ ≤ 0.125,
32
3.2. Numerical results and discussion
Figure 3.4: Period-doubling bifurcations for different values of the control parameter.
Plots are shown for the first symmetry region of both phase spaces. Simulations were
obtained for γ˜ = 0.001, ε = 0.012 (L = 1µm, hz = 40 mT).
33
Chapter 3. Nonlinear dynamics of a suspended nanowire
Figure 3.5: Chaotic whirling motion of the suspended carbon nanotube A. In the pre-
crisis regime, (a) and (b) depict the two symmetric attractors in the phase spaces for
the Y component of the displacement. In (a) red lines indicate the dynamical evo-
lution of a point with initial conditions in the first region of symmetry, region I. Its
symmetric dual is shown in (b), where blue lines represent the dynamical flowing of
the system for initial conditions in the second region of symmetry. For the Z com-
ponent of the nanotube motion, (c) and (d) are the symmetric plots for the chaotic
attractors. As before, red in (c) and blue lines in (d) account for the evolution of a
point with initial conditions in the first and second region of symmetry, respectively.
For the plots γ˜ = 0.001, ε = 0.012, (L = 1µm, hz = 40 mT).
34
3.2. Numerical results and discussion
Figure 3.6: Chaotic whirling motion of the suspended carbon nanotube B. In the crisis
regime, red lines in (a) and blue lines in (b) show the single chaotic attractor formed by
the two symmetric chaotic attractors joined by a periodic orbit on the basin boundary
that separates them, the plot is for the dynamics in the Y component. Similarly, for
Z-direction, red lines in (c) and blue lines in (d) indicate the attractor merging crisis.
For the plots γ˜ = 0.001, ε = 0.012, (L = 1µm, hz = 40 mT).
35
Chapter 3. Nonlinear dynamics of a suspended nanowire
the two original stable points and the two unstable ones move in phase space and
approach each other. Finally, when the control parameter ranges from 0.125 to 0.140,
the bistable and the created unstable solutions coalesce into two unstable solutions.
In this stage, there is only one stable and three unstable stationary points.
It turns out that for κ > 0.140, the system leaves the regime of equilibria and
enters into the one of periodic solutions. In fact, at the critical control parameter κ ∼
0.150, there is an exchange of stability from the unique stable equilibrium to a stable
limit cycle. This transformation in phase space is performed through a Poincaré-
Andronov-Hopf bifurcation and the computed results show that the stable limit cycle
grows in phase space for 0.150 ≤ κ ≤ 1. Hereupon, further changes in the control
parameter will be reflected in the periodicity of the limit cycle as it can clearly be
seen in Fig. 3.4. In this figure, the nanotube dynamics undergoes successive period-
doubling cascade bifurcations in the amplitude modulation when varying κ. Due to
the symmetric character of the solutions, the results are only plotted in region I.
Period-doubling bifurcations pave the way for chaotic dynamics [92] and for the
control parameter κ ∼ 1.20; the system is already within this regime. As it is shown
in Figs. 3.5(a)-3.5(d), two chaotic attractors coexist and they are symmetric images of
each other in the phase space. In Figs. 3.5(a) and 3.5(b), the phase portrait for the
slowly varying amplitudes (Ar, Ai) for the Y component of the nanotube displace-
ment is presented. In Fig. 3.5(a), red lines describe the dynamical evolution to the
chaotic attractor for a point with initial conditions in the first region of symmetry of
the phase space, region I. In this case, the attractor and its basin of attraction are
completely located in this part of the phase portrait. Its symmetric dual is shown in
Fig. 3.5(b), where the chaotic attractor (depicted in blue) and its basin of attraction
are found to be in region number II; this plot was obtained by a simple study of the
evolution of a point with initial conditions in the second region of symmetry, region
II. Similarly, the qualitative description of the Z component of the nanotube dis-
placement is also analyzed through the phase flow of the slowly varying amplitudes
(Br, Bi). This is shown in Figs. 3.5(c) and 3.5(d), where the symmetric character of the
chaotic attractors is noticeable. As before, red lines indicate the dynamics of a point
with initial conditions in region I and blue ones refer to the complex behavior of a
point with initial conditions in the second region of symmetry, II. However, there is
a difference with respect to the phase portraits for (Ar, Ai). Now, the chaotic attrac-
tors are not completely located in each region of symmetry: they cross the symmetry
line (that divides the phase portrait into two parts) and enter into the other region,
nevertheless, the chaotic attractors and their symmetry are very well defined.
The occurrences of sudden qualitative changes of chaotic attractors, as a parame-
ter of the system is varied, are denoted crises [45, 46]. In particular, an attractor merg-
ing crisis takes place for the studied system when the control parameter reaches the
critical value κc ' 1.23. In this phenomenon, the two symmetric attractors simultane-
ously touch a periodic trajectory or orbit on the basin boundary that separates them
and merge together to form a single multipiece chaotic attractor [47,48]. The merging
crisis regime can be visualized in Figs. 3.6(a)-3.6(d) where the phase portraits for the
nanowire amplitudes suggest that the initially disconnected chaotic attractors, in the
pre-crisis state, placed in the symmetric regions I and II are now joined. Further-
more, the flow diagrams for initial conditions in region I (red) and region II (blue)
indicate that the system dynamically evolves between the two regions, i.e., the flow
36
3.3. Final remarks
lines generated by Eqs. (3.10) and (3.11) are sensitive to the initial values and dense in
both regions, where the multipiece single attractor can be readily identified. In order
to make the attractor merging crisis clear, we study the dynamics of the system for
κ = 1.25. The phase portraits for the Y direction have been split for two cases. Red
lines in Fig. 3.6(a) account for the dynamics of a point with initial conditions in the
first region of symmetry in the pre-crisis regime, while blue lines in Fig. 3.6(b) de-
scribe the dynamical evolution of a point initiated in the second region of symmetry,
region II. In a similar manner, for the Z component, red lines in Fig. 3.6(c) repre-
sent the evolution of a point with initial conditions in region I and Fig. 3.6(d), the
evolution of a point with initial conditions in region II.
By increasing further the control parameter, numerical studies indicate that the
single attractor formed during the attractor merging crisis is suddenly destroyed
(boundary crisis) and then, for parameter values above the crisis, points initialized in
the region previously occupied by the former attractor appear to move in the region
chaotically, but only for a finite time (chaotic transient) [45, 46]. This phenomenon
occurs at κ ∼ 1.6, however, this value for the control parameter corresponds to a cur-
vature radius of about RT = 41 nm, which is outside the range of curvature radius for
MFM tips experimentally reported, 50 − 70 nm [93]. Due to this fact, the analysis of
the transient chaos is not discussed in detail.
3.3 Final remarks
In this chapter we have considered a voltage-biased nanoelectromechanical Joseph-
son junction, where a suspended nanowire is serving as a weak link between two
superconducting electrodes, in an inhomogeneous magnetic field. For our case study,
we have assumed that the magnetic field is generated by an MFM cantilever tip and
the nanowire is in the form of a metallic carbon nanotube. In such a scenario, the inho-
mogeneity of the field in conjunction with the Josephson current flowing through the
tube gives rise to a Laplace force that induces the nanoresonator to perform a whirling
movement. We have studied the time evolution of the amplitude and relative phase of
this non-planar whirling motion and demonstrated that at the parametric resonance,
their coupled dynamics exhibits a rich dynamical behavior characterized by multista-
bility, limit cycles, and chaos. These stationary regimes depend on the degree of the
magnetic field inhomogeneity, which in the present case, is related to the curvature
radius of the magnetic cantilever tip.
The experimental implementation of the system considered in the article is plau-
sible in light of current state-of-the-art nanofabrication techniques. For instance, Pil-
let and collaborators designed and constructed a superconducting hybrid nanostruc-
ture that comprises a carbon nanotube suspended between two superconductors [94].
Concerning the MFM cantilever tip, the experimental results reported by Matveev
and co-workers in Ref. [93] suggest that it is possible to fabricate cantilever tips coated
by a magnetic film with a curvature radius in the range of 50− 70 nm with maximum
magnetic fields in the range of 40 − 80mT. Due to a growing interest in complex be-
havior in nanodevices [85], the nonlinear and nonplanar phenomena exhibited by the
system studied here have potential applications in signal processing, chaotic encryp-
tion, and random number generation [95, 96].
37
CHAPTER 4
Parametric effects in a shuttle system
In this chapter we investigate theoretically the dynamics of a spatially symmetric
shuttle-system subjected to an ac gate voltage. We demonstrate that in such a system
parametric excitation gives rise to mechanical vibrations when the frequency of the
ac signal is close to the eigenfrequency of the mechanical subsystem. These mechani-
cal oscillations result in a dc shuttle current in a certain direction due to spontaneous
symmetry breaking. The direction of the current is defined by the phase shift be-
tween the ac gate voltage and the parametrically excited mechanical oscillations. The
material discussed in this chapter summarizes the findings discussed in Paper IV.
4.1 System and equations of motion
To describe the new shuttling mechanism, we consider a system schematically de-
picted in Fig. 4.1 where a single-level quantum dot (D) is connected via elastic links
to the left (L) and right (R) electrodes. The characteristic distance between the elec-
trodes and the quantum dot at equilibrium position is d. In this setup the dot is
acting as a nano-oscillator, and the deviation of the dot from its equilibrium position
is denoted by x(t). Both electrodes are grounded, i.e., VL = VR = 0, while a signal
VG = V stG + V acG cos(ωGt) is applied to the gate (G). To analyze the electro-mechanical
phenomena in such a structure, in the most simplest approximation, we describe the
dynamics of the central island by Newton’s equation,
x¨+Q−1ω0x˙+ ω20x =
α
me
2n(t)x. (4.1)
Here, m is the mass, ω0 is the eigenfrequency, and Q is the quality factor of the oscil-
lator. In Eq. (4.1), the parameter α = [1/(2C2(0))]∂2C(x)/∂x2|x=0, where C(x) is the
effective capacitance of the dot. We consider a symmetric situation CT (x) = CT (−x)
and one can estimate α ≈ d−3. Note that in this geometry there is no force acting on
the grain if it is in its equilibrium position. The population of the dot n(t) = 0, 1 is
controlled by the stochastic evolution of the charge.
We focus on the case in which the mechanical vibration frequency of the dot is
very low in comparison to the tunneling rates between the quantum dot and elec-
trodes and the electric force is much smaller than the mechanical one, αe2/mω20 ≡ ² ¿
1. In such conditions, the force generated by the stochastic variable n(t) can be taken
into account only on average by substituting in Eq. (4.1) its mean value 〈n(t)〉 = P (t).
The variable P (t) represents the probability to find one electron in the quantum dot at
time t. As a consequence, the electronic state of the central island can be described in
38
4.1. System and equations of motion
Figure 4.1: Schematic diagram of the three-terminals shuttle device investigated in
this work. A quantum dot D can oscillate between two grounded metallic leads L,R.
The dot is capacitevely coupled to a gate electrode G, to which a voltage VG is ap-
plied. Electron tunneling takes place between the dot and the leads. IL and IR are the
currents between the left and right leads L,R and the dot D.
terms of this probability. In accordance with Fig, the time evolution of the probability
P (t) is,
dP
dt =
(dP
dt
)
in
−
(dP
dt
)
out
=
[(dP
dt
)
in,L
+
(dP
dt
)
in,R
]
−
[(dP
dt
)
out,L
+
(dP
dt
)
out,R
]
. (4.2)
In this case, each independent term in the last equation is defined as [97],
(dP
dt
)
in,L
= (1− P )ΓL(x)f(x, t),
(dP
dt
)
in,R
= (1− P )ΓR(x)f(x, t),
(dP
dt
)
out,L
= PΓL(x)[1− f(x, t)],
(dP
dt
)
out,R
= PΓR(x)[1− f(x, t)]. (4.3)
Finally, the kinetic equation for the time evolution of the probability P is recast in the
following master equation
P˙ = [ΓL(x) + ΓR(x)](f(x, t)− P ) . (4.4)
Here, the position-dependent tunneling rates between the left/right electrodes and
the dot are ΓL,R(x) = Γ0e∓x/λ with λ the tunneling length, f(x, t) = f(ED(x, t)) =
39
Chapter 4. Parametric effects in a shuttle system
[1 + e(ED−µ)/kBT ]−1 is the Fermi-Dirac thermal distribution, where µ is the chemical
potential of the leads. The energy of the electron inside the dot is ED = εα + EC ,
where εα is the energy associated to space quantization and EC(x, t) = e2/2C(x) −
eβVG(t), is the electrostatic energy with β ≈ 1 being the transmission coefficient.
Considering µ = εα + e2/2C(0) and small displacements of the dot, x ¿ d, we can
rewrite ED(x, t)− µ = −(α/2)e2n(t)x2 − eVG(t).
4.2 DC shutle current
The variation in time of the number of electrons in the leads depends on the applied
oscillating voltage VG(t) and the position of the dot x(t). Therefore, the instantaneous
current through the system, averaged over fast fluctuations due to the discrete nature
of charge tunneling, is [49, 66]
Ia(t)
e = [ΓL(x(t))− ΓR(x(t))]{f(x(t), t)− P (t)}, (4.5)
and the dc component of this instantaneous current can be calculated as
Idc
e = − limT→∞
2Γ0
T
∫ T
0
dt sinh(x(t)λ )[f(x(t), t)− P (t)] ,
= lim
T→∞
1
Tλ
∫ T
0
dt x˙(t)P (t)cosh2(x(t)/λ) . (4.6)
Here, in writing Eq. (4.6) we use Eq. (4.4). From this expression one can observe that
the dc current between the leads is defined by the correlations between the velocity
and population of the dot.
4.3 Analysis and discussion
In light of the previous section, in order to study the dc current through the system,
one should first analyze the dynamical system defined by Eqs. (4.1) and (4.4). We can
perform a perturbative analysis of these equations exploiting the small value of the
parameters ω0/Γ0 w ωG/Γ0 ¿ 1 and ²,Q−1 ¿ 1. In doing so, we can assume
P (t) = f(x, t), (4.7)
in the leading order of the parameter ωG/Γ0. Substituting this relation in Eq. (4.1), we
obtained a non-linear and time-dependent equation for x
x¨+Q−1ω0x˙+ ω20x = ²ω20f(x, t)x, (4.8)
where f(x, t) is a periodic function of time. From this equation, one can find that
the mechanical subsystem may experience a parametric instability if |ωG − ω0| ¿ ω0,
when considering the second harmonic term in the Fourier expansion of f(x, t) [98].
Note that Eqs. (4.4) and (4.8) are invariant under the transformation
[x(t), P (t)] → [−x(t), P (t)], (4.9)
40
4.3. Analysis and discussion
and it is clear that the only static stationary solution x(t) = 0 is invariant under such
transformation. If mechanical excitation takes place, it will result in, at least, two
different stable periodic (with period TG = 2pi/ωG) solutions:
[xst(t), Pst(t)] (4.10)
[x˜st(t), P˜st(t)] ≡ [−xst(t), Pst(t)] (4.11)
From Eq. (4.6), it immediately follows that these two different stationary regimes gen-
erate shuttle currents in opposite directions. Which regime arises from the ac-voltage
switching will depend on spontaneous forces accompanying transient process. It is
worth mentioning that in our considerations we have not taken into account noise
forces. Such forces will result in fluctuations of the amplitude and phase around the
stationary values 1 and the dc current should be the average over these fluctuations.
When fluctuations are much less than the average values, they will modify very lit-
tle the results discussed here. However, strong fluctuations will lead to transitions
between stationary points and, by this, restore the symmetry in the system.
To analyze the regime of parametric excitations one can use the ansatz
x(t) = A(t) cos(ωGt+ χ(t)) , (4.12)
where the amplitude A(t) and χ(t) (the phase shift between the mechanical and gate
voltage oscillations) are supposed to be slowly-varying functions of time: A˙/A, χ˙ w
²ωG, ωG/Q. It is convenient to introduce the following dimensionless variables: τ =
ωGt, ξ = x/λ, and E = A2/2λ2 . Then, after substituting the ansatz given by Eq. (4.12)
into Eq. (4.8) and averaging over the fast oscillations [99], one obtains the following
coupled differential equations for E(τ) and χ(τ),
∂E
∂τ =
∂H
∂χ −Q
−1E, (4.13a)
∂χ
∂τ = −
∂H
∂E . (4.13b)
Here, H is the generating Hamiltonian function,
H(E,χ) = (ω˜0 − 1)E + ²2ηpi
∫ pi
−pi
dθ ln{1 + e[ηE cos2(θ)+νst+νac cos(θ−χ)]}, (4.13c)
with ω˜0 = ω0/ωG, η = αe2λ2/2kBT , νst = eV stG /kBT , νac = eV acG /kBT . Different
stationary oscillation regimes of the dot (labeled by the subscript i) are defined by the
different stationary solutions of Eqs. (4.13), i.e., Ei = const, χi = const.
In order to find the dc current corresponding to a given stationary regime of oscil-
lations, 2Idc(Ei, χi) ≡ IieωG, we substitute Eq. (4.7) and Eq. (4.12) into the expression
for the dc current given by Eq. (4.6). As a result, it reads
Ii =
∫ √2Ei
−√2Ei
dξ
cosh2(ξ)
sinh[biυi(ξ) sin(χi)]
cosh[ai(ξ)] + cosh[biυi(ξ) sin(χi)] , (4.14a)
1Concerning this issue we would like to emphasise that the amplitude of shot noise fluc-
tuations at the driving frequency ωG is small due to the fact Γ0 À ωG and its contributon is
negligible.
41
Chapter 4. Parametric effects in a shuttle system
Figure 4.2: Threshold ac voltage ν∗ac as a function of the static dc voltage νst. Red
markers correspond to results obtained from numerical integration of Eqs. (4.1) and
(4.4), while blue lines refer to Eq. (4.16). The plot is calculated for a gold quantum
dot of radius r = 4 nm and mass m = 5×10−21 kg with ω0 = 10 GHz, Q = 1000 ,
d ∼ 2 nm, λ = 0.1 nm, Γ0 = 100 GHz, ωG = 10 GHz and T = 10 K. Consequently,
² = 0.1, α = 8.37, ω˜ ∼ 1.
with
ai(ξ) = ηξ2 + νst + biξ cos(χ) , bi = νac/
√
2Ei. (4.14b)
Here, υi(ξ) is the modulus of the dot velocity as a function of its position given by
υi(ξ) =
√
2Ei − ξ2. From Eqs. (4.14), we can conclude that the dc current solely exists
at non-zero amplitude of oscillation, while its sign (symmetry breaking signature) is
controlled by the phase difference χi, sgn(Ii) ∝ sin(χi).
To proceed further, we consider the case of exact resonance,
ωG = ω0
{
1 + (²/4pi)
∫ pi
−pi
dθ[e−νst + eνac cos(θ−χ)]−1
}
. (4.15)
In Eq. (4.15), we have taken into account the renormalization of the frequency due
to Coulomb interactions, which is proportional to ². In this situation, the solution
E = 0 is a stationary point. However, in the frame of a perturbative analysis for
small values of E in Eqs. (4.13), one finds that this solution is unstable (parametric
mechanical instability) if the condition,
1
Q² ≤
sin(2χi)
4pi
∫ pi
−pi
dθ cos(2θ) tanh(νst + νac cos(θ)2 ), (4.16)
is fulfilled. In the above instability criterion, χi is defined through the relation,
H(E,χi)/∂E|E=0 = 0 . (4.17)
From Eq. (4.16) one can find that there is a critical amplitude for the ac voltage,
ν∗ac(νst), above which the system becomes excited. Moreover, if the static voltage
tends to zero, νst → 0, the critical ac voltage becomes infinite, ν∗ac → ∞. This thresh-
old ac voltage is shown in Fig. 4.2, where red markers correspond to results obtained
from numerical integration of Eqs. (4.1) and (4.4) while blue lines refer to Eq. (4.16).
42
4.3. Analysis and discussion
Figure 4.3: Stationary phase (a) and amplitude (b) of the system as a function of the
applied static dc voltages νst = 2 (orange square) and νst = −2 (blue circles) for
different ac voltages νac. The phase is almost zero for the positive dc voltage while it
is non-vanishing for the negative one. As a result, current transport is more feasible
at negative dc voltages (see Eqs. (4.14)). The plot is calculated for ² = 0.1, α = 8.37,
ω˜ ∼ 1.
Due to periodicity of the generating Hamiltonian function,
H(E,χ) = H(E,χ+ pi), (4.18)
the stationary solutions of Eqs. (4.13) come in pairs, to any solution
Si = {Ei, χi}, (4.19)
corresponds a conjugated solution
S¯i = {Ei, χi + pi}. (4.20)
This fact is a clear manifestation of the symmetry properties of Eqs. (4.4) and (4.8)
discussed above. Please notice that the pi-dephasing in both solutions is responsible
for the sign of the current, e.g., a signature of the symmetry breaking.
In the non-excited regime, i.e., νac(νst) < ν∗ac(νst), the system defined by Eqs. (4.13)
possesses four formal stationary points: S1 = {0, pi/4}, S2 = {0, 3pi/4} and their con-
jugates. For νst > 0, the stationary points [S1, S¯1] and [S2,S¯2] are stable and unstable,
respectively. In the opposite case, νst < 0, these points exchage stability.
In the regime of oscillations, νac(νst) > ν∗ac(νst), besides the stationary points
[S1, S¯1, S2, S¯2], two more points, [S3 = {E3, χ3}, S¯3 = {E3, χ3 + pi}], appear on the
phase diagram. The original stationary points S1, S2 (and their conjugates) become
unstable while the new solutions are stable.
In Fig. 4.3, the phase shift and amplitude of the stable periodic solution S3 = {E3, χ3}
are shown as a function of the applied ac voltage (νac) for two different static dc volt-
ages: νst = 2 (blue circles) and νst = −2 (orange squares). From this graph, one can
observe that the phase shift and, as a consequence, the dc current (see Eqs. (4.14)) are
43
Chapter 4. Parametric effects in a shuttle system
Figure 4.4: Contour plot of the dc current as a function of the applied voltages, I3 =
I3(νst, νac). Current through the nanostructure is significant for negative dc voltage
values, νst. In the plot ² = 0.1, α = 8.37, ω˜ = 1.
almost zero for the positive dc amplitude. However, the plot indicates that the phase
is nearly χ ∼ pi/2 and a non-zero dc current is flowing through the nanostructure for
the negative dc voltage.
We also investigated the behavior of the dc current as a function of the applied
voltages, I3 = I3(νst, νac), and the results are displayed in Fig. 4.4. From the contour
map, the assymetric behavior of the dc current with respect to the static dc voltage
νst is evident. Therefore, in the light of results previously discussed, the values of the
applied voltages should be choosen in order to maximize the charge transport, i.e.,
look for a stationary phase χ3 = pi/2, this condition is attainable for negative values
of νst as it is shown in Fig. 4.4.
4.4 Final Remarks
To conclude, we have investigated the possibility to generate a shuttle dc current in a
completely symmetric single-dot shuttle system. We have demonstrated that, in this
scheme, despite the lack of a bias voltage, a shuttle dc current can still be detected.
This charge transport is achieved by applying an ac voltage to a gate electrode which
controls the electronic population of a metallic island and, in this form, also the stiff-
ness of the resonator. As a result, the metallic dot experiences a parametric mechanical
instability at the resonant frequency. We have found that parametric excitation gives
rise to two different regimes of sustained mechanical oscillations characterized by the
same amplitudes but different phases (they differ by pi). These mechanical vibrations
result in shuttle transportation of electrons through the nanostructure. In the phe-
nomena under consideration, the shuttle current is controlled by the phase shift and,
as a consequence, the two distinct stationary regimes of oscillations generate shuttle
currents in opposite directions. Which regime arises from the ac-voltage switching
will depend on spontaneous forces accompanying transient process.
44
CHAPTER 5
Summary
This final Chapter presents brief summaries of the appended papers.
Paper I - Cooling of a suspended nanowire by an ac Josephson current flow
Paper I focuses on the description of a new type of cooling mechanism for a sus-
pended nanowire, in the form of a metallic carbon nanotube, serving as a weak link
between two voltage-biased superconducting leads. The possibility to cool the quan-
tized vibrations relies on the transfer of mechanical energy from the flexural vibra-
tions of the nanowire to the electronic quasiparticle in the leads. Cooling is accom-
plished by the absorption of a quantum of mechanical energy that promotes tran-
sitions from the lower to the upper Andreev level. The electromechanical coupling
required for the cooling mechanism can be achieved by applying an external mag-
netic field perpendicular to the axis of the carbon nanotube. In this situation, the
mechanical and electronic degrees of freedom are coupled through the Lorentz force.
The cooling process is analyzed in the framework of the density matrix formalism
and the scattering matrix approach. The study indicates that the nanoresonator can
be cooled down to the ground state.
Paper II - Voltage-driven superconducting weak link as a refrigerator for cooling
of nanomechanical vibrations
Paper II extends the analysis of Paper I by considering dc current measurements as
a probe for the stationary distribution of the population of the vibrational modes. In
this paper it is shown that the stationary population of the vibrational modes of the
nanowire can be directly measured since it scales linearly with the dc current. This ex-
perimental test can be accomplished, for instance, by changing the external pressure
of the environment which implies the change of the quality factor of the system. In
the limit of high quality factor the dc current is inversely proportional to the quality
factor. In the limit of low quality factor the current is independent of the quality factor.
Paper III - Dynamics of a suspended nanowire driven by an ac Josephson current
in an inhomogeneous magnetic field
Paper III considers the same voltage-biased nanoelectromechanical system studied in
Papers I-II, but now extends the analysis to a case in which the system is subjected to
a nonuniform magnetic field. It is assumed that the magnetic field is generated by a
magnetic force microscope (MFM) cantilever tip in the form of a wedge. In this situa-
45
Chapter 5. Summary
tion, a nonlinear coupling between the Josephson current and the magnetic field gen-
erates an oscillatory driving force that induces a nonplanar motion of the nanowire.
Numerical studies of the time evolution of the amplitude and relative phase of this
whirling motion have demonstrated that at the parametric resonance, their coupled
dynamics presents a rich dynamical behavior characterized by: multistability, peri-
odic motion, period doubling, and chaos. These stationary regimes contingent on the
degree of the magnetic field inhomogeneity, which in the present case, is related to
the curvature radius of the magnetic cantilever tip.
Paper IV - Parametric excitation of a dc shuttle current via spontaneous symmetry
breaking
Paper IV analyzes the dynamics of a spatially symmetric shuttle-system, i.e., no ap-
plied bias voltage, subjected to an ac gate voltage. In such a system, it has been found
that parametric excitation gives rise to mechanical vibrations when the frequency of
the ac signal is close to the eigenfrequency of the mechanical subsystem. The most im-
portant outcome of this research is that these induced mechanical oscillations result
in a dc shuttle current in certain direction due to spontaneous symmetry breaking. In
this case, the direction of the current (symmetry breaking) is defined by the phase shift
between the ac gate voltage oscillations and the parametrically excited vibrations.
46
CHAPTER 6
Acknowledgements
I would like to express my gratitude to my advisor Leonid Gorelik for guided me
during the years as a graduate student. I also would like to thank my examiner Pro-
fessor Mats Jonson and Professor Robert Shekhter for all their help. I also would like
to give a big thank to Bea, Johanna, Rose-Marie, and Magnus Karlsteen for making
my work at the University of Gothenburg so nice. Thank you to all my partners from
Chalmers and GU.
A big thank you to the CMT group: Caroline, Gustav, Fabio, Juan, Danko, Hee-
Chul, Robert, Yury, Weihua, Daniel, Tora for sharing your experiences. A big thank
you to Anders, with whom I have shared office. I really enjoyed our discussions
about music, SHP, and Italy. Thank you Tomasz for all the support, patience and
friendship. Thank you Aurora for showing us Norway and sharing great moments
here in Gothenburg.
I also would like to thank Alessandro Scorrano from La Sapienza University of
Rome: Thank you very much for fruitful scientific collaborations.
Life is Sweden has been so enjoyable thanks to Edwin and Brigida and their beau-
tiful family, Nerina and David, Kasper and Vanessa, Don Ener (thank you for your
suggestions, support and help), and Jerker. Thank you.
Sharing is caring. A big thank you to: Ale, Vero, William, Job, Eliana, Jadwiga, Te-
nil, Lilianne, Gaby and her parents, Jorge and family, David, Simon and all members
of the REX group.
To my friends in Colombia Néstor, Oscar, Jhon Carlos Molina: Thank you.
I would like to give a big thank you to my family. This thesis is completely dedi-
cated to you. Your support and encouragement were very important in finishing this
process.
Gracias especialmente a mi tía Lucila por todo el amor, apoyo, soporte, confianza
y fé incondicional. Si existe alguien que en la distancia nos ha acompañado durante
todo éste largo trayecto ha sido mi tía Lucila. Gracias por las visitas al ejército, gracias
por las vacaciones (especialmente a San Andrés), gracias por todo......... Tía la quiero
muchisiisissisisisimo y siempre puede contar con nosotros.
Sin lugar a dudas, ésta tesis es dedicada a mi familiares (Aza - Moreno Castillo).
Gracias por todo el apoyo.
Sandra, mi vida entera, mi todo, mi princesita preciosa, mi mundo, la vida misma
hecha mujer. Tú eres la fuerza de cada día, eres la prueba más clara del amor de Dios,
eres la vida. No existen palabras para expresarte cuánto te amo. Una vida entera no
es suficiente para decírtelo. Sin embargo, tú me lo dices con una sonrisa y un beso
cada mañana..... Y yo descubro que tú amor es como el oxígeno, que llena de luz mi
vida; que me da vida y la felicidad se dibuja en tus ojos. Gracias, Gracias te amo.
47
Appendix 1: Adiabatic condition for steady level
population
In order to avoid Landau-Zener transitions at φ(t) = pi, the voltage V must be below
a certain value which can be estimated using the following arguments: The transi-
tion probability from the lower Andreev level to the upper one in the Landau-Zener
formalism at the resonant point is
P ∼ exp
[(2∆0
√
R)2/(∆0eV
)]−1 , (1)
where R is the reflection coefficient due to the impurities in the carbon nanotube and
∆0 is the superconducting gap. The condition for no transitions from Eq. (1) reads
P ∼ exp
[(2∆0
√
R)2/(∆0eV
)]−1 ¿ 1 . (2)
The last equation implies that
(2∆0
√R)2
∆0eV ≥ 1 , (3)
and it is possible to conclude that
V 5 VC = 4∆0R/e , (4)
where Vc is the maximum allowed voltage.
48
Appendix 2: Rotating Wave Approximation
The rotating wave approximation (RWA) was used in Chapter 2.6 to determine the
terms in the interaction Hamiltonian that describe Andreev level transitions through
the absorption of quanta of mechanical energy. Here, the procedure will be the ex-
plained in more detail.
The interaction Hamiltonian of the system is
Hˆint(t) = ∆0Φsin(φ(t)/2)︸ ︷︷ ︸
g(φ(t))
[bˆ† + bˆ]τˆx . (5)
By rewriting the Pauli matrix τˆx as
τˆx =
[(0 1
0 0
)
+
(0 0
1 0
)]
, (6)
the interaction Hamiltonian given by Eq. (5) can be restated as
Hˆint(t) = g(φ(t))(bˆ† + bˆ)
[(0 1
0 0
)
+
(0 0
1 0
)]
. (7)
By expanding all the products in the last equation it reads
Hˆint(t) = g(φ(t))
[
bˆ†
(0 1
0 0
)
+ bˆ†
(0 0
1 0
)
+ bˆ
(0 1
0 0
)
+ bˆ
(0 0
1 0
)]
. (8)
It is possible to elucidate the meaning of each term in Eq. (8) by letting them operate
on a general state
|η〉 =
(1
0
)
|n〉+
(0
1
)
|m〉 . (9)
The first term in the right hand side of Eq. (9) accounts for a state in the upper Andreev
level with n vibrons and the second one for a state in the lower Andreev level with m
vibrons. Expanding Hˆint(t)|η〉 term by term
1.
g(φ(t))bˆ†
(0 1
0 0
)[(1
0
)
|n〉+
(0
1
)
|m〉
]
= g(φ(t))√m+ 1
(1
0
)
|m+ 1〉 . (10)
The last equation describes the process in which the system is taken from the
lower level to the upper level and a vibron is created.
49
Chapter 6. Acknowledgements
2.
g(φ(t))bˆ†
(0 0
1 0
)[(1
0
)
|n〉+
(0
1
)
|m〉
]
= g(φ(t))√n+ 1
(0
1
)
|n+ 1〉 . (11)
In this case the system is taken from the upper to the lower level and a vibron
is created.
3.
g(φ(t))bˆ
(0 1
0 0
)[(1
0
)
|n〉+
(0
1
)
|m〉
]
= g(φ(t))√m
(1
0
)
|m− 1〉 . (12)
Here, the system is taken from the lower level into the upper level and a vibron
is absorbed.
4.
g(φ(t))bˆ
(0 0
1 0
)[(1
0
)
|n〉+
(0
1
)
|m〉
]
= g(φ(t))√n
(0
1
)
|n− 1〉 . (13)
This last term describes the situation when the system is taken from the upper
level to the lower level and a vibron is annihilated.
From this analysis it is concluded that the terms describing transitions through the
absorption channels are
bˆ†
(0 0
1 0
)
and bˆ
(0 1
0 0
)
. (14)
After this procedure the interaction Hamiltonian becomes
Hˆint(t) = g(φ(t))
[
bˆ†
(0 0
1 0
)
+ bˆ
(0 1
0 0
)]
,
= ∆0Φsin(φ(t)/2)
( 0 bˆ
bˆ† 0
)
. (15)
Therefore, the total effective Hamiltonian in Eq. (2.17) is
Hˆeff (t) = Hˆmech + Hˆele + Hˆint,
= ~ωbˆ†bˆ+∆0
√
1−D sin2(φ(t)/2)
(1 0
0 −1
)
+∆0Φsin(φ(t)/2)
( 0 bˆ
bˆ† 0
)
,
(16)
and the Schrödinger equation becomes
i~∂|Φ(t)〉∂t = Hˆeff (t)|Φ(t)〉 . (17)
If the state of the system |Φ(t)〉 is transformed via
|Φ(t)〉 → |Φ(t)〉 exp(−iωbˆ†bˆ) , (18)
50
it is found that the Schrödinger equation, Eq. (17), transforms into
i~∂|Φ(t)〉∂t = Hˆeff (t)|Φ(t)〉 ,
=
( E(t) g(φ(t))bˆ exp(−iωt)
g(φ(t))bˆ† exp(iωt) −E(t)
)
|Φ(t)〉 . (19)
Where E(t) is given by Eq. (2.16). By performing again a new state transformation
|Φ(t)〉 → |Φ(t)〉 exp(−i(ω/2)σˆzt) , (20)
and substituting it in Eq. (19), the Schrödinger equation becomes
i~∂|Φ(t)〉∂t =
( E(t)− ~ω/2 ∆0Φsin(φ(t)/2)bˆ
∆0Φsin(φ(t)/2)bˆ† −E(t) + ~ω/2
)
|Φ(t)〉 . (21)
From Eq. (21), the expression for the effective Hamiltonian given in Eq. (2.18) is ob-
tained:
Hˆeff (t) =
( E(t)− ~ω/2 ∆0Φsin(φ(t)/2)bˆ
∆0Φsin(φ(t)/2)bˆ† −E(t) + ~ω/2
)
. (22)
This Hamiltonian was the initial expression for evaluating the probabilities for An-
dreev levels scattering in Chapter 2.6.
51
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