Download Current fluctuations in single electron devices

Università degli studi di Genova
Facoltà di Scienze matematiche, fisiche e naturali
Dipartimento di fisica
Tesi di dottorato in fisica
XIX ciclo
2 aprile 2007
Current fluctuations in single
electron devices
Federica Haupt
Relatore: Prof. Maura Sassetti
Correlatore: Prof. Rosario Fazio
Contents
Introduction
iii
1 Single electron tunneling
1.1 Basic ingredients for single charge tunneling
1.2 Transport through a quantum dot . . . . .
1.3 Experimental realizations . . . . . . . . . .
1.3.1 Semiconductors 2DEG structures . .
1.3.2 Single molecule devices . . . . . . .
1.3.3 Carbon nanotubes . . . . . . . . . .
1.3.4 One-dimensional quantum dots . . .
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1
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2 Noise in mesoscopic systems
2.1 Types of noise . . . . . . . . . . . . . . . . . . . . . .
2.2 Shot noise in single electron tunneling devices . . . .
2.3 Shot-noise enhancement . . . . . . . . . . . . . . . .
2.3.1 Shot-noise in Luttinger liquid quantum dots .
2.3.2 Shot-noise in nanoelectromechanical systems
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16
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3 The sequential tunneling regime
3.1 A general model for Coulomb blockade
3.2 The sequential tunneling regime . . . .
3.2.1 Matrix formalism . . . . . . . .
3.3 Current and shot noise . . . . . . . . .
3.3.1 The zero frequency limit . . . .
3.3.2 The two states regime . . . . .
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27
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4 Shot noise in a 1D ring
4.1 A one dimensional quantum ring . . . . . . . . . . . . . .
4.2 The Luttinger liquid model . . . . . . . . . . . . . . . . .
4.2.1 Luttinger liquid with periodic boundary conditions
4.2.2 Luttinger liquid with open boundary conditions . .
4.3 Model and Hamiltonian . . . . . . . . . . . . . . . . . . .
4.4 The sequential tunneling regime . . . . . . . . . . . . . . .
4.4.1 Tunneling rates . . . . . . . . . . . . . . . . . . . .
4.4.2 Charge and orbital tunnel currents . . . . . . . . .
4.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Zero magnetic flux . . . . . . . . . . . . . . . . . .
4.5.2 Non-zero magnetic flux . . . . . . . . . . . . . . .
4.5.3 Monte Carlo simulation . . . . . . . . . . . . . . .
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5 Shot noise in NEMS
5.1 Introduction . . . .
5.2 Model . . . . . . .
5.3 Rate Equation . .
5.4 Results . . . . . . .
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57
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Conclusions
72
A Classical suppression of shot noise
74
B The current correlation function
76
C The polaron transformation
79
D The transition rates
81
ii
Introduction
“Noise is the signal” was a saying of Rolf Landauer [1], one of the founding fathers
of mesoscopic physics. What he meant is that fluctuations in time of a measured
quantity can be a valuable source of information that is not present in the time
averaged value. Even if from the engineering point of view noise is essentially a
nuisance which has to be as much as possible minimized, a physicist may actually
delight in noise.
The unique informative role of noise has been known since long time in the
quantum optics comunity. It was back in 1909 when Albert Einstein first realized
that the electromagnetic fluctuations are different if the energy is carried by waves
or by particles [2]. The magnitude of energy fluctuations scales linearly with the
averaged energy for classical waves, but it scales with the square root for classical
particles. Since the photon is neither a classical wave nor a classical particle, the
two contributions coexist. Fifty years later, with the Hambury Brown and Twiss
experiment, it was discovered that intensity fluctuations provide information on the
statistics of a photon’s source. Originally intended as a novel interferometric mean
to determine the diameter of stars, the Hambury Brown and Twiss experiment is
best-known today as the first experimental demonstration of photon bunching [3].
Since electrons share the particle-wave duality with photons, one might expect
fluctuations in the electrical current to play a similar diagnostic role. Current
fluctuations due to discreteness of the electrical charge are known as “shot noise”.
This term was introduced in 1918 by Walter Schottky, drawing an analogy between
the electrons moving in a vacuum tube and the small pellets of lead that hunters
use for a single charge of a gun [4]. Although the first experimental observation
of shot noise dates back to the 1920’s, the interest on the electronic shot noise has
progressed more slowly than for photons. Much of the information it contains has
been appreciated only recently from experiments on mesoscopic devices.
Mesoscopic physics is a branch of condensed matter physics which studies the
electrical properties of system with typical dimensions at the borderline between the
macroscopic scale of the world we live in and the microscopic scale in which each
atom is separately considered [5]. In mesoscopic systems, the electrons maintain
their phase coherence over distances larger than the sample size, leading to interference effects which cannot be described classically. Moreover, in these devices
the confining potential changes over length scales which are comparable with the
electron wavelength, so that quantized states are formed. But at the same time,
these devices contain from several thousand to millions of atoms so that statistical
properties -such as temperature and distribution functions- still have a meaning.
Moreover, mesoscopic devices can be connected and probed through macroscopic
wires and amplifiers, leading to classical measurements of classical currents. Even if
some mesoscopic effects are observable on the macroscopic scale - as for example the
quantum Hall effect- most studies are carried out on devices of submicron dimensions. For this reason, the experimental research in mesoscopic physics has always
been closely related with the development of sophisticate lithographic and crystal
growth techniques. Fundamental research has also benefited much from massive
iii
Introduction
iv
industrial research and development efforts towards miniaturization of integrated
electronic circuit based on semiconducting materials. Moreover, in recent years
mesoscopic physics has developed into a interdisciplinary field including physics,
chemistry and biology. In fact, in the relentless search for smaller electronic devices,
the idea of using single organic molecules, nanotubes or DNA as active components
in electrical circuits has become an important topic.
Although an experimental break-through at one Kelvin may not have a direct
impact on the commercial technology, the cumulative knowledge and experience
gained through work at the mesoscopic scale can provide insight for conceptually
new electronic devices. A notable example is the quantum dot, or single electron
device, which exploits the quantum phenomenon of tunneling to control and measure
the flow of single electrons inside a circuit [6]. A quantum dot is an extremely precise
electrometer, thousand times more sensible to charge variations than the state-ofthe-art conventional transistors. For this reason, dots have important applications
in metrology and in ultra-sensitive analog applications. As an example, coupling a
single electron device to a nanoscale mechanical resonator, a displacement sensitivity
of the order of 10−15 m has been recently achieved [7]. To put this number into
perspective, these devices are able to detect displacement as small as ten-millionth
of the size of the hydrogen atom.
The electronic properties of mesoscopic system are routinely investigated by
measuring the current flowing through the device in the presence of an applied
voltage. However, in the last decade current measurements have often be sided
by noise measurements, which have shown to be an highly efficient investigation
tool. Indeed, noise in the electrical current contains information on the underlying
transport processes which are not contained in the average current. As an example, Schottky’s seminal idea of extracting the charge of the current carriers from
a shot noise measurement [4], has been the lever to measure the fractional charge
of Laughling’s quasiparticles in the fractional quantum Hall effects [9]. Moreover,
shot noise measurements are the most promising candidates to show experimental
signatures of fractional statistics. Electrical noise has also become an alternative
and very accurate method to determine the temperature of electrons in a solid. In
the field of quantum computing, noise has been proposed as a tool to investigate
the correlations caused by the entanglement. The shot noise has also proved to
be very sensitive to the presence of interactions. As an example, the correlations
induced by strong Coulomb interaction generally lead to a detectable suppression
of the shot noise in a quantum dot.
The effects of the electron-electron interaction become prominent in one dimension. Truly one-dimensional electronic system can be created either by using
sophisticated lithographic techniques or by growing carbon nanotubes. Unlike in a
usual three dimensional conductor, in one dimension electronic interactions affect
substantially the properties of the ground state and of the low-energy excitations.
For this reason, one dimensional devices are expected to show unique transport
features. The Luttinger liquid model serves as a theory for interacting electrons
in one dimension [11]. It is characterized by power-law correlation functions with
an interaction-dependent exponent. This exponent show up in all the experimental
accessible quantities, which manifest a power-law behavior.
Beside electronic interaction, the properties of a mesoscopic device are strongly
influenced also by the electron-phonon interaction. In solid state devices electrons
generally interact with a continuous phonon-bath which is a major source of decoherence and relaxation. The situation is radically different for a molecular device.
In this case the phonon spectrum is discrete and, to a good approximation, at low
energies the electrons can be considered as coupled to a single phonon mode.
In this thesis we study how the electron-electron interaction and the electron-phonon
v
Introduction
interaction affects the current noise in a single electron device. We employ the Luttinger liquid description to treat electronic interaction in one dimension and an
Anderson-Holstein-type Hamiltonian to harness the electron phonon interaction.
We focus on the sequential tunneling regime and we calculate the current and the
shot noise employing a rate equations approach. The layout of this thesis is as follow:
Chapter 1. In the first chapter we discuss the most prominent features which characterize a single electron device. We first analyze the circumstances under which
charge quantization effects are important and then we introduce the fundamental
concepts of Coulomb blockade and Coulomb oscillations within the framework of
the costant interaction model. Moreover, we discuss how the exceptional charge
sensitivity of a single electron transistor can be transmuted into ultra-sensitive
displacement detection. Subsequently we shortly describe a number of different
realizations of single electron devices, ranging from semiconductor based quantum
dots, to carbon nanotubes and molecular devices. Particular attention is devoted
to carbon nanotubes and cleaved-edge overgrowth heterostructures since they represent two of the most important realizations of one-dimensional electronic system.
Chapter 2. A short introduction to current fluctuations will be given in this chapter. We first summarize the generic characteristics of noise in a mesoscopic device,
discussing the differences between shot noise, thermal noise and quantum noise,
their distinct physical origin and the regimes where each of them become dominant. Then we focus on the zero frequency shot noise of a single electron device.
We discuss the suppression of the shot noise induced by the charging energy, and
then we describe mechanism known as dynamical channel blockade, which is instead
responsible for a strong enhancement of the noise.
Chapter 3. In this chapter we develop the theoretical tools required to evaluate
the current and the current noise in the sequential tunneling regime. We introduce
a general model for a Coulomb blockade mesoscopic system, discussing the characteristic energy scales and the conditions under which transport through the system
can be considered as a sequential tunneling process. In the sequential tunneling
regime, the dynamics of the system can be described in terms of Markovian rate
equations. We show how the rate equations formalism allows to derive simple expression for the average current and for the shot noise in a compact matrix notation.
Chapter 4. In order to introduce the peculiarities of a one dimensional quantum
ring, at the beginning of this chapter we give a brief introduction on the main
features of the Luttinger liquid (LL) model, considering both the case of periodic
and open boundary conditions. Afterwards, we study current fluctuations in a 1D
quantum ring, connected via tunnel contacts to external leads, in the presence of
interactions. This geometry permits to analyze the interplay between interaction,
Coulomb Blockade and Aharonov-Bohm interference. Signatures of this interplay
reflect in a wealth of the interaction-induced noise features. We characterize the
noise by considering both the charge current noise (related to the fluctuations of the
tunneling current) and the magnetization current noise (related to the fluctuations
of the persistent current). We study the noise both analytically and numerically
and we support our observations with a Monte Carlo simulation.
Chapter 5. In this chapter we discuss how phonon relaxation influence the transport properties of a single electron transistor coupled to a mechanical oscillator.
We find that finite relaxation rates affects the dynamics in a highly non trivial
way. Both current and noise can be either enhanced or suppressed by relaxation,
depending on the electron-phonon coupling and on the considered voltage range. In
Introduction
vi
particular, for voltages higher than a certain critical value, the Fano factor can be
even suppressed below 1/2. This suppression is observed in a completely incoherent
regime as a consequence of the interplay between vibration assisted tunneling and
direct relaxation of different vibrational states.
Finally, we give a summary of the conclusions which can be drawn from this thesis.
Chapter 1
Single electron tunneling
In the last two decades, an impressive number of theoretical and experimental works
have been devoted to the study of single electron quantization effects in solid state
devices [6, 12, 13]. Charge quantization effects are really nothing new: already in
1911 Millikan observed the effects of single electrons on the falling of oil drops [14].
Nevertheless, quantization effects play little role in conductors, where charge can
be safely regarded as a continuous quantity. Because the wavefunctions of electrons
in conductor are extended over macroscopic distances, the charge in any small volume is not quantized. Therefore, to observe charge quantization effects it is first
necessary to localize the electrons to tiny regions of space. Progress in nanolithography and thin-film processing have opened up the possibility to confine electrons
in sub-micron regions, allowing for the experimental observation of single electron
quantization effects in solids. In the next section we examine the circumstances under which charge quantization effects are important. In section 1.2 we discuss the
characteristics of transport through a Coulomb blockaded island at the light of the
constant interaction model. Finally, in section 1.3 we will discuss a number of experimental realizations of quantum dots, ranging from semiconductor heterostructure,
to carbon nanotubes and molecular devices.
1.1
Basic ingredients for single charge tunneling
Let us consider a charge transport experiment in which a voltage difference is applied
to two electrodes separated by an insulating gap. In the middle of the gap lies a
third electrode which we call “island”, since it is surrounded by the insulator or
also “quantum dot”, meaning that the electrons are confined in all three spatial
directions. To travel from the source to the drain the electrons must go through
the island. We assume that particle exchange occurs by quantum tunneling. This
process is so fast that we can consider that the electrons are traversing the insulating
gap one at a time. The key point is that during the journey from the source to
the drain, the electron necessarily makes the charge of the dot vary by e. This is
a tiny amount of charge if we consider an ordinary electronic device, 1 however, if
the island is sufficiently small, the variation of the electrostatic potential due to the
presence of an excess electron can be large enough to react back on the tunneling
probabilities [15].
The change of the electrostatic potential is conveniently expressed in terms of
the capacitance C of the island. An extra charge e changes the potential of the dot
by the charging energy EC = e2 /C. When this energy scale exceeds the thermal
1 As a example, each charge packet in a charge coupled device (CCD) is composed of about 10 6
electrons.
1
Single electron tunneling
2
energy kB T , the quantum dot acquires a sensitivity to charge variations at the level
of a single electron. A first condition to observe charge quantization effects is then
e2 /C kB T.
(1.1)
The second requirement is that the resistance of the tunneling barriers have to be
much bigger than the resistance quantum h/e2 ≈ 25.813 kΩ
RT h/e2 .
(1.2)
This condition is obtained by requiring the RC-time constant for an electron to
tunnel off of the island to be great enough that the energy uncertainty is much
smaller than the charging energy, i.e. h/(RT C) EC . This condition essentially
ensures that the wave function of an excess electron in the island is localized there.
If RT is small on the scale provided by h/e2 , charging effects are suppressed because
of quantum fluctuations.
To summarize, the two conditions to observe effects due to the discrete nature of
charge are
RT h/e2 ,
2
e /C kB T.
(1.3)
(1.4)
When both of them are satisfied, transport through the island is dominated by
charging effects and one speaks about a Coulomb blockaded mesoscopic system.
The first criterion Eq.(1.3) can be met by weakly coupling the island to the source
and drain, the second Eq.(1.4) by making the dot small. Modern experimental
techniques permits to create electronic islands whose size is comparable with the
Fermi wavelength of the host material. In this case, electrons occupy discrete “orbital like” quantum levels and have a discrete excitation spectrum. For this reason
quantum dots are sometimes also called artificial atoms [16]
1.2
Transport through a quantum dot
In a typical transport experiment, a quantum dot is operated in a three terminal
geometry, see Fig.(1.1). The dot is cupled by tunnel barriers to a source and a drain
Figure 1.1: Schematic set up for a quantum dot connected to source, drain and gate
electrodes.
3
1.2 Transport through a quantum dot
leads, which in turn are connected to macroscopic current and voltage meters. A
third “gate” electrode is capacitively coupled to the island, and can be used to
tune the electrostatic potential of the dot with respect to the reservoirs. As we will
discuss in this section, in this configuration a quantum dot acts as a single electron
transistor (SET). A convenient way to study the transport properties of a SET is
the costant interaction model [13,17,18]. This relays on two important assumptions.
(i) The Coulomb interaction among the electrons in the dot and between the
electrons in the dot and those in the environment is parametrized by a single,
costant capacitance C. This capacitance can be thought of as the sum of the
capacitances between the dot and the source, Cs , the drain, Cd , and the gate
Cg , i.e C = Cs + Cd + Cg (see Fig. 1.1).
(ii) The discrete energy spectrum does not depend on the number of electrons in
the dot.
Under these assumptions the total ground state energy for N electron in the dot at
zero temperature is given by
U (N ) =
N
i2 X
X
1 h
e(N − N0 ) −
n
C i Vi +
2C
n=1
(1.5)
i=s,d,g
where e is the charge of the electron, and N0 is the number of electrons in the dot in
an electrically neutral situation. The terms Ci Vi (i = s, d, g), represent the charge
induced on the dot by the applied external voltages. The last term of Eq. 1.5 is a
sum over the occupied single particle energy levels n . The energy levels depend on
the characteristics of the confinement potential.
To describe transport experiments, it often is more convenient to refer to the
electrochemical potential µ(N ) which is, by definition, the minimal energy required
for adding the N th electron to the dot
X C i Vi
1
µ(N ) ≡ U (N ) − U (N − 1) = EC N − N0 −
− EC
+ N
2
e
(1.6)
i=s,d,g
where EC = e2 /C is the charging energy. The electrochemical potentials for the
transition between ground states with different electron number N are shown in
Fig. 1.2(a). The discrete levels are spaced by the so-called addition energy
Eadd (N ) ≡ µ(N + 1) − µ(N ) = EC + ∆.
(1.7)
The addition energy consit of a purely electrostatic part, the charging energy E C ,
plus the energy spacing between two discrete quantum levels ∆ = N +1 − N . The
addition energy is large for small capacitance and/or large energy splitting ∆.
In linear response regime µs ≈ µd , a non-zero addition energy can lead to a
blockade for tunneling of electrons on and off of the dot. Infact, at zero temperature
the N -th electron can tunnel into the dot from the ith-electrode only if energy
is conserved, i.e. µ(N ) ≤ µi (i = s, d). Analogously, it can tunnel off only if
µ(N ) ≥ µi . Therefore, if the electrochemical potentials of the dot and the leads are
arranged as in Fig.1.2a, tunneling is interdicted and the number of electrons into
the dot is fixed to N − 1. So, for µ(N − 1) < µs , µd < µ(N ), electron transport is
blocked and this situation is known as Coulomb blockade.
Coulomb blockade can be removed by changing the gate voltage to align µ(N )
into the bias window (see Fig.1.2(b))
µd ≤ µ(N ) ≤ µs .
(1.8)
Single electron tunneling
4
Figure 1.2: Schematic diagrams of the electrochemical potential of a quantum dot
for different electron numbers. (a) The chemical potentials of the dot are misaligned
with respect to µS and µD . The number of electrons in the dot is fixed to N − 1
due to Coulomb blockade. (b) As µ(N ) falls into the bias window, the number of
electron can oscillate between N and N − 1 resulting in a single-electron current.
(c) Both the transition between the N − 1 and the N ground states (black line) and
the transition to an N -electron excited state (grey line) are energetically allowed.
(d) The electrochemical potentials for N and N + 1 electrons are both comprised
into the bias window. In this case two electrons can tunnel into and out of the dot
at the same time. After Ref. [35].
In this case, an electron can tunnel from the source electrode into the dot and then
it can tunnel off to the drain. Now, a new electron can tunnel-in and repeat the
cycle N − 1 → N → N − 1. This cycle is known as single electron tunneling.
By measuring the current while sweeping the gate voltage a trace as shown in
Fig. 1.3a is obtained. Each peak occurs when an electrochemical potential of the
dot fall into the bias window and the dot can oscillate between two adjacent charge
states. Viceversa, in the valleys between the peaks the number of electrons on
the dot is fixed because of Coulomb blockade. The distance in gate voltage ∆Vg
between two peaks can be readily obtained from Eq.(1.6) by imposing the condition
µ(N )|Vg = µ(N + 1)|Vg +∆Vg . This leads to
e∆Vg =
C
(∆ + EC ).
Cg
(1.9)
Therefore ∆Vg gives information about the energy spectrum of the dot.
A second way to lift Coulomb blockade is to increase the source-drain voltage
V (see Fig. 1.2c). Again, current can flow only when an electrochemical potential
level falls within the bias window. By increasing V until both the ground state as
well as an excited state transition fall into the bias window, an electron can choose
to tunnel not only through the ground state, but also through an excited state of
the N -th electron dot. This is visible as a change in the total current. In this
way, excited-state spectroscopy can be performed. How exactly the current changes
depends on the tunnel rates of the two different paths.
In a typical experiment, the current I or the differential conductance ∂I/∂V ,
is measured while sweeping the bias voltage for a series of different values of the
gate voltage. Such a measurement is shown schematically in Fig. 1.3b. Inside
the diamond-shaped region, the number of electron is fixed because of Coulomb
blockade and no current can flow. Outside the diamonds, Coulomb blockade is
lifted and single-electron tunneling is possible. For larger bias voltage even doubleelectron tunneling is possible (see Fig. 1.2d). Excited states are revealed as changes
in the current, i.e. peaks or dips in the differential conductance. From such a
“Coulomb diamond”, the charging energy as well as the energy of the excited states
can be read off immediately.
5
1.2 Transport through a quantum dot
Figure 1.3: Transport through a quantum dot. (a) Coulomb peaks in current versus
gate voltage in the linear response regime (µs ≈ µd ). (b) Coulomb diamonds in differential conductance, ∂I/∂V . versus V and Vg . The edges of the diamond shaped
region (black) correspond to the onset of the current. Diagonal lines emanating
from the diamonds (gray) mark the onset of transport through excited states.After
Ref. [35].
SET as a displacement detector
Up to now we have discussed how a quantum dot is operated as single electron
transistor. This is the most sensitive electrometer, with a sensitivity which been
shown to be as low as few 10−5 e Hz−1/2 , i.e. a charge variation of 10−5 e can be
detected in a measurement time of 1s [19, 20]. This exceptional sensitivity can be
exploited to design ultra-sensitive displacement detectors. The basic idea is frailly
simple [21]: a mechanical resonator coated with a thin metallic layer and placed next
to a SET acts as a mechanically-compliant gate electrode. When a fixed voltage is
applied to this electrode, the motion of the resonator modulates the electrochemical
potential of the SET and hence the tunneling current.
Today nanolithografic techniques allow to create quantum dots which are closely
integrated with a sub-micron size mechanical resonators [22–24]. Because of their
small size, these resonators have small inertial masses (10−15 −10−17 kg) and vibrate
at frequencies ranging from few MHz up to around a GHz [25].
By coupling a SET to 116-MHz doubly coupled suspended beam (see Fig1.4), a
record displacement sensitivity of 10−15 m Hz−1/2 has been achieved at the temperature of 30 mK. To put this number into perspective, this device is able to detect
displacements as small as ten-millionth of the size of an hydrogen atom. This sensitivity is roughly a factor 100 larger than the zero-point displacement uncertainty
of the mechanical beam.
The SET displacement detector belongs to the class of devices known as nanoelectromechanical systems (NEMS). Besides having important applications as ultrasensitive measuring devices [7, 26, 27], NEMS are interesting dynamical systems in
their own right. Infact, not only the motion of the resonator affects the tunneling
current but also, because of the small mass of the oscillator and the strong electrostatic coupling to the SET, individual electrons travelling through the SET can
give significant displacement “kicks” to the mechanical resonator. This back-action
leads to fluctuations in the oscillator position and to damping [28].
Single electron tunneling
6
Figure 1.4: (a) Scanning electron micrograph of the device used in Ref. [7]. The
substrate and the suspended beam are fashioned from GaAs (blue regions). The
SET and beam electrode are thin aluminum layers (yellow regions), with aluminum
oxide forming the tunnel barriers. The beam is located 0.25 µm away from the
island electrode (scale bar, 1 µm). The measured fundamental flexural frequency
for in-plane motion is 116 MHz. (b) A schematic of the mechanical and electrical
operation of the device.
1.3
Experimental realizations
A quantum dot is a general kind of system and many different sizes and materials can
be used to realize it: single molecules trapped between electrodes [29,30], metallic or
superconducting nanoparticles, self-assembled quantum dots, semiconductor lateral
or vertical dot [12, 16] or carbon nanotubes between closely spaced electrodes [31,
32]. In the remaining of this chapter, we will shortly discuss of a few of these
experimental realizations. This is by no means intended to be an extended review,
we are merely interested in introducing the typical length and energy scales of these
devices and the main physical issues that characterize them.
1.3.1
Semiconductors 2DEG structures
Fabrications of semiconductor based quantum dots starts with a semiconductor
heterostructure, a sandwich of different semiconductor material (see Fig. 1.5). These
layers, most commonly GaAs and AlGaAs, are grown one on top of each other using
molecular beam epitaxy (MBE). Because these two semiconductor have nearly the
same crystal structure, MBE growth results in very clean crystals. By doping the
n − AlGaAs layer with Si, free electrons are introduced. As a consequence, the
shape of the conduction band is modified and an almost triangular potential well
develops on the GaAs side of the interface (inversion layers). Electrons accumulate
in this potential well, forming a thin (∼ 10nm) sheets of electrons that can move
only along the interface. Due to the confinement, the single particle energy spectrum
splits in discrete subbands
~2 k 2
+ n ,
(1.10)
~k,n =
2m∗
where m∗ = 0.067 me is the effective mass of electrons in GaAs and ~k ≡ (kx , ky ),
being z the heterostructure growth direction. If the Fermi energy is such that only
the lowest subband is populated, ε0 < EF < ε1 , then the system behaves effectively
as two-dimensional. Assuming typical parameters for the GaAs/AlGaAs structures,
the Fermi energy and the average level spacing can be estimated as EF ∼ 10meV and
∆ ∼ 20meV respectively [34]. Therefore, at temperatures kB T ∆−EF ∼ 100K
7
1.3 Experimental realizations
Figure 1.5: Different layers of semiconductor in a GaAs/Alx Ga1−x As heterostructure. Due to the different band gap of GaAs and Alx Ga1−x As, a triangular potential
well is formed on the GaAs side of the interface. The conduction band diagram as
well as the carrier concentration have to be calculated solving selfconsistently the
Poisson and the Schrödinger equation. At low temperatures kB T ∆ − EF only
the energy ε0 of the first subband lies below the Fermi energy, so that the system
behave effectively as two-dimensional. After [33].
the system is “frozen” into the lower subband n = 0 and is dynamically a 2D system.
The so created two-dimensional electron gas (2DEG) has several properties desirable for studying mesoscopic effects. Infact they can have a high mobility and a
relatively low electron density (105 −106 cm2 /Vs and ∼ 3×1015 m−2 , respectively).
The low electron density results in a large Fermi wavelength (∼ 40 nm) and a large
screening length, which allows us to locally deplete the 2DEG with an electric field.
Two-dimensional electron gases can be easily given an arbitrary shape using
lithographic techniques. This is achieved either by permanently etching a portion
of the 2DEG, or by using metallic gates deposited on the top of the nanostructure.
To fabricate these gates, the surface of the heterostructure is first covered by a
layer of organic resist. Then the gate pattern is defined by writing in the electron-
Single electron tunneling
8
Figure 1.6: Fabrication of metal electrodes by electron beam lithography. (a) A
specific patter is written into the resist-layer with an electron beam. (b) A resistmask is formed over the heterostructure surface after the resist which was impressed
by the electron beam has been removed. (c) Metal evaporation. (d) After lift-off,
only the metal electrodes remain on the surface. After Ref. [35]
sensitive resist with a focused electron beam. This locally breaks up the polymer
chains, so that the exposed parts can be removed by a developer. In the next step,
metal is evaporated and, because of the resist mask, it can make contacts with the
heterostructure only where the resist has been exposed and removed. In the last
step, the remaining resist is removed with acetone and metal electrodes are left
at the places that were exposed to the electron beam. These fabrication steps are
summarized in Fig. 1.6.
The electron beam can accurately write very small patterns with a resolution of
about 20 nm, allowing to make very complicated structures. By applying negative
voltages to the gates, the 2DEG is locally depleted, creating one or more islands
that are isolated from the large 2DEG reservoirs (see Fig. 1.7). These islands are
quantum dots.
To create freely suspended nanostructures, as the one showed in Fig. 1.4 electron
beam lithography is combined to a sequence of reactive ion etching and dilute HF
wet etching to remove selectively portions of GaAs [36, 37].
Another promising technique is atomic force microscope (AFM) nanolithography, where the tip of an AFM is used to define neat in-plane gates by local oxidation
of the GaAs cap layer [38]. Combining this technique with additional metallic top
gates an an exceptional high tunability can be reached.
a
b
Figure 1.7: Planar quantum dot. a Quantum dot defined by 5 metallic gates fabricated on the surface of a GaAs based heterostructure (after Schönenberger’s group
homepage http://pages.unibas.ch/phys-meso/). b By applying negative voltages to
metal electrodes on the surface of the heterostructure, the underlying 2DEG can be
locally depleted. By choosing a suitable geometry for the gates, the electrons can
be confined to one or even to zero dimensions.
9
1.3.2
1.3 Experimental realizations
Single molecule devices
The idea of using single molecules as functional devices in electrical circuits trace
back in 1974 to the pioneering work by Aviram and Ratner [39]. As active electronic
units, single molecule offer several unique properties. First of all, the size of most
of the simple molecules is within several nanometer, resulting into a typical energy
scale of ∼eV for the electronic spectrum. Secondly, molecules are self-assembling,
which is very useful in fabricating electronic devices at such a small scale. Finally,
another great advantage is the huge variety of molecules with different functionalities and thus the possibility to design the electrical properties of a device through
chemical synthesis. The main challenge in molecular electronics is to “wire up”
Figure 1.8: Experimental data by H. Park [29] for electronic transport through
a C60 transistor. The device is formed by a single-C60 molecule placed between
two closely spaced Au-electrode. Upper panel: Current-voltage characteristic for
different values of the gate voltage Vg . The current is strongly suppressed near zero
bias followed by step-like current jumps at higher voltages. Lower panel: Differential
conductance ∂I/∂V for four different devices as a function of gate and bias voltage.
Dark regions correspond to the conductance gap and bright lines to peaks in the
differential conductance. White arrows mark the position of vibrational sidebands.
Measurements were taken at T = 1.5K.
a single molecule. In recent years, a number of ingenious experimental schemes
have been developed. The tip of a scanning tunneling microscope (STM) can be
Single electron tunneling
10
employed as an electrode to measure the current through a molecule which rests
on a substrate [40]. Othervise, a planar configuration can be employed, situating
the molecule between two closely spaced electrodes realized by means of breakjunctions [41] or electromigration [42]. The molecule may or may not be chemically
bound (e.g. by thiol groups) to the electrodes.
Up to today, a large variety of molecules, ranging from H2 [30] to DNA [43]
have been investigated experimentally in two-terminal [30,43,44] and three-terminal
configurations [29, 40, 45–47]. In many of these experiments unambiguous evidence
of Coulomb blockade has been reported [29, 45, 46] as well as the observation of
Kondo effects [46, 47].
An important distinction between transport through semiconductor quantum
dots and single molecules lies in the coupling to phonon degrees of freedom. While
charge carriers typically interact with a continuum of phonon modes in quantum
dots, molecules are characterized by a discrete spectrum of vibrational modes. A
prominent effect of the coupling of the electrons to discrete phonon modes, is the
appearance of vibrational side bands in the current-voltage characteristics [29,30,45]
(see Fig. 1.8).
1.3.3
Carbon nanotubes
Carbon nanotubes (CN) are macro-molecules with a hollow cylindrical shape, made
up only of carbon atoms. Since their discovery, back in 1991 [48], they have been at
the center of an intense research activity because of their outstanding mechanical
and electrical properties.
Roughly speaking, a CN can be viewed as a sheet with an hexagonal lattice
of carbon atoms (graphene) which is rolled up to form a seamless cylinder. From
the electronic point of view, graphene is a gapless semiconductor [49]. When a
graphene sheet is rolled up to form a nanotube, its 2D-dimensional band structure
split up into one-dimensional subbands, because of the boundary conditions around
the circumference. Because of the symmetry and unique electronic structure of
graphene, the geometric structure of the nanotube strongly affects its electrical
properties [49].
The direction along which the sheet is fold can be indicated by the “folding
~ = n~a1 + m~a2 , where ~a1 and ~a2 are the elementary translation vector on
vector” C
the basis of the honeycomb lattice (see Fig.1.9). If m = 0, the nanotubes are called
“zigzag”, if n = m they are called “armchair”, otherwise they are called “chiral”.
For a given (n, m)-nanotube, if n − m = 3j, j ∈ N the tube acts as a 1D metal,
with a Fermi velocity vf = 8 × 105 m/s comparable to typical metals. Moreover
it has a linear dispersion relation and a flat density of states around the Fermi
energy. Otherwise, if m − n 6= 3j the CN is a semiconductor with a gap between
the filled hole states and the empty electron states. The bandgap is predicted to be
∼0.9eV/d[nm], where d is the diameter of the tube. These remarkable theoretical
predictions has been verified scanning tunneling microscopy experiments [50, 51].
When metallic electrodes are attached to a nanotube, the properties of the resulting device depends crucially on the quality of the contacts. With good contacts,
a metallic nanotube behaves as perfect electronic wire and it becomes a ballistic
conductor with a conductance almost approaching the theoretical maximum value,
4e2 /h ≈ 155µS [52, 53]. Viceversa, if the the contacts between the CN and the
electrodes are poor, a tunnel barrier is formed at each contact and electron transport occur via quantum tunneling. In this case, the portion of tube between the
electrodes forms a quantum dot [31, 32]. As an example, in Ref. [31] transport
measurements through ropes of nanotubes bridging 200nm-spaced contacts have
evidenced a gap in the I − V curves at low temperature and a series of sharp
peaks in the conductance as a function of a gate voltage Vg . These observations
11
1.3 Experimental realizations
Figure 1.9: The graphene honeycomb lattice and its basis ~a1 ,~a2 . The folding vector
~ = n~a1 + m~a2 . The vector T~ denote the tube axis. Folding directions for zigzag
is C
and armchair nanotubes are also shown.
are consistent with single-electron transport through a segment of a single nanotube
with a typical addition energy of ∼ 10 meV and an average level spacing of ∼ 3meV.
Carbon nanotubes are also ideal systems for exploring electromechanical effects
because they have a low mass, are easily scalable through their length and have a
Young’s modulus of the order of TPa [54], which is almost one order of magnitude
higher than that of silicon. Moreover, they can be defect free and their electronic
properties are nowadays well understood [49, 55].
The idea of using a doubly clamped suspended carbon nanotube as a nanoelectromechanical system was first envisioned by Sapmaz et al. [56] and then and
then implemented experimentally by Sazanova et al. [57] and Witkamp et al. [58].
The basic idea is to actuate a nanotube by applying an ac-voltage to the gate,
which induces a driving force on the tube due to the displacement-dependent gate
capacitance. The CN starts to oscillate when the driving frequency approaches a
mechanical eigenfrequency. In a semiconducting CN this lead to a modulation of
the conduction, therefore the nanotube can be employed to detect its own motion.
Furthermore, other experiments on suspended carbon nanotubes have evidenced
the interplay between mechanical and electrical degrees of freedom also in the absence of an actuating gate voltage [59, 60]. Electrons tunneling onto the nanotube
excite discrete vibrations by spending some of their excess energy provided by the
bias voltage. This leads to the appearance of a series of peaks in the conductance
(phonon-sidebands) which forms an harmonic low-energy spectrum. By analyzing
the energy scale and the size-dependence of this spectrum, electron coupling to the
radial breathing mode (RBM) [59,61] and to the longitudinal (stretching) mode [60]
have been demonstrated. Moreover, the measurements show that free-hanging tubes
operate in the under-damped regime of low dissipation. For the fundamental bending mode the reported quality factor is about 300 [58]; for the RBM it is estimated
as high as 10000, corresponding to a phonon lifetime of the order of 10 ns [59].
Single electron tunneling
1.3.4
12
One-dimensional quantum dots
The electronic properties of one-dimensional (1D) conductors have generated much
interest. The reason for this excitement lies in their very rich phase diagram and the
prediction that in a 1D system the Coulomb interaction should lead to a strongly
correlated electron gas, called a Luttinger (LL) liquid instead of the weakly interacting quasi-particles described as a Fermi-liquid in conventional metals [62]. The
reason is that, unlike their higher dimensions counterpart, in 1D conductors the
electroninc interaction is poorely screened. Consequently, in 1D the fermionic quasi
particle excitations that are characteristic of Fermi liquids become unstable and collective density fluctuations constitute the stable elementary excitations [63]. The
difference between a Luttinger and a Fermi liquid becomes striking already in the
presence of a single impurity. According to the Landauer’s theory, at zero temperature the conductance of a single non-interacting channel with a barrier is given
by G = |T |2 e2 /h, where |T | is the transmission probability through the barrier.
This result holds even at finite temperature, given that |T | is approximately energy
independent. Viceversa, Luttinger liquid theory predict a vanishing conductance at
T = 0 even for the smallest barrier, and a power law dependence G(T ) ∝ T α at
finite temperature T [64].
Carbon nanotubes are one of the most promising candidates for a detailed study of
1D physics. In fact, as we will see in Sect. 4.2, the Luttinger liquid model assumes a
linear dispersion relation. In metallic CN this is an unusually good approximation
as their dispersion relation is linear over a large energy range of the order of several
hundreds meV [49]. Moreover, due to the enormous subband separation (of the order
of eV), metallic nanotubes can be considered truly one dimensional conductors even
at room temperature.
Experimental evidence of Luttinger liquid behavior in metallic nanotubes comes
from tunneling experiments in ropes of carbon nanotubes between metal electrodes
in the Coulomb blockade regime [65]. It has been shown that the conductance
and the differential conductance scale as power laws with respect to temperature
and bias voltage, respectively and that the functional form and the exponents are
in good agreement with theoretical predictions. In this case, the one-dimensional
CN-quantum dot was coupled directly to 2D-dimensional non interacting leads (the
metal electrodes). Differently, by employing AFM manipulation, it is possible to
create 1D quantum dots connected to leads which are themself one-dimensional. In
fact, the tip of an AFM can be used to introduce local structural defects (buckles)
in the CN, which behaves very much like electronic tunnel junctions. When two
buckles are introduced along the nanotube, a quantum dot embedded in a onedimensional conductor is formed [66]. An example of a similar quantum dot is
presented in Fig. 1.10. This technique allows to reduce the size of the quantum dot
to few tens of nm. As an example, the distance between the two buckles in Fig. 1.10
is approximately 25nm. Consequently, the dot has a very high charging energy of
about 120 meV, allowing for the observation of Coulomb blockade peaks up to room
temperature. By studying the temperature dependence of the conductance peaks,
striking evidence of power law scaling for the peak area and for the peak maximum
have been found.
Signatures of Luttinger liquid behavior have also been reported in semiconductor
devices created by means of the cleaved edge overgrowth technique (CEO) [67].
This is an highly sophisticated molecular beam epitaxy technique, developed at
Bell Labs to create low dimensional quantum structures with atomic precision [68].
The starting point to create a CEO-wire is a thin layer of GaAs sandwiched between two thicker layer of AlGaAs (Fig.1.11a). An high quality 2DEG is created at
13
1.3 Experimental realizations
) Zero-bias
conductance versus
gate voltage for a
single conductance
peak. Data are shown
4, 20, 30, 40,
70, 80, and 90 K. The
solid lines are fits to
/
)
)] [as deT
dependence of the
w
G
Fabrication of a
room-temperature single-electron transistor
within an individual
metallic carbon nanotube by manipulation
A)
Nanotube between Au
electrodes on top of a
substrate with
a gate-independent resistance of 50 kilohm.
T
, which displays a linear behavior.
Figure 1.10: Top: atomic force microscopy images of a single wall carbon nanotube
(blue) on top a Si/SiO2 substrate (green), contacted to metallic gates (brown). After
the deposition of the nanotube (A, white bar: 200nm), whose radius is 20-50 nm,
two buckles are created by using the AFM tip (B, C), thus defining a short region
in between, of length ≈20 nm (white bar: 20nm). Bottom left: linear conductance
peaks for different temperature values ranging from 4K to 90K – dots: experimental
data, solid line: theoretical fit. Bottom right: maximum (right scale) and area (left
scale) for the linear conductance peaks as a function of the temperature in double
logarithmic scale, shows clearly a power-law behavior. (After [66]).
the interface between the semiconductors by an appropriate doping scheme. The
specimen is then cleaved in ultra high vacuum and another doped AlGaAs layer is
overgrown over the smooth cleavage plane (Fig.1.11b). This introduces electrons at
the edge of the GaAs (Fig.1.11d), creating one or more edge states along the cleave.
Electrons in such edge states are firmly confined along two directions (y and z in
Fig. 1.11): on three sides, confinement is due to atomically smooth barriers and, on
the fourth side, to the strong triangular potential of the cleaved-edge modulation
doping sequence. Strong overlap between the 2DEG and the edge states couples
both systems intimately along the entire rim. The edge state can be decoupled by
applying a negative voltage to a top gate T , which was evaporated on top of the
sample before the cleave. As VT is biased increasingly negative, the 2DEG splits in
two sheets and a 1D wire is formed in the depletion region (Fig.1.11c). Because of
the strong confinement in the y and z, the energy spectrum of the wire presents a
series of 1D sub-bands, with a mean level spacing of ∼ 20meV (∼ 2K). By tuning
the top voltage to increasingly negative values, the electron density in the wire can
be reduced until only a single subband is populated. Conductance measurements
in this low density regime show an abrupt decrease of the conductance, followed
by a series of almost equally spaced sharp resonances (see inset in Fig. 1.13).
These interpreted as Coulomb blockade peaks through a 1D electronic island which
is formed when the highest peaks of the impurity potential cross the Fermi energy
(see Fig. 1.12). In fact, despite the care taken in all the fabrication steps, some
impurities are unavoidably present along the wire and are seen by the electron as
a disordered potential. When the electron density is so low that the highest peak
of this potential exceeds the Fermi energy, the wire is split into two parts and the
conductance drops to zero. By reducing the density even further, a second barrier
crosses the Fermi energy forming a 1D island and transport can then occur through
Single electron tunneling
14
Figure 1.11: The creation of a CEO sample and the experimental setup for transport
measurements. Insets d, e and f show a sequence of schematic cross sections of
charge distribution in the wire region for different VT . The side gate S, primarily
serves to vary the electron density along the entire edge. The top gate affects the
electron density only in the wire region. For strongest 1D confinement, the top
gate is biased negatively and the side gate strongly positively, pushing the electrons
aganist the cleaved edge of the quantum well. (After [67])
Figure 1.12: Sketch of the impurity potential in a CEO wire. In the ultra-low
density regime, the highest peaks of the impurity potential can exceed the Fermi
energy. As soon as two peaks cross the Fermi energy, a quantum dot is formed.
15
1.3 Experimental realizations
resonant states. In this case, the system consists of a 1D quantum dot connected
to interacting LL leads. Signatures of Luttinger liquid behavior were indeed found
in the temperature dependence of the linewidth Γi of the resonant states. In contrast to the conventional Coulomb blockade theory [12], where Γi is temperature
independent, in this case Γi decrease as a power law of temperature (see Fig 1.13),
in agreement with the theoretical predictions of Ref. [69].
(b)
Conductance
Γi (arb. units)
(2e2 /h)
(a)
Top gate voltage [V]
Deduced temperature
Figure 1.13: Coulomb oscillations in a CEO-wire. a Main panel: Linear conductance
as a function of the top-gate voltage for the CEO wire studied in Ref. [70]. Inset: A
zoom-in of the conductance in the subthreshold region. A series of nearly equidistant
peaks is clearly observable. Those peaks have been interpreted as the Coulomb
oscillations of a 1D-quantum embedded into the wire. b Intrinsic linewidth Γi
vs. temperature for two resonant peaks (peak # 1 and peak # 2 of the inset). Γ i
decreases as a power law of temperature, indicating a Luttinger liquid behavior [69].
The dashed line are power law fit to the data Γi ∝ T 1/g−1 . (After Ref. [70])
Chapter 2
Noise in mesoscopic systems
In recent years, much attention has been devoted to the statistical analysis of the
current fluctuations in mesoscopic system. Initially this study has been confined
to the second moment in the current fluctuations distribution, which goes under
the name of noise [71, 72]. Probably the most paradigmatic example of the effectiveness of the noise as an investigation tool is the direct observation of fractional
quasiparticle charges in the fractional quantum Hall effect [8–10]. In this chapter,
a quick overview of the properties of noise in electronic systems will be given. In
section 2.1, we discuss the main sources of noise in a electrical conductor and the
different regimes where they are dominant. Afterwords, we focus on the effects of
the interactions on the shot noise of a single electron device, analyzing both the
suppression due to Coulomb blockade (section 2.2), and the enhancement due to
dynamical channel blockade (section 2.3)
2.1
Types of noise
Noise represents the spontaneous, dynamical fluctuations of a physical quantities
away from its time averaged mean value. These stem from the thermal motion of
matter and the discreteness of its structure and form a stochastic random process
For an electrical system, not all sources of noise are inherent to the device or to
the electron transport processes. These generate extrinsic noise and can be reduced
or even eliminated by employing better device designs and improved fabrication
technologies [71]. In this case the study of noise can help developing higher quality
devices. On the other hand, intrinsic noise, which is a characteristic of the device,
cannot be reduced and sets an upper limit for the accuracy that can be achieved in
measurements. For quantum systems, the presence of noise represents an important
source of decoherence. From this point of view, the intrinsic noise can be regarded
as a nuisance. But, at the same time, the study of electrical fluctuations is an
important tool to investigate the electronic properties of mesoscopic devices [72]. If
we consider the electrical current through such a device, the intrinsic fluctuations
contain information about the interactions that are regulating the transport.
From the theoretical point of view, electrical noise is usually described in terms
of time dependent current fluctuations in a conductor with a fix, non fluctuating
voltage between the contacts. In the stationary regime, a detailed description of
the noise in time domain is given by the correlation function
CI (t) = h∆I(t + t0 )∆I(t0 )i,
(2.1)
where ∆I(t) = I(t) − hIi and hIi is the mean current (see Fig. 2.1). Here and in
the following, the brackets h· · ·i denote an average over an ensamble of identical
16
17
2.1 Types of noise
Figure 2.1: Time dependent fluctuations of the current around its average value
hIi.
physical systems or, equivalently, an average over the initial time t0 (ergodicity
assumed). Equivalently, noise might be represented in the frequency domain by the
power spectral density (the power of noise per unit frequency)
Z ∞
SI (ω) =
dt eiωt CI (t).
(2.2)
−∞
If the current I(t) is a classical observable, the correlator is real and symmetric, i.e.
C(t) = C(−t) and so is the power density S(ω) = S(−ω). It is therefore enough to
consider a symmetrized spectrum
Z ∞
dt eiωt CI (t), ω > 0
(2.3)
S(ω) = SI (ω) + SI (−ω) = 2
−∞
defined for positive frequencies ω. This is also the quantity that is detected in
standard, low frequencies noise measurement. In the quantum limit, the spectrum is
no longer symmetric SI (ω) 6= SI (−ω) and this classical description is not anylonger
valid. However, as in the rest of this thesis we will consider only the zero-frequency
current noise, we refer to Ref. [5,73,74] for for a detailed analysis of the asymmetric
quantum noise.
The zero-frequency component of the symmetrized noise spectral density
S ≡ S(0).
(2.4)
is interesting for several reasons. First, it is the easiest to access experimentally.
Infact, the frequency dependence of the shot noise is governed by microscopic rates,
whose frequency are typically measured in units of THz. On the other hand, current
and noise measurements are most usually performed ad much lower frequencies (in
the range of several kHz to several 100 kHz), so that one is justified in taking the
limit ω → 0 in the calculation of the shot noise. Second, the noise versus voltage
curve have a rich structure, which is particularly revealing when rescaled by the
average current. This can be used either to fit the parameter in the model more
accurately or to provide a consistency check for parameters already determined from
the average current. However, not all types of electrical noise are informative. The
fluctuation current in a conductor in thermal equilibrium merely tell us the value
of the temperature of the electrons T . To get more out of noise one has to bring
the electrons out of thermal equilibrium. Before getting into that, let us us say a
bit more about thermal noise.
Thermal noise
For kB T eV , thermal agitation is the main cause of current fluctuations in a
conductor. These fluctuations are called thermal noise or also Johnson-Nyquist
Noise in mesoscopic systems
18
noise, after the experimentalist [75] and the theorist [76] who first investigated it.
At equilibrium V ≈ 0, thermal noise is directly related to the conductance G of the
system by the fluctuation dissipation theorem [71, 77]
S = 4kB T G.
(2.5)
Therefore, the investigation of thermal noise does not provide more information than
the investigation of the linear conductance. Eq.(2.5) is valid as long as ω k B T .
At very high frequencies ω ≥ kB T , vacuum fluctuations contribute to equilibrium
fluctuations too. In this case, Eq.(2.5) has to be changed by replacing kB T with
the expression ω/2 coth(ω/2kT T ), so that noise is no longer frequency independent
(“white”) but increases linearly with ω.
Shot noise
At low temperature kB T eV , the main source of fluctuations in the electrical
current is the randomness inherent in the transmission of discrete charge quanta.
These kind of fluctuations are known as shot noise and, unlike thermal noise, they
carry information on the temporal correlation of the electrons which are not contained in the conductance [72, 73, 78].
We start considering the situation in which the electrons are transmitted through
the device randomly and independently of each other. In this case, which is,
typical of a vacuum tube or of a tunnel junction, electron transfer can be described as
P a Poisson process [71, 79]. In other words, the current is expressed as
I(t) = e δ(t − tn ), where tn are uncorrelated random quantities, with the average
interval τ between the arrivals of consequent electrons. In this case the average
current is hIi = e/τ , while the current noise is
S = 2ehIi ≡ SP oisson
(2.6)
and it does not depend on frequency. This formula was first derived by Walter
Schottky in 1918 [4], who proposed it as an alternative method to measure the
charge of the electron.
The possibility, envisioned by Schottky, of measuring the charge of the current
carriers from the shot noise power, has been fully appreciated only recently in
systems in which the granularity of the current is not the elementary charge e
(e.g. in the case of the fractional Hall effect or in superconductors). In this case,
Eq.(2.6) becomes S = 2qhIi, where the electron charge e is replaced by an effective
charge q. Shot noise measurement performed in the fractional quantum Hall regime
have allowed the direct observation of the fractional charge corresponding to the
quasiparticles [8–10]. In the case of superconductor-normal metal junctions, an
enhancement of the shot noise by a factor 2 was observed [80], which is attributed
to the q = 2e effective charge transfer in Andreev reflection process.
The Schottky formula Eq.(2.6), relays on the hypothesis that transport is a
Poissson process. However, in general the electrons in a conductor are not independent and correlations cause the shot noise to deviate from Eq. 2.6. Deviations from
the Poissonian value are expressed by the Fano factor F, which is defined as
F ≡
S
.
2ehIi
(2.7)
A main source of correlations is the Pauli principle. In a vacuum tube or in a
tunnel junction, the mean occupation of a state is so small that the Pauli principle
is inoperative, but this is not so in the general case. In a system of non interacting
electrons, an efficient way of accounting for the correlations induced by the Pauli
19
2.1 Types of noise
principle is Landauer’s scattering approach [81, 82]. According to the Landauer
formula, the time-averaged current hIi is given by
hIi =
N
2e2 X
V
Tn
h
n=1
(2.8)
where, V is the applied voltage, 2e2 /h is the conductance quantum (including a
factor two for spin). and Tn are the transmission probabilities (0 ≤ Tn ≤ 1) of the
N independent channels which characterize the system (formally, the Tn ’s are the
eigenvalues of the product t·t† , where t is the transmission matrix of the conductor).
At zero temperature, the noise is related to the transmission probabilities by [83,84]
S = 2e
N
2e2 X
V
Tn (1 − Tn ).
h
n=1
(2.9)
The factor (1 − Tn ) describes the reduction of noise due to Pauli principle. The
shot noise formula Eq.(2.9) has a simple statistical interpretation. Consider first a
one-dimensional conductor, i.e. a device with a single channel with transmission
probability T1 . Electrons in a range eV above the Fermi level, enter the conductor
at a rate eV /h. In a time τ̄ , the number of attempt transmissions is τ̄ eV /h. At zero
temperature, there are no fluctuations in this number, since each occupied state
contains exactly one electron (Pauli principle). Fluctuations in the transmitted
charge Q arise because the transmission attempts are successful with a probability
T1 different from 1 or 0. The statistics of Q in not Poissonian anylonger but rather
binomial. The mean-squared fluctuations h∆Q2 i of the charge for the binomial
statistics is given by
h∆Q2 i = e2 (τ̄ eV /h)T1 (1 − T1 ).
(2.10)
The relation S = (2/τ̄ )h∆Q2 i between the mean-squared fluctuation of current and
the transmitted charge gives back exactly Eq.(2.9) for a single channel. Since fluctuations in different channels are independent, the multi-channel version is simply
a sum over channels (as an aside, a classical derivation of the T (1 − T ) suppression
of the shot noise is given in appendix A).
In a tunnel barrier, electrons have a very small probability of being transmitted.
In the Landauer formalism, this fact is implemented by the condition Tn 1, for
all n. In this case it is easy to see that Eq.(2.9) simply reduce to the Poissonian
limit S = SP oisson .
The quantum shot noise formula has been tested experimentally in a variety of
systems [85, 86]. As an example, Fig.2.2 shows a comparison between theoretical
predictions and experimental measurements for the current and the Fano factor in
a quantum point contact, which is a narrow constriction in a 2DEG. By applying a
negative voltage to a metallic gate, the 2DEG is depleted and a narrow channel can
be formed by an opening in the gate. By changing the gate voltage one can adjust
the width of the constriction, which in turn determine the number N of discrete
channels contributing to the current. According to Eq.(2.8), each open transmission
channel give a discrete contribution to the average current, which is then expected
to show a staircase behavior as a function of the gate voltage. Viceversa, the shot
noise shows peaks in between the conductance plateaux, and it is vanish on the
plateaux since fully open (or closed) channels do not contribute to Eq.(2.9).
Nowadays, shot noise measurement is routinely employed in the study of atomic
size contact (e.g. break junction) as, combined with conductance measurements, it
allows to identify uniquely “mesoscopic PIN code” of the junction, i.e. the number
of open channels and their transmission (the ) [87].
Shot noise can also be employed to probe particle statistics. Bosons emitted
by a thermal source tend to bunch (more of them are emitted at the same time)
20
S/S0
S/S0
Noise in mesoscopic systems
Figure 2.2: (a) Theoretical predictions [84] for the conductance G (dashed line)
and shot-noise power S (full line) of a point contact versus Fermi energy. (b)
Experimentally observed G and S versus gate voltage Vg at T = 0.4K (experimental
data from the Reznikov’s group). After [78].
resulting in a super-Poissonian statistics [3]. Due to Pauli principle, a fermionic
thermal source emits particles separately (anti-bunching), leading to sub-Poissonian
statistics [88]. When a thermal stream of particle is separated at a beam splitter,
the correlations between the two outgoing streams depend on the particles being
bosons or fermions. By using a Hanbury-Brown and Twiss geometry, positive correlations were measured in the case of photons [3, 88] and negative correlations for
electrons [89, 90].
In a macroscopic devices, like macroscopic metallic wires, the granularity of the
charge flow is smeared out by inelastic scattering of the electrons with environment.
This is the reason way a macroscopic conductor shows thermal noise, but not shot
noise.
Before finishing this section, it is worthy to stress that thermal noise and shot
noise are not two different processes, but rather two different limits of the same
physical effect. The crossover from thermal noise Eq.(2.5) to shot noise Eq.(2.9)
generally depends on the transmission eigenvalues.
1/f noise
Beside thermal and shot noise, in a real device the are other sources of noise which
are not related to the transmission properties. The most common example is 1/f
noise. These type of fluctuations are caused by slow changes in the device resistance and they are found in almost any conducting material. Their spectrum is
proportional to the square of the injected DC current and increase as an inverse
power of the frequency. The exponent is often very close to unity, hence the name
“1/f ” noise. The origin of the resistance fluctuations is not universal and, despite
of number of experimental and theoretical work devoted to this subject, for most
system is still unknown.
In this thesis we will not deal at all on 1/f noise, even if it can be truly important
in nowadays mesoscopic devices (for an extensive review on 1/f noise see [5,71]). We
merely remark that, as 1/f noise can obviously not be avoided in a low frequency,
any measurement in units of shot noise has to be made at frequencies above those
where 1/f noise is significant (typically several kHz). This is still in the zerofrequency limit as far as the shot noise is concerned, since its dependence on the
frequency is governed by microscopic rates whose frequencies are typically measured
in units of THz. We will be therefore justified in taking the limit ω → 0 in the
calculation of the shot noise
21
2.2
2.2 Shot noise in single electron tunneling devices
Shot noise in single electron tunneling devices
Beside Pauli principle, another important source of correlations among electrons
is Coulomb interaction. An energy scale of Coulomb repulsion is the charging
energy EC = e2 /C of a single electron inside a conductor with capacitance C. In
open conductors, where C is large, charging effects are expected to be negligible.
However for close conductors, such as a double barrier junction or a quantum dot,
EC can be as large as kB T . In this case case, as we have seen in Chapter 1, charging
have a pronounced influence on the transport properties of the device. The theory
of the shot noise in single electron devices has been developed both with quantum
mechanical approaches [91, 92] and by means of semiclassical derivations based on
the rate equations technique [93,94]. Interestingly, both approaches predict identical
shot noise results, implying that the shot noise is not sensitive to quantum coherence
in single electron structures.
The first measurement of the shot noise in a Coulomb blockaded system was
reported by Birk et al. [95] in metal nanoparticle. In this experiment the doublebarrier junction is formed by positioning the tip of a scanning-tunneling microscope
above a metal nanoparticle on an oxidized substrate. Because of the tiny size of
the particle it is Ec ≥ 1000kB T at T = 4K. The relative heights of the two tunnel
barriers can be modified by changing the tip-particle distance. The experimental
results in the case of strongly asymmetric junctions (R2 ≈ 100R1 , where 1 and
2 stand for the tip-particle and the particle-substrate junction, respectively) are
represented in Fig. 2.3.
The I−V current display a stepwise increase of the current with voltage (rotating
by 90 the plot yields to the usual presentation of the “Coulomb staircase”). At small
voltage, the current is vanishingly small I ≈ 0 because of Coulomb blockade. At
each subsequent step in I, the number of excess electrons on the particle increase
or decreases by one. The corresponding measured shot noise oscillates along with
the step structure in the I − V curve. The full Poissonian value S = SP oisson is
reached at each plateau of constant I (arrow A in Fig. 2.3), while it is suppressed
for current steps in between plateaux (arrow B in Fig. 2.3). The experimental data
are in excellent agreement with the theory of [93].
A qualitative understanding for the shot noise suppression caused by charging
effects can be given by the two state model. Because of the large charging energy
EC kB T , at each step only the two charge states with the energetically most
favored number of electrons on the particle are considered. In this case, tunneling
always occurs alternately across the two junctions. After an electron has been added
to the particle by tunneling across the junction 1, this junction is blocked until the
electron is tunnels out through junction 2. On a current plateau in the Coulomb
staircase, the number of excess electrons is constant for most of the time, only
disturbed during very short instants. In other words, for most of the time electron
transport is blocked by the more resistive barrier R2 , which inhibits electrons to
tunnel out of the particle. Once an electrons leave the dot, another one quickly
takes its place tunneling from the tip of the STM. In this case the tunneling events
are solely determined by junction 2, hence the shot noise is the same as for a single
tunnel junction, S = SP oisson . The situation is completely different for voltages
where the Coulomb staircase show a step. Here two charge states are degenerate,
which result in similar tunneling probabilities for the two junctions. As both barriers
are on average alternately blocked during equal times, tunneling become correlated
causing a suppression of shot noise with a maximum of 1/2.
It is worth noticing that measuring the shot noise of a quantum dot using conventional noise measurement techniques is very challenging. This is because of the
very low currents and the corresponding low noise levels in these systems. Indeed,
Noise in mesoscopic systems
22
Figure 2.3: Left panel:Current and noise measurements in a metallic quantum dot.
Top panel. I-V characteristic for system formed by the tip of an STM positioned
over a metallic nanoparticle on substrate. The thick line is the experimental curve at
T = 4.2K, while the thin line represent the theoretical prediction. The “Coulomb
staircase”, i.e. stepwise increase of the current with increasing voltage is clearly
distinguishable. Lower panel. Zero-frequency shot noise as a function of the current.
The dashed lines correspond to the maximal and minimal values S = 2eI and
S = eI. Right panel. Measured Fano factor F = SI /2eI as a function of the
current I. In both panels, diamonds: experimental data, solid line: theory by [93].
After [95].
after the breakthrough represented by Ref. [95], in the next decade few other experimental observations of the shot noise in dots have been reported (e.g. [96, 97]).
An alternative way to investigate current fluctuations was introduced by Levitov
et al. [98] and it is known as full counting statistics (FCS). This method relies on
the evaluation of the probability distribution function of the number of electrons
through a conductor within a given time period. In addition to the current and
to the shot noise, which are the first and the second moment of this distribution,
this methods give access to higher order moments. The most intuitive method
for measuring the FCS of electron transport is to count electrons passing one by
one through the conductor. Very recently, real-time detection of single electron
tunneling through a quantum dot has been achieve, allowing to measure the full
counting statistics of the corresponding transport process [99]. The shot noise could
be extracted from the measured FCS, and it showed a clear suppression below the
Poissonian limit as a function of the asymmetry of the tunneling barriers.
2.3
Shot-noise enhancement
As we have seen in the previous section, for electron transport through quantum
dots, the noise is typically of sub-Poissonian nature, i.e. F < 1. This is due to
the Coulomb blockade, which enhance the correlation between the electrons and
thereby reduces the noise. However, when several channels with different coupling
strengths contribute to the electron transport, interactions can lead to more complex
processes and to an enhancement of the noise [100, 101].
To explain these results let us consider the case in which transport through a
quantum dot can occur both via the ground A state and via an excited state B
(e.g. Fig. 1.2c). The tunnel coupling between the dot and the leads is given by the
overlap of the dot and leads electronic wave functions. Since the wave functions
corresponding to A and B may have different spatial distributions, it can happen
that one of the two states, say A, is coupled to the leads much weaker than the
23
2.3 Shot-noise enhancement
other one, B, so that its contribution to the current and noise is negligible. Because
of Coulomb blockade, the dot may only hold one or zero excess electrons. This
in turn implies that transport through the two states A and B is correlated: if A
is occupied, current through B is blocked, while if A is empty, current through B
proceeds in an ordinary way. Thus, if tunnel rates for A and B are of the order
of ΓA and ΓB , respectively, transport through the system proceeds in bunches of
ΓB /ΓA 1 electrons and the Fano factor achieve large super–Poisson values.
This mechanism, which takes the name of dynamical channel blockade, was first
introduced by Safonov et al. [102] to explain the super-Poissonian noise observed in
tunneling through localized states. We stress that in order to observe bunching of
electrons in a quantum dot it is necessary that the charging energy is greater than
the separation between the energy levels relevant for transport. Theoretical predictions of super–Poissonian noise exist for sequential tunneling regime in quantum
dots with ferromagnetic leads [103, 104] (if both leads are partially polarized, say
spin-up, then spin-up electrons tend to tunnel in bunches, and spin-down electrons
block the current for a long time), in a magnetic field [105]. Moreover, general
dynamical channel blockade have been predicted for sequential tunneling in single [106,107] and in double quantum dots [108,109], and in quantum dots where the
level coupling is mediated by non-equilibrium plasmons [110] (see below). Finally,
Ref. [111] predicted Super-Poissonian noise in the inelastic cotunneling regime. Recently super-Poissonian noise was experimentally observed by Onac et al. [112] in a
carbon nanotube quantum dot and by Gustavsson et al. in a semiconductor quantum dot [113]. In Ref. [112] a Fano factor up to F = 3 was observed inside the
Coulomb blockade diamonds, which is therefore noise was observed inside the diamonds and is therefore associated with inelastic cotunneling. Viceversa, in Ref. [113]
the super-Poissonian Fano factor was observed at the onset of the conduction region, so that it is related to sequential tunneling through an excited state and it
suggests a long relaxation time for the involved excited state.
As suggested by the above list of references, the shot noise of a quantum dot has
been widely studied in the last decade. In the remaining part of this section we will
only shortly revised those results which are more closely related to the topics dealt
with in my thesis.
2.3.1
Shot-noise in Luttinger liquid quantum dots
An intriguing issue is the influence of Luttinger liquid correlations on the shot noise
of a one-dimensional (1D) quantum dot, such as those presented in Sec. 1.3.4. As
we will discuss in detail in Chap. 4, electronic interactions plays a unique role in
one dimension. Here, the usual Landau-Fermi liquid theory breaks down and it has
to be replaced by the Luttinger liquid model [11]. Luttinger liquids have a number of peculiar properties. As an example, their elementary excitations cannot be
described in term of fermionic quasiparticles but rather of collective charge density
waves (plasmons) which have a bosonic nature. Because of this, Luttinger liquids
are characterized by power law correlations functions with interaction dependent
exponent. Intriguingly, this same exponent shows up also in most of the measurable
observables.
Signatures of Luttinger liquid behavior were recently found in the shot noise of
a one-dimensional quantum dot. As an example, studying the dependence Fano
factor on the applied voltage, Braggio et al. [114] have found an oscillating behavior
similar to the one observed in the non interacting case. However, both the position of the deeps and the asymptotic value of the Fano factor for large V , depends
strongly on the strength of the interaction (see Fig.2.4). This dependence allows to
extract the value of the Luttinger liquid parameter without resorting to any fitting
Noise in mesoscopic systems
24
procedure [114]. In this work, fast relaxation of the plasmonic excitations in the
Figure 2.4: Fano factor F of a one dimensional quantum dot as a function of the
applied voltage, for T = 0 and strong asymmetry of the tunneling barriers. Different
curves correspond to different strength of the electronic interaction: g = 1 for non
a interacting system and g < 1 for repulsive interactions. Color legend: g = 1red,
g = 0.8 black, g = 0.6 blue and g = 0.4 gray. After [114].
quantum dot was assumed, implying that excitations created by one tunneling electron do not influence subsequent tunneling events (see also discussion in Sec.4.4.1).
The opposite case was lately addressed by Kim et al. [110], who have focused on the
properties and the consequences of the non-equilibrium distribution of the plasmon
in the dot. This is the case when the plasmonic excitations in the quantum dot
redistribute only via the single-electron tunneling events through tunnel barriers.
They found that while the average electric current is only weakly affected by the non
equilibrium properties of the plasmons [115], the shot noise show distinctly different features [110]. In particular, the shot noise can turn to super-Poissonian values,
and the enhancement is more severe in the strong interaction limit (see Fig. 2.5a).
The origin of this enhancement is again connected to a dynamical channel blockade
Figure 2.5: (a) Fano factor F of a one dimensional quantum dot with nonequilibrium plasmonic excitations, for T = 0 and strong asymmetry of the tunneling
barriers. εp is excitation energy of the plasmonic excitations. Different curves correspond to different strength of the electronic interaction: g = 1 green, g = 0.7 black,
g = 0.5 red and g = 0.3 blue. (b) In the presence of relaxation of the plasmonic
excitations, the enhancement of the shot noise is strongly suppressed. Here γp is a
phenomenological relaxation rate. Different curves correspond to different values of
γp (in units of the average tunneling rate Γ0 ): γp = 0 black, γp = 0.1 red, γp = 1
green and γp = 104 blue. After [110].
effect. In fact, for sufficiently high voltages, the two transport channels
N → N +1→N
N → N + 1∗ → N
(the asterisk stand for a transition occurring through an excited state in the dot)
compete in transport. As we have seen above, if the transition rate for one of the
25
2.3 Shot-noise enhancement
channels is much stronger than other one, then a super-Poissonian Fano can be
achieved. This enhancement of the Fano factor is intimately related to the presence of non-equilibrium excitations. In the presence of relaxation of the plasmonic
excitations, the super-Poissonian Fano factor is strongly suppressed (see Fig. 2.5b).
Eventually, when the relaxation rate becomes comparable with the tunneling rate,
the Fano factor is always sub-Poissonian and it recovers the characteristic behavior
discussed by Braggio et al. [114].
2.3.2
Shot-noise in nanoelectromechanical systems
Another class of devices in which non-equilibrium effects are expected to play an
important role are nanoelectromechanical systems (NEMS). NEMS are single electron devices in which the electronic degrees of freedom are strongly coupled to a
mechanical degree of freedom (see Sect. 5.1). Electrons tunneling onto a NEMS
can excite discrete vibrational modes (phonons) by spending some of the excess
energy provided by the bias voltage. Thereby, electron transport through will tend
to drive the vibrational mode out of equilibrium, if the quality factor of the latter
is sufficiently high. It is then a physically relevant question whether the vibrational
energy is reduced by relaxation processes induced by coupling to an external environment [116] or rather because tunneling itself [117–119]. Up to now, theoretical
works have focused mostly on the case of negligible relaxation, taking the opposite
case of strong relaxation as a reference term. It has been predicted that the shot
noise would show a radically different behavior in this two cases, both for weak
and strong electro-mechanical coupling [119, 120]. Of particular interest is the case
of strong coupling. In fact, a giant enhancement of the Fano factor has been predicted in the case of strong electro-mechanical coupling and negligible relaxation of
the vibrational excitations, as opposed to sub-Poissonian noise for equilibrated (i.e.
thermally distributed) excitations, see Fig.2.6 [120]. The origin of the giant Fano
Figure 2.6: Fano factor F as a function of bias voltage (in units of the frequency ω 0 of
the vibrational mode) for a NEMS with strong electro-mechanical coupling. Results
are shown for both non-equilibrated and equilibrated vibrational excitations. The
combined effect of the strong coupling and weak phonon relaxation leads to a giant
enhancement of the Fano factor. After [120]
factor stem from the combined effects of the suppression of the tunneling rates due
to the strong electro-mechanical coupling and the weak relaxation of the vibrational
excitations. This interplay is responsible for an avalanche like transport of electrons
Noise in mesoscopic systems
26
through the device [120–122]. However, as we will discuss in Chapter 5, this peculiar transport regime is strongly inhibited even by weak relaxation, leading to a
suppression of F .
Chapter 3
The sequential tunneling regime
In this chapter we develop the theoretical tools required for the study of current
and noise in the sequential tunneling regime. In section 3.1 we introduce a general
model for a Coulomb blockade mesoscopic system, discussing the characteristic
energy scales and the conditions under which transport through the system can
be considered as a sequential tunneling process in section 3.2. In the sequential
tunneling regime, the dynamics of the system can be studied in terms of a Markovian
rate equation for the occupation probability of the island. In section 3.3 we derive
the general expressions for the stationary current and for the current noise within
the rate equation formalism. Throughout this chapter we keep wording and notation
as general as possible to stress the wide applicability of this formalism, which forms
the basis for most of the following chapters.
3.1
A general model for Coulomb blockade systems
In general terms, a Coulomb blockade system can be described as a small electronic
island (“dot”) coupled to leads by tunneling barrier. The important point is that
to travel from one lead to the other the electrons must go through the island, as
there is no direct coupling between the two external electrodes. The corresponding
Hamiltonian can be written as:
H = Hd + Hleads + HT .
(3.1)
Here, Hd is the Hamiltonian of isolated the electronic island and it must contain
all degrees of freedom which are suspected to be relevant for transport. Quite
in general, it can be written as the sum of a single particle part and of an interaction part Hd = Hd0 + Hdint . The non-interacting part is simply given by
P
the single particle spectrum of the island Hd0 = νd ενd c†νd cνd , where c†νd is the
creation operator for state νd , while the interaction part Hdint is specific of the
considered model. Importantly, since we are considering the isolatedP
island, Hd
must be electron-number conserving i.e. [Hd , Nd ] = 0, where Nd = νd c†νd cνd .
Hence, each many-particle eigenstate of Hd has a well defined electron number, i.e
Hd |αi = Eα |αi ⇒ Nd |αi = Nα |αi.
The second term in Eq.(3.1) is the Hamiltonian for the leads, which we treat as
electronic reservoirs. As the left and right leads are uncoupled we can write Hleads =
H1 + H2 . We denote with |Li i the eigenstates of Hi and with ELi the corresponding
energy. We assume that the leads are at any time in thermal equilibrium with the
local chemical potential µ1,2 . In other words, we assume that the probability to
27
The sequential tunneling regime
28
Figure 3.1: Schematic picture of a electronic island connected to source and drain
leads by tunneling barriers.
find the state |Li i is given by the Boltzmann distribution function.
WLβi =
1 −β(EL −µi NL )
i
i
e
Z
(3.2)
where β = 1/kB T , NLi is the number of electrons into the state |Li i and Z is the
grancanonical partition function for the considered lead. The bias window between
the leads is fixed by external voltage µ1 − µ2 = eV . Furthermore, we assume the
density of the states of the leads to be flat around the Fermi energy. The reservoirs
Hamiltonian can additionally incorporate a bosonic heath bath to take into account
a dissipative environment [123].
Finally, the last term in Eq.(3.1) is the tunneling Hamiltonian HT which couples
the electronic island to the leads:
X (i)
X (i)
(i)
(i) ∗
(tνi,νd c†νi cνd + tνi,νd
c†νd cνi )
(3.3)
HT ,
HT =
HT =
i=1,2
νd,νi
where c†νi creates an electron in the state νi into the ith-lead. Because of dot-leads
coupling, the eigenstates of the electronic island acquire a finite lifetime τ = 1/Γ,
where Γ is a characteristic rate for tunneling. Equivalently, one can says that
there is a broadening of eigenstates of the dot proportional to Γ. Regarding H T
as a perturbation, this rate can be evaluated by means of Fermi’s golden rule. In
particular, the rate for the transition between two states of the dot |αi → |βi is
given by
X X β
(i)
(3.4)
WLi |hβ, L0i |HT |α, Li i|2 δ(Eα − Eβ + ELi − EL0i ),
Γα→β = 2π
i=1,2 L ,L0
i
i
where the sum is taken over all the possible initial and final states for the reservoirs
and each contribution is weighted with the probability WLβi to find the initial state
|Li i. Importantly, to the lowest order in HT , only transitions between states with
neighboring charge are allowed, i.e. it must be Nβ = Nα ± 1. Moreover, tunneling
at the different barriers gives two independent contributions to the total transition
(1)
(2)
rate which can then be written as Γα→β ≡ Γα→β + Γα→β .
3.2
The sequential tunneling regime
The general model we have introduced in the previous section, is characterized by
several energy and associated time scales Their relation one to each other determines
the underlaying transport regime. There are basically three energy scales one has
to consider: the level broadening Γ induced by tunneling, the typical energy separation ∆ between the many-body eigenstates of the electronic island and finally the
temperature T , which determines the equilibrium distribution in the reservoirs.
29
3.2 The sequential tunneling regime
We are interested in the sequential tunneling regime, which means that the electrons
travel through the dot via subsequent incoherent tunneling events. For this to be
true, the average time between two tunneling events 1/Γ must be much longer than
the coherence time in the reservoir. As the time scale over which excitations in the
reservoirs are thermalized is τeq = 1/kB T , we require Γ kB T . This condition
guarantees that the excitations in the reservoirs created by a tunneling event are
thermalized before the next electron can tunnel. In this case, it is justified to
consider transport in the lowest order in HT and to disregard any higher order
contribution. Moreover, since we are not interested into the interference effects
given by mixing of different states of the dot due to tunneling, we demand Γ ∆.
In this case the states of the island remains well separates.
When tunneling defines the lowest energy scale into the problem, i.e. Γ kB T, ∆, the dot can be described it in terms of a distribution function Pα that gives
the probability of finding the island in a particular state |αi. In equilibrium this is
nothing but the Boltzmann distribution function. However, when a voltage bias is
applied across the system, the distribution function is not thermal anylonger and it
has to be determined self-consistently from the solution of the rate equations [13,18]
X
d
Pα (t) =
{Pβ (t)Γβ→α − Pα (t)Γα→β } .
dt
(3.5)
β
where the transition rates Γα→β are given in Eq.(3.4). These equations are rather
self-explanatory: the second term on the right-hand side gives the rate at which the
state |αi decays. It is proportional to the probability that the system is in state |αi
in the first place, multiplied by the rates for transitions from |αi to any other state
|βi. The first term gives the rate for the opposite process, i.e. the rate at which
the system ends up in state |αi given that it started in some other state |βi.
The rate equations must include all the states relevant to the dynamics of the
system. The phase space Ω one has to consider depends on external parameters
such as the voltage and temperature. In general, the rate equations have to be
solved numerically and the set Ω have to be determined self-consistently by means
of an iterative procedure which determines also the stability of the solution itself.
To obtain an analytical solution of Eq.(3.5) one has to restrict to situations in which
there are only few states involved into transport.
As a final remark, we stress that the rate equations are to be understood as
semiclassical equations because we assumed that in between tunneling events the
state if the system can be described by classical probabilities Pα . A rigorous derivation of Eq.(3.5) can be achieved through the density-matrix formalism [124, 125].
Its starting point is the formulation of the von-Neumann equation for the reduced
density matrix of the island. In general, this is a complicated integro-differential
equation. The crucial simplification arises from the assumption that the electronic
relaxation in the leads is fast compared to the tunneling dynamics, i.e kB T . In
this case, the Born-Markov approximation can be employed and the equation of
motion becomes an ordinary differential equation, the master equation for the reduced density matrix. Assuming Γ ∆, coherences between different states |αi are
negligible, and the master equation reduces to the rate equation. This represents
the evolution of diagonal part of the reduced density matrix. 1
1 In literature, the distinction between the terms “master equation” and “rate equation” is
slightly inconsistent. The rate equation it is often called Pauli master equation [125, 126], or even
simply master equation [127]. Following Ref. [124], we refer to the equation of motion of the
reduced density matrix as master equation and to its to diagonal part as rate equation.
The sequential tunneling regime
3.2.1
30
Matrix formalism
It is convenient to re-arrange the occupation probabilities Pα (t) into a vector P(t)
and to express the rate equations Eq.(3.5) in terms of a matrix equation of the form
d
P(t) = M P(t),
dt
(3.6)
where M is coefficient matrix with elements
Mαβ = Γβ→α − δα,β Γout
β ,
Γout
β =
X
Γβ→α .
(3.7)
α
From Eq.(3.6) one can immediately see that the rate equations are formally solved
by
P(t) = eMt P(0).
(3.8)
where the vector P(0) is determined by the boundary conditions at t = 0. Interpreting this expression in terms of Bayes theorem, one sees that the “evolution”
operator eMt determines the conditional probability to find the system in some
state at time t given a certain initial condition
[ eMt ]αβ = P (α, t|β, 0).
(3.9)
A direct consequence of the definition Eq.(3.7) is
X
Mαβ = 0
∀ β.
(3.10)
α
This sum rule, which ensures the conservation of probability (i.e. the sum of Pα (t)
is independent of time), in turn implies that the matrix M is singular and therefore
it has a zero eigenvalue. In the following we will always assume the zero eigenvalue
to be non degenerate. This implies that there are no metastable states in which
the system can get stuck, which is a reasonable hypothesis for many physical systems. In this case, the corresponding eigenvector P(st) gives the steady probability
distribution.
M P(st) = 0,
(3.11)
3.3
Current and shot noise
Although only the net transition rate Γα→β enter in the rate equation, the current
depends on whether an electron moves to the right or to the left. As an example, if
we fix the sign of the current to be positive for electron transfer from left to right,
then a tunnel event in which an electron enter into the dot through the first junction
gives a positive contribution to the current while it gives a negative contribution if
it comes across the second. To take into account the difference in sign for the two
process, we introduce two new matrices, I (i) (i = 1, 2), which contains the rates for
tunneling across the first and second barrier with the appropriate sign

(i)

if Nα = Nβ + 1
 Γβ→α
(i)
(i)
Iαβ = (−1)i+1
(3.12)
−Γ
if
Nα = N β − 1
β→α


0
otherwise
With this definition, the tunnel current across each junction can be written as
X (i)
hI (i) (t)i = eTr I (i) P(t) ≡ e
Iαβ Pβ (t),
(3.13)
αβ
31
3.3 Current and shot noise
where the brackets h·i denotes the statistical average over the states of the system
and the “trace” of a vector is defined as the sum over its components. In particular,
the stationary current is given by
hI (i) i = eTr I (i) P(st) .
(3.14)
Due to charge conservation, the average current across the two junctions must be
equal in the stationary case, hI (1) i = hI (2) i = hIi.
The same formalism can be applied to calculate the current noise
ij
S (ω) = 2
Z
+∞
−∞
dt eiωt h∆I (i) (t)∆I (j) (0)i.
(3.15)
Following Ref. [93], we obtain the following general expression for the correlation
function of current fluctuation
h∆I (i) (t)∆I (j) (0)i = e2 Cxij (t) + Cxji (−t) + δij Cai ,
(3.16)
where the first two terms contains the correlation between two different tunneling
events
Cxij (t) = θ(t)Tr δI (i) eMt δI (j) P(st)
with δI (i) = I (i) − hI (i) i/e
(3.17)
and the third is the self-correlation term
Cai = δ(t)Tr |I (i) |P(st) ,
(3.18)
σ(t) = θ(t) eMt
(3.19)
(a detailed derivation of the previous expression is given in Appendix B). To evaluate the Fourier transform of Eq.(3.16), it is convenient to introduce the operator
which is a sort of retarded Green’s function for the rate equation. In Fourier space,
the equation of motion for the operator σ(ω) becomes a simple algebraic equation
−iω σ(ω) = M σ(ω) + 1, which has the solution
σ(ω) = −
1
.
M + iω1
(3.20)
The current noise can now be readily evaluated. By substituting Eq.(3.16) into
Eq.(3.15) and taking Eq.(3.20) into account we come to the following result:
with
S ij (ω) = 2 e2 sijx (ω) + sjix (−ω) + δij Cai ,
(3.21)
sijx (ω) = Tr δI (i) σ(ω)δI (j) P(st)
(3.22)
Eq.(3.21) represents the general expression for the current noise in the rate-equation
formalism. At finite frequency, this expression is very convenient for numerical
calculations as the inverse of the matrix M + iω1 can be easily evaluated with
standard numerical tools [128].. However, some care has to be taken in the zero
frequency limit.
The sequential tunneling regime
3.3.1
32
The zero frequency limit
As the matrix M is singular, the operator σ(ω) does not exist for ω = 0. Nevertheless, it can be shown that the quantity
sijx (ω = 0) = −Tr δI (i) M−1 δI (j) P(st) ,
(3.23)
is well defined and that it can be evaluated by means of standard numerical techniques.
To show this, let us first introduce a little bit of technical notation. Let R(M) be
the range of M and N (M) its kernel, i.e.
R(M) = {v ∈ Rn : v = M u for some u ∈ Rn },
N (M) = {v ∈ Rn : M v = 0},
(3.24)
(3.25)
where n is the dimension of the phase space of the island. Since we assumed the
existence of a unique stationary solution P(st) , M is a matrix of rank n − 1 and
N (M) = {αP(st) , α ∈ R}.
Even if the operator M is not invertible, the quantity M−1 v exist if v is a
vector in the range of M, and correspond to the preimage of v under M,
v ∈ R(M) ⇒ M−1 v = {u ∈ Rn : M u = v}.
(3.26)
This set can always be written as M−1 v = y + N (M), where y is an arbitrary
solution of the linear system My = v.
It is easy to see that the range of M is given by the subset of traceless vectors.
Infact, because of sum rule Eq.(3.10), each vector in R(M) is traceless, hence
R(M) ⊆ Ω ≡ {v ∈ Rn : Tr[v] = 0}. The subset Ω has dimension n − 1, being
defined by a single condition. But, since rank(M) = n−1, this is also the dimension
of R(M) and therefore it must be R(M) = Ω.
Let’s now go back to Eq.(3.23). As δI (j) P(st) is by construction a traceless
vector, the quantity M−1 δI (j) P(st) is well defined and is given by M−1 δI (j) P(st) =
y + αP(st) , where α ∈ R and y is a solution of the linear system My = δI (j) P(st) .
This system can be easily solved by means of standard techniques (e.g. singular
value decomposition). Therefore we can write Eq.(3.23) as
sijx (ω = 0) = −Tr δI (i) (y + αP(st) ) = −Tr δI (i) y ,
(3.27)
from which we see that sijx (ω = 0) is also uniquely defined.
To summarize, the zero-frequency current noise in the rate equation formalism is
given by the following expression
S i,j = 2e2 δi,j Tr |I (j) |P(st)
(3.28)
−2e2 Tr δI (i) M−1 δI (j) P(st) + Tr δI (j) M−1 δI (i) P(st) ,
given that the symbol M−1 is not be misinterpreted as the inverse of the (singular)
M.
3.3.2
The two states regime
The electron transport involving only the lowest energy states of a quantum dot
have been deeply studied by several authors (see, for instance [93,94]). Nevertheless,
for later reference we introduce the discussion of the shot noise with the two-state
process, since it can be easily solved analytically and since, it provides a reasonable
33
3.3 Current and shot noise
approximation for many physical systems at low energies. For definiteness, let us
call |0i and |1i the two states involved into transport and ε their energy difference,
i.e ε = E1 − E0 . The rate matrix M is then simply given by
−Γ0→1
Γ1→0
M=
(3.29)
Γ0→1 −Γ1→0
(1)
(2)
(1)
(2)
where Γ0→1 = Γ0→1 + Γ0→1 and Γ1→0 = Γ1→0 + Γ1→0 are the total rates for the
transitions |0i → |1i and |1i → |0i respectively. According to Eq.(3.12), “current
matrices” I (1) and I (2) are defined as
#
#
"
"
(2)
(1)
0
Γ
0
−Γ
1→0
1→0
.
(3.30)
,
I (2) =
I (1) =
(2)
(1)
−Γ0→1
0
Γ0→1
0
The stationary current can be straightforwardly evaluated by employing Eq.(3.14),
yielding
(1)
(2)
e Γ0→1 Γ1→0 (1 − e−βeV )
hIi = (1)
.
(3.31)
(2)
Γ0→1 (1 + eβ∆U (1) ) + Γ1→0 (1 + e−β∆U (2) )
Here we have made use of the detailed balance relation
(i)
(i)
(i)
Γ0→1 = e−β∆U Γ1→0
(3.32)
where ∆U (i) = ε − µi is the change in the free energy associated with the tunneling
into the dot across the ith-junction. At zero temperature, transitions can occur only
in “one direction”, which is fixed by the external bias voltage. For definiteness, in
the following we assume eV > 0 so that the only non vanishing rates at T = 0 are
(2)
(1)
Γ0→1 and Γ1→0 . In this case Eq.(3.31) reduces to
(1)
hIiT =0 =
(2)
e Γ0→1 Γ1→0
(1)
(2)
Γ0→1 + Γ1→0
.
(3.33)
Analogously, taking into account Eq.(3.28), one can evaluate the zero frequency
current noise and the Fano factor, obtaining
(1)
F =1−
(2)
2Γ0→1 Γ1→0
(1)
(2)
[Γ0→1 (1 + eβ∆U (1) ) + Γ1→0 (1 + e−β∆U (2) )]2
.
(3.34)
From this expression one can easily convince oneself that in the two state regime
the noise is always sub-Poissonian and bigger than SP oisson /2, i.e.
1/2 ≤ F ≤ 1.
(3.35)
At zero temperature, the expression for the Fano factor becomes particularly plain
(1)
FT =0 =
(2)
(Γ0→1 )2 + (Γ1→0 )2
(1)
(2)
[Γ0→1 + Γ1→0 ]2
.
(3.36)
We wish to emphasize that although Eq.(3.34) and Eq.(3.36) look simple, they
(1)
(2)
contain a great deal of structure because the rates Γ0→1 and Γ1→0 depend on
voltage (see e.g.. Ref. [93, 114]). It is convenient to re-express Eq.(3.36) as
FT =0 =
1
(1 + α2 )
2
(3.37)
The sequential tunneling regime
34
where α is a function which takes into account the asymmetry between the tunneling
rates
(1)
(2)
Γ0→1 − Γ1→0
α ≡ (1)
.
(3.38)
(2)
Γ0→1 + Γ1→0
Equation (3.37) has a clear physical interpretation: in the case of strongly asymmetric barriers α = ±1, the Fano factor reaches the full Poissonian limit F = 1
as transport is essentially dominated by a single barrier (the more opaque). Viceversa, when the two tunneling rates become comparable (α ≈ 0), tunneling events
at different barriers become correlated, leading to a suppression of the shot noise.
(1)
(2)
The maximal suppression occur for perfectly symmetric junctions Γ0→1 = Γ1→0 .
It is important to stress that α is a function both of the intrinsic transparency of
the barriers and of the applied voltages, so that it is generally possible to tune
experimentally α between -1 and 1. Indeed, very recently the shot noise of a semiconductor quantum dot have been measured as a function of α, showing an excellent
agreement between experimental data and the prediction of Eq.(3.37), see Fig. 3.2
Figure 3.2: Fano factor of a semiconductor quantum dot as a function of the asymmetry of the tunneling rates α. The experimental data show an excellent agreement
with the theoretical prediction (solid line). After [99]
Chapter 4
Shot noise in a 1D ring
In this chapter we want to study the shot noise of a one-dimensional quantum ring
[129]. Possible experimental realizations of this system include [130, 131]. The 1D
ring geometry allows to study the interplay between Fermi statistics, interactions,
and Coulomb Blockade. Signatures of this interplay , which were for instance found
in the behavior of the linear conductance [115], will reflect in the richness of the
interaction-induced noise features analyzed below. In section 4.2 we give a brief
introduction to the Luttinger liquid model. Subsequently, we describe the system
under investigation in section 4.3 and in 4.4 we generalize the technique developed
in Chap.3 to evaluate the current noise to the case of the magnetization current.
In section 4.5.1 we present results in the absence of external magnetic flux, while
in section 4.5.2 the effects of the latter is briefly investigated. In section 4.5.3 we
present an interpretation of the results in terms of a Monte Carlo simulation of the
transport dynamics in the system.
4.1
A one dimensional quantum ring
The system we want to study is schematically depicted in Fig. 4.1. It is composed
+
g
g`
g`
Φ
R
−
Figure 4.1: Schematic description of the one-dimensional ring under investigation.
The ring has radius R and the one-dimensional leads are characterized by the interaction parameter g` , while the ring by the parameter g.
of a one-dimensional (1D) quantum ring of radius R, connected by tunneling barriers to two external leads, which we assume to be also 1D. As we have briefly
discussed in section 1.3.4, one-dimensional electronic systems are expected to show
35
Shot noise in a 1D ring
36
unique transport features [64]. In fact, unlike in two or three dimensions, where
the Coulomb interaction affects the transport properties only perturbatively, in 1D
it completely modifies the ground state from its well-known Fermi liquid form, and
the Fermi surface is qualitatively altered even for weak interactions. Today, it is
well established theoretically that the low-temperature transport properties of interacting 1D-systems are described in terms of a Luttinger liquid (LL) rather than
a Fermi liquid [62, 132].
Fermi vs Luttinger liquids
In three dimensions interacting Fermions are well described by the Landau’s theory
for the Fermi liquid [133]. The key observation is that macroscopic properties
involve only excitations of the system on energy scales small compared to the Fermi
energy so that the state of the system can be specified in terms of its ground
state, i.e. its Fermi surface, and its low-lying elementary excitations, a rarefied
gas of quasiparticles. These quasiparticles are a superpositions of single particle
excitations and particle-hole pairs which evolve continuously out of the states of
the free Fermi gas when the interactions are switched on. They are in one-toone correspondence with the bare particles (adiabatic continuity), possess the same
quantum numbers and obey the same statistics, but their dynamical properties are
renormalized by the interaction. Moreover, beacuse of the interaction they have a
finite lifetime τ . However at T = 0, as the Fermi surface is approached τ diverges
according to τ ∼ (E − EF )−2 , so that the quasiparticles are robust aganist small
displacements away from EF . This scenario emerges because at low temperature
the phase space for scattering particles is severly restricted by Fermi statistics and
only a fraction T /TF 1 participate in the scattering process.
The concept of quasiparticles breaks down in one dimension. In fact, in 1D the
lifetime of the quasiparticles becomes strictly zero τ = 0, undermining the very same
idea of adiabatic inclusion of the interaction. Starting with the early work of Mattis
and Lieb [132], it has become clear that in one dimension the elementary excitations
are collective density fluctuations. The correlations between these excitations is
anomalous and show up as interaction-dependent non universal power-laws in many
physical quantities, where those of ordinary metals are characterized by universal
(interaction independent) powers.
These properties are generic for one-dimensional fermionic systems, but are particularly prominent in the Luttinger liquid model. This is a theory based on the
exact diagonalization of the interacting Hamiltonian and on the bosonization of
the fermionic operators. All the correlation functions of the LL can be computed
exactly, so that one has direct access to all the physical properties of interest.
4.2
The Luttinger liquid model
In this section we follow closely Ref. [11, 134] in giving a brief introduction to the
Luttinger liquid model. For sake of simplicity, we consider only a spinless model,
referring to Ref. [11] for a exhaustive analysis of the spin degrees of freedom.
4.2.1
Luttinger liquid with periodic boundary conditions
In the absence of interactions, the Hamiltonian of a system of spinless fermions
confined on a ring of radius R is simply
H0 =
X
k
εk c†k ck ,
εk = k 2 /2m
(4.1)
37
4.2 The Luttinger liquid model
where k = n/R (n ∈ Z) because of the periodic boundary conditions. If we include
the electronic interaction, the total Hamiltonian becomes H = H0 + Hint , where
Hint =
1 X
V (q)c†k1 +q c†k2 −q ck2 ck1
2L
(4.2)
k1 ,k2 ,q
represents the most generic translationally invariant interaction among the electrons. Under certain assumptions (see below), the interaction term can be treated
exactly, and the total Hamiltonian H can be diagonalized in terms of density operators which obey a boson statistics. This approach, which form the basis of the
Luttinger liquid description, relays on three fundamental approximations:
(i) The spectrum of the free electrons is linearized around the Fermi energy,
forming two distinct linear branches which we call “right-movers” and “leftmovers”.
(ii) The two branches are both extended to infinity, i.e. −∞ < k < ∞. This
introduce an infinite number of states with negative energy which, in order to
obtain physically meaningful results, needs to be all occupied. The presence
of these unphysical states is not expected to affect the low-energy physics of
the model (|ω| EF , |q| kF ).
(iii) Solely forward scattering is considered, i.e. electrons cannot exchange branch
as a consequence of the interaction.
With these assumptions, the kinetic energy and the interaction term become:
X
H0 =
vF (rk − kF ) : c†r,k cr,k :
(4.3)
k,r=±
and
Hint =
1 X
V (q){: ρr (q)ρr (−q) : + : ρr (q)ρ−r (−q) :}
4πR q,r=±
(4.4)
respectively. Here, c+,k (c†+,k ) and c−,k (c†+,k ) are the annihilation (creation) operators for the right- and left-moving branch, respectively (ck ≈ c+,k + c−,k ), and
X †
ρr (q) =
cr,k+q cr,k
(4.5)
k
is the density fluctuation operator for the rth-branch (r = ±). The “comma” denote
normal ordering, i.e. : A :≡ A − hAi0 , being hAi0 the expectation value with respect
to the ground state of H0 . The normal ordering
convention is necessary to avoid
P
reference to the total number of particles k hc†r,k cr,k i which is ill-defined. Acting
on the ground state, the ρr (q) behave either as creation or annihilation operator,
depending on the sign of q, i.e. ρr (−rq)|0i = 0, for q > 0. Moreover, the states
ρr (rq)|0i and H0 obey a simple commutation relation
[H0 , ρr (q)] = vF rq ρr (q).
(4.6)
which means that the states ρs (q)|0i are eigenstates of the linearized Hamiltonian
Eq.(4.3) with energy vF q.
Bosonization
There are three important steps in achieving a complete solution of the model:
(i) the realization that due to the infinite dispersion, the ρr (q) obey exact boson
commutation relation [132];
Shot noise in a 1D ring
38
(ii) a representation of the free Hamiltonian Eq(4.3) as a bilinear in these boson
operator [132];
(iii) the explicit√construction
of a boson representation for the fermion operators
P
ψr (x) = 1/ 2πR k cr,k eikx [62].
The fact that the density operators obey Bose-like commutation relations can be
easily shown by writing the commutator in terms of normal ordered operators
X †
[cr,k+q cr,k+q0 − c†r,k+q−q0 cr,k ]
[ρr (q), ρr0 (−q)] = δr,r0
k
= δr,r0
X
k
: c†r,k+q cr,k+q0 − c†r,k+q−q0 cr,k :
+ δr,r0 δq,q0
X
k
[hc†r,k+q cr,k+q i0
−
(4.7)
hc†r,k cr,k i0 ]
= −δr,r0 δq,q0 rqR
One can safely change the summation variable in the second line of Eq.(4.7) because
the operators are normal ordered; the two terms add up to zero, leaving only the
contribution of the third line.
This result, combined with Eq.(4.6), allows to represent the free fermion Hamiltonian H0 , as a bilinear operator of the bosons ρr (q). In fact, it can be shown that
Eq.(4.3) is completely equivalent to a the following form [132],
H0 =
vF X
vF
: ρr (q)ρr (−q) : + [N 2 + J 2 ]
2R
4R
(4.8)
r,q6=0
where N ≡ N+ + N− and J = N+ − N −, and Nr ≡ ρr (q = 0) is the number of
particles added to the ground state in the rth-branch. In other words, N measures
the total number of excess electrons and J the left-right asymmetry of the fermions
added to the reference state (N = N0 and J = 0). Obviously, N and J obey the
selection rule (−1)N = (−1J ). The first term of Eq.(4.8) represents the energy of
density fluctuations at fixed electron number, while the second term stands for the
energy of N particles added to the system and put into the lowest available states
above the Fermi energy. One can easily convince oneself that Eq.(4.3) and Eq.(4.8)
have the same spectrum. That the multiplicities of the levels are also equal can be
proved by calculating the gran partition function both in the fermion Eq.(4.3) and
in the boson Eq.(4.8) representation. Thus the fermionic and the bosonic spaces
are identical.
A completely satisfactory boson solution of the non interacting problem also
require an explicit representation of the fermion operators ψr (x) in terms of the
bosons ρr (q). A detailed derivation of this procedure is given in literature [11, 62,
135]. while here we will merely summarize its main ingredients. To this end, we
introduce the field operators
φ(x)
Π(x)
= −
=
i X 1 −α|q|/2−iqx x
e
ρ+ (q) + ρ− (q) − N
2R
q
2R
(4.9)
q6=0
J
1 X −α|q|/2−iqx e
ρ+ (q) − ρ− (q) +
2πR
2πR
(4.10)
q6=0
where α is a cutoff parameter which has to be set to zero at the end of any calculation. The fields φ and Π obey canonical boson commutation relations
φ(x), Π(x0 ) = iδ(x − x0 )
(4.11)
39
4.2 The Luttinger liquid model
The single electron operator can be expressed in terms of the fields φ and Π as [62]
eir(kF −1/2R)x †
√
Ur exp{−i[rφ(x) − Θ(x)]}
α→0
2πα
ψr (x) = lim
(4.12)
where Ur† is an operator which decrease the particle number in the rth-branch by
one unit and
Z x
i X 1 −α|q|/2−iqx
x
Θ(x) = π
Π(x0 )dx0 =
e
(4.13)
[ρ+ (q) − ρ− (q)] + J
2R
q
2R
−∞
q6=0
The structure of Eq.(4.12) may be intuitively justified by noting that the field
operator φ is related to the local particle density via
∂φ/∂x = −πρ(x)
(4.14)
where ρ0 = kF /π is the average particle density in the ground state. Therefore,
introducing a particle at site x creates a kink of amplitude π in the field φ, i.e.
at points on the left of x, φ has to be shifted by π. This shift is operated
by the
Rx
exponential of the conjugated momentum operator Π, i.e. by exp[i −∞ dx0 Π(x0 )].
This operator commutes with itself hence, to construct a fermionic field, one has
to restore anticommutation by multiplying it with exp[irφ(x)]. Finally, the unitary
ladder operators Ur , which increase by unity the number of electrons, are necessary
because the bosons fields ρr (q 6= 0) conserve the total number of particles. Moreover
these operators, which take the name of Klein factors, assure the anticommutation
of fermionic fields in different branches [135].
Diagonalization
While the fermionic and bosonic representations are true alternatives for the non
interacting problem, the success of bosonization is related to the fact that the
bosonic one become more “natural” once interaction are introduced. Under the
assumption of considering only forward scattering, the interaction Hamiltonian can
be also written as a bilinear operator of the ρr (q)’s, see Eq.(4.4), so that the total
Hamiltonian reads
1 X
V (q) π X : ρr (q)ρr (q) : +
vF +
V (q) : ρr (q)ρ−r (−q) :
H =
2R
2πvF
4πR
r,q6=0
r,q6=0
i
vF h
V (0) 2
N + J2 .
+
1+
(4.15)
4R
πvF
In this bosonic form, the problem of a system of interacting electrons can be solved
exactly by means of a canonical transformation. In fact, Eq.(4.15) can be diagonalized by a Bogoliubov transformation [132]
H̃ = eiS He−iS
S=
1 X ξ(q)
[ρ+ (q)ρ− (−q) − ρ− (q)ρ+ (−q)]
R p>0 q
(4.16)
where the angles ξ(q) are defined by the condition
−1/2
V (q)
.
g(q) ≡ e2ξ(q) = 1 +
πvF
(4.17)
The density operators transforms as
ρ̃r (q) = eiS ρr (q)e−iS = ρr (q) cosh ξ(q) + ρ−r (q) sinh ξ(q)
(4.18)
Shot noise in a 1D ring
40
and the diagonal form of the Hamiltonian is
H̃ =
1 1 X
vN N 2 + v J J 2
v(q) : ρr (q)ρr (−q) : +
2R
4R
(4.19)
q6=0
with
v(q) =
vF
g(q)
vN =
vF
= 0)
g 2 (q
vJ = v F
(4.20)
For repulsive interactions, it is g(q) < 1 while for attractive interactions g(q) >
1. Importantly, there are three different velocities in the problem: v(q) is the
(renormalized) sound velocity and governs the bosonic excitations, vN is related to
the charge excitations and measures the shift in the chemical potential upon varying
the Fermi wave vector δµ = vN δkF and, finally, vJ measures the energy required
to create a persistent current into the ring. Because of the interaction they are all
different.
Under the Bogoliubov transformation, the fermionic field transform as, ψ̃r (x) ∝
exp[i(Θ̃(x) − rφ̃(x)], where
p
i X g(q) −α|q|/2−iqx x
φ̃(x) = −
e
ρ+ (q) + ρ− (q) − N
(4.21)
2R
q
2R
q6=0
Θ̃(x)
=
x
1
i X
p
e−α|q|/2−iqx [ρ+ (q) − ρ− (q)] + J
2R
2R
q g(q)
(4.22)
q6=0
(the zero modes Nr are not changed by the canonical transformation).
It is convenient to introduce bosonic operators
s
1 X
θ(rq)ρr (q)
a†q =
R|q| r
(4.23)
which satisfy canonical commutation relations aq , a†q0 = δq,q0 . With this definitions
the Hamiltonian of the Luttinger model Eq.(4.19) becomes
H̃ =
X
q6=0
ω(q)a†q aq +
1 vN N 2 + v J J 2
4R
(4.24)
where we have introduced ω(q) = vF |q|/g(q). As long as one is interested only in the
low energy properties of the model, q ≈ 0, a reasonable assumption is to linearize the
excitation spectrum ω(q) → vF |q|/g, where g ≡ g(q = 0) is the effective interaction
parameter. From now on we will consider only this long wavelength limit and we
will treat the interaction as q-independent. In terms of the operators aq , a†q the
electronic field operator become
eirkF x † irNr x/R i$r (x)
ψr (x) = lim √
Ur e
e
α→0
2πα
(4.25)
where
$r (x) =
1
√
iX 1
p
e−iqx a†q − eiqx aq √ + r sgn(q) g .
2
g
R|q|
(4.26)
q6=0
Persisten currents
The Luttinger liquid model description can be used straightforwardly to obtain
the current induced in a strictly one-dimensional ring threaded by a magnetic flux
41
4.2 The Luttinger liquid model
Φ [136]. In the one-dimensional geometry, the vector field can be removed entirely
from the Hamiltonian via a gauge transformation which leads to the boundary
condition
ψ(2πR) = e−i2πΦ/Φ0 ψ(0)
(4.27)
for the fermion field operator, where Φ0 = hc/e is the flux quantum. This is achieved
by replacing
Φ
(4.28)
Π(x) → Π(x) −
πRΦ0
in the bosonization formula Eq.(4.12). The J-dependent part of the Hamiltonian
then becomes
Φ 2
vJ J −2
(4.29)
HJ =
4R
Φ0
giving rise to a number current
j=
vJ Φ
Φ0 ∂E
=
J −2
2π ∂Φ
2πR
Φ0
(4.30)
At equilibrium J is chosen so as to minimize the energy. Given that at constant particle number J can only change by two units, one can easily see that the equilibrium
(persistent) current has a periodicity Φ0 and reach the maximum value vJ /2πR at
Φ = Φ0 /2, giving rise to the familiar Aharanov-Bohm oscillations for the current as
a function of the flux.
4.2.2
Luttinger liquid with open boundary conditions
Up to now we have considered the case of a Luttinger liquid with periodic boundary
conditions. Even if this prescription fits perfectly with the idea of a quantum ring,
it is not suited to describe a “segment” of Luttinger liquid, for which open boundary
conditions have to be employed [137–140].
Non interacting electrons
Consider a system of electrons confined in the finite region [0, L] by infinite walls.
This condition imposes the electronic field to vanish at the boundary, i.e
ψ(0) = ψ(L) = 0.
(4.31)
p
P
In this case, the Fourier transform of ψ(x) takes the form ψ(x) = 2/L k sin(kx)ck ,
with k, = πn/L (n ∈ N) and the Fermi surface consist of a single point k = kF .
Close to this point the fermion field can be expanded as
ψ(x) ≈ eikF x ψ+ (x) + e−ikF x ψ− (x)
where
−i r X irqx
e
ckF +q .
ψr (x) = √
2L q>0
(4.32)
(4.33)
Unlike the case of periodic boundary conditions, the left and right moving fields are
not independent but satisfy the condition
ψ− (x) = −ψ+ (−x)
(4.34)
so that one can actually work with one of the two fields only, say ψ+ (x). The
annihilment condition ψ(L) = 0 implies that the right moving field ψ+ (x) have to
satisfy the relation ψ+ (L) = ψ+ (−L). Therefore we can regard ψ+ as defined on
the whole x axes but periodic with period 2L, i.e. ψ+ (x) = ψ+ (x + 2L).
Shot noise in a 1D ring
42
The linearized Hamiltonian for the free electrons can then be re-written as
Z L
Z L
†
†
†
H0 =
dxvF (ψ+ i∂x ψ+ − ψ− i∂x ψ− ) =
dxvF ψ+
i∂x ψ+ .
(4.35)
0
−L
The fermion field ψR (x) can be bosonized in the same spirit as in the case of periodic boundary conditions. However, different boundary conditions yield different
relations between the electron field operator and the boson operators [135,137–139].
In the present case, the appropriate expression for the field ψ+ (x) is
ψ+ (x) = lim √
α→0
1
† iπN x/L iΛ(x)
U+
e
e
.
2πα
(4.36)
Here, N is the number of extra electrons with respect to the ground state and the
phase fiel Λ(x) is given by the expression
r
X
π −αq/2 iqx
e
e aq + e−iqx a†q
(4.37)
Λ(x) =
qL
q>0
where aq are canonical Bose operators, q = πn/L and α−1 is a momentum cut-off
The phase field Λ(x) satisfies periodic boundary conditions Λ(x) = Λ(x + 2L) and
is related to the density of right moving electrons by
ρ+ (x) =
1
N
+
∂x Λ(x).
2L 2π
(4.38)
The corresponding expressions for ψ− (x) and ρ− (x) can be readily obtained by
employing ψ− (x) = −ψ+ (−x) and ρ− (x) = ρ+ (−x).
The bosonized form of the kinetic energy is
Z L
X
πvF 2
q a†q aq +
dxρ+ (x)ρ+ (x) = vF
H0 = πvF
N
(4.39)
2L
−L
q>0
Interaction effects
Now we turn to consider the electronic interaction
Z
1
dxdx0 ψ † (x)ψ † (x0 )V (x − x0 )ψ(x0 )ψ(x).
Hint =
2
(4.40)
As in the case of periodic boundary conditions, we restrict only to forward scattering
and, additionally, we assume short range interactions V (x − x0 ) = V0 δ(x − x0 ). In
this case the interaction term simply becomes
Z L
Z
X
V0 L
dxρ+ (x)ρ− (x).
(4.41)
Hint =
dx
ρr (x)ρr (x) + V0
2 0
0
r=±
In order to make use of the above bosonization procedure, one has to express also
Hint in terms of the right moving field ψ+ alone, at the cost of introducing a non
local interaction term
Z L
Z L
dx ρ+ (x)ρ+ (−x).
(4.42)
dx ρ+ (x)ρ+ (x) + V0
Hint = V0
−L
−L
As Hint is quadratic in the electron densities, it takes a simple form in terms of
bosonic operators. By employing Eq.(4.38) we come to the following expression
i V
1
V0 X h †
0
q aq aq − (aq aq + a†q a†q ) +
N2
(4.43)
Hint =
2π q>0
2
4L
43
4.3 Model and Hamiltonian
The total Hamiltonian H = H0 + Hint can be diagonalized in a standard way by
means of the Bogoliubov rotation
aq → ãq = cosh(ξ) aq − sinh(ξ) a†q ,
with
−1/2
V0
e2ξ = 1 +
≡g
πvF
(4.44)
Note that whereas the rotation angle ξ is defined as in the case of periodic boundary
conditions, there is an important difference in sign.
The transformed Hamiltonian reads
X
πvN 2
N
(4.45)
H̃ =
ω(q)a†q aq +
2L
q>0
with ω(q) = vF q/g and vN = vF /g 2 . Comparing this expression with Eq.(4.24)
one sees that there are two important differences: first, the sum over momenta is
restricted to positive values only, second, there is a single zero-mode term, i.e. the
topologicalPcurrent J is not conserved anylonger. Both are a consequence of the
condition r ψr (rx) = 0.
Under the Bogoliubov transformation the field ψ+ (x) Eq.(4.36) transforms as
ψ+ (x) → ψ̃+ (x) = lim √
α→0
with
Λ̃(x) =
4.3
1
† iπN x/L
U+
e
exp{iΛ̃(x)}
2πα
√
i
1 1 1 h√
g + √ Λ(x) −
g − √ Λ(−x) .
2
g
g
(4.46)
(4.47)
Model and Hamiltonian
We can now consider in more detail the system represented in Fig. 4.1. We model the
system as two semi-infinite LL leads weakly coupled to a one-dimensional quantum
ring. As discussed above, we have to employ periodic boundary conditions and
open boundary conditions for the ring and the leads, respectively. The wires are
weakly coupled to the ring so that single electron tunneling is the dominant charge
transport mechanism.
The total Hamiltonian of the system is then given by the sum of the bosonized LL
Hamiltonians describing the isolated ring and leads and the tunneling Hamiltonian,
which takes into account single-electron hops through the two junctions
H = Hring + Ht + Hleads ,
Hleads = H1 + H2
(4.48)
Using the bosonization technique described in the previous section, the LL
Hamiltonian can be expressed in terms of creation and annihilation operators for
collective excitations. For the semi-infinite leads it reads
Z ∞
Hi =
dq Ω(q)a†i (q)ai (q),
for i = 1, 2
(4.49)
0
where Ω(q) = vF /g` q is the continuous dispersion relation for the bosonic modes,
the operators a†i (q), a†i (q) creates and annihilate a plasmonic excitation in the i-th
lead, and g` is the Luttinger interaction parameter. The leads are adiabatically
connected to reservoirs which keep them in internal equilibrium. The difference of
their chemical potential is controlled by the external bias voltage µ2 − µ1 = eV .
Specifically, we assume symmetric voltage drop on the tunneling barriers so that it
is µi = (−1)i eV /2.
Shot noise in a 1D ring
44
While the zero modes play no role for the semi-infinite leads, they need to be
accounted for in the case of the ring, which yields
Hring
2 X
EN
Φ
EJ
2
ω(q) a†q aq .
+
=
J −2
(N − Ng ) +
2
2
Φ0
(4.50)
q6=0
where N = N+ + N− represents the total number of excess charges in the ring and
J = N+ − N− represents the imbalance between right- and left-moving electrons.
Obviously, the zero modes are related by the condition N + J = even. Furthemore,
Eq.(4.50) takes into account the presence of an external magnetic flux Φ piercing
the ring and the extra charge eNg induced by an external gate capacitively coupled
to the ring. The charging energy EN represent the minimum energy cost to add an
excess electron into the system. A theoretical estimate for EN is provided directly
within the LL model (EN ∼ vF /2Rg 2), however as the charging energy is also
strongly influenced by the coupling to external circuit, long range interactions and
screening effects, in the following we regard EN as a free parameter which sets
the highest energy energy scale into the problem. The orbital addition energy is
EJ = vF /2R EN and it stand for the energy required to create a persistent
current into the ring. Finally, the last term in (4.50) is the contribution due the
bosonic modes, with a†q (aq ) the creation (annihilation) operator for an excitation
with energy ω(q) = v|q|, where q = n/R (n ∈ Z) because of the periodic boundary
conditions, and v = vF /g is the group velocity of the collective excitations in the
ring.
The tunneling events between the ring and the leads are described by the Hamiltonian
i
X h
ti ψ (i)† (xi )ψ (ring) (xi ) + h.c. ,
(4.51)
Ht =
i=1,2
where ψ (ring)† (xi ), ψ (ring) (xi ) are the electron creation and annihilation operators of
the ring, and ψ (i)† (xi ) and ψ (i) (xi ) are the corresponding fields near to the boundary
of the ith-leads.
4.4
The sequential tunneling regime
The eigenstates of the isolated ring are identified by the following set of quantum
numbers
|Si = |N, J, {lq }i
(4.52)
where N and J are respectively the total number and the left-right asymmetry of
the electrons added to the ground state, and {lq } is the set the occupation numbers
of the plasmonic excitations at different q (lq = 0, 1, 2, . . . ). In the following we will
often refer to J as the angular momentum of the ring, as it is intimately related to
the expectation value of the axial angular momentum.
At fixed flux and gate voltage, the total energy of the state |Si is given by
2
EN
Φ
EJ
2
U(S) =
J −2
+εq.
(N − Ng ) +
2
2
Φ0
(4.53)
Since the excitation spectra are linear in wave number, the energy P
of the plasmonic
modes depends only on the total number of excitation quanta q = n∈Z n ln , being
ε = v/R = vF /Rg the excitation energy. As we consider only repulsive interactions
in the ring, i.e. g < 1, we obtain the following “hierarchy” of energy scales
2EJ ≤ ε EN .
(4.54)
45
4.4.1
4.4 The sequential tunneling regime
Tunneling rates
The occupation probabilities of the quantum states of the ring change via electron
tunneling events across the two junctions. In the weak coupling regime, i.e. when
the level broadening due to tunneling is much smaller than temperature and level
spacing Γ kB T, EJ , the electron tunneling is the source of small perturbation to
the three isolated Luttinger liquids. In this regime, we calculate the transition rates
between two eigenstates of Hring to the lowest non-vanishing order in the tunneling
amplitudes. In this golden rule approximation, we integrate out the leads degrees of
freedom since they are assumed to be in thermal equilibrium, so that the transition
rates are given as a function of the state variables and the energies of the ring
only [69]
Z ∞
(i)
(i)
2
Γ|Si→|S 0 i = ti ΨS,S 0
dτ e−i∆U τ hψ (i)† (xi , τ )ψ (i) (xi , 0)ith
(4.55)
−∞
(i)
where the coefficient ΨS,S 0 stand for the matrix element
†
ΨS,S 0 = δ∆N,1|hS|ψ (ring) (xi )|S 0 i|2 + δ∆N,−1|hS|ψ (ring) (xi )|S 0 i|2
(4.56)
and ∆U (i) is the change in the Gibbs free energy associated with the tunneling
across the ith-barrier
∆U (i) = U(S 0 ) − U(S) − µi ∆N,
(4.57)
being ∆N = N 0 − N the difference between the number of the electrons in the ring
in the states |S 0 i and |S 0 i. The thermal averages are calculated with respect to
the uncoupled Hamiltonian Hi and ψ (i)† (xi , τ ) = exp(iHi τ )ψ (i)† (xi ) exp(−iHi τ ).
Working out the lead contribution, one comes to [69]
Z ∞
π
γ(E)
(4.58)
dτ e−iEτ hψ (i)† (xi , τ )ψ (i) (xi , 0)ith =
αωc
−∞
where ωc is an high energy cutoff and the function γ(E) is given by
γ(E) =
1 e−βE/2 1
βE 2 2π 1/g` −1
+i
Γ
2π Γ(1/g` )
2g`
2π
β ωc
(4.59)
where Γ(x) is the Euler Gamma function. The function γ(E) represent the local
tunneling density of states at the boundary of the leads. In the case of non interacting leads g` = 1, γ(E) simply reduces to the Fermi distribution function (remember
|Γ(1/2+iy)|2 = π/ cosh(πy)). At zero temperature, γ(E) is proportional to a power
law of energy
1/g` −1
1
|E|
(4.60)
lim γ(E) = θ(−E)
T →0
Γ(1/g` ) ωc
vanishing for positive energy gain, as required by energy conservation.
Let us now turn to the ring contribution. Up to now, we have regarded the bosonic
modes in the ring as infinitely stable states. Even if this assumption is perfectly
consistent with the plasmons being eigenstates of Hring , in realistic systems this
is usually not the case. In fact, the coupling of the system to the environment,
such as external circuit or background charge in the substrate, leads to relaxation
towards equilibrium. In the following we will assume fast relaxation of the plasmonic
excitations in the ring, implying that excitations created by one tunneling event
do not influence subsequent tunneling events. In other words, we assume that the
Shot noise in a 1D ring
46
tunneling rate is much smaller than the rate of relaxation of the bosonic excitations,
so that the plasmon relax between each tunneling event according to a thermal
distribution. Therefore the only dynamical variables are N and J. We can define
an effective tunneling rate as
X β X (i)
(i)
Γ|N,Ji→|N 0,J 0 i
W{lq }
Γ|N,J,{lq }i→|N 0 ,J 0 ,{l0 }i
(4.61)
q
{lq }
{l0q }
β
where W{l
is the thermal probability distribution with respect to Hring . Working
q}
out this sum, one comes to the final form for the transition rates for kB T < ε
(i)
Γ|N,Ji→|N 0 ,J 0 i
=
(i)
Γ0
∞
X
n=−∞
an γ(∆U + nε − µi ∆N )
(4.62)
where ∆U is the energy difference between the states |N 0 , J 0 i and |N, Ji
h h
Φ 1i
1i
+ EJ ∆J J − 2
+ .
∆U = EN ∆N (N − Ng ) +
2
Φ0
2
(4.63)
Note that considering Ht to the lowest order allows only for ∆N = N 0 − N = ±1
and ∆J = J 0 − J = ±1. The weights an comes from the discrete spectral density
of the ring and are given by [141, 142]
i
1
1 −βε h (0)
(0)
(0)
+
an = a(0)
g
+
e
a
+
a
−
2a
(4.64)
n
n
n+1
n−1
2
g
(0)
where aq
are the ring weights at zero-temperature
(g+1/g)/2 Γ ((g + 1/g)/2 + n)
−ε/ωc
θ(n) .
a(0)
n = 1−e
Γ ((g + 1/g)/2) n!
(4.65)
(i)
The costant (i.e. independent on the interaction and energy) prefactor Γ0 ∝ t2i
defines the bare tunneling rate through the ith-barrier. At low temperature the
tunneling rate Eq.(4.62) exhibits a step-like structure as a function of the external
bias voltage: a new step open each time µi ∆N ≥ ∆U + nε. In this case, an
electron entering (or leaving) the ring can spend some of its excess energy to excite
n excitation quanta. The collective modes will then quickly relax before the next
tunneling event takes place. The height of each step depends on n and on the
interaction parameter g, see Eq.(4.65). In the absence of interaction in the ring
(g = 1) the energy of the plasmon is exactly twice the orbital addition energy
(0)
ε = 2EJ , and the weights Eq.(4.65) become independent of n, i.e an ≈ (ε/ωc ).
4.4.2
Charge and orbital tunnel currents
Given the transition rates, we can build up the rate equations for the probabilities
PN J (t) of occupation of the ring states |N, Ji
X d
PN 0 J 0 (t)Γ|N 0 ,J 0 i→|N,Ji − PN J (t)Γ|N,Ji→|N 0 ,J 0 i
PN J (t) =
dt
0
0
(4.66)
N ,J
P
(i)
where Γ|N,Ji→|N 0,J 0 i =
i=1,2 Γ|N,Ji→|N 0 ,J 0 i is the total rate for the transition
|N, Ji → |N 0 , J 0 i. Introducing a compact labeling for the states |N, Ji ≡ α, we can
re-write the rate equation Eq.( 4.66) as
∂t P(t) = MP(t)
(4.67)
47
4.4 The sequential tunneling regime
where P(t) a vector representing the occupation probabilities of the various states
of the ring and the transition matrix M is


if Nα = Nβ ± 1 and Jα = Jβ ± 1
Γβ→α
P
(4.68)
Mαβ = − α6=β Γβ→α if Nα = Nβ and Jα = Jβ


0
elsewhere
where we have explicitly taken into account the selection rule ∆N = ±1 and ∆J =
±1, which comes from treating Ht to the lowest order.
Let us look more closely at the different tunneling events which can occur in
our system. Assume that the ring is initially in the state |N, Ji and consider a
transition in which an electron tunnels into the ring, N → N + 1 . This electron
can either join the right- or the left-moving branch of the linearized spectrum of
the ring, giving a positive or a negative contribution to the angular momentum J,
respectively. Analogously, an electron leaving the ring, N → N − 1, can either
increase or decrease J depending on the branch where it comes from. Therefore,
one is left with four classes of events, as summarized in Table 4.1.
∆N
+1
+1
-1
-1
∆J
+1
-1
+1
-1
Type of event
In +
In −
Out +
Out −
Table 4.1: Table of the four classes of events, determined by the possible tunneling
processes. The notation “In +” stands for a electron entering into the ring and
joining the right moving branch. Analogously, “Out −” represent an electron leaving
the ring from the left-moving branch, and similarly for all the other types of events.
Each tunneling event can take place at both barriers.
Importantly, each tunneling event can take place at both barriers. We can then
generalize the definitions of current matrices given in Chapter 3 to take into account
the variations of J. To this end we introduce the matrix I (i),N , with elements
 (i)

if Nα = Nβ + 1
Γβ→α
(i),N
(i)
i+1
Iαβ = (−1)
(4.69)
−Γβ→α if Nα = Nβ − 1 .


0
otherwise
which is related to the charge tunneling current through the ith-barrier, and the
matrix I (i),N :
 (i)

if Jα = Jβ + 1
Γβ→α
(i),J
(i)
i+1
Iαβ = (−1)
(4.70)
−Γβ→α if Jα = Jβ − 1


0
otherwise
which is connected to the angular tunneling current instead. The stationary currents are then simply given by
hIν(i) i = eTr[I (i),ν P(st) ],
ν = N, J
(4.71)
where P(st) is the stationary solution of the rate equations Eq.(4.68). We now define
the zero-modes matrices N and J , which are expressed in components by
Nαβ = Nα δαβ ,
Jαβ = Jα δαβ
(4.72)
Shot noise in a 1D ring
48
so that the expectation value for the number of electrons into the ring and for the
angular momentum at time t are given by
hN (t)i = Tr[N P(t)],
hJ(t)i = Tr[J P(t)].
(4.73)
deriving these expressions with respect to t, and observing that N and J satisfying
the commutation relations
[N , M] = I (1),N − I (2),N
[J , M] = I (1),J − I (2),J
(4.74)
we come to the following couple of continuity equations
d
hN (t)i
dt
d
e hJ(t)i
dt
(1)
(2)
(4.75)
(1)
(2)
(4.76)
= hIN (t)i − hIN (t)i,
e
= hIJ (t)i − hIJ (t)i.
From these expressions one sees immediately that the stationary currents are independent of the barriers
hIν(1) i = hIν(2) i .
(4.77)
Moreover, Eq.(4.76) allows for physical interpretation of hIJ i. As hIN i is related to
˙ i.e.
the variation of the number of electrons into the dot, hIJ i is connected to hJi,
to variations of the persistent current of the ring, due to tunneling.
Following the same steps described in Appendix B, we can use the elements defined
above to evaluate the correlation function of current fluctuations (ν = N, J)
ji
i
(4.78)
h∆Iν(i) (t)∆Iν(j) (0)i = e2 Cx ij
ν (t) + Cx ν (−t) + δij Ca ν
where the cross correlation term is
(i) Mt
δIν(j) P(st) ]
Cx ij
ν (t) = θ(t)Tr[δIν e
(i)
(i)
(4.79)
(i)
with δIν = Iν − hIν i/e and
Ca iν = Tr{|Iν(i) |Pst }
(4.80)
is the autocorrelation part. The noise spectral density Sνij (ω) is then simply given
by the Fourier transform of Eq. (4.78). In particular, the zero frequency noise is
given by
Sνij (ω = 0) = 2e2 δi,j Tr |Iν (j) |P(st)
(4.81)
(i)
(j) (st)
(j)
(i) (st)
2
−1
−1
−2e Tr δIν M δIν P
+ Tr δIν M δIν P
,
where the symbol M−1 has to be dealt with as explained Chapter 3.
Because of the continuity equations, at zero frequency the noise spectral density
Sνij (ω = 0) is independent of the barrier index Sνij (ω = 0) = Sνkl (ω = 0) ∀i, j, k, l;
therefore we adopt the simplified notation Sνij (ω = 0) ≡ Sν . For the charge tunneling current, it is customary to refer to the Fano factor
FN ≡
SN
.
2ehIN i
(4.82)
Analogously, we define the Fano factor for the angular current as
FJ ≡
SJ
.
2ehIN i
(4.83)
which is related to the fluctuations of IJ . Note that we have to define FJ with
respect to hIN i because, in the absence of an external magnetic field it is hIJ i = 0
because of time reversal symmetry.
49
4.5
4.5 Results
Results
Assuming the charge addition energy EN to be the largest energy scale into the
problem kB T, eV EN , the electron transport across the ring is governed by
just two charge states N and N + 1. This condition, however, does not exclude
the possibility of exciting states with an high angular momentum, by means of a
sequence of transitions such as
In,±
Out,∓
In,±
|N, 0i −→ |N + 1, ±1i −→ |N, J ± 2i −→ |N + 1, J ± 3i · · ·
(4.84)
where we have assumed N to be even. It is useful to notice that, in this two chargestate (2CS) regime, any state of the ring is fully specified by the angular momentum
J only. Infact, because of the constrain (−1)N = (−1)J , it is sufficient to give J
in order to identify the corresponding N . For definiteness, from now on we will
assume N to be even and we will adopt the simplified notation
|Ji ≡ |N, Ji,
(4.85)
Importantly, in the 2CS regime the electron transport transport forms a one-step
Markov process, which means that transitions are only possible between neighboring states |Ji ↔ |J + 1i. A remarkable property of one-step process, is that the
stationary occupation probability P(st) obeys the balance condition
(st)
PJ
(st)
ΓJ→J±1 = PJ±1 ΓJ±1→J
(4.86)
which allows us to write down immediately the stationary occupation probability
(st)
PJ
(st)
= P0
J−1
Y
j=0
(st)
where the constant P0
4.5.1
Γj→j+1
,
Γj+1→j
(st)
(st)
P−J = P0
J−1
Y
j=0
Γ−j→−(j+1)
Γ−(j+1)→−j
(4.87)
is fixed by the normalization condition Tr[P(st) ] = 1.
Zero magnetic flux
Before discussing the transport properties of the ring, it is worthy to look at the
phase space of our system. As we have discussed in Chapter 3, the number of states
involved into transport depends on the temperature, on the applied voltages and,
in the present case, on the magnetic flux Φ. At low temperature kB T EJ , to a
good approximation there is only a finite number of states involved in the transport
J ∈ {Jmin , . . . , Jmax }, where Jmin(max) (V, Ng , Φ) can be determined solely on the
base of energetic consideration.
We start considering the case where there is no external magnetic field, Φ = 0.
In this case, it is ΓJ→J 0 = Γ−J→−J 0 (J > 0) because of the degeneracy of the
states |Ji and | − Ji. Therefore it is also Jmin = −Jmax . From Eq.(4.86), we see
that in the stationary regime a certain state |J + 1i participate to the dynamic of
the system only if: (i) the previous state |Ji is occupied; (ii) the transition rate
ΓJ→J+1 is non vanishing. Taking into account dependence of the tunneling rates
Eq.(4.62) on the external voltages, we can subdivide the (V, Ng )-plane according
to the different possible transitions (remember, µi = (−1)i eV /2). For example,
the transition |0i → |1i through the right barrier is possible only in the half plane
defined by the condition
(2)
Γ|0i→|1i 6= 0
⇔
Ng > −eV /2EN + N̄
(4.88)
Shot noise in a 1D ring
50
where N̄ = N + (EN + EJ )/2EN . Analogously it is
(1)
Γ1→0 6= 0
⇔
Ng < +eV /2EN + N̄
(4.89)
6= 0
⇔
Ng < +eV /2EN + N̄ − 2EJ /EN
(4.90)
6= 0
⇔
Ng > −eV /2EN + N̄ − 2EJ /EN
(4.91)
6= 0
⇔
..
.
Ng > −eV /2EN + N̄ + 2EJ /EN
(4.92)
(1)
Γ1→2
(2)
Γ2→1
(2)
Γ2→3
and so on. For definiteness, from now on we assume V > 0. A little care have to be
taken while considering transitions between states with high angular momentum.
Consider for instance the transition |2i → |3i across the right barrier. Such an event
can take place only if the state |2i is occupied. In other words, the transition |2i →
|3i is relevant (i.e. it contributes to the tunneling current) when both Eq.(4.92) and
Eq.(4.88 – 4.91) are satisfied. This give rise to the characteristic fish-bone structure
of Fig. 4.2. Here, we have denoted with different colors regions characterized by
different Jmax . As an example, in the yellow region it is Jmax = 1, which means that
there are only three states accessible to the system {|0i, | ± 1i}, while in the cyan
region it is Jmax = 2, and the relevant states are {|0i, |±1i|±2i}. In the following we
will refer to the yellow and cyan regions as three- and five-states region, respectively.
The boundaries of zones with different Jmax corresponds to the conductance peaks
∂I/∂V of an ideal current measurement. Similar considerations holds for transitions
a
0
→
3
→
→
2→ 0→
1
1
→
2
1 →2 →
b
Ng
→
1→
→
Ng
V
→
V
Figure 4.2: (a) Scheme of the relevant transport regions in the (V, Ng ) plane for
Φ = 0 and N even. Solid lines denote detectable conductance peaks, the involved
|J| values are denoted. The black lines represent transitions which do not involve
plasmonic excitations. The green line represents the transition 2 → 1 with the excitation of one plasmon. Yellow region: three-states regime Jmax = 1. Cyan region:
5 states regime Jmax = 2. (b) Same scheme as in (a), for 0 < Φ < Φ0 /2. Each
transition line is now split in two (for the sake of simplicity, the plasmon-related
green lines have been omitted). Arrows indicate the lines evolution increasing Φ.
involving an excited state. As an example, the transition |2i → |1i involving a
plasmon excitation into the final state, which we indicate as |2i → |1∗ i, can occur
only if Eq.(4.88 – 4.91) are all satisfied (i.e. if the initial state |2i is populated) and
if the bias voltage provides the necessary energy
(2)
Γ|2i→|1∗ i 6= 0
⇔
Ng > −eV /2EN + N̄ + (ε − 2EJ )/EN .
(2)
(4.93)
Note that for 0.5 < g < 1 the onset of the transition Γ|2i→|1∗ i cut the five states
region into two zones which we call I and II. For a non interacting ring g = 1,
there is only zone II, as the plasmonic excitations becomes degenerate with the
topological currents, and finally for g < 0.5 we have only zone I.
51
4.5 Results
We now focus on the regions of the (V, Ng ) plane where Jmax ≤ 2. In this case, the
transition matrix M is a 5 × 5 tridiagonal matrix and rate equations and Eq.(4.71)
and Eq.(4.81) can be evaluated analytically. Even in this simple case the expression for current and for the noise at finite temperature are extremely cumbersome,
therefore in the following we will consider only the zero-temperature limit T = 0.
In this case, we obtain
(2)
hIN i = e
(2)
(1)
(1)
2Γ0→1 Γ2→1 (Γ1→0 + Γ1→2 )
(1)
(2)
(2)
(1)
(4.94)
(1)
Γ1→0 Γ2→1 + 2Γ0→1 (Γ1→0 + Γ1→2 )
for the charge currents while, as expected, hIJ i = 0. The charge Fano factor is
(1)
FN = 1 + 2
(1)
(2)
(2)
(2)
(2)
(1)
(2)
(2)
(1)
Γ1→0 Γ1→2 (2Γ0→1 − Γ2→1 )2 − 2Γ0→1 Γ2→1 (Γ1→0 Γ2→1 + 2Γ0→1 Γ1→2 )
(1)
(2)
(2)
(1)
(1)
[Γ1→0 Γ2→1 + 2Γ0→1 (Γ1→0 + Γ1→2 )]2
(4.95)
and the one related to the angular tunnel current is
FJ = 1 − 2
(1)
(1)
(1)
(1)
(1)
(1)
Γ1→2 Γ1→0 − Γ1→2
Γ1→0 Γ1→0 + Γ1→2
.
(4.96)
Comparing these expression with the exact numerical result, we have found that
(2)
they describe very accurately the temperature regime kB T ≤ 0.1EJ . For Γ1→2 =
0 (three states regime), they give back the well known results for a degenerate
electronic single level [143, 144]
(2)
hIN i = e
(1)
2Γ0→1 Γ1→0
(2)
(1)
2Γ0→1 + Γ1→0
(2)
,
FN =
(1)
(2Γ0→1 )2 + (Γ1→0 )2
(2)
(1)
(2Γ0→1 + Γ1→0 )2
(4.97)
and FJ = 1. An important difference between the three- and the five-states regime
in that while in the first case in is always 0.5 ≤ FN ≤ 1, the second addend in
Eq.(4.95) may be positive, increasing the value of the Fano above the Poissonian
limit.
It is useful to introduce the average dwell time for the Jth-state, i.e. the average
time spent by the system in the state |Ji. In our model, a reasonable estimate for
the dwell times τ|J| is given by
τ0 =
1
(2)
2Γ0→1
,
τ1 =
1
(1)
Γ1→0
+
(1)
Γ1→2
,
τ2 =
1
(2)
Γ2→1
.
(4.98)
Moreover, we define the adimensional parameters
(1)
r1 =
(2)
Γ1→2
r2 =
(1)
Γ1→0
2Γ0→1
(4.99)
(2)
Γ2→1
In terms of these quantities, the expressions for the charge and the angular Fano
factor Eq.(4.95) and Eq.(4.96) become
FN = 1 + 2
FJ
r1 (1 − r2 )2 − (1 + r1 r2 )(1 + r1 )τ1 /τ0
= 1 − 2r1
[1 + r1 r2 + (1 + r1 )τ1 /τ0 ]
1 − r1
.
1 + r1
2
,
(4.100)
(4.101)
From Eq.(4.101), one sees that the orbital Fano factor depends only on r1 , which
represents the “populating ratio” between the states |J| = 2 and J = 0. In other
Shot noise in a 1D ring
52
words r1 measures the transition probability per unit of time for the transition
|1i → |2i with respect to the one of |1i → |0i. In order to achieve FJ > 1 it is
necessary to have what we call an inversion of the populating ratio r1 > 1, i.e. the
transition |1i → |2i leading to the more energetic state have to be stronger than
the one going to the ground state |1i → |0i. Note that the r1 > 1 does not imply an
(st)
(st)
invertion of populations P2 > P1 . Infact, from Eq.(4.87) one can easily see that
(st)
(st)
(st)
(st)
the ratio P2 /P0 depend on both r1 and r2 . Indeed, it is 2P2 /P0 = r1 r2 .
The parameter r2 is defined by the ratio between the dwell times τ0 and τ2
r2 = τ2 /τ1 ,
(4.102)
and it plays a fundamental role in determining the properties of the charge Fano
factor FN . From Eq.(4.100) we see that in order to observe FN > 1, it have to be
r2 6= 1 (and r1 6= 0). As we will discuss below, having τ2 6= τ2 is a signature of the
electronic interactions in the ring. Furthermore , it is necessary to have asymmetric
tunneling barriers. More precisely, to have FN > 1, the asymmetry of the barriers
(1)
(2)
A ≡ Γ0 /Γ0 , have to be bigger than a certain critical value defined by
(2)
Ac =
(i)
(i)
2Γ̄0→1
(1)
Γ̄1→0
+
(1)
Γ̄1→2
(1 + r1 r2 )(1 + r1 )
,
r1 (1 − r2 )2
(4.103)
(i)
(i)
where Γ̄α→β ≡ Γα→β /Γ0 are the the normalized rates. Note that because of Γ̄α→β ,
the critical asymmetry strongly depends on the voltages and on the interaction
parameters g and g` , see Fig 4.4. Note that in the A Ac , one has τ1 τ0,2 ,
so that the transport dynamics is dominated by the two last time scales τ0 and τ2 .
Before discussing the characteristics of the critical asymmetry, let us look in more
detail to the properties of the Fano factors FN and FJ .
1.6 FN
a
1
0.8
2 FJ
b
1
0.6
0
(eV − 2.43EJ )/ε
2.5
Figure 4.3: Fano factors as a function of (eV − 2.43EJ )/ε, where V varies according
to Ng = 1/2 + 0.93EJ /EN − eV /2EN , with kB T = 0.02EJ and A = 20. (a) FN for
an interacting ring, g = 0.7, and different interactions in the leads: g` = 1 (red),
0.9 (cyan), 0.5 (green), 1.2 (blue); magenta: g = g` = 1. (b) FJ , parameters and
colors as in (a).
In Fig. 4.3, the values of FN (a) and FJ (b), calculated along the line Ng =
1/2+0.93EJ /EN −eV /2EN (red dash-dot line in Fig. 4.2a), are shown as a function
of the offset adimensionalized voltage (eV −2.43EJ )/ε. Both exhibit clear jumps as a
consequence of the excitation of plasmonic modes in the transitions |J| = 1 → J = 0
and |J| = 1 → |J| = 2. In the absence of interactions (g = 1 = g` , magenta curves)
the plasmonic modes are degenerate ε = 2EJ . Since E1→2 = E1→0 + 2EJ and
53
4.5 Results
E2→1 = E0→1 + 2EJ , from (4.62) it is clear that r1 < 1 and r2 = 1, leading
always to at most Poissonian noise. With an interacting ring, the degeneracy is
lifted (ε > 2EJ ) so that it is possible to have r2 6= 1. In this case it is possible to
achieve FN > 1, if A > Ac . The super-Poissonian charge noise is robust against
the leads interactions as can be seen in Fig. 4.3 (a). In particular, and in striking
contrast with the behavior of FJ , the charge noise is not sensitive to the presence
of repulsive g` < 1 or attractive g` > 1 interactions in the leads. On the contrary,
for the magnetization noise, it is crucial to have g` > 1 in order to obtain superPoissonian values.
For an interacting ring, in region I it is always r2 = 1. Moving along the red
dash-dot line Ng = 1/2+0.93EJ /EN −eV /2EN , one can identify two energy regions
σ± (n)
σ− (n) if (n − 1)ε < eV − 2.43EJ < (n − g)ε
σ+ (n)
if (n − g)ε < eV − 2.43EJ < nε .
In these two regions, for r1 one has
(
1
in σ− (n)
r1 =
n/(n + 1) in σ+ (n)
In σ− (n), for A Ac and g` = 1, the Fano factors reach the asymptotic limits
FN = 1 + 2/9 and FJ = 1. By tuning the voltage V , one periodically enters the
regions σ+ (n), where the Fano factor depends on the details of the interactions. For
noninteracting leads one has
FN = 1 +
2n(n + λ)
,
(3n + λ)2
FJ = 1 −
2nλ
.
(n + λ)(2n + λ)
(4.104)
with λ = (g +g −1 )/2. For interacting leads, one finds smoothed steps in FN and FJ ,
with a power law behavior as a function of V . For particularly strong interactions
g . 0.5, the smoothing of the step-like structure is almost complete. It is interesting
to notice the strong increase of FN , as compared to the suppression of FJ (green
curves). In Fig. 4.4, the critical asymmetry Ac is plotted in the plane XY , where
0.08 a
A∞
Y
0.03
b
Y
0
1
0
X
0.2
Figure 4.4: Color-contour plot of Ac for g = 0.7, g` = 0.8, kB T = 0.02EJ in the
XY plane (see text). (a) Region I, A∞ = 4100. (b) Region II, A∞ = 2.6 105 .
X = 1/2 − Ng − 1.5EJ /EN + eV /2EN and Y = −1/2 + Ng − 0.5EJ /EN + eV /2EN .
Note that Y = 0 corresponds to the transition line 0 → 1, Y = 0.03 to the line
E2→1 = ε, and X = 0 to the line 1 → 2 (cf. Fig. 4.4 (a)). In the panels (a) and
(b), the regions I and II are shown. In both regions, near certain lines parallel
to the X axis, for g` it is possible to have r2 = 1. Here, Ac diverges and no
super-Poissonian charge noise can be achieved. Away from these lines, increasing
Shot noise in a 1D ring
54
the voltage, Ac decreases because of the increasing number of excited plasmons
present in the output transition. The latter decrease the dwell time τ1 increasing
(1)
the ”output” rates Γ1→0,2 . For sufficiently high voltages, one can eventually reach
the condition Ac < 1.
4.5.2
Non-zero magnetic flux
When Φ 6= nΦ0 /2 (n ∈ Z), the degeneracy of the states with ±J is lifted. As a
consequence, the transition lines J → J 0 split in two and move in opposite directions
increasing Φ, with a separation proportional to EJ Φ/Φ0 . In Fig. 4.2 (b) an example
of such a situation is shown. The plane (V, Ng ) is subdivided in many different
regions, in each of which different values of Jmin and Jmax are found. This makes
an analytical diagonalization of the matrix M̄ extremely difficult and we have to
resort to numerical evaluations. Many interesting results are found. We restrict
our attention on the 0 < Φ < Φ0 /2 regime. The results shown in Fig. 4.5 represent
0.5
1.21
Φ
FN
a
0
0.5
FJ
Φ
0
0.86
1
0
(eV − 3.2EJ )/ε
b
1
0.6
Figure 4.5: Density plot of FN (a) and FJ (b) as a function of (eV − 3.2EJ )/ε
(voltage moving according to Ng = 1/2 + 1.7EJ /EN − eV /2EN ) and Φ, in units
Φ0 . Parameters are: g = 0.8, g` = 1, kB T = 0.02EJ and A = 20.
FN and FJ evaluated along a diagonal line which lies inside zone II at Φ = 0 and
are specular with respect to the Φ = Φ0 /2 when Φ0 /2 < Φ < Φ0 . As a whole, the
results are periodic, with period Φ0 . Increasing Φ, many moving transition lines
cross the diagonal, fixed one. Therefore, it is possible to study the correlation effects
on the noise spectrum in a wide range of transport regions. As Fig. 4.5 confirms,
the noise exhibits an extremely rich dynamics by tuning the flux. At Φ = 0 one
has A < Ac , so that FN < 1. However, increasing the flux, super-Poissonian charge
noise is reached for Φ ≈ 0.4Φ0 , signalling the crossover to a transport regime where
A > Ac . The onset of this region is at Φ∗ = 0.35Φ0, given by the intersection of
the line along where V is varied, Ng = 1/2 + 1.7EJ /EN − eV /2EN , with the upper
moving transition line 2 → 1 with one plasmon, Ng = 1/2 + EJ /2EN (4/g − 3 +
4Φ/Φ0 )−eV /2EN , i.e. Φ∗ = [1.7+(3−4/g)/2]Φ0/2. The possibility of crossing over
between sub- and super-Poissonian behavior as a function of the flux is a signature
of the presence of interactions. Indeed, in a noninteracting ring we always find subPoissonian behavior, regardless of the number of states supporting the transport.
Concerning the magnetization noise, interactions in the ring and finite flux are not
enough to induce FJ > 1 (Fig. 4.5 (b)).
4.5.3
Monte Carlo simulation
As discussed in section 4.5.1, the interaction-induced separation of the time scales
τ0 and τ2 is responsible for the super-Poissonian behavior of SN . In order to better
55
4.5 Results
τ0
τ1
1
2
1
τ1
→
2
1
0
τ0
→
→
0
1
→
→
1
0
1
→ a
0
understand this mechanism we set-up a BKL Monte Carlo simulation [145] in the
T = 0 regime. An effective time evolution for the system is modeled by a sequence
of discrete steps. At the k-th step, the ring makes the transition Jk → Jk+1 . The
transition is supposed to be instantaneous, while the average time between the k-th
and the k + 1-th steps is simply the dwell time τ|Jk | . The situation is schematically
τ2
τ
Figure 4.6: Schematic description of the Monte Carlo simulation. The subsequent
simulation steps are denoted by red crosses. An example of possible transitions and
corresponding dwell times is displayed (see text).
depicted in Fig. 4.6. The k-th transition is performed according to a conditioned
probability given by
X
ΓJk →Jk+1
ΓJk →J 0
J 0 =Jk ±1
(we omit here the barrier indexes for ease of notation) with −2 ≤ Jk , J 0 , Jk+1 ≤ 2
and |Jk+1 − Jk | = 1. It is possible to show that the average time spent in each
state converges, in the asymptotic (k → ∞) limit, to the stationary occupation
probability as calculated by solving the rate equation 4.66 in the t → ∞. In the
a
b
c
d
τ /τ0
Figure 4.7: Output sequences of a Monte Carlo simulation for tunneling events at
junction 2, with eV = 3EJ , Ng = 0.48, kB T = 0.02EJ and A = 20. Black (white)
dots denote a clockwise (anticlockwise) entering electron. Colored bars are the
sequence of the orbital value |J|, green: oscillations 0 → ±1 → 0; red: oscillations
±2 → ±1 → ±2; (a) g = g` = 1; (b) g = 0.7, g` = 1; (c) g = 0.7, g` = 0.5; (d)
g = 0.7, g` = 1.2.
following, we consider A Ac , Φ = 0 and V > 0 in the 5 states region. Figure 4.7
shows typical outcomes of the simulation, represented as sequences of tunneling
events for clockwise (black dots) and anti-clockwise (white dots) electrons tunneling
through the input barrier. Since τ1 τ0,2 , the ring state ”oscillates” alternatively
around J = 0 and |J| = 2. The colored bars describe these oscillations: green
represents the transitions 0 → ±1 → 0, red represents ±2 → ±1 → ±2. We denote
these two sequences as S0 and S2 . The average time interval between tunneling is τ0
inside S0 and τ2 inside S2 and. Correspondingly, the average number of transitions
in S0 is n0 = (1 + r1 )/r1 with an average duration time T0 = n0 τ0 , while in S2
we have n2 = (1 + r1 ) with an average duration time T2 = n2 τ2 . It is interesting
to notice that T2 represents the average sequential-tunneling relaxation time of the
Shot noise in a 1D ring
56
states |J| = 2 [146]. Without interactions, Fig. 4.7 (a), the tunneling events are
uniformly distributed since (τ0 = τ2 ). In the interacting case, the removal of the
degeneracy of τ0 and τ2 is reflected in a bunching tendency of the tunneling events
as can be clearly seen in panels (b,c,d) of Fig. 4.7. The bunching can obviously be
present either in S0 – panels (b,d) – or in S2 – panel (c). In all of these cases we
have FN > 1. Quite generally, while with superpoissonian noise we always find a
bunching of the tunneling events, the converse can be false: if A < Ac the bunching
might still occur but the interplay of the two barriers with comparable transparency
gives rise to negative correlations and a depression of the noise below the Poissonian
limit, thus giving FN < 1. The interpretation of FJ is distinctly different (Fig. 4.7
(d)). Here, the time scales τ0,2 do not play any role. The important parameter is the
number of events n2 inside the sequence S2 in which the ring oscillates around the
excited state J = 2. The condition FJ > 1 is fulfilled only for n2 > 2, independent
of the bunching mechanism, which in principle could even be absent.
Chapter 5
Shot noise in NEMS
In this chapter we study the effect of relaxation on the transport properties of a
nanoelectromechanical system (NEMS) [147]. We describe the NEMS as a quantum
dot coupled to an oscillator with incoherent dynamics and we consider the sequential tunneling regime. We show that relaxation induces qualitative changes in the
transport properties of the dot, depending on the strength of the electron-phonon
coupling and on the applied voltage. In particular, critical thresholds in voltage
and relaxation are found such that a suppression below 1/2 of the Fano factor is
possible. The Chapter is organized as follows. The model Hamiltonian is defined in
Sec. 5.2, while in Sec. 5.3 we introduce the rate equation and the formal expressions
for the current and the noise. In Sec. 5.4 numerical results for the current and Fano
factor are presented: in particular, the suppression of the Fano factor is discussed
in detail for a wide range of parameter. Finally, analytic expressions for the current
and the Fano factor are derived within a toy model employing few phononic states.
5.1
Introduction
Nanoelectromechanical systems (NEMS), are MEMS scaled to the submicron dimensions. Micromechanical systems (MEMS) have been studied for decades [148,
149], with an increasing interest because of growing commercial applications. Nowadays, micromechanical devices are routinely employed in a wide field ranging from
ink-jet printers, accelerometers, scanning probe microscopy and optical communications. Reducing the size of mechanical devices to the nanoscale may provide a
revolution in applications such as sensors, medical diagnostics, display and data
storage [22, 23]. Indeed, with resonant frequencies from kilohertz to gigahertz, low
dissipation and small masses [25], NEMS are the ideal candidate to ultra-sensitive
detection [7,26,27,150]. Moreover, their dimension not only make them susceptible
to local forces, but also make it possible to integrate and tightly couple them to a
variety of electronic structures, such as quantum dots or point contacts [36]. From
the fundamental point of view, the dynamics of nanoelectromechanical systems
have attracted a considerable interest because of the novel transport mechanisms
they can give rise to [151, 152] and their interesting non linear and chaotic properties [153, 154]. Furthermore, when cooled to ultralow temperatures, high-frequency
resonators are expected to display quantum mechanical behavior so that NEMS
might be used to explore the cross over from quantum to classical behavior in mechanical systems [155].
A prerequisite for attaining the ultimate potential from NEMS is displacement
sensing, that is reading out the motion of the device induced by an applied stimulus.
Perhaps not surprisingly, transduction schemes which are important for MEMS do
57
Shot noise in NEMS
58
not prove optimal for nanoscale devices. For example, the sensitivity of opticalinterferometry approaches is severely limited by diffractions, since the dimension of
NEMS are small compared to the wavelength of light.
Some of the most successful transduction efforts are focused on coupling a
nanomechanical system to quantum dots or single electron transistors (SET) used
in configurations where the motion of the mechanical part modulate the electron
transport properties. This can be achieved by coating the mechanical resonator by
a thin metallic layer, and placing it next to a dot [7, 27]. The resonator acts as a
capacitor whose presence affects the current flowing through the SET. The capacitance, and therefore the current, depend on the position and hence on the motion
of the oscillator. The exceptional charge sensitivity of the SET allows to motion
detection with a sensitivity down to the quantum limit (see discussion on Sec. 1.2).
However, because of the small inertial mass of the oscillator, the movement of the
electrons through the SET acts back on the resonator affecting its dynamics in a
important way. Intriguingly, in these devices the tunneling of a single electron may
induce a displacement of the movable structure.
Another class of systems for which the electro-mechanical coupling plays a fundamental role are molecular devices (see Sec. 1.3.2). This coupling originates from
the fact that, when a molecule is employed as an active part in an electrical circuit, its nuclear configuration needs to adjust to the addition or removal of electrons. Because of this, electron transfer is generally accompanied by excitation
or de-excitation of molecular vibrations (phonons). Recent experiments on singlemolecule junctions have shown that these excitations are reflected in the current
voltage characteristics as vibrational sidebands [29, 40, 45]. As an example, signatures of the center of mass mode of a C60 molecule between gold electrodes have
been recently reported [29], whereas in a experiment on C140 dimers, vibrational
sidebands due to an internal mode have been observed [45]. Moreover, as electrons
tunneling on to the molecule can excite vibrational modes by spending some of their
excess energy provided by the external bias, they will tend to drive the vibrations
out of equilibrium, if the quality factor of the vibrational mode is sufficiently large.
Such nonequilibrium vibrations have so far been observed in at least one experiment on a suspended carbon nanotube as absorption satellites of Coulomb blockade
peaks [59]
Despite the obvious differences in design and characteristic energy scales, electron transport through NEMS and single molecule devices shows many common
features. Therefore the acronym NEMS is customarily used for both class of devices.
On the theoretical side, NEMS are often described in terms of a very simple phenomenological model, based on the following series and simplifications:
(i) Transport is assumed to be dominated by tunneling through a single electronic level with single-particle energy ε. Due to coulomb interaction, double
occupation of the level is associated with an additional charging energy U > 0.
(ii) Only a single vibrational mode with frequency ω0 is taken into account within
the harmonic approximation.
(iii) The coupling between the charge on the dot and the mechanical mode is
assumed to be linear
The key features of this model, which is also known as the Anderson-Holstein model,
are schematized on Fig. 5.1. Even within these rude simplifications, many peculiar features such as negative differential conductance [156–158], shuttling instability [151, 159] and strong mechanical feedback [160] have been predicted in the case
of an underdamped oscillator.
59
5.2 Model
Figure 5.1: Potential surfaces for a NEMS-model featuring a single, spin-degenerate
electronic orbital coupled to a single vibrational mode. The model Hamiltonian
approximates the potential surfaces by a harmonic-oscillator potential. The electronphonon coupling, parametrized by the coupling strength λ, corresponds to a
charge-dependent x-shift of the potential surfaces.
5.2
Model
According to the Anderson-Holstein model, the Hamiltonian of a NEMS is given by
Hs = Hn + Hb + Hn,b
(5.1)
where Hn , Hb and Hn,b are respectively given by
Hn
Hb
= ε n + U n ↑ n↓
= ω0 (b† b + 1/2),
(5.2)
(5.3)
Hn,b
= λ ω0 (b† + b) n.
(5.4)
The operators nσ = d†σ dσ and n = σ nσ represent the spin-resolved and the total
occupation number of the single level. The single particle energy ε is measured
with respect to the zero bias Fermi energy, and it may be tuned by applying a
voltage Vg to the gate electrode ε = ε(Vg ). Vibrational excitations with frequency
ω0 are created b† and their ground state is defined as the zero-phonon state when
n = 0. The frequency of the oscillator ω0 can range from the tens of MHz of a
nanometrical cantilever [25] to a dozen of THz in the case of molecular devices
or suspended nanotubes. [45, 59] The coupling betwen vibrational and electronic
degrees of freedom is described by the term Hn,b . The dimensionless parameter λ
represents the strength of the e–ph interaction. For example, λ ∼ 1 was reported
for the C60 devices [29] and for suspended carbon nanotubes, [60] while values of λ
between 0.4 and 3 have been found in different C140 samples. [45] We are interested
in the case where the NEMS is coupled both to external electronic leads and to a
dissipative environment. The total Hamiltonian then reads
P
H = Hs + Hleads + Ht + Henv + Hb,env .
(5.5)
Shot noise in NEMS
60
The Hamiltonian for the non interacting leads is
X X
Hleads =
εi,k c†i,kσ ci,kσ
(5.6)
i=1,2 k,σ
where c†i,kσ (c†i,kσ ) is the creation (annihilation) operator for an electron with momentum k and spin projection σ in lead i (i = 1, 2). For vanishing bias voltage
V = 0, both leads assume the same Fermi energy, which we define as the zeropoint of our energy scale. At finite bias, the chemical potentials of the two leads
are are shifted and their difference is fixed by µ1 − µ2 = eV . The exact voltage
splitting betwen the left and right junction depends on the details of the junction
capacitance’s [161]. In general, the voltage splitting may be described in terms of
a capacitance dependent splitting parameter 0 ≤ η ≤ 1 such that
µ2 = (1 − η)eV.
µ1 = ηeV,
(5.7)
However, as the case of asymmetric voltage splitting η 6= 1/2 can always be compensated for by tuning the gate voltage, in the following we may restrict our discussion
to the case of symmetric voltage drop µ1,2 = ±eV /2.
The coupling between the NEMS and the leads is determined by the tunneling
Hamiltonian
XX
Ht =
ti (c†i,kσ dσ + d†σ ci,kσ ),
(5.8)
i=1,2 k,σ
The strength of this coupling is parametrized by the tunneling matrix element ti .
For simplicity, in the following we will assume symmetric barriers t1 = t2 ≡ t0 .
The coupling between the oscillator and the environment is included as a linear
coupling to a bath of harmonic oscillators in the spirit of the Caldeira and Legget
theory [123]
X
(5.9)
Henv =
ωj (a†j aj + 1/2),
j
Hb,env
=
X
χj ωj (a†j + aj )(b† + b).
(5.10)
j
Here a†j are the creation operators of the bosonic bath modes and χj are the coupling
constats for the oscillator-bath interaction. The environmental coupling is usefully
characterized by its spectral function [162]
X
J (ω) = 2π
ωj2 χ2j δ(ω − ωj ).
(5.11)
j
Polaron transformation
The coupling term Hn,b can be eliminated from Hs by means of a canonical transformation similar to the Lang-Firsov transformation [163, 164] (see Appendix 5.12 for
details). This transformation corresponds to a basis change to polaron-type quasiparticles, i.e. electrons surrounded by clouds of vibrations. Importantly, because
of the coupling term Hb,env , the transformation must include both the operators of
the oscillator and those of the environment [116]
X
Ō = eAn Oe−An ,
A = κ(b† − b) − 2κ
χj (a†j − aj ),
(5.12)
j
where
κ=
1−4
λ
.
2
j χj ωj /ω0
P
(5.13)
61
5.3 Rate Equation
Applying this transformation to the total Hamiltonian Eq.(5.5) we obtain
H̄ = H̄n + Hb + Hleads + Henv + H̄t + Hb,env
where
H̄t =
X
t0 (c†k,i e−A d + d† eA ck,i ).
(5.14)
(5.15)
k,i=1,2
and H̄n = ε̄ n + Ū n↑ n↓ with
ε̄ = ε − λκω0 ,
Ū = U − 2λκω0 .
(5.16)
Hence, the electron–phonon coupling term Hn,b is eliminated at the cost of introducing displacement operators in the tunneling matrix elements t0 → t0 e−A . Moreover,
the single-particle energy and the charging energy are renormalized (polaron shift).
The renormalization of the charging energy U is irrelevant since we assume this to
be the largest energy in the problem, however, let us note that for weak Coulomb
interaction the polaron shift can even result in a negative effective Ū [120, 165].
From now on, we will proceed with the transformed Hamiltonian.
5.3
Rate Equation
There are four different energy scales relevant for the analysis of transport in the
Anderson-Holstein model: the phonon energy ω0 , the bare tunneling rate Γ =
2πνt20 (with ν the density of states of the leads), the relaxation rate Γrel which
characterizes the coupling to the dissipative environment and the temperature T .
Ifthe phonon energy is large compared to the level broadening induced by tunneling Γ, then different vibrational states do not get mixed and the system completes many oscillation cycles between two tunneling events. On the other hand,
for phonon energies small compared to the level width a strong mixing between
vibrational states occurs and coherences between them must be taken into account.
A scenario typical of this limit is the case of the quantum shuttling in nanoelectromechanical devices [159, 166].
Analogously, due to the presence of the environment, the discrete vibrational
excitations become broadened by Γrel . The ratio Q = ω0 /Γrel , which is known as
the quality factor, characterizes the energy loss of the vibrational mode per period
(1/ω) due to damping. For large quality factors the broadening of the vibrational
excitations can be neglected.
Finally, the temperature determines the time scale τeq = 1/kB T over which excitations in the leads decay . As we have seen in Chapter 3, the sequential tunneling
regime requires τeq to be the shortest energy scale into play. Therefore, in the following we will always assume Γ, Γrel ω0 kB T which is a typical experimental
situation [29, 45, 59]. In this regime, the coupling Hamiltonian Hint = H̄t + Hb,env
can be treated perturbatively and the calculation of transport characteristics may
be carried out within the framework of the rate equation, as discussed in Chapter 3.
The transition rates can be evaluated to the lowest order in Hint by means of Fermi’s
golden rule (see appendix D)
Γ|ii→|f i
= 2π|hf |Hint |ii|2 δ(Ei − Ef )
= 2π|hf |H̄t |ii|2 δ(Ei − Ef ) + 2π|hf |Hb,env |ii|2 δ(Ei − Ef ).
This leads to two different kind of contributions: the tunneling rates proportional
to t20 , and relaxation rates, which depend on χ2j .
Shot noise in NEMS
62
Tunneling rates. The tunneling Hamiltonian H̄t is responsible for the transfer of
one electron from a lead into the electronic level or vice versa. Because of the e–ph
coupling, the electron transfer is generally accompanied by excitation (or excitation)
of molecular vibrations. Assuming the electrons in the leads are at equilibrium with
their chemical potential, one obtains the following expressions
Γi 0→σ
= Γ Xl0 l f µ(l0 − l) − µi ,
(5.17)
l→l0
σ→0
0
Γi l→l0 = Γ Xl0 l 1 − f µ(l − l ) − µi ,
(5.18)
where f (x) is the Fermi function and µ(l 0 − l) = ε̄ + ω0 (l0 − l) is the addition
energy of the NEMS, i.e. the energy required to add an electron into the system
while changing the vibrational state l → l 0 . The coefficient Xll0 denotes the matrix
element for the transition l → l 0 , and is given by
Xll0 = |hl|e−λ(b
†
−b) 0 2
|l i| ,
(5.19)
These terms are called Franck–Condon (FC) factors and they will discussed in detail later in this chapter. Importantly, even if the displacement operator in H̄T Eq.
(5.15) includes also the momenta of the environment, the tunneling rates depends
only on λ(b† − b). This is a consequence of treating the Hint to the lowest order in
the coupling constant t0 , χi . Therefore, as the tunneling rates are already ∝ |t0 |2 ,
we have to be careful to disregard all terms which includes χi .
Relaxation rates The relaxation rates represent transitions between vibrational
excitations without change of the electronic state
βω0 rel
Γrel
Γ(l−1)→l = l
l→(l−1) = e
J (ω0 )
,
1 − e−βω0
(5.20)
where J (ω0 ) is the spectral density of the phonon bath Eq. (5.11), evaluated at the
frequency of the oscillator. Note that treating Hb,env at second order allows only
transitions between neighboring states (i.e. |l 0 − l| = 1). Transitions with |l 0 − l| ≥ 1
can be included considering different relaxation mechanisms [120, 156].
Given the tunneling and the relaxation rates, we can write down explicitly the rate
equation for the occupation probability for the eigenstate of H̄s
X
X
d
0→σ
rel
P0l =
[Pσl0 Γ σ→0
[P0l0 Γrel
l0 →l − P0l Γ l→l0 ] +
l0 →l − P0 l Γl→l0 ]
dt
0
0
σ,l
l
X
X
d
rel
σ→0
[Pσ l0 Γrel
[P0 l0 Γ l0→σ
Pσ l =
0 →l − Pσ l Γ l→l0 ] +
l0 →l − Pσ l Γl→l0 ]
dt
0
0
l
(5.21)
l
0
ν→ν
=
where Pνl (ν = 0, ±σ) is the occupation probability of the state |ν, li and Γ l→l
0
P ν→ν 0
is
the
total
tunneling
rate.
Γ
0
i i l→l
Since all the transition rates do not depend on the spin projection σ, we may
introduce the total probability P1 l = P↑l + P↓l for having one electron into the level
and sum Eq.(5.21) over σ. This result in spin averaged rate equations
X
X
d
0→1
rel
P0 l =
[P1 l0 Γ l1→0
[P0 l0 Γrel
0 →l − P0 l dΓ l→l0 ] +
l0 →l − P0 l Γl→l0 ]
dt
0
0
l
l
X
X
d
rel
0→1
1→0
[P1 l0 Γrel
[P0 l0 dΓ l0 →l − P1 l Γ l→l0 ] +
P1 l =
l0 →l − P1 l Γl→l0 ]
dt
0
0
l
(5.22)
l
1→0
σ→0
0→1
where we have called Γi 0→σ
l→l0 ≡ Γi l→l0 and Γi l→l0 ≡ Γi l→l0 and where the factor d
takes in account the spin degeneracy. We have introduced the notation d in order
63
5.3 Rate Equation
to have the possibility to disregard the spin degeneracy by simply inserting d = 1.
From Eq.(5.22) is evident that, in the case of strong electronic interaction, a spindegenerate electronic level is formally equivalent to a spinless single level, apart for
an effective doubling of the tunneling-in rates [167].
For sake of simplicity, in the following we will consider the spinless case d = 1,
however our results can be readily generalized to the case of a spin-degenerate level.
In the remaining of this section we will shortly discuss two of the main features of
the rate equation Eq.(5.22). One is the dependence of the Franck-Condon factors
Xll0 on the strength of the e–ph coupling λ, the other is the harmonic spectrum of
the oscillator, which allows for vibration assisted cascades.
Cascades and accumulation of vibrational energy
Due to the harmonic spectrum, the addition energy for the transition |0, li ↔ |1, l 0 i
depends only on the change of the difference ∆l = l 0 − l and not on the individual
vibrational numbers l, l 0 . This means that everytime that the addition energy µ(∆l)
comes into the bias window, all the transitions |0, li ↔ |1, l + ∆li for all l and fixed
∆l become possible. This is an intrinsic property of the harmonic spectrum and
gives rise to vibrational assisted cascades. As an example, imagine that the addition
energies µ(0) and µ(1) lie within the bias window. An electron tunneling onto the
NEMS can therefore excite the vibrational mode by one quanta by the transition
|0, 0i → |1, 1i. When it tunnels off, it can leave the system in the excited state
by the transition |1, 1i → |0, 1i. If we assume that the vibrational energy does not
relax between tunneling events, this excitation is conserved until the next tunneling
event which may even even increase the excitation by the transition |0, 1i → |1, 2i.
The system can then reach a highly excited state via subsequent tunneling events
|0, 0i → |1, 1i → |0, 1i → |1, 2i → |0, 2i → |1, 3i → |0, 3i → |1, 4i → . . .
The amplitude of this cascade depends on the details of the FC factors and on all
other possible transitions.
Franck–Condon factors
Even if the addition energy depends only on the difference ∆l, the tunneling rates
acquire a strong dependence on the vibrational numbers l, l 0 because of the Franck–
Condon factors Eq.(5.19). The properties of the FC factors are best elucidated by
†
expressing the matrix element hl 0 |e−λ(b −b) |li in coordinate representation as the
overlap of two displaced harmonic oscillator wave functions φl and φl0
X
ll0
Z
= +∞
−∞
dx φ∗l (x
2
√
2
2
0 l< ! |l−l0 |
0
+ 2λ x0 )φl (x) = e−λ λ2|l−l | Ll< (λ2 )
l> !
(5.23)
p
where x0 = 1/M ω0 is the the zero-point displacement of the oscillator and l< =
min{l, l0 }, l> = max{l, l0} and Lnl (x) is a generalized Laguerre polynomial. The
coupling parameter λ however determines the shift between the two equilibrium
positions of the oscillator. However, from Eq.(5.23) we see that the relevant scale
is not λ but λ2 as one could have expected observing that λ2 is the elastic energy
in units ω. 1
1 In
other words, λ2 ω is the elastic energy gained by the system when it is charged because of
the deformation.
This can be easily seen imagining a classical spring which is elongated by an
√
amount 2λ. Then the potential energy that the spring gains is exactly λ2 ω.
Shot noise in NEMS
64
The FC factors are symmetric under the exchange of indices Xll0 = Xl0 l and
obey the following sum rule
X
X
Xll0 =
Xll0 = 1
(5.24)
l
l0
which sets an upper bound for the current at large bias. A consequence is the
inequality Xll0 ≤ 1 for all l, l0 .
The behavior of the FC factors depends crucially on the strength of the e–ph
coupling. Roughly speaking, one can distinguish three regimes:
(i) λ2 1.
Transitions which conserve the vibrational number have a dominant amplitude
up to high excitations l, l 0 . For λ → 0 the FC factors are given by Xll0 = δll0
due to the or thonormality of the harmonic oscillator eigenfunctions.
(ii) λ2 ∼ 1.
Transitions slightly changing the vibrational number are favored but the global
maximum of the FC factors still lies at small vibrational numbers. For λ2 = 1
it is X01 = X00 = 1/e, while for λ2 < 1 it is X01 < X00 and for λ2 > 1 it is
X01 > X00 .
(iii) λ2 1.
Transitions which change l considerably are strongly favored and the maximum FC factors lie at l = 0, l 0 = λ2 . For small l, l0 a gap opens in which the
FC factors increase exponentially with increasing l + l 0 .
To summarize, we can say that the effect of the e–ph interaction on transport is two
fold: on one hand it suppresses the effective tunneling rate (Xll0 ≤ 1, ∀l, l0 ) and on
the other, it induces a highly non trivial dependence on the phononic indices l, l 0 .
5.4
Results
As discussed in Chapter 3, within the rate equation approach the current and
the current noise can be evaluated by means of standard techniques [94]. During
numerical calculations, we introduce a cut-off lmax 1 for the vibrational number.
In this case the occupation probability vector P is a vector of length 2lmax and the
transition matrix M is a 2lmax × 2lmax matrix (see Chapter 3 for the definitions
of P and M). The phonon cut-off has to be chosen quite carefully because, due to
the harmonic spectrum, for weak relaxation the possibility of vibrational cascades
exists, i.e. in principle there is an infinite number of molecular states. To choose
lmax , we check the convergence of the probabilities.
Full solution of the rate equation
The dynamics of the system is characterized by two competing time scales: the
average time spent by an electron in the dot τel and the phonon relaxation time τph .
If τel τph , the vibrational excitations tend to relax between each tunneling event
(eq)
to the thermal Bose distribution Pl
= e−βlω0 (1 − e−βω0 ). In this limit, charge
(eq)
and vibrational degrees of freedom decouple Pnl = Pn Pl
and the dynamics of
the system reduces to an effective two–state sequential tunneling process. [116] The
analytic expressions for current and noise are well known [72] and, for kB T eV ,
are respectively given by
I (eq) = e
Γ̃1 Γ̃2
,
Γ̃1 + Γ̃2
F (eq) =
Γ̃21 + Γ̃22
.
(Γ̃1 + Γ̃2 )2
(5.25)
65
5.4 Results
λ=1
λ = 16
Xll0
λ = 0.04
l0
l0
l
l0
l
l
Figure 5.2: Franck–Condon factors for weak (λ2 = 0.04), intermediate (λ2 = 1) and
strong (λ2 = 16) e–ph coupling. Panels in the top and middle row show the FC
factors as a function of the initial and final phonon state l and l 0 respectively. The
graphs on the bottom visualize the displacement of the harmonic potential surfaces
for the neutral and the charged system.
P
P
Here Γ̃1 = Γ(0) l al f1 (lω0 ) and Γ̃2 = Γ(0) l al [1 − f2 (−lω0 )] are the renormalized
rates for tunneling in and out of the dot and the coefficients al are Poissonian weight
2
factors 2 al = θ(l)e−λ λ2l /l!. In this case the smallest possible value of the Fano
factor is F (eq) = 1/2.
Vice versa, if τel τph the tunneling electrons drive the vibrations out of equilibrium and peculiar features such as negative differential conductance (NDC) [156,
158, 168] and super–Poissonian shot–noise [120] have been predicted.
2
−1
−1
In our model, a rough estimate of τel
is given by τel
= Γe−λ , i.e. by the effec−1
tive transparency of the barrier set by the e–ph coupling, while τph
is determined
−1
by environment spectral density τph = J (ω0 ). It is useful to define a dimensionless
parameter for the relaxation strength
w = J (ω0 )/Γ(0) .
(5.26)
In terms of w, the condition for equilibrated phonons τel τph reads w exp(−λ2 ).
It is then evident that the e–ph coupling defines a characteristic scale for relaxation:
the stronger is the coupling, the more sensitive is the system to phonon relaxation.
(st)
(st)
(st)
This is reflected by the stationary phonon distribution Pl
= P0 l + P1 l . For
(st)
(eq)
increasing relaxation strength w, Pl
tends monotonically to Pl
but with λ2 –
depending speed (see Fig.5.3). For strong e–ph coupling, the phonon distribution
is narrow already in the non–relaxed case [119] w = 0 and it reaches equilibrium
for values of w which are sensibly smaller than for weak λ2 .
2 The
general
expression ` for
the
factors
is:
al
=
´weight
exp[−λ2 coth(βω0 /2)] exp[lβω0 /2]Il λ2 / sinh(βω0 /2) , where Il is the modified Bessel function of first kind.
Shot noise in NEMS
66
1
w=0
w=1
w = 10
w=∞
w=0
w=1
w = 10
w=∞
(st)
Pl
0
1
(st)
Pl
0
0
6 0
l
6 0
l
l
6 0
l
6
(st)
Figure 5.3: Stationary phonon probability distribution Pl for different values of w.
Upper panels: λ2 = 0.4, lower: λ2 = 7. The rightmost panels (w = ∞) represent the
(eq)
thermal Bose distribution Pl
= e−βlω0 (1−e−βω0 ). Other parameters: eV = 3 ω0 ,
ε̄ = 0 and kB T = 0.02 ω0.
w
w
w
w
w
100
F
=0
= 0.1
=1
= 10
=∞
10
1
0.5
0
2
4
6
8
10
12
14
eV /ω0
Figure 5.4: Fano factor as a function of voltage for λ2 = 16 and for different values of
the relaxation strength w. The dotted line w = ∞ corresponds to case of thermally
distributed phonons. Other parameters: ε̄ = 0 and kB T = 0.02 ω0.
67
5.4 Results
(st)
(eq)
Since Pl
converges monotonically to Pl
for growing w, one expects most of
the features of the non equilibrated case to be washed out by increasing relaxation.
This is particularly evident in the case of the giant Fano factor observed at low
voltages for strong interaction (λ2 1), which is strongly suppressed even by weak
relaxation (see Fig. 5.4). This behavior can be easily understood observing that
F 1 depends dramatically on the non equilibrium distribution of the vibrational
excitations induced by tunneling. [120, 121] In fact, for large λ2 transitions between
low lying phonon states are exponentially suppressed (see Eq. (5.23)). Therefore,
the main contribution to the current comes from high excited vibrational states
(states with large l) but at low voltages the occupation probability of those states
is strongly suppressed. [119, 158] These conditions leads to avalanches of tunneling
processes which, in turn, are responsible for the huge values of F . [120, 121] Direct
phonon relaxation inhibits this mechanism reducing even further the occupation
of states with large l and, consequently, both the current and the Fano factor are
strongly suppressed. For very strong relaxation (w → ∞), F → 1/2 as one would
expect for equilibrated phonon on resonance (ε̄ = 0).
Similarly, relaxation has a destructive effect on NDC (not shown) as this is also
a consequence of the peculiarity of the nonequilibrium phonon distribution induced
by tunneling itself. [158, 168]
One could be tempted to conclude that considering explicitly the effects of relaxation simply results in an “interpolating” behavior between the opposite limits
of no relaxation and thermally distributed phonons. However, we find that finite
relaxation rate can induce unexpected features.
Let’s first consider the case of moderate coupling λ2 = 3. In Fig. 5.5 we plot
the Fano factor as a function of voltage for different values of w and for ε̄ = 0. It
0.7
w
w
w
w
w
0.6
=0
=5
= 12
= 100
=∞
F
0.5
0.4
0
2
4
6
8
10
12
14
16
18
20
eV /ω0
Figure 5.5: Fano factor as a function of voltage at λ2 = 3 and for different values of
the relaxation strength w. The dotted line w = ∞ corresponds to case of thermally
distributed phonons. Other parameters: ε̄ = 0 and kB T = 0.02 ω0.
appears that F has a non systematic dependence on w: it can be either enhanced or
suppressed by relaxation depending on the considered voltage range. For eV < 6ω 0
it is always F ≥ 1/2. In particular, for eV < 2ω0 it is F = 1/2 as the tunneling
electrons cannot excite vibrations and the system behaves as an ordinary single
level. More interestingly, for eV > 6ω0 , relaxation can suppress F even below 1/2.
In this voltage range, then, relaxation of the mechanical mode induces a tendency
to ordered transfer of electrons through the SET.
It is worthwhile to stress that such a suppression of current fluctuations is not
merely an effect of the coupling of the SET to the oscillator, but it is a consequence of
Shot noise in NEMS
68
a non trivial interplay between vibration–assisted tunneling and phonon relaxation
which induces correlation between different current pulses via emission-absorption
of phonons.
20
20
0.6
λ2 = 0.4
w
0.6
λ2 = 1
w
←
←
0.48
0
0
eV /ω0
15
20
λ2 = 3
0.46
0
0
eV /ω0
0.86 20
w
15
λ2 = 7
1.5
w
←
0.43
0
0
eV /ω0
20
0.41←
0
0
eV /ω0
20
Figure 5.6: Density plot of the Fano factor as a function of bias V and relaxation
w for different values of λ. In all the panels: dark gray F < 1/2, medium gray
F = 1/2 (indicated by the arrow in the color map ) and light gray F > 1/2. The
white line, corresponding to F = 1/2, represents wt (V ). The black line in the 4th
panel delimits the region where noise is superpoissonian, F > 1. Other parameters:
¯ = 0 and kB T = 0.02 ω0.
This peculiar behavior is observable in a wide range of parameter. Infact, from
a numerical analysis, it emerges that F < 1/2 can be found for any value of the
e–ph coupling. In particular, we observed that it exists a voltage threshold set by
the e–ph interaction
eVt (λ) = 2ω0 int[λ2 ]
(5.27)
such that, for V > Vt (λ), relaxation larger than a certain threshold value wt (V, λ)
suppresses the Fano factor below 1/2. This is shown in Fig. 5.6, which represents a
grayscale plot of the Fano factor in the (V, w)-plane, for different values of λ2 . The
white contour line corresponds to F = 1/2 and separates two different regions in
the (V, w)-plane: the one to the right of the contour, where F < 1/2 and the other
one where F > 1/2. In other words, the white line denotes wt as a function of V
at given λ2 . The threshold voltage Vt (λ) corresponds to the position of the vertical
asymptote of wt (V, λ). For λ2 < 2 the critical voltage coincides with the onset
of vibration assisted tunneling eVt (λ) = 2ω0 ; vice versa for strong e–ph coupling
(λ2 1), Vt (λ) becomes very large and this is why F is always higher than 1/2 in
Fig. 5.4.
The minimal value assumed by the Fano factor Fmin depends itself on the e–ph
coupling (see Fig. 5.7). For weak coupling, Fmin differs only slightly from 1/2. For
69
5.4 Results
stronger coupling (λ2 > 1) it decreases logarithmically and it only reaches the value
Fmin ∼ 0.4 for considerably strong interactions. Note that each point in Fig. 5.7
corresponds to different values of voltage and relaxation strength, as the position
of Fmin in the (V, w)-plane depends on λ2 . The inset shows the voltage Vmin where
the minimum is found.
0.52
0.5
0.48
45
0.46
Fmin
0.44
Vmin
0.42
0.40
0
0
14
λ2
0.38
10−3
10−2
10−1
1
10
λ2
Figure 5.7: Main panel: Fmin as a function of λ2 . Each point corresponds to
different values of w and V . Inset: voltage Vmin where the minimum is found as a
function of λ2 .
Finally, let’s observe that for λ2 > 2, the threshold voltage Vt (λ) corresponds
to the onset of the transition l : 0 → int[λ2 ]. In this case Vt (λ) is a characteristic voltage also for the current which can be either suppressed or enhanced by
relaxation depending on V being smaller or larger than Vt (λ) (see Fig. 5.8). Relaxation contributes to populate the low lying phonon states and then, at low voltages,
it inhibits the current as the transitions between those states have exponentially
suppressed rates. However, for V > Vt (λ) the transition l : 0 → int[λ2 ] is allowed
and, as it correspond the greatest Franck–Condon factor, [158] it gives a substantial
contribution to the current. In this case relaxation has the opposite effect and it
sustains the current “feeding” the population of the vibrational ground state. For
V ∼ Vt (λ), these two mechanisms coexist and, consequently, the current depends
only weakly on relaxation (see inset in Fig. 5.8).
This observation fits nicely what is reported in literature. [119, 120] In fact, for
λ2 < 2 the critical voltage is smaller than the energy required to have phonon–
assisted tunneling and then the current is enhanced by phonon relaxation at any
voltage, consistently to what was observed in Ref. [119] Vice versa for very strong
e–ph coupling the enhancement of the current due to relaxation can be hardly
seen [120] as Vt (λ) shifts to very large voltages.
All the previous results are not qualitatively modified by considering a spindegenerate electronic level. As we have discussed above,in the case of strong electronic interaction, a spin-degenerate electronic level is formally equivalent to a spinless single level, apart from an effective doubling of the tunneling-in rates. [167]
In particular, at low temperature the analytic expressions for current and noise
in the limit of strong relaxation, when the system behaves as a single spin de(eq)
(eq)
generate electronic level, become: [144] Isd = e 2Γ̃1 Γ̃2 /(2Γ̃1 + Γ̃2 ) and Fsd =
(eq)
(4Γ̃21 + Γ̃22 )/(2Γ̃1 + Γ̃2 )2 The lowest value for Fsd , in this case is not 1/2 but 5/9.
Performing numerical analysis we have found in the case of a spin degenerate
electronic level qualitatively the same behavior observed in the spinless case, i.e. the
Fano factor shows a non monotonous dependence on the strength of the relaxation
rate. Infact, without relaxation the coupling to an harmonic oscillator generally
Shot noise in NEMS
70
0.4
w
w
w
w
w
0.3
I
=0
=1
= 10
= 100
=∞
0.2
0.22
0.1
0.21
4
eV /ω0
6
0
0
2
4
6
8
10
eV /ω0
Figure 5.8: Current as a function of voltage for λ2 = 3 and for different values of
the relaxation strength w.The dotted line w = ∞ corresponds to case of thermally
distributed phonons. Other parameters ε̄ = 0 and kB T = 0.02ω0 . Inset: zoom of
the plateau around eV = 5ω0 . Current in units eΓ.
leads to an increase of the current noise with respect to the uncoupled electronic
level. However, relaxation of the phononic mode can introduce negative correlation
between different current pulses via emission-absorption of phonons and lead to
a suppression of the Fano factor below 5/9, which is the minimal expected value
without e–ph interaction. Moreover, as for the spinless-case, we have found that
such a suppression is observable only if the source drain voltage V > Vt (λ) - see
Eq. (5.27) -. For certain choices of the parameters we also registered F < 1/2, but
for the spin-degenerate case such a value does not represent any relevant limit.
Toy model
To get some insight in these results, we focus on the low voltage region eV < 4ω 0
and we consider a toy model with only four accessible state, i.e. n = 0, 1 and
l = 0, 1. In this case analytical expressions for current and noise can be derived.
For sake of simplicity, we report only the solutions on resonance (ε̄ = 0) and at zero
temperature
I = eΓ(0)
hX
00
2
+ θ(eV − 2ω0 )
X01 (w + 2X01 − ∆) i
,
2(w + 2X01 )
(5.28)
and
F =
1 θ(eV − 2ω0 )X01 ∆[w2 + w(2X01 − ∆) − X01 ∆]
−
,
2
(w + 2X01 )2 K
(5.29)
where ∆ = X00 − X11 and K = [w(X01 + X11 + ∆) + X01 (2X01 + 2X11 + ∆)].
From Eq. (5.28) it is easy to show that the current is an increasing function
of w only for ∆ > 0 (that is, for λ2 < 2, see Eq. (5.23)). Vice versa, for ∆ < 0
(λ2 > 2) the current decreases for increasing relaxation, in agreement with what
was previously observed. Moreover Eq. (5.29) tells that ∆ > 0 (λ2 < 2) is the
necessary condition to have F < 1/2 in the region 2ω0 < eV < 4ω0 . In fact only in
this case, it exist a threshold value for relaxation
2wt (λ) = ∆ − 2X01 +
q
2 ,
∆2 + 4X01
(5.30)
71
5.4 Results
such that for w > wt the Fano factor is smaller than 1/2. For stronger e–ph
coupling λ2 > 2 (∆ < 0), it is always F > 1/2. This confirms the numerical
estimate eVt (λ) = 2ω0 as the threshold voltage for any λ2 < 2.
Despite the coarseness of the model, Eq. (5.29) accords qualitatively with the
exact numerical solution for eV < 4ω0 (see Fig. 5.9). The agreement is reasonably
good even in the case of weak e–ph coupling, where the phonon distribution is mostly
broadened [119] and one expects the four state approximation to be more inaccurate.
A better agreement can be obtained considering a six state model with n = 0, 1 and
l = 0, 1, 2 but, in this case, the analytic solutions become quite cumbersome and we
don’t report them here for simplicity. The agreement of the four states model with
numerical result suggests that F < 1/2 rather depends on the interplay between
relaxation and vibration–assisted tunneling, than on the possibility to access an
high number of vibrational states.
0.68
0.6
0.62
F
F
F
λ2 = 3
λ2 = 1
λ2 = 0.4
0.5
0.5
0.5
0.44
0.46
0
w
50
0
w
50
0
w
50
Figure 5.9: Fano factor as a function of relaxation strength w for eV = 3ω 0 and
for different values of the e–ph interaction. Red (solid) line: exact numerical result;
blue (dash-dotted) line: result for the four states model - Eq. (5.29); green (dashed)
curve: result for a six states model with n = {0, 1} and l = {0, 1, 2}. Dotted line:
F = 1/2. Other parameters ¯ = 0 and kB T = 0.02ω0 .
Conclusions
In this thesis we have analyzed the effects of two different kind of interactions,
namely the electron-electron interaction and the electron-phonon interaction, on
the shot noise of a single electron device. While studying the influence of the electronic interaction, we have considered a one-dimensional quantum ring connected to
leads by tunneling barriers, as this peculiar geometry allows to analyze the interplay
between Luttinger liquid correlations, Coulomb blockade and Aharanov-Bohm interference. Subsequently, we have focused our attention on the interaction between
a single electronic level and a local phonon mode. This peculiar kind of coupling is
typical of molecular transistors and, more in general, of a number of nanoelectromechanical systems operating in the Coulomb blockade regime. Our main results are
summarized in the following.
Quantum ring. We have investigated the influence of the electronic interaction
on the transport properties of a one dimensional quantum ring. In addition to
the charge tunnel-current, we have considered the angular tunnel-current related to
variations of the persistent current into the ring induced by tunneling. The shot
noise of the charge and angular currents show distinctly different properties. In the
presence asymmetric tunnel barriers, the interactions can drive the shot noise of the
charge current to super-Poissonian values. In this case, a Fano factor larger than 1
indicates bunching and super-Poissonian statistics. Viceversa the angular current
noise is insensitive to the asymmetry of the barriers and to bunching effects, while
it is closely related to the occupation dynamics of states with large angular momentum. The Fano factor of the angular current can be larger than one only in the
presence of attractive interaction in the leads, when an inversion of the populating
ratio is achieved. In order to test these effects experimentally, one needs to separate
from the total noise the magnetization current contributions. This might be done
by measuring the magnetization of a single-wall nanotube ring or semiconductor
ring attached to the leads.
Nanoelectromechanical system. While in the case of the quantum ring we have
assumed an equilibrium distribution of the plasmonic excitations, in the case of a
nanoelectromechanical system we have concentrate on non-equilibrium effects and,
in particular, on how relaxation of the phononic excitations influence the current
shot noise. We have found that while the occupation probability distribution of
the states of the system evolves monotonically towards thermal equilibrium for
increasing relaxation strength, the Fano factor exhibits an highly non–monotonous
behavior. Moreover, for finite relaxation rates it can be even suppressed below 1/2,
which is the minimum expected value in the case of fully relaxed excitations. This
relaxation–induced tendency to order of the electronic transfer through the dot
is unexpected, since we are dealing with an oscillator with incoherent dynamics
coupled to a quantum dot in the sequential tunneling regime. We have analyzed
in detail the onset of this behavior as a function of relaxation, electron-phonon
interactions, and external voltages. We have found that for any value of the electron72
73
Conclusions
phonon coupling, it exists a critical value such that for voltages larger than this
threshold a suppression of the Fano factor below 1/2 is possible. At low voltages,
these results are qualitatively predicted by a four states toy model.
Appendix A
Classical suppression of shot
noise
The shot noise of a single barrier can be understood from simple classical arguments
treating transmission through the barrier as a stochastic process with probability
T of success.
Consider a stream of particles incident on a barrier. If the particles behave
independently, the probability that nT charges are transmitted through the barrier
given that there were n incoming ones, is expressed by a binomial distribution:
n
P (nT |n) =
(A.1)
T nT (1 − T )n−nT .
nT
The average number of transmitted particles hnT i is simply given by
hnT i =
=
∞ X
n
X
n=0 nT =0
∞
X
nT P (nT |n)Pτ (n)
Pτ (n)
n=0
n
X
nT =0
nT
n
T nT (1 − T )n−nT = hniτ T,
nT
(A.2)
(A.3)
where Pτ (n)is the probability
P of having exactly n incident particles in the measurement time τ and hniτ = n nPτ (n).
Following the same steps, one can evaluate the variance h(∆nT )2 i obtaining:
h(∆nT )2 i = hniτ T (1 − T ).
(A.4)
Knowing hnT i and h(∆nT )2 i it is immediate to turn to current and current fluctuations. Infact the mean current flowing through the barrier is
hIi =
q
hnT i = qνT ,
τ
(A.5)
where ν = hniτ /τ is the mean rate of incidence. Moreover, in the limit of an
infinitely long time of measurement the zero frequency shot-noise is simply related
to the variance of the number of transmitted particles:
2q 2
2q 2
h(∆nT )2 i =
hniτ T (1 − T ).
τ →∞ τ
τ
S = lim
(A.6)
Expressing S in term of the average current Eq.(A.5)
S = 2qhIi(1 − T ),
74
(A.7)
75
it emerges clearly that the shot-noise of a tunnel barrier is always smaller than to
the Poisson limit SP oisson = 2qhIi. In particular, the shot-noise vanish in the limit
of a truly ballistic system (T = 1). The shot-noise measure the extra randomness
introduced into the flow of particles by the transmission process; if T = 1 there is
no randomness and therefore there is no shot noise.
In the opposite limit of very small transparency T , the binomial distribution
Eq.(A.1) can be approximated by the Poisson distribution and the Schottky formula
S = 2qhIi is recovered again.
Finally, we remark that even if the Schottky problem [4] (i.e. electron motion in
vacuum tubes) is often referred as ballistic, it is in fact a problem in which carriers
have been emitted by a source into the vacuum by tunneling through a barrier with
very small transparency.
This simple derivation fails for small biases eV < kB T as we have not included
thermal fluctuations and we assumed that all the noise is due to current flowing
through the sample. Consequently, Eq.(A.7) has to be thought of as the limit of
the shot-noise for very large voltages (eV kB T ), where thermal fluctuations are
not important.
Appendix B
The current correlation function
In this appendix we review Korotkov’s technique [94] for computing the current
shot noise in the rate eqution formalism.
As discussed in Chapter 3, in the sequential tunneling regime the dynamics of
the dot can be described in terms of a distribution function Pα (t) hat gives the
probability of finding the island at time t in a particular state |αi. The time
evolution of Pα (t) forms a Markov process and the corresponding rate equation is
given in Eq.(3.5). The conditional probability to find the system in the state |αi at
time t given that it was in |βi at t = 0 is given by
P (α, t|β, 0) = eMt αβ
(B.1)
where M is the “evolution” matrix defined in Eq.(3.7). Given P (α, t|β, 0), we can
easily evaluate the correlation function of any observable relative to the dot X (i.e.
of any observable which is only a function of the state of the dot Xα ≡ X(|αi) ).
As an example, if we consider the number N of electrons into the dot we have
X
hN (t)N (0)i =
Nα Nβ P (α, t|β, 0)Pβ0 , t > 0
(B.2)
α,β
and the solution for t < 0 is determined by the symmetry hN (t)N (0)i = hN (0)N (−t)i.
However, an expression simlar to Eq.(B.2) for h∆I (i)(t) I (j)(0) i is not valid as
the current is not a function of the state of the dot. Hence the current correlator
has to be calculated anew, taking into account that in the current in the sequential
tunneling regime is generated by discrete charge transfer process. The current across
each junction may then be written as a sum of δ-like contributions
X X (i)
(i,α,β)
I (i) (t) = e
ςαβ δ(t − tk
),
(B.3)
k
α,β
(i,α,β)
where tk
(k = 1, 2, . . . ) represent the times at which the transition |αi → |βi
(i)
through the i-th junction take place and the quantity ςαβ ≡ (−1)i (Nα − Nβ ) stands
for the number of electrons transfered through the barrier in that particular transition. Its sign reflects whether the transfer occurs in the positive or negative direction. As in the sequential tunneling regime transitions are possible only between
(i)
states with neighbouring charges, it is ςαβ ∈ {−1, 1}.
If we take the average of this expression over the stationary state and we compare
with Eq.(3.14), we obtain the relation
X
(i,α,β)
(i)
hδ(t − tk
)i = Γα→β Pα0 .
(B.4)
k
76
77
Let us now turn to the current-current correlator. When substituing Eq.(B.3) into
the correlator hI (i) (t0 )I (j) (t)i, we obtain a multiple sum where each summand is
proportional to the average of the product of two δ-functions
X X X (i) (j)
(i,α,β)
ςαβ ςγδ hδ(t0 − tk
)δ(t − tq(j,γ,δ) )i.
(B.5)
hI (i) (t0 )I (j) (t)i = e2
k,q α,β γ,δ
(i,α,β)
As the set of all times tk
forms the time lattice of the Markov process, the times
(i,α,β)
(j,γ,δ)
are considered to be pairwise different, i.e. tk
6= tq
whenever {k, i, α, β} 6=
{q, j, γ, δ}. In other words, as in the sequential tunneling regime the probability
of simultaneous tunneling events is equal to zero, different tunneling events occur
at different times. The correlator can then be separated into auto-correlation and
cross-correlation contributions
hI (i) (t0 )I (j) (t)i = hI (i) (t0 )I (j) (t)ia + hI (i) (t0 )I (j) (t)ix
(B.6)
where
XX
(i)
= e2 δ i,j
hI (i) (t0 )I (j) (t)ix
= “terms with tk
k
α,β
(i)
(i,α,β)
ςαβ ςαβ hδ(t0 − tk
hI (i) (t0 )I (j) (t)ia
(i,α,β)
(i,α,β)
)δ(t − tk
6= tq(j,γ,δ) ”
)i, (B.7)
(B.8)
We begin considering the self-correlation of a given tunnel event with itself. Using
(i,α,β)
(i,α,β)
(i,α,β)
the fact that hδ(t0 − tk
)δ(t − tk
)i = δ(t0 − t)hδ(t − tk
)i and applying
Eq.(B.4), we obtain
X (i)
hI (i) (t0 )I (j) (t)ia = e2 δi,j δ(t0 − t)
Γα→β Pα0 .
(B.9)
α,β
(i)
since (ςαβ )2 = 1. attenzione!!!! Dire che stiamo considerando il regime stazionario
e che quindi Pα (t) → Pα0
The cross-correlation terms now involve only distinct processes at different times.
These contributions can be treated in the following way. Let us denote with
dQ(i) (t) = I (i) (t)dt the charge passed through the i-th junction in the interval
(t, t + dt). The cross-correlation term can then be written as
hI (i) (t)I (j) (0)ix =
hQ(i) (t)Q(j) (0)i
dt0 dt
(B.10)
The product dQ(i) (t)dQ(j) (0) is non vanishing only if we have a tunneling event in
junction i during (t, t + dt) and in junction j during (0, dt0 ). The average value
hdQ(i) (t)dQ(j) (0)i is then given by
X X (i) (j)
(i)
(j)
hdQ(i) (t)dQ(j) (0)i = e2
ςαβ ςγδ P (ςαβ , t; ςγδ , 0),
(B.11)
α,β γ,δ
(i)
(j)
where P (ςαβ , t; ςγδ , 0) is the joint probability for having the transition |αi → |βi at
the i-th junction at time t0 and the transition |γi → |δi at the j-th junction at the
initial time. As transport is a stationary Markov process, it is given by
(i)
(j)
(i)
(j)
P (ςαβ , t; ςγδ , 0) = Γα→β dtP (α, t0 − t|β, 0)Γγ→δ dt0 Pγ0 ,
(B.12)
where we have assumed t > 0. This leads to
X X (i) (j) (i)
(j)
hI (i) (t)I (j) (0)ix = e2
ςαβ ςγδ Γα→β P (α, t|β, 0)Γγ→δ Pγ0 ,
(B.13)
α,β γ,δ
The current correlation function
78
for t > 0. The expression for t < 0 can be readily obtained by exploiting the
symmetry
hI (i) (t)I (j) (0)i = hI (i) (0)I (j) (−t)i.
(B.14)
Let us emphasize that in Eq.(B.13), the initial state for the evolution (during time
t) is not γ but δ. One can say that the current at t = 0 behaves as an operator
which change the state of the dot.
We can write the the previous results in a compact form by adopting the notations of Chaper 3
(i)
(i)
(i)
P (α, τ |β, 0) = eMτ α,β
Iαβ = ςαβ Γα→β ,
(B.15)
With this definitions the current-current correlator becomes
hI (i) (t)I (j) (0)i = δi,j δ(t)Tr[ |I (i) | P0 ]
(i) Mt (j)
+θ(t)Tr[ I e
I P0 ] + θ(−t)Tr[ I
(B.16)
(j) −Mt (i)
e
I P0 ]
The correlation function for current fluctuations h∆I (i) (t)∆I (j) (0)i can be simply
obtained by replacing I (i) with δI (i) ≡ I (i) − hI (i) i/e in the previous expression.
Appendix C
The polaron transformation
In this appendix we give a detailed derivation of the canonical transformation
Eq.(5.12), which we introduced to eliminate the e–ph coupling Hn,b .
Without coupling to the environment Hb,env , the problem is formally analogous to the independent boson model [164], and the system Hamilonian Eq.(5.2) is
diagonalized by the Lang-Firsov transformation
H̄ = eS He−S
(C.1)
with S = λ(b† − b)n. In the presence of Hb,env this transformation have to be generalized to include also the momentum operators of the environment. We consider
the following form for the generator of the transformation
X
κj (a†j − aj ),
(C.2)
S = An,
A = κ(b† − b) +
j
where κ, κj ∈ R. With this definition, we calculate the transformation of the
relevant annihilation operators applying the Baker-Campbell-Hausdorff formula
eA Be−A =
∞
X
1
1
1
[A, B]m ≡ B +[A, B]+ [A, [A, B]]+ [A, [A[A, B]]]+. . . (C.3)
m!
2
3!
m=0
In this way we obtain
∞
X
Am
[n, dσ ]m = dσ e−A
m!
m=0
d¯σ
= eAn dσ e−An =
b̄
= eAn b e−An =
āj
= eAn aj e−An =
c̄i,kσ
= eAn ci,kσ e−An = ci,kσ
∞
X
κm n m †
[(b − b), b]m = b − κ n
m!
m=0
∞
m
X
κm
j n
[(a†j − aj ), aj ]m = aj − κj n
m!
m=0
(C.4)
(C.5)
(C.6)
(C.7)
where we have used [n, dσ ]m = (−1)m dσ and [b† − b, b] = −1. The corresponding
creation operators are imediately obtained by taking the Hermitian conjugate
d¯†σ = d†σ eA ,
b̄† = b† − κ n,
ā†j = a†j − κj n.
(C.8)
Inserting these expressions into the transformed Hamiltonian
H̄ = eAn Ha−An = H̄s + H̄leads + H̄t + H̄env + H̄b,env
79
(C.9)
The polaron transformation
80
one obtains
H̄s
H̄leads
H̄t
= Hd + Hb + ω0 (λ − κ)(b† + b)n + ω0 κ2 n2
= Hleads
X
=
t0 (c†k,i e−A d + d† eA ck,i )
(C.10)
(C.11)
(C.12)
k,i=1,2
H̄env
H̄b,env
= Henv −
X
= Hb,env −
j
ωj [κj (a†j + aj )n − κ2j n2 ]
X
j
(C.13)
χj ωj [2κj (b† + b)n + 2κ(a†j + aj )n − 4κκj n2 ] (C.14)
Is now a matter of simple algebra to show that all the e–ph terms are eliminated
by choosing
λ
P
and
κj = −2χj κ
(C.15)
κ=
1 − 4 j χ2j ωj /ω0
This way, one comes to the final result
H̄ = H̄d + Hb + Hleads + H̄t + Henv + Hb,env
(C.16)
where we have included all the terms proportional to n2 = n + 2n↑ n↓ into H̄d =
ε̄n + Ūn↑ n↓ ,renormalizing the the single-particle and of the charging energy, i.e.
ε → ε̄ = ε − λκω0 and U → Ū = U − 2λκω0 .
This is exactly Eq(5.14).
Appendix D
The transition rates
In this appendix we evaluate the transition rates induced by Hint = H̄t + Hb,env
to the lowest order in the coupling constant ti , χi . For convenience, we report here
the Hamiltonian of the NEMS in the canonically transformed base
H̄ = H̄s + Hleads + Henv + H̄t + Hb,env
(D.1)
with
H̄s
=
Hleads
=
H̄d + Hb = ε̄n + Ū n↑ n↓ + ω0 (b† b + 1/2)
X X
εi,k c†i,kσ ci,kσ
(D.2)
(D.3)
i=1,2 k,σ
Henv
X
=
ωj (a†j aj + 1/2),
(D.4)
j
Ht
XX
=
ti (c†i,kσ e−A dσ + d†σ eA ci,kσ )
(D.5)
i=1,2 k,σ
Hb,env
X
=
χj ωj (a†j + aj )(b† + b).
(D.6)
j
We treat the interaction term Hint = H̄t + Hb,env as a perturbation to H0 =
H̄s + Hleads + Henv . The transition betwen two eigenstates of H0 can be evaluated
to the lowest order in Hint by means of Fermi’s Golden rule
Γ|ii→|f i = 2π|hf |Hint |ii|2 δ(Ei − Ef ).
(D.7)
In the absence of coupling, the eigenstates of H0 are simply given by the tensor
product of the eigenstates of Hs , Hleads and Henv . We denote the eigenstates of
H̄s as |ν, li, with ν ∈ {0, σ, 2} and l ∈ {0, 1, 2, . . . }, and we call Eν,l corresponding
eigenenergies. Analogously, we indicate with |Li, |Ri and EL , ER the eigenstates
and the eigenenrgies of Hleads and Henv . For definiteness, let us call |ii = |ν, li|Li|Ri
and |f i = |ν 0 , l0 i|L0 i|R0 i. Writing down |hf |Hint |ii|2 explicitely
|hf |Hint |ii|2 =
|hf |H̄t |ii|2 + |hf |Hb,env |ii|2
+ hi|H̄t |f ihf |Hb,env |ii + hi|Hb,env |f ihf |H̄t |ii
(D.8)
one can easily convince oneself that there are only two non vanishing contributions,
namely |hf |H̄t |ii|2 and |hf |Hb,env |ii|2 . The mixed terms vanish as the number of
particles cannot be conserved. i.e. hf |H̄t |ii ∝ δν 0 ,ν+σ and hi|Hb,env |f i ∝ δν 0 ,ν
therefore hi|Hb,env |f ihf |H̄t |ii = 0.
81
The transition rates
82
Relaxation rates
Let us start considering the transitions induced by |hf |Hb,env |ii|2 , which we call
relaxation rates
2
Γrel
(D.9)
|ii→|f i = 2π|hf |Hb,env |ii| δ(Ei − Ef ).
Actually, we are only interested in transitions between two different states of the
system, while the state of the leads and of the environment remains unobserved. The
total rate for the transition |ν, li → |ν 0 , l0 i is obtained from Eq.(D.9) by summing
over all the initial and final states for the environment and the leads, and weighting
each contribution with the probability to find the initial state occupied
XX β β
Γrel
WL WR |hν 0 , l0 , L0 , R0 |Hb,env |ν, l, L, Ri|2δ(Ei − Ef )
|ν,li→|ν 0 ,l0 i = 2π
L,L0 R,R0
(D.10)
where Ei = Eν,l + EL + ER and Ef = Eν 0 ,l0 + EL0 + ER0 and where WLβ , WRβ are
the equilibrium distributions for the leads and for the bosonic environment. The
modulus squared of matrix element is given by
|hν 0 , l0 , L0 , R0 |Hb,env |ν, l, L, Ri|2
X
χ2j ωj2 |hν 0 , l0 , L0 , R0 |(a†j + aj )(b† + b)|ν, l, L, Ri|2
=
j
=
X
j
χ2j ωj2 |hR0 |(a†j + aj )|Ri|2 |hl0 |(b† + b)|li|2 δν 0 ,ν δL0 ,L
= δν 0 ,ν δL0 ,L [(l + 1)δl0 ,l+1 + l δl0 ,l−1 ]
X
j
χ2j ωj2 |hR0 |(a†j + aj )|Ri|2
from which we see that considering Hb,env to second order allows only transitions
between states with neighbooring vibrational number l ↔ l ± 1. Let us focus on the
transition l → l − 1. Inserting this expression into Eq.(D.10) we obtain
X
X β
χ2j ωj2
WR |hR0 |(a†j + aj )|Ri|2 δ(ω0 + ER − ER0 ).
Γrel
|ν,li→|ν,l−1i = 2π l
j
R,R0
To evaluate the contribution of the bosonic environment we exploite the fact that
the eigensates |Ri can be written in terms of the occupation numbers nj of each of
the vibrational mode of the environment ωj , i.e. |Ri = |n0 , n1 , . . .i (nj ∈ {0, ∞}),
and that the occupation probability factorize WRβ = Πj Wnβj with Wnβj = e−βωj nj /Z.
This way we obtain
X β
WR |hR0 |(a†j + aj )|Ri|2 δ(ω0 + ER − ER0 ) =
R,R0
=
∞
X
1 −βωj nj
1
δ(ω0 − ωj )
e
[nj δ(ω0 + ωj ) + (nj + 1)δ(ω0 − ωj )] =
Z
1 − e−βωj
n =0
j
Collecting results, the rate for the transiton |ν, li → |ν, l − 1i is given by
Γrel
|ν,li→|ν,l−1i = 2π l
∞
X
χ2j ωj2
j=0
1
δ(ω0 − ωj ).
1 − e−βωj
Introducing spectral density of the the environmental coupling
X
J (ω) = 2π
ωj2 χ2j δ(ω − ωj )
j
(D.11)
(D.12)
83
we come to final result Eq.(5.20)
Γrel
l→l−1 = l
J (ω0 )
.
1 − e−βω0
(D.13)
where we have eliminated the reduntant index ν for simplicity. Following the same
steps one can calculate the rate for the upward transition l → l + 1, obtaining
Γrel
l→l+1 = (l + 1)
J (ω0 )
.
eβω0 − 1
(D.14)
Comparing this result with Eq.(D.13) we see that the relaxation rates obey the
detailed balance
βω0 rel
Γrel
Γl−1→l .
(D.15)
l→l−1 = e
Tunneling rates
Let now turn to evaluate the tunneling rates. A little care has to be taken while
considering |hf |H̄t |ii|2 , as the renormalized tunneling matrix element ti e−A contains
terms of any order in the couplig constant χj (see Eq.(5.12)). Writing e−A in powers
of χj one obtains
ti e−A = ti e−λ(b
†
−b)
1 − 2λ
X
j
χj (a†j − aj ) + . . . .
(D.16)
Since we are treating Hint to the lowest order in the coupling constants ti , χj , when
evaluating the tunneling rates we have to discard all the terms of Eq.(D.16) which
contains χj . Therefore we are left with
2
XX
†
†
ti hf |(c†i,kσ e−λ(b −b) dσ + d†σ eλ(b −b) ci,kσ )|ii δ(Ei − Ef )
Γ|ii→|f i = 2π i=1,2 k,σ
XX †
†
= 2π
t2i |hf |c†i,kσ e−λ(b −b) dσ |ii|2+|hf |d†σ eλ(b −b) ci,kσ )|ii|2 δ(Ei − Ef ).
i=1,2 k,σ
(D.17)
From this expression we see that there are four kind of independent contributions,
which correspond to an electron tunneling into or out of the island from the two
different barriers. In other words, to the lowest order in ti , there is no coherence
between the two tunneling junctions.
Analogously to what we have done for ralaxation rate, to obtain the total rate for
the transition |ν, li → |ν 0 , l0 i we have to sum Eq.(D.17) over all initial, weighted
with the probability to find these states, and over all final states.
For definiteness, we consider the transition |σ, li → |0, l 0 i through the first junction.
X X β βX
†
WL WR
t21 |h0, l0 , L0 , R0 |c†1,kσ0 e−λ(b −b) dσ0 |σ, l, L, Ri|2 ×
Γ1 σ→0
l→l0 = 2π
L,L0 R,R0
k,σ 0
× δ(ε̄ + ω0 (l − l0 ) + EL − EL0 + ER − ER0 )
X
X
†
= 2πt21 |hl0 |e−λ(b −b) |li|2
WLβ1 |hL01 |c†1,kσ0 |L1 i|2 δ(µ(l − l0 ) +EL1−EL01 )
k L1 ,L01
(D.18)
(1)
0
where we have introduced the notation Γ1 σ→0
l→l0 ≡ Γ|σ,li→|0,l0 i and µ(l − l ) ≡ ε̄ +
ω0 (l −l0 ). The only possible states with non vanishing matrix element hL01 | c†1,kσ |L1 i
are |L1 i = | . . . , 0kσ , . . .i and |L01 i = | . . . , 1kσ , . . .i where this notation means that
The transition rates
84
the state with wavevector k and spin projection σ is unoccupied in |L1 i while it is
occupied in |L01 i. The occupation of states with other quantum numbers is arbitrary.
Therefore, we obtain
X
2
0 −λ(b† −b)
Γ1 σ→0
|li|2
[1 − f (ε1,k − µ1 )] δ(µ(l − l0 ) − ε1,k ) (D.19)
l→l0 = 2π t1 |hl |e
k
where we have exploited the fact that the probability WLβ1 factorize, i.e. WLβ1 =
Πkσ Wnβ1,kσ and that W1β1,kσ = f (ε1,k − µ1 ) and W0β1,kσ = 1 − f (ε1,k − µ1 ), since
we have assumed the leads to be in equilibrium with the local chemical potential.
Replacing the sum over k with an integral over energy, we obtain
Z
σ→0
2
0 −λ(b† −b)
2
Γ1 l→l0 = 2π t1 |hl |e
|li|
dερ1 (ε)[1 − f (ε − µ1 )] δ(µ(l − l0 ) − ε) (D.20)
P
where ρ1 (ε) = k δ(ε−ε1,k ) is the density of the states (DOS) of the lead. Assuming
the DOS to be constant over the relevant energy interval ρ1 (ε ≈ ρ1 ), we come to
the final simple result
0
Γ1 σ→0
l→l0 = Γ1 Xll0 [1 − f (µ(l − l ) − µ1 )]
where Γ1 = 2π t21 ρ1 is the bare tunneling and Xll0 = hl0 |e−λ(b
Condon matrix element.
(D.21)
†
−b)
|li|2 is the Franck-
The remaining tunneling rates can be evaluated in a similar fashion. This way we
obtain
0→σ
Γi l→l
0
Γi σ→2
l→l0
Γi σ→0
l→l0
2→σ
Γi l→l
0
= Γi Xll0 f (µ(l0 − l) − µi )],
(D.22)
= Γi Xll0 f (µ(l0 − l) + Ū − µi ),
(D.23)
= Γi Xll0 [1 − f (µ(l − l ) − µi )],
(D.24)
0
= Γi Xll0 [1 − f (µ(l − l0 ) + Ū − µi )].
(D.25)
Franck-Condon factors
Now, we are only left with the evaluation of the Franck-Condon factors
Xll0 = hl0 |e−λ(b
†
−b)
|li|2 .
(D.26)
†
To evaluate the matrix element hl 0 |e−λ(b −b) |li, it is convenient to “disentangle” the
†
displacement operator e−λb eλb by means of the relation
1
eA+B = eA eB e− 2 C ,
C = [A, B]
(D.27)
which holds if [A, C] = [B, C] = 0. This way we obtain
hl0 |e−λ(b
†
−b)
1
2
†
|li = e− 2 λ hl0 |e−λb eλb |li.
(D.28)
We can then easily calculate
s
∞
l
X
X
λm m
l!
λm
e |li =
b |li =
|l − mi
m!
m! (l − m)!
m=0
m=0
s
l0
X
l0 !
(−λ)n
0 −λb
hl0 − n|
hl |e
=
0
n!
(l
−
n)!
n=0
λb
(D.29)
(D.30)
85
Eploiting the orthogonality relation
hl0 − n|l − mi = δl0 −n,l−m
(D.31)
and assuming l0 > l one obtains
0
hl |e
−λ(b† −b)
|li = e
−λ2 /2
= e−λ
2
/2
(−λ)
l0 −l
0
(−λ)l −l
r
r
l
l0 !
l! X
(−λ2 )m
0
l ! m=0
m!(l − m)!(l0 − l + m)!
(D.32)
l! l0 −l 2
L (λ )
l0 ! l
where Lkn (x) is the associated Laguerre polynomial
Lkn (x) =
k
X
(−x)m
m=0
m!(n + k)!
.
(n − m)!(k + m)!
†
The expression for l > l 0 can be obtained using hl 0 |e−λ(b
the end, we get the following result for arbitrary l, l 0
s
hl0 |e−λ(b
†
−b)
|li = e−λ
2
/2
0
[sgn(l − l0 )λ]|l −l|
−b)
|li = hl|eλ(b
l< ! |l0 −l| 2
L
(λ )
l> ! l <
(D.33)
†
−b) 0 ∗
|l i . In
(D.34)
where l< = min{l, l0 } and l> = max{l, l0}. Taking the modulus squared of Eq.(D.34)
we get the final expression for the Franck-Condon factors
Xll0 = |hl0 |e−λ(b
†
−b)
2
0
|li|2 = e−λ λ2|l −l|
l< ! |l0 −l| 2 2
|L
(λ )| .
l> ! l <
(D.35)
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