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Preface
This thesis work was carried out in Quantum Computing and Devices
group, QCD Labs, at the Department of Applied Physics, Aalto University
during 2012-2016. We collaborated tightly with an Australian research
group based at the University of New South Wales, Sydney. The earliest
publication was carried out in collaboration with PICO group at Aalto
University.
I started in QCD Labs as a summer student in 2009. The group leader
Prof. Mikko Möttönen supervised me remotely from Australia. After one
year of searching myself somewhere else, I returned to make my Master’s
thesis and later on this PhD thesis in QCD Labs. Thank you Mikko for
all the guidance and supervision during these years. I also would like to
express my gratitude to Prof. Matti Kaivola for supervision and providing
the infrastructure as the head of the department of applied physics.
During the years of working I have met many awesome colleagues both
in- and outside the laboratory — many of them which I consider great
friends. Especially, I would like to thank Dr. Kuan Yen Tan for introducing me to the charge pumping project and lab work in general. He
also built mostly the cryostat, which I shamelessly hogged for almost four
years. My gratitude goes also to soon-to-be-PhD Joonas Govenius who I
have teased with myriad of questions related to technology, science, computers, and life in general. Dr. Russell Lake and M. Sc. Matti Partanen I
thank for all the great and unforgettable years together in the lab.
Many thanks to all the wonderful — old and current — members of
QCD Labs: Sofia, Shumpei, Kostya, Tuomas, Emmi, Santeri, Roope, Joni,
Pekko, Ivan, Ville, Elsi, Jukka, Juha, Philip, Harri, Janne, Jani, Paolo,
and Yuchiro. You all keep up the good spirit in our group. Also thanks to
the summer students I supervised: Kukka and Akseli. I also acknowledge
i
Preface
M. Sc. Mate Jenei, a brilliant PhD student, who will continue my legacy.
I express my gratitude to the collaborators in Sydney: Prof. Andrew
Dzurak, Dr. Alessandro Rossi, B. Sc. Yuxin Sun, B. Sc. Ruichen Zhao,
Dr. Kok Wai Chan, Dr. Giuseppe Tettamanzi, Prof. Sven Rogge, and.
Dr Fay Hudson. I thank also to all the members of our, beloved, neighbouring PICO group. Especially, I acknowledge Prof. Jukka Pekola, Dr.
Olli-Pentti Saira, and Dr Yong-Soo Yoon for collaboration and supervision.
I also thank Doc. Matthias Meschke, M. Sc. Anna Feschenko, Dr. Alexander Savin, Markku Korhonen, and Jari Isomäki for participating in building the cryostat. I thank Dr. Ilkka Iisakka and Prof. Dmitri Averin for
collaboration as well as Dr. Antti Kemppinen, Dr. Janne Lehtinen, and
M. Sc. Emma Mykkänen for inspiring discussions. For financial support
I acknowledge Jenny and Antti Wihuri Foundation, Emil Aaltonen Foundation, Education Network in Condensed Matter and Materials Physics,
National Graduate School in Materials Physics, Academy of Finland, and
European Research Council.
Of course life would be nothing without excellent friends. For the great
company I thank Joonas, Sini, Juho, Minna, Otto, Antti, Michi, Kai, Miska,
Matti, Aavee, Artti, Anu, and Jukka. Special thanks also to my flatmates
Teppo and Toni. Finally, I would like to thank my brother Kimmo and my
parents Leena and Jukka for all the love and support over the years. You
are the best!
Espoo, May 9, 2016,
Tuomo Tanttu
ii
Contents
Preface
i
Contents
iii
List of Publications
Author’s Contribution
v
vii
1. Introduction
1
2. Single-electron tunneling effects
7
2.1 Tunnel junction . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.2 Coulomb blockade and single-electron transistor . . . . . . .
8
3. Sample fabrication and measurement set-up
13
3.1 Fabrication processes . . . . . . . . . . . . . . . . . . . . . . .
15
3.2 Measurement set-up . . . . . . . . . . . . . . . . . . . . . . .
15
4. Single-electron pumping and current metrology
19
4.1 Requirements for the SI ampere and the closure of QMT . .
19
4.2 Electron pumping in silicon quantum dots . . . . . . . . . . .
20
4.3 Electrical confinement and the accuracy of the dot . . . . . .
23
4.4 Electron counting and its consistency with the direct output
current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
4.5 Three-waveform bidirectional pumping . . . . . . . . . . . .
29
5. Jarzynski equality and Crooks fluctuation theorem
5.1 Non-equilibrium fluctuation relations . . . . . . . . . . . . .
31
31
5.2 Experimental study of fluctuation relations in a single-electron
box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
iii
Contents
6. Summary and conclusions
37
References
41
Errata
51
Publications
53
iv
List of Publications
This thesis consists of an overview and of the following publications which
are referred to in the text by their Roman numerals.
I A. Rossi, T. Tanttu, K. Y. Tan, I. Iisakka, R. Zhao, K. Chan, G. Tettamanzi, S. Rogge, A. Dzurak, and M. Möttönen. An accurate singleelectron pump based on a highly tunable silicon quantum dot. Nano
Letters, 14, 6,3405–3411 (7 pages), May 2014.
II A. Rossi, T. Tanttu, F. E. Hudson, Y. Sun, M. Möttönen, and A. Dzurak.
Silicon metal-oxide-semiconductor quantum dots for single-electron pumping. The Journal of Visualized Experiments, 100, e52852 (11 pages),
article video available at http://www.jove.com/video/52852/silicon-metaloxide-semiconductor-quantum-dots-for-single-electron, May 2015.
III T. Tanttu, A. Rossi, K. Y. Tan, K.-E. Huhtinen, K. W. Chan, M. Möttönen, and A. Dzurak. Electron counting in a silicon single-electron pump.
New Journal of Physics, 17, 10, 103030 (6 pages), September 2015.
IV T. Tanttu, A. Rossi, K. Y. Tan, A. Mäkinen, K. W. Chan, M. Möttönen,
and A. Dzurak. Three-waveform pumping of single electrons with a silicon quantum dot. Submitted to Scientific Reports, arXiv:1603.01225 (9
pages), March 2016.
V O.-P. Saira, Y. Yoon, T. Tanttu, M. Möttönen, D. V. Averin, and J. P.
Pekola. Test of the Jarzynski and Crooks fluctuation relations in an
v
List of Publications
electronic system. Physical Review Letters, 109, 18, 180601 (5 pages),
October 2012.
vi
Author’s Contribution
Publication I: “An accurate single-electron pump based on a highly
tunable silicon quantum dot”
The author carried out the experiments and the data analysis together
with the first author and helped in the preparation of the manuscript.
Publication II: “Silicon metal-oxide-semiconductor quantum dots for
single-electron pumping”
The author wrote a part of the manuscript, composed three published figures, and commented on the screenplay of the corresponding video, where
he was a member of the cast.
Publication III: “Electron counting in a silicon single-electron pump”
The author carried out most of the experiments and the data analysis,
wrote the manuscript, and made the video abstract for the manuscript.
Publication IV: “Three-waveform pumping of single electrons with a
silicon quantum dot”
The author carried out most of the experiments and data analysis and
wrote the manuscript.
vii
Author’s Contribution
Publication V: “Test of the Jarzynski and Crooks fluctuation
relations in an electronic system”
The author actively participated in carrying out the measurements together with the first author, helped with the analysis, and commented on
the manuscript.
During the PhD studies the author has also contributed to an additional
article [1], not included to this thesis.
viii
1. Introduction
In 1909, Robert Millikan and Harvey Fletcher obtained experimental evidence suggesting that there is a smallest possible quantum of charge,
which they proposed to be the charge of the electron [2]. Nowadays the
absolute magnitude of electron charge is referred to as the elementary
charge since it is the smallest observed solitary charge, and hence it is
considered as one of the fundamental constants of nature.
Electrons, which all have the same negative charge, repel each other due
to the Coulomb interaction. This effect can be utilized in the nanometer
scale systems, where the interaction strength can be dominant for two individual electrons. Coulomb blockade, an effect owing to the discreteness
of charges and their mutual repulsion, enables us to control the positions
of individual electrons precisely in condensed matter systems [3, 4]. This
effect allows, for instance, us to implement accurate single-charge sensors and sources of quantized current [5], both of which can be used for
metrological purposes.
The modern International System of Units (SI) has been developed since
the 19th century. Today the SI has seven base units: second (time), kelvin
(temperature), meter (length, distance), kilogram (mass), ampere (electrical current), mole (amount of substance), and candela (luminous intensity) [6]. All other units in this system such as the volt (voltage) and the
pascal (pressure) can be derived from these base units. Initially the metric system, established in 1791 in France, had two units: the meter and
the kilogram. The second, established already in the medieval times, was
added to the system in 1832. In the late 19th century, also the ampere
was adapted, followed by the mole, the candela, and the kelvin in the
20th century.
Initial definitions for the meter, second, kilogram, and kelvin were based
1
Introduction
on the dimensions and dynamics of the earth and the properties water. At
the end of the 19th century, the meter and the kilogram were redefined to
be equal to the length and mass of model artifacts located in Paris. The
kilogram is still based on this more than 100-year-old model artifact, the
weight of which is known to drift over time [7]. The second was originally
defined as a fraction of a day and the kelvin was based on the absolute
temperature scale invented by Lord Kelvin.
Later on the development of measurement techniques and the need for
more accurate and mobile standards have lead to the redefinition of the
meter and the second. The second is today defined as a multiple of a period
of certain radiation from a caesium-133 atom. Since 1983, the definition of
the meter has been based on a fixed value for the speed of light in vacuum
and on the definition of the second.
Initially, the ampere was defined by the magnetic field which is produced by current flowing in a circular conductor. In 1946, the ampere
was redefined in the SI system based on Ampere’s law: two thin parallel
conductors, one meter apart from each other, have exactly one ampere of
current if they affect each other by a force that equals to 2 × 10−7 N/m.
Since the definition of the ampere is based on a classical effect related to
force which is derived from the kilogram, meter, and second, it is vulnerable to all uncertainties that occur in these standards, especially in the
kilogram. The relative uncertainty in the implementation of the present
SI ampere is roughly 10−7 . As is already the case for the meter, there
has also been major effort for the redefinition of the kilogram and the
kelvin to be based on fixed natural constants: the Planck constant and
the Boltzmann constant, respectively [8]. It would be therefore tempting
to redefine the ampere in such a way that it would also be based on a
fundamental constant, in this case on the elementary charge [9].
Today the so-called conventional electrical units provide electrical standards widely used in the industry [10]. These units are based on the
quantum Hall effect (QHE) yielding a standard for resistance, and on the
Josephson effect (JE) yielding a standard for voltage [11, 12]. Discoveries
of both of these effects have been awarded with Nobel prize. Conventional
electrical units are based on the von Klitzing constant RK and the Josephson constant KJ which are related to QHE and JE, respectively. In the
conventional electrical units, the constants RK−90 and KJ−90 are fixed to
certain values defined in 1990. However, the most recent measurements
2
Introduction
Quantum Hall effect
Josephson effect
Single-electron pumping
Figure 1.1. Quantum metrology triangle that relates three quantum mechanical effects
and their constants. Single-electron pumping (elementary charge, e) gives a
quantum current, quantum Hall effect (von Klitzing constant, h/e2 , where
h is the Planck constant) gives resistance, and Josephson effect (Josephson
constant, 2e/h) gives voltage.
have revealed that these constants differ from RK by 2 × 10−8 and from KJ
by 6 × 10−8 [13]. The closure of the so-called quantum metrology triangle
(QMT) depicted in Fig. 1.1 is an experiment that would check the consistency of the theories and constants related to QHE and JE [9, 14]. In order to close the metrology triangle we require a quantized single-electron
current source which can output an integer n number of electrons with
a fixed frequency f , yielding total dc current I = nef . Furthermore, it
should provide output current of at least hundreds of picoamperes with
relative error of 10−8 or lower [15–17].
The idea of single-electron current sources was suggested in 1986 [18].
The first single-electron current sources, based on metallic structures,
were demonstrated in 1990 [19–22]. The first semiconductor-based quantum dot source was presented a year later [23]. These sources yielded
currents of a few picoamperes with the relative uncertainty of per cents.
As soon as 1994, the first single-electron source with relative uncertainty
below 10−6 was demonstrated [24] and two years later with uncertainty
of roughly 10−8 [25]. These sources are based on multiple normal-metal
islands and their output current is limited to few picoamperes. Other
kind of quantized current sources were suggested in order to increase the
3
Introduction
output current. Semiconductor devices using surface acoustic waves can
produce a high current but due to, for example, heating they have not
been accurate enough [26–28]. Other suggestions such as normal-metal–
superconductor hybrid turnstiles [29–32], quantum-phase-slip superconducting devices [33–36], and impurities in semiconductors [37–42] have
been proposed. At the moment, the most promising candidates are devices based on semiconductor quantum dots [43–58].
The two most widely studied platforms for semiconductor based quantum dot pumps are silicon metal–oxide–semiconductor (MOS) devices and
GaAs/AlGaAs-heterostructures. A direct current of 87 pA with 0.2 parts
per million (ppm) has been achieved in GaAs/AlGaAs heterostructure quantum dots [49]. Also many other high-accuracy, high-yield quantum dot
architectures made out of this material have been demonstrated [43, 47].
In this thesis, we concentrate on silicon quantum dot pumps that provide an interesting alternative for the realization of the quantum ampere [37–39, 57, 58]. The benefits of the silicon-based pumps are that certain type of devices exhibit suppressed 1/f noise and absence of large
background charge jumps [59–63]. The MOS technology is also widely
used and well-known for the semiconductor industry making the manufacturing of these devices scalable.
In this thesis, our aim is to show that silicon quantum dots are an excellent candidate for the realization of the quantum ampere. We present a
highly tunable device architecture which can be utilized for single-electron
pumping with radio frequency (rf) driving voltages. This architecture allows the electrical enhancement of the pumping process as well as electron counting and bidirectional pumping of the electrons.
In Publication I, we demonstrate that the lateral electrical confinement
of the dot can be used to increase the charging and single-particle excitation energies. Subsequently, this result has been confirmed in other
structures [52]. We also show that a current of 80 pA with uncertainty below 30 ppm, can be obtained with our pump. In Publication II, we discuss
the details of the sample fabrication as well as the measurement set-up
used in most of the experiments. Publication III introduces an electron
counting scheme in our pump architecture. Furthermore, we show that
the average charge transported in the counting experiment matches the
direct current using identical waveforms to drive the pump. In Publication IV, we present a quantum dot set-up with full rf control over the dot
4
Introduction
potential and the surrounding potential barriers. We demonstrate that
by tuning only the phase of one rf drive it is possible to perform bidirectional pumping. We can also improve the pumping process by increasing
the amplitude of the exit barrier.
In this thesis, we also discuss the statistics of single electron transport in metallic structures. In driven nanometer scale systems the socalled Jarzynski equality provides a strong statement relating the work
done on the system to equilibrium variables [64, 65]. The Crooks fluctuation theorem describes the probability of the work done in certain paths
in the phase space [66–68]. Previously, these non-equilibrium theorems
have been studied, for example, in molecular folding/unfolding experiments [69–76]. In Publication V, we experimentally study the Jarzynski
equality and the Crooks fluctuation theorem in a single-electron system.
We find that both of them are consistent with the experimental results.
This thesis is organized as follows. In Chapter 2, we discuss the basics
of single-electron tunneling in condensed matter systems. In Chapter 3,
we introduce the employed fabrication processes and experimental setups. The basic operation principles of this single-electron pump as well as
the main results of this thesis are discussed in Chapter 4. Chapter 5 is
devoted to the Jarzynski equality and Crooks fluctuation theorems, especially in condensed-matter systems. Chapter 6 includes the conclusions of
this thesis and future prospects.
5
Introduction
6
2. Single-electron tunneling effects
Control of individual electrons is the key element in charge pumping and
sensing discussed in this thesis. A tunnel junction, Coulomb blockade,
and single-electron transistor (SET) introduced in this chapter are powerful tools for controlling electrons in condensed-matter systems.
2.1
Tunnel junction
A tunnel junction is typically a capacitive, insulating barrier between two
electrodes. The barrier is so thin that the wave functions of electrons
can penetrate throughout the barrier enabling tunneling through the barrier [3]. Symbols for normal-metal–insulator–normal-metal (NIN) and
superconductor–insulator–normal-metal (SIN) tunnel junction are shown
in Fig. 2.1(a). Tunneling resistance is a quantity which describes how
transparent this potential barrier is. The wider and higher is the potential barrier for the electrons, the higher is the tunneling resistance.
If the insulating layer is oxide, it is not possible to tune the transparency
of the barrier after the oxidation. However, there are systems, such as
quantum dots, where it is possible to tune the tunneling resistance with a
voltage applied to a gate on top of the barrier [77]. In the case of an NIN
junction, the tunneling rate through the junction in one direction is given
by [78–80]
Γ+ (ΔV ) =
1
e 2 RT
∞
−∞
dEf (E, T )[1 − f (E − eΔV, T )],
where e is the elementary charge, f (E, T ) =
1
eE/(kB T ) +1
(2.1)
is the Fermi func-
tion, RT is the tunneling resistance, kB is the Boltzmann constant, T is
the temperature, and ΔV is the voltage between the electrodes.
7
Single-electron tunneling effects
(a)
Qj1 , Cj1 Qj2, Cj2
(c)
(b)
Qj Qg
RT
-
+
Vd /2
- +
Vg
A
Cj C g
Qg
Cg
Vg
Vd /2
Figure 2.1. (a) Symbols for normal-metal–insulator–normal-metal (top) and
superconductor–insulator–normal-metal (bottom) tunnel junctions. (b) Conducting island with tunnel junction and gate capacitance and total
charge Qj − Qg . (c) Schematic circuit diagram of a single-electron transistor
with two tunnel junctions (Cj1 and Cj2 ) which are capacitively coupled to a
gate Cg , with gate voltage Vg which can be used to control the current flowing
through the transistor.
2.2
Coulomb blockade and single-electron transistor
The Coulomb blockade is the key concept in single-charge pumps and in
many other single-electron devices. It can typically be described by classical Coulomb repulsion of electrons in small conductive regions. Let us
consider a simple system presented in Fig. 2.1(b). A conductive island has
a gate capacitance Cg , with charge Qg , gate voltage Vg , and a tunnel junction Cj with charge Qj . The total charge of the island is Q = Qj −Qg = −ne,
where e is the elementary charge and n is the number of excess electrons
in this island. The gate voltage may be expressed in terms of the charges
as Vg = Qg /Cg + Qj /Cj . We solve Qg and Qj and obtain [4, 80]
Q j = Cj U g ,
(2.2)
Qg = Cg (Vg − Ug ),
(2.3)
and
where Ug = (−ne+Cg Vg )/CΣ , is the potential of the island and CΣ = Cj + Cg .
The total energy stored in the system is
E(n) =
Q2j
2Cj
+
Q2g
0
+ Ebatt
− Qbatt Vg ,
2Cg
(2.4)
0
where Ebatt
is the offset energy of the battery and Qbatt is the charge that
has flown through the battery with respect to offset. The total charge
flown through the battery to have in total n excess electrons in the island
is Qbatt = Cg ne/CΣ . Thus we can rewrite the energy of Eq. (2.4) as [4, 80]
E(n) = EC (ng − n)2 −
8
e2 n2g Vg ng e
,
+
CΣ
2
(2.5)
Single-electron tunneling effects
where EC = e2 /(2CΣ ) is the charging energy and ng = Vg Cg /e is the gate
charge.
Let us consider the electrochemical potential of the island, i.e., the energy required to add an electron to the island. The electrochemical potential for nth electron is
μ(n) =E(n) − E(n − 1) = EC (2n − 2ng + 1).
(2.6)
The electrochemical potential difference between n + 1 and n states is
μ(n + 1) − μ(n) = 2EC .
(2.7)
Note that the charging energy describes the electrochemical potential difference between two adjacent charge states of the island and that the gate
voltage can be used to linearly shift all the electrochemical potentials as
described.
Let us consider a single-electron transistor depicted in Fig. 2.1(c), where
Coulomb blockade can occur. A conductive island is connected to the
source and drain leads with tunnel junctions having capacitances Cj1 and
Cj2 and charges Qj1 and Qj2 . If a small bias voltage Vd is applied between
the leads, the current through the system is heavily dependent on the
electrochemical potential of the conductive island. As above this potential can be controlled with gate voltage Vg connected to capacitor Cg with
charge Qg .
Figure 2.2 illustrates the electrochemical potential configuration of the
SET with two different Vg values. Figure 2.2(a) depicts the Coulomb blockade, where the electrochemical potential of n + 1 electrons in the island is
above the Fermi level [81] of the source and the electrochemical potential
of n electrons is below the Fermi level of the drain. In this situation, only
the thermal excitations of electrons would yield a current through the system. Therefore, at low temperatures, there is essentially no current, even
though there is a finite bias between the source and the drain. However, it
is possible to tune Vg so that the electrochemical potential for some number of electrons in the island is between the Fermi levels of the source and
the drain. In this case, depicted in Fig. 2.2(b), the electrons can tunnel
from the source to the island and from the island to the drain without
the need for any excitations and there is a notable current through the
transistor.
If we increase Vd such that Vd >
2EC
e ,
there is always at least one elec-
trochemical potential level between the Fermi levels of the leads and the
9
Single-electron tunneling effects
(a)
(b)
CgVg
2EC
Source
eVd
eVd
Drain
Drain
Source
island
island
Figure 2.2. Coulomb blockade. Schematic illustration of the electrochemical potential
configuration of the single-electron transistor presented in Fig. 2.1(b). The
filled states up to the Fermi level of the leads are denoted by blue and yellow indicates the tunnel barriers. The electrochemical potential of the island
can be tuned with the gate voltage Vg in Fig. 2.1(c). Here, eVd defines the
electrochemical potential difference between the leads. The black solid lines
indicate the electrochemical potential levels of certain charge state. (a) Since
there are no available electrochemical potential levels between the Fermi levels of the leads, current cannot flow through the island. The transistor is in
the Coulomb blockade. (b) With the proper Vg , the electrochemical potential
for n electrons in the dot is between the Fermi levels of the leads and the
electrons can tunnel from the source to the drain through the island.
current can flow. If, however, |Vd | <
2EC
e ,
the value of Vg defines whether
we have current or not. We can measure the current through the system
depicted in Fig. 2.1(c) as a function of Vg and Vd . A schematic diagram
of such measurement outcome is shown in Fig. 2.3(a). The blue regions
in the figure are referred to as Coulomb diamonds. In these regions, the
Coulomb blockade takes place whereas in other regions it is absent.
In Fig. 2.3(b), we show the result of a Coulomb diamond experiment.
The differential conductance of a single-electron transistor is shown in the
logarithmic scale as a function of Vg and Vd . Here, the Coulomb diamonds
are well visible. This measurement was carried out with the quantum
dot device used in most of the publications of this thesis. We observe
that the diamonds enlarge with decreasing gate voltage. This indicates
that we are heavily depleting the dot. In fact, there are no conduction
electrons in the dot at the lowest gate voltages, which is indicated by the
non-closing diamond at the lowest gate voltages. Also the charging energy,
roughly corresponding to the height of the diamond, increases towards
the last electron. With lower gate voltage, the size of the dot is smaller,
and hence it has less capacitance yielding a higher charging energy. In
addition to the simple circuit model discussed above, the heights of the
Coulomb diamonds are affected by the quantization of the energy levels
in the potential well.
The Coulomb blockade is a very powerful tool for controlling individual
10
Single-electron tunneling effects
(a)
ΔVd=2EC/e
I=0
Vd
0
I≠0
Vd (mV)
(b)
Vg
10.0
8.0
6.0
4.0
2.0
0.0
Log10(dI/dV/Ω)
-10
-11
n=0
n=1
n=2
n=3 n=4
-12
-2.0
-4.0
-6.0
-8.0
0.62
-13
0.64
0.66
0.68
Vg(V)
0.70
-14
Figure 2.3. Coulomb diamonds. (a) Schematic current through a single-electron transistor in the parameter space spanned by the source–drain bias voltage (Vd )
and the gate voltage (Vg ). In the blue diamond-shaped regions, referred to as
Coulomb diamonds, the current through the single-electron transistor vanishes. In these regions, the Coulomb blockade takes place as in Fig. 2.2(a).
In the orange region, the current does not vanish and we are in the configuration of Fig. 2.2(b). (b) Differential conductance of the single-electron
transistor dI/dV in the logarithmic scale as a function of source–drain bias
voltage and the gate voltage of the dot in the sample used in Publications
(I)–(IV). The green lines are guides to the eye to show the regions where we
have Coulomb blockade. Data is taken from Ref. [82].
11
Single-electron tunneling effects
electrons. Single-electron transistors employing this effect can be used to
detect the relocation of individual electrons [3, 4]. We exploit this phenomenon in Publication III and Publication V to sense the charge state of
a conductive island. The Coulomb blockade is also extremely important
in the charge pumping schemes — it prevents undesired electrons from
entering the dot during the pump process.
12
3. Sample fabrication and
measurement set-up
In this thesis, we discuss single-electron pumping in silicon quantum dots.
The only exception is Publication V, where we study the statistics of electron transport in a superconductor–normal-metal hybrid structure. The
principles to control electrons are the same in both device types. In this
chapter and in Publication II we discuss the metal–oxide–semiconductor
(MOS) technology used to fabricate the quantum dot devices [83] and the
two-angle shadow evaporation technique used to fabricate the sample in
Publication V. The sample fabrication in all of the publications was carried out by the collaborators. Furthermore, we describe the measurement
set-up used in the silicon pump experiments.
A schematic cross sections of an SET based on MOS and metallic structure are presented in Fig. 3.1. In a silicon quantum dot SET, a conductive
two-dimensional electron gas (2DEG) is induced at the interface between
Si and SiO2 with the gates on the top of the oxide. The transparency of
the tunnel barriers can be tuned with the gate voltages. The higher the
voltage in the gates is, the more transparent the barriers are. It is also
possible to tune the potential of the dot. Furthermore, we can suppress
the size of the dot, which results to a very high charging energy. This
can be utilized in the charge pumping schemes. The charging energy of
our silicon quantum dot can be up to several meV, whereas in metallic
islands, the charging energy is usually of the order of hundreds of μeV.
In the metallic SETs, the barriers are defined by metal-oxides and their
tunneling resistances are fixed after the oxidation of the sample.
13
Sample fabrication and measurement set-up
(a)
+ + +
+
+ + +
SiO2
Doped region
Si
2 DEG
I
(b)
I
SiO2
Si
Figure 3.1. Single-electron transistor cross sections. (a) Sample structure of a silicon
based single-electron transitor. A positive voltage applied to the green gates
induces a two-dimensional electron gas of conduction electrons at the interface between Si and SiO2 . The red gates induce the tunneling barriers between the dot (in the middle) and the source and drain. By tuning the voltage
on these gates, we can tune the tunnel resistance of the sample. (b) Sample
structure of a metallic single-electron transistor on a silicon wafer. In this
structure, the electrons move in the metallic films. The metal oxide (orange)
works as a tunnel junction between the leads and the conductive island. Both
(a) and (b) correspond to the circuit diagram shown in Fig. 2.1(c)
14
Sample fabrication and measurement set-up
3.1
Fabrication processes
The fabrication of our silicon quantum dot sample consists of two parts:
microfabrication and nanofabrication, both of which are presented in Publication II. In the microfabrication process, the ohmic contacts and the
bonding pads for the source and drain are fabricated as well as the thin
gate oxide. Ohmic contacts are formed in regions with heavy phosphorus
doping under the SiO2 , and the bonding pads are formed by metal deposition on top of this region. The thin gate oxide is formed by first etching a
region to fully remove the initial thick layer of SiO2 and then by oxidizing
a 8-nm-thick SiO2 layer on it.
The gate pattern of the device is fabricated to the thin oxide region in
a nanofabrication process. A process for an individual gate layer is depicted in Fig. 3.2. First, the oxide is covered with polymethylmethacrylate (PMMA) resist. Electron beam lithography is used to pattern the
gates to the resist. The resist exposed by electrons is selectively removed
by a solvent. The aluminium gates are formed with metal evaporation.
After the lift-off process where all the resist is removed, the remaining Al
gates are oxidized. This process is repeated three times in order to make
a three-layer device.
The double angle shadow evaporation used to fabricate the sample used
in Publication V is rather similar to the gate layer evaporation process
presented above [80, 84]. In this process, we use two different resists:
methylmetacrylate (MMA) and PMMA on top of it. The lithography is
carried out in the same way as above, but instead of evaporating just one
layer, we evaporate two: an aluminium and a copper layer using different
angles. Between the evaporations, the tunnel junctions are formed by
oxidizing the first layer. In the end, we lift off the resists.
3.2
Measurement set-up
In Publication II, we also discuss the measurement set-up used in Publications I, III, and IV. The measurements in all publications are carried out
below 1-K temperature in a self-made plastic dilution refrigerator (PDR)
utilizing a liquid helium bath. The schematics of the dilution refrigerator
are presented in Fig. 3.3(a). The working principle of a dilution refrigerator is based on a phase separation in a mixture of 3 He and 4 He. The
15
Sample fabrication and measurement set-up
Individual gate layer
2.4.1. Spin resist
PMMA resist
SiO
SiO
22
Si
2.4.4. Evaporaon
2.4.2. E-beam lithography
2.4.3. Development
Resist
SiO2
Resist
SiO2
Si
Si
2.4.5. Li-off
2.4.6. Oxidaon
Al
AlxOy
Resist
SiO2
SiO2
SiO2
Si
Si
Si
Figure 3.2. Gate patterining process for an individual silicon quantum dot gate layer.
Polymethylmethacrylate is used as a resist. The stages are: resist spinning,
electron beam lithography, resist developement, aluminum evaporation, resist lift-off, and oxidation. The figure is reprinted from Publication II with
permission.
operation is based on the circulating 3 He. In the refrigerator, the 3 He is
condensed and then directed to mixing chamber where it dilutes to the
4 He
rich phase. This dilution process is endothermic and it absorbs heat
from the surroundings. In the still of the refrigerator, 3 He is pumped out,
after which the same gas is condensed again. The base temperature of
our PDR is 100 mK. In some experiments, the temperature increases up
to 800 mK due to the rf power applied to the sample.
The sample is attached to a sample holder with a custom-made printed
circuit board (PCB) and connected to it with Al bond wires. The gates
that require rf driving are connected to bias tees with rf and dc components. Coaxial cables are used for rf lines and twisted pair loom lines for
the dc lines. A transimpedance amplifier and optoisolation are used before we record the signal with a multimeter. The dc lines are connected to
floating voltage sources connected through 1:5 voltage dividers. The driving waveforms are generated with an arbitrary waveform generator and
attenuated by 10 dB at 4 K to reduce thermal noise from the room temperature. The schematics of the room temperature connections and the
cryostat wiring are presented in Fig. 3.3(a) and connections to the sample
and the PCB are presented in Fig. 3.3(b).
16
Sample fabrication and measurement set-up
(a)
(b)
Figure 3.3. Schematic measurement set-up for the charge pumps. (a) Schematics of the
dilution refriregator used in all of the publications together with the schematics of the room temperature components and cryostat wiring. (b) Schematics
of the PCB including bias tee, bonded sample together with zoomed sample bonding pads, and SEM image of the sample. The figure is adapted
from Publication II with permission.
17
Sample fabrication and measurement set-up
18
4. Single-electron pumping and current
metrology
The Coulomb blockade can be employed to pump individual electrons with
rf driving voltages applied on the device. In such an operation, the output
current can be written as I = nef , where e is the elementary charge,
f is the pumping frequency, and n is the average number of electrons
transported per cycle. The target is to establish a plateau in the output
current, i.e., a region where n is quantized very close to an integer value
and insensitive to changes in the system parameters. Several challenges
have to be overcome before we can have a satisfactory realization of the
new quantum ampere and close the quantum metrology triangle.
4.1
Requirements for the SI ampere and the closure of QMT
In the SI, it is desirable to have the ampere defined based on the fixed elementary charge. For this definition to be reasonable, we have to demonstrate an implementation traceable from the definition to the real measurement equipment. A real-life implementation of this emerging SI ampere and the closure of QMT are pursued by measuring the current produced by the pump with an ultrastable low-noise current amplifier (ULCA)
or with a cryogenic current comparator (CCC). Specifically ULCA is a
room temperature transimpedance amplifier designed in PhysikalischTechnische Bundesanstalt, Germany. It is designed to amplify low current signals with amplification 109 V/A with relative uncertainty of 10−7
that can be calibrated with respect to quantum Hall and Josephson effects [16, 17]. On the other hand, CCC is a cryogenic device that utilizes
properties of superconductors. It is designed to compare currents from
two different sources with relative uncertainty of 10−9 [85]. In quantum
metrology triangle experiments, the current from the current pump is
19
Single-electron pumping and current metrology
compared to the current produced with the Josephson voltage standard
and a resistor calibrated with respect to the quantum Hall effect.
For practical realizations and for the closure of QMT, three requirements
need to be fulfilled [5,15]. Firstly, the relative uncertainty of the quantum
charge pump needs to be at least in the level 10−8 , i.e., roughly the same
relative error as that of the voltage and resistance standards based JE
and QHE, respectively. This uncertainty would also be below the relative
uncertainty of the present definition of the ampere, being roughly 10−7 .
Secondly, we need to have an output current of hundreds of picoamperes
to reduce the time required for the QMT measurement to be reasonable.
Thirdly, we need a way to measure the accuracy of the pump directly with
a counting experiment. We can not rely on the accuracy measured based
on JE or QHE. A counting experiment is performed with a single electron
charge sensor integrated near the pump. In the counting experiment,
the charge state is monitored with a charge sensor and individual electrons are moved with the charge pump. The signal of the sensor indicates
whether the electron has been pumped successfully or not.
4.2
Electron pumping in silicon quantum dots
In Chapter 3, we present the structure of the silicon quantum dot device. The silicon pump discussed in Publications I–IV has the same architecture depicted in a scanning electron microscope (SEM) image in
Fig. 4.1(a). The pumping in the silicon devices is based on the gates that
control the potential of the barriers and the potential of the quantum dot.
The potential of the quantum dot is controlled with the plunger gate
(PL). Two barrier gates, the left barrier gate (BL) and the right barrier
gate (BR), are used to control the transparency, i.e., the tunneling resistance of the tunnel junctions between the dot and the source and between
the dot and the drain, respectively. We can also tune the potential well
of the dot in the lateral direction perpendicular to electron transport with
the confining gates (C1 and C2). The source lead gate (SL) and the drain
lead gate (DL) are used to induce the 2DEG leads. These leads are connected to the ohmic contacts leading to the bonding pads. The switch barrier (SB) is used to turn on and off the channel to the drain. By depleting
the channel to the drain, we can create a large reservoir dot, the charge
20
Single-electron pumping and current metrology
(a)
ABL
VBL
APL
ABR
VBR VC2
VPL
BL BR C2
PL
DL
SL
-I
C1
y
x
(b)
z
500 nm
B1 B2
VC1
TG
SB
V
(d)
30 nm
SL BL PL BR DL
x
(c) E
F
QD
SB
8 nm
RES
Figure 4.1. Sample and pump plateaux. (a) Scanning electron micrograph of the quantum dot device used in most of the publications together with measurement
set-up for the direct current from the pump. The position of the pump dot
is denoted with red. The single-electron transistor dot (blue) used in Publication III is capacitively coupled to the reservoir dot (green). (b) Schematic
cross section of the sample along x-axis and (c) the potential landscape for
the electrons along the two dimensional electron gas denoted with orange.
(d) Pumping plateaux. Current through the pump as a function of plunger
dc voltage (VPL ) is presented. The red lines indicate the expected current at
different plateaux.
21
Single-electron pumping and current metrology
(a)
(b)
(c)
(e)
(f)
(I)
(II)
(III)
(d)
(I)
(II)
(III)
Figure 4.2. (a) Pumping process for two-drive single-electron pumping. The process consists of loading (I), trapping (II), and unloading (III) phases. (b)–(f) Error
mechanisms. (b) Non-adiabatic excitation escape. (c) No electrons loaded
from the source. (d) More than one electron is transported per cycle. (e)
Electron is loaded from the drain. (f) Electron does not unload to the drain.
state of which we can monitor. The charge sensor SET that is capacitively
coupled to the reservoir has three gates. The top gate (TG) induces the
sensor dot and the source and drain leads of the sensor. The sensor barrier left gate (B1) and the sensor barrier right gate (B2) form the potential
barriers around the dot. In our measurement set-up, we either apply two
(BL and PL) or three (BL, PL, and BR) rf driving voltages to the pump.
The pumping protocol in the case of rf drives on BL and PL is presented
in Fig. 4.2(a). The protocol consists of three phases. In the loading phase
(I), the potential of the entrance barrier and the dot is lowered such that
an electron enters the dot from the source. In the trapping phase (II), the
entrance barrier is closed and the electron is captured into the dot. In the
unloading phase (III), the potential of the dot is raised such that the electron exits from the dot to the drain. Ideally, we transport a single electron
per cycle. Example data on the pumped current as a function of PL dc
voltage is presented in Fig. 4.1(d). Here, we observe the quantization of
the current up to five plateaux. The nth plateau corresponds to n number
of electrons transported per cycle.
There are several error mechanisms, due to which we might transport
22
Single-electron pumping and current metrology
an undesired number of electrons per cycle. Some examples of these mechanisms are depicted in Fig. 4.2(b)–(f). If the ramp of the barrier potential
is fast, it might be that the electron enters non-adiabatically into an excited state and escapes back to the source (b) [86]. It is also possible that
in the loading phase no electrons enter the dot, and hence no electrons
are transported (c). We may also transport unintentionally two electrons
per cycle instead of one (d), load the electron from the drain instead of the
source (e), or fail to transport the electron to the drain in the unloading
process (f). There is a certain probability for each type of error to occur.
The probabilities define the accuracy of our pumping process.
Several theoretical studies on single-electron pumping are presented in
the literature [5, 87–92]. For the semiconductor quantum dots, the most
widely used model is the so-called decay cascade model [91, 92]. Here, it
is assumed that in the loading phase the dot is accurately initialized to
an integer number of electrons. When the barrier is ramped up, some
of the electrons tunnel out back the source. Here, this out tunneling is
the dominating error mechanism. Accordingly, the average number of
electrons transported per cycle near the first plateau can be expressed as
nave (Vd ) =
2
exp[− exp(−aVd + Δi )],
(4.1)
i=1
where a and Δi are fitting parameters and Vd is the dot dc potential. Parameters Δi correspond to the positions of the plateaux and a describes
the sharpness of the rise to the plateau. This function can be generalized
for multiple current plateaux. The decay cascade model is widely used in
semiconductor quantum dots mainly because of its simplicity and because
it is one of the few models available.
4.3
Electrical confinement and the accuracy of the dot
In Publication I, we demonstrate that we can improve the pumping process with the electrical confinement using gates C1 and C2. Earlier similar effects arising from an externally applied magnetic field on the confinement and accuracy of semiconductor quantum dots have been studied [47, 93–95]. With the help of the increased electrical confinement, we
improve the accuracy of our pump. We obtain 29-ppm uncertainty in single electron transport at 500 MHz pumping frequency.
23
Single-electron pumping and current metrology
The effect of the confinement is studied by investigating the effect of
the C2 dc voltage, VC2 , to the current plateau and to the non-adiabatic
excitations observed in our pump. In Fig. 4.3(a), we observe a widening
of the plateau in PL dc voltage, VPL , as we decrease VC2 . The fits of the
decay cascade model to the experimental data in Fig. 4.3(b) show that
the plateau also becomes flatter with lower VC2 , implying that it is less
sensitive to changes in the system parameters. The theoretical pumping
∗ of the plateau is given by the parameter
error at the flattest point VPL
∗ ) also decreases as a function of V . The non-adiabatic
p = 1 − nave (VPL
C2
excitation observed in Fig. 4.3(c) is studied to show the effect of the confinement to the excited state. The non-adiabatic excitation process is illustrated in Fig. 4.2(b) [86]. Figures 4.3(d) and 4.3(e) show that the energy of the excitation, ΔE, shifts and that it becomes less probable as we
decrease VC2 . This decrease of the excitation probability is due to the increased confinement of the dot: the excitation energy becomes higher, and
hence the excitation probability naturally decrease in a given temperature.
The measurement set-up used to measure the accuracy of the pump is
presented in Fig 4.4(a). We use a high-accuracy voltage source with voltage output, Vcal , calibrated with respect to a Josephson voltage standard
and a high-accuracy resistor, R = 1.000037 GΩ, calibrated with respect to
a quantum Hall resistance standard with the relative uncertainties of 2.5
ppm and 5.7 ppm, respectively. The accuracy of the pump is measured by
setting Vcal to be such that Vcal /R ≈ ef . Thus, if the pump performs perfectly, the null current, Inull , vanishes. First, we carry out a measurement
near the middle of the first pumping plateau shown in Fig. 4.4(b). For
higher accuracy, we measure Inull when the pump and the voltage source
are turned off, and when they are turned on. Traces of these on and off
cycles are depicted in Fig. 4.4(c). The differences of the averages of the
on and off cycles are presented in Fig 4.4(d). We find that by taking into
account the uncertainty of the voltage source and of the resistor, and the
statistical uncertainty, the total uncertainty of our pump output is 29 ppm
with 80 pA of output current. Although there are no data points inside the
uncertainty window, but most of their one-sigma errorbars overlap with
the window, thus rendering our result statistically plausible. Even though
accurate, it is not enough for the QMT measurement. However, we do not
employ here all the tunability options that our architecture allows such
24
Single-electron pumping and current metrology
(a)
(c)
(d)
(b)
(e)
Figure 4.3. (a) Contours of the pumped current with 100-MHz frequency as a function of
the confining gate C2 voltage, VC2 , and plunger gate dc voltage, VPL . Inset:
The pumped current at the first plateau as a function of pumping frequency.
(b) Pumped current as a function of VPL with diffent voltages on C2 together
with the decay cascade fit. (c) Pumped current and its differential conductance Gdiff as a function of VPL . Inset: Non-adiabatic excitation process due to
fast ramping speed. (d) The derivative of the pumped current as a function of
VC2 and VPL referenced to the position of the ground state (GS). The excited
state (ES) shifts away from GS as we increase the confinement. (e) Occupation probabilities of GS and ES as functions of VC2 . The figure is reprinted
from Publication I with permission.
25
Single-electron pumping and current metrology
Figure 4.4. (a) Schematic measurement set-up for the high-accuracy measurement. Calibrated voltage, Vcal , and high accuracy resistor, R, are used to generate current IR that equals the expected quantized current ef from the pump. The
null current is measured with a transimpedance amplifier. (b) Null current
as a function of plunger dc voltage, VPL . The red dashed line indicates the
offset of the measurement set-up. (c) Representative trace of null current
during on and off cycles. The off cycle (blue dots) corresponds the measurement of the offset as both the pump and the voltage source are off. During the
on cycle (red dots) the pump operates at 500-MHz frequency and the voltage
source is on such that IR = ef . (d) Difference of the average offset and the
average of the null current during the on cycles as a function of VPL with 1σ
errorbars. The figure is reprinted from Publication I with permission.
26
Single-electron pumping and current metrology
as three driving voltages. Thus the accuracy can still be improved in the
future.
4.4
Electron counting and its consistency with the direct output
current
There are several different ways to implement counting experiments in
charge pumps. One option is to pump electrons back and forth to a reservoir island which is capacitively coupled to a charge detector [25, 96–98].
Series of pumps and a detector coupled to each island in between the
pumps have also been studied [48,99]. Recently, charge counting based on
the direct monitoring of the pump dot has also been demonstrated [50].
In Publication III, we demonstrate an electron counting scheme where
we cyclically empty the reservoir. We also find that the electron counting
scheme and the direct current produced by our pump are consistent. The
idea of the scheme is that we pump several electrons from the source to
a large reservoir dot and detect its charge state with a quantum dot SET.
From the SET signal, we determine how many electrons we pump to the
reservoir with each pulse. First, we examine, how many electrons we can
pump to the reservoir without causing too much back action due to the
accumulation of electrons in the dot. As shown in Fig. 4.5(a) we can pump
up to 50 electrons to the reservoir before the transition probability for one
electron decreases below 99%. To avoid any significant back action, we
initialize the reservoir after every 22 pulses by lowering the potential of
BL, BR, and PL such that electrons can flow from the reservoir back to
the source.
Due to the weak coupling of the SET and the reservoir, there is considerable uncertainty in the charge detection: the distributions of signal
strengths for different number of electrons transported per cycle overlap,
as visible in Fig. 4.5(b). It is therefore impossible differentiate with high
confidence between n and n + 1 transported electrons for an individual
pumping cycle. Note that this difficulty decreases the counting fidelity
and calls for improvement in the case of the quantum metrology triangle
experiment. However, for a large number of repetitions we can define the
probability Pi for i number of electrons transported in each cycle. We do
this by fitting the distribution of the acquired signal strength with a sum
27
Single-electron pumping and current metrology
(a)
(b)
Number of Counts
1.00
P1
0.96
0.92
0.88
0
40
80
120
Number of Pulses
150
n=0
n=1
n=2
n=3
100
50
σ
0
-1
(c)
3.2 σ
0
1
2
3
Signal Area (pC)
4
1.10
3
ncount, ndc
1.05
1.00
ndc, ncount, Pn
0.95
ndc
0.90
2
0.51
0.52
0.53
VPL(V)
0.54
P0
0.55
P1
P2
P3
P4
ncount
1
0
0.50
0.55
VPL(V)
0.60
Figure 4.5. (a) Back action due to the accumulation of the electrons in the reservoir. The
probability for transporting one electron per cycle as a function of the number of pumping pulses. (b) Histogram of the charge detector signal at the
middle of different pumping plateaux together with Gaussian fits. Adjacent
histograms considerably overlap due to the weak coupling of the detector. (c)
Consistency check for counting and direct-current experiments. The probabilities Pi for i electrons trasferred per pulse and the average number of
electrons transported ncount = i iPi in the counting experiment and in the
direct current experiment, ndc = IP /ef , as functions of the plunger dc voltage, VPL . Inset: Detailed comparison of ncount and ndc at the first plateau.
Orange and black dashed lines correspond to the averages of the counting
experiment and direct current experiment, respectively.
28
Single-electron pumping and current metrology
of Gaussian distributions that have the same midpoints and variances as
in Fig. 4.5(b) that have been obtained in the middle points of the plateaux.
For the consistency check, we first measure the direct output current
in a wide range of VPL values. Then we determine Pi :s with an electron counting scheme in the same range. In Fig. 4.5(c), we show the
average number of electrons transported from the counting experiment
ncount = i iPi as well as that from the direct current ndc = IP /ef . These
two quantities are in very good agreement with each other. A more detailed comparison is carried out at the first plateau shown at the inset of
Fig. 4.5(c). We find that these two schemes agree within the experimental
uncertainty: ndc = 1.000 ± 0.006 and ncount = 0.998 ± 0.004.
Our experiment suffers from two shortcomings: the integration time of
the charge sensor signal is relatively long, 750 ms, and the signals for
different transitions overlap, making it impossible to reliably distinguish
between two adjacent charge transitions for an individual pumping event.
Both of these problems can be solved with an improved architecture where
the sensor is more strongly coupled to the reservoir. This may be achieved,
for instance, with a metallic SET sensor on top of the reservoir dot. Another option is to have more than one detector and to utilize cross correlations.
4.5
Three-waveform bidirectional pumping
In Publications I–III, we use at most two rf waveforms for electron pumping. A measurement scheme with full rf control over all the barriers and
the quantum dot potential is presented in Fig. 4.1. In Publication IV, we
demonstrate that such set-up can be used to perform bidirectional pumping by only changing the phase of PL rf drive. To find a suitable operation
point, we first pump with two rf drives: one on the entrance barrier and
one on PL. Then we introduce a third drive to the exit barrier and study
the maximum plateau length in VPL as a function of the amplitude on
the exit barrier. As shown in Fig. 4.6(a), the maximum plateau length
increases roughly linearly as a function of the exit barrier amplitude.
We set barrier right amplitude, ABR , to correspond the maximum plateau
length observed in Fig. 4.6(a) and measure the pumped current as a function of VPL and the PL drive phase difference with repect to BL drive,
29
Single-electron pumping and current metrology
(a)
(b)
(c)
(d)
Figure 4.6. (a) Plateau length in the plunger dc voltage as a function of the amplitude of
the left and right barrier drive as indicaed. (b) Pumped current as a function
of the plunger dc voltage and the plunger drive phase difference with respect
to barrier-left drive. (c) Absolute value of the pumped current along the cross
sections (red and blue dashed lines) in (b). (d) Cross section along the green
dashed line in (b).
φBL-PL . As shown in Fig. 4.6(b), we have pumping plateaux in both directions. More careful examination of the data [see Figs. 4.6(c) and 4.6(d)]
reveals that the absolute magnitude of current in each direction is the
same within the experimental accuracy. We also find a large range of
possible PL phases which we can utilize in bidirectional pumping.
This bidirectional pumping can be used to upgrade the electron counting
scheme presented in Publication III into an error counting scheme where
we pump electrons in and out of the reservoir [25]. Previously performing
this kind of error counting has been difficult to implement in semiconductor devices due to the asymmetries of the samples and the pumping protocols. There have been previous studies on this kind of a scheme in silicon
devices [97, 98] but the transport frequency in these experiments was of
the order of Hz. The bidirectional pumping protocol presented here allows
significantly faster switching between the pumping directions. With the
help of the third drive, we can potentially improve also the output current
and the accuracy compared with Publication I.
30
5. Jarzynski equality and Crooks
fluctuation theorem
In the macroscopic world, most phenomena appear deterministic, whereas
in microscopic systems the probabilistic nature behind the phenomena is
more pronounced. For instance, we know in advance how much gasoline
we need for a 100 km drive on a flat road with a certain car. The work
gained from the chemical and thermodynamic processes inside the car
engine is predictable. However, if we decrease the size of the engine or
the system that outputs the work, we observe at some point, usually at
the nanometre scale, that the work obtained from a certain protocol is not
the same for each repetition but rather a random variable. These variations of work are due to microscopic thermodynamic fluctuations which
can perturb the protocol. See Refs. [100–103], for more details of the thermodynamic and statistical physics concepts discussed in this chapter.
5.1
Non-equilibrium fluctuation relations
Even though it is not possible to predict the work W done on the system at
the microscopic scale, we know something about its probability distribution. The so-called Jarzynski equality relates the exponential expectation
value of the work done in the protocol and the equilibrium variables of the
system at the beginning and end of the protocol [64,65]. The requirements
for the Jarzynski equality to hold are that initially the system should be
in equilibrium with its environment and that it can relax spontaneously
to the equilibrium state with its environment after the drive. Consider
a system with external control parameters that can be driven from initial values A to final values B. In this case, the Jarzynski equality is
31
Jarzynski equality and Crooks fluctuation theorem
expressed as
eW β = eΔF
A→B β
,
(5.1)
where β = 1/(kB T ), T is the temperature of the environment, the average over the microscopic realizations is denoted by ·, and ΔF A→B is the
Helmhotz free-energy difference between configurations A and B. A notable aspect of the Jarzynski equality is that the system does not need to
be in thermal equilibrium except in the beginning of the protocol, which
is not the case for most thermodynamic relations [80]. Importantly, the
Jarzynski equality may be applied in molecular biology: the Helmholtz
free energy difference of certain DNA configurations can be obtained by
measuring the work performed in folding/unfolding experiment [69–76].
The Jarzynski equality has also been studied widely in numerical simulations [76, 104–106]. Note that the Jarzynski equality yields the second
law of thermodynamics in the form [80]
W ≥ ΔF A→B .
(5.2)
It is possible that for an individual trajectory the work done is less than
ΔF A→B [107]. However, since we cannot predict which repetitions of
the process have this property and the average is greater or equal than
ΔF A→B , the second law is not violated.
Soon after Jarzynski published his equality another non-equilibrium relation was published: the Crooks fluctuation theorem [66–68]. This theorem states that
PA→B (W )
= eW β ,
PB→A (W )
(5.3)
where PA→B (W ) is the probability for work W applied in the drive from
state A to B and PB→A (W ) is the probability for the drive from state B
to A [80].
5.2
Experimental study of fluctuation relations in a single-electron
box
In Publication V, we study the Jarzynski equality in a single-electron box.
The studied system consists of two metallic islands, a normal-metal and
a superconducting one, connected with a tunnel junction. An SEM image of the sample and the measurement set-up are presented in Fig. 5.1.
32
Jarzynski equality and Crooks fluctuation theorem
The charge state of the normal-metal island is monitored with a nearby
integrated SET. Our control parameter is the voltage difference between
the islands. Our protocol is a sinusoidal drive between two adjacent gate
charges of the system. During the drive, electrons may tunnel from one
island to the other. The charge sensor registers the transitions between
these two states. The voltage difference between the islands and the tunneling direction at the tunnel time instant defines the heat dissipated in
a single tunnel event Qi . The total heat dissipated Q is the sum of all the
Qi :s during the drive. According to the first law of thermodynamics we
have Q = W − ΔU , where ΔU is the change of the internal energy in the
system. The theoretical study of the system shows [108, 109] that in our
set-up ΔU and ΔF essentially vanish. Thus the Jarzynski equality can
be written in terms of heat as
eQβ = 1,
(5.4)
and the Crooks fluctuation relation as
P (Q)
= eQβ .
P (−Q)
(5.5)
We measure the dissipated heat for several hundreds of thousands of
repetitions of the drive protocol and compile histograms to estimate the
probability distributions for the dissipated heat. The probability distributions for the heat Q dissipated during the protocol, P (Q), are presented
in Fig. 5.2(a) together with P (Q)/P (−Q) as a function of Q. The data is
collected from hundreds of thousands of repetitions of the cycle. The experimental results follow the theoretical predictions accurately. We find
that for these probability distributions the expectation values eQβ are
1.033 ± 0.043 (1-Hz drive), 1.032 ± 0.043 (2-Hz drive), and 1.044 ± 0.044
(4-Hz drive). Hence the Jarzynski equality holds within the experimental accuracy. Figure 5.2(a) shows that the experimental results also follow
the Crooks fluctuation theorem. In Fig. 5.2(b) and 5.2(c), we show the first
and the second cumulants of Q/EC as functions of the driving frequency
at different bath temperatures together with corresponding theoretical
predictions. Also here the theoretical predictions are consistent with the
experimental data.
After these pioneering experiments the work with the fluctuation theorems in condensed matters has been carried on both theoretically and
[5,110–115] and experimentally [116–121]. In Ref. [116], a similar system
33
Jarzynski equality and Crooks fluctuation theorem
Figure 5.1. (a) False-color SEM image of the sample used in Publication V. The superconducting island is on the left (blue) and the normal-metal island is on the right
(orange). Between the normal-metal and superconducting islands, there is
a tunnel junction. At the top right, we have the single-electron transistor
(green) that is used to monitor the charge state of the normal-metal island.
(b) Schematic circuit diagram of the sample and the measurement set-up.
The charge state of the system is controlled by sinusoidally changing external voltage Vg . The figure is reprinted from Publication V with permission.
was studied where the Jarzynski equality does not hold any more due to
the overheating of the superconducting island, but a more general theorem, the integral fluctuation theorem [106, 122–124], was shown to be applicable. Furthermore a Szilard engine [117] and Maxwell’s demon [118,
119] have been realized in single-electron devices. In Refs. [120, 121], the
fluctuation theorems in GaAs/AlGaAs semiconductors are studied experimentally. There is an exciting opportunity for the future to study the
fluctuation theorems in our silicon quantum dot architecture. The control
over the transparency of the two barriers and the dot provides flexibility
compared with previous studies.
34
Jarzynski equality and Crooks fluctuation theorem
Figure 5.2. (a) Probability distributions for the amount of heat dissipated during a single
control protocol for driving frequencies 1 Hz (black), 2 Hz (red), and 4 Hz
(blue). Solid lines are the exact theoretical predictions and the dashed lines
show the Monte Carlo simulations where the finite bandwidth of the SET
sensor is taken into account. Inset: Probability of heat produced divided
by the corresponding probability of the reversed event to study the Crooks
fluctuation theorem for each repetition frequency. (b) First and (c) second
cumulants of the probability distribution of dissipated heat as functions of
the driving frequency with different bath temperatures. The solid lines are
exact theoretical predictions and circles are experimental results. The figure
is reprinted from Publication V with permission.
35
Jarzynski equality and Crooks fluctuation theorem
36
6. Summary and conclusions
In this thesis, we discuss single-electron pumping in silicon quantum dots
and statistics of work and heat in electron transport. We introduce a silicon quantum dot device that can be used as a single-electron pump for future metrological purposes. We also study the Jarzynski equality and the
Crooks fluctuation theorem in a hybrid normal-metal–superconducting
structure.
In Publication I, we demonstrate that it is possible to improve the pumping process with the confining gates integrated into our system. This was
done by studying the sensitivity of the quantized current to system parameters at different confining gate voltages as well as by studying the
non-adiabatic excitations observed in the course of pumping. With the
help of a high-accuracy resistor and voltage source, we obtain 80-pA current at 29-ppm uncertainty with our silicon quantum dot.
The device discussed in this thesis is a metal–oxide–semiconductor quantum dot. The techniques used to fabricate the sample are described in
detail in Publication II. Furthermore, our measurement set-up, including
the cryostat, wiring of the sample, measurement equipment, and methods
to characterize the device, are discussed.
In Publication III, we perform electron counting in our quantum dot
architecture. We confirm that the direct current produced by our pump
and the electron counting scheme are consistent. However, the coupling
of the charge sensor is weak and should be increased for faster and more
accurate counting.
In Publication IV, we demonstrate bidirectional pumping with fast and
convenient switching with the help of a third waveform. We show that the
plateau length in the plunger voltage can be increased by increasing the
amplitude of this third waveform. Subsequently, we measure the pumped
37
Summary and conclusions
current as a function of the plunger dc voltage and phase of the plunger
drive. We find a parameter region where we can perform bidirectional
pumping simply by changing the phase of the drive.
This thesis also includes a study of the Jarzynski equality and the Crooks
fluctuation theorem in a single-electron box. In Publication V, we introduce a system with two islands, one being normal-metal and the other superconducting, with a tunnel junction between them. We study the heat
dissipated during a driving protocol of the system. The probability distributions for the heat are consistent with the theoretical expectations and
they agree well with the fluctuation relations. In the future, it would be
interesting to study the non-equilibrium fluctuation relations in the silicon quantum dots similar to one presented in Publications I–IV.
The tunable silicon quantum dot introduced in this thesis provides a
promising candidate for the realization of the quantum ampere. To this
end we list the following future objectives:
• Increase the capacitive coupling between the reservoir and the charge
sensor to perform more reliable counting experiments. For instance,
one could fabricate a metallic single-electron transistor on top of the
reservoir dot or have more detectors and perform cross-checking [48].
• Upgrade the electron counting scheme to an error counting scheme and
determine the accuracy of the pump with an error counting experiment.
In error counting experiment, electrons are pumped in and out of the
reservoir and we only detect the rare errors that occur in this process.
This could be done with the design presented in this thesis.
• Exploit the tunability of the silicon pump and reduce the uncertainty of
the pump to 10−8 with more than 100 pA of dc current.
• Close the quantum metrology triangle. Employ a current measurement
set-up that is calibrated with respect to the Josephson effect and the
quantum Hall effect. Compare the resulting current with the output
current of the pump that is verified to operate at the uncertainty level
of 10−8 .
The next General Conference on Weights and Measures (CGPM) meeting,
38
Summary and conclusions
where new SI standards will potentially be defined, will be held in August
2018. It is possible that already then the definition of the ampere will be
changed. Should this happen, the scalable production of the realizations
of this standard becomes very important: it should be possible to produce
reliable current standards for the metrology institutes around the world.
Here, silicon platforms may be excellent; they are already widely utilized
by the semiconductor industry, perhaps providing a more convenient route
to scalability than GaAs/AlGaAs devices. In conclusion, it seems that current pumps play a significant role in the future current metrology. The
device and methods presented in this thesis provide an important development of silicon current pumps.
39
Summary and conclusions
40
References
[1] M. Partanen, K. Y. Tan, J. Govenius, R. E. Lake, M. K. Mäkelä, T. Tanttu,
and M. Möttönen. Quantum-limited heat conduction over macroscopic distances. Nature Phys., 12:478, 2016.
[2] R. A. Millikan. On the elementary electrical charge and the avogadro constant. Phys. Rev., 2:109, 1913.
[3] D. V. Averin and K. K. Likharev. Mesoscopic Phenomena in Solids. Nature
Publishing Group, Amsterdam, 1991.
[4] T. Heikkilä. The Physics of Nanoelectronics. Oxford University Press, 1st
edition, 2013.
[5] J. Pekola, O.-P. Saira, V. Maisi, A. Kemppinen, M. Möttönen, Y. Pashkin,
and D. Averin. Single-electron current sources: Toward a refined definition
of the ampere. Rev. Mod. Phys., 86:1421, 2013.
[6] Bureau International des Poids et Mesures. The International System of
Units (SI). STEDI Media, Paris, 8th edition, 2006.
[7] T. J. Quinn. The kilogram: the present state of our knowledge. IEEE
Trans. Instrum. Meas., 40:81, 1991.
[8] M. Gläser, M. Borys, D. Ratschko, and R. Schwartz. Redefinition of the
kilogram and the impact on its future dissemination. Metrologia, 47:419,
2010.
[9] M. J. T. Milton, J. M. Williams, and A. B. Forbes. The quantum metrology
triangle and the redefinition of the SI ampere and kilogram; analysis of a
reduced set of observational equations. Metrologia, 47:279, 2010.
[10] P. Giacomo. News from the BIPM. Metrologia, 25:113, 1988.
[11] K. v. Klitzing, G. Dorda, and M. Pepper. New method for high-accuracy
determination of the fine-structure constant based on quantized Hall resistance. Phys. Rev. Lett., 45:494, 1980.
[12] B. D. Josephson. Possible new effects in superconductive tunneling. Phys.
Lett., 1:251, 1962.
41
References
[13] N. Fletcher, G. Rietveld, J. Olthoff, I. Budovsky, and M. Milton. Electrical
units in the new si: Saying goodbye to the 1990 values. NCSLI Measure J.
Meas. Sci, 9(3):30–35, 2014.
[14] M. W. Keller. Current status of the quantum metrology triangle. Metrologia, 45:102, 2008.
[15] N. Feltin and F. Piquemal. Determination of the elementary charge and
the quantum metrological triangle experiment. Eur. Phys. J. Spec. Top.,
172:267, 2009.
[16] D. Drung, C. Krause, U. Becker, H. Scherer, and F. J. Ahlers. Ultrastable
low-noise current amplifier: A novel device for measuring small electric
currents with high accuracy. Rev. Sci. Instrum., 86:024703, 2015.
[17] D. Drung, M. Götz, E. Pesel, and H. Scherer. Improving the traceable measurement and generation of small direct currents. IEEE Trans. Instrum.
Meas., 64:3021, 2015.
[18] D. V. Averin and K. K. Likharev. Coulomb blockade of single-electron tunneling, and coherent oscillations in small tunnel junctions. J. Low Temp.
Phys., 62:345, 1986.
[19] L. J. Geerligs, V. F. Anderegg, P. A. M. Holweg, J. E. Mooij, H. Pothier,
D. Esteve, C. Urbina, and M. H. Devoret. Frequency-locked turnstile device for single electrons. Phys. Rev. Lett., 64:2691, 1990.
[20] L. J. Geerligs, S. M. Verbrugh, P. Hadley, J. E. Mooij, H. Pothier, P. Lafarge,
C. Urbina, D. Esteve, and M. H. Devoret. Single Cooper pair pump. Z.
Phys. B, 85:349, 1991.
[21] H. Pothier, P. Lafarge, P. F. Orfila, C. Urbina, D. Esteve, and M. H. Devoret.
Single electron pump fabricated with ultrasmall normal tunnel junctions.
Physica B, 169:573, 1991.
[22] H. Pothier, P. Lafarge, C. Urbina, D. Esteve, and M. H. Devoret. Singleelectron pump based on charging effects. Europhys. Lett., 17:249, 1992.
[23] L. P. Kouwenhoven, A. T. Johnson, N. C. van der Vaart, C. J. P. M. Harmans, and C. T. Foxon. Quantized current in a quantum-dot turnstile using oscillating tunnel barriers. Phys. Rev. Lett., 67:1626, 1991.
[24] J. M. Martinis, M. Nahum, and H. D. Jensen. Metrological accuracy of the
electron pump. Phys. Rev. Lett., 72:904, 1994.
[25] M. W. Keller, J. M. Martinis, N. M. Zimmerman, and A. H. Steinbach. Accuracy of electron counting using a 7-junction electron pump. Appl. Phys.
Lett., 69:1804, 1996.
[26] J. M. Shilton, D. R. Mace, V. I. Talyanskii, Yu. Galperin, M. Y. Simmons,
M. Pepper, and D. A. Ritchie. High-frequency single-electron transport in
a quasi-one-dimensional GaAs channel induced by surface acoustic waves.
J. Phys. Condens. Matter, 8:L531, 1996.
42
References
[27] V. I. Talyanskii, J. M. Shilton, M. Pepper, C. G. Smith, C. J. B. Ford, E. H.
Linfield, D. A. Ritchie, and G. A. C. Jones. Single-electron transport in a
one-dimensional channel by high-frequency surface acoustic waves. Phys.
Rev. B, 56:15180, 1997.
[28] J. Cunningham, V. I. Talyanskii, J. M. Shilton, M. Pepper, M. Y. Simmons,
and D. A. Ritchie. Single-electron acoustic charge transport by two counterpropagating surface acoustic wave beams. Phys. Rev. B, 60:4850, 1999.
[29] J. P. Pekola, Juha J. Vartiainen, M. Möttönen, O.-P. Saira, M. Meschke,
and D. V. Averin. Hybrid single-electron transistor as a source of quantized
electric current. Nature Phys., 4:120, 2008.
[30] V. F. Maisi, Yu. A. Pashkin, S. Kafanov, J. S. Tsai, and J. P. Pekola. Parallel
pumping of electrons. New J. Phys., 11:113057, 2009.
[31] A. Kemppinen, S. Kafanov, Yu. A. Pashkin, J. S. Tsai, D. V. Averin, and
J. P. Pekola. Experimental investigation of hybrid single-electron turnstiles with high charging energy. Appl. Phys. Lett., 94:172108, 2009.
[32] A. Kemppinen, M. Meschke, M. Möttönen, D. V. Averin, and J. P. Pekola.
Quantized current of a hybrid single-electron transistor with superconducting leads and a normal-metal island. Eur. Phys. J. Spec. Top., 172:311,
2009.
[33] J. E. Mooij and Yu. V. Nazarov. Superconducting nanowires as quantum
phase-slip junctions. Nature Phys., 2:169, 2006.
[34] J. J. Vartiainen, M. Möttönen, J. P. Pekola, and A. Kemppinen. Nanoampere pumping of Cooper pairs. Appl. Phys. Lett., 90:082102, 2007.
[35] M. Möttönen, J. J. Vartiainen, and J. P. Pekola. Experimental determination of the Berry phase in a superconducting charge pump. Phys. Rev.
Lett., 100:177201, 2008.
[36] O. V. Astafiev, L. B. Ioffe, S. Kafanov, Yu. A. Pashkin, K. Yu. Arutyunov,
D. Shahar, O. Cohen, and J. S. Tsai. Coherent quantum phase slip. Nature,
484:355, 2012.
[37] G. Yamahata, K. Nishiguchi, and A. Fujiwara. Gigahertz single-trap electron pumps in silicon. Nat. Commun., 5:5038, 2014.
[38] G. C. Tettamanzi, R. Wacquez, and S. Rogge. Charge pumping through a
single donor atom. New J. Phys., 16:063036, 2014.
[39] G. Yamahata, T. Karasawa, and A. Fujiwara. Gigahertz single-hole transfer in Si tunable-barrier pumps. Appl. Phys. Lett., 106:023112, 2015.
[40] G. P. Lansbergen, Y. Ono, and A. Fujiwara. Donor-based single electron
pumps with tunable donor binding energy. Nano Lett., 12:763, 2012.
[41] B. Roche, R.-P. Riwar, B. Voisin, E. Dupont-Ferrier, R. Wacquez, M. Vinet,
M. Sanquer, J. Splettstoesser, and X. Jehl. A two-atom electron pump. Nat.
Commun., 4:1581, 2013.
43
References
[42] G. P. Lansbergen, R. Rahman, C. J. Wellard, I. Woo, J. Caro, N. Collaert,
S. Biesemans, G. Klimeck, L. C. L. Hollenberg, and S. Rogge. Gate-induced
quantum-confinement transition of a single dopant atom in a silicon finfet.
Nature Phys., 4:656, 2008.
[43] M. D. Blumenthal, B. Kaestner, L. Li, S. Giblin, T. J. B. M. Janssen, M. Pepper, D. Anderson, G. Jones, and D. A. Ritchie. Gigahertz quantized charge
pumping. Nature Phys., 3:343, 2007.
[44] M. R. Connolly, K. L. Chiu, S. P. Giblin, M. Kataoka, J. D. Fletcher,
C. Chua, J. P. Griffiths, G. A. C. Jones, V. I. Fal’ko, C. G. Smith, and T. J.
B. M. Janssen. Gigahertz quantized charge pumping in graphene quantum
dots. Nature Nanotech., 8:417, 2013.
[45] P. Mirovsky, B. Kaestner, C. Leicht, A. C. Welker, T. Weimann, K. Pierz, and
H. W. Schumacher. Synchronized single electron emission from dynamical
quantum dots. Appl. Phys. Lett., 97:252104, 2010.
[46] S. P. Giblin, S. J. Wright, J. D. Fletcher, M. Kataoka, M. Pepper, T. J.
B. M. Janssen, D. A. Ritchie, C. A. Nicoll, D. Anderson, and G. A. C. Jones.
An accurate high-speed single-electron quantum dot pump. New J. Phys.,
12:073013, 2010.
[47] S. P. Giblin, M. Kataoka, J. D. Fletcher, P. See, T. J. B. M. Janssen, J. P.
Griffiths, G. A. C. Jones, I. Farrer, and D. A. Ritchie. Towards a quantum
representation of the ampere using single electron pumps. Nat. Commun.,
3:930, 2012.
[48] L. Fricke, M. Wulf, B. Kaestner, V. Kashcheyevs, J. Timoshenko,
P. Nazarov, F. Hohls, P. Mirovsky, B. Mackrodt, R. Dolata, T. Weimann,
K. Pierz, and H. W. Schumacher. Counting statistics for electron capture
in a dynamic quantum dot. Phys. Rev. Lett., 110:126803, 2013.
[49] F. Stein, D. Drung, L. Fricke, H. Scherer, F. Hohls, C. Leicht, M. Götz,
C. Krause, R. Behr, E. Pesel, K. Pierz, U. Siegner, F.-J. Ahlers, and H. W.
Schumacher. Validation of a quantized-current source with 0.2 ppm uncertainty. Appl. Phys. Lett., 107:103501, 2015.
[50] S. P. Giblin, P. See, A. Petrie, T. J. B. M. Janssen, I. Farrer, J. P. Griffiths,
G. A. C. Jones, D. A. Ritchie, and M. Kataoka. High-resolution error detection in the capture process of a single-electron pump. Appl. Phys. Lett.,
108:023502, 2016.
[51] M. Jo, T. Uchida, A. Tsurumaki-Fukuchi, M. Arita, A. Fujiwara, Y. Ono,
K. Nishiguchi, H. Inokawa, and Y. Takahashi. Fabrication and singleelectron-transfer operation of a triple-dot single-electron transistor. J.
Appl. Phys., 118:214305, 2015.
[52] M. Seo, Y.-H. Ahn, Y. Oh, Y. Chung, S. Ryu, H.-S. Sim, I.-H. Lee, M.-H.
Bae, and N. Kim. Improvement of electron pump accuracy by a potentialshape-tunable quantum dot pump. Phys. Rev. B, 90:085307, 2014.
[53] S. d’Hollosy, M. Jung, A. Baumgartner, V. A. Guzenko, M. H. Madsen,
J. Nygård, and C. Schönenberger. Gigahertz quantized charge pumping in
44
References
bottom-gate-defined InAs nanowire quantum dots. Nano Letters, 15:4585,
2015.
[54] A. Fujiwara and Y. Takahashi. Manipulation of elementary charge in a
silicon charge-coupled device. Nature, 410:560, 2001.
[55] A. Fujiwara, N. M. Zimmerman, Y. Ono, and Y. Takahashi. Current quantization due to single-electron transfer in Si-wire charge-coupled devices.
Appl. Phys. Lett., 84:1323, 2004.
[56] A. Fujiwara, K. Nishiguchi, and Y. Ono. Nanoampere charge pump by
single-electron ratchet using silicon nanowire metal-oxide-semiconductor
field-effect transistor. Appl. Phys. Lett., 92:042102, 2008.
[57] K. W. Chan, M. Möttönen, A. Kemppinen, N. S. Lai, K. Y. Tan, W. H. Lim,
and A. S. Dzurak. Single-electron shuttle based on a silicon quantum dot.
Appl. Phys. Lett., 98:212103, 2011.
[58] X. Jehl, B. Voisin, T. Charron, P. Clapera, S. Ray, B. Roche, M. Sanquer, S. Djordjevic, L. Devoille, R. Wacquez, and M. Vinet. Hybrid
metal-semiconductor electron pump for quantum metrology. Phys. Rev.
X, 3:021012, 2013.
[59] N. M. Zimmerman, B. J. Simonds, A. Fujiwara, Y. Ono, Y. Takahashi, and
H. Inokawa. Charge offset stability in tunable-barrier Si single-electron
tunneling devices. Appl. Phys. Lett., 90:033507, 2007.
[60] E. Hourdakis, J. A. Wahl, and N. M. Zimmerman. Lack of charge offset
drift is a robust property of Si single electron transistors. Appl. Phys. Lett.,
92:062102, 2008.
[61] N. M. Zimmerman, W. H. Huber, B. Simonds, E. Hourdakis, A. Fujiwara,
Y. Ono, Y. Takahashi, H. Inokawa, M. Furlan, and M. W. Keller. Why
the long-term charge offset drift in Si single-electron tunneling transistors
is much smaller (better) than in metal-based ones: Two-level fluctuator
stability. J. Appl. Phys., 104:033710, 2008.
[62] P. J. Koppinen, M. D. Jr. Stewart, and N. M. Zimmerman. Fabrication
and electrical characterization of fully CMOS-compatible Si single-electron
devices. IEEE Trans. Electron Devices, 60:78, 2013.
[63] N. M. Zimmerman, C.-H. Yang, N. S. Lai, W. H. Lim, and A. S. Dzurak.
Charge offset stability in Si single electron devices with al gates. Nanotechnology, 25:405201, 2014.
[64] C. Jarzynski. Nonequilibrium equality for free energy differences. Phys.
Rev. Lett., 78:2690, 1997.
[65] C. Jarzynski. Equilibrium free-energy differences from nonequilibrium
measurements: A master-equation approach. Phys. Rev. E, 56:5018, 1997.
[66] G. E. Crooks. Nonequilibrium measurements of free energy differences
for microscopically reversible Markovian systems. J. Stat. Phys., 90:1481,
1998.
45
References
[67] G. E. Crooks. Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences. Phys. Rev. E, 60:2721,
1999.
[68] G. E. Crooks. Path-ensemble averages in systems driven far from equilibrium. Phys. Rev. E, 61:2361, 2000.
[69] J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco, and C. Bustamante. Equilibrium information from nonequilibrium measurements in an experimental
test of Jarzynski’s equality. Science, 296:1832, 2002.
[70] F. Douarche, S. Ciliberto, A. Petresyan, and I. Rabbiosi. An experimental
test of the Jarzynski equality in a mechanical experiment. Europhys. Lett.,
70:593, 2005.
[71] D. Collin, F. Ritort, C. Jarzynski, S. B. Smith, I. Tinoco, and C. Bustamante.
Verification of the Crooks fluctuation theorem and recovery of RNA folding
free energies. Nature, 437:231, 2005.
[72] I. Junier, A. Mossa, M. Manosas, and F. Ritort. Recovery of free energy
branches in single molecule experiments. Phys. Rev. Lett., 102:070602,
2009.
[73] G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, and D. J. Evans. Experimental demonstration of violations of the second law of thermodynamics
for small systems and short time scales. Phys. Rev. Lett., 89:050601, 2002.
[74] J. Gore, F. Ritort, and C. Bustamante. Bias and error in estimates of equilibrium free-energy differences from non-equilibrium measurements. Proc.
Natl. Acad. Sci. USA, 100:12564, 2003.
[75] F. Ritort, C. Bustamante, and I. Tinoco Jr. A two-state kinetic model for
the unfolding of single molecules by mechanical force. Proc. Natl. Acad.
Sci. USA, 99:13544, 2002.
[76] A. Alemany, M. Ribezzi, and F. Ritort. Recent progress in fluctuation theorems and free energy recovery. AIP Conf. Proc, 1332:96, 2011.
[77] A. Fujiwara, H. Inokawa, K. Yamazaki, H. Namatsu, Y. Takahashi, N. M.
Zimmerman, and S. B. Martin. Single electron tunneling transistor with
tunable barriers using silicon nanowire metal-oxide-semiconductor fieldeffect transistor. Appl. Phys. Lett., 88:053121, 2006.
[78] R. Liboff. Introductory Quantum Mechanics. Addison-Wesley, Reading,
2002.
[79] A. Kemppinen. Tunnel junction devices for quantum metrology. PhD thesis, Helsinki University of Technology, 2009.
[80] T. Tanttu. Non-equilibrium fluctuation relations in driven single-electron
devices. Master’s thesis, Aalto University, 2012.
[81] N. W. Ashcroft and N. D. Mermin. Solid state physics. Brooks/Cole, Belmont, 1st edition, 1975.
46
References
[82] K.-E. Huhtinen. Counting charge transitions in a single-electron pump,
Bachelor’s thesis, aalto university, 2014.
[83] S. M. Sze. Physics of Semiconductor Devices. Wiley Interscience, New York,
1969.
[84] S. Franssila. Introduction to microfabrication. John Wiley & Sons, West
Sussex, England, 2004.
[85] I. K. Harvey. A precise low temperature dc ratio transformer. Rev. Sci.
Instrum., 43:1626, 1972.
[86] M. Kataoka, J. D. Fletcher, P. See, S. P. Giblin, T. J. B. M. Janssen, J. P.
Griffiths, G. A. C. Jones, I. Farrer, and D. A. Ritchie. Tunable nonadiabatic
excitation in a single-electron quantum dot. Phys. Rev. Lett., 106:126801,
2011.
[87] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs. Quantized
Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett.,
49:405, 1982.
[88] Y. Levinson, O. Entin-Wohlman, and P. Wölfle. Pumping at resonant transmission and transferred charge quantization. Physica A, 302:335, 2001.
[89] V. Kashcheyevs, A. Aharony, and O. Entin-Wohlman. Resonance approximation and charge loading and unloading in adiabatic quantum pumping.
Phys. Rev. B, 69:195301, 2004.
[90] V. Kashcheyevs and J. Timoshenko. Quantum fluctuations and coherence
in high-precision single-electron capture. Phys. Rev. Lett., 109:216801,
2012.
[91] B. Kaestner and V. Kashcheyevs. Non-adiabatic quantized charge pumping with tunable-barrier quantum dots: a review of current progress. Rep.
Prog. Phys., 78:103901, 2015.
[92] V. Kashcheyevs and B. Kaestner. Universal decay cascade model for dynamic quantum dot initialization. Phys. Rev. Lett., 104:186805, 2010.
[93] J. D. Fletcher, M. Kataoka, S. P. Giblin, S. Park, H.-S. Sim, P. See, T. J.
B. M. Janssen, J. P. Griffiths, G. A. C. Jones, H. E. Beere, and D. A. Ritchie.
Stabilization of single-electron pumps by high magnetic fields. Phys. Rev.
B, 86:155311, 2012.
[94] B. Kaestner, C. Leicht, V. Kashcheyevs, K. Pierz, U. Siegner, and H. W.
Schumacher. Single-parameter quantized charge pumping in high magnetic fields. Appl. Phys. Lett., 94:012106, 2009.
[95] S. J. Wright, M. D. Blumenthal, G. Gumbs, A. L. Thorn, M. Pepper, T. J.
B. M. Janssen, S. N. Holmes, D. Anderson, G. A. C. Jones, C. A. Nicoll, and
D. A. Ritchie. Enhanced current quantization in high-frequency electron
pumps in a perpendicular magnetic field. Phys. Rev. B, 78:233311, 2008.
[96] R. L. Kautz, M. W. Keller, and J. M. Martinis. Leakage and counting errors
in a seven-junction electron pump. Phys. Rev. B, 60:8199, 1999.
47
References
[97] G. Yamahata, K. Nishiguchi, and A. Fujiwara. Accuracy evaluation of
single-electron shuttle transfer in Si nanowire metal-oxide-semiconductor
field-effect transistors. Appl. Phys. Lett., 98:222104, 2011.
[98] G. Yamahata, K. Nishiguchi, and A. Fujiwara. Accuracy evaluation and
mechanism crossover of single-electron transfer in Si tunable-barrier turnstiles. Phys. Rev. B, 89:165302, 2014.
[99] Lukas Fricke, Michael Wulf, Bernd Kaestner, Frank Hohls, Philipp
Mirovsky, Brigitte Mackrodt, Ralf Dolata, Thomas Weimann, Klaus Pierz,
Uwe Siegner, and Hans W. Schumacher. Self-referenced single-electron
quantized current source. Phys. Rev. Lett., 112:226803, 2014.
[100] R. Pathria. Statistical Mechanics. Butterworth-Heinemann, Cornwall,
2000.
[101] J. Arponen. Statistinen Fysiikka. LIMES, Helsinki, 2009.
[102] L. Reichl. A Modern Course in Statistical Physics. Wiley-VCH, New York,
2nd edition, 1998.
[103] K. Sekimoto. Stochastic Energetics. Springer, Heidelberg, 2010.
[104] S. Park, F. Khalili-Araghi, E. Tajkhorshid, and K. Schulten. Free energy
calculation from steered molecular dynamics simulations using Jarzynski’s equality. J. Chem. Phys., 119:3559, 2003.
[105] D. West, P. Olmsted, and E. Paci. Free energy for protein folding from
nonequilibrium simulations using the Jarzynski equality. J. Chem. Phys.,
125:204910, 2006.
[106] S. Vaikuntanathan and C. Jarzynski. Escorted Free Energy Simulations: Improving Convergence by Reducing Dissipation. Phys. Rev. Lett.,
100:190601, 2008.
[107] D. J. Evans, E. G. D. Cohen, and G. P. Morriss. Phys. Rev. Lett., 71:2401,
1993.
[108] J. P. Pekola and O.-P. Saira. Work, free energy and dissipation in voltage
driven single-electron transitions. J. Low Temp. Phys., 169:70, 2012.
[109] D. V. Averin and J. P. Pekola. Statistics of the dissipated energy in driven
single-electron transitions. EPL, 96:67004, 2011.
[110] M. Borrelli, J. Koski, S. Maniscalco, and J. P. Pekola. Fluctuation relations
for driven coupled classical two-level systems with incomplete measurements. Phys. Rev. E, 91:012145, 2015.
[111] F. W. J. Hekking and J. P. Pekola. Quantum jump approach for work and
dissipation in a two-level system. Phys. Rev. Lett., 111:093602, 2013.
[112] R. Sánchez and M. Büttiker. Detection of single-electron heat transfer
statistics. Europhys. Lett., 100:47008, 2012.
[113] S. Deffner and A. Saxena. Quantum work statistics of charged Dirac particles in time-dependent fields. Phys. Rev. E, 92:032137, 2015.
48
References
[114] A. Kutvonen, T. Ala-Nissilä, and J. Pekola. Entropy production in a nonMarkovian environment. Phys. Rev. E, 92:012107, 2015.
[115] T. Albash, D. A. Lidar, M. Marvian, and P. Zanardi. Fluctuation theorems
for quantum processes. Phys. Rev. E, 88:032146, 2013.
[116] J. V. Koski, T. Sagawa, O-P. Saira, Y. Yoon, A. Kutvonen, P. Solinas, M. Möttönen, T. Ala-Nissila, and J. P. Pekola. Distribution of entropy production
in a single-electron box. Nature Phys., 9:644, 2013.
[117] J. V. Koski, V. F. Maisi, J. P. Pekola, and D. V. Averin. Experimental realization of a Szilard engine with a single electron. Proc. Natl. Acad. Sci.
USA, 111:13786, 2014.
[118] J. V. Koski, V. F. Maisi, T. Sagawa, and J. P. Pekola. Experimental observation of the role of mutual information in the nonequilibrium dynamics of a
maxwell demon. Phys. Rev. Lett., 113:030601, 2014.
[119] J. V. Koski, A. Kutvonen, I. M. Khaymovich, T. Ala-Nissilä, and J. P. Pekola.
On-chip maxwell’s demon as an information-powered refrigerator. Phys.
Rev. Lett., 115:260602, 2015.
[120] B. Küng, C. Rössler, M. Beck, M. Marthaler, D. S. Golubev, Y. Utsumi,
T. Ihn, and K. Ensslin. Test of the fluctuation theorem for single-electron
transport. J. Appl. Phys., 113:136507, 2013.
[121] A. Hofmann, V. F. Maisi, C. Rössler, J. Basset, T. Krähenmann, P. Märki,
T. Ihn, K. Ensslin, C. Reichl, and W. Wegscheider. Equilibrium free energy
measurement of a confined electron driven out of equilibrium. Phys. Rev.
B, 93:035425, 2016.
[122] U. Seifert. Entropy production along a stochastic trajectory and an integral
fluctuation theorem. Phys. Rev. Lett., 95:040602, 2005.
[123] T. Speck and U. Seifert. Integral fluctuation theorem for the housekeeping
heat. J. Phys. A, 38:L581, 2005.
[124] U. Seifert. Stochastic thermodynamics, fluctuation theorems and molecular machines. Rep. Prog. Phys., 75:126001, 2012.
49
References
50
Errata
In Publication III, the black filled triangle in Fig 4(b) should be Δ.
51
Errata
In Publication III, the black filled triangle in Fig 4(b) should be Δ.
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