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(,#ą " W B J OT B OB U () ' . ,) & ) !#ąŗ & %. ,) (#& -% (. ŗ ăÿüĚ ĀúĚ ĀĂýüĚ ăŗ ăÿüĚ ĀúĚ ĀĂýýĚ Āŗ * 4 #/ Q B J OF U U V ăāĂĚ * 4 #/ Q E G ăāĂĚ þăýþŗ þăýþŗ þăþüŗ * 4 4 / ûāăăĚ * 4 4 / Q B J OF U U V ûāăăĚ * 4 4 / Q E G ûāăăĚ & -#(%#ŗ + VM L B J T VQ B J L L B 4 J W VN»»S » ûûāŗ & -#(%#ŗ 1 B J OP Q B J L L B 7 VP T J üúûĀŗ . *ĆĞĞ/,(ĄðĞĆĆăāĂĚ ăÿüĚ ĀúĚ ĀĂýýĚ Āŗ VS O ". 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. 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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. 51 9HSTFMG*agidcj+ I S BN9 7 89 5 2 6 0 6 832 9( p ri nt e d ) 7 89 5 2 6 0 6 8336( p d f ) I S BN9 I S S N L1 7 9 9 4 9 34 I S S N1 7 9 9 4 9 34( p ri nt e d ) I S S N1 7 9 9 4 9 4 2( p d f ) A a l t oU ni v e r s i t y S c h o o lo fS c i e nc e D e p a r t me nto fA p p l i e dP h y s i c s w w w . a a l t o . f i A al t o D D1 0 3/ 2 0 1 6 T o day t h ede finit io no ft h eampe rein t h e I nt e rnat io nalSyst e mo fU nit s is base do n c l assic alph ysic s.I t s pre c ise impl e me nt at io n is ve ry c h al l e nging.T h e re h as be e n gl o bale ff o rtt ore de finet h eampe re base do nt h equant iz at io no ft h ee l e c t ric c h arge .So c al l e dc h argepumps po t e nt ial l y pro videa dire c tand pre c iseimpl e me nt at io n o ft h is e me rging de finit io n.I nt h is t h e sis, w e int ro duc ea sil ic o nbase d quant um do t syst e mt h atc an beuse d as a c h argepump. West udy t h eac c urac yo ft h epump, c o unt individuale l e c t ro ns, and impl e me nt bidire c t io nalpumping.T h e sere sul t s pave t h ew ay fo rt h equant um ampe re . BU S I N E S S+ E C O N O M Y A R T+ D E S I G N+ A R C H I T E C T U R E S C I E N C E+ T E C H N O L O G Y C R O S S O V E R D O C T O R A L D I S S E R T A T I O N S