Silicon Quantum Electronics

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Cover: Transport spectroscopy data of a single arsenic donor embedded in a silicon field effect transistor, measured at 0.3 Kelvin. The signal is the channel current versus gate voltage, with each trace recorded at a constant (negative) source/drain bias voltage. The steps in current, such as in the red trace, are due to the quantum mechanical excited states of the donor atom. The red line, on the right side of the front cover, shows the signal from the donor in the ionized state. Then, moving to the left, via the spine to the back cover, the donor undergoes charge transitions from the ionized state to the neutral state and finally to the singly charged state.

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Silicon quantum electronics

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. ir. K. C. A. M. Luyben, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op donderdag 20 december 2012 om 12.30 uur door

Jan Verduijn

natuurkundig ingenieur geboren te Rotterdam, Nederland.

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Dit proefschrift is goedgekeurd door de promotor: prof. dr. S. Rogge

Samenstelling van de promotiecommissie:

Rector Magnificus voorzitter

prof. dr. S. Rogge University of New South Wales, promotor prof. dr. H. W. M. Salemink Technische Universiteit Delft

prof. dr. L. M. K. Vandersypen Technische Universiteit Delft prof. dr. P. M. Koenraad Technische Universiteit Eindhoven

dr. A. J. Ferguson University of Cambridge

dr. F. A. Zwanenburg Universiteit Twente

prof. dr. H. W. Zandbergen Technische Universiteit Delft, reservelid

Printed by: GVO drukkers & vormgevers B.V. | Ponsen & Looijen, Ede Cover design: Jan Verduijn

An electronic version of this dissertation is available at http://repository.tudelft.nl/.

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1 Entering the quantum world

The invention of the transistor started a revolution that greatly improved the quality of life for mankind. Scientists and researchers are continuously looking for technologies that may eventually have a similar impact on our daily lives. This chapter puts the work of this thesis in a historical perspective and presents some of the recent developments. Hereby it is attempted to show how this work contributes to the development of new technologies and the understanding of fundamental laws of nature. This chapter concludes with a section that provides and overview of the chapters that follow.

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2 Entering the quantum world

1.1 _{Improving devices}

M

iniaturization of logic circuits was first made possible with the inven-tion of a working solid-state transistor. The inveninven-tion of the transistor was followed by the crucial development of a fabrication process for circuits that integrated all components on a singe piece of material. This sparked the developments that lead to present day, pcs, laptops, smart phones, game consoles, etc. All of this took place at an unimaginable speed. The first demon-stration of a working transistor was at Bell labs in 1947. This was followed by the development of a practical implementation in the years thereafter, also mainly at Bell labs [1]. The concept of a small fully electrical switch was then combined with the idea to fabricate many individual devices in a circuit on a single piece of material (chip), the integrated circuit. For technological reasons silicon became the dominant material to make these circuits in these years. Today, silicon is still used for the vast majority of integrated circuits.

Since the early 70s these developments have resulted in a doubling the number of components within integrated circuits, compared to their size, about every 18 months. Gordon Moore, a co-founder of Intel, a company that is still one of largest producers of integrated circuits in the world, noted this exponential trend in 1965, and therefore it is named after him as “Moore’s Law”. Moore’s Law is an economic law and there is no physical reason why it should be true. But the rate at which computers, smart phones and other devices that rely on integrated circuits have improved is still exponential at the moment. There are, however, physical limits to these improvements in the conventional way. Laws of nature put bounds to this development and will eventually bring it to an end.

FinFET TFET

a b

Figure 1.1: Cross-sectional transmission electron micrograph image of a FinFET and TFET device, see text. a) The FinFET is for the first time used by Intel in their latest processor because of the superior performance over conventional planar device designs. The device studied in chapter 5 of this thesis is virtually identical to the shown device. Im-age adapted from: [2] b) A radically different physical effect is used for switching in the TFET: tunneling of electrons. Image adapted from “IMEC Scientific Report 2011’’, url: http://www.imec.be

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1.1. Improving devices 3 To sustain the increase in computational power of integrated circuits, the smallest feature size of the individual components in the circuit have been shrunken in an exponential way as well [3]. In the early 70s the size of the smallest features was 10 – 20 µm, whereas in Intel’s latest processors they are 22 nm. This is roughly a 10,000-fold decrease in about forty years. A decrease in the size of the features increases the number of possible devices per unit area present on a chip. This, in turn, increases the number of computational operations per unit time. In addition, the speed of the devices increases and dissipation per device decreases. Today, the main factor limiting scaling is the dissipation per unit area on a chip. Therefore, especially in the last ten to fifteen years, many new materials have been introduced to follow Moore’s law and create reliably working devices at the same time. Before that time, there had been no significant changes to the technology as developed in the early days. The industry has also started to work actively on the development of transistors with a non-conventional geometry, as well as devices that use fundamentally different physical principles to operate. Trusting these mea-sures will help them to keep improving performance, Intel has announced in their latest R&D pipeline that they plan to introduce a technology with 5 nm minimum feature size around 2020. At the moment however, there exists no technology that can achieve this reduction in size and can be used in mass production.

A successful new design that is radically different from the conventional planar geometry is the FinFET (Fin Field Effect Transistor), Figure 1.1a. This transistor design is used in Intel’s latest processors, introduced in early 2012. It basically consists of a thin wire fabricated on a substrate with a gate that wraps around the wire to form a channel region. By applying a voltage on this gate the electric current in the wire can be switched on and off. Because of the three-dimensional design, which provides improved electrostatic control over the channel compared to conventional planar FETs, the switching voltages and on/off current ratio are both reduced. Therefore, the power dissipation per device is reduced and the switching speed can be increased [4, 5]. The physical principle on which this switching is based is, however, still the same as in the early 70s.

A device that uses a radically different physical operation principle and is currently considered a promising candidate for future devices, is the TFET (Tunneling Field Effect Transistor), Figure 1.1b. This device relies on a quan-tum mechanical effect for its operation; tunneling of electrons. Just like a FET, a voltage on the gate switches the current through the device on and off. One advantage of using electron tunneling is that, in contrast to conventional FETs, the voltage range over which the device switches is, in principle, not limited by the temperature of the device. Therefore, lower switching voltages can be used and, consequently the power dissipation in the device can be up to a 100-fold reduced [6]. It should be noted here, that for the development

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4 Entering the quantum world of a practically useful TFET some challenges, such as a high enough contrast between the on and off state, still need to be overcome. But many of these challenges are currently being addressed by introducing new materials and improving designs [6].

At the same time, scientist and researchers are exploring the use of new materials and physical principles, not only to improve conventional computer architectures, but there is also a search for radically different ways to perform computations. A new paradigm for computation that has seen tremendous progress towards a practical implementation in the last fifteen years is the quantum computer [7].

1.2 _{Quantum computation}

The development of a quantum computer is only partly driven by the never-ending need for better devices, as it can probably not replace today’s com-puters. Quantum computers promise a huge increase in the computational efficiency for certain problems that take a very long time to solve on a conven-tional computer, such as searching large databases and breaking encryption codes. Besides this, a quantum computer will be able to perform quantum simulations that are impossible to do on classical computers [8]. Quantum simulations would, for example, help a researcher finding a new drug by de-signing a chemical reaction by an exact quantum simulation, instead of empir-ically having to find the reaction path. The power of the quantum computer lies in the fact that it can perform computations in a highly parallel way. The quantum mechanical principles on which such a computer is based allows to encode information and perform computational operations on a large amount of information in parallel. Instead of using a 0s and 1s, which naturally map to the “off” and “on” state of a transistor, a quantum computer uses quantum bits or qubits.

A qubit can be set up to be partly in a state ∣0⟩ and partly in a state ∣1⟩ at the same time. Furthermore, states of different qubits can be entangled with each other. This means that the outcome of a measurement on one of the entangled qubits will influence the outcome of a measurement on the other, irrespective of how far the qubits are apart. There are detailed theoretical ideas that show how these properties of the quantum world can be used in a clever way [8]. The challenge for experimentalists is to demonstrate real working implementations, which turns out to be not so easy.

In principle, many physical systems can be used as a qubit. Examples that have been demonstrated to work are isolated atoms, photons and spins in the solid state [7]. For all these systems the basic criteria that allow the construction of a quantum computer from these qubits are not at all, or only partly, met at the moment [10]. One of these criteria is that the qubit has to hold the information as long as it takes to perform the computation. This is a major challenge, because quantum information is very sensitive to a

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pro-1.2. Quantum computation 5 Coupler Cc = 15 fF Qubit A 100 µm SQUID Bias Qubit B Qubit C Qubit D

Figure 1.2: An example of a quantum device consisting of four flux qubits (A, B, C and D) that can be entangled with the couplers. This device has been used to demonstrate three-qubit entanglement. Image adapted from [9]

cess that causes loss of information, called decoherence. In other words, the computation has to be finished well within the typical time scale on which de-coherence takes place, i.e. the de-coherence time. Dede-coherence can be prevented if the qubit is isolated from the outside world. But this is in contradiction with the requirement of actually performing a computation, because this requires access to the quantum states in which the information is encoded. Thus, this links the states to the outside world and causes decoherence. Another chal-lenge is to link qubits to each-other to build a network that can perform more complex computational tasks than a single qubit. This requires a reliable way of transportation of the quantum information from one stationary qubit to another distant qubit.

Spins in the solid state have proven to maintain their quantum state for an extremely long time, up to minutes have been observed [11]. Especially spins in silicon are promising candidates. The main reason for this is that the long coherence times of a naturally occurring system can be combined with advanced fabrication techniques. Furthermore, it is possible to make silicon very pure and thereby removing sources of distortion (decoherence) of the quantum state. Another technological reason is that the years of development to improve the performance of silicon devices has generated an enormous base of knowledge from which can be drawn to create devices for quantum appli-cations. At the same time, however, it is difficult to manipulate spins at a speed that allow a sufficient number of operations to be performed before the

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6 Entering the quantum world information is lost by decoherence. Since it is a natural system, it is not easy to change this property. A different approach is to design a device which can mimic a spin state.

A spin is a naturally occurring property of elementary particles. An ex-ample of a device that simulates a spin state in the solid state is the super-conducting flux qubit. This system consists of relatively large ring in which the quantum information is encoded in clockwise and anti-clockwise running currents. Fast manipulation of the ‘spin’ in this system is easier than for nat-ural spins in the solid state, because of the freedom to engineer the qubits and optimize of fast manipulation. Furthermore, it has been shown that they can be coupled, see Figure 1.2. Their main limitation is the coherence time, which is much shorter than natural spins in solids.

Recently, a new idea, that may help to overcome limitations of naturally occurring spins as well as engineered spins, sparked the interest of many re-searchers: the topological quantum computer. The idea is to use a certain type of quantum state that is robust to decoherence in a way similar to a knot in a rope. Such a state can not be ‘unknotted’ by simply twisting and pulling. A more complex operation is needed to unknot the rope or to change the knot from one type into another. Driven by this idea, recently an important goal has been achieved, discussed in the next section.

1.3 _{Designing particles}

Engineered solid state devices can be used to design quantum mechanical de-grees of freedom that are not readily available in nature, as well as tailor quan-tum systems to requirements for specific applications. Using the right mix of materials and the right geometry, new quantum states can emerge under the right conditions. Such a state can also be pictured as a particle, or more pre-cisely, a quasi-particle. A quasi-particle is a composite particle build up out of elementary particles (atoms and electrons) of which the device consists. An intriguing example of a quasi-particle that is really and ‘engineered’ particle, which has not been found as an elementary particle in nature, is the Majorana Fermion [12], see Figure 1.3. A Majorana Fermion is the particle (or quantum state) that can be used to ‘knit’ quantum states together, as explained in the previous paragraph, but its discovery is also relevant to fundamental physics. In 1937 the Italian physicist Ettore Majorana predicted the existence of a certain type of particle from a then recently developed theory for fundamental particles. This particle is now called a Majorana Fermion. Special about the Majorana Fermion is that it is its own anti-particle. Furthermore, it has been suggested that the knotted quantum states, that are particularly robust to decoherence, can be made with it. Since the prediction by Majorana, physi-cists have searched for this particle in nature, but have found no sign of it so far. Very recently, however, the first experimental evidence of a Majorana Fermion was observed in a superconducting device [12], see Figure 1.3. Even

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1.4. This thesis 7 S N eV

*

1 2 3 4 b 1 μm

B

B 1 m 1 2 3 4 N S a

Figure 1.3: a) A quasi-particle can be created with the right device design under certain conditions. b) By combining a superconductor labeled (S) and normal conductor labeled (N) in the same structure, a Majorana Fermion appears at each side denoted by the red “∗”. Image adapted from [12]

though this experiment still needs to be confirmed, and the properties that make the observation interesting from a fundamental point of view and for quantum computation have not been probed directly yet, this experiment is a nice illustration of how combining and structuring materials allow one to create objects in solids that do not exist in nature.

1.4 _{This thesis}

Silicon FinFET devices are in this thesis used to study quantum effects. The main focus will be on devices with dopants embedded in them. A dopant is a naturally occurring impurity that can hold a single electron and thereby localizes a single spin. This makes it suitable to serve as a building block for a quantum device. The main objective of this thesis is not to directly work on the development of a physical implementation of a qubit in silicon, but rather to aid the understanding of a physical system that has favorable properties for the implementation of several proposed quantum device schemes.

Chapter 2 starts by discussing the fabrication and basic characteristics of devices studied in this thesis. This is followed by an overview of the basic theory of single electron transport and the effect correlations and interference that play a role in the presented experiments. To illustrate the usefulness of donors for quantum device applications, chapter 2 finishes with some exam-ples of qubit implementations and coherent control. Chapter 3 discusses the level spectrum of donors confined in silicon nano-structures by outlining an effective mass theory that is used to calculate the level spectrum. In chapter 4, a detailed analysis of the quantum mechanical level spectrum of an elec-tron localized near an interface and subject to an electric fields is presented. By comparing devices with and without donors, the influence of the donors and the electric field is studied. In chapter 5 this is further detailed and the

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8 Entering the quantum world experimental observation of a controlled transition of a electron from a donor to a nearby interface is presented. Chapter 6 discusses an interference effect associated with transport mediated by donors and in chapter 7 it is shown that similar interference effects persists in the presence of many-body correlations.

References

[1] The Silicon Engine: A Timeline of Semiconductors in Computers, url: http://www.computerhistory.org/semiconductor/timeline.html, accessed on 13 September 2012.

[2] Roche, B. et al. A tunable, dual mode field-effect or single electron tran-sistor. Appl. Phys. Lett. 100, 032107 (2012).

[3] Ferain, I., Colinge, C. A. & Colinge, J.-P. Multigate transistors as the future of classical metal–oxide–semiconductor field-effect transistors. Na-ture 479, 310–316 (2011).

[4] Hu, C. et al. FinFET-a self-aligned double-gate MOSFET scalable to 20 nm. IEEE Transactions On Electron Devices 47, 2320–2325 (2000). [5] Yang, F.-L. et al. 5nm-Gate Nanowire FinFET. In Digest of Technical

Papers, Symposium on VLSI Technology, 196–197 (IEEE, 2004).

[6] Ionescu, A. M. & Riel, H. Tunnel field-effect transistors as energy-efficient electronic switches. Nature 479, 329–337 (2011).

[7] Buluta, I., Ashhab, S. & Nori, F. Natural and artificial atoms for quantum computation. Reports on Progress in Physics 74, 104401 (2011).

[8] Nielsen, M. Quantum computation and quantum information (Cambridge University Press, Cambridge New York, 2010).

[9] Neeley, M. et al. Generation of three-qubit entangled states using super-conducting phase qubits. Nature 467, 570 (2010).

[10] DiVincenzo, D. P. The physical implementation of quantum computation. Fortschritte der Physik 48, 771–783 (2000).

[11] Steger, M. et al. Quantum information storage for over 180s using donor spins in a28_{Si “semiconductor vacuum”. Science 336, 1280–1283 (2012).}

[12] Mourik, V. et al. Signatures of majorana fermions in hybrid superconductor-semiconductor nanowire devices. Science 336, 1003–1007 (2012).

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2 Quantum electronics and transport

Silicon is a material that has some major advantages over other materials for quantum electronics applications. First of all, due to the wide use of silicon for conventional electronics applications, there is a large base of knowledge available about the fabrication of devices as well as extremely high quality raw material. Processing equipment and fabrication techniques for both electronic and optical applications are readily available. Furthermore, there are some more fundamental advantages to the choice for silicon, such as the possibility to remove the nuclear spins from the silicon. These nuclear spins are disad-vantageous for spin-based quantum applications because they cause the loss of quantum information that is encoded in spin states. In this thesis transport spectroscopy is used to study single donors in silicon. The transport is studied in two ways: it is used as a spectroscopic tool to probe the electronic properties of donors, and to study the transport of electrons.

In this chapter, the basics of electron transport at low temperature via quantum states are outlined. First, the main features and fabrication of the devices that serve as a platform are explained. Then the basic concepts of single electron transport are discussed, followed by some examples of the effect of interference and correlations on the transport. Finally, the properties of donors in the context of donor-based quantum computation are highlighted.

Section 2.1 and 2.2 of this chapter have some overlap with the chapter titled “Single dopant transport” in “Single-Atom Nanoelectronics” to be published 31st March 2013 by Pan Stan-ford Publishing, ISBN: 978-981-4316-31-6

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10 Quantum electronics and transport

2.1 _{Devices and fabrication}

W

ith single dopant transport spectroscopy at low temperature it has be-come possible to study the properties of individual dopants directly [1–6]. This opportunity initially emerged as a natural consequence of the ef-fort of the semiconductor industry to overcome the fundamental limitations of classical device geometries. Very recently however, an atomic-scale fabri-cation technique using STM lithography on a surface passivated with atomic hydrogen was successfully used to fabricate a single dopant transport device [6]. Another important milestone in the study of single dopants is the read-out and coherent manipulation of the electron spin bound to a single phosphorus donor [7, 8]. In this thesis, the FinFET (Fin Field Effect Transistor) is used as a platform to study single dopant transport, see Figure 2.1. This device was developed by the semiconductor industry as an alternative to the conventional planar field-effect transistor geometry. The fin-geometry of the device has favorable properties that delay CMOS (Complementary Metal Oxide Semi-conductor) scaling problems, such as short channel effects and drain induced barrier lowering [9]. It has also enabled the investigation of the electron oc-cupation of single donor atom [1], the excited state spectra [10], the binding energy [5] and the effect of a nearby interface on the electron wave function. [10–13].

Buried SiO2 Si Substrate Wafer SiN Spacer

SiO2 Gate Dielectric Channel drain source Poly-Si Gate source drain gate 200 nm x y z b a

Figure 2.1: a) SEM image of a FinFET device. b) A schematic source/drain cross-section device.

The devices studied in this thesis are MOSFETs (Metal Oxide Semicon-ductor Field Effect Transistors) and have been fabricated at a CMOS foundry on 200 mm wafers. The reason for this choice is two-fold. (i) It is more diffi-cult to achieve the degree of control in fabrication processes in a smaller-scale research cleanroom compared to a highly optimized and standardized foundry

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2.1. Devices and fabrication 11 fabrication process. (ii) A large number of devices with small intentional varia-tions in characteristics can be fabricated with high precision in a reproducible way. There are, however, disadvantages to this choice as well. First of all, it is often quite involved to modify individual fabrication steps as this may require re-optimization of the process flow as a whole. Second, the turnover time is relatively long. To fabricate one batch of wafers can take up to several months, excluding the time spent on process development. Despite these diffi-culties, fabricating single dopant devices and single electron devices, even with multiple gates, with a foundry-based CMOS process seems to be a promising route toward large-scale integration of quantum devices [14]. The fabrication process consists of many fabrication steps (>150) and therefore, instead of describing the fabrication, the aspects important for single dopant transport are outlined here. Also, some experimental techniques that give insight in the mode of transport of nano-scale devices will be briefly discussed.

FinFET devices consist of a nano-wire that is etched out of a SOI film (silicon on insulator film), see Figure 2.1b. The insulator consists of a buried silicon dioxide that is about 150 nm thick for modern devices. Because of this thin oxide, the substrate wafer can be used as a back gate at low temperature [15]. A 5 nm thick silicon dioxide layer is electrically isolating the channel from the polycrystalline silicon gate. This conservative choice in terms of gate material, oxide material and thickness, was made to minimize gate leakage and charge noise. For a nano-scale device with dimensions ≲ 60nm the gate leakage is typically less than 10 pA with 1 Volt on the gates. The source and drain contacts are formed by implanting the device with a high dose of arsenic which is activated in a subsequent anneal process. During the source/drain implantation, the channel region is protected by the gate and the silicon nitride (SiN) spacers next to the gate, see Figure 2.1b. This results in an underlapping structure with good gate control and low access resistance to the channel. The term ‘underlap’ is used when the highly doped source and drain contacts do not extend all the way under the gate, but there is a small portion of the wire next to the gate that is left undoped (or has the same doping as the channel). Especially at low temperature, the access regions determine the size of the tunnel barrier to the part of the channel under the gate. Since the gate control over the access regions is weak, large access regions will result in the formation of a quantum dot (potential dip) centered between the source and drain contacts that can confine electrons. Small access regions, on the other hand, will result in a structure with a single tunnel barrier controlled by the gate [16]. The latter structure is particularly useful to study single dopants.

To fabricate FinFET devices for single dopant transport, two strategies have been shown to work, see Figure 2.2. One possibility is to make use of in-diffusion of dopants from the source/drain contacts into the channel region. In this case the transport is direct resonant tunneling from source via a dopant to the drain [5] or mediated by a transport channel that is formed at the

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channel-12 Quantum electronics and transport

electron energy

conduction band edge

50 nm 300 nm s s d d g g a b d c 30 nm Gate Source Drain Ids (µ A) −2 V_g(V) −1 0 1 2 10−7 10−5 10−3 10−1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 G (e₂ /h ) −108 −50 −24 0

Ionization energy (meV)

−100 −50 0 50 100 V (mV)d −1.4 −1.2 −1.0 −0.8 −0.6 −0.4 Vg (V)

Figure 2.2: a) Stability diagram, showing Coulomb diamonds (white on this color scale) that are distorted by the presence of a dopant in the barrier region. b) This dopant it not carrying any current, but influences the Coulomb blockade of the quantum dot beneath the gate. c) Direct transport can be achieved if the channel of the device is very short (see inset of d) and is used, in this case, to determine the ionization energy. d) In these devices the dopant is visible in the current versus gate voltage trace, even at room temperature. The red and blue lines correspond to devices with dopants in the central region and the dashed line to a device without. Images a) and b) are adapted from [17] and c) and d) from [5].

gate dielectric interface [1, 18]. Especially in devices with wide spacers, this strategy very often leads to dopants that indirectly influence the transport. This results in a modification of the tunnel barrier to the quantum dot formed under the gate in a series configuration [19], or by modifying the spacing of the Coulomb oscillations in a side-coupled configuration [17, 20], see for example Figure 2.2a and b. Another strategy is to dope the silicon film before etching the channel. This will result in randomly distributed dopants in the channel, that can induce resonant tunneling to single dopants if the channel is short enough. Fairly high doping concentrations will need to be used, because the used devices typically have a gate length ≲ 60 nm. For example, a doping concentration of 1 ⋅ 1017_cm−3_{results in on average one dopant in a 22 × 22 × 22}

nm3 _{cube and a concentration of 1 ⋅ 10}18_cm−3 _{results in one dopant every}

10 × 10 × 10nm3_.

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2.1. Devices and fabrication 13 can be used to obtain information about the shape of the potential landscape. This, however, probes only local properties as the transport will take place in regions with the lowest potential. Instead, studying thermally assisted trans-port as a function of temperature has proven to be a very powerful method to get insight in the various modes of operation of nano-scale FinFETs [18, 21, 22]. This technique has the advantage that it can be directly applied to a nano-scale FET device and no special test structures need to be used, as is very often the case with other methods.

a c b 0 200 400 0 20 40 60 Eb [meV] Vg [mV]

Figure 2.3: a) A thermionic measurement provides information about the barriers formed in a device. b) The conductance versus gate voltage at small source/drain bias, c) and at higher bias of a similar device, confirms the formation of residual barriers on both sides of the gate. Images adapted from [16].

Thermally assisted transport, or thermionic transport, allows for the deter-mination of the barrier height as well as the effective cross-section of the device channel participating in the transport. With this technique, the barrier height is almost directly related to temperature and gate voltage, and is therefore straightforward to extract from experimental data alone. The active cross-section is more difficult to extract. Details of the band structure need to be known to extract quantitative results. This is not straightforward, especially for nano-scale devices, since the band structure is warped by the nano-scale confinement [21]. Figure 2.3 shows an example of the barrier height extracted by fitting conductance data versus gate voltage and temperature to an analytic formula [18, 23]. This clearly demonstrates that increasing the gate voltage first lowers the barriers in a linear fashion, red line in Figure 2.3a. But at larger gate voltages residual barriers are formed that are more or less constant for Vg>0.1V. These barriers are formed in the access regions under the spacers which are only weakly influenced by the gate. Figure 2.3b and c confirm this scenario. Clear Coulomb blockade due to electrons confined in the channel under the gate is observed in this regime, see also section 2.2.

Much progress has been made in recent years with the fabrication of de-vices in CMOS processes as well as the understanding of microscopic device properties. Single dopant transport and, very recently, also transport through

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14 Quantum electronics and transport a system of tunnel-coupled dopants [24], has become reasonably straightfor-ward. Also, the fabrication of devices with multiple gates in which coupled quantum dots can be formed has been demonstrated with this technology [25]. Beside low temperature transport, thermionic transport has proven to be a useful tool to study nano-scale devices directly. In the next section transport spectroscopy of single donors in FinFET devices will be discussed.

2.2 _{Transport spectroscopy of single donors}

To explain the basic ideas of transport spectroscopy at low temperature, elec-tronic transport through quantum dots is discussed. Elecelec-tronic transport through quantum dots is very similar to transport through dopants and many concepts can be transferred to single dopant transport. There are, however, some important differences, such as the nature of the confining potential, which causes deviations from the constant interaction model, commonly used to un-derstand transport through quantum dots. Therefore, before single electron transport in general is explained, some of these differences are discussed. For a more complete overview of transport in quantum dots please refer to Kouwen-hoven et al. [26] or Hanson et al. [27].

2.2.1 Single donor transport

Single donor transport can draw on a large number of ideas developed in the context of transport through quantum dots in metals and semi-conductors [28]. The most important difference, however, is the nature of the confining poten-tial. A quantum dot is formed by gates that define the potential landscape and therefore the size of the dot. Due to variations in the fabrication, each quantum dot will therefore be different and the electrical characteristics vary. Dopants, on the other hand, are all identical since the confining potential is defined by the charged nucleus, see Figure 2.4. When the donor is ionized, it is customary to denoted it as the D+ state, the neutral state of the system,

the nucleus with one electron bound to it, is called the D0 _{and the charged}

two-electron state D−. A bulk donor cannot bind more than two electrons.

The imperfect screening of the nuclear charge by the first electron allows the binding of a second electron [29] with a binding energy of a few meV, but excited states are unbound with respect to the conduction band minimum. This is another important difference between donors and quantum dots. De-pending on the size and strength of the confining potential, quantum dots can accommodate up to several thousands of electrons.

A consequence of the two-electron system being confined in a small region is that the energy difference between the D0_{and D}−state, the charging energy, is

relatively large. This is advantageous for device applications involving the D0

state, because it makes this state relatively stable with respect to changes in the local chemical potential that may be induced with a nearby gate. It also allows such a device to operate at temperatures ≲4 K, while for quantum dots often

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2.2. Transport spectroscopy of . . . 15

+

dopant

a b

quantum dot

Figure 2.4: The confining potential and single particle excitation spectrum of a) a dopant and b) a typical quantum dot.

lower temperatures are required. Another advantage for high temperature operation is the large splitting between the ground state and the first excited state, 11.7 meV for phosphorus and 21.1 meV for arsenic [30]. This large splitting provides a well isolated ground state in which the electron spin can be used to encode quantum information. First of all the quantum information encoded in the spin of the ground state will not very easily be lost due to thermal excitation to one of the excited states. In addition, a strong microwave field at high frequencies can be applied to a spin in the orbital ground state without inducing transitions to excited states.

It should be noted however, that if electrical addressing of single dopants is required, gates need to be placed in close vicinity of the dopants. This affects many of the above mentioned favorable characteristics. In chapter 3 the electronic structure of confined donors is discussed. In nano-structures that are used to get access to single donors, electric fields and the presence of metallic gates and dielectric interfaces generally decrease the splitting between excited states, and may change the binding energy [12, 13].

2.2.2 Single electron transport

Single electron transport spectroscopy relies on the Coulomb blockade effect. The Coulomb blockade effect is a natural consequence of the fact that electrons repel each other because of their equal charge. When a region is connected to metal contacts with a small enough conductance, this leads to quantized charging. A small conductance in this context means smaller than the conduc-tance quantum G0=e2/h = 38.6 µS or, equivalently, a resistance larger than 25.9 kΩ [28]. A three-terminal configuration is the simplest way to change the potential independently from the chemical potential of the contacts. For an accessible overview of single electron tunneling please refer to Thijssen and van der Zant [31].

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inter-16 Quantum electronics and transport action model. In this model the energy to charge the system with exactly one additional electron is given by:

EC= e2

C (2.1)

where C is the total capacitive coupling to the charged region. The most sig-nificant contributions to the total capacitance are usually assumed to be the geometric capacitance due to the presence of gates to change the electrostatic potential and source and drain electrodes to allow charge transport, thus the total capacitance is given by C = Cg+Cs+Cd, with Cs, Cdand Cg the source,

drain and gate capacitances, see Figure 2.5a. Note that the charging energy is assumed to be independent of the number of electrons N within this approx-imation. An additional requirement for charge transport is that the source and drain are tunnel coupled to the localized state. The electric gate is used to change the potential of the confined region by applying a gate voltage Vg

and thereby increasing or decreasing the number of electrons of the ground state. These electrons are supplied by the source and drain contacts. Besides the capacitances, Cs,d, the source and drain contacts are characterized by a

tunnel coupling, Γs,d, see Figure 2.5b. Because of the capacitive coupling,

the potential of the source and drain contacts will also influence the potential of the confined region. Therefore, this can also cause a change in the num-ber of electrons. A stability diagram is a convenient way to plot transport spectroscopy data, see Figure 2.5c. Electron transport is allowed in the grey regions where electrons tunnel sequentially. In the diamond-shaped regions (white) the number of electrons is stable and sequential electron tunneling is forbidden due to Coulomb blockade.

0 N=0 N=1 N=2 V_g V_sd E_C E_x V_sd Cd, Γd C_g Cs, Γs EC, Ex V_g I_d a b

Figure 2.5: a) Circuit of a quantum dot or dopant (open circle) tunnel coupled to a source and drain and just capacitively coupled to a gate. b) A stability diagram shows Coulomb blockade and can be used to extract values for the charging energy EC(see text) and the excited state energy Ex.

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2.2. Transport spectroscopy of . . . 17 equations. Sequential tunneling considers the individual tunnel events as being uncorrelated, once the electron has tunneled across a barrier, it will not “re-member” anything of its past. Within this framework, obtaining expressions or simulating transport is straightforward using the master equation approach. To calculate steady state transport (DC), the procedure is to first compute the mean occupation of the localized state and then calculate the stationary cur-rent. The rate at which the occupation of a state coupled to two leads changes is given by: dρo dt = ∑_i (Γ s,in i +Γ d,in i )ρe− (Γ s,out i +Γ d,out i )ρo (2.2) with Γin,out

i the tunnel rate in and out of the i

th _{state, and ρ}

o and ρe are the

total probability of finding the system respectively occupied or empty. Once ρo

(= 1 − ρe) is known, the current can be calculated by considering the transport

across one of the barriers on the left or right (s or d), see Figure 2.5a. The current across the left barrier (source side), given by:

I(t) = Is(t) = e ∑

i

[Γs,in_i ρe(t) − Γ s,out

i ρo(t)] (2.3)

with e one electron charge. Note that in a steady state situation this is equal to the current across the right barrier (drain side). For simple systems in a steady state, i.e. dρo/dt = 0, this can be done analytically. One useful example

is a simple system with Coulomb blockade and thermally occupied leads. In this case, the current at small source/drain bias voltage, Vsd, and transport

through a single quantum state, develops resonance given by [32]: I = Vsd e2 4kT ΓsΓd Γs+Γd cosh−2(αV′ g/2kT ) (2.4) with V′

g the detuning with respect to the level and α the conversion factor

from gate voltage to energy given by eCg/C, also referred to as the capacitive

coupling. This equation is valid when a single quantum level is occupied and the thermal energy, kT , is larger than the level spacing (explained below) and eVsd, but much smaller than the charging energy EC. The characteristic

temperature dependence of the resonance height and the width can be used to prove that the transport is mediated by a single quantum level only [33]. Equation 2.4 was derived under the assumption that only a single level (the ground state) contributes to the transport. If excited states also take part, single electron tunneling exhibits some qualitative differences [33].

Confinement of electrons in a small region results in discrete quantum states with a well-defined energy. This is typically the case for dopant states and small quantum dots in semiconductors. An electron tunneling onto such a system will have to occupy one of these states. As a consequence of the dis-crete excitation spectrum, the sequential current shows structure as a function

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18 Quantum electronics and transport of the voltages on the contacts. When a quantum level becomes accessible and contributes the current, the current changes in a stepwise manner. This enables the determination of the excited state spectrum of localized states of dopants or quantum dots. For that purpose, it is most of the time convenient to represent the spectroscopy data as the differential conductance, G, which is the derivative of the current with respect to the source/drain bias voltage, dId/dVsd. Figure 2.5c schematically illustrates how the excited states show up

in the transport spectroscopy data of a single donor embedded in a nano-scale transistor. At the finite experimental temperature, the thermal occupation of the leads limits the resolution of transport spectroscopy. It broadens the tran-sitions in current, or differential conductance resonances, by the characteristic energy kT , because the thermal environment supplies or absorbs energy to or from the tunneling electrons.

Up till now, only sequential tunneling has been discussed. In the model of sequential tunneling, individual tunneling events are uncorrelated and fol-low Poissonian statistics. In general, this means that there is no coherence. Even though the sequential tunneling model gives good insight, very often its approximations are not sufficient to explain certain features of the transport. In certain cases electrons can interfere in a way similar to a double slit inter-ference experiment, or form collective states that mediate transport. In the following section three examples of this will be discussed, the Fano effect for electron transport, the Aharonov-Bohm effect and the Kondo effect.

2.2.3 _{Interference effects and correlations}

Fano resonances can be observed even when interactions are not important. In that case electrons interfere with themselves. The symmetric Breit-Wigner or Lorentzian resonance can be considered a special case of a Fano resonance, with a large so-called Fano factor [34], discussed in more detail below. Meso-scopic systems that are likely to exhibit a Fano-type resonance, must have a transport channel with a rapidly varying phase and at least one other channel with a slow phase variation. For example, an open channel with a side-coupled localized state [35] or a system with one open channel in parallel with a res-onant channel, see Figure 2.6, can show Fano-type transport. Theoretically, Fano-like transport can be obtained if the interference between an open non-resonant channel and a non-resonant channel is considered, where tnand trare the

non-resonant and resonant transmission amplitudes respectively, ∣tn+tr∣2= ∣βe−iθ+γ

iΓ/2 iΓ/2 + ∣

2

. (2.5)

Here, θ is the acquired phase, Γ is the decay rate of the resonance and the detuning of the resonance from the Fermi energy, β and γ are the respective transmission amplitudes of the non-resonant and resonant paths. The acquired phase, θ, can be just fixed by the geometry or can be tunable by, for example,

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2.2. Transport spectroscopy of . . . 19 the Aharonov-Bohm effect as discussed below. The decay rate depends on the tunnel coupling of the localized state that induces the resonance (rapid phase change). Depending on this coupling the degree of hybridization of localized state with the continuum environment, broadens the resonance and decreases the life time.

5 2.5 0 -2.5 -5 0 1 2 3 4 B a b c

Figure 2.6: Examples of systems in which a Fano effect can be observed are a a side-coupled localized state and b) a localized state embedded in an Aharonov-Bohm ring arrangement coupled to leads. The black arrows indicate the possible paths (only to first order) from the left to the right lead. c) A plot of the Fano-like behavior of Eq. 2.5 for different values of θ with β = γ = Γ = 1.

The phase across the resonance changes rapidly by π, over a range ∼ Γ. Note that, even though this has not been explicitly taken into account in Eq. 2.5, also electrons through the resonant path may acquire a geometrical phase. This, however, results in just an overall phase offset and results in the same resonance. Figure 2.6 shows plots of Eq. 2.5 for different values of θ. The most commonly used form of the of Fano formula for a general complex parameter qis:

∣t()∣2=

∣ + qΓ/2∣2

(Γ/2)2₊2 (2.6)

This is a phenomenological form and detailed modeling is in general required to related q to physical properties of the system. Even though Eq. 2.5 cannot be derived exactly from Eq. 2.6, the role of tan(θ) in Eq. 2.5 is similar to q in the standard form Eq. 2.6. For a Fano effect it is important that electrons preserve coherence while traversing the structure to interfere with themselves or with other electrons. Therefore, the observation of a Fano resonance is direct evidence that coherence is preserved. As mentioned before, apart from the geometrical phase electrons acquire while traversing the structure, the phase can be tuned by the Aharanov-Bohm (AB) effect [36].

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20 Quantum electronics and transport Electrons traversing a loop with magnetic field, B, applied perpendicular to the loop, will acquire a phase which is proportional to the applied magnetic field. The phase difference for electrons with the same start and end points will have acquired a phase given by [37]:

∆ϕ = A ⋅ Be ̵ h=2π

A ⋅ B

Φ0 (2.7)

where A is the area enclosed by both paths and Φ0 =e/hthe flux quantum. The conventional manifestation of the AB effect in a mesoscopic ring is the si-nusoidal modulation of the current as a function of magnetic field. Embedding a system, such as a dopant or quantum dot, in an AB ring geometry is one of the most direct methods available to study coherent transport phenomena. Such a system allows to monitor the phase of electrons as they interfere with themselves or with other electrons. And local electrical gates on the embed-ded system allow for the investigation of the energy spectrum. Tuning the magnetic field changes the phase through the AB effect, see from Eq. 2.7. Be-sides single electron coherence, coherence can also be carried by a correlated many-body electron state. Many-body effects in electron transport arise at low temperature due to interactions, as a result of a reduced disturbance by the thermal environment. The Kondo effect, discussed below, is an example of such a correlation effect mediated by virtual intermediate states.

Virtual intermediate states can mediate transport in the Coulomb blockade region of dopants or quantum dots and generate considerable current. Intu-itively, tunneling via virtual states can be understood as the electron allowing to gain energy ∆E for a short time ∆t while still satisfying the Heisenberg uncertainty relation ∆E∆t ≳ ̵h/2. Here, the time ∆t is set by the tunnel rate Γs,d ≈1/∆tand ∆E is the energy needed to occupy the virtual intermediate state. In order to preserve energy, these processes involve simultaneous tunnel-ing of multiple electrons, and the tunnel probability is therefore higher order in the tunnel rate. For this reason these tunnel prosesses are called cotunnel-ing processes. Only for systems that have a relatively large tunnel coupling to both leads higher order tunneling processes can generate a net measurable current. In the following, the main characteristics of a higher order tunneling effect, the Kondo effect, will be briefly discussed. An overview of transport regimes in the Coulomb blockage region is published by Pustilnik & Glazman [38].

Traditionally, the Kondo effect was observed in experiments on metals with magnetic impurities. It was found that the resistance unexpectedly increased below a critical temperature [39]. In transport through quantum dots [40] and donors [41] on the other hand, the Kondo effect enhances the transport in the Coulomb blockade regime. It emerges as a resonance in differential conductance around zero source/drain bias, see Figure 2.7. The Kondo effect can be seen as the coherent addition of elastic cotunneling processes, see Figure

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2.2. Transport spectroscopy of . . . 21 2.7a. These particluar tunneling processes involve a spin-flip and thus leave the system in a different spin state. Collectively, these processes form a transport channel due to the formation of a many-body state in which electrons in the contacts screen the magnetic moment of a electron localized on the quantum dot or donor and form a spin singlet state.

a b 360 380 400 420 440 -6 -4 -2 0 2 4 6 0 1 2 3 4 VG (mV) VSD (mV) dI/dV (μS)

Figure 2.7: a) The Kondo can be seen as a many-electron spin singlet state that forms between the localized electron spin and the spins in the leads. b) This can mediate transport in the Coulomb blockade regime and shows up as a resonance at zero source/drain bias indicated by the arrow. Image b adapted from [41]

Characteristics that are generally seen as the experimental prove of a Kondo effect are [40]: (i) a resonance in differential conductance at zero bias, (ii) the logarithmic increase of the zero bias conductance below the Kondo temperature TK and (iii) the splitting of the zero bias resonance by Vsd = ±gµBB in a

magnetic field B. Because the energy difference equal to the Zeeman energy has to be supplied at finite magnetic field, the Kondo resonance splits in two resonances at finite bias. When the Zeeman energy exceeds the width of the Kondo resonance, kTK, the current is quenched completely. Since the binding

energy of the spin singlet Kondo ground state is ∼ kTK, thermal fluctuations

will destroy the Kondo state for T > TK. As the temperature is lowered below

the Kondo temperature, coherence increases and the ground state forms until it saturates as T → 0. This zero-temperature regime is called the unitary limit of the Kondo effect.

For quantum device applications tunable interactions between states are necessary to manipulate the quantum information that is encoded in them. At the same time it is important that the coupling of the system to a thermal environment is weak as this can cause the decoherence of the quantum state. In the next section a brief overview of the usefulness of donors for quantum device applications are given. To illustrate the current status of the field, some examples of coherent manipulation of donor states are discussed.

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22 Quantum electronics and transport

2.3 _{Single donors as qubits}

A quantum bit, or qubit, takes advantage of the fact that quantum states can be built from arbitrary superpositions of states that can be entangled. The vast majority of information is nowadays stored in the form of classical bits, 0s and 1s, in some physical system. This is convenient because many systems in nature can easily be made bi-stable and information can be processed us-ing simple switches that switch ‘on’ and ‘off’. There is however a large base of knowledge being build up on how a quantum system may be able to per-form some tasks in a much more efficient way than ever can be done using a classical system. Some first implementations have been demonstrated on a small scale in various physical systems [42]. Promising candidate systems for a practically useful quantum computer are for example real atoms or ions (in the gas phase), superconducting devices, semi-conductor quantum dots and donor spins in the solid state. Below, first a brief explanation of a qubit and a prototypical implementation of a quantum computer as proposed in 1998 by Kane will be given. Then two examples of coherent manipulation are high-lighted; manipulation of donor spins, and coherent manipulation donor-bound charge. Finally, an overview of work in the following chapters, which aids to the advance of the implementation of a donor-based quantum computer, is given. Please refer to the review by Buluta et al. [42] for a comprehensive overview of physical implementations of qubits.

In a qubit information is stored in two basis states ∣0⟩ and ∣1⟩ by bring-ing the qubit in a superposition of these two states, α∣0⟩ + β∣1⟩. Here, the normalization condition ∣α∣2_{+ ∣β∣}2

= 1 must be satisfied at all times. Such a state can be visualized in a convenient way by a unit vector directed to a point on the surface of a sphere, called the Bloch sphere, see Figure 2.8a. By performing rotations about the origin, for example about the x-, y- and z-axis, any arbitrary linear combination can be made. A computational task can be performed by initializing multiple qubits in some initial state, then performing some rotations on individual qubits that may be dependent on the state of the other qubits and eventually read off the outcome [43]. During this process a fundamental requirement is that the coherence of the qubits is preserved [43]. Figure 2.8b shows the signal of coherent rotations of the nuclear spin asso-ciated with a phosphorus donor in a silicon crystal. Donors (in silicon) can be used to build a quantum computer in so-called Kane quantum computers [46]. The physical implementation of the Kane quantum computer is schematically illustrated in Figure 2.9. Kane envisioned that donor-bound electrons can be used to mediate the interaction between the donor nuclear spins [46]. Using electric gates to manipulate the donor-bound electrons located just beneath the surface in combination with a global microwave field and a static magnetic field, quantum computations can be performed, see Figure 2.9b. The electric gates are used to switch the interactions between individual donors as well as to control the interactions between the electron spin and the donor nuclear

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2.3. Single donors as qubits 23 0 100 200 300 400 500 -10 -5 0 5 10 ES E In te ns ity (a u) Time (ns) in-phase (X) quadrature (Y) x y |0 |1 B1 M a b

Figure 2.8: a) A point on the Bloch sphere is a convenient way to visualize arbitrary rotations of the state of a qubit. Image adapted from [44]. b) Coherent rotations of the phosphorus-31 nuclear spin embedded in a silicon crystal about the z-axis, as apparent from the time dependence of the projections of the qubit state on the x- and y-axis. Image taken from [45]

spin [44]. The read-out is performed by charge sensing using a spin-to-charge conversion process [8, 47]. Donors |1 |0 Donor architecture Control ga tes 31_P 31_P 31_P 31_P 20 nm Si SiO2 _{R ead-out} e– e– e– e– Si conduction band 31_P _e– a b

Figure 2.9: a) The nuclear-electron spin system of a phosphorus donor (blue and red arrows respectively) forms a natural basis for a qubit with extremely long coherence times. b) Placing several dopants close to an interface to allow electrical manipulation of the donor-bound electron is the prototypical architecture of a donor-based quantum computer. Images adapted from [44]

A spin-1/2 system, which has only two eigenstates, such as the spin of a single electron and the nuclear spin of phosphorus-31 atom, form a natural well-defined state space for a qubit. Silicon has further the advantage that it can be purified to one of the purest materials that exists, even on isotopic level [48]. A typical wafer on which semiconductor devices are fabricated has a purity better than 10−9_{. Because the spin states are so well isolated, the spin}

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24 Quantum electronics and transport spin [49] and up to several minutes for a nuclear spin state of a donor [50]. To create well separated spin states, a static magnetic field is applied, see Figure 2.9a. Using a resonant microwave excitation, interactions between the four spin states of the electron-nucleus system can be switched on and off. By switching the interaction on for only a certain amount of time and varying the phase of the microwave field, coherent rotations on the Bloch sphere can be performed. Morton et al. have demonstrated a nuclear spin memory in a phosphorus doped silicon crystal using these techniques [51].

There is no complete consensus yet on what is required from a physical sys-tem to be suitable to be practically useful for quantum computation, but one generally accepted requirement is that at least around 104_{– 10}6_{operation have}

to be performed within the time the coherence of the complete system is pre-served [52, 53]. This can be achieved by choosing a system with an extremely long coherence time, such as donor spins, or by performing the rotations suffi-ciently fast. Next, a qubit is discussed where much faster manipulation of the states is possible than for a donor-spin based qubit.

Using electrical pulses to tune the potential difference of two donors that form a donor molecule, Dupont-Ferrier et al. show that the wave function of a single electron can be manipulated [54], see Figure 2.10. The donors used here are arsenic donors, a shallow donor in silicon, embedded in a nano-wire transistor that allows electrical gating. When the donor ground-states are tuned in resonance with each other, the one-electron ground state is a linear combination of the states on the left and right donor, labeled ∣L⟩ and ∣R⟩. By rapidly tuning the energy of state ∣L⟩ by a certain amount below ∣R⟩ and vice versa, the charge density alternates between being located on the left and right, while preserving the coherence between ∣L⟩ and ∣R⟩. Changing the amount of detuning results in oscillations in the measured current through the system. This principle is called Landau-Zener-Stuckelberg interferometry. Please refer to [55] for a more complete overview of experiments and a detailed explanation of the principle.

The main significance of the charge manipulation experiment by Dupont-Ferrier et al. as discussed above, lies in the fact that these manipulation can, in principle, be performed extremely fast by using shallow donors in silicon. In contrast to the case of donor spins, where the weak coupling to the environ-ment helps to preserve coherence but limits the manipulation speed, electrical charge control can take advantage of a much stronger coupling. The price to pay for the stronger coupling is that the coherence time is much shorter, measured as only 0.3 ns for the system in Figure 2.10. Manipulation speeds can in principle be much larger. Because the ground state is well isolated from the first excited state for shallow donors in silicon, excitation pulses with har-monic components up to a frequency of ∼ 1 THz can be used. This frequency corresponds to a splitting of ∼ 4 meV as observed in the experiment discussed here. The manipulation speed exceeds the speed at which donor spins can

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2.3. Single donors as qubits 25 μs μd Ground state Excited state D D b a

Figure 2.10: a) A double donor system coupled to source and drain reservoirs can act a two-level system that allows for extremely fast electrical manipulation. The excited state is never occupied during the manipulation described in the text. b) Oscillations in the current reveal that rapid manipulation of the donor levels results in the shift of the charge density of the one-electron state from left to right and back as a function of detuning and pulse height. Images adapted from: [54]

be manipulated by many orders of magnitude. It should be noted however that despite the speed, only a couple of hundred rotations can be performed within the coherence time and therefore this needs to extended, or instead, a larger splitting is necessary. But encouraging prospects are offered by the fact that, for example, the bulk splitting for phosphorus is known to be 11.67 meV [30], and a value of 10 meV has been observed for a gated phosphorus donor [24]. This should allow to use frequencies up to ∼ 12 THz. With the measured coherence time of 0.3 ns, about 1,000 manipulations can be performed, still below the 104 _{operations typically required for a useful qubit. Therefore,}

de-spite the fast manipulation, there is the need to extent the coherence time of this system. This experiment is a nice illustration of fast coherent rotations and provides a powerful tool to study the dynamics of the double donor two-level system. However, swapping the states ∣L⟩ and ∣R⟩ only corresponds to a rotation about x (or y), and is therefore by itself not directly useful to perform arbitrary rotations on the Bloch sphere.

Relevance of the work in this thesis to quantum computation– Instead of directly presenting qubit implementations, this thesis describes several results that may add to the understanding of donor-based quantum device applica-tions. Chapter 3 and 4 discuss how the one-electron D0 _{states (ground state}

and excited states) of donors are modified when they are embedded in a nano-structure. The ability to gate donors, and thereby adiabatically tune the electron wave function, is an essential aspect of a scalable quantum device architecture. Chapter 5 is concerned with the tunability of the ground state,

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26 Quantum electronics and transport but focusses on a different aspect relevant for quantum computation: namely, how the ground state is affected by the band structure of silicon and how new quantum numbers occur for donor atoms in silicon as compared to an atom in vacuum. Chapter 6 and 7 are concerned with coherent interactions of donor-bound electrons with their environment. Exploring these interactions may result in finding mechanisms useful for quantum computation as well as a better understanding of effects that can cause decoherence when an isolated system couples to the outside world.

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Silicon Quantum Electronics

Silicon quantum electronics

Proefschrift

Jan Verduijn

Contents

1

Entering the quantum world

1.1

Improving devices

M

1.2

Quantum computation

1.3

Designing particles

*

*

B

1.4

This thesis

References

2

Quantum electronics and transport

2.1

Devices and fabrication

W

2.2

Transport spectroscopy of single donors

+

2.3

Single donors as qubits

References

_{Improving devices}

_{Quantum computation}

_{Designing particles}

_{This thesis}

_{Devices and fabrication}

_{Transport spectroscopy of single donors}

_{Single donors as qubits}