Single Atom Electronics

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Single Atom 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 vrijdag 14 september 2012 om 12:30 uur door

Jan Andries Mol

natuurkundig ingenieur geboren te ’s-Gravendeel, Nederland.

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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. ir. L.M.K. Vandersypen Technische Universiteit Delft Prof. dr. P.M. Koenraad Technische Universiteit Eindhoven Prof. dr. ir. H.J.W Zandvliet Universiteit Twente

Prof. dr. F. Remacle Université de Liège

Dr. A.F. Otte Technische Universiteit Delft

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

Published by: Jan Mol Cover design by: Jan Mol

ISBN 978-94-6191-389-0

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

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1

Introduction

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1.1 _History

In 1926 Schrödinger developed the quantum mechanical treatment of the hy-drogen atom following a description based on wavefunctions. Shortly there-after in 1928, Bloch applied this theory to electrons in a periodic potential, thus laying the foundation for solid-state physics. Initially solid-state physics was treated with some disdain by certain physicists since its predictions strongly depended on material purity. In a reaction to Peierls work on residual resis-tance Pauli remarked

"order of magnitude physics . . . a dirt effect and one should not wallow in dirt" [1].

Yet it is precisely a dirt effect that lays at the heart of the information tech-nology revolution that has taken place in the last six decades. Following the invention of the point-contact transistor by Bardeen and Brattain in 1947, Schockley realized that by forming the emitter, collector and base as three different layers sandwiched in a single piece of semiconductor would eliminate the fragile point contact. The three layers were created by doping the semi-conductor with different foreign atoms, or dopants. The resulting transistor (see Fig. 1.1) formed the basis for todays multi-billion dollar semiconductor industry.

Doping, in conjunction with lithography, lead to the development of an in-tegrated circuit (IC) on a monolithic chip of semiconductor material by Kilby in 1958. In 1965, following the introduction of the first commercially available ICs Moore observed that the number of transistors that can be placed on an IC doubles every two years [2]. Moore’s prediction still holds today. Between 1970 and 2011 the gate length of metal-on-insulator field-effect-transistors (MOS-FETs) decreased from 10 µm to 28 nm (see Fig. 1.2). Consequently, the number of dopant atoms per transistor channel decreased from 106_{− 10}9 _{to 10}_{− 100.} Recent studies have shown the transport through the localized quantum states of single dopant atoms in commercial FETs [3].

1.2 _{Paradigm shift}

Device-to-device variability due to the discrete nature of doping is one of the seemingly fundamental obstacles preventing further scaling of semiconductor

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1.2. Paradigm shift 3

Figure 1.1: The first junction transistor developed in 1948. From: LUCENT TECHNOLOGIES INC./BELL LABS.

devices. However, this technological barrier can be turned into an advantage by embracing the quantum nature of single dopant atoms. Shallow impurities that bind a single electron or hole are the solid-state analogues to the hydrogen atom. Many well-known effects from atomic physics, such as the Stark and Zeeman effect, find their direct analogue in dopant atoms [4]. The 1/r confining potential of a dopant atom does not rely on fabrication parameters, this makes dopants ideal reproducible building blocks for nano-electronics. Moreover, coherent quantum states of donor-bound electron spins exist for as long as a second in purified silicon [5], making them natural candidates for quantum bits (qubits). Qubits store a richer form of information than ones and zeroes of a classical bit. As a consequence, devices that can process quantum information can possibly solve certain classes of problems exponentially faster than classical devices [6]. Recent studies have demonstrated that it is conceptually possible to place single dopant atoms with atomic precision, paving the way towards single dopant based electronics [7]. Although there are still many technological obstacles ahead it is likely that single atom electronics will be the ultimate scaling of Moore’s law. It is, at least from a symbolic perspective, a beautiful

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101 102 103 104 101 102 103 104 105 106 107 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 Tr an sistor d en sity (m m -2) Year AMD IBM Intel Motorola Minim um fe at ure siz e ( nm ) All manufacturers

Figure 1.2: Between 1970 and 2011 the gate length of silicon MOSFETs has de-creased from 10 µm to 28 nm. o illustrate, an inter-dopant distance of 28 nm corre-sponds to a doping density of 2 × 1017cm−3. From [8].

coincidence that a 100 years after Schrödinger gave a quantum mechanical description of the hydrogen atom and Bloch lay the foundations for solid-state physics, the minimum feature size of silicon MOSFETs will approach the effective Bohr radius of a single dopant, the solid-state analogue to the hydrogen atom.

1.3 _{This thesis}

This thesis will describe a number of electronic transport measurements of single dopant atoms in silicon. By drawing parallels between transport mea-surements in nanoscale devices and scanning tunneling spectroscopy (STS) measurements [9] a comprehensive physical picture of single dopant transport is developed. Quantitative scanning tunneling spectroscopy measurements of important properties such as energy spectra, wavefunctions and carrier life-time, of single dopant bound electron and hole states will be presented. There will be special emphasis on the possibilities of using STS in combination with bottom-up doping engineering [10]. The focus of all experiments will be on the use of single dopant atoms for future electronic devices, either for classical or quantum computation.

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1.3. This thesis 5 followed by an overview of measurement techniques for electrical detection of single dopants. Further, a review of the current status of deterministic doping techniques, whereby dopants are placed with (sub-)nanometer precision, will be discussed. Chapter 2 will finalize with a description of dopant based quan-tum computation.

Scanning tunneling spectroscopy experiments on subsurface acceptors and donors will be discussed in Chapter 3, 4 and 5. Chapter 3 will elude on the measured effect of a semiconductor-dielectric interface on the ionization energy of subsurface boron acceptors. Since interface enhanced ionization en-ergy could lead to dopant deactivation in nanostructures this is a problem that has strong implications for the further down-scaling of conventional device di-mensions.

Chapter 4 describes a quantitative measurements of the ground-state splitting of acceptors near the interfere. These measurements demonstrate a high de-gree of control over the splitting. Moreover, transitions between states in the resulting Kramers doublets could be driven by an oscillating electric field due to spin-orbit coupling in the valence band, which might have implications for the use of acceptors for dopant-based quantum computation.

Electronic transport measurements of phosphorus donors, deliberately placed 5 monolayers beneath the surface of a p-type substrate, are describe in Chap-ter 5. Transport through the localized donor state is limited by carrier re-combination. Minority-carriers (electrons) that are injected into the localized donor state need to recombine with bulk majority-carriers (holes), essential forming an atomic p-n junction.

Chapter 6 is devoted to the implementation of a classical logic operation, a full addition, on a circuit containing single atom transistor (SAT). This oper-ation can be performed using only a fraction of the transistors required in a conventional CMOS circuit thanks to the charge and energy quantization in the SAT.

The final chapter of this thesis (Chapter 7) will give an overview of the current status of the field of single atom electronics and will provide a future outlook.

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References

[1] Pauli, W. 1930–1935 Scientific Correspondence with Bohr, Einstein, Heisenberg and Others, vol. 2,4 (Springer, 1985).

[2] Moore, G. E. Cramming more components onto integrated circuits, Reprinted from Electronics, volume 38, number 8, April 19, 1965, pp.114 ff. IEEE Solid-State Circuits Newsletter 20, 33–35 (2006).

[3] Lansbergen, G. et al. Gate-induced quantum-confinement transition of a single dopant atom in a silicon FinFET. Nature Physics 4, 656–661 (2008).

[4] Koenraad, P. M. & Flatté, M. E. Single dopants in semiconductors. Nature Materials 10, 91–100 (2011).

[5] Tyryshkin, A. M. et al. Electron spin coherence exceeding seconds in high-purity silicon. Nature Materials 11, 143–147 (2011).

[6] Deutsch, D. Quantum Theory, the Church-Turing Principle and the Uni-versal Quantum Computer. Proceedings of the Royal Society A: Mathe-matical, Physical and Engineering Sciences 400, 97–117 (1985).

[7] Fuechsle, M. et al. A single-atom transistor. Nature Nanotechnology 7, 242–246 (2012).

[8] 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).

[9] Wiesendanger, R. Scanning Probe Microscopy and Spectroscopy: Meth-ods and Applications (1994).

[10] Ruess, F. et al. Toward atomic-scale device fabrication in silicon using scanning probe microscopy. Nano Letters 4, 1969–1973 (2004).

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2

Background

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2.1 _Introduction

This chapter will focus on single dopants in silicon. A detailed review on single dopants in silicon and other semiconductor materials can be found in Ref. [1]. Section 2.2 will provide a theoretical description of the properties of shallow impurities in silicon followed by a discussion on the effects of an interface on these properties. An overview of experimental methods for detections of single dopants is discussed in section 2.4, followed by a discussion on the current status of deterministic doping techniques in section 2.5. This chapter will conclude with a description of dopant based qubits in section 2.6.

2.2 _{Shallow dopants in silicon}

The nature and concentration of impurities determines, for a large part, the type and magnitude of conductivity in semiconducting materials. The oper-ation of semiconducting devices relies heavily on the ability to controllably introduce known types of impurities into the host material. Over the past sixty years substitutional and interstitial foreign atoms (dopants), complexes, vacancies and interstitial host atoms have been extensively investigated. From a fundamental point of view shallow impurities that bind one electron or hole can be considered as a solid state analogue to the hydrogen atom. Effects that are well-known from atomic physics (e.g. Stark shift and Zeeman splitting) have been studied to great extend and a detailed review on "hydrogenic" im-purities can be found in Ref. [2]. This section will give a general description of shallow dopants in silicon followed by a discussion on the effect of the presence of an interface on the properties of dopant bound states.

Four out of the five outermost electrons in the 3s2_3sp3 _{states of a} phos-phorus atom (group V) in silicon (group IV) form covalent bonds with the neighboring silicon atoms. The fifth electron does not participate in the for-mation of covalent bonds and is donated to the conduction band. The donated electron is bound to the P+ _{ion by Coulomb attraction, analogues to the} hy-drogen atom, the potential at a distance r from the impurity is given by:

Ui(r) = −e 2 4π0Sir

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2.2. Shallow dopants in silicon 9 where a static dielectric constant Si adjustment of the charge density of the silicon lattice in the field of the positively charged donor. Electrons in a periodic potential behave as particles with an effective mass m∗ different, and usually smaller, than the free-electron mass m. Under the assumption of a static dielectric constant and an isotropic effective mass, the donor bound electron will have hydrogen-like states with energies

En= −m

∗_e4 32π22

02Si̵h2n2

, (2.2)

where n= 1, 2, 3, . . .. In contrast to group V impurities group III impurities, such as boron, accept an electron from the silicon valence band. Following similar arguments, the states of a hole bound to an acceptor in silicon can be described in the hydrogenic model.

Although the simple hydrogenic model gives an intuitive picture of the bound electron and hole states [3, 4], there are two important extensions to the model. Firstly, the model does not describe the potential correctly. The fact that arsenic has a binding energy of 54 meV versus 45 meV for phosphorus even though they are both hydrogenic impurities is due to the potential at r< a, where a is the lattice spacing. For small r the charge of the nucleus is not screened by the electrons in the host lattice and therefore the potential will be more strongly attractive. The potential at r< a depends on the chemical nature of the dopant atom, i.e. the nuclear charge, and is not captured by the hydrogenic model. Secondly, in silicon, like in many semiconductors, the effective mass m∗ of electrons and holes are not scalars but tensors, reflecting the nature of the conduction band or valence band respectively.

Figure 2.1 shows the band structure of silicon. The conduction band of silicon has six minima along the [100] and equivalent axes. These minima located at k0= 0.85 × 2π/a, are commonly known as "valleys". Each valley is characterized by an effective mass m∗

∥= 0.98m along the direction of the axis on which the valley is located, and m∗⊥= 0.19m perpendicular to the axis, as indicated in Fig. 2.2. From effective mass theory [6] it follows that the ground state of the donor bound electron can be written as

Ψ(r) = 6 ∑ j=1

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−15 −10 −5 0 5 Energy (eV) Wavevector k

Figure 2.1: The band structure of silicon calculated using a fully atomistic sp3s∗d5 tight-binding method [5]. The top of the valence band is located at the Γ-point (k = 0). The conduction band has six minima located along the [100] and equivalent axes at k0=0.85 × 2π/a.

where αjare numerical coefficients and ϕ(r) the Bloch functions. The envelope wavefunctions Fj(r) satisfy the hydrogen-like Schrödinger equation, taking into account the anisotropic effect mass. Because the donor is located at a substitutional site in the tetrahedral silicon lattice the total wavefunction has Td symmetry, i.e the total wavefunctions are invariant under any operations, including reflections, which take a regular tetrahedron into itself. The 1s ground-state multiplet is reduced to a singlet 1s(A1), a doublet 1s(E) and a triplet 1s(T2) in the irreducible tetrahedral representations of Td. For r< a the nucleus is not screened by other electrons and the potential will be much stronger than that described by Eq. 2.1, as a consequence, because the different states have different charge densities at r< a, the six-fold degeneracy of the ground state is lifted (see Fig. 2.2). Since only the 1s(A1) singlet has non-zero charge density at the nucleus this state will be most strongly effected by this species-dependent effect. As this effect depends on the chemical species of the donor it is sometimes referred to as "chemical splitting" or more often "valley-orbit splitting".

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2.2. Shallow dopants in silicon 11

(a) (b) conduction band

valley-orbit splitting

Figure 2.2: (a)Schematic depiction of the six conduction band minima of silicon. (b) Predicted energy spectrum of a bulk donor in silicon. The six-fold degeneracy of the hydrogenic 1s state is lifted due to vally-orbit splitting.

Acceptor bound states reflect the nature of the valence band. There are two important properties of the valence band in tetrahedrally semiconductors that are reflected in the acceptor wavefunction. (i) The valence band is degenerated at the center of the Brillouin zone. (ii) As a result of the degeneracy the valence bands are warped and there exists no simple effective mass tensor. In other words, the valence band can not be simply described by a parabolic E− k dispersion relation.

The initially three-fold degenerate top of the valence band at k= 0 is lifted by spin-orbit coupling. Because of spin-orbit interaction, spin and angular mo-mentum are no longer good quantum numbers and the bands are characterized by their "pseudo-angular momentum" J. Two bands with J= 3/2 remain de-generate at k= 0, whereas the split-off band with J = 1/2 shifts λ ∼ 43 meV downward. States associated with J= 3/2 are denoted by p3/2 in Fig. 2.3 and those associated with J= 1/2 by p1/2.

Since the impurity potential possesses the Tdsymmetry of the impurity site, the acceptor wavefunction must transform according to the operations of the double group Td[7]. This introduces coupling between the angular momentum L of the envelope wavefunction and the pseudo-angular momentum J of the Bloch functions [8]. Since L is no longer a good quantum number and states

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with ∆L= 0, ±2 are mixed, the acceptor wavefunctions are characterized by their "total pseudo-angular momentum" defined by

F = L + J, (2.4)

analogues to the total angular momentum L+ S in the case of spin-orbit coupling in atomic physics. Note that the pseudo-angular momentum of the Bloch function plays the role of spin in this case. As in atomic physics, the states of a shallow acceptor can be labeled with spectroscopic notation. States from the p3/2 band with L = 0 are denoted by nS3/2, where the subscript denotes F . States nP1/2, nP3/2and nP5/2have angular momentum L= 1, and so on.

The 1S3/2 ground-state of the acceptor bound hole in silicon is four-fold degenerate and has Γ8 symmetry under Td, reflecting the nature of the p3/2 valence band. Symmetry breaking, for example due to an external electric field or strain, splits the ground-state into two Kramers doublets with mJ = ±3/2 and mJ= ±1/2, as illustrated in Fig.2.3. Under a magnetic field time-reversal symmetry is broken and the two-fold degeneracy of the Kramers doublets is lifted. This illustrates the spin-orbit-like behavior of the acceptor bound hole states, with the notion that here it is the pseudo-angular momentum of the Bloch functions that fulfills the role that the electron spin plays in atomic physics.

2.3 _{Dopants near an interface}

In the previous section, donors and acceptors have been discussed in terms of bulk properties. However, all experiments discussed in this thesis deal with dopants either in nanoscale devices or close beneath the surface. This para-graph will give an overview of recent theoretical and experimental studies of the effect of an interface, i.e. a potential barrier and an abrupt change in the dielectric constant, on the properties of dopants in silicon.

Three effects influence the properties of shallow dopants near the interface. (i) There is an abrupt transition in the dielectric constant at the interface. As a consequence, the impurity potential is either stronger or weaker screened, depending on material with which the silicon is interfaced. Hence the electron

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2.3. Dopants near an interface 13 (a) (b) valence band E field, strain B field

Figure 2.3: (a)Energy dispersion at the top of the valence band of silicon. The valence band is degenerate at k = 0 and the bands are warped as a result of this degeneracy. (b) The four-fold degeneracy of the acceptor ground state is lifted by an external electric field or strain. The resultant Kramers doublets can be further split by an external magnetic field.

or hole will be weaker or stronger bound to the impurity. (ii) The formation of a potential barrier at the interface will shift the bound states upward in energy. (iii) Strain induced by surface reconstruction will break the tetrahedral sym-metry of the silicon lattice and will consequently split orbital degenerate states. Recent results of transport measurements of silicon nanowires suggest that dielectric mismatch at the interface leads to an increase in the ionization energy which in turn results in dopant deactivation and higher resistively for wires with a small diameter [9]. Evidence for increasing ionization energy has also been suggested from room temperature threshold shifts in nanoscale field-effect transistors [10]. However, theoretical models of shallow donor states near an interface predict that the effect of dielectric mismatch, although appreciable, competes with and is largely canceled by quantum confinement at the inter-face [11–13]. In the same theoretical framework the effect of the interface on charging energy of shallow donors have been investigated [14, 15]. Moreover, strong effects of hybridization between donor states and interface-like states on the energy spectra of both the one- and two-electron states have been ob-served [16, 17] and are well predicted within a tight-binding framework [18]. Furthermore, interactions between two or more donors have been theoretically

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investigated [19–21].

The effect of an interface on the states of a shallow acceptor is illustrated by solving the Schrödinger equation

(HLK+ Ui)Ψ = EΨ, (2.5)

on a grid. Here, HLK is the Luttinger-Kohn Hamiltonian, Ui the impurity potential a Ψ(r) the total wavefunction defined as

Ψ(r) = ∑ mJ

FmJ(r)∣

3

2, mJ⟩, (2.6)

where FmJ is the envelope wavefunction belonging to the Bloch wavefunctions

∣3

2, mJ⟩ with J = 3/2 and mJ = −3/2, . . . , +3/2. Spin-orbit interaction in the valence band is included in the 4× 4 Luttinger-Kohn Hamiltonian [2]:

HLK(k) = ⎛ ⎜⎜ ⎜⎜ ⎝ P+ Q −S R 0 −S∗ _P_{− Q} ₀ _R R∗ 0 P− Q S 0 R∗ S∗ P+ Q ⎞ ⎟⎟ ⎟⎟ ⎠ , (2.7) where P = ̵h 2 2m0 γ1(k2x+ k2y+ k2z) (2.8) Q = ̵h 2 2m0 γ2(k2x+ k2y− 2kz2) (2.9) R = ̵h 2 2m0 √ 3[γ2(k2y− k 2 x) + 2iγ3kxky] (2.10) S = ̵h 2 2m0 2√3γ3(kx+ ky)kz (2.11) (2.12) and k is replaced by−i∇ [6]. The attractive impurity potential Ui is taken to be hydrogenic as in Eq. 2.1. Moreover, the effect of dielectric mismatch be-tween the silicon and the vacuum at the interface is accounted for by employing the method of image charges [22]. Schechter [7] and Lipari and Baldereschi [8]

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2.3. Dopants near an interface 15 x y z 0 0 1 1 -1 bulk interface no image charge image charge 0 2 4 6 0 20 40 Depth (nm) Energy (meV) (a) (b) (c) 60

Figure 2.4: (a) Energy of the split ground state as a function of dopant depth. (b) The wavefunction of the acceptor bound ground state for an acceptor near the interface and a bulk acceptor. The interface causes the wavefunction to shift away from the acceptor leading to a decrease in binding energy. The position of the acceptor is indicated by the circle. (c) Difference in probability density between the two state belonging to the two Kramers doublets. The different z-components of the d-like orbital part of the wavefunctions causes the mF ±1/2 states to be stronger perturbed by the interface, resulting in the splitting of the ground state.

solved Eq. 2.5 by expanding the envelope wavefunction FmJ(r) in spherical

terms and obtained the results discussed in the previous section. However, solving the Schrödinger equation on a Cartesian grid allows for the investi-gation of effects that could otherwise not be studied, i.e. confinement at the interface and the effect of dielectric mismatch.

The ground-state of a bulk boron acceptor is four-fold degenerate with a binding energy of 45 meV above the valence band edge [2]. The degenerate ground state manifold is comprised of the total pseudo-angular momentum states mF = ±3/2 and mF = ±1/2. An electric field, strain or interfacial con-finement can split the ground state degeneracy into two Kramers doublets. Figure 2.4 shows the energy spectrum as a function of the dopant depth, calculated using the effective mass treatment. If the dielectric mismatch is

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neglected, i.e. in the absence of an image charge, a monotonic decrease of the binding energy is observed as the dopant gets closer to the interface. The acceptor states are shifted towards the bulk valence band (i.e. binding energy decreases) primarily due to the additional quantum confinement by the inter-face. The calculated charge densities in Fig. 2.4(b) illustrate how the bulk wave functions get perturbed by the potential barrier at the boundary, leading to the reduction of the binding energy. The splitting of the four-fold degen-erate ground state can also be understood in terms of this boundary effect. When the difference in charge density between the mF± = 3/2 and mF = ±1/2 , as plotted in Fig. 2.4(c), is considered it is clear that the orbital part of the wavefunction of the mF± = 1/2 doublet gets more strongly perturbed by the interface barrier than the orbital part of the mF = ±3/2 wavefunction, which leads to the splitting into the Kramers doublets.

When dielectric mismatch is taken into account, two competing effects in-fluencing the binding energy are observed. The attractive potential of the image charge of the nucleus enhances the binding energy as the dopant depth, z0 decreases, whereas the effect of the interface barrier lowers the binding en-ergy. Since the former effect decays as 1/z0 and the latter exponentially, the two competing effects result in a maximum binding energy at z0 ∼ 2.4 nm. In addition, the image charge of the hole bound to the acceptor repels the hole from the interface, thereby further reducing the enhancement effect on the binding energy due to the image charge of the nucleus. Moreover, an en-hancement in the ground-state splitting energy, compared to the zero image charge case, is observed as the distance between the acceptor and the interface decreases. This enhancement of the splitting is caused by the electric field due to the nucleus image charge.

Chapter 3 of this thesis will elaborate on the measurements of the presence of an interface on the ionization energy of sub-surface acceptors in silicon. Furthermore, Chapter 4 will describe the experimental verification of interface induce ground state splitting of acceptors in silicon.

2.4 _{Electrically detecting single dopants}

Generally speaking there are two approaches to electrical detection of single dopant atoms. The first approach relies on nanoscale two- or three-terminal

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2.4. Electrically detecting single dopants 17 (a) (b) drain gate source μ_D μ_S dopant μ E_C 180 220 260 −20 0 20 −10 Current (nA) 10 Bias v oltage (mV ) Gate voltage (mV) 0 µ = µS µ = µD

Figure 2.5: (a) Schematic depiction of the resonant tunnel through a localized dopant state in a nano-scale three-terminal device. When the electrochemical poten-tial of the dopant, µ, is outside the bias window defined by µS and µD transport is

Coulomb blocked. Each time a localized state of the dopant enters the bias window there is a stepwise increase in the tunnel current. (b) Measured source-drain current through a localized dopant state as function of gate voltage and bias voltage. Solid lines indicate the gate and bias voltage for which µ = µS and µ = µDbetween which

transport can occur. The dotted line indicates the voltages for which an excited state enters the bias window, leading to an increase in the tunnel current.

devices in which the transport is dominated by a single dopant. In the second approach a scanning probe is used to locally detect single dopants, either using scanning tunneling or scanning capacitance techniques. This section will elucidate and draw comparisons between both approaches.

Resonant transport through the localized states of individual arsenic [16, 23] and phosphorus [10] dopants has been measured in nano sized silicon field-effect-transitors (FinFETs) and single boron and platinum impurities have been detected in Schottky barrier metal-oxide- semiconductor field-effect tran-sistors (SBMOSFET) [24–27]. Deliberately placed phosphorus atoms have been studied in nanoscale double-gated field-effect-transistors [28] as well as in bottom-up fabricated single atom transistors [29] (details of deterministic doping are discussed in section 2.5).

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Although experimental details differ for these measurement they all rely on resonant tunneling and Coulomb blockade for the detection of single dopant atoms. There exist well-formed theoretical descriptions of single electron trans-port through localized states coupled to source/drain and gate electrodes that were developed in conjunction with the first experiments on quantum dot structures. A brief overview of the application of this physical model to single dopant transport will be given in the following paragraph. An extensive review on single electron transport can be found in reference [30].

Figure 2.5(a) shows the schematic depiction of a donor that is tunnel cou-pled to a source- and drain-electrode. The potential U on the donor is modu-lated by the gate voltage VG as

U(N) = [eN − CGVG]2/2C + ∑ N

EN,l, (2.13)

where N is the number of electrons on the donor, C = CG+ CS + CD the sum of the gate- source and drain-capacitance, and EN,l the single-particle level spectrum. The first term of Eq. 2.13 describes the electrostatic energy of the donor, whereas the second term describes the energy associated with the discrete quantum states of the donor. The electrochemical potential µN of the donor is defined as

µ= U(N) − U(N − 1) = (N − 1/2)EC− e(CG/C)VG+ EN,l, (2.14) where EC = e2/C is the electrostatic energy required to add one extra elec-tron to the donor. Whenever µ resides in the energy window defined by the electrochemical potential of the source- and drain electrode µS and µD, i.e. µD< µ < µS or µS < µ < µD electrons can flow via the localized donor states from source to drain or vice versa. When the bias voltage VSD is increased more than one single-particle level may reside in the bias window. Each time an excited state l enters the bias window there is an additional channel for transport resulting in a step in the source-drain current ISD. As a result, the source-drain current, as shown in figure 2.5(b) is a direct measure for the energy spectrum of the donor.

Alternative to measuring resonant transport through the localized states of individual dopant atoms, charge sensing experiments involving a single or few

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2.4. Electrically detecting single dopants 19 impurities have been performed using both top-down [31, 32] and bottom-up [33] type devices. In these experiments the island of a single-electron tran-sistor (SET) is capacitively coupled to a single or few dopant atoms. As a consequence, the electrochemical potential of the SET island is modulated by the charge state of the dopant atom(s). The electrochemical potential of the dopant atoms is controlled by a plunger gate and the dopants can load and unload electrons from a reservoir. Each charging event of a single impurity is registered a modulation of the source-drain current of the SET. In contrast to transport measurements of single dopants, where > 1 × 107 _{electron per} sec-ond need to tunnel through the localized dopant state in order to measure an appreciable current, an electron may reside indefinitely long on a dopant atom in a charge sensing experiment, as the measured current does not pas through the localized state. As a result, coherent quantum states of the impu-rity bound electron can be initialized, manipulated and subsequently read-out [32,33] (see Section 2.6).

Unlike the previously mentioned experiments in nanoscale devices, where a single or few dopant atoms at a fixed distance from source-, drain- and gate-electrodes were studied, scanning probe techniques allow for the investigation of many individual subsurface impurities in addition to the ability to change the coupling between the dopant atoms and the electrode (tip). However, the physical principles for the electrical detection of single dopants using scanning probe techniques, i.e. resonant tunneling and charge sensing, are similar to those employed in nanoscale devices.

Subsurface boron [34,35] acceptors and arsenic [35] beneath the Si(100):H surface have been observed using scanning tunneling microscopy (STM) and their topographic features are understood in terms of scattering theory of sub-surface impurities [35,36]. Boron and phosphorus dopants have also been ob-served beneath the Si(111):H surface [37,38]. Unlike for the cleaved GaAs(100) surface, surface states appear in the band gap of Si(100) that pin the Fermi level and thereby obscure the effects of electrically active dopants. Nonethe-less, buried phosphorus donors beneath the clean Si(100) have been observed [39] where the Fermi level is pinned at the surface due to the presence of surface states. In these experiments the charge of the ionized nucleus of the subsurface dopants is "sensed" by measuring the change in the local density of states due to the Coulomb potential of the charged nucleus. Figure 2.6(a) shows

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filled-(a) (b)

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Figure 2.6: (a) Filled-state (valence band states) image of a step-edge on a clean, 2 × 1 reconstructed Si(100):H surface. Sub-surface acceptors appear as broad hillocks (indicated by arrows) superimposed on the atomic 2 × 1 dimer-rows. (b) A zoomed-in filled-state image of a single sub-surface acceptor clearly illustrates how the acceptor induced feature extends over several dimer rows. (c) In the empty-state (conduction band states) image of the same sub-surface acceptor the acceptor induced feature appears as a dip superimposed on the dimer-rows.

state STM topography of a subsurface boron acceptor beneath the Si(100):H surface. Since the surface is hydrogen terminated there are no surface states within the band gap that pin the Fermi level and the conduction and valence bands will bend at the interface as a function of applied bias voltage. This effect, called tip induced band bending, is crucial in the interpretation of scan-ning tunneling microscopy and spectroscopy data [40]. In filled-state imaging a negative sample bias is applied and electrons tunnel from the valence band of the sample to the tip. The negatively charge nucleus of the subsurface boron acceptor leads to an increase in the local density of valence band states. Since the tunnel current I is proportional to the integrated LDOS and the tip height in turn is proportional to the tunnel current, the subsurface acceptor is ob-served as a height increase superimposed on the atomic surface reconstruction.

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2.4. Electrically detecting single dopants 21 In contrast, a decrease in the local density of conduction band states due to the negatively charged acceptor nucleus results in a decrease in topography height in empty-state imaging (see Fig. 2.6(b)), i.e. when a positive sample bias is applied and electrons tunnel from the tip to the sample conduction band.

Charge state manipulation of individual dopant atoms has been demon-strated using low temperature (4 K) scanning tunneling spectroscopy (STS) on Si donors in beneath the cleaved gallium arsenide (GaAs) (110) surface [41– 43]. In these experiments the charge state of the donors depend on the applied sample voltage and the position of the tip relative to the donor. The charging event of the individual donors is registered as an increase in the tunnel current to the conduction band. Controlled charge state manipulation had also been observed by applying the same technique to donors in ZnO [44].

In addition to charge state observation and manipulation, resonant trans-port has been measured through localized states of subsurface Mn [45–51] and Zn [52, 53] acceptors beneath the GaAS(110) surface using low temperature STS.

The STS experiments described in this thesis will be approached from a "transport measurement’" point-of-view. It will be shown that clear parallels can be drawn between the transport measurements in two- or three-terminal devices and scanning tunneling spectroscopy measurements.

Figure 2.7(a) shows a schematic depiction of the resonant tunnel process through the localized state of a subsurface acceptor. The electrochemical po-tential µ of the acceptor is determined by the applied bias voltage V . Whenever µ resides in the bias window defined by the electrochemical potential of the tip and the sample, µT and µS, electrons can flow through the localized state of the acceptor. Each time an excited state l enters the bias window there is an additional channel for transport resulting in a step in the tunnel current I. Figure 2.7(b) shows the differential conductance dI/dV as function of applied voltage and position in the presence of a subsurface boron acceptor beneath the Si(100):H surface. As opposed to dopants in three-terminal devices, there is generally no "third" terminal in STS experiments. Therefore, µ can not be brought in and out of resonance independent of the bias voltage. However, in STS there is the freedom to change the position of the tip with respect to the dopant atom, making it possible to spatially probe the extend of the wave

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(a) (b) dopant TIBB sample vacuum tip eV μ_S E_V μ_T μ Position (nm) Voltage (V) 0 5 10 15 20 0 1.2 0.8 0.4 -0.4 Normalized Conductance (1/V) 0 9 resonant tunneling charge sensing

Figure 2.7: (a) Schematic depiction of the resonant tunnel process through a local-ized dopant state in an scanning tunneling spectroscopy measurement. Tip induced band bending controls the electrochemical potential, µ of the dopant. Resonant tunneling through the dopant occurs when µ is within the bias window defined by µT and µS. Each time a localized state enters the bias window there is a stepwise

increase in the tunnel current. (b) Measured normalized conductance through a sub-surface dopant as function of vertical tip position x and bias voltage V . The charge of the ionized dopant, located at x = 15 nm, is "sensed" by the shift in the valence band states. Moreover, resonant tunneling through the localized acceptor state is measured in the band gap as tip induced band bending brings µ in resonance with µS.

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2.5. Deterministic doping 23 functions of the localized states.

As well as imaging the spatial extend of localized dopant state, the spatial degree of freedom in STM/STS enables the possibility to study the interaction between dopant atoms and their environment. The influence of the interface [41, 50], tip [52, 53], strain and electric field [47, 48], charged vacancies [54] and other dopant atoms [55] on the properties of subsurface atoms have been studied extensively using STS. The effect of neighboring donors on the charg-ing energy of individual Si donors in GaAs [56] and individual B acceptors in Si [57] has also been investigated using a capacitance-based scanning-probe technique.

Chapter 3, 4 and 5 will elaborate on the use of low temperature scanning tunneling spectroscopy to investigate individual subsurface boron and phos-phorus dopants beneath the hydrogen passivated silicon (100) surface. By applying transport models which are well-known from electron transport in nanoscale devices a quantitative understanding of single dopant transport is obtained. Moreover, low temperature STS is combined with deterministic δ-doping described in Section 2.5 in order to study donors that were deliber-ately placed 5 monolayers beneath the surface on a p-type substrate. Finally, a classical logic operation implemented on a single arsenic donor embedded in a nanoscale FET will be discussed in Chapter 6.

2.5 _{Deterministic doping}

The reduction in the size of conventional transistors will in the near future face the barrier that on the sub-nano length scale matter is discrete. The ran-dom and discrete nature of doping leads to variability in device characteristics [58], which is a major problem in device down-scaling. Emerging determin-istic doping techniques aim to mitigate statdetermin-istical fluctuations in doping of nanoscale devices by placing dopant atoms with (sub-)nanometer precision, while at the same time providing a significant potential for solid-state dopant based quantum computers (see Section 2.6). In this section both top-down and bottom-up deterministic doping techniques will be discussed.

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has been demonstrated [59,60]. The impact of a single ion in this top-down approach can be detected from electron-hole pairs [59,61], secondary electrons [62–64] or modulation of the drain current [60, 65, 66]. The latter was also employed to detect single ion impact in a nano-scale FET [67].

Atomically precise dopant placement can be achieved by means of bottom-up STM lithography. In this approach a hydrogen monolayer on the silicon (100) surface is selectively desorbed using the STM tip in ultra-high vacuum (UHV) [68–70]. Subsequently the exposed silicon surface is exposed to molec-ular phosphine (PH3). After the PH3 has chemisorbed to the surface [71,72], the phosphorus atoms are incorporated by heating the sample. Finally, the sample is encapsulated [73] and electrodes are patterned to the device [74– 77]. This technique has thus far culminated in the fabrication of atomically precise SETs [78], single donors coupled to charge detectors [33], atomic scale nanowires [79] and single-atom transistors [29].

Moreover, this technique has been successfully translated to germanium substrates [80,81].

Chapter 5 demonstrates that the bottom-up dopant engineering approach can be used to study single donors in a low density delta-layers using low temperature scanning tunneling spectroscopy.

2.6 _{Single dopants as qubits}

Classical information is encoded in a bit that can be either in the classical state ∣0⟩ or ∣1⟩. By contrast, quantum information is represented by the phase ϕ of a quantum bit (qubit) which is in coherent quantum state∣0⟩ + eiϕ_{∣1⟩. As a} result of this, devices that are able to process quantum information should be capable of fundamentally solving certain classes of problems more efficiently than conventional (classical) devices [82].

Although there exist many exciting theoretical proposals, the experimental realization of a fully operational thousand-qubit quantum computer remains a distant goal. Prototypical qubit systems include natural atoms, such as neutral atoms [83] or ions [84], and artificial atoms, such as µm-sized superconducting circuits [85] or spins in solids [86]. An excellent overview of the characteristics of these qubit systems can be found in Ref. [87]. When comparing natural

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2.6. Single dopants as qubits 25 atoms and artificial atoms there are two important aspects to be considered: controllability and scalability. The fact that natural atoms of the same species are indistinguishable, unlike artificial atoms, is a great advantage in terms of controllability. While thousands or millions of neutral atoms can be loaded in optical lattices, individual addressing has not been fully demonstrated yet [88]. Artificial atoms on the other hand can, in principle, be integrated into circuits in large numbers, making them more suitable in terms of scalability. Dopant atoms in a semiconductor host combine the advantages of natural atoms and artificial atoms in the sense that dopant atoms of the same species are indistinguishable [89] and can, in principle, be placed in large numbers on a chip (see Section 2.5).

Dopant atoms naturally exhibit coherent quantum behavior and therefore form a obvious building blocks for a solid-state quantum computer. Silicon is an attractive host material for dopant-based quantum computer, not in the least because it is by far the most important and well developed material in the microelectronics industry. The choice of states upon which the qubit is encoded depends on how long a coherent superposition of this state can be maintained and how rapidly this state can be coherently manipulated. In classical electronics information carried by the electron charge. Coherent su-perpositions of charge states in silicon quantum dots can be maintained for up to 200 ns [90]. However, coherent states of donor-bound electron spins can exist for over 100 µs [91] to a few ms [92] in bulk natural silicon, making spin the prime candidate to carry quantum information. Even though silicon has a low natural abundance of nuclear spins (∼5%29_{Si) hyperfine coupling to the} nuclear spin bath remains a dominant decoherence mechanism. By removing this source decoherence, i.e. by using a highly purified28_{Si substrate,} deco-herence times in excess of 1 s can be achieved [93]. Alternatively, hole spins in silicon are coupled more weakly to the nuclear spin bath due to the atomic p3/2 of the valence band. Although spin-orbit coupling might make hole spins more susceptible to decoherence due to electron-phonon coupling it does allow for rapid electrical manipulation of the hole spin. Coherence times of 2.6 µs have been observed for acceptor-bound hole states in silicon [94]. A detailed review on electron-spin based silicon quantum electronics can be found in Ref. [95]. The remainder of this section will discus how the spin degree of freedom of single dopant atoms in silicon can be utilized to perform qubit operations.

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Control gates SiO₂ Si 31_P 31P e- e -(a) (b) 31_P _e -J A -J A J

Figure 2.8: (a) Schematic depiction of a donor-based quantum computer. The con-trol gates tune the hyperfine interaction (A-gates) of individual donors and exchange coupling (J -gates) between neighboring donors. (b) Energy levels of a 31P bound electron. Coherent transitions between the ∣0⟩ and ∣1⟩ states can be driven by ESR.

Figure 2.8 shows the envisioned donor-based silicon quantum computer architecture [96] which consists of an individual 31_{P donors placed 20 nm} apart. Gates on top of the donors, so-called A-gates control the hyperfine interaction between the electron spin and the 31_{P nuclear spin. Qubit-qubit} exchange coupling is controlled by J-gates in-between neighboring donors. The electron spin can be manipulated using well-known electron spin resonance (ESR) techniques. Transitions between the spin-up and spin-down state are driven by applying a microwave field in resonance with the Larmor frequency of the electron spin. In order to address individual qubits they can be tuned in and out of resonance either by Stark shifting the hyperfine coupling [97] or by locally straining the substrate [98].

The donor-bound electron spin can be read-out using a so-called spin-to-charge conversion technique. When a single donor is placed in close vicinity of a single electron transistor (SET), the SET current will depend upon the charge state of the donor as this will directly influence the chemical potential of the SET island. By gating the donor such that the electron can only tunnel of the donor when it is in the spin-up state the current SET current will depend on the spin state of the donor-bound electron. If the electron is in a spin-down state it will not be able to tunnel of the donor leaving the SET current constant. In contrast, if the electron is in a spin-up state it will be able to tunnel of the donor, leaving the donor ionized and thus changing the SET current. As soon as an electron tunnels back onto the donor the SET

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2.6. Single dopants as qubits 27 current will return to its original value. Recent studies have demonstrated this single-shot read-out of a single electron spin state in silicon [32].

In addition to the electron spin, the donor nuclear spin can be used to store quantum information (see Fig. 2.8(b)). Due to their weaker magnetic moment coherent nuclear spin states are longer lived than electron spin states [89,99]. By using a combination of resonant microwave pulses and radio frequency pulses the electron spin state may be coherently swapped with the nuclear spin state, providing a way to store the electron spin state in the nuclear spin state ’memory’ [100].

In summary, donors in silicon provide a robust platform for electron spin qubits. Both the donor electron spin and nuclear spin degree of freedom can be manipulated using well-known resonance techniques. Moreover, the spin state of a single electron can be read out in a single-shot measurement.

Recent studies have highlighted the possible advantages of using the hole spin to encode quantum information [101–108]. A major source of spin deco-herence is hyperfine interaction with the nuclear spin bath. There are three terms in the hyperfine interaction, (i) coupling between the electrons orbital angular momentum and the nuclear spin, (ii) dipole-dipole interaction that couples electron and nuclear spins at a distance and (iii) a scalar interaction due to the classically forbidden overlap between the electron wavefunction and the nucleus, the so-called contact term. The atomic p3/2nature of the valence band strongly reduces the, generally dominant, contact term of the hyperfine interaction as the atomic p3/2 orbitals have zero probability density at the nucleus. Moreover, strong coupling between spin and momentum allows for local electric field manipulation of the hole spin states via electron dipole spin resonance (EDSR) [109,110].

As is the case for donors, acceptors form a naturally robust 1/r confining potential for holes. Analogues to the donor-based quantum computer archi-tecture an acceptor-based quantum computer would consist of individual ac-ceptors, for instance boron, that are controlled using top-gates. In addition to exchange interaction, qubit-qubit coupling could be mediated by dipole-dipole interaction [111]. The four-fold degenerate ground state of the acceptors split under influence of an electric field, as a consequence the top-gates can control the ground state splitting of individual acceptors. The states in the resulting Kramers doublets are protected against decoherence by time-reversal

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symme-try [112] and Zeeman split under a magnetic field. Transitions between the states in a Kramers doublet of an individual acceptor can be driven by a local AC electric field applied to the top-gate over the acceptor.

Single-shot spin read-out of a single acceptor spin could be performed in an analogues fashion to the read-out of a single donor spin. By tuning the electrochemical potential of the acceptor such that tunneling is only possible from one of the states of the Kramers doublets, the acceptor spin state can be read-out using a charge detector such as a SET.

Chapter 4 of this thesis will demonstrate a large degree of control over the ground state splitting of individual boron acceptors beneath the silicon surface using low temperature scanning tunneling spectroscopy. The ability to controllably isolate Kramers doublets of an individual acceptor might provide the pathway towards a scalable acceptor-based quantum computer.

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3

The ionization energy of

subsurface acceptors

The understanding of the electronic properties of dopants near an interface is one of the key challenges for doping nano-scale devices. We have determined the effect of dielectric mismatch and quantum confinement on the ionization energy of individual subsurface acceptors beneath the hydrogen passivated sili-con (100) surface. Dielectric mismatch between the vacuum and the silisili-con at the interface results in an image charge of the acceptor nucleus which enhances the binding energy of sub-surface acceptors. Quantum confinement, on the other hand, reduces the binding energy. Using scanning tunneling spectroscopy we measure resonant transport through the localized states of individual accep-tors. Thermal broadening of the conductance peaks provides a direct measure for the absolute energy scale. Our data unambiguously demonstrates that these effects compete and that as a result the bulk ionization energy is retained even for acceptors less than a Bohr radius from the interface.

This chapter has been submitted for publication as: J.A. Mol, J. Salfi, J.A. Miwa, M.Y. Simmons and S.Rogge. Interplay between quantum confinement and dielectric mismatch for ultra-shallow dopants.

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