MANIPULATING AND CHARACTERIZING SPIN QUBITS BASED ON ...

MANIPULATING AND CHARACTERIZING SPIN QUBITS BASED ON

DONORS IN SILICON WITH ELECTROMAGNETIC FIELD

A Thesis

Submitted to the Faculty

of

Purdue University

by

Yu Wang

In Partial Fulfillment of the

Requirements for the Degree

of

Master of Science in Electrical and Computer Engineering

August 2015

Purdue University

West Lafayette, Indiana

ii

To my beloved wife and my parents.

iii

ACKNOWLEDGMENTS

I would like to thank my advisor, Professor Gerhard Klimeck who gave me this

precious opportunity to work as a member of his nanoelectronic modeling group. I

began appreciating his work and achievements when I was an undergraduate student.

I would like to express my sincere gratitude to Professor Supriyo Datta and Dr.

Rajib Rahman for being my advisory committee members. I would like to express my

appreciation to Professor Datta, from whom I gained a lot of knowledge on quantum

transport. Specially, I would like to thank Dr. Rahman who is guiding my research

work on a daily basis. He led me into the world of solid-state quantum computing.

He is always patient and open to discussions. I benefit a lot from his knowledge,

personality and his way of thinking and solving problems.

I would like to thank Professor Michelle Simmons and Professor Lloyd Hollenberg

for valuable guidance and discussions during our weekly remote meetings. I would

also like to thank the colleagues in UNSW for cooperation, discussions and their

insights from an experimental perspective, and their help when I was visiting Sydney.

I would like to thank Yaohua Tan for his selfless help and insights on almost

everything I started to work on. I would like to thank Yu He for his help with fixing

problems when I began to work in this group. I would like to thank Yuling Hsueh,

Rifat Ferdous, Archana Tankasala, Chinyi Chen for valuable cooperation, help and

discussions. I would like to thank Zhengping Jiang, Xufeng Wang, Pengyu Long, Kai

Miao and all the other colleagues in Network for Computaitional Nanotechnology for

friendship and help.

Last but not the least, I would like to thank my wife and my parents for their love

and unconditional support. Without them, I cannot fulfill this important step in my

life.

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TABLE OF CONTENTS

Page

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Motivation for quantum computing . . . . . . . . . . . . . . . . . . 1

1.2 Silicon spin qubits and STM lithography technique . . . . . . . . . 2

1.3 ESR and NMR technique . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Objectives of this Work . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 QUBITS BASED ON ELECTRON AND NUCLEAR SPINS OF DONOR-CLUSTERS IN SILICON . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Electronic structure of P donors in silicon . . . . . . . . . . 10

2.2.2 Effective spin Hamiltonian . . . . . . . . . . . . . . . . . . . 12

2.2.3 Time evolution of spin states driven by electromagnetic field 14

2.3 Electron spin qubits based on donors in silicon . . . . . . . . . . . . 14

2.3.1 A single P-donor electron spin qubit in silicon . . . . . . . . 14

2.3.2 A donor-cluster electron-spin qubit in silicon . . . . . . . . . 16

2.4 Nuclear spin readout with ESR measurements . . . . . . . . . . . . 18

2.5 Importance of nuclear spin orientation in coupled electron qubits . . 19

3 A METROLOGY BASED ON ESR FOR A DONOR-CLUSTER QUBIT 22

3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Detection of donor number . . . . . . . . . . . . . . . . . . . . . . . 26

3.3 Detection of donor location . . . . . . . . . . . . . . . . . . . . . . 28

v

Page

3.4 Detection of electron number . . . . . . . . . . . . . . . . . . . . . 29

3.5 Anisotropy of the ESR spectra of donor-clusters in silicon . . . . . . 31

3.5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.5.2 Detection of different donor locations . . . . . . . . . . . . . 32

3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4 TWO-QUBIT CONTROL WITH EXCHANGE AND ESR . . . . . . . . 35

4.1 Proposed two-qubit device structure . . . . . . . . . . . . . . . . . . 35

4.2 Two-qubit swap gate based on exchange . . . . . . . . . . . . . . . 36

4.3 Individual qubit control with ESR . . . . . . . . . . . . . . . . . . . 39

5 SUMMARY AND OUTLOOK . . . . . . . . . . . . . . . . . . . . . . . . 41

LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

vi

LIST OF TABLES

Table Page

2.1 The binding energies of 1s states of a P donor in silicon [33] . . . . . . 11

4.1 The truth table of a logical SWAP gate . . . . . . . . . . . . . . . . . . 36

4.2 The values of exchange interaction and electron g-factor when exchange isswitched off and on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.3 The values of Fermi-contact hyperfine when exchange is switched off andon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.4 The values of the zz component of anisotropic hyperfine tensor when ex-change switched is off and on . . . . . . . . . . . . . . . . . . . . . . . 38

vii

LIST OF FIGURES

Figure Page

1.1 STM image of the central device region acquired during hydrogen lithog-raphy, showing a four-terminal dot device with source (S) and drain (D)leads and two in-plane gates (G1, G2). The bright patches, which havebeen outlined for clarity, indicate hydrogen-desorbed regions where phos-phorus donors will eventually be incorporated from the gaseous precursormolecule phosphine (PH3). (adopted from [28].) . . . . . . . . . . . . . 4

1.2 Schematic of the chemical reaction to deterministically incorporate a singlephosphorus atom into the surface. Saturation dosing of a three-dimerpatch (I) at room temperature (RT) followed by annealing to 350 oC allowssuccessive dissociation of PH3(II–IV) and subsequent incorporation of asingle phosphorus atom in the surface layer, ejecting a silicon atom in theprocess (V) (adopted from [30]). . . . . . . . . . . . . . . . . . . . . . . 5

1.3 (a) A Bloch sphere of a spin state. (b) Schematic diagram of Zeeman split-ting and Larmor frequency. (c) An example of ESR spectrum (adoptedfrom [32]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1 (a) A P donor substitutes a silicon atom in the crystal lattice. (b) Thevalley-orbit (VO) effect on the 1s energy states of a P donor in silicon. gdenotes the degree of degeneracy. . . . . . . . . . . . . . . . . . . . . . 10

2.2 The effect of Fermi contact hyperfine interaction in a single P donor insilicon on the ESR driving frequency. . . . . . . . . . . . . . . . . . . . 15

2.3 The effect of anisotropic hyperfine interaction in a donor-cluster in siliconon the ESR driving frequencies. . . . . . . . . . . . . . . . . . . . . . . 17

2.4 The steps to manipulate and readout a nuclear spin qubit. . . . . . . . 18

2.5 Two Zeeman-split nuclear spin levels for storing qubit information. . . 19

2.6 A singlet-triplet Bloch sphere. . . . . . . . . . . . . . . . . . . . . . . . 20

viii

Figure Page

3.1 (a) Central part of the device (in yellow), patterned by desorbing thehydrogen mask (in blue), showing the SET island (D1), the few-donorquantum dot (D2), and the innermost sections of the source and the drainleads of the SET. Spin dependent tunnelling between D1 and D2 is shownschematically. (b) The overall device, showing the layout of the two gates(G1,G2) and the extensions of the S and D leads. The white box showsthe area shown in a. (c) Close-up image of D2, overlaid by a grid withdimer-row spacing, revealing the exposed silicon dimers. (d,e) Schematicrepresentation of D2, showing sites suitable for P donor incorporation ingreen and the rest in yellow. The set of four white circles show two possibleconfigurations of the P donors. (adopted from reference [43]) . . . . . . 23

3.2 (a) The binding energy as a function of electron number and donor numberin donor clusters in silicon (adopted from [25]) (b) Tunnel coupling betweentwo donor/donor-clusters as a function of separation in the [100] direction.(c) Exchange coupling between two donor/donor-clusters as a function ofseparation in the [100] direction. . . . . . . . . . . . . . . . . . . . . . . 24

3.3 The flow chart of a metrology based on ESR technique. . . . . . . . . . 25

3.4 (a) Single-electron probability density distributions of the ground states(in nm−3). The four figures show the 1-4 P donor cases respectively. (b)ESR transitions of samples with 1-3 P donors. The thick (thin) arrowsindicate the orientations of the nuclear (electron) spins. (c) The ESRfrequencies of samples with 1-4 P donors. . . . . . . . . . . . . . . . . . 27

3.5 (a) Single-electron probability density distributions of the ground statesof two donor cases (in nm−3). The four figures (from left to right) showthe D1-D4 configurations respectively. (b) The Fermi contact hyperfineconstant as a function of two donor separation in the [110] direction cor-responding to (a). (c) The ESR frequencies of D1-D4 configurations in(a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.6 (a) Single-electron probability density distribution of the third electronin D4 with Hartree self-consistent field solution (in nm−3). (b) The ESRfrequencies of the third electron in D4, compared with the case of only 1electron in D4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.7 (a) Three 2P donor-clusters with different configurations. (b) The defini-tion of the in-plane B-field orientation. . . . . . . . . . . . . . . . . . 32

3.8 The ESR frequencies for the ⇓⇓↑ - ⇓⇓↓ transitions as a function of the in-plane B-field direction for the samples A, B and C shown in Fig. 3.7(a). 33

4.1 Proposed two-qubit STM-patterned device structure. . . . . . . . . . . 35

ix

Figure Page

4.2 Modeling of a two-qubit SWAP gate in a 2P-1P system. . . . . . . . . 38

4.3 Modeling of individual qubit control in the coupled 2P-1P qubits. . . . 40

x

ABSTRACT

Wang, Yu MSECE, Purdue University, August 2015. Manipulating and Character-izing Spin Qubits Based on Donors in Silicon with Electromagnetic Field. MajorProfessor: Gerhard Klimeck.

Single donors in semiconductors are promising candidates for spin qubits and have

attracted a lot of interest in the past decades. Quantum dots defined by single donors

in silicon as spin qubits are expected to have homogeneous performance as they are

highly localized and the electron wave function is confined by natural Coulomb po-

tential. Also, scanning tunneling microscopy (STM) lithography technique allows

positioning donors deep into the silicon lattice with atomic precision, which makes it

promising to fabricate a scalable solid-state quantum computer. The qubit informa-

tion, i.e. “0” and “1”, can be encoded in the electron spins or the nuclear spins in a

donor or a donor-cluster (several closely packed donors in silicon).

In this work, it is demonstrated that the manipulation and readout of a qubit de-

fined in the electron spins of donor-clusters in silicon can be realized with electron spin

resonance (ESR) technique. For donor-cluster qubits fabricated with STM lithogra-

phy technique, there exist configuration uncertainties, i.e. the number of donors, the

number of electrons and donor locations, which lead to different quantum confine-

ment. As a result, factors that can affect qubit performance such as binding energy,

tunnel coupling, and exchange coupling can vary significantly. Hereby, a characteriza-

tion technique based on the spin structure and ESR technique is developed to identify

the configurations of donor-clusters and to determine the corresponding manipulation

frequencies of ESR signals for single and multiple coupled qubits. Qubit operations

with electrical and magnetic control are also demonstrated in a two-donor-cluster-

xi

qubit system. This work is useful for understanding and designing experimental

two-qubit devices in silicon.

1

1. INTRODUCTION

1.1 Motivation for quantum computing

A quantum computer is a computation system that makes direct use of quantum-

mechanical phenomena to perform operations on data [1]. Quantum bits (qubits) are

the elementary units of a quantum computer. Because qubits are entangled and each

can be in a superposition “0” and “1” at the same time, they are superior to classical

bits in terms of storage and computation. For example, in n classical bits, only one

value in the range of 0 to 2n-1 can be stored and only one arithmetic-logic-unit (ALU)

operation can be performed at one time. However, benefiting from superposition and

entanglement, 2n values can be stored and 2n ALU operations can be performed in

n qubits simultaneously. As a result, an exponential increase is expected in terms of

storage and computation speed compared to classical computation.

Quantum computation and quantum algorithm provide efficient solutions to classes

of problems that are intractable for classical computation, such as large-integer fac-

torization, database search, simulations in many-body chemical systems and material

science. For example, the most popular public-key crypto-system RSA is based on the

assumption that large integer is impossible to be factorized into two prime numbers

efficiently. But using a quantum algorithm, Shor’s algorithm, a large integer can be

factorized in polynomial time [2].

Since quantum computation provides a way to solve extremely difficult problems

for classical computing, a lot of effort has been made to build a quantum computer.

Though it may still need many years to come to a final solution like the complementary

metal-oxide-semiconductor (CMOS) technology in classical computing, much progress

have been made in different proposals of implementing qubits. Solid-state quibts are

2

one of the promising prototypes for a scale-up quantum computer architecture and

will be introduced in the next section.

1.2 Silicon spin qubits and STM lithography technique

To eventually build a scalable quantum computer, a lot of research work has been

concentrating on integrating qubits into semiconductors through quantum dots [3–6].

A quantum dot is an artificial atom where the electron wave function is confined in all

the three dimensions and the energy spectrum is discrete and quantized as a result.

The qubit information, i.e. “0” and “1”, can be encoded in two charge states (a charge

qubit) or two spin states (a spin qubit) in a quantum dot. A charge qubit is sensitive

to charge defects and noise in the host environment, which is almost impossible to

get rid of in a semiconductor host and in a qubit device with interfaces and gates,

etc. As a consequence, the lifetime of a charge qubit is usually short, which limits the

number of qubit operations. However, this can be improved by encoding information

in spin states. Spin states can be nearly immune to charge fluctuations as long as

spin-orbit coupling is weak in the host material, such as Si, the most commonly used

semiconductor in industry. As a result, the lifetime of a spin qubit can be much longer

than that of a charge qubit.

Then there two questions to be addressed here to build a spin qubit: (1) which

semiconductor is appropriate to serve as the host material; (2) how to realize 3D

quantum confined states that serve as quality qubits [7]. For the first question, GaAs

and silicon are two common materials to host a spin qubit. But with regards to GaAs,

the spins of its nuclei are 3/2 that can interacts with electron spin through hyperfine

and dipolar interactions, so the spin coherence time in GaAs is around 200µs [8].

While for silicon, this can be improved because nuclear spin in Si-28 is zero and a

silicon host can be purified to almost zero-nuclear-spin-isotopes [9]. As a result, the

spin coherence time in silicon is usually much longer than in GaAs, which makes

silicon an ideal environment to host a spin qubit. The qubit coherence time has been

3

shown to be as long as 28 ms in a silicon quantum dot [10] and >35 s in a P donor

in silicon [11].

As to the second question, a three-dimensional confinement can be realized in

various ways. One of the state-of-art approaches is to confine quantum dots with top

gates at a heterostructure interface, like InGaAs/GaAs [12,13], SiGe/Si [14] or metal-

oxide-semiconductor [15] structures. The benefit from interface quantum dots is that

the orbital splittings and valley splittings can be changed electrically, which provides

freedom of tunability in qubit operations. However, they are sensitive to interface

roughness and fluctuations, e.g. charge defects [16] that undermine the qubit relia-

bility and performance. It is also challenging to make gate confinement identical for

different dots since the actual metal gate length is hard to control, which may ham-

per building a scalable quantum computer with qubits having uniform performance.

Another approach to introduce a three-dimensional confinement in semiconductors is

to make use of donors. A donor potential is a Coulomb potential of a positive charge

screened by the dielectric constant of the host material. Because the donor potential

is naturally occurring and identical between dopants of the same species, the qubit

performance can be guaranteed to be homogeneous.

From the argument above, spin qubits based on donors in silicon are promising

candidates for a scalable quantum computer. Several proposals have been put forward

on quantum computing based on donors in silicon [17–19], which are pursued by a

lot of research work both in theory [20–22] and in experiments [23–25]. In addition,

advances in STM lithography technique has pushed it a step forward [26]. Recent

progress in STM lithography technique allows incorporating single phosphorus (P)

atoms deep into the silicon lattice, which prevents the qubits from the interface and

surface fluctuations, making them less prone to environmental noise [27] and giving

rise to more stable performances. More importantly, donors can be placed with

atomic precision, which makes this fabrication technique promising in implementing

qubits with uniform performance, and helps limiting discrepancies between fabricated

devices and actual designs. In addition, the whole qubit device structure comprising

4

in-plane gates and leads [28] as well as metallic wires [29] can also be patterned

by the STM lithography technique. As a result, with this fabrication technique, it

is promising to eventually build all the components in an all-in-plane and scalable

quantum computer architecture. Fig. 1.1 shows an example of a device with planar

geometries patterned by the STM lithography technique [28]. The highlighted regions

are densely doped layers as metallic gates and leads by P donors, where the inter-

donor separations are typically less than 1 nm. In these regions, one of every four

silicon atoms can be substituted by a P atom.

Fig. 1.1. STM image of the central device region acquired during hy-drogen lithography, showing a four-terminal dot device with source(S) and drain (D) leads and two in-plane gates (G1, G2). The brightpatches, which have been outlined for clarity, indicate hydrogen-desorbed regions where phosphorus donors will eventually be incorpo-rated from the gaseous precursor molecule phosphine (PH3). (adoptedfrom [28].)

In the following, the experimental procedure of placing a P donor in the silicon

lattice with atomic precision will be briefly described. At first, a sample of single-

crystal silicon with the (100) surface passivation by hydrogen atoms is prepared.

Then the hydrogen atoms in the targeted region to place a P donor are removed by

5

the STM tip. After that, phosphine (PH3) is saturatedly dosed in this region. Fig.

1.2 depicts the process of the chemical reaction to deterministically incorporate a

single phosphorus atom into the surface in reference [30], which claims the accuracy

of placing a P donor in silicon with ±1 lattice site (±3.8 A). After the incorporation

of the P donor, temperature is increased to make the remaining hydrogen atoms on

the surface form hydrogen molecules that are then vaporized. At last, silicon lattice

is grown on top of the original surface such that the P donor is buried deep in the

crystal lattice.

Fig. 1.2. Schematic of the chemical reaction to deterministically incor-porate a single phosphorus atom into the surface. Saturation dosingof a three-dimer patch (I) at room temperature (RT) followed by an-nealing to 350 oC allows successive dissociation of PH3(II–IV) andsubsequent incorporation of a single phosphorus atom in the surfacelayer, ejecting a silicon atom in the process (V) (adopted from [30]).

1.3 ESR and NMR technique

From the above section, we know that a spin qubit in silicon has long lifetime

and can be fabricated with high quality with STM lithography. In this section, the

approach to manipulate a spin qubit, i.e. to make transitions between the spin-

up and the spin-down states, will be introduced. Electron spin resonance (ESR) and

6

nuclear magnetic resonance (NMR) are two commonly used techniques to manipulate

electron-spin qubits and nuclear-spin qubits, respectively [10,31]. They are based on

a similar mechanism which will be explained in the following.

Fig. 1.3. (a) A Bloch sphere of a spin state. (b) Schematic diagramof Zeeman splitting and Larmor frequency. (c) An example of ESRspectrum (adopted from [32]).

The spin-up and the spin-down states of an electron or a spin-1/2 nucleus can form

a two-level system. These two states and their quantum mechanical superpositions

can be represented by a Bloch sphere as shown in Fig. 1.3 (a). If a constant magnetic

field is applied, e.g. in the z direction as shown, the degeneracy of the two states

is broken due to the Zeeman effect, and the energy splitting between them can be

increased with the magnitude of the magnetic field, as shown in Fig. 1.3 (b). With

the presence of the constant magnetic field, the transition between the two spin

states can be driven by an alternating (ac) electromagnetic field applied orthogonal

to constant magnetic field as shown in Fig. 1.3(a). If the angular frequency of

the ac electromagnetic field is in resonance with the energy splitting divided by the

reduced Planck constant h, the system will evolve between the two spin states. Such

a resonance frequency is called Larmor frequency and can be measured in an ESR or

7

NMR experiment. An example of ESR measurement is provided in Fig. 1.3(c) where

the electron spin-up to electron spin-down transition frequencies can be measured [32].

Such a spectrum can help determine the driving frequencies for the manipulation of

spin qubits.

1.4 Objectives of this Work

As for a spin qubit based on a P donor in silicon, if the qubit information is

encoded in the two electron-spin states, it can be manipulated with an ESR signal.

Since the nuclear spin of a P atom is 1/2, it can also act as an nuclear spin qubit

which can be operated by an NMR signal. A spin qubit can also be realized in a

donor-cluster which is a small patch of a few closely packed donors in silicon, where

additional addressability is provided because additional nuclear spins are involved.

The resonance frequency of an electron spin qubit or a nuclear spin qubit in a donor

or a donor-cluster in silicon cannot be derived only from the Zeeman splitting as

shown in Fig. 1.3(b). Hyperfine interactions, dipolar interactions and even electron-

electron interactions also need to be considered. These types of interactions are

system-dependent so that the driving frequencies can be different from one another

depending on the detailed electronic structure, whereby a characterization technique

can be developed to figure out the configuration information of a donor-cluster. After

resolving the ESR frequencies of individual donor or donor-clusters, we can couple

them and perform multi-qubit operations with ESR technique, combined with tunable

electron-electron exchange interaction [22].

Based on the statements above, this work will focus on manipulating and char-

acterizing spin qubits based on P donors in silicon with ESR technique. In chapter

2, manipulation and readout of electron spin qubits and nuclear spin qubits based

on single donors and donor-clusters in silicon using ESR and NMR signals will be

demonstrated. In chapter 3, a metrology based on ESR technique for a donor-cluster

qubit in silicon will be proposed and a characterization technique based on anisotropic

8

hyperfine interactions will be put forward. Operations with ESR technique in donor-

based two qubits will be illustrated in chapter 4. Summary and outlook will be given

in chapter 5.

9

2. QUBITS BASED ON ELECTRON AND NUCLEAR

SPINS OF DONOR-CLUSTERS IN SILICON

2.1 Introduction

A qubit is basically a two-level quantum system. The two levels in a spin qubit are

a spin-up state and a spin-down state where “0” and “1” are encoded, respectively. A

static magnetic field can break the degeneracy of the two spin-states, which makes “0”

and “1” distinguishable. The transition between the two spin states can be driven by

an electromagnetic field, with which the manipulation of a spin qubit can be realized.

The type of a spin qubit determines the type of driving signal. An electron spin qubit

can be manipulated with an ESR signal and a nuclear spin qubit an NMR signal.

The driving frequency of the signal is system-dependent due to different electronic

structure and interactions with the background spins. In this chapter, modeling

methods are introduced to solve the electronic structure of P donors in silicon and to

determine the operation frequency of ESR/NMR signals to manipulate a spin qubit

based on P donors in silicon. An electron spin qubit and a nuclear spin qubit based on

two recent experiments will be explained with theory. Spin qubits based on a single P

donor and a P donor-cluster in silicon will be theoretically studied by investigating the

effects of Zeeman effect, Fermi contact hyperfine interaction and anisotropic hyperfine

interaction. Nuclear spin readout of a donor and a donor-cluster with ESR technique

will be introduced.

10

2.2 Method

2.2.1 Electronic structure of P donors in silicon

A P atom is a group-V atom that has five valence electrons. When it substitutes

a silicon atom in the crystal lattice, four of the valence electrons form covalent bonds

with the four nearest-neighbor silicon atoms shown as Fig. 2.1(a). As a result, there

is a net positive charge that provides a Coulomb potential, similar to a hydrogen

nucleus. At a cryogenic temperature, the fifth valence electron is bound by this

Coulomb potential whose spin can act as the two qubit states. To understand and

accurately model its electronic structure is of great importance for P-donor qubit

device design.

g=6

g=1

g=3g=2

VO

1s:A1

1s:T2

1s:E

1s(no VO)

(a) (b)

P

Si Si Si

Si

SiSiSi

Si

e

Fig. 2.1. (a) A P donor substitutes a silicon atom in the crystal lattice.(b) The valley-orbit (VO) effect on the 1s energy states of a P donorin silicon. g denotes the degree of degeneracy.

Unlike the electron of a hydrogen atom in vacuum, the fifth electron of the P

donor in silicon is not only affected by the Coulomb potential, but also by the back-

11

ground silicon. Since silicon has six equivalent conduction band valleys, the 1s state

of the bound electron should have six-fold degeneracy at first glance. But the six-fold

degeneracy is lifted due to valley-orbit (VO) coupling. The donor potential gives rise

to strong coupling between different valleys. After valley re-population, three types

of 1s states are formed that are named as A1, T2 and E, and have 1, 3, and 2 degrees

of degeneracy, respectively, as illustrated in Fig. 2.1(b). The corresponding binding

energies of these 1s states are listed in Table 2.1 (adopted from [33]).

Table 2.1The binding energies of 1s states of a P donor in silicon [33]

State Binding energy (meV)

1s(A1) 45.59

1s(T2) 33.89

1s(E) 32.58

To theoretically capture the electronic structure of P donors in silicon and the

VO effect, an accurate modeling method is required. The full Hamiltonian of silicon

and donor atoms are represented by sp3d5s* atomic basis with nearest neighbor in-

teractions and spin-orbital coupling. This has been implemented in a nanoelectronic

modeling tool (NEMO-3D) [34] and is utilized in this work. The electron wavefunction

is solved from the Hamiltonian that is expressed as

H = HSi +HD, (2.1)

where HSi represents the TB Hamiltonian of the silicon lattice, and HD denotes the

P-donor Hamiltonian. A P donor is represented by the Coulomb confinement of a

positive charge screened by silicon dielectric constant:

HD =q2

4πε0εSi|~R− ~R0|, (2.2)

where ε0 is the vacuum permittivity, εSi is the relative dielectric constant of silicon.

~R denotes the donor electron position and ~R0 denotes the donor location. A cut-off

12

potential along with on-site orbital energy corrections is used at the donor site to

model the effect of central cell correction [35]. When ~R = ~R0, HD = 3.78 eV is used

to capture the correct energy of the 1s(A1) state. This approach has successfully

reproduced the full energy spectrum of a single P donor in silicon, as well as its

valley-orbit splitting [36], Stark effects and Fermi contact hyperfine interaction [21].

In addition, the real-space electron probability density distribution and momentum-

space valley dispersion by this model also showed excellent agreement with recently

reported STM imaging experiments [24]. Moreover, the model can be extended to

the many-particle regime by Hartree self-consistent field (SCF) approximation when

the donor/donor-cluster is occupied by multi-electrons. The effective single-electron

Hamiltonian can be written as

H = HSi +HD +He−e, (2.3)

where He−e is the mean-field potential energy:

He−e =

∫q2n( ~R′)

4πε0εSi|~R− ~R′|, (2.4)

which is solved self-consistently with the electron probability density n( ~R′):

n( ~R′) = |Ψ( ~R′)|2, (2.5)

where Ψ( ~R′) is the inner electron wavefunction at ~R′. This method is able to capture

the experimental D− (the second electron bound to a single P donor) binding energy

[37] and can reproduce experimentally measured spin-lattice relaxation times [38].

2.2.2 Effective spin Hamiltonian

With an accurate model of P-donor electronic structure in silicon ready, the next

step is to theoretically describe the ESR phenomenon of a donor or a donor-cluster.

Physically, the spin system of a donor-cluster can be described as m nuclear spins

and n electron spins localized in a silicon crystal. The effective spin Hamiltonian can

be expressed as

Hspin = Hzeeman +Hhyperfine, (2.6)

13

In ESR/NMR experiments or spin qubit operations, a static external magnetic field

~B is applied on the device sample to polarize spins, and the energies of different spin

configurations will split due to the Zeeman effect, so that polarized spin states are

formed to encode qubit information. The Zeeman term, Hzeeman is expressed as

Hzeeman =n∑

i=1

geµB~B · ~Si −

n∑j=1

gnµn~B · ~Ij, (2.7)

where ~Si(~Ij) denotes the spin operator of the ith electron (the jth nuclei), ge(gn)

is the electron (nuclear) gyromagnetic ratio, µB(µn) is the Bohr magneton (nuclear

magneton). The second term, Hhyperfine, is the hyperfine interaction between electron

spins and the nuclear spins of a donor-cluster, which is

Hhyperfine =n∑i

m∑j

Aij~Ij · ~Si +

n∑i

m∑j

~Ij ·Dij · ~Si, (2.8)

where Aij represents the Fermi contact hyperfine coupling, which is proportional to

the ith electron probability density at the jth donor site. Dij is the anisotropic

hyperfine (or magnetic dipolar) interaction tensor. Aij and Dij are expressed as

Aij =8π

3γIγSh

2|Ψi( ~Rj)|2, (2.9)

Dij = 〈Ψ|3rirj − r2δij

r5|Ψ〉, (2.10)

where γI and γS are the nuclear and electron gyromagnetic ratio, respectively. Ψ is

the donor-cluster electron wavefunction, ~Rj is the position vector of the jth nuclei,

and ri,j = (x, y, z) are the electron position vector components. To obtain Aij and

Dij requires calculations of the electron wavefunctions. The atomistic TB method

as introduced in section 2.2.1 is utilized. When there is multi-electron occupation,

a Hartree SCF method is applied to approximate it as an effective single-electron

problem. In an effective single-electron regime, n in the equations above always

equals 1 as a result. The spin structure of a donor/donor-cluster in silicon, as well as

its ESR and NMR frequencies can be calculated from the eigenvalues of the effective

spin Hamiltonian Hspin. Then the targeted qubit operations can be performed by

ESR/NMR signals with the obtained frequencies.

14

2.2.3 Time evolution of spin states driven by electromagnetic field

As introduced in Chapter 1, an ac electromagnetic field orthogonal to the static

B-field can drive the transition between two spin states. In this case, the magnetic

field in equation 2.7 is not a constant anymore, but a function of time. The static

component is assumed to be applied in the z direction as always, and the time-

dependent B-field is applied in the x-y plane and is assumed to be circularly polarized.

Such a B-field can be expressed as:

~B = ~xBac0cos(ωt)− ~yBac0sin(ωt) + ~zBz, (2.11)

where Bac0 is the magnitude of the ac B-field, ω is the chosen angular Larmor fre-

quency of two spin states, and t stands for time.

The spin state |Ψ(t)〉 at the time node t is obtained by solving time-dependent

Schrodinger equation with Hspin:

|Ψ(t)〉 = U(t, t0)|Ψ(t0)〉, (2.12)

where |Ψ(t0)〉 is the initial spin state (at time t0) and U(t, t0) is the unitary time

propagator:

U(t, t0) = e−iHspin(t−t0)/h. (2.13)

2.3 Electron spin qubits based on donors in silicon

2.3.1 A single P-donor electron spin qubit in silicon

A spin qubit can be implemented in the spin states of the electron bound to a

P donor in silicon, where one electron spin and one nuclear spin are involved in the

spin system, i.e. m = 1 and n = 1 in equation 2.7 and 2.8. The spin structure of

this system, the ESR driving frequencies and the effect of hyperfine interaction will

be explained.

Fig. 2.2 shows the spin structure of the 1P1e (one P donor and one electron

in silicon) system, where a static magnetic field is applied in the z direction whose

15

Fig. 2.2. The effect of Fermi contact hyperfine interaction in a singleP donor in silicon on the ESR driving frequency.

magnitude is 1.77 T in line with reference [32]. As a response, the spins are polarized

and separated into four spin states corresponding to the full combination of “electron

+ nuclear” spin configurations. The blue lines (dashed and solid) denote states with

an electron up-spin and the red lines (dashed and solid) are for states with an electron

down-spin. The double arrows indicate allowed transitions that can be driven by ESR

signals between states with different electron spins but same nuclear spins. γESR1,2

denotes the driving frequencies. If only the Zeeman effect is considered, the spin levels

are shown as the dashed lines in the figure. This results in a single driving frequency,

i.e. γESR1 = γESR2 (= 49.5824 GHz) as shown in the figure. In this scenario, single

frequency is detected because the electron spin and the nuclear spin are decoupled.

But in reality, there is a hyperfine interaction between the electron spin and the

nuclear spin. If it is taken into account, the spin levels are modified and shifted by

different amounts, which lead to two different ESR driving frequencies shown in the

bottom left corner of this figure. The dipolar interaction here results in changing the

16

spin levels less than 1 Hz from calculation so it can be ignored in this system. This is

because the wavefunction has cubic symmetry so that the anisotropic hyperfine term

vanishes (Eq. 2.10). The values of the two frequencies and their splitting are shown on

the right hand side, which have a good agreement with the experimental measurement

demonstrated in reference [32]. The electron qubit can be manipulated with these two

frequencies depending on the nuclear spin orientation. After manipulation, the spin

can be readout with single shot and high fidelity through a nearby single-electron-

transistor (SET) via spin-to-charge transition as demonstrated in references [39,40].

2.3.2 A donor-cluster electron-spin qubit in silicon

A single-electron spin qubit can also be implemented in a donor cluster in sili-

con, whose manipulation follows the similar fashion of the 1P1e system. There are

two major differences between a single donor and a donor-cluster system. First, the

quantum confinement is different due to additional positive charges within a small

region, which leads to different electron probability concentration on the donor sites

and different the hyperfine couplings. Second, more nuclear spins are introduced so

the number of the spin states will increase, and the ESR spectrum will be different.

Here a 2P1e system is demonstrated as an example, where the two donors are sepa-

rated by ∼1.5nm. The detailed donor locations (D4 in Fig. 3.5) will be shown in the

next chapter. Here we focus on the derivation of its spin structure and corresponding

ESR driving frequencies.

The left part of Fig.2.3 displays the calculated spin structure and ESR spectrum

by considering only the Zeeman effect and Fermi contact hyperfine interactions. The

static magnetic field is kept the same (1.77 T). As shown in this figure, the number

of states is doubled by introducing one more nuclear spin into the 1P1e system, and

correspondingly four transitions are allowed at frequencies γESR1−4, which provides

additional addressability compared to a 1P1e qubit. It should be noticed that γESR2

and γESR3 almost have the same value and agree with the ESR frequency without

17

Fig. 2.3. The effect of anisotropic hyperfine interaction in a donor-cluster in silicon on the ESR driving frequencies.

considering the hyperfine interaction in the 1P1e system. This is because the corre-

sponding two nuclear spins are anti-parallel and so the hyperfine coupling effects from

them are canceled with each other. If the anisotropic hyperfine interaction is further

considered, one can find that the spin levels and ESR peaks at γESR1 and γESR4 are

shifted by about 100kHz, while γESR2 and γESR3 are only shifted by less than 1kHz as

shown in the right part of Fig. 2.3. As can be seen, the effect of anisotropic hyperfine

interaction is more prominent in a 2P1e system than that in a 1P1e system due to

different quantum confinement and different electron wavefunction distribution. The

cubic symmetry of the wavefunction no longer holds for a 2P1e donor-cluster and the

anisotropic hyperfine term is playing a role. Moreover, to manipulate more coherently

the electron spin qubit in a 2 P donor-cluster, the ESR driving frequencies γESR2 and

γESR3 are more favored in order to offset the electron-nuclei interactions.

18

2.4 Nuclear spin readout with ESR measurements

Nuclear spins of P donors in silicon can also be taken advantage of to make spin

qubits. One of the benefits of using nuclear spin as the host of qubit information

is that nuclear spins usually have extremely long coherence time. It was recently

demonstrated that quantum information can be stored in the nuclear spin of a ionized

P donor in silicon-28 over 39 min even at room temperature and 3 hours at cryogenic

temperature [41]. Experimentalists have also demonstrated the manipulation with

NMR and readout with ESR of a nuclear spin qubit based on a single-P donor in

silicon [32]. It needs to be highlighted that its readout fidelity can reach as high as

99.99%, which is the highest reported so far for any solid-state qubits. The main

dephasing channel of a nuclear spin qubit in qubit manipulation is the hyperfine

coupling to electron spins. As a result, the coherence time is longer if the qubit is

ionized. This experiment also showed that the coherence time of a ionized P donor

nuclear spin qubit was made about three times longer than that of a charge neutral

donor [32].

Fig. 2.4. The steps to manipulate and readout a nuclear spin qubit.

Fig. 2.4 illustrates the steps to manipulate and readout a nuclear spin qubit. After

initialization, the qubit is ionized to gain access to longer coherence times during

manipulation by NMR signals. After manipulation, the electron is loaded back for an

ESR measurement. The measured ESR frequency is dependent on the nuclear spin

orientation as explained in the last section, whereby the nuclear spin can be readout.

For a single-P donor nuclear spin qubit, if it is ionized and exposed to a static

magnetic field, its spin structure is simply formed as a two level system, i.e. nuclear

spin-up state and spin-down state as shown in Fig. 2.5 where B = 1.77 T is kept the

same. The Zeeman splitting can be easily solved from the second term of equation

19

Fig. 2.5. Two Zeeman-split nuclear spin levels for storing qubit information.

2.7, and the NMR driving frequency is γNMR = 30.5059MHz. After manipulation,

the electron is reloaded to the donor to readout the spin of the nucleus by conducting

an ESR experiment. By referring to Fig. 2.2, if the measured ESR frequency is γESR1

(= 49.6413 GHz), then the spin of the nucleus is up. If it is γESR2 (= 49.5237 GHz),

then the spin of the nucleus is down. By taking advantage of hyperfine splitting in

ESR spectrum, a nuclear spin can be readout with the assist of an electron spin.

The nuclear spins in a donor-cluster can also be readout by following the same

manner. Take the 2P cluster mentioned in section 2.3.2 as an example, by referring to

Fig. 2.3, if the two spins of the nuclei are both up, then the ESR frequency is γESR1.

If they are both down, then the ESR frequency is γESR4. If they are anti-parallel,

then the ESR frequency is γESR2 or γESR3.

2.5 Importance of nuclear spin orientation in coupled electron qubits

The control of nuclear spins is important when two separated donors or donor-

clusters are coupled to make an electron singlet-triplet qubit or two electron-spin

qubits. Similar to a single spin qubit, in a singlet-triplet qubit, the two-electron

singlet and triplet define the qubit states, and all their superpositions can be denoted

on a Bloch sphere, as shown in Fig. 2.6. The south pole and the north pole are

the singlet state |S〉 and the triplet state |T 〉, respectively, and on the equator are

| ↑↓〉, | ↓↑〉 and their superpositions. To describe such a system with effective spin

20

|S>

|T>

|↑↓> |↓↑>

J

���Beff

Fig. 2.6. A singlet-triplet Bloch sphere.

Hamiltonian, one needs to add up the effective spin Hamiltonian for each qubit, plus

a term for electron-electron exchange interaction Hexchange:

Hexchange = J ~S1 · ~S2, (2.14)

where J is the inter-qubit exchange energy between two exchange-coupled electrons.

It can be solved with a full configuration interaction method based on atomistic TB

wavefunctions [16]. The singlet state and the triplet state are split by the exchange

energy J which is responsible for the evolution on the equator. | ↑↓〉 and | ↓↑〉

are split by the effective local Zeeman energy difference which is responsible for the

evolution between |S〉 and |T 〉 [42]. The hyperfine interactions in donors/donor-

clusters create an effective magnetic field which depends on the local number of nuclei,

their orientations and their locations. The net hyperfine coupling is determined by the

orientations of the nuclear spins as explained in section 2.3, and the local difference in

coupling leads to an effective magnetic field difference or an effective Zeeman energy

difference. As a result, the control of the transition between a singlet and a triplet

state can be realized by controlling the nuclear spin orientations. To be specific, if the

two nuclear spins are anti-parallel, then the Zeeman energy difference is large, which

is roughly 117 MHz, i.e. the hyperfine interaction of a single P donor. Then the

evolution between |S〉 and |T 〉 is switched on. If they are parallel, the Zeeman energy

21

difference is small and the evolution is switched off. As stated in the above section,

NMR can control the nuclear spin and hereby control the singlet-triplet operation.

The role of NMR in coupled donors/donor-clusters qubits depends on specific design

of qubits, and needs to be further studied.

22

3. A METROLOGY BASED ON ESR FOR A

DONOR-CLUSTER QUBIT

3.1 Motivation

A single P atom in silicon can act as either an electron or a nuclear spin qubit with

high readout fidelity as introduced in chapter 2. However, both experiments [32, 40]

were conducted with top-down implantation approach. It is challenging to precisely

control the number of P atoms and their locations in the silicon crystal with this

approach. Single atom patterning by scanning tunneling microscope (STM) makes it

easier to control the placement of the P donors with atomic precision. In addition,

P-donor clusters provide additional addressability to single P donors [43]. Single-

shot electron spin readout for a P-donor cluster with high fidelity as well as long T1

time (∼2s) has been demonstrated in reference [43]. Here by increasing the donor

number for a single electron qubit, we have recently shown that the T1 time can be

further increased [38]. However, there is still uncertainty in the number of donors,

number of electrons and donor locations in an ultra-small donor island as shown in

Fig. 3.1 (adopted from reference [43]). As can be seen in Fig. 3.1c, the ultra-small

island contains ∼4 P donors with unknown donor locations. Two possibilities of their

locations are given in d and e. Such uncertainty within atomic scale can strongly

affect quantum confinement, and as a result a lot of crucial factors affecting qubit

operations can vary from one cluster to another.

Fig. 3.2 shows the configuration effect on the parameters that are relevant to qubit

operations. Fig. 3.2(a) shows the binding energies of a P donor or a P donor-cluster

in bulk silicon as a function of electron number and donor number [25]. Binding

energy is a metric of how easy it is to ionize an electron from a system. It has

a significant effect on charge stability and largely influences the gate voltage bias

23

Fig. 3.1. (a) Central part of the device (in yellow), patterned bydesorbing the hydrogen mask (in blue), showing the SET island (D1),the few-donor quantum dot (D2), and the innermost sections of thesource and the drain leads of the SET. Spin dependent tunnellingbetween D1 and D2 is shown schematically. (b) The overall device,showing the layout of the two gates (G1,G2) and the extensions of theS and D leads. The white box shows the area shown in a. (c) Close-upimage of D2, overlaid by a grid with dimer-row spacing, revealing theexposed silicon dimers. (d,e) Schematic representation of D2, showingsites suitable for P donor incorporation in green and the rest in yellow.The set of four white circles show two possible configurations of theP donors. (adopted from reference [43])

range for qubit operations. As can be seen, with different donor number, electron

number and donor locations, the binding energy of a donor/donor-cluster varies from

∼450 meV to ∼2 meV. Measuring binding energy could be a characterization scheme

to figure out the configuration information of a donor-cluster. Binding energy is

usually obtained by current measurements. For the device in reference [25], binding

energies are measurable because there is current flow through the donor-clusters.

However, binding energy is hard to measure in devices for spin readout, e.g. the

24

Separation in [100] (nm)

Exch

ange

(H

z)

Bin

din

g en

ergy

(m

eV)

(a)

(b) (c)Electron number, n

Fig. 3.2. (a) The binding energy as a function of electron numberand donor number in donor clusters in silicon (adopted from [25]) (b)Tunnel coupling between two donor/donor-clusters as a function ofseparation in the [100] direction .(c) Exchange coupling between twodonor/donor-clusters as a function of separation in the [100] direction.

device shown in Fig. 3.1 [43]. There is no current flow through the donor-cluster so

this characterization scheme could fail.

The configurations of donor-clusters can also largely affect the interactions be-

tween the two qubits when they are coupled. Fig. 3.2(b) and (c) show that the tun-

nel coupling and the electron-electron exchange energy as a function of inter-qubit

separation along the crystallographic axis [100] for different configurations. The tun-

nel coupling indicates the speed of an electron shuttling from one qubit to another,

which is a critical parameter for designing a charge qubit [12]. Also, tunnel coupling

indicates the exchange energy of a singlet-triplet qubit when the system is detuned

to a charge transition point [42], which is also a parameter of great significance that

indicates the speed of operation. Exchange energy is also a crucial parameter in

qubit operations. In the popular Kane quantum computer architecture [17], the nu-

25

clear spins are readout through electron-electron exchange. In a singlet-triplet qubit,

exchange energy distinguishes the two-electron singlet and triplet states where the

qubit information is encoded. In two-qubit operations, exchange energy itself realizes

a logical two-qubit SWAP gate and also plays important roles in other types of qubit

gates, such as controlled-NOT (CNOT) and controlled-rotation (CROT) [22,44].

Quantum confinement

Donor-cluster configuration

Hyperfine interaction A

Electron spin resonance(ESR) spectrum

(� ∝ |�(��)|�)

Fig. 3.3. The flow chart of a metrology based on ESR technique.

In short, tunnel coupling and exchange coupling are of great importance in types

of qubits and qubit operations, and can vary by orders of magnitude between different

configurations as shown in Fig. 3.2(b) and (c). Therefore, without knowing the con-

figuration information of a donor-cluster qubit, understanding an experiment could

be difficult. Therefore, a metrology is needed to characterize donor-cluster qubits.

In this chapter, we propose a unique metrology technique to extract the configura-

tions of donor clusters based on ESR technique, which is developed from different

hyperfine interactions induced by different quantum confinement in donor-clusters

as illustrated in Fig. 3.3. To be specific, different donor-cluster configurations, i.e.

26

different electron number, donor number and donor locations, give rise to different

quantum confinement, which leads to different electron wavefunction concentration

at the donor site ~Rj. Hyperfine interaction of donor j is proportional to the electron

probability density at the donor site ~Rj. And different hyperfine interaction leads to

different ESR spectrum that can be directly measured in experiments. Consequently,

hyperfine interaction build a bridge between a donor-cluster configuration and its

measurable ESR spectrum. Thus, the ESR spectrum casts light on the configuration

of the donor-cluster, whereby a metrology can be developed.

In the following, we will demonstrate how this metrology works theoretically on

different device samples. For tight-binding wave function calculations, the whole

simulation domain is set to be 30nm x 30nm x 30nm of a regular silicon crystal

lattice in NEMO-3D, so that the Coulomb potential given by donors placed at the

center of the domain decays to almost zero at the boundaries. This eliminates any

artificial surface effects. The P-donor cluster is placed in the central region of the

middle (001) atomic plane. A static magnetic field B = 1.5 T is applied in the [001]

direction so that the electron spin can be polarized. The ESR spectra are obtained

from the corresponding effective spin Hamiltonians by following equation 2.6.

3.2 Detection of donor number

Fig. 3.4(a) shows the single-electron probability density distributions for 1 to 4 P

donors in the central plane (the X-Y plane at Z = 15.2 nm). Fig. 3.4(b) demonstrates

the spin levels calculated by diagonalizing the effective spin Hamiltonian as shown

in equation 2.6, which have been converted into the unit of frequency by dividing by

h, Planck’s constant. The spin frequencies are well-separated into two bands, which

indicates the electron spin has been polarized by the static magnetic field (1.5 T). One

can see the number of levels increases with the number of donors on the ultra-small

island, which is the combined effects of the Zeeman splittings of multiple nuclear spins

and the hyperfine interaction of the electron with multiple nuclear spins. Thus, the

27

Fig. 3.4. (a) Single-electron probability density distributions of theground states (in nm−3). The four figures show the 1-4 P donor casesrespectively. (b) ESR transitions of samples with 1-3 P donors. Thethick (thin) arrows indicate the orientations of the nuclear (electron)spins. (c) The ESR frequencies of samples with 1-4 P donors.

number of nuclear spins determines the number of spin frequencies. As introduced

in chapter 2, after spin polarization, an a.c. magnetic field is applied perpendicular

to the static B-field to measure the ESR frequencies. Once the frequency of the

a.c. B-field is in resonance with the difference between two electron spin frequencies,

namely ↑ and ↓ for the same nuclear spin configuration, there will be a transition as

shown by long arrows in Fig. 3.4(b), which can be detected in an ESR experiment. In

theory, the ideal ESR spectrum can be obtained by simply finding the gaps between

each electron ↑ and ↓ within a particular nuclear spin configuration, solved from the

effective spin Hamiltonian. Fig. 3.4(c) shows the ESR spectra of devices with 1-4 P

donors on the island. It is obvious that the number of ESR frequencies goes up with

the number of donors. Therefore, the donor number can be extracted by comparing

28

and matching the number of ESR frequencies between theoretical and measured ESR

spectra.

3.3 Detection of donor location

For a single electron spin qubit, devices with same number of donors will give the

same number of resonance frequencies in ESR spectra because the total number of

spins, i.e. m + 1, does not change. However, the ESR spectra of different P-donor

locations are still different from each other. Hyperfine coupling between electron spin

and nuclear spin can change evidently even if the location variations of donors are

within one or several unit cells. This is because hyperfine coupling is proportional to

electron probability density on the donor site. The quantum confinements and the

on-site wave function concentrations can vary apparently from one configuration to

another with spatial variations of ∼1-2 nm.

Here we take 2P donor clusters as an example. Fig. 3.5(a) shows the single-

electron probability density distributions of four different 2P donor locational con-

figurations, labeled as D1-D4. From left to right, the separation of the two donors

are increased by a step of ∼3.84 angstroms in the [110] direction, from D1 to D4.

And in the [110] direction, two donors are ∼3.84 angstroms apart, which is fixed for

all the four cases. From the color bar of each plot, we can see the on-site electron

probability density decreases with the distance between two donors. This will also

lead to hyperfine coupling decreasing with distance, which is shown in Fig. 3.5(b).

Fig. 3.5(b) gives the values of Fermi contact hyperfine coupling of one of the two

donors, and the other shares the same value because of system symmetry. The values

of the anisotropic hyperfine couplings are not shown since they are over 3 orders of

magnitude smaller than Fermi contact hyperfine coupling. It has little effect on ESR

frequencies. Fig. 3.5(c) shows the corresponding ESR spectra of the four configura-

tions. The ESR peaks are more apart if the donors are more closely packed. Thus,

29

Fig. 3.5. (a) Single-electron probability density distributions of theground states of two donor cases (in nm−3). The four figures (from leftto right) show the D1-D4 configurations respectively. (b) The Fermicontact hyperfine constant as a function of two donor separation inthe [110] direction corresponding to (a). (c) The ESR frequencies ofD1-D4 configurations in (a).

devices with the same number of donors but different donor locations can also be

distinguished by this metrology.

3.4 Detection of electron number

So far, we have studied cases that have only one electron bound to a donor-cluster.

However, this is not often the case due to the strong Coulomb confinement given by

the positive charges of donor nuclei, and thus usually several electrons can be bound.

For example, two electrons can even be bound to a single P donor in silicon [30].

The number of electrons can be even or odd, which will give different scenarios. We

30

are mostly interested in resonance effects with effective 1/2 spins in the donor-cluster

system as a spin qubit, so only odd number of electrons is considered in the following.

In the modeling and simulation, the multi-electron interactions are approximated

with Hartree SCF scheme as introduced in section 2.2. As to multi-electron occu-

pations, compared with the one-electron case, the unpaired electron feels the extra

Coulomb repulsion from other electrons, so the wave function is expected to spread

out spatially and therefore be less concentrated at the donor sites. Effectively for the

electron net spin, Fermi contact hyperfine couplings will become smaller.

Fig. 3.6. (a) Single-electron probability density distribution of thethird electron in D4 with Hartree self-consistent field solution (innm−3). (b) The ESR frequencies of the third electron in D4, com-pared with the case of only 1 electron in D4.

Here we show the D4 case (in Fig. 3.6 (a)) with 3 electrons as an example to

verify the argument above. Fig. 3.6(a) shows the probability density distribution of

the third electron solved with Hartree self-consistent field approximation. It is more

spread out in the same region and has lower on-site concentrations by comparing with

the single-electron probability density of D4 (shown in Fig. 3.5(a)). With two more

electrons, consequently the Fermi contact hyperfine decreases to 10.5 MHz from 101.8

MHz. The ESR frequencies of the third electron in D4 is compared with that of 1

31

electron in D4 in Fig. 3.6(b). So the ESR resonance peaks are closer if the number

of bound electrons in a particular donor cluster is larger.

One may ask that larger separation can also result in a decrease in hyperfine

interaction which adds to a possibility of a hyperfine decrease. However, from the

magnitude of hyperfine interaction, by placing the two donors apart can only result in

a decrease of a factor of ∼3 at most, as indicated in Fig. 3.5(b). If the two donors are

placed further apart, they will tend to act as single P donors in bulk silicon because

of smaller tunnel coupling between their wavefunction, and as a result the hyperfine

interactions will converge to the bulk value 117 MHz. While by adding two extra

electrons the hyperfine interaction is decreased by about one order of magnitude.

Therefore, the electron number in a donor cluster can also be detected with this

metrology.

3.5 Anisotropy of the ESR spectra of donor-clusters in silicon

3.5.1 Background

The spin structure and ESR spectrum of a single P donor in bulk silicon is isotropic

because its wavefunction has cubic symmetry and the anisotropic part of the hyper-

fine interaction vanishes. However, for a donor-cluster in silicon, the bound electron

is shared by the donors and form molecular states. As a result, the cubic symmetry

is broken and the anisotropic hyperfine term plays a role. Enlightened by electron

nuclear double resonance (ENDOR) technique proposed by Feher [45], a characteri-

zation technique can be developed by taking advantage of the anisotropy of hyperfine

interaction of a donor-cluster.

ENDOR is a powerful technique to probe environmental nuclear spins with high

precision. It was employed for detecting environmental Si-29 nuclei experimentally

[46] and predictively combined with atomistic TB simulations [47]. In this section,

the anisotropy of hyperfine interaction is utilized to detect the locations of P donors in

silicon. The difference between a P donor and a Si-29 atom is that a P donor provides

32

a Coulomb confinement such that the electron wavefunction is redistributed, and thus

the anisotropic hyperfine interaction is different. This can be well captured by the

atomistic TB approach as illustrated in chapter 2 and is shown in the above sections.

To make the effect of anisotropic hyperfine interaction prominent, in the following

section, a static B-field is not only applied in the [001] direction, but also in the (001)

plane whose in-plane direction can be changed to detect the anisotropy of the ESR

spectra of a donor-cluster.

3.5.2 Detection of different donor locations

P donorSi atom

[100]

[010]

In-plane B-field = 1.5 T

�

A B C(a)

(b)

[100]

[010]

Fig. 3.7. (a) Three 2P donor-clusters with different configurations.(b) The definition of the in-plane B-field orientation.

2P donor-clusters with different donor locations are taken as an example to demon-

strate the technique. Three different samples are studied as shown in Fig. 3.7(a),

labeled as A, B and C. All the donors are placed on the same (001) atomic plane

in line with STM lithography fabrication technology introduced in Chapter 1. The

33

two P donors are separated along the [100] direction in sample A, along the [110]

direction in sample C, and in between in sample B (same as the D2 configuration in

Fig. 3.5(a)). An out-of-plane static magnetic field of 1.5 T is applied in the [001]

direction, whose magnitude (1.5 T) and direction is fixed all the time. In addition, an

in-plane magnetic field 1.5 T is applied in the (001) atomic plane whose magnitude

is also fixed but the direction can be changed. The direction is defined by the angle

θ as indicated in Fig. 3.7(b).

A

B

C

⇓⇓↑

⇓⇓↓

Fig. 3.8. The ESR frequencies for the ⇓⇓↑ - ⇓⇓↓ transitions as afunction of the in-plane B-field direction for the samples A, B and Cshown in Fig. 3.7(a).

ESR experiments can be conducted under different B-field orientations and the

ESR frequency as a function of θ can be obtained [45, 46]. Following the same fash-

ion and the theory illustrated in section 2.2, the ESR frequencies of the ⇓⇓↑ - ⇓⇓↓

transitions are calculated for sample A, B and C, respectively. The in-plane B-field

direction angle θ is varied from 0o to 360o and the ⇓⇓↑ - ⇓⇓↓ transition frequencies

of sample A, B and C as a function of θ are plotted in Fig. 3.8. As can be seen, the

34

⇓⇓↑ - ⇓⇓↓ transition frequencies of sample A, B and C have obvious differences in θ

dependence due to different anisotropy of hyperfine interactions. The fist minimum

frequency of sample A, B and C occurs at ∼90o, ∼60o and ∼45o, respectively. In

other words, the angular dependence pattern shifts leftward gradually from θ = 0o to

θ = 45o. Therefore, donor locations of a donor-cluster can also be detected from the

in-plane B-field dependence of ESR frequencies.

3.6 Conclusion

In conclusion, we propose a metrology technique to characterize spin qubits defined

by phosphorus donor-clusters in silicon. The number of donors, donor locations and

number of electrons can be detected by the ESR technique. The electron wave func-

tions are calculated with atomistic tight-binding approach, and Hartree self-consistent

field theory is employed to approximate electron-electron interactions. We found that

the number of ESR frequencies increases with the number of donors. Also, the sep-

arations of ESR frequencies are larger if intra-cluster donor distances are smaller.

In addition, ESR frequencies are closer for multi-electron occupation than single-

electron occupation. Due to the broken cubic symmetry and the effect of anisotropic

hyperfine interaction, the anisotropy of ESR spectra of donor-clusters with different

donor locations are shown to be different. This metrology is useful for extracting the

configurations of donor-clusters and thus helpful for designing solid-state qubits.

35

4. TWO-QUBIT CONTROL WITH EXCHANGE AND

ESR

4.1 Proposed two-qubit device structure

SET

D

S

� �

G1

G2

ESR/NMR

Qubit1

Qubit2

readout

Fig. 4.1. Proposed two-qubit STM-patterned device structure.

As introduced in chapter 2, ESR and NMR signals can be used to manipulate and

readout individual P-donor spin qubits, which have been addressed in this work. As

for quantum computing, individual qubits are not isolated, but coupled and entangled

so that multi-qubit operations can be performed. Two-qubit logical gates are basic

operational components that are needed to implement various quantum algorithms.

Single-qubit and two-qubit operations are needed to function coordinately in order to

perform different qubit gates. For two-electron spin qubits, ESR signals and electron-

36

electron exchange interactions are need to complete two-qubit operations. By taking

advantage of the current STM lithography technique and P-donor qubits in silicon,

a two-qubit device structure is designed as shown in Fig. 4.1. The yellow regions are

delta-doped layers in silicon patterned by STM lithography technique. S (source) -

SET - D (drain) forms a single electron transistor which is used to readout individual

electron spin of qubit 1 and qubit 2. G1 and G2 are the detuning gates that can

control the exchange energy. i is an ac current that can generate electromagnetic

fields, i.e. the ESR/NMR signals to manipulate individual spins of qubit 1 or 2.

Experimentally a nanowire was fabricated with the STM lithography technique and

has metallic behavior at cryogenic temperature [29], which is possible to carry an ac

current. An STM-patterned device that combines the SET and the nanowire as well

as the electrostatic gates is also feasible in experiments as shown in Fig. 4.1. Instead

of the in-plane ESR line, a metallic wire can also be fabricated on the top surface of

the device to carry an ac current as shown in reference [10]. Taking advantage of a

donor-cluster, a two-qubit SWAP gate and individual operations will be demonstrated

based on a 2P-1P system in the following, where the 2P configuration is chosen as

”D4” in Fig. 3.5.

4.2 Two-qubit swap gate based on exchange

Table 4.1The truth table of a logical SWAP gate

Input Output

0 0 0 0

0 1 1 0

1 0 0 1

1 1 1 1

37

A SWAP gate logically exchange the values of two-electron spin qubits as shown

in the truth table 4.1, where “0” is encoded in ↑ and “1” is encoded in ↓. As a result,

a SWAP gate can be realized simply through electron-electron exchange interaction.

Since any qubit operations should be able to be switched on and off, a tunable ex-

change interaction is needed to implement a SWAP gate and other two-qubit gates

like controlled-NOT (CNOT), as shown in reference [44]. As demonstrated in our

recent work [48], the exchange tunability can be improved by orders of magnitude by

using a donor-cluster in two qubits instead of using two single donors. In line with

reference [48], we use a 2P donor-cluster as qubit 1 and a single donor as qubit 2

which are separated by 15 nm in Fig. 4.1.

Table 4.2The values of exchange interaction and electron g-factor when ex-change is switched off and on

J (MHz) gx(e1,2P) gz(e1,2P) gx(e2,1P) gz(e2,1P)

J-on 1.6 1.9999634888 1.9973471512 1.9999676342 1.9953300615

J-off 2623.1 1.9999634108 1.9973385324 1.9999671454 1.9952311606

Table 4.3The values of Fermi-contact hyperfine when exchange is switched off and on

Ae1(2P1)(MHz) Ae1(2P2)(MHz) Ae1(1P)(MHz)

J-on 95.7 107.2 1.2× 10−5

J-off 101.6 102.1 1.4× 10−8

Ae2(2P1)(MHz) Ae2(2P2)(MHz) Ae2(1P)(MHz)

J-on 8.6× 10−5 1.3× 10−4 114

J-off 6.7× 10−3 5.8× 10−3 117.1

38

Table 4.4The values of the zz component of anisotropic hyperfine tensor whenexchange switched is off and on

Dzz,e1(2P1)(kHz) Dzz,e1(2P2)(kHz) Dzz,e1(1P)(kHz)

J-on 55.1 53.2 0.099

J-off 54.3 54.3 0.098

Dzz,e2(2P1)(kHz) Dzz,e2(2P2)(kHz) Dzz,e2(1P)(kHz)

J-on 0.086 0.1 0.6

J-off 0.099 0.11 0.31

Fig. 4.2. Modeling of a two-qubit SWAP gate in a 2P-1P system.

To see how the two qubits evolve under the SWAP operation dynamically, equation

2.13 needs to be solved. The effective spin Hamiltonian contains Zeeman, hyperfine

and exchange terms as illustrated in Chapter 2, but without an ac magnetic field.

By pulsing the bias of G1 and G2 (see Fig. 4.1), the evolution can be switched

39

on and off. The exchange-on value is chosen where the (2,0) charge configuration

has not been reached to avoid indistinguishability of the electrons. It should be

noticed that, not only the exchange interaction, but also the Fermi-contact hypefine

interaction constant Aj, anisotropic hyperfine interaction Dij, and electron g-factor ge

(in equations 2.7,2.8) are functions of the gate biases due to Stark effect. In Table 4.2,

4.3 and 4.4, these terms are listed with exchange switched off and on, simulated with

atomistc TB method introduced in Chapter 2. As can be seen, anisotropic hyperfine

interaction and g-factors are insensitive to the detuning electric field. With these

parameters ready, the effective spin Hamiltonian can be constructed. For simplicity,

the effective spin Hamiltonian is projected onto a fixed nuclear spin configuration,

⇑⇓ − ⇓, to form an electron-spin-only Hamiltonian, and then the time evolution of

the spin states is solved from equation 2.13. At time t = 0, the initial two-qubit state

is ↑↓ and exchange is switched off. Exchange is switched on at t = 3 ns and the

two-qubit state starts to evolve between ↑↓ and ↓↑. If the initial state is ↑↑ or ↓↓,

it will stay at the initial state and not change with time (not shown). Therefore, a

SWAP gate can be realized in a 2P-1P system.

4.3 Individual qubit control with ESR

In the following, individual qubit control of the coupled two qubits will be shown.

To simulate this procedure, equation 2.13 is also utilized to solve the time evolution

of the two-qubit state. In the effective spin Hamiltonian, exchange is switched off and

a circularly polarized ac B-field is applied as stated in Chapter 2, and its magnitude

is assumed to be 0.3 mT. If its angular frequency is chosen as the angular Larmor

frequency of the ↓↓ - ↓↑ transition, then this transition can be driven by the ac B-

field. The left figure in Fig. 4.3 demonstrates the time evolution of the two-qubit

states under such an ac B-field with the initial state being ↓↓. If the first spin is ↑,

then the two-qubit state will not change since it is out of resonance with the ac B-

field. Moreover, if we choose the angular frequency of the ac B-field to be the angular

40

Larmor frequency of the ↓↓ - ↑↓, the control qubit (the first qubit) can be manipulated

as shown in the right figure in Fig. 4.3. Benefiting from the asymmetry of the 2P-1P

system, the two transition frequencies mentioned above are distinguishable and can

be easily resolved in experiment, as shown in the two figures in Fig. 4.3. We also

observe that no mixing between these two types of transitions happens. In conclusion,

an asymmetric 2P-1P system provides large exchange tunability and helps address

individual spins without mixing.

Fig. 4.3. Modeling of individual qubit control in the coupled 2P-1P qubits.

The driving frequency of the ac B-field also depends on the nuclear spin configu-

ration as indicated in chapter 2 and also in reference [22]. Nuclear spin flipping could

introduce decoherence in manipulation. In addition, exchange energy dependence of

detuning gate bias depends on the detailed configurations of qubit 1 and 2, as well

as the inter-qubit distance. To optimize two-qubit gate operations, pulse sequence

of gate bias and ESR signals need to be designed with care. These aspects and a

full-fledged design of such two-qubit devices need to be further studied in details in

the future.

41

5. SUMMARY AND OUTLOOK

Phosphorus donors in silicon are promising candidates for a solid-state quantum com-

puter unit. Homogeneous performance among P-donor qubits are expected because

they provide natural Coulomb confinement. Moreover, as spin qubits, the coherence

time is long due to the weak spin-orbit coupling in silicon and a nuclear spin free

environment in purified silicon. Taking advantage of recent advances STM lithog-

raphy technique, P donors can be placed deep into the silicon lattice with atomic

precision. This guards qubit from interface and surface noise and makes fabrication

highly consistent with design.

This work focuses on manipulating and characterizing spin qubits based on P

donors in silicon with electromagnetic fields. Manipulation and readout of electron

and nuclear spins in single P donors and P donor-clusters with ESR and NMR tech-

niques are theoretically described in details. A P donor-cluster can act as an excellent

qubit which has a long coherence time and provides additional addressability, but

there are configuration uncertainties in terms of the number of donors, the number

of electrons and donor locations, which are difficult to detect directly in experiments.

Using ESR technique, a characterization technique is proposed to distinguish different

donor-clusters in silicon, which is useful for silicon qubit design and operation. More-

over, since the next milestone of donor-based quantum computing is to experimen-

tally demonstrate two-qubit logical gates, a STM-based device structure is proposed.

Two-qubit SWAP gate operation based on exchange interaction, and individual qubit

manipulation with ESR signals are demonstrated. Therefore, this work is useful for

understanding the cutting edge experiments and paving the way for designing future

two-qubit devices.

In the future, to finally realize two-qubit or general multi-qubit operations, several

detailed and important issues need to be tackled and I would like to share some my

42

insights here. First, to optimize the performance of donor-based two-qubit devices,

how to control exchange energy and gain enough tunability needs to be studied in

an engineering perspective [48]. Second, to realize different two-qubit gate operations

with high control fidelity, the control pulse sequence needs to be carefully designed

and optimized [50]. And the decoherence mechanism from the donor nuclei and the

environment needs to be studied in more details to eventually reach the fault-tolerant

requirement. Last but not the least, the current tunneling-based spin readout could

introduce decoherence sources, e.g. the single-electron transistor nearby. Optical

readout through donor bound exciton could be non-destructive and a better choice

[49]. In a word, we need to overcome obstacles before realizing donor-based quantum

computing, but we believe it is not a daydream anymore and will come true with our

continuous effort.

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43

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Licensed content publication Nature Nanotechnology

Licensed content title Spectroscopy of few­electron single­crystal silicon quantum dots

Licensed content author Martin Fuechsle, S. Mahapatra, F. A. Zwanenburg, Mark Friesen, M.A. Eriksson et al.

Licensed content date May 23, 2010

Volume number 5

Issue number 7

Type of Use reuse in a dissertation / thesis

Requestor type academic/educational

Format print and electronic

Portion figures/tables/illustrations

Number offigures/tables/illustrations

1

High­res required no

Figures Fig. 1.1

Author of this NPG article no

Your reference number None

Title of your thesis /dissertation

Manipulating and characterizing spin qubits based on donors in siliconwith electromagnetic field

Expected completion date Jul 2015

Estimated size (number ofpages)

44

Total 0.00 USD

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Terms and Conditions for Permissions

Nature Publishing Group hereby grants you a non­exclusive license to reproduce thismaterial for this purpose, and for no other use,subject to the conditions below:

1.  NPG warrants that it has, to the best of its knowledge, the rights to license reuse of thismaterial. However, you should ensure that the material you are requesting is original toNature Publishing Group and does not carry the copyright of another entity (as credited inthe published version). If the credit line on any part of the material you have requestedindicates that it was reprinted or adapted by NPG with permission from another source, thenyou should also seek permission from that source to reuse the material. 

2.  Permission granted free of charge for material in print is also usually granted for anyelectronic version of that work, provided that the material is incidental to the work as awhole and that the electronic version is essentially equivalent to, or substitutes for, the printversion.Where print permission has been granted for a fee, separate permission must beobtained for any additional, electronic re­use (unless, as in the case of a full paper, this hasalready been accounted for during your initial request in the calculation of a print run).NB: Inall cases, web­based use of full­text articles must be authorized separately through the 'Useon a Web Site' option when requesting permission. 

3.  Permission granted for a first edition does not apply to second and subsequent editions andfor editions in other languages (except for signatories to the STM Permissions Guidelines, orwhere the first edition permission was granted for free). 

4.  Nature Publishing Group's permission must be acknowledged next to the figure, table orabstract in print. In electronic form, this acknowledgement must be visible at the same timeas the figure/table/abstract, and must be hyperlinked to the journal's homepage.

5.  The credit line should read:Reprinted by permission from Macmillan Publishers Ltd: [JOURNAL NAME] (referencecitation), copyright (year of publication)For AOP papers, the credit line should read:Reprinted by permission from Macmillan Publishers Ltd: [JOURNAL NAME], advance onlinepublication, day month year (doi: 10.1038/sj.[JOURNAL ACRONYM].XXXXX)

Note: For republication from the British Journal of Cancer, the following creditlines apply.Reprinted by permission from Macmillan Publishers Ltd on behalf of Cancer Research UK:[JOURNAL NAME] (reference citation), copyright (year of publication)For AOP papers, thecredit line should read:Reprinted by permission from Macmillan Publishers Ltd on behalf of Cancer Research UK:[JOURNAL NAME], advance online publication, day month year (doi: 10.1038/sj.[JOURNALACRONYM].XXXXX) 

 6.  Adaptations of single figures do not require NPG approval. However, the adaptation should be

credited as follows:

Adapted by permission from Macmillan Publishers Ltd: [JOURNAL NAME] (reference citation),copyright (year of publication)

Note: For adaptation from the British Journal of Cancer, the following credit lineapplies.Adapted by permission from Macmillan Publishers Ltd on behalf of Cancer Research UK:[JOURNAL NAME] (reference citation), copyright (year of publication) 

7.  Translations of 401 words up to a whole article require NPG approval. Please visithttp://www.macmillanmedicalcommunications.com for more information.Translations of upto a 400 words do not require NPG approval. The translation should be credited as follows:

Translated by permission from Macmillan Publishers Ltd: [JOURNAL NAME] (referencecitation), copyright (year of publication).

Note: For translation from the British Journal of Cancer, the following credit lineapplies.Translated by permission from Macmillan Publishers Ltd on behalf of Cancer Research UK:[JOURNAL NAME] (reference citation), copyright (year of publication)

We are certain that all parties will benefit from this agreement and wish you the best in theuse of this material. Thank you.

Special Terms:

v1.1Questions? [email protected] or +1­855­239­3415 (toll free in the US) or+1­978­646­2777.

NATURE PUBLISHING GROUP LICENSETERMS AND CONDITIONS

Jun 23, 2015

This is a License Agreement between Yu Wang ("You") and Nature Publishing Group("Nature Publishing Group") provided by Copyright Clearance Center ("CCC"). The licenseconsists of your order details, the terms and conditions provided by Nature PublishingGroup, and the payment terms and conditions.

All payments must be made in full to CCC. For payment instructions, please seeinformation listed at the bottom of this form.

License Number 3654881143146

License date Jun 23, 2015

Licensed content publisher Nature Publishing Group

Licensed content publication Nature Nanotechnology

Licensed content title A single­atom transistor

Licensed content author Martin Fuechsle, Jill A. Miwa, Suddhasatta Mahapatra, Hoon Ryu,Sunhee Lee et al.

Licensed content date Feb 19, 2012

Volume number 7

Issue number 4

Type of Use reuse in a dissertation / thesis

Requestor type academic/educational

Format print and electronic

Portion figures/tables/illustrations

Number offigures/tables/illustrations

1

High­res required no

Figures Fig. 1.2

Author of this NPG article no

Your reference number None

Title of your thesis /dissertation

Manipulating and characterizing spin qubits based on donors in siliconwith electromagnetic field

Expected completion date Jul 2015

Estimated size (number ofpages)

44

Total 0.00 USD

Terms and Conditions

Terms and Conditions for Permissions

Nature Publishing Group hereby grants you a non­exclusive license to reproduce thismaterial for this purpose, and for no other use,subject to the conditions below:

1.  NPG warrants that it has, to the best of its knowledge, the rights to license reuse of thismaterial. However, you should ensure that the material you are requesting is original toNature Publishing Group and does not carry the copyright of another entity (as credited inthe published version). If the credit line on any part of the material you have requestedindicates that it was reprinted or adapted by NPG with permission from another source, thenyou should also seek permission from that source to reuse the material. 

2.  Permission granted free of charge for material in print is also usually granted for anyelectronic version of that work, provided that the material is incidental to the work as awhole and that the electronic version is essentially equivalent to, or substitutes for, the printversion.Where print permission has been granted for a fee, separate permission must beobtained for any additional, electronic re­use (unless, as in the case of a full paper, this hasalready been accounted for during your initial request in the calculation of a print run).NB: Inall cases, web­based use of full­text articles must be authorized separately through the 'Useon a Web Site' option when requesting permission. 

3.  Permission granted for a first edition does not apply to second and subsequent editions andfor editions in other languages (except for signatories to the STM Permissions Guidelines, orwhere the first edition permission was granted for free). 

4.  Nature Publishing Group's permission must be acknowledged next to the figure, table orabstract in print. In electronic form, this acknowledgement must be visible at the same timeas the figure/table/abstract, and must be hyperlinked to the journal's homepage.

5.  The credit line should read:Reprinted by permission from Macmillan Publishers Ltd: [JOURNAL NAME] (referencecitation), copyright (year of publication)For AOP papers, the credit line should read:Reprinted by permission from Macmillan Publishers Ltd: [JOURNAL NAME], advance onlinepublication, day month year (doi: 10.1038/sj.[JOURNAL ACRONYM].XXXXX)

Note: For republication from the British Journal of Cancer, the following creditlines apply.Reprinted by permission from Macmillan Publishers Ltd on behalf of Cancer Research UK:[JOURNAL NAME] (reference citation), copyright (year of publication)For AOP papers, thecredit line should read:Reprinted by permission from Macmillan Publishers Ltd on behalf of Cancer Research UK:[JOURNAL NAME], advance online publication, day month year (doi: 10.1038/sj.[JOURNALACRONYM].XXXXX) 

 6.  Adaptations of single figures do not require NPG approval. However, the adaptation should be

credited as follows:

Adapted by permission from Macmillan Publishers Ltd: [JOURNAL NAME] (reference citation),copyright (year of publication)

Note: For adaptation from the British Journal of Cancer, the following credit lineapplies.Adapted by permission from Macmillan Publishers Ltd on behalf of Cancer Research UK:[JOURNAL NAME] (reference citation), copyright (year of publication) 

7.  Translations of 401 words up to a whole article require NPG approval. Please visithttp://www.macmillanmedicalcommunications.com for more information.Translations of upto a 400 words do not require NPG approval. The translation should be credited as follows:

Translated by permission from Macmillan Publishers Ltd: [JOURNAL NAME] (referencecitation), copyright (year of publication).

Note: For translation from the British Journal of Cancer, the following credit lineapplies.Translated by permission from Macmillan Publishers Ltd on behalf of Cancer Research UK:[JOURNAL NAME] (reference citation), copyright (year of publication)

We are certain that all parties will benefit from this agreement and wish you the best in theuse of this material. Thank you.

Special Terms:

v1.1Questions? [email protected] or +1­855­239­3415 (toll free in the US) or+1­978­646­2777.

NATURE PUBLISHING GROUP LICENSETERMS AND CONDITIONS

Jul 07, 2015

This is a License Agreement between Yu Wang ("You") and Nature Publishing Group("Nature Publishing Group") provided by Copyright Clearance Center ("CCC"). The licenseconsists of your order details, the terms and conditions provided by Nature PublishingGroup, and the payment terms and conditions.

All payments must be made in full to CCC. For payment instructions, please seeinformation listed at the bottom of this form.

License Number 3663791457465

License date Jul 07, 2015

Licensed content publisher Nature Publishing Group

Licensed content publication Nature

Licensed content title High­fidelity readout and control of a nuclear spin qubit in silicon

Licensed content author Jarryd J. Pla, Kuan Y. Tan, Juan P. Dehollain, Wee H. Lim, John J. L.Morton et al.

Licensed content date Apr 17, 2013

Volume number 496

Issue number 7445

Type of Use reuse in a dissertation / thesis

Requestor type academic/educational

Format print and electronic

Portion figures/tables/illustrations

Number offigures/tables/illustrations

1

High­res required no

Figures Fig. 1.3

Author of this NPG article no

Your reference number None

Title of your thesis /dissertation

Manipulating and characterizing spin qubits based on donors in siliconwith electromagnetic field

Expected completion date Jul 2015

Estimated size (number ofpages)

44

Total 0.00 USD

Terms and Conditions

Terms and Conditions for Permissions

Nature Publishing Group hereby grants you a non­exclusive license to reproduce thismaterial for this purpose, and for no other use,subject to the conditions below:

1.  NPG warrants that it has, to the best of its knowledge, the rights to license reuse of thismaterial. However, you should ensure that the material you are requesting is original toNature Publishing Group and does not carry the copyright of another entity (as credited inthe published version). If the credit line on any part of the material you have requestedindicates that it was reprinted or adapted by NPG with permission from another source, thenyou should also seek permission from that source to reuse the material. 

2.  Permission granted free of charge for material in print is also usually granted for anyelectronic version of that work, provided that the material is incidental to the work as awhole and that the electronic version is essentially equivalent to, or substitutes for, the printversion.Where print permission has been granted for a fee, separate permission must beobtained for any additional, electronic re­use (unless, as in the case of a full paper, this hasalready been accounted for during your initial request in the calculation of a print run).NB: Inall cases, web­based use of full­text articles must be authorized separately through the 'Useon a Web Site' option when requesting permission. 

3.  Permission granted for a first edition does not apply to second and subsequent editions andfor editions in other languages (except for signatories to the STM Permissions Guidelines, orwhere the first edition permission was granted for free). 

4.  Nature Publishing Group's permission must be acknowledged next to the figure, table orabstract in print. In electronic form, this acknowledgement must be visible at the same timeas the figure/table/abstract, and must be hyperlinked to the journal's homepage.

5.  The credit line should read:Reprinted by permission from Macmillan Publishers Ltd: [JOURNAL NAME] (referencecitation), copyright (year of publication)For AOP papers, the credit line should read:Reprinted by permission from Macmillan Publishers Ltd: [JOURNAL NAME], advance onlinepublication, day month year (doi: 10.1038/sj.[JOURNAL ACRONYM].XXXXX)

Note: For republication from the British Journal of Cancer, the following creditlines apply.Reprinted by permission from Macmillan Publishers Ltd on behalf of Cancer Research UK:[JOURNAL NAME] (reference citation), copyright (year of publication)For AOP papers, thecredit line should read:Reprinted by permission from Macmillan Publishers Ltd on behalf of Cancer Research UK:[JOURNAL NAME], advance online publication, day month year (doi: 10.1038/sj.[JOURNALACRONYM].XXXXX) 

 6.  Adaptations of single figures do not require NPG approval. However, the adaptation should be

credited as follows:

Adapted by permission from Macmillan Publishers Ltd: [JOURNAL NAME] (reference citation),copyright (year of publication)

Note: For adaptation from the British Journal of Cancer, the following credit lineapplies.Adapted by permission from Macmillan Publishers Ltd on behalf of Cancer Research UK:[JOURNAL NAME] (reference citation), copyright (year of publication) 

7.  Translations of 401 words up to a whole article require NPG approval. Please visithttp://www.macmillanmedicalcommunications.com for more information.Translations of upto a 400 words do not require NPG approval. The translation should be credited as follows:

Translated by permission from Macmillan Publishers Ltd: [JOURNAL NAME] (referencecitation), copyright (year of publication).

Note: For translation from the British Journal of Cancer, the following credit lineapplies.Translated by permission from Macmillan Publishers Ltd on behalf of Cancer Research UK:[JOURNAL NAME] (reference citation), copyright (year of publication)

We are certain that all parties will benefit from this agreement and wish you the best in theuse of this material. Thank you.

Special Terms:

v1.1Questions? [email protected] or +1­855­239­3415 (toll free in the US) or+1­978­646­2777.

NATURE PUBLISHING GROUP LICENSETERMS AND CONDITIONS

Jun 23, 2015

This is a License Agreement between Yu Wang ("You") and Nature Publishing Group("Nature Publishing Group") provided by Copyright Clearance Center ("CCC"). The licenseconsists of your order details, the terms and conditions provided by Nature PublishingGroup, and the payment terms and conditions.

All payments must be made in full to CCC. For payment instructions, please seeinformation listed at the bottom of this form.

License Number 3654890696216

License date Jun 23, 2015

Licensed content publisher Nature Publishing Group

Licensed content publication Nature Communications

Licensed content title Spin readout and addressability of phosphorus­donor clusters insilicon

Licensed content author H. Büch, S. Mahapatra, R. Rahman, A. Morello, M. Y. Simmons

Licensed content date Jun 18, 2013

Volume number 4

Type of Use reuse in a dissertation / thesis

Requestor type academic/educational

Format print and electronic

Portion figures/tables/illustrations

Number offigures/tables/illustrations

1

High­res required no

Figures Fig. 3.1

Author of this NPG article no

Your reference number None

Title of your thesis /dissertation

Manipulating and characterizing spin qubits based on donors in siliconwith electromagnetic field

Expected completion date Jul 2015

Estimated size (number ofpages)

44

Total 0.00 USD

Terms and Conditions

Terms and Conditions for Permissions

Nature Publishing Group hereby grants you a non­exclusive license to reproduce this

material for this purpose, and for no other use,subject to the conditions below:

1.  NPG warrants that it has, to the best of its knowledge, the rights to license reuse of thismaterial. However, you should ensure that the material you are requesting is original toNature Publishing Group and does not carry the copyright of another entity (as credited inthe published version). If the credit line on any part of the material you have requestedindicates that it was reprinted or adapted by NPG with permission from another source, thenyou should also seek permission from that source to reuse the material. 

2.  Permission granted free of charge for material in print is also usually granted for anyelectronic version of that work, provided that the material is incidental to the work as awhole and that the electronic version is essentially equivalent to, or substitutes for, the printversion.Where print permission has been granted for a fee, separate permission must beobtained for any additional, electronic re­use (unless, as in the case of a full paper, this hasalready been accounted for during your initial request in the calculation of a print run).NB: Inall cases, web­based use of full­text articles must be authorized separately through the 'Useon a Web Site' option when requesting permission. 

3.  Permission granted for a first edition does not apply to second and subsequent editions andfor editions in other languages (except for signatories to the STM Permissions Guidelines, orwhere the first edition permission was granted for free). 

4.  Nature Publishing Group's permission must be acknowledged next to the figure, table orabstract in print. In electronic form, this acknowledgement must be visible at the same timeas the figure/table/abstract, and must be hyperlinked to the journal's homepage.

5.  The credit line should read:Reprinted by permission from Macmillan Publishers Ltd: [JOURNAL NAME] (referencecitation), copyright (year of publication)For AOP papers, the credit line should read:Reprinted by permission from Macmillan Publishers Ltd: [JOURNAL NAME], advance onlinepublication, day month year (doi: 10.1038/sj.[JOURNAL ACRONYM].XXXXX)

Note: For republication from the British Journal of Cancer, the following creditlines apply.Reprinted by permission from Macmillan Publishers Ltd on behalf of Cancer Research UK:[JOURNAL NAME] (reference citation), copyright (year of publication)For AOP papers, thecredit line should read:Reprinted by permission from Macmillan Publishers Ltd on behalf of Cancer Research UK:[JOURNAL NAME], advance online publication, day month year (doi: 10.1038/sj.[JOURNALACRONYM].XXXXX) 

 6.  Adaptations of single figures do not require NPG approval. However, the adaptation should be

credited as follows:

Adapted by permission from Macmillan Publishers Ltd: [JOURNAL NAME] (reference citation),copyright (year of publication)

Note: For adaptation from the British Journal of Cancer, the following credit lineapplies.Adapted by permission from Macmillan Publishers Ltd on behalf of Cancer Research UK:[JOURNAL NAME] (reference citation), copyright (year of publication) 

7.  Translations of 401 words up to a whole article require NPG approval. Please visithttp://www.macmillanmedicalcommunications.com for more information.Translations of upto a 400 words do not require NPG approval. The translation should be credited as follows:

Translated by permission from Macmillan Publishers Ltd: [JOURNAL NAME] (reference

citation), copyright (year of publication).

Note: For translation from the British Journal of Cancer, the following credit lineapplies.Translated by permission from Macmillan Publishers Ltd on behalf of Cancer Research UK:[JOURNAL NAME] (reference citation), copyright (year of publication)

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Jun 23, 2015

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License date Jun 23, 2015

Licensed content publisher Nature Publishing Group

Licensed content publication Nature Nanotechnology

Licensed content title Spin blockade and exchange in Coulomb­confined silicon doublequantum dots

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Manipulating and characterizing spin qubits based on donors in siliconwith electromagnetic field

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Adapted by permission from Macmillan Publishers Ltd: [JOURNAL NAME] (reference citation),copyright (year of publication)

Note: For adaptation from the British Journal of Cancer, the following credit lineapplies.Adapted by permission from Macmillan Publishers Ltd on behalf of Cancer Research UK:[JOURNAL NAME] (reference citation), copyright (year of publication) 

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Translated by permission from Macmillan Publishers Ltd: [JOURNAL NAME] (referencecitation), copyright (year of publication).

Note: For translation from the British Journal of Cancer, the following credit lineapplies.Translated by permission from Macmillan Publishers Ltd on behalf of Cancer Research UK:[JOURNAL NAME] (reference citation), copyright (year of publication)

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Special Terms:

v1.1Questions? [email protected] or +1­855­239­3415 (toll free in the US) or+1­978­646­2777.