Download optical pumping of single donor-bound electrons in zinc

OPTICAL PUMPING OF SINGLE DONOR-BOUND ELECTRONS
IN ZINC SELENIDE AND SILICON
A DISSERTATION
SUBMITTED TO THE DEPARTMENT OF PHYSICS
AND THE COMMITTEE ON GRADUATE STUDIES
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
Darin Jay Sleiter
August 2012
© 2012 by Darin Jay Sleiter. All Rights Reserved.
Re-distributed by Stanford University under license with the author.
This work is licensed under a Creative Commons AttributionNoncommercial 3.0 United States License.
http://creativecommons.org/licenses/by-nc/3.0/us/
This dissertation is online at: http://purl.stanford.edu/rm270nf6941
ii
I certify that I have read this dissertation and that, in my opinion, it is fully adequate
in scope and quality as a dissertation for the degree of Doctor of Philosophy.
Yoshihisa Yamamoto, Primary Adviser
I certify that I have read this dissertation and that, in my opinion, it is fully adequate
in scope and quality as a dissertation for the degree of Doctor of Philosophy.
David Goldhaber-Gordon
I certify that I have read this dissertation and that, in my opinion, it is fully adequate
in scope and quality as a dissertation for the degree of Doctor of Philosophy.
Jelena Vuckovic
Approved for the Stanford University Committee on Graduate Studies.
Patricia J. Gumport, Vice Provost Graduate Education
This signature page was generated electronically upon submission of this dissertation in
electronic format. An original signed hard copy of the signature page is on file in
University Archives.
iii
iv
Abstract
The spin of single electrons bound to donors in semiconductor materials are promising
candidates for quantum bits implementations. These electrons have been shown to
be very homogenous and have extremely long decoherence times in a bulk semiconductor environment. In this work, I have studied two donor systems as quantum bit
candidates, with a focus on using optical pumping methods to initialize and measure
the electron and nuclear spins of the donor system.
The first donor system, fluorine in zinc selenide, is a very optically bright system which is a particularly good candidate for quantum repeater technologies. The
relatively large electron binding energy leads to a stable qubit at low temperatures,
and the potential for isotopic depletion of nuclear spin from the host semiconductor
crystal suggests very long decoherence times can be achieved. In this work, I confirm
the isolation of a single F-bound electron, and present results on the use of resonant
optical pumping to initialize the electron to a particular spin state. These results
open the door for optical control of the electron spin as a qubit.
The second donor system, phosphorus in silicon, is the semiconductor system with
the longest published decoherence times, obtained for the nuclear spin of the donor.
Due to the long excited-state lifetime of the donor optical transitions, the linewidth
of the transition is narrower than the hyperfine splitting, allowing optical access to
the donor nuclear spin. However, to date, single phosphorus donors have not been
optically isolated. In this work, I present a theoretical description of a hybrid optical
and electrical device for the measurement of a single phosphorus donor nuclear spin.
If experiments can confirm the properties of this device, this measurement technique
would provide a key element for a silicon-based quantum computer.
v
Acknowledgements
I have been extremely fortunate to have had a great deal of help and support from
numerous people throughout my time at Stanford, and this work certainly would not
have been possible without them.
I would first like to thank my advisor, Yoshihisa Yamamoto, for the opportunity he
gave me to work in his group on what I find to be such a fascinating area of research.
The freedom he has given me to work on a variety of topics, and to plan and execute
research on my own, has helped me develop my research skills, while his support
and insight into complex quantum behavior has kept the projects on a successful
path. I would also like to thank the other members of my reading committee: Jelena
Vuckovic and David Goldhaber-Gordon, both of whose research has been beneficial
to my understanding of the physics involved in the systems I’ve studied.
During the first few years of my time at Stanford, Thaddeus Ladd was a great
mentor to me, and I am very thankful for the opportunity to have worked with him.
His understanding of physics, ability to explain it, skills as an experimentalist, and
knowledge of how to conduct research are all skills that I greatly admire, and he really
helped me get a grip on the research process. I am also very grateful to a number of
other students and researchers with whom I’ve had the opportunity to work closely.
I’d like to thank Na Young Kim for her interest and excitement for the silicon project,
as well as for her organizational skills. I very much enjoyed working with her, and
hope she will keep the project going. I’d like to thank Susan Clark and Kaoru Sanaka,
who I worked closely with on the zinc selenide project. I had a great time working
many hours in the lab with Susan and am thankful for her contagious enthusiasm
for the project. I enjoyed working with Kaoru as well, whose determination for the
vi
project I admire.
I’d like to thank Kristiaan de Greve and Peter McMahon for their useful discussions and enjoyable distractions in lab, as well as Katsuya Nozawa and Tomoyuki
Horikiri for their help and guidance in getting the silicon project started. I would
also like to thank the other members of the Yamamoto group, present and past,
all of whom have contributed to my time here, either through direct research, or
through fruitful and enjoyable interactions: Leo Yu, Zhe Wang, Cody Jones, Wolfgang Nitsche, Shruti Puri, Kai Wen, David Press, Georgios Roumpos, Kai-Mei Fu,
Qiang Zhang, Shinichi Koseki, Chandra Natarajan, Shelan Tawfeeq, Jung-Jung Su,
and many others.
I am very thankful for the chance to work with Alex Pawlis, through our zinc
selenide collaboration. The samples he grew gave us the best opportunity for developing the system, and his understanding of the system and continued improvement
of the growth process were crucial. I’d like to thank Michael Thewalt for his deep
insight and understanding into the phosphorus donor system in silicon, and to Sven
Rogge for his devices and for collaborating with us on the preliminary silicon device
experiments.
Many thanks to the numerous people who helped keep the Yamamoto group,
Ginzton, and the physics department running smoothly. I am especially very grateful
to Yurika Peterman and Rieko Sasaki for all their help in running the Yamamoto
group, I always enjoyed the opportunity to visit their office and talk with them.
I am thankful to Maria Frank for running the physics office and handling all the
paperwork and requirements, and to the Ginzton office for approving all the liquid
helium orders and other requests necessary to keep the experiments running. I’m also
very grateful to Mike Schlimmer for keeping everything in the building running well
and for helping solve building issues, and to Larry Randall for all his help solving
computer and electronics issues.
I am so very thankful to all my friends, at Stanford and elsewhere, for keeping me
going through their continued support, distractions, and great times. I have been very
fortunate to meet such a fantastic group of friendly people at Stanford, particularly
the guys in the physics program, who have made my time here so enjoyable. While the
vii
number of people who have impacted me are more numerous than can reasonably be
listed here, I’d like to thank three guys in particular, Dan Walker, Phil Van Stockum,
and John Ulmen, who I had the opportunity to live with and who have been a part
of the vast majority of my best memories at Stanford.
Finally, I am extremely thankful to my family for all their support. To my parents,
Cathy and Jay, for their support and confidence in me, and for always teaching me
while growing up that I need to balance working hard with playing hard. My father’s
insistence that I never ‘baby the equipment’ has given me the confidence to push the
limits, and my mother’s reason and kindness has kept me (moderately) levelheaded.
To my siblings, Bryan and Kristi, and Lauren and Kevin, thank you for keeping me
down to earth and for all the fun times we had together when we’re able to get away
from work. And last, but not least, to my best friend and girlfriend, Jackie, thank
you for giving me a reason and a purpose to finish grad school and I can’t wait to
start on our next adventure together.
viii
Contents
Abstract
v
Acknowledgements
vi
1 Introduction
1
1.1
Quantum bits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
Qubit requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3
Quantum key distribution . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3.1
Quantum repeaters . . . . . . . . . . . . . . . . . . . . . . . .
6
1.4
Qubit candidates . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.5
Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2 Neutral donor system
13
2.1
Effective mass theory . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.2
Donor-bound exciton state . . . . . . . . . . . . . . . . . . . . . . . .
17
2.3
Optical transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.4
Nuclear coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.5
Spin qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
3 Fluorine donors in ZnSe
27
3.1
Single donor isolation . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
3.2
Optical spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
3.2.1
Magnetophotoluminescence . . . . . . . . . . . . . . . . . . .
34
3.2.2
Single photon source . . . . . . . . . . . . . . . . . . . . . . .
35
ix
3.3
Single donor confirmation . . . . . . . . . . . . . . . . . . . . . . . .
39
3.4
Optical pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
3.4.1
Time averaged optical pumping . . . . . . . . . . . . . . . . .
44
3.4.2
Time resolved optical pumping . . . . . . . . . . . . . . . . .
48
F:ZnSe outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
3.5
4 Phosphorus donors in Si
55
4.1
P:Si optical system . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
4.2
Photoluminescence excitation spectroscopy . . . . . . . . . . . . . . .
59
4.3
Zero-field splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
4.3.1
Strain Model . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
4.3.2
Fitting Results . . . . . . . . . . . . . . . . . . . . . . . . . .
63
Electrical detection . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
4.4.1
Quantum Hall charge sensor . . . . . . . . . . . . . . . . . . .
67
4.4.2
Description of device and measurement scheme
. . . . . . . .
68
Device physics and simulation . . . . . . . . . . . . . . . . . . . . . .
71
4.5.1
Donor electron ground state . . . . . . . . . . . . . . . . . . .
72
4.5.2
Edge Channel Scattering . . . . . . . . . . . . . . . . . . . . .
78
4.5.3
Ionization and recapture . . . . . . . . . . . . . . . . . . . . .
82
4.5.4
Optical transition . . . . . . . . . . . . . . . . . . . . . . . . .
84
4.5.5
Photoconductivity . . . . . . . . . . . . . . . . . . . . . . . .
85
4.5.6
Device prospects . . . . . . . . . . . . . . . . . . . . . . . . .
85
Preliminary experiments with FinFET device . . . . . . . . . . . . .
86
4.6.1
Device structure and behavior . . . . . . . . . . . . . . . . . .
86
P:Si outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
4.4
4.5
4.6
4.7
5 Conclusion and Outlook
91
A Selection rules
94
A.1 F:ZnSe selection rules . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
A.2 P:Si selection rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
x
B Experimental setup
106
B.1 F:ZnSe experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
B.2 P:Si experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Bibliography
111
xi
List of Tables
1.1
Comparison between three qubit candidates . . . . . . . . . . . . . .
11
4.1
Lifetimes of decay mechanisms in P:Si . . . . . . . . . . . . . . . . .
57
4.2
Fitting parameters for the strain model . . . . . . . . . . . . . . . . .
63
4.3
Energy fitting parameters . . . . . . . . . . . . . . . . . . . . . . . .
66
xii
List of Figures
1.1
Representation of possible cbit and qubit states . . . . . . . . . . . .
3
1.2
Diagram of the BB84 QKD protocol . . . . . . . . . . . . . . . . . .
5
1.3
Function of a quantum repeater . . . . . . . . . . . . . . . . . . . . .
7
2.1
Substitutional donor . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.2
Neutral donor system . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.3
Lambda system used for optical control . . . . . . . . . . . . . . . . .
26
3.1
F:ZnSe sample structure . . . . . . . . . . . . . . . . . . . . . . . . .
29
3.2
Example bulk F:ZnSe spectra . . . . . . . . . . . . . . . . . . . . . .
31
3.3
Example 4 nm quantum well F:ZnSe spectra . . . . . . . . . . . . . .
32
3.4
Example 2 nm quantum well F:ZnSe spectra . . . . . . . . . . . . . .
33
0
3.5
D X transition at 7 T . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
3.6
Hanbury-Brown-Twiss experiment . . . . . . . . . . . . . . . . . . . .
36
3.7
Example g 2 (τ ) data . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
3.8
Hong-Ou-Mandel experiment . . . . . . . . . . . . . . . . . . . . . . .
38
3.9
Spectra of a confirmed single donor . . . . . . . . . . . . . . . . . . .
39
3.10 Power saturation of a confirmed single donor . . . . . . . . . . . . . .
40
3.11 Zeeman splitting of a confirmed single donor . . . . . . . . . . . . . .
41
3.12 g 2 (τ ) of a confirmed single donor . . . . . . . . . . . . . . . . . . . .
42
3.13 Diagram of the optical pumping scheme
. . . . . . . . . . . . . . . .
44
3.14 Resonant and power dependent optical pumping behavior . . . . . . .
47
3.15 Time-dependent optical pumping behavior . . . . . . . . . . . . . . .
48
3.16 Representation of the Monte-Carlo simulation . . . . . . . . . . . . .
49
xiii
4.1
P:Si optical system . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
4.2
Zero-field splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
4.3
Magnetic field dependent transition energies . . . . . . . . . . . . . .
65
4.4
Measurement device schematic . . . . . . . . . . . . . . . . . . . . . .
68
4.5
Quantum Hall effect within the measurement device . . . . . . . . . .
69
4.6
Eigenstate energies . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
4.7
Potential energy countour plot . . . . . . . . . . . . . . . . . . . . . .
76
4.8
Edge state scattering . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
4.9
FinFET device schematic . . . . . . . . . . . . . . . . . . . . . . . . .
87
4.10 Temperature dependent I-V curves . . . . . . . . . . . . . . . . . . .
88
4.11 FinFET switching behavior . . . . . . . . . . . . . . . . . . . . . . .
89
A.1 Hole wavefunction in a quantum well . . . . . . . . . . . . . . . . . .
95
A.2 F:ZnSe Zeeman splitting . . . . . . . . . . . . . . . . . . . . . . . . .
96
A.3 P:Si Zeeman splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
A.4 High-field P:Si spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 104
B.1 F:ZnSe experimental setup . . . . . . . . . . . . . . . . . . . . . . . . 107
B.2 P:Si experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . 109
xiv
Chapter 1
Introduction
Quantum information is the relatively new field which uses the mathematics of quantum mechanics to perform mathematical operations that are not available when using
the mathematics of classical mechanics. As the computers used today (classical computers) use physical classical states to represent mathematical numbers and classical
interactions to perform mathematical operations, quantum computers use physical
quantum states to represent mathematical numbers and quantum interactions to perform mathematical operations. While large-scale quantum computers do not yet exist,
researchers have discovered algorithms for quantum computers that can solve certain
problems much faster than is possible on a classical computer [1, 2, 3, 4].
In addition to enabling quantum computers, quantum mechanics can be used to
distribute cryptographic keys securely (quantum key distribution) [5], and to simulate the behavior of complex quantum interactions in one system using simpler and
well-understood quantum interactions in another system (quantum simulator) [1].
The potential benefits of quantum simulators, quantum key distribution, and quantum computers currently drive a large amount of international research into various
types of quantum systems to see which system can most easily be used as the basic
representation of quantum information.
1
2
CHAPTER 1. INTRODUCTION
1.1
Quantum bits
The simplest representation of quantum information is a two-state quantum system,
where one state represents the value 0 and the other state represents the value 1. In
analogy with the classical bit (cbit), such systems are called quantum bits, or qubits.
The main difference between classical bits and quantum bits is that while a classical
bit can only be in state 0 or state 1, the quantum bit can be in state |0i, state |1i, or
a superposition of both states α |0i + β |1i.
This superposition state can be thought of as what makes quantum information so
powerful. In particular, it enables “quantum parallelism”, where a single computation
can be performed on a superposition of all possible discrete inputs at the same time.
In classical computation, only one input can be sent through the processor at a time.
However, obtaining the result of a particular computation is a bit more complex in
a quantum computer. A measurement of the final state of the qubits generally gives
the result of the computation of one input value rather than the result of all the
inputs. Fortunately, due to the wave nature of quantum mechanics, the inputs can
be superimposed in a way that the final states constructively interfere to produce
the desired answer. This is how some of the known quantum algorithms are able to
compute the answer.
The cbit is often described by a two-state switch, either off or on (Fig. 1.1(a)).
The qubit, on the other hand, has an infinite number of possible superposition states.
Its SU(2) symmetry implies that any qubit state can be described as
|ψi = cos (θ/2) + eiφ sin (θ/2) ,
(1.1)
where θ and φ are the two degrees of freedom available to the state. This equation
describes the surface of a sphere, which leads to the description of a qubit state by
a position on the so-called Bloch Sphere. As shown in Fig. 1.1(b), the north pole
represents the state |1i, the south pole represents the state |0i, and any other location
on the surface of the sphere represents a superposition state.
The qubit is not the only possible basic building block of a quantum computer.
Other systems containing three or more states have been proposed, such as the qutrit
1.2. QUBIT REQUIREMENTS
3
Figure 1.1: Representation of the possible states for (a) cbits, and (b) qubits.
(three-level system) or qudit (n-level system). These other quantum information
building blocks are essentially just as useful as qubits, but are more complex to work
with. This thesis will focus only on qubit implementations.
1.2
Qubit requirements
The first description of the necessary requirements in order for a qubit to be a practical basis for a quantum computer was suggested by David DiVincenzo [6]. These
requirements are:
1) a scalable physical system with well characterized qubits,
2) the ability to initialize the state of the qubits,
3) decoherence times much longer than the gate operation time,
4) a universal set of quantum gates, and
5) a qubit-specific measurement capability.
These are often supplemented by a variety of more specific criterion for particular
qubit implementations. However, these criteria are a good starting place, and have
yet to be fully met by any qubit candidate system. There are many qubit candidate
systems currently being studied, from single photon states [7] to individual ions [8] to
superconducting currents [9]. Each candidate fits differently with this set of criteria,
and there is no clear best candidate at the moment. For instance, the first successful
4
CHAPTER 1. INTRODUCTION
quantum algorithm was performed using nuclear magnetic resonance (NMR) on an
ensemble of molecules where the nuclear spins within each molecule served as the
quantum bits [10]. In this system, criteria (2)-(5) were met, but the lack of scalability
(more bits would require increasingly large molecules, each with different resonances)
makes it an unreasonable candidate system. Other systems, such as semiconductor
systems, have found scalability to be easy, but two-qubit gates and decoherence time
have been difficult to achieve. However, each of these criteria are required in order to
have a large-scale quantum computer.
1.3
Quantum key distribution
In addition to the benefits of quantum computation, quantum mechanics also provides a method to create shared information between two separated locations through
a quantum channel while making it impossible (according to the laws of quantum
mechanics) for a third party to intercept this information undetected. This is called
quantum key distribution (QKD), and generally places less stringent requirements on
qubit systems than required for quantum computation.
In its simplest implementation, QKD requires a well defined qubit that can be
initialized and measured and can be easily transported between two locations, called
a flying qubit. QKD systems already exist and are even sold commercially. These
systems use photon states for the qubit basis, and transport the qubits through optical
fibers between two locations. One particular implementation I will describe here uses
the polarization of single photons as the qubit states. This protocol is called BB84 [5].
The BB84 system consists of an optical fiber between two parties, who we can call
Alice and Bob. Alice encodes a random string of classical bits into the polarization
of single photons that she creates and sends to Bob through the fiber. For each
photon qubit, she randomly chooses the basis she uses to encode the classical bit,
selecting either rectilinear (horizontal |Hi & vertical |V i), or diagonal (diagonal |Di
& antidiagonal |Ai). If she chooses rectilinear, then she encodes the classical bit
according to |Hi = 0, |V i = 1. If she chooses diagonal, she encodes the bit according
to |Di = 0, |Ai = 1 (see Fig. 1.2).
1.3. QUANTUM KEY DISTRIBUTION
5
Figure 1.2: Diagram representing the BB84 QKD protocol. Single photons are sent
down an optical fiber from Alice to Bob. Alice encodes each bit in one of the two
polarization bases, randomly selected. Bob detects the photons he receives in one of
the polarization bases, again randomly selected.
Next, Bob detects each photon he receives using a random basis. Following this,
Alice and Bob compare the creation and measurement basis for each photon over an
open classical channel. If Bob detects a photon in the same basis in which it was
created (50% of the time on average), he measures the same bit as was encoded by
Alice. However, if he chooses a different basis from which it was created, he measures
a random bit value compared to what was encoded. Alice and Bob keep the bit values
of the photons which were created and measured in the same basis as their new shared
cryptographic key, and discard the other bits.
Since Alice and Bob only share the creation and measurement basis over the
open classical channel, they transmit no information about the actual shared key bit
values that an eavesdropper, Eve, could use to learn anything about the key. On the
other hand, if Eve listens in on the quantum channel (the optical fiber), she will be
unable to gain any information without modifying the quantum states of the photon
qubits. Due to the quantum no-cloning theorem, she is unable to make copies of
the single photon states, and her only options are to attempt to detect the photon
polarization without destroying the photon, or detect the photon and then send a
new photon with the same state as she detected. Eve does not know the basis in
which the photons were created, and so 50% of the time she measures in the wrong
6
CHAPTER 1. INTRODUCTION
basis, gaining no information. Furthermore, each time she measures in the wrong
bases, she unavoidably modifies the quantum state sent to Bob.
Any modification of the quantum states between Alice and Bob can easily be
detected by comparing some of the shared bits over the open classical channel. On
average, 25% of the bits Eve attempts to detect will introduce an error in Bob’s bits.
Thus, if Alice and Bob determine they have an unusually large error rate, they know
someone must be attempting to listen. However, if the error rate is normal, they know
that they have a completely secure shared bit string which they can use to encrypt
information over a classical channel.
The quantum key distribution networks that exist today all rely on the transmission of photons in optical fibers. While the systems work, they all share one limitation:
lossy fibers. Classical optical networks get around this using repeaters, where an optical signal is periodically amplified in order to overcome any loss. Unfortunately,
since quantum mechanics forbids the copying of quantum states, we cannot amplify
a quantum signal. For this reason, current QKD systems are limited to 100-200 km
transmission distances for reasonable bitrates due to the loss of the single-photon
flying qubits. Longer transmission distances will require the development of a new
type of repeater.
1.3.1
Quantum repeaters
Quantum repeaters are different from classical repeaters in that they do not amplify
a signal. Instead, they serve to build up entangled resources over long distances,
which can then be used to transfer qubit states from one location to another through
quantum teleportation. The qubit requirements for quantum repeaters are greater
than that required for general QKD, and in fact requires all of the elements listed in
Sec. 1.2 for general quantum computers. However, the scale of the system required for
quantum repeaters is usually considered to be much less than that of a full quantum
computer, and so is a good intermediate technological goal, somewhere in between
the simple QKD protocols and a full quantum computer system.
1.3. QUANTUM KEY DISTRIBUTION
7
Figure 1.3: Schematic of how a quantum repeater would work. (a) Alice, Bob, and
the Repeater all send single photons entangled with their stationary qubits through
optical fibers. (b) These states can be re-written as a sum of Bell states between
the stationary qubits and between the flying qubits. After photon detection, the
stationary qubits are projected into one of the Bell states. (c) The quantum repeater
performs entanglement swapping by performing a Bell-state measurement of its two
stationary qubits, projecting Alice and Bob’s qubits into an entangled Bell state.
Quantum repeaters require two types of qubits: flying qubits (such as single photons, as used in QKD), and stationary qubits. The flying qubits are used to transfer
quantum information between repeaters, while the stationary qubits are used to store
the quantum information. The protocol works as follows (shown in Fig. 1.3).
Alice and Bob both have a stationary qubit which they entangle to a flying qubit,
which we will assume is a single photon. Thus, Alice and Bob start with qubit states
1
|ΨA i = √ (|0i1 |Hi1 − |1i1 |V i1 ) ,
2
1
|ΨB i = √ (|0i4 |Hi4 − |1i4 |V i4 ) ,
2
(1.2)
(1.3)
8
CHAPTER 1. INTRODUCTION
where |0i and |1i refer to the stationary qubit, and |Hi and |V i refer to the flying
qubit. A quantum repeater between Alice and Bob has two pairs of stationary and
flying qubits
1
|ΨR1 i = √ (|0i2 |Hi2 − |1i2 |V i2 ) ,
2
1
|ΨR2 i = √ (|0i3 |Hi3 − |1i3 |V i3 ) .
2
(1.4)
(1.5)
Alice and Bob’s flying qubits fly towards the repeater, while the repeater’s flying
qubits fly towards Alice and Bob (Fig. 1.3(a)). At this point, the states can be
rewritten as
1
(|0i1 |Hi1 − |1i1 |V i1 ) (|0i2 |Hi2 − |1i2 |V i2 ) ,
2
1
=
[(|0i1 |0i2 + |1i1 |1i2 ) (|Hi1 |Hi2 + |V i1 |V i2 )
4
+ (|0i1 |0i2 − |1i1 |1i2 ) (|Hi1 |Hi2 − |V i1 |V i2 )
|ΨA i |ΨR1 i =
(1.6)
− (|0i1 |1i2 + |1i1 |0i2 ) (|Hi1 |V i2 + |V i1 |Hi2 )
− (|0i1 |1i2 − |1i1 |0i2 ) (|Hi1 |V i2 − |V i1 |Hi2 )] ,
1 + + =
Φ 12 φ 12 + Φ− 12 φ− 12
2
− Ψ+ 12 ψ + 12 − Ψ− 12 ψ − 12 ,
(1.7)
(1.8)
where the capital Ψ and Φ refer to the Bell states for the stationary qubits, and the
lower-case ψ and φ refer to the Bell states for the flying qubits. The qubits states
between the repeater and Bob can be described in the same way,
|ΨB i |ΨR2 i =
1 + + Φ 34 φ 34 + Φ− 34 φ− 34
2
− Ψ+ 34 ψ + 34 − Ψ− 34 ψ − 34 .
(1.9)
The photons from Alice and the repeater meet in the middle of the optical fiber on
a beamsplitter, and will travel along paths 1’ and 2’ towards single-photon counters
1.3. QUANTUM KEY DISTRIBUTION
9
(Fig. 1.3(b)). After the beamsplitter, the four flying-qubit Bell states transform into
+
φ
12
−
φ
12
+
ψ
12
−
ψ
12
1
([|HHi10 + |V V i10 ] |0i20 − |0i10 [|HHi20 + |V V i20 ])
2
1
→
([|HHi10 − |V V i10 ] |0i20 − |0i10 [|HHi20 − |V V i20 ])
2
1
→ √ (|HV i10 |0i20 − |0i10 |HV i20 )
2
1
→ √ (|V i10 |Hi20 − |Hi10 |V i20 ) ,
2
→
(1.10)
(1.11)
(1.12)
(1.13)
where |P1 P2 i refers to two photons in a particular mode with the specified polarizations P1 and P2 (H or V ) and |0i refers to zero photons in a particular mode. In
principle, all four of these states are distinguishable. In practice, using only two polarization and number insensitive detectors, as shown in Fig. 1.3, only the |ψ − i state
can be distinguished (|ψ + i can also be distinguished if four detectors are used). |ψ − i
is the only state that has a photon on each path, so a coincidence of detection events
in both detectors is a measurement of the photon state |ψ − i. This then projects the
stationary qubits into the entangled state |Ψ− i.
A beam splitter and pair of detectors between Bob and the repeater perform the
same entanglement procedure on their side. This procedure can be repeated on each
side until a coincidence detection event indicates a successful entanglement. Once
both Alice and Bob have their stationary qubit entangled with a repeater qubit, the
entanglement swapping phase begins. At the start of this phase, the four stationary
10
CHAPTER 1. INTRODUCTION
qubit states can be written as
− −
1
Ψ
Ψ
=
(|0i1 |1i2 − |1i1 |0i2 ) (|0i3 |1i4 − |1i3 |0i4 ) ,
34
12
2
1
=
[− (|0i1 |0i4 + |1i1 |1i4 ) (|Hi2 |Hi3 + |V i2 |V i3 )
4
+ (|0i1 |0i4 − |1i1 |1i4 ) (|Hi2 |Hi3 − |V i2 |V i3 )
(1.14)
− (|0i1 |1i4 + |1i1 |0i4 ) (|Hi2 |V i3 + |V i2 |Hi3 )
− (|0i1 |1i4 − |1i1 |0i4 ) (|Hi2 |V i3 − |V i2 |Hi3 )] , (1.15)
1
=
− Φ+ 14 Φ+ 23 + Φ− 14 Φ− 23
2
(1.16)
+ Ψ+ 14 Ψ+ 23 + Ψ− 14 Ψ− 23 .
The quantum repeater next measures its two stationary qubits (2 & 3) in the Bell
state basis. This can be accomplished, for instance, by applying a CNOT 2-qubit
gate followed by a Hadamard 1-qubit gate and a measurement in the {|0i , |1i} basis.
When this occurs, Alice and Bob’s qubits (1 & 4) are projected into an entangled
Bell state (Fig. 1.3(c)). This procedure is called entanglement swapping, where the
entanglement between qubits 1 & 2 is swapped for entanglement between qubits 1 &
4 by consuming the entanglement between qubits 3 & 4.
Now that Alice and Bob have an entangled qubit pair, they can simply measure
their own qubit in the {|0i , |1i} basis. Based upon the information of the measured
state of the repeater qubits (which the repeater sends to Alice and Bob over an open
classical channel), Alice and Bob know the measured value of the other person’s
qubit without communicating any information. After running this entire procedure
numerous times, they will have built up a sequence of shared bits that only they
know. As with the simple QKD scheme from Sec. 1.3, Alice and Bob can openly
share a few of these bits to check their error rate and determine if a third party was
attempting to get in the middle of their entanglement procedure.
Once a quantum repeater can be built, multiple repeaters can be chained together in series to extend the range of quantum key distribution essentially without
limit. Since the entanglement creation along individual optical paths can be done
asynchronously and in parallel, the time to create entangled pairs between Alice and
1.4. QUBIT CANDIDATES
11
Bob will depend upon the length of the individual segments instead of on the total
distance between them (assuming the entanglement swapping procedure takes negligible time). Quantum repeater technology will be critical in enabling long-distance
quantum key distribtution.
1.4
Qubit candidates
There are many candidate systems for qubit implementations currently under investigation. The best candidates for quantum repeater technology are those that easily
interact with optical photons. Thus, for this application, the leading three candidate
systems at the moment are trapped ions in a vacuum [8, 11, 12], nitrogen vacancies
(NV) centers in diamond [13, 11, 14, 15, 16], and self-assembed InGaAs quantum
dots [17, 18, 19, 11, 20, 21, 22, 23, 24, 25]. Great progress has been made with each
of these systems, however, none of them have met all of the requirements necessary
for a quantum repeater. A summary of eight important requirements and how each
system performs for each is shown in Table 1.1 below.
Optical quantum
efficency
Homogeneity
Decoherence
time
Initialization
1-qubit control
2-qubit
interaction
Projective
measurement
Fabrication &
integration
Trapped ion
100%
NV center
3% [13]
InGaAs QD
97% [18]
1:1
15 s [11]
1,000:1 [13]
2 ms [11]
10,000:1 [19]
3 µs [11]
99% in 10 ms [12]
99% in 1 µs [12]
Phonon coupling
5 µs [14]
48 ns [14]
Nuclear spin
coupling
Destructive, low
success rate
Difficult, cavity
integration
possible
92% in 13 ns [19]
94% in 4 ps [19]
Theory
Cycling, slow
Difficult
Destructive, low
success rate
Easier, cavity
integration
possible
Table 1.1: Comparison between the three qubit candidates. Homogeneity is indicated
by the ratio between the ensemble linewidth and the homogeneous linewidth.
12
CHAPTER 1. INTRODUCTION
Trapped ion systems have perfect optical quantum efficiency in addition to unequalled homogeneity and long decoherence times, but qubit control is relatively slow,
and it is very difficult to fabricate and integrate trapped ion systems into deployable
technology due to the addition requirements of cooling and trapping the ions. Nitrogen vacancies are reasonably homogeneous, have long decoherence times coupled
with fast gate operations, but have a low quantum efficiency and it is very difficult
to fabricate devices made out of diamond. InGaAs QDs, on the other hand, have
high quantum efficiency, very fast control times, and devices are easy to fabricate,
but they suffer from inhomogeneity and decoherence, and have not yet achieved any
sort of two-qubit gate experimentally.
In order for a qubit system to enable quantum repeater technology, it must score
well in each of these categories. While a great deal of progress has been made in
reaching this goal since the idea of quantum key distribution was first presented, each
system still has a lot more that needs to be accomplished.
1.5
Thesis outline
This thesis will focus on a fourth qubit candidate system, which has the potential
to avoid some of the biggest obstacles impeding progress in the other systems. This
system is the neutral donor system. Chapter 2 will introduce the neutral donor system
and discuss the important characteristics that make it a good qubit system. Chapter
3 will present fluorine donors in zinc selenide as a particularly good candidate for
quantum repeater technology and present our experimental results on the optical
pumping of a single fluorine donor for qubit initialization. Chapter 4 will present
phosphorus donors in silicon as a good candidate for quantum computer technology
and present theoretical results on a technique for measuring a single phosphorus
donor, as well as discuss some related preliminary experimental results. Chapter 5
will then conclude and present an outlook for neutral donor systems.
Chapter 2
Neutral donor system
Isolated semiconductor spins are natural qubits due to their well defined quantum
states. These individual spins can be isolated in a number of ways, but most of the
methods boil down to creating a local potential energy minimum which can localize
the spin, and making the energy levels of the spin system distinguishable from the
energy levels of neighboring spins. The most common isolation systems are quantum
dots, electric gates, and semiconductor impurities. The focus of the remainder of this
thesis will be on one particular type of semiconductor impurity, a single donor.
A substitutional donor in a semiconductor crystal provides a single electron which
is located in the conduction band at room temperature. However, the donor creates
a shallow attractive potential which at sufficiently low temperatures can trap that
electron in a localized state just below the conduction band edge. This is referred
to as a donor-bound electron, or a neutral donor state D0 (Fig. 2.1). Effective
mass theory allows us to compute the energy of these states and approximate the
wavefunction of the bound electron, as will be shown in Sec. 2.1.
In addition to an electron, the donor can also trap an exciton, a quasiparticle
composed of an electron and a hole bound together. This excited state is called the
donor-bound exciton state D0 X (Sec. 2.2). Optical transitions between D0 and D0 X
states can be used to interact with and manipulate the donor-bound electron spin.
13
14
CHAPTER 2. NEUTRAL DONOR SYSTEM
Figure 2.1: A substitutional donor within the semiconductor crystal with a single
bound electron.
2.1
Effective mass theory
A semiconductor crystal is a complicated system, involving many interacting electrons and nuclei. Fortunately, effective mass theory [26, 27, 28] allows us to greatly
simplify the Hamiltonian of the system in order to predict the behavior of electrons in
particular states. The full Hamiltonian for a defect-free crystal and a single electron
can be described by the equation
H0 = −
h̄2 2
∇ + Vp ,
2m0
(2.1)
where m0 is the free electron mass and Vp is the periodic potential of the semiconductor crystal lattice.
To apply effective mass theory, we first use Bloch’s theory to rewrite the wavefunction in the form of a sum of Bloch waves times an envelope function,
Ψ=
X
ak,n eik·r ψk,n (r),
(2.2)
k,n
where ψk,n (r) is a periodic Bloch function that has a periodicity equal to that of Vp .
2.1. EFFECTIVE MASS THEORY
15
Due to the linear nature of the Hamiltonian, we can go through the following derivation using just one particular Bloch wavefunction, and the results will be accurate for
a sum of Bloch wavefunctions. The Bloch wavefunction form allows us to separate
the component of the Hamiltonian describing the periodic potential from the other
components that are of interest. Then,
h̄2 2
H0 Ψ =
−
∇ + Vp eik·r ψk,n (r)
2m0
h̄2 2 ik·r
ψk,n (r) + Ep,i (k)Ψ,
∇e
=
−
2m0
(2.3)
(2.4)
where
Ep,i (k)Ψ =
h̄2
2
−
−2ik · ∇ψk,n (r) + ∇ ψk,n (r) + Vp ψk,n (r) eik·r .
2m0
(2.5)
Now, all of the complication of the periodic potential is contained within Ep,i (k),
which depends upon both the momentum k and the band index n. For many semiconductors, this energy has a minimum around k = 0, allowing us to approximate it
as
Ep,n (k)Ψ ≈ En + αn k2 Ψ.
We can then rewrite the full Hamiltonian as
h̄2 2 ik·r
2 ik·r
H0 Ψ = En Ψ + −
∇ e + αi k e
ψk,n (r)
2m0
h̄2 2 ik·r
= En Ψ + − ∗ ∇ e
ψk,n (r),
2m
(2.6)
(2.7)
(2.8)
where we combine the term describing the momentum of the free-particle wave with
the term describing the 2nd order approximation of Ep,n (k), giving the system an
effective mass m∗ . Now all of the effect of the periodic crystal potential is contained
within En and m∗ , and the Bloch wave can be ignored, resulting in an effective
Hamiltonian Heff and and an envelope wavefunction Ψeff for a particular band n
16
CHAPTER 2. NEUTRAL DONOR SYSTEM
(assuming k is near the band minimum),
Heff Ψeff
Ψeff
h̄2 2
=
En −
∇ Ψeff ,
2m∗
X
=
ak eik·r .
(2.9)
(2.10)
k
The full Hamiltonian for an electron in a semiconductor crystal has now been
reduced to the Hamiltonian for a free particle of mass m∗ and a base energy En .
Eeffective mass theory has allowed us to greatly simplify the interactions of an electron. It is important to recognize that m∗ is not the real mass of the electron, but
determines the relationship between the crystal momentum of the electron k and its
energy when k is near the band minimum. Furthermore, we have assumed that the
electrons are non-interacting. This is most valid at the top of the valence band and
in the conduction band, and those are the only electron states we will be considering.
Note that we performed this derivation under the assumption that the band minimum
was at k = 0, but in fact a similar derivation can be performed when the minimum
is not at k = 0.
Next, we can add in the potential VD created by replacing a crystal nucleus with
the donor nucleus. We will only be considering the lowest-energy conduction band
and the highest-energy valence band, and so in this case, the screening of the electrons
in the inner-most valence bands or shells causes the potential of the donor nucleus to
resemble that of a single positive charge,
VD = −
e2
,
4πr
(2.11)
where is the static dielectric constant of the semiconductor.
The effective Hamiltonian for an electron in the conduction band in the presence
of a donor is now
Heff
h̄2 2
e2
= Ec −
∇
−
,
2m∗
4πr
(2.12)
which resembles the Hamiltonian for a hydrogen atom. In fact, the solutions are
identical after the substitution of m∗ for m0 , for 0 , and adding the energy of the
2.2. DONOR-BOUND EXCITON STATE
17
conduction band minimum Ec . This tells us that a donor provides localized electron
states with envelope functions resembling hydrogen wavefunctions:
Ψeff = ψnlm (r, θ, φ),
1 e4 m∗
.
E = Ec + 2 2
n 2h̄ (4π)2
(2.13)
(2.14)
The electron also has an effective Bohr radius,
a∗ =
h̄2 4π
.
m∗ e2
(2.15)
The lowest energy state, where n = 1, l = 0, and m = 0, is generally referred to as
the neutral donor state D0 .
Effective mass theory in fact turns out to give a reasonable approximation for the
binding energy of an electron in many systems (the difference in energy between the
conduction band edge and the donor-bound electron state). However, the approximation does not accurately describe the interactions very near to the donor where
the inner-shell electrons exist, and so effective mass theory is not exact. A correction
term is needed to account for the difference in energy between the experimentally
measured energy and the energy predicted by effective mass theory. This correction
is called the central cell correction [27], and is different for each donor system. The
central cell correction is large for so called deep donors, which have a large binding
energy and therefore a significant portion of the electron wavefunction overlaps with
the central cell region in the immediate proximity of the donor. Shallow donors, on
the other hand, have a small central cell correction and are therefore well described
by effective mass theory. Both fluorine in zinc selenide and phosphorus in silicon are
considered shallow donors, and are well approximated by effective mass theory.
2.2
Donor-bound exciton state
The donor-bound exciton state D0 X is composed of an exciton bound to a neutral
donor. An exciton is a quasiparticle formed when an electron in the conduction band
18
CHAPTER 2. NEUTRAL DONOR SYSTEM
and a hole in the valence band bind to one another, as shown in Fig. 2.2. The hole
itself is a quasiparticle, resulting from the cooperative behavior of the electrons in a
valence band missing an electron, but can be described as a particle with a charge
of +e, an effective mass of m∗h , and a spin opposite that of the unoccupied valence
band electron state. The exciton is a bound state between an electron and a hole,
and can also be approximated by effective mass theory [29]. Since the exciton is a
bound state, the energy to create it is slightly less than the energy required to take an
electron from the valence band and put it into the conduction band. The difference
between these energies is the exciton binding energy EX . The exciton also has an
effective radius, which for semiconductor systems is generally spread over many unit
cells and as a Wannier exciton [30].
Figure 2.2: Diagram of the D0 ground state with a single bound electron, and the
D0 X state with an additional electron-hole pair. An optical transition connects the
two states.
Most excitons are free excitons, and are able to move around within the semiconductor crystal. A neutral donor, however, provides a potential which can bind
a single exciton. In this state, two electrons and a hole are both localized around a
single donor. It might seem unclear why a neutral particle to another neutral particle,
2.2. DONOR-BOUND EXCITON STATE
19
but it could be described as analogous to how two atoms with unpaired electrons will
form a covalent bond by sharing electrons.
Computation of the binding energy of an exciton to a neutral donor is a very
difficult task due to the 4-body nature of the state. However, an experimental relation
called Hayne’s rule [31] has been very successful in predicting D0 X binding energies.
Hayne’s rule says that the binding energy of the exciton to a neutral donor should be
proportional to the binding energy of the electron to the donor in the D0 state,
ED0 X ≈ aED0 ,
(2.16)
where a is the proporionality constant and is generally different for each semiconductor.
The two electrons in the D0 X state form a spin singlet in order to exist in the
same spatial wavefunction state, and so the hole determines the spin of the D0 X
state. Since the electrons are in the bottom of the conduction band, their Bloch
wavefunctions share the symmetry of the s-orbitals for atomic electrons, with zero
units of orbital angular momentum. The holes, on the other hand, are at the top
of the valence band, and share the symmetry of p-orbitals, with one unit of orbital
angular momentum. Therefore, spin-orbit coupling determines the effective spin of
the hole states by combining the 1/2 unit of spin angular momentum with the one
unit of orbital angular momentum.
The total spin 3/2 manifold is higher in the valence band than the total spin
1/2 due to the energy of the spin orbit coupling, and therefore a total spin 3/2 hole
requires less energy to create. This means that the total spin 1/2 D0 X states are
noticeably higher in energy than the the total spin 3/2 D0 X states, and can often
be unbound states that play no role in the system. The total spin 3/2 manifold is
broken up into spin projection ±3/2 and ±1/2 states. The ±3/2 states have a larger
effective mass than the ±1/2 states, and therefore the former are called heavy-hole
(HH) states while the latter are called light-hole (LH) states.
The D0 X state can be created by bringing a valence electron into the conduction
band. The electron can then bind with the hole it left behind, creating an exciton,
20
CHAPTER 2. NEUTRAL DONOR SYSTEM
and then the exciton can further bind to the neutral donor. Thus the energy of the
transition between the D0 state and the D0 X state is given by
E(D0 → D0 X) = EG − EX − ED0 X ,
(2.17)
where EG is the band gap between the minimum in the conduction band and the
maximum in the valence band. In general, the D0 → D0 X transition energy is just
below that of the band gap, which is in the optical energy range. For many semiconductors the D0 X state can be created resonantly by the absorption of a photon
of the correct energy, assuming spin selection rules are followed and momentum is
conserved.
2.3
Optical transitions
Both the D0 and D0 X states are composed of a spatial wavefunction and a spin state:
|Ψi = |ψ(r)i |si .
(2.18)
Neglecting the nuclear spin for now, there are two D0 states, corresponding to the two
electron spin states |↑i and |↓i. The relative energy of these two states is determined
by the Zeeman splitting due to an applied magnetic field B,
∆E = 2ge µB S · B,
(2.19)
where ge is the electron g-factor and µB is the Bohr-magneton.
The D0 X states, however, consist of four states, corresponding to the two heavy
hole spin states |±3/2i and the two light hole spin states |±1/2i. Due to spin orbit
coupling, the spatial wavefunction and the spin state are entangled, and so the relative
energies between these states depend on perturbations to the spatial wavefunction in
addition to the Zeeman splitting. Two such perturbations important for this thesis
are strain in the crystal structure, and confinement due to a quantum well.
We can calculate which D0 → D0 X transitions are allowed and the relative rates
2.3. OPTICAL TRANSITIONS
21
of these transitions by looking at the electric dipole matrix element between the
states. Fermi’s Golden Rule [32] tells us that the rate of transition for an oscillating
perturbation is proportional to the square of the inner product of the perturbing
Hamiltonian H 0 between the initial and final states:
2
Ri→f ∝ |hΨf | H 0 |Ψi i| .
(2.20)
For the case of stimulated optical transitions of electrons, the perturbing Hamiltonian
is the electric dipole operator
H 0 = er · E
(2.21)
where E is the complex electric field vector amplitude. In the case of a transition
between D0 X and D0 , we have
RD0 →D0 X ∝ |eE · hΨD0 X | r |ΨD0 i|2 .
(2.22)
All of the D0 states share the same 1s envelope function |φ100 i and s-like Bloch
∗
i, and can be written as
wavefunction |ψn00
(
∗
|φ100 i |ψn00
i |↑i
1
ΨD0 , ±
,
=
∗
2
i |↓i
|φ100 i |ψn00
(2.23)
where 100 or n00 indicate the quantum numbers nlm of the hydrogenic wavefunctions
and the
∗
indicates that the Bloch wavefunctions are similar but not identical to the
hydrogenic wavefunctions. In the D0 X states, the two electrons form a spin singlet
within the spatial wavefunction given above,
1
∗
|φ100 i |ψn00
i √ (|↑i |↓i − |↓i |↑i) ,
2
(2.24)
while the hole determines the particular D0 X state. The hole envelope function is not
the same as we computed for the electron in D0 , especially since it is attracted to the
electrons but repelled from the nucleus. However, like the D0 states, all D0 X states
share the same envelope function, which we will notate as |φhh i. Instead, each hole
22
CHAPTER 2. NEUTRAL DONOR SYSTEM
state has a unique combination of Bloch wavefunctions:

∗

i |↑i
|φhh i |ψn11


q
q


1
2
∗
∗

|φhh i
|ψn11 i |↓i + 3 |ψn10 i |↑i
ΨD0 X , ± 3 ± 1 =
q
q 3 .
1 ∗
2
∗

2 2

|φ
i
|ψ
i
|↓i
ψ
|↑i
+
hh
n10
n1−1

3
3


∗ 
|φhh i ψn1−1 |↓i
(2.25)
It is important to note that the inner product in Eq. 2.22 is not an inner product
between the electron in the D0 state and the hole in the D0 X state, but is rather
an inner product between the initial and final state of the electron that makes the
jump from the valence band to the vacant donor-bound state. So a transition from
ΨD0 , + 1 to ΨD0 X , + 3 should be computed by taking the matrix element between
2
2
the valence band electron state ΨD0 X , − 23 (since a negative electron spin corresponds
to a positive hole spin) and the donor state ΨD0 , − 1 .
2
In order to compute the inner product between the D0 and D0 X states, we need
to know the orientation of the spatial wavefunction with respect to the electric field
of the incident light. In the case of D0 , the orientation will be determined by the
characteristics of the semiconductor, such as strain or quantum wells, and is unaffected
by a magnetic field. However, in the case of D0 X, where spin-orbit coupling exists, the
orientation can also be effected by an applied magnetic field. Therefore, the allowed
transitions and their respective rates largely will depend upon the characteristics of
the particular semiconductor sample. Computation of the transition rates for F:ZnSe
and P:Si, in particular, are discussed in Secs. A.1 and A.2, respectively.
However, even without detailed calculations, we can gain a bit of insight into which
transitions would be forbidden by noting that the spin components |si commute with
r. This means that spin must always be conserved (although total angular momentum
is not necessarily conserved). Conservation of momentum is also required, although
no explicit momentum terms or operators appear in Eq. 2.21.
In a direct bandgap semiconductor, such as ZnSe or GaAs, absorption and emission of photons is the dominant way to create excitons. In an indirect bandgap semiconductor, such as Si, this is generally considered “forbidden” without the emission
2.4. NUCLEAR COUPLING
23
or absorption of an accompanying phonon due to the lack of momentum conservation. However, a zero-phonon transition is possible in the case of a donor-bound
exciton, even though it is much slower than the non-radiative or phonon-assisted
transitions. The existence of this transition could be thought of as a result of the
momentum-position uncertainty principle. When the exciton is localized to a donor,
it’s momentum becomes more uncertain and the slight overlap between the momentum distribution of the D0 X state and the D0 state makes the transition possible.
The mathematics of this momentum overlap would be contained within the Bloch
wavefunctions, which are not explicitly calculated here.
So far we have been considering the lowest energy bound electron states (D0
states). However, bound neutral donor excited states do exist in some circumstances,
which have envelope function that look like 2s or 2p orbitals. Transitions between a
D0 X state and an excited neutral donor state D0e have traditionally been called TES
transitions, for two-electron satellite. The name came from the fact that researchers
assumed the transition was associated with two electrons bound to the donor. Transitions to D0e states have slightly lower energy than transitions to D0 states, and the
difference can be computed using Eq. 2.14. A TES transition can be observed in Fig.
3.2.
2.4
Nuclear coupling
Up until now, we have been ignoring the spin of the donor nucleus. However, in some
semiconductor systems, the nuclear spin plays an important role in determining the
states due to the hyperfine coupling
HHF = AIN · Se .
(2.26)
24
CHAPTER 2. NEUTRAL DONOR SYSTEM
In the absence of a magnetic field, the donor-bound electron spin and the nuclear
spin couple into a set of spin triplets and a spin singlet in the D0 state
|1, 1i = |↑e i |↑n i
|1, 0i =
√1
2
(|↑e i |↓n i + |↓e i |↑n i) , |0, 0i =
√1
2
(|↑e i |↓n i − |↓e i |↑n i) .
(2.27)
|1, −1i = |↓e i |↓n i
However, if the Zeeman splitting (Eq. 2.19) is much stronger than the hyperfine
coupling, then electron and nuclear spin states are decoupled:
|↑e i |↑n i , |↑e i |↓n i
|↓e i |↑n i , |↓e i |↓n i
.
(2.28)
The hyperfine coupling constant A is dominated by the point-contact hyperfine
term for the D0 states since both the s-like Bloch wavefunction and the s-like envelope
wavefunction for the electron overlaps the donor nucleus. However, for the D0 X states
with p-like Bloch wavefunctions, the hole wavefunction does not overlap the donor
nucleus and the point-contact term is zero. Since the other terms that contribute to
hyperfine coupling are orders of magnitude smaller than the point-contact term, the
hyperfine coupling between the hole spin in D0 X and the nuclear spin is generally
negligible.
Due to the fact that the hyperfine constant A is generally quite small, the hyperfine
split states of D0 are usually not optically resolvable since the width of the optical
transition (determined by the D0 X → D0 transition rate) is broader than the the
hyperfine splitting. However, as will be discussed in Sec. 4.1, Si is a very interesting
semiconductor since its indirect bandgap produces a very slow transition rate, and the
resulting optical linewidth is far narrower than the hyperfine splitting. This nuclear
distinguishability provides an interface to interact optically with the donor nucleus.
2.5. SPIN QUBIT
2.5
25
Spin qubit
The spin of an electron bound to a donor can serve as a natural qubit by defining
|↑i as |0i and |↓i as |1i. Semiconductor qubit implementations in general hold many
advantages over other qubit systems, such as ease of fabrication and scalability, integration with classical electronics, and a wealth of industry experience working with
those materials. While solid state systems are very complex and provide many challenges due to many-body interactions, defects, and disorder, a number of experiments
have shown measurement and coherent control of spin qubits in semiconductor devices [33, 34, 35, 25, 36, 37, 38, 39]. In addition to this, donor-bound electrons have
been shown to have high homogeneity and extremely long decoherence times in very
pure systems [40].
The electron spin can be directly manipulated through the use of resonant microwave radiation. However, manipulation this way is not particularly fast (µs timescale),
and it is difficult to isolate the radiation to one qubit. Therefore, systems which can
be manipulated optically have an immediate advantage in the speed of gate operations and the scalability of the system due to the strong and localized electric fields
available at optical frequencies. In these interactions, a third, excited state |ei is
required, which is optically connected to the two qubit states. This forms what is
called a lambda-system, as shown in Fig. 2.3.
For electrons trapped in InGaAs quantum dots, the excited state is the trion state,
which is composed of the trapped electron plus an exciton [19]. For donor-bound
electrons, the excited state is the D0 X state. Spin rotations have been performed in
both systems utilizing ultrafast two-photon stimulated Raman transitions [41, 19].
These experiments result in ps gate operation times, allowing for orders of magnitude
more gate operations within the decoherence time of the systems.
A great deal of progress has been made on quantum dot qubits (Sec. 1.4), but the
system has disadvantages in homogeneity and decoherence. Donor systems, on the
other hand, are behind in terms of single spin control accomplishments, but have the
promise of increased homogeneity and reduced decoherence, and so are a worthwhile
system to research.
26
CHAPTER 2. NEUTRAL DONOR SYSTEM
Figure 2.3: Simple diagram of the lambda system used for optically manipulating an
electron spin.
The first optical spin rotations of donor-bound electrons were performed on silicon
donors in GaAs [41]. Unfortunately, researchers were not able to isolate individual
Si donors and the spin rotations suffered from decoherence, both of which were most
likely caused by the small binding energy of the exciton to the neutral donor. Fluorine
donors in zinc selenide have a relatively large exciton binding energy, and so there is
hope that spin rotations will work in this system. Approximately half of my research
time has been spent on the F:ZnSe system, and in Chapter 3, I discuss our results on
the isolation and optical pumping of individual F donors.
Although phosphorus donors in silicon was the first donor system proposed for
quantum information purposes [42], Si is extremely optically inefficient due to it’s
indirect band gap. Therefore, electrical methods for interacting with the donor must
be used, at least in part. Despite this disadvantage, the homogeneity and decoherence
times in this system are unprecedented, and so a great deal of work on this system is
ongoing. The other half of my research time has been spent on the P:Si system, and
so in Chapter 4, I will discuss our research on a novel way to combine optical pumping
and electrical detection in order to detect the nuclear spin of a single P donor.
Chapter 3
Fluorine donors in ZnSe
Single electrons bound to fluorine donors in zinc selenide have many potential advantages over other qubit candidates. As with other donor systems, F-bound electrons in
ZnSe are very homogeneous. It has been recently shown that an ensemble of fluorine
donors in bulk ZnSe features long electron-spin dephasing times T2∗ , greater than 30 ns
for temperatures up to 40 K [43]. Furthermore, photons emitted from the D0 → D0 X
transition of two different donors are identical [44], and can be entangled [45]. ZnSe
is a direct bandgap semiconductor, and so radiative recombination is the dominant
mechanism for decays between D0 X and D0 . This optical transition has a quantum
efficiency very close to unity [46], and can function as a source of triggered single photons [44]. The optical dipole coupling is particularly strong for the F:ZnSe system,
leading to fast optical decay times and strong interaction with applied laser fields.
Zinc and selenium atoms both have isotopes with zero nuclear spin: zinc has 96%
natural abundance and selenium has 94% natural abundance of zero-spin isotopes.
This is a very big advantage over other materials for which there are no zero-spin
isotopes, such as III-V semiconductors, where the nuclear spin of the host semiconductor crystal can couple to the electron spin and cause decoherence. In fact, this is
the leading decoherence-causing mechanism for single electrons in InGaAs quantum
dots [47]. In ZnSe, isotopic purification can be used to deplete nuclear spin from
the ZnSe crystal, and is expected to greatly increase decoherence times. This technique has been successfully used in both diamond and silicon [48, 40]. Fluorine, on
27
28
CHAPTER 3. FLUORINE DONORS IN ZNSE
the other hand, has 100% natural abundance of spin-1/2 isotopes. The spin of a F
nucleus could therefore be used as a long lived quantum memory which is naturally
coupled to the electron spin.
A further advantage is the ability to implant F donors using ion implantation.
This technique could eventually be used to deterministically place single qubits in
specified locations through the use of implantation through a mask and single-impact
registration [49]. Systems such as self-assembled InGaAs quantum dots have been
difficult to grow in specified locations [50], and would therefore be harder to turn into
a scalable technology. Ion implantation would make the F:ZnSe system much more
scalable. F has been successfully implanted and has been shown to take on the role
of individual donors [51].
All of these properties make flourine-bound electrons in ZnSe strong candidates
for quantum information processing, particularly for quantum repeater technology. In
this chapter, I will explain the optical properties of F-bound electrons, describe how
we were able to isolate single F donors, present experimental results demonstrating the
first optical control of a single donor-bound electron, and discuss future experiments
and the outlook for the system.
3.1
Single donor isolation
In our samples, the F donors were isolated by means of a quantum well and mesa
structuring, as described in Ref. [52] and shown in Fig. 3.1. A ZnSe quantum well
confined between ZnMgSe cladding layers serves to isolate F donors to a 2D plane,
while mesa etching further confines the donors to disk-like quantum well regions.
Based on the areal doping density of the F within the sample and the size of the
mesas, we can control how many donors exist within a particular mesa.
After this, the quantum well, and therefore the F donors with optical transitions
below the ZnSe bandgap, are confined to ∼100 nm diameter mesas, which are separated by 10 µm.
The samples were grown using molecular-beam epitaxy (MBE) on top of a GaAs
substrate. Unfortunately, good single-crystal ZnSe substrates do not exist, and so
3.1. SINGLE DONOR ISOLATION
29
Figure 3.1: Mesa structure used to isolate individual donors. Donors are δ-doped
within the center of the ZnSe quantum well. After growth, the sample is etched
down through the quantum well to the surface, leaving behind 100 nm-diameter mesa
structures.
GaAs was chosen due to the similarity of the lattice constants (5.67 Å for ZnSe [53]
and 5.65 Å for GaAs [54]). On top of that, a layer of ZnSe was grown, usually 20 nm
in thickness, which helps the lattice matching and the adhesion between the GaAs
and the ZnMgSe. The cladding layers of Zn1−x Mgx Se generally had up to 17% Mg
and were grown to be a few 10s of nm thick. Between the two ZnMgSe cladding layers
is the ZnSe quantum well with a thickness of 1-10 nm.
The F were placed within the quantum well by either turning on the F source for
a short time during the epitaxial growth, resulting in a δ-doped layer of F, or through
ion implantation following the growth. The epitaxially doped F were located only
within the quantum well, and for many of the samples studied for this thesis, the areal
density was close to 3 × 1010 cm−2 . The flourine placed using ion implantation had a
distribution of depths, and so only a fraction of the F end up within the quantum well.
30
CHAPTER 3. FLUORINE DONORS IN ZNSE
Fortunately, the only donors with optical transitions within the region of energies we
have been studying are the F located within ZnSe, and any F that end up within
ZnMgSe play no role in our studies. The resulting areal density within the quantum
well was similar to the density of the δ-doped samples. Following the growth and
implantation, the samples were generally annealed in order to reduce the amount
of defects with the sample. Implantation creates many defects, and so annealing is
usually required for those samples.
Structuring was done using electron-beam lithography and wet etching in order
to fabricate the mesa structures. Mesas with 100 nm diameter and a F concentration
of 3×1010 cm−2 within the quantum well result in 2.4 donors on average per mesa.
The mesas were separated by 10 µm, ensuring we could isolate optical emission from
individual mesas. After structuring, a layer of silicon dioxide or silicon nitride was
usually deposited on top of the entire sample and acts as a passivation coating. Due to
the lattice mismatch and varying thermal expansion coefficients between the layers in
the sample, samples would often acquire dislocations or fractures after a few thermal
cycles between room temperature and measurement temperatures near 4 K. This
decreases the optical efficiency of emission from the donors. The passivation coating
serves to improve the durability of the samples.
3.2
Optical spectra
The bandgap in ZnSe is wide and has an energy near of 2.8 eV in bulk, which corresponds to photons in the blue with a wavelength around of 440 nm. A single electron
bound to a fluorine donor in ZnSe has a binding energy of 29.3 meV in bulk [55, 56, 57].
This relatively large binding energy means that the electron is usually bound even
up to room temperature. The D0 X state has a binding energy of 5 meV in bulk [58],
which means that the donor will only trap an exciton at temperatures significantly
below 60 K. The D0 X → D0 transition in bulk at 4K has an energy of 2.800 eV [58].
These emission binding energies are only accurate for bulk ZnSe, and can be noticeably blue-shifted when the fluorine donors are located within the quantum well used
in our samples.
3.2. OPTICAL SPECTRA
31
The D0 X binding energy can be measured using photoluminescence spectroscopy.
For these experiments, we used an above-band laser (a laser with photon energies
larger than the bandgap of ZnSe) to illuminate the sample. This creates many free
carriers, both electrons and holes, within the sample. Most of these will combine into
excitons and later recombine, emitting a photon. Occasionally an exciton will relax
into the D0 X state before recombining, and so it will release a photon with slightly
lower energy, where the difference is the D0 X binding energy. The emitted photons
are then collected and sent to a spectrometer. Fig. 3.2 shows an example bulk ZnSe
spectra.
Figure 3.2: Example spectra of a bulk F:ZnSe sample. Here, both the HH and LH
free exciton lines are visible. Slightly lower in energy than the free excitons is the
strong D0 X transition peak. Even lower energy is the two-electron-satellite transition
(Sec. 2.3), which is due to D0 X transition into an excited state of the neutral donor.
When the quantum well thickness approaches the effective bohr radius of the
donor-bound electron, 3.59 nm for F:ZnSe [41] (see Eq. 2.15), the quantum well
starts to compress the electron wavefunction in one dimension, forcing it to spread
out in the plane of the quantum well. This tends to increase the energy of the
D0 state. However, the quantum well also effects the conduction band free-electron
32
CHAPTER 3. FLUORINE DONORS IN ZNSE
wavefunctions, which begins to look like a particle-in-a-box wavefunction along the
growth direction of the quantum well. This increases the energy of the free-electron
wavefunctions as well. Except in the case of a very narrow quantum well, the increase
of the free-electron energy is greater than that of the D0 energy, resulting in an increase
of the electron binding energy to the F donor [41]. This same effect happens with the
free excitons and donor-bound excitons, resulting in an increase in the D0 X binding
energy. The location of the F within the quantum well also has an effect on these
binding energies [41]. See Figs. 3.3 and 3.4 for example 4 nm and 2 nm quantum well
luminescence.
Figure 3.3: Example spectra for a single mesa with a 4 nm thick quantum well. The
entire spectra is blue-shifted compared to the bulk spectra shown in Fig. 3.2. The
set of peaks at higher energy are due to HH free excitons. The large single peak at
lower energy is likely due to one or more donors, based upon the energy and width
of the peak.
The quantum well also serves to provide enough strain that the LH D0 X states
are split off from the HH D0 X states, due to the Pikus & Bir strain Hamiltonian [59].
This puts the HH states higher in the valence band than the LH states, resulting in
a smaller D0 X → D0 transition energy. This splitting is large enough that emission
from the LH D0 X states are never observed from quantum well samples. Therefore,
3.2. OPTICAL SPECTRA
33
Figure 3.4: Example spectra for a single mesa with a 2 nm thick quantum well.
The spectra is blue-shifted even further in the narrower quantum well. The narrow
quantum well also amplifies the relaxation of excitons into the neutral-donor state,
increasing the amplitude peak corresponding to that transition.
we only need to worry about the spin-3/2 states, and the full optical system that we
consider in most situations is two D0 states and two D0 X states, both defined by their
spin states.
The minimum width of the optical transition is determined by the lifetime of
excited state with the relation
∆ν =
1
2πτ
(3.1)
where ∆ν is the linewidth in Hz, and τ is the lifetime in s. This lifetime-limited
transition has a Lorentzian shape, since the fourier transform of an exponential decay
is a Lorentzian curve. However, the transition can be broadened by inhomogeneity,
such as the distribution of emission energies from a large ensemble of donors, or by
the time averaged emission of a donor that has a fluctuating transition energy. This
inhomogeneous broadening generally results in a Gaussian shaped curve rather than
a Lorentzian curve.
34
CHAPTER 3. FLUORINE DONORS IN ZNSE
3.2.1
Magnetophotoluminescence
When placed in a magnetic field, both the D0 X and D0 states split due to Zeeman
splitting. The orientation of the magnetic field with respect to the quantum well
growth direction plays a big role in determining the wavefunction of the donor states,
which determine the transition selection rules. There are two orientations we work in.
In the first, the magnetic field is parallel to the growth direction, and in the second,
the magnetic field is perpendicular to the growth direction. Due to the fact that our
optical axis is always parallel to the growth direction, the first orientation is Faraday
geometry, and the second is Voigt geometry.
Using magnetophotoluminescence, we can show that the D0 X optical transitions
do in fact follow the selection rules discussed in Sec. 2.3 and computed in Sec. A.1.
The experiments are the same as the photoluminescence experiments, using aboveband laser illumination and a spectrometer for collecting photon emission, with the
addition of an applied magnetic field and polarization-selective measurements. The
polarization sensitivity is created by using a wave-plate in front of a polarizing beam
splitter (PBS) with the H-polarization output of the PBS leading to the spectrometer.
To detect V-polarization photons |V i, we use a half-wave plate (HWP) with the fastaxis at an angle of 45◦ to the horizontal, which changes |V i into |Hi before the PBS.
Using the HWP at 0◦ , |Hi remains |Hi, thus measuring H-polarization photons. We
can also measure in the circular basis |σ + i/|σ − i using a quarter wave plate (QWP)
with the fast axis at +45◦ /-45◦ with respect to the horizontal.
Using a magnetic field of 7 T, we can observe D0 X transitions with the expected
polarization in either Faraday or Voigt geometry, as shown in Fig. 3.5. By fitting
the splitting between the peaks as a function of magnetic field, we can determine
the g-factor of the electron and the hole. Since the hole spin is unable to align with
the magnetic field in Voigt geometry, the magnetic field does not cause any Zeeman
splitting, and so the hole g-factor is approximately zero. By comparing the energy of
four optical transitions in Voigt geometry, the splitting between the two D0 and two
D0 X states can be determined. This determines both the electron g-factor, ge , and
the hole g-factor in a magnetic field perpendicular to the growth direction, gh⊥ . Since
the electron wavefunction is not confined by the quantum well, the electron g-factor
3.2. OPTICAL SPECTRA
35
is the same in both Faraday and Voigt geometry. Therefore, we can compare the
electron g-factor from Voigt geometry to the splitting of the two optical transitions in
Faraday geometry in order to determine the hole g-factor in a magnetic field parallel
k
to the growth direction, gh .
Figure 3.5: Spectral data showing the D0 X transition of a single donor at 7 T in both
Faraday and Voigt geometry. The points are the data, and the lines are Gaussian
curve fits. For the Faraday plot, the blue curve is R circular polarization, and the red
curve is L circular polarization. For the Voigt plot, the blue curve is H polarization,
and the red curve is V polarization.
The g-factor also depends upon the thickness of the quantum well, in addition to
the orientation. This is because the g-factors for ZnSe are slightly different than those
in ZnMgSe, and so if the electron or hole wavefunction leaks out into the quantum well
barriers, the g-factor will change. In the case of the hole, the spin-orbit coupling is also
changed as the wavefunction is compressed in the quantum well, further modifying
the g-factor.
3.2.2
Single photon source
Since the donor emits one and only one photon for each decay from the D0 X state
and since the donor can only trap one exciton, it makes for a great single photon
36
CHAPTER 3. FLUORINE DONORS IN ZNSE
source. When the D0 X state is exciting with an above-band pulsed laser, the donor
is a triggered single-photon source. Each pulse creates many excitons, and one of
them can bind to the donor. This exciton will eventually decay, emitting a single
photon. However, the exciton lifetime is shorter than the D0 X lifetime [44], and so
the remaining excitons will decay as free excitons before the D0 X emits the photon.
Therefore, there are no remaining excitons to recreate the D0 X state after it emits
the first photon, and there will be at most one photon emitted by the donor for each
excitation pulse.
The single-photon nature of the donor system can be confirmed by determining
the two-photon correlation function g 2 (τ ),
2
g (τ ) =
a† (t)a† (t + τ )a(t + τ )a(t)
ha† ai2
.
(3.2)
This function can be measured using a Hanbury-Brown-Twiss (HBT) experiment [60],
as depicted in Fig. 3.6.
Figure 3.6: Simple diagram of the Hanbury-Brown-Twiss experiment. A pulsed laser
(purple) excites the D0 X state within the sample. The emitted photons are collected
and sent to a beam splitter. If there is at most one photon at any time, it is impossible
to detect a photon on each detector at the same time.
In this experiment, the sample is excited by an above-band pulsed laser. The
emission is filtered by frequency so that all of the photons from the pump laser are
3.2. OPTICAL SPECTRA
37
blocked, and the only part of the spectrum detected is the narrow frequency range
associated with the D0 X transition. We can then imagine a regular train of single
photons emitted from the sample and heading to the beam splitter. When the photons
hit the beamsplitter, they have a 50% probability of being sent to either SPCM. When
the SPCM detects a photon, a timer starts. The timer is stopped when the second
SPCMs detects a photon. A histogram of the time between photon arrivals produces
a plot of g 2 (τ ) (after normalizing the plot by setting g 2 (τ ) = 1 for τ far from zero
delay). Since a single photon cannot be split and detected by both SPCMs at the
same time, a single photon source will always have g 2 (0) = 0.
Unfortunately, in a real experiment, it is often hard to filter out all the photons
at other energies, such as those that come from other light sources in the room and
especially those from any laser incident on the sample. This will result in the g 2 (0)
dip not going all the way to zero. The threshold for confirming a single photon source
is for g 2 (0) to be less than 0.5, since 0.5 would correspond to the case where two
photons were always emitted together. If the measured dip is less than 0.5 by a few
standard deviations, we can be quite confident that the emitter is a single photon
source. Fig. 3.7 shows a measurement of g 2 (τ ) using the experiment described above,
resulting in g 2 (0) = 0.25. From this result, we know that the photoluminescence
Figure 3.7: Example g 2 (τ ) data for a single donor. At τ = 0, the normalized dip
drops to 0.25. This figure is taken from Ref. [52].
38
CHAPTER 3. FLUORINE DONORS IN ZNSE
Figure 3.8: Simple diagram of the Hong-Ou-Mandel experiment. In this case, two
different donors are excited by the pulsed laser. The emitted photons are collected
and sent to the same beam splitter. If identical photons arrive at the same time, they
will bunch together and travel to the same detector.
emission we were detecting was coming from a single photon source.
A slight variation of the above experiment can be used to show that the emitted
photons are indistinguishable. This is called the Hong-Ou-Mandel (HOM) experiment [61], and is depicted in Fig. 3.8. When two perfectly indistinguishable photons
hit a beam splitter at the same time from opposite sides, they will bunch together
and always exit the beam splitter on the same side.
1
c† c† − d† d†
a† b† |0i = (c† + d† )(c† − d† ) |0i =
|0i .
2
2
(3.3)
Therefore, at zero time delay, g 2 (0) should go to zero for indistinguishable photons.
However, in reality the photons will not always arrive at the same time even if
they are detected at the same time due to the finite lifetime of the excited state
3.3. SINGLE DONOR CONFIRMATION
39
causing jitter in the photon arrival time and the detector jitter reducing the precision of the measurement of the arrival time. In addition, the center frequency or
wavepacket shape can vary slightly. These effects result in a reduced indistinguishability. In experiments, photons from two different fluorine donors have shown good
indistinguishability [44]. As shown in Sec. 1.3.1, the interference of indistinguishable
photons can be used to entangle qubits separated over macroscopic distances. The
first step of this scheme, which is the post-selected entanglement of photons from a
pair of donors, has also recently been demonstrated [45].
3.3
Single donor confirmation
For the optical pumping experiments described in the next couple of sections, a
mesa containing a single F donor needed to be found. To confirm the presence of a
single isolated F donor, we used four experiments on the same mesa. The first was to
confirm using a standard photoluminescence setup (Sec. B.1) that the spectra exhibit
a sharp peak within the expected range of separation from the free excitons where
Figure 3.9: Spectra of the mesa used for the optical pumping experiments. The strong
peak is within the energy range expected for the D0 X transition of a donor, and has
been confirmed to in fact contain a single donor.
40
CHAPTER 3. FLUORINE DONORS IN ZNSE
we’d expect to find D0 X transitions. Since the precise D0 X binding energy depends
upon the exact thickness of the quantum well and the location of the donor within the
quantum well [41], that range is approximately 5-15 meV for a 2 nm thick quantum
well. We know from studies across many samples that the number of these lines per
mesa depends on the F doping concentration. So if we do find peaks in this region,
we believe they are related to the F-donor. A spectra from the particular mesa used
for the optical pumping experiments is shown in Fig. 3.9.
After finding a mesa with the required optical transition, we determined the dependence of the height of the interesting peak on the laser power, and compared that
to the power dependence of the free-exciton emission. This experiment was done with
the above-band pulsed laser. The number of free excitons created should be roughly
proportional to the laser pump power, as shown in Fig. 3.10. However, due to the
finite lifetime of the D0 X state and the fact that only a single exciton can bind to the
Figure 3.10: Plot showing the relationship between laser pumping power and the
amplitude of certain optical transition peaks for mesa with the confirmed single donor.
The peaks which we have associated with FE-HH have a linear dependence, while the
peak associated with the donor saturates at higher pump powers.
3.3. SINGLE DONOR CONFIRMATION
41
donor at a particular time, there can be at most one photon emitted from the donor
per pulse. At low pump powers, the probability of an exciton relaxing into the D0 X
state before emitting a photon is roughly proportional to the pump power. However,
at high pump powers, the probability will saturate at 100%, resulting in a saturation
of the D0 X transition peak, as observed in Fig. 3.10. This saturation indicates that
the source of the transition is a state with a finite lifetime and a finite number of
occupiable states.
The third experiment was to use magnetophotoluminescence to determine the
polarization of the transitions and the electron and hole g-factors, in order to compare
them to the expected transitions and g-factors for the flourine-bound D0 X transition.
At a magnetic field of 7 T, the transitions have the expected polarization in both
Faraday and Voigt geometry Fig. 3.11. By finding the positions of the peaks at
various fields, we can also determine the electron and hole g-factors by comparing to
the expected Zeeman splitting Fig. A.2. From Voigt geometry, we determined the
electron and HH perpendicular g-factors to be |ge | = 1.3 ± 0.3 and |gh⊥ | = 0.0 ± 0.1.
k
From Faraday geometry, we determined that |3gh − ge | = 0.9 ± 0.1. These g-factors
and polarizations agree very well with previous results, strongly indicating that the
Figure 3.11: Zeeman splitting of the confirmed single donor, in both Faraday and
Voigt geometry.
42
CHAPTER 3. FLUORINE DONORS IN ZNSE
Figure 3.12: g 2 (τ ) data histogram for the confirmed single donor. The black and red
are the bin counts, and the blue bars are summed counts from the red regions. The
dip at g 2 (0) reaches 0.22±0.06, indicating a single emitter.
source of the peak of interest is the D0 X transition.
The final experiment was to do the HBT experiment described in Sec. 3.2.2 in
order to confirm that the sharp transition came from a single photon source, which
would indicate that there is only one donor, as opposed to multiple donors with
overlapping emission. After summing together the time bins corresponding to the
arrival of photons after each above-band pulse and then normalizing, the resulting
histogram shows a strong g 2 (0) dip down to 0.22±0.06 (Fig. 3.12), indicating that it
is in fact a single photon source.
These four experiments together, performed on the same mesa, give us very high
confidence that the optical transition of interest is the D0 X → D0 transition of a
single electron bound to a flourine donor.
3.4. OPTICAL PUMPING
3.4
43
Optical pumping
Optical pumping is the name for a process that moves population from one quantum
state to another through optical excitation. Perhaps the most common example is
certain types of lasers where population inversion is achieved by optically exciting
the electrons to the higher energy state. Optical pumping was first developed by
Alfred Kastler in the 1950s [62]. His work eventually won him the 1966 Nobel Prize
for the ”for the discovery and development of optical methods for studying Hertzian
resonances in atoms” [63]. For the work presented in this thesis, optical pumping was
used to put the electron into one desired spin state, which serves to initialize the spin
qubit.
In particular, optical pumping was used to move electron population from one
D0 spin state to the other through excitation to one of the D0 X states, as shown
in Fig. 3.13. The sample is in a 7 T field, configured in Voigt geometry. We have
named the two D0 states |0i and |1i, and the two D0 X states are |e0 i and |e1 i. A
continuous-wave (CW) laser is applied resonantly to the transition from |1i to |e1 i. If
the electron begins in state |1i, it will be excited to |e1 i, where it will then decay back
into either |1i or |0i with equal probability. If it falls into |1i, it will be re-excited to
|e1 i, and will eventually end in |0i. If the electron begins in |0i, the laser is far off
resonance with any of the optical transitions connected to that state, and the electron
will remain in state |0i. The signature of an optical pumping event is a single photon
emitted from the |e1 i → |0i transition. By collecting this photon, we know that the
electron has been pumped from |1i into |0i.
Once the electron is in state |0i, we need a method to reset the experiment so
that we can run it again. This could be performed by turning off the CW opticalpumping laser for a time longer than the T1 relaxation time of the donor-bound
electron. Unfortunately, this is very slow and is expected to be at least on the order
of ms, as is the case for other trapped electron systems. For our experiments, we
instead used a second, pulsed above-band laser to reset the experiments. Every time
this spin-randomization laser pulse hits the sample, it creates a significant number of
excitons, one of which will occasionally bind to the donor. The donor will then be
44
CHAPTER 3. FLUORINE DONORS IN ZNSE
Figure 3.13: Diagram of how optical pumping works in the F:ZnSe system. A resonant
laser pumps an electron from |1i into |e1i. From there, it can fall back into |1i and be
repumped, or it can fall into |0i and emit a single photon. Once in |0i, the electron
remains there because the laser is no longer resonant. The single photon emitted from
the transition between |e1i and |0i can be collected to count optical pumping events.
put into either |e0 i or |e1 i, with approximately equal probability since they are very
nearly degenerate. Following this, the electron will decay into either |0i or |1i with
equal probability. At this point, the electron spin has been randomized and a new
experiment can begin.
3.4.1
Time averaged optical pumping
By running many experiments and accumulating photons from the individual optical
pumping events, we can determine whether or not optical pumping is occurring.
For each experiment, there is a 25% chance that a photon will be emitted from the
|e1 i → |0i transition due to spontaneous emission just after the spin-randomization
pulse. In the absence of optical pumping, this is the only way a photon can be emitted
from that transition. There is also a 50% chance that the electron will decay to |1i
following the above-band pulse. If optical pumping is strong enough, an electron
will always be optically pumped within the 13 ns window between pulses, emitting a
photon along the |e1 i → |0i transition. Therefore, 75% of the experiments will emit
3.4. OPTICAL PUMPING
45
a single photon from the correct transition with optical pumping, compared to only
25% in experiments without optical pumping. The ratio γ between the count rate of
single photons when the optical-pumping laser is on versus when it is off is a good
measure of optical pumping, and it can take a value between 1 (no optical pumping
occurring) and 3 (saturated optical pumping).
The exact value of γ will depend upon the relationship between the rate of optical
pumping and the average experiment length. The CW optical-pumping laser is always
kept on. Therefore the rate of optical pumping, or the rate at which an electron in
|1i is excited to |e1 i is proportional to the power of the optical-pumping laser P1 ,
R = αP1 ,
(3.4)
where the proportionality constant α can be determined experimentally.
The timing of each individual optical pumping experiment, on the other hand, is
determined by the repetition rate of our pulsed spin-randomization laser. Every 13 ns,
a picosecond above-band (∼410 nm) laser pulse incident on the sample has a chance
to excite the D0 X state. If this occurs, then the spin resets and a new experiment
begins. If not, the experiment continues until a subsequent pulse resets the spin. The
experiment length is always a multiple of 13 ns, but the average experiment length,
determined by the power of the spin-randomization laser P2 , follows the equation
T =
13 ns
,
1 − β P2
(3.5)
where β, again, can be determined experimentally. Together, R and T determine γ:
γ = 1 + 2 1 − e−T R .
(3.6)
We have measured these relations experimentally, using the setup shown in Sec.
B.1. Photons from the |e1 i → |0i transition are selectively filtered by polarization,
using a polarizing beam splitter (PBS), and by frequency, using a pair of optical gratings and slits. This serves to separate the signal photons from the scattered photons
due to the optical-pumping laser. Unfortunately, this separation is not perfect, as
46
CHAPTER 3. FLUORINE DONORS IN ZNSE
the photon energies are separated by only 150 GHz. After filtering, the photons are
sent to either a spectrometer or a single photon counting module (SPCM). While
the SPCM has timing resolution and better quantum efficiency, the additional grating and slits of the spectrometer significantly reduced the background noise due to
photons scattered from the optical-pumping laser.
In one set of experiments using the spectrometer, we measured 30.2±0.8 counts/sec
with only the spin-randomization laser on, 13.0±0.6 counts/sec with only the opticalpumping laser on, 95.0 ± 3.0 counts/sec with both lasers on, and a background of
−0.2 ± 0.1 counts/sec with both lasers off (the spectrometer calibration allows for
negative count rates). The optical pumping ratio γ from these measurements was
therefore equal to (95.0 − 13.0)/(30.2 + 0.2) = 2.7 ± 0.1, indicating near-saturation
optical pumping.
In the above explaination, we have only been pumping on the |1i → |e1 i transition,
but we can achieve optical pumping by tuning the CW laser into resonance and setting
the correct polarization for any of the four transitions. By monitoring the coupled
transition, we achieved an optical pumping ratio γ larger than 2.3 for each of the four
optical transitions. This shows the fully connected nature of the optical system.
We have also measured the power-dependence of each laser when pumping on the
|1i → |e1 i transition, which follow the expected relations in Eqs. 3.4-3.6, as is plotted
in Fig. 3.14(c)-(d). In Fig. 3.14(c), the power of the spin-randomization laser was
held fixed while the power of the optical-pumping laser was varied. In Fig. 3.14(d),
the power of the optical-pumping laser was held fixed while the power of the spinrandomization laser was varied. The plotted curves are fitted using Eq. 3.6 with α
and β as fitting parameters, and show very good agreement to with the data.
Optical pumping is a resonant effect, as presented in Fig. 3.14(b). In this set
of data, the pulsed laser was held fixed while the optical-pumping laser was set at
different wavelengths with constant pumping power. This is essentially an absorption linewidth measurement. The best-fit curve is a Gaussian with a full-width at
half-maximum (FWHM) of 58 GHz. This is 25 times larger than the lifetime-limited
linewidth of 2.3 GHz (for a 70 ps lifetime [44]). Since we know that the source of the
3.4. OPTICAL PUMPING
47
Figure 3.14: (a) optical pumping scheme. (b) Confirmation of the resonant behavior
of optical pumping. The optical pumping ratio γ is plotted vs. the laser pump
wavelength. Blue data points are experimental values, while the red curve is a bestfit Gaussian curve. (c) Power dependence of the optical-pumping laser while the
spin-randomization laser power is held constant. (d) Power dependence of the spinrandomization laser with the optical-pumping laser power held fixed. The red curve
in both (c) and (d) is a theory fit using Eq. 3.6.
transition is a single donor, the Gaussian shape and large linewidth suggest fast spectral diffusion is occurring, resulting in time-ensemble averaged measurements. The
cause of the spectral diffusion has not been systematically studied, but we hypothesize it is caused by Stark shifting from charge fluctuations due to the combination of
48
CHAPTER 3. FLUORINE DONORS IN ZNSE
surface charge traps created through mesa structuring and the large number of free
carriers created with the above-band laser pulses.
3.4.2
Time resolved optical pumping
We also observed the time-dependent behavior of the optical pumping experiments
using an SPCM, as shown in Fig. 3.15. The data show an initial pulse due to
photons collected from the spontaneous emission from the |e1 i → |0i transition after
the spin-randomization pulse, followed by 13 ns of optical pumping. The width of
the initial pulse is determined by the timing resolution of the SPCM rather than by
the excited state lifetime of the D0 X state. The signature of optical pumping is the
negative slope in long tail following the initial pulse. This indicates that the average
population in state |1i is decreasing over time, implying that the average population
in |0i is increasing.
Figure 3.15: Time-dependent optical pumping behavior. (a) blue data measured with
just the spin-randomization laser on, and (b) blue data obtained with both lasers on.
In both plots, the red data is from a Monte-Carlo simulation averaged over 10x the
number of experiments used in the blue data.
3.4. OPTICAL PUMPING
49
In order to observe the optical pumping within a 13 ns time window, we had
to pump with high power from both lasers. Unfortunately, this created a lot of
background signal, as shown in Fig. 3.15. However, the difference between the data
obtained with and without optical pumping is statistically significant even despite
this large background, and indicates with high probability that optical pumping is
occurring.
Figure 3.16: Representation of the transitions simulated in the Monte-Carlo simulation. On the left is the initial spin-randomization event at t = 0 (purple laser), and
the four spontaneous emission paths (in blue and red). On the right are the four
stimulated absorption and emission paths due to the resonant laser (in blue and red).
The rates of these are determined by the laser detuning from a particular transition,
as well as the polarization of the transition and laser.
To determine the statistical significance, we modeled the time-dependent optical
pumping behavior and simulated a range of optical pumping rate. The data were
modeled using a Monte-Carlo simulation which followed the state trajectory of the
donor optical system both with and without optical pumping. The simulation first
initialized the electron into |e0 i or |e1 i with equal probability at t = 0. It was then
transferred between the 4 states according to the allowed transition rates, and the
time of jumps between |e1 i and |0i was recorded as well as the final electron state
after 13 ns (Fig. 3.16). The allowed transitions were:
(1) spontaneous emission from either D0 X state |ei i into either D0 state |ji with
50
CHAPTER 3. FLUORINE DONORS IN ZNSE
equal rate determined by the D0 X state lifetime of τ = 70 ps [44],
Rsp (|ei i → |ji) =
1
.
2τ
(3.7)
(2) stimulated emission along all four transitions due to the optical-pumping laser,
∆2
Rst (|ei i ↔ |ji) = RXe− 2σ2 ,
(3.8)
where R is the effective optical pumping rate and X accounts for the relative polarization of the optical-pumping laser and the optical transition. σ is the standard
deviation associated with the 58 GHz Gaussian linewidth determined in Fig. 3.14(b)
and ∆ is the energy detuning between the laser and the optical transition. The laser
was on resonance with the |e1 i → |0i transition and had a polarization of |Hi with
an experimentally measured extinction ratio of 2000. The detuning ∆ from the other
transitions was determined by the measured electron and hole g-factors.
Following the simulation of the photon emission times, the simulated emission
times were convolved with the measured timing resolution distribution of the SPCM
in order to accurately simulate the effect of the SPCM. And following that, a flat background was added to the simulation, equal to the mean of the measured background
caused by scattering of photons from the optical-pumping laser into the SPCM. This
background was also added to the experimental data obtained when the opticalpumping laser was off, in order to better compare the data sets.
The Monte-Carlo simulation was used to create a series of simulated data sets
by varying two parameters: the total number of simulated experiments N , and the
optical pumping rate R.
The maximum-likelihood method was then used to find the best fit to the experimental data, and the χ2 test was used to determine an interval of confidence [64].
The maximum-likelihood computation takes each time bin i and computes the probability P (R, N |xi , yi ) that the measured photon counts xi and the simulated photon
counts yi could both be measurement results of the same ‘true’ photon count rate
F (R, N, i). Assuming a Poisson noise distribution, we take the standard deviation of
√
the probability distribution that xi was a measurement of F (R, N, i) to be xi , and
3.4. OPTICAL PUMPING
51
likewise for yi . The resulting probability distribution for a single time bin is
2
1
e−(yi −xi ) /2(xi +yi ) .
P (R, N |xi , yi ) = p
2π (xi + yi )
(3.9)
The product of these probabilities for each time bin then gives the likelihood of the
experimental and simulated data coming from the same model. The log of this is
most commonly used, called the log-likelihood.
L̂ =
N
−1
X
ln [P (R, N |xi , yi )]
(3.10)
i=0
N −1
1X
= −
2 i=0
!
(yi − xi )2
+ ln [2π (xi + yi )]
(xi + yi )
(3.11)
According to the maximum-likelihood method, the model which has the largest
log-likelihood is the best-fit model. From this analysis, we obtained a maximum
likelihood pumping rate of140 MHz. We can infer from this best-fit simulation that
the rate of transition from |1i to |0i was 1/15 ns−1 , and that after 13 ns of optical
pumping, the electron ends in state |0i in 79% of the experiments.
To determine a confidence interval, we used the χ2 test. First, we compute the
test statistic X 2 of the model fit to the data
2
X =
N
−1
X
i=0
(yi − xi )2
.
xi + y i
(3.12)
Next, we compare this to the standard χ2 cumulative distribution function with N −M
degrees of freedom where N is the number of time bins, and M is the number of fitting
parameters plus one (in this case, N − M = 3).
Z
C=
X2
χ2 (x, N − M )dx.
(3.13)
0
This gives us the level of confidence C that one particular model with a test statistic
of X 2 fits to the experimental data.
52
CHAPTER 3. FLUORINE DONORS IN ZNSE
The χ2 test gives us 95% confidence that the pumping rate is between 8 and
750 MHz. At even slower pumping rates, the likelihood drops sharply, giving 99%
confidence that the pumping rate is greater than 6 MHz. This tells us that our optical
pumping data is significantly different than the data without optical pumping, and
that our model explains the difference. These results indicate that we have succesfully
demonstrated optical pumping of a single F-bound electron through the D0 X excited
state into one desired spin state.
3.5
F:ZnSe outlook
While a 79% probability of initializing the spin to the correct state is not good
enough for any sort of practical quantum computation, these results are a proof of
concept that can be improved upon. The optical pumping rate, and therefore the
resulting population, was limited by the available power of the optical-pumping laser
incident on the sample in our setup, and can likely be increased in future experiments.
Simulations with higher optical pumping powers suggest that the rate of transition
into |0i could be increased to greater than 1 ns−1 , which is eventually limited by the
lifetime of the D0 X state. At this initialization rate, it would only take 4 ns to have
a better than 99% chance of initializing the spin to the correct state.
Now that we are able to initialize the spin state, and measure the relative occupation of each spin state, the next step towards complete control of the electron
quantum state is to use fast optical pulses to control the spin state through a twophoton stimulated Raman transition[41]. This has been successfully implemented in
InGaAs quantum dots [25] and for a large ensemble of electrons bound to Si donors
in GaAs [41]. The results in this chapter show that electrons bound to F in a ZnSe
quantum well have the same structure of ground states and excited states as those
used for the stimulated Raman transition in InGaAs quantum dots, and so the same
procedure should work for F:ZnSe.
Another line of research will need to determine the source of the relatively large
inhomogeneous linewidth discussed in Sec. 3.4.1, and what can be done to avoid this
broadening. We suspect that the source of this inhomogeneous linewidth could be
3.5. F:ZNSE OUTLOOK
53
Stark shifting of the donor states due to interaction between the large number of free
carriers created by the above-band laser pulse and a series of charge traps and local
defects created by the mesa structuring. Our collaborator at the University of Paderborn who grows our samples, Dr. Alexander Pawlis, is developing samples that use a
metal mask with small holes to isolate donors in place of the mesa structuring. This
type of structure was used to improve the decoherence times of InGaAs quantum dots
over those samples where the quantum dots are located within micropillar structures,
and we believe that it will help our inhomogeneous broadening. In addition, if spin
rotation using a detuned optical pulse can be achieved, then the above-band laser
will not be required for future experiments since the spin can be reset by rotating it
from |0i to |1i.
A third line of research will be to develop the implantation technology. If a
photoresist mask is used to only allow ions to hit the sample in specified regions, then
there is no need for mesa etching. If an in situ single-impact registration technique
can also be developed for this system, then we would be able to deterministically
implant single qubits in specified locations. This would be a very important step
forward in the scalability of the system.
A fourth line of research will be to experiment with the interaction between the
electron and the nuclear spin of the fluorine donor. The extremely long decoherence
times measured for P donor nuclei in Si imply that the F nuclei could serve as longlived quantum memories. Perhaps this will require a combination of RF control of
the nuclei with optical control of the electron, or perhaps new all-optical techniques
can be developed to control the nucleus only through interaction with the electron.
The results presented in this chapter are important for the F-donor system because
they demonstrate the first practical method for controlling the quantum state of a
single donor-bound electron. While the F:ZnSe system is far behind other systems,
such as InGaAs quantum dots, the results are quickly catching up. While took 8 years
for quantum dots to go from the first evidence of the lambda system to complete
optical control (between Ref. [20] and Ref. [23]), took only 3 years to accomplish in
the F:ZnSe system (from Ref. [44] to the work presented here). In addition to moving
relatively quickly, electrons bound to flourine donors in ZnSe have many potential
54
CHAPTER 3. FLUORINE DONORS IN ZNSE
advantages over other systems, as discussed at the start of this chapter. A great deal
of advances will need to be made before this system is viable for any sort of quantum
information processing, but F:ZnSe remains a good qubit candidate.
Chapter 4
Phosphorus donors in Si
Silicon crystal technology is the most advanced of any semiconductor material, and
crystals can be grown with extremely low density of defects and impurities. As a
result of this purity, spin qubits in silicon by far have the longest relaxation times of
any other semiconductor system. Phosphorus donors in particular have gained a lot
of attention, starting with the seminal Kane proposal for using the 31 P nuclear spin in
Si as a potential semiconductor qubit [42]. More recently, P nuclear spins have been
utilized to store quantum information for more than 180 s [65], at least two orders of
magnitude longer than the best results in diamond [66], 5 orders of magnitude longer
than electrons in InGaAs quantum dots [11], and even one order of magnitude longer
than trapped ion systems, the purest of systems [67]. As with F in ZnSe, both the
donor-bound electron spin and the nuclear spin associated with P in Si can be used
as quantum bits. The donor-bound electron also has an extremely long decoherence
time, greater than 2 s [40].
The ultra-pure local environment for P donors in Si has been described as a
“semiconductor vacuum” [65], comparing the purity of the environment to the vacuum
used in trapped ion systems. This can be understood in the context of the effective
mass theory in Sec. 2.1, where a perfect crystal produces a perfectly periodic potential
that allows the electrons to act as free particles in a vacuum. The presence of any
impurities would interact with the P donor and its electron, leading to decoherence.
After reducing the number of intrinsic and extrinsic defects in Si crystals, the
55
56
CHAPTER 4. PHOSPHORUS DONORS IN SI
biggest factor in increasing decoherence times has been isotopic purification. As
discussed for ZnSe, Si has isotopes with zero nuclear spin, such as
28
Si. Through
isotopic purification, the nuclear spin of the host crystal can be depleted. Samples of
the highest-purity silicon available with 99.995%
28
Si [65] have become available as
a serendipitous side-effect of creating a better weight standard for the kilogram [68].
These samples are the ones in which extraordinarily long decoherence times have been
achieved.
Standard techniques for manipulation of the nuclear or electron spin of a P donor
utilize nuclear magnetic resonance (NMR) techniques, for the nuclear spin, or electron
spin resonance (ESR), for the electron. While these techniques can produce very
high fidelity spin rotations [65], they are based on radio frequency (RF) or microwave
frequency (MW) radiation and are significantly slower than the optical techniques
used for InGaAs quantum dots [25], operating on µs to ms timescales, instead of on
ps to ns timescales. Fortunately, the exceedingly long decoherence times achieved in
Si compensate for these longer gate operation times.
Perhaps the largest difficulty with qubits based on P in Si is in isolation of a
single P donor. Nuclear spin states have only been measured in large ensembles using
either magnetic resonance techniques [69] or, more recently, optical spectroscopy of
31
P donor-related transitions [70]. Unfortunately, NMR and ESR techniques cannot
be used to manipulate individual donors in a scalable manner, so optical or electrical
methods must be pursued.
A few recent proposals outline how a single donor spin could be measured either
electrically [71] or optically [72]. These proposals face a large number of hurdles,
beginning with the difficulty of isolating a single donor. We may compare these prior
works by dividing each proposal into two phases: a “pump” phase, in which spinselective transitions are driven, and a “detection” phase, in which a scattering process
reveals the result of the pump phase.
Optical techniques excel in the pump phase due to the easily distinguishable
hyperfine-split optical transitions in isotopically purified
28
Si [70]. However, optical
detection is very challenging because of the extremely inefficient radiative recombination due to the indirect bandgap in Si, requiring heroic efforts in cavity quantum
4.1. P:SI OPTICAL SYSTEM
57
electrodynamics to enhance the weak emission of a single donor [72]. Electrical methods have achieved great success in the detection component of the measurement [39],
but electrical scattering combined with MW driving introduces many noise processes,
quickly relaxing the measured spin [71]. Microwave fields are also difficult to localize
to a single device, an important consideration for future quantum computers.
In this chapter, I will discuss the P:Si optical system and some of its interesting
features and results. I will then explain a method for isolating a single P impurity and
measuring the electron and nuclear spin using a hybrid optical and electrical scheme,
and end with some preliminary experimental results suggesting that this scheme could
be successful if implemented.
4.1
P:Si optical system
Silicon has an indirect bandgap. As mentioned in Sec. 2.3, this means that optical
dipole coupling between the D0 and D0 X states is very small, leading to a very long
optical lifetime of the D0 X state (the D0 X → D0 optical transition was explained in
Sec. 2.3). Most photon emission is paired with a phonon emission in order to make
up the momentum mismatch, but occasionally the donor will make a transition along
the no-phonon (NP) line. The most dominant decay mechanism, however, is Auger
recombination followed by ionization of the donor. Table 4.1 lists the lifetimes of
these three decay mechanisms from the D0 X state [73].
Decay Mechanism
Auger recombination
Phonon-assisted optical recombination
No-phonon optical recombination
lifetime
272 ns
600 µs
2 ms
Table 4.1: Contribution of the three main decay mechanisms to the D0 X lifetime [73].
P impurities have an electron binding energy of 29 meV at low temperature, and
the binding energy of the exciton to the neutral donor is 5 meV. This results in nophonon (NP) optical transitions with a wavelength of 1078 nm, near the bandgap.
One of the most interesting and useful features of the P:Si optical system is that
58
CHAPTER 4. PHOSPHORUS DONORS IN SI
the long decay lifetime and high purity of isotopically purified Si samples lead to
extremely narrow D0 X → D0 NP optical linewidths. When emission is coupled with
a phonon, the emitted photon has noticeably less energy, and the exact energy will
depend upon the type of phonon. The NP transition is very narrow compared to
the phonon-assisted sidebands, which are broadened by the distribution of phonon
energies, and therefore contain much less information.
The relation between the decay lifetime and the homogeneous optical linewidth
is given by Eq. 3.1. This 0.6 MHz NP linewidth is only four times smaller than
the narrowest experimentally measured homogenous linewidth of 2.4 MHz [74]. Due
to the high purity of available Si samples, a large ensemble of P donors has an
inhomogeneously-broadened linewidth as narrow as 36 MHz [70]. This is noticeably
smaller than the hyperfine splitting of 60 MHz, and results in optically distinguishable
nuclear spin states, even in large ensembles.
The main difference between the optical spectra for the P:Si samples we will
consider here and that for the F:ZnSe samples discussed in Sec. 3.2 is that the P:Si
samples are bulk, unstrained samples. Therefore, both the HH and LH D0 X states
are important, and the hole wavefunction always aligns with any reasonably large
applied magnetic field. This results in four D0 X states. Due to the visible hyperfine
splitting, there are four D0 states as well, as explained in Sec. 2.4.
At large magnetic fields, this results in 12 allowed transitions between the eight
states, as computed in Sec. A.2 and shown in Fig. 4.1. In natural Si, the transitions
are very broad, and no hyperfine splitting is visible. However, when looking at isotopically purified samples, all 12 individual peaks can be resolved. The polarization
and relative intensities of these transitions are also explained in Sec. A.2.
As a result of the indirect bandgap, photoluminescent spectroscopy is not as useful
for quantum information as it was for ZnSe since very little photoluminescence is
emitted from the Si sample. However, resonant absorption can still be utilized for
measurements of the donor system, as explained in Sec. 4.4.1.
4.2. PHOTOLUMINESCENCE EXCITATION SPECTROSCOPY
59
Figure 4.1: (a) Optical transition between the D0 and D0 X states. The electrons form
a singlet. (b) In a magnetic field, the D0 X state is split into the four Zeeman levels
of the spin- 23 hole, while the D0 state is split into two electron Zeeman levels, each
split again by the hyperfine coupling to the nuclear spin. Spin selection rules lead to
12 allowed optical transitions.
4.2
Photoluminescence excitation spectroscopy
Optical pumping is a useful tool for Si, despite the indirect bandgap. One example
of this is with photoluminescence excitation (PLE) spectroscopy. PLE is a really
fantastic spectroscopic tool. It is an absorption method, where a narrow-linewidth
CW laser is scanned while the emission of photons at another energy is collected.
The resulting relation between the pump laser wavelength and the emission rate into
the other energy band produces a very good spectra absorption plot. For example,
the data in Fig. 1 of Ref. [70] was obtained using PLE spectroscopy.
In this case, we are interested in pumping on the D0 X NP transition, while monitoring the phonon-assisted sidebands. For silicon, we can put in a lot of power to
60
CHAPTER 4. PHOSPHORUS DONORS IN SI
get a high absorption rate despite the indirect bandgap, and at the same time we
can take advantage of the stronger emission into the phonon-assisted sidebands. The
resolution is determined by the linewidth of the resonant laser, and the sensitivity is
determined the rate of emission into the phonon-assisted sidebands. Without PLE,
we wouldn’t know how narrow the D0 X NP transition really is.
PLE is fundamentally an optical pumping experiment. The donor-bound electron
is pumped into the D0 X state, just like for ZnSe in Sec. 3.4. However, when the
electron decays into the D0 state, it emits both a photon and a phonon. The lower
energy photon can easily be distinguished from a pump photon, and the emission rate
will be proportional to the absorption rate.
Professor Thewalt at Simon Fraser University has performed the highest resolution optical spectroscopy of donor and acceptor impurities in Si using PLE [70, 74].
His experiments using isotopically purified Si allowed him to determine the hyperfine coupling constant between the P donor and the donor-bound electron spin to a
precision many orders of magnitude higher than previous results. While doing these
measurements, he came across an unexpected splitting of the D0 X NP line at zero
field which was in addition to the known hyperfine splitting. Using some data taken
while I visited his lab in 2007, we were able to work out the probable source of this
zero-field splitting. These results are explained in the next section.
4.3
Zero-field splitting
The samples used for this measurement were so pure that they should have been
essentially strain-free. Therefore, the only observable splitting at zero field would
be the hyperfine splitting. However, the data showed not two, but four peaks. The
four were divided into two pairs of lines, where the lines within a pair were split by
an energy equal to the hyperfine splitting. The cause for the splitting between the
pairs was not clear, but it was observed that this splitting was dependent upon the
concentration of P doping in the sample. The higher the doping concentration, the
larger the splitting. Two example plots are shown in Fig. 4.2 here, and Fig. 3 of
reference [70].
4.3. ZERO-FIELD SPLITTING
61
Figure 4.2: PLE spectra showing the zero-field splitting of the D0 X no-phonon transition. Four peaks are shown here (the center peak appears broadened because it
contains two overlapping peaks), which are split first by the hyperfine splitting, and
second by some unknown splitting with approximately equal magnitude.
As it turns out, this zero-field splitting can be explained very well by a small
amount of non-isotropic strain. The exact mechanism for creating this strain is not
known, but it seems to be caused by the P doping. In order to model this splitting,
we fit a strain model to a set of PLE data taken from the same sample at varying
magnetic field. The model parameters obtained from fitting the strain model to the
PLE data at non-zero magnetic field led to a splitting at zero field that matched the
observed splitting.
4.3.1
Strain Model
To model the splitting, we composed a separate Hamiltonian for the excited D0 X
states and the ground D0 states, and then calculated energies and probabilities of
transitions between these states. For the D0 state, our hamiltonian is comprised of
the electron and nuclear Zeeman terms in addition to the hyperfine point contact
interaction term,
HD0 = µB ge S · B − µn gn I · B + AS · I.
(4.1)
62
CHAPTER 4. PHOSPHORUS DONORS IN SI
Here, µB = 13996 MHz/T is the Bohr magneton and µn = 7.6226 MHz/T is the
nuclear magneton. ge and gn are the electron and nuclear g-factors, respectively, and
A is the hyperfine constant. S and I are the electron and nuclear spin operators,
respectively, and B is the magnetic field vector.
For the D0 X state, we used the Pikus and Bir effective strain Hamiltonian for
J = 3/2 holes [59] in addition to the Zeeman terms. In this excited state, the two
electrons combine into a spin singlet, and only the hole is affected by the magnetic
field. We assume that the hyperfine coupling between the hold and the nucleus in
the excited state is negligible since there is no point contact term for the p-like hole
wavefunction.
HD0 X = −µB gh1,h3 J · B − µn gn I · B + a(exx + eyy + ezz ) + b[(Jx2 − J 2 /3)exx + c.p.] (4.2)
Here, c.p. means cyclic permutation of {x, y, z}. gh1 and gh1 are the g-factors for
the spin-1/2 and spin-3/2 holes, respectively, and exx , eyy , and ezz are the diagonal
elements of the strain tensor (we assume there is no sheer-strain). J is the spin-3/2
hole spin operator, and a and b are the deformation potentials for silicon, −2.4 × 109
MHz and −5.3 × 108 MHz, respectively [59].
Any isotopic strain (exx = eyy = ezz ) is indistinguishable from a constant energy
shift equal magnitude for all transitions. While the PLE data used for the fitting was
sensitive to relative energy, the data does not have good absolute energy resolution.
Therefore, for the purposes of fitting, the sum of the strain was artificially set to zero
to avoid any unnecessary free-parameters.
With the addition of strain, the four optical transitions which are normally considered to be “forbidden”, since they correspond to a change of two units of angular
momentum, become allowed. This is due to mixing between the HH and LH states
in the presence of strain. This brings the total number of optical transitions from 12
to 16.
By solving these Hamiltonians (Eqs. 4.1 & 4.2), the energy levels of each state
can be found, and the energies of all 16 nuclear-spin-preserving optical transitions can
be obtained. Furthermore, the eigenstates of the Hamiltonians in combination with
4.3. ZERO-FIELD SPLITTING
63
spin selection rules from Fermi’s golden rule determine the relative optical transition
amplitudes (see Sec. A.2).
4.3.2
Fitting Results
Nineteen sets of PLE data taken at magnetic fields varying from 1711 G to 0 G were
individually fit to this model. The magnetic field axis was defined to be the z-axis. To
obtain best fits, 16 Lorentzian peaks at the relative energies produced by the model
were summed and compared to the PLE data. In addition to the strain model, which
determined transition energies, a number of other fitting parameters were necessary to
model data. An absolute amplitude and a background intensity were fit independently
to each of the PLE plots to account for variations in experimental setup such as
varying collection efficiency and background light. Relative PLE peak heights were
modeled by including the relative populations of the ground states pi . The geometry
was somewhere in between Faraday and Voigt, and the exact angle between the laser
axis and the magnetic filed was not measured, so an additional polarization angle
fitting parameter, θ, was added to account for the transition selection rules in the
Energy Fitting Parameters:
electron g-factor
ge
spin-3/2 hole g-factor
g3h
spin-1/2 hole g-factor
g1h
hyperfine coupling constant
A
diagonal strain tensor elements
exx , eyy , ezz
Amplitude Fitting Parameters:
polarization angle
θ
Lorentzian linewidth
γ
population of the ground states p1 , p2 , p3 , p4
total transition amplitude
N
background intensity
I
Table 4.2: Fitting parameters for the strain model. The model was fit to each set
of PLE data by varying these parameters. Note that the hyperfine coupling A and
the strain components also slightly affect the transition amplitudes by modifying the
eigenstates. The sum of the populations of the four ground states was constrained to
be 1 (p1 + p2 + p3 + p4 = 1), and the same linewidth γ was used for all 16 transitions.
64
CHAPTER 4. PHOSPHORUS DONORS IN SI
two geometries (Sec. A.2). A Lorentzian linewidth γ, equal for all peaks, was also
included. The nuclear g-factor did not modify the eigenstate energies sufficiently
enough to be observable within our data, so gn was fixed to 1.13 in computations and
not varied during the fitting [42]. The fitting parameters are summarized in Table
4.2.
From these fits, the energy of the 16 optical transitions was extracted from each
PLE data set corresponding to a particular magnetic field value. These are plotted
as diamonds in Fig. 4.3. The full-width-half-maximum (FWHM) linewidth observed
in fitting was 60 ± 10 MHz across all 19 data sets.
We next fit those transition energies as a function of magnetic field to the same
model as above (excluding the amplitude fitting parameters). In this case, one set of
energy fitting parameters was fit to the data from all magnetic fields. The resulting
energies are shown as lines and circles in Fig. 4.3, and the best-fit parameters are
given in Table 4.3. Based on the agreement between the model and the data, we
believe this zero-field splitting is caused by a very small amount of non-isotropic
strain on the order of 10−8 .
This is an extremely sensitive measurement of strain, only made possible by the
extremely narrow linewidth of the D0 X NP transition. It is unclear why the strain is
dependent on the P-doping concentration and what is the exact nature of the source
of this strain. Negative strain along one axis and positive strain along the other
might seem to suggest a directional force such as gravity. However, for a sample 1
cm in length, the expected strain due to gravity would be over an order of magnitude
smaller than that observed here, and gravity would not explain the dependence on
phosphorus concentration. Data fitting across many samples of varying concentration
and other properties would have would have to be performed in the future in order
to determine the exact cause of the strain. Unfortunately, while these result could be
considered novel, the small strain observed is most likely is not important enough to
pursue an explanation.
4.4. ELECTRICAL DETECTION
65
(a) Full Field
(b) Low Field
Figure 4.3: (a) Plot of transition peaks obtained from fitting to data, along with
the model fit to the peak positions. (b) Zoomed plot showing the low magnetic field
range.
4.4
Electrical detection
PLE spectroscopy has great resolution, but still suffers from silicon’s indirect bandgap.
Only one out of every 2000 excitation events results in the emission of a photon. The
66
CHAPTER 4. PHOSPHORUS DONORS IN SI
Energy Fitting Parameters:
ge
1.97
g3h
0.85
g1h
1.30
A
121.1 MHz
exx
−4.6 × 10−8
eyy
4.3 × 10−8
ezz
4.5 × 10−9
Table 4.3: Best-fit values of the fitting parameters for the strain model fit to the
transition energies of all 19 PLE plots.
rest of the events result in the emission of a free electron into the conduction band.
For this reason, electrical detection has the potential to be much more sensitive than
optical detection.
There are two main methods of electrical detection. The first is to detect the free
electron kicked out of the D0 state through Auger recombination. In bulk, this is a
photoconductivity measurement. The pump part of the measurement is the same as
in PLE, where a resonant laser is scanned across the region of interest and resonantly
excites the D0 X state. However, instead of detecting photon emission, a change in the
conductivity of the sample as a function of pump wavelength produces an absorption
spectra. Fig. 4 of Ref. [70] shows an example photoconductivity spectra. In that case,
the sample was p-type (boron doping), and so the photo-emitted electrons ionize the
acceptors, resulting in a negative correlation between the photoconductivity of the
sample and the absorption spectra. Measurements must be averaged over either an
ensemble of donors, or over multiple emission and absorption events from a single
donor in order to obtain an adequate signal-to-noise ratio.
The second method of electrical detection is to detect the ionized donor. Since
the ionized donor can affect many electrons, it can result in a more sensitive method
for measurement. In particular, by constraining the flow of electrons within a narrow
interaction region near the donor, the conductivity of those electrons can be strongly
affected by the ionization of the donor. Two methods for constraining the electron
channel will be discussed here. The first is to use a quantum point contact and the
integer quantum Hall effect, which will be discussed in theory. The second is to use a
4.4. ELECTRICAL DETECTION
67
FinFET transistor, for which some preliminary experimental results will be presented.
Either scheme, when coupled with the nuclear-spin selective optical pumping, can be
used as a projective non-demolition measurement of a single phosphorus nuclear spin.
4.4.1
Quantum Hall charge sensor
In this section, we propose a novel scheme that employs the advantages of both
optical and electrical measurement techniques in order to overcome the difficulties of
each. By combining the hyperfine selectivity of optical pumping with the sensitivity
of electrical detection, our proposed measurement device can perform deterministic
quantum non-demolition projective measurements of a single
31
P donor nuclear spin
in Si. One scheme of note that combines optical pumping with electrical detection
together, the “optical nuclear spin transistor”, was previously mentioned in [70].
We extend this scheme by using a quantum Hall bar device instead of a normal
transistor to perform the electrical measurement. By employing the integer quantum
Hall effect (IQHE) in a conductance plateau region, electrical noise due to defects and
background magnetic fields are suppressed. Additionally, a quantum point contact
(QPC) makes the electrical device sensitive to only the small volume surrounding the
single
31
P donor, isolating the desired signal from noise due to other impurities and
the electrical contacts.
Besides offering a measurement technique which overcomes difficulties of existing
measurement proposals, the IQHE may also introduce a possible method for performing two-qubit gates between two donor nuclei [75], where the extended-state edge
channels can coherently couple two donors. In addition to the possibilities for gate
and measurement operations, this scheme has many advantages in terms of controllability and integration. Many of the donor interaction parameters can be electrically
controlled by the Hall bar device, such as the strength of the interaction between
the edge channels and the donor. Furthermore, this device can be fabricated consistent with current CMOS fabrication techniques and is easily integrated with other
electronics on the same chip, as opposed to other systems such as diamond.
In Sec. 4.4.2 we explain the measurement scheme and device structure, and then
68
CHAPTER 4. PHOSPHORUS DONORS IN SI
in Sec. 4.5 we elaborate on the particular interactions and effects occurring within
our system, discuss our simulation of the device physics, and explain how many of
the interactions can be tuned by the device parameters.
4.4.2
Description of device and measurement scheme
Our device is composed of three main components (Fig. 4.4): a basic silicon metal
oxide semiconductor field-effect transistor (MOSFET) Hall bar device, a single P
donor, and a QPC which surrounds the donor. The measurement also employs one
or two external narrow linewidth continuous wave (CW) lasers which can be tuned
to the set of neutral donor to donor bound exciton optical transitions. We will
first describe the system in an ideal case, where many of the complex interactions
within semiconductors are ignored (these effects are discussed in Sec. 4.5). For now,
Figure 4.4: A device schematic showing the important components. The device is
comprised of a MOSFET Hall bar on a p-type substrate with a global gate to create
and tune the inversion layer. Above the donor and global gate is the QPC gate (blue),
separated from the rest of the device by an oxide layer (green). An aperture in the
global gate (not shown) allows optical illumination in the proximity of the donor while
preventing illumination of the source, drain, and measurement electrodes.
4.4. ELECTRICAL DETECTION
we assume a perfect
28
Si crystal with only one
69
31
P donor, we ignore the effects of
the substrate, oxide, and electric field on the donor and likewise neglect spin-spin
scattering between conduction electrons and the donor electron.
The basis of the measurement device is a MOSFET Hall bar on a p-type substrate
which exhibits the IQHE when placed in a static perpendicular magnetic field at
low temperatures. In the IQHE, the transverse conductance across the Hall bar as
a function of magnetic field becomes quantized and exhibits conductance plateaus
(with different filling factors ν). The longitudinal conductance down the length of
the Hall bar is equal to zero during these plateaus and exhibits sharp peaks during
the transitions. This conductance quantization is a result of the transformation of
the momentum plane-wave eigenstates of the 2DEG at zero field into edge channels
at nonzero field, localized at discrete distances from the boundaries of the 2DEG
(Fig. 4.5(a)). Both the distance between edge channels and the width of the channels
p
are determined by the magnetic length lb = h̄/m∗ ωc where ωc = eB/m∗ is the
electron cyclotron frequency, m∗ is the electron effective mass in silicon, and B is the
Figure 4.5: IQHE edge channels. (a) Only one occupied edge channel (yellow) tunnels
into the QPC. (b) The edge channels are transmitted in the D+ (ionized) case, while
(c) the edge channels reflect from the donor in the D0 (neutral) case.
70
CHAPTER 4. PHOSPHORUS DONORS IN SI
magnetic field [76]. The edge channels have energy spacing h̄ωc , which is significantly
larger than the energy width of each edge channel at low temperature determined
by the Fermi-Dirac distribution. Consequently, the 2DEG conductance is quantized
whenever the Fermi energy falls between two edge channel energies, which is described
as having an integer filling factor ν.
Our scheme takes advantage of the IQHE phenomena to improve the robustness
and decrease the noise in our measurement device. By operating within the ν = 1
filling factor regime, only one edge channel is populated. Due to the chirality of the
edge channels, forward and backward propagating channels lie spatially separated, on
opposite sides of the Hall bar. Since the next edge channel is separated significantly
in energy from the first (∼ 0.7 meV at 2 T field, corresponding to ∼ 8 K), at low
enough temperatures, electron scattering from one edge state to another by impurities
or other defects is negligible, and the measured conductance is insensitive to these
defects. Similarly, at the ν = 1 conductance plateaus, it is insensitive to magnetic
field variations, further reducing noise.
While we want the measurement to be insensitive to most defects in the device,
we need the device to interact with one particular defect, a P donor which is located
just below the inversion layer in the center of the Hall bar. This donor could be
implanted using single ion implantation through the optical aperture in the global
gate [77], or be placed via registered STM techniques [78]. To enable the coupling
between the Hall bar and the donor, a QPC is located above the donor, insulated
from the global gate by the oxide layer (Fig. 4.4). With an appropriate potential, the
QPC restricts the edge channels and forces the two states with opposite wavevectors
to slightly overlap, allowing scattering from one edge state to the other. Due to its
proximity to the 2DEG, the donor can scatter the edge channels within the QPC
region, and the scattering rate will be different when the donor is ionized (D+ state)
compared to when it is neutral (D0 state), as shown in Fig. 4.5 (b) and (c). Edge
state resonant scattering from a single impurity within a QPC has previously been
observed and successfully simulated [79]. However, this effect was only observed near
a transition between plateau regions and only when the Fermi energy was resonant
with the impurity state, and a change in scattering due to the ionization of the
4.5. DEVICE PHYSICS AND SIMULATION
71
impurity was not studied.
We can ensure that the donor remains neutral in the absence of excitation-causing
radiation by setting the Fermi energy above the ground state energy of the neutral
donor near the 2DEG while keeping the device cold. We can then control the ionization state of the donor through the use of a narrow linewidth CW laser tuned to one or
more of the D0 →D0 X transitions of the P donor, described in Sec. 4.1 (Fig. 4.1(a)).
When tuned to one of these transitions, a laser will selectively ionize the donor only
if it has the particular nuclear spin and electron spin corresponding to the ground
state of that transition, otherwise leaving the donor neutral. If we excite a pair of
transitions with the same nuclear spin state but opposite electron spin states (using
either a pair of lasers or one alternating between these transitions), the ionization
will only be dependent upon the nuclear spin state. This ionization will modify the
transport of the edge channels through the QPC until the donor recaptures another
electron from the 2DEG.
The nuclear spin state will have negligible probability of flipping throughout hundreds of thousands of repetitions of this process [72], so the laser(s) can re-ionize the
donor again, followed by electron recapture, and so on. By monitoring the transverse
and longitudinal conductivity across the Hall bar, we should observe a random telegraph signal if the donor is in one particular nuclear spin state, and no change in
conductivity if the donor is in the other spin state. If the change in conductivity is
very large, a single-shot measurement would also be possible. Either method results
in a deterministic quantum non-demolition measurement of the single donor nuclear
spin.
4.5
Device physics and simulation
In this section, we will discuss the important physical effects that occur within our
device, how we model these effects, and how to tune the device parameters to make
the measurement feasible. Considerations include the effect of the oxide interface on
the donor-bound electron, scattering of the edge channels due to the donor, ionization
and re-capture rates for the donor, and optical linewidths and hyperfine splitting with
72
CHAPTER 4. PHOSPHORUS DONORS IN SI
oxide-modified states.
4.5.1
Donor electron ground state
The presence of the oxide layer next to the donor by necessity will modify the donor
electron state, and the distance from the inversion layer to the donor is an important
quantity. If the donor is too close to the oxide, the inversion layer will strip the
electron from the donor. However, if the donor is too far from the oxide, the donor
potential cannot affect edge channel scattering in the QPC. Ideally, the combination
of the donor and the 2DEG potentials will induce a donor-electron ground state which
is partially located at the donor position and partially located within the inversion
layer. Recent work on single As donors near an oxide layer suggests that the desired
hybridization regime is obtainable [80].
In order to model the donor and 2DEG, we have numerically calculated approximate eigenstates of the system’s Hamiltonian. We first construct an effective Hamiltonian for the system, making a number of simplifying assumptions. We assume a
homogeneous, perfect crystal with a single effective mass m∗ (obtained by the geometric average of the effective mass along the three principal axes in Si), and at this
time we ignore valley-orbit coupling and spin effects. Our Hamiltonian has the form
1
H=
2m∗
h̄
∇ + eA
i
2
+ V2deg + Vqpc + Vd .
(4.3)
We take the vector potential to be A = Bxŷ, and have used a field of 2 T in our
calculations.
V2deg defines the potential for the 2DEG inversion layer created by the global gate
and the oxide. We take the origin to be located at the position of the donor and
the positive z-axis to be perpendicular to and point towards the oxide interface. We
approximate this potential as
V2deg =

−2γ
exp [(z−z0 )]/d]
1+exp [(z−z0 )/d]
for z < z0

Vb
for z > z0
,
(4.4)
4.5. DEVICE PHYSICS AND SIMULATION
73
where z0 is the distance from the donor to the oxide, d is the width of the inversion
layer (taken to be 5 nm in our calculations), γ is the depth of the interface potential
(taken to be 15 meV), and Vb is the energy difference between the Si band edge and
the oxide band edge (3 eV).
Vqpc is the potential term for the QPC channel. We have modeled Vqpc as a
parabolic potential along the x-axis, which is perpendicular to the direction of the
edge state propagation. We assume that the QPC is centered on the donor and that
the potential is uniform along the length of the QPC channel in the y-direction, giving
us
1
Vqpc = m∗ ωq2 x2 .
2
(4.5)
Here, ωq defines the strength of the QPC confinement, chosen to produce a parabolic
potential of 0.26 µeV/nm2 in our calculations.
Vd is the effective donor nucleus potential, which is approximated as
Vd = −
h̄2
β
p
,
∗
∗
ma
r2 + rs2
(4.6)
where r2 = x2 + y 2 + z 2 , m∗ is the effective mass averaged over the three directions
(0.33me , where me is the bare electron mass), a∗ is the effective Bohr radius in Si
from effective mass theory (20 Å) and β ' 1.26 is a correction to effective mass theory
which gives the correct bulk binding energy [81]. rs is a phenomenological screening
distance which simulates inner-shell electron screening (taken to be 5 Å).
We next use this effective Hamiltonian to calculate approximate eigenstates of
the system, by constructing a set of basis states and diagonalizing the Hamiltonian in
this basis. We expect that our eigenfunctions will be hybridized wavefunctions with
a donor-electron-like component and an edge-channel-like component. We employ
two separate orthonormal sets of basis states, one of approximate eigenstates for
the donor electron and one of approximate eigenstates for the 2DEG edge states,
and orthogonalize the combination of basis states using the standard Gram-Schmidt
orthonormalization procedure. By using the combined basis, our basis states will
match the shape of our eigenstates well, aiding the numerical computations.
74
CHAPTER 4. PHOSPHORUS DONORS IN SI
For the donor-electron-like basis states, we simply use normalized hydrogenic
wavefunctions centered on the donor, notated as |ψnlm i with
ψnlm = Rnl (r)Ylm (θ, φ).
(4.7)
While these donor states are not exact eigenstates of Vd , they are similar enough to
form a good basis set for the donor-electron-like portion of the wavefunction. In the
absence of an oxide, the donor electron energy is approximated by the energy of these
states in a Coulomb potential,
Enlm
h̄2 β 2 1
= − ∗ ∗2 2 .
2m a n
(4.8)
For the edge-channel-like basis states, we use the normalized product of HermiteGaussian functions in the x-direction, momentum plane-wave functions in the ydirection, and Airy functions in the z-direction, notated as |φpqk i where
1
φpqk = Xp (x − xk )Zq (z − z0 ) √ eiky
Lc
2
1 exp − (x − xk ) /2σx
(x − xk )
Xp (x − xk ) = √ p
Hp
σx
2 p!
(σx2 π)1/4
1
(z − z0 )
0
Zq (z − z0 ) = 0 0 √ Ai αq −
.
σz
Ai (αq ) σz
(4.9)
(4.10)
(4.11)
Here, αq0 is the qth root of the Airy function (Ai) and Lc is the channel length, which
defines the set of allowed wavevectors and is taken to be 300 nm. The widths σx and
σz are given by
r
h̄
m∗ ωt
2 13
h̄ d
σz =
,
γm∗
σx =
(4.12)
(4.13)
and xk indicates the displacement of the chiral edge states from the center of the
4.5. DEVICE PHYSICS AND SIMULATION
75
QPC
xk = −
where ωt =
h̄ky ωc
,
m∗ ωt2
(4.14)
p
ωq2 + ωc2 is the combination of the QPC frequency ωq with the cyclotron
frequency of the electron in the magnetic field ωc = eB/m∗ .
The products of the Hermite-Gaussians in the x-direction and the momentum
eigenstates in the y-direction are exact edge-state-like eigenstates of the combination
of Vqpc and a magnetic field along the z-axis. The Airy functions are not exact
eigenstates of V2deg , but are eigenstates of the triangle potential Vtri where
Vtri

−γ(1 + (z − z0 )/2d) f orz < z0
,
=

∞
f orz > z0
(4.15)
which is approximately equal to V2deg close to the oxide [82]. As a result, the states are
very similar to the eigenstates of the 2DEG potential. Therefore, this combination of
Figure 4.6: Energies of eigenstates as the distance between the donor and oxide
is varied, measured with respect to the binding energy of P in bulk Si. Solid lines
indicate donor-like states |Ψm i and dotted lines indicate 2DEG-like states |Φn i. Anticrossing and crossing points are indicated by symbol ‘o’ and ‘x’ respectively.
76
CHAPTER 4. PHOSPHORUS DONORS IN SI
the Hermite-Gaussian and plane-wave in the xy-plane and the Airy functions along
the z-direction form a good basis set for the 2DEG edge-state-like portion of the
wavefunction. In the absence of the donor, the energy of these states is approximated
by their energy in the triangle potential Vtri and the QPC potential Vqcp , giving us
Epqk
h̄2 k 2 ωq2
h̄2
1
+
−
= h̄ωt p +
α0 − γ.
2
2m∗ ωt2 2m∗ σz2 q
(4.16)
Our two sets of basis states are not orthogonal, which is necessary for Hamiltonian
diagonalization. Therefore, we choose a subset of basis functions from each type of
state and orthogonalize them using the standard Gram-Schmidt orthonormalization
procedure. After this, we can rewrite our Hamiltonian in terms of these new basis
states and diagonalize it to find the hybridized eigenstates of the donor electron. Some
of these states will be more donor-like, and we will label those states as |Ψm i. The
states that are more 2DEG-like will be labeled as |Φn i. Fig. 4.6 shows the energies
of a few of these eigenstates as a function of the distance between the donor and the
oxide, and Fig. 4.7 shows the wavefunction in blue for the lowest energy donor-like
Figure 4.7: Contour plot of the potential energy in the xy plane, for a donor that is 10
nm from the oxide. Plot also shows the amplitude of three eigenstate wavefunctions:
the lowest donor-like state in blue, a forward-propagating edge state in red, and a
backward-propagating edge state in green.
4.5. DEVICE PHYSICS AND SIMULATION
77
state |Ψ0 i at a donor-oxide distance of 10 nm.
From these simulation results, we can draw a number of conclusions. We notice
that the binding energy of the lowest D0 -like state is not significantly modified by
the presence of the oxide unless the donor is less than 5 nm from the oxide. This is
important for two reasons. First, the P donor will continue to be capable of binding
an electron while close to the oxide and the 2DEG. Second, the fact that the electron
binding energy changes only very slightly suggests that the donor will also bind an
exciton in the presence of the oxide.
Theoretical approximations [83, 84] and indirect experimental evidence [85] indicate that the Bohr radius of the donor-bound-exciton in bulk Si is in the range of
3.5-5.0 nm. This implies that the D0 X state will not significantly overlap with the
oxide, and interaction with the 2DEG potential will be the dominant perturbation
affecting the state. Since it is very challenging to calculate the binding energy of the
complex four-quasi-particle D0 X state in the presence of the 2DEG potential, we will
use Hayne’s rule as an approximation (Eq. 2.16). For Si, Hayne’s rule says that the
proportionality constant is 1/10, ED0 X ' ED0 /10 [31]. Although the donor is not in
a bulk environment, the similarity of the D0 and D0 X Bohr radii implies that the
two states experience alike environments, supporting the validity of the approximation. Following this rule, the exciton binding energy of 5 meV will vary only slightly,
allowing the donor to continue to bind an exciton.
Analysis of the wavefunction of the D0 lowest-energy eigenstate also shows that
the wavefunction amplitude at the position of the donor is not significantly modified
unless the donor is less than 5 nm from the oxide. This is important because it tells
us that the hyperfine coupling between the donor electron and the nucleus should
still be close to the bulk value, as discussed in Sec. 4.5.4. For this reason and the
two above, we expect that the optical properties of the D0 X state such as the optical
transition linewidth will be similar to that observed in bulk, which is also discussed
in Sec. 4.5.4.
This simulation has given us an important lower-bound to the distance of the
oxide from the donor. The next section will provide an upper-bound, at which point
we’ll discuss the tolerances of this distance, and what methods we have to tune our
78
CHAPTER 4. PHOSPHORUS DONORS IN SI
device in order to detect a donor at various depths.
4.5.2
Edge Channel Scattering
Outside of the QPC, edge channel states are spatially separated, and we assume
there is negligible scattering from one state to another. However, within the QPC,
the tightly confined parabolic potential causes the edge states to overlap, allowing
mixing. By tuning the magnetic field and the QPC voltage, we can allow only the
lowest Landau level (ν = 1) to tunnel through the QPC. Within the confinement of
the QPC, the forward- and backward-propagating edge channels will overlap with the
donor potential, where they will be scattered with some amplitude. This amplitude
will change when the donor becomes ionized, which can be detected by monitoring
the conductance through and reflection from the QPC.
The scattering potential in the ionized donor case is simply Vd from Eq. 4.6. For
the neutral donor case, we must use the donor-bound-electron potential Ve in addition
to the donor potential Vd . This potential describes the Coulomb interaction, including
exchange terms, between the donor-bound-electron in the lowest energy state having
wavefunction |Ψ0 i (as computed in Sec. 4.5.1), and a second electron. This potential
can be written as
Ve
h̄2 β
=
m ∗ a∗
Z
Z
dx
0
dx
|xi hΨ0 |x0 i hx0 |Ψ0 i hx|
|x − x0 |
|xi hx|Ψ0 i hΨ0 |x0 i hx0 |
−
.
|x − x0 |
(4.17)
For this computation, we have assumed the electrons are spin polarized as expected
for the ν = 1 regime.
To calculate the scattering amplitudes, we use the edge-channel basis states from
Eq. 4.9. We assume that the occupied edge channel states are well represented by
the lowest energy states, with p = q = 0 and k determined by the Fermi energy. The
scattering amplitude matrix elements for the scattering from a forward-propagating
edge channel with wavevector +k to a backward-propagating edge channel with the
4.5. DEVICE PHYSICS AND SIMULATION
79
same energy and wavevector −k are
Sk0 = hφ00−k | Vd + Ve |φ00+k i
(4.18)
Sk+ = hφ00−k | Vd |φ00+k i .
(4.19)
The squares of these scattering amplitudes are shown for three different wavevectors in Fig. 4.8. The difference between the squared matrix elements in the neutral
case (dashed lines) and the ionized case (solid lines) shows the first-order change in
scattering due to ionization of the donor, and gives strong indication that this change
will be substantial for donor depths less than about 20 nm.
Figure 4.8: Amplitude squared of edge state scattering between edge states off the
neutral donor within the QPC as a function of the distance between the oxide and the
donor. Scattering for edge states with three different wavevector in both the ionized
(solid line) and neutral (dotted line) donor cases is shown.
The fact that the interaction with the donor is relatively strong suggests that a
significant portion of a quantum of conductance would be reflected by the presence of
the donor. Furthermore, we see that the scattering amplitudes can be tuned by many
orders of magnitude by adjusting the wavevector (or equivalently the Fermi energy),
80
CHAPTER 4. PHOSPHORUS DONORS IN SI
from negligible scattering with large k, to strong scattering with smaller k.
We can also gain some insight from the relationship between the scattering matrix
elements in the two cases. First, we notice there are two interaction regimes. When
the donor is close to the oxide, where the exchange interaction is strongest, the
edge channels are scattered more by the presence of the donor electron than by the
donor alone. When far from the oxide, where Coulomb interaction dominates over
the exchange interaction, the edge channels are scattered more by the ionized donor
than by the neutral donor. In each regime, the scattering rates differ between the two
cases by a few orders of magnitude, which suggests that there will be a large difference
in conductivity, a necessity for the success of this measurement scheme. In order to
achieve both strong scattering and a large difference in scattering between the ionized
and neutral donor cases, we will likely work in the exchange-interaction-dominated
region because the absolute strength of the scattering is much stronger there. The
crossover between regimes happens when the scattering from the neutral and ionized
donor is equivalent, around 20 nm (this exact value varies with wavevector). This
sets an upper limit on the depth of the donor where the measurement could still be
successful.
When scattering rates are small (i.e. when the donor depth or the wavevector k
is large) or the interaction time is very short, these scattering matrix elements can be
combined with the Landauer-Buttiker formalism to estimate conductivity [86]. This
formalism requires the calculation of a reflection coefficient Rk , which indicates the
fraction of the edge-channel with a particular wavevector k that scatters from the
QPC. We estimate this by taking the scattering rate from Fermi’s golden rule and
multiplying by the interaction time ti , which produces
Rk =
2
2πti
ρ(Ek ) Sk+,0 .
h̄
(4.20)
Here, ρ(Ek ) is the 1D density of states of the edge-channels with wavevector k confined
to the channel length Lc from Eq. 4.9,
1
ρ(Ek ) =
h̄
r
m∗
2Ek
Lc
2π
ωt
ωq
,
(4.21)
4.5. DEVICE PHYSICS AND SIMULATION
81
and Ek is the energy component of the edge-channel states in Eq. 4.16 which is
determined by k,
h̄2 k 2
Ek =
2m∗
ωq
ωt
2
.
(4.22)
The interaction time ti is estimated by dividing the interaction length Li by the group
velocity of the edge-channel state vg , which is also obtained from Eq. 4.16. From this
approximation, we find
Li
ti =
= Li
vg
r
m∗
2Ek
ωt
ωq
.
(4.23)
where we have taken Li to be twice the effective bohr radius (' 40 Å) for our calculations.
This approximation allows an estimation of Rk for both the neutral and ionized
donor cases; for example, at a relatively large wavevector k = 2π/50 nm−1 and a
donor-oxide distance of 10 nm, we estimate that Rk changes from 0.03 to 0.0002
upon ionization of the donor. The corresponding transverse conductance values can
be obtained from the Landauer-Buttiker formalism,
0,+
Gxy
=
e2 Lc Li k 2 0,+ 2
,
Sk
h 4Ek 2
(4.24)
and are 1 MΩ−1 for the neutral donor and 0.008 MΩ−1 for the ionized donor, a
difference which can be measured by current equipment. For smaller k values, the
scattering matrix elements are substantially higher, in many cases causing the firstorder reflection coefficient to exceed unity. This indicates the need for inclusion of
higher-order interference terms within the formalism. For these k values, even larger
conductance changes are expected.
In Sec. 4.5.1, we gave an effective lower-limit to the depth of the donor of 5 nm,
and in this section we found an upper limit of around 20 nm. Note that this is only for
a single set of device parameters, and in an actual experiment we will have a range
of methods for tuning the interaction. Using the global gate and the background
p-doping concentration, we will be able to tune the depth and width of the 2DEG.
By varying the source-gate voltage and background p-doping, we can tune the Fermi
82
CHAPTER 4. PHOSPHORUS DONORS IN SI
energy of the current-carrying edge-state wavevector, and by tuning the QPC voltage,
we can ensure only a single edge channel is mixed within the QPC. This gives us a
very large parameter space which should encompass the desired interaction regime
for a broad range of donor depths.
4.5.3
Ionization and recapture
The ionization rate of laser excitation and subsequent Auger recombination is limited
by the lifetime of the D0 X state, which is 272 ns in bulk and has been shown experimentally to vary with binding energy Ed as τ ∝ Ed−3.9 [73]. If our donor is in the
optimal range of 5 to 20 nm, the donor binding energy varies only a small percentage
from the bulk value, and so we expect that the excited state lifetime when the donor
is near the oxide should not differ significantly from the lifetime in bulk.
After the donor is ionized, an edge state electron can be scattered into the donorbound-electron state, re-neutralizing the donor. Since the donor ground state is
lower in energy than the edge state which lies at the Fermi energy, this process is
not reversible at low temperature, and the donor remains neutral until re-ionized
by radiation. This recapture is important for the measurement scheme, as it allows
the laser to re-ionize the donor to continue the cycle, producing a random telegraph
signal. In particular, the recapture time is crucial; if it is much faster than a few
nanoseconds, the ionized state will come and go too fast for detection. If it is too
slow, then the time required to repeat the cycle maybe become prohibitively long.
Unfortunately, the strongly-coupled, many-body nature of the relevant interactions
makes the type of perturbative approaches we have used so far ineffective to estimate
this recapture rate. Experiments in optical spectroscopy in the presence of aboveband carriers [70], however, suggest that this recapture rate could be made to fall
into a range reasonable for an effective measurement with appropriate tuning of the
bias current.
Acceptors in the p-type substrate could affect electron recapture by the donor due
to their ability to strip an electron from a nearby donor. However, this will not be a
significant obstacle due to the availability of free electrons in the 2DEG that can bind
4.5. DEVICE PHYSICS AND SIMULATION
83
to acceptors located near the oxide. After accepting an electron, these impurities will
no longer prevent the donor from binding another electron. Acceptors located too far
away from the oxide could strip electrons from neighboring donors, but we are not
interested in donors further than 20 nm from the oxide. Acceptors closer than this
distance should have an interaction with the electron bath comparable to that of the
donor, allowing them to bind an electron from the 2DEG instead of from the donor.
Successful optical spectroscopy has been performed on the D0 → D0 X transition in
p-type Si when free electrons are made available to the acceptors [70].
Any ionization process which competes with the Auger decay will manifest itself
as noise in this measurement. Two such ionization processes are thermal ionization
and field impact ionization due to the electric field near the oxide. Since all of the
experiments will be conducted below 4 K, the donor binding energy of 45 meV is
much larger than kb T (' 0.3 meV), making thermal ionization negligible.
Field impact ionization of the donor electron is a concern, as that can occur at
fields of 400 V/cm [87], which is about half of the field felt by a donor 20 nm from
the oxide. However, the field creating the 2DEG is only felt in the near vicinity of the
oxide and does not have enough distance to give a free carrier the energy necessary
to ionize the donor. Furthermore, most of the free carriers in bulk are frozen out at
low temperatures.
Field impact dissociation of the exciton is a much greater concern, as this can occur
at 50 V/cm [87]. However, instead of ionizing the donor, this effect would prevent
the donor from being ionized. As with impact ionization of the donor electron, the
field range is very small and carriers will be mostly frozen out at low temperature,
greatly reducing the rate of this effect. Future devices could be optimized to reduce
the 2DEG field strength in order to limit this effect.
The electric field itself can also pull an electron from the donor or pull apart the
exciton in the D0 X state. However, field ionization of the neutral donor is a negligible
effect below 35 kV/cm [88] and dissociation of free excitons should not occur below
5 kV/cm [89]. Despite the 2DEG potential and the slightly larger D0 X radius, these
effects should be negligible for appropriate donor depths.
84
CHAPTER 4. PHOSPHORUS DONORS IN SI
4.5.4
Optical transition
The narrow linewidth of the D0 → D0 X transition is a crucial requirement for the
success of our proposed experiment. In order to selectively ionize the donor in a
particular spin state, the linewidth must be smaller than the hyperfine interaction
energy. Fortunately, a bulk sample within a strong magnetic field has a hyperfine
splitting of 60 MHz and the best measurement of the donor-bound-exciton transition
homogeneous linewidth is 2.4 MHz [74]. However, this linewidth could increase in the
presence of the oxide due to excited-state-lifetime changes or additional oxide defects.
Near the oxide, we do not expect lifetime shortening to increase the linewidth
beyond 2.4 MHz, since that value is already four times the lifetime-limited linewidth,
and as discussed in Sec. 4.5.3, this lifetime should not be significantly modified by
the presence of the oxide. Oxide defects, however, could broaden the linewidth due
to mechanisms such as spectral diffusion of oxide defects. As long as these defects do
not shift the donor states excessively fast or strong, the linewidth should be similar
to the bulk case.
The hyperfine splitting will change slightly due to the oxide since the splitting
is proportional to the square of the electron wavefunction amplitude at the position
of the donor nucleus [81], and that amplitude varies with the donor-oxide distance
because the donor-bound electron wavefunction shifts slightly into the 2DEG when
the donor is close to the oxide. However, from analyzing the hybridized donor-boundelectron wavefunction we calculated in Sec. 4.5.1, this amplitude does not change by
more than 4% for a donor further than 5 nm from the oxide, resulting in less than a
10% change in the hyperfine splitting.
The oxide will cause strain, which is known to broaden the transition linewidths
in PL spectroscopy. However, this broadening is inhomogeneous and the linewidth
of the single donor should not be broadened by the strain. Strain does cause shifting
of the bound-exciton state energies, but that does not affect the splitting between a
hyperfine-split pair of transitions.
Central-cell corrections to the binding energy of the donor-bound-electron could
also cause shifting of the transition energies, but this effect should be small since the
amplitude of the electron wavefunction throughout the central-cell region similarly
4.5. DEVICE PHYSICS AND SIMULATION
85
does not change significantly. Furthermore, any ground or excited state energy shifts
due to central-cell corrections, strain, band bending, or other effects (such as the DC
Stark effect) that do not cause broadening can easily be compensated for by tuning
the laser. For this reason, and because the hyperfine splitting should remain significantly larger than the bulk homogeneous linewidth, selective ionization of individual
donor states should still be possible in the presence of the oxide, even with moderate
broadening due to nearby defects.
4.5.5
Photoconductivity
A major concern for photoconductivity measurements in Si and other semiconductors
is free electron creation from illuminated metallic leads. Photocurrent originating
from these metallic leads will greatly increase background noise. Due to the rather
long wavelength of the excitation photons, even a diffraction limited laser spot incident
on the device would likely cause a significant amount of background photocurrent.
For this reason, the metallic global gate will also perform duty as a beam block to
protect the metallic leads from the laser illumination in addition to being used to
tune the 2DEG (see Fig. 4.4). A small aperture will be created in the global gate
above the position of the donor to protect the leads in proximity to the donor. The
global gate and the QPC gates will be illuminated, but they are electrically isolated
from the source, drain, and measurement electrodes, and will not directly produce
noise. An antireflection coating on the back of the sample may also be required to
reduce reflection from the back of the sample.
4.5.6
Device prospects
In summary, by combining optical pumping and electronic detection methods together
into one system, we are able to take advantage of the benefits of each method in order
to overcome the difficulties presented by semiconductor systems and create a realistic
measurement device. This measurement scheme is a deterministic non-demolition
measurement of the nuclear spin of a single
31
P in Si. If single-shot measurement is
achieved, then by optically pumping on a pair of transitions beginning in the same
86
CHAPTER 4. PHOSPHORUS DONORS IN SI
electron spin state but opposite nuclear spin state we can also perform a deterministic
measurement of the donor-bound-electron spin, but in this case it is a destructive
measurement.
A number of other factors will contribute to this measurement scheme. In particular, we have not included the critical details of the Si band structure. The different
valley-orbit states make the scattering problem more complicated, reduce the symmetry, and introduce constraints on the donor placement. Further, the inversion layer is
treated here as an empty, triangle-like potential well, but in reality there is a bath of
electrons in this potential, and the many-body effects of screening and spin-spin scattering will play an important role inside the QPC. The present discussion is intended
to introduce the principle of our measurement scheme; more detailed calculations
including these important effects are left as future work.
4.6
Preliminary experiments with FinFET device
Without a collaboration to design and fabricate the Hall bar device introduced in
this chapter, the project was put on hold indefinitely. However, a chance discussion
with Professor Sven Rogge in 2010, who was at Delft University of Technology (he
is currently at the University of New South Wales in Melbourne), resuscitated the
project. He is currently working on designing and fabricating Hall bar devices to
begin experimentation, but before those devices are ready, he suggested that optical
pumping could be just as effective for spin measurement in his FinFET devices [80].
The results presented in this section are preliminary results using FinFET devices
from Professor Rogge’s research group. The data exhibit the sort of behavior under
optical pumping we would expect to see with the photo-ionization of single P donors
within or near the channel.
4.6.1
Device structure and behavior
The devices of interest, shown in Fig. 4.9 and also in Fig. 1(a) of Ref. [80], are
FinFET devices, so named because of the fin-like shape of the gate. The channel
4.6. PRELIMINARY EXPERIMENTS WITH FINFET DEVICE
87
is the small region of the path between the source and drain that is below the gate
fin. Depending on the device, the channel can either be slightly p-doped with boron,
n-doped with phosphorus, or nominally undoped. A typical might have a channel
with dimensions of L = 60 nm × W = 50 nm × H = 8 nm.
Figure 4.9: Schematic drawing of the FinFET devices used in this work. The channel
dimensions are determined by the width W, the Length L, and the height H.
Under normal behavior, the FinFET devices act just like a normal field-effect
transistors. When a positive electric potential is created between the gate and channel, the electric field from the gate creates a channel for holes and electrons to move
At very low temperatures, around 4 K, the distribution of occupied electron states
around the Fermi energy becomes very narrow, resulting in a sharp transistor turn-on,
and the appearance of fluctuations in the density of states, which in part determines
conductivity, that are washed out at room temperature. A representative set of curves
taken at various temperatures from room temperature down to 4 K are shown in Fig.
4.10. These curves show the current between source and drain, Isd as a function of
the potential between the gate and the drain, Vgd .
If a donor is near the channel and becomes ionized, we would expect that the
electric potential from the charged donor D+ would affect the conductivity through
88
CHAPTER 4. PHOSPHORUS DONORS IN SI
Figure 4.10: Temperature dependence I-V curves for a FinFET device. Roomtemperature data is shown in red, while 4 K data is shown in blue and magenta.
Curves from yellow to cyan were taken at decreasing temperature.
the channel. Whether the conductivity increases or decreases depends upon the distance between the donor and the channel, similar to what was computed for the Hall
bar device in Sec. 4.5.2. After ionization, the donor will eventually recapture an
electron and the conductivity would return to its original value. In our preliminary
experiments, we have seen behavior in phosphorus-doped devices that appears to be
consistent with this explanation.
In Fig. 4.11, we show a series of data that was taken with fixed Vsd = 10 mV and
Vgd = 500 mV (above threshold). The only thing that was varied was the presence
or absence of laser light incident on the device. The laser wavelength (1065 nm) was
larger in energy than the bandgap, so as to create many excitons which could bind
to a P donor and ionize it. In the data, large, discrete jumps are observed when the
laser is on. When the laser is blocked, these jumps are not observed. Most of the
jumps are of identical amplitude between two levels, suggesting that the source of the
jumps is a switching between two discrete states of some impurity or complex.
This switching behavior was not seen in the boron-doped devices, and while the
4.6. PRELIMINARY EXPERIMENTS WITH FINFET DEVICE
89
Figure 4.11: Data taken with fixed Vsd of 10 mV and Vgd of 500 mV. The curve in
red was taken first with the laser blocked. The yellow curve was next, with the laser
blocked for 60 s, and then unblocked for 60 s. The green data was taken with the
laser on for the entire time, and the data in blue was taken with the laser on for 60
s, and then blocked for 60 s. The time between consecutive curves was a few minutes
each. Note that jumps with an amplitude of approximately 5 nA only occur with the
laser on.
undoped devices did show some laser-dependent switching, it often showed a gradual
decay when the laser was turned off, instead of the sharp transitions shown in Fig.
4.11. Different phosphorus doped devices showed possible switching at various speeds,
consistent with a distribution of coupling to the channel. This switching also appeared
to be both wavelength and power dependent.
We have only studied a few devices and have only used above-band pumping,
as opposed to resonant pumping, so we cannot say definitively what is the origin of
these jumps. We can only say that they seem to be consistent with the ionization and
reneutralization of individual phosphorus donors near the FinFET channel. I hope
that future experiments with new devices and an attempt at resonant pumping can
determine the source of this switching.
90
CHAPTER 4. PHOSPHORUS DONORS IN SI
4.7
P:Si outlook
While the preliminary results with the FinFET devices are very exciting, a Hall bar
device has two main advantages over using a FinFET. The first advantage is in the
homogeneity of the surrounding environment. In a FinFET, the environment near the
donor is a complex heterostructure with varying doping concentrations and strain.
The Hall bar device, on the other hand, can be a more uniform structure near the
donor with constant doping and a flat oxide layer. It is the electric potential provided
by the top gate and QPC that controls the interaction, rather than physical structuring. This has the potential to reduce the number of defects and inhomogeneities
which can cause decoherence in the system.
The second advantage is that the sensing edge channel is a coherent state, as
opposed to the incoherent sensing current traveling through the FinFET. Coherence
lengths of over 20 µm have been measured for individual IQHE edge channels [90].
This first suggest that a coherent interaction between the edge channel and the donorbound electron could be utilized to perform local, single-qubit gate operations. If
single-qubit gates are possible, then this interaction very naturally extends to twoqubit gates by putting two donors within the coherence length of the edge channel.
If the initialization technique described in Ref. [91], the ensemble ESR and NMR
control described in Ref. [92], and the dynamic decoupling described in Ref. [65]
could be combined with single- and two-qubit electron gates utilizing the coherent
edge channel and with the measurement technique proposed here, we would have all
the basic building blocks for a cluster state quantum computer [93]. Of course, the
details of gate fidelity and operation time, decoherence times, and device layout would
need to be worked out before an estimate of the feasibility of such an architecture
could be made. However, the potential of the system warrants continued research
into building and testing the Hall bar measurement device.
Chapter 5
Conclusion and Outlook
The main goal of the work presented in this thesis has been to further the development of quantum bits implementations based upon neutral donors in semiconductor
systems. These systems are seen as important for quantum information research due
to their combination of atomic-like and semiconductor-like properties. Individual
donors in bulk are highly homogeneous like atomic systems, yet they reside within
the tuneable environment of a semiconductor crystal like quantum dots. The hope is
that they can provide a middle ground between other quantum bit implementations
while avoiding the major obstacles presented to those systems.
In particular, electrons bound to fluorine donors in zinc selenide are good qubit
candidates for quantum repeater technology. They have a strong optical dipole transition and high quantum efficiency, and emit photons entangled to the electron spin
when excited into the D0 X state. They are more homogeneous than other semiconductor systems, and can be fabricated to avoid the decoherence due to nuclear
spin in the host crystal through isotopic purification. Furthermore, ion implantation
technology could lead to deterministically placed single qubits, and semiconductor
fabrication makes it easy to integrate electronics within a device.
The work presented in Chapter 3 shows, firstly, that the single-donor D0 X to D0
optical transitions matches with the theoretical expectation in both Faraday and Voigt
geometry, and forms a connected lambda system which is appropriate for ultra-fast
optical spin rotations. Second, measurements on ion-implanted devices confirms that
91
92
CHAPTER 5. CONCLUSION AND OUTLOOK
optically-active fluorine donors can be successfully implanted in devices. Finally, the
results presented here demonstrate optical pumping of a single donor-bound electron
in order to initialize the spin into a particular state. 79% initialization was achieved
within 13 ns, which can likely be improved by future experiments. In addition, optical
pumping provides a method for single spin measurement and opens the door for spin
manipulation using ultrafast optical pulses.
Electrons bound to phosphorus donors in silicon, on the other hand, are great
qubit candidates for quantum computers due to their extremely long lifetimes and
strong homogeneity. Silicon is the most mature semiconductor system used today, and
as a result, has the greatest capabilities for device fabrication of any semiconductor.
In addition, the nuclear-spin resolution of the D0 X to D0 transition opens the door
to individual donor nuclear-spin access. Furthermore, the natural coupling between
the electron spin and the nuclear spin provides opportunities for two-qubit gates.
The work discussed in Chapter 4 presents a novel approach to phosphorus qubits
by combining optical pumping with electrical detection in the quantum Hall regime.
Optical pumping allows the local excitation of donors in a particular nuclear spin
state, which then results in the ionization of the donor. Electrical detection using
a quantum Hall bar with a quantum point contact isolates the region of sensitivity
to the vicinity of the phosphorus donor, and can greatly enhance the measurement
signal while reducing sensitivity to other impurities. Preliminary results using FinFET devices indicate that a single impurity could have a large effect upon electrical
conduction through a nanoscale channel.
There are a number of future experiments that could be undertaken in both the
F:ZnSe system, and the P:Si system, as discussed in Sec. 3.5 and Sec. 4.7, respectively.
Most important for F:ZnSe is to attempt spin rotation. If successful, many of the
techniques used for InGaAs quantum dots can likely be quickly reproduced. Most
important for P:Si will be to find the resonance of the D0 X to D0 transition, which
will confirm that the conduction random telegraph signal is in fact due to phosphorus.
Then, quantum Hall bar devices should be fabricated and tested to see if they meet
their potential.
Both donor systems have a good deal of room for advancement. This work just
93
scratches the surface of the potential each system holds. Furthermore, this is only one
particular type of qubit candidate system. There are many other candidate systems
not mentioned here, and there are sure to be surprise developments in a variety of
systems before any is determined to be qualified for quantum information processing.
I hope that the work presented here can play a small role in pushing forward the
limits of what is capable in the field of quantum information, and I hope it will only
be a small achievement compared to what is accomplished in the future.
Appendix A
Selection rules
A.1
F:ZnSe selection rules
Due to the fact that the s-like Bloch wavefunction for the D0 states has no angular
momentum, there is no spin-orbit coupling, and the spin will always align with the
magnetic field. However, the p-like Bloch wavefunction for the D0 X state does couple
to the hole spin. In the case of bulk ZnSe, the wavefunction can orient itself in any
direction, and so both the spin and the wavefunction will align along the magnetic
field. However, in a narrow quantum well, the wavefunction strongly aligns with the
quantum well, as in Fig. A.1. If the wavefunction were to orient itself in another
direction, a large component of the wavefunction spreads into the larger-bandgap
barrier, thus increasing the energy of the state. This confinement energy depends on
the thickness of the quantum well.
In Faraday geometry, this confinement does not matter, as the magnetic field is
aligned with the growth direction. However, in Voigt geometry, the quantum well and
the magnetic field compete to orient the hole spin. At very strong magnetic fields
or very wide quantum wells, the spin could force the wavefunction to reorient along
the magnetic field. However, at lower magnetic fields or in narrow quantum wells,
the confinement energy is larger than the Zeeman energy, and so the wavefunction
and spin align along the growth direction. The largest magnetic field available to
our experiment was 7 T, and at this magnetic field, the hole wavefunction is almost
94
A.1. F:ZNSE SELECTION RULES
95
Figure A.1: Representation of the hole wavefunction in the D0 X state when confined
within a quantum well. The shaded area shows the wavefunction, while the arrow in
the center indicates the spin.
completely aligned along the growth direction in the 2 nm quantum well samples.
However, in larger quantum wells more than 4 nm thick, the hole wavefunction can
be more aligned along magnetic field axis.
From the Zeeman splitting for the D0 and D0 X states in Faraday geometry, we
know that the level structure looks like what is shown in Fig. A.2(a). We can
determine the transition selection rules by using Eq. 2.22 and the D0 wavefunctions
from Eq. 2.23 and the HH D0 X wavefunctions from Eq. 2.25.
Thinking carefully about the transition from ΨD0 X , + 32 to ΨD0 , + 12 , the electron
transition is actually from the 2nd electron in the D0 X state ΨD0 , − 21 to the unoc
cupied valence state, which has the opposite spin as the hole, ΨD0 X , − 32 . Therefore,
the transition rate can be computed by
2
3
1
R ∝ eE · ΨD0 X , − r ΨD0 , − .
2
2
(A.1)
Inserting the appropriate wavefunctions from Eqs. 2.23 & 2.25, we find
2
∗ ∗
hφhh | r (|φ100 i |ψn00
R ∝ eE · h↓| ψn1−1
i |↓i) ,
(A.2)
96
APPENDIX A. SELECTION RULES
Figure A.2: The Zeeman splitting of the states and allowed optical transitions in (a)
Faraday geometry and (b) Voigt geometry.
where the |ψ ∗ i terms represent the Bloch functions and the |φi terms represent the
envelope function. If we make the assumptions that the envelope function does not
change significantly across the volume of a nuclear site, and that the Bloch function
components from neighboring nuclear sites do not overlap, we can simplify this to
2
∗ ∗
r |ψn00
R ∝ |h↓ | ↓i|2 |hφhh |φ100 i|2 eE · ψn1−1
i .
(A.3)
The second term in Eq. A.3 is the overlap between the envelope function for the
donor-bound electron and the bound heavy-hole. This is the same for any transition
between D0 X and D0 , and must be nonzero since the optical transitions have in fact
been observed. We will absorb it into the proportionality constant from here on.
The first term tell us that any transition must be spin-conserving, since the electric dipole operator commutes with the spin states, and the final term tells us that the
Bloch function determines the polarization of the radiation that can interact with the
transition. Since the Bloch functions share the symmetries of the hydrogen wavefunctions, we can apply the selection rules used in transitions between Hydrogen states to
the case of the D0 X transition. These rules tell us that the only allowed transitions
are when ∆l = ±1 and ∆m = 0 or ± 1. Fortunately, Eq. A.3 satisfies both of these
A.1. F:ZNSE SELECTION RULES
97
equations.
The next step is to determine the polarization of the interacting radiation by
computing the three inner products,
∗
∗
x |ψn00
i
ψn1−1
∗ ∗
ψn1−1 y |ψn00
i
∗ ∗
i
ψn1−1 z |ψn00
(A.4)
(A.5)
(A.6)
As before, we will define the growth direction as ẑ, while x̂ and ŷ are both in the
plane of the quantum well. Symmetry determines that the angular component of the
wavefunctions must be described by spherical harmonics Ylm (θ, φ). Therefore,
∗
|ψnlm
i = Fnl (r)Ylm (θ, φ).
(A.7)
By integrating over the angular components, we can determine the vector of the
electric field E.
∗
∗
x |ψn00
ψn1−1
i
Z
∞
∗
Fn1
(r)Fn0 (r)r3 dr ×
0
Z π Z 2π
Y1−1∗ (θ, φ)Y00 (θ, φ) cos(φ)dφ sin2 (θ)dθ (A.8)
0
0
r
Z π Z 2π r
3
1
iφ
∝
sin(θ)e
cos(φ)dφ sin2 (θ)dθ
(A.9)
8π
4π
0
0
r
Z π
Z 2π
3
3
=
sin (θ)dθ
cos(φ)eiφ dφ
(A.10)
32π 2 0
0
r
1
=
.
(A.11)
6
=
Likewise,
∗
y
ψn1−1
∗
|ψn00
i
Z
π
Z
∝
0
r
= i
2π
Y1−1∗ (θ, φ)Y00 (θ, φ) sin(φ)dφ sin2 (θ)dθ
(A.12)
0
1
,
6
(A.13)
98
APPENDIX A. SELECTION RULES
and
∗ ∗
i ∝
ψn1−1 z |ψn00
Z
π
Z
0
2π
Y1−1∗ (θ, φ)Y00 (θ, φ)dφ cos(θ) sin(θ)dθ
(A.14)
0
= 0.
(A.15)
Putting these inner products back into Eq. A.3, we obtain
R ∝ |Ex + iEy |2 ,
(A.16)
which tells us that the emitted photon from the ΨD0 X , + 32 to ΨD0 , + 12 transition
will have right-circular polarization, or |σ + i. As required, this transition conserves
angular momentum: the system loses +1 unit of angular momentum and is carried
by the |σ + i photon, which also has +1 unit of angular momentum.
If we consider the ΨD0 X , + 3 to ΨD0 , − 1 transition, we find
2
2
2
∗ ∗
r |ψn00
R ∝ |h↓ | ↑i|2 |hφhh |φ100 i|2 eE · ψn1−1
i .
(A.17)
The spin component h↓ | ↑i = 0 since spin must be conserved, telling us that this
transition is not allowed. The transition from ΨD0 X , − 3 to ΨD0 , + 1 is also not
2
2
allowed for the same reason.
The last transition, ΨD0 X , − 23 to ΨD0 , − 21 gives us
2
∗ ∗
r |ψn00
R ∝ |h↑ | ↑i|2 |hφhh |φ100 i|2 eE · ψn1+1
i
2
Z π Z 2π
∝ E ·
Y1+1∗ (θ, φ)Y00 (θ, φ)r̂dφ sin(θ)dθ
0
0
r !2
r
1
1
x̂ + i
ŷ = E · −
6
6
∝ |Ex − iEy |2 .
(A.18)
(A.19)
(A.20)
(A.21)
Therefore, this transition couples to left-circular polarization, or |σ − i.
We have just computed which transitions are allowed in Faraday geometry as well
as the polarization of those transitions (Fig. A.2(a)). There are only two transitions,
A.1. F:ZNSE SELECTION RULES
99
which have an equal transition rate and opposite circular polarization.
Next, we can perform the same computation for Voigt geometry, shown in Fig.
A.2(b). Let us assume that the quantum well confinement is stronger than the magnetic field, so that both the wavefunction and the spin of the hole are aligned along
the growth direction, while the magnetic field is now pointing in another direction.
Since the spin is perpendicular to the magnetic field, the Zeeman interaction contributes nothing to the energy of the D0 X state. However, it does serve to mix the
normally degenerate HH exciton states,
r 3
3
1 |ΨD0 X , +i =
Ψ 0 ,+
+ ΨD0 X , −
2 DX 2
2
r 3
1 3
|ΨD0 X , −i =
Ψ 0 ,+
− ΨD0 X , −
.
2 DX 2
2
(A.22)
(A.23)
In the D0 ground states, the electron spin is free to rotate, and aligns along the
magnetic field, which can define to be pointing along the x-axis.
∗
|ΨD0 , +i = |φ100 i |ψn00
i |↑x i
r 1
1
1 Ψ 0, +
+ ΨD0 , −
=
2 D
2
2
∗
|ΨD0 , −i = |φ100 i |ψn00 i |↓x i
r 1 1
1
=
Ψ 0, +
− ΨD0 , −
,
2 D
2
2
(A.24)
(A.25)
(A.26)
(A.27)
where we have just rewritten |↑x i and |↓x i in terms of |↑z i and |↓x i, which is the basis
we are using for the D0 X states.
Next, using Eq. 2.22 again, we can compute the transition rates between the
100
APPENDIX A. SELECTION RULES
states
R ∝ |eE · hΨD0 X , ±| r |ΨD0 , ±i|2 .
(A.28)
3
1
3
ΨD0 X , + ± ΨD0 X , − r ×
= eE ·
2
2
2
2
ΨD0 , + 1 ± ΨD0 , − 1 (A.29)
2
2 e
3 3 1
1
= E · ΨD0 X , + r ΨD0 , +
± ΨD0 X , − r ΨD0 , −
2
2
2
2
2
2
1
3 1 3 ± ΨD0 X , − r ΨD0 , +
.(A.30)
± ΨD0 X , + r ΨD0 , −
2
2
2
2 These four inner products are exactly the ones we computed for Faraday geometry.
We already know that the last two are equal to 0, since they do not conserve spin.
The first two inner products are equal in magnitude but opposite in direction. For
the transition |ΨD0 X , +i to |ΨD0 , +i, we get
"r
#2
r
1
1
R ∝ E ·
(x̂ + iŷ) +
(−x̂ + iŷ) 6
6
∝ |Ey |2 .
(A.31)
(A.32)
In our setup, this corresponds to vertical polarization, |V i. A transition from |ΨD0 X , −i
to |ΨD0 , −i also produces |V i, with equal magnitude. The cross transitions, |ΨD0 X , +i
to |ΨD0 , −i and |ΨD0 X , −i to |ΨD0 , +i, are also allowed, with a rate
"r
#2
r
1
1
(x̂ + iŷ) −
(−x̂ + iŷ) R ∝ E ·
6
6
∝ |Ex |2 .
(A.33)
(A.34)
This corresponds to horizontal polarization, |Hi.
In Voigt geometry, all four transitions are allowed, and all have an equal rate. Two
of the transitions have |V i polarization, while the other two have |Hi polarization,
as shown in Fig. A.2(b).
A.2. P:SI SELECTION RULES
A.2
101
P:Si selection rules
The P:Si samples of interest are not confined by a quantum well, and so the hole is
free to align its spin with the magnetic field. Therefore, we only need to compute one
set of transition rates.
The total spin 1/2 states are split off by spin-orbit coupling, so we are still only
considering the total spin 3/2 hole states. However, without significant strain, both
the HH and LH states are important here. For the purpose of this calculation, I’ll
assume there is zero strain. The relative states and transitions are shown in Fig. A.3.
Figure A.3: The Zeeman splitting of the states and allowed optical transitions.
As with the previous section, we’ll begin with Eq. 2.22, the D0 wavefunctions
from Eq. 2.23 and the HH D0 X wavefunctions from Eq. 2.25. The addition of the
102
APPENDIX A. SELECTION RULES
∗
∗
LH states means that we need to also compute hψn10
| x |ψn00
i,
∗
∗
hψn10
| r̂ |ψn00
i
Z
π
2π
Z
Y10∗ (θ, φ)Y00 (θ, φ)r̂dφ sin(θ)dθ
∝
=
0
0
r
1
ẑ.
3
(A.35)
(A.36)
Therefore, we have the three inner products:
r
r
1
1
∗
∗
r̂ |ψn00
ψn1+1
x̂ + i
ŷ
i ∝ −
6
6
r
1
∗
∗
hψn10
ẑ
| r̂ |ψn00
i ∝
3
r
r
∗ 1
1
∗
ψn1−1 r̂ |ψn00 i ∝
x̂ + i
ŷ.
6
6
(A.37)
(A.38)
(A.39)
Remembering that a transition from ΨD0 X , + 32 to ΨD0 , + 12 is really an electron
transition from ΨD0 , − 1 to ΨD0 X , − 3 , etc, we can compute possible transitions:
2
2
3
1
1
R ΨD0 X , ± , ±
→ ΨD0 , ±
2 2
2
2
1
3
1
∝ eE · ΨD0 X , ± ± r ΨD0 , ± 2 2
2
(A.40)
Due to the hyperfine coupling in the D0 state (Sec. 2.4), there are two regimes
to consider. In the high field, the electron and nuclear spins are separable, and the
nuclear spin has no effect on the optical transitions. Therefore, both nuclear spin
A.2. P:SI SELECTION RULES
103
states will have equal transition rates, and we only have to compute 8 transitions.
1
3
→ ΨD0 , +
R ΨD0 X , +
2
2
1
3
R ΨD0 X , +
→ ΨD0 , −
2
2
1
1
R ΨD0 X , +
→ ΨD0 , +
2
2
1
1
→ ΨD0 , −
R ΨD0 X , +
2
2
1
1
R ΨD0 X , −
→ ΨD0 , +
2
2
1
1
R ΨD0 X , −
→ ΨD0 , −
2
2
3
1
R ΨD0 X , −
→ ΨD0 , +
2
2
3
1
R ΨD0 X , −
→ ΨD0 , −
2
2
∝
1
|−Ex + iEy |2
6
∝ 0
2
|Ez |2
9
1
∝
|−Ex + iEy |2
18
1
|Ex + iEy |2
∝
18
2
|Ez |2
∝
9
∝
∝ 0
∝
1
|Ex + iEy |2 .
6
(A.41)
(A.42)
(A.43)
(A.44)
(A.45)
(A.46)
(A.47)
(A.48)
It is interesting to note that if the incident light is averaged over the Ex , Ey , and Ez
(if for instance, the light bounces around within the sample, essentially randomizing
the direction of incidence), then these equations reproduce the 3:2:1 ratio of PL
observed in spectra, as in Fig. A.4.
If, on the other hand, the field is very low, then the electron and nuclear spin form
into a singlet and 3 triplet states.
|ΨD0 , S0 i =
|ΨD0 , T+1 i =
|ΨD0 , T0 i =
|ΨD0 , T−1 i =
r 1 1
1
Ψ 0, +
|N ↓i − ΨD0 , −
|N ↑i
2 D
2
2
ΨD0 , + 1 |N ↑i
2
r 1 1
1
Ψ 0, +
|N ↓i + ΨD0 , −
|N ↑i
2 D
2
2
ΨD0 , 1 |N ↓i .
2
(A.49)
(A.50)
(A.51)
(A.52)
104
APPENDIX A. SELECTION RULES
Figure A.4: High-field splitting of the P:Si system. The approximate ratio between
the 1st, 3rd, and 5th pairs of peaks is 1:2:3, agreeing with the calculated ratio. The
same is seen in the ratio between the 6th, 4th, and 2nd pairs.
Therefore, the 16 optical transition rates are
3
R ΨD0 X , +
→ |ΨD0 , S0 i
2
3
R ΨD0 X , +
→ |ΨD0 , T+1 i
2
3
R ΨD0 X , +
→ |ΨD0 , T0 i
2
3
R ΨD0 X , +
→ |ΨD0 , T−1 i
2
1
R ΨD0 X , +
→ |ΨD0 , S0 i
2
1
R ΨD0 X , +
→ |ΨD0 , T+1 i
2
1
R ΨD0 X , +
→ |ΨD0 , T0 i
2
1
→ |ΨD0 , T−1 i
R ΨD0 X , +
2
1
|−Ex + iEy |2
12
1
|−Ex + iEy |2
∝
6
1
∝
|−Ex + iEy |2
12
∝
∝ 0
1
|Ex − iEy + 4Ez |2
36
2
∝
|Ez |2
9
1
∝
|−Ex + iEy + 4Ez |2
36
1
∝
|−Ex + iEy |2
18
∝
(A.53)
(A.54)
(A.55)
(A.56)
(A.57)
(A.58)
(A.59)
(A.60)
A.2. P:SI SELECTION RULES
1
→ |ΨD0 , S0 i
R ΨD0 X , −
2
1
R ΨD0 X , −
→ |ΨD0 , T+1 i
2
1
→ |ΨD0 , T0 i
R ΨD0 X , −
2
1
R ΨD0 X , −
→ |ΨD0 , T−1 i
2
3
→ |ΨD0 , S0 i
R ΨD0 X , −
2
3
R ΨD0 X , −
→ |ΨD0 , T+1 i
2
3
→ |ΨD0 , T0 i
R ΨD0 X , −
2
3
R ΨD0 X , −
→ |ΨD0 , T−1 i
2
105
∝
∝
∝
∝
∝
1
|Ex + iEy − 4Ez |2
36
1
|Ex + iEy |2
18
1
|Ex + iEy + 4Ez |2
36
2
|Ez |2
9
1
|Ex + iEy |2
12
∝ 0
1
|−Ex − iEy |2
12
1
∝
|Ex + iEy |2 .
6
∝
(A.61)
(A.62)
(A.63)
(A.64)
(A.65)
(A.66)
(A.67)
(A.68)
Using these equations, we can take the polarization axis in either Voigt or Faraday
geometry in order to predict the particular transition rates we should observe.
Appendix B
Experimental setup
B.1
F:ZnSe experiments
The F:ZnSe experiments were all performed using an Oxford Spectromag cryostat,
which allowed us to keep a sample at liquid helium temperatures for months at a
time. The Spectromag contains a 7 T superconducting magnet, and has windows for
optical access in both Voigt geometry and Faraday geometry. The sample holder was
placed on a sample rod which could be oriented in either direction.
The sample holder contained a set of three Attocube nano-positioners that allow
us to move the sample along the x-, y-, and z-axes with a range of motion of a few
milimeters. This allows us to position a sample behind a fixed objective lens that is
mounted on the sample holder.
The spectromag was mounted on top of an optical isolation table, along with the
laser sources, detectors, and all other optical elements. The layout is shown in Fig.
B.1. Three laser sources were used: (1) a fixed-wavelength diode laser at 408nm,
used for alignment and simple spectroscopy. (2) a Spectra-Physics Matisse Ti:Saph
tunable CW laser followed by a WaveTrain resonant doubling cavity, which provided a
wavelength tunability range that would encompass above band pumping and all nearband edge resonances for all samples studied, approximately 420-450 nm. This laser
was used for resonant pumping. (3) a Coherent Mira Ti:Saph pulsed laser followed
by a doubling crystal with a repetition rate of approximately 76 MHz and a pulse
106
B.1. F:ZNSE EXPERIMENTS
107
Figure B.1: Experiment setup for the F:ZnSe experiments.
width of a few picoseconds. This laser was generally tuned near 410 nm and used for
pulsed above-band excitation.
Each of these lasers was independently controlled for power, collimation, and
polarization. Following this, the three lasers were combined into a single beam path
that could be directed into either the Faraday or Voigt geometry window using nonpolarizing beam splitters (NPBS). The collimation of the beam and the distance
between the objective lens is adjusted so that the laser spot size on the sample is near
the diffraction limit of around a few hundred nm. When the beams are well collimated,
we can ensure that the lasers will only hit a single mesa, which is separated by 10
µm from the nearest neighboring mesa. The laser light that is transmitted through
both NPBSs is directed onto an optical power meter, which allows us to determine
the excitation power.
This constitutes the entire optical illumination side of the setup. The collection
path utilizes the same objective lens to collimate the emission, which is then separated
from the illumination path by the NPBS. Following the NPBS is a 420 nm low-pass
108
APPENDIX B. EXPERIMENTAL SETUP
filter which transmits any photoluminescence in the 430-440 nm range, but blocks any
scattered above-band laser light near 410 nm. The photoluminescence is then sent
through a pair of confocal lenses with a pinhole in the focal plane (on a flip-mount),
which blocks any emission that does not originate in the near vicinity of the mesa of
interest.
Following the pinhole is a mirror on a flip-mount that directs emission to a CCD
camera. This camera allows us to image the photoluminescence coming from an area
of approximately 5 µm × 5 µm (when the pinhole is flipped out). This allows us
to find particular mesas and orientation markers when moving the sample using the
piezo stages.
The next set of optical elements controls the polarization of the detected emission.
By using a half-wave plate (HWP) and a quarter-wave plate (QWP), the polarization
can be rotated by any desired unitary transformation. Following this, a polarizing
beam splitter (PBS) transmits only the horizontal component H of the incident light.
Thus, any desired polarization can be selected for detection by using the HWP and
QWP to transform the desired polarization into H, which will then pass through the
PBS.
Beyond the PBS, the emission can be sent down a variety of paths, depending
upon the combination of flip-mount states selected, and is eventually sent either to a
spectrometer (Princeton Instruments - 750 mm), or a pair of single photon counting
modules (MPD PDM devices). The first set of flip-mirrors switch between a straight
path and a path which utilizes an optical grating and a slit in order to provide a degree
of frequency selection. The second set of flip-mirrors switches between a direct path
to the spectrometer and a path that leads to the third set of flip-mirrors. The third
set switches between a straight path and a second optical grating and slit, which can
be used to provide even stronger frequency selection. Following this, the fourth set of
flip-mirrors switches between a path to the SPCMs and a path to the spectrometer.
Both sets of gratings and slits have an experimentally measured FWHM of approximately 0.2 nm at 432 nm. The SPCMs have a timing resolution of approximately
250 ps and a detection quantum efficiency of approximately 32% at 432 nm. The
spectrometer has a resolution of approximately 0.03 nm at 432 nm.
B.2. P:SI EXPERIMENTS
109
Figure B.2: Experiment setup for the P:Si experiments.
The largest source of noise in the optical pumping experiments was scattered
photons from the optical-pumping laser. Even after the pinhole, cross polarization,
and both gratings, the background count rate from these scattered photons was approximately equal to the count rate from the signal photons. However, substantial
improvements in the signal-to-noise ratio could not be gained without the addition
of further frequency selection.
B.2
P:Si experiments
The P:Si experimental setup, shown in Fig. B.2 is noticeably simpler than that used
for F:ZnSe due to the fact that there is no need for the collection of optical emission.
The preliminary experiments were performed using a continuous flow liquid-helium
cryostat (Oxford Optistat). A tunable diode laser (Sacher Lasertechnik external
cavity) was used to tune the excitation laser from 1065 nm to 1105 nm. The power
was controlled with a variable neutral-density filter, and the polarization was not
controlled. The illumination beam was large in order to easily illuminate the FinFET
device.
110
APPENDIX B. EXPERIMENTAL SETUP
The FinFET electrical properties with and without optical illumination were measured using a Semiconductor Parameter Analyzer (Hewlett Packard 4155A), which
allowed us to control the gate and bias voltages, and measure the source-drain and
gate-leakage currents.
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