Optical Pumping Of Single Donor-bound Electrons In Zinc Selenide And Silicon

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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 = − h ¯2 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,   h ¯2 2 H0 Ψ = − ∇ + Vp eik·r ψk,n (r) 2m0   h ¯ 2 2 ik·r ψk,n (r) + Ep,i (k)Ψ, ∇e = − 2m0 (2.3) (2.4) where  Ep,i (k)Ψ = 
h ¯2 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   h ¯ 2 2 ik·r 2 ik·r H0 Ψ = En Ψ + − ∇ e + αi k e ψk,n (r) 2m0   h ¯ 2 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   h ¯2 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 h ¯2 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 2¯ h (4π)2 (2.13) (2.14) The electron also has an effective Bohr radius, a∗ = h ¯ 2 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. ˚ for ZnSe [53] GaAs was chosen due to the similarity of the lattice constants (5.67 A and 5.65 ˚ A 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. ˆ = L 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 = h ¯ /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 h ¯ ωc , 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∗  h ¯ ∇ + eA i 2 + V2deg + Vqpc + Vd . (4.3) We take the vector potential to be A = Bxˆ y, 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 = − h ¯2 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 ˚) and β ' 1.26 is a correction to effective mass theory from effective mass theory (20 A which gives the correct bulk binding energy [81]. rs is a phenomenological screening distance which simulates inner-shell electron screening (taken to be 5 ˚ A). 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 h ¯ 2β 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 h ¯ m∗ ωt  2  13 h ¯ 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 = h ¯ ky 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   h ¯ 2 k 2 ωq2 h ¯2 1 + − =h ¯ ωt 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 h ¯ 2β = 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 . h ¯ (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 ) = h ¯ 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, h ¯ 2k2 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 ˚ A) 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 zˆ, while xˆ and yˆ 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)ˆrdφ sin(θ)dθ 0 0 r ! 2 r 1 1 ˆ+i ˆ x y = 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 · (ˆ x + iˆ y) + (−ˆ x + iˆ y) 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 (ˆ x + iˆ y) − (−ˆ x + iˆ y) 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 | ˆr |ψn00 i Z π 2π Z Y10∗ (θ, φ)Y00 (θ, φ)ˆrdφ sin(θ)dθ ∝ = 0 0 r 1 zˆ. 3 (A.35) (A.36) Therefore, we have the three inner products: r r 1 1 ∗ ∗ ˆr |ψn00 ˆ+i ˆ ψn1+1 x y i ∝ − 6 6 r 1 ∗ ∗ hψn10 zˆ | ˆr |ψn00 i ∝ 3 r r ∗ 1 1 ∗ ˆ+i ˆ. ψn1−1 ˆr |ψn00 i ∝ x y 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. 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