Development of nanoscale electron paramagnetic resonance using a single nitrogen-vacancy center in diamond

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Content DEVELOPMENT OF NANOSCALE ELECTRON PARAMAGNETIC RESONANCE USING A SINGLE NITROGEN-V ACANCY CENTER IN DIAMOND by Chathuranga Abeywardana A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (CHEMISTRY) December 2017 Copyright 2017 Chathuranga Abeywardana This dissertation is dedicated to my parents Sumithra and Gamini Abeywardana, to my person, my wife Sewwandi. ii Acknowledgements First and foremost, I would like to thank my mentor Dr. Susumu Takahashi for his invaluable advices, encouragement, understanding and patience. If it were not for his guidance this thesis work wouldn’t have been a reality. I would also like to extend my gratitude to my defense committee members, Dr. Alex Benderskii and Dr. Vitaly Kresin and my qualifying committee members Dr. Chao Zhang, Dr. Daniel Lidar and Dr. Peter Qin for their unstinting support and invaluable feedback. I thank supports from the National Science Foundation (DMR-1508661 and CHE-1611134), for my Ph.D. research. Im extremely thankful to our collaborators, Dr. George Christou at University of Florida for providing Mn 3 SMM samples, Dr. Kai-Mei Fu group at University of Washington for kindly sharing their experience in diamond annealing and annealing initial diamond samples and Dr. Ania Jayich at University of Santa Barbara for her invaluable consulta- tion about NV coherence measurements. Special token of thanks also goes to, USC Chemistry Department Staff Michele Dea, Magnolia Benitez and Katie McKissick for their support during my time at USC. I iii would like to thank Allan Kershaw for his assistance with cryogenic related technical arrangements, Dr. Frank Devlin for spectroscopy instrumentation and Donald Wiggins and USC machine shop staff for their awesome machining jobs. I would like to thank my former colleagues, Dr. Franklin Cho, Dr. Viktor Stepanov and Dr. Rana Akiel for being wonderful lab mates and their fruitful input on my work. I also thank the current lab members, Zaili (Sharon) Peng, Benjamin Fortman and Laura Mugica and wish them good luck with their graduate studies. Furthermore, I would like to thank my family and friends for their love and encour- agement. Special thanks to my sister, Tharindu for standing by my side since my child- hood. Finally, I would like to express my heartfelt gratitude to my parents for giving me the best education possible so that I could celebrate this moment today and to my wife Sewwandi, for her immense support, patience and great understanding. Thank you! iv Table of Contents Dedication ii Acknowledgements iii List of Figures viii Abbreviations x Abstract xii Chapter 1: Introduction 1 1.1 Evolution of Magnetic Resonance . . . . . . . . . . . . . . . . . . . . 1 1.2 Spin Hamiltonian and Interactions . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Electron Zeeman interaction . . . . . . . . . . . . . . . . . . . 5 1.2.2 Zero-field splitting . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.3 Hyperfine interaction . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.4 Nuclear Zeeman interaction . . . . . . . . . . . . . . . . . . . 9 1.2.5 Nuclear quadrupole interaction . . . . . . . . . . . . . . . . . . 10 1.2.6 Magnetic interactions . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Foundation of EPR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.1 Classical picture of magnetic resonance . . . . . . . . . . . . . 13 1.3.2 Quantum picture of magnetic resonance . . . . . . . . . . . . . 14 1.3.3 Spin relaxations . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3.4 EPR Instrumentation . . . . . . . . . . . . . . . . . . . . . . . 19 Chapter 2: Nitrogen-vacancy center in diamond and optically detected mag- netic resonance 22 2.1 Diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 NV Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2.1 Electronic Structure . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2.2 Spin Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2.3 Optical Properties . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2.4 Optically Detected Magnetic Resonance of NV . . . . . . . . . 26 v 2.3 ODMR experiments of the NV center . . . . . . . . . . . . . . . . . . 27 2.3.1 ODMR system . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3.2 Imaging and CW ODMR . . . . . . . . . . . . . . . . . . . . . 29 2.3.3 Photon statistic measure of a single NV . . . . . . . . . . . . . 33 2.4 Pulsed ODMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.4.1 Pulsed ODMR overview . . . . . . . . . . . . . . . . . . . . . 34 2.4.2 Rabi nutations . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4.3 Pulse ODMR of a single NV . . . . . . . . . . . . . . . . . . . 39 2.4.4 T 1 relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.4.5 Ramsey measurement . . . . . . . . . . . . . . . . . . . . . . . 41 2.4.6 Spin echo measurement . . . . . . . . . . . . . . . . . . . . . . 43 2.4.7 Dynamical Decoupling sequences . . . . . . . . . . . . . . . . 45 2.4.8 NV-based EPR . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Chapter 3: Nanoscale NV-based EPR of several spins in diamond 51 3.1 Scope and Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Chapter 4: NV-based EPR of a few external spins 65 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2 NV fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.3 Covalent attachment of Nitroxide spin . . . . . . . . . . . . . . . . . . 70 4.4 EPR spectroscopy of Nitroxide spin labels using NV . . . . . . . . . . 71 4.5 Magnetic field sensitivity calculations . . . . . . . . . . . . . . . . . . 75 4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Chapter 5: HF EPR study of spin coherence in a Mn 3 single-molecule magnet 78 5.1 Introduction to coherence in SMM . . . . . . . . . . . . . . . . . . . . 78 5.2 High field - high frequency EPR spectrometer . . . . . . . . . . . . . . 79 5.2.1 Rotational sample holder . . . . . . . . . . . . . . . . . . . . . 83 5.3 Mn 3 SMM Experiment and Results . . . . . . . . . . . . . . . . . . . . 85 5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Chapter 6: ODMR at High Frequency and High Magnetic fields 93 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Chapter 7: Conclusion 102 Chapter A: Pulse measurement normalization 104 vi Chapter B: Theory of ESEEM 106 Chapter C: FID and SE decay functions 114 Chapter D: Characterization of NV centers in various diamonds 119 D.0.1 CVD-grown diamond . . . . . . . . . . . . . . . . . . . . . . . 119 D.0.2 Summary ofT 2 of implanted NVs . . . . . . . . . . . . . . . . 125 Bibliography 126 vii List of Figures 1.1 Evolution of magnetization vector in Bloch sphere . . . . . . . . . . . . 17 1.2 Schematic diagram of a typical EPR instrument . . . . . . . . . . . . . 19 2.1 Characteristic features of NV center . . . . . . . . . . . . . . . . . . . 25 2.2 Schematic of ODMR experimental setup . . . . . . . . . . . . . . . . . 28 2.3 FL image of diamond and FL fluctuation dependence on integration time 30 2.4 Positioning of NV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.5 CW ODMR of a single NV . . . . . . . . . . . . . . . . . . . . . . . . 32 2.6 Photon Statistic measurement of a single NV . . . . . . . . . . . . . . . 33 2.7 Overview of pulse measurement sequence . . . . . . . . . . . . . . . . 35 2.8 FL time measurement of a single NV . . . . . . . . . . . . . . . . . . . 37 2.9 Rabi measurement of a single NV . . . . . . . . . . . . . . . . . . . . 38 2.10 pulsed ODMR of 14 NV . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.11 T 1 measurement of a single NV . . . . . . . . . . . . . . . . . . . . . . 41 2.12 Ramsey measurement of an NV center . . . . . . . . . . . . . . . . . . 42 2.13 Spin echo measurement of an NV center . . . . . . . . . . . . . . . . . 44 2.14 CPMG measurement on an NV center . . . . . . . . . . . . . . . . . . 46 2.15 NV-based EPR spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 48 2.16 NV-based EPR Rabi measurements . . . . . . . . . . . . . . . . . . . . 50 3.1 CW EPR spectrum of bulk type Ib diamond crystal taken at 8 T . . . . . 52 3.2 ODMR experiment of NV1 used in NV-based EPR experimet . . . . . . 54 3.3 NV-based EPR experiment using DEER spectroscopy . . . . . . . . . . 57 3.4 Simulation of the NV-based EPR signals to estimate detected number of spins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.1 Summary of SRIM calculation and FL images at different depths . . . . 66 4.2 FL images of diamond after each annealing process . . . . . . . . . . . 67 4.3 A stack of FL images at different depth taken before and after NV fab- rication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.4 Pulsed ODMR of 15 NV . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.5 NV-based EPR of Nitroxide attached diamond crystal . . . . . . . . . . 73 viii 4.6 NV-based EPR measurement using narrower excitation bandwidth with Nitroxide attached diamond crystal . . . . . . . . . . . . . . . . . . . . 74 4.7 Nitroxide spectrum analysis. . . . . . . . . . . . . . . . . . . . . . . . 75 4.8 Minimum detectable dipolar field versus coherence time of the NV . . . 76 5.1 Overview of HF EPR spectrometer . . . . . . . . . . . . . . . . . . . . 81 5.2 Circuit diagram of the high frequency, high-power transmitter and re- ceiver system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.3 Sample rotating stage for HF EPR spectrometer . . . . . . . . . . . . . 83 5.4 Calibrating rotational sample stage using angle dependence of diamond CW EPR spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.5 Schematic structure and CW HF EPR measurements of Mn 3 crystals . . 87 5.6 Spin echo measurements of Mn 3 single crystals . . . . . . . . . . . . . 89 5.7 Temperature dependence of the spin decoherence for Mn 3 samples A and B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.1 Schematic diagram of HF ODMR setup . . . . . . . . . . . . . . . . . 95 6.2 Magnetic field dependence of the NV center FL intensity . . . . . . . . 96 6.3 Angle and magnetic field dependence of the NV populations ofm s states 97 6.4 ODMR data taken with HF ODMR setup . . . . . . . . . . . . . . . . . 99 A.1 Pulse measurement normalization scheme . . . . . . . . . . . . . . . . 104 B.1 Basis and Spin matrix operators used in ESEEM derivation . . . . . . . 108 B.2 Diagonalization unitary operator angle relations . . . . . . . . . . . . . 110 D.1 Optical microscope image of CVD grown surface. . . . . . . . . . . . . 120 D.2 FL image of CVD grown layer . . . . . . . . . . . . . . . . . . . . . . 121 D.3 CW ODMR of the selected FL spot . . . . . . . . . . . . . . . . . . . . 122 D.4 Pulsed ODMR of the selected FL spot . . . . . . . . . . . . . . . . . . 123 D.5 SE and NV-based EPR taken with the selected FL spot . . . . . . . . . 124 D.6 Statistical summary of T 2 measured from implanted NVs at different depths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 ix Abbreviations AC Alternating Current AOM Acoustic Optic Modulator APD Avalanche Photodiode BNC Bayonet Neill-Concelman CPMG Carr-Purcell-Meiboom-Gill CVD Chemical Vapor Deposition CW Continuous Wave DAQ Data Acquisition Card DC Direct Current DD Dynamic Decoupling DEER Double Electron Electron Resonance EG Electric Grade ELDOR Electron Electron Double Resonance EPR Electron Paramgentic Resonance ESEEM Electron Spin Echo Envelop Modulation FID Free Induction Decay FL Fluorescence FWHM Full Width at Half Maximum GPIB General Purpose Interface Bus HBT Hanbury-Brown-Twiss HF High Frequency x HPTH High Pressure High Temperature IF Intermediate Frequency LHe Liquid Helium LNA Low Noise Amplifier LO Local Oscillator MR Magnetic Resonance MRFM Magnetic Resonance Force Microscopy MRI Magnetic Resonance Imaging MW Microwave ND Nano Diamond NMR Nuclear Magnetic Resonance NV Nitrogen Vacancy OA Optical Attenuators ODMR Optically Detected Magnetic Resonance OU Ornstein-Uhlenbeck QTM Quantum Tunneling of Magnetization RF Radio Frequency RO Readout SE Spin Echo SMM Single Molecular Magnet SNR Signal to Noise Ratio SPS Single Photon Sources SRIM Stopping and Range of Ions in Matter TCSPC Time Correlated Single Photon Counting TTL Transistor-Transistor Logic ZFS Zero Field Splitting xi Abstract Magnetic resonance techniques, such as nuclear magnetic resonance (NMR) and elec- tron paramagnetic resonance (EPR), are powerful and versatile analytical methods used in science. These techniques can be used to probe the local structure and dynamic prop- erties of various systems including biological molecules. However, one draw back of the MR techniques is their intrinsically low sensitivity. This will limit its applications to an- alyze samples with very small volumes. For instance, more than 10 10 electron spins are typically required to observe EPR signals at room temperature. An instrument capable of single spin EPR and NMR will open up unprecedented opportunities. For instance, it will enable determination of atomic structure of each individual biomolecule, which will transform structural biology. A nitrogen-vacancy (NV) center consisting of a substitutional nitrogen next to a vacancy is a paramagnetic color center in diamond. Because of unique optical and spin properties of the NV center including stable fluorescence, long spin coherence, optical readout of the spin state and ability to initialize the spin state optically, an NV is a promising candidate for fundamental studies in quantum information processing and nanoscale sensing. This thesis discusses the development of nanoscale EPR spectroscopy using a single NV center in diamond. In chapter 1, I introduce basic concepts of MR. Physical, optical xii and spin properties of NV centers in diamond are introduced in Chapter 2. Moreover, Chapter 2 describes optically detected magnetic resonance spectroscopy to identify a single NV center and to measure the spin coherent state of the NV center as well as double electron-electron resonance spectroscopy to perform EPR using the NV center (NV-based EPR). In chapter 3 and 4, I discuss the demonstrations of NV-based EPR spectroscopy. Chapter 3 discusses nanoscale NV-based EPR of several paramagnetic spins inside the diamond crystal. Chapter 4 presents NV-based EPR of a single or a few radicals located outside the diamond. In Chapter 5 and 6, I discuss the experiments at high magnetic fields. In Chapter 5, the investigation of spin decoherence of Mn 3 molecular nanomagnets using high-frequency EPR spectroscopy is discussed. The de- velopment of NV-based EPR spectrometer is discussed in Chapter 6. Finally, Chapter 7 summarizes the work presented in this thesis. xiii Chapter 1: Introduction 1.1 Evolution of Magnetic Resonance The first observation of magnetic resonance (MR) phenomena was reported in 1938 by Isidor Isaac Rabi (American physicist, 1898-1988) where his team observed res- onant absorption of electromagnetic waves from both Li and Cl nuclear spins in LiCl molecules. 1 The experimental setup consists of a LiCl beam source, a molecular detector and a pair of magnets. The magnets are placed in series to provide the opposite direc- tion of inhomogeneous magnetic fields (magnetic field gradients). When LiCl molecular beam is injected into the magnet system, the beam is deflected due to the interaction be- tween the magnetic moments in LiCl and the magnetic field. However, the setup was adjusted such that the total deflection is zero by canceling the deflection from the first magnet by the deflection from the second magnet. Therefore, the detector located after the second magnet records a high intensity of LiCl beam. Then, for the observation of MR, a stage to provide a static magnetic field (B 0 ) and an alternating magnetic field (B 1 (t)) was inserted between the first and second magnets in the system to orient the magnetic moment of LiCl molecules through MR. When the the frequency of the alter- nating magnetic field satisfies the MR condition, the inserted stage flips the magnetic moment of LiCl (i.e. + z ! z ), therefore, the system does not hold the condition of 1 Chapter 1. Introduction 2 the total defection cancellation anymore, then a lower intensity is recorded by the detec- tor. In the experiment, the beam intensity was recorded as a function of the frequency of the alternating magnetic field, which displayed the MR absorption spectrum at the predicted frequency successfully. This remarkable discovery stimulated other scientists to find MR in other systems. One example is quantitative determination of the neutron moment reported by L.W. Alvarez and F. Bloch in 1940 using the molecular beam MR technique. 2 Rabi was awarded Nobel prize in 1944 under the title of “for his resonance method for recording the magnetic properties of atomic nuclei”. As it was for many, World War II was a dark period of time for MR resonance as well. However, a new era of MR began after this period due to the tremendous improvement and availability of microwave (MW) instrumentation. First ever successful MR discovery of electron spins, so called Electron Paramagnetic Resonance (EPR) was observed by the soviet scientist E.K. Zavoisky in 1944 at Kazan State University. 3 In this experiment, he introduced a magnetic field modulation technique to increase the MR detection sensitivity with a broadband excitation, then he successfully observed the change of intensity of the paramagnetic absorption as a function of applied static magnetic field for CuCl 2 2H 2 O powder sample. In 1950s Norman F. Ramsey (American scientist, 1915-2011) published a theoretical paper of MR. 4, 5 The paper described a way to detect MR signals using a short interval of oscillatory fields (pulse fields). This was a new and innovative idea inspired by Rabi’s molecular beam MR experiment. The advantage of this method was its ability to acquire a narrower resonance absorption. This work initiated the so-called pulsed MR which is a standard technique to measure Nuclear Magnetic Resonance (NMR) now. Ramsey was awarded Nobel prize in 1989 for this work. Soon after Ramsey’s work, E.L. Hahn came up with the genius idea of NMR spin- echo (SE) technique. 6 He used pulses of resonant electromagnetic radiation to refocus Chapter 1. Introduction 3 spin magnetization. From this point onward there have been tremendous progress in MR techniques (specially, NMR), and the fundamentals of MR have also been applied to various fields of sciences and engineering. One such example is magnetic resonance imaging (MRI) which plays a significant role in medical diagnosis by capturing images of anatomy. MRI uses pulse radiation to excite nuclear transitions and localize signals in space using magnetic field gradients while recording relaxation times of nuclei, mostly the hydrogen nuclei of water and fat. Therefore, MR is a technique that has a broader impact reaching beyond basic scientific research. The work presented in this thesis relates to EPR. Detection of unpaired electrons in transition metal ions, free radicals and defect centers in semiconductors can be obtained through EPR spectroscopy. In chemistry, EPR is used in applications such as, study- ing kinetics of radical reactions, 7 spin-trapping to characterize short lived free radicals 8 and to monitor polymer degradation via detecting radical formation under exposure to radiation. 9 In physics EPR is used to measure magnetic susceptibility in material since an object with a different magnetic susceptibility than the surrounding medium perturbs the static magnetic field which will change the features of the EPR signal. 10 EPR is also employed in crystallography to detect and characterize the defects in crystals 11 and mea- sure crystal fields in single crystals. 12 EPR is also widely used in biology and medicine as well. Studying confirmational changes in bio-molecules (such as DNA, proteins) by measuring the distances among spin labels is a common application of EPR that is applied in biology. 13 Additionally, EPR is used to study photosynthesis by detecting the radical formation during electron transport. 14 EPR is successfully applied to study metalloprotein structures 15, 16 and in oximetry, to measure blood oxygen levels. 17 Chapter 1. Introduction 4 1.2 Spin Hamiltonian and Interactions The energy of a spin state in a paramagnetic system which has an electron spinS andm number of nuclear spinI is often described by the static spin Hamiltonian 18 as follows. H 0 =H EZ +H ZFS +H HF +H NZ +H NQ +H SS = B B 0 ! g e ^ S + ^ S ! D ^ S + m X i=1 ^ S ! A i ^ I i N g n;i B 0 ^ I i + ^ I i ! P i ^ I i +H SS ; (1.1) where ^ S and ^ I represents the electron and nuclear spin operators respectively. The elec- tron Bohr magneton and the nuclear magneton are denoted by B and N respectively. g e is the electron g factor andg n is the nuclear g factor. The zero field splitting tensor defined as D. A and P represents the hyperfine tensor and the nuclear quadrupole ten- sor respectively. Equation 1.1 includes;H EZ : the electron Zeeman interaction;H ZFS : the zero-field splitting;H HF : the hyperfine coupling between the electron spin and the m nuclear spins;H NZ : the nuclear Zeeman interaction;H NQ : the nuclear quadrupole iterations for spins withI > 1=2;H SS : weak coupling with neighboring electron spins and nuclear spins. In these formulas energies are in SI unit Joule and interaction terms are presented in descending order of the interaction strength. A Hamiltonian of a spin system consisting of n electron spins and m nuclear spins spans in a Hilbert space with n H dimensions given by n H = n Y i=1 (2S i + 1) m Y i=1 (2I i + 1): (1.2) Chapter 1. Introduction 5 1.2.1 Electron Zeeman interaction The interaction between an electron spin and external magnetic fieldB 0 (B x 0 ;B y 0 ;B z 0 ) is described by the electron Zeeman term (equation 1.3) which is often a dominant term in the spin Hamiltonian. The electron Zeeman Hamiltonian inH EZ is given by, H EZ = e 2m e B 0 gS = B h B 0 gS = e B 0 S; (1.3) where B is the Bohr magneton, m e is the rest mass of the electron, e is the ele- mentary charge of the electron and h is the Planck constant. Isotropic g-tensor (g e = 2.0023193043617) or electron gyromagnetic ratio e ( e =jejg e = (2m e ) = g e B =} =228.201495164 GHzT 1 ) for an isolated electron spin is one of the most precisely determined physical constants. 19 The external magnetic fieldB 0 are orientation depen- dent. Thereforeg is defined by a general form of a tensor with the components, g = 0 B B B B B B @ g xx g xy g xz g yx g yy g yz g zx g zy g zz 1 C C C C C C A : (1.4) Chapter 1. Introduction 6 g-tensor can be diagonalized via the Euler angle transformation of the magnetic field vector into the molecular coordinate system to yield g diag: = 0 B B B B B B @ g x 0 0 0 g y 0 0 0 g z 1 C C C C C C A : (1.5) For a system with cubic symmetry, g x = g y = g z ; for a system with axial symmetry, g x = g y = g ? andg z = g k , and for a system with orthorhombic symmetry,g x 6= g y 6= g z . A deviation of theg values (equation 1.7) from the g e value is caused by a spin-orbit coupling. This effect is described by the Hamiltonian H =H Z +H LZ = B B 0 ^ L +g e ^ S + ^ L ^ S; (1.6) whereH Z is the electron Zeeman term andH LS is the spin-orbit interaction with being the spin-orbit coupling constant. We consider an effective Hamiltonian using the perturbation theory. The first order perturbation is zero since orbital angular momentum is quenched. The second order perturbation term is non-zero because admixture of the orbital angular momentum to the spin is caused by the interaction of excited states and ground state. 20 Therefore, the resulting Hamiltonian is given by, H = B B 0 g e ! I + 2 ! | {z } g ^ S + 2 ^ S ! ^ S + 2 B ^ B 0 ! ^ B 0 : (1.7) ! is a symmetric tensor with the elements ! = X n6=0 h 0 j ^ Lj n ih n j ^ Lj 0 i 0 n ; (1.8) Chapter 1. Introduction 7 wherej 0 i andj n i represents the wave-functions of the ground state and the nth ex- cited state respectively. The corresponding energies are 0 and n . The first term in equation 1.7 is the effective Zeeman term as given in the equation 1.3. The second term represents the zero-field splitting due to crystal field and further explained in the next section. This will be non-zero for a spin system withS > 1=2. The third term does not have any effect on MR spectrum as it introduce an equal offset on all the spin levels. 1.2.2 Zero-field splitting For a system with a spin-orbit interaction and a spin-spin interaction, it is often conve- nient to represent the system using their total momentumS(> 1=2). In the representa- tion withS, the interaction term is given by the zero-field splitting term. It is expressed using following term, H ZFS = ^ S ! D ^ S; (1.9) where ^ S is the total spin operator and ! D is the traceless zero-field splitting tensor. In the principal axes system of the ! D tensor, equation 1.9 is also written as H ZFS =D x ^ S 2 x +D y ^ S 2 y +D z ^ S 2 z =D ^ S 2 z 1 3 S(S + 1) +E ^ S 2 x ^ S 2 y ; (1.10) whereD = 3D z =2,E = (D x D y )=2 and (S i ) i=x;y;z represents the spin operators in Cartesian coordinate system. D = E = 0 for a system with cubic symmetry. For a system with axial symmetry, D6= 0;E = 0. For system with symmetries lower than axial symmetry,D6= 0,E6= 0. Chapter 1. Introduction 8 1.2.3 Hyperfine interaction The third term in equation 1.1 represents the interaction between an electron and nuclear spins which is called the hyperfine coupling. The hyperfine interaction consists of two terms. Isotropic or Fermi contact interaction;H F and electron-nuclear dipole-dipole coupling;H D . The Fermi contact interaction is written as H F =a iso ^ S ^ I; (1.11) where a iso = 2 0 3 g e B g n N j 0 (0)j 2 ; (1.12) denotes the isotropic hyperfine coupling constant andj 0 (0)j 2 is the electron density at the nucleus. g N is the nuclear g factor and N is the nuclear magneton. The second contribution of the hyperfine interaction is from electron-nuclear dipole-dipole coupling and is expressed as H D = 0 4 g e B g n N " 3( ^ Sr)( ^ Ir) r 5 ^ S ^ I r 3 # ; (1.13) wherer is the radius vector representing the displacement between electron and nuclear spins. Since electron has a certain distribution which depends on the orbital, equa- tion 1.13 has to be integrate over the spatial distribution of electron. This will yield the Hamiltonian term for an-isotropic dipole-dipole coupling 21 H D = ^ S ! A D ^ I; (1.14) Chapter 1. Introduction 9 where ! A D is the traceless dipolar coupling tensor with elements (A D ) ij = 0 4 g e B g n N 0 3r i r j ij r 2 r 5 0 : (1.15) Using equation 1.11 and equation 1.13, the hyperfine interaction term (the third term in equation 1.1) is written as H HF = ^ S ( ! A D + ! a iso ) ^ I = ^ S ( ! A ) ^ I: (1.16) ! A tensor can be diagonalized in Cartesian coordinate system. A dig: = 0 B B B B B B @ A xx 0 0 0 A yy 0 0 0 A zz 1 C C C C C C A : (1.17) Similar to the isotropic g-tensor introduced in section 1.3 the hyperfine interaction terms for an axial symmetry system has;A xx =A yy =A ? andA zz =A k . 1.2.4 Nuclear Zeeman interaction Analogous to the electron Zeeman interaction, the interaction between nuclear spin I and the external magnetic fieldB 0 is described by the following nuclear Zeeman inter- action H NZ = N m X i g n;i B 0 ^ I i : (1.18) The nuclear spin quantum numberI and the nuclear g-factor,g n are inherent properties of a nucleus. The nuclear Zeeman interaction is considered to be isotropic in most Chapter 1. Introduction 10 of EPR experiments and it virtually has no effect on the EPR transition energy unless nuclear Zeeman interaction and hyperfine interaction are in same order of magnitude. When they are in the same order of magnitude, so-called spin-flip transitions may occur due to the nuclear Zeeman interaction. Such transitions are observed as weak satellite lines on either side of the allowed EPR transitions. 1.2.5 Nuclear quadrupole interaction A non-spherical charge distribution in nuclei withI 1 gives rise to a nuclear electrical quadrupole momentQ. This moment interact with an electric field gradient generated by electrons and nuclei in the vicinity. This contribution is described by H NQ = m X I i >1=2 ^ I i ! P i ^ I i ; (1.19) where ! P is the traceless nuclear quadrupole tensor and the Hamiltonian is also written as H NQ =P x I 2 x +P y I 2 y +P z I 2 z = e 2 qQ 4I(2I 1) (3I 2 z I(I + 1) 2 +(I 2 x I 2 y ) ; (1.20) wheree is the charge of an electron,q is the electric field gradient and = (P x P y )=P z is asymmetry parameter withjP z jjP y jjP x j and 0 1 whereP z being the largest term in the quadrupole tensor which can be expressed asP z = e 2 qQ=(2I(2I 1)). 1.2.6 Magnetic interactions While electron spins located in the same atom is often described by the total spin with S > 1=2 as discussed in section 1.2.2 it may be more convenient to describe a pair of Chapter 1. Introduction 11 weakly interacting spins by their individual spin Hamiltonian (H 0 (S 1 )andH 0 (S 2 )) and additional terms representing the coupling between them 22 H SS =H 0 (S 1 ) +H 0 (S 2 ) +H exch +H dd ; (1.21) whereH exch is the exchange term andH dd is the dipole-dipole interaction term. In the presence of significant orbital overlap between two species, unpaired electrons can be exchanged among them. The exchange coupling is divide into isotropic and anisotropic contributions which are characterized using the exchange coupling tensorJ. The exchange coupling term in equation 1.21 is in the form H exch = ^ S 1 ! J ^ S 2 =J 12 ^ S 1 ^ S 2 : (1.22) WhenJ 12 is positive, the ground state is a triplet state and excited state is a singlet state. This is opposite whenJ 12 is negative. Positive value ofJ 12 represents the max- imum multiplicity in ground state (ferromagnetic coupling) and negative value of J 12 represents anti-ferromagnetic coupling. 23 Hamiltonian term which represents dipole coupling for the central electron spin to n N number of electron spins in the bath which is analogous to equation 1.13 is given by 20, 22 H dd = 0 4 g e 2 B n N X i=1 g N i " ^ S ^ S N i r 3 i ( ^ Sr i )( ^ S N i r i ) r 5 i #! ; (1.23) where r i represents the vector connecting i-th bath electron spin ( ^ S N i ) to the central electron spin ( ^ S). (g N i ) represents theg values of bath spins. Expanding above equa- tion in terms of the components of spin operator and the spatial vector will lead to Chapter 1. Introduction 12 equation 1.9. 24 In the high field approximation where the dipoles align parallel toB 0 , equation 1.23 becomes 20 H dd = 0 4 g e 2 B n N X i=1 3 cos 2 i 1 r 3 i ^ S z g N i ( ^ S N z ) i ; (1.24) where i is the angle between the direction of the central dipole and the displacement vector from the central dipole to thei th coupled dipole. Thus, the interaction energy of two dipoles ofS = 1=2 under high field approximation is written as E dd = 0 4 g e 2 B 1 r 3 1 3 cos 2 : (1.25) 1.3 Foundation of EPR In a case where the magnetic field is aligned along the z-direction thatB x = B y = 0, the electron Zeeman term (equation 1.3) is given by, H =g e B S z B 0 ; (1.26) where B z = B 0 (conventionally used in EPR). The energies of the system shown in equation 1.26 are given by Ej"i = 1 2 g e B B 0 Ej#i = + 1 2 g e B B 0 ; (1.27) whereEj"i andEj#i denotes the eigenvalues of the magnetic moments aligned par- allel and anti-parallel with respect to the applied magnetic field, respectively. Once the frequency of MW excitation equals to the energy gap between the two states Chapter 1. Introduction 13 (E = E j"i E j#i), a resonant absorption take place. Thus, the frequency of MW excitation required to excite the resonance transition is hf =g e B B 0 f = 2 B 0 : (1.28) The frequency (! 0 =f 2) is defined as the Larmor frequency. 1.3.1 Classical picture of magnetic resonance A simple picture of MR absorption can be visualized by considering equation of motion for a magnetization vectorM subjected to an external magnetic fieldB. The equation of motion and their solutions are given in following equations. 24 dM x dt =! 0 M y ; dM y dt =! 0 M x ; dM z dt = 0; M x =M ? 0 cos(! 0 t); M y =M ? 0 sin(! 0 t); M z =M k 0 : (1.29) Above equations describe precession of the magnetization about z-axis with the fre- quency of ! 0 . Assume we apply a small oscillatory magnetic field B 1 (B 1x = B 1 cos(! mw t);B 1y = B 1 sin(! mw t);B 1z = 0) and (B 1 B 0 ) at the frequency of ! mw , then consider the system in a rotating frame which rotates withB 1 . In the frame, the magnetization vector will see a small constant magnetic field inxy plane. So the equation of the motion is given by, 18 dM x dt =(! 0 ! mw )M y dM y dt = (! 0 ! mw )M x ! mw M z dM z dt =! mw M y ; (1.30) Chapter 1. Introduction 14 where (! 0 ! mw ) represents the resonance offset. In the presence of the applied B 1 field, effective precession frequency of magnetization vector can be written as ! eff = q (! 0 ! mw ) 2 +! 2 1 ; (1.31) where! 1 =g e B B 1 =~. Under the resonance condition (! mw =! 0 ),! eff ! 1 . So the magnetization continuously nutates from its +z projection toz projection along the direction of the appliedB 1 field passing throughxy plane. TheB 1 field is provided thorough electromagnetic radiation (MW or RF). 1.3.2 Quantum picture of magnetic resonance In presence ofB 1 (B 0 ), equation 1.26 becomes H =H 1 +H 0 =g e B B 1 (S x cos! mw t +S y sin! mw t) +g e B B 0 S z : (1.32) Suppose = c " " +c # # is the eigenfunction of the above Hamiltonian. Substituting in time dependent wave equation (H =i~(d =dt)) we can obtained H 1 (c " " +c # # ) =i~ " @c " @t +i~ # @c # @t : (1.33) Multiplying equation 1.33 by # and " separately we get c " h#jH 1 j"ie i(Ej"iEj#i)t ~ =i~ @c # @t c # h"jH 1 j#ie i(Ej"iEj#i)t ~ =i~ @c " @t : (1.34) Chapter 1. Introduction 15 Matrix elements in the above equation is written as follows by considering the full form ofH h#jH 1 j"i =g e B B 1 hD #j ^ S x j" E cos! mw t + D #j ^ S y j" E sin! mw t i =g e B B 1 [cos! mw t +i sin! mw t] = 1 2 g e B B 1 e i!mwt h"jH 1 j#i = 1 2 g e B B 1 e i!mwt : (1.35) The intensity of an EPR signal is proportional to the transition probability which is given by the square of the matrix elements (equation 1.35). With the use of equation 1.35 and the initial condition (t = 0,c " = 0 andc # = 1), equation 1.34 is solved forc " c " (t) = g e B B 1 2i~ e i(! 0 !mw)t 1 i(! 0 ! mw ) : (1.36) Hence the probability to find the system inj"i is given as c " c " (t) = g e B B 1 2~ 2 sin 2 1 2 (! 0 ! mw )t 1 2 (! 0 ! mw ) 2 : (1.37) Equation 1.37 indicates the transition probability has a maximum when ! mw = ! 0 which is the resonance condition. 1.3.3 Spin relaxations Spins will find their way to the thermal equilibrium distribution by being able to ex- change energy with the environment or the lattice (in solids). This method of relaxation is known as the spin-lattice relaxation or the longitudinal relaxation which is often de- noted by the symbol T 1 . Origin of this relaxation is the absorption or the stimulated Chapter 1. Introduction 16 emission of phonons which are energy quanta of lattice vibrations. There are few mech- anisms to explain theT 1 relaxation. Namely, Direct process where one phonon at reso- nance frequency is absorbed or emitted; Raman process where the spin system absorbs a phonon with a frequency that is higher than resonance frequency and transit to a virtual energy level and fall back to the ground state; Orbach process where the Raman process involves when an actual low-lying spin level exists. Longitudinal relaxation which leads to the spin flip influences the correlation of the phase of precession between flipping spin and the other spins. This will contribute to transverse relaxation which is typically denoted byT 2 . Following section explains the effect of relaxation processes on the spin dynamics. Bloch equations given in equation 1.30 is now modified to the following by consid- ering longitudinal relaxationT 1 and transverse relaxationT 2 dM x dt =(! 0 ! mw )M y M x T 2 dM y dt = (! 0 ! mw )M x ! mw M z M y T 2 dM z dt =! mw M y M z M 0 T 1 : (1.38) A simple pulse experiment is to observe a free evolution of the spin after application of a resonant MW pulse. Assume that the MW pulse was applied along x axis with length t p with frequency! mw . To find components of the magnetization vector at the end of the pulse, the equation 1.30 has to integrate with the initial conditionsM z = M 0 and M x =M y = 0 att = 0. This leads to Chapter 1. Introduction 17 M x =M 0 sin(! 1 t p ) M y =M 0 sin(! 1 t p ) M z =M 0 cos(! 1 t p ) ; (1.39) where angle = ! 1 t p is the flip angle. As an example when! mw = ! 0 ift p is long enough to flip magnetization toy axis or a complete=2 flip angle such MW pulse is defined as=2 pulse. Immediately after the MW pulse, the components of the magneti- zation vector are represented by equation 1.39. The free evolution of the magnetization is considered as rotation about z axis as illustrated in Fig. 1.1. y z x M 0 α σ -M 0 sinαcosσ M 0 sinαsinσ M 0 cosα M 0 sinα Figure 1.1: Evolution of magnetization vector in Bloch sphere after MW pulse of length t p which corresponds to flip angle = ! 1 t p and = (! 0 ! mw ). Therefore phase accumulation during free evolution timet would bet. Chapter 1. Introduction 18 The components of the magnetization after the free evolution time t is written as M x (t) =M 0 sin() sin((! 0 ! mw )t) M y (t) =M 0 sin() cos((! 0 ! mw )t) M z (t) =M 0 cos() : (1.40) Similarly, the solutions of equation 1.38 are obtained as M x (t) =M 0 sin() sin((! 0 ! mw )t) exp t T 2 M y (t) =M 0 sin() cos((! 0 ! mw )t) exp t T 2 M z (t) =M 0 (M 0 M z ) exp t T 1 ; (1.41) which includes decay of the magnetization govern by the relaxation processes. In con- ventional EPR detection combination of the components of the transverse magnetization is measured as complex transverse magnetization which is defined asM x +iM y . How- ever, ODMR experiments may measure the longitudinal magnetization via spin popula- tion dynamics. In such a case it is necessary to transfer the transverse components of the magnetization to the longitudinal magnetization using an additional pulse (probably =2) at the end of sequence. Such a case will be explained in the section 2.4.5. Furthermore, considering the bloch equations (section 1.3.3) and the relaxation mechanisms, absorption lineshape of an EPR spectrum is write as 24 L(! mw ) = A M 0 (1=T 2 ) ! 2 1 (T 1 =T 2 ) + (1=T 2 ) 2 + (! 0 ! mw ) 2 ; (1.42) whereA is the parameter depends on instrumental factors,T 1 is the longitudinal relax- ation andT 2 is the transverse relaxation.M 0 is the magnetization at thermal equilibrium. Chapter 1. Introduction 19 As were in equation 1.37, the functionL(! mw ) has a maximum when! mw = ! 0 . For a largeB 1 fields and longerT 1 , intensity of the functionL(! mw ) reduces due to the sat- uration. Therefore, the function in equation 1.42 will reduces to the lorentzian function as! 2 1 (T 1 =T 2 ) term is neglected. 1.3.4 EPR Instrumentation Excitation Detection Magnet Isolator MW Source Directional Coupler Circulator Terminated load Reference Arm Sample in Resonace Cavity Modulation coil Pre- Amplifier Lock - in Computer Variable attenuator Modulation Freq. Generator Figure 1.2: Schematic diagram of a typical EPR instrument This section explains fundamentals of EPR spectrometer. Figure 1.2 shows a schematic of a typical EPR instrument (such as X-band EPR spectrometer). The MW bridge carries both MW source used in excitation (radiation source) and the detector. Early days vacuum tube called “Klystron” 21 used to synthesize MW. Now, solid state devices such as “Gunn diodes” is often used. The isolator located after the MW source protects the synthesizer from any reflected MW. Then the MW excitation splits in to two by directional coupler. One path will directs to the cavity via the circulator and other path goes to the reference arm. The MW power in these paths can controlled via Chapter 1. Introduction 20 variable attenuators. An additional phase shifter located in the reference arm is used to set a defined phase relationship with respect to reflected signal. Thus, a phase sensitive detection mechanism can be employed. The circulator only passes reflected MW signals (remaining excitation radiation after absorption) coming from the cavity towards the detector. The cavity or MW resonator may be employed to enhance the strength of magnetic field associate with MW radiation. However, EPR resonance can also be obtained without cavity as seen in high frequency EPR instrument (section 5.2). An iris is used to control the radiation passes in to the cavity from waveguide. The impedance matching between the cavity and the waveguide is also accomplished with the iris. When the iris it set at perfect matching all the MW radiation is stored in the cavity and dissipate as heat but no power is reflected back. At the resonance condition, MW is absorbed in the sample. This changes the coupling to the cavity and the cavity will be no longer critically coupled. As a result of this, MW will be reflected back as magnetic field sweeps through a resonant transition. The reflected MW signals are modulated by a small Helmholtz coil located outside the cavity. This produces an ac magnetic field (often 1 kHz - 100 kHz) along the axis of the static magnetic field. The combined reflected signal and reference signal directs to the detection. A Schot- tky diode which converts MW power to an electric current is the most commonly used detector in modern EPR instruments. The MW power coming from the reference arm act as a bias to keep Schottky diode in linear region for optimal performance. In the ex- periment with the field modulation, at the resonance, the intensity of the reflected MW is modulated with the same frequency as the modulation frequency. A lock-in detec- tion compares the modulated signal with the reference from the field modulation and Chapter 1. Introduction 21 produces a DC signal proportional to the amplitude of the modulated signal and sup- presses other components different from modulation frequency. The spin sensitivity of the EPR spectrometer depends largely on the sensitivity of the MW detector used in the system (e.g. mixer and Schottky diode). A typical spin sensitivity at room temperature is 10 10 spins. Chapter 2: Nitrogen-vacancy center in diamond and optically detected magnetic resonance In this chapter, I introduce a nitrogen-vacancy (NV) defect center in diamond and opti- cally detected magnetic resonance (ODMR) spectroscopy. As I discussed in section 1.2, the spin sensitivity of EPR is limited10 10 spins at room temperature due to the photon sensitivity of the MW detector used in the EPR spectrometer. One way to improve the sensitivity is the use of a sensitive spin sensor. A single electron spin detection has been demonstrated using magnetic resonance force microscopy (MRFM), 25 however MRFM detection is sensitive only at very low temperature ( sub Kelvin) due to its strong cou- pling to surrounding environments. An NV center in diamond is a promising candidate for applications of room temperature magnetic sensing with single spin sensitivity. My Ph.D. research explores a method of EPR detection using a single NV center (NV-based EPR) to achieve the spin sensitivity to the level of a single electron spin. Here, I intro- duce physical and electrical properties of NV centers, ODMR techniques for a single NV detection and NV-based EPR experiment. 22 Chapter 2. Nitrogen-vacancy center in diamond and optically detected magnetic resonance 23 2.1 Diamond Diamond is one of the most common materials used in modern day jewelry industry. Physical properties of diamond is also extreme. It is an excellent thermal conductor and one of the hardest materials. In fact, diamond has been known as a hard material for a very long time. In 250 BC in India, diamond had already been used to drill holes in beads. Furthermore, a recent research showed that diamond was used as an abrasive material in 4500 year ago in China. 26 Moreover electronic and optical properties of diamond is also found to be unique. There have been extensive research on diamond for future applications in electronic devices, optical communication and quantum in- formation science recently. The electronic and optical properties highly depend on the impurity contents in diamond. Therefore, extensive search of impurities had been per- formed and there are many impurities in the diamond lattice which had already been identified. 11 2.2 NV Center In my Ph.D. research, I used an NV center in diamond as a magnetic sensor for nanoscale EPR. An NV center is a paramagnetic impurity in diamond which consists of a substi- tutional nitrogen impurity atom next to a vacancy in the diamond lattice. 2.2.1 Electronic Structure Six electrons are involved in electronic structure of the NV center. The substitutional nitrogen atom provides two electrons and three electrons are coming from the dangling Chapter 2. Nitrogen-vacancy center in diamond and optically detected magnetic resonance 24 bonds of three carbon atoms surrounding the vacancy. An extra electron which is cap- tured from the lattice, typically from nitrogen donors, makes the negatively charged NV center (NV ). In the ground state, two electrons in higher-energy lying states form a S = 1 spin triplet state while the other four electrons in lower-lying states form the closed shell so that the spin of the ground state is S = 1. 27 The orbitals of the two unpaired electrons are mostly located between the nitrogen atom and vacancy. 28 2.2.2 Spin Properties As described in the section 2.2.1, an NV center in diamond isS = 1 spin system. The spin Hamiltonian of the NV center in the unit of Hz is given by, H NV =D S 2 z 2 3 + g NV B h B 0 S; (2.1) whereD is the zero-field splitting term,g NV is the g-factor, B is the Bhor magneton,h is the Planck constant,B 0 is an external magnetic field andS is the spin operator. The magnetic dipole interaction between the two unpaired electron spins in the NV center lifts the degeneracy of m s =1 and m s = 0 states, resulting the zero-field splitting term (D) in the Hamiltonian. For the ground stateD = 2.87 GHz. As illustrated in Fig. 2.1 (a) inset, the degeneracy inm s =1 is lifted by applying a magnetic field. The NV is sensitive to this field dependence which makes the platform to all NV based sensing applications. 2.2.3 Optical Properties Figure 2.1(a) shows a simplified energy level diagram of an NV center which contains three electronic levels, namely the triplet ground state, the triplet excited state and the Chapter 2. Nitrogen-vacancy center in diamond and optically detected magnetic resonance 25 (b) Conduction band Valance band Ground State Excited State m S = 0 m S = ±1 m S = 0 m S = ±1 Singlet State m S = 0 5.5 eV 1.945 eV Magnetic Field = 0 Magnetic Field = B m S = 0 m S = ±1 m S = 0 m S = +1 m S = −1 D = 2870 MHz 2γB } (a) Optical Excitation 532nm (c) 0 55 20 40 counts/ms 1 μm FL Intensity (arb. units) Frequency (MHz) 2800 2850 2900 2950 2750 D = 2870 MHz 2γB Figure 2.1: Characteristic features of the NV center. (a). Energy level diagram of the NV center. Curvy arrows indicates radiative spin selective optical transi- tions. Solid arrows represents non radiative transitions involves the singlet state. Inset shows the splitting of ground state spin sub levels under applied magnetic field B. (b).Spatial FL map of diamond. (c). ODMR spectrum of NV center. Upper trace was taken without applied magnetic field. Single peak at 2870 MHz represents the zero-field splitting of NV center. Lower trace was taken with 2 mT magnetic field applied alongh111i direction. Lower frequency peak cor- responds tom S = 0$ m S =1 EPR transition and higher frequency peak corresponds tom S = 0$m S = +1 EPR transition. singlet metastable state. Both the ground and excited states are further split into three spin sub-levels. The energy difference between the ground and excited states (the zero- phonon line) corresponds to 1.945 eV (638 nm). At room temperature, FL spectrum of Chapter 2. Nitrogen-vacancy center in diamond and optically detected magnetic resonance 26 the NV center is typically broaden and ranged from 600 nm to 800 nm, due to vibrational side bands. The life time of the excited state is about 13 ns and the relaxation time from the singlet state to the ground state is about few hundred nanosecond (approximately 250 ns). 29 The singlet state plays a key role in NV’s unique spin transitions. Most of the pop- ulation in the excited state m s =1 decays to the singlet state due to differences in the cross relaxation rates. The population in singlet state slowly decays into the ground statem s = 0 non radiatively. An electron in the excited statem s = 0 decays into the ground state through fast radiative transition. This spin-state dependent dynamics create difference in the FL intensity betweenm s = 0 andm s =1 states. The FL contrast between m s = 0 and m s =1 is typically up to 30%. Section 2.4.1 contains more details about these FL timing experiments. Moreover, as shown in Fig. 2.1(b), a spatial map of NV centers can be obtained by recording the FL emission. 2.2.4 Optically Detected Magnetic Resonance of NV As described in section 2.2.3, the FL intensity of an NV center is related to the spin state, therefore the EPR transition of NV centers is possible to be detected though changes of the FL intensity using ODMR spectroscopy. ODMR tend to be much more sensitive than EPR. In fact, ODMR detection of a single NV center at room temperature has been demonstrated in 1997 by Gruber et. al. 30 In ODMR measurement, EPR of an NV center is recorded as changes of the FL intensity due to population changes betweenm s = 0 and m s =1 states. Figure 2.1(c) show the FL intensity of a single NV center as a function of the applied MW frequency. Once MW excitation resonate with m s = 0 to m s = 1 transition, the population in m s = 0 reduces. As the result, decrease in FL intensity observed. This ODMR is a characteristic feature of the NV center. Chapter 2. Nitrogen-vacancy center in diamond and optically detected magnetic resonance 27 Figure 2.1(c) shows the ODMR signals without and with application of the magnetic field. An external field B results in splitting ofm s =1 states. 2.3 ODMR experiments of the NV center In this section I describe the experimental setup that I used to perform ODMR on a single NV center. The detection of the FL signals from a diamond sample and confirmation of a single NV centers will be discussed. 2.3.1 ODMR system To perform the ODMR experiment with NV centers in diamond, I built a confocal micro- scope. The confocal microscope allows us to illuminate and record an emission signal from a diffraction-limited volume so that it is intrinsic to address a single NV center in diamond. Figure 2.2 shows a schematic of latest version of the setup used to take the low magnetic field data presented in this thesis. A diode pumped solid state laser of 532 nm with 100 mW power from CrystalLaser R was used to excite the optical transitions of the NV . An Acoustic Optic Modulators (AOM) (ISOMET 1250) was used to produce the laser pulses. We employed two AOMs in a series configuration, to increase the optical isolation between on and off states of the AOM. Currently we have more than 66 dB isolation between on and off state. Impor- tance of this high isolation will be further discussed in section 2.4.1. Then the excitation beam focuses onto the NV using the objective lens. This microscope is employed with either dry objective or wet objective lenses. As shown in Fig. 2.2, the diamond sample is mounted on a glass cover slip which is placed on a copper PCB board with a transmis- sion line engraved. A thin gold wire with 20m diameter lies on the diamond surface Chapter 2. Nitrogen-vacancy center in diamond and optically detected magnetic resonance 28 for MW access to NV centers. The sample positions was controlled using a XYZ piezo controller which has the translational range of 20m. Laser (532 nm) AOM 1 AOM 2 APD 1 APD 2 TCSPC Objective X Y Z Piezo Stage B 0 {100} L1 L2 L3 L4 LF OA DM L5 FF BS M1 M2 DAQ Pulse Blaster COM Computer MW1 (SG386) I Q MW2 (SG386) PIN PIN COMBINER AMP ATTN(50dB) 50Ω Sample holder B 0 {111} PCB Diamond Coverslip Au wire (20 μm) Ag paste Figure 2.2: Schematic of the ODMR system based on confocal microscope. Green line denotes the excitation pathway and red line shows the pathway of the FL emission. Yellow lines represent TTL signal connections and DAQ inputs and outputs. Opaque maroon dashed lines shows communication connection between devices and computer. L1-L4 denotes the lens used in excitation path- way and L5 is the tube lens compatible to Nikon objectives. Focal lengths of L1,L3 are 50 mm and L2,L4 are 200 mm. Focal length of L5 is 200 mm. Laser filter (LF) will filter 532 nm laser line with 2 nm spectral width. Optical atten- uators (OA) with different optical densities were used as required to keep NV FL emission just below the saturation and maximize FL contrast (FL - Back- ground). Chapter 2. Nitrogen-vacancy center in diamond and optically detected magnetic resonance 29 The FL emission is collected with the same objective lens and passed along the same pathway up to the dichroic mirror where the FL is separated from the excitation. We employ a FL filter (FF) to remove the reflected green light (transmission of the green light is less than 20%) and a tube lens (L5) focuses FL to a multimode fiber optical cable which is then connected to the APD. The optical cable has a 105 m diameter and acts as the pin hole in the confocal which determines the spatial resolution of the microscope. The APD output was counted by a DAQ counter (National Instrument PCIe 6321). The beam splitter (BS) is used to split the FL signals when auto-correlation (AC) measurement takes place. In this case, two APDs are connected to a time correlated single photon counting (TCSPC) module (PicoHarp 300) to perform AC measurements. AC experiment with a single NV center is discussed in following sub-section 2.3.3. For MW excitation, we employ two independent MW sources which ranges from DC to 4 GHz. IQ modulation of MW source 1 (MW1) is activated and used to control the relative phases between MW pulses. Additional PIN switch at MW1 output was used to enhance the MW power isolation between MW on and MW off states. The intrinsic isolation of MW1 is about40 dB and the PIN switch adds additional35 dB isolation. MW2 was used to excite other spins during the double electron-electron resonance (DEER) measurements (section 2.4.8). Then two MW propagation paths were coupled using a combiner and amplified with about a 45 dB (10 W) amplifier which is then connected to the sample holder. 2.3.2 Imaging and CW ODMR A raster scanning technique is used to record the FL signals from a diamond samples (Fig. 2.3 (a)). In the imaging, the diamond sample is moved along the positive X and Chapter 2. Nitrogen-vacancy center in diamond and optically detected magnetic resonance 30 positive Y direction using the peizo stage with 10 ms stepping intervals. In the FL imag- ing, the TTL counter is typically gated for 25 ms. The integration time of the measure- ment is proportional to the signal-to-the noise ratio. Figure 2.3 (b) shows FL intensity with various integration time. As shown in Fig. 2.3 (b), fluctuation of the FL intensity is very pronounced with the integration time of<15 ms. In most of experiments, we used the integration time of 25 ms. 1 μm 5 95 65 35 FL Intensity (Counts/ms) (a) (b) 80 75 70 FL Intensity (counts/ms) 65 60 Tracking Time (minutes) 1.7 2.5 3.3 4.2 0.0 5.0 0.8 25ms 15ms 5ms 1ms Figure 2.3: (a). FL image of diamond. 3 3m 2 area scanned with 60 pixels per axis. Circled FL spot was confirmed as a single NV center. (b). Fluctuation of FL intensity at different integration times. After the FL imaging, we set the focal point of the confocal microscope to one of the FL centers for AC and ODMR experiment. However the position of the focal point may drift during the experiment due to thermal drifts and mechanical drifts of the piezostage. To get rid of such fluctuations, a tracking mechanism was implemented. The principle of the tracking is to keep the maximum FL intensity by continuously adjusting the X,Y and Z piezo positions. In practice, we measure the FL intensities with three different X,Y and Z points, then choose the point with the maximum FL as the center of the FL signal. Then we repeat the measurement to keep tracking the FL center. This keeps the NV at the center of tracking cube all the time as far as correct step size is used and any external fluctuations are smaller than the range of tracking. Chapter 2. Nitrogen-vacancy center in diamond and optically detected magnetic resonance 31 FL Intensity (Counts/ms) 0 60 40 20 1 μm (a) (b) 40 30 20 10 0 Relative peizo position (μm) -2 -1 0 1 -4 2 -3 50 FL Intensity (Counts/ms) 3 X data Y data Z data X Fit Y Fit Z Fit X FWHM = 258 ± 3 nm Y FWHM = 292 ± 1 nm Z FWHM = 793 ± 10 nm Figure 2.4: NV positioning mechanism. (a). FL image of the NV used to demonstrate positioning. (b). Scatters show positioning data collected by mov- ing each X,Y and Z piezo positioners. This data was collected from the NV circled in (a). Solid lines show the Gaussian fits for each. For X and Y 3.2m range was scanned with 40 points whereas Z was scanned with 6.4m with 80 points. FWHM for each fit is given in the inset. During this measurement NV was continuously excited with green laser and FL was integrated for 20 ms for each point. In the pulse measurements (section 2.4) piezo positions was adjusted to bring the NV in to focal volume in order to maximize FL signals. We named this process as ‘positioning’. As required positioning was automatically implemented during the pulse measurement and maintaining maximum FL throughout the entire measurement. Figure 2.4 shows data collected with positioning. Once the FL imaging is measured, we move the focal point to the NV of interest. First the X direction was scanned through the NV . Red scatters in Fig. 2.4(b) shows the FL intensity of the selected NV in the Fig. 2.4(a). The data was fitted with a Gaussian function to find the maximum position of the FL, then the X piezo position was set to the maximum. This process was repeated for the Y and Z directions respectively to find the center of FL. The width of the FL distribution represents the resolution of the confocal microscope. Chapter 2. Nitrogen-vacancy center in diamond and optically detected magnetic resonance 32 Frequency (MHz) 2850 2900 2950 2750 2800 1.0 0.9 0.8 0.7 1.0 0.9 0.8 Frequency (MHz) 2850 2900 2950 2750 2800 FL Intensity (Kcounts ) 3.4 3.2 3.0 2.8 2.6 2.4 Normalized FL Intensity (arb. units) |m S = 0〉 |m S = ±1〉 ↔ |m S = 0〉 |m S = −1〉 ↔ |m S = 0〉 |m S = +1〉 ↔ B = 0 T B ~ 2 mT No MW MW Normalized data 0.6 Figure 2.5: CW ODMR of a single NV center. CW ODMR spectrum of the NV center selected in Fig. 2.3. Upper panel shows the normalized CW ODMR spectrum at zero magnetic field corresponding tojm S = 0i$jm S =1i transition. Inset shows the raw data of CW ODMR spectrum at zero magnetic field. Two data points, with MW and without MW were collected per frequency step. Laser was on for the whole time and MW either ‘ON’ or ‘OFF’ during the integration time. Lower panel shows the CW ODMR spectrum at magnetic field of strength 2 mT applied along NV axis. Low frequency peak represents thejm S = 0i$jm S =1i transition and higher frequency peak denotes jm S = 0i$jm S = +1i transition. Spectrum resolution is 2 MHz. 250 ms integration time was used to acquire one frequency point and average of five frequency sweeps were obtained. After finding a FL emitter, we perform cw ODMR experiment to check whether the FL emitter is from an NV center or not. The first step is to run CW-ODMR without any external magnetic field and obtain resonance at 2.87 GHz. Figure 2.5 upper panel shows the ODMR of the selected FL center without applied magnetic field. The FL data was collected by sweeping the MW frequency. The heat generated from the MW excitation Chapter 2. Nitrogen-vacancy center in diamond and optically detected magnetic resonance 33 causes FL spot to drift. This drift was corrected by taking two data points per each frequency. One point with MW and the other point without MW. The inset in the upper panel of Fig. 2.5 shows raw data taken with and without MW. The ratio of these two traces is used to measure the contrast of the ODMR signal as shown in Fig. 2.5(upper). Application of a magnetic field along the NV axis splits the degeneratedjm S =1i spin sub levels, as shown in Fig. 2.5 lower panel. Moreover, the CW ODMR signal is often broaden by the power saturation and it may not have enough resolution to resolve the hyperfine coupling and the inhomogeneous linewidth. The spectral resolution can be improved using pulsed ODMR described in section 2.4.1. 2.3.3 Photon statistic measure of a single NV 2 1 0 g 2 (τ) Time,τ (μs) 0.1 0.2 0.3 0.4 0.0 0.5 (b) APD 2 TCSPC BS APD 1 Delay D photon 1 photon 2 τ start stop τ+D 0.1 0.2 0.3 0.4 0.0 0.5 [τ + D] (μs) (a) Figure 2.6: Photon Statistic measure of a single NV . (a). Schematic of exper- imental setup in HBT configuration. Additional delay was introduced in ‘stop’ channel by extending the length of BNC cable which carries the TTL signals from APD2 to TCSPC. (b). Background corrected autocorrelation data of the NV shown in Fig. 2.3(a). Delay of 26.1 nm was obtained by adding 5 m addi- tional length of BNC cable in ‘stop’ channel. 256 ps width of bin size was used in this measurement. After verifying the FL signal from NV centers, we employ the AC measurement to check whether the FL signal is from a single NV center or multiple NV centers. For the AC Chapter 2. Nitrogen-vacancy center in diamond and optically detected magnetic resonance 34 experiment, we use Hanbury-Brown-Twiss (HBT) 31 configuration which employs two APDs that compensate a deadtime of each detector when measuring the arriving time delay between two consecutive photons. 32 Our HBT setup is illustrated in Fig. 2.6(a). HBT setup allows to record the second order photon autocorrelation function, g 2 () = hI PL (t)I PL (t +)i hI PL (t)ihI PL (t +)i ; (2.2) where I PL (t) is the FL intensity at time t and represents the time interval between photon arrivals. The brackets denotes the time average. The intensity of g 2 at zero time delay depends of the photon statics govern by the number of quantum emittersn, g 2 (0) = 1 1=n. 32 A single quantum emitter shows characteristic anti-bunching (zero probability to emit two photons simultaneously) dip (g 2 (0) = 0) as shown in Fig. 2.6 (b). Two quantum emitters generateg 2 (0) = 1=2 dip (dotted line in Fig. 2.6 b). Background FL intensity may causeg 2 (0) to have higher value than zero. Overall, the observation of ODMR andg 2 (0)<0.5 confirm the identification of a single NV center. 2.4 Pulsed ODMR Next, I explain pulsed ODMR measurements of a single NV center. The pulsed mea- surement is a core for EPR experiment with a single NV spin sensor. 2.4.1 Pulsed ODMR overview A schematic of control sequences used in pulse experiments is shown in Fig. 2.7. In the beginning of the sequence DAQ counter was read and record the value 0 R1 0 (Fig. 2.7(a)). Then NV spin was polarized intom s = 0 state by applying the initialization laser pulse with approximately 2 s duration. Characterization the AOM delay will be discussed Chapter 2. Nitrogen-vacancy center in diamond and optically detected magnetic resonance 35 AOM TTL MW TTL DAQ Counter Readout Initialization Readout MW pulse sequence (time or freq. sweep) R1 R2 DAQ Counter gate P(m s = 0) Reference P(m s = ±1) Reference Repeat ~(10 6 ) ∆ AOM ∆ AOM Positioning as required N number of sweeps (a) (b) (c) (d) Figure 2.7: Overview of pulse measurement sequence. (a). DAQ counter readout control sequence which also defined as the counter update. TTL signlas were routed to counter update port to readout the counter and record asR1 and R2. (b). TTL signal train that controls the AOM. (c). MW control sequence. For time sweep measurements pulse width of TTL signals were changed us- ing PulseBlaster for fixed frequency and power. Usually RF output of the MW source was controlled via PIN switch to acquire better isolation than triggering the MW source itself. Frequency sweep measurement were done with fixed TTL pulse length and frequency is controlled though GPIB. (d). TTL signals to control the counter gate. AOM delay time was introduced here to synchronize counter gating and actual readout laser pulse. For each references and signal separate counter was gated. In these measurement all together three counters were used. Common counter update signal was used to readout all three coun- ters and record readings separately. For one variable change, either time or frequency, counters were updated after repeating (b)-(d) many times and move to the next variable value. After sweeping desired range entire measurement has been restarted from the beginning and keep continuing until better SNR is achieved. later in this section. Software delay was introduced to resolve AOM delay. Then rele- vant MW pulse sequence was applied. Then resultant spin population was captured by Chapter 2. Nitrogen-vacancy center in diamond and optically detected magnetic resonance 36 applying a short readout laser pulse of 300 ns and gating the DAQ counter for same time period. Finally, DAQ counter was read at the end and record the 0 R2 0 values. Difference betweenR1 andR2 dived by the total integration time (300 ns number of repetitions) is the measured FL rate. We also collected two references in addition to the signal. One of the references reads the population ofm s = 0 just after the initialization pulse and that can be defined as P (ms = 0) = 1. The second reference is taken by applying a resonant pulse after the initialization. This change the NV state from m s = 0 to m s =1. This value was defined asP (ms = 0) = 0. For the continuous acquisition of the signal, reference 1 and reference 2, we use three counters in the DAQ card where the counters are gated separately at the timing of the signal, reference 1 and reference 2 measurements. The references are used to remove systematic and random noises in the measurement. In addition, we use the reference intensity to monitor the FL cen- ter positioning so that we can keep the FL center within the excitation spot during the experiment by performing the positioning when the reference intensity becomes low. The FL timetrace measurement shown in Fig. 2.8 was performed to determine pulse sequence parameters that include AOM delay and optimal readout window to maximize ODMR contrast. The AOM delay is the time gap between the rising edge of the TTL signal that triggers AOM driver and the rising edge of the actual optical pulse. This delay is unique to an AOM device. The pulse sequence in Fig. 2.8(a) is used to obtain the data shown in Fig. 2.8(b). First, data was collected with the pulse sequence mentioned as “No MW” (Fig. 2.8(a)) and then measurement was repeated with the pulse sequence mentioned as “With MW” where a resonant pulse was applied after the initialization. As the result of the resonant pulse, population in the m s = 0 state drops as shown in Fig. 2.8(b). The readout window described in Fig. 2.7 is determined by taking the Chapter 2. Nitrogen-vacancy center in diamond and optically detected magnetic resonance 37 Counts (arb. units) 1.0 0.5 0.0 Time (μs) 1 2 3 4 0 (b) No MW MW 0 10 20 30 Contrast (%) Time (μs) 1.5 1.8 2.1 2.4 1.2 2.7 3.0 (c) AOM delay (∆ AOM ) AOM TTL MW TTL Init. RO Init. RO t ̟ TCSPC 3 μs 3 μs 2 μs 2 μs (a) No MW With MW Figure 2.8: FL time measurement of a single NV to characterize AOM delay and optimize readout window. (a). TTL pulse sequence used in the measure- ment to gate AOM, trigger TCSPC and gate MW excitation. Shaded area in AOM sequence represent the actual position of laser pulse caused by AOM de- lay. (b). Normalized FL intensity as a function of time for with and without application of resonant pulse. 128 ps width of bin was used. Timet = 0 to rising edge gap is due to the AOM delay which was measured as 1.15s. (c). Ratio of two traces given in (b). Maximum contrast of ODMR reached 30%. Shaded area represent 300 ns readout window. ratio between two traces in Fig. 2.8(b). As shown in Fig. 2.8(c) shaded area with 300 ns width denotes maximum contrast of 30%. 2.4.2 Rabi nutations With application of a magnetic field where the degeneracy ofm s =1 states is lifted, jm S = 0i$jm S =1i andjm S = 0i$jm S = +1i transitions of NV is con- sidered as a two level system. Population oscillation (so called Rabi nutations) between Chapter 2. Nitrogen-vacancy center in diamond and optically detected magnetic resonance 38 jm S = 0i andjm S = +1i orjm S =1i is induced by driving the transition with the ap- plication of a resonant MW pulse. Figure 2.9 shows the population of them S = 0 state as a function of the length of the resonant pulse (t p ). As shown in the figure, the popu- lation oscillates by increasingt p . From the period of the oscillations, we can calibrate pulse parameters required for pulsed ODMR. Further discussion about this calibration is in appendix A. 0 100 Pulse width t p (ns) 200 300 400 500 600 P (m s = 0) 1.0 0.8 0.6 0.4 0.2 0.0 0 10 20 30 ODMR contrast (%) Data Fit t ̟ = 110.5 ± 0.2 ns T d = 1.5 ± 0.2 μs AOM TTL MW TTL Initialization Readout t p 6 dBm 3 dBm 0 dBm -6 dBm √P MW (√W) 0.00 0.02 0.04 0.06 f Rabi (MHz) 0 3 6 9 Contrast (%) 0 20 0 20 0 20 0 20 Pulse width t p (μs) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 (a) (b) Data Fit Figure 2.9: (a). Rabi oscillation experiment at B 0 = 2:06 mT and the mi- crowave frequency of 2.812 GHz corresponding tojm S = 0i$jm S =1i transition performed on the NV shown in Fig. 2.3(a). Square dots connected with lines indicate the measurement and solid line indicates the fit. The os- cillation frequency (f Rabi ) of 4.5 0.2 MHz (corresponding pulse is t = 110.5 ns) and the decay time (T d ) of 1.5 0.2 s were obtained from a fit to 1=2[cos(2f Rabi t P ) exp((t P =T d ) 2 ) + 1]. The top panel shows the pulse sequence with excitation laser (Initialization), microwave (MW TTL) and FL measurement (Readout). Left y-axis represents the calibration based on popu- lation of m S = 0 and right y-axis represent the ODMR contrast. (b). Upper panel: Rabi oscillation experiment performed at different MW powers clearly shows oscillation period gets longer as power reduces. Lower panel: Rabi fre- quency (f Rabi ) as a function of square root of output power of MW ( p P MW ) source. Data was fitted with a linear trend line with a zeroy intercept. Chapter 2. Nitrogen-vacancy center in diamond and optically detected magnetic resonance 39 The oscillation frequency of the Rabi nutation (f Rabi ) depends on B 1 field at NV . Power of MW (P MW ) determines the strength ofB 1 field. Therefore,f Rabi / p P MW as shown in Fig. 2.9(b) lower panel. For a fixed MW power, the duration of the resonance pulse which requires to flip the spin is defined as the pulse since it is a half of the Rabi period. As introduced in section 1.3.3, the flip angle is given by the pulse duration and the Rabi frequency ast p ==(2f Rabi ). Therefore this rotation is similar to the rotation of the magnetization vector of an effective spin 1/2 system about an axis inxy plane. Relative phases among rotation axis is selected by the phase of MW radiation. Such applications with composite pulse sequence will be discussed in section 2.4.7. 2.4.3 Pulse ODMR of a single NV As I mentioned in section 2.2.4, the ODMR signal of a single NV center is due to the population change of the m s = 0 state. In cw ODMR measurement, a strong change is achieved by saturating the NV’s spin transition, therefore cw ODMR signal tends to be a power-broaden spectrum. However, this is not the case for pulsed ODMR. In the case of pulsed ODMR, the linewidth of the signal is limited by the excitation bandwidth so that the spectral linewidth can be improved by adjusting the excitation bandwidth. Here I discuss a way to improve the ODMR spectral resolution to resolve the hyper- fine coupling between NV’s electron and nitrogen nuclear spins. Figure 2.10(a) shows the pulse sequence of the pulsed ODMR and data from a single NV center. In the pulsed ODMR experiment, NV is first initialize tom S = 0 state by applying long laser pulse (see Fig. 2.7(b)). Then MW pulse of fixed length (probably pulse, t ) is ap- plied with variable frequency (Fig. 2.7(c)). Finally remaining population is readout by applying readout laser pulse (Fig. 2.7(b),(d)). In comparison, Fig. 2.10(b) shows the power-broaden CW- ODMR spectrum of the same NV . Chapter 2. Nitrogen-vacancy center in diamond and optically detected magnetic resonance 40 2806 2808 2810 2812 2814 2816 1.00 0.99 0.98 0.97 Contrast Frequency (MHz) 14 NV 14 ∆ = 2.2 MHz 14 ∆ 14 ∆ AOM TTL MW TTL Initialization Readout t π 2800 2810 2820 Frequency (MHz) 0.95 0.90 0.85 Contrast Data (a) (b) Figure 2.10: Pulsed ODMR for 14 NV shows hyperfine splitting due to the nuclear spin in N isotope of the NV . (a). Data was taken from an NV center in type 1b diamond crystal. Top panel shows the pulse sequence used in the experiment. MW pulse (t ) of 800 ns length was used with -18 dBm MW1 source output. Separation of peaks ( 14 ) agree with the 14 N hyperfine splitting. (b). CW ODMR data of the same NV . Blue vertical dashed lines represents the range of frequency scanned in the pulse ODMR measurement shown in (a). MW1 source output was -10 dBm. 2.4.4 T 1 relaxation The pulse sequence for the T 1 measurement is shown in Fig. 2.11(a) inset. First, the NV center is initialized tom S = 0 state by a long initialization pulse (see Fig. 2.11). After a time T , the remaining population in m S = 0 is record by applying readout laser pulse (RO). A reference named as ‘min’ is also collected after the ‘signal’ with the application of resonant pulse which flips NV population tom S =1 orm S = +1 (see Fig. 2.11(b)). Plotting deference between signal and min represents the longitudinal relaxation. This method eliminates the thermal and mechanical fluctuations which may occur during the delay time T specially when it gets longer. In addition to that T 1 Chapter 2. Nitrogen-vacancy center in diamond and optically detected magnetic resonance 41 measurements of ensemble of NV centers which is excited by pulse can be obtained using this method . 33 Data Fit 0 5 T (ms) 10 15 20 25 [Signal - min] (arb. units) 800 600 400 200 0 T 1 = 3.95 ± 0.24 ms AOM TTL MW TTL Init. RO Init. RO t ̟ Counter T T signal min 3000 2500 0 2 4 6 T (ms) Counts (arb. units) (b) (a) Max Signal Min Figure 2.11: T 1 measurement of the single NV shown in Fig. 2.3(a). (a). Inset shows the pulse sequence used in the experiment. ‘Signal’ was taken without MW pulse and ‘min’ was collected with the application of resonant pulse. Difference between signal and min was plotted against variable timeT . Data was fitted with a single exponential functionA exp (T=T 1 ) andT 1 was extracted as 3.950.24 ms. (b). Raw data shows the signal and two references. 2.4.5 Ramsey measurement Free induction decay (FID) measurement was performed using the MW pulse sequence given in the Fig. 2.12 top panel. As shown in Fig. 2.12 top panel, the pulse sequence used in the measurement is=2=2 where represents the resonant MW pulse which flips the NV spin fromm S = 0 tom S =1 or +1 and the free evolution time is denoted by. During the free evolution time, the coherence evolves in transverse plane under the interactions from surrounding spins. The last=2 pulse will transfer the coherence to initial statem S = 0. Chapter 2. Nitrogen-vacancy center in diamond and optically detected magnetic resonance 42 MW TTL τ π 2 π 2 AOM TTL Initialization Readout P (m s = 0) 1.00 0.75 0.50 0.25 0 0 1 τ (μs) 2 3 4 5 Data Fit T * 2 = 0.87 ± 0.06 μs Figure 2.12: Ramsey (FID) measurement data of the single NV . Top panel shows the MW pulse sequence used in the measurement. Data was fitted to the functionFID() = 1=2 1=2 cos(2A 15N ) exp((=T 2 ) 2 ) whereA 15N = 3:2MHz is the hyperfine splitting of NV coming from 15 N nuclear spin and extractedT 2 as 0.87s. The FID decay in electron spins have been successfully described by treating the bath as a classical noise field (B n (t)) where B n (t) was modeled by the Ornstein- Uhlenbeck (O-U) process with the correlation function C(t) = hB n (0)B n (t)i = b 2 exp(jtj= C ), where the spin-bath coupling constant (b) and the rate of the spin flip- flop process between the bath spins (1= C ). 34, 35 The FID decay due to the O-U process is given by, FID(t) = 1 2 1 2 [cos(2A 15NV z t)] exp[(b C ) 2 ( t C +e t= C 1)]; (2.3) whereA 15NV z = 3.2 MHz is the hyperfine coupling of NV center with 15 N isotope. The derivation of this equation will be discussed in the appendix C. In the quasistatic limit (whereb c 1) equation 2.3 is simplified to the form given in the Fig. 2.12 caption and T 2 = p 2=b. Chapter 2. Nitrogen-vacancy center in diamond and optically detected magnetic resonance 43 2.4.6 Spin echo measurement Hahn echo method is used to measure decoherence time (T 2 ) of a single NV center. 6 Conventional Hahn echo is modified in NV application by adding additional=2 pulse at the end of the sequence. The purpose of this modification is to transfer the spin coherence into the spin population. As shown in Fig. 2.13 top, the pulse sequence used in the measurement is=2=2 where represents the resonant MW pulse which flips by 90 and the free evolution times are denoted by. The first =2 pulse followed by the laser initialization converts the spin population to the coherence between the effective two level system. This operation corresponds to the flip of the longitudinal magnetization vector in the Bloch sphere to the transverse plane. Then during the first fixed free evolution time coherence evolves in transverse plane. Fanning out of the magnetization or the FID described under the Ramsey experiment take place during the free evolution time due to the interactions from surrounding spins. Application of pulse refocuses the coherence by removing the dephasing caused by static magnetic shifts and last =2 pulse will transfer the coherence to initial state m S = 0 (assume phase of all three MW pulses are same. Changing phases may bring the system to m S =1 orm S = +1 depends on the excited EPR transition). Decay of the echo can be seeing by recording the FL intensity as a function of 2 as shown in Fig. 2.13. T 2 can be extracted by fitting the decay to an exponential function which can be modeled considering the properties of the surrounding spin bath as explained in the section 2.4.5. The SE decay due to the O-U process is given by, SE() = 1 2 + 1 2 exp[(b C ) 2 ( 2 C 3e (2)= C + 4e (2)=(2 C ) )]: (2.4) Chapter 2. Nitrogen-vacancy center in diamond and optically detected magnetic resonance 44 In the quasistatic limit (whereb c 1) indicating slow bath dynamics, equation 2.4 is simplified to the form given in Fig. 2.13 caption whereT 2 = 3 p 12 c =b. The derivation of this equation will be discussed in the appendix C. SE in combination with the FID is a great tool to extract the dynamics of the surrounding spin bath and will be further discussed in the chapter 3. P (m s = 0) 1.00 0.75 0.50 0.25 0.00 0 2τ (μs) 50 100 Fit Data Sim. 1.0 0.5 0 10 20 30 0 2τ (μs) P (m s = 0) MW TTL τ π 2 π π 2 τ T 2 = 46 ± 2 μs AOM TTL Initialization Readout Figure 2.13: Spin echo measurement data of a single NV . Population ofm s = 0 plotted against total free evolution time 2. Dashed line shows the simulated decay curve of the functionSE() = 1=2 + 1=2 exp((2=T 2 ) 3 ) whereT 2 was 46 s. Upper inset shows the MW pulse sequence used in SE measurement. Lower inset shows the first 30s of SE data which shows the strong modulation coming from the surrounding nuclear spins. Modulation in the SE data shown in Fig. 2.13 is called electron spin echo envelop modulation (ESEEM) and due to couplings to the surrounding nuclear spins. Data in Fig. 2.13 inset was simulated with by considering the presence of both 15 N nuclear spin and 13 C nuclear spin bath. The modulation function for a system consistsS = 1=2 and Chapter 2. Nitrogen-vacancy center in diamond and optically detected magnetic resonance 45 I = 1=2 is given by S i (2) = h 1 2 B i ! i N = i i 2 sin 2 ( i ) sin 2 i i , where ! N is the Larmor frequency of thei th nucleus, i and i are the NMR frequencies of thei th nucleus andB is a function of the hyperfine coupling of electron spin to thei th nucleus. 36–38 In the present case we have to consider the modulation effects coming from both 15 N nuclear spin and 13 C nuclear spin bath. Therefore, the decay of the SE is written as SE() = 1=2 + 1=2 S 13C S 15N exp((2=T 2 ) 3 ) where, S 13C is the modulation due to the 13 C nuclear spin bath and S 15 N is the modulation due to the 15 N nuclear spin in the NV . The modulation due to the 15 N nuclear spin is negligible when the applied magnetic field is perfectly aligned with the quantization axis of the NV . However, in the experiment there is a minor misalignment between the applied magnetic field and the NV axis. Therefore, the modulation due to the 15 N nuclear spin is considerable. Appendix B contains the derivation of modulation function of the spin echo envelope. 2.4.7 Dynamical Decoupling sequences A dynamical decoupling sequence such as Carr-Purcell-Meiboom-Gill (CPMG) se- quence is capable to suppress the decoherence of NV centers by decoupling the NV centers from noisy magnetic fluctuations. 39, 40 Figure 2.14 shows the applications of the the Carr-Purcell-Meiboom-Gill (CPMG) pulse sequence. 42, 43 The MW pulse sequence of CPMG is shown in the inset of Fig. 2.14. The CPMG pulse sequence consists of a series of the rephasing pulses (N represents number of pulses) which suppresses the noise responsible the SE de- cay. 90 degree phase difference between=2 and pulses is introduced to minimize the error accumulation during the series of pulses when the first=2 pulse is not per- fect. As shown in Fig. 2.14, the coherence time of the NV center becomes longer while Chapter 2. Nitrogen-vacancy center in diamond and optically detected magnetic resonance 46 0 10 20 30 Total Evolution time (μs) P(m s =0) (arb. units) Spin Echo CPMG-4 CPMG-16 CPMG-32 τ τ CPMG-N N (π/2) y (π/2) y π x Sim. MW TTL Figure 2.14: Decoherence time (T d of the NV center measured by SE, CPMG- 4, CPMG-16 and CPMG-32 sequences. The pulse sequence is given in the inset. SE consists of a single -pulse (N = 1) and CPMG-N consists of N number of -pulses. Lengths of /2 and pulses were 44 ns and 88 ns, respectively. Dashed lines shows the simulations of decay envelope using 1=2 + 1=2 exp[(t=T d ) 3 ], where t and T d represents the total evolution time and decoherence time, respectively. The obtainedT d values are 1.7 0.1, 3.1 0.1, 6.4 0.3, 7.2 0.3 s for SE, CPMG-4, CPMG-16 and CPMG-32, respectively. Reprinted figure with permission from the reference( 41). the number of the pulses (N) in CPMG increases. With the application of the CPMG sequence with 32 pulses (CPMG-32), we observed that the coherence time (T d ) was extended from 1.7s (SE) to 7.2s (CPMG-32). 41 Chapter 2. Nitrogen-vacancy center in diamond and optically detected magnetic resonance 47 2.4.8 NV-based EPR Double electron-electron resonance (DEER) spectroscopy (or electron-electron double resonance (ELDOR)) is a well-known technique to measure the magnetic dipole cou- pling between spins and often used for biological systems to study confirmations and conformational changes. In my Ph. D. work, we employ DEER spectroscopy to per- form NV-based EPR of a few spins. Figure 2.15(a) and (b) shows the pulse sequence used in NV-based EPR spec- troscopy, 44–46 which was adopted from DEER spectroscopy used in EPR. 18 The se- quence consists of the spin echo sequence for an NV center and an additional-pulse at a different microwave frequency (denoted as MW2 in Fig. 2.15(b)). When the MW2 pulse is resonant with surrounding electron spins near the NV center, the magnetic mo- ments of the surrounding spins are flipped, and this alters the magnetic dipole field experienced by the NV center from the surrounding spins during the second half of the spin echo sequence. The alteration causes a shift in the Larmor frequency of the NV center, which leads to different phase accumulation of the Larmor precession during the second half of the sequence from the first half as shown in Fig. 2.15(b) inset. As the result, the NV center suffers phase-shift in the echo signal and the reduction of FL in- tensity from the original spin echo signal is observed. Figure 2.15(c) shows the results of the measurement. The SE intensity was recorded as a function of frequency with and without MW2 to distinguish reduction in echo amplitudes clearly. The intensity of NV-based EPR can be used to estimate the effective dipolar field at the NV center since the intensity is proportional to cos (!) (see Fig. 2.15(b)). ! = g NV B B dip;eff =~ whereg NV = 2:0028 is theg-value of the NV center, ~ is the reduced Planck constant and = 2:43s in this measurement. Using the above expressions,B dip;eff is extracted Chapter 2. Nitrogen-vacancy center in diamond and optically detected magnetic resonance 48 NV SE Bath spins π/2 π π MW2 τ τ ω’ Δωτ = (ω 0 −ω’)τ δB δB’ (a) (b) (c) 0.8 0.7 0.6 0.5 750 800 850 900 950 MW2 Frequency (MHz) P(m s =0) SE Amp. Signal Figure 2.15: NV-based EPR measurement. (a). SE pulse sequence and evolu- tion of magnetization vector during the SE sequence. (b). Additional pulse defined as MW2 to excite surrounding paramagnetic species. In the real ex- periment MW2 is applied 12 ns after the pulse in SE sequence to avoid the overlap between the two frequencies. Upper inset shows change in accumulated phase and resultant echo amplitude reduction. Lower inset shows a simple car- toon that represents the flip of bath spin due to the MW2 . (c). Results of the DEER measurement obtained using the combined pulse sequence given in (a) and (b), shows a clear reduction in the spin echo amplitude. Signal represents the NV-based EPR spectrum and SE Amp. denotes the amplitude of SE without MW2 . Pulse parameters used in this measurement were NV=2 = 36 ns; NV = 72 ns; = 2.43s; MW2 = 44 ns. Chapter 2. Nitrogen-vacancy center in diamond and optically detected magnetic resonance 49 as 1.54 T. In addition, peak position of the NV-based EPR signal can be used to es- timate theg value of the surrounding spin bath using the equation 1.28. In the present case g was extracted as 2.0063. We assumed the origin of this NV-based EPR signal corresponds toS = 1/2 surface impurity spins. Furthermore, pulse width of the MW2 pulse ( MW2 ) can be optimized by performing the Rabi measurement on the observed signal. Figure 2.16 shows the Rabi data collected with the NV-based EPR signal shown in Fig. 2.15(c). As shown in Fig. 2.16(a) pulse sequence, Rabi was measured by vary- ing the pulse width of resonant MW2 pulse (t p MW2 ) at a fixed frequency of MW2 for a fixed free evolution time. Amplitude of SE was also collected without applying res- onant MW2 pulse. Figure 2.16(b) shows the raw data of the measurement performed on the NV-based EPR signal shown in Fig. 2.15(c). In addition to Rabi and SE ampli- tudes, max and min references are also collected as shown in Fig. 2.16(b). Rabi data was normalized using max and min references as shown in Fig. 2.16(c). Instead of using conventional SE sequence, the dynamical decoupling pulse se- quences can be also used in NV-based EPR measurements (section 4.4). In dynamical decoupling based DEER, the total phase acquired by the NV is B eff N where, B eff is the effective dipolar field change caused by spin label at the location of the NV; N is the number of pulses applied in the dynamical decoupling sequence and is the free evolution time. The essence of this DEER sequences is one of the main tools I used to perform NV-based EPR spectroscopy shown in the chapter 3 and chapter 4. Chapter 2. Nitrogen-vacancy center in diamond and optically detected magnetic resonance 50 AOM TTL Init. RO τ (π/2) (π/2) π MW1 TTL MW2 TTL Init. RO t p_MW2 P(m s = 0) Reference P(m s = ±1) Reference τ τ (π/2) (π/2) π τ 0 100 200 t p_MW2 (ns) 0.8 0.7 0.6 P(m s =0) (c) (b) 0 100 200 t p_MW2 (ns) Max Min DEER Rabi SE 15 14 13 FL Intensity (Kcounts) 12 (a) DEER Rabi Fit Figure 2.16: NV-based EPR Rabi measurements. (a). Basic pulse sequence used in the measurement. Rabi measurement was taken by varying the width of the resonant MW2 pulse (t p MW2 ). Frequency of MW2 was 863 MHz. was 2.43s. Amplitude of SE was recorded without MW2 pulse. (b). Raw data of the Rabi measurements. (c). Normalized data of the Rabi measurements. Data was fitted with a simple cosine function with an exponential envelope to extract the period. Estimated MW2 pulse was 44 ns. Chapter 3: Nanoscale NV-based EPR of several spins in diamond Materials presented in this chapter can also be found in the article titled Electron spin resonance spectroscopy of small ensemble paramagnetic spins using a single nitrogen- vacancy center in diamond by Chathuranga Abeywardana, Viktor Stepanov, Franklin H. Cho and Susumu Takahashi in Journal of Applied Physics 120, 123907 (2016) (Reprinted with permission. Copyright [2016], AIP Publishing LLC.) 3.1 Scope and Experiment Here we discuss on nanoscale EPR spectroscopy using a single NV center in diamond. Using a 230 GHz EPR spectrometer, we first perform ensemble EPR of the sample crys- tal and find that a major paramagnetic impurity in the sample crystal is a substitutional single nitrogen center (N spin; also known as P1 center). Next we perform ODMR mea- surements of a single NV center using the ODMR setup described in section 2.3.1. We then carry out set of pulsed ODMR measurements on the single NV center to investigate static and dynamic properties of bath spins surrounding the NV center. We find hetero- geneity in nanoscale bath spin properties by investigating several single NV centers in the same crystal. We also employ DEER spectroscopy to measure EPR signals in the localized environment using the NV center. The detected bath spins are identified as N 51 Chapter 3. Nanoscale NV-based EPR of several spins in diamond 52 spins from the analysis of the observed EPR spectrum. Based on the intensity of the observed NV-based EPR signal, the detected magnetic field is equivalent to that from a single S = 1=2 spin with the distance of7 nm. Finally, we discuss the number of spins detected by the NV-based EPR spectroscopy. By comparing the experimental result with simulation, we estimate the number of the detected spins to be 50 spins. 3.2 Results and Discussion Figure 3.1: (a) Picture of the type-Ib diamond used in the investigation. (b) cw EPR spectrum of N spins measured using the 230 GHz EPR spectrometer. The spectrum was obtained by a single scan at 0.2 mT/s with field modulation of 0.03 mT at 20 kHz. We studied a single crystal of high-temperature high-pressure type-Ib diamond, which is commercially available from Sumitomo electric industries. The size of the diamond crystal is 1:5 1:5 1 mm 3 (see Fig. 3.1(a) inset). The concentration of N Chapter 3. Nanoscale NV-based EPR of several spins in diamond 53 spins is 10 to 100 ppm, corresponding to 4 10 15 to 4 10 16 N spins existing in dia- mond. First, using a 230 GHz EPR spectrometer, 47 we measured ensemble EPR of the sample diamond crystal at room temperature to characterize its bulk properties where the magnetic field was applied along theh111i-direction of the single crystal diamond in the measurement. As shown in Fig. 3.1(b), continuous-wave (cw) EPR spectroscopy showed that five EPR spectra corresponding to the N spins are drastically stronger than the remaining signals, which indicates that the number of N spins dominates the spin population in the sample. Moreover, no EPR signals from NV centers and other para- magnetic impurities were observed in the EPR measurement because of their low con- centration in the sample crystal. We used reported Hamiltonian and its parameters in literature to simulate N spin spectrum as shown in Fig. 3.1(b) dotted red line. The spin Hamiltonian of N spin is given by, H N = B g N S N B 0 +S N $ A N I N N n g N n I N B 0 +P N z (I N z ) 2 ; (3.1) where B is the Bohr magneton,B 0 is the external magnetic field, g N x;y = 2:0024 and g N z = 2:0025 are theg-values of the N electron spin,S N andI N are the electronic and nuclear spin operators, respectively. $ A N is the anisotropic hyperfine coupling tensor to 14 N nuclear spin (A N x = A N y = 82 MHz andA N z = 114 MHz). 48, 49 The gyromagnetic ratio of 14 N nuclear spin ( N n g N n =h) is 3.077 MHz whereh is the Planck constant. The last term of the Hamiltonian is the nuclear quadrupole couplings where P N z is5:6 MHz. 11 Fig. 3.1(b) confirms the agreement between the experimental data and simulated EPR spectrum using Eq. (3.1) which confirms substitutional Nitrogen as the dominant impurity in the diamond sample. Figure 3.2 shows ODMR measurements of a single NV center in the diamond crys- tal. The ODMR experiment was performed using a home-built confocal microscope Chapter 3. Nanoscale NV-based EPR of several spins in diamond 54 (b) 1 μm NV 1 10 15 20 25 (a) (counts/ms) (f) 0 2 4 2τ (μs) 0.5 1.0 0 200 400 600 800 0.0 1.0 P (m S =0) Pulse length t P (ns) Exp. Fit f Rabi = 7.4 ± 0.2 MHz FL Sig. Exc. RO Init. MW t p (c) P (m S =0) T 2 = 1.2 μs Exp. Fit FL Sig. Exc. RO Init. MW τ τ π π/2 π/2 0.0 1.0 P (m S =0) FL Sig. Exc. RO Init. MW t π/2 π/2 Exp. Fit 0.0 0.2 0.4 t (μs) 0.6 (e) b = 30 ± 4 M rad MW Frequency (GHz) B 0 = 35.7 mT 0 -1 ↔ FL intensity (counts/ms) 1.80 1.85 1.90 28 26 24 22 Exp. Fit 0.0 0.4 0.0 0.5 τ (μs) 1.0 1.5 g 2 (τ) (d) τ C = 144 ± 39 μs exp(-(2τ/T 2 ) 3 ) Figure 3.2: ODMR experiment of NV1. (a) FL image of the diamond crys- tal. The scanned area is 33m 2 . The color scheme for FL intensity is shown in the legend. NV 1 is circled. (b) the autocorrelation data of NV1. (c) cw ODMR experiment. The solid line indicates a fit to the Lorentzian function. (d) Rabi oscillation experiment at B 0 = 35:7 mT and the microwave frequency of 1.868 GHz. Square dots connected with lines indicate the measurement and solid line indicates the fit. The oscillation frequency (f Rabi ) of 7.4 0.2 MHz and the decay time (T d ) of 0.77 0.04 s were obtained from a fit to 1=2[cos(2f Rabi t P ) exp((t P =T d ) 2 ) + 1]. The inset shows the pulse sequence with excitation laser (Exc.), microwave (MW) and FL measurement (FL). (e) FID measurement atB 0 = 35:7 mT and 1.868 GHz. The length of/2-pulse was 34 ns. FL intensity decay was monitored as a function oft. The inset shows the pulse sequence used in the measurement. (f) SE experiment atB 0 = 35:7 mT and 1.868 GHz. The lengths of/2- and-pulse used were 34 and 68 ns, respectively. FL intensity decay was monitored as a function of 2. The inset shows the pulse sequence used in the measurement.b and C were extracted by fitting FID and SE data with Eqs. (3.2) and (3.3). For pulsed ODMR measure- ments, laser initialization pulse (Init.) of 2 s, and laser read-out pulse (RO) and FL measurement pulse (Sig.) of 300 ns were used. Also the FL intensity was normalized and re-scaled into them S = 0 state population of NV 1. system. 50 For microwave excitations, two microwave synthesizers, a power combiner, and a 10 watt amplifier were connected to a 20m-thin gold wire placed on a surface of the diamond sample. First a FL image of the diamond was recorded in order to map out FL signals in the diamond crystal (see Fig. 3.2(a)). After choosing an isolated FL spot, Chapter 3. Nanoscale NV-based EPR of several spins in diamond 55 we carried out the autocorrelation and cw ODMR measurements in order to identify FL signals from a single NV center. As shown in Fig. 3.2(b), the autocorrelation measure- ment of the chosen FL spot revealed the dip at zero delay which confirmed the FL signals originated from a single quantum emitter. In addition, cw ODMR measurement of the selected single NV center was performed with application of the external magnetic field (B 0 ) of 35.7 mT along theh111i axis. As shown in Fig. 3.2(c), the reduction of the FL intensity was observed at the microwave frequency of 1.868 GHz corresponding to the ODMR signal of them S = 0$1 transition of the NV center. Thus, the observations of the autocorrelation and cw ODMR signals confirmed the successful identification of the single NV center (denoted as NV 1). Next, we performed pulsed ODMR measurements of NV 1. In the pulsed measure- ments, a NV center was first prepared in them S = 0 state by applying an initialization laser pulse, and the microwave pulse sequence was applied for the desired manipulation of the spin state of the NV center, then the final spin state was determined by applying a read-out laser pulse and measuring the FL intensity. In addition, the FL signal intensity was mapped into the population of the NV’s m S = 0 state (P (m S = 0)) using two references (the maximum and minimum FL intensities corresponding to the m S = 0 and m S =1 states, respectively). 40 Moreover, the pulse sequence was averaged on the order of 10 6 -10 7 times to obtain a single data point. First, the Rabi oscillation mea- surement was performed at B 0 = 35:7 mT and 1.868 GHz, which corresponds to the m S = 0$1 transition of NV 1. As the microwave pulse length (t P ) was varied in the measurement, pronounced oscillations ofP (m S = 0) was observed as shown in Fig. 3.2(c). By fitting the observed Rabi oscillations to the sinusoidal function with the Gaussian decay envelope, 35 the lengths of=2- and-pulses for NV 1 were determined to be 34 and 68 ns, respectively. Second, FID was measured using the Ramsey fringes. Chapter 3. Nanoscale NV-based EPR of several spins in diamond 56 The pulse sequence are shown in the inset of Fig. 3.2(e). As seen in Fig. 3.2(e), the FL intensity of NV 1 was recorded as a function of free evolution time (t) and FID was observed in the range oft 100 ns. Third, we carried out the SE measurement. Fig- ure 3.2(f) shows the FL intensity as a function of free evolution time (2). As shown in the inset of Fig. 3.2(f), the applied microwave pulse sequence consists of the con- ventional spin echo sequence, widely used in EPR spectroscopy, 18 and an additional =2-pulse at the end of the sequence to convert the resultant coherence of a NV center into the population of them S = 0 state. 51 The FID and SE decay in electron spin baths have been successfully described by treating the bath as a classical noise field (B n (t)) whereB n (t) was modeled by the Ornstein-Uhlenbeck (O-U) process with the correla- tion functionC(t) =hB n (0)B n (t)i = b 2 exp(jtj= C ), where the spin-bath coupling constant (b) and the rate of the spin flip-flop process between the bath spins (1= C ). 34, 35 The FID and SE decay due to the O-U process is given by, FID(t) = 1 2 1 6 [1 + 2 cos(2A NV z t)] exp[(b C ) 2 ( t C +e t= C 1)]; (3.2) and SE() = 1 2 + 1 2 exp[(b C ) 2 ( 2 C 3e (2)= C + 4e (2)=(2 C ) )]; (3.3) whereA NV z = 2.3 MHz is the hyperfine coupling of NV center. 11 Derivation of above equations can be found in appendix C. In the quasi-static limit (b C 1) indicating slow bath dynamics, SE(t) exp[b 2 t 3 =(12 C )] = exp[(t=T 2 ) 3 ] where T 2 is the spin decoherence time. 34, 35, 52 By fitting both FID and SE data (Fig. 3.2(e) and (f)) simultaneously with Eqs.(3.2) and (3.3), we obtained b and C to be 30 4 (M rad) and 144 39s, respectively. The result indicates that the surrounding spin bath is in Chapter 3. Nanoscale NV-based EPR of several spins in diamond 57 the quasistatic limit (b C = 4320 1), therefore,T 2 (= (12 c =b 2 ) 1=3 ) of NV 1 is 1.2 s. Using the previous study, 52 the local concentration of the bath spins around NV 1 is estimated to be 20 ppm. (b) MW2 Frequency (f MW2 ) (GHz) Exc. MW Init. RO FL τ τ MW2 (a) t MW2 π π/2 π/2 Sig. (c) 1.0 P (m S =0) NV 3 0 4 8 12 16 NV 2 0.2 0.4 0 2 4 6 8 2τ (μs) P (m S =0) 0 0.5 0.0 2τ (μs) 0.5 1.0 τ C = 966 ± 244 μs 0 2 4 6 8 0.2 0.4 NV 4 P (m S =0) 0 0.5 0.0 2τ (μs) 0.5 1.0 0.5 P (m S =0) 0 0.2 0.4 0.5 0.0 b = 34 ± 8 M rad τ C = 3908 ± 1170 μs b = 31 ± 6 M rad 0.9 1.0 1.1 0.8 P (m S =0) 0.5 0.6 0.7 Exp. Sim. B 0 =35.7 mT NV 1 0.9 MW2 Frequency (f MW2 ) (GHz) 0.8 0.5 0.6 0.7 0.9 1.0 1.1 0.3 0.4 0.5 0.6 P (m S =0) B 0 = 36.1 mT 0.9 1.0 1.1 1.2 B 0 = 36.1 mT B 0 = 16.2 mT 0.9 Exp. Sim. Exp. 0.8 0.5 0.6 0.7 0.9 0.8 0.5 0.6 0.7 0.9 Sim. Exp. Sim. τ C = 259 ± 16 μs b = 38 ± 9 M rad Figure 3.3: NV-based EPR experiment using DEER spectroscopy. (a) The pulse sequence of the DEER spectroscopy. In addition to the spin echo sequence of the NV center (denoted as MW), another microwave pulse (denoted as MW2) was applied 12 ns after the-pulse in MW to rotate the surrounding spins. (b) The obtained EPR spectrum. In the measurement, = 500 ns andt MW2 = 90 ns were used. The simulated EPR spectrum is shown in the dotted line. (c) SE, FID and NV-based EPR data of other three single NV centers (NV 2-4). Furthermore, we performed NV-based EPR using DEER spectroscopy atB 0 = 35:7 mT (f MW1 =1.868 GHz corresponding to them S = 0$1 transition). Figure 3.3(a) Chapter 3. Nanoscale NV-based EPR of several spins in diamond 58 shows the pulse sequence used in the DEER measurements, 44–46 which was adopted from the field of EPR. 18 The DEER sequence consists of the spin echo sequence for a NV center and an additional-pulse at different microwave frequency (denoted as MW2 in Fig. 3.3(a)). When the additional-pulse is resonant with surrounding electron spins near the NV center, the magnetic moments of the surrounding spins are flipped, and this alters the magnetic dipole field experienced by the NV center from the surrounding spins during the second half of the spin echo sequence. The alteration causes a shift in the Larmor frequency of the NV center, which leads to different phase accumulation of the Larmor precession during the second half of the sequence from the first half. As a result, the NV center suffers phase-shift in the echo signal and reduction of FL intensity from the original spin echo signal is observed. As shown in Fig. 3.3(b), we monitored the spin echo intensity of NV 1 at a fixed as a function off MW2 , and observed clear intensity reductions at five frequencies (f MW2 = 0.90, 0.92, 1.01, 1.10, and 1.12 GHz). In the measurement, = 500 ns andt MW2 = 90 ns were chosen to maximize the NV- based EPR signals. As shown in Fig. 3.3(b), the resonant frequencies of the observed NV-based EPR signals were in a good agreement with the EPR of N spins calculated from Eq. (3.1), which confirms the observation of N spin EPR spectrum. The DEER intensity (I DEER ) represents the change of the SE intensity given by the change in the effective dipolar field (B dip;eff ). Assuming that -flip of the N spins is instantaneous (i.e. the effect of a finite width of the pulse was not considered),I DEER = cos(g NV B (2)B dip;eff =~) whereg NV = 2:0028 is theg-value of the NV center, 11 ~ is the reduced Planck constant, and = 500 ns in the present case. In addition, including the effects ofT 2 decay, the NV-based EPR signal inP (m S = 0) is given by, I NVEPR = 1=2(1 +I DEER SE(2)): (3.4) Chapter 3. Nanoscale NV-based EPR of several spins in diamond 59 Using Eq. (3.4), we obtained that the intensity of the NV-based EPR atf MW2 = 0:90 GHz (I NVEPR 0:625) corresponds toB dip;eff 6T. This magnetic field is equiva- lent to the dipole magnetic field from aS = 1=2 single spin with the NV-spin distance (d) of7 nm (B dip = 0 =(4)(3n(n m) m)=d 3 ) with n==m,jmj = g B m S ,g = 2 andm S = 1/2). In addition to the investigation of NV 1, we have studies other NVs in the same diamond crystal. Figure 3.3(c) shows the set of FID, SE and DEER data from other three NV centers. The results from NV 1-4 were different because of their heterogeneous nanoscale local environments. As shown in Fig. 3.3(c), we found that the obtained C values were varied widely from 144s to 3908s. On the other hand, the variation of b = 30-38 M rad is much smaller than C . As the result,T 2 of NV 1-4 ranges from 1.2 to 3.4s. We also noticed that all NV 1-4 are in the quasistatic limit (b C 1). As shown in Fig. 3.3(b) and (c), the NV-based EPR spectra of NV 1-4 were quite different. The NV-based EPR spectra from NV 1, 2 and 4 displays five EPR signals whereas NV 3 only shows three visible signals at 0.381, 0.474 and 0.553 GHz. Possible reasons of the difference are heterogeneity of the number and the spatial configuration of N spins around the NV centers, and an uneven distribution of the N spin orientations due to a small number of N bath spins. Finally, we analyze the intensity of the observed NV-based EPR signal to estimate the number of the detected N spins. Our analysis here focuses on the signal atf MW2 = 0:90 GHz corresponds to thejm S =1=2;m I =1i$jm S = 1=2;m I =1i transition of N spins oriented along theh111i direction (Fig. 3.3(b)). In the analysis, we simulate DEER intensities by calculating the effective magnetic dipole field at the NV center from the surrounding N spins (B dip;eff ). First step of the simulation was to generate a model configuration of the NV center and N spins in a diamond lattice, and Chapter 3. Nanoscale NV-based EPR of several spins in diamond 60 x (nm) y (nm) z (nm) 0 100 -100 0 100 -100 100 0 -100 (a) (b) N:[111] and m I =–1 NV N:others (c) (d) <1 1 2 4 8 16 32 64 ≥128 0 0.3 <n N,90% > 80 40 0.5 0.7 0.6 0.4 0 <n N,90% > 80 40 0.3 0.5 0.7 0.6 0.4 I NVEPR 0.3 0.5 0.7 0.6 0.4 0 <n N,90% > 80 40 <1 1 2 4 8 16 32 64 ≥128 <1 1 2 4 8 16 32 64 ≥128 0 50 <n N,90% > Occurence 100 0 120 0 0 50 <n N,90% > Occurence 140 0 0 50 <n N,90% > Occurence T 1n = 10 ms T 1n = 1 s T 1n = 100 s I NVEPR I NVEPR Figure 3.4: Simulation of the NV-based EPR signals. (a) Overview of the simulation model. Diameter of the simulated lattice is 600a wherea = 0:375 nm is the lattice constant of diamond. Red sphere denotes the NV center placed at the origin of the lattice. Blue spheres represent randomly distributed N spins with 20 ppm concentration. Green spheres are a subset of N spins which are in theh111i orientation withm I =1. N spins at or next to the origin, next to other N spins, and overlap to other N spins, were suppressed in the simulation. (b-d) Intensity plots ofhn N;90% i versusI NVEPR forT 1n = 1 s, 10 ms, and 100 s, respectively. 10 4 spatial configurations were simulated for each T 1n . The color scheme for intensity is shown in the legend. Vertical solid lines indicate the intensity of the observed EPR signal, namely I NVEPR = 0.60-0.65. The insets show the occurrence of spatial configurations of the simulated lattice that resulted in 0:60 < I NVEPR < 0:65 as a function ofhn N;90% i. The number of such configurations were1200,1300, and 400 forT 1n = 1 s, 10 ms, and 100 s, respectively, which translate to 413 % out of total 10 4 configurations. this was done by placing the NV center at the origin of the diamond lattice and assigning the positions of N spins randomly in the lattice sites (see Fig. 3.4(a)). Based on theT 2 Chapter 3. Nanoscale NV-based EPR of several spins in diamond 61 measurement (T 2 = 1.2s 20 ppm),18,000 N spins were placed in the simulated diamond lattice with a diameter of200 nm. Next, we randomly chose the orientation of N spins where only a quarter of N spins were assigned along theh111i direction and the rest of N spins were assigned along the other three directions, i.e., [ 111], [1 11], and [11 1]. We then assigned the nuclear spin value (m I ) of either 1, 0, or1 to all N spins with equal probability. For the simulation, we only considered the contribution coming from the N spins oriented along theh111i direction withm I =1 (The signal atf MW2 = 0:90 GHz in Fig. 3.3(b)). Thus,B dip;eff was computed from only 1=12-th of all N spins in the simulated lattice on average (1500 N spins). Finally we assigned the electron spin value (m S ) of either 1=2 or1=2 with equal probability to the N spins. B dip;eff is given by the sum of individual dipolar field from each N spin as, B dip;eff = n N X i=1 B dip;i (3.5) whereB dip;i = 0 4 (3cos 2 i 1) r 3 i g N B m i S is the dipolar field strength ofi-th N spin at the NV center, n N is the number of the N spins oriented along theh111i direction with m I =1, 0 is the vacuum permeability, r i and i are the magnitude and polar an- gle of the vector that connects the NV center andi-th N spin, respectively, andm i S is the electron spin value of i-th N spin. The mutual flip-flops within N electron spins was also not considered because of a low rate of the spin flip-flop process during the measurement, i.e. C (Fig. 3.2(e) and Fig. 3.3). We then calculated the number of N spins (n N;90% ) out ofn N to obtain more than 90% of theB dip;eff by successively adding individual dipolar field term from each N spin in descending order of magni- tude (i.e., whenjB dip;eff P n N;90% j=1 B dip;j j=jB dip;eff j 0:1). In addition, we took into Chapter 3. Nanoscale NV-based EPR of several spins in diamond 62 account for the electron and nuclear spin relaxations of N spins during the DEER mea- surement time ( 100 s with 10 7 averaging) by statistically re-assigning the electron and nuclear spin values (m S andm I ) according to the electron and nuclear longitudinal relaxation times (T 1e andT 1n ), respectively. AlthoughT 1e has been investigated previ- ously (T 1e 10 ms), 49, 53 T 1n has not been reported before to the best of our knowledge. In the simulation, we usedT 1e = 10 ms and consideredT 1n to be 10 ms, 1 s and 100 s, therefore the assignment ofm I happens 10 2 times forT 1n = 1 s (10 4 and 1 times for T 1n = 10 ms and 100 s, respectively) and a total of 10 4 iterations (100 s /T 1e ) occurs during the simulation. Then, what we finally computed were the average of 10 4 values of n N;90% and I DEER (i.e.,hn N;90% i = P 10 4 k=1 n k N;90% andhI DEER i = P 10 4 k=1 I k DEER ), therefore, by rewriting Eq. (3.4), the simulated NV-based EPR signal is given by, I NVEPR = 1=2(1 +hI DEER iSE(2)): (3.6) In addition, in order to see the spatial configuration dependence onhn N;90% i and I NVEPR , we repeated the procedure described above for 10 4 spatial configurations and calculated 10 4 values ofhn N;90% i andI NVEPR . Figure 3.4(b)-(d) shows the simulated result ofhn N;90% i as a function of I NVEPR (Eq. (3.6)) for T 1n = 1 s, 10 ms, and 100 s, respectively, from 10 4 spatial configura- tions. As shown in Fig. 3.4(b), the number of the N spin that contributes to the NV- based EPR signal (hn N;90% i) depends on the detectedI NVEPR . We first noted that, for all 10 4 configurations we simulated, only a small portion (60) out of1500 N spins contributes to more than 90 % of B dip;eff on average. In the case of homogeneously distributed N spins, the 60 spins are located in a sphere with70 nm diameter. More- over, when the NV-based EPR intensity is large (I NVEPR 0:5),hn N;90% i is smaller becauseB dip;eff in such spatial configurations is only dominated by a smaller ensemble Chapter 3. Nanoscale NV-based EPR of several spins in diamond 63 of the N spin located in the vicinity of the NV center. In addition, there was little ob- servation ofhI DEER i < 0 (i.e., I NVEPR < 0:5) because a small probability exists for the NV center to be coupled with a single or a few N spins with the 20 ppm N concen- tration and the resultanthI DEER i is a weighted sum of many oscillatory functions with different frequencies. On the other hand, when the NV-based EPR intensity is small (hI DEER i 1) (i.e. I NVEPR 0.73),hn N;90% i is larger because the N spins in such spatial configurations spread uniformly and the N spins located farther away from the NV also contribute toB dip;eff . The experimentally observed NV-based EPR intensity was 0:625 0:025 (see Fig. 3.3(b)). The inset of Fig. 3.4(b) shows a histogram of the occurrence ofhn N;90% i from the simulation that yieldedI NVEPR = 0:6-0.65. The oc- currence was in the range of 7 42 forT 1n = 1 s. Moreover, as shown in Fig. 3.4(c), hn N;90% i withT 1n = 10 ms was similar to the result withT 1n = 1 s (The occurrence was in the range of 7 45). On the other hand, in the case ofT 1n = 100 s, the distribution was slightly different and the occurrence was in 2 28 as shown in Fig. 3.4(d). Thus, from the simulation, the number of N spins in the present NV-based EPR measurement is estimated to be 50 spins. This corresponds to improvement of EPR spin sensitivity by 10 9 . 3.3 Summary In summary, we presented nanoscale EPR spectroscopy using a single NV center in diamond. First, we demonstrated the identification of nanoscale spin baths surrounding a single NV center and the investigation of static and dynamic properties of the bath spins using Rabi, FID, SE measurements as well as NV-based EPR spectroscopy. We also performed the investigation with several other single NV centers in the diamond sample and found that the properties of the bath spins are unique to the NV centers. Finally, Chapter 3. Nanoscale NV-based EPR of several spins in diamond 64 by analyzing the intensity of the NV-based EPR signal using the computer simulation, we estimated the detected spins in the DEER measurement to be 50 spins. Thus, the investigation demonstrated ability of NV based EPR to quantitatively detect bath spins when it is more than few spins. Chapter 4: NV-based EPR of a few external spins 4.1 Introduction This chapter discusses NV-based EPR spectroscopy of external electron spins located on the diamond surface. First, I discuss the fabrication of NVs closer to diamond surface (few nanometers deep). Next, NV based EPR measurements of external spins are pre- sented. Finally, sensitivity estimations being discussed based on the observed coherence times and SNR of fabricated NVs. 4.2 NV fabrication Fabricated NV centers have a couple of important advantages over native NV centers. The main advantage is fabrication allows to confine NV centers closer to the diamond surface within few nanometers. This allows to use fabricated NV centers to employ as sensors to detect external paramagnetic spins. Common practice is to use 15 N isotope (I = 1/2) because the low natural abundance of 15 N makes us easy to distinguish from 14 N isotope (99.6%, I = 1). In the experiment, we verify the hyperfine coupling of 15 NV to identify the fabricated NV using pulsed EPR measurement as discussed under section 2.4.3. (see Fig. 4.4). There are two methods to fabricate NVs. First method is 65 Chapter 4. NV-based EPR of a few external spins 66 (a) (b) 0 10 20 Depth (nm) 5 keV 0 +10 -10 Lateral distance (nm) 30 1 10 100 Energy (keV) 1000 1000 100 10 1 Average depth (nm) 7 0 Diamond Surface Average longitudinal depth Longitudinal straggle Figure 4.1: Implantation simulation and FL images at different depths. (a). Summary of the results obtained from SRIM simulation for 5keV implantation energy. 7 degrees tilt angle was used in the simulation. (b). Nominal average implanted depth and longitudinal straggle as a function of implantation energy. to fabricate a thin layer of diamond with appropriate NV concentration using a chemi- cal vapor deposition (CVD) method. In this method, the NV concentration is controlled through the Nitrogen concentration of the feeding gases. We have such diamond through the collaboration with Prof. Itoh group in Keio University. ODMR data taken from this diamond can be found in appendix D. The second method is to implant 15 N + ions into the diamond crystal, then subsequently to anneal the crystal at high temperature. The penetration depth of the implanted 15 N + ions is controlled by the implantation energy hence the nominal depth of fabricated NVs is aimed by adjusting the implantation en- ergy. Figure 4.1(a) shows simulated trajectories of implanted 15 N + ions inside the dia- mond crystal for the 5 KeV implantation energy. The simulation was done using SRIM package. 54 The implantation energy dependence on the depth and straggle (straggle is the square root of the lateral variance) is shown in the Fig. 4.1(b). Tilt angle of 7 de- grees was used in actual implantation to prevent channeling effect which is defined as the process that disturbs the trajectory of a charged particle in a crystal. However, SRIM simulation does not account for ion channeling effect. Chapter 4. NV-based EPR of a few external spins 67 0 20 40 60 75 counts / ms (a) (b) (c) (d) diamond diamond 2μm 850 ° C annealed 465 ° C annealed Center Corner Figure 4.2: FL images of diamond after each annealing process. (a). FL image taken after 850 C annealing closer to center of the diamond. (b). FL image taken after 465 C annealing closer to center of the diamond. Background is about 8 counts/ms and FL intensity is about 50 counts/ms. (c) and (d). FL image taken after 850 C and 465 C annealing from one of the corner, respectively. Implantation is followed by high temperature annealing at 850 C for one hour which facilitates moving vacancies around. Before the annealing diamond was cleaned by ultrasonicating 30 minutes in acetone followed by 15 minutes in Ethanol. Then rinsed with DI water and ultrasonicated in saturated solution of KOH for 5 minutes. After that crystal was rinsed with DI water followed by ethanol and let it dry in the oven. High temperature annealing was done under the continuous flow of 95% : 5% mix- ture of Ar : H 2 gas. Diamond crystal was placed on the quartz tube of the furnace without any crucible to provide better air flow around the crystal surfaces. The diffu- sion process of vacancies during high temperature annealing can be explained by the Chapter 4. NV-based EPR of a few external spins 68 Arrhenius lawD = D 0 exp(E a =kT ), whereD is the diffusion coefficient,E a is the activation energy associated with the vacancy diffusion, k and T are the Boltzmann constant and the annealing temperature, respectively. D 0 is defined as the pre-factor. It is reported that near surface vacancy diffusion pre-factor as D 0 = 3:6 10 6 cm 2 s 1 which is higher than in the bulk. Reported values for activation energies are in the range of 1.7 - 4 eV. 55 Once the diffusion coefficientD is known diffusion distanced can be simply estimated asd p Dt wheret is the annealing time. For the above mentioned annealing conditionsd is estimated as 154 nm by considering the lowest limit ofE a . In addition, the activation energy of substitutional nitrogen impurity is about 6.3 eV and immobile up to 2600 C where as activation energy of an NV center is approximated as 4.5 eV and immobile up to 1700 C. 56 After high temperature annealing diamond was annealed at 465 C for 24 hours to remove any graphitic layer formed during high temperature annealing. Summary of FL images after each annealing step are given in Fig. 4.2. According to the manufacture’s specifications of these electric grade diamond crys- tals from element six, it may contains very few native NVs. However, as shown in Fig. 4.3(a), FL images does not shows any native NVs. As shown in Fig. 4.3(b), a stack of confocal images taken at different depths after both annealing process confirm NV centers are located only in the vicinity of surface. In comparison, type 1b diamonds contain NV centers trough out the bulk crystal (Fig. 4.3(c)). The density of the NVs can be controlled by the dose of the implantation which defined as the number of impurity atoms per cm 2 . Considering the NV density in FL images (Fig. 4.2(b) and Fig. 4.3(b)) the concentration of NV centers is estimated as 0.3 ppb with 1% yield of formation. Chapter 4. NV-based EPR of a few external spins 69 Figure 4.3: A stack of FL images at different depth taken before and after NV fabrication. (a). FL images taken at different depths of diamond crystal before implantation. No NVs were observed. (b). FL images taken at different depths of diamond crystal after implantation and both annealing. 5 keV implantation energy and does of 10 9 nitrogen per cm 2 . Z = 0 m corresponds to surface. NVs appeared only closer to the surface. (c). A stack of FL images taken with type 1b diamond crystal. FL from NVs can be seeing at any depth. Furthermore, the consistency of nitrogen nuclear spin was confirmed by performing the pulsed ODMR measurement. Figure 4.4 shows the results of pulsed ODMR exper- iment taken with fabricated NVs. This signal is different from the data presented in Fig. 2.10 due to the different nuclear spin values and the hyperfine interaction of 14 N and 15 N nuclear isotopes. A summary ofT 2 measurements of implanted NVs is given in the appendix D. Chapter 4. NV-based EPR of a few external spins 70 2948 2952 2956 2960 1.00 0.95 0.90 Contrast Frequency (MHz) 15 NV 15 ∆ 15 ∆ = 3.2 MHz Figure 4.4: Pulse ODMR spectrum of an 15 NV in implanted diamond crystal with 15 N isotopes . MW pulse of 1000 ns length was used with -24 dBm MW source output. Separation of peaks ( 15 ) agree with the 15 N hyperfine splitting. 4.3 Covalent attachment of Nitroxide spin Covalent attachment of Nitroxide spin labels on to nanodiamond surface using Copper Click chemistry has been shown. 57 Same procedure as in reference 57 was followed here to attach nitroxide spin labels to the bulk diamond which NV centers were fabricated. First, hydroxylation was performed on bulk diamond crystal after cleaning the crys- tal with 9:1 mixture of concentrated Sulphuric and Nitric acids for 72 hours at 90 C. Then crystal was thoroughly cleaned with DI water. Next diamond was placed in a stir- ring suspension of dry 6 mL of THF and added 1 mL of 1.0 M BH 3 -THF dropwise. Then the mixture was heated to 60 C and stirred for 24 h. After cooling down, mixture was hydrolyzed with 2 M HCl and rinsed the diamond with DI water and acetone. Chapter 4. NV-based EPR of a few external spins 71 Next Silanization was carried out obtain bromide-functional diamond surface. Dia- mond crystal was placed in 10 mL of anhydrous toluene under nitrogen atmosphere and heated to 80 C. 0.5 mL of 3- bromopropyltrichlorosilane was added dropwise and the mixture was kept at 80 C for 24 hours. Then crystal was rinsed with toluene to remove unreacted 3- bromopropyltrichlorosilane. Then Br- group was replace by Azide (-N 3 ) group. Diamond was placed in saturated solution of NaN 3 (25 mg in 10 mL of dimethylformamide) and stirred the mixture for 24 h at 80 C. Then crystal was cleaned in DI water followed by ethanol. Nitroxide spin labels (4-hydroxy-TEMPO) with alkyne group were taken from the same batch used in reference 57. Finally the Click reaction was performed. To a flask containing azide functional diamond in anhydrous acetonitrile (550L), 50 mM CuI (200L) and 50 mM alkyne- Nitroxide (200 L) with 50 L of triethanolamine (TEA) added and flushed with Ni- trogen. Then the mixture was stirred for 48 h at room temperature. Then the diamond crystal was sonicated in anhydrous acetonitrile several times (8 times) until the EPR signal of free Nitroxide in the wash disappeared. In our study, 58 by analyzing the EPR intensity of the attached nitroxide radicals after the reaction, we estimated the number of the attached molecule to be 2900 per 100 nm ND, corresponding the average distance between them to be 3.3 nm. 4.4 EPR spectroscopy of Nitroxide spin labels using NV DEER measurements on NV centers were carried out once the covalent attachment of Nitroxide spins is done. As shown in Fig. 4.5(a), SE measurement was performed first. Then the DEER measurement was performed as explained in section 2.4.8. Figure 4.5(b) shows the results of the DEER measurement. As show in the Fig. 4.5(b), a pronounce Chapter 4. NV-based EPR of a few external spins 72 single broader peak was observed where nitroxide spectrum was expected. The observed NV-based EPR spectrum is analyzed considering the presence of a single nitroxide spin. The nitroxide spectrum is simulated by the Hamiltonian, H NS = B g NS S NS B 0 +S NS $ A N SI NS NS n g NS n I NS B 0 +P NS z (I NS z ) 2 ; (4.1) where B is the Bohr magneton,B 0 is the external magnetic field,g NS x = 2:0085,g NS y = 2:0056 and g NS z = 2:0033 are the g-values of the nitroxide electron spin, S NS and I NS are the electronic and nuclear spin operators, respectively. $ A NS is the anisotropic hyperfine coupling tensor to 14 N nuclear spin (A NS x = 6:5 MHz,A NS y = 5:6 MHz and A NS z = 37 MHz). 58 The gyromagnetic ratio of 14 N nuclear spin ( NS n g NS n =h) is 3.077 MHz where h is the Planck constant. The last term of the Hamiltonian is the nuclear quadrupole couplings whereP NS z is5:6 MHz. 11 The observed spectrum is narrower than expected hyperfine splitting of nitroxide spin labels. Therefore, powder spectrum of nitroxde was considered as shown by the green simulation in Fig. 4.5(b). The effect of broadening of the powder spectrum com- ing from the larger excitation bandwidth due to the shorter MW2 pulse is shown by the simulated blue line in Fig. 4.5(b). Next, DEER measurement was carried out with relatively lower MW2 power to reduce the excitation bandwidth. Figure 4.6 shows the results of DEER experiment performed with lower MW2 power where MW2 was 114 ns. CPMG16 (i.e. CPMG pulse sequence with 16 pulses for both MW1 and MW2) based DEER sequence was used in this measurement. (see Fig. 4.6(a)). In addition toP (m s = 0) andP (m s =1) references, the echo amplitude of CPMG16 with fixed without application of MW2 was taken as the third reference (Fig. 4.6(a)). The echo amplitude reference was useful to see features in EPR spectrum Chapter 4. NV-based EPR of a few external spins 73 0.8 0.7 0.6 0.5 750 800 850 900 950 MW2 Frequency (MHz) P(m s =0) SE Amp. Signal Nitroxide sim. Exc. BW sim. 1.0 0.8 0.6 0.4 P(m s =0) 0.2 0 4 8 12 16 20 24 2τ (μs) SE Data (b) (a) Figure 4.5: NV-based EPR of Nitroxide attached diamond crystal. (a). SE of the NV . Blue line shows the simulation of decay envelope [0:5 + 0:5 exp ((t=T 2 ) 3 )] with T 2 = 14 s. (b). NV-based EPR spectrum measured using the DEER pulse sequence (see Fig. 2.15(a)). SE Amp. data points represent the amplitude of the spin echo for fixed without application of the MW2 pulse. Signal represents the amplitude of the spin echo for fixed with application of the MW2 pulse. Green line shows the simulated powder spectrum of Nitroxide. The blue line shows the simulated powder spectrum of nitroxide with broadening effect imposed by the excitation bandwidth. Pulse parameters used in the experiment were, NV/2 = 36 ns; NV = 72 ns; MW2 = 44 ns; = 2.43s. Calculated magnetic field strength based on NV resonance positions was 30.55 mT. specially when signals are noisy. Figure 4.6(b) shows the coherence decay measured with CPMG16 of the NV that the nitroxide spectrum was observed. Figure 4.6(c) shows the results of the NV-based EPR measurement. Three pro- nounced dips in the NV-based EPR signal are visible. Inset of Fig. 4.6(c) represents the raw data collected from all four counters. The signal and CPMG16 Amp. were normalized using the ‘max’ and ‘min’ references to map the signal intensities to the population ofm s =1 state as shown in main body of Fig. 4.6(c). The signal and CPMG16 Amp. data shown in the Fig. 4.6(c) inset was used in the following analysis. The ratio of signal and CPMG16 Amp. data was normalized be- tween 0 and 1 as show in Fig. 4.7. The cartoon in Fig. 4.7 inset represents the simple Chapter 4. NV-based EPR of a few external spins 74 0 10 20 30 Total Evolution time (μs) P (m s = -1) 1.0 0.9 0.8 0.7 0.6 0.5 40 50 T 2 = 23 μs Data Sim. 700 750 800 850 MW2 Frequency (MHz) 900 P (m s = -1) 0.9 0.8 0.7 0.6 0.5 Signal CPMG16 Amp. Max Min Signal CPMG16 Amp. 700 750 800 850 MW2 Frequency (MHz) 900 7.5 6 Counts x 10 4 (arb. units) AOM TTL Init. RO τ 2τ CPMG-16 (π/2) y (π/2) y π x MW1 TTL MW2 TTL Init. RO (π/2) y (π/2) y π x τ 2τ CPMG-16 (a) (b) (c) π MW2 P(m s = 0) Reference P(m s = ±1) Reference Figure 4.6: NV-based EPR measurement with Nitroxide attached diamond crystal. (a). Pulse sequence based on CPMG16 used to perform DEER mea- surement. (b). CPMG16 data of the NV . Blue line shows the simulation of decay envelope [0:5 + 0:5 exp ((t=T 2 ) 3 )] withT 2 = 23s. (c). EPR spectrum measured using the pulse sequence in (a). CPMG16 Amp. data points represent the amplitude of the spin echo for fixed without application of the MW2 pulses. Signal represents the amplitude of the spin echo for fixed with appli- cation of the MW2 pulse. Pulse parameters used in the experiment were, NV /2 = 32 ns; NV = 64 ns; MW2 = 114 ns; = 420 ns;. Calculated magnetic field strength was 28.46 mT. Inset shows the raw data collected from all four counters. model of nitroxide spin considered in this analysis. Since the three peaks are well sep- arated it is possible that the angle is acute. The side peaks move towards the center peak as gets closer to the right angle. Simulated nitroxide spectrum was convoluted with a sinc function to include the broadening coming from the excitation bandwidth considering the pulse width ( MW2 ) Chapter 4. NV-based EPR of a few external spins 75 0 1.0 0.8 0.6 0.4 0.2 MW2 Frequency (MHz) Normalized intensity (arb. units) 700 750 800 850 900 Data Sim. B 0 NS θ diamond Figure 4.7: Nitroxide spectrum analysis. Simulation shows the spectrum of a single nitroxide spin label with = 30 degrees. used in the DEER experiment. The agreement between the experimental data and simu- lation was optimized by reducing the 2 . As shown in Fig. 4.7 the best fit was obtained when = 30 . Therefore, observed NV-based EPR signal is possibly from a single or a few nitroxide spin labels covalently attached on the diamond surface. 4.5 Magnetic field sensitivity calculations In Chapter 3, I discussed estimation of dipole fields based on the intensity of the DEER signal. In this section I discuss the minimum amplitude (SNR to be 1) of the dipole field that can detect by these fabricated NVs considering observed coherence times and noise in the measurements. Variation of the noise in SE and CPMG measurements is about 0.08 in the unit of P(m s = 0). Using the equation ofI DEER giving in chapter 3, the effective dipolar field (B dip;eff ) which corresponds to NV-based EPR of 0.08 in the unit of P(m s = 0) is calculated as a function of the total free evolution time ( 0 ) andT 2 was assumed to be three times longer than 0 . The results of this calculation is shown in Fig. 4.8. It is possible to detect any diploar field lies upper side of this curve for a Chapter 4. NV-based EPR of a few external spins 76 1 B dip (μT) 10 0.1 1 10 10 2 10 3 τ' (μs) 10 -1 10 -2 10 -3 10 -4 10 4 Electron spin, 5 nm Nuclear spin, 5 nm Electron spin, 10 nm Nuclear spin, 10 nm Sim. Figure 4.8: Calculated minimum detectable dipolar field as a function of the total free evolution time ( 0 ). The Intensity of NV-based EPR is assumed as 0.08 in the unit of P(m s = 0) based on the noise in our pulse measurement. Blue and green solid lines (dashed lines) shows the strength of the dipolar field created by a single electron spin and a single nuclear spin located 5 nm (10 nm) away parallel to the NV , respectively. particular 0 . Blue and green solid lines in Fig. 4.8 shows the required coherence time of the NV to detect the dipole field strength of an electron and a nuclear spin located 5 nm away from the NV respectively. Dashed lines shows the case when the dipole field strength of an electron and a nuclear spin located 10 nm away from the NV respectively. 4.6 Summary NV centers closer to the diamond surface was fabricated successfully using implantation and annealing techniques. Nitroxide spin labels were covalently attached on the bulk Chapter 4. NV-based EPR of a few external spins 77 diamond surface. Successful detection of covalently attached nitroxide spin label was obtained with NV-based EPR spectroscopy. Chapter 5: HF EPR study of spin coherence in a Mn 3 single-molecule magnet Materials presented in this chapter can also be found in the article titled Spin coherence in a Mn 3 single-molecule magnet by Chathuranga Abeywardana, Andrew M. Mowson, George Christou and Susumu Takahashi in Applied Physics Letters 108, 042401 (2016) (Reprinted with permission. Copyright [2016], AIP Publishing LLC.) 5.1 Introduction to coherence in SMM Single-molecule magnets (SMMs) are nanoscale magnets that possess large magnetic moments and an anisotropy energy barrier between their spin-up and spin-down states at the molecular level. The energy barrier prevents spin reversal, leading to slow mag- netization relaxation and hysteresis (bistability) at low temperatures. 59, 60 The quantum mechanical nature of their nanomagnetism also emerges at low temperatures, with be- haviors such as quantum tunneling of magnetization (QTM) 61–63 and quantum phase interference of two tunneling paths (Berry phase). 64–66 Various types of SMMs with 78 Chapter 5. HF EPR study of spin coherence in a Mn 3 single-molecule magnet 79 different sizes of magnetic moments and energy barriers have been synthesized, includ- ing SMMs made from several transition metal ions, 59, 60, 67–72 a dimer of SMMs, 73, 74 and mononuclear SMMs based on lanthanides. 75–78 The nanomagnetism and spin physics of SMMs have been extensively investigated on large ensembles of SMMs. 59–64, 79–81 In ad- dition, it has been demonstrated that an individual SMM or a small ensemble of SMMs can be placed on a surface with some retention of their magnetic behavior; 82–88 there- fore, SMMs are also candidates for potential applications in dense quan-tum memory, quantum computing, and molecular spintronics. 87, 89, 90 In spite of wide interest in the quantum nature of SMMs, decoherence effects that ul- timately limit such behavior have yet to be fully understood. Until now, coherent manip- ulation of spin states in SMMs has been experimentally demonstrated only in a very few cases, including Fe 8 , 91, 92 V 15 , 93 Fe 4 , 94 and Cr 7 M (M=Ni and Mn) 95 systems. In particu- lar, even though Mn-based SMMs have been extensively studied for over two decades, no coherent manipulation on Mn-based SMMs has been reported to date. Recent in- vestigations have shown that there are three main decoherence mechanisms present in SMMs: spins can couple locally (1) to phonons (phonon decoherence); (2) to many nu- clear spins (nuclear decoherence); and (3) to each other via dipolar interactions (dipolar decoherence). 91, 96, 97 In particular, the long range nature of dipolar interactions is a major problem in many SMMs. Interestingly, recent experimental investigations have demonstrated that dipolar decoherence is significantly suppressed using high frequency electron paramagnetic resonance (HF-EPR) spectroscopy at low temperature. 91, 92, 98 5.2 High field - high frequency EPR spectrometer This section will describe the operation of custom built high field, high frequency (HF) EPR spectrometer at USC. This spectrometer can operate both continuous and pulse Chapter 5. HF EPR study of spin coherence in a Mn 3 single-molecule magnet 80 modes at 107-120 GHz and 215-240 GHz frequency range, 0-12 T magnetic field and room temperature to about 1.52K temperature. 47, 99 As shown in Fig. 5.1, spectrometer consists of solid state transmitter (700 mW at 115 GHz and 100 mW at 230 GHz), a quasioptical system, a receiver, a phase sensitive superheterodyne detection system which outputs I and Q components of the d.c. signal, a cryostat operates with 4 He and a cryogenic free super conducting magnet. Figure 5.2(a) shows the circuit diagram of high power HF transmitter system which consists of two MW synthesizers with base frequency of 24 GHz and 12 GHz to get out- put frequencies in the range of 215240 and 107120 GHz, respectively , directional couplers, isolators, PIN switches with short rising time about 12 ns which can be con- trolled by TTL signals from pulse generator, a power combiner, an amplifier and series of frequency multipliers contain one active frequency tripler and three passive frequency doublers. The complete system was custom built by Virginia Diodes, Inc. (VDI). Out- put of transmitter is a single mode rectangular waveguide which is then connected to corrugated horn to output linearly polarized Gaussian wave denoted by red double ar- row in Fig. 5.1. To cancel frequency dependence of the quasioptics, Gaussian waves are periodically focuses by right-angle ellipsoidal mirrors with a focal length of f = 254 mm. A quasioptical system is used to guide the HF MW radiation to the sample and also guide the EPR signals to detection system. Quasioptical system contains corrugated horns, rotating wire grid polarizer to control transmitting power (attenuation is > 30 dB), fixed wire grid polarizer and Faraday rotator to prevent standing waves that can originate by reflection (purple dashed double arrow represents the reflections directed to the absorbed). Then Gaussian excitation coupled to corrugated wave guide that that Chapter 5. HF EPR study of spin coherence in a Mn 3 single-molecule magnet 81 Transmitter Receiver Detection system Quasioptical system 4 He cryostat Cryogen-free superconducting magnet PC control Lock-in amplifiers Oscilloscope Corrugated horns Pulse generator IQ mixer 3 GHz IF LO Sig I Q 3 GHz ref Ref 1 Ref 2 Pulse triggering Microwave signals (arrowhead indicates direction) Absorber Rotating wiregrid polarizer Ellipsoidal mirror Fixed wiregrid polarizer Faraday rotator Sample LNA Corrugated waveguide Data acquisition via GPIB, LAN, or USB connection Figure 5.1: Overview of HF EPR spectrometer. First, the receiver down con- verts the EPR signals to an intermediate frequency (IF) of 3 GHz. Then the signal will amplified by a low noise amplifier (LNA) followed by the second amplifier. Using references (Ref 1 and Ref 2) obtained from transmitter and re- ceiver, a 3 GHz reference was produced. A IQ mixer and 3 GHz reference was used to down convert IF signal to I and Q components of d.c. signals. A 12.1 T superconducting magnet is cooled by a closed-cycle pulse tube cryocooler system. A 4 He cryostat was used to perform low temperature experiments. A computer was used in data acquisition and controlling electronic instrumenta- tion. Reprinted figure with permission from ref [ 99]. act as the sample probe. Superheterodyne detection (converts received signal to IF) is used to detect EPR signals. As shown in Fig. 5.2(b), a sub-harmonically pumped mixer is used in down Chapter 5. HF EPR study of spin coherence in a Mn 3 single-molecule magnet 82 High-power frequency multiplier chain 10 MHz ref 9-11 GHz synthesizer 8-10 GHz synthesizer Directional couplers Isolators PIN switches Power combiner X3 X2 X2 X2 Ref 1 107-120 GHz output 215-240 GHz output Amplifier Active frequency tripler Passive frequency doublers 10 MHz ref 2-20 GHz synthesizer Isolator PIN switch Single pole double throw switch X3 X2 X2 X2 Amplifier Active frequency doubler Directional coupler Ref 2 Passive frequency doubler X3 Passive frequency tripler Passive frequency tripler Subharmonically -pumped mixers Passive frequency doubler IF (3GHz) IF (3 GHz) Signal (107-120 GHz) Signal (215-240 GHz) SMA connection Rectangular waveguide connection TTL signal (a) (b) Figure 5.2: Circuit diagram of the high frequency, high-power transmitter (a) and receiver (b) system. Reprinted figure with permission from ref [ 99]. conversion configuration to produce 3 GHz IF signals by mixing HF local oscillator signal (LO) and EPR signal. HF LO is somewhat similar to synthesizer. A single-pole double-throw switch is used to select operating frequency range. Finally IF signals at 3 GHz are further down converted to I and Q d.c. signals by mixing with 3 GHz reference produced by Ref.1 and Ref.2. In CW measurement intensities of frequency modulated I and Q components (typical modulation frequency is about 20 kHz) are measured using lock-in amplifiers while in pulsed EPR a digital scope (sampling rate of 500 MHz) samples the transient I and Q signals. Pulsed EPR sensitivity of HF EPR spectrometer was estimated to be 2 3 10 10 spins at 2 K temperature where polarization is 88% , and 2 3 10 12 spins at 300 K temperature where polarization is 0.9% for 115 GHz. For 230 GHz spin sensitivity was estimated to be 0:9110 10 spins at 2 K temperature Chapter 5. HF EPR study of spin coherence in a Mn 3 single-molecule magnet 83 and 5 6 10 11 spins at 300 K temperature where spin polarization is 99% and 1.8%, respectively. 99 5.2.1 Rotational sample holder Ability to collect EPR spectra as a function of angle with respect to external magnetic field would provide additional information to determine Hamiltonian parameters. There- fore, a mechanism to rotate the sample was implemented. As shown in Fig. 5.3, a com- bination of a worm and a worm gear confined in an Aluminum gear box housing was designed to build the rotating sample holder. (a) Grear box housing Modulation Coil holder Pedestal Worm shaft Worm gear shaft Worm gear Worm (b) 1mm Figure 5.3: Schematic of HF EPR sample rotating stage. (a). 3D drawing of the rotating gear box attached to modulation coil holder. Magnetic field is parallel to ^ z - axis and ^ y - is the axis of rotation. (b). Gear box housing that holds the worm (sketched in blue) and the worm gear (sketched in orange). Green arrows denote the direction of rotation. Worm shaft comes out of the magnet bore though vacuum tight O-ring and connected to a graduated dialer. End of the worm gear shaft (along ^ y - axis) that located at the center of magnetic field used to mount the sample. Gear ratio of worm to worm gear is 36 . Chapter 5. HF EPR study of spin coherence in a Mn 3 single-molecule magnet 84 Shaft connected to the worm gear carries the sample in to the center of the magnetic field. Performance of the rotating sample stage has been verified with a type 1b diamond crystal and summary of the experiment results taken at room temperature are given in Fig. 5.4. Figure 5.4(a) shows the geometry of P1 center in lab frame when the angle is 0 . Each color along bond represents the simulated spectra for particular orientation as shown in Fig. 5.4(b) and Fig. 5.4(c). Figure 5.4(b) shows the data collected by rotating the worm shaft by 360 where worm gear rotates by 36 . Dotted lines shows the simu- lated peak positions for each orientation as a function of angle. As shown in Fig. 5.4(c) the positions of peak extracted from data are plotted against the angle of rotation. Solid lines in Fig. 5.4(c) denoted the simulated position of peaks for each orientation. Ob- served angle dependence of peak positions are well agreed with the simulated positions as shown in Fig. 5.4(c). Moreover, accuracy of the angle control is estimated as 2.3 degrees. Same measurement was performed at low temperatures ranging from 6 K to 30 K and confirmed the performance of the stage at low temperatures. Chapter 5. HF EPR study of spin coherence in a Mn 3 single-molecule magnet 85 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 8.202 8.204 8.206 8.208 8.210 Magnetic Field (T) Rotation Angle (Degrees) |-1/2,-1〉↔|+1/2,-1〉 |-1/2,0〉↔|+1/2,0〉 |-1/2,+1〉↔|+1/2,+1〉 Data Sim. Ori. 1 Sim. Ori. 2 Sim. Ori. 3 Sim. Ori. 4 B 0 (z) ^ (y) ^ 0° (b) (a) (c) 0 36 72 108 144 180 216 252 288 324 360 Rotation Angle (Degrees) 8.2040 8.2035 8.2030 8.2025 8.2105 8.2100 8.2095 Magnetic Field (T) |-1/2,-1〉↔|+1/2,-1〉 |-1/2,+1〉↔|+1/2,+1〉 Ori. 1 data Ori. 2 data Ori. 3 data Ori. 4 data Sim. Ori. 1 Sim. Ori. 2 Sim. Ori. 3 Sim. Ori. 4 Figure 5.4: (a). Geometry of P1 center with respect to magnetic field at 0 position. Colors of bonds are selected to match with the simulation. (b). The EPR spectra (orange horizontal lines) of P1 centers taken by rotating the type 1b diamond crystal 360 full rotation with 36 steps at room temperature. Vertical color lines shows the simulated resonance positions as a function of angle for each orientation of the P1 center. On the top corresponding transitions are la- beled usingjm S ;m I i notation representing electron spin state and nuclear spin state, respectively. (c). Peak positions of the EPR spectrum as a function of an- gle of rotation. Scatters shows the peak positions extracted from the data shown in (b). Solid lines shows the simulated results of peak positions. Upper panel and lower panel shows the resonance position ofj 1=2;1i$j + 1=2;1i andj 1=2; +1i$j + 1=2; +1i transitions, respectively. 5.3 Mn 3 SMM Experiment and Results We investigate spin coherence in single crystals of the SMM [Mn 3 O(O 2 CEt) 3 (mpko) 3 ](ClO 4 ), abbreviated Mn 3 , which has a ground state spin of S = 6. 100, 101 In addi- tion, quantum mechanical couplings between different Mn 3 SMMs have recently been Chapter 5. HF EPR study of spin coherence in a Mn 3 single-molecule magnet 86 demonstrated with synthesis of a covalently linked dimer 102 and tetramer 103 while re- taining the intrinsic magnetic properties of each Mn 3 SMM. Therefore, the Mn 3 system is clearly a potentially great testbed for investigating quantum coherence in Mn-based SMMs. The investigation was performed with continuous-wave (cw) and pulsed EPR spectroscopy at 230 GHz. Using 230 GHz cw EPR spectroscopy, we first identified the EPR transition between them S =6 andm S =5 states, and then performed spin echo measurements to probe the coherence in Mn 3 SMMs. At resonance, the energy difference between them S =6 andm S =5 states is 11 K, so Mn 3 spins are almost completely polarized to them S =6 ground state below 2.0 K. This complete polar- ization significantly reduces the dipolar decoherence. At 1.6 K, the spin decoherence time (T 2 ) of Mn 3 was measured to be 205 ns. Upon raising the temperature to 2.2 K, the T 2 decreased by nearly one order of magnitude. As temperature increases, so do magnetic fluctuations caused by the magnetic dipole interaction between SMMs, thus T 2 is reduced. Excellent agreement between the ob-served temperature dependence of T 2 and the dipolar decoherence model strongly supports that a major source of the spin decoherence is dipolar decoherence. In addition, we will show that there exist other decoherence sources which limit the maximumT 2 to be 260 ns. The Mn 3 SMM consists of three Mn III atoms each with spin S = 2. These are ferromagnetically coupled to each other to give the total spin of S = 6. A schematic of the molecular structure is shown in Fig. 5.5(a). The following Hamiltonian is sufficient to describe the magnetic properties of Mn 3 SMMs. H = B gSB 0 +DS 2 z +E S 2 x S 2 y : (5.1) The first term of the Hamiltonian (equation 5.1) is the electronic Zeeman interaction, where B is the Bohr magneton, g is the isotropic g-factor which equals assume to Chapter 5. HF EPR study of spin coherence in a Mn 3 single-molecule magnet 87 8 10 Magnetic field (T) Sim. Intensity (arb. units) Exp. T = 10 K 6 4 1:-6↔-5 2:-6↔-5 (a) (c) 1 mm (b) Figure 5.5: (a) A schematic of the structure of Mn 3 . (b) A photo of Sample A. (c) 230 GHz EPR cw spectrum of Mn 3 . Them S =6$5 transitions from each orientation are indicated by arrows (labeled 1 and 2). In the simulation, = 75 and = 0 for Group 1 and = 69 and = 74 for Group 2, where and are azimuthal and polar angles of the molecular axis, respectively (B 0 is along the z-axis). be 2.00, S is the spin operator, and B 0 is the applied magnetic field. The second and third terms represent zero-field interaction terms corresponding to axial and rhombic anisotropies, which areD = -10 GHz andE = 0.3 GHz, respectively. 100 Higher order terms in the spin Hamiltonian have been excluded in this study. It is important to note that having a large negativeD-value and a high spin quantum number (S = 6) leads to large zero-field splittings within Mn 3 (110 GHz for the m S =6 and5 states). In Chapter 5. HF EPR study of spin coherence in a Mn 3 single-molecule magnet 88 addition, it has been shown in previous studies that a single crystal of Mn 3 consists of two spin sub groups where the easy axes of each group are oriented with an angle of 70 degrees 41 . In the present study, two single crystals of Mn 3 SMMs were investigated (called here samples A and B) using a 230 GHz cw and pulsed EPR spectroscopies. The 230 GHz EPR spectrometer is based on a 100 mW solid-state source, quasi-optics, a su- perheterodyne detection system and a 12.1 T superconducting magnet. Details of this spectrometer can be found elsewhere. 47 Both crystals were slab-shaped (see Fig. 5.5(b) for Sample A) and placed on a conducting end plate in the sample holder where the orientation of the external magnetic field (B 0 ) is perpendicular to the crystal plane. Fig- ure 5.5(c) shows a 230 GHz cw EPR spectrum taken at 10 K. The applied microwave excitation power and the field modulation intensities were carefully tuned to prevent distortions in the EPR lineshape. As shown in Fig. 5.5(c), the simulated EPR spectrum using equation 5.1 agrees fairly well with the experimental spectrum. The two EPR signals at 9.6 T and 8.9 T (labeled as 1 and 2) originate from the m S =6$5 transition of the two different spin groups in the Mn 3 crystal. Next, we investigated spin decoherence in Mn 3 using 230 GHz pulsed EPR spec- troscopy. For the measurement, the magnetic field was first set to 9.6 T, which corre- sponds to them S =6$5 transition of Group 1, as shown in Fig. 5.5(c), then we applied the spin echo sequence (=2--- echo) where is the free evolution time of spins. As shown in Fig. 5.6(a), the echo signal was clearly observed at 2 = 0.8 s, which confirms the first successful observation of coherence in a Mn-based SMM. In addition, we measured the echo intensity as a function of 2 to determine the spin de- coherence time (T 2 ). As shown in Fig. 5.6(b), the observed decay of the echo intensity was well represented by a single exponential function. We therefore extracted the spin Chapter 5. HF EPR study of spin coherence in a Mn 3 single-molecule magnet 89 (a) (c) Echo Intensity (arb. units) Time (μs) 2τ = 0.8 μs T = 1.69 K Magnetic Field (T) 0.5 1.0 8.8 9.2 9.6 10.0 T = 1.74 ± 0.03 K (b) 0.4 0.8 1.2 1.6 2.0 2τ (μs) T 2 = 173 ± 8 ns T = 1.69 ± 0.03 K 1 2 0.0 0.0 Echo Intensity (arb. units) Echo Intensity (arb. units) 8.5 9.0 9.5 10.0 Magnetic Field (T) (d) Echo Intensity (arb. units) T = 1.58 ± 0.03 K Sample B Sample A Sample A Sample A Figure 5.6: Spin echo measurements of Mn 3 crystals. (a) The echo signal at 1.69 K. The optimal pulse widths in the spin echo sequence were found to be =2 = 60 ns and= 90 ns by adjusting the pulse length to maximize the echo intensity. Shown pulse widths are highly exaggerated due to the free induction decay (FID). This snapshot of the oscilloscope trace was taken at 2= 800 ns. The signal was averaged 16 times. (b) Spin echo decay at 1.69 K. Intensity of the echo peak plotted as a function of 2. Data were fitted by a single expo- nential andT 2 was extracted as 173 ns. (c) Echo-detected field sweep data from Sample A. Spin echo intensity at 2= 700 ns recorded as magnetic field sweeps. For this measurement=2 and pulses were 100 ns and 150 ns, respectively. m S =6$5 transitions are labeled by arrows. (d) Echo-detected field sweep data from Sample B. Spin echo intensity at 2= 500 ns recorded as mag- netic field sweeps. For this measurement=2 and pulses were 80 ns and 100 ns, respectively. Them S =6$5 transitions are labeled by arrows. decoherence time (T 2 ) to be 173 ns by fitting the decay to the single exponential func- tion (exp(2=T 2 )). Echo-detected field sweep measurements were also performed by Chapter 5. HF EPR study of spin coherence in a Mn 3 single-molecule magnet 90 measuring the echo intensity as a function of magnetic field to verify the EPR transi- tions. Figure 5.6(c) shows echo-detected field sweep measurements taken at 1.74 K. As shown in Fig. 5.6(c), two pronounced peaks were observed at 9.6 T and 8.9 T (labeled 1 and 2, respectively) for Sample A, which are consistent with the m S =6$5 transitions. Similarly, we observed two pronounced echo-detected field sweep signals from Sample B as shown in Fig. 5.6(d). Finally, in order to identify the major sources of spin decoherence in Mn 3 , we stud- iedT 2 as a function of temperature. As shown in Fig. 5.7(a),T 2 at 1.58 K was measured to be 205 6 ns for Sample A, then T 2 decreases rapidly as the temperature was in- creased. T 2 was extracted up to 2.2 K (T 2 = 59 30 ns). Above 2.2 K,T 2 became too short to measure due to the limitation in the time resolution of the EPR spectrometer. The temperature dependence of T 2 is summarized in Fig. 5.7(a). At low temperature (k B T << h Larmor where k B is the Boltzmann constant and T is temperature), the dipolar decoherence is caused by the interaction between the k = 0 magnon, which is excited by the microwave excitation, and a thermally excited magnon (k6= 0 magnon); therefore, the decoherence rate (1/T 2 ) is highly dependent on the population of the ther- mally excited magnon. At 230 GHz and such low temperatures, the Mn 3 polarization is above 99 % which almost eliminates the thermally excited magnon. Using the previ- ously reported model 91, 96 the temperature dependence of the decoherence rate (1/T 2 ) is given by, 1=T 2 =Aexp (h Larmor =k B T ) + ; (5.2) where the first term is the dipolar decoherence rate,A is a constant, and the second term is the residual decoherence rate, i.e., phonon and nuclear decoherence. As shown in Fig. 5.7(b), a fit of the temperature dependence to equation 5.2 agrees well and the analysis indicates that the dipolar decoherence is dominant in the higher temperature Chapter 5. HF EPR study of spin coherence in a Mn 3 single-molecule magnet 91 50 100 150 200 T 2 (ns) 1.6 1.8 2.0 2.2 Temperature (K) Sample A Sample B Sample A Sample B Residual Dipolar Total 1/T 2 (μs -1 ) Temperature (K) 5 10 (a) (b) 2.4 2.0 1.6 Figure 5.7: Temperature dependence of the spin decoherence for samples A and B. (a) Spin decoherence time (T 2 ) as a function of temperature. The mea- surements were performed at 9.6 T and 8.9 T for samples A and B, respectively. (b) The rate of spin decoherence as a function of temperature. The long-dashed line shows the contribution to spin decoherence from the residual decoherence sources, whereas the short-dashed line denotes the dipolar contribution. The solid line shows the total decoherence from these two contributions. regime. The obtained value ofA is 1785 149 (s 1 ). It is important to note that a similar strength of the dipolar decoherence was found in a single crystal of Fe 8 SMMs (A 3000 s 1 ) . 91 On the other hand, the decoherence is limited by the residual source at low temperatures. The amount of the residual decoherence determines the Chapter 5. HF EPR study of spin coherence in a Mn 3 single-molecule magnet 92 magnitude of the spin decoherence time by quenching the dipolar decoherence. As shown in Fig. 5.2(b), we obtained 267 36 ns for the residual decoherence time (1/). 5.4 Summary In summary, we have investigated spin coherence in Mn 3 SMMs. Using the HF EPR spectroscopy, we successfully revealed coherence in Mn 3 SMMs by suppressing the dipolar decoherence. In addition, temperature dependence ofT 2 showed that the domi- nant source of the decoherence is the dipolar decoherence and the decoherence time can be extended to 267 ns by quenching the dipolar decoherence. Chapter 6: ODMR at High Frequency and High Magnetic fields Materials presented in this chapter can also be found in the article titled High-frequency and high-field optically detected magnetic resonance of nitrogen-vacancy centers in di- amond by Viktor Stepanov, Franklin H. Cho, Chathuranga Abeywardana and Susumu Takahashi in Appl. Phys. Lett. 106 , 063111 (2015) (Reprinted with permission. Copy- right [2015], AIP Publishing LLC.) 6.1 Introduction In this chapter, I present the development of a HF ODMR system to investigate NV cen- ters in diamond. Similar to NMR spectroscopy, HF spectroscopy significantly increases the spectral resolution of ODMR and NV-based EPR techniques including DEER and ENDOR, thus highly advantageous in distinguishing target spins from other species (e.g., impurities existing in diamond) for NV-based MR spectroscopy. In addition, HF MR spectroscopy can produce extremely high spin polarization at low tempera- tures, which improves the signal-to-noise ratio of NV-based MR measurements on en- sembles of target electron spins and increases spin coherence of target spins signifi- cantly. 91, 92, 98, 104 The HF ODMR employs a confocal FL imaging system, a HF excita- tion component and a 12.1 T superconducting magnet. The HF excitation component is 93 Chapter 6. ODMR at High Frequency and High Magnetic fields 94 a part of the HF EPR spectrometer (see section 5.2) operating in the frequency range of 107-120 GHz and 215-240 GHz, 47 therefore the system is capable of performing in-situ HF ODMR and HF EPR experiments. Here, I also present HF ODMR measurements of a single NV center in diamond. First, a single NV center is identified by FL imaging, autocorrelation, and field-dependent FL measurements. Then, we perform continuous- wave (cw) ODMR at the external magnetic field of 4.2 T and microwave frequency of 115 GHz. We also demonstrate the coherent manipulation of a single NV center spin at 4.2 T by performing Rabi oscillations, pulsed ODMR, and spin echo measurements. 6.2 Experimental setup Figure 6.1 shows an overview of the HF ODMR system consisting of a NV detection system, a HF excitation component, a microscope system, and a sample stage. A cw 532 nm laser in the NV detection system is employed for optical excitation of NV centers. The cw excitation laser first couples to an acousto-optic modulator (AOM) for pulsed operations, then is transmitted through a single-mode optical fiber to the bottom of a 12.1 T superconducting magnet where the excitation laser couples to the microscope system. As shown in Fig. 1, the microscope system consists of a microscope objective (Zeiss) for both the optical excitation and collection of NV’s FL signals, a z-translation stage (Attocube), and a corrugated waveguide (Thomas-Keating) for the HF microwave excitation. The HF microwave excitation component is a part of our HF EPR spectrom- eter. 47 The output frequency of the HF microwave source is continuously tunable in the range of 107-120 GHz and 215-240 GHz. The sample stage is supported by the z- translation stage. The z-direction of the sample position is adjusted by the z-translation stage, and the xy-direction of the laser excitation volume is controlled by a combination of a steering mirror and a pair of lenses. The HF ODMR employs a confocal microscope Chapter 6. ODMR at High Frequency and High Magnetic fields 95 HF Source 107-120GHz Quasioptics Corrugated waveguide 12.1 T magnet Lenses Objective lens Steering Mirror Sample Laser AOM APD1 NV Detection Laser FL Microscope System HF Excitation Peizo (Z) Diamond Optical Fibers FL Image 5 66 25 45 Intensity (counts/ms) 2μm APD2 BS Figure 6.1: Overview of the HF ODMR system. The HF source in the HF ex- citation component is tunable continuously in the range of 107-120 GHz and 215-240 GHz. HF microwaves are guided by quasioptics and a corrugated waveguide. The NV detection system consists of a 532 nm cw diode-pumped solid state laser, an AOM, fiber couplers, optical filters, a beam splitter (BS), and APDs. The excitation laser is applied to NV centers through a microscope objective located at the center of the 12.1 T superconducting magnet and the FL signals of NV centers are collected by the same objective. The FL signals are fil- tered by optical filters in the NV detection system. For autocorrelation measure- ments, the FL signals are split into two and detected by two separate APDs. The microscope system consists of a microscope objective, a z-translation stage, and the corrugated waveguide. The sample stage is supported by the z-translation stage. Lower left panel shows a FL image taken on a type 1bh111i diamond and the NV used in rest of the section was circled. Chapter 6. ODMR at High Frequency and High Magnetic fields 96 system to detect FL signals of NV centers. The FL signals are collected by the same ob- jective, then are transmitted to the detection system by a 50m multi-mode optical fiber cable that enables confocal FL imaging. Finally the FL signals are filtered by optical filters and detected by avalanche photodiodes (APDs) in the NV detection system. (a) (c) NV (b) 50 100 X (µm) Y (µm) 5 0 5 0 0 1 2 3 g 2 (τ) Delay time τ (ns) 0 2 4 6 8 10 12 16 20 FL intensity (arb. units) Magnetic field (T) FL changes at level anti-crossing 0.05 0.10 0.15 NV 20 0 10 FL intensity (counts/ms) Single quantum emitter 0 0.1 0.2 0.3 30 40 50 {100} Diamond (d) Magnetic field (T) FL intensity (arb. units) Figure 6.2: (a) FL intensity image of a type-Ib diamond crystal. The scanning area is 5 5m 2 . Solid circle indicates a single NV center that was used in the subsequent measurements. (b) Autocorrelation curve observed from the the single NV center. The observation ofg 2 ( = 0) < 0:5 confirms the detection of the single NV center. (c) Magnetic field dependence of the single NV center FL intensity. The field was applied along theh111i axis of the diamond within 8 degrees. The inset shows reduction of FL intensity at0.05 and 0.1 T due to LAC of the excited and ground states of the NV center, respectively. (d). Mag- netic field dependence of the FL intensity taken with a NV inf100g diamond. Angle between magnetic field and NV axis is about 54.7 degrees. Figure 6.3 shows the results of simulated population of NV spin states using the ground state HamiltonianH =g B BS +D(S 2 z S(S + 1)=3); where B is the Bohr magneton,g is the electron g-factor,S is the total spin (S = 1),B is the magnetic field and D is the ground state ZFS (2.87 GHz). General form of wave function of state i (i = 1,2,3) for an any angle can be written asj i i = a i j 1i +b i j0i +c i j + 1i. As Chapter 6. ODMR at High Frequency and High Magnetic fields 97 an example when there is no misalignment wavefunction of m s = 0 can be written asj ms=0 i = a 0 j 1i +b 0 j0i +c 0 j + 1i wherea 0 andc 0 become zero. Populations were calculated after obtaining eigenvectors. Figure 6.3 (a) shows there is a population 0.6 0.8 1.0 Population (arb. units) 0 0.2 0.4 0 0.1 0.2 0.3 Magnetic field (T) 0.4 0.5 1 0 8 0 54.7 0 0.5 1.0 Contrast (arb. units) 0 0 20 40 Angle (degrees) 60 80 (a) (b) |−1〉 |0〉 |+1〉 0.6 0.8 1.0 Contrast (arb. units) 0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 Contrast (arb. units) 0 2 4 6 Magnetic field (T) 8 10 0 2 4 6 8 10 Magnetic field (T) B 0 = 4 T Figure 6.3: Angle and magnetic field dependence of the NV populations of m s states. (a). Population of each state as a function of magnetic field. Solid line shows the simulation results when NV axis is 8 degrees misaligned from magnetic field whereas dashed lines represents when the angle is 54.7 degrees. Inset shows the zoomed data from 0 T to 0.5 T. (b). Magnetic field dependence on the contrast of populations calculated by taking the difference betweenj0i andj 1i populations for three different angels. Inset shows the contrast as a function of misalignment angle calculated at 4 T. mixing due to the level anti crossing (LAC) closer to 0.1 T when NV axis has a small misalignment (8 degrees) to the magnetic field. But not for a larger angle (54.7 degrees in the case of diamond withf100g faces). Therefore, FL dependence of magnetic field data shown in Fig. 6.2(a) and (b) agrees with the simulation. Figure 6.3(b) shows the calculated contrast based on population differences which determines the contrast of ODMR signals. This results reveals how important the alignment of NV with respect to the magnetic field as it greatly affects the ODMR contrast. As shown in Fig. 6.3(b) inset, at 4 T magnetic field, the contrast with the misalignment angle of 54.5 becomes Chapter 6. ODMR at High Frequency and High Magnetic fields 98 very small, thus it is very challenging perform ODMR with NVs in a diamond crystal withf100g faces. 6.3 Results and Discussion Using the HF ODMR system, we performed FL measurements on a single crystal of type-Ib diamond (1.5 1.5 1.1 mm 3 , Sumitomo Electric Industries). First, as shown in Fig. 6.2(a), FL imaging was carried out to map FL signals from the diamond crys- tal. After choosing a well-isolated FL peak (Fig. 6.2(a)), we performed anti-bunching measurement and observed the autocorrelation signal which verifies that the FL signal was due to a single quantum emitter (Fig. 6.2(b)). Then, the FL intensity was moni- tored continuously as we applied the external magnetic field from 0 to 10 T. As shown in the inset of Fig. 6.2(c), we observed two dips of the FL intensity at0.05 and 0.1 T, originating from the level anti-crossing (LAC) in NV’s optically excited and ground states, respectively. 105 Therefore the observation of the LAC and autocorrelation signal confirmed the detection of a single NV center. In addition, as shown in Fig. 6.2(c), we found that the FL intensity is stable in high magnetic fields up to 10 T. FL dependence on magnetic field show in Fig. 6.2(d) was taken from an NV which is aligned along one of the [ 111]; [1 11]; [11 1] crystal axis. Next, we demonstrated ODMR measurements of the single NV center at the mi- crowave frequency of 115 GHz. First, we performed cw ODMR spectroscopy of the single NV center with applications of cw microwave and laser excitations. As shown in Fig. 6.4(a), the observed FL signals as a function of magnetic field showed a dip of the FL intensity at 4.2022 T, which corresponds to the m S = 0 and m S =1 transition of the NV center. Theg-factor of the NV center was estimated to be 2.0027-2.0041 by taking into account for the strength of magnetic field at the resonance and uncertainty Chapter 6. ODMR at High Frequency and High Magnetic fields 99 Magnetic field (T) (c) Pulse length t P (ns) 2000 4000 Exc. MW Init. RO Sig. FL t P f Rabi = 0.8 MHz Exp. Fit FL intensity (arb. units) 4.201 4.202 4.203 Exc. MW Init. RO Sig. FL t MW Exp. Fit FL intensity (arb. units) FL intensity (arb. units) Magnetic field (T) (a) 200 400 600 FL intensity (arb. units) 2τ (ns) (d) Exc. MW Init. RO Sig. FL π/2 π/2 π τ τ Exp. Fit T 2 = 325±20 ns 4.201 4.202 4.203 Exp. Fit 4.11 4.12 Intensity ↔ 0 -1 ↔ 0 -1 ↔ 0 -1 ↔ 0 -1 (b) Figure 6.4: (a) cw ODMR measurement of the single NV center at 115 GHz. The ODMR signal of the single NV center was observed at 4.2022 T. The solid line indicates a fit to the Gaussian function. The inset shows the in-situ EPR measurement of single-substitutional nitrogen impurities in the diamond. (b) Rabi oscillation experiment. The frequency of the observed Rabi oscillations was 0.8 MHz. The inset shows the applied pulse sequence consisting of the initialization (Init.) and readout (RO) pulses by the 532 nm laser (Exc.), mi- crowave pulse (MW) of lengtht P , and FL signals (Sig.). Init. = 4s and RO = Sig. = 300 ns were used in the measurement.t P was varied. (c) Pulsed ODMR as a function of magnetic field. The solid line indicates a fit to the Gaussian function. The inset shows the pulse sequence. Init. = 4s, RO = Sig. = 300 ns, andt MW = 500 ns were used in the measurement. (d) The spin echo mea- surement to determine the spin decoherence time (T 2 ) of the single NV center. The solid line indicates a fit to exp((2=T 2 ) 3 ). The inset shows the pulse se- quence. Init. = 4s, RO = Sig. = 300 ns,=2 = 250 ns, and = 600 ns were used in the measurement. was varied. The pulse sequence was repeated on the order of 10 6 times to obtain a single point in all measurements. Chapter 6. ODMR at High Frequency and High Magnetic fields 100 of the orientation of magnetic field (< 8 degrees). The calibration of magnetic field was done by in-situ ensemble EPR measurement of single-substitutional nitrogen impurities (see the inset of Fig. 6.4(a)). 98 Then, we carried out pulsed experiments at 115 GHz and 4.2022 T. In the pulsed experiments (section 2.4), the NV center was first prepared in them S = 0 spin sublevel with the application of the laser initialization pulse, then the microwave excitation pulse sequence was applied. The final state of the NV center was determined by measuring its FL intensity with the application of the laser readout pulse. In all pulsed measurements, the FL intensity was normalized by the reference signal, which is the FL intensity measured without the microwave pulse sequence in or- der to cancel noises associated with laser intensity fluctuations and thermal/mechanical instability of the setup. In addition, each data point was obtained by repeating the pulse sequence and averaging FL signals on the order of 10 6 times. As shown in Fig. 6.4(c), Rabi oscillations of the same single NV center were observed by varying the duration of the single microwave pulse (t P ). Figure 6.4(b) shows pulsed ODMR signals of the single NV center as a function of magnetic field. The observed full-width at half-maximum was 0.29 mT (8 MHz) which is typical for NV centers in type-Ib diamond crystals. 50 Next, we measured the spin decoherence time (T 2 ) of the single NV center at 4.2022 T. As shown in the inset of Fig. 6.4(d), the applied microwave sequence consists of the spin echo sequence and an additional=2 pulse that converts the resultant coherence of the NV center into them S = 0 state. By fitting the observed FL decay to exp((2=T 2 ) 3 ), T 2 of the single NV center was determined as 325 20 ns (see Fig. 6.4(d)). 6.4 Summary In summary, we presented the development of the HF ODMR system, which enables us to perform the ODMR measurements of a single defect in solids at high frequencies Chapter 6. ODMR at High Frequency and High Magnetic fields 101 and high magnetic fields. Using the HF ODMR system, we demonstrated cw ODMR, Rabi oscillation, pulsed ODMR, and spin echo measurements of a single NV center in a type-Ib diamond crystal at the microwave frequency of 115 GHz and magnetic field of 4.2 T. Chapter 7: Conclusion In conclusion, this dessertation discussed about the development of NV based EPR spec- troscopy to detect a few spins inside and outside of diamond. The development was done for experiments at low and high magnetic fields. In Chapter 2, I described the experimental setup. I also explained the ODMR tech- niques which are routinely performed for NV-based EPR. The principle of the technique and the example results are also shown in the explanation. In Chapter 3, I presented nanoscale EPR spectroscopy of substitutional Nitrogen im- purities in bulk diamond using a single NV center. Identification of nanoscale spin baths surrounding a single NV center and the investigation of static and dynamic properties of the bath spins using Rabi, FID, SE measurements as well as NV-based EPR spec- troscopy were presented. With the investigation of several other single NV centers in the diamond sample I showed that the properties of the bath spins are unique to the NV centers. Finally, by analyzing the intensity of the NV-based EPR signal using the com- puter simulation, we estimated the detected spins in the DEER measurement to be 50 spins. Thus, the investigation demonstrated ability of NV-based EPR to quantitatively detect bath spins when it is more than few spins. Fabrications of NV centers closer to the diamond surface using implantation and annealing techniques was presented in the Chapter 4. These fabricated NVs exhibit 102 Chapter 7. Conclusion 103 favorable optical and spin properties to employ them as a spin sensor to detect paramag- netic spins located on the diamond surface. Using these diamond samples, a successful detection of covalently attached nitroxide spin label was obtained with NV-based EPR spectroscopy. In Chapter 5, I presented successfully observation of coherence in Mn 3 SMMs by suppressing the dipolar decoherence using the HF EPR spectroscopy. Even though, Manganese based SMMs are the largest family of SMMs this is the first ever coherence observation from any Manganese SMM reported to date. Furthermore, temperature dependence of T 2 of Mn 3 showed that the dominant source of the decoherence is the dipolar decoherence and the decoherence time can be extended to 267 ns by quenching the dipolar decoherence. Chapter 6 presented the performance of ODMR measurements of a single defect in solids at high frequencies and high magnetic fields. Using the HF ODMR system, we demonstrated cw ODMR, Rabi oscillation, pulsed ODMR, and spin echo measurements of a single NV center in a type-Ib diamond crystal at the microwave frequency of 115 GHz and magnetic field of 4.2 T. Appendix A: Pulse measurement normalization 6000 5000 4000 Counts (arb. units) 0 100 Pulse width (ns) 200 300 400 500 600 700 max i signal i min i P(m s =0) = 1 P(m s =0) = 0 Norm. i Normalization Scheme max signal min max fit min fit Figure A.1: Raw data of the rabi measurement shown in Fig. 2.9(a). Max (population ofm s = 0 is 1) reference is the FL readout followed by initializa- tion and Min reference (population of m s = 0 is 0 or m s =1 is 1) is the FL readout after applying a resonant MW pulse. Signal represents the rabi measurement where width of the resonant MW pulse is being changed. norm i = signal i min i max i min i : (A.1) 104 Appendix A. Pulse measurement normalization 105 Scheme given in Fig. A.1 with equation A.1 was used to normalize the pulse mea- surements data. As shown by the dashed lines in Fig. A.1, max and min references were fitted with a linear fit functions separately.max i andmin i represents the values of max and min extracted from the fit for i th data point, respectively. signal i denotes the i th data point of the actual signal. APD output was internally rerouted to three counters in the PCI DAQ card and trig- gered separately as which reference or the signal to record (refer Fig. 2.7). During DEER measurements additional fourth counter was used to record the intensity of SE (or CPMG) amplitude. Then, SE (or CPMG) amplitude is also normalized using max and min references as explained above. This will map NV-based EPR and SE (or CPMG) amplitude into population ofm S = 0. Appendix B: Theory of ESEEM Explicit analytical expression for electron spin echo envelope modulation (ESEEM) first introduced by L.G. Rowan in 1965. 36 Later on W. B. Mims introduced a generalized approach to obtain the analytical expression. 37 Here I explained the derivation of an- alytical expression using density matrix formalism adapted from above two references for S =1/2 and I = 1/2 system. Evolution of a density matrix ^ subjected to a HamiltonianH which governs the successive intervals of nutations and free precessions is described by, @ ^ @t = i } [^ ;H] (B.1) To calculate electron spin echo amplitude, above equation has to be integrate to find density matrix at the echo observation time ( ^ e ) starting from initial density matrix ( ^ 0 ) before pulses. Equation B.1 readily integrated for time independent operator (H =H 0 ). So density matrix ^ e at the end of time periodt f can be expressed as follows. ^ e (t f ) = ^ R t ^ (t i ) ^ R 1 t (B.2) where ^ e (t i ) is the density matrix at the beginning of intervalt i . Operator ^ R t can be written as, 106 Appendix B. Theory of ESEEM 107 ^ R t = exp iH 0 (t f t i ) } (B.3) ^ e can be calculated by partitioning the pulse sequence in to nutation periods and free precession intervals during pulses. In spin echo measurement during free precession interval, exponential operator ^ R can be written as, ^ R = exp iH 0 } (B.4) The Hamiltonian acting on the system during the MW pulses is a sum of time depen- dent HamiltonianH 1 and time independent spin HamiltonianH 0 .H 1 takes following form in the rotating frame where time dependence is removed, H 1 = }! 1 ^ S x (B.5) where! 1 is the flipping frequency as defined in the section 1.3.3. We assume the application of pulses are instantaneous. Therefore during application of pulse evolution of the system underH 0 can be neglected for simplicity. Therefore exponential operator during pulses can be written as, ^ R P = exp iH 1 t P } (B.6) wheret P is the pulse width (section 1.3.3). Using equations B.5 and B.6 consider- ing exponential expansions ^ R P can be simplified as follows, 37 ^ R P = exp(i! 1 t P ^ S x ) = cos ! 1 t P 2 2i ^ S x sin ! 1 t P 2 (B.7) Now density matrix at the time of echo observation can be written as, Appendix B. Theory of ESEEM 108 ^ e = ^ R ^ R t PII ^ R ^ R t PI ^ 0 ^ R 1 t PI ^ R 1 ^ R 1 t PII ^ R 1 (B.8) where ^ 0 is the equilibrium density matrix, ^ R t PI and ^ R t PII are the nutation operators corresponding to=2 and pulses respectively. Amplitude of spin echo is then proportional to component of the magnetization along y axis and it can be expressed as, 36 E() = Tr(^ e ^ S y ) (B.9) Figure B.1 contains the basis functions and matrix representation of spin operators used in the following derivation. S x = 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 1 2 |αα> = 1 0 0 0 S y = 0 0 -1 0 0 0 0 -1 1 0 0 0 0 1 0 0 i 2 S z = 1 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 -1 1 2 I x = 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 1 2 I y = 0 -1 0 0 1 0 0 0 0 0 0 -1 0 0 1 0 i 2 I z = 1 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 -1 1 2 |αβ> = 0 1 0 0 |βα> = 0 0 1 0 |ββ> = 0 0 0 1 |m s m i > basis spin operators Figure B.1: Basis and Spin matrix operators used in ESEEM derivation Hamiltonian acting on the system during free precession can be written as follows with respect to lab frame, 36 Appendix B. Theory of ESEEM 109 H L 0 = }! e ^ S z +}! N ^ I z +A ^ S z ^ I z +B ^ S z ^ I x (B.10) where! e and! N represents the Larmor frequency of electron and nucleus respec- tively. A = A zz andB = p A 2 zx +A 2 zy relate to secular and pseudo secular hyperfine coupling. ? Above Hamiltonian can be transformed to rotating frame using the unitary operator ^ U R = exp(i}! e ^ S z t) and Hamiltonian in rotating frame can be written as follows (resonance conditioned was assumed) H R 0 = }! N ^ I z +A ^ S z ^ I z +B ^ S z ^ I x (B.11) Matrix for operator giving in equation B.4 can be simply evaluate whenH 0 is diag- onal. Therefore, Hamiltonian given in equation B.11 has to be diagonalize. This can be done dividing 4x4 matrix in to four 2x2 matrices and then diagonalize 2x2 matrices lie along the diagonal. Or this can be done using following unitary transformation. ? ^ U D = 2 6 6 6 6 6 6 6 6 6 6 4 cos( =2) sin( =2) 0 0 sin( =2) cos( =2) 0 0 0 0 cos( =2) sin( =2) 0 0 sin( =2) cos( =2) 3 7 7 7 7 7 7 7 7 7 7 5 (B.12) where tan( ) =B=(A + 2}! N ) and tan( ) =B=(A 2}! N ). Figure B.2 shows graphical relations between and . Using ^ U D Hamiltonian in equation B.11 can be diagonalized as follows Appendix B. Theory of ESEEM 110 -B A 2ħω N η α η β η β − η α √B 2 + (A-2ħω N ) 2 = λ β √B 2 + (A+2ħω N ) 2 = λ α H H = -4Bħω N √B 2 + (A+2ħω N ) 2 sin(η β − η α ) = -Bħω N λ α λ β cos(η β − η α ) = B 2 + A 2 - 4(ħω N ) 2 4λ α λ β Figure B.2: Diagonalization unitary operator angle relations H D 0 = ^ U D H R 0 ^ U y D H D 0 = 2 6 6 6 6 6 6 6 6 6 6 4 1 = =2 0 0 0 0 2 = =2 0 0 0 0 3 = =2 0 0 0 0 4 = =2 3 7 7 7 7 7 7 7 7 7 7 5 (B.13) where 1 to 4 denotes the four eigenvalues. = p (A=2 +}! N ) 2 +B 2 =4 and = p (A=2}! N ) 2 +B 2 =4. Same transformation should performed on theH 1 to find the transformedH T 1 . In order to findH T 1 it is only required to transform ^ S x as follows Appendix B. Theory of ESEEM 111 ^ S T x = ^ U D ^ S x ^ U y D ^ S T x = 1 2 2 6 6 6 6 6 6 6 6 6 6 4 0 0 cos 2 sin 2 0 0 sin 2 cos 2 cos 2 sin 2 0 0 sin 2 cos 2 0 0 3 7 7 7 7 7 7 7 7 7 7 5 (B.14) Similarly, ^ S T y = ^ U D ^ S y ^ U y D ^ S T y = i 2 2 6 6 6 6 6 6 6 6 6 6 4 0 0 cos 2 sin 2 0 0 sin 2 cos 2 cos 2 sin 2 0 0 sin 2 cos 2 0 0 3 7 7 7 7 7 7 7 7 7 7 5 (B.15) Substituting equation B.14 and B.15 in to equation B.7 we can construct the matrix operator for ^ R t PI and ^ R t PII . Appendix B. Theory of ESEEM 112 ^ R PI = exp(i 2 ^ S T x ) = cos 4 2i ^ S T x sin 4 ^ R PI = 1 p 2 2 6 6 6 6 6 6 6 6 6 6 4 1 0 2i cos 2 2i sin 2 0 1 2i sin 2 2i cos 2 2i cos 2 2i sin 2 1 0 2i sin 2 2i cos 2 0 1 3 7 7 7 7 7 7 7 7 7 7 5 ^ R PII = exp(i ^ S T x ) = cos 2 2i ^ S T x sin 2 ^ R PII =2i 2 6 6 6 6 6 6 6 6 6 6 4 0 0 cos 2 sin 2 0 0 sin 2 cos 2 cos 2 sin 2 0 0 sin 2 cos 2 0 0 3 7 7 7 7 7 7 7 7 7 7 5 (B.16) Matrix representation of equation B.4 can be expressed as, ^ R = 1 2 2 6 6 6 6 6 6 6 6 6 6 4 exp(i 1 ) 0 0 0 0 exp(i 2 ) 0 0 0 0 exp(i 3 ) 0 0 0 0 exp(i 4 ) 3 7 7 7 7 7 7 7 7 7 7 5 (B.17) The density matrix at thermal equilibrium ^ 0 can be represented in high temperature approximation as, ? ^ 0 = ^ S z (B.18) Appendix B. Theory of ESEEM 113 Now we can find ^ e which is given in equation B.8 by using equations B.16 , B.17 and B.18. Then ^ e and equation B.15 can be substitute in to equation B.9 to find the amplitude of SE signal. Final answer can be expressed as E() = 1 sin 2 ( ) 2 1 cos( ) cos( ) + 1 2 cos(( + )) + 1 2 cos(( )) (B.19) Using trigonometric relations equation B.19 can be further simplified to more com- pact form as used in section 2.4.6. E() = 1 2 sin 2 ( ) sin 2 2 sin 2 2 (B.20) sin ( ) can be find in Fig. B.2. Above equation was used in the section 2.4.6 asS i (2) wherei represents the labels for 13 C and 15 N. Appendix C: FID and SE decay functions This section summarizes the analytical expressions of coherence decay envelopes for FID and SE considering the process of spectral diffusion (SD). SD referes to the fluc- tuation of qubit Zeeman frequency (ZF) through the spin resonance line. ZF of the qubit is B eff and B eff includes the local magnetic fields arising from the environ- ment. B eff = B z = A z I z +a k S z k , whereA z is the hyperfine component coming from Nitrogen nuclear spin of NV ,I z is the nuclear spin value,S z k is the electron spin value of k th bath spin anda k relates to coupling strength of NV andk th bath spin. This is defined asa k = (g nv g p1 2 B =r 3 k )(1 3(n z k ) 2 ) since it has a dipolar origin. 35 General form of coherence envelop can be expressed as 106 E(t) = exp Z 1 1 d!S(!)F (t;!) (C.1) whereS(!) is the noise spectrum which can be defined asS(t) = R e i!t S(!)d! and F (t;!) is the filter function which depends on the relevant pulse sequence. Equation C.2 includes the filter functions for FID and SE (Hahn echo) 106 . 114 Appendix C. FID and SE decay functions 115 F FID (;!) = 1 2 sin 2 (!=2) (!=2) 2 F SE (2;!) = 1 2 sin 4 (!=2) (!=4) 2 (C.2) Electron spin bath which governs the coherence decay has been explained by treating the bath as a classical noise field. The noise fieldB(t) was modeled by O-U (Ornstein- Uhlenbeck) process which is stationary, GaussMarkov process, with correlation function S(t), 34 S(t) =hB(0)B(t)i =b 2 exp(jtj= c ) (C.3) where c is the correlation time of the bath spins which measures the flip-flop rate be- tween bath spins, also defined as the memory of the environmental noise. Mean of the distribution isA z I z (forI z = 1 it is +A z ;A z and 0 with equal probability). b 2 is the variance that can be expressed asb 2 = (1=4) P k a 2 k . Fourier transforming the equation C.3, the resulting noise spectrum can be expressed as S(!) = b 2 c 1 (! c ) 2 + 1 (C.4) Lets consider the FID decay first. Substituting equation C.2 and C.4 in to equa- tion C.1, content of the exponential term can be written as, I = Z 1 1 b 2 c 2 1 (! c ) 2 + 1 sin 2 (!t=2) (!=2) 2 d! (C.5) Appendix C. FID and SE decay functions 116 Denominator of the above equation can be factorized as follows 1 ! 2 [(! c ) 2 + 1] = A! +B ! 2 + C! +D (! c ) 2 + 1 A = 0;B = 1;C = 0;D = 2 c ; 1 ! 2 [(! c ) 2 + 1] = 1 ! 2 2 c (! c ) 2 + 1 (C.6) Therefore equation C.5 can be rewrite as I = b 2 c 2 Z 1 1 sin 2 (!t=2) (!=2) 2 d! 4 2 c Z 1 1 sin 2 (!t=2) (! c ) 2 + 1 (C.7) Equation C.7 can be divide in to two components as follows I 1 = Z 1 1 sin 2 (!t=2) (!=2) 2 d! using the substitutiony =!t=2 = 2t Z 1 1 sin 2 y y 2 dy = 2t (C.8) I 2 = Z 1 1 sin 2 (!t=2) (! c ) 2 + 1 = Z 1 1 e i!t=2 e i!t=2 2i 2 1 (! c ) 2 + 1 d! = 1 4 Z 1 1 e i!t (! c ) 2 + 1 d! 2 Z 1 1 d! (! c ) 2 + 1 + Z 1 1 e i!t (! c ) 2 + 1 d! (C.9) Equation C.9 can be solve easily by dividing it in to three parts Appendix C. FID and SE decay functions 117 I 21 = Z 1 1 e i!t (! c ) 2 + 1 d! !!! = Z 1 1 e i!t (! c ) 2 + 1 d! = Z 1 1 e i!t (! c ) 2 + 1 d! =I 23 I 22 = Z 1 1 d! (! c ) 2 + 1 = 1 c tan 1 (! c ) 1 1 == c I 21 +I 23 = 2 Z 1 1 e i!t (! c ) 2 + 1 = 2 b 2 c Z 1 1 S(!)e i!t d! = 2 c e t=c (C.10) Hence solution ofI 2 (equation C.9) can be written as I 2 = 1 4 2 c e t=c 2 c (C.11) Combining equations C.8 and C.11 solution ofI (equation C.7) can be written as I = b 2 c 2 2t + 2 c 2 c e t=c 2 c I =b 2 c t + c e t=c 1 (C.12) Finally, decay of FID can be written as Appendix C. FID and SE decay functions 118 E FID (t) = expfIg = exp b 2 2 c t= C + e t=c 1 (C.13) Following the similar derivation decay envelope of SE can be obtained. Only differ- ence is equation C.9 will contain 4 th power ofsin instead of a square. So decay of SE can be written as E SE (2) = exp b 2 2 c 2= C 3 + 4e =c e 2=c (C.14) Modulation in FID signal is coming from the static component of the B z which is govern by the hyperfine interaction to Nitrogen nuclear. Assuming instantaneous =2 pulse, acquired phase can be written as = R t 0 B z dt 0 = I z R t 0 A z dt 0 = I z A z t. Population ofm s = 0 can be then express as 35 P = +1 X Iz=1 P Iz cos(A z I z t) = 1 3 cos [1 + 2 cos(A z t)] (C.15) whereP Iz is the polarization of the nuclear spin which can be assumed as equally populated in the magnetic fields we usually apply. 107 For 15 N equation C.15 becomes P = cos [2 cos(A z t)]. Appendix D: Characterization of NV centers in various diamonds D.0.1 CVD-grown diamond A CVD-grown diamond contains a layer with a high N concentration. The thickness of the layer is 90 nm. The CVD-grown diamond was fabricated through collaboration with Prof. K. Itoh (Keio University) and Dr. H. Watanabe (AIST). 119 Appendix D. Characterization of NV centers in various diamonds 120 Magnification = 10X 40X E6, EG (100) N doped CVD (~90nm) Figure D.1: Optical microscope image of CVD grown surface taken with 10 times magnification. Top right panel shows the schematic of the diamond. Ni- trogen doped CVD layer of 90 nm thickness was grown on element six EG diamond substrate. Nitrogen/Carbon ratio was about 6%. Triangular shape structures are on the substrate surface. These are formed during polishing pro- cess of the substrate prior to CVD growth. Lower panel shows zoomed out with 40X magnification. Small black dots are not belong to diamond and coming from some impurities in the microscope eye piece. Appendix D. Characterization of NV centers in various diamonds 121 114 0 Counts / ms 2μm Figure D.2: FL image of CVD grown layer. OD 0.6 was used. The FL spot circled was used to collect data presented in rest of this section. NVs were always appeared as groups. No single NVs observed from total area of 0.148 mm 2 . FL spots identified as NV (not single) through ODMR measurement has the density of 1 FL spot per 85m 2 . For implanted diamonds with with 10 10 per cm 2 dose, this number was 1 FL spot that gave NV ODMR per 0.6m 2 . Appendix D. Characterization of NV centers in various diamonds 122 ODMR Contrast MW Frequency (GHz) 1.00 0.95 0.90 2.75 2.80 2.85 2.90 2.95 3.00 ~2 mT Orientation 1 Orientation 2 Figure D.3: CW ODMR of the selected FL spot at2 mT applied alongh111i crystal axis. Four peaks represents two transitions per orientation. Orientation 1 is the NVs align with [111] crystal axis and orientation 2 is the NVs align with other three crystal axes. This obviously confirms the FL spot contains multiple NVs. Appendix D. Characterization of NV centers in various diamonds 123 ODMR Contrast 1.00 0.90 MW Frequency (MHz) 2628 2631 2634 2637 15Δ Figure D.4: Pulsed ODMR of the selected FL spot at8.5 mT for orientation 1,jm s = 0>$jm s =1> transition. Separation between peaks agrees with the 15 N hyperfine splitting. Appendix D. Characterization of NV centers in various diamonds 124 P(m s =0) 1.00 0.75 0.50 0.25 2τ(μs) 0 50 100 0.8 0.7 0.6 0.5 P(m S =0) 0.6 0.7 0.8 MW2 Frequency (GHz) SE amp. no π MW2 SE amp. with π MW2 τ = 18μs 15 N A ⊥ = 113.8 MHz A || = 159.7 MHz (a) (b) Figure D.5: (a). SE measurement of the selected NV performed at 24.43 mT. =2 and pulses used in SE sequence was 32 ns and 64 ns, respectively. Output power of MW source 1 was 6 dBm. (b). DEER measurement of the selected NV . was 18s and MW2 was 200 ns. Output power of MW source 2 was -3 dBm. Peak positions of the observed spectrum agrees with the 15 N P1 centers. Appendix D. Characterization of NV centers in various diamonds 125 D.0.2 Summary ofT 2 of implanted NVs Figure D.6 shows the statistical summary of T 2 measurements taken with NV centers implanted in different depths. The total number of NVs measured is mentioned in each figure. Figure D.6 (a) and (b) shows the data collected with two diamond crystals im- planted with 3 keV and 10 keV implantation energies with the fluence of 10 10 per cm 2 , respectively. The significant difference in theseT 2 distributions is most probably due to the impurity spins located on the diamond surface. Figure D.6 (c) shows the summary ofT 2 measured with a diamond crystal implanted with 5 keV implantation energy and fluence of 10 9 per cm 2 . This crystals was measured again after attaching nitroxide spin labels and results are summarized in Fig. D.6 (d). 1 0 3 2 4 T 2 (μs) 1 2 Number of NVs 3 keV 20 60 40 80 T 2 (μs) 100 120 0 1 2 Number of NVs 3 10 keV 0 1 2 Number of NVs 3 0 20 10 30 T 2 (μs) 40 50 5 keV 5 keV Nitroxide attached 12 NVs 15 NVs 10 NVs 25 NVs 4 12 8 16 T 2 (μs) 20 0 1 2 Number of NVs 3 4 5 (a) (b) (c) (d) Figure D.6: Statistical summary ofT 2 measured from implanted NVs at dif- ferent depths. Bibliography [1] Rabi, I. I.; Zacharias, J. R.; Millman, S.; Kusch, P. 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Asset Metadata

Creator Abeywardana, Chathuranga (author)

Core Title Development of nanoscale electron paramagnetic resonance using a single nitrogen-vacancy center in diamond

Contributor Electronically uploaded by the author (provenance)

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Chemistry

Publication Date 11/03/2017

Defense Date 10/16/2017

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag EPR,magnetic resonance,nitrogen vacancy (NV),OAI-PMH Harvest,optically detected magnetic resonance (ODMR),single-molecule magnet (SMM)

Language English

Advisor Takahashi, Susumu (committee chair)

Creator Email cabeywar@usc.edu,gaminiabey2@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-449742

Unique identifier UC11263226

Identifier etd-Abeywardan-5869.pdf (filename),usctheses-c40-449742 (legacy record id)

Legacy Identifier etd-Abeywardan-5869.pdf

Dmrecord 449742

Document Type Dissertation

Rights Abeywardana, Chathuranga

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA

Abstract (if available)

Abstract Magnetic resonance techniques, such as nuclear magnetic resonance (NMR) and electron paramagnetic resonance (EPR), are powerful and versatile analytical methods used in science. These techniques can be used to probe the local structure and dynamic properties of various systems including biological molecules. However, one draw back of the MR techniques is their intrinsically low sensitivity. This will limit its applications to analyze samples with very small volumes. For instance, more than 10¹⁰ electron spins are typically required to observe EPR signals at room temperature. An instrument capable of single spin EPR and NMR will open up unprecedented opportunities. For instance, it will enable determination of atomic structure of each individual biomolecule, which will transform structural biology. ❧ A nitrogen-vacancy (NV) center consisting of a substitutional nitrogen next to a vacancy is a paramagnetic color center in diamond. Because of unique optical and spin properties of the NV center including stable fluorescence, long spin coherence, optical readout of the spin state and ability to initialize the spin state optically, an NV is a promising candidate for fundamental studies in quantum information processing and nanoscale sensing. ❧ This thesis discusses the development of nanoscale EPR spectroscopy using a single NV center in diamond. In chapter 1, I introduce basic concepts of MR. Physical, optical and spin properties of NV centers in diamond are introduced in Chapter 2. Moreover, Chapter 2 describes optically detected magnetic resonance spectroscopy to identify a single NV center and to measure the spin coherent state of the NV center as well as double electron-electron resonance spectroscopy to perform EPR using the NV center (NV-based EPR). In chapter 3 and 4, I discuss the demonstrations of NV-based EPR spectroscopy. Chapter 3 discusses nanoscale NV-based EPR of several paramagnetic spins inside the diamond crystal. Chapter 4 presents NV-based EPR of a single or a few radicals located outside the diamond. In Chapter 5 and 6, I discuss the experiments at high magnetic fields. In Chapter 5, the investigation of spin decoherence of Mn₃ molecular nanomagnets using high-frequency EPR spectroscopy is discussed. The development of NV-based EPR spectrometer is discussed in Chapter 6. Finally, Chapter 7 summarizes the work presented in this thesis.