Magnetic-resonant detection of hyperpolarized noble gas field enhancement and multiphoton transition in organic polymers

In this work, nuclear magnetic resonance (NMR) and electron paramagnetic resonance (EPR) schemes have been implemented in two different projects in order to study magnetic resonance phenomena under unconventional drive and investigate hyperpolarization spin-exchange effects. In the realm of EPR, we investigate a second-order electrically detected magnetic resonance (EDMR) that appears at g=1 of a free charge carrier (electron or hole) in an organic polymer semiconductor environment. The two-photon transition interpretation is adopted for this observation. We study helicity dependence of this effect, using different polarizations of radiation and show the disappearance of this resonance with a circularly polarized excitation field under high-drive conditions as a way to rule out any technical artifact attribution. Magnetic resonance was induced at very low static magnetic fields in order to reach the high-drive regime, a regime where the amplitude of the resonant radiation is on the order of the Zeeman field. For the detection of magnetic resonance under this condition where spin polarization is almost vanishing, we used spin-dependent recombination currents in organic light-emitting diodes (OLEDs). In the study of hyperpolarization effects on the spin interaction, we report a measurement of the dimensionless enhancement factor 0κ for the Rb-129Xe pair commonly used in spin-exchange optical pumping to produce hyperpolarized 129Xe. 0κ characterizes the amplification of the 129Xe magnetization contribution to the Rb electronic effective field, compared to the case of a uniform continuous medium in classical magnetostatics. The measurement is carried out in Rb vapor cells containing both 3He and 129Xe and relies on the previously measured value of 0κ for the Rb-3He pair. The measurement is based on the optically detected frequency shift of the 87Rb EPR hyperfine spectrum caused by the SEOP nuclear polarization and subsequent sudden destruction of nuclear polarization of both species and a comparison of NMR signals for the two species acquired just prior to the EPR frequency shift measurements. We find 0(RbXe)5188κ=±, in good agreement with previous measurements and theoretical estimates but with improved precision.

MAGNETIC-RESONANT DETECTION OF HYPERPOLARIZED NOBLE GAS FIELD ENHANCEMENT AND MULTIPHOTON TRANSITION IN ORGANIC POLYMERS by Adnan Nahlawi A dissertation submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Department of Physics and Astronomy The University of Utah December 2019 Copyright © Adnan Nahlawi 2019 All Rights Reserved The University of Utah Graduate School STATEMENT OF DISSERTATION APPROVAL Adnan Nahlawi The dissertation of has been approved by the following supervisory committee members: Christoph Boehme and by , Chair 8/22/2019 Brian Saam , Member 9/9/2019 Jordan Gerton , Member 8/22/2019 Robert Wayne Springer , Member 8/22/2019 Eun-Kee Jeong , Member 9/4/2019 Christoph Boehme the Department/College/School of Date Approved Date Approved Date Approved Date Approved Date Approved , Chair/Dean of Physics and Astronomy and by David B. Kieda, Dean of The Graduate School. ABSTRACT In this work, nuclear magnetic resonance (NMR) and electron paramagnetic resonance (EPR) schemes have been implemented in two different projects in order to study magnetic resonance phenomena under unconventional drive and investigate hyperpolarization spin-exchange effects. In the realm of EPR, we investigate a second-order electrically detected magnetic resonance (EDMR) that appears at g= 1 of a free charge carrier (electron or hole) in an organic polymer semiconductor environment. The two-photon transition interpretation is adopted for this observation. We study helicity dependence of this effect, using different polarizations of radiation and show the disappearance of this resonance with a circularly polarized excitation field under high-drive conditions as a way to rule out any technical artifact attribution. Magnetic resonance was induced at very low static magnetic fields in order to reach the high-drive regime, a regime where the amplitude of the resonant radiation is on the order of the Zeeman field. For the detection of magnetic resonance under this condition where spin polarization is almost vanishing, we used spin-dependent recombination currents in organic light-emitting diodes (OLEDs). In the study of hyperpolarization effects on the spin interaction, we report a measurement of the dimensionless enhancement factor κ 0 for the Rb-129Xe pair commonly used in spin-exchange optical pumping to produce hyperpolarized 129Xe. κ 0 characterizes the amplification of the 129Xe magnetization contribution to the Rb electronic effective field, compared to the case of a uniform continuous medium in classical magnetostatics. The measurement is carried out in Rb vapor cells containing both 3He and 129Xe and relies on the previously measured value of κ 0 for the Rb-3He pair. The measurement is based on the optically detected frequency shift of the 87Rb EPR hyperfine spectrum caused by the SEOP nuclear polarization and subsequent sudden destruction of nuclear polarization of both species and a comparison of NMR signals for the two species acquired just prior to = 518 ± 8 , in good agreement the EPR frequency shift measurements. We find κ 0 (RbXe) with previous measurements and theoretical estimates but with improved precision. iv To my father Imad and my mother Zeina, who made me the confident thinker I am today. TABLE OF CONTENTS ABSTRACT....................................................................................................................... iii LIST OF FIGURES ......................................................................................................... viii ACKNOWLEDGMENTS .................................................................................................. x Chapters 1 INTRODUCTION ........................................................................................................... 1 1.1 References ............................................................................................................ 5 2 THEORETICAL BACKGROUND ................................................................................. 7 2.1 Semi-Classical Picture of Magnetic Resonance .................................................. 7 2.1.1 Relaxation Processes. ............................................................................... 9 2.1.2 Precession .............................................................................................. 11 2.1.3 Radiation Field and Rotating Wave Approximation (RWA) ................ 14 2.1.4 Radiation Absorption and Radiation-Induced Emission........................ 16 2.2 The Quantum Mechanical Picture of Magnetic Resonance ............................... 20 2.2.1 Rotating Coordinate Transformation Method ........................................ 20 2.2.2 Time-Dependent Perturbation Method .................................................. 22 2.3 Electrically Detected Magnetic Resonance ....................................................... 25 2.3.1 Polaron Pair Dynamics .......................................................................... 26 2.3.2 Transitions and Selection Rules ............................................................. 29 2.3.3 Radiation Polarization and π -Photon .................................................... 31 2.4 Multiphoton Transition ...................................................................................... 33 2.5 Large Driving Field Regime .............................................................................. 36 2.6 Optical Pumping ................................................................................................ 40 2.7 Spin Exchange Between Alkali-Metal Atoms and Noble Gas .......................... 42 2.8 References .......................................................................................................... 46 3 EXPERIMENTAL METHODS AND PREPARATIONS ............................................ 49 3.1 OLED Fabrication .............................................................................................. 49 3.1.1 ITO Patterning ....................................................................................... 51 3.1.2 Dicing ..................................................................................................... 51 3.1.3 Solution Preparation............................................................................... 53 3.1.4 Layer Deposition .................................................................................... 54 3.2 Vapor Cell Preparation ...................................................................................... 56 3.3 References .......................................................................................................... 58 4 IDENTIFICATION OF MULTIPHOTON TRANSITIONS BETWEEN MAGNETIC DIPOLE STATES USING ELECTRICALLY DETECTED MAGNETIC RESONANT EXCITATION WITH VARIABLE DRIVE-FIELD HELICITIES ...... 59 4.1 Introduction ........................................................................................................ 59 4.2 Theory ................................................................................................................ 61 4.3 Experiment ......................................................................................................... 64 4.3.1 B0 Setup.................................................................................................. 66 4.3.2 B1 Setup.................................................................................................. 68 4.3.3 The Sample ............................................................................................ 71 4.4 Results and Analysis .......................................................................................... 74 4.4.1 EDMR Spectrum .................................................................................... 74 4.4.2 B1 Polarization ....................................................................................... 76 4.4.3 g=1 Resonance ....................................................................................... 77 4.5 Discussion and Conclusion ................................................................................ 79 4.6 References .......................................................................................................... 80 5 HIGH-PRECISION DETERMINATION OF THE FREQUENCY-SHIFT ENHANCEMENT FACTOR IN Rb-129Xe .................................................................. 82 5.1 Introduction ........................................................................................................ 82 5.2 Theory ................................................................................................................ 84 5.3 Experiment ......................................................................................................... 87 5.3.1 NMR ...................................................................................................... 88 5.3.2 Optically Detected EPR ......................................................................... 92 5.3.3 Stabilization of Applied Magnetic Field ................................................ 93 5.4 Results and Analysis .......................................................................................... 96 5.4.1 Flip-Angle Measurement ....................................................................... 96 5.4.2 NMR Signal Acquisition........................................................................ 98 5.4.3 EPR Frequency Shift Acquisition .......................................................... 99 5.5 Discussion and Conclusion .............................................................................. 101 5.6 References ........................................................................................................ 104 6 CONCLUSION ............................................................................................................ 106 6.1 References ........................................................................................................ 108 APPENDIX: THROUGH-SPACE FIELD OF THE CELL PULL-OFF ....................... 109 vii LIST OF FIGURES Figures 2.1 Energy levels EU and EL of a single electron ............................................................. 8 2.2 Sketches of the precessional motion of three physical objects ................................... 12 2.3 Plots of M y for a spin ensemble ............................................................................... 20 2.4 Energy levels of a spin-1/2 particle (namely, an electron) as a function of B0 .......... 23 2.5 Plot of the transition probability (from Eq. (47)) as a function of the radiation ......... 25 2.6 Energy diagram of a polaron pair ............................................................................... 30 2.7 Classification of σ and π type photon ...................................................................... 34 2.8 Illustration of σ + and σ − single transitions (emission and absorption) ..................... 35 2.9 Energy diagram of a polaron pair showing the allowed and forbidden transitions .... 38 2.10 Sketch of 87 Rb optical pumping using a simplified energy diagram ....................... 41 2.11 Sketch of 87 Rb optical pumping taking into consideration the nuclear spin............... 43 3.1 Sketch showing the lateral structure of the used OLEDs ........................................... 50 3.2 A photo of the used glass substrate with the lithographically patterned ITO............. 52 3.3 Schematic of a gas-handling system with a vapor cell photo ..................................... 56 4.1 Energy term diagrams and vectorial sketch of fields .................................................. 62 4.2 EDMR measurements .................................................................................................63 4.3 Radiofrequency domain electron spin resonance spectrometer .................................. 65 4.4 LabVIEW program ......................................................................................................67 4.5 Sketch of the RF perpendicular coils along with the driving sinewaves .................... 69 4.6 RF coils and a probe head .......................................................................................... 70 4.7 I-V curve for a SY OLED measured with a source-meter .......................................... 72 4.8 Model of spin-dependent recombination .................................................................... 73 4.9 EDMR spectra ............................................................................................................. 75 4.10 EDMR spectra at different B1 powers ...................................................................... 78 5.1 3He (top) and 129Xe (bottom) NMR signals ................................................................ 89 5.2 87 Rb EPR frequency shift as a function of time ........................................................ 90 5.3 Schematic of experimental apparatus ......................................................................... 91 5.4 Optically detected 87 Rb EPR hyperfine spectrum ..................................................... 94 5.5 Current-stabilizing circuit design ................................................................................ 95 5.6 Signal intensity as a function of pulse number ........................................................... 98 5.7 Calculated values of κ 0 with the corresponding error bars ...................................... 101 A.1 Sketch of the vapor cell modeled as a sphere of radius R ....................................... 110 ix ACKNOWLEDGMENTS First, I would like to thank Christoph Boehme for his guidance and support throughout the past couple of years. His kindness and energy continue to fascinate and encourage me to pursue my passion for science. I would also greatly acknowledge Brian Saam for his continuing care and support. I appreciate his effort in keeping our promising research work alive despite the inconvenient hurdles. I would also like to thank the rest of my dissertation committee, Jordan Gerton, Wayne Springer, and Eun-Kee Jeong (E.K.), for their support, especially during the advising meetings. Much gratitude also goes to E.K. for the discussions that allowed me to broaden my skills and knowledge. To my parents, Imad and Zeina, thank you for believing in me and always encouraging me to set my aspirations high. Thank you for being there with me through the highs and lows even when we are oceans apart. To my younger siblings, Acile and Mohammed, thank you for brightening my day every time I talk to you. Your achievements and potential keep me optimistic about what the future holds. To my sister Layan and brother-in-law Sharief, thank you for giving me all the pieces of advice and experiences you have gathered in your academic journeys that made mine easier. To my in-laws, especially my father-in-law Dean, thank you for your support, encouragement, and your unwavering confidence in me. For the one that made the last couple of years bearable, my wife, Laiyan, thank you for taking care of me and being by my side during all the ups and downs. Thank you for sharing your happiness and cheerfulness when I couldn’t find mine. Thank you for being my confidant and my lighthouse. Finally, and above all, I thank God for giving me the strength and the supportive people around me to successfully conclude this exciting chapter of my life. xi CHAPTER 1 INTRODUCTION Magnetic resonance is a spectroscopic technique with which we can learn about the composition and structure of matter at atomic and molecular scales. The technique is also considered as one of the best ways to study fundamental concepts of quantum mechanics and see them in action [1]. Magnetic resonance is realized through the interaction of electromagnetic radiation (most commonly oscillating magnetic field) with magnetic moments of matter arising from electron or nuclear spins. When the radiation frequency corresponds to the magnetically induced energy splitting between the spin states, absorption of radiation takes place provided that the transition satisfies the governing selection rules. In most cases, the transition energy is determined by the Zeeman splitting due to an externally applied static magnetic field with small perturbations occurring due to the particular nature of a paramagnetic system, i.e., due to influences of spin-orbit coupling, hyperfine couplings (the interaction of electron and nuclear spins), dipolar, quadrupolar, exchange interactions, etc. [1, 2]. It is the ability of magnetic resonance to reflect the quantitative influence of all these perturbations that makes this spectroscopic technique so powerful and widely applied. Electron paramagnetic resonance (EPR) and nuclear magnetic resonance (NMR) 2 are two types of techniques that implement the magnetic resonance phenomenon through electron and nuclear spins, respectively. EPR and NMR are conceptually analogous and rely fundamentally on the same theory. However, they have many practical and technological differences [3]. These differences mainly arise from the fact that the mass of the electron is ~2000 times less than the mass of the smallest nucleus (proton). As a result, the magnetic moment of the electron turns out to be ~2000 times bigger than the one for the proton. That means that for the same static magnetic field, the Zeeman splitting in the case of EPR is ~2000 times larger than the splitting in the case of NMR. Thus, the frequency of radiation needed in EPR, for technologically straightforwardly available magnetic fields on the order of T, is in the microwave range (most commonly X-band) [4] while frequencies needed for NMR are typically in the radiofrequency range. For instance, for a typical static field of 300 mT, the EPR driving frequency is ~8 GHz, whereas the NMR driving frequency needed is only ~13 MHz. Dealing with two different ranges of radiation frequency requires different spectrometer hardware. For instance, NMR spectrometers use coax cables and regular tuning capacitors and excitation coils while EPR spectrometers need waveguides and microwave bridges and cavities [5, 6]. Moreover, for NMR, one usually varies the frequency while keeping the magnetic field at a constant value. The opposite is most commonly done in EPR spectrometers [4]. Alternatively, one can see the difference between EPR and NMR as follows: for a fixed, predetermined radiation frequency, the NMR static magnetic field needs to be much larger than the static field in the case of EPR. That explains why superconducting magnets are widely used in NMR spectrometers while copper-coil-based electromagnets are very common in EPR [7]. In this work, both NMR and EPR schemes have been implemented in different 3 experiments in order to investigate quantum mechanical effects like the magnetization enhancement within Rb-129Xe pairs in a gaseous state as well as multiphoton transitions of charge carrier spin pair states in a polymer semiconductor environment under extreme conditions, conditions under which the excitation radiation becomes on the order of the static magnetic field. The enhancement factor κ 0 , which we studied here, characterizes the amplification of the 129 Xe magnetization contribution to the Rb effective field that, in turn, produces hyperpolarized 129 Xe through spin-exchange optical pumping (SEOP) [8]. The study is based on the optically detected (with Faraday rotation) frequency shift of the 87 Rb EPR hyperfine spectrum caused by subsequent sudden destructions of nuclear polarization of 3 He and 129 Xe. These shifts are compared to NMR signals for the two noble species acquired just prior to the EPR frequency shift measurements. Whether using microwave or radiofrequency as the excitation radiation in EPR and NMR, respectively, the radiation power level also plays a significant role in the behavior of the system. In particular, high-power excitation regime provides interesting insights about unusual EPR conditions, where the excitation power becomes on the order of the static field, and the rotating wave approximation (RWA) breaks down. In this regime, second-order effects, like multiphoton transitions, start to emerge [9]. However, power saturation (vanishing spin polarization) and other technical limitations (regarding the power of the radiation sources and heating problems for example) make it hard to operate EPR experiments in such a regime [10]. The same is not true for optical spectroscopy where technologically available intense irradiation (e.g., Laser) has been widely used without causing a significant temperature rise of the spin system. This is due to the multitude of the 4 relaxation processes available in the case of the large visible-light transitions like luminescence [1]. In order to overcome the experimental challenges like the unavailability of large enough radiation sources for the static fields typically used in EPR spectrometers, we try to detect magnetic resonance at a very low static field. For the detection of magnetic resonance at a low static field where spin polarization is almost vanishing, we use a previously demonstrated electrically detected magnetic resonance (EDMR) spectroscopy scheme [11], where spin-dependent currents are measured in an organic light-emitting diode (OLED). We investigate a second-order resonance that appears at g=1 of the free electron indicating a two-photon transition. In order to study any helicity dependence of such effect, we also used different polarizations (linear and circular) of radiation and showed a considerable difference between them at high power. In essence, the two projects pursued in the course of this dissertation aimed at the study of magnetic resonance under generally unconventional EPR conditions, i.e., very low static magnetic field and electron spin excitation frequencies. Both studies have led to new insights into the quantum mechanics of magnetic resonance in the nonlinear, nonperturbative regime. This dissertation is divided into six chapters. This introduction represents Chapter 1. Chapter 2 is a theoretical background of all the physical concepts needed to understand the experiments discussed later. Chapter 3 is a note on experimental methods and sample preparation procedures inevitable to the realization of the experiments. Chapters 4 and 5 give a complete description and analysis of the two projects that form the core of this dissertation; “Identification of multiphoton transition between magnetic dipole 5 states using electrically detected magnetic resonant excitation with variable drive-field helicities” and “High-precision determination of the frequency-shift enhancement factor in Rb-129Xe”, respectively. The final chapter is a summary of the results and conclusions drawn from the current work. 1.1 References [1] J. A. Weil, and J. R. Bolton, Electron paramagnetic resonance: elementary theory and practical applications (John Wiley & Sons., 2007). [2] C. P. Slichter, Principles of magnetic resonance (Springer Science & Business Media, 2013). [3] M. H. Levitt, Spin dynamics: Basics of nuclear magnetic resonance (John Wiley & Sons., 2001). [4] W. R. Hagen, Biomolecular EPR spectroscopy (CRC press, 2008). [5] D. D. Wheeler, and M. S. Conradi, Practical exercises for learning to construct NMR/MRI probe circuits, Concept. Magn. Reson. A 40(1), 1-13 (2012). [6] S. K. Misra, and J. Freed, Multifrequency electron paramagnetic resonance. Theory and Applications (Wiley-VCH Verlag & Co, Weinheim, 2011). [7] M. Brustolon, Electron paramagnetic resonance: a practitioner's toolkit (John Wiley & Sons., 2009). [8] S. R. Schaefer, G. D. Cates, T. R. Chien, D. Gonatas, W. Happer, and T. G. Walker, Frequency shifts of the magnetic-resonance spectrum of mixtures of nuclear spinpolarized noble gases and vapors of spin-polarized alkali-metal atoms, Phys. Rev. A 39(11), 5613 (1989). [9] A. Abragam, Principles of nuclear magnetism (Oxford Univ. Press, London/New York, 1961). [10] S. Jamali, G. Joshi, H. Malissa, J. M. Lupton, and C. Boehme, Monolithic OLEDmicrowire devices for ultrastrong magnetic resonant excitation, Nano Lett. 17(8), 4648-4653 (2017). 6 [11] D. R. McCamey, H. A. Seipel, S. Y. Paik, M. J. Walter, N. J. Borys, J. M. Lupton, and C. Boehme, Spin Rabi flopping in the photocurrent of a polymer light-emitting diode, Nat. Mater. 7(9), 723 (2008). CHAPTER 2 THEORETICAL BACKGROUND In the following, we derive the theory of magnetic resonance for the case of EPR, where the magnetic moment arises from electron spin. The same derivation holds true for the case of NMR, where the magnetic moment arises from a nuclear spin (e.g., proton spin). Nuclear magnetic moments point in the same direction as the nuclear spins. In contrast, electron spins point in a direction that is opposite to its spin, due to the electron’s negative charge, which makes the nuclear spin and electron spin states correspond to the opposite energy levels [1]. 2.1 Semi-Classical Picture of Magnetic Resonance   We consider an electron of spin S placed in a static magnetic field B0 . The   magnetic moment µ due to the electron spin can either be aligned or anti-aligned with B0 (corresponding to the two eigenstates of the electron spin, called “up” and “down”). The magnetic dipole interaction between the electron’s dipole and the magnetic field results in two energy levels that correspond to these two configurations. The energy levels, illustrated in Fig. 2.1 are here referred to as EL and EU , where EL designates the lower    energy level when µ is aligned (while S is anti-aligned) with B0 and EU designates the 8 Figure 2.1 Energy levels EU and EL of a single electron along with the corresponding     magnetic moment ( µ U and µL ) and spins ( S U and S L ) direction with respect to the static  magnetic field, B0 .    upper energy level when µ is anti-aligned (while S is aligned) with B0 . The energy difference between EU and EL is called Zeeman splitting. It depends linearly on B0 : EU − EL = g e β e B0 [1], where g e is the g-factor and β e is the Bohr magneton. The electron can make transitions between the two Zeeman energy levels through different mechanisms: photon scattering processes such as spontaneous radiative emission, radiation absorption, and stimulated emission due to resonant and non-resonant scattering, e.g., with lattice vibrations (phonons) and also interactions with neighboring electron and nuclear spin. All these mechanisms together are referred to as spin-relaxation processes. Since the probability for spontaneous photon emission is proportional to the cube of the energy difference between EU and EL , and it is only significant for a large energy gap in the optical frequency ranges [2], it is, therefore, going to be ignored in this discussion. In the following, we’re going to discuss several other spin relaxation processes and radiationinduced absorption and emission. 9 2.1.1 Relaxation Processes The Relaxation processes are responsible for the return of spin ensembles from nonthermal equilibrium to thermal equilibrium states. Consider a spin ensemble consisting of N noninteracting spins with s = 1 2 and, thus, two spin eigenstates. At thermal equilibrium, the number nL0 of spins in the lower energy state exceeds the number nU0 of spins in the upper state according to the Boltzmann distribution [1]: nU0 = exp  − ( EU − EL ) kT  = exp(− g e β e B0 kT ) . nL0 Equivalently, nL0 = N e + ge βe B0 2 kT e + ge βe B0 2 kT + e − ge βe B0 2 kT and nU0 = N e − ge βe B0 2 kT e + ge βe B0 2 kT + e − ge βe B0 (1) 2 kT . The thermal magnetization of the system, defined as the macroscopic sum of all the magnetic moments per unit volume, is given by: nL0 − nU0 µ, M0 = V where = µ µ= z (2)  ge βe and the z-axis is taken to be along B0 . The randomly-oriented x 2 and y components of the magnetic moments cancel out: +g β B  Ng e β e  e e e 0 ⇒ M0 =   +g β B  2V  e e e 0 2 kT 2 kT − e − ge βe B0 + e − ge βe B0 2 kT 2 kT , (3) Ng e 2 β e 2 B0 which comes down to Curie law when approximated for small B0 [3]: M 0 = . 4VkT  At equilibrium, the magnetization vector M would be pointing upward along the direction  M y 0 and = M z M0 . of B0 = : M x 0,= 10 Assume that the system is now disturbed into a nonequilibrium state such that nL and nU are the new numbers of spins in the lower and upper states, respectively. Denote n nL − nU and define ω↓ to be the probability for an electron the number difference by = to get relaxed downward and ω↑ the probability for an electron to get relaxed upward. Hence, the differential equation describing the dynamics of the system is: dn = 2(ω↓ nU − ω↑ nL ) dt (4) Using the equilibrium condition ω↑ nL0 = ω↓ nU0 , the solution to Eq. (4) can be written as: (n 0 − n)t = (n 0 − n)t =0 exp(− t T1 ) , ( (5) ) 0 nL0 − nU0 and where n 0 is the number difference at equilibrium n= = T1 1 (ω↓ + ω↑ ) is called the spin-lattice relaxation time constant. In terms of the magnetization: = M z M 0 (1 − e −t T1 ) (6) M − M0 dM z = − z dt T1 (7) or in the differential form: The system can also be disturbed such that M x and M y become nonzero. In this case, the relaxation of the x and y components of the magnetization is called transverse spin-spin relaxation, and it is similarly given by [4]: M xy = M xy0 e −t T2 , (8) or in the differential form: dM xy dt = − M xy T2 , (9) 11 where M xy0 is the x or y component of the magnetization at t = 0 and T2 is the spin-spin relaxation time constant. Putting Eq. (7) and Eq. (9) together, we get the set of equations governing the relaxation dynamics of the system, including both the longitudinal spin-lattice and the transverse spin-spin relaxation times: Mx  dM x  dt = − T 2   dM y M = − y  T2  dt  dM z M − M0 = − z  T1  dt (10) 2.1.2 Precession Before Larmor precession is discussed, let us first discuss the nature of mechanical precession: We consider a cylinder of mass m, radius r, and length l. The cylinder is tilted by an angle ϕ from the vertical direction and it is spinning about its central axis with an  angular speed ωs , as shown in Fig. 2.2(a). The tilt results in a torque τ about the pivot P  exerted by the weight mg of the cylinder that is exerted on its center of mass. As a result of the concurrent spinning and torque, the cylinder experiences a precession around the vertical direction. The rate of precession is calculated as follows; see Fig. 2.2(b): ωp = ∆θ . ∆t (11) ∆L We can write ∆θ = where L is the magnitude of the angular momentum due to the L sin ϕ spinning. Substituting ∆θ in Eq. (11) gives: 12 Figure 2.2 Sketches of the precessional motion of three physical objects. (a) A tilted rotating cylinder of mass m, and a tilt angle ϕ . ωs is the rotation frequency and ωp is the  precession frequency. (b) A spinning magnet with angular momentum L . (c) A particle   with spin S in a magnetic field B0 . ωp = τ ∆L = , L sin ϕ ∆t L sin ϕ (12) 1 2 l where τ = mg   sin ϕ , and= L I= mr ωs . sω s 2 2  On the other hand, when a magnet of magnetic moment µ is placed in a magnetic   field B0 (see Fig. 2.2(c)), a torque τ is exerted on the magnet trying to align it with the      field ( ϕ is the angle between µ and B0 ): τ =µ × B0 =µ B0 sin ϕ τˆ , where τ min = 0 for ϕ= 0° or 180° , and τ max = µ B for ϕ = 90° or 270° .  Now, if the magnet is also spinning around its axis with angular momentum L , the  torque exerted by the magnetic field B0 tends to make the magnet precess instead of getting it aligned. The concurrence of the angular momentum and the magnetic moment gives rise  to a precession around the magnetic field B0 . 13 The rate of precession can be calculated using Eq. (12): ωp = τ L sin ϕ , where now τ = µ B0 sin ϕ : µB ⇒ ωp =0 . L (13)    Finally, a particle with spin S is placed in a magnetic field B0 . The spin S is an intrinsic angular momentum that gives rise to a magnetic moment as well [5]:  gβ   − e eS= −γ S , µ= (14)  where γ is the gyromagnetic ratio. Having both angular momentum and magnetic moment,  the spin experiences a precession about the magnetic field B0 called Larmor precession [1]; see Fig. 2.2(c). Larmor frequency can be calculated using Eq. (12): ωp = τ L sin ϕ , B0 sin ϕ γ SB0 sin ϕ and here L = S . = where now τ µ= ⇒ ωL = γ B0 (15) The same relation in Eq. (15) with more details can be derived as follows: The torque   exerted by the magnetic field B0 on the magnetic moment µ is given by:    τ= µ × B0 (16) The torque is also defined as the rate of change of angular momentum (here it is the spin  S ):  dS τ = dt  (17)  Setting Eq. (16) and Eq. (17) equal and substituting S for its expression from Eq. (14), we get the equation of motion for the precession of the magnetic moment: 14    dµ = γ B0 × µ dt  d µx  dt = −γ B0 µ y   d µy = γ B0 µ x .   dt  d µz  dt = 0  (18)  In terms of the magnetization M within an ensemble of spins, we obtain the system of ordinary differential equations:    dM = γ B0 × M dt  dM x  dt = −γ B0 M y   dM y = γ B0 M x   dt  dM z  dt = 0  (19) whose solution is given by [1]:  M x = ( M sin ϕ ) cos ( γ B0t )   M y = − ( M sin ϕ ) sin ( γ B0t ) ,   M z = M cos ϕ (20) This is again a precession motion around the z-axis with frequency, ωL = γ B0 , the spin system’s Larmor frequency. 2.1.3 Radiation Field and Rotating Wave Approximation (RWA)  Consider a spin system in a static magnetic field B0 in the z-direction, with an applied radiation field whose polarization plane is perpendicular (transverse radiation). Classically, radiation can be represented by an electromagnetic wave with an oscillating   electric field E1 and magnetic field B1 . For charged spin systems that produce a magnetic  moment, the oscillating magnetic field B1 is the effective part of the radiation [3], while 15 the interaction of the electric field can be ignored. The interaction of the oscillating  magnetic field B1 (say given by: B1 cos ωt xˆ ) with the spin system can be more easily studied using the rotating wave approximation (RWA) in the rotating frame of Larmor precession provided that B1 is small [1]. In fact, for small magnetic field amplitude B1 , the  magnetic field vector B1 can be approximated with a rotating (instead of oscillating) vector of magnitude B1 2 , and frequency ω. RWA can be represented as follows: B1 B B B cos ωt xˆ + 1 cos ωt xˆ + 1 sin ωt yˆ − 1 sin ωt yˆ 2 2 2 2 B B = 1 ( cos ωt xˆ + sin ωt yˆ ) + 1 ( cos ωt xˆ − sin ωt yˆ ) 2 2 B1 cos ωt xˆ = (21) This is the sum of two counter-rotating vectors of magnitude B1 2 and frequency ω . In a reference frame that rotates around the lab frame at the Larmor frequency, the precessing spin appears stationary, and the behavior of the rotating and counter-rotating magnetic fields would depend on their frequency ω compared to the Larmor frequency ωL . Here, we consider two cases: ω  or  ωL and ω  ωL .  If ω  or  ωL , the stationary spin doesn’t sense the presence of the rotating B1  due to the different time scales involved. Hence, B1 has no effect on the behavior of the spin. If ω  ωL , in the rotating frame (i.e., with respect to the spin), the corotating part of  B1 appears stationary in the perpendicular plane (see Fig. 2.2(c)) whereas the counter rotating part of B1 rotates with frequency 2ωL with respect to the spin. For B1  B0 , one  can show that the counter-rotating part of B1 doesn’t affect the behavior of the spin and it can be ignored [1], i.e., only one photon helicity is effective; see Sec. 2.3.3 for a discussion 16 about photon helicity and polarization. Hence, for small value of B1 , the oscillating magnetic field, B1 cos ωt xˆ , can be approximated by B1 ( cos ωt xˆ + sin ωt yˆ ) . For 2 simplicity, it is common to start with an oscillating field 2 B1 cos ωt xˆ so that RWA gives B1 ( cos ωt xˆ + sin ωt yˆ ) without the factor ½. 2.1.4 Radiation Absorption and Radiation-Induced Emission The rotating orientation of the magnetic field vector within the radiation field becomes static, from the viewpoint of the rotating frame described in the last paragraph. Thus, the description of the spin system’s propagation in the radiation fields within the rotating frame becomes identical to a spin propagating in a static magnetic field. In the  rotating frame, the spin precesses around B1 with frequency ωR . This frequency, also  called Rabi frequency, depends on the amplitude B1 of the radiation field and is therefore given by [5]: ωR = γ B1 . (22) From the lab frame point of view, the spin orientation in presence of a static magnetic field and a perpendicular oscillating magnetic field appears to be spiraling around the direction  of B0 , due to the simultaneous influence of the Larmor precession and the Rabi precession. Rabi precession, also referred to as a spin flip, implies quantum mechanical transition of the spin system to a higher energy state (radiation absorption), or to a lower energy state (radiation-induced emission) [3]. In order to get an expression for the macroscopic magnetization of a group of spins during the absorption (and the emission) as a function of 17 the contributing factors, we derive the magnetization equations of motion as follows: Using RWA, the total magnetic field is given by:    B = B0 + B1 = B0 zˆ + B1 ( cos ωt xˆ + sin ωt yˆ ) , (23)    where at t = 0 , B1 is pointing along the x-axis. Replacing B0 in Eq. (19) with B from Eq. (23), and adding the relaxation terms discussed previously, we get a complete set of Bloch equations: M  dM x −γ B0 M y + γ B1M z sin ωt − x  dt = T2      dM y M dM = γ B × M + ( Relaxation Term )  = γ B0 M x − γ B1M z cos ωt − y (24) dt dt T 2   dM z M − M0 = γ B1M y cos ωt − γ B1M x sin ωt − z  T1  dt   describing the time evolution of the magnetization M in a static field B0 , under magnetic  excitation by a radiation field B1 , and in presence of relaxation processes ( T1 and T2 ). To solve this equation, we consider the left-hand side of Eq. (19): The total derivative of the magnetization in the lab frame can be written as the sum:   dM ∂M   = + Ω× M . dt ∂t (25) of two terms: (i) the derivative of the magnetization with respect to a rotating frame and  (ii) the cross product of the rotation angular frequency Ω by the magnetization [1]. Substituting this expression in Eq. (19), we get:    ∂M   + Ω× M = γ B0 × M ∂t     ∂M ⇒ = γ B0 − Ω γ × M ∂t ( ) (26) 18  This equation is the same equation governing the behavior of a magnetization M in a      solitary static field B0 provided that B0 is replaced by an effective field, Beff= B0 − Ω γ .  Using this trick, we now know how to transform the static field B0 from the lab frame into     a rotating frame with angular frequency Ω : B0 → B0 − Ω γ . Next, we want to find out  how a rotating field B1 with frequency ω transforms when moving from lab frame into a  rotating frame with angular frequency Ω . It is trivial that if Ω =ω , then B1 would appear constant along its initial direction: B1 cos ωt xˆ + B1 sin ωt yˆ → B1 xˆ . Implementing these transformations in Eq. (24), we obtain in the rotating frame the following: M  ∂M x −γ ( B0 − ω γ ) M y − x  ∂t = T2  ∂M y M =γ ( B0 − ω γ ) M x − γ B1M z − y ,  T2  ∂t  ∂M z M − M0 = γ B1M y − z T1  ∂t (27) where the partial derivative notation indicates the reference to the rotating frame. Eq. (27) without time dependence on the right-hand side looks much simpler than Eq. (24) and steady-state solutions can be found by setting each equation equal to zero. The solutions are given by:  2  M x = γ B1 2M20 ∆ω2 T22 1 + ∆ω T2 + γ B1 T1T2   γ B1M 0T2  M y = − 1 + ∆ω 2T2 2 + γ 2 B12T1T2   M 0 (1+∆ω 2 T2 2 ) M =  z 1 + ∆ω 2T2 2 + γ 2 B12T1T2 (28) 19 where ∆ω= γ B0 − ω . For our purposes, M y is one of the most important magnetization components. It can be thought of as a measure of the net radiation absorption. When plotted, M y gives the so-called absorption spectrum. M y has a maximum at ω = γ B0 , which indicates the resonance absorption condition. As we can see in Fig. 2.3(a), upon increasing B1 , the absorption initially increases linearly with B1 until it reaches a point where it saturates and then starts to slowly decrease sat (power broadening regime). The saturation B1 is given by B1 = 1 γ T1T2 , which implies that the larger T1 (or T2 ) is, the smaller B1sat would be. In other words, for spin systems with slow relaxation mechanisms, it is very likely to reach the power broadening regime even with low-power B1 . This behavior is due to a limited ability of the system to dissipate energy from its spins to its surrounding (lattice) as thermal motion, and it is alternatively referred to as heating of the spin system [3]. For systems with fast relaxation mechanisms relative to B1 (i.e., B1  1 γ T1T2 ), the term γ 2 B12T1T2 in M y is negligibly small. Hence, the absorption spectrum would be a Lorentzian function; see Fig. 2.3(b): My = γ B1M 0T2 , 1 + ∆ω 2T22 (29) with the absorption line height linear in B1 and FWHM = 2 T2 . However, if γ 2 B12T1T2 is not negligible, then the expression for M y becomes more complicated and generally depends on B1 in the way shown in Fig. 2.3(a). 20 Figure 2.3 Plots of M y for a spin ensemble. (a) M y under magnetic resonance as a function of B1 showing the maximum at 1 γ T1T2 and extending over the power broadening region. (b) EPR absorption spectrum as a Lorentzian function for the case of fast relaxation. 2.2 The Quantum Mechanical Picture of Magnetic Resonance 2.2.1 Rotating Coordinate Transformation Method   Consider an electron of spin S placed in a static magnetic field B0 = B0 zˆ . A small  rotating magnetic = field B1 B1 ( cos ωt xˆ + sin ωt yˆ ) with frequency ω is applied perpendicular to  B0 . The total magnetic field is then given by:  B = B1 cos ωt xˆ + B1 sin ωt yˆ + B0 zˆ .   gβ   The spin Hamiltonian of the system is [5]: Ĥ =− µ ⋅ B , where µ = − e eS  ⇒ Hˆ= γ B0 S z + γ B1 ( S x cos ωt + S y sin ωt ) The corresponding Schrödinger equation is:  = −γ S : (30) 21 ∂ −i ψ =γ B0 S z + γ B1 ( S x cos ωt + S y sin ωt )  ψ ∂t  Using transformation into rotating coordinates, ψ Writhing Eq. (31) in terms of ψ ′ (31) transforms to ψ ′ = eiωt S z  ψ . and multiplying by eiωt S z  from the left, the Schrodinger equation in the rotating coordinates becomes: ∂ ψ ′ −i = ( γ B0 − ω ) S z + γ B1S x  ψ ′ , ∂t  (32) where the transformed Hamiltonian is now independent of explicit time and defined as: Hˆ ′ = ( γ B0 − ω ) S z + γ B1S x . (33) The solution to Eq. (32) is obviously given by: ˆ′ ψ ′(t ) = e-(i  ) H t ψ ′(0) . (34) Yet when we move out of the rotating coordinates, we obtain: ˆ′ ψ (t ) = e-iωt S  e-(i  ) H t ψ ′(0) . z (35) Let us now find the expectation value of the z-component of the magnetic moment, µ z of a single spin. For simplicity, take the resonance case, where ω = γ B0 . Hence, Hˆ ′ = γ B1S x -( i  )ωtS z -( i  )γ B1tS x e ψ ′(0) : and ψ (t ) = e µ z (t ) = ψ (t ) µ z ψ (t ) = ψ ′(0) e(i  )γ = ψ (0) e(i  )γ B1tS x B1tS x e(i  )ω z µ z e-(i  )ω z e-(i  )γ tS tS B1tS x ψ ′(0) (36) µ z e-(i  )γ B tS ψ (0) 1 x ( i  )γ B1tS x S z= e −(i  )γ B1tS x S z cos γ B1t + S y sin γ B1t , we get: Using the following relation [5]: e = µ z (t ) µ z (0) cos γ B1t + µ y (0) sin γ B1t , (37) 22 where the oscillating magnetic moment in the z-direction corresponds to the Rabi  precession about B1 with frequency ωR = γ B1 . 2.2.2 Time-Dependent Perturbation Method The same result with more details can be shown using the time-dependent perturbation theory. In this case, the Hamiltonian can be written as: ˆ Hˆ + Hˆ , H = 0 1 ( (38) ) ˆ γ B S cos ωt + S sin ωt is treated as the perturbation due where Hˆ 0 = γ B= 1 x y 0 S z and H1 to the small B1 . The eigenstates of the unperturbed Hamiltonian Ĥ 0 are: ψ α = α e −iEα t  ψβ = β e (39) − iE β t  1 where α and β are the eigenstates of S z with mS = ± , respectively: 2 1 Sz α = +  α 2 1 Sz β = −  β 2 with Eα = (40) 1 1 γ B0 and Eβ = − γ B0 ; see Fig. 2.4. Assuming that ψ α and ψ β form a 2 2 complete orthonormal basis, an arbitrary state ψ can exist and it can be expressed as: = ψ cα (t ) ψ α + cβ (t ) ψ β . Substituting this expression of ψ get: in the Schrödinger equation: (41) i ∂ ψ = − Ĥ ψ , we ∂t  23 Figure 2.4 Energy levels of a spin-1/2 particle (namely, an electron) as a function of B0 along with the corresponding spin eigenstates. In the absence of the magnetic field, the spin system’s eigenenergies would not undergo Zeeman splitting. dc dcα i ψα + β ψ β = − cα (t ) Hˆ 1 ψ α + cβ (t ) Hˆ 1 ψ β  dt dt ( ). (42) Multiplying by ψ α and ψ β , we get differential equations that govern the evolution of cα and cβ : dcα i -i ( E − E )t  = − cβ α Hˆ 1 β e α β  dt dcβ i i ( E − E )t  = − cα β Hˆ 1 α e α β dt  (43) γ B0 . α Ĥ1 β and β Ĥ1 α can be evaluated as follows: where Eα − Eβ = = α Hˆ 1 β * = β Hˆ 1 α γ B1 ( α S x β cos ωt + α S y β sin ωt ) = γ B1 ( cos ωt + i sin ωt ) = γ B1eiωt Eq. (43) becomes: (44) 24 dcα i ω −γ B t = −iγ B1e ( 0 ) cβ dt dcβ -i ω −γ B t = −iγ B1e ( 0 ) cα dt (45) Eqs. (45) can be combined into one second-order differential equation in cα : d 2 cα dc + i (ω − γ B0 ) α + γ 2 B12 cα = 0 2 dt dt (46) For example, we find the solution to Eq. (46) for the case where the system is initially (at t = 0 ) in the ψ β state and calculate the probability to be found in the ψ α state at t > 0 . The current situation can be represented by the following initial conditions: cα (0) = 0 and cβ (0) = 1 . Integrating Eq. (46), we get the following solution [1]:  −i (ω -γ B0 ) t   −iγ B1  cα  =  sin ( Ωt )  exp  2  Ω     γ 2 B12  2 cα cα* = ⇒ Transition Probability =  2  sin ( Ωt )  Ω  where = Ω (47) γ 2 B12 + ( γ B0 − ω ) 4 . The transition probability (absorption or emission) in 2 Eq. (47) is an oscillating function with frequency Ω representing the Rabi oscillation. At ω = γ B0 , the Rabi oscillation frequency becomes Ω Res = γ B1 , and the oscillation amplitude becomes the largest indicating the magnetic resonance condition; see Fig. 2.5. Moreover, ±1 . This from Eq. (44), one can infer the following transition selection rule: ∆mS = selection rule is trivial in the case of a single spin system as there are only two energy states. However, as we’ll see later in Sec. 2.3.2, this selection rule has less trivial consequences in the case of a two-spin system. 25 Figure 2.5 Plot of the transition probability (from Eq. (47)) as a function of the radiation frequency ω . The plot shows a maximum at the resonance frequency γ B0 . 2.3 Electrically Detected Magnetic Resonance In the previous sections, we discussed the theoretical background of magnetic resonance in a general NMR and EPR environment that involves the study of the interaction between electromagnetic radiation and individual magnetic moments in an external static magnetic field. Magnetic resonance can also be studied in an organic diode to investigate the nature of charge carriers transport and recombination involved in such devices. Electrically detected magnetic resonance, EDMR, is one of the techniques used to observe magnetic resonance in organic-based diodes [6]. Unlike NMR and EPR, EDMR does not require net spin magnetization of the system as it involves the measurement of spin-dependent recombination current that is governed by spin-permutation symmetry within spin pairs formed by recombining charge carriers. Because of this, EDMR can be performed at high temperatures and, simultaneously, at very low static magnetic field. In an organic semiconductor-based diode, electrons and holes can be injected into a single semiconductor material. Due to Coulombic interaction, the two types of charge carriers 26 (electrons and holes), which both exist in highly localized yet mobile states, get attracted and start to move towards each other until they form bound pair states called polaron pairs [6]. From a spin perspective, polaron pairs can exist in any of four spin states (e.g., the four combinations of spin up and spin down if spin-spin coupling within the pair is weak), which are always linear combinations of the one singlet and three triplet pair states. Subsequent to its formation, a polaron pair can dissociate or recombine. Due to spin angular momentum conservation caused by the weak spin-orbit coupling found in organic semiconductors, a polaron pair can recombine into a doubly occupied electron state with singlet character. Such a transition will happen at a much higher rate though if the polaron pair before the transition has a singlet spin state rather than a triplet spin state. Transitions between singlet and triplet states can be accomplished using a magnetic-resonance-induced Rabi precession within a single polaron pair. Hence, a current change through the diode is detected every time the dynamic equilibrium of pair generation, dissociation, and recombination are disturbed by a magnetic resonance measurement. 2.3.1 Polaron Pair Dynamics The dynamics of a system under EDMR can be described by studying the quantum mechanical evolution of a polaron pair whose Hamiltonian is determined by two interacting  spin-½ particles in a static magnetic field B0 , under the influence of an oscillating magnetic  field B1 [6]: Hˆ = Hˆ 0 + Hˆ ′ + Hˆ 1 (t ) , (48) 27 where     • Ĥ 0= γ e B0 ⋅ Se + γ h B0 ⋅ S h is the Hamiltonian of the Zeeman interaction with the static   field B0 . Taking the z-direction to be along B0 , Ĥ 0 can be written as: = Hˆ 0 γ e B0 Sez + γ h B0 S hz (49) = ωe Sez + ωh S hz where ωe and ωh are the Larmor frequencies for the electron and the hole, respectively. The frequencies are taken to be different to account for the different local hyperfine fields experienced by the electron and the hole.     • Hˆ ′ = − J Se ⋅ S h − D 3Sez S hz − Se ⋅ S h is the spin-spin interaction Hamiltonian with J and ( ) D the exchange and dipolar coupling constants, respectively [6].     • Hˆ 1 (t )= γ e B1 ⋅ Se + γ h B1 ⋅ S h is the time-dependent Hamiltonian of the interaction with the  oscillating magnetic field B1 . For small B1 , Hˆ 1 (t ) can be considered as a time-dependent perturbation. The eigenstates of Ĥ 0 form an orthonormal basis called the product basis: ↑↑ ≡ ↑e ↑h , ↑↓ ≡ ↑e ↓h , ↓↑ ≡ ↓e ↑ h , ↓↓ ≡ ↓e ↓ h (50) where ↑e , ↓e , ↑h and ↓h are the eigenstates of Sez and S hz as follows: 1 Sez ↑e = +  ↑e 2 1 Sez ↓e = −  ↓e 2 1 S hz ↑h = +  ↑h 2 1 S hz ↓h = −  ↓h 2 (51) 28 ↑ and ↓ are equivalent to the S , mS notation, where S = 1/ 2 and mS = ±1/ 2 , respectively. The time-independent part Hˆ 0 + Hˆ ′ of the Hamiltonian can be represented by a nondiagonal matrix in the product basis, which shows that the product states are not all eigenstates of Hˆ 0 + Hˆ ′ . By matrix diagonalization and using the coordinate-rotation method, we find the eigenstates and the eigenvalues of Hˆ 0 + Hˆ ′ corresponding to the different energy levels of a polaron pair in a weak radiation field: 1 E1 =− J + D −  (ωe + ωh ) 2 E2 =− D − ω∆ E3 =− D + ω∆ (52) 1 E4 =− J + D +  (ωe + ωh ) 2 where = ω∆ ( J + D)  2 2 ( ω − ωh ) + e 4 2 . The corresponding eigenstates of Hˆ 0 + Hˆ ′ are given by: 1 = ↓↓ 2 = − sin φ ↑↓ + cos φ ↓↑ = 3 cos φ ↑↓ + sin φ ↓↑ (53) 4 = ↑↑ J +D 1 where φ = arcsin   . Notice that the states ↑↑ and ↓↓ are already eigenstates  2 ω  ∆  of Hˆ 0 + Hˆ ′ and they are independent of the spin-spin interaction. However, as the spinspin interaction increases, the eigenstates 2 and 3 change continuously from product 29 states to fully coupled states. For example, in the absence of spin-spin interaction, J= D= 0 and φ = 0 . In this case, we get: 2 = − sin(0) ↑↓ + cos(0) ↓↑ = ↓↑ 3 = cos(0) ↑↓ + sin(0) ↓↑ = ↑↓ (54) For extremely large spin-spin interaction, J and D → ∞ and φ → π 4 . In this case, we get after normalization: = 2 = 3 2 2 ↑↓ − ↓↑ 2 2 2 2 ↑↓ + ↓↑ 2 2 (55) where the two product states occur equally admixed. The eigenstates in this limit can be denoted by F , mF where F is the total spin quantum number of the polaron pair ( F = Se ± S h = 1 or 0) : 1 = 1, −1 ≡ T− , 2 = 1, 0 ≡ S , 3 = 0, 0 ≡ T0 , 4 = 1, +1 ≡ T+ (56) As shown previously in the case of single spin system, the application of a weak oscillating  magnetic field B1 leads to transitions between the spin eigenstates provided that the energy  difference between the relevant states matches the frequency of B1 . However, here we have more than one possible transition; see Fig. 2.6. Hence, we need to study the system selection rules and determine the allowed and forbidden transitions. 2.3.2 Transitions and Selection Rules hν Efinal − Einitial , where ν is the frequency of the The resonance condition = radiation, can be thought of as a conservation of energy condition. In addition, conservation 30 Figure 2.6 Energy diagram of a polaron pair for an arbitrary strength of the spin-spin interaction. The diagram also shows the allowed transitions in the case of σ + - and π radiation. of angular momentum is manifested by the transition probability between an initial and a final spin states that is proportional to ψ initial Ĥ1 ψ final 2 . Consider here the common case    where B1 is linearly-oscillating along a direction perpendicular to B0 (say B1  xˆ ;  B1 = 2 B1 cos ωt xˆ ) and the spin-spin interaction is extremely large. Hˆ 1 (t ) becomes: = Hˆ 1 (γ S x e e + γ h S hx ) 2 B1 cos ωt . By writing Sex and S hx in terms of the ladder operators Se± and S h± between two coupled states F , mF and F ′, mF′ , and having these states as a superposition of the product states, we arrive at the following selection rule [3]: ∆mF = ±1 (57) The transitions ∆mF = 0 or ± 2 are forbidden. This selection rule can be more understood by taking the limit where the spin-spin interaction is zero. In this case, sandwiching Sex 31 and S hx between two product states mSe , mSh and mS′e , mS′h leads to the following selection rule: ∆mSe = ±1 & ∆mSh = 0 or ∆mSe = 0 & ∆mSh = ±1 , (58) which can be understood in terms of conservation of the angular momentum. The absorption of linearly-oscillating transverse radiation takes place only if the radiation flips the spin of the electron or the hole upward or downward, each one at a time. This is the case of the so-called σ + -photon radiation. This leads to four allowed transitions between the polaron states, as shown in Fig. 2.6. 2.3.3 Radiation Polarization and π -Photon   Consider now the case where B1 is linearly-oscillating along the direction of B0  Hˆ ( γ S ( B  zˆ ) . Hˆ (t ) becomes:= 1 1 1 e z e + γ h S hz ) 2 B1 cos ωt . In the limit of negligible spin-spin interaction, the eigenstates are the product states and the selection rules come down to evaluating: mSe , mSh Sez mS′e , mS′h and mSe , mSh S hz mS′e , mS′h . This obviously leads to the following selection rules: ∆mSe= 0 & ∆mSh= 0 (59)  which means that no single spin flips are allowed. Hence, parallel B1 radiation cannot be absorbed in this case [3]. However, for a nonzero spin-spin interaction (which is practically the case at all times), the selection rules in Eq. (59) take a new form that involves the appropriate quantum numbers in the new regime. To illustrate this case, we take the limit of large spin-spin interaction where the coupled states F , mF are good eigenstates. 32 Evaluating: F , mF Sez F ′, mF′ and F , mF S hz F ′, mF′ leads to the following selection rule: ∆mF = 0, (60) which now allows for a transition between 1, 0 and 0, 0 ; see Fig. 2.6. This transition can be thought of as a simultaneous electronic and hole spin flips, where there is no net angular momentum transfer between the radiation and the spin system but only energy. This is the case of the so-called π -photon radiation as opposed to σ -photon radiation   treated in the previous paragraph. Hence, when B1  B0 , absorption of radiation may occur, but it gives rise to a spectral line of relatively low intensity as this transition is only one of   its type (contrasted with four transitions in the case of B1 ⊥ B0 ). At this point, one can gain more insight by thinking in terms of individual photons that make up the radiation field. The photon’s spin angular momentum component along its direction of motion can take on only one of two values: + or − . Hence, a photon can occur in either of two helicity states: positive or negative. Alternatively, photon’s spin angular momentum can be visualized as a clockwise- or counterclockwise-rotating quantity according to the right-hand rule with respect to the spin vector. Thus, helicity can also be referred to as right-hand or left-hand circular helicity. There is no photon with zero helicity [3]. The term polarization is only used with statistically large groups of photons (i.e., radiation); circularly-polarized radiation refers to phase-correlated photons traveling in the same direction with the same circular helicity (right-hand or left-hand), and linearlypolarized radiation refers to an admixture of an equal number of photons with right-hand and left-hand circular helicity. 33 In EPR experiments, linearly-polarized radiation is equivalent to a linearlyoscillating magnetic field. As seen in the selection rules section, depending on the direction of the field oscillation relative to the axis of the electron spin, the absorption of radiation can either transfer energy and angular momentum (photon type called σ + or σ − ) or transfer only energy (photon type called π ). Hence, π -type photon is only defined after getting involved in a transition [3]. The π -photon is still a photon of either helicity (positive or negative, right-hand or left-hand) but that induces a special type of transition. Fig. 2.7 shows how different radiation field directions, with respect to the static field, define the σ and π types of a photon, taking into account the direction of photon travel. 2.4 Multiphoton Transition As the radiation field intensity is increased, nonlinear behavior of the interaction between the radiation and the spin system emerges, and the rotating wave approximation becomes inaccurate, i.e., both photon helicities become effective. Using different techniques for solving Schrödinger’s equation like second quantization or Floquet perturbation theory, one can go beyond the rotating wave approximation and discover a richer theory of the magnetic resonance phenomenon including multiphoton transitions, Dicke effect, Bloch-Siegert shifts, Raman scattering, harmonic generation, etc. [7]. Here, we discuss the first two effects. In this section, we focus our attention on the multiphoton effects and discuss the Dicke effect in the following section. Multiphoton magnetic resonance has been known since the beginning of this century [8]. 34 Figure 2.7 Classification of σ and π type photon according to the direction of the radiation    field B1 and the photon propagation K with respect to the static magnetic field B0 . Although it had been widely studied in optical spectroscopy [9], multiphoton magnetic resonance has been rarely applied as a spectroscopic tool in EPR and NMR [7]. One of the earliest pieces of evidence for multiphoton transitions was observed from experiments with optically pumped atoms. Nuclear resonances were seen every time the Zeeman splitting approached an integer multiple of the frequency of the radiation field [10, 11]. Later, two-photon transitions were reported in cw EPR experiments, where parallel radiofrequency and perpendicular microwave fields were used simultaneously to irradiate free radicals in solids [12]. In most of the magnetic resonance experiments, only one photon is involved in each of the electronic transitions. The frequency and the helicity ( σ + or σ − ) of the photon involved satisfy conservation of energy as well as total angular momentum during any transition. For instance, due to the opposite directions of electron and proton magnetic moments, only σ + -photons can be involved in any EPR transition whereas NMR transitions involve only σ − -photons; see Fig. 2.8(a) and (b) for an illustration of single transitions in two-level EPR 35 Figure 2.8 Illustration of σ + and σ − single transitions (emission and absorption) in twolevel EPR (a) and NMR systems (b), respectively. (c) Two σ + and one σ − photons, each with energy equal to one third the Zeeman splitting, inducing a multiphoton transition in a two-level spin system. (d) Two-photon transition involving the essential π -photon in a two-level spin system. and NMR systems, respectively. Without loss of generality, we restrict our following discussion to transitions in EPR only. In addition to single photon transitions, several photons of different helicities can cooperate in one transition to meet the requirement of energy and angular momentum conservation. For example, Fig. 2.8(c) shows two σ + and one σ − photons, each with energy equal to one third the Zeeman splitting, inducing a transition in a two-level spin system. This specific combination of helicity and energy of each of the three photons insures the conservation of energy and angular momentum [7]. In general, for a transition with ∆m =±1 , a transverse radiation field can only induce a multiphoton transition that involves an odd number (2n + 1) of photons, with (n + 1) σ + and (n) σ − -photons. There is always one extra σ + -photon that compensates for the change in the electron spin angular momentum (±) before and after the transition. 36 On the other hand, for a misaligned (nontransverse) radiation field, both even and odd number of photons can be involved in a multiphoton transition. In the case of an even number of photons, π -photons are also needed to satisfy the conservation of total angular momentum (electron + photons); see Fig. 2.8(d) for an example of a two-photon transition in a two-level spin system. Intermediate spin states are also required to accomplish any multiphoton transition. The intermediate states can be real or even virtual based on the “dressed atoms” formalism [13, 14]. The Rabi frequency ωR( N ) , corresponding to a multiphoton magnetic resonance transition, with N photons, is proportional to B1N B0 N −1 [15]. This shows a slower Rabi oscillation than the fundamental one for a single-photon transition, where ωR = γ B1 . The photons involved in a multiphoton transition can be of the same or different energies. In the latter case, several radiation sources with different frequencies can be simultaneously used to irradiate the spin system. In the former case, scattering of a spin system with two identical photons is enabled by the presence of very large photon densities, i.e., the applied radiation intensity is very high. 2.5 Large Driving Field Regime In order to give more insights on the nonlinear characteristic of the interaction between radiation and spins, we study the dependence of the system states and the spindependent recombination current in organic diodes on the strength of the driving magnetic field. We consider two spin-1/2 particles (electron and hole polaron pair) driven by a   transverse oscillating magnetic field B1 in a static magnetic field B0 . As seen in Sec. 2.3.1, the effect of spin-spin interaction is predominantly to change the system states from the 37 product of independent states to coupled states. We assume here weakly interacting electron and hole spins and ignore their interaction term in the Hamiltonian. The Hamiltonian of the system is then given by [16]: Hˆ = ωe Sez + ωh S hz + 2ωR ( Sex + S hx ) cos ωt , (61) = ω0 + δ e,h , ω0 = γ B0 is the Larmor frequency, δ e,h is the average hyperfine field where ωe,h (in frequency unit) sensed by the electron and the hole, respectively (assume for simplicity that δ e > δ h ), ωR = γ B1 is the Rabi frequency, and ω is the frequency of the oscillating magnetic field. Solving the eigenvalue equation of the Hamiltonian, we get, for i = 1, 2,3, 4, the eigenstates ϕi ( B1 ) and the eigenvalues Ei ( B1 ) as a function of the driving field strength B1 . As seen in Fig. 2.9, the eigenstates of the polaron pair change continuously  from the product states in the case of extremely weak B1 to new combinations of coupled  states in the case of strong B1 . Similarly, the corresponding energy eigenvalues change between two sets of extreme values. In fact, for B1  δ b where δ b  δ e − δ h γ is the average difference in the distribution of hyperfine fields sensed by the electron and the hole, we get: ϕ1 ( small B1 ) = ↓↓ , ϕ2 ( small B1 ) = ↓↑ , ϕ3 ( small B1 ) = ↑↓ , ϕ4 ( small B1 ) = ↑↑ ,   (ωe + ωh ) = −ω0 − (δ e + δ h ) , 2 2   E2 ( small B1 ) = − (ωe − ωh ) = − (δ e − δ h ) , 2 2   E3 ( small B1 ) = (ωe − ωh ) = (δ e − δ h ) , 2 2   E4 ( small B1 ) = (ωe + ωh ) = ω0 + (δ e + δ h ) . 2 2 E1 ( small B1 ) = − 38 Figure 2.9 Energy diagram of a polaron pair showing the allowed and forbidden transitions between the prevailing eigenstates for the extreme conditions B1  δ b and B1  δ b . The diagram also illustrates the subsequent electron-hole recombination processes. These product states are commonly replaced by the coupled states ( T+ , T− , T0 and S ) in many references, either by expressing them in terms of ( T+ , T− , T0 , and S ) as a change of basis or by assuming a nonvanishing interaction between spins [6, 16, 17]. On the other hand, for B1  δ b , we get: 1 E1 ( large B1 ) = −  ω 0 − ω R , 2 , S= E2 ( large B1 ) 0, ( ) ϕ1 ( large B1 ) = T+ + T− − 2 T0 , ϕ2 ( large B1 ) 1 2 1 T+ + T− + 2 T0 , = 2 ϕ3 ( large B1 ) = ( T+ − T− ) , ϕ4 ( large B1 ) ( ) 0, E3 ( large B1 ) = E4 ( large B1 ) = ω0 + ωR . As seen before, the effect of an oscillating magnetic field is to induce transitions between the different states of the system according to the prevailing selection rules. However, as B1 increases, new selection rules evolve, leading to a change in the allowed transitions and, thus, the charge carrier recombination rates. 39 The dynamics of the polaron pair ensemble under the influence of an oscillating magnetic field, including the relaxation processes and recombination from singlet states, can be described by the following Liouville equation [6]: i  ρˆ , Hˆ  + S [ ρˆ ] + R { ρˆ − ρˆ 0 } , ρˆ =   (62) where ρ̂ is the density operator describing a two-spin-1/2 system, S describes the external changes of the ensemble, and R describes the spin relaxation processes. Solving Eq. (62), we get the decay rates of the Hamiltonian eigenstates as a function of B1 , given as the imaginary part of the Liouville equation eigenvalues. As shown before, the radiation-induced transitions tend to suppress the trapping ability of the T+ and T− states by mixing down the trapping states with the recombining triplet and singlet states [18]. This leads to a detectable current change in the organic diode. By incorporating the recovered B1 -dependent decay rates in the calculation of the spin-dependent recombination current change, the Dicke effect is shown as follows [19]: (1) The current change goes linearly with B1 as long as B1  δ b . This effect arises from the increasing efficiency of developing resonant Rabi oscillations in one of the polaron-pair partners under locally different hyperfine fields. (2) Upon increasing B1 beyond a certain limit, the transitions between the singlet and all other triplet states become strongly quenched (see Fig. 2.9) leading to the saturation in the current change and eventually the return to the same current that was in the absence of the driving field. (3) The new eigenstates, prevailing under the strong driving field B1  δ b , acquire different lifetimes, which means a change in the recombination rates. For instance, 40 the lifetime of the singlet state is cut in half and the nearest triplet state, 1 ( T+ − T− ) , acquires a long lifetime, which makes it a third trapping mode that 2 doesn’t recombine as is the case for the other two triplet states. The presence of three trapping modes now instead of two leads to further quenching in the recombination rates, which results in a current change in the opposite direction compared to the sign of the low-intensity EDMR signal. The strong driving field effects, explained in (2) and (3), arise from the fact that both polaron-pair partners indistinguishably and collectively resonate with the strong driving field that exceeds the individual spin state inhomogeneous broadening arising from the local hyperfine fields [17]. 2.6 Optical Pumping The alkali-metal Rubidium (Rb) is widely used in optical pumping experiments due to its experimentally feasible vapor temperature (~100 °C) and its suitable D1 resonance frequency that matches commonly-used solid-state diode lasers [20]. A simplified version of the energy-level diagram for 87Rb is given in Fig. 2.10. Here, we ignore the Rb nuclear spin. The ground state is the degenerate state 5S1/2 with both spin-up and spin-down configurations. Similarly, the first excited state is the 2-fold degenerate state 5P1/2. The energy splitting between the ground state and the first excited state corresponds to 376.24 THz or the so-called Rb D1 frequency. In order to spin-polarize Rb vapor, the latter is irradiated with a circularly-polarized D1 light. As a result, the Rb valence electron gets excited from the state 5S1/2 into the 5P1/2. In order to satisfy the conservation of total angular momentum (electron + photon), the excited electron can only transition between 41 Figure 2.10 Sketch of 87 Rb optical pumping using a simplified energy diagram that ignores the nuclear spin. Due to angular momentum conservation, a D1 σ − light excites the Rb valence electron exclusively out of the m j = 1 2 ground state (5S1/2) into the m j = −1 2 excited state (5P1/2). The electrons subsequently get relaxed nonradiatively after getting mixed up between the m j = ± 1 2 excited states by collisional means. Continuous repetition of the cycle results in a population imbalance in the ground state, i.e., the polarization of Rb. two states of opposite spin direction (i.e., from spin-up to spin-down state in the case of σ − light or from spin-down to spin-up state in the case of σ + light). The excited electrons get rapidly mixed up between spin-up and spin-down excited states due to mutual collisions between Rb atoms and collisions with a buffer gas. Subsequently, the equally populated excited states get relaxed nonradiatively into the ground states, and a population imbalance starts to build up after continuous repetition of the pumping cycle. The nonradiative characteristic of the relaxation is essential to avoid radiation trapping, which ruins the desired depopulation of the ground state. This is achieved by adding nitrogen to the gaseous sample under study. The N2 molecules have rotational vibration energy close to D1 and can be used to carry away the energy of the excited Rb atoms. Eventually, the system reaches a dynamic equilibrium between the absorption and relaxation rates. Depending on the helicity of the radiation, Rb optical pumping leads to the accumulation of the electrons in 42 either the m j = 1 2 or the m j = −1 2 states, which is equivalent to Rb polarization in either direction. In reality, 87Rb has a nuclear spin of 3/2. Fig. 2.11 shows the energy diagram of 87 Rb taking into consideration the nuclear spin and an external static magnetic field. The external field is commonly used with optical pumping to prevent stray fields from randomizing the direction of spins and consequently ruining the polarization. In Fig. 2.11, the Rb ground and first excited states are shown to be split into two hyperfine manifolds corresponding to the total angular momentum quantum number F=1 and F=2 [21]. Each hyperfine manifold is, in turn, divided into multiple Zeeman levels according to the azimuthal quantum number mF . The Zeeman splitting is small compared to the hyperfine splitting. Following the same reasoning we used in the simplified case, the total angular momentum (electron + photon) has to be conserved. This condition is realized here through ±1. Moreover, the radiation frequency has to be broader than the the selection rule: ∆mF = hyperfine splitting in order to avoid selective excitation into a nonextreme state. After repeated cycles of absorption and nonradiative relaxation according to the selection rules, we end up with spin-polarized Rb atoms where an excess of valence electrons accumulates in one of the extreme states mF = 2 or mF = −2 . 2.7 Spin Exchange Between Alkali-Metal Atoms and Noble Gas When a spin-polarized alkali-metal vapor (e.g., polarized Rb vapor by optical pumping) is put with a noble gas (e.g., Xe gas) in the same gaseous sample, the noble gas nuclei start exchanging spin angular momentum with the valence electron of the alkali- 43 Figure 2.11 Sketch of 87 Rb optical pumping taking into consideration the nuclear spin and an external static magnetic field (~30 G). The Rb ground and first excited states are split into two hyperfine manifolds, F= 1 and F= 2. Each hyperfine manifold is divided into multiple Zeeman levels with different mF . Due to angular momentum conservation, a D1 σ − light excites the Rb valence electron from a ground state in the 5S1/2 manifold to an −1. The electrons subsequently get relaxed nonexcited state in the 5P1/2, only if ∆mF = radiatively back into the ground states. Continuous repetition of the cycle results in the accumulation of the electrons in the mF = −2 extreme ground state (encircled with red), i.e., the polarization of Rb. 44 metal atoms and align their spins in the same direction. This spin alignment eventually develops nuclear spin polarization few orders of magnitude larger than the thermal one. The result is a hyperpolarized noble gas. This spin-exchange interaction is among a number of different interaction processes that take place simultaneously in the gaseous sample and include collisions with the container walls, spin-rotation interaction, and diffusion through field gradients. The spin exchange is realized through a through-space magnetic dipole interaction and a Fermi-contact interaction. The Fermi-contact interaction is due to the penetration of the Rb valence electron into the site of the noble gas nucleus. The Fermicontact contribution to the spin exchange dominates the through-space interaction, which has a much smaller effect in this case [22]. The Hamiltonian of the Fermi-contact hyperfine   interaction between the electron spin S of the alkali-metal atom and the nuclear spin K of   the noble gas nucleus can be written as: H = α K .S . The coupling constant α is given by: α= µ 8π 2 ge µ B K ψ ( R) , 3 K (63) 2 where ψ ( R) is the probability density of the valence electron of the alkali-metal atom at the position R of the noble gas nucleus, g e is the Lande g factor, µ B is the Bohr magneton, and µ K is the noble gas nuclear magnetic moment. By averaging the Hamiltonian over many collisions, one can show that the Fermi-contact interaction makes the alkali-metal valence electron experience an additional effective magnetic field due to the polarized noble-gas nuclei:  δB =  8π µ K κ AX [ X ] K , 3 K (64) 45 where [ X ] is the noble gas number density, and κ AX is the enhancement factor that we’ll  discuss in the following paragraph. This effective field δ B results in an alkali-metal EPR frequency shift that can be detected experimentally: = ∆ν 1 1   1 g e µ B 8π µ K = µ. δ B κ AX [ X ] K z , h ( I + 1 2) h ( 2 I + 1) 3 K (65)  where I is the nuclear spin of the alkali-metal nucleus and µ is the magnetic moment of the alkali-metal valence electron. κ AX is a dimensionless factor that describes the magnetic field enhancement due to the quantum mechanical Fermi-contact interaction between the noble gas and the alkali-metal atoms. If we ignore this type of microscopic interaction, κ AX would be equal to 1 and the magnetic field increment would only be due to the classical field of a magnetized sphere (macroscopic effect). Fermi-contact interaction can be divided into two types: interaction between free pairs (binary collisions) and interaction between bound pairs (van der Waals molecules). The contribution to the enhancement factor from the latter depends on the mean lifetime of the van der Waals molecules which, in turn, depends on the noble gas pressure in the cell [22]. At high enough pressure, the van der Waals interaction fully contributes to the enhancement and κ AX reaches its upper limit κ 0 . At low pressure, the van der Waals contribution is suppressed by the long molecular lifetime during which large precessions of the electron spin and nuclear spin about the molecular angular momentum take place. Those large precessions result in a reduction of the effective magnetic field increment and lead to the suppression of the enhancement factor by κ1 in the extreme case (i.e., κ AX= κ 0 − κ1 ). In general, κ AX = (κ 0 − κ1 ) + ε AXκ1 , where ε AXκ1 is the contribution from van der Waals molecules ( ε AX = 0 for minimum contribution and 1 for maximum contribution). Under most experimental conditions of 46 temperature and pressure, the van der Waals contribution to the enhancement factor is much smaller than that of binary collisions, which makes the κ AX generally given to a good approximation by κ 0 (κ 0  κ 0 − κ1 ). Eq. (65) can also be written in the following form: 1 g µ 8π κ0M 0 , ∆ν = e B h ( 2 I + 1) 3 (66) where M 0 is the noble gas longitudinal magnetization and it can be related to the noble gas polarization PX through the following relation: M 0 = µX [ X ]PX . Eq. (66) can be used to calculate κ 0 knowing both the EPR frequency shift of the alkali metal and the corresponding noble gas polarization. It can also be used to calculate noble gas polarization if we can measure the corresponding EPR frequency shift and we know κ 0 in advance. Hence, having a precise and accurate value of κ 0 for different noble-gas-alkali-metal systems is highly required. 2.8 References [1] C. P. Slichter, Principles of magnetic resonance (Springer Science & Business Media, 2013). [2] C. J. Foot, Atomic physics (Oxford University Press, 2005). [3] J. A. Weil, and J. R. Bolton, Electron paramagnetic resonance: elementary theory and practical applications (John Wiley & Sons., 2007). [4] M. H. Levitt, Spin dynamics: basics of nuclear magnetic resonance (John Wiley & Sons., 2001). [5] A. Abragam, Principles of nuclear magnetism (Oxford Univ. Press, London/New York, 1961). 47 [6] C. Boehme, and K. Lips, Theory of time-domain measurement of spin-dependent recombination with pulsed electrically detected magnetic resonance, Phys. Rev. B 68(24), 245105 (2003). [7] I. Gromov, and A. Schweiger, Multiphoton resonances in pulse EPR, J. Magn. Reson. 146(1), 110-121 (2000). [8] M. Göppert‐Mayer, Über elementarakte mit zwei quantensprüngen, Ann. Phys. 401(3), 273-294 (1931). [9] M. Ito and N. Mikami, Multiphoton spectroscopy, Appl. Spectrosc. Rev. 16(2), 299352 (1980). [10] J. Margerie and J. Brossel, Transitions à plusieurs quanta électromagnétiques, C. R. Acad. Sci. Hebd. Seances Acad. Sci. 241(4), 373-375 (1955). [11] J. Winter, Spectroscopie Hertzienne-etude de transitions faisant intervenir plusieurs quanta entre deux niveaux atomiques, C. R. Acad. Sci. Hebd. Seances Acad. Sci. 241(4), 375-377 (1955). [12] J. Burget, M. Odehnal, V. PetŘíček, and L. Trlifaj, Double quantum transitions in free radicals, Czech. J. Phys. B 11(10), 719-728 (1961). [13] S. A. Al'tshuler and B. M. Kozyrev, Electron paramagnetic resonance in compounds of transition elements (Keterpress Enterprise, Jerusalem, 1974). [14] C. Cohen-Tannoudji, Optical pumping and interaction of atoms with the electromagnetic field. Atoms in Electromagnetic Fields (World Scientific Publishing Co. Pte. Ltd. 2004). [15] J. Romhányi, G. Burkard, and A. Pályi, Subharmonic transitions and Bloch-Siegert shift in electrically driven spin resonance, Phys. Rev. B 92(5), 054422 (2015). [16] R. C. Roundy, and M. E. Raikh, Organic magnetoresistance under resonant ac drive, Phys. Rev. B 88(12), 125206 (2013). [17] S. Jamali, G. Joshi, H. Malissa, J. M. Lupton, and C. Boehme, Monolithic OLEDmicrowire devices for ultrastrong magnetic resonant excitation, Nano Lett. 17(8), 4648-4653 (2017). [18] R. K. Malla, and M. E. Raikh, Spin dynamics and spin-dependent recombination of a polaron pair under a strong ac drive, Phys. Rev. B 96(8), 085311 (2017). [19] D. P. Waters, G. Joshi, M. Kavand, M. E. Limes, H. Malissa, P.L. Burn, J. M. Lupton, and C. Boehme, The spin-Dicke effect in OLED magnetoresistance, Nat. Phys. 11(11), 910 (2015). 48 [20] Z. Ma, Frequency shifts during spin-exchange optical pumping of 3He and 129Xe and applications of hyperpolarized 129Xe, Ph.D. thesis, The University of Utah, 2012. [21] Z. L. Ma, E. G. Sorte, and B. Saam, Collisional He 3 and Xe 129 Frequency Shifts in Rb–Noble-Gas Mixtures, Phys. Rev. Lett. 106, 193005 (2011). [22] S. R. Schaefer, G. D. Cates, T. R. Chien, D. Gonatas, W. Happer, and T. G. Walker, Frequency shifts of the magnetic-resonance spectrum of mixtures of nuclear spinpolarized noble gases and vapors of spin-polarized alkali-metal atoms, Phys. Rev. A 39(11), 5613 (1989). CHAPTER 3 EXPERIMENTAL METHODS AND PREPARATIONS In this chapter, the methods for the physical systems that are utilized for the experiments described in Chapters 4 and 5 are discussed, i.e., it is outlined how the organic semiconductor samples and devices used for the observation of spin-dependent electronic currents that is described in Chapter 4 were made, and also how the vapor cells used for the optical spin pump experiments described in Chapter 5 were prepared. Although not directly related to the understanding of the experiments, the experimental methods described in the following have been essential for the accomplishment of the measurements. We start with the OLED fabrication, including template preparation and layer deposition. We then discuss the procedure by which vapor cells are fabricated and filled with various amounts of gases. 3.1 OLED Fabrication For the observation of charge carrier spin-permutation symmetry-dependent electric currents, bipolar injection devices based on a polymer semiconductor with weak spin-orbit coupling were prepared. Such devices, inherently built in order to facilitate charge carrier recombination, are essentially organic light-emitting diodes (OLEDs). The 50 devices used for the experiments reported in the following chapters were fabricated on a narrow (3.2 × 50mm) glass substrate. The devices themselves consist of an organic polymer layer (either Poly[2-methoxy-5-(2-ethylhexyloxy)-1,4-phenylenevinylene], MEH-PPV, or the super yellow light-emitting PPV copolymer, SY-PPV) were deposited between two gate dielectric layers of the polystyrene sulfonate doped hole conductor poly(3,4-ethylenedioxythiophene) (PEDOT:PSS) and Calcium, respectively. All polymer materials used in this dissertation were provided by Sigma Aldrich Corporation. The three layers are sandwiched between two conductive electrodes where Aluminum serves as the cathode and Indium Tin Oxide (ITO) as the anode [1]. The active area of the device is a 2 × 3mm pixel at the end of the substrate (see Fig. 3.1) defined by the overlap between the photolithographically patterned ITO and the masked evaporated aluminum [1]. See Chapter 4 for details about the electrical properties of the used OLEDs as well as the adopted model of the spin-dependent processes within the device. Figure 3.1 Sketch showing the lateral structure of the used OLEDs and how the ITO (purple) and the Aluminum (dark gray) electrodes are designed to sandwich the three intermediate layers (yellow) and connect to the device contact area. 51 3.1.1 ITO Patterning For the preparation of the lateral device structure, a 0.7mm-thick 50 × 75mm glass slide first prepared with a 120-nm thick layer of indium tin oxide (ITO) is provided by SPI supplies. The slide is first cleaned, i.e., sonicated in acetone for 5 minutes and isopropyl alcohol (IPA) for another 5 minutes, then dried out with nitrogen or placed on a hot plate for 5-10 minutes at 110 °C. The slide is then spin-coated with hexamethyldisilazane (DMHS) at 4000 rpm for 2 minutes followed by the C1813 positive photoresist at 3000 rpm for 1 minute. In order to dry the photoresist, the slide is then softly baked at 110 °C for 1 minute [2]. After loading the photolithography mask into the aligner (Suss MA 1006), the substrate is placed and aligned with the mask using the aligner microscope. The photoresist is then exposed to UV light with hard contact for 15 seconds at a separation of < 100 µ m . After removing it from the aligner, the substrate is submerged in “AZ developer 1:1” for 1 minute then rinsed thoroughly with deionized (DI) water and dried with nitrogen. For our experiments, we prepared a (½ HCl: ½ water) etch solution and immersed the substrate therein for 2.5-3 minutes. After rinsing thoroughly with DI water and drying with nitrogen, the etched area was tested for infinite resistance with a DMM. The substrate was finally cleaned with acetone and IPA preparing it for dicing. 3.1.2 Dicing As mentioned above, the OLED devices used in the EDMR experiments were fabricated on a narrow (3.2 × 50mm) glass template. In order to prepare such templates with the given dimensions, the previously prepared and lithographically patterned 52 substrate (see Fig. 3.2) was diced by a Disco (DAD 641) dicing saw. The dicing saw used a Disco diamond blade (P1A850 SDC320R10MB01) to perform 23 equidistant cuts along the width of the substrate yielding 22 individual matchstick-like templates. For the dicing procedure, the substrate was placed face down at the center of a tape applicator and filmframed. The framed substrate is then loaded into the dicing machine and aligned with the machine blade using the high-magnification camera and the ITO cross marks. After finishing the cuts, the frame was pulled out, dried with nitrogen, and cut out around the glass substrate with a regular utility knife. Figure 3.2 A photo of the used glass substrate with the lithographically patterned ITO before dicing. One of the repeated patterns is marked for emphasis. 53 3.1.3 Solution Preparation The active recombination layers in the OLED devices discussed in the next chapter were made predominantly from SY-PPV. This layer can be deposited by spin coating from a solution prepared in a glove box with inert nitrogen atmosphere. The preparation procedure for a homogeneous solution with a polymer concentration appropriate for thin-film deposition is described. This procedure also ensures minimal contamination of the polymer film: 1) Open the SY mother bottle and use the lid to pour a small amount of SY powder. 2) Put the small amount in a glass vial, close the mother bottle tightly, and return it to storage as soon as possible. 3) Distribute a small amount of SY powder to several glass vials and weigh each carefully (subtract the vial weight). 4) Make sure to have in each vial an amount small enough so that there is enough room for the solvent and big enough so that the weighing and volume measuring has a good precision (try to weigh 5-10 mg). 5) Take one of the vials and store the other vials carefully. 6) Put the vial in a holder stand in order to avoid getting knocked over accidentally. 7) Put a stirring bar in the vial. 8) Open a toluene bottle (close it quickly when you’re done). 9) Use the 1000 μL pipette with a new tip to add the calculated amount (rule of three) of toluene that gives the right concentration (5.5 g/L) of the solution. 54 10) Close the vial tightly and put it on the stirring plate between 350 and 700 rpm without heat and leave it overnight (make sure the stirring speed keeps the solution at the bottom of the vial without splashing over the wall of the container). 11) Once the solution becomes homogeneous, we can use it for device fabrication. 3.1.4 Layer Deposition Once the SY solution and the device templates are ready, the OLED device fabrication can take place: 1) Take a few glass templates and clean them in Acetone then IPA using an ultrasonic cleaner for 5 minutes each. 2) Use the Novascan UV/Ozone cleaner to clean the samples from carbonated contaminants by exposing them to UV light and O2 for 15 minutes. This step improves the adhesion of PEDOT:PSS to the ITO. 3) Prepare a PEDOT:PSS syringe and filter. Spin coat three drops of PEDOT:PSS at 200 rpm for 10 seconds then 3000 rpm for 40 seconds. 4) Bake the samples at 120 °C for 10 minutes (optional) 5) Put the samples into the glove box (inert nitrogen environment) after purging three times in the antechamber. 6) Spin coat the samples with toluene for cleaning purposes (optional) and make sure they are dry. 55 7) Take the previously prepared SY solution and spin coat the sample with SY (~10 μL on the pipette with a new tip) on the active region at 1500-2000 rpm for 40 seconds. 8) Bake the sample in a petri dish at 100 °C for 10 minutes. 9) Refill the evaporation bell jar and open it when it reaches atmospheric pressure. 10) Rotate the mask holder to a good orientation to insert the shadow mask. 11) Put in the evaporation shadow mask (used to define the geometrical shape of the evaporated metal) along with the samples. 12) Close the bell jar and start evacuation until the pressure is < 5 ×10−6 bar. 13) Under vacuum, evaporate 7nm of Ca then 150 nm of Al under the following Ca/Al electrodes settings:  Ca layer evaporation rate: 0.4 A/s without ramp.  Al layer evaporation rate: 1 A/s for 25 nm thickness, ramp for 3 minutes then 5 A/s up to 150 nm. 14) When it is done, start the refill and take out the samples. 15) Encapsulate the device with a few drops of epoxy (Huntsman, Araldite AW106 + hardener HV953). 16) Check the IV curve. We should get a diodic curve that looks like the one shown with the sample description in Chapter 4. As a rule of thumb, the resistance of the device should be measured to be around 2 M Ω . 56 3.2 Vapor Cell Preparation In this section, we describe the procedure with which the sample cells used in the measurement of κ 0 were handled and filled with various gas pressures. The sample cells are Pyrex glass spheres connected to a Pyrex tubing with a U-bend on one end and Tshape on the other; see Fig. 3.3. The relatively large U-bend serves as a kinetic trap that is easily held at low enough temperature to keep the melting Rb away from the vacuum system. The T-shape on the other end is called the retort. It is closed from the bottom and opened from the top. The retort is used to drop and hold the solid Rb ampule [3]. Figure 3.3 Schematic of a gas-handling system with a vapor cell photo. (left) The system is used to handle and fill sample cells with desired gas pressures with the minimum amount of contamination. The cells are part of a Pyrex manifold (shown in blue) that attaches to the vacuum system. After a Rb ampule is dropped into the retort, the manifold is sealed by melting the Pyrex at the inlet. The manifold is then evacuated with the Turbo pump while being heated for several days. Melted Rb is chased into the nearest cell with a softly-heating torch. A previously measured volume of the manifold is filled with a previously calculated pressure of Xe. The Xe gas is then released and collected into the cell immersed in LN2. Similarly, 3He and N2 are then consecutively allowed into the same cell. Finally, while held under LN2, the cell is pulled off of the manifold by melting its stem at subatmospheric pressure. (right) A photo of the vapor cell 204B showing its dimension, the pull-off stem, and the silvery Rb solid on the inside of the cell wall. 57 The entire Pyrex manifold is attached to the gas-handling system with a Cajon fitting. The retort is then sealed by melting the Pyrex at the top end, and the air inside the manifold is removed using the vacuum pump. To make sure the manifold is completely clean from any remaining residuals, the entire manifold was heated to about 130 °C by wrapping it with a heat tape for a few days or until the pressure gauge on the turbo pump stabilizes. At this point, we are ready to start filling the cell with Rb. By softly heating the retort with a torch, we chase the melted Rb that starts to escape to cooler spots of the manifold ending up in the nearest cell. After chasing the Rb into the nearest cell, the manifold is left to cool down for a few minutes. The cell should be ready now to receive the desired gases. Xe was first introduced by initially filling a previously measured volume at a previously calculated pressure. The pressure is calculated such that the amount of Xe contained in the premeasured volume will eventually give the desired pressure inside the cell. The Xe gas is then released into the manifold by opening the corresponding valves while one cell is held immersed in LN2. When Xe is fully absorbed into the extremely cold cell, all the pressure gauges read zero. Similarly, 3He and N2 are then consecutively added. In order to reach a homogeneous gas mixture inside the cell, the manifold was blown with consecutive hot and cold air near the U-bend for ~15 minutes. As the last step, the cell is removed from the manifold by melting its Pyrex stem while keeping the cell under LN2. The total pressure of the manifold was always less than atmosphere. This has the advantage of making the cell pull-off easier as the melted Pyrex tends to collapse inwards. Each manifold provides us with four cells. The same procedure 58 is repeated for each cell. 3.3 References [1] D. R. McCamey, H. A. Seipel, S. Y. Paik, M. J. Walter, N. J. Borys, J. M. Lupton, and C. Boehme, Spin Rabi flopping in the photocurrent of a polymer light-emitting diode, Nat. Mater. 7(9), 723 (2008). [2] S. Jamali, G. Joshi, H. Malissa, J. M. Lupton, and C. Boehme, Monolithic OLEDmicrowire devices for ultrastrong magnetic resonant excitation, Nano Lett. 17(8), 4648-4653 (2017). [3] Z. Ma, Frequency shifts during spin-exchange optical pumping of 3He and 129Xe and applications of hyperpolarized 129Xe, Ph.D. thesis, The University of Utah, 2012. CHAPTER 4 IDENTIFICATION OF MULTIPHOTON TRANSITIONS BETWEEN MAGNETIC DIPOLE STATES USING ELECTRICALLY DETECTED MAGNETIC RESONANT EXCITATION WITH VARIABLE DRIVE-FIELD HELICITIES 4.1 Introduction In a typical EPR experiment, a linearly oscillating driving field B1 is used to drive spins into magnetic resonance at a specific frequency determined by the Zeeman splitting static field B0 and other perturbations. The linear polarization of a small B1 can be treated as a circular one using the rotating wave approximation (RWA) [1, 2]. However, this approximation becomes inaccurate at high power of B1, where secondorder effects start to emerge, like Bloch-Siegert shift and multiple photon transitions [3]. The breakdown of the RWA, at high B1, i.e., under application of high radiation power, suggests that there is a detectable difference in the response of the spin system under magnetic resonance when linear or when purely circular B1 fields are applied. To scrutinize those effects, we carry out a magnetic resonance experiment using different polarizations (linear, elliptical, and circular) of B1 under nonlinear conditions. In order to ensure those conditions where the Bloch-siegert shift and other nonlinear magnetic resonance effects occur, we have applied strong driving fields B1 that 60 are on the order of the applied static magnetic field B0. Doing so is experimentally challenging due to the unavailability of large enough radiation sources for the static field (on the order of T) which is typically used in EPR spectrometers. Hence, we go the opposite way and carry out magnetic resonance at a very low static magnetic field B0, in the lower mT range. For the detection of magnetic resonance at low B0, where spin polarization is all but vanishing, we use previously demonstrated electrically detected magnetic resonance a (EDMR) spectroscopy scheme, where permutation-symmetry sensitive electron-hole pair transitions are probed, and the resulting spin-dependent recombination currents are measured in an organic light-emitting diode (OLED) [4]. Magnetic resonance under extreme conditions has been previously studied using high power linearly-polarized B1 [5, 6]. However, elliptically and circularly polarized B1 were not incorporated in these experiments. In fact, in addition to the Bloch-Siegert shift effects, Jamali et al. have seen a second resonance at exactly twice the B0 field of the main resonance [7], yet they attributed this resonance to a second harmonic caused by the nonlinearity of the radiation source and, thus, to an experimental artifact. This conclusion was entirely plausible for this singular observation that was made under maximum power conditions. However, this observed signal can just as well be attributed to a second-order quantum optical effects such as two-photon transitions that become significant when B1 becomes large [8, 9, 10]. Given these two fundamentally different yet plausible physical explanations for this double magnetic field spin resonance, an experiment allowing for a test of these two hypotheses is presented in this chapter. 61 4.2 Theory Without any physical interpretation, the second resonance can be called a g=1 resonance because using the same RF frequency ν ′ = ν and having the static field B0′ twice as big as the static field B0 at the main resonance implies that the g-factor has to be equal to 1: hν = g e β e B0 hν ′ = g e′ β e B0′ with g e = 2 (1) (2) but ν ′ = ν and B0′ = 2 B0 ⇒ g e′ = 1, where the prime designates the second resonance. So how could a two-photon EPR transition possible? Our system can be described by two spin-1/2 particles (an electron and a hole). In a static magnetic field, Zeeman splitting takes place, and four energy levels occur (three levels if the higher-order interactions, which split the intermediate levels, are neglected) corresponding to the four combinations of up and down spins or the four eigenstates of a spin system with = s 1= and s 0 ; see Fig. 4.1(a). Quantum mechanically, magnetic resonance is just the absorption process of a photon with a frequency equal to the Zeeman splitting between the two relevant energy levels. In the case of the g=2 resonance, this photon has to be a σ + photon for positive B0 and σ − photon for negative B0 in order to satisfy energy and angular momentum conservation at the same time. Both σ + and σ − photons exist in the case of linearly polarized B1 because linear polarization consists of negative and positive helicities at the same time. However, in the case of purely circular polarization, only one helicity is present: 62 Figure 4.1 Energy term diagrams and vectorial sketch of fields. (a) Term diagram of a twospin-1/2 system (e.g., an electron and a hole) for the g=2 resonance (left) involving one σ + photon and the g= 1 resonance (right) involving two photons, σ + and π . (b) Vectorial representation showing B1 and its components along the static field B0 and the effective field Beff due to the hyperfine fields Bhyp. (c) Sketch of linearly polarized B1 as a superposition of right-hand σ + and left-hand σ − circular polarization. either the σ + or the σ − [11]. This explains the fact that two g=2 peaks appear in a linearly polarized B1 driven EDMR measurement whereas only one peak appears in a circularly polarized B1 driven EDMR measurement; see Fig. 4.2. For the g= 1 resonance, the static field is twice as big as the field at the main resonance. This means the Zeeman energy splitting is now doubled compared to the conditions for the g=2 magnetic resonance. Using the same drive field frequency, we now need two photons to fulfill energy conservation for the transition. To conserve angular momentum, those photons cannot be of the same helicity but one photon has to be a σ ± , and the other has to be a π photon [12]. In a π -type transition, no angular momentum transfer occurs but only energy [8]. By definition, a π -photon is a photon with different k vector where B1 is parallel to B0, rather than perpendicular, as it is the case for σ ± photons. 63 Figure 4.2 EDMR measurements. (a) Plot of EDMR spectra obtained for SY-PPV OLED devices at room temperature for negative and positive static magnetic fields B0 . The spectra are measured with different B1 helicities (linear, elliptical, and circular polarization). B1 helicity is given by the phase shift angle between the two sinewaves fed simultaneously into the two perpendicular coils. The angle here is measured directly from the relative areas of the peaks. (b) The corresponding Lissajous figures along with the phase shift angles measured using the method explained in Fig. 4.5. 64 Generally, π -photons cannot induce a spin reorientation. However, at strong B1, simultaneous up and down spin flips can occur, which gives rise to a spectral line of extremely low intensity. In our setup, the RF coils are oriented such that B1 is always perpendicular to B0. So, where does the π -photon come from? In addition to the externally applied magnetic field B0 , there is a randomly oriented hyperfine field due to the nuclear spin surrounding the electrons and holes. In fact, in organic semiconductors, a lot of hydrogen atoms (i.e., protons with spin 1/2) are present. These give rise to local nuclear fields or the so-called hyperfine fields. Due to the long spin relaxation times of nuclear spins, these fields can be considered static in the dynamic range of electron spins. Thus, the electron spin can be considered to be exposed to an additional effective static magnetic field that produces a new quantization axis with regard to which B1 can be split into a perpendicular as well as a parallel component, as depicted in Fig. 4.1(b). When a circularly polarized B1 is applied instead of a linear B1, π -photons and one type of σ ± photons should still exist, yet the g = 1 resonance should change in magnitude. In contrast, if g = 1 resonance were to be caused by RF amplifier nonlinearities, no changes between linear and circular excitation would be observable. This is the reasoning that is used in the following for the study of the underlying physical nature of the electrically detected g = 1 resonance in OLED devices. 4.3 Experiment We built a radiofrequency-domain electrically detected magnetic resonance setup that allows for high power driving fields with arbitrary polarization; see Fig. 4.3. 65 Figure 4.3 Radiofrequency domain electron spin resonance spectrometer that allows for driving fields with arbitrary polarization; model/manufacturer designations appear in the text. A homogeneous and stable static magnetic field B0 is delivered by an electromagnet with a bipolar power supply (i) in a current control mode. B0 can be continuously swept from negative to positive and actively corrected with a feedback loop including an actively reading Gaussmeter. The RF coils (g and h) delivering B1 are built in a perpendicular arrangement such that B1 lies in a plane perpendicular to B0. The coils are simultaneously driven by the same RF source (a) using a power splitter (d). After the splitting, the RF frequency is phase shifted using a phase shifter (e), and amplitude equalized using a potentiometer (f). An OLED sample is used as a spin probe. The OLED is biased with a battery (c), and the current through is fed to a preamplifier than a lock-in and recorded on a PC. 66 4.3.1 B0 Setup The static magnetic field B0 is generated by an iron-core electromagnet (Varian V3603) driven with a 400-W bipolar power supply (Kepco BOP 20-20D) operating in a current control mode. Operated with a water-cooling system, the electromagnet coils could be driven with a maximum current of 160 A delivering up to 1 T of a static magnetic field. However, for a low-frequency/field spectroscopy, where B0 is in the mT range, the coils current used is below 3 A, and the water-cooling system is not needed. The power supply current output is regulated to better than 10 mA for line voltage variation and load resistance and temperature changes. As a result, the generated magnetic field is stabilized to better than 0.1 mT. The power supply current output is controlled linearly by a zero to ±10 V DC signal applied to the front panel current programming input terminals. The DC signal is provided by a computer through a National Instrument input/output interface (NI PCI-6221). The generated magnetic field is continuously swept at a speed of 0.2 mT/s through the positive and negative ranges passing through zero with no discontinuity effects. The magnetic field is actively monitored with a Gaussmeter (F. W. Bell 5080) and actively corrected through a feedback loop implemented using a LabVIEW Proportional-IntegralDerivative (PID) control program; see Fig. 4.4. The power supply is designed mainly for resistive loads; however, for the small inductance (~30 mH) of our electromagnet and low currents used in these measurements, the current oscillation is negligible. 67 Figure 4.4 LabVIEW program. (a) The front panel of the LabVIEW program. The program implements two main tasks: field control and signal detection. The front panel includes three graphs ( B0 vs. Time, Signal vs. Time, and Signal vs. B0 ), a panel for setting the center field and the sweep width and speed, and a panel for signal detection settings (Filter, modulation frequency, etc.). (b) Part of the program block diagram responsible for setting the starting field point through a PID function reading the Gaussmeter input and controlling the power supply. (c) Part of the program block diagram responsible for the field sweep. (d) Part of the program block diagram responsible for the signal detection through the lockin amplifier. 68 4.3.2 B1 Setup The goal was to build a setup capable of generating both a linearly-oscillating and a rotating driving field B1 with different helicities in the RF range. This is accomplished by generating two perpendicular oscillating fields of equal frequency ν and amplitude B , but different phase shifts ϕ . For instance, when the two oscillating fields are in phase (ϕ = 0°), the resulting B1 field is linearly oscillating between ± 2B at 45° from each perpendicular direction. When the two oscillating fields are 90° out of phase, the resultant B1 field is circularly rotating with amplitude B . A phase shift between 0° and 90° results in an elliptically rotating B1 field; see Fig. 4.5. The RF coils consist of two identical solenoids built in a perpendicular arrangement. The solenoid is made of 18-AWG magnet wire. Each one is 0.65-in long with 6 turns and 0.2-in diameter; see Fig. 4.6. The RF coils are simultaneously driven by two sinewaves from the same RF source (HP Agilent 8656B) using a hybrid coupler (Anareen, model 10011-3) as an equal power splitter. The phase shift between the two outputs of the splitter is adjusted using different coaxial cable lengths to the coils and also fine-tuned using a phase shifter (Sage laboratories, model 6501) installed on one of the outputs. The amplitude of the two sinewaves is equalized using a 1- kΩ potentiometer (Bourns, 3549H). For high-power measurements, the RF source was followed by a 100-W RF amplifier (ENI, model 5100). To monitor the different polarizations of B1, the voltage across a 2.75- Ω sensing resistor in series with each coil is measured on an oscilloscope (Agilent MSO6104A), and Lissajous figures are plotted; see Fig. 4.2 and Fig. 4.5. The RF sinewave is amplitude-modulated for the resultant magnetic resonance signal to be eventually detected using a lock-in amplifier referenced at the internally generated 69 Figure 4.5 Sketch of the RF perpendicular coils along with the driving sinewaves. The coils are driven with two sinewaves ( V1 and V2 ) of the same frequency and amplitude with a certain angle phase shift ( ϕ= ϕ x − ϕ y ). Depending on the phase shift, the resultant B1 field could be either linear, elliptical, or circular. The phase shift angle is given by ϕ = arcsin h2 . h1 70 Figure 4.6 RF coils and a probe head. (a) Photos of different RF coil arrangement designs for the generation of elliptically and circularly polarized EPR excitation. The 3rd photo also shows the sample holder with the biasing contact pins. The combination is conventionally called probe head. (b) Probe head design that was used in the work presented here. 71 modulation frequency (typically 10 kHz). 4.3.3 The Sample For spin-dependent recombination current measurements presented in this chapter, we used OLEDs, which consisted of an active organic polymer layer (SY-PPV) that was deposited between two charge carrier injection layers (PEDOT: PSS and Calcium for holes and electrons, respectively); see the inset of Fig. 4.7 and Fig. 3.1. The three layers are sandwiched between two conductive electrodes (Aluminum as the cathode and Indium Tin Oxide, ITO, as the anode) that inject electrons and holes, respectively. The gate dielectric layers are specifically chosen to help the injection of the electrons and holes into the active organic layer and prevent high gate leakage currents. This structure is a diode structure or more specifically, due to its radiative electroluminescent behavior, an organic lightemitting diode (OLED) for which a diodic I -V curve, like the one shown in the main panel of Fig. 4.7, is essential to verify its proper functionality. The injected electrons and holes (spin-1/2 particles) behave as free charge carriers within the organic layer that can form pairs due to Coulombic interactions; see Fig. 4.8. The excited electron-hole pair cascades down the states until it reaches a metastable polaron state (either triplet or singlet) [4]. This so-called polaron pair (PP) can dissociate back into free charge carriers conducting electric current or recombine forming a tightly-bound excitonic state that decays to ground state emitting light. The dissociation rates and recombination rates depend on whether the PP is in a triplet or singlet configuration. In the absence of any perturbation and in the presence of some relaxation processes, the whole system reaches a dynamic equilibrium (steady state) where the electric current and emitted light intensity become constant. Using a 72 Figure 4.7 I-V curve for a SY OLED measured with a source-meter (Keithley 2400). Operation point is typically around (2.6 V, 20 μA) . The inset shows the layer structure of an OLED sample; an organic polymer layer (SY-PPV) is squeezed between 2 conductive electrodes (Al and ITO). The Ca and the PEDOT: PSS, in between, are electron and hole injector layers, respectively. When the sample is biased, electron-hole recombination takes place and causes a current through the device. The glass substrate forms mechanically stable support for the above layers. 73 Figure 4.8 Model of spin-dependent recombination involving the excess electrons and holes within the active organic layer of an OLED device. First, the injected electrons and holes (spin-1/2 particles) behave as free charge carriers then form pairs due to Coulombic interactions, so-called polaron pairs (PPs). These PPs cascade down the energy states until they reach a metastable polaron pair state (either triplet PPT or singlet PPS). PPs can dissociate and thereby generate free charge carriers. They can also recombine under formation of excitonic states [singlet excitons (SE) or triplet excitons (TE)], which quickly decays to the ground state S0. The dissociation rate coefficients for singlet and triplet PPs, dS and dT, respectively, are mutually different. Similarly, the recombination rate coefficients for singlet and triplet PPs, kS and kT, respectively, are mutually different, too. Thus, using magnetic resonance (MR), one can manipulate the singlet and triplet populations, causing measurable changes in electric current and emitted light. 74 magnetic resonance scheme, one can then rotate the orientation of the electron spin or the hole spin, and hence manipulate the singlet and triplet populations, leading to measurable changes in electric current and even emitted light. It is this effect that forms the basis for magnetic resonance detection at nearly arbitrarily low RF frequencies and static magnetic fields. In our experiment, we bias the sample with a 6-V battery tapped with a potentiometer. The bias voltage is usually set to 2-3 V. The current through the device (typically 20 μA ) is initially converted into a proportional voltage using a low-noise current preamplifier (SRS SR570) then fed into a lock-in amplifier (Zurich HF2LI) to improve the measurement sensitivity and eventually recorded on a PC. 4.4 Results and Analysis 4.4.1 EDMR Spectrum Fig. 4.9 shows an electrically detected magnetic resonance (EDMR) spectrum given as the difference between the OLED current change with and without RF excitation as a function of the static field B0. The application of an RF frequency causes two recognizable resonance peaks in the positive and negative regions of B0. In addition to these magnetic resonance effects, organic semiconductors also show a conductivity response to a weak DC magnetic field in the form of resistivity or what we call organic magnetoresistance represented by the inverted bell-shaped baseline. Taking the difference of the first two curves shows the pure magnetic resonance spectrum filtered out of the magnetoresistance baseline. The same EDMR spectrum can also be measured with much better sensitivity (better SNR) using a phase-sensitive detection. At an RF frequency of 75 Figure 4.9 EDMR spectra. (a) 20-scan average of the change in the OLED current as a function of the static magnetic field B0 with and without RF radiation at 92 MHz. B0 is continuously swept over the negative and the positive regions showing a magnetoresistance behavior of the device. The difference between the two curves gives an EDMR spectrum with a dip in each region of B0. (b) 3-scan average of an EDMR spectrum (absolute value of the current change as a function of B0) measured at 92 MHz for negative and positive static field using an RF amplitude-modulated lock-in detection. The phase sensitive detection shows significant improvement to the signal to noise ratio and filters out the magnetoresistance baseline. 76 92 MHz, the magnetic resonance peak occurs at 3.2 mT (and -3.2 mT), which satisfies the magnetic resonance relation: hν = g e β e B0 with g e  2 . This implies that the measured magnetic species are nearly-free electrons and holes within the OLED device. 4.4.2 B1 Polarization In order to verify control over the helicities of B1, we conducted an EDMR experiment with different drive field polarization at low B1. Fig. 4.2 shows EDMR spectra for negative and positive B0 acquired with seven different polarization of B1 . The polarization of B1 is given by the angle phase shift between the RF sinewaves driving the RF coils and measured using the sensing resistors as explained in the previous section. The phase shift angle is given by: sin ϕ = h2 , h1 (3) where h1 and h2 are the ellipse large and small heights, respectively, as shown in Fig. 4.5. For near-linear polarization, the direction of the static field does not affect the resonance. However, if we go to the positive or negative helicities, only positive or negative static field causes the resonance. For instance, circular polarization shows only one peak, either in the negative or the positive region of B0. Linear polarization shows two peaks of equal amplitude in both the negative and positive regions. Elliptical polarization shows unequal peaks. Having this obvious relation between B1 polarization and the peak intensity, one can find the polarization of B1 right from the EDMR spectrum by taking the ratio of the peak areas: 77 sin ϕ = Ab − As , As (4) where Ab and As are the areas of the big and small peak, respectively. The agreement between the two phase-angle methods reduces towards the extreme helicities due to B1 inhomogeneity and the nonlinearity of the peak intensity. 4.4.3 g=1 Resonance In order to study the g= 1 resonance dependence on B1 helicities, we compare EDMR spectra for near-linear and near-circular polarization at increasing powers of B1; see Fig. 4.10. Starting with a linearly polarized B1, at low power ( B1 < 176 μT ), only the g= 2 resonance appears. However, if we keep increasing B1 power, at some point (176 < B1 < 340 μT) , the g= 1 resonance kicks in indicating a two-photon transition. As mentioned before, this is a second-order effect that gives rise to a low-intensity spectral line. To emphasize its presence, the g= 1 peak is 20-times magnified. The power dependence of this peak is studied at nine different powers of B1. A quadratic function shows a better fit to the data than a linear function. Both fits are contrasted on the same plot for comparison; see the middle inset of Fig. 4.10(a). For a circularly polarized B1, we only see g=2 resonance regardless of the power of B1, which may indicate the absence of the right π -photon needed for a 2-photon transition. We paid careful attention to make sure that the highest power spectra for linear and circular polarizations were measured at virtually the same B1 power. This way we excluded the fact that the absence of the g=1 peak with circular polarization is due to the lower power of B1. Of course, this is only true up to the error in the measured power ( ±40 μT ) of B1. 78 Figure 4.10 EDMR spectra at different B1 powers. (a) EDMR spectra extending across a wide range of B0 that encompasses the g=2 and g=1 resonance positions. The spectra are measured using near-linearly (3° ± 1°) polarized B1 at 125 MHz with three different powers. At high B1 power, a small g=1 resonance (also shown with 20 times vertical magnification) appears on either side of the spectra indicating a two-photon transition as depicted by the term diagram in the upper right corner. The middle inset is a plot of the intensity of the g=1 resonance peak as a function of B1. The plot also shows a linear and quadratic fit of the data points. (b) EDMR spectra extending across the g=2 and g=1 resonance positions and measured using near-circularly (85° ± 5°) polarized B1 with three different powers. Only the g=2 resonance appears regardless of the power of B1. This indicates the dominance of 1-photon transition depicted in the term diagram. 79 4.5 Discussion and Conclusion The g= 1 magnetic resonance observed for SY-PPV OLED devices at room temperature can be quantum mechanically interpreted as a two-photon transition [1]. This is reasonably verified by (1) the fact that the Zeeman energy splitting at the g=1 resonance is twice as big as the splitting at the main resonance while the excitation field frequency is the same (this means two photons are now needed to satisfy energy conservation during the transition), and (2) the dramatic difference between the application of linear or circular polarized excitation in terms of the g=1 resonance peak. This difference wouldn’t be there in the case of an experimental artifact. Fig. 4.10 shows that the g=1 resonance only appears as a low-intensity line even at high excitation power. The high excitation power, needed to make the g=1 resonance appear, is mainly due to the fact that multiphoton transitions are enabled by the presence of very large photon densities. The low-intensity of the spectral line even at high power is not unexpected due to the second-order behavior of such resonance [2]. The conclusion about the helicity dependence of the g=1 resonance highly relies on the fact that the highest power spectra for linear and circular polarizations are measured at virtually the same B1 power. However, one can argue that this is only true up to the error in the measured power ( ±40 μT ) of B1 but this is relatively small and not expected to cause a critical difference. Theoretically speaking, there doesn’t seem a reason that prevents the g=1 resonance from appearing when using a circularly polarized B1. In fact, π -photons should exist in a circularly polarized B1 as well [3]. However, the magnitude of the resonance is expected to change due to the difference in the number density of each type of photons in the case of different polarization. 80 The excitation field polarization has also been shown to have an effect on the main resonance at low power. With bipolar static field sweep and near-linear polarization, the magnetic resonance equally appears for both negative and positive static magnetic field. However, if we go to extreme helicities, the resonance only appears for positive or negative static field at a time. 4.6 References [1] C. P. Slichter, Principles of magnetic resonance (Springer Science & Business Media, 2013). [2] A. Abragam, Principles of nuclear magnetism (Oxford Univ. Press, London/New York, 1961). [3] J. Romhányi, G. Burkard, and A. Pályi, Subharmonic transitions and Bloch-Siegert shift in electrically driven spin resonance, Phys. Rev. B 92(5), 054422 (2015). [4] C. Boehme, and K. Lips, Theory of time-domain measurement of spin-dependent recombination with pulsed electrically detected magnetic resonance, Phys. Rev. B 68(24), 245105 (2003). [5] D. P. Waters, G. Joshi, M. Kavand, M. E. Limes, H. Malissa, P.L. Burn, J. M. Lupton, and C. Boehme, The spin-Dicke effect in OLED magnetoresistance, Nat. Phys. 11(11), 910 (2015). [6] S. Jamali, Detection of strong magnetic resonant drive effects using spin-dependent electronic transition rates in organic semiconductor materials, Ph.D. thesis, The University of Utah, 2018. [7] S. Jamali, G. Joshi, H. Malissa, J. M. Lupton, and C. Boehme, Monolithic OLEDmicrowire devices for ultrastrong magnetic resonant excitation, Nano Lett. 17(8), 4648-4653 (2017). [8] J. A. Weil, and J. R. Bolton, Electron paramagnetic resonance: elementary theory and practical applications (John Wiley & Sons., 2007). [9] R. C. Roundy, and M. E. Raikh, Organic magnetoresistance under resonant ac drive, Phys. Rev. B 88(12), 125206 (2013). 81 [10] R. K. Malla, and M. E. Raikh, Spin dynamics and spin-dependent recombination of a polaron pair under a strong ac drive, Phys. Rev. B 96(8), 085311 (2017). [11] Gromov, and A. Schweiger, Multiphoton resonances in pulse EPR, J. Magn. Reson. 146(1), 110-121 (2000). [12] A. Schweiger, and G. Jeschke, Principles of pulse electron paramagnetic resonance (Oxford University Press on Demand, 2001). CHAPTER 5 HIGH-PRECISION DETERMINATION OF THE FREQUENCY-SHIFT ENHANCEMENT FACTOR IN Rb-129Xe This chapter is a preprint of a paper submitted to Physical Review A in the year 2019 with coauthors Z. L. Ma, M. S. Conradi, and B. Saam. 5.1 Introduction Almost six decades have passed since the first report of a hyperpolarized noble gas [1], but the field continues to grow and evolve, both in terms of the basic physics and the manifold applications. Nuclei of the stable spin-1/2 isotopes, 3He [2] and 129Xe [3], may be polarized by spin-exchange optical pumping (SEOP) [4], a two-stage process of angular momentum transfer. The ground-state electron spins of an alkali-metal vapor are polarized by the absorption of circularly polarized resonant light at the D1 transition (5S1/2 → 5P1/2 in 87Rb); subsequent collisions of the polarized alkali-metal atoms with noble gas atoms mediate an interatomic Fermi-contact interaction through which electron and nuclear spin is exchanged. The result is an ensemble of noble-gas atoms with a nuclear-spin polarization in the range of 10-100%, several orders of magnitude beyond the thermal-equilibrium value at room temperature in even the largest laboratory magnetic fields. Hyperpolarized noble 83 gases (in some cases along with a cohabitating polarized alkali-metal vapor) are used in sensitive magnetometry [5], inertial guidance [6], and the search for physics beyond the Standard Model [7-9]. They are additionally used as a sensitive signal source in magnetic resonance imaging [10, 11]. EPR of the alkali-metal hyperfine structure can be a sensitive embedded probe of the magnetic field generated by the evolving noble-gas nuclear magnetization in a SEOP cell. The same Fermi-contact interaction that transfers spin to the noble gas also produces a shift in the alkali-metal EPR frequency ν A that is proportional to the nuclear magnetization [12]: 1 8π  dν ∆ νA =  A h 3  dB   M X κ AX  (1) [X] (2) where the nuclear magnetization M X is given by: M X = µX Kz K In Eqs. (1) and (2), [X], µX , and K are the noble-gas number density, magnetic moment, and spin, respectively; h is Planck’s constant, B is the applied magnetic field, and κ AX > 0 is a dimensionless factor specific to each alkali-metal/noble-gas pair that parameterizes the enhancement of the noble-gas magnetic field sensed by the alkali-metal electrons. The enhanced field results from the quantum mechanical overlap of the electron wave function at the noble-gas nucleus, time-averaged over many collisions. A value of unity for κ AX corresponds to the hypothetical case where the electron classically overlaps a continuous uniform noble-gas magnetization having the same value as that calculated by Eq. (2) for discrete noble-gas atoms. The enhancement is about 5 in the case of Rb-3He [13] and about 84 500 in the case Rb-129Xe [14], owing to xenon’s much greater atomic number. In this work, we present a precise measurement of κ 0 for the Rb-129Xe pair; the limit κ AX → κ 0 obtains for sufficiently high third-body gas pressure (see Sec. 5.2 below). Our measurement is based on a ratiometric comparison of the optically detected 87Rb EPR = 518 ± 8 , is in good agreement frequency shifts due to 129Xe and 3He. Our result, (κ 0 ) RbXe with a previous measurement, 493 ± 31 , also based on the measured ratio of 129Xe to 3He NMR frequency shifts [14]. It is also in good agreement with a recent theoretical prediction, 588 ± 50 , based on detailed electronic-structure calculations [15]. A precise measurement of (κ AX ) RbXe is relevant to better understanding of SEOP physics for the RbXe pair [16] and vital for understanding and correcting systematic shifts in several tests of fundamental symmetries that feature alkali-metal and/or noble-gas magnetometers and comagnetometers [17]. It can also be used to calibrate in situ polarimetry of hyperpolarized 129 Xe as it is produced by SEOP for the various applications. 5.2 Theory For an applied magnetic field B0 where the alkali-metal hyperfine splitting is large compared to the electron Zeeman splitting, the effective alkali-metal gyromagnetic ratio ±1 transitions within the same hyperfine manifold, correct to linear terms in B0 for ∆mF = is [18]:  g s µ B  2m F g s µ B  dν A  = B + Ο( B 2 ) + ⋅⋅⋅  , 1    A(2 I + 1)  dB  I ±1 2 2 I + 1   (3) where I is the alkali-metal nuclear spin, g s = −2.0023 is the free electron g-factor, and A 85 is the alkali-metal hyperfine coupling strength. The total angular momentum quantum number is F= I ± 1 2 , corresponding to the two hyperfine manifolds, and m F is the mean value of the two magnetic quantum numbers for the neighboring levels involved in the transition. Eq. (3) represents the quadratic Zeeman splitting into 4I hyperfine spectral lines, one for each F , m F pair. Under our experimental conditions, a highly polarized vapor means that the Rb atoms are pumped into the end states mF = ± F of the I + 1 2 manifold by σ ± polarized light. Spin exchange is mediated through both binary Rb-Xe collisions and the formation of longer-lived Rb-Xe van der Waals molecules [19]. As described in detail by Schaefer et al. [12], the enhancement factor may be written as: κ AX = (κ 0 − κ1 ) + ε AXκ1 (4) where 0 < ε AX < 1 characterizes the fractional suppression of the enhancement that occurs as the mean lifetime of RbXe van der Waals molecules increases, i.e., as the mean precession angle φ of the coupled angular momenta about the molecular magnetic field during a RbXe van der Waals molecular lifetime approaches and exceeds one radian [20]. At sufficiently large third-body pressure (short molecular lifetime), ε AX → 1 and κ AX → κ 0 . According to Zeng et al. [20]: φ= p0 , p (5) where p is the total gas pressure and p0 is a pressure that depends on gas composition and characterizes the transition from short to long molecular lifetime. For the Cs-Xe pair, p0 = 384 torr if the third-body gas is He [21]; we assume that this number is not very 86 different for the Rb-Xe pair. (For N2 as the third body, the difference is about 30% [12].) All of our cells contain at least four times this characteristic pressure of He. From Tab. II in Schaefer et al. [12], the ratio κ1 κ 0 ≈ 0.08 for the Rb-Xe pair; from Eq. 8 in Ref. [12], we calculate ε AX > 0.995 for all of our cells. We conclude that for our experiments, the true value of κ AX deviates from κ 0 by less than 0.1% in all cases, and we treat the effect of κ1 as negligible. In Rb vapor cells containing both 3He and 129Xe, one can perform rapid consecutive measurements of the EPR frequency shifts due to each noble-gas species (see Sec. 5.3). Using Eq. (1), the ratio of these shifts is: ∆ν Xe (κ 0 ) RbXe M Xe = ∆ν He (κ 0 ) RbHe M He (6) The ratio M Xe M He can be determined by NMR measurements of free-induction decay (FID) signals for the two species. Using the previously measured value of = (κ 0 )RbHe 4.52 + 0.00934T , where T is the temperature in °C [13]. Eq. (6) can then be used to determine (κ 0 )RbHe . Both the EPR frequency shift and the NMR signal intensity depend only on the product of the nuclear polarization and the noble-gas density, i.e., the magnetization M in Eq. (2); therefore, the measurement of (κ 0 )RbHe requires no knowledge of the polarization of either noble-gas species and is insensitive to uncertainties in the cell pressures listed in Table 5.1. The accuracy of the measurement comes down to (1) the degree to which we ensure the same initial nuclear magnetization for the EPR and NMR measurements (separately for both 3He and 129Xe), and (2) the degree to which the recorded measurement of the EPR and NMR signals are proportional to those 87 Table 5.1 Gas composition (in torr at 20 °C) of the three cells used in this work. Cell volume is shown in parenthesis below the cell designation. Cell 203C (7.75 cm3) Cell 204F (7.61 cm3) Cell 205B (7.1 cm3) Helium 2185 ± 85 1639 ± 65 2103 ± 80 Xenon 41 ± 2 22 ± 1 18 ± 1 Nitrogen 45 ± 2 34 ± 1 43 ± 2 2271 ± 85 1695 ± 65 2164 ± 80 Total magnetizations. 5.3 Experiment We performed experiments on three spherical vapor cells of similar volume made of borosilicate (Pyrex) glass; they were filled with several tens of Torr of N2 and Xe gases (the Xe is enriched to 90% 129Xe), about 2000 Torr 3He and a few milligrams of Rb (see Table 5.1). The oven temperature for SEOP ranged between 125 °C and 175 °C, as determined by a resistive thermometric device (RTD) affixed directly to the cell, although laser heating likely still causes the actual interior cell temperature to be somewhat warmer. In general, both 3He and 129 Xe were polarized to their respective maximum (saturation) values; we note that this typically took many hours for 3He and minutes for 129Xe, and that the latter may be polarized at lower temperatures and thus polarized to saturation several times in one experiment with little to no effect on the 3He polarization. In all cases, 3He and 129 Xe were polarized into the low-energy Zeeman state using σ − light. The SEOP pump laser was a 30-watt diode-laser array model A317B (QPC Lasers), externally tuned to the 795 nm D1 resonance and narrowed to ≈ 0.3 nm with a Littrow cavity [22]. This was 88 an older laser with at least many hundreds of hours of use, and the total output power had deteriorated somewhat; the narrowed output incident on the cell was typically 10-12 W. There are two distinct types of measurements made: (1) Separate measurements of the 3He and 129 Xe NMR signal intensities acquired with a Redstone NMR spectrometer (Tecmag); these measurements are made sequentially with a known RF-excitation (flip) angle at one Larmor frequency, adjusting the applied magnetic field B0 to the corresponding value for each nucleus (see Sec. 5.3.1); typical data is shown in Fig. 5.1. (2) Sequential optically detected (Faraday rotation) EPR measurements of the 87Rb frequency shift made at the same stable value of applied magnetic field (26.5 G); one made after destruction of the 3He magnetization and one made after destruction of the 129 Xe magnetization; the destruction is accomplished with several near-90° resonant RF pulses from the Redstone spectrometer, which is switched between the respective Larmor frequencies; typical data is shown in Fig. 5.2. Fig. 5.3. is a schematic diagram for the entire experiment, showing alignments of the pump and probe lasers, main-field and excitation coils, and the detection and amplification chains for both NMR and EPR. 5.3.1 NMR The NMR probe is a parallel-capacitance-tuned pair of coils (3 cm diam) separated by about 5 cm, each with multiple layered windings of 25/45 Litz wire (coil inductance was ≈ 30 μH). Since the goal here is to measure the signal ratio for 129Xe and 3He at known flip angles for each nucleus, we sought to eliminate all other sources of instrumental discrepancies and systematic effects by using the same cell, along with the same NMR 89 Figure 5.1 3He (top) and 129Xe (bottom) NMR signals. The left side shows the time-domain free-induction decay (FID) and the right side shows the corresponding Fourier transform relative to the carrier frequency. Signals were acquired at the same frequency (31.25 kHz) with the same RF power and excitation pulse length. 90 Figure 5.2 87 Rb EPR frequency shift as a function of time, measured relative to the initial recorded frequency f 0 = 18.6 MHz . At t = 0 , both 3He and 129Xe were fully polarized. At ≈ 90 s , the 3He magnetization was selectively destroyed with a comb of resonant ′ . At ≈ 290 s , the (85.6 kHz) NMR pulses causing a frequency-shift ∆ν He 129 Xe magnetization was similarly destroyed, yielding a frequency shift ∆ν Xe . The 129 Xe magnetization recovered in a few minutes to its original value; the 3He magnetization showed no appreciable recovery throughout the experimental run. 91 Figure 5.3 Schematic of experimental apparatus; model/manufacturer designations not noted here appear in the text. (a) The NMR spectrometer transmits excitation pulses to and receives the subsequent FID signal from a tuned coil placed near the cell. (b) Optically detected EPR consists of (1) The 795 nm pump laser with 10-12 W of power narrowed to ∼0.3 nm. (2) An 80-mW probe laser detuned by ≈ 1 nm that probes the Rb magnetization via Faraday rotation; the transmitted light (intensity modulated at the ≈ 19 MHz EPR frequency) is focused onto a fast photodiode. (3) A two-turn tuned EPR coil, located coaxially but outside the NMR coil, is driven by a voltage-controlled oscillator (VCO) through a ZHL-32A RF amplifier (Mini-Circuits). The ZAD-1 RF mixer (Mini-Circuits) homodynes the photodiode signal with the VCO output; the difference signal is fed to a model 186A lock-in amplifier (Princeton Applied Research) referenced to a 100-Hz sine wave that also modulates the VCO frequency with an amplitude much smaller than the transition linewidth. The derivative (error) signal at the lock-in output locks the frequency to the peak of the 87Rb hyperfine resonance. (4) A precision counter records the output frequency from the VCO and sends it to a PC-type computer running a LabVIEW program for display and analysis. (c) The applied magnetic field (26.5 G) is generated by a 60-cmdia. Helmholtz pair, the current stabilizer described in Fig. 5.5, and a 0.10-Ω monitor resistor. The voltage across the monitor resistor is used to correct the residual long-term current drift. 92 spectrometer, frequency, probe and amplifier settings for both measurements. For our operating Larmor frequency of 31.25 kHz, this required adjusting B0 to 26.5 G for 129Xe and 9.6 G for 3He. A home-built pre-amplifier coupled the probe to the spectrometer; signals were displayed and analyzed using the NTNMR (Tecmag) software. The spectrometer was also used (in transmit mode only) for rapid destruction of 129Xe and 3He polarizations at the frequencies 31.25 kHz and 85.6 kHz, respectively, as further discussed in the next section. 5.3.2 Optically Detected EPR EPR of 87Rb was optically detected in a manner similar to that described by Chann et al. [23], and a double-homodyning scheme was used to lock the EPR frequency to the measured magnetic field. A 80-mW external-cavity-tuned probe laser, model DL-7140201S (Sanyo), was directed transverse to the main applied magnetic field through the cell and then detected by a fast (2-ns rise time) silicon PIN photodiode, model 54-520 (Edmund Optics). The laser is detuned ≈ 1 nm from the 87Rb D2 resonance at 780 nm, where Faraday rotation of the plane of polarization has a well-characterized linear dependence on the 87Rb spin polarization along the propagation axis of the probe beam. The probe beam diameter is a few millimeters, small compared to the cell diameter. A weak cw RF excitation generates a steady-state precessing transverse 87 Rb magnetization at frequencies corresponding to the hyperfine resonances ( ≈ 19 MHz at 26.5 G for 87Rb). The resonance intensity at each hyperfine frequency was measured by converting the light-polarization modulation to a light-intensity modulation: a linear polarizer oriented at 45° to the nominal plane polarization of the probe laser is placed in front of the photodiode detector. The 93 detector output was homodyned with the RF source using a voltage-controlled oscillator (VCO) model 80 (Wavetek). The VCO could be swept with a voltage ramp across all of the resonances to generate the full hyperfine spectrum, where we note that the spectrum under optical pumping conditions is asymmetric: most of the intensity resides in the F=2 end resonance that corresponds to the helicity of the D1 pumping light; see Fig. 5.4. Operating the VCO with a low-frequency (≈ 100 Hz) modulation having an amplitude small compared to the resonance width generated the derivative of the more intense end resonance; the homodyned photodetector output was fed to a lock-in amplifier (model 186A; Princeton Applied Research) referenced to the modulation frequency; see Fig. 5.3. The zero-crossing of the derivative signal can be used in a loop fed back to the VCO to lock the EPR frequency, which was read out on a precision frequency counter, model 53220A (Agilent). In an otherwise stable applied field, the locking circuit could reproducibly follow both gradual and sudden changes in the noble-gas nuclear magnetization. 5.3.3 Stabilization of Applied Magnetic Field The measured EPR frequency shifts due to nuclear polarization were 1-10 kHz; we thus needed to stabilize the applied magnetic field B0 to much better than a part in 104. This was accomplished with the home-built current-stabilization circuit designed by one of us (M.S.C.) and shown in Fig. 5.5. Most commercial DC power supplies have some difficulty driving inductive loads in current-control mode. The supply used here (model 6267B; Hewlett-Packard) was run in voltage-control mode in series with a power MOSFET and a 250 W 0.5 Ω resistor (Vishay-Dale) having a temperature coefficient of 10 ppm/°C. 94 Figure 5.4 Optically detected 87 Rb EPR hyperfine spectrum under optical pumping conditions at 26.5 G from cell 204A; peaks are labeled by F , m F . A voltage-controlled oscillator (VCO) swept with a wide voltage ramp across the resonances generates the full hyperfine spectrum. Here, most of the intensity resides in the F = 2, m F = − 3 2 transition, because the σ − pumping light drives population towards the 2, −2 state. The F = 1 transitions are barely discernable (180° out of phase with F = 2) to either side of the 2, − 1 2 peak. 95 Figure 5.5 Current-stabilizing circuit design: a power MOSFET (A) (IRL2910PbF) is placed in series with a high-power shunt resistor (B) (Vishay-Dale 250 W, .5 Ω, 100 ppm/°C) and a high-precision variable resistor (E) (Vishay Accutrim 1240, 0-500 Ω, 10 ppm/°C). The shunt resistor is submerged in mineral oil to minimize temperature change due to air flow. The voltage across the resistors is compared to a stable voltage reference (F) (MAX6341) and fed into the gate terminal of the transistor through a noninverting OP27 operational amplifier (G) in a negative feedback mode. The gate voltage controls the drain-source current, which is approximately the same as the one flowing through the coils (C). The 1-μF capacitor is used to damp any current oscillation from the coils. The crossed diodes are used to protect the Op-Amp. (H) is the 0.10 Ω sensing resistor used to correct the current drifts. 96 The resistor was immersed in mineral oil for better thermal stability. The current through the MOSFET, running well out on the at portion of the I-VDS curve, was controlled via the gate input generated by comparing the voltage across the 0.5 Ω resistor with a stable voltage reference, thus stabilizing the current through the Helmholtz coils. The Helmholtzcoil current was additionally monitored with a precision digital multimeter (model 2000, Keithley) measuring the voltage across a 0.10 Ω stable resistor in series with the coils only. The monitored stabilized current was used to correct for baseline drifts in frequencyshift data acquired overlong time periods (many tens of minutes to hours). By running at certain times in the day (usually in the middle of the night) and employing these fieldstabilization and monitoring schemes, the rms frequency noise (typically integrated over 1.5 seconds) was reduced to ≤ 100 Hz , or about one part in 2 ×105 of the EPR frequency. Data were acquired and stored with the help of LabVIEW software (National Instruments) on a PC-type computer. 5.4 Results and Analysis An experimental run consists of (1) an NMR flip-angle measurement, (2) a measurement of the relative intensity of 129 Xe and 3He NMR signals, and (3) a measurement of the relative size of the corresponding 87Rb EPR resonance shifts due to the 129 Xe and 3He magnetizations. 5.4.1 Flip-Angle Measurement At some time after both 3He and 129Xe polarizations have reached their maximum (saturation) values ( > 20 h of SEOP), we started an experimental run by measuring the 97 magnetization ratio with NMR at 31.25 kHz. The measured initial height S of the freeinduction decay (FID) is proportional to the longitudinal magnetization from which it was generated through excitation at some flip angle θ . The magnetization ratio just prior to excitation is given by: M Xe S Xe sin θ He = M He S He sin θ Xe (7) Had we been able to operate in the regime where θ  1 radian for both nuclei, then the sine ratio in Eq. (7) could be replaced by a factor of the ratio of gyromagnetic ratios γ He γ xe [24], but this is not the case, and we had to carry out frequent and precise measurements of the flip angles used. This was done in a separate measurement by polarizing the 3He to saturation and then applying a long series of identical 80-μs pulses at 31.25 kHz, spaced 55 ms apart. The FID acquired after each pulse was Fourier transformed and the peak area was measured; see Fig. 5.6. The magnetization after the nth pulse was: n M ( n) M M 0 e n ln(cosθ ) = = 0 cos θ (8) Defining b ≡ ln ( cos θ ) , we fit data such as that shown in Fig. 5.6 to the function aebn with fitting parameters a and b then calculated: θ = arccos(eb ) (9) Provided we used all of the same electronics and the exact same gains and other settings, the 129Xe flip angle is given by: θ Xe = γ Xe θ γ He He The output of the NMR pulse amplifier is monitored with an oscilloscope before and after every NMR measurement to ensure that the pulse characteristics in the flip-angle 98 Figure 5.6 Signal intensity as a function of pulse number for eighty consecutive 80-μs pulses at the 3He Larmor frequency, 31.25 kHz; the rate is one pulse every 55 ms. The 3He FID after each pulse is collected, digitized, and Fourier transformed. The peak area is plotted vs. the number n of pulses and fit to an exponential decay (red line); see Eq. (8). measurement remain consistent over the course of the entire experimental run. 5.4.2 NMR Signal Acquisition The 129Xe NMR signal in our cells, particularly at low flip angles, is quite weak but recovers in minutes with SEOP. The 3He signal is strong but needs tens of hours to recover if destroyed. The EPR frequency shifts from 3He are also smaller than those for 129Xe. We needed to choose a 3He flip angle low enough to record the NMR signal and still have plenty of magnetization left to record a significant EPR frequency shift when the remaining magnetization was destroyed. If we wanted to use the exact same pulse (frequency, power, 99 and duration) for both species, we thus needed to acquire the 129Xe FID at an even smaller flip angle and then average many such acquisitions. Starting at full polarization for both 3He and 129Xe and with the SEOP pump laser on continuously, we apply N = 100 pulses 50 ms apart at the 129Xe Larmor frequency of 31.25 kHz in a field B0 = 26.5 G. The acquired FIDs are added together and Fourier transformed. We make the assumption that there is no significant SEOP occurring over the total acquisition time of 5 s. Due to the magnetization destroyed after each pulse, the total measured signal S N = ∑ n =1 S nXe needed correction to represent the signal SXe after the first N measurement: S Xe = Immediately after the 129 ∑ SN N −1 cos n θ n =0 (11) Xe NMR measurement, the applied field is lowered to 9.6 G by reducing the current in the power supply, corresponding to a 3He Larmor frequency of 31.25 kHz. A single 80-μs pulse is applied, and S He is immediately recorded for 3He without signal averaging. The RF pulse length, frequency, and power are unchanged between the 3He and 129 Xe NMR measurements, as are the amplifier gain settings on the receive side. 5.4.3 EPR Frequency Shift Acquisition After the NMR measurements, the 129 Xe magnetization is allowed to recover by SEOP (typically requiring no more than a few minutes). The applied magnetic field is returned to 26.5 G and stabilized using optically detected 87Rb EPR, as described in Sec. 5.3.2. A baseline EPR frequency was established prior to destroying the remaining 3He 100 magnetization with a rapid series of large-angle pulses at 85.6 kHz. A new baseline was ′ is recorded. We use the prime because this shift must established and the total shift ∆ν He still be corrected for the magnetization lost from the one RF pulse used previously to acquire a single FID: ∆ν ′ ∆ν He = He cos θ He (12) The 3He magnetization, once destroyed, did not recover to any significant degree during the remainder of the experimental run due to the low spin-exchange rate; this was verified at the very end of the run by repeating the above steps and noting no significant frequency shift. A few minutes after the 3He measurement, the 129 Xe polarization is similarly destroyed with a series of pulses at 31.25 kHz. Because the 129Xe magnetization recovers quickly via SEOP (within 3 min), we repeat its destruction and recovery five times for each experimental run, averaging the results to obtain the 129 Xe frequency shift ∆ν Xe . Unlike the case of 3He, this shift requires no correction, since the starting point is full polarization prior to each destructive series of pulses. See Fig. 5.2. There was a total of 48 experimental runs with three different sample cells (Table 5.1) and four different temperatures. The value of κ 0 for Rb-Xe and associated uncertainty was calculated using Eq. (6) for each run; results are shown in the plot in Fig. 5.7. Table 5.2 shows the uncertainty-weighted average values categorized according to both temperature and the sample cell used. The weighted average value for all measurements yields (κ 0 )RbXe = 518 ± 8 (green line in Fig. 5.7), where 8 is the standard deviation of the mean (dark green region) and the light green region ( ± 59) corresponds to the sample 101 Figure 5.7 Calculated values of κ 0 with the corresponding error bars from 48 measurements of three cells at four temperatures. The uncertainty-weighted average of κ 0 with its standard error is shown (dark green band) along with the weighted standard deviation (±59) for the whole set of data (light green band). standard deviation. 5.5 Discussion and Conclusion There appears to be little correlation of the measured value of κ 0 with the three different cells used. Cell 204F has the lowest total pressure and a 5% smaller value for κ 0 than the other two cells. As discussed in Sec. 5.2, we would not expect this to be due to κ1 in Eq. (4), as there is enough total gas pressure in all of these cells to assure that the 102 Table 5.2 Number of measurements (in bold) shown with the uncertainty-weighted average (with the standard deviation of the mean) of (κ 0 ) RbXe for each cell at each temperature measured. 125 °C 150 °C 165 °C 175 °C Average Cell 203C 3 596 ± 21 10 519 ± 13 … 4 537 ± 49 17 532 ± 14 Cell 204F 1 585 ± 53 12 490 ± 15 12 523 ± 21 … 25 501 ± 12 Cell 205B 5 527 ± 26 … 1 520 ± 29 … 6 526 ± 22 Average 9 551 ± 20 22 505 ± 10 13 523 ± 19 4 537 ± 49 48 518 ± 8 Van der Waals molecules are in the short-lifetime limit [19]. Any temperature dependence would appear to be weak at best; this is not unexpected due to the steep core wall of the RbXe van der Waals potential and the strong dependence of the contact interaction on the distance of closest approach. One could argue for a slight downward trend with increasing temperature, but the trend reverses for the (relatively few) data points taken at 175 °C. Looking at the overall scatter in the data, the 1.5% relative uncertainty in the weighted average may be a bit optimistic. If so, the source of any systematic error is likely to come from the NMR ratio measurement, since the frequency-shift measurements have better SNR and involve less post-processing to arrive at the 129Xe-3He signal ratio. Despite our careful efforts to calibrate the NMR equipment, drifts in sensitivity during the measurements due to thermally sensitive components or to inhomogeneities in the transverse excitation field cannot be completely ruled out. 103 To put our measurement into context with those preceding it, we first note that the result agrees well statistically with the results of Ma et al. [14], who measured (κ 0 ) RbXe = 493 ± 30 using an entirely different experimental method (comparing the NMR frequency shifts for 3He and 129Xe in the presence of a polarized 87Rb vapor), although they similarly relied on the previous κ 0 result for 87Rb-3He [13]. The present result is somewhat smaller than the recent theoretical/computational results of Hanni et al. [15] who = 588 ± 50 , but the discrepancy is not alarming considering the determined (κ 0 ) RbXe uncertainties. Earlier estimates and measurements by Schaefer et al. [12] were in the range of 650 to 750, but had much larger uncertainties, and so are also not inconsistent with our result. With appropriate rearrangement of Eqs. (1) and (2), our result can be used to calibrate the measured EPR frequency shift to the absolute PXe = K z 129 Xe polarization K . This relationship also shows that the largest measured EPR shifts in our cells, combined with knowledge of the Xe density, sets a lower bound for (κ 0 ) RbXe . 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Saam, Collisional 3He and 129Xe frequency shifts in Rb– noble-gas mixtures, Phys. Rev. Lett. 106, 193005 (2011). [15] M. Hanni, P. Lantto, M. Repisk`y, J. Mareˇs, B. Saam, and J. Vaara, Electron and nuclear spin polarization in Rb-Xe spin-exchange optical hyperpolarization, Phys. Rev. A 95, 032509 (2017). [16] A. Korver, D. Thrasher, M. Bulatowicz, and T. G. Walker, Synchronous spinexchange optical pumping, Phys. Rev. Lett. 115, 253001 (2015). [17] J. C. Allred, R. N. Lyman, T. W. Kornack, and M. V. Romalis, High-sensitivity atomic magnetometer unaffected by spin-exchange relaxation, Phys. Rev. Lett. 89, 130801 (2002). [18] G. Breit and I. I. Rabi, Measurement of nuclear spin, Phys. Rev. 38, 2082 (1931). [19] W. Happer, E. Miron, S. Schaefer, D. Schreiber, W. A. van Wijngaarden, and X. Zeng, Polarization of the nuclear spins of noble-gas atoms by spin exchange with optically pumped alkali-metal atoms, Phys. Rev. A 29, 3092 (1984). [20] X. Zeng, Z. Wu, T. Call, E. Miron, D. Schreiber, and W. Happer, Experimental determination of the rate constants for spin exchange between optically pumped K, Rb, and Cs atoms and 129Xe nuclei in alkali-metal–noble-gas van der Waals molecules, Phys. Rev. A 31, 260 (1985). [21] J. Hsu, Z. Wu, and W. Happer, Cs induced 129Xe nuclear spin relaxation in N2 and He buffer gases, Phys. Lett. A 112, 141 (1985). [22] B. Chann, I. Nelson, and T. G. Walker, Frequency-narrowed external-cavity diodelaser-array bar, Opt. Lett. 25, 1352 (2000). [23] B. Chann, E. Babcock, L. W. Anderson, and T. G. Walker, Measurements of 3He spin-exchange rates, Phys. Rev. A 66, 032703 (2002). [24] B. T. Saam and M. S. Conradi, Low frequency NMR polarimeter for hyperpolarized gases, J. Magn. Reson. 134, 67 (1998). CHAPTER 6 CONCLUSION As discussed in Chapter 4, the g=1 magnetic resonance of a free charge carrier in an organic polymer can be interpreted as a two-photon transition [1]. This is reasonably verified by the fact that the Zeeman energy splitting at the g=1 resonance is twice as big as the splitting at the main resonance but the excitation field frequency is the same. This means two photons are now needed to make the transition. Chapter 4 also shows that the g=1 resonance only appears (as a low-intensity line) when the excitation radiation field is linearly polarized at high power. The low-intensity of the spectral line even at high power is not unexpected due to the second-order behavior of such resonance [2]. The conclusion about the helicity dependence of the g=1 resonance highly relies on the fact that the highest power spectra for linear and circular polarizations are measured at virtually the same B1 power. However, one can argue that this is only true up to the error in the measured power ( ±40 μT ) of B1 but this is relatively small and not expected to cause a critical difference. Theoretically speaking, there doesn’t seem a reason that prevents the g=1 resonance from appearing when using a circularly polarized B1. In fact, π -photons should exist in a circularly polarized B1 as well [3]. However, one could hypothesize that having π -photons for linear polarization or for circular polarization is 107 practically not identical and that the studied transition is somehow governed by a selection rule, which is only satisfied by the π -photons of a linearly polarized field. The excitation field polarization has also been shown to have an effect on the main resonance at even low power. With bipolar static field sweep and near-linear polarization, the direction of the static field doesn’t affect the resonance. However, if we go to the positive or negative helicities, only positive or negative static field causes the resonance. In Chapter 5, the measured value of κ 0 appears to be just a little correlated with the different cells used. We report the lowest total pressure in Cell 204F, which also has a 5% smaller value for κ 0 than the other two cells. The temperature dependence appears to be very weak; this is most likely due to the steep core wall of the RbXe van der Waals potential and the strong dependence of the contact interaction on the distance of closest approach. Although only 1.5% relative uncertainty was reported in the weighted average of the κ 0 scatter, systematic errors could lead to a higher uncertainty. The source of systematic errors is mainly from the NMR ratio measurement where thermally sensitive components in the NMR equipment or inhomogeneities in the transverse excitation field cannot be ruled out. The frequency-shift measurements have better signal to noise ratio then NMR and involve less post-processing to arrive at the 129Xe-3He signal ratio. Our determined value of κ 0 agrees with the results of Ma, et. al. [4], who measured (κ 0 ) RbXe = 493 ± 30 using an entirely different experimental method, but relying on the same κ 0 result for 87Rb-3He [5]. The current result is somewhat smaller than the recent computational results of Hanni et al. [6]: (κ 0 )RbXe = 588 ± 50 and the earlier estimates by Schaefer et al. [7]: 650-750; but the discrepancy is not inconsistent considering the large 108 uncertainties. 6.1 References [1] I. Gromov, and A. Schweiger, Multiphoton resonances in pulse EPR, J. Magn. Reson. 146(1), 110-121 (2000). [2] J. Romhányi, G. Burkard, and A. Pályi, Subharmonic transitions and Bloch-Siegert shift in electrically driven spin resonance, Phys. Rev. B 92(5), 054422 (2015). [3] J. A. Weil, and J. R. Bolton, Electron paramagnetic resonance: elementary theory and practical applications (John Wiley & Sons., 2007). [4] Z. L. Ma, E. G. Sorte, and B. Saam, Collisional 3He and 129Xe frequency shifts in Rb– noble-gas mixtures, Phys. Rev. Lett. 106, 193005 (2011). [5] M. V. Romalis and G. D. Cates, Accurate 3He polarimetry using the Rb Zeeman frequency shift due to the Rb−3He spin-exchange collisions, Phys. Rev. A 58, 3004 (1998). [6] M. Hanni, P. Lantto, M. Repisk`y, J. Mareˇs, B. Saam, and J. Vaara, Electron and nuclear spin polarization in Rb-Xe spin-exchange optical hyperpolarization, Phys. Rev. A 95, 032509 (2017). [7] S. R. Schaefer, G. D. Cates, T.-R. Chien, D. Gonatas, W. Happer, and T. G. Walker, Frequency shifts of the magnetic-resonance spectrum of mixtures of nuclear spinpolarized noble gases and vapors of spin-polarized alkali-metal atoms, Phys. Rev. A 39, 5613 (1989). APPENDIX THROUGH-SPACE FIELD OF THE CELL PULL-OFF The pull-off is modeled as a cylinder of radius r and length 2L, centered at the origin of a cylindrical coordinate system; see Fig. A.1. According to Caciagli et al. [1], the magnetic field due to such cylinder with a uniform magnetization in the x-direction, is given by: = H ρ (ρ ,ϕ , z) where β ± = (k ) and P4= with γ = z±L ( z ± L) 2 + ( ρ + r ) 2 M 0 r cos ϕ [ β + P4 (k+ ) − β − P4 (k− )] , 2πρ , k±2 = (1) ( z ± L) 2 + ( ρ − r ) 2 ( z ± L) 2 + ( ρ − r ) 2 γ γ 2 (τ − κ ) + (γ 2τ − κ ) − κ + (κ − ε ) 2 2 1− γ 1− γ 1− k 2 ρ −r and κ , ε , and τ are the elliptic integrals of the first, second, and third kind, ρ +r respectively. H ϕ and H z are ignored here because they’re going to be zero in our case. We want the H-field at the center of the spherical cell. For a point along the axis of the cylinder, Eq. (1) can be simplified by taking the limit when ρ = 0 and ϕ = 0 : M  z+L z−L H ρ (0, 0, z ) = − 0 − 4  ( z + L) 2 + r 2 ( z − L) 2 + r 2  ,   (2) 110 Figure A.1 Sketch of the vapor cell modeled as a sphere of radius R connected to a cylinder of radius r and length 2L where R = 5r and L = 3r . The cylinder is centered at the origin of a cylindrical coordinate system. with H ϕ (0, 0, z ) = 0 and H z (0, 0, z ) = 0 . According to the measured dimensions of the vapor cell, the radius of the cylinder is r = 0.2 R and its length is 2 L = 1.2 R , where R is the radius of the sphere. In terms of the cylinder radius r, we get: R = 5r and 2 L = 6r ⇒ L = 3r . The center of the spherical cell is located 8r above the center of the cylinder, i.e., z = 8r . Substituting 3r for L and 8r for z in Eq. (2), we get the magnetic field at the center of the cell: H pull = −0.0038M 0 . 111 Point dipole approximation: The magnetic field along the z-axis for a point dipole located at the origin and pointing in the x-direction is given by: H ρ (0, 0, z ) = − 1 µcyl . 4π z 3 (3) The pull-off can be approximated by a point dipole at its center with magnetization µcyl = M 0 (π r 2 )(2 L). Substituting the expression of µcyl in Eq. (3), we get: M 0r 2 L 1 H ρ (0, 0, z ) = − . 2 z3 (4) Eq. (4) can also be derived by taking [1]: ρ , z  r , L in Eq. (1) or r , L → 0 then setting ρ = 0 and ϕ = 0 . Taking L = 3r and z = 8r , the magnetic field at the center of the spherical cell will be given by: H pull = −0.0029 M 0 MKS to CGS unit conversion: The conversion from MKS to CGS for the H-field and the magnetization M, is given by: H= H MKS × CGS M CGS = 4π , 103 M MKS . 103 So, the equation: H MKS = −0.0029 M MKS can be converted to CGS as follows: 112 4π 103 M  . = − ( 0.0029 × 4π )  MKS 3   10  = −0.037 M CGS H CGS = −0.0029 M MKS × ________________________________________________________________________ [1] A. Caciagli, R. J. Baars, A. P. Philipse, and B. W. Kuipers, Exact expression for the magnetic field of a finite cylinder with arbitrary uniform magnetization, J. Magn. Magn. Mater. 456, (2018), p.423-432.