Microscopic dynamics of isolated lithium in crystalline solidsrevealed by nuclear magnetic relaxation and resonance of 8LibyRyan Michael Lund McFaddenB.Sc. (Hons.), Mount Allison University, 2013A Thesis Submitted In Partial Fulfillment OfThe Requirements For The Degree OfDoctor of PhilosophyinThe Faculty of Graduate and Postdoctoral Studies(Chemistry)The University of British Columbia(Vancouver)April 2020© Ryan Michael Lund McFadden, 2020The following individuals certify that they have read, and recommend to the Faculty of Graduateand Postdoctoral Studies for acceptance, the dissertation entitled:Microscopic dynamics of isolated lithium in crystalline solids revealed by nuclearmagnetic relaxation and resonance of 8Lisubmitted by Ryan Michael Lund McFadden in partial fulfillment of the requirements for thedegree of Doctor of Philosophy in Chemistry.Examining Committee:W. Andrew MacFarlane Department of ChemistryResearch Supervisor University of British ColumbiaRobert F. Kiefl Department of Physics & AstronomyCo-Supervisor University of British ColumbiaCarl A. Michal Department of Physics & AstronomySupervisory Committee Member University of British ColumbiaSuzana K. Straus Department of ChemistryUniversity Examiner University of British ColumbiaChadW. Sinclair Department of Materials EngineeringUniversity Examiner University of British ColumbiaH. Martin R. Wilkening Institute for Chemistry and Technology of MaterialsExternal Examiner Graz University of TechnologyAdditional Supervisory Committee Members:Grenfell N. Patey Department of ChemistrySupervisory Committee Member University of British ColumbiaiiAbstractThis thesis reports measurements on the dynamics of isolated lithium in single crystalmaterials using ion-implanted 8Li β-detected nuclear magnetic resonance. From spin-latticerelaxation and resonance measurements, we identify the kinetic parameters describing the ion’ssite-to-site hop rate— the elementary process in long-range solid-state diffusion— and comparethe results with theoretical work in the literature, as well as experiments at higher concentration.In addition to these “ionic” details, the nuclear magnetic resonance probe provides informationon the electronic properties of the host, whose most intriguing features are also discussed.In the one-dimensional ion conductor rutile TiO2, we find two sets of thermally activateddynamics: one below 100K and another at higher temperatures. We suggest the low temperatureprocess is unrelated to lithium motion, but rather a consequence of electron polarons in thevicinity of the implanted 8Li+. Above 100K, Li+ undergoes diffusion as an isolated uncom-plexed cation, characterized by an activation energy and prefactor that are in agreement withmacroscopic diffusion measurements, but not with theory.In Bi2Te2Se, a topological insulator with layered tetradymite structure, implanted8Li+ under-goes ionic diffusion above 150K, likely in the van der Waals gap between adjacent Te planes. Acomparison with structurally related materials reveals the mobility of isolated Li+ is exceptional.At lower temperature, we find linear Korringa-like relaxation, but with a field dependent slopeand intercept, accompanied by an anomalous field dependence to the resonance shift. Wesuggest that these may be related to a strong contribution from orbital currents or the magneticfreezeout of charge carriers in this heavily compensated semiconductor.In the doped tetradymite topological insulators Bi2Se3:Ca and Bi2Te3:Mn, the onset of lithiumdynamics is suppressed to above 200K. At low temperatures, the nuclear magnetic resonanceproperties are those of a heavily doped semiconductor in the metallic limit, with Korringarelaxation and a small, negative, temperature-dependent Knight shift. From this, we make adetailed comparison with isostructural Bi2Te2Se.iiiLay SummaryIon transport in solids is crucial for the operation of many practical devices, such as lithium-ion batteries. At the microscopic level, this transport can be described in terms of the elementarytranslation motion between neighbouring empty sites in a crystal lattice. When many ions aremoving at once, their local arrangement can greatly influence their trajectories, resulting in acomplex situation that is challenging to understand. A much simpler situation, wherein onlya single isolated ion moves, is much easier to consider theoretically; however, this situationis nearly impossible to study experimentally. Using a low-intensity radioactive Li+ beam, weimplanted lithium in crystalline solids, easily achieving the isolated limit, and measured themobility of Li+. Results from these novel experiments were compared against existing theoreticalwork, as well as other experiments at higher concentrations.ivPrefaceThe work in this thesis is based on the synergistic collaboration between several researchgroups at the University of British Columbia and TRIUMF, as well as other contributors fromacross the globe. The unique experiments described herein required large accelerator facilities,where access to available instrument time is both highly competitive and limited. As a pre-requisite, all such experiments required the successful evaluation of a research proposal andallotment of time at the facility. Scheduled experiments were run around-the-clock (i.e., 24 ha day) for blocks of time on the order of a week or two, typically several times annually. Thelabour of running the measurements was, consequently, split into shifts and shared among theexperimenters. In all the work described herein, I participated in the data collection, with morespecific contributions listed below.Chapter 3 is based on the publications:R. M. L. McFadden, D. L. Cortie, D. J. Arseneau, T. J. Buck, C.-C. Chen, M. H. Dehn,S. R. Dunsiger, R. F. Kiefl, C. D. P. Levy, C. Li, G. D. Morris, M. R. Pearson, D.Samuelis, J. Xiao, J. Maier, and W. A. MacFarlane, β-NMR of 8Li+ in rutile TiO2, J.Phys.: Conf. Ser. 551, 012032 (2014)R. M. L. McFadden, T. J. Buck, A. Chatzichristos, C.-C. Chen, K. H. Chow, D. L.Cortie, M. H. Dehn, V. L. Karner, D. Koumoulis, C. D. P. Levy, C. Li, I. McKenzie,R. Merkle, G. D. Morris, M. R. Pearson, Z. Salman, D. Samuelis, M. Stachura, J. Xiao,J. Maier, R. F. Kiefl, andW. A. MacFarlane,Microscopic dynamics of Li+ in rutile TiO2revealed by 8Li β-detected nuclear magnetic resonance, Chem. Mater. 29, 10187 (2017)The original research proposal was written by W. A. MacFarlane, with all subsequent progressreports written by me andW. A. MacFarlane. The experiments were designed by me, in consul-tation with W. A. MacFarlane and R. F. Kiefl. I wrote the code and performed all of the analysis.The manuscript was written by me andW. A. MacFarlane. I wrote the first draft. All coauthorsparticipated in the discussion of the results and the revision of the manuscript.Chapter 4 is based on the work published in:R. M. L. McFadden, A. Chatzichristos, K. H. Chow, D. L. Cortie, M. H. Dehn, D. Fuji-moto, M. D. Hossain, H. Ji, V. L. Karner, R. F. Kiefl, C. D. P. Levy, R. Li, I. McKenzie,G. D. Morris, O. Ofer, M. R. Pearson, M. Stachura, R. J. Cava, andW. A. MacFarlane,Ionic and electronic properties of the topological insulator Bi2Te2Se investigated viaβ-detected nuclear magnetic relaxation and resonance of 8Li, Phys. Rev. B 99, 125201(2019).The original research proposal and progress reports were written byW. A. MacFarlane, who alsodesigned the experiments. I wrote the code and performed all of the analysis. The manuscriptvwas written by me and W. A. MacFarlane. I wrote the first draft. All coauthors participatedin the discussion of the results and the revision of the manuscript. The Bi2Te2Se crystal wassynthesized by H. Ji at Princeton University under the supervision of R. J. Cava.Chapter 5 is based on the draft manuscript:R. M. L. McFadden, A. Chatzichristos, D. L. Cortie, D. Fujimoto, Y. S. Hor, H. Ji, V. L.Karner, R. F. Kiefl, C. D. P. Levy, R. Li, I. McKenzie, G. D. Morris, M. R. Pearson, M.Stachura, R. J. Cava, and W. A. MacFarlane, Local electronic and magnetic propertiesof the doped topological insulators Bi2Se3:Ca and Bi2Te3:Mn investigated using ion-implanted 8Li β-NMR, arXiv:1911.12212 [cond-mat.mtrl-sci].The original research proposal and progress reports were written by W. A. MacFarlane. Theexperiments were designed by me andW. A. MacFarlane. I wrote the code and performed all ofthe analysis. The manuscript was written by me andW. A. MacFarlane. I wrote the first draft. Allcoauthors participated in the discussion of the results and the revision of the manuscript. TheBi2Se3:Ca and Bi2Te3:Mn crystals were synthesized by Y. S. Hor and H. Ji at Princeton Universityunder the supervision of R. J. Cava.The respective publishers grant permission to reproduce these aforementioned works in thisthesis, as well as other previously published figures.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAcronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Dynamics of ions in solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1 Diffusion in a lattice: discrete random walks . . . . . . . . . . . . . . . . 31.1.2 Temperature dependent kinetics . . . . . . . . . . . . . . . . . . . . . . 51.2 Measurement techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.1 NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.2 β-NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3 Thesis organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 Principles of ion-implanted β-NMR . . . . . . . . . . . . . . . . . . . . . . . . . 182.1 Fundamental properties of β-decay . . . . . . . . . . . . . . . . . . . . . . . . . 202.2 The TRIUMF implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2.1 Isotope production & transport . . . . . . . . . . . . . . . . . . . . . . . . 262.2.2 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2.3 Ion-implantation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2.4 β-NMR & β-NQR spectrometers . . . . . . . . . . . . . . . . . . . . . . . 352.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.3.1 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.3.2 Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41vii2.4 Data & Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 Microscopic dynamics of Li+ and electron polarons in rutile TiO2 . . . . . . . 493.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3 Results and analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.3.1 Spin-lattice relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.3.2 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.6 Supplemental material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.6.1 Biexponential relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.6.2 The 4c site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714 Ionic and electronic properties of the topological insulator Bi2Te2Se . . . . . . 744.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.3 Results and analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.3.1 Spin-lattice relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.3.2 Modelling relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.3.3 Resonance spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.4.1 High temperature lithium-ion diffusion . . . . . . . . . . . . . . . . . . 884.4.2 Low temperature electronic properties . . . . . . . . . . . . . . . . . . . 924.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.6 Supplemental material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.6.1 Implantation profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.6.2 Helicity-resolved resonance spectra . . . . . . . . . . . . . . . . . . . . . 994.6.3 8Li+ sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005 Local ionic, electronic, and magnetic properties of the doped topological in-sulators Bi2Se3:Ca and Bi2Te3:Mn . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.3 Results and analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.3.1 Bi2Se3:Ca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.3.2 Bi2Te3:Mn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.4.1 Dynamics of the Li+ ion . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.4.2 Electronic effects at low temperature . . . . . . . . . . . . . . . . . . . . 1195.4.3 Magnetism in Bi2Te3:Mn . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1235.6 Supplemental material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1235.6.1 8Li+ implantation profiles . . . . . . . . . . . . . . . . . . . . . . . . . . 1235.6.2 Helicity-resolved resonances . . . . . . . . . . . . . . . . . . . . . . . . . 1246 Summary & outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127viiiBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131A β-NMR probe ion concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . 158B Continuum diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163C Empirical compensation “laws” . . . . . . . . . . . . . . . . . . . . . . . . . . . 165Colophon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167ixList of TablesTable 1.1: Expected asymptotic limits of the spin-lattice relaxation rate 1/𝑇1, depend-ing of the dimensionality of the fluctuations causing relaxation. . . . . . 13Table 2.1: Properties of radionuclides used in β-detected nuclear magnetic resonanceexperiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Table 4.1: Properties of host nuclei in Bi2Te2Se relevant to nuclear magnetic reso-nance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Table 4.2: Results from the analysis of the 8Li spin-lattice relaxation 1/𝑇1 peaks inBi2Te2Se using Equations (1.3), (1.15) and (4.1). . . . . . . . . . . . . . . . 83Table 5.1: Arrhenius parameters obtained from the analysis of the temperature de-pendence of 1/𝑇1 in Ca doped Bi2Se3. . . . . . . . . . . . . . . . . . . . . 110Table 5.2: Results from the analysis of the 8Li resonance in Bi2Te3:Mn at high andlow temperature with 𝐵0 = 6.55 T ∥ (001). . . . . . . . . . . . . . . . . . . 115Table 5.3: Korringa analysis of Bi2Se3:Ca and Bi2Te2Se at 6.55 T and low temperature. 121xList of FiguresFigure 1.1: Sketch of a discrete random walk in one dimension. . . . . . . . . . . . 4Figure 1.2: Sketch of a mobile ion in a periodic one-dimensional potential. . . . . . 7Figure 1.3: Example of diffusion-induced spin-lattice relaxation. . . . . . . . . . . . 14Figure 1.4: Example of motional narrowing of a nuclear magnetic resonance linefrom translational motion. . . . . . . . . . . . . . . . . . . . . . . . . . . 16Figure 2.1: Radioactive decay scheme for 8Li. . . . . . . . . . . . . . . . . . . . . . . 21Figure 2.2: Parity violation in the β-decay of the positive muon μ+. . . . . . . . . . . 22Figure 2.3: Angular distribution of β-emissions from spin-polarized 60Co nuclei. . . 24Figure 2.4: Schematic of the low-energy beamline at TRIUMF’s Isotope Separatorand ACcelerator facility. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Figure 2.5: Electronic structure of the neutral 8Li atom and its optical pumpingscheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Figure 2.6: Comparison of the 8Li nuclear spin-polarization obtained from opticalpumping with thermal equilibrium. . . . . . . . . . . . . . . . . . . . . 30Figure 2.7: Typical alkali vapour cell bias scan used to spin-polarize a 8Li+ beam. . . 31Figure 2.8: Typical stopping profiles for an implanted ion simulated using the Stop-ping and Range of Ions in Matter Monte Carlo code. . . . . . . . . . . . 33Figure 2.9: Typical 8Li+ beamspot at the low field spectrometer. . . . . . . . . . . . 34Figure 2.10: Schematic of TRIUMF’s high-field β-detected nuclear magnetic reso-nance spectrometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Figure 2.11: Schematic of TRIUMF’s low-field β-detected nuclear quadrupole reso-nance spectrometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Figure 2.12: Illustration of the temporally inhomogeneous (statistical) error bars in apulsed spin-lattice relaxation measurement. . . . . . . . . . . . . . . . . 43Figure 2.13: Comparison of 8Li spin-lattice relaxation data in common oxide insulatorsnear room temperature at high field. . . . . . . . . . . . . . . . . . . . . 45Figure 3.1: The rutile TiO2 crystal structure. . . . . . . . . . . . . . . . . . . . . . . 50Figure 3.2: 8Li spin-lattice relaxation data in rutile TiO2 at high and low field. . . . . 54Figure 3.3: Results from the analysis of the 8Li spin-lattice relaxation measurementsin rutile TiO2 using at high- and low-field. . . . . . . . . . . . . . . . . . 55Figure 3.4: 8Li resonance spectra in rutile TiO2 with 𝐵0 = 6.55 T ∥ (100). . . . . . . . 58Figure 3.5: Results for the analysis of the 8Li resonance measurements at high fieldin rutile TiO2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60xiFigure 3.6: Arrhenius plot of the 8Li fluctuation rate in TiO2 extracted from spin-lattice relaxation and nuclear magnetic resonance measurements. . . . . 62Figure 3.7: Arrhenius plot of the Li+ diffusion coefficient 𝐷 in rutile TiO2 estimatedfrom the 8Li+ hop rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Figure 3.8: Summary of the sources of dynamics in rutile TiO2 revealed by8Li β-detected nuclear magnetic resonance. . . . . . . . . . . . . . . . . . . . 69Figure 3.9: Results from the analysis of the 8Li spin-lattice relaxation measurementsin rutile TiO2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Figure 3.10: View of the near-neighbour atoms to interstitial lithium in the rutile TiO24𝑐 site. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Figure 4.1: Crystal structure of Bi2Te2Se. . . . . . . . . . . . . . . . . . . . . . . . . 75Figure 4.2: 8Li spin-lattice relaxation data in Bi2Te2Se at high and low magnetic fields. 79Figure 4.3: Field dependence of 8Li spin-lattice relaxation data in Bi2Te2Se at 10K. . 80Figure 4.4: The spin-lattice relaxation rate for 8Li in Bi2Te2Se at high magnetic fieldswith 𝐵0 ∥ (001) using a stretched exponential analysis. . . . . . . . . . . 81Figure 4.5: Temperature dependence of the 8Li+ 1/𝑇1 in Bi2Te2Se at lowmagnetic field. 82Figure 4.6: Field dependence of the intercept and slope describing the low tempera-ture electronic relaxation in Bi2Te2Se. . . . . . . . . . . . . . . . . . . . . 84Figure 4.7: 8Li nuclear magnetic resonance spectra in Bi2Te2Se with 6.55 T ∥ (001). . 85Figure 4.8: Fit results for 8Li resonance in Bi2Te2Se at high field with 𝐵0 ∥ (001). . . 87Figure 4.9: Meyer-Neldel plot of the 8Li+ Arrhenius activation energy 𝐸𝐴 and prefac-tor 𝜏−10 pairs in Bi2Te2Se. . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Figure 4.10: Arrhenius plot of the 8Li fluctuation rate in Bi2Te2Se extracted from posi-tions of 1/𝑇1(𝑇max). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Figure 4.11: Arrhenius plot of the diffusion coefficient 𝐷 for 8Li+ in Bi2Te2Se. . . . . 92Figure 4.12: Comparison of the 8Li spin-lattice relaxation rate 1/𝑇1 in Bi2Te2Se withother materials where electronic relaxation dominates. . . . . . . . . . . 95Figure 4.13: Temperature dependence of the 8Li spin-lattice relaxation rate 1/𝑇1 inBi2Te2Se at 𝐵0 = 2.20 T ∥ (001) for two implantation energies. . . . . . . 97Figure 4.14: Stopping distribution and range for 8Li+ implanted in Bi2Te2Se. . . . . . 100Figure 4.15: Typical helicity-resolved 8Li resonance spectra in Bi2Te2Se. . . . . . . . . 101Figure 4.16: High-symmetry sites within the van der Waals gap of Bi2Te2Se. . . . . . 102Figure 5.1: Typical 8Li spin-lattice relaxation data in Ca doped Bi2Se3 at high and lowmagnetic field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Figure 5.2: Temperature and field dependence of the 8Li spin-lattice relaxation rate1/𝑇1 and stretching exponent 𝛽 in Ca doped Bi2Se3. . . . . . . . . . . . . 109Figure 5.3: 8Li resonance spectra in Bi2Se3:Ca at low magnetic field. . . . . . . . . . 111Figure 5.4: The resonance amplitude as a function of temperature in Ca doped Bi2Se3at 15mT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112Figure 5.5: Typical 8Li spin-lattice relaxation data in Mn doped Bi2Te3 at high andlow magnetic field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114Figure 5.6: Temperature dependence of the 8Li spin-lattice relaxation rate 1/𝑇1 inMn doped Bi2Te3 at high and low field. . . . . . . . . . . . . . . . . . . . 115Figure 5.7: Typical 8Li resonances in Mn doped Bi2Te3 and Ca doped Bi2Se3 at highmagnetic field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116xiiFigure 5.8: Stopping distribution for 8Li+ implanted in the doped topological insula-tors Bi2Se3:Ca and Bi2Te3:Mn. . . . . . . . . . . . . . . . . . . . . . . . . 124Figure 5.9: Stopping details for 8Li+ implanted in Bi2Se3:Ca. . . . . . . . . . . . . . . 125Figure 5.10: Typical helicity-resolved 8Li spectra in Bi2Se3:Ca, revealing the fine struc-ture of the line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126FigureA.1: Scaled three-dimensional 8Li+ implantation profile used to determinethe typical (peak) 8Li+ concentration in a β-detected nuclear magneticresonance experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160Figure B.1: Sketch of a (radio)tracer diffusion measurement. . . . . . . . . . . . . . 164FigureC.1: Illustration of the Meyer-Neldel rule for a kinetic process. . . . . . . . . 166xiiiGlossaryC General purpose compiled programming language developed since the early 1970s. It isregarded as the classical systems programming language, but its speed and general capabil-ities make it well-suited for a diverse range of applications, including scientific computing.Its popularity and ubiquity have made it the lingua franca in computer science. xiv, xv, 45,47, 167C++ General purpose compiled programming language derived from C and developed sincethe late 1970s. It allows the simultaneous use of high-level abstractions, through object-oriented techniques, and low-level programming interfaces, without sacrificing perfor-mance. xiv, xv, 47, 48, 80, 108, 167CERN Large physics laboratory near the France-Switzerland border operated by the EuropeanOrganization for Nuclear Research. As home to the world’s largest and most powerfulparticle accelerator, the laboratory’s main area of research is particle physics; however, italso hosts research programmes in nuclear and materials science. xiv, xv, 47Fortran General purpose compiled programming language developed since the 1950s. It findsheavy use for numeric computations, especially in the scientific community. xiv, xv, 47Gaussian Spectral lineshape that follows a normal distribution: 𝐺(𝑥) ∝ exp[−0.5(𝑥/𝜎)2]. xv,161, 164helicity The projection of a particle’s spin S onto its momentum p. More precisely, the scalarprojection ℎ = p ⋅ S/|p|, which adopts maximum/minimum values of ±1 for parallel/an-tiparallel orientations. 27, 28Lorentzian Spectral lineshape that follows a Cauchy distribution: 𝐿(𝑥) ∝ 1/(𝑥2 + 𝛾2). xv, 11,57, 59, 61, 109, 111, 114, 116, 126Matplotlib Graphics library implemented in Python for creating publication quality plots withtight integration in the SciPy ecosystem. 167MINUIT Numerical minimization routines in developed at CERN during the 1970s. Originallywritten in Fortran, modern implementations are available in C++ as part of the ROOTproject. 47, 53, 80, 86, 108, 167xivMonte Carlo A computing method based on random sampling named named after the casinoin Monaco. This numeric procedure is typically used to solve problems with a probabilisticinterpretation, especially where analytic approaches are not possible. xi, 32, 33, 52, 99,100, 106, 123–125, 159NumPy Software library for efficient handling of arrays in Python and part of the SciPy ecosys-tem. 167OpenMP A collection of compiler preprocessor directives and software library for developingroutines which run in parallel. The programming interface, available for Fortran, C, andC++, greatly simplifies the complexity of the task. 48Python An object-oriented scripting language with software libraries well-suited for scientificapplications (e.g., reading, manipulating, and plotting data). xiv, xv, 161, 167ROOT An object orientated data analysis framework written in C++ and developed at CERNsince the 1990s. It is the de facto standard software used by the nuclear and particle physicscommunities. xiv, 47, 53, 80, 86, 108, 167SciPy Python software library providing convenient access to fundamental algorithms forscientific computing. xiv, xv, 161, 167TRIUMF Canada’s particle accelerator centre, located in Vancouver, BC. Historically knownas the TRi-University Meson Facility, it is now owned and operated by a consortium ofCanadian universities. As a facility for internal and external users, the laboratory hostsmajor research efforts in physical, accelerator, and life sciences. v, xi, xix, 19, 20, 25, 26, 28,35–37, 44, 45, 52, 77, 106Voigt Spectral lineshape that is a convolution of Gaussian and Lorentzian profiles. Also knownas the Faddeeva function, it is the real part of the scaled complex complementary errorfunction. 31xvAcronyms1D one-dimensional. iii, xi, 3, 7, 12, 13, 17, 49, 59, 67, 69, 128, 1602D two-dimensional. 12, 13, 44, 82, 89, 90, 95, 97, 105, 109, 110, 118, 129, 160, 1613D three-dimensional. xiii, 11, 13, 24, 44, 59, 74, 78, 97, 98, 109, 110, 118, 119, 160, 161ARPES angle-resolved photoemission spectroscopy. 75, 96, 105β-NMR β-detected nuclear magnetic resonance. iii, x–xiii, 2, 15–20, 23–25, 28, 30, 32, 34–38,41–45, 51, 52, 68, 69, 76, 77, 88, 93, 98, 105, 106, 118, 123, 127–130, 158, 160–162β-NQR β-detected nuclear quadrupole resonance. xi, 34, 36, 37BCA binary collision approximation. 32BPP Bloembergen-Purcell-Pound. 11, 12, 81, 83, 84, 108, 113, 118, 127, 129BSC Bi2Se3:Ca. iii, x, xii, xiii, 17, 105–108, 111, 113–118, 120, 121, 123–127, 129BTM Bi2Te3:Mn. iii, x, xiii, 17, 105, 106, 113–119, 122–124, 127, 129, 130BTS Bi2Te2Se. iii, x, xii, 17, 74–85, 87, 88, 90–103, 105, 107, 108, 111–121, 123, 127, 129CCD charge-coupled device. 33–35, 159, 161CGS centimetre-gram-second system of units. 113CMMS Centre for Molecular and Materials Science. xix, 35, 45CPU central processing unit. 48CREATE Collaborative Research and Training Experience Program. xixCW continuous wave. 39, 40, 52, 78, 90, 159DC direct current. 34DFT density functional theory. 50, 64, 116EFG electric field gradient. 13, 15, 38, 39, 43, 56, 57, 61, 64–66, 71, 72, 79, 84, 86, 88, 90, 98,100–103, 110, 111, 116–118, 122xviENDOR electron nuclear double resonance. 51, 65, 66EOM electro-optic modulation. 28EPR electron paramagnetic resonance. 50, 51, 65, 68, 119, 121, 122FCC face-centred cubic. 12FWHM full width at half maximum. 16, 112, 115, 160, 161HB hole-burning. 15, 128HV high-voltage. 35HWHM half width at half maximum. 31IS impedance spectroscopy. 8, 61, 62ISAC Isotope Separator and ACcelerator. xi, 19, 26–28, 35, 77, 106ISOL isotope separation online. 25, 26IsoSiM Isotopes for Science and Medicine. xixKWW Kohlrausch-Williams-Watts. 44LE-μSR low-energy muon spin rotation. 76LLZO Li7La3Zr2O12. 13, 14LSAT (La,Sr)(Al,Ta)O3. 53m/q mass-to-charge ratio. 26MD molecular dynamics. 64MIT metal-insulator transition. 94MNR Meyer-Neldel rule. xiii, 7, 67, 83, 89, 128, 165, 166MPMS magnetic property measurement system. 106MUD MUon Data. 45μSR muon spin rotation/relaxation/resonance. 2, 18, 19, 23, 25, 35, 39, 41, 44, 45, 48, 71, 105,118, 122, 158NBM neutral beam monitor. 30NMR nuclear magnetic resonance. iii, x–xii, 2, 8–11, 13–16, 18, 19, 37–44, 51, 56, 57, 59, 61,62, 64, 65, 67, 68, 71, 75–77, 79, 81–85, 90, 91, 93–95, 97–99, 101, 105, 106, 108, 112, 115,117–124, 126, 128–130NQR nuclear quadrupole resonance. 36xviiNSERC Natural Sciences and Engineering Research Council of Canada. xixOA optical absorption. 8, 61, 62PEO polyethylene oxide. 16ppm parts per million. 41, 57, 86, 112, 120QL quintuple layer. 74, 75, 81, 102–104, 118RF radio frequency. 9, 35, 39–42, 56, 57, 66, 99, 122RF-μSR radio frequency muon spin rotation. 66RIB radioactive ion beam. xix, 19, 25–28, 31, 32, 35, 36RKKY Ruderman–Kittel–Kasuya–Yosida. 122S/N signal-to-noise ratio. 13, 18, 19, 41SAE spin-alignment echo. 15SLR spin-lattice relaxation. iii, x–xii, 9–15, 17, 40–45, 48, 51–55, 57, 62, 69, 70, 75, 78–81, 83, 86,88, 95, 97, 98, 106–109, 111, 113–120, 122, 123, 125, 127–129, 159SRIM Stopping and Range of Ions in Matter. xi, 32, 33, 52, 99, 100, 106, 123–125, 159–161SSR spin-spin relaxation. 9, 14STIX Scientific and Technical Information eXchange. 167STM scanning tunnelling microscopy. 15STS scanning tunnelling spectroscopy. 75, 105TI topological insulator. iii, xiii, 74–76, 78, 98, 104–106, 108, 116, 123, 124, 129, 130TMD transition metal dichalcogenide. 75, 101, 102, 104, 129TSS topological surface state. 74–76, 95, 99, 100, 104–106, 119, 123, 129, 130TST transition state theory. 5, 165UHV ultra-high vacuum. 35, 37, 40, 106vdW van der Waals. iii, xii, 74, 75, 77, 81, 83, 85, 88, 90, 100–105, 116–119, 122, 123, 125, 129,130VESTA Visualization for Electronic and STructural Analysis. 75, 102, 167YBCO YBa2Cu3O6+x. 33YSZ yttria-stabilized zirconia. 53xviiiAcknowledgementsFirst and foremost, I would like to thank my supervisor W. A. MacFarlane. I am indebted tohis patience and timely guidance, as well as the freedom he allowed during the pursuit of myacademic interests. Similarly, I would like to thank R. F. Kiefl, my second supervisor, whose sageadvice and input has been invaluable. To both: I am deeply appreciative of their enthusiasm forscience and general good humour — a combination which has made for a truly enriching andenjoyable tenure at the University of British Columbia. I am also thankful for the input from mythesis committee members throughout my studies.I am honoured to have worked alongside so many exceptional colleagues and collaborators.In particular, I would like to thank: D. L. Cortie, M. H. Dehn, V. L. Karner, A. Chatzichristos,D. Fujimoto, and J. O. Ticknor. Specifically, I am much obliged for: all of our stimulating andenlightening discussions; the sharing in the burden of labour; and all of the pleasant digressionsfrom scholarship. I am especially grateful for the guidance from D.L.C. during the early stagesof my studies.None of this thesis would be possible without the hard work of TRIUMF’s staff, especiallythose in the Centre for Molecular and Materials Science (CMMS). This thesis has benefitedenormously from the efforts of: G. D. Morris, I. McKenzie, S. R. Dunsinger, D. J. Arseneau, B.Hitti, D. Vyas, R. Abasalti, C. D. P. Levy, M. R. Pearson, R. Li, and M. Stachura. In particular, Iwould like to thank: G.D.M. and S.R.D. for my first “lesson” in radioactive ion beams (RIBs);D.J.A. andM.R.P. for their unwaveringwillingness to lend a hand; andM.S. for all of her guidanceand encouragement. I would also like to acknowledge the hospitality of the TRIUMF CMMSand thank them for providing both ample office space and a fast desktop computer.I gratefully acknowledge the financial support from my Natural Sciences and EngineeringResearch Council of Canada (NSERC) Collaborative Research and Training Experience Program(CREATE) Isotopes for Science and Medicine (IsoSiM) fellowship over the majority of this work.Last, but certainly not least, I would like to thank all of my friends and family for theirunconditional love and support throughout this endeavour, especially during its latter stages!xixChapter 1IntroductionThis thesis is about the study of the elementary, microscopic motions of Li+ in crystallinematerials. The sum of these individual atomistic displacements can give rise to long-rangeion transport, such as diffusion or conductivity (when driven by a gradient of the electrostaticpotential), a quality that is crucial for the operation and modelling of many functional devices(e.g., rechargeable batteries [1] or iontronics [2]). In real devices, thismacroscopic transport isoften limited by imperfections of the solid host (e.g., defects, interfaces, grain boundaries, etc.),but in order to clarify the origin of the bottleneck, amicroscopic understanding of ionic mobilityis required.While most practical devices relying on ion transport contain a large number of mobileions (i.e., at the atomic % level), at high concentrations the interactions between, as well as thelocal configurations of, the mobile ions become influential on their transport. For example,the repulsive Coulomb interaction between cations traversing through a solid will bias theirtrajectory, leading to correlatedmotion, making it challenging to simulate from first principles(see e.g., [3, 4]). On the other hand, as the concentration of the diffusing species is decreased tothe dilute limit, ion-ion correlation vanishes, greatly simplifying the transport dynamics; however,infinitely dilute conditions are nearly impossible to realize and study in most experiments. Sucha situation is precisely what the experiments in this thesis aim to probe. Here, we study thedynamics of isolated Li+ in the dilute limit. This situation is amenable to simulation with theory— the foothold for understanding the basic bulk behaviour before considering other influences1(e.g., lattice imperfections, correlated motion, etc.).Specifically, this thesis applies β-detected nuclear magnetic resonance (β-NMR) of the ion-implanted β-emitter 8Li [5] to study themobility of the simple (closed shell) light ion Li+ in highlyordered crystalline hosts. This unique spectroscopic approach is akin to conventional nuclearmagnetic resonance (NMR) using stable nuclei [6, 7], which is well-suited for the study of atomicmotions in solids (see e.g., [8–11]); however, the use of high-energy radioactive decay products inthe β-NMRdetection scheme (see e.g., [5, 12]) affords it with the necessary sensitivity to probe theultra-dilute limit. In this respect, it is quite similar to muon spin rotation/relaxation/resonance(μSR), which uses the positive muon μ+ as a probe [13–15]. A key distinction between 8Li β-NMRand μSR is that, in the former, the implanted 8Li acts as both the dopant and the NMR probe,providing a direct measure of Li+ dynamics.1 The short nuclear lifetime 𝜏𝛽 = 1.21 s [18] of8Liensures its maximum concentration never exceeds ∼1014 cm−3 (see Chapter 2 and Appendix A),providing experimental access to a regime where all the diffusers are effectively isolated.In the rest of this chapter, we begin with a short introduction to the mobility of ions in solidsin Section 1.1. In Section 1.2, an overview of NMRmethods for probing ionic motion is given.Finally, an outline of the organization of the remainder of the thesis in presented in Section 1.3.1.1 Dynamics of ions in solidsIt may at first seem surprising that crystals — the most stable and rigid physical phaseof matter — can support diffusive motion. To be clear, most of the atoms in a solid are notundergoing appreciable translational motion, otherwise the crystal would melt! That said,no crystal is perfectly rigid and it is well-known that all atoms in a crystal vibrate about theirequilibrium positions, giving rise to quantized normal modes or phonons. A key distinctionbetween vibrational and translational forms of motion is the relatively low energy cost fordisplacement and the zero average net displacement for the former. While the two are bothdriven by random thermal excitations of the host “bath”, diffusive motion involves a non-zerodisplacement of an atom.1Of course, μSR has been used to study ion dynamics in solids (see e.g., [16]); however, its sensitivity is primarilyproximal and complicated by the muon’s own propensity to mobilize at relatively low temperatures (see e.g., [17]).2While diffusion in all media requires vacant space in the immediate vicinity of the migratingatom (i.e., for it to move into), solids constrain these available sites based on the framework ofstatic atoms. The availability of these sites necessitates the presence of point defects — atomisticimperfections in the crystal, which are always present (at some level) to entropically lower thematerial’s overall free energy. The most common defects in solids are vacancies, consistingof an empty site nominally occupied by an atom, and insterstitials, where an atom resides inthe unoccupied space between lattice sites. At the microscopic scale, diffusion in a crystallinematerial is composed of elementary hops wherein an atom passes through either site. Themigration scheme for both defects is direct; an empty site and sufficient energy is all that isrequired for transport.2 These discrete displacements are a random, stochastic process, whichwe consider further in the next section.1.1.1 Diffusion in a lattice: discrete randomwalksA simple mathematical treatment of the motion of atoms in crystal is the discrete randomwalk [19], wherein the mobile ions occupy sites on a discrete lattice in space and undergostochastic hopping. An illustration of the one-dimensional (1D) discrete random walk is givenin Figure 1.1. Here, the walker has an average residence time 𝜏 in each site, correspondingto a hop rate 𝜏−1 for traversing the lattice. By analogy with a real solids, the jump length ℓ isapproximately fixed, defined by the distance separating nearest-neighbour sites. Superimposedon these translational motions are the vibrational displacements of the diffusing atom withinits potential well before making its jump to the next site. As we shall see later, these latticevibrations play an important role on the overall site-to-site hop rate 𝜏−1.With the basic ideas of a discrete random walk sketched above, we consider the diffusivity ofthemigrating species, which depends on the quantities that describe the elementary jump processfor the walker. Explicitly, the connection is given by the Einstein-Smoluchowski expression [20,21]:𝐷∗ =ℓ22𝑑𝜏−1, (1.1)2Note that other, more complicated diffusion mechanisms are possible (e.g., interstitialcy, where two (or more)atoms move together simultaneously); however, they are unimportant for this work and will not be consideredfurther.3Time tPosition rresidence time jump length lattice vibrations1D discrete random walkFigure 1.1: Sketch of a discrete random walk in one dimension. At each site, the walker residesfor an average time 𝜏 before making a stochastic jump to a vacant neighbouring site separated bya distance ℓ. The small displacements much shorter than ℓ represent the harmonic vibrationalmotion of the walker in the potential well at each lattice site. While the residence time at eachposition is drawn here to be identical, in reality it will vary according to a probability distribution.where 𝐷∗ is the diffusivity or diffusion coefficient, ℓ2 is the mean-squared jump length, 𝜏−1 is thehop rate (i.e., the reciprocal of the mean residence time), and 𝑑 = 1, 2, 3 is the dimensionality ofthe lattice. These ideas were first considered by Einstein [20] and Smoluchowski [21] for thecase of Brownian motion in fluids, but remain applicable to the solid state. In the limit of manysmall steps, the theory of discrete random walks crosses over to continuum diffusion, familiarfor fluid phases (see Appendix B).For interstitial diffusers at dilute concentrations, Equation (1.1) holds; however, for otherdiffusion mechanisms (or higher mobile ion concentrations), the motion of the walkers becomesbiased and the true diffusivity 𝐷 is related to 𝐷∗ by:𝐷 = 𝑓𝐷𝐷∗, (1.2)where 𝑓𝐷 is the so-called correlation factor, whose value depends on the geometric details of iontransport [22] and interactions among the diffusing species. In the work presented here, it is4taken that 𝑓𝐷 ≈ 1 and the terms 𝐷/𝐷∗ are synonymous.1.1.2 Temperature dependent kineticsAsmentioned earlier, diffusive motion is driven by random thermal excitations which supplythe mobile ion with enough energy to surmount a potential barrier to reach an adjacent vacantsite. In many ways, atomistic diffusive motion is quite similar to a chemical reaction, wherethe reaction rate constant often increases exponentially with increasing temperature, followinga simple Arrhenius relation [23] that is the result of a Boltzmann distribution of energies inthermal equilibrium at a temperature 𝑇. The hop rate in solid state diffusion also generallyfollows such a relation:𝜏−1 = 𝜏−10 exp (−𝐸𝐴𝑘𝐵𝑇) , (1.3)where 𝐸𝐴 and 𝜏−10 are the Arrhenius activation energy and prefactor, respectively. Consequently,by analogy with Equation (1.1), the diffusivity follows a similar temperature dependence:𝐷 = 𝐷0 exp (−𝐸𝐴𝑘𝐵𝑇) , (1.4)where the prefactor 𝐷0 can be written as:𝐷0 =ℓ22𝑑𝜏−10 . (1.5)The Arrhenius temperature dependence of 𝜏−1 (or 𝐷), can be understood in terms of ideasdeveloped from transition state theory (TST), which describe how reaction rate constants dependon temperature (see e.g., [24]). Within thermodynamic rate theory, the atomic hop rate 𝜏−1 isgiven by [25, 26]:𝜏−1 = 𝜏−10 exp (−Δ𝐻𝑚𝑘𝐵𝑇) , (1.6)where Δ𝐻𝑚 is the enthalpy of migration and the prefactor 𝜏−10 is given by:𝜏−10 = ̃𝜏−10 exp (Δ𝑆𝑚𝑘𝐵) . (1.7)5Here, Δ𝑆𝑚 is the migration entropy and̃𝜏−10 =∏3𝑁𝑖 𝜈𝑖∏3𝑁−1𝑖 𝜈𝑖, (1.8)where 𝜈𝑖 are the vibrational modes in the equilibrium (numerator) and saddle-point (denomina-tor) configurations, respectively. The interpretation of ̃𝜏−10 (and by extension 𝜏−10 ) is that thisprefactor or “attempt” frequency is related to the vibrational mode of the mobile atom alongthe migration pathway [26]. Based on a simple sinusoidal lattice potential between adjacentequilibrium sites [25], this “fundamental” migration frequency can be estimated, within theharmonic approximation, as:̃𝜏−10 ≈√𝑉02𝑚ℓ2, (1.9)where𝑉0 is the potential barrier,𝑚 is themass of the diffusing atom, and ℓ is the distance betweenequilibrium sites. This simple picture is compatible with the findings from many experiments;a simple Arrhenius analysis of 𝜏−1 vs. 1/𝑇 yields prefactors typically in the range of 1012 s−1 to1014 s−1, in good agreement with values predicted by Equation (1.9) (see Figure 1.2) [25].The goal then of any measurement of solid state ionic mobility is thus to quantify 𝐸𝐴 and 𝜏−10(or 𝐷0), from which all other quantities (with some elementary knowledge of the lattice) can bederived. Unsurprisingly, analyzing experimental data using amodel of the form of Equation (1.3)is most useful in practice, and valid so long as no curvature in a plot of 𝜏−1 vs. 1/𝑇 is apparent(e.g., from multiple diffusion mechanisms or transport mediated by quantum tunneling3).While the determination of 𝐸𝐴 is usually quite robust, as pointed out by Hänggi et al. [30]:In rate measurements, just as with theory, the detailed prefactor behavior of the rateis naturally more difficult to extract from measured data sets.Key to this difficulty is estimating the entropic contribution to 𝜏−10 (exp(Δ𝑆𝑚) in Equation (1.7)).Using themodel in Equation (1.9), such an estimate is possible from the ratio of the experimentaland predicted prefactors [31]. Note that, for reasonable values of barriers and jump lengths,3For all but the lightest isotopes, quantum mechanical effects, such as tunnelling, are unimportant and themigration process can be treated as a purely classical traversal of a potential barrier. For example, quantum effectsare important for the muon at low-temperature [28, 29]; however, these are completely suppressed for the (relatively)much heavier lithium isotopes.61 0 1 2x /0.00.20.40.60.81.0V(x)/V0A0.0 0.2 0.4 0.6 0.8 1.0Potential barrier V0 (eV)0.51.01.52.02.5Hop length (Å)Mass m=8.0225 u0.5×1013 s11.0×1013 s12.0×1013 s14.0×1013 s 1B0246810V 0/(2m2 ) (1013 s1 )Figure 1.2: Sketch of a mobile ion in a periodic 1D potential. (A) The migration path of an ionin a solid, approximated by a 1D sinusoidal potential with a barrier height 𝑉0 and a jump distanceℓ. The solid and dashed segments of the line denote the potential between neighbouring sites andthe periodicity of the potential, respectively. (B) The attempt frequency 𝜏−10 (i.e., the Arrheniusprefactor) for a mobile ion traversing a 1D potential barrier within the (classical) harmonicapproximation [25], given by Equation (1.9). The magnitude of the prefactor is representedby the cividis colourmap [27]. For typical values of ℓ (∼1Å) and 𝑉0 (∼0.5 eV) in solids, 𝜏−10remains on the order of ∼1013 s−1. The mass𝑚 = 8.0225 u used here corresponds to that of theradionuclide 8Li.Equation (1.9) invariably yields ̃𝜏−10 ∼ 1013 s−1 (see Figure 1.2). It is not uncommon, however,to find experimental prefactors which are different (i.e., larger or smaller) by several orders ofmagnitude. Such prefactor “anomalies” [32] imply either large or small migration entropies,depending on the scale of the anomaly. Alternatively, one complication that can arise is thestrong correlation between the measured activation barriers and prefactors, due to the relativelyfew degrees of freedom available to paramaterize the kinetic process (cf. Equation (1.3)). Thiscorrelation can manifest as an empirical compensation “law” known colloquially as the Meyer-Neldel rule (MNR) [33], wherein for related measurements of a single kinetic process, 𝜏−10 (or𝐷0) increases exponentially with increasing 𝐸𝐴. A more detailed account is given in Appendix Cand we shall consider the influence of such effects later in Chapters 3 and 4.71.2 Measurement techniquesQuite commonly, ion diffusion is studied using techniques sensitive to ionic motions overmacroscopic length scales, such as electrical conductivity (e.g., impedance spectroscopy (IS)) orvarious means of measuring concentration profiles within a solid (e.g., optical absorption (OA)).Such macroscopic methods, however, yield only the overall diffusivity of all migration pathways,making no microscopic distinction among them. Their results could thus be dominated bysmall, highly mobile regions of a sample that are not representative of its intrinsic behaviour.Similarly, conductivity measurements become increasingly difficult if the material is a mixedionic/electronic conductor, where it can be challenging to isolate the respective contributions.Key to the study of microscopic diffusion is the ability to distinguish between such regions (andcontributions). NMR techniques are one approach well-suited to just such a task [8–11, 34–38].Comprehensive treatments of the subject can be found in several authoritative monographs(e.g., [6, 7]) and only the essentials will be given here.1.2.1 NMRNMR is based on the Boltzmann polarization of nuclear moments of stable nuclei, whichproduces a macroscopic magnetization M in a static applied magnetic field B0. At normaltemperatures,M is very small, but measurable. For a collection of nuclei with nuclear spin 𝐼and number density 𝑁, the (thermal) equilibrium magnetizationMeq is given by:Meq ≈ 𝑁𝛾2ℏ2𝐼(𝐼 + 1)3𝑘𝐵𝑇B0, (1.10)which is just the familiar 1/𝑇 (nuclear) Curie law and a macroscopic measure of the degreeof nuclear spin-polarization. The remaining factors are the reduced Planck constant ℏ, thenuclear spin quantum number 𝐼, and the gyromagnetic ratio 𝛾, which defines the proportionalitybetween 𝐵0 and the NMR (or Larmor) frequency 𝜔0 for a given nucleus:𝜔0 = 2𝜋𝜈0 = 𝛾𝐵0. (1.11)8In the case of purely magnetic interactions, 𝜔0 is a measure of the energy splitting between the2𝐼 + 1 Zeeman magnetic sublevels levels 𝑚𝑖 (with 𝑖 = −𝐼, −𝐼 + 1, … , 𝐼 − 1, 𝐼), typically in theradio frequency (RF) range. Inside a material, local electromagnetic fields modify the preciseenergy splitting of these levels. These small perturbations Δ𝜔 provide a sensitive measure of theNMR probe’s immediate environment, including any dynamic modifications to it.In a typical NMR measurement, a large homogeneous B0 ≳ 1 T is applied along the ̂𝑧direction, producing a magnetizationM =Meq parallel to B0 (i.e., an𝑀eq𝑧 ≠ 0). Subsequently,this longitudinal component of M is manipulated and its time-evolution monitored, usually intwo steps:1. resonantly tiltingM out of equilibrium using a (pulsed) transverse RF field B1(𝑡);2. and detectingM(𝑡) (after a time-delay 𝑡) as it “relaxes” back to equilibrium.The time evolution of the reorientation of M can be described classically by the phenomenologi-cal Bloch equations [39], written here in vector form:4dMd𝑡= 𝛾 (M × B) − ?̃? ⋅ (M −Meq) , (1.12)where B = B0 + B1(𝑡) and ?̃? is the relaxation matrix:?̃? =⎛⎜⎜⎜⎝1/𝑇2 0 00 1/𝑇2 00 0 1/𝑇1⎞⎟⎟⎟⎠.Here, 𝑇1 and 𝑇2 are the (longitudinal) spin-lattice relaxation (SLR) and (transverse) spin-spinrelaxation (SSR) times, respectively, which are the key timescales in the recovery ofM(𝑡) →Meq.Most important in the study of ionic motion is the quantification of 𝑇1 at different 𝑇 and 𝜔0 (𝐵0),which we now consider further.4Note that Equation (1.12) is a macroscopic model of the collective behaviour of nuclei. To obtain a purelymicroscopic picture, one requires more general treatment, such as the density matrix of the spin system [6, 7]. Insuch treatments, it is possible to generalize Equation (1.12) to higher orders of spin interactions (see e.g., [40, 41]);however, most importantly, the physical picture provided by Equation (1.12) remains unchanged.9Spin-lattice relaxationSLR involves the exchange of energy between a nuclear spin and its surrounding environment(i.e., the “lattice”). 1/𝑇1 denotes the rate at which this exchange takes place, resulting fromtransitions between magnetic sublevels, as the spin system returns to (thermal) equilibrium.More specifically, 𝑇1 is the characteristic 1/𝑒 time in the return of a non-equilibrium state tothermal equilibrium. In a SLR measurement, the goal is to quantify the recovery time 𝑇1, whichis dependent on the dynamical properties of the material that cause the local field at the NMRprobe site to fluctuate. When these fluctuations have spectral weight at 𝜔0, the NMR frequencyat the specific applied field, transitions among the magnetic sublevels ensue, effectively re-orientating the spin ensemble back to equilibrium. Many types of dynamic phenomena areeffective at causing relaxation, each with their own signature temperature and/or frequencydependence. For example, in a metal, spin-flip scattering with (polarized) conduction electronsleads to “Korringa relaxation” with a characteristic 𝑇-linear dependence to 𝑇1(𝑇) [7, 42]. Quitedistinct from these origins is relaxation induced by the motion of ions, which we consider inmore detail below.5When, for example, an NMR probe atom undergoes site-to-site hopping, the motion fromthe random walk causes stochastic changes (i.e., fluctuations) to its local (electro)magneticfield.6 While the chief determinant of the local field is 𝐵0, the sites traversed by the probedictate the subtle perturbations Δ𝜔 to its Zeeman levels. Independent of the precise interactionmechanism (e.g., magnetic dipolar or electic quadrupolar), these fluctuations can be describedby a correlation function 𝐺(𝑡), which describes how the local field (or Δ𝜔) varies with time froma specified initial condition (see e.g., [44]). As nearly all NMR interactions depend on the relativepositions of interacting spins, any motion — including ionic diffusion — will randomly alter thelocal field. This correlation function can be written quite generally as:𝐺(𝑡) = ⟨Δ𝜔(0) ⋅ Δ𝜔(𝑡)⟩ = 𝐺(0) ⋅ 𝑔(𝑡), (1.13)5Beyond the account given here, more comprehensive treatments can be found elsewhere (see e.g., [6–11, 34–38]).6In contrast to translational motion, changes to the local field by lattice vibrations are ineffective at causing SLR,as their frequency range is too high. In order to be effective, a Raman-like process for phonon relaxation is required(see e.g., [43]).10where ⟨⋯⟩ denotes the ensemble average, which, for simplicity, we separate into time-dependentand time-independent components 𝑔(𝑡) and 𝐺(0), respectively. Noting that longitudinal fluc-tuations won’t cause SLR, we restrict ourselves to the transverse component of the local field.Implicitly, 𝐺(𝑡) encapsulates the temporal and spatial information of the dynamic process caus-ing SLR by way of how the motion modulates the spin interactions described in the NMRHamiltonian. If the dynamics are caused by a “stationary” random process, 𝐺(𝑡) is a monotonicfunction that decreases from 𝐺(0) → 0, where 𝐺(0) is the strength of the fluctuating interaction.To connect these fluctuations to SLR, it is necessary to consider their spectral density 𝐽(𝜔), whichis directly obtained from the Fourier transform of 𝑔(𝑡):𝐽(𝜔) = ∫∞−∞𝑔(𝑡) exp (−𝑖𝜔𝑡) d𝑡.When 𝐽(𝜔) has weight at 𝜔0, the fluctuations become effective at inducing transitions betweenZeeman levels, resulting in SLR. Alternatively stated, the SLR rate 1/𝑇1 is driven by the com-ponent of the fluctuation spectrum whose Fourier component matches the NMR frequency,or1/𝑇1 ∝ 𝐽(𝜔0).In the simplest case, the decay of 𝑔(𝑡) (and 𝐺(𝑡)) is exponential with a time constant 𝜏𝑐, thecorrelation time. Assuming this simplest form, together with three-dimensional (3D) isotropicfluctuations, leads to the Bloembergen-Purcell-Pound (BPP) result [34], with:𝐽3D𝑛 (𝜔0) ≈𝜏𝑐1 + (𝑛𝜔0𝜏𝑐)𝛼 , (1.14)where the exponent 𝛼 = 2. Clearly, 𝐽3D𝑛 is maximized when 𝑛𝜔0𝜏𝑐 = 1. In the case of SLR fromionic diffusion, 𝜏−1𝑐 is, to an excellent approximation, a measure of the site-to-site hope rate 𝜏−1.If 𝜏−1𝑐 follows an Arrhenius temperature dependence (cf. Equation (1.3)), the ideal BPP responseyields, in an Arrhenius plot of the SLR rate, a symmetric Lorentzian shape (i.e., a “rooftop”curve), whose asymptotic slopes are proportional to the activation energy 𝐸𝐴 characteristic of 𝜏−1𝑐 .The low- and high-𝑇 flanks of the 1/𝑇1 maximum are referred to as the slow- and fast-fluctuationlimits, corresponding to whether 𝜏−1𝑐 ≪ 𝜔0 or 𝜏−1𝑐 ≫ 𝜔0. By changing the 𝜔0 (i.e., 𝐵0 used in11the experiment), one can shift the temperature of the 1/𝑇1 peak position, though this principallyonly modifies the values of 1/𝑇1 when 𝜏−1𝑐 ≪ 𝜔0. An example of this is shown in Figure 1.3.Quite commonly, deviations from the idealized BPP result are encountered and these depar-tures reveal additional information about the dynamics causing SLR. For example, structuraldisorder in the host material or Coulomb interactions between (mobile) ions can lead to correla-tions between successive site-to-site hops, resulting in a reduction in the quadratic dependenceof 𝜔0 in the low-𝑇 regime predicted by Equation (1.14) [45, 46]. Such correlations reduce theapparent slope of the SLR peak’s low-𝑇 flank, resulting in a BPP peak that is asymmetric with1 ≤ 𝛼 < 2. In the highly crystalline, dilute-limit studies to be shown here, we do not expectdeviations from 𝛼 = 2, but this provides a means of confirming such local character. Alterna-tively, when the nature of the fluctuations causing SLR are dimensionally constrained (e.g.,from motion confined to a plane or tunnel) [47], 𝐽𝑛 can have a form considerably different fromEquation (1.14).7 In the case of two-dimensional (2D) diffusion, a phenomenological expressionintroduced by Richards [35, 50], which is correct in the asymptotic limits [51], has the form:𝐽2D𝑛 (𝜔0) ≈ 𝜏𝑐 ln [1 + (𝑛𝜔0𝜏𝑐)−𝛼] , (1.15)where 1 ≤ 𝛼 ≤ 2 [52], similar to the (generalized) BPP case in Equation (1.14). While no analyticform of 𝐽𝑛 for the case of 1D fluctuations has been derived, its asymptotic limits are known [51],and a summary of the limiting behaviour for diffusive 1/𝑇1 in different dimensions is given inTable 1.1. Most importantly, the effect of dimensionally confined fluctuations on SLR show uponly on the high-𝑇 flank of the 1/𝑇1 peak, distinct from the influence of disorder/correlationeffects.While the details of ionic motion dictate the exact form of 𝐽𝑛 (e.g., dimensional character,correlations, etc.), the most salient features of all reasonable models remain largely independentof such factors; 1/𝑇1(𝑇, 𝜔0) is peaked at a temperature 𝑇max(𝜔0)when 𝜏−1 ≃ 𝜔0. It is permissiblethen to simply identify 𝑇max from a series of 1/𝑇1(𝑇)measurements, where 𝜏−1 may be equated7The deviation in 𝐽𝑛 from the ideal BPP result in Equation (1.14) can be traced to the fact that, for spatiallyrestricted fluctuations,𝐺(𝑡) in Equation (1.13) deviates from a simple exponential. This can be derived from detailedmodelling of the spin interactions occurring during a random walk on a given lattice (cf. diffusion on a plane [48]vs. diffusion on a face-centred cubic (FCC) lattice [49]). For example, in the case of diffusion on a plane, theautocorrelation functions are thought to be power laws decaying as 1/𝑡𝑥 instead of exponentials [48].12Table 1.1: Expected asymptotic limits of the SLR rate 1/𝑇1, depending of the dimensionality ofthe fluctuations causing relaxation [35, 51]. While the limiting behaviour in the slow-fluctuatingregime, expected at low temperatures, is dimension independent, different behaviour is expectedwhen the fluctuations are faster at high temperatures. After [37].Dimensionality 𝜔0𝜏𝑐 ≫ 1 (low temperature) 𝜔0𝜏𝑐 ≪ 1 (high temperature)3D 1/𝑇1 ∝ 𝜏−1𝑐 𝜔−𝛼0 1/𝑇1 ∝ 𝜏𝑐2D 1/𝑇1 ∝ 𝜏−1𝑐 𝜔−𝛼0 1/𝑇1 ∝ 𝜏𝑐 ln (1/𝜔0𝜏𝑐)1D 1/𝑇1 ∝ 𝜏−1𝑐 𝜔−𝛼0 1/𝑇1 ∝ 𝜏𝑐/√𝜔0𝜏𝑐with 𝜔0. This approach has the advantage of being (minimally) model-dependent and applicablewhen the analytic form of 𝐽𝑛 is not known a priori (e.g., for SLR from 1D diffusion [51]). For acomplete description of 𝜏−1, it is thus always preferable to measure 1/𝑇1(𝑇, 𝜔) over as wide atemperature and frequency range as possible.8 This allows for 𝜏−1 to be determined over thewidest possible dynamic range. A nice example of this is shown for the garnet-type fast Li+conductor Li7La3Zr2O12 (LLZO) in Figure 1.3, where the mobility of Li+ situated in (distorted)octahedral intersticies dominate its conductivity. As an Arrhenius plot, the connection betweenthe 1/𝑇1 maxima and the absolute hop rate is illustrated in Figure 1.3.While NMR SLR is well-suited to probe relatively fast atomic motion, complementary in-formation on slower motion can be gained from features of the NMR spectrum, which we nowconsider briefly.Motional narrowingComplementary information can be obtained from motion induced changes to the NMRlineshape. In the low temperature limit, where all translational motion is effectively suppressed,one obtains the ideal static resonance lineshape. Its detailed form is characteristic of the latticesite, with features such as: magnetic dipolar broadening [58, 59] from, for example, the nuclei ofneighbouring atoms; and quadrupolar splitting [60] from a finite electric field gradient (EFG) in anon-cubic environment. As the temperature increases and the hop rate exceeds the characteristicfrequency Δ𝜈 of these spectral features, usually on the order of kHz for Li isotopes, dynamic8The NMR signal-to-noise ratio (S/N) is proportional to the square of the NMR frequency (i.e., 𝐵0), so it isgenerally only pragmatic to do experiments at high field. So-called 1/𝑇1𝜌 measurements, using another (transverse)locking probe field are used to interrogate slower motions with faster SLR (see e.g., [7, 10, 53–55]).13Figure 1.3: Example of diffusion-induced SLR. (A) Arrhenius plot of the 7Li NMR relaxationrates 𝑅𝑖 ≡ 1/𝑇𝑖 in LLZO determined by Kuhn et al. [56]. Values for the SLR rate 𝑅1, the SLR ratein the rotating reference frame𝑅1𝜌, and the SSR rate𝑅2 obtained at different NMR frequencies𝜔0(indicated in the inset) are shown. Here, the solid lines denote a fit to amodel describing 𝑅𝑖(𝑇, 𝜔),described elsewhere [10, 56]. The position of the 1/𝑇𝑖 maximum shifts to higher temperatureas the (effective) NMR frequency is raised. (B) Arrhenius plot comparing the Li+ hop rate 𝜏−1extracted from NMR [56] and ionic conductivity [56, 57] measurements in LLZO. The verticaldashed lines show the connection of the 1/𝑇𝑖 maximum position and the extracted 𝜏−1. Theagreement between the techniques demonstrates that NMR is sensitive to the elementarymotionsresponsible for long-range Li+ diffusion. Adapted from [10]. Copyright © 2012, Elsevier B. V.14averaging yields substantially narrowed spectra with sharper structure (see e.g., [61–63]). Inslightly more precise terms, this phenomenon, collectively known as “motional narrowing”,occurs when𝜏−1 ≳ 2𝜋Δ𝜈. (1.16)A typical example of motional narrowing in a solid, shown as the 6Li NMR in the glassy Li+conductor, LiAlSi2O6 [62], is given in Figure 1.4. To extract quantitative information on thekinetics of the process causing the narrowing, several approaches are available [6, 34, 64, 65];however, the results are often model-dependent — much more so than for SLR. A simpleapproach successful in practice is to identify the temperature of the inflection point alongthe narrowing “curve” and estimate 𝜏−1 as 2𝜋Δ𝜈 [6]. Together with SLR, motional narrowingprovides access to the rate kinetics spanning nearly six decades (see e.g., Figure 1.3B).91.2.2 β-NMRLike conventional NMR, β-NMRallows you to infer themotion of ions through the dynamicaleffects on the local electromagnetic fields experienced by probe nuclei [37, 67]. While theframework for studying atomic motion is largely the same, NMR is inapplicable to situationswhere the number of probes are dilute (i.e., below ∼1017 in a typical sample). Conversely, asalient feature of β-NMR is its enhanced sensitivity (per nucleus) [5, 12], allowing it to uniquelyaccess this concentration regime in solids.10Historically, β-NMR has been used successfully to study Li+ in materials already containingLi atoms using activation by absorption of polarized neutrons [67].11 For example, experiments inmetal alloys [70–72], lithiated graphites [73–80], glassy Li2O·3 B2O3 [81, 82], and the superionicconductor Li3N [83, 84], were shown to be in excellent agreement with results from conventionalNMR [85–88]. This approach cannot be used to study extrinsic Li+ conductors (i.e., those which9Note that there are other NMR methods available which can extend this dynamic range to even slower motions(i.e., sub-kHz rates). For example, spin-alignment echos (SAEs) can be used measure atomic jump rates betweensites with inequivalent EFGs (see e.g., [38]) and resonant hole-burning (HB) can be used to follow spectral diffusionof the probe nuclei (see e.g., [66]). However, neither technique is used in this thesis and the methods will not beconsidered further.10While it is true that this situation can be probed directly on surfaces (e.g., with scanning tunnelling microscopy(STM)), this is environmentally distinct from inside a solid [68, 69].11Of course other β-NMR probes have been used successfully to study atomic motion in solids; however, for brevity,we shall not list them here.15Figure 1.4: Example of motional narrowing of a NMR line from translational motion. (A)Temperature dependence of the 6Li NMR lineshape in glassy LiAlSi2O6. As the temperatureincreases, the line sharpens and its width decreases. (B) Temperature dependence of the 6LiNMR full width at half maximum (FWHM) in glassy LiAlSi2O6. A low temperature rigid latticelinewidth of ∼3 kHz narrows substantially between 300K to 400K (highlighted in grey). Theinflection point at𝑇 ≈ 375K along the narrowing curve can be identifiedwhere 𝜏−1 ≈ 2 × 104 s−1[see Equation (1.16)]. Adapted from [62]. Copyright © 2008, American Physical Society.nominally contain no lithium) or materials in the form of thin films. In this thesis, this limitationis overcome by use of an ion beam of 8Li [5], wherein, the implanted 8Li+ behaves both as thedopant and the probe. The first hint that the dilute limit may be different from “bulk” conditionscame in the polymer electrolyte polyethylene oxide (PEO) [89], where the addition of a lithium-salt was shown to appreciably inhibit the dilute limit Li+ dynamics. Further experiments inPEO suggested this was related to the ionicity of the lithium-salt [90], but this situation could bequite different in a host whose immobile sublattice is both more rigid and ordered.Besides this sensitivity to “ionic” material properties, the light interstitial probe 8Li is alsosensitive to the host’s electronic properties with a characteristically small hyperfine coupling [5],similar to μ+ [91, 92]. Indeed, a principal use of 8Li β-NMR to date has been to study the electronic16and magnetic properties of solids [5], which give rise to additional contributions to the SLR andmodifications to the resonance line. Generally, these features are distinct from those originatingfrom ionic diffusion and the two contributions may be separated. In the materials studied inthis thesis, characterizing the low-temperature electronic properties is an important step in thequantification of the ion dynamics taking place at elevated temperatures. As will be seen lateron, these electronic properties are also of interest and, accordingly, Chapters 3 to 5 each containa discussion of the most intriguing features.1.3 Thesis organizationThe remainder of this thesis is organized as follows: first, a detailed account of the ion-implanted β-NMR technique is given in Chapter 2; results in the 1D ion conductor rutile TiO2are given in Chapter 3; in Chapter 4, results on the ionic electronic properties of the layeredmaterial Bi2Te2Se (BTS) are given; and, in Chapter 5, the influence of doping in Bi2Se3:Ca (BSC)and Bi2Te3:Mn (BTM), both structurally related to BTS, are presented. Finally, a summary of themain findings and an outlook for future work is given in Chapter 6.17Chapter 2Principles of ion-implanted β-NMRβ-detected nuclear magnetic resonance (β-NMR) is a sensitive and versatile nuclear tech-nique, used here as a microscopic probe of condensed matter at the nanoscale [5, 12]. It is akinto conventional nuclear magnetic resonance (NMR) of stable nuclei [6, 7], wherein the nuclearspins of a probe isotope are used to interrogate electromagnetic fields inside a host material. Inboth NMR and β-NMR, the goal of any experiment is to characterize the spectral and temporal(or even spatial) dependence of these “internal” fields, which are connected to the fundamentalproperties of the system under study.By-and-large, the machinery of NMR and β-NMR is quite similar; however, a key differenceis how these internal fields are detected. Instead of using the weak (macroscopic) magnetismfrom the dipole moments of the probe nuclei, β-NMR uses the high-energy β-emissions from anunstable (i.e., radioactive) isotope. This is made possible by the anisotropic property of β-decay,wherein the β-ray emission direction is correlated with the orientation of the nuclear spin at themoment of decay. In this respect, β-NMR is quite similar to muon spin rotation/relaxation/reso-nance (μSR) [13, 14], which uses (almost exclusively) the positive muon μ+ as its “sensor”. Themain advantage of this detection scheme is the tremendous boost in sensitivity it affords. Forexample, in a typical conventional NMR experiment, about ∼1017 nuclear spins are needed toachieve a reasonable signal-to-noise ratio (S/N), whereas in β-NMR (or μSR) ∼107 (cumulative)decay events are often sufficient to produce a spectrum. This ∼1010 enhancement in sensitivity(per nucleus) uniquely allows β-NMR to interrogate material situations where NMR is effectively18impossible. Moreover, as the radioactive probes are extrinsic to the material under study, thetechnique can be used to investigate anymaterial (i.e., not only those containing elements with“good” NMR isotopes).Despite these advantages, the use of β-NMR has not flourished compared to similar nuclearmethods, due to the significant infrastructure required for a working implementation. Specifi-cally, the necessity of large reactor or accelerator facilities to produce the probe radionuclides, inboth high quantity and isotopic purity, is perhaps the limiting factor. While this requirement isshared by μSR, by contrast, it has benefited from decades of use and refinement, thanks to theglobal proliferation of so-called meson “factories” over the last half-century [93, 94]. Interestin β-NMR is, however, growing thanks to modern radioactive ion beam (RIB) user facilities —principally the Isotope Separator and ACcelerator (ISAC) at TRIUMF [95, 96].In principle, any radionuclide with a non-zero nuclear spin that undergoes β-decay canbe used in a β-NMR experiment. In fact, the primary historic use of β-NMR was to measurethe electromagnetic moments of short-lived nuclei [97, 98], whose values could be used totest theories on nuclear structure and search for physics beyond the standard model [99, 100].For condensed matter applications, the requirements are more stringent; the probe must becompatible with performing high-throughputmeasurements and produce a large S/N . In general,it is desirable that the probe [101, 102]:1. have a high production efficiency;2. be easy to spin-polarize;3. have a large β-decay asymmetry;4. have a small mass and charge;5. have a small (non-zero) nuclear spin;6. and have a nuclear lifetime comparable to the lifetime of spin-polarization decaying toequilibrium, which is usually on the order of (or less than) ∼1 s.While the first three criteria can be considered essential, the remaining items are merely ideal.Naturally, many radionuclides are a reasonable fit and a several possibilities are listed in Table 2.1;however, considering all of the above qualities, 8Li is really the best and most widely used probe,making it the tool of choice in this thesis.19Table 2.1: Properties of radionuclides used in β-NMR experiments. Here, 𝐼 is the nuclearspin quantum number, 𝛾 is the gyromagnetic ratio, 𝑄 is the electric quadrupole moment, 𝜏𝛽 isthe nuclear lifetime, and 𝑅𝑝 is the approximate production yield at TRIUMF. For comparison,the properties of the positive muon μ+ are also included. Probes used in condensed matterexperiments are indicated with a☆.Probe 𝐼 𝛾/2𝜋 (MHzT−1) 𝑄 (mb) 𝜏𝛽 (s) 𝑅𝑝 (s−1)μ+ 1/2 135.54 0 0.000 002 2 107 ☆8Li 2 6.3016 32.6 1.21 108 ☆9Li 3/2 17.46 −31.5 0.257 107 ☆11Be 1/2 −25.636 0 19.9 10612B 1 7.645 13.2 0.0202 - ☆15O 1/2 10.8 0 176 10619O 5/2 4.67 3.7 38.2 10420F 2 7.98 56 16.1 10417Ne 1/2 12.0 0 0.158 10419Ne 1/2 −28.7 0 24.9 10731Mg 1/2 −13.47 0 0.334 105 ☆In this chapter, the principles of β-NMR will be detailed. The relevant fundamentals willfirst be discussed in Section 2.1, followed by the specific implementation used in this thesis inSection 2.2. An explanation of the types of experiments is given in Section 2.3, and the chapterconcludes with a discussion of the elements crucial to the data analysis in Section 2.4.2.1 Fundamental properties of β-decayEssential to β-NMR are certain details of radioactive β-decay. A nucleus undergoes β-decaywhen the charge state of one of its nucleons (i.e., a proton p+ or neutron n0) is transmutedin order to achieve a more stable nuclear configuration. Formally, a progenitor nucleus X istransformed into a progeny nucleus Y such that𝐴𝑍X →𝐴𝑍±1Y,where 𝑍 is the atomic number and 𝐴 is the atomic mass number of the parent X. Severalquantities are conserved in the decay process, such as the mass number, electrical charge, andlepton number. To conserve charge, an electron e– (or positron e+) — for historic reasons called20Figure 2.1: Radioactive decay scheme for 8Li. For each nuclear level, the spin and parity 𝐼𝜋(left), as well as energy in keV (right), are indicated. With a half-life 𝜏1/2 ≈ 838ms [18],8Li in itsground state undergoes β– decay (𝑄𝛽− ≈ 16MeV) to the ∼3MeV excited state of8Be, yielding anend-point energy of ∼13MeV for the reaction. The decay has a nearly ∼100% branching ratio,with a comparative half-life log10 𝑓𝜏1/2 of ∼5.6. Following the β-decay, the8Be progeny decaysalmost instantaneously into two α-particles (with 𝜏1/2 ≈ 8.19 × 10−17 s). Adapted from [103].Copyright © 1999, Wiley-VCH.a β± particle — is spontaneously created and emitted from the nucleus.1 Simultaneously, an“invisible” neutrino νe (or antineutrino ̄νe) is emitted to conserve lepton number. For example,the (proton deficient or neutron rich) 8Li nucleus undergoes β–-decay:83Li →84Be + e– + ν̄e, (2.1)yielding 8Be (in an excited nuclear state), an electron, and an electron antineutrino.β-decay always produces a three-body final state; the energy released by the decay process,𝑄𝛽±, is distributed among the decay products. This produces a continuous energy spectrum forthe emitted β-particles up to an end-point energy, defined by the difference in 𝑄𝛽±, the rest-massof the emitted (anti)neutrino, and (if applicable) the state of the progeny. Note that, due toconservation of momentum, there is a finite energy associated with the recoil of the decayprogeny. It is, however, very small and can be ignored in most cases. The 𝑄𝛽− for the decay of8Liin Equation (2.1) is ∼16MeV (see Figure 2.1), which contributes to its short radioactive lifetime𝜏𝛽 ≈ 1.21 s [18]. Note, however, that the end-point energy for the reaction is ∼13MeV, due tothe decay to the excited 8Be state.β-decay, unlike all other radioactive decay processes, is governed by the weak interaction,1Note that so-called electron capture β-decay — the inverse of this process — wherein an electron is “taken” fromatomic orbitial, is also possible, but not considered here.21Figure 2.2: Parity violation in the β-decay of the positive muon μ+ [14]. First, mirror images ofparticle collisions in a container are shown on top. As both situations are observable in reality,their translation motion is deemed a parity conserved process (i.e., the total energy is the same).Similar mirror images of the positron e+ emissions from a muon whose spin s𝜇 is oriented.Since spin is chiral, its handedness is not preserved upon reflection. If parity was conserved,the direction of the β-emissions would be identical in both images; however, this is not whatis observed experimentally [104, 105]. Consequently, parity is violated in β-decay. Adaptedfrom [14]. Copyright © 1999, Taylor & Francis.which is unusual in the context of the fundamental physical forces. Unlike in gravitational,electromagnetic, and strong (nuclear) interactions, parity—which was thought to be a universalconservation law— is not conserved! Simply put, any process that is not invariant under a mirroror inversion coordinate transformation does not conserve parity. For example, the interactionsdescribing “classical” collisions of dilute gaseous particles inside a sealed container are preservedupon a change in coordinate system. By contrast, for a collection of oriented nuclei undergoingβ-decay, the emission directions of the β-particles are not preserved under a mirror-plane (orinversion) symmetry operation. This is illustrated pictorially in Figure 2.2 for the decay of μ+.The theoretical prediction that β-decay may violate parity was revolutionary [106]. More22surprising still, was the confirmation of such predictions only a year later in the decay of 60Copolarized at very low temperature [104] and in the decay of the positive muon μ+ [105], whichis produced spin-polarized naturally from positive pion π+ decay. The pragmatic consequenceof parity violation is simple; the direction of β-emissions from an ensemble of spin-polarizednuclei (or particles) are probabalistically correlated with the spin-orientation at the instant ofdecay. The great implications of this discovery were quickly realized, even beyond the realm ofthe particle physics community. Within the following years, the basic principles of the techniquewere laid [107] and the first experiments were performed [108]. In parallel, the entire field ofμSR was born, with statements like [105]:…polarized positive and negative muons will become a powerful tool for exploringmagnetic fields in nuclei…, atoms, and interatomic regions.clearly coming true in time [13–15]. Of course, μSR is just a special case of β-NMR and it too, inspite of technical barriers, prevailed as a fruitful tool for materials science [5, 12] — albeit at amuch slower pace.The complete, general theory describing the details of (allowed) β-decay transitions can befound elsewhere [99, 100, 109, 110], but, for our purpose, the main property is the directionaldistribution of the β-rays for an ensemble of “oriented” nuclei. The emission probability𝑊 isgiven by the simple expression:𝑊(𝜃) = 1 +𝑣𝑐𝐴𝑃 cos 𝜃, (2.2)where 𝜃 is the angle between the spin direction and the β-electron emission direction, 𝑣/𝑐 is theβ-electron velocity (as a fraction of the speed of light 𝑐), 𝑃 is the nuclear spin-polarization, and𝐴𝛽 is the β-decay anisotropy coefficient, denoting how asymmetric the β-emissions are. Notethat since the electron rest mass𝑚0𝑐2 is only 511 keV, it is often a reasonable approximation that𝑣/𝑐 ∼ 1 for all but the lowest energy βs (i.e., those with a kinetic energy less than the rest mass).When 𝑄𝛽 ≫ 𝑚0𝑐2, such low energy βs only account for a tiny fraction of the full β-spectrum(e.g., in 8Li). Moreover, a real experiment is usually desensitized to the low-energy range as theseelectrons are more easily shielded and deposit less energy in the detectors.An example showing the angular distribution described in Equation (2.2) for oriented 60Co23Figure 2.3: Angular distribution of β-emissions from spin-polarized 60Co nuclei [111]. Theasymmetric (cardoid- or limaçon-like) shape of the distribution demonstrates the probabilisticcorrelation between the emission direction and the orientation of a polarized nuclei. Note thatthe plotted values account for the distribution in the emitted β velocities [cf. Equation (2.2)].The low value of 𝐴exp in the “thick” source is attributed to an increase in scattering of the βs bythe host cobalt atoms. Adapted from [111]. Copyright © 1980, Elsevier B. V.nuclei [111] is shown in Figure 2.3. Excellent agreement, at the % level, is attained betweentheory and experiment, confirmed by subsequent more detailed measurements [112]. It is thissimple expression on which all the capabilities of β-NMR are based.While the connection to the fundamental properties of β-decay are clear, notice that, fora measurement at fixed 𝜃, Equation (2.2) also offers a path to monitor the polarization of theβ-emitters. In a nuclear physics experiment, this is just a detail, but, for condensed matterapplications, this is the connection to the surrounding environment! In a real experiment, allonemeasures are the “counts” (i.e., decay events registered) in a detector𝑁𝜃 at fixed 𝜃, subtendingsome finite solid angle Ω of the full three-dimensional (3D) emission distribution. Based onEquation (2.2), if we consider a measurement using two identical detectors at angles 0° and180°, we ought to obtainmaximally different count rates for the opposed detectors. A little more24explicitly:𝑊(0)𝑊(𝜋)∝𝑁0𝑁𝜋=1 +𝑣𝑐𝐴𝑃1 −𝑣𝑐𝐴𝑃,which is easily rearranged to give:𝑣𝑐𝐴𝑃 =𝑁0 − 𝑁𝜋𝑁0 + 𝑁𝜋. (2.3)This normalized difference in counts between two anti-parallel detectors is referred to as the(experimental) asymmetry and is the hallmark measurable quantity in any β-NMR (or μSR)experiment. The clear benefit of this construct is the proportionality to the spin polarizationof the β-decaying probe. Most importantly, the relation provides a direct experimental way tomonitor 𝑃—amicroscopic quantity— through the relatively easy-to-measure high-energy decayproducts of a collection of unstable nuclei.While this framework was established almost immediately after parity violation was dis-covered [107], it is technically challenging to put into practice. The main bottleneck of anyimplementation is the production and preparation of the probe radionuclides. One approach,made possible by high-intensity RIB facilities, is using the isotope separation online (ISOL)method, wherein the radionuclides are simultaneously generated and extracted from a “genera-tor” and delivered to the experiment as an ion beam. All preparation of the probe spin-statestakes place during the beam delivery. This is the approach taken at TRIUMF and used in thisthesis. It is outlined in the following section.2.2 The TRIUMF implementationThe key components of TRIUMF’s implementation of the β-NMR technique can be brokendown into: the production and collection of the probe radioisotopes; the polarization of theprobe nuclear spin; implantation of the probe ions in a sample under study; and the detectionand measurement of the β-NMR signal. Each of these will be discussed in separate sectionsbelow.252.2.1 Isotope production & transportAt TRIUMF, a ∼500MeV H– cyclotron is used to produce up to a ∼100 µA p+ beam. One ofseveral p+ beams are delivered to the ISAC facility where it interacts with one of two dedicatedisotope production target stations. For 8Li production, the most commonly used productiontarget material is Ta. Specifically, a typical target consists of a stack of D-shaped Ta metal foilshoused within a Ta tube 19mm in diameter and 20 cm long. The (radio)isotopes are producedby nuclear reactions (e.g., fragmentation and spallation) caused by the incident high-energyprotons, creating a radioactive “soup” inside the target foils (i.e.,many species are produced, notjust the one desired). Invariably, at least one of the produced species needs to be extracted andthis is achieved through the ISOL method.The easiest way of extracting a radioisotope is by out-diffusion from the target foils. To ensurefast diffusion, the target container is heated to very high temperature (i.e., above 2000K) limitedby the mechanical stability of Ta.2 The heat is supplied resistively, but a substantial portionresults from the energy deposition of the p+ beam (up to 10 kW). Only a fraction of all producedisotopes are extracted and delivered to experiments. Once a radionuclide has escaped the foil,it effuses along a 3mm transfer tube, where it undergoes adsorption-desorption cycles, eachtime with a finite probability of undergoing surface ionization. A Re foil is typically used alongthe transfer tube, due to its high work function ∼4.8 eV. For elements with low first ionizationpotentials (e.g., alkali metals), the surface ionization method is simple and nearly 100% efficientat yielding monovalent cationic species.Once the radioisotopes are ionized, they are transported to the experiment. The ions areaccelerated out of the transfer tube with a large electrostatic bias typically on the order of ∼20 kV(and a maximum of 60 keV). The extracted RIB is nearly mono-energetic, with only a smallenergy spread (∼1 eV). The beam is then “filtered” through a high-resolution magnetic dipolemass separator [113], where it is purified based on mass-to-charge ratio (m/q). For heavierisobars (i.e., different elements with the same mass number), further measures may be necessaryto achieve an isotopically pure beam; however, for light elements (e.g., Li), contamination isnot an issue.3 The beam is then transported through a high vacuum beamline (∼10−6 Torr)2If the target tube and foils melt it no longer works!3This is particularly true for 8Li, since there are no stable species with𝐴 = 8.26using electrostatic optical elements [114] to the experiment station. The nuclei at this stage are,however, unpolarized and require additional preparation to be useful in the experiment. Thiscrucial step will be discussed in the following section.2.2.2 PolarizationDuring the beam transport, a crucial step is the polarization of the RIB [115], wherein thenuclear spins of the ions become oriented. This is achieved through optical pumping [116], wherelaser light is used to induce a high degree of electronic polarization, which, as a consequenceof the coupling of (total) electronic and nuclear angular momentum, is also transferred to thenuclei. In practice, this is done using circularly polarized resonant laser light that’s co-linearwith the RIB and a dedicated setup for this exists at ISAC [117]. The detailed mathematics ofthe pumping process are described elsewhere [116] and the following discussion will be limitedto the case of Li, which has the simplest alkali scheme [115].4The process consists of several steps. The 8Li+ beam is first passed through an alkali vapourcharge-exchange cell, where it is neutralized to atomic lithium (i.e., 8Li0).5 Following neutral-ization, any remaining 8Li+ is electrostatically removed (i.e., steered into a beam dump), whilethe neutralized fraction of the beam continues to drift down the beamline though a 1.9m longoptical pumping region. This section of the beamline is shown in Figure 2.4.Wenowconsider the pumping of 8Li0. The neutral atoms interactwith the counter-propagatinglight tuned to 671 nm, corresponding to the 2𝑆1/2 →2𝑃1/2 D1 electronic transition, illustrated inFigure 2.5. The coupling of 8Li0’s total electronic angular momentum (𝐽 = 1/2) to its nuclearspin (𝐼 = 2) results in the splitting of both the 2𝑆1/2 and2𝑃1/2 levels into 𝐹 = 3/2 and 𝐹 = 5/2hyperfine states. A small longitudinal holding field (typically ∼1mT) parallel to the beam liftsthe degeneracy of the 2𝐹+1magnetic hyperfine sublevels𝑚𝐹 and the helicity 𝜎± of the circularlypolarized laser light ensures absorptive transitions of Δ𝑚𝐹 = ±1. Any resonant transition issubsequently followed by spontaneous emission (exited state lifetimes of ∼50 ns) with the usualselection rules (i.e., Δ𝑚𝐹 = 0,±1) over the µs transport times through the pumping region.4Variations on the approach described here can, however, be used for elements in other groups on the periodictable (e.g., 11Be+ [118, 119] or 31Mg+ [120]).5Historically, Na was used for this task; however, a switch to Rb was made in ∼2015, after it was found to have anequivalent neutralization efficiency, but with the benefit of vaporizing at far lower temperatures.27Figure 2.4: Schematic of the low-energy beamline at TRIUMF’s ISAC facility (ca. 2003) [102].Shown is the polarizer section [117] upstream of the dedicated high [121–123] and low [122–124]field β-NMR spectrometers. Note that, since ∼2015, Na has been supplanted with Rb in theneutralizer cell. Adapted from [102]. Copyright © 2003, Elsevier B. V.This results in the 𝑚𝐹 sublevel populations being biased to either the highest or lowest rungof the 𝑚𝐹 states (see Figure 2.5). At typical transport energies, the interacting neutral atomsundergo 10 to 20 absorption/emission cycles, which is sufficient to produce a high degree ofelectronic and nuclear polarization, the latter being on the order ∼70% [125–127]. Note thatthis is orders of magnitude higher than the thermal equilibrium value achievable with all butthe most extreme temperatures and fields (see Figure 2.6). Equally important is the axis of theproduced polarization, which is defined by the laser’s helicity direction and is accurately parallelto the beam and the holding field in this region.In a real experiment, the polarizing laser is operated at a fixed bandwidth (∼1MHz) andelectro-optic modulation (EOM) is used to broaden the laser’s spectral width to cover the range ofall hyperfine transitions (cf. Figure 2.5). While the nominal transition frequencies are predictable,as a consequence of the Doppler effect, they depend intimately on the precise energy of the RIB.In practice, one sweeps through a small range of biases (i.e., on the order of ∼100V) just upstreamof the neutralizer cell to find the setpoint that induces optimal resonant overlap between 8Li0and the laser light. This is determined unambiguously by the emergence of non-zero β-decayasymmetry, and a typical scan is shown in Figure 2.7. Once determined, the optimal bias is setand held constant for the remainder of the experiment.28Figure 2.5: Top: Electronic structure of the neutral of 8Li atom. Shown qualitatively (from leftto right) are the energy levels for the 𝑛 = 2 atomic orbitals, their splitting into distinct electronicstates for different total angular momentum 𝐽 = 𝐿 + 𝑆, and the hyperfine splitting 𝐹 = 𝐽 + 𝐼 ofthe electronic levels by the 8Li nuclear spin. The rightmost plot shows the magnetic splittingof hyperfine levels in the 2𝑆1/2 atomic ground state. Adapted from [128]. Copyright © 1995,American Institute of Physics. Bottom: Optical pumping scheme for the 8Li0 D1 transition using𝜎+ laser light. Here, an exciting Δ𝑚𝐹 = +1 transition (thick arrow) is followed by spontaneousemission (thin arrow) with a Δ𝑚𝐹 of −1, 0, or +1. After several absorption/emission cycles, thepopulation of atoms accumulate in the𝑚𝐹 = +5/2 state, eventually approaching 100% (in theabsence of any relaxation mechanism). Note that energy differences in the𝑚𝐹 sublevels havebeen omitted for clarity. Adapted from [115]. Copyright © 2008, American Institute of Physics.29100 101 102Temperature (K)10 710 510 310 1P zI z/I8Li: I=2,2 =6.3016 MHz T1B0=102 TB0=101 TB0=100 TB0=10 1 TB0=10 2 TopticalpumpingFigure 2.6: Comparison of the 8Li nuclear spin-polarization obtained fromoptical pumpingwiththermal equilibrium. The polarization fromoptical pumping (∼70% [127]) is orders of magnitudelarger than the equilibrium value, and independent of both temperature and field. Only in verylarge applied fields at extremely low temperatures does the equilibrium polarization 𝑃eq𝑧 becomecomparable to the optically pumped state. In the temperature and field range accessible in theseβ-NMR experiments (3K to 317K and up to 6.55 T), to an excellent approximation, 𝑃eq𝑧 ≈ 0 (seeSection 2.3.2).Following optical pumping, the neutral beam is re-ionized by passing through a helium gascell. For the charge “liberation” reactionLi0 +He(g) → Li+ + e– +He(𝑔),the process has an efficiency of ∼50% [131] and, most importantly, does not disturb the nuclearspin-polarization.6 The charged beam is then electrostatically steered through a 45° bendfollowed by another 45° switchyard (see Figure 2.4), where it may be “kicked” to one of twodedicated spectrometers [5, 121, 122, 124]. The fraction of the beam that fails to be re-ionizedcontinues to drift into the aptly named neutral beammonitor (NBM), comprised of a Pd foil target,used as an independent diagnostic for the experimental stations (e.g., it was used to produce the6It does, however, increase the beam emittance slightly.30202Asymmetry (%)Positivehelicity( +)Negativehelicity( )0123Asymmetry (%)Combinedhelicities20 40 60 80 100 120 140Bias (V)0.20.00.2A± (%) DifferencesFigure 2.7: Typical alkali vapour cell bias scan used to spin-polarize a 8Li+ beam. Here, anincident 20 keV 8Li+ beam is Doppler-shifted by the bias voltage (ordinate) onto resonance withcollinear𝜎± circularly polarized laser light. This is achieved by a small deceleration bias upstreamof the alkali vapour cell (see Figure 2.4) where the beam gets neutralized. Spin-polarization isevidenced by the non-zero β-decay asymmetry. The solid coloured lines show fits to a sum of sixVoigt lines [129, 130], corresponding to the expected number of 8Li0D1 hyperfine transitions (seeFigure 2.5). The baseline asymmetry, corresponding to zero polarization, is indicated by dashedblack lines. The optimum bias is indicated by the vertical grey band, whose width representsthe peak half width at half maximum (HWHM). Small, systematic differences between the twohelicities are shown in the bottom panel, which are minimized at the optimum bias.spectra in Figure 2.7). The foil has a central aperture 5mm in diameter for transmission of thelaser beam [117].With the polarized RIB now delivered up to the experiment, it is ready for implantation in a“sample” to be studied.312.2.3 Ion-implantationIon-implantation is the means in which the probe nuclei are introduced into the materialunder investigation. This is specific to this type of β-NMR; other approaches may use in situneutron activation, placing the probes randomly over the full volume of typical targets [12], or thehigh-energy recoils from a target foil which implant with a range of much higher energies [132].In contrast to these methods, the essentially mono-energetic beams used here allow for maximalcontrol over the stopping range of the probes. In general, this provides a potentially importantdegree of freedom to a series of measurements that is unachievable by other techniques. Thepenetration depth of the incident ions is dependent on the incident kinetic energy of the RIB,which is defined by the extraction bias of the isotope production target. Further control over thestopping depth is achieved with a high-voltage deceleration bias, made possible by the electricallyisolated spectrometer platforms [121, 122, 124].The stopping of ions in solids is a stochastic process; each ion follows a discrete path as itscatters off atoms in the hostmaterial. These atomic collisions can be statistically simulated usingMonte Carlo codes, which allow one to predict the profile of the stopped ions. For example, theStopping and Range of Ions in Matter (SRIM) software package [133], which treats all scatteringevents within the binary collision approximation (BCA), is known to give reliable estimates ofthe implantation profile for light atoms and is used here. In these simulations, the beam andtarget properties can be flexibly specified (e.g., isotope, energy, target thickness and density,etc.) From the simulation, one obtains the stopping profile of the probe in the target material,which is, like the energy distribution of the products from β-decay (see e.g., Equation (2.1)), acontinuous distribution. The distribution details, in the nomenclature of the ion-implantationliterature, such as the range and straggle (i.e., the mean and standard deviation), are highlysensitive to the properties of the beam and target materials. Generally though, a positivelyskewed asymmetric distribution is obtained for light ions at keV energies in most solids. Typicalimplantation profiles simulated using SRIM are shown in Figure 2.8.It has been shown that results from BCA Monte Carlo codes are in good agreement withexperiment, especially those using implanted hyperfine probes. For example, this has beendemonstrated for μ+ (treated as a light H+ ion) in noble metal [135] and Al [136] films on320 50 100 150 2000.000.010.020.030.04〈z〉 = 25± 12 nm〈z〉 = 96± 28 nmz (nm)ρz(nm−1 )5 keV20 keVFigure 2.8: Typical stopping profiles for an implanted ion simulated using the SRIMMonteCarlocode [133]. Shown are profiles at two energies for 106 8Li+ ions implanted in TiO2, histogrammedwith 1 nm wide bins. The ion range and straggle (i.e., the mean and standard deviation) at eachenergy are indicated. Adapted from [134]. Copyright © 2014, the Author(s).SiO2, based on the contrast between μ+ and Mu precession signals. In YBa2Cu3O6+x (YBCO), acomparison wasmade by a direct imaging approach using an inhomogenousmagnetic field [137].For 8Li, a comparison of stopping fractions has been done for Al and Au films [138] using thenormalized resonance amplitude. This approach, however, unlike the case of μ+, dependson details of the resonance technique which complicates the comparison (see Section 2.3.1).Despite this difficulty, reasonable agreement was obtained. Another confirmation of the SRIMprofiles for 8Li comes from the measurement of the magnetic penetration depth in the Meissnerstate of NbSe2 [139], which compares well with other methods. Ways to directly measurethe implantation profile have also been considered when the incoming ions are not mono-energetic [132].Just as important as how deep the probe ions penetrate is themacroscopic position of thebeam on the target. In almost all experiments, a well-focused beam spot centred on the sampleis desirable, preferably without any diffuse “halo”. An approximate image of the beam canbe obtained using materials that scintillate under ionizing radiation. For a 8Li+ beam, the α-emissions from 8Be progeny are particularly useful to this end. A “snapshot” of the beam profileis obtained from the scintillating region using a fixed charge-coupled device (CCD) camera. In33-Al2O3 (0001)Tune: 141107_010920 keV 8Li+HV Bias = 0 kVB0=20 mT12.5 mm12.5 mmFigure 2.9: Typical 8Li+ beamspot at the low field spectrometer. Shown is a CCD image of a12.5mm × 12.5mm × 0.5mm α-Al2O3 (0001) single crystal irradiated with a DC8Li+ beam overa 10 s exposure time. The bright outline inside the dashed box corresponds to a 8mm × 8mmarea of the crystal backside visible while mounted in the spectrometer’s cold-finger cryostat. Thebeamspot corresponds to a 20 keV 8Li+ transport energy (for a 0 kV platform bias) in a 20mTmagnetic field perpendicular to the beam.fact, defective α-Al2O3 turns out to be a good scintillator [140, 141] — a fortuitous convenience,as its often used as a substrate for mounting small crystals. A typical beamspot on the β-detectednuclear quadrupole resonance (β-NQR) spectrometer obtained with a CCD camera under adirect current (DC) 8Li+ beam is shown in Figure 2.9.For experiments aiming to measure a depth-dependent quantity, one would additionally likea beamspot that is equally focused and at the same position for all implantation energies. Inpractice, both the macroscopic position and profile of the beamspot can change when beamenergy is modified, creating a further systematic that is convoluted with any depth dependence.The extent of this depends on the quality of the ion beam “tune”. The main results in this thesisare, however, not concerned with depth dependent quantities and the impact of the use ofdifferent beam energies is unimportant.With the beam energy and position set, the β-NMR experiment can proceed. Before consid-ering the experimental details, it is instructive to review the technical capabilities of the facilityspectrometers, which are described in the following section.342.2.4 β-NMR & β-NQR spectrometersTRIUMF is home to two dedicated β-NMR spectrometers [121–124], each residing in thelow-energy branch of its ISAC facility [95, 96]. The instruments are maintained by TRIUMF’sCentre for Molecular and Materials Science (CMMS) [123], which also oversees the lab’s μSRprogram. More complete descriptions can be found elsewhere [5, 121–124], but the essentialdetails are described here.Each instrument is equipped with: a pair of scintillation detectors for counting β-rays; ahelium-flow cold-finger cryostat, providing stable temperature operation between 3K to 317K; amagnet for applying a homogeneous longitudinal field; and a small (approximately Helmholtz)coil for producing transverse radio frequency (RF) magnetic fields. The sample environmentoperates at ultra-high vacuum (UHV) (i.e., ∼10−10 Torr), which is necessary to prevent build-upof residual gas adsorbents at cryogenic temperatures. As a corollary, all studied materialsmustbe compatible with UHV. The UHV chambers are accessed through load-locks using separatevacuum systems and the time-duration of a sample removal/reinsertion cycle is typically about∼1 h (i.e., the UHV is not vented during a sample change). Upstream of the spectrometers,additional vacuum pumps and small apertures are used to transition the beamline pressuredown to the required UHV. Both spectrometer platforms are electrically isolated, allowing themto be biased to high-voltage (HV), enabling controlled deceleration of the incoming RIB. Thefull deceleration is achieved within the last few cm of the ion beam flight path. Note that thereare difficulties associated with sustaining a HV bias while operating a high magnetic field >1 T(e.g., the production of a corona discharge) and the β-NMR spectrometer requires extensiveHV “conditioning” in advance of use. The high quality RIBs at ISAC are nearly monoenergeticwith minimal transverse momentum, providing focused beamspots at the sample. Beamspotsfrom scintillation can be imaged using a CCD camera mounted on a beamline viewport. Typicalbeamspots are∼2mm in diameter and samples as small as 5mm × 5mm are routinely measured(without the use of any beam collimation). Key to controlled deceleration during experimentsis the stability of the HV bias, which can be achieved up to ∼30 kV, providing RIB energies onthe order of ∼30 keV down to ∼100 eV. This translates roughly to mean stopping ranges on theorder of ∼100 nm down to a only a few nm below the sample surface.35Figure 2.10: Schematic of TRIUMF’s high-field β-NMR spectrometer [121–123]. Adaptedfrom [121]. Copyright © 2004, American Physical Society.With these basic elements common to both spectrometers, where the instruments differ isin their geometry and range of applied fields. A high field spectrometer, “β-NMR”, is equippedwith a 9 T superconducting solenoid on axis with the incoming RIB. The field is thus appliednormal to a sample’s surface. Its detectors are positioned (inside the vacuum of the beamline)asymmetrically about the sample position (one ∼10 cm downstream and the other ∼75 cmupstream). Experiments rely on the high field to focus the βs in the upstream detector, whichgives the two detectors similar effective solid angles, even though they differ substantially inzero field. Consequently, measurements at fields <1 T suffer from low counts, but only in the“backward” detector. The β-NMR spectrometer’s cryostat admits samples with lateral dimensionsup to 8mm × 12mmwith a typical thickness of 0.5mm. A schematic of the instrument is shownin Figure 2.10.The other spectrometer covers a complementary range of lower fields (0mT to 24mT) usinga normal conducting Helmholtz magnet. Additionally, it is equipped with auxiliary trim coilsin orthogonal directions, which can be used to accurately zero the (vector) magnetic field atthe sample position (see e.g., [142]), allowing for pure nuclear quadrupole resonance (NQR)experiments to be performed. Consequently, the instrument is called “β-NQR”. While thelongitudinal geometry is strictly required at high fields to limit the deflection of the beam,this is not the case in (sub-)mT fields. Consequently, the polarization (and primary field) aretransverse to the incoming beam (i.e., parallel to the sample surface). In contrast to the highfield instrument, the detectors are located outside the beamline, but positioned symmetrically36Figure 2.11: Schematic of TRIUMF’s low-field β-NQR spectrometer [122–124]. Adaptedfrom [139]. Copyright © 2009, American Physical Society.about the sample. While the high field instrument uses a single sample manipulator rod, theβ-NQR spectrometer uses a ladder which can house up to four samples at a time. One may,by simply changing the ladder position, perform measurements on different samples withoutbreaking the UHV. The procedure is quick, typically taking <1min. Samples with maximumlateral dimensions 12.5mm × 12.5mm are compatible with the β-NQR cryostat. Figure 2.11shows a schematic of the instrument.2.3 ExperimentsBy direct analogy with conventional NMR [6, 7], there are two primary types of β-NMRexperiments: resonance and relaxation. Each mode provides access to different details about thelocal electromagnetic fields inside a sample, which can be connected to its fundamental materialproperties. Distinct to ion-implanted β-NMR is the way the measurements are performed, whichis described in the subsequent sections.2.3.1 ResonanceIn many ways, β-NMR is very similar to stable isotope NMR [6, 7]. The nuclear spin sensesthe local magnetic fields via the Zeeman interaction and their time-averages contribute to the37resonance shift and lineshape. In addition, since 8Li possesses a non-zero nuclear quadrupolemoment, the nuclear spin is coupled to the local electric field gradient (EFG) [6, 7, 60],𝑒𝑞 =𝜕2𝑉𝜕𝑥𝑖𝜕𝑥𝑗,a tensor that is zero under cubic (or higher) symmetry. When the EFG tensor is non-vanishing,the quadrupolar interaction splits the resonance into a set of 2𝐼 satellites. The integer spin 𝐼 = 2(nonexistent in conventional NMR) of 8Li has the important consequence that the quadrupolarspectrum has no “main line” at the Larmor frequency,7 determined by the gyromagnetic ratio 𝛾 ofthe NMR nucleus and the (dominant) static applied magnetic field 𝐵0, given by Equation (1.11).In contrast, for the more familiar case of half-integer 𝐼, the𝑚 = ±1/2 transition yields, to firstorder, a line at 𝜈0 unperturbed by quadrupole effects [60]. While the quadrupole interaction isoften themost important perturbation to the nuclear spin energy levels in nonmagneticmaterials,for 8Li it is still relatively small (in the kHz range), because its nuclear electric quadrupolemoment 𝑄 is small, compared, for example, to 209Bi (see Table 4.1).In a β-NMR (or NMR) spectrum, the scale of the quadrupolar splitting is given by thequadrupole frequency [60, 143]:8𝜈𝑞 =𝑒2𝑞𝑄8ℎ. (2.4)As the splitting is typically small relative to the NMR frequency (i.e., 𝜈𝑞 ≪ 𝜈0), it can be treatedaccurately as a perturbation. Explicitly, the satellite positions can be written as [60, 143]:𝜈𝑚−1→𝑚 = 𝜈0 +∑𝑛𝜈(𝑛)𝑚−1→𝑚, (2.5)where 𝜈0 is the NMR frequency, given in Equation (1.11), and 𝜈(𝑛)𝑚−1→𝑚 are the 𝑛th-order contri-butions to the 𝑚 − 1 → 𝑚 sublevel transition. These corrections, up to second-order, can bewritten as [143]:𝜈(1)𝑚−1→𝑚 = 𝜈𝑞√63(1 − 2𝑚) 𝑓0 (2.6)7This is also true in the NMR of 𝐼 = 1 nuclei, such as 2H, 6Li, or 14N.8Note that this definition of 𝜈𝑞 is not unique and differs with those by other authors by a factor of two (seee.g., [6, 60]); however, in Equation (2.4), 𝜈𝑞 corresponds to the absolute (inner) satellite spacing about the resonancecentre-of-mass (to first-order).38for the first-order contribution and𝜈(2)𝑚−1→𝑚 = −2𝜈2𝑞9𝜈0{[24𝑚 (𝑚 − 1) − 4𝐼 (𝐼 + 1) + 9] |𝑓1|2 + [12𝑚 (𝑚 − 1) − 4𝐼 (𝐼 + 1) + 6] |𝑓2|2}(2.7)for the second-order treatment.9 The factors 𝑓𝑛 in Equations (2.6) and (2.7) scale themagnitude ofthe splitting according to the symmetry and orientation of the EFG tensor relative to the externalfield. More precisely, these factors can be expressed in terms of the polar and azimuthal angles(𝜃 and 𝜙) between 𝐵0 and the EFG principal axis, along with the EFG asymmetry parameter𝜂 ∈ [0, 1]. In the common case where the EFG has axial symmetry, 𝜂 = 0 and the angular factorsin Equations (2.6) and (2.7) can then be written as [143]:𝑓0 =√32(3 cos2 𝜃 − 1) , (2.8)|𝑓1|2 = −34(−3 cos4 𝜃 + 3 cos2 𝜃) , (2.9)and|𝑓2|2 =32(38cos4 𝜃 −34cos2 𝜃 +38) . (2.10)The definitions in the above expressions are tied to the convention |𝑉𝑍𝑍| ≥ |𝑉𝑌𝑌| ≥ |𝑉𝑋𝑋| thatlabels the EFG principle axes.Distinct from conventional NMR, the probe is extrinsic to the host, and its lattice site isnot known a priori. Like the implanted positive muon in μSR, the 8Li+ ion generally stops in ahigh-symmetry site in a crystalline host. Some site information is available in the resonancespectrum, since the local field and EFG depend on the site, but generally one has to combine thisinformation with knowledge of the structure and calculations to make a precise site assignment(see e.g., the case of the muon [145]).Resonances are typically acquired in a continuous 8Li+ beam with a continuous wave (CW)transverse RF magnetic field stepped slowly through the 8Li Larmor frequency. In this mea-surement mode, the spin of any on-resonance 8Li is rapidly precessed by the RF field (i.e., theparticipating magnetic sublevel populations become equalized in the limit of saturation), result-9Note that third-order expressions have also been derived [144], but are unnecessary here.39ing in a loss in the average time-integrated asymmetry. If the resonance is traversed too quickly(relative to 𝜏𝛽), an asymmetric lineshape results due to the slow recovery of the polarization (i.e.,a history dependence). Dealing with this polarization “drag” complicates the analysis and thissituation is generally avoided. In the rare circumstance that the fast sweeping is desired, it ispossible to encapsulate the form of the skewed line, provided the RF sweep rate is known [146].Unique to this mode of measurement, the resonance amplitudes are determined by severalfactors, some quite distinct from conventional pulsed RF NMR. First, the maximum amplitude isdetermined by the baseline asymmetry [147], which represents a time integral of the spin-latticerelaxation (SLR). It also depends on the magnitude of the RF magnetic field 𝐵1 relative to thelinewidth, since the RF will only precess 8Li within a frequency window of width ∼ 𝛾𝐵1. Theresonance amplitude may be enhanced by slow spectral dynamics occurring up to the secondtimescale, since the RF is applied at a particular frequency for an integration time of typically1 s, and any 8Li that are resonant during this time will be precessed by it. Quadrupole satelliteamplitudes are reduced by the simple fact that saturating a single quantum transition (i.e.,Δ𝑚 = ±1) can, at most, reduce the asymmetry by 25% for 𝐼 = 2 [127]. Unsplit resonances can,in contrast, be much larger, since all the Δ𝑚 = ±1 transitions are resonant at the same frequencyand, if saturated, the RF will precess the full polarization giving the full amplitude equal to theoff-resonance asymmetry.In spite of these complexities, the main strength of the CWmethod is that it provides themost expedient way to measure a resonance; it provides a measure of the line with the greatestcounting statistics within the shortest possible measurement time. The importance of thispragmatism is paramount given the finite allotment of beamtime. A typical measurement takes∼30min to acquire, though quadrupole split spectra can take much longer.It is preferable to acquire resonances in the dedicated high-field spectrometer [121, 122],where, at 𝐵0 > 1 T,10the resonance frequency can be precisely calibrated, typically using anindependent measurement in single crystal MgO (100) at room temperature [148]. This choiceof reference is arbitrary,11but sensible; MgO is a wide band gap insulator with cubic rocksaltstructure, whose (bulk) magnetic susceptibility is weakly diamagnetic [150] and the internal10In this range, first-order treatment of quadrupolar effects is sufficient for 8Li.11Note that the aqueous salt solutions typically used as references in conventional NMR (see e.g., [149]) areincompatible with the required beamline UHV.40field experienced by an implanted probe is expected to be essentially zero. This calibration thusprovides access to the average local field at the site of the nucleus — the NMR (relative) shift:𝛿 = 106 (𝜈0 − 𝜈MgO𝜈MgO) , (2.11)where 𝜈0 and 𝜈MgO are the resonance frequencies in the sample and MgO, respectively. Typically,𝛿 is reported in parts per million (ppm). This field is the sum of several contributions, onebeing demagnetization for the host material, which generates a relative shift in the resonanceposition, but can be easily corrected for given the knowledge of the sample’s geometry and (bulk)susceptibility [151].2.3.2 RelaxationAs with conventional NMR, the SLR is determined by fluctuations at the Larmor frequency— typically in the RF range. Specific to the β-NMR (or μSR) mode of detection, the rangeof measurable 𝑇1 relaxation times are determined by the radioactive lifetime 𝜏𝛽. As a rule ofthumb, measurable 𝑇1 values lie in the range 0.01𝜏𝛽 to 100𝜏𝛽 [12]. Near the upper end of thisrange, the spin relaxation is very slow and exhibits little or no curvature on the timescale of themeasurement. One can still measure the relaxation rate from the slope, but it is significantlycorrelated to the initial amplitude of the relaxing polarization signal. At the other end of thetimescale, most of the polarization vanishes before radioactive decay of the probe, resulting in astep or “missing fraction” of the signal.In conventional NMR, the S/N is proportional to the square of the Larmor frequency, favour-ing high applied fields. Consequently, for practical reasons, NMR is often done at a singlefixed field in the range of ∼10 T. In contrast, the signal in β-NMR is independent of frequency,and the field can easily be varied. This can be useful, for example, in identifying relaxationmechanisms with distinct field dependencies. In addition, this enables β-NMR in the realm oflow applied fields (up to 10s of mT). As the applied field approaches zero, fluctuations of thestable magnetic nuclei of the host often become the dominant source of relaxation. At such lowfields, distinction of different nuclei by their Larmor frequency is suppressed, and the isolated8Li begins to resonantly lose its spin polarization to the bath of surrounding nuclear spins (see41e.g., [139, 152]). Effectively, this simply appears as another relaxation mechanism active onlyat low fields. The extent of the low field regime depends on the moment, density, and NMRproperties of the host lattice nuclei.SLR measurements are performed by monitoring the transient decay of spin-polarizationboth during and following a short (e.g., 4 s) pulse of beam [153, 154]. During the pulse, thepolarization approaches a steady-state value, while after the pulse, it relaxes to ∼0. This is anapproximation, but reasonable for spins in thermal equilibrium over the conditions accessiblein β-NMR experiments (see Figure 2.6), where 𝑃eq is essentially zero on the scale of the highlypolarized initial state from optical pumping. Note that the sharp “kink” at the edge of thebeam pulse (see e.g., Figures 2.12 and 2.13) is characteristic of β-NMR SLR data acquired in thismanner. Unlike conventional NMR, no RF field is required for the SLR measurements, as theprobe spins are implanted in a spin state already far from equilibrium. As a result, it is generallyboth faster and easier to measure SLR than the resonance; however, as a corollary, this typeof relaxation measurement has no spectral resolution and represents the spin relaxation of allthe 8Li — even ones whose resonance cannot be resolved. A typical SLR measurement takes∼20min.Accounting for the polarization during and after the beam pulse is, at face value, a compli-cation; however, using this portion of the data makes maximal use of a spectrum’s countingstatistics, which are highly inhomogenous with time (see Figure 2.12). Similarly, it is straightfor-ward to deal with the distribution of probe arrival times. Considering all 8Li arriving at time𝑡′ < 𝑡, the spin-polarization 𝑃(𝑡) at time 𝑡 can be written as [153, 154]:𝑃(𝑡) = 𝑃0 ×⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩∫𝑡0exp [− (𝑡 − 𝑡′) /𝜏𝛽] 𝑅 (𝑡, 𝑡′) d𝑡′∫𝑡0exp [−𝑡′/𝜏𝛽] d𝑡′, 𝑡 ≤ Δ∫Δ0exp [− (Δ − 𝑡′) /𝜏𝛽] 𝑅 (𝑡, 𝑡′) d𝑡′∫Δ0exp [−𝑡′/𝜏𝛽] d𝑡′, 𝑡 > Δ(2.12)where 𝑃0 is the degree of polarization at 𝑡 = 0 determined by the optical pumping of8Li prior to420 2 4 6 8 10 12Time (s)051015Asymmetry (%)Beam On Beam OffBi2Te2Se:28 keV 8Li+7.5 mT (001)fitdata [10 K]0.00.20.40.60.81.0Normalized Statistical WeightFigure 2.12: Illustration of the temporally inhomogeneous (statistical) error bars in a pulsedβ-NMR SLR measurement. The statistical weight of each point 1/𝛿2𝑖 — see Equation (2.15) — isrepresented by the cividis colourmap [27], showing amaximumnear the end of the beam pulse at4 s, and minima at the edges of the spectrum’s time-window. This temporal dependence is quiteunlike similar nuclear techniques, where the statistical uncertainty usually grows monotonicallywith time. For clarity, the spectrum has been binned by a factor of 20.implantation, Δ is the duration of the 8Li+ beam pulse, and 𝑅 (𝑡, 𝑡′) is the relaxation function.Quite generally, 𝑅 (𝑡, 𝑡′) can be written as a sum of decaying exponentials, where a singleexponential is the simplest casewith a single SLR rate 1/𝑇1.12 The latter is, however, the exception,rather than the rule; the non-zero quadrupole moment of 8Li makes it sensitive to fluctuations inthe local EFG, which, a priori, can produce biexponential relaxation for spin 𝐼 = 2 [71, 155, 156],even for a single implantation site.13 More common is the case that the SLR is well-described by a12Note that using a simple exponential for 𝑅(𝑡, 𝑡′) implies that 𝑃(𝑡) in Equation (2.12) eventually decays from𝑃0 to zero; however, in reality, 𝑃(𝑡) returns to a thermal equilibrium value 𝑃eq that is close, but not equal to zero(otherwise NMR wouldn’t work)! Nevertheless, as is shown in Figure 2.6, the (implicit) approximation that 𝑃eq ≈ 0is reasonable for the conditions accessible in these β-NMR experiments (see Section 2.2.4). More explicitly, for 8Liat 6.55 T, 𝑃eq is in the range of 6.6 × 10−4 to 6.2 × 10−6 (for 3K to 317K), which is clearly close to zero, especiallycompared to the 𝑃0 ≈ 0.7 [125–127].13It is amusing to note that it was once pointed out [155]:For 𝐼 = 2, the longitudinal and transverse relaxation transients are also the sums of two exponentials.Since there are no stable nuclei with spin of two, the results in this case are of no importance.Of course, at this point in time (ca. 1970) β-NMR using 8Li had not yet flourished, so the assertion is understandable.43(phenomenological) stretched exponential or Kohlrausch-Williams-Watts (KWW) function [157–161]:𝑅 (𝑡, 𝑡′) = exp {− [𝜆 (𝑡 − 𝑡′)]𝛽} , (2.13)where 𝜆 ≡ 1/𝑇1 is the relaxation rate and 0 < 𝛽 ≤ 1 is the stretching exponent. This form of𝑅 (𝑡, 𝑡′) is quite general and can arise from a continuous distribution of exponential relaxationtimes [160], with special cases of 𝛽 = 1/2 [162, 163] or 𝛽 = 1/3 [84] coming from SLR that isinhomogenously averaged in 3D or two-dimensional (2D) limits. The approach is often usedin conventional solid-state NMR [164], though chiefly as an expedient simplification of themulti-exponential magnetization transients, particularly when quadrupolar interactions arepresent [165]. In disordered or glassy materials like polymers a distribution of relaxation timesis expected and a stretched exponential works works well [89, 90, 166–169]. While it may not bea priori obvious why a relaxation transient appears stretched, the function has the following ad-vantages as a phenomenological model: 1) it accurately captures the characteristic 1/𝑒 relaxationtime; and 2) it contains relatively few degrees of freedom (i.e., less than a biexponential model).Given these complexities, some care is requiredwhen choosing a particular𝑅 (𝑡, 𝑡′). Generallythough, the salient features in the raw data prevail, independent of what model is chosen for theanalysis (see e.g., Figure 2.13). Consequently, a model’s self-consistency may be checked againstwhat is plainly apparent in the data.2.4 Data & AnalysisQuite distinct from more common techniques, there is no de facto standard software widelyavailable for the processing and analysis of ion-implanted β-NMR data — commercial or other-wise. For comparison, the similar, but more widely used μSR has a rich set of tools available fortaming (or wrangling with) their data (e.g., WIMDA [170], Musrfit [171],14or Mantid [172]).While this situation is improving [173], the lack of “standard” options usually requires develop-ment of custom software to tackle even relatively routine analyses. Here this process is brieflydelineated, starting first with the storage format.14Note that Musrfit [171] is capable of handling a subset of the β-NMR data taken at TRIUMF; however, itsfunctionality does not cover all work presented in this thesis.440 2 4 6 8 10 12Time (s)0.00.20.40.60.81.0Normalized Asymmetry20 keV 8Li+, 6.55 T, 295 KMgO (100)-SiO2 (0001)TiO2 (001)TiO2 x (100)-Al2O3 (0001)YSZ (001)Figure 2.13: Comparison of 8Li SLR data in common oxide insulators near room temperatureat high field. Notice the dramatic difference in the haste of the SLR in each material, which isclearly distinguished prior to considering any detailed analysis. The solid black lines denote fitsto a biexponential model [see Equation (3.1)] convolved with a 4 s square beam pulse [153, 154].For clarity, the spectra have been binned by a factor of 20.The archival record of all β-NMR experiments performed at TRIUMF are stored in thelaboratory’s MUon Data (MUD) format, which are binary encoded — a relic of from the dayswhen disk space was expensive — and require a C library for reading/writing. In brief, the filesconsist of a table of histograms containing the count rate in each detector as a function of anindependent variable (e.g., time, frequency, etc.), as well as some header information describingthe measurement. Only a summary of the raw decay events is stored (i.e., as a pre-binnedhistogram), rather than a discrete array of every detected decay event. This has the advantage ofgreatly reducing the size of the data files (each measurement, colloquially abstracted as a single“run”, occupies only 50 kB to 500 kB of storage space), but limits the upper resolution of the data(i.e., the bin size). All μSR and β-NMR data taken at TRIUMF are publicly available through anonline repository maintained by the CMMS group.The objective of any detailed analysis is to accurately model the data and encapsulate how it45evolves (e.g., with temperature, field, etc.). This requires finding the best fit parameters popt andtheir associated uncertainty estimates 𝛿popt for one (or several) model functions. Accountingfor the uncertainty of each point in a spectrum (reliably determined by counting statistics) iscritical to this task. Like in all radioactive techniques, the statistics per point are inhomogeneousacross a spectrum (see e.g., Figure 2.12), and, as such, they must be weighted appropriately inthe analysis. For a general non-linear least-squares fit, the quantity 𝜒2, which is the weightedsum of the squared fit residuals, is minimized. Explicitly, for a set of 𝑁 pairs of experimentaldata points (𝑥𝑖, 𝑦𝑖), where 𝑖 denotes the index,𝜒2 =𝑁∑𝑖(𝑦𝑖 − 𝑓(𝑥𝑖;p)𝛿𝑖)2. (2.14)Here, 𝑓(𝑥𝑖;p) is the fit function, which is dependent on a vector of variable parameters p. Thesquared residuals are weighted by the effective variance 𝛿2𝑖 , defined as (see e.g.: [174]):𝛿2𝑖 = 𝛿𝑦2𝑖 + (𝜕𝑓(𝑥𝑖;p)𝜕𝑥)2𝛿𝑥2𝑖 , (2.15)where 𝛿𝑥𝑖 and 𝛿𝑦𝑖 as the uncertainties in 𝑥𝑖 and 𝑦𝑖, respectively. In essence, this approach servesto transform the uncertainty in the 𝑥 coordinate to the 𝑦 coordinate, at the expense of someadditional complexity in the fitting procedure. For instance, this transforms a linear problemto one that is non-linear. Additionally, an analytic form for 𝜕𝑓(𝑥𝑖;p)/𝜕𝑥may not be available,requiring the partial derivative to be estimated numerically. Notice, however, that when |𝛿𝑥𝑖| or|𝜕𝑓(𝑥𝑖;p)/𝜕𝑥| become small, 𝛿2𝑖 ≈ 𝛿𝑦2𝑖 , which is the usual weighting in a least-squares analysis.When performing any analysis, it is beneficial to constrain the vector of variable parametersp to an acceptable minimum number. This not only increases the available degrees of freedomin the fit, but often reduces the uncertainty in the extracted parameter values. Similarly, it alsoavoids overparameterization. While the values of most parameters depend on the particularconditions of a measurement (e.g., temperature, magnetic field, etc.), some are independent ofsuch quantities and common to subset of experiments. It is desirable that these parameters beshared between 𝜒2 calculations for different measurements, such that an overall optimum valueis determined. This so-called “global” fitting procedure ensures that an optimum in precision46is achieved from a fit, within the confines of using a particular model. The global 𝜒2, 𝜒2global,following Equation (2.14), is simply:𝜒2global =𝑛∑𝑖𝜒2𝑖 =𝑛∑𝑖𝑁∑𝑗(𝑦𝑖𝑗 − 𝑓(𝑥𝑖𝑗;p′𝑖)𝛿𝑖𝑗)2, (2.16)where 𝑛 is the number of data sets with indices 𝑖, 𝑁 is the number of data points with index 𝑗of the 𝑖th data set, and p′𝑖 is vector of fit parameters for the set 𝑖. p′𝑖 is determined by a mappingfunction Λ that translates ap′𝑖 = Λ(𝑖;pglobal), (2.17)wherepglobal denotes the global list of fit parameters. The complexity in this seemingly straightfor-ward task reduces to bookkeeping; the total number of global shared and locally free parametersmust be defined and organized such that Λ correctly maps them between the individual 𝜒2calculations.The global fitting procedure usually requires a considerable number of numeric computationsand model iterations before converging to an optimum answer. It is therefore favourable toimplement such a routine in a compiled programming language, such as Fortran, C, or C++. Thework in this thesis uses custom C++ code that interfaces with the fitting classes implementedwithin ROOT [175] — an object-orientated data analysis framework developed at CERN high-energy physics experiments. The minimization of 𝜒2 is carried out by ROOT’s implementationof the MINUIT algorithms [176, 177], which were originally written in Fortran. Parameteruncertainties are computed using the MINOS algorithm [178], wherein the parameter spaceis naively scanned about 𝜒2min for positions where 𝜒2(𝑝′𝑖) − 𝜒2min(𝑝𝑖) = 1. The approach hasthe advantage of making no assumptions about the shape of the minimum near 𝜒2(popt) andaccounts for the fact that, in general, parameter uncertainties are asymmetric. In the workpresented here, this is mainly a precaution; the parabolic errors are generally in good agreementwith their asymmetric counterparts and are used throughout the remainder of this thesis.When the fit function has an analytic form, the calculation time of 𝜒2global can be relativelyexpedient. Where the process slows is if there are an extraordinary number of data points to47evaluate.15 Using a compiled language (e.g., C++), allows for computer architecture optimizedmachine code to be generated, speeding up fitting appreciably. On modern machines withmulti-core central processing units (CPUs), the fitting procedures can be further optimizedby recognizing that the calculation of 𝜒2global in Equation (2.16) is an “embarrassingly parallel”problem, whose work can in principle be easily split between all available CPU cores. Thatis, since all of the individual 𝜒2𝑖 calculations are independent, they can each be computedconcurrently. Modern software tools, like OpenMP [179], make this task straightforward for theamateur programmer. For the work in this thesis, calculations following Equation (2.16) wereparallelized over the outer double sum.16 Independent of the precise implementation details,with compiled and parallelized code, it is possible to perform sophisticated data treatment ina reasonable time. For example, a global fit to a set of ∼30 SLR measurements on a CPU with4 cores can be achieved within a few min, provided a good initial guess to the solution wasprovided at the outset.In some cases, however, the fit function does not have a closed analytic solution andnumericalmethods (e.g., numerical integration of Equation (2.12)) must be used in its evaluation. Thisballoons computation times from minutes to days, making the process of testing fit modelsslow and arduous. With careful selection of the numerical algorithm (e.g., evaluating stretchedexponential SLR during the beam pulse using double-exponential integration techniques [180,181]), one can alleviate this tedium tremendously and reduce computation times by nearly afactor of 10.15In the notation of Equation (2.16), the total number of points to evaluate is∑𝑛𝑖𝑁𝑖, where both 𝑛 and𝑁may belarge.16Note that this approach differs somewhat from the implementation used in Musrfit [171], which parallelizes thecalculations of the inner double sum. The difference in approaches reasonable; μSR experiments often have orders ofmagnitude more points-per-run, making the inner sum the bottleneck in the fitting procedure.48Chapter 3Microscopic dynamics of Li+ andelectron polarons in rutile TiO23.1 IntroductionThe mobility of lithium ions inserted into rutile TiO2 is exceptionally high and unmatchedby any other interstitial cation [182, 183]. Even at 300K, the Li+ diffusion coefficient is as largeas 10−6 cm2 s−1 [184], exceeding many state-of-the-art solid-state lithium electrolytes [185, 186].Moreover, thismobility is extremely anisotropic [184], and rutile is a nearly ideal one-dimensional(1D) lithium-ion conductor. This is a consequence of rutile’s tetragonal structure [187], whichhas open channels along the 𝑐-axis that provide a pathway for fast interstitial diffusion (seeFigure 3.1). This has, in part, led to a keen interest in using rutile as an electrode in lithium-ionbatteries [188], especially since the advantages of nanosized crystallites were realized [189].Simultaneously, much effort has focused on understanding the lithium-ion dynamics [190–202];however, many underlying details in these studies are inconsistent with available experimentaldata. For example, a small activation energy of around 50meV is consistently predicted [192–194,196, 198–202], but measured barriers are greater by an order of magnitude [184, 203, 204]. Thisdisagreement is troubling considering the simplicity of both the rutile lattice and the associatedLi+ motion. One explanation for the discrepancy is that most experimental methods sense themacroscopic ion transport, while theory focuses on elementary microscopic motion. The two49Rutile TiO2P42=mnm (136)a = b = 4:593Åc = 2:959ÅAB CFigure 3.1: The rutile TiO2 crystal structure [187]. (A) Titanium-centred conventional unitcell showing titanium (blue) and oxygen (red) ions. Bonds (grey) are drawn to emphasizecoordination. (B) View along the 4-fold 𝑐-axis, revealing the channels for fast interstitial Li+diffusion. Lithium (green) are shown in the channel centre (4𝑐 sites) surrounded by the blueTi-centred octahedra. (C) Off-axis view revealing the 1D channels. The structures were drawnusing CrystalMaker [205]. Adapted from [206]. Copyright © 2017, American Chemical Society.would only be related if themacroscopic transport were not strongly influenced by crystal defects,as might be expected due to the highly one-dimensional mobility. Thus, for a direct comparisonwith theory it is important to have microscopic measurements of the Li+ dynamics.The electronic properties of rutile are also of substantial interest [207, 208]. While it isnatively a wide band-gap (3 eV) insulator, it can bemade an 𝑛-type semiconductor by introducingelectrons into vacant titanium 3𝑑 𝑡2𝑔 orbitals, reducing its valence from 4+ to 3+. This is easilyachieved through optical excitation, extrinsic doping, or by oxygen substoichiometry. Ratherthan occupying delocalized band states, these electrons form small polarons, where the Ti3+ ionis coupled to a substantial distortion of the surrounding oxygen octahedron. Polaron formationin rutile is not predicted by naive density functional theory (DFT), and to obtain it one mustintroduce electron interactions [209]. Recently, the polaron has been studied optically [210] andusing electron paramagnetic resonance (EPR) [211]. Compared to delocalized band electrons,50polaron mobility is quite limited and often exhibits thermally activated hopping. Calculationspredict that the polaronmobility, like interstitial Li+, is also highly anisotropic, with fast transportalong the 𝑐-axis stacks of edge sharing TiO6 octahedra [196, 209, 212, 213]. Importantly forour results, the positive charge of an interstitial cation like Li+ can bind the polaron into aLi+-polaron complex, effectively coupling the electronic and ionic transport [214]. Even beforethis complex was observed by EPR and by electron nuclear double resonance (ENDOR) [215],its effect on the mobility of Li+ was considered theoretically [196, 216].To study lithium-ion dynamics in rutile, a technique sensitive to the local environment ofLi+ is desirable. Nuclear magnetic resonance (NMR) is a sensitive microscopic probe of matterwith a well-developed toolkit for studying ionic mobility in solids [8–11, 34–38]. In particular,spin-lattice relaxation (SLR) measurements provide a means of studying fast dynamics. Asdescribed in Section 1.2.1, they are sensitive to the temporal fluctuation in the local fields sensedby NMR nuclei, which induce transitions between magnetic sublevels and relax the ensemble ofspins towards thermal equilibrium. When these stochastic fluctuations induced by, for example,ionic diffusion, have a Fourier component at the Larmor frequency 𝜔0 (typically on the orderof MHz), the SLR rate 𝜆 ≡ 1/𝑇1 is maximized. Complementary information can be obtainedfrom motion induced changes to the resonance lineshape. In the low temperature limit, thestatic NMR lineshape is characteristic of the lattice site, with features such as the quadrupolarsplitting andmagnetic dipolar broadening from the nuclei of neighboring atoms. As temperatureincreases and the hop rate exceeds the characteristic frequency of these spectral features, dynamicaveraging yields substantially narrowed spectra with sharper structure. This phenomenon iscollectively known as “motional narrowing” and is sensitive to slow motion with rates typicallyon the order of kHz. Together, SLR and resonance methods can provide direct access to atomichop rates over a dynamic range up to nearly 6 decades.Here, we use β-detected nuclear magnetic resonance (β-NMR) [5, 12] to measure the Li+dynamics in rutile. Short-lived 8Li+ ions are implanted at low-energies (∼20 keV) into single crys-tals of rutile, and their NMR signals are obtained by monitoring the 8Li nuclear spin-polarizationthrough the anisotropic β-decay. 1/𝑇1 measurements reveal two sets of thermally activateddynamics: one low-temperature process below 100K and another at higher temperatures. Thedynamics at high temperature are due to long-range Li+ diffusion, in agreementwithmacroscopic51diffusion measurements, and corroborated by motional narrowing of the resonance lineshape.We find a dilute-limit activation barrier of 0.32(2) eV, which is consistent with macroscopicdiffusivity, but inconsistent with theory. We suggest that the dynamics below 100K and its muchsmaller activation barrier are related to the low-temperature kinetics of dilute electron polarons.The rest of this chapter is organized as follows: experimental details are given in Section 3.2;the results and analysis are described in Section 3.3; a detailed discussion that includes compari-son to the extensive literature is presented in Section 3.4; and, finally, a concluding summary isgiven in Section 3.5. Some further detail on the SLR model and the candidate site for interstitialLi+ is also given in Sections 3.6.1 and 3.6.2, respectively.3.2 Experimentβ-NMR experiments were performed at TRIUMF in Vancouver, Canada. A low-energy∼20 keV hyperpolarized beam of 8Li+ was implanted into rutile single crystals mounted in oneof two dedicated spectrometers [5, 121, 122, 124]. The incident ion beam had a typical fluxof ∼106 ions/s over a beam spot ∼3mm in diameter. At these implantation energies, the 8Li+stop at average depths of at least 100 nm [134], as calculated by the Stopping and Range ofIons in Matter (SRIM) Monte Carlo code [133] (see Figure 2.8). Spin-polarization was achievedin-flight by collinear optical pumping with circularly polarized light, yielding a polarizationof ∼70% [127]. The TiO2 samples were one-side epitaxially polished (roughness < 0.5 nm),commercial substrates with typical dimensions 8 × 10 × 0.5mm3 (Crystal GmbH, Berlin). Allthe samples were transparent to visible light, but straw-coloured, qualitatively indicating a minoroxygen deficiency [217]. Details of the SLR and continuous wave (CW) resonance measurementsare given in Section 2.3. A typical SLR measurement took about 20min, while a resonancemeasurement required about 1 h.523.3 Results and analysis3.3.1 Spin-lattice relaxationTypical SLR data at high and low field are shown in Figure 3.2 for several temperatures. Thespectra have been normalized by their apparent 𝑡 = 0 asymmetry (i.e., 𝐴0), as determined froma global fitting procedure described below. In high magnetic fields, the relaxation is remarkablyfast compared to other oxide insulators,1consistent with an earlier report in an intermediatefield of 0.5 T [218]. It is also immediately evident that the SLR rates are strongly dependent onboth temperature and field; the rate of relaxation increases monotonically as the magnetic fieldis decreased towards zero (see Figure 3.2). At fixed field, however, the temperature dependenceof the relaxation is nonmonotonic, and there is at least one temperature where the relaxation rateis locally maximized. Moreover, the temperature of the relaxation rate peak is field-dependent,increasing monotonically with increasing field.To make these observations quantitative, we now consider a detailed analysis. The relaxationis not single exponential at any field or temperature, but a phenomenological biexponentialrelaxation function, composed of fast and slow relaxing components yields a good fit.2 For an8Li+ ion implanted at time 𝑡′, the spin polarization at time 𝑡 > 𝑡′ follows [153, 154]:𝑅(𝑡, 𝑡′) = 𝑓slow exp {−𝜆slow (𝑡 − 𝑡′)} + (1 − 𝑓slow) exp {−𝜆fast (𝑡 − 𝑡′)} , (3.1)where the rates are 𝜆fast/slow ≡ 1/𝑇fast/slow1 , and 𝑓slow ∈ [0, 1] is the slow relaxing fraction. Wediscuss possible origins for biexponential 𝑅(𝑡, 𝑡′) in Section 3.6.1.With this model, all the data at each field are fit simultaneously with a shared commoninitial asymmetry (𝐴0) using the MINUIT [176] minimization routines within ROOT [175] tofind the optimum global nonlinear least-squares fit. Notice that the statistical error bars arehighly inhomogenous with time, characteristic of the radioactive 8Li decay. During the beam1See, for example, SrTiO3 [146], MgO [148], LaAlO3 [219], yttria-stabilized zirconia (YSZ) [220], or (La,Sr)(Al,Ta)O3(LSAT) [220]. This is also illustrated in Figure 2.13.2We emphasize that, in contrast to the results presented in Chapters 4 and 5, the stretched exponential discussedin Section 2.3.2 does not adequately encapsulate the 8Li SLR in TiO2. Generally, when the SLR is described by abiexponential with components that are sufficiently different in rate and amplitude (like they are in rutile), a fit usingEquation (2.13) adopts a 𝛽 ≲ 0.2, causing the model to: 1) be highly divergent at early times and miss the data; and2) yield an optimum rate and amplitude that both are unphysically large.53BeamOnBeamOff25K50K79K295K0 5 100:00:20:40:60:81:0Time (s)NormalizedAsymmetry20 keV, 6:55T ‖ (100)!0=2ı ≈ 41MHzBeamOnBeamOff5K30 K90K300K0 5 10Time (s)20 keV, 10mT ⊥ (100)!0=2ı ≈ 63 kHzFigure 3.2: 8Li SLR data in rutile TiO2with 𝐵0 = 6.55 T ∥ (100) [left] and 10mT ⟂ (100) [right].The relaxation becomes faster with decreasing magnetic field and is a non-monotonic functionof temperature. The solid lines are a fit to biexponential relaxation function [Equation (3.1)]convolved with a 4 s square beam pulse [153, 154]. These fit curves were obtained from a globalfitting procedure where a temperature independent overall amplitude 𝐴0 is shared for all spectraat each field, as described in the text. The obtained 𝐴0s are used to normalize the displayedspectra, which have been binned by a factor of 20 for clarity. Adapted from [206]. Copyright ©2017, American Chemical Society.pulse, the uncertainty decreases with time as the statistics increase, reaching a minimum at thetrailing edge of the 4 s beam pulse. Following the pulse, the error bars grow exponentially asexp(𝑡/𝜏𝛽) (see e.g., Figure 3.2). Accounting for this purely statistical feature of the data is crucialin the analysis. A subset of the results are shown as solid coloured lines in Figure 3.2. The fitquality is good in each case (global ̃𝜒2 ≈ 1.1). The relaxation rates of the two components arevery different with 𝜆fast > 20𝜆slow, and the analysis distinguishes them clearly. Typical initialasymmetries 𝐴0 of ∼10% and ∼9% were obtained at high and low fields, respectively.The main fit results are shown in Figure 3.3. Consistent with the qualitative behaviour of the540.00.51.0f slow101 10210−210−1100Temperature (K)1=Tslow1(s−1)10 mT ⊥ (100)6:55 T ‖ (100)Figure 3.3: Results from the analysis of the 8Li SLR measurements in rutile TiO2 using Equa-tion (3.1) at high- and low-field. Shown are the fraction of the slow relaxing component 𝑓slow(top) and the slow relaxation rate 1/𝑇slow1 (bottom). 𝑓slow is surprisingly both temperature- andfield-dependent, but increases towards 1 by 300K. The qualitative features of the SLR data inFigure 3.2 can be seen clearly in the two field-dependent rate maxima in 1/𝑇slow1 . The solid redline is drawn to guide the eye. Adapted from [206]. Copyright © 2017, American ChemicalSociety.spectra in Figure 3.2, the relaxation rate exhibits two maxima at each field. This is most apparentin the slow relaxing component at high magnetic field, while at low field, the low temperaturepeak is substantially broadened. Generally, a maximum in the relaxation rate occurs whenthe average fluctuation rate matches the Larmor frequency [8–11, 34–38], while the detailedtemperature dependence 𝜆(𝑇) depends on the character of the fluctuations.Though the two relaxing components share similarities in their temperature dependence (seeFigure 3.3 and Figure 3.9 from Section 3.6.1), we emphasize that the slow relaxing componentis the more reliable. Even though the sample is much larger than the incident ion beamspot,backscattering can result in a small fraction of the 8Li+ stopping outside the sample whichtypically produces a correspondingly small fast relaxing asymmetry [5]. At high field, where558Li+ relaxation is generally slow, most materials show such a fast relaxing component easilydistinguishable from the features of interest; however, when quadrupolar relaxation is present,which results inmultiexponential relaxation for high-spin nuclei [71, 155, 156] (see Section 3.6.1),distinguishing the background contribution from an intrinsic fast component becomes difficult.At low fields, a background is even harder to isolate as 8Li+ relaxation is typically fast underthese conditions. Therefore, even though the slow component is a minority fraction at low field(see Figure 3.3), we assert that it is the more reliable.3 The fact that, as discussed in the followingsections, we are able to reproduce material properties observed with other techniques is strongconfirmation of the appropriateness of this choice.3.3.2 ResonanceWe now turn to the measurements of the 8Li resonance spectrum at 𝐵0 = 6.55 T. As expectedin a noncubic crystal, the NMR spectrum is split into a multiplet pattern of quadrupole satellitesby the interaction between the 8Li nucleus and the local electric field gradient (EFG) charac-teristic of its crystallographic site. As seen in Figure 3.4A, the resonance lineshape changessubstantially with temperature. At 10K, it is broad with an overall linewidth of about 40 kHz,near the maximummeasurable with the limited amplitude radio frequency (RF) field of thisbroadband spectrometer [121, 122]. Some poorly resolved satellite structure is still evidentthough. As the temperature increases, the intensity of the resonance increases considerablywith only limited narrowing. Above 45K, however, the resonance area decreases dramaticallyand is minimal near 100K. As the temperature is raised further, the quadrupolar splitting be-comes more evident, especially above 130K. Moreover, the sharpening of these spectral featurescoincides with a reduction in the breadth of line, along with another increase in signal intensity.These high-temperature changes are qualitatively consistent with motional narrowing for a mo-bile species in a crystalline environment. By room temperature, the spectrum is clearly resolved(see Figure 3.4B) into the expected pattern of 2𝐼 single quantum (|Δ𝑚| = 1) quadrupole satel-lites interlaced with narrower double-quantum transitions occurring at the midpoint between3In Section 3.6.1, we consider possible origins for the strong field dependence of 𝑓slow.56neighbouring single quantum satellites.4 The well-resolved quadrupolar structure indicates awell-defined time-average EFG experienced by a large fraction of the 8Li at this temperature. Atall temperatures, the centre of mass of the line is shifted to a lower frequency relative to 8Li+ inMgO at 300K. Note that the resonances in Figure 3.4 are all normalized to the off-resonancesteady state asymmetry, which accounts for all the variation of signal intensity due to SLR [147].As described in Section 2.3.1, the scale of the quadrupolar interaction is given by thequadrupole frequency 𝜈𝑞 in Equation (2.4). From the spectra, the splitting is on the order of a fewkHz, small relative to the Larmor frequency. In this limit, the single quantum satellite positionsare given accurately by first-order perturbation theory [Equations (2.5), (2.6) and (2.8)] [60, 143].The spectrum thus consists of 4 satellites split symmetrically about 𝜈0. Unlike the more commoncase of half-integer spin, there is no unshifted “main line” (the𝑚 = ±1/2 transition). The satel-lite intensities are also different from conventional NMR, being determined mainly by the highdegree of initial polarization that increases the relative amplitude of the outer satellites [127].We now consider a detailed analysis of the resonances. In agreement with an earlier re-port [134], the anti-symmetry in helicity-resolved spectra [5] reveals the resonance is quadrupolesplit at all temperatures; however, below 100K the splitting is not well resolved, and an attemptto fit the spectra to a sum of quadrupole satellites proved unsuccessful. Instead, we use a singleLorentzian in this temperature region to approximate the breadth of the line. At higher tem-peratures, where the satellite lines become sharper, a sum of Lorentzians centred at positionsgiven by Equations (2.5), (2.6) and (2.8) (including interlacing double-quantum transitions closeto room temperature) [5] with all 𝑚-quanta satellites sharing the same linewidth. From thefits, we extract: the central frequency 𝜈0; the quadrupole splittings 𝜈𝑞; and the overall/satellitelinewidths. Note here that 𝜈𝑞 is the directly measured satellite splitting, following Equation (2.4),with an assumed temperature-independent angular factor 𝑓0 = 1. From 𝜈0, we calculate thefrequency shift 𝛿 relative to 8Li+ inMgO at 300K in parts permillion (ppm) using Equation (2.11).Additionally, the normalized resonance area was estimated following a procedure that removedany effect of SLR on the line intensity using a baseline estimation algorithm [223]. This allowedfor a common integration scheme, independent of a particular fit model.4These double quantum transitions are a non-linear effect of the RF field, occurring in quadrupole split resonanceswhen𝐵1 is relatively large (𝐵1 ∼ 0.5G for the spectra in Figure 3.4) and when 𝜈𝑞 relatively small (see e.g., [221, 222]).57A−40 0 4010K25K45K65K85K0 − MgO (kHz)−40 0 40130K158K250K315K0 − MgO (kHz)B−30 −20 −10 0 10 20 30300KDQTDQTDQT0 − MgO (kHz)Figure 3.4: 8Li resonance spectra in rutile TiO2 with 𝐵0 = 6.55 T ∥ (100). (A) Temperaturedependence of the resonance. Note that both the vertical and horizontal scales are the samefor each spectra, which are offset for clarity. The zero-shifted position is taken as the resonancefrequency of 8Li in MgO at 300K. The lineshape changes substantially with temperature. Noticethat it is most intense at 25K, but quickly diminishes as the temperature is raised. At highertemperatures, motional narrowing is apparent and the quadrupolar structure becomes clearlyvisible. (B) High-resolution spectrum at 300K. The four quadrupolar statellite transitions areclearly visible, including three narrow double-quantum transitions (DQTs) [positions indicatedby DQT], each occurring at the midpoint between neighbouring single-quantum satellites (seee.g., [221, 222]). The solid orange line is a fit described in the text. Adapted from [206]. Copyright© 2017, American Chemical Society.58The results of this analysis are shown in Figure 3.5. Though the scatter in the quantitiesextracted are largest near 100K, where the resonance is weakest, the qualitative trends notedabove are evident. At low temperatures, the resonance is bothwidest andmost intense, narrowingonly modestly approaching 100K. The resonance area is clearly largest around 25K, but quicklydiminishes to aminimumaround 100K. At higher temperatureswhere the quadrupolar structureis better resolved, the satellites have a nearly temperature independent width of ∼4.8 kHz.The apparent 𝜈𝑞 reduces gradually from ∼3.8 kHz to nearly half that value by 200K. Thisreduction in splitting coincides with an increase in area, consistent with the picture of motionalaveraging of the quadrupolar interaction [63, 224]. This implies a fluctuation rate on the orderof ∼2 × 104 s−1 by ∼140K (𝑇MN in Figure 3.5). The resonance shift 𝛿 is both small and negativeat all temperatures, gradually increasing towards zero as the temperature is raised.3.4 DiscussionThe remarkably fast and strongly temperature dependent spin-lattice relaxation at high mag-netic field implies an exceptional relaxation mechanism for 8Li in rutile distinct from other oxideinsulators [146, 148, 219, 220]. The occurrence of a 𝑇1 minimum indicates some spontaneousfluctuations are present that are: 1) coupled to the nuclear spin; and 2) their characteristic ratesweeps through the NMR frequency at the temperature of the minimum. Diffusive motion of8Li+ through the lattice provides at least one potential source of such fluctuations, as is wellestablished in conventional NMR [8–11, 34–38]; however, without assuming anything about theparticular fluctuations, we extract the temperatures 𝑇min of the two 𝑇1 minima (see Figure 3.3),by simple parabolic fits, at several magnetic fields corresponding to NMR frequencies spanningthree orders of magnitude. This approach has the advantage of not relying on any particularform of the NMR spectral density function 𝐽(𝜔𝐿) [225], which is proportional to 1/𝑇1. Note thatthere is a clear field dependence to the high-temperature flanks of the 𝑇1 minima in Figure 3.3(i.e., they do not coalesce at high temperatures). This indicates that the fluctuations are notthe result of a three-dimensional (3D) isotropic process [34], but rather one that is spatiallyconfined to lower dimensions [35, 51]. Note that for a 1D process (as might be expected inrutile), one needs to account for the characteristic non-Debye (i.e., non-Lorentzian) fluctuation59Single LorentzianQuadrupolar Satellites02040FWHM(kHz)fi−1(TMN) ≈ 2× 104 s−1TMN ≈ 141K0246 q(kHz)−100−500‹(ppm)0 100 200 3000123Temperature (K)Norm.Area(kHz)Figure 3.5: Results for the analysis of the 8Li resonancemeasurements at high field in rutile TiO2.Shown (from top to bottom) are the temperature dependence of the: linewidths, quadrupolefrequency, resonance shift, and integrated area. In qualitative agreement with the normalizedspectra, the resonance is broadest at low temperatures, and gradually reduces in breadth asthe temperature is raised. Above 100K the quadrupole satellites become resolved and theirsplitting is gradually reduced by a factor of ∼2 by 200K. An estimate of the motional narrowingtemperature 𝑇MN and an associated hop rate 𝜏−1(𝑇MN) is indicated. The line is clearly mostintense at low temperatures, as indicated by the resonance area, even though it is broadest here.Adapted from [206]. Copyright © 2017, American Chemical Society.60spectrum [226, 227], where only the asymptotic form is known [35, 51].Identifying the inverse correlation time of the fluctuations 𝜏−1(𝑇min) with the NMR fre-quency 𝜔0 [Equation (1.11)], we construct an Arrhenius plot of the average fluctuation rate inFigure 3.6. The value of this approach is evident in the linearity of the results which indicates twoindependent types of fluctuations each with a characteristic activated temperature dependence.To this plot, we add the estimate of the fluctuation rate causing motional narrowing of theresonance spectra (see 𝜈𝑞(𝑇) in Figure 3.5), where 𝜏−1(𝑇MN)matches the static splitting of theline, further expanding the range of 𝜏−1. That this point lies along the steeper of the two lines is astrong confirmation that the same fluctuations responsible for the high temperature 𝑇1minimumcause the motional narrowing. Fits to a simple Arrhenius relationship given by Equation (1.3)yield: 𝜏−10 = 1.23(5) × 1010 s−1 and 𝐸𝐴 = 26.8(6)meV for the shallow slope, low-temperaturefluctuations; and 𝜏−10 = 1.0(5) × 1016 s−1 and 𝐸𝐴 = 0.32(2) eV for the steep high temperaturefluctuations.The motional narrowing above 100K is clear evidence that the corresponding fluctuationsare due to long-range diffusive motion of 8Li+. Unlike liquids, where motion causes the broadsolid state lines to collapse to a single narrow Lorentzian, fast interstitial diffusion in a crystalaverages only some of the features of the lineshape (see e.g., the 7Li NMR spectra in Li3N [228]).In particular, since the quadrupole splitting (the major spectral feature of the 8Li resonance inrutile), is finite at every site, fast motion between sites results in an averaged lineshape consistingof quadrupole satellites split by an average EFG of reduced magnitude, see Figure 3.4. From thiswe conclude that the rate 𝜏−1 for the steep high temperature fluctuations in Figure 3.6 should beidentified with the rate of activated hopping of 8Li+ between adjacent sites — the elementaryatomic process of diffusion in a crystal lattice. Further confirmation of this identification comesfrom the excellent agreement of the activation energy with macroscopic diffusion measurementsbased on optical absorption (OA) [184] and impedance spectroscopy (IS) [203].For a closer comparison with these experiments, we convert our hop rates to diffusivity viathe Einstein-Smoluchowski expression, given by Equation (1.1), using 𝑑 = 1 and 𝑓𝐷 = 1. Using𝑙 ≈ 1.5Å, based on the ideal rutile lattice (details of the precise site of 8Li+ are discussed below),we compare the results to𝐷measured by othermethods in single crystal [184], thin film [229, 230],and nanocrystalline [203] rutile in Figure 3.7. The agreement in activation energy is apparent61EA ≈0:027eVfi −10≈1:2×10 10s −1EA≈0:32eVfi −10≈1:0×1016s −10 10 20 30 404567891000=T (K−1)log10` fi−1=[s−1]´/ T slow1 (Tmin), B0 ‖ (100)/ T slow1 (Tmin), B0 ⊥ (100)q(TMN), B0 ‖ (100)/ Arrhenius FitFigure 3.6: Arrhenius plot of the 8Li fluctuation rate in TiO2 extracted from SLR and NMRmeasurements. Two thermally activated processes can be identified: one below 100K with ashallow slope (blue points); and one at higher temperatures with a steep slope (red points). Thesolid lines are fits to Equation (1.3), with the activation energy 𝐸𝐴 and prefactor 𝜏−10 indicatedfor each process. Adapted from [206]. Copyright © 2017, American Chemical Society.in the similarity of the slopes, but our 𝐷 is somewhat larger than the macroscopic diffusivitydue to a larger prefactor. In our measurements 8Li+ is essentially in the dilute limit, while thebulk measurements have much higher concentrations. One might expect repulsive Li+ Li+interactions would inhibit ionic transport, and yield a smaller macroscopic 𝐷; however, theagreement in 𝐸𝐴 with themacroscopic𝐷, where the (Li/Ti) concentration is as high as 30% [203],implies the barrier is very insensitive to concentration, probably due to strong screening of theCoulomb interaction by the high dielectric response of rutile [231]. Alternatively, Equation (1.1)shows that either overestimating the jump distance 𝑙 or the presence of correlated hopping (thatreduces 𝑓𝐷 from unity) [22] would lead to an overestimate of 𝐷 and might account for some ofthe discrepancy. Note that isotopic mass effects on 𝐷 are expected to be negligible [232].While our 𝐸𝐴 agrees well with macroscopic measurements using OA [184] and IS [203], it62EA ≈0:33eVOptical AbsorbanceEA ≈0:35eVImpedanceSpectroscopyImpedance SpectroscopyCyclic VoltammetryEA ≈0:32eV8Li ˛-NMR0 2 4 6 8 10−14−12−10−8−6−4−21000=T (K−1)log10` D=[cm2s−1]´/ Single CrystalNanoparticle/ Thin FilmArrhenius FitFigure 3.7: Arrhenius plot of the Li+ diffusion coefficient 𝐷 in rutile TiO2 estimated fromthe 8Li+ hop rate extracted in Figure 3.6 using the Einstein-Smoluchowski expression [Equa-tion (1.1)] with 𝑓 = 1, 𝑑 = 1, and 𝑙 = 1.5Å. Literature values obtained from rutile singlecrystals [184], nanoparticles [203], and thin films [229, 230] are shown for comparison. Notethat the concentration of lithium varies greatly in the reported values. While a clear deviationin the magnitude of 𝐷 is observed for different forms of rutile, the similarity of 𝐸𝐴 implies acommon diffusion mechanism. Adapted from [206]. Copyright © 2017, American ChemicalSociety.disagrees with theory by nearly an order of magnitude [192–202]. Generally𝐸𝐴 is amore robustlydetermined quantity than the absolute value of 𝐷 at a single temperature, which may exhibitdependence on both sample [184] and measurement technique (as is apparent in e.g., Figure 3.7).Moreover, several calculations also predict a strong 𝐸𝐴 dependence on Li+ concentration [195,198, 200], inconsistent with our results. While a concentration dependence to 𝐷 has beenobserved [184, 203], it must find an explanation other than a change in 𝐸𝐴. From this weconclude that some ingredient is missing in the theoretical treatments, possibly related to thelattice relaxation around interstitial Li+ that has a strong effect on the calculated barrier [193, 194].Our result is also inconsistent with the suggestion [196] that the higher barrier is characteristic63of diffusion of the Li+-polaron complex instead of simply interstitial Li+. We discuss this pointat more length below.We turn now to the fluctuations that predominate below 100K and cause the low temperature𝑇1minimum. While, we cannot be as conclusive about their origin, we delineate some interestingpossibilities. In contrast to the long-range diffusive behaviour at higher temperature, the smallactivation energywe find is in the range of barriers obtained frommolecular dynamics (MD) [196,198, 199, 202] and DFT [192–194, 198, 200, 201] calculations for interstitial Li+; however, thisappears to be coincidental, since the absence of motional narrowing in this temperature range isinconsistent with long-range motion.On the other hand, the relaxation may be caused by some highly localized Li+ motion at lowtemperature. Local dynamics of organic molecules in solids are well-known, for example therotation of methyl groups of molecules intercalated into crystalline hosts, where they can causesome limited dynamic averaging of the NMR lineshape [224] and relaxation [233]. Analogouseffects are found for some point defects in crystals. For example, a small substitutional cationmay adopt one of several equivalent off-centre sites surrounding the high symmetry site of thelarge missing host cation, and subsequently hop randomly among these sites within the anioniccage (e.g., Ag+ in RbCl [36]).To expand further on this possibility, we now consider the 8Li+ site in rutile in more detail.When Li is introduced either thermally or electrochemically, it is known to occupy the openchannels along the 𝑐-axis. Two high-symmetry sites are available here: theWyckoff 4𝑐 site withina distorted oxygen octahedron; and the 4𝑑 quasi-tetrahedral site, but the precise location remainscontroversial and may depend on Li concentration [189, 234, 235]. Although ion implantation isfar froma thermal process, the implanted ion often stops in themost energetically stable site in theunit cell. From first-principles, the lowest energy site for isolated Li+ is 4𝑐 along the centre of the𝑐-axis channel [196, 198, 200, 201], see Figure 3.1B andC. In disagreementwith these calculations,an off-centre site near 4𝑐 has been predicted [192–194], but this seems unlikely given the modestsize of the quasi-octahedral cage compared to the Li+ ionic radius [236]. Metastable sites outsidethe channels (in the stacks of TiO6 octahedra) have substantially higher energies [192–194, 202]and would also be characterized by much larger EFGs and quadrupole splittings (in Section 3.6.2we consider the prospects for using the quadrupole splitting to determine the 8Li+ site). If, on64the other hand, 8Li+ stops at a metastable site along the channels, such as 4𝑑, it would have avery small barrier to moving to the nearest 4𝑐 site. Thus, while we cannot rule out some localmotion of 8Li+ at low temperature, we regard it as unlikely. Moreover, it is not clear how localmotion could account for the temperature evolution of the resonance area, whose main featureis a peak in intensity below 50K, see Figure 3.4.We now consider another source of low 𝑇 fluctuations, namely the electron polaron, which,for simplicity, we denote as Ti′Ti. The polaron is only slightly lower in energy (0.15 eV) than thedelocalized electronic state (e′) at the bottom of the rutile conduction band [207, 212]. We canwrite the localization transition as:Ti×Ti + e′ ⇌ Ti′Ti.Having localized, the polaron can migrate in an activated manner, with a calculated 𝐸𝐴 thatmay be as low as ∼30meV for adiabatic hopping along the 𝑐-axis stacks of TiO6 octahedra [196,209, 212, 213]. Note that polaron localization also results in the formation of a local electronicmagnetic moment— the polaron is a paramagnetic defect— as is clearly confirmed by EPR [211].At low temperature, the polaron is likely weakly bound to other defects such as an oxygenvacancies, from which it is easily freed [237, 238]. If the one dimensionally mobile polaron andinterstitial Li+ on adjacent sublattices come into close proximity, they may form a bound state:Lii + Ti′Ti ⇌ Lii Ti′Ti,that is a charge-neutral paramagnetic defect complex that has been characterized by EPR andENDOR [215]. The complex is predicted to be quite stable [196], but its EPR signal broadensand disappears above about 50K [215]. The complex is also expected to be mobile via a tandemhopping process [196].8Li+ bound to a polaron will have a very different NMR spectrum than the isolated interstitial.The 3+ charge of the nearby Ti′Ti will alter the EFG and modify the quadrupole splitting, but themagnetic hyperfine field of the unpaired electron spin is an even larger perturbation, so strong infact, that complexed 8Li+will not contribute at all to the resonances in Figure 3.4, since, based on65the ENDOR [215] their resonance frequency is shifted by at least 350 kHz. For this reasonwe alsoexclude the possibility that the high temperature dynamics corresponds tomotion of the Lii Ti′Ticomplex. There is no evidence that its spin polarization is wiped out by fast relaxation whichwould result in a missing fraction in Figure 3.2. However, if immediately after implantationthe 8Li+ is free for a time longer than the period of precession in the RF field (∼1 kHz), it willcontribute to the resonance before binding with a polaron. Similarly, if the Lii Ti′Ti complexundergoes cycles of binding and unbinding at higher temperature, provided it is unbound forintervals comparable to the precession period, it will participate in the resonance. In analogywith the closely related technique of radio frequency muon spin rotation (RF-μSR) [239], onecan thus use the resonance amplitude of the diamagnetic 8Li+ in Figure 3.4 to follow kineticprocesses involving the implanted ion (see e.g., the hydrogenic muonium defect in silicon [240]).Along these lines, we suggest that the nonmonotonic changes in resonance amplitude at lowtemperature reflect dynamics of the Li+-polaron complexation. The sample in ourmeasurementsis nominally undoped, and we expect the main source of polarons is oxygen substoichiometry.From its colour [217], it could have oxygen vacancies at the level of 0.1% or less and polaronsresulting from these vacancies are known to be mobile at quite low-𝑇 (i.e., below 50K) [237, 238].Alternatively, polarons may result from electron-hole excitations created by the implantation of8Li+.The large increase in resonance area between 10 and 25K implies some form of slow dynam-ics on the timescale of the 8Li lifetime 𝜏𝛽. This could be a modulation of the EFG, but more likelyit is a magnetic modulation related to the polaron moment as it mobilizes. This is not motionalnarrowing, but rather a slow variation in the resonance condition, such that the applied RFmatches the resonance frequency for many more 8Li at some point during their lifetime. Theincrease of intensity then corresponds to the onset of polaron motion, while the loss in intensitywith increasing temperature is due to formation of the Li+-polaron complex, and the fluctuationsfrom this motion also become fast enough to produce the 𝑇1 minimum. The complex does notnecessarily survive to high temperatures [215], though, and based on the resonance intensity,the motionally narrowed quadrupolar split resonance at high temperature corresponds to nearlyall of the 8Li. The 𝐸𝐴 from the low temperature slope in Figure 3.6 is remarkably compatiblewith the thermal instability of the intrinsic (unbound) polaron in rutile [196, 209, 211–213, 215],66consistent with this picture.Aside from the activation energies, the prefactors 𝜏−10 fromEquation (1.3)may provide furtherinformation on the processes involved. For atomic diffusion, the prefactor is often consistentwith a vibrational frequency of the atom in the potential well characteristic of its crystalline site,typically 1012–1014 s−1. Prefactor anomalies refer to any situation where 𝜏−10 falls outside thisrange [32]. From Figure 3.6, we see that 𝜏−10 for the high temperature dynamics is anomalouslyhigh, while for the low temperature process it is anomalously low. Within thermodynamicrate theory [25, 26], the prefactor is given by Equation (1.7). For closely related processes, Δ𝑆is not independent of 𝐸𝐴, giving rise to the Meyer-Neldel rule (MNR) and (enthalpy-entropy)correlations between the Arrhenius slope and intercept [31, 33, 241] (see Appendix C), butindependent of such correlations, a prefactor anomaly may simply result from Δ𝑆𝑚/𝑘𝐵 beingsubstantially different from ∼1.We first consider the high temperature prefactor, noting that the bulk diffusivity also showsan unusually large𝐷0 [184]. Prefactors of this magnitude are uncommon, but not unprecedented.For example, 7Li NMR in LiF at high temperature yields a comparably large 𝜏−10 for vacancydiffusion in the “intrinsic” region [242]. Similarly, a large prefactor is observed from 19F NMR insuperionic PbF2 [243]. The latter case was attributed, not to motion of an isolated fluoride anion,but rather to the total effect of all the mobile interstitial F–, whose concentration is also activated.This may also explain the LiF prefactor, but it clearly does not apply to the extrinsic implanted8Li+ in the dilute limit. With the advent of sensitive atomic resolution probes of surfaces inthe past few decades, a very detailed picture of diffusion on crystal surfaces has emerged [68,69], which can help to refine our ideas about bulk diffusion. For example, in some cases, theArrhenius prefactor of adatoms diffusing along a step edge is significantly enhanced over a flatterrace [244]. Like the channels in rutile, step edges consist of a 1D array of vacant sites, but thedirect relevance is not clear, since the adatoms are generally far from the dilute limit. We suggestthat the most reasonable explanation of the high 𝜏−10 for long-range diffusion of Li+ in rutile is alarge Δ𝑆𝑚 which can result from a ballistic picture of hopping [245, 246]. Similarly, a (Li/Ti)concentration dependence of Δ𝑆𝑚 may contribute to the observed concentration dependence of𝐷 [184, 203].Amore common and widely discussed case is a small prefactor as we find at low temperature.67Low prefactors are often encountered in superionic conductors both in NMR [35, 247] andtransport measurements, where they have been attributed to a breakdown of rate theory [248]or to low dimensionality that is often found for these structures [50] (the latter certainly appliesto rutile). However, as argued above, the low temperature 8Li relaxation likely reflects polarondynamics rather than Li+motion. Evidence for this comes from the low temperature evolution ofthe electronic conductivity of lightly deoxidized rutile that shows a resistivity minimum at about50K [238]. The complex low temperature behavior of rutile probably combines polaron bindingto defects [237, 238] with intrinsic polaronic conductivity and the instability to delocalize [212].Prefactors for defect-bound polarons at low temperature are significantly lower than our 𝜏−10 [237].It would be interesting to compare our prefactor with the activated disappearance of the EPR [211,215] to test the connection between these two phenomena with very similar activation energies.3.5 ConclusionIn summary, using low-energy ion-implanted 8Li β-NMR, we have studied the dynamicsof isolated 8Li+ in rutile TiO2. Two sets of thermally activated dynamics were found: onebelow 100K; and one at higher-temperatures. At low temperature, an activation barrier of26.8(6)meV is measured with an associated prefactor of 1.23(5) × 1010 s−1. We suggest this isunrelated to Li+ motion, and rather is a consequence of electron polarons in the vicinity of theimplanted 8Li+ that are known to become mobile in this temperature range. Above 100K, Li+(not polaron complexed) undergoes long-range diffusion, characterized by an activation energyand prefactor of 0.32(2) eV and 1.0(5) × 1016 s−1, in agreement with macroscopic measurements.A cartoon illustrating these findings is shown in Figure 3.8. These results in the dilute limitfrom a microscopic probe indicate that Li+ concentration does not limit the diffusivity even upto high concentrations, but that some key ingredient is missing in the calculations of the barrier.Low temperature polaronic effects may also play a role in other titanate Li+ conductors, such asthe perovskites [249] and spinels [250]. The present data, combined with EPR and transportstudies, will further elucidate their properties.68100K8Li+ DiffusionEA ≈ 0:32 eVPolaron DynamicsEA ≈ 0:027 eV8Li+Ti3+ Ti4+e-1000=Tlog10`fi−1´Figure 3.8: Summary of the sources of dynamics in rutile TiO2 revealed by8Li β-NMR, shown asan Arrhenius plot of the fluctuation rate 𝜏−1. Below 100K, we suggest the low-barrier dynamicsare associated with the mobility of dilute electron polarons in the vicinity of implanted 8Li+. Athigher temperatures, 8Li+ undergoes long-range diffusion (as an uncomplexed cation) withinthe 1D crystallographic tunnels parallel the rutile 𝑐-axis. Adapted from [206]. Copyright © 2017,American Chemical Society.3.6 Supplemental material3.6.1 Biexponential relaxationFollowing the analysis described in Section 3.3.1, the fast relaxing component extractedfrom fitting the SLR measurements at high- and low-field to a biexponential relaxation func-tion, defined in Equation (3.1), is shown in Figure 3.9. While some of the qualitative featuresseen in the slow component at low temperatures are apparent in 1/𝑇fast1 , they are much lesspronounced at high field (see Figure 3.3). The monotonic increase in 1/𝑇fast1 above ∼150Kcontrasts the behaviour of the slow component and dominates over any local maxima that maybe present. Interestingly, the ratio of relaxation times 𝑇fast1 /𝑇slow1 varies only weakly with temper-ature and remains field-independent over much of the temperature range. Deviations from afield-independent ratio occur near the local maxmina in 1/𝑇slow1 , clearly visible in Figure 3.3.We now consider what might produce the biexponential relaxation 𝑅(𝑡, 𝑡′) in Equation (3.1).A fraction of the implanted 8Li+ stopping at a metastable crystallographic site at low temper-ature would show distinct resonance and relaxation (e.g., 8Li+ in simple metals [147]). Thiswould, however, be independent of applied field and would exhibit a very different temperaturedependence from the in-channel diffusing site. As we find no clear evidence for multiple sites inthe resonance analysis or field-independent activated modulation of the SLR rates, we concludea secondary site cannot be the source of the biexponential relaxation.690:000:020:040:06Tfast1=Tslow1101 102100101102Temperature (K)1=Tfast1(s−1)10 mT ⊥ (100)6:55 T ‖ (100)Figure 3.9: Results from the analysis of the 8Li SLR measurements in rutile TiO2 using Equa-tion (3.1) at high- and low-field, described in the text. Shown are the ratio of the fast/slowrelaxation times (top) and the fast relaxation rate (bottom). The ratio varies weakly with tem-perature and is field independent for much of it, deviating only in the vicinity of 𝑇slow1 (𝑇min).Some of the features clearly seen in 1/𝑇slow1 are apparent in 1/𝑇fast1 (cf., Figure 3.3), but muchless pronounced, especially at high field. Adapted from [206]. Copyright © 2017, AmericanChemical Society.If quadrupolar fluctuations are the dominant source of relaxation, as would be expected for8Li+ diffusion, then on the low temperature side of the 𝑇1 minimum, where the fluctuations areslow compared to 𝜔𝐿, the relaxation may be intrinsically biexponential for spin 𝐼 = 2 [71, 155,156]. However, these Redfield-theory calculations of 𝑅(𝑡, 𝑡′) differ in several key assumptionsfrom our situation, specifically: 1) the initial state of the optically polarized 8Li spin is quitedifferent [127]; 2) we are not always in the extreme high field limit; and 3) one dimensionalhopping yields a non-Debye fluctuation spectrum [226, 227]. Moreover, above the 𝑇1 minimum,where the fluctuations are fast, the biexponential should collapse to a single exponential [156],which is certainly not the case here, particularly for the low field data. It remains to be seenwhether a suitably modified theory along these lines could account for the field dependence of70the biexponential at high temperatures.In contrast, at low temperature, we suspect the relaxation has a significant, possibly domi-nant, contribution from the polaron magnetic moment. Here, the spectrum of magnetic fieldfluctuations is naturally field dependent, yielding a strongly field dependent 𝜆. In this case,some of the fast component at low fields would cross over to the high field slow component as𝜆(𝐵0) decreases with increasing field. This may contribute to the pronounced change in the slowrelaxing fraction 𝑓slow from high to low field, whose behaviour is reminiscent of longitudinal fielddecoupling often observed in muon spin rotation/relaxation/resonance (μSR). A detailed fielddependence of the relaxation, specifically in the region of ∼150mT [215], would be necessary toto confirm this.The character of the relaxation (magnetic, quadrupolar or mixed) can also be tested by com-parison with another Li isotope, such as the spin-3/2 9Li as has been recently demonstrated [251–253]. It is important to recall that the amplitude of the relaxing signal is temperature independenteven at low field, meaning that we do not have a significant fraction of the signal that is so fastrelaxing that it is lost (wiped out) as has been seen in 7Li NMR from polarons in the perovskiteLi3xLa2/3–xTiO3 [249]. We find no evidence for a missing fraction to the level of 10% of the totalsignal, but can’t rule out a small missing component at the level of a few %. Thus if a significantfraction of 8Li+ forms the bound complex here, then its relaxation remains measurable even atlow field.3.6.2 The 4c siteThe quadrupole splittings depend sensitively on the 8Li+ site and its symmetry, and with theangular dependence of the splittings (see Section 2.3.1) combined with calculations of the EFG(including lattice relaxation), one might be able to make a site assignment. Here we set out afew properties of the most likely site (4𝑐) to make some initial observations based on the sitesymmetry in an ideal lattice.The 4𝑐 site is coordinated by two near-neighbour and four more distant oxide ions in ashortened octahedron. As can be seen in Figure 3.10A, the axis of the two nearest-neighbouroxide ions alternates from one site to the next along 𝑐 by 90°. Beyond the first coordination,71there are two nearest neighbour Ti on opposite sides, and the Ti Li Ti direction also alternatesbetween [100] and [010] along the channel, shown in Figure 3.10B. Overall, the site has 2/𝑚symmetry with the two-fold axis parallel to 𝑐. This symmetry is too low to yield an axial EFG, so𝜂 is non-zero, and may even approach ∼1; however, if we consider fast hopping along the 𝑐-axis,and the alternating character of the adjacent sites, one expects that the time-average EFG willbecome (four-fold) axisymmetric. This should be the case at room temperature, where there isclear evidence for Li+ diffusion, and a simple test of an axial angle dependence could confirmthis.The characteristics of this site are also important in determining properties of the Li+-polaroncomplex. Here, one of the two neighbouring titanium is 3+, rather than 4+, which furtherlowers the site symmetry and alters the EFG. Calculations suggest that the polaron is mainlymobile along the stacks of adjacent edge-sharing TiO6 octahedra in the 𝑐 direction [196, 209, 212,213] and it is unable to move to the other Ti neighbour on the far side of the Li site. Notice thatif the polaron does hop to the next Ti along the stack, it is not the near neighbour of the adjacentLi site, but of the second nearest Li site along the chain. This is easily seen by the alternation ofTi Li Ti mentioned above (see Figure 3.10B). If the bound complex moves in tandem, withthe polaron remaining in a single TiO6 stack, then fast motional averaging should not result inan axisymmetric EFG, in contrast to the case of free Li+ diffusion.72ABFigure 3.10: View of the near-neighbour atoms to interstitial lithium (green) in the rutile TiO24𝑐 site, showing their 90° alternating coordination axis between neighbouring 4𝑐 sites along the𝑐-axis channels. The nearest-neighbour atoms are indicated here by connecting grey cylinders.(A) The two nearest-neighbour oxygen (red) atoms. (B) Two nearest-neighbour titanium (blue)atoms. The structures were drawn using CrystalMaker [205]. Adapted from [206]. Copyright ©2017, American Chemical Society.73Chapter 4Ionic and electronic properties of thetopological insulator Bi2Te2Se4.1 IntroductionBismuth chalcogenides with the formula Bi2Ch3 (Ch = S, Se, or Te) are narrow gap semi-conductors that have been studied for decades for their thermoelectric properties. They crys-tallize in the layered tetradymite structure [254, 255], consisting of stacks of strongly boundCh Bi Ch Bi Ch quintuple layers (QLs) loosely coupled by van derWaals (vdW) interactions(see Figure 4.1). More recently, interest in their electronic properties has exploded [256] follow-ing the realization that strong spin-orbit coupling and band inversion combine to make themthree-dimensional (3D) topological insulators (TIs) [257], characterized by a gapless topologicalsurface state (TSS). Electronically, this family of TIs is characterized by a relatively insulatingbulk and a robustly conductive surface, with greater contrast in conductivity between the tworegions significantly facilitating identification and study. The prevalence for self-doping inbinary chalcogenides (e.g., Bi2Se3 or Bi2Te3) often yields crystals far from insulating in the bulk,masking the signature of the conductive surface state. This has been mitigated, for example, inthe most widely studied tetradymite TI Bi2Se3 with Ca doping to suppress the more usual 𝑛-typeconductivity [258, 259]. On the other hand, the stoichiometric ordered [260, 261] ternary linecompound Bi2Te2Se (BTS) exhibits a much lower conductivity thanks to its fortuitous crystal74unit cellTeBiSevan der Waalsgap (2.698 Å)Bi2Te2Sespace group:R3m (#166)quintuplelayer(7.319 Å)30.051 ÅFigure 4.1: Crystal structure of BTS [261], consisting of Te Bi Se Bi Te layers. These QLsare weakly coupled through vdW interactions, giving rise to a characteristic gap between adjacentTe planes. This atomic arrangement is analogous to transition metal dichalcogenides (TMDs),where it is possible to insert foreign atoms and smallmolecules within the vdWgap. The structurewas drawn using Visualization for Electronic and STructural Analysis (VESTA) [273]. Adaptedfrom [274]. Copyright © 2019, American Physical Society.chemistry [261], with anti-site defects (e.g., bismuth substitution on a tellurium site) playing animportant role [262]. Indeed, BTS crystals with a characteristic bulk band gap of ∼0.3 eV [263]have demonstrated such desired large bulk resistivities at low temperatures [261, 264–266]. Agreat deal is known about its surface properties, with angle-resolved photoemission spectroscopy(ARPES) revealing the characteristic linear dispersion about the Dirac point [262, 267–272];however, the material is not an ideal TI due to the close proximity of the Dirac point to the topof the bulk valence band [255].With the novel electronic structure of the Bi chalcogenides evident primarily in the TSS,much inquiry has focused on surface sensitive probes, such as ARPES, scanning tunnellingspectroscopy (STS), and transport. On the other hand, nuclear magnetic resonance (NMR) iswell-known to reveal electronic properties in metals through hyperfine coupling of the nucleusto surrounding electron spins that gives rise to the Knight shift, a measure of the Pauli spinsusceptibility, and the Korringa spin-lattice relaxation (SLR) [7, 42]. Theory predicts dramaticeffects in such quantities for the TSS [275], but NMR is generally a bulk probe with very little75Table 4.1: Properties of host nuclei in BTS relevant to NMR. For each isotope, the nuclearspin 𝐼, gyromagnetic ratio 𝛾/2𝜋, quadrupole moment 𝑄, and natural abundance are listed. Forcomparison, properties of the β-detected nuclear magnetic resonance (β-NMR) probe nucleus8Li are included in the bottom row.Isotope 𝐼 𝛾/2𝜋 (MHzT−1) 𝑄 (mb) Natural abundance (%)77Se 1/2 +8.156 785 0 7.6123Te 1/2 −11.234 910 0 0.89125Te 1/2 −13.545 426 0 7.07209Bi 9/2 +6.962 477 −516 1008Li 2 +6.3016 +32.6 0sensitivity to the surface. Despite this, a considerable body of conventional NMR in the Bi2Ch3TIs has accumulated. Since these results are closely related to the present study, we give a briefsummary.All elements in BTS have NMR-active isotopes (see Table 4.1). The most conspicuous fea-ture of 100% abundant 209Bi NMR is the strong quadrupolar interaction. However, the broadquadrupole pattern shows clear evidence of a shift related to carrier density, indicating a verystrong hyperfine coupling [276], confirmed more recently by very high field NMR [277]. The be-haviour of 1/𝑇1 is less consistent, with results ranging from𝑇-linear to nearly𝑇-independent [276,278] at low temperature. The low abundance (∼7%) puremagnetic spin 1/2 probes 77Se and 125Teshow a small 𝑇-independent shift, and Korringa relaxation at low temperature in orientationallyaveraged powder spectra [279]. This Korringa relaxation is enhanced in nanocrystals and wasattributed to the TSS [280], but given the evidence for aging effects that stabilize a conventionalmetallic surface state [281], this connection remains unclear. More recent work in single crystalshas identified distinct resonances from inequivalent chalcogen planes, and a more detailedanalysis of the chalcogen NMR is required [282–284].While a great deal is known about the TSS from surface sensitive probes, little is knownabout how this behaviour transitions to the bulk as a function of depth below the crystal surface.We plan to address this question using highly spin-polarized radioactive ions, specifically, usingdepth-resolved β-NMR [5] with short-lived 8Li as the probe nucleus, similar to low-energymuon spin rotation (LE-μSR) [15]. As a first step, here we present results using relatively highimplantation energies, 5–28 keV, corresponding to 8Li+ implantation depths below the TSS. We76find that below ∼150K 8Li experiences strongly field dependent relaxation and resonance shifts.The relaxation is 𝑇-linear and reminiscent of a Korringa mechanism, but conventional theoryfor NMR cannot account for its field dependence, which we suggest is due to magnetic carrierfreeze out. At higher temperatures, an additional relaxation mechanism is observed, originatingfrom ionic diffusion of isolated 8Li+ in the vdW gap.The remainder of the chapter is organized as follows: a brief description of the experimentis given in Section 4.2, followed by the results and analysis in Section 4.3. A detailed discussionis given in Section 4.4, focusing on the high temperature dynamics of the isolated 8Li+ ion in thevdW gap (Section 4.4.1), and the electronic properties of BTS giving rise to the field dependentrelaxation and resonance shifts at low temperature (Section 4.4.2). Finally, a concluding summarycan be found in Section 4.5.4.2 ExperimentA single crystal of BTS was grown as detailed by Jia et al. [261] and taken from the moreinsulating section of a larger boule. This insulating character was confirmed by 4-wire transportmeasurements, which revealed an increase in resistivity by a factor of ∼3 below 100K, charac-teristic of the more insulating BTS compositions [261]. At low temperature, the resistivity ismaximized at ∼0.1Ω cm. Prior to the β-NMR experiments, the BTS crystal, with dimensions5 × 4 × 0.1mm3, was cleaved in air and affixed to a sapphire plate using Ag paint (SPI Supplies,West Chester, PA) for mounting on a cold finger cryostat.β-NMR experiments were performed at TRIUMF’s Isotope Separator and ACcelerator (ISAC)facility in Vancouver, Canada. A low-energy highly polarized beam of 8Li+ was implanted intothe BTS single crystal within one of two dedicated spectrometers [5, 121, 122, 124]. The incident8Li+ ion beam had a typical flux of ∼106 ions/s over a beam spot ∼2mm in diameter. At theimplantation energies 𝐸 used here (5 keV to 28 keV), the ions stop at average depths >30 nm(see Figure 4.14 in Section 4.6.1). Spin-polarization was achieved in-flight by collinear opticalpumping with circularly polarized light, yielding a polarization of ∼70% [127], and monitoredafter ion-implantation through the anisotropic β-decay emissions of 8Li. Specifically, the β-countrates in two opposed scintillation counters were recorded, whose asymmetry is proportional77to the average longitudinal nuclear spin-polarization [5, 12], with the proportionality factordepending on the experimental geometry and the details of the β-decay. Details of the SLRand continuous wave (CW) resonance measurments are given in Section 2.3. A typical SLRmeasurement took ∼20min, while resonance measurements typically took ∼30min to acquire.4.3 Results and analysis4.3.1 Spin-lattice relaxationTypical 8Li SLR data at high and lowmagnetic field in BTS are shown in Figure 4.2 for severaltemperatures, where the spectra have been normalized by their 𝑡 = 0 asymmetry (𝐴0). It isimmediately evident that the SLR rates are strongly dependent on both temperature and field. Inhigh fields, the relaxation is relatively slow, but comparable to that observed in some elementalmetals [5] and semimetals [285]. Surprisingly, the relaxation is faster than in the normal state ofthe structurally similar NbSe2 [286], despite a very much smaller carrier density. However, it isalso slower than in the 3D TI Bi1–xSbx [285]. At all temperatures, the relaxation rate increasesmonotonically with decreasing magnetic field. Even at very low temperatures near∼10K, wheremost excitations are frozen out, the SLR remains substantial, as shown in Figure 4.3. Therelaxation rate at low field is orders of magnitude faster than in Tesla fields, suggesting theimportance of low field relaxation from the host lattice nuclear moments [139, 152]. On top ofthe field dependence, there is also a strong temperature dependence; the relaxation rate increaseswith increasing 𝑇, but this trend is non-monotonic and at least one temperature exists (per field)where the rate ismaximized.We now consider a detailed analysis to quantify these observations. First, we remark thatthe relaxation is non-exponential at all temperatures and fields. The precise origin for thisremains unclear. While it is well-known that magnetic relaxation of quadrupolar spins ismultiexponential, this is not the case here, as the initial state following optical pumpinghas purelyvector (i.e., Zeeman) polarization,1 yielding single exponential magnetic relaxation, similar to1That is, the magnetic sublevel populations 𝑝𝑚 vary linearly in𝑚, as opposed to, for example, quadratically inthe case of quadrupolar alignment. This description of spin orientation (as well as other higher order terms) can beobtained by decomposing the spin system’s density matrix in terms of irreducible spherical tensor operators ̂𝑇𝑘,𝑞 (seee.g., [12, 40, 41, 71, 287–289]). Specifically, vector/Zeeman polarization corresponds to the ̂𝑇𝑘=1,𝑞=0 term.780 5 10Time (s)0.00.20.40.60.81.0Normalized AsymmetryBeamOnBeamOff2.20 T (001)0 13.86 MHz6 K87 K173 K295 K0 5 10Time (s)BeamOnBeamOff7.5 mT (001)0 47.26 kHz10 K100 K210 K297 KFigure 4.2: 8Li SLR data in BTS at high and low magnetic fields. The characteristic kink at𝑡 = 4 s corresponds to the trailing edge of the 8Li+ beam pulse. The 8Li SLR is strongly field-dependent, increasing with decreasing magnetic field, and increases non-monotonically withincreasing temperature. The solid black lines are the result of a global fit to all spectra at agiven field to a stretched exponential relaxation function, Equation (2.13), convolved with thesquare beam pulse [153, 154]. The spectra have been binned by a factor of 20 for clarity. Adaptedfrom [274]. Copyright © 2019, American Physical Society.the conventional NMR case where all the quadrupole satellites are simultaneously saturated (seee.g., [290]). If, however, the relaxation is quadrupolar (due to a fluctuating electric field gradient(EFG)), then in the slow limit for 𝐼 = 2, the relaxation is biexponential [71, 155, 156]. In light ofthis, we model the spin relaxation using a stretched exponential, consistent with the approachadopted in conventional NMR in similar materials [276, 279, 280, 291]. Explicitly, for a 8Li+ ionimplanted at time 𝑡′, the spin polarization at time 𝑡 > 𝑡′ follows Equation (2.13). where 𝜆 ≡ 1/𝑇1is the relaxation rate and 0 < 𝛽 ≤ 1 is the stretching exponent. We find this to be the simplestmodel that fits the data well across all temperatures and fields. Using Equation (2.13) convolvedwith the 4 s beam pulse [153, 154], SLR data grouped by magnetic field 𝐵0 and implantationenergy 𝐸 were fit simultaneously with a shared common initial asymmetry 𝐴0(𝐵0, 𝐸). Notethe statistical uncertainties in the data are strongly time-dependent and accounting for this is790 2 4 6 8 10 12Time (s)0.00.20.40.60.81.0Normalized Asymmetry6.55 T (001)2.20 T (001)20 mT (001)7.5 mT (001)2.5 mT (001)Beam On Beam OffT=10 KFigure 4.3: Field dependence of 8Li SLR data in BTS at 10K. The characteristic kink at 𝑡 = 4 scorresponds to the trailing edge of the 8Li+ beam pulse. Even at low temperature, where mostexcitations are frozen out, the 8Li SLR is strongly field-dependent, increasing with decreasingmagnetic field. This field dependence suggests the importance of fluctuations from host latticenuclear moments [139, 152]. An implantation energy of 28 keV was used for each measurement,except at 2.20 T, which was recorded at 20 keV 8Li+. The solid black lines are the result of a globalfit to all spectra at a given field to a stretched exponential relaxation function, Equation (2.13),convolved with the square beam pulse [153, 154]. The spectra have been binned by a factor of 20for clarity.crucial in the analysis. To find the global least-squares fit, we used custom C++ code leveragingthe MINUIT [176] minimization routines implemented within ROOT [175]. The fit quality isgood in each case ( ̃𝜒2global ≈ 1.02) and a subset of the fit results is shown in Figure 4.2 as solidblack lines. The large values of 𝐴0 extracted from the fits (∼10% for 𝐵0 ≥ 2.20 T and ∼16% for𝐵0 ≤ 20mT) are consistent with the full beam polarization, implying that there is no appreciablemissing fraction due to a very fast relaxing component. 𝐴0 extracted from these fits are used tonormalize the spectra in Figure 4.2. For all the fits, the stretching exponent 𝛽 ≈ 0.5, with a weaktemperature dependence: in high field, decreasing slightly at temperatures below ∼200K.800 50 100 150 200 250 300Temperature (K)0.000.050.100.150.200.250.301/T 1 (s1 )28 keV6.55 T10 keV3.60 T5 keV2.20 T1/T1 a+ bT 1/T1 Jn(n 0)Figure 4.4: The SLR rate for 8Li in BTS at high magnetic fields with 𝐵0 ∥ (001) using a stretchedexponential analysis. 1/𝑇1 is strongly field-dependent, increasing with decreasing magnetic field,but also increases non-monotonically with temperature. At each field, a clear 1/𝑇1 maximumcan be identified where the average fluctuation rate of the dynamics inducing relaxationmatchesthe 8Li Larmor frequency. The solid black lines are fits to the model in Equations (1.3), (1.15)and (4.1), consisting of a linear 𝑇-dependence with a non-zero intercept and a term due to ionicdiffusion. The highlighted grey region denotes the lower limit of measurable 1/𝑇1 due to the8Liprobe lifetime. Adapted from [274]. Copyright © 2019, American Physical Society.4.3.2 Modelling relaxationWe now consider a model of the temperature dependence of the measured 1/𝑇1. We interpretthe relaxation peak as a Bloembergen-Purcell-Pound (BPP) peak [34], where the rate of afluctuating interaction with the nuclear spin sweeps through the Larmor frequency 𝜔0 = 𝛾𝐵0at the rate peak [34, 35, 225]. As we discuss in more detail below, we attribute this fluctuationto diffusive motion of interstitial 8Li+ in the vdW gap between QLs in BTS. It is clear that thepeaked relaxation adds to an approximately linearly temperature dependent term, reminiscentof Korringa relaxation characteristic of NMR in metals [7, 42], and probably of electronic origin.810 50 100 150 200 250 300Temperature (K)05101520253035401/T 1 (s1 )28 keV7.5 mT1/T1 maximumFigure 4.5: Temperature dependence of the 8Li+ 1/𝑇1 in BTS at low magnetic field [7.5mT ⟂(001)]. The relaxation rate is orders of magnitude larger than at high fields, with both a largerapparent 𝑇-linear slope and intercept. There is some evidence of a small 1/𝑇1 peak (see inset)near ∼150K superposed on the dominant linear dependence. Above 200K, 1/𝑇1 increasessubstantially, suggestive of another relaxation mechanism absent at higher fields. Adaptedfrom [274]. Copyright © 2019, American Physical Society.Based on this, we use the following model:1/𝑇1 = 𝑎 + 𝑏𝑇 + 𝑐 (𝐽1 + 4𝐽2) . (4.1)The first two terms in Equation (4.1) account for the linear 𝑇-dependence 𝜆e ≡ 𝑎 + 𝑏𝑇 and theremaining terms 𝜆diff ≡ 𝑐 (𝐽1 + 4𝐽2) describe the peak. 𝜆diff consists of a coupling constant, 𝑐,proportional to the mean-squared transverse fluctuating field and the 𝑛-quantum NMR spectraldensity functions, 𝐽𝑛 [225]. In this context, 𝐽𝑛 is a frequency dependent function peaked at 𝑇max,which occurs when the fluctuation rate driving relaxation matches ∼ 𝑛𝜔0. While the choiceof a precise form of 𝐽𝑛 depends in detail on the dynamics, we use the empirical expression ofRichards [35] given by Equation (1.15), which gives the correct asymptotic limits for relaxationproduced by two-dimensional (2D) fluctations [51], originating, for example, from diffusion of82Table 4.2: Results from the analysis of the 8Li SLR 1/𝑇1 peaks in BTS, shown in Figures 4.4and 4.5, using Equations (1.3), (1.15) and (4.1). Here, ∡ denotes the orientation of the BTStrigonal 𝑐-axis with respect to the applied field𝐵0 and𝐸 is the8Li+ implantation energy. Values forthe coupling constant 𝑐, prefactor 𝜏−10 , and activation energy 𝐸𝐴 are indicated. For comparison,the kinetic parameters extracted from fitting 𝜔0(𝑇max) to Equation (1.3) in Figure 4.10 (describedin Section 4.4.1) are shown in the bottom row. The good agreement in these values, independentof the analysis details, indicates a single common dynamic process. Differences in the parameterpairs 𝜏−10 and 𝐸𝐴 may be attributed to the empirical Meyer-Neldel rule (MNR) [33], which isevident when plotted on a semi-logarithmic scale (see Figure 4.9). Adapted from [274]. Copyright© 2019, American Physical Society.𝐵0 (T) ∡ 𝐸 (keV) 𝑐 (106 s−2) 𝜏−10 (1012 s−1) 𝐸𝐴 (eV)6.55 ∥ 28 1.88(2) 1.02(13) 0.156(3)3.60 ∥ 10 5.08(10) 0.7(2) 0.153(6)2.20 ∥ 20 2.86(4) 28(7) 0.235(5)2.20 ∥ 5 7.23(16) 1.2(5) 0.173(9)0.0075 ⟂ 28 0.07(1) 1 0.164(3)0.8(3) 0.185(8)8Li+ confined to the vdW gap. As mentioned previously, we make the usual assumption that the𝜏𝑐 in Equation (1.15) follows an Arrhenius temperature dependence given by Equation (1.3).Fits of the 1/𝑇1 data to the model given by Equation (4.1) with Equations (1.3) and (1.15)are shown in Figures 4.4 and 4.5 as solid black lines, clearly capturing the main features. Whilethe kinetic parameters determining the position and shape of the SLR peaks differed somewhatat different fields (see Table 4.2), good overall agreement was found from this analysis, and weconsider the details further below in Section 4.4.1. As anticipated, the slope and intercept of𝜆e were both strongly field dependent. At low fields, the intercept 𝑎 is quite large and varies as𝜔−20 (Figure 4.6), behaviour characteristic of relaxation due to host nuclear spins. Just as in thecase of 8Li in NbSe2 [139], this field dependence can be described by a simple BPP model [34]given by Equation (1.14). A fit to this model, shown as a solid red line in the upper panel inFigure 4.6, yields a coupling constant ̄𝐶20 = 1.5(3) × 106 s−2 and 𝜏𝑐 = 3.2(7) × 10−5 s. We notethat the magnitude of 𝐶20 is compatible with the observed8Li resonance linewidth, discussedin Section 4.3.3, and that 𝜏𝑐 is remarkably close to the209Bi NMR 𝑇2 time in Bi2Se3 [278]. It is,however, not clear whether the intercept at 2.20 T is a remnant of this low field relaxation, or isdue to an electronic mechanism [292]. In any case, it is small and we will not consider it further.In contrast, the slope 𝑏 varies as 𝜔−1.50 with a pronounced anisotropy between orientations830510a (s1 )B0 (001)B0 (001)105 106 107 1080 (s 1)10 410 310 2b (s1 K1 )b 1.51±0.040B0 (001)B0 (001)10 2 10 1 100 101B0 (T)1 2 30 (108 s 1)0.000.02a (s1 )Figure 4.6: Field dependence of the intercept 𝑎 and slope 𝑏 describing the low temperatureelectronic relaxation in Equation (4.1). (Top panel): The 𝑇 → 0 intercept of 1/𝑇1 falls withincreasing field as 1/𝜔20, behavior consistent with a low field relaxation due to fluctuating hostlattice nuclear spins [139]. The solid red line indicates a fit to a simple BPP model [34] givenby Equation (1.14), yielding a coupling constant ̄𝐶20 = 1.5(3) × 106 s−2 and correlation time𝜏𝑐 = 3.2(7) × 10−5 s, the latter being remarkably close to the 209Bi NMR 𝑇2 time in Bi2Se3 [278].The inset shows a zoom of the small high field intercepts, at the lower limit measurable dueto the 8Li probe lifetime highlighted in grey. (Bottom panel): Field dependence of the lowtemperature slope of 1/𝑇1, exhibiting a strong orientation dependence, but a common fielddependence ∝ 𝜔−1.50 , indicated by the solid coloured lines. Adapted from [274]. Copyright ©2019, American Physical Society.(Figure 4.6). We return to these points below in Section 4.4.2.4.3.3 Resonance spectraWe now turn to the 8Li resonance in BTS. Typical spectra at high field are shown in Figure 4.7.Consistent with a non-cubic crystal, the 8Li NMR is quadrupole split (as described in Section 2.3).This is confirmed by the helicity-resolved spectra (see Figure 4.15 in Section 4.6.2) which showopposite satellites in opposite helicities [5]. The EFG that produces the splitting is characteristicof the 8Li site in the crystal. From the spectra, it is relatively small, on the order of a few kHz.84MgO reference41.20 41.25 41.30 41.35Frequency (MHz)050100150200250300Temperature (K)B0=6.55T (001)Normalized Asymmetry (a.u.)Figure 4.7: 8Li NMR spectra in BTS with 6.55 T ∥ (001) shown as solid black lines. The verticalscale is the same for all the spectra, but the baselines are offset to match the absolute temperatureindicated on the left. The spectra reveal a small, temperature-independent quadrupole splitting,centred about an unsplit Lorentzian line, whose amplitude grows above 100K, becoming thedominant feature at higher temperatures. The solid coloured lines are global fits to a sum ofthis Lorentzian plus 2𝐼 = 4 quadrupolar satellites of the quadrupole split resonance (see textfor further details). The reference frequency, determined by the 8Li resonance position in MgO(100) at room temperature, is indicated by the vertical dashed grey line. Note the small negativeshift of the BTS resonances as the temperature is lowered. Adapted from [274]. Copyright ©2019, American Physical Society.In addition, a second unsplit line is evident near the midpoint of the quadrupole pattern. Thisline must be due to a distinct 8Li site with a small, unresolved quadrupolar splitting. A similarunsplit resonance was observed in the vdW layered material NbSe2 [286], which was thought tooriginate from implanted 8Li+ stopped at an interstitial site within the vdW gap.This evidence for two sites suggests that our single component relaxation model in Sec-tion 4.3.2 is too simple. However, more complicated relaxation functions suffer from over-parametrization, and we have retained the single stretched exponential to represent the overallaverage relaxation of 8Li in BTS.The resonance was found to evolve substantially with temperature, as shown in Figure 4.7.85Here, the spectra have been normalized to the off-resonance steady-state asymmetry to accountfor the variation of intensity due to SLR [147]. While it is apparent that the satellite intensitiesand splitting remain nearly temperature independent, the amplitude of the central line increasessignificantly above ∼100K, becoming the dominant feature by room temperature. Additionally,a small shift in the resonance centre-of-mass frequency can be seen, increasing in magnitudewith decreasing temperature.To quantify these observations, we now consider a detailed analysis, noting first that thescale of the quadrupolar splitting is given by the quadrupole frequency 𝜈𝑞 [60], defined inEquation (2.4). As the splitting is small (i.e., 𝜈𝑞 ≪ 𝜈0), the satellite positions are given accuratelyby Equation (2.5) using the first-order corrections in Equations (2.6) and (2.8) [60]. From 𝜈0, weadditionally calculate the frequency shift, 𝛿, in parts per million (ppm) using Equation (2.11)and the position in MgO at 300K with 𝐵0 ∥ (100) as the reference [148].The helicity-resolved spectra were fit using the quadrupole splitting above with 𝜈0 and𝜈𝑞 as free parameters, in addition to line widths and amplitudes. Similar to the SLR data inSection 4.3.1, a global fitting procedure was used by way of ROOT’s [175] implementation ofMINUIT [176]. The two helicities of each spectrum were fit simultaneously with resonancepositions and widths as shared parameters. The fits were constrained such that the centre-of-mass frequency 𝜈0 was shared between the unsplit Lorentzian and quadrupole satellites. Anydifference in centre of mass of the split and unsplit lines was too small to measure accurately,and the two lines shift in unison with temperature, as can be seen in Figure 4.7. In addition, weassume the EFG principal axis is along the 𝑐-axis and 𝜂 = 0, making the angular factor unity.This assumption does not affect accurate extraction of the splitting frequency, but precludesunambiguous identification of the EFG tensor elements. Based on a simple point charge modelof the lattice (see Section 4.6.3), all reasonable interstitial 8Li+ sites retain the 3-fold rotationaxis of the hexagonal unit cell, supporting this simplification. Changes to the satellite splittingpattern from preliminary measurements at low field with 𝐵0 ⟂ 𝑐 are consistent with theseassumptions.The main parameters extracted from this analysis are shown in Figure 4.8. Consistent withthe two main qualitative features of the spectra, both the amplitude and shift show substantialchanges with temperature. In the top panel of Figure 4.8, above ∼150K, the shift is nearly8650050Shift (ppm)10 keV3.60 T20 keV2.20 T28 keV6.55 TT* 180 K0 50 100 150 200 250 300Temperature (K)0102030NormalizedAmplitude (%)centralLorentzianquadrupolesatellitesFigure 4.8: Fit results for 8Li resonance in BTS at high field with 𝐵0 ∥ (001), following the fittingprocedure described in the text. The upper panel shows the resonance shift 𝛿 relative to MgO.Above ∼150K, the shift is nearly field independent and scattered about 0; however, at lowertemperatures, the trends at each field diverge, revealing a field dependent shift whose magnitudeis maximized at the lowest temperatures. The lower panel shows the change in amplitude of thecentral Lorentzian and quadrupole satellites (averaged over all four lines), normalized by theoff-resonance baseline [147]. The amplitude of the unsplit line grows above ∼100K, contrastingthe temperature independence of the quadrupole satellite amplitudes. The inflection point 𝑇∗of this trend is indicated by the dashed vertical black line. The solid black lines are drawn toguide to the eye. Adapted from [274]. Copyright © 2019, American Physical Society.field independent and centred around ∼0 ppm. Shifts of this magnitude are difficult to quantifyaccurately and the scatter in the values is of the same order as systematic variations in the lineposition, making differences of this order not very meaningful. In contrast, at lower tempera-tures, the shift trends for each field diverge, revealing a significant field-dependent shift whosemagnitude is maximized at the lowest temperature.The changes in amplitude of central Lorentzian and quadrupole satellites (averaged overall four lines), normalized by the off-resonance baseline [147], appear in the bottom panel ofFigure 4.8. Consistent with Figure 4.7, the amplitude of the unsplit line grows above ∼100Kand approaches saturation near room temperature, in contrast to the temperature insensitivity87of the satellite amplitudes. From the smooth growth of the unsplit line amplitude, we identifythe inflection point 𝑇∗ ≈ 180K of the trend, as indicated in Figure 4.8. The other spectralparameters are nearly independent of temperature. The line widths are about ∼6 kHz, with thecentral Lorentzian narrowing slightly with increasing 𝑇. This narrowing is likely the cause of itsincrease in amplitude. Similarly, 𝜈𝑞 ≈ 7.4 kHz was characteristic of the splittings over the entiremeasured temperature range.Based on these results at the lowest measured temperatures, where dynamic contributionsto the resonance are absent, we estimate a 1:1 relative occupation for 8Li+ in the two sites.Note, however, that changes in amplitude, up to our highest measured temperature 317K, areinconsistent with a site change, where the growth of one amplitude is at the expense of the other(see e.g., the 8Li β-NMR in Ag [121] or Au [293]).4.4 DiscussionWith the main results presented in Section 4.3, the remaining discussion is organized asfollows: in Section 4.4.1, we consider the dynamics causing the high temperature 1/𝑇1 peaks,while Section 4.4.2 considers the electronic properties of BTS giving rise to relaxation andresonance shifts at low temperature.4.4.1 High temperature lithium-ion diffusionThe most likely source of the relaxation at high temperatures is diffusive motion of 8Li+.While we cannot rule out some local stochastic motion within a cage, or motion of anotherspecies in the lattice, given the demonstrated ability to chemically insert Li at room temperaturein isostructural bismuth chalcogenides [294, 295], we expect a low barrier to interstitial diffusionfor any implanted 8Li+ in the vdW gap. Stochastic motion causes the local magnetic field andEFG to become time dependent causing relaxation [34, 35]. From the SLR, we can thus obtaininformation on the kinetic parameters of the diffusion. From the model of 1/𝑇1(𝑇) introducedin Equation (4.1) from Section 4.3.2, we obtain the kinetic parameters listed in Table 4.2. Thesuccess of the model is demonstrated by the good self-agreement in the barrier 𝐸𝐴 and prefactor880.10 0.15 0.20 0.25 0.30EA (eV)101110121013101410 (s1 )MN=0.022 ± 0.002 eVTiso=254 ± 27 KMeyer-Neldel fitJn(n 0) fitsimple parabolic fitFigure 4.9: Meyer-Neldel plot of the 8Li+ Arrhenius activation energy 𝐸𝐴 and prefactor 𝜏−10pairs from Table 4.2, showing the exponential increase in 𝜏−10 with increasing 𝐸𝐴. The linearityof the data on this scale suggests of a common origin for the kinetics. From the fit to the datashown in black, the compensation energy ΔMN and isokinetic temperature 𝑇iso are identifiedand indicated in the inset.𝜏−10 , indicating that an activated dynamic process with a 2D spectral density provides a singlecommon source for the observed 𝜆diff. Thus, the correlation rate 𝜏−1𝑐 in Equation (1.3) representsthe atomic hop rate 𝜏−1. The small barrier, on the order of ∼0.2 eV, is consistentwith expectationsfor an isolated interstitial ion, where Coulomb Li+ Li+ repulsion is negligible. Similarly, the𝜏−10 s on the order of ∼1012 s−1 are compatible with typical optical phonon frequencies, as is oftenthe case for mobile ions in a lattice. The relatively small deviations in 𝐸𝐴 and 𝜏−10 obtained atdifferent fields may be ascribed to the empirical MNR, where 𝜏−10 increases exponentially withincreasing 𝐸𝐴, as is often observed for related kinetic processes [33]. This relationship is evidentwhen the values in Table 4.2 are plotted on a semi-logarithmic scale, as shown in Figure 4.9 (seealso Appendix C).In general, the exponent 𝛼 appearing in 𝐽𝑛 from Equation (1.15) can vary from 1 to 2 [52],with deviations from 2 reflecting correlated dynamics that can arise from, for example, Coulomb89interactions with other ions acting to bias the probe ion’s trajectory [45]. Such correlations affectthe shape the 1/𝑇1 peak, yielding a characteristic asymmetry with a shallower slope on the low-𝑇side. In contrast, the high symmetry about 𝑇max is consistent with uncorrelated fluctuationsdriving relaxation, as expected for isolated 8Li undergoing direct interstitial site-to-site hopping.As further confirmation of the appropriateness of the form of 𝐽𝑛, we consider an alternativeapproach agnostic to these details. At each field, we determine the temperature 𝑇max of the 1/𝑇1peak using a simple parabolic fit (after removal of the 𝑇-linear contribution). This approach hasthe advantage that it does not rely on any particular form of 𝐽𝑛, and we recently used it to quantifydiffusion of isolated 8Li+ in rutile TiO2 [206]. Finally, for each 𝑇max we assume 𝜏−1𝑐 matchesthe Larmor frequency 𝜔0. The results are shown in the Arrhenius plot in Figure 4.10, wherethe linearity of the data, spanning three orders of magnitude, demonstrates the consistencyof the approach. The Arrhenius fit shown yielded an activation energy 𝐸𝐴 = 0.185(8) eV andprefactor 𝜏−10 = 8(3) × 1011 s−1, in good agreement with the values from the analysis using the2D 𝐽𝑛. Noting that this result lies in the middle of range reported in Table 4.2, we take it as thebest determination of the 8Li+ hop rate.Another well-known signature of diffusion in NMR is motional narrowing. When thediffusive correlation rate exceeds the characteristic static frequency width of the line, the localbroadening interactions are averaged and the line narrows. In the context of dilute interstitialdiffusion in a lattice, the primary quadrupolar interaction may, however, not be averaged tozero, since, in the simplest case, each site is equivalent and characterized by the same EFG. Weobserve a slight narrowing and a large enhancement in the amplitude of the unsplit resonancewith an onset in the range 100K to 120K, consistent with where the extrapolated 𝜏−1𝑐 wouldbe in the kHz range of the linewidth. With the CW resonance measurement, we often find thechange in amplitude is more pronounced than the width [296].Using the 8Li+hop rate fromabove, we convert 𝜏−1 to diffusivity via theEinstein-Smoluchowskiexpression, given by Equation (1.1), where 𝑑 and 𝑓𝐷 are assumed to be 2 and 1, respectively,to compare with other measurements of interstitial ionic diffusion in related materials. Using𝑙 ≈ 4.307Å, the distance between neighbouring 3𝑏 sites in the vdW gap in the ideal BTS lat-tice (see Figure 4.16 in Section 4.6.3), we estimate 𝐷 for 8Li+, finding a value on the order of10−7 cm2 s−1 at 300K. An Arrhenius plot comparing the diffusivity of 8Li+ in BTS with other903 4 5 6 7 81000/T (K 1)1051061071081091 (s1 )EA=0.185 ± 0.008 eV10 = (8 ± 3)× 1011 s 1Arrhenius fit±1 uncertainty1/T1(Tmax) : B0 (001)1/T1(Tmax) : B0 (001)300 250 200 175 150 125T (K)Figure 4.10: Arrhenius plot of the 8Li fluctuation rate in BTS extracted from the NMR frequencydependent positions of 1/𝑇1(𝑇max) in Figures 4.4 and 4.5. The solid red line is a fit to Equation (1.3)with the activation energy𝐸𝐴 and prefactor 𝜏−10 indicated in the inset and the fit’s±1𝜎 uncertaintyband is highlighted in grey. The kinetic parameters extracted from this minimally model-dependent analysis are in good agreement with those obtained from fits of the 1/𝑇1 data toEquations (1.3), (1.15) and (4.1), given in Table 4.2. Adapted from [274]. Copyright © 2019,American Physical Society.cations in structurally related materials [294, 297–301] is shown in Figure 4.11. Note that mostof these diffusion coefficients were determined using either radiotracer [297, 298] or transientelectrochemical [294, 299, 300] techniques. For ℎ-Li0.7TiS2, we used the Li+ hop rate determinedby 7Li NMR [301] and Equation (1.1), taking 𝑓𝐷 = 1, 𝑑 = 2, and 𝑙 = 2.14Å (the distance between1𝑏 and 2𝑑 sites) [302].It is clear that the mobility of isolated 8Li+ is exceptional; our estimate for 𝐷 greatly exceedsthat of lithium in the well-known fast ion conductor ℎ-LixTiS2 [301, 302]. Similarly, the lithiumdiffusion coefficient in lithium intercalated Bi2Se3 is considerably slower [294], possibly due toLi+ Li+ interaction. Interestingly, the mobility of Cu in isostructural Bi2Te3 is also extremelyhigh, as revealed by 64Cu radiotracer [297] and electrochemical methods [300]. Lastly, we notethat similarly large𝐷 values were reported recently for 8Li+ in the one dimensional ion conductor91Li+ inLi0.1Bi2Se30 2 4 61000/T (K 1)10 1110 1010 910 810 710 610 510 4D (cm2 s1 )Cu+ in CuxBi2Te3EA=0.21 eVAg+ inAgxBi2Te3EA=0.45 eV8Li+ in Bi2Te2SeEA=0.185 eV(this work)Li+ inh-Li0.7TiS2EA=0.41 eVLi+ inLi0.6NbSe2EA=0.52 eV1000 500 300 200 150T (K)Figure 4.11: Arrhenius plot of the diffusion coefficient 𝐷 for 8Li+ in BTS. Values for 𝐷, shownin black, were estimated using the Einstein-Smoluchowski expression [Equation (1.1)], taking𝑓 = 1, 𝑑 = 2, and 𝑙 = 4.307Å, with the values of 𝜏−1 taken from Figure 4.10. For comparison,diffusion coefficients for Li+ [294, 299, 301], Ag+ [298], and Cu+ [297, 300] in structurally relatedmaterials are also included. It its clear that the diffusivity of isolated 8Li+, free from repulsive Li+-Li+ interactions, is exceptional, significantly exceeding that in the well-known fast ion conductorℎ-LixTiS2 [301, 302]. Adapted from [274]. Copyright © 2019, American Physical Society.rutile TiO2 [206] and we speculate that the exceptional mobility may be generic for isolated Li+(i.e., at infinitely dilute concentrations) in ion conducting solids.2 It would be interesting totest this conjecture against detailed ab initio calculations. Understanding the mobility of diluteintercalates, a simple theoretical situation difficult to interrogate experimentally, remains offundamental interest [303].4.4.2 Low temperature electronic propertiesWe turn now to the low temperature results, below the diffusion-related peak, in the range 5Kto 150K. The diffusive contribution to the relaxation rate is falling exponentially with reduced2That is, the mobility of isolated Li+ may be exceptional relative to more concentrated conditions in materialswhose structures are amenable to macroscopic ion transport.92temperature below the peak, and the remaining low temperature relaxation must thus have adistinct origin. In this regime, the relaxation rate is linear in 𝑇, with a slope that is significantlyfield dependent. In this range there is also a 𝑇-dependent resonance shift that is also fielddependent.In metals, the dominant features of NMR (and β-NMR) are due to coupling between the con-duction electron and nuclear spins, giving rise to the Korringa 𝑇-linear 1/𝑇1 and a 𝑇-independentKnight shift proportional to the Pauli susceptibility [7, 42]. The apparent Korringa-like depen-dence in Figures 4.4 and 4.5 is thus surprising, since ideal BTS is a narrow gap semiconductor,with an energy gap 𝐸𝑔 ≈ 0.3 eV [263]. In comparison to metals, NMR in semiconductors is muchless well-known, largely because the coupling to the electron spins is much less universallydominant, and often other channels, such as phonons, compete with electronic effects andcomplicate the interpretation (e.g., in elemental Te [304] and InSb [305], with similar 𝐸𝑔 to BTS).However, while calculations and experiments agree on 𝐸𝑔 in BTS [263, 267], it is abundantlyclear from both experiment [306] and theory [262, 307] that it is unlikely to exist as an intrinsicsemiconductor. Rather, it is significantly self-doped by native defects, such as chalcogenidevacancies (donor) and Bi/Te antisite defects (acceptor). Among the tetradymites, BTS is relativelyhighly insulating, but this does not indicate a paucity of native defects, but rather a coincidentalnear compensation between the 𝑛- and 𝑝-types. Brahlek et al. have made the point that thesematerials are, in fact, so highly doped that they are metallic, but are poor conductors due to disor-der [308]. Such disorder is clearly evident in our data, contributing to the stretched exponentialcharacter of the spin relaxation and the line broadening.The origin of the linearity of the Korringa law is evident from a derivation based on aFermi golden rule approach to the spin-flip scattering of conduction electrons by the hyperfineinteraction with the nuclear spin. The sum over electronmomenta when converted to an integralover energy yields:1𝑇1=2𝜋ℏ𝐴2∫𝜌2(𝐸)𝑓(𝐸) [1 − 𝑓(𝐸)] d𝐸, (4.2)where 𝐴 is the hyperfine coupling energy, 𝑓(𝐸) is the Fermi-Dirac distribution, 𝜌(𝐸) is theelectronic density of states, and the integral is over all electron energies in the conduction band.In a broad band degenerate metal, where the Fermi level 𝐸𝐹 ≫ 𝑘𝐵𝑇, the density of states 𝜌(𝐸) is93practically constant over the range where the Fermi factor is nonzero, and the integral is givento an excellent approximation by:1𝑇1=2𝜋ℏ𝐴2𝜌2(𝐸𝐹)𝑘𝐵𝑇. (4.3)The Korringa slope is thus determined by the square of the product of 𝐴 and 𝜌(𝐸𝐹). In fact, oninspection of the 8Li 𝜆e in BTS, we find it comparable to wide band metals with vastly highercarrier densities [121, 147, 152, 286, 293, 309–311] (see Figure 4.12). While the coupling 𝐴 forimplanted 8Li+ in BTS is not known, it is unlikely to compensate for the much lower 𝜌(𝐸𝐹), toyield a comparable Korringa slope. Note that the Korringa law is remarkably robust to disorderand, for example, applies in the normal state of the alkali fullerides where the mean free path iscomparable to the lattice constant (the Ioffe-Regel limit) [312]. However, in highly disorderedmetals, the slope is strongly enhanced [313–315]. Such an enhancement may account for thesubstantial slope we observe. However, Equation (4.2) indicates that if 𝜌(𝐸) has significantstructure on scales comparable to 𝑘𝐵𝑇, as it might in a narrow impurity band, its detailed formand nondegeneracy can render the 𝑇-dependence nonlinear [316], as found, for example, nearthe metal-insulator transition (MIT) in doped silicon [317]. Thus, the evident linearity is stillsurprising. Moreover, in metals, the Korringa slope does not depend on magnetic field.In doped Si, near theMIT, a field dependence of the enhanced Korringa slope has been foundat mK temperatures [319], where it was attributed to the occurrence of uncompensated localizedelectron spins, probably on some subset of more isolated neutral P donors. Such moments arealso evident in the NMR of the dopant nuclei [320]. However, in BTS (in contrast to Si), due tothe high dielectric constant and low effective mass,magnetic carrier freezeout [321, 322] mayaccount for the diminished slope at high fields. In this case, the field localizes the carriers so theyno longer participate in the conduction band, correspondingly reducing the Korringa slope. Thismay account, in part, for the significant positive magnetoresistance in BTS [265, 323]. Similarly,the field and temperature dependent shifts in Figure 4.8 may reflect a constant (diamagnetic)contribution (cf. the 125Te shift [279]), in addition to a positive hyperfine field related to thecarriers, which diminishes with localization. However, the localized electrons may well provideadditional (more inhomogeneous) relaxation and resonance broadening. We know of no case940 50 100 150 200 250 300Temperature (K)0.000.050.100.150.200.250.300.351/T 1 (s1 )Bi2Te2Se2.20 T NbSe23.00 TPt4.10 TAg15 mTFigure 4.12: Comparison of the 8Li SLR rate in BTS with other materials where electronicrelaxation dominates. The rates in NbSe2 [286], Ag [147], and Pt [311] were reported previously,with additional estimates shown here for NbSe2 obtained using the resonance baseline asymme-try [147, 286]. The solid black lines are fits to the data. The bump in the Ag 1/𝑇1 around 125Kis due to 8Li transitioning from a metastable interstitial to a substitutional site [121, 147, 318].The temperature dependence of the relaxation rate in BTS contrasts the structurally similarNbSe2, which, in spite of a much higher carrier concentration, shows a shallower Korringaslope and the absence of diffusion-induced 1/𝑇1 peak; however, common to both materials is anapparent apparent finite 1/𝑇1(𝑇 = 0K) intercept, even at relatively high magnetic fields. Thehighlighted grey region denotes where 1/𝑇1 ≤ 10−2/𝜏𝛽 and is difficult to measure during the8Liprobe lifetime [12].where NMR has been used to study magnetic freezeout. In order to test this idea, it will beessential to compare results on different samples of BTS and related materials (e.g., Ca-dopedBi2Se3).Above, we only considered bulk origins for the 𝜆e. While we do not expect any directcoupling to the TSS at the implantation energies used, it is important to consider the effects ofband bending at the surface. If the bulk electronic bands are bent downward at the surface, aconventional 2D electron gas may be stabilized [281]. Metallic screening will confine this surfaceregion to a few nm from the surface. In the opposite case, upward bending makes the surface95region more insulating, and the dielectric screening is much weaker, producing a depletionregion on the scale of µm, as has been demonstrated by ionic liquid gating [324]. In the lattercase, the acceptor band is depopulated by the surface dipole. Calculations suggest this is not thecase [325], but there is substantial evidence for time-dependent band bending from ARPES [271].Generally, this downward bending is found to produce a more metallic surface. With its exposureto air prior to the measurements, we assume that our sample’s surface is passivated and ismetallic, typical of an “aged” surface, so we can safely neglect possible depth dependence to thecarrier concentration at the implantation energies used. This is consistent with the absence ofan appreciable implantation energy dependence in 𝜆e at 2.20 T (see Figure 4.13).3We have, so far, focused exclusively on the interaction between 8Li and the electron spins.We should also consider its interaction with their orbital currents. Note that at 100K 1/𝑇1 inBTS is comparable to semimetallic bismuth [285], with a similar carrier density, but a muchlonger mean free path. In Bi, due to its strong orbital diamagnetism, we suggested that orbitalfluctuations might be responsible for the fast relaxation of 8Li and its concentration dependencein BixSb1–x solid solutions [285]. This mechanism is due to fluctuating electronic currents, so itis naturally related to the conductivity 𝜎. Specifically [326]:(1𝑇1)orb∝ 𝑘𝐵𝑇∫Re {𝜎⟂(𝑞, 𝜔0)}𝑞2d3𝑞, (4.4)where𝜎⟂(𝑞, 𝜔) is the generalizedwavevector (𝑞) and frequency dependent conductivity transverseto the nuclear spin. Clearly, this relaxation is enhanced by higher conductivity; however, inmetals, it has long been recognized that orbital relaxation is usually much weaker than thespin-related Korringa relaxation discussed above.For interstitial 8Li+ in a layered conductor, the hyperfine coupling 𝐴 for interstitial 8Li can beparticularly weak (e.g., NbSe2 [286]). In these circumstances, it is possible that orbital relaxationwill dominate. Unlike the contact coupling 𝐴, orbital fields fall as 1/𝑟3, where 𝑟 is the distancebetween the probe spin and the fluctuating current, so all nuclei in the material, and potentially3We note that there is an apparent depth dependence in the diffusive contribution to 1/𝑇1 in Figure 4.13. At lower8Li+ implantation energies, the 1/𝑇1 peak appears larger, though its position remains unchanged. The precise originfor this is unclear. One possibility, however, is an enhancement in the number of structural defects close to the BTSsurface, which could inflate the magnitude of the fluctuating field (i.e., the coupling term 𝑐 in Equation (4.1)). Notethat the values of 𝑐 for the different data series in Figure 4.13 only differ by a factor of ∼2.5 (see Table 4.2), implyingthe defect enhancement isn’t dramatic, consistent with the depth-independent electronic 1/𝑇1.960 50 100 150 200 250 300Temperature (K)0.000.050.100.150.200.250.301/T 1 (s1 )20 keV2.20 T5 keV2.20 T1/T1 a+ bT 1/T1 Jn(n 0)Figure 4.13: Temperature dependence of the 8Li SLR rate 1/𝑇1 in BTS at 𝐵0 = 2.20 T ∥ (001)for two implantation energies. The virtually identical relaxation rates below 150K suggest theelectronic contribution to 1/𝑇1 is depth independent over the range of8Li+ implantation energiesused here. However, the magnitude of the 1/𝑇1 peaks are visibly depth dependent, though theirpositions remain unchanged. The origin for this remains unclear.even nuclei in close proximity, will sense spatially extended orbital fluctuations. For example,this mechanism has been studied as a proximal source of decoherence in spin based qubitdevices [327]. Lee and Nagaosa have explicitly considered the case of orbital relaxation in a2D layered metal [328]. They find a weak logarithmic singularity in 1/𝑇1 in the clean limit thatis cut off by a finite mean free path [329]. Similar to the Korringa rate, the orbital relaxationrate is linear in 𝑇 for a broad band metal. Recently, this approach was generalized to the caseof massive Dirac-like electrons in 3D, appropriate to semimetallic Bi1–xSbx [326, 330]. Theyfind 𝑇-linear relaxation when the chemical potential is outside the gap [326], but within thegap, an anomalous dependence that is explicitly field dependent via the NMR frequency 𝜔0 as𝜆orb ∝ 𝑇3 ln(2𝑘𝐵𝑇/ℏ𝜔0). While this is not what we find, it does suggest that if orbital relaxationis effective here, it may exhibit some unexpected field dependence in the inhomogeneousmetallic state hypothesized for the tetradymites [331]. Similarly, the existence of orbital current97fluctuations implies additionally a fluctuating EFG. This gives rise to a quadrupolar contributionto the orbital relaxation mechanism, which has been treated in detail for 3D Dirac electronsystems [326]. While it is unclear if the result applies directly to our situation, this quadrupolarcontribution to 1/𝑇1 was concluded to be negligible.In fact, some features of our data do suggest the importance of orbital effects. One is thesimilarity of shifts of the quadrupolar split resonance and the unsplit Lorentzian. If, as seems tobe reasonable, these resonances originate in different lattice sites of 8Li+, then one would expectdifferent hyperfine couplings and different (spin) shifts. If the shift is rather orbital in originwith significant contribution from long length-scale currents, one would expect the same shiftfor any site (and even any nucleus) in the unit cell. In general, however, one would expect bothspin and orbital couplings, and nuclei of such different species, such as 209Bi or 8Li, would likelydiffer.We have ruled out a number of possibilities, but we do not have a conclusive explanation ofthe interesting features of the data at low temperatures. At this point, it is worth noting thatconventional NMR in the tetradymite TIs are also characterized by highly variable power law𝑇-dependent relaxation [276, 278, 332] whose dependence on magnetic field has largely notbeen explored.4.5 ConclusionUsing temperature and field dependent ion-implanted 8Li β-NMR, we studied the hightemperature ionic and low temperature electronic properties of BTS. Two distinct thermalregions were found; above ∼150K, the isolated 8Li+ probe undergoes ionic diffusion with anactivation energy 𝐸𝐴 = 0.185(8) eV and attempt frequency 𝜏−10 = 8(3) × 1011 s−1 for atomicsite-to-site hopping. A comparison of the kinetic details with other well-known Li+ conductorssuggests an exceptional mobility of the isolated ion. At lower temperature, field dependentrelaxation and resonance shifts are observed. While the linearity in temperature of the SLR rateis reminiscent of a Korringa mechanism, existing theories are unable to account for the extent ofthe field dependence. We suggest that these may be related to a strong contribution from orbitalcurrents or the magnetic freezeout of charge carriers in the heavily compensated semiconductor.98Field dependent conventional NMR of the stable host nuclei, combined with the present data,will further elucidate their origin.4.6 Supplemental material4.6.1 Implantation profilesAs mentioned in Section 4.2, 8Li+ implantation profiles in BTS were predicted using theStopping and Range of Ions in Matter (SRIM) Monte Carlo code [133]. At each implantationenergy, stopping events were simulated for 105 ions, with their resulting histogram representingthe predicted implantation profile. From the profiles shown in Figure 4.14, we calculate, in thenomenclature of ion-implantation literature, the range and straggle (i.e., the mean and standarddeviation) at each simulated energy. At the implantation energies used here (5 keV to 28 keV),the incident 8Li+ ions typically stop on average >30 nm below the crystal surface, depths wellbelow where the TSS is expected to be important.4.6.2 Helicity-resolved resonance spectraTypical helicity-resolved resonance spectra are shown in Figure 4.15, demonstrating thetwo key features in the line’s fine structure. A quadrupolar splitting on the order of severalkHz, clearly evidenced by the asymmetric shape about the resonance centre-of-mass in eachhelicity, can be associated with the outermost satellite lines. Note that the satellite intensitiesare different from conventional NMR and are determined mainly by the high degree of initialpolarization, which increases the relative amplitude of the outer satellites [127], with theirprecise (time-average) values depending on the relaxation details [147]. Secondly, anothersignificantly smaller quadrupolar frequency can be ascribed to a “central” Lorentzian-like line,analogous to what was observed in the structurally similar NbSe2 [286]. Note that there is nounshifted𝑚±1/2 ↔ 𝑚∓1/2 magnetic sublevel transition, in contrast to spin 𝐼 = 3/27Li. The radiofrequency (RF) amplitude dependence and the absence of other multiquantum lines indicatethat it is also not a multiquantum transition. Instead it must originate from the overlap of thefour unresolved satellites with a small quadrupole splitting [285]. Such a feature, in a noncubic990.00 0.01 0.02Stopping probability (nm 1)050100150200250300Depth (nm)8Li+Bi2Te2Se(7.76 g cm 3)5 keV10 keV20 keV28 keV0 10 20 308Li+ Energy (keV)020406080100120Depth (nm)RangeStraggleFigure 4.14: Stopping distribution and range for 8Li+ implanted in BTS calculated using theSRIMMonte Carlo code [133]. The histogram profiles, shown on the left, are generated fromsimulations of 105 ions. The ion range and strange at each implantation energy are shown onthe right. All measurements in this study correspond to average stopping depths ≥30 nm belowthe crystal surface, well below where the TSS is expected to be important. Adapted from [274].Copyright © 2019, American Physical Society.layered crystal (see Figure 4.1 in Section 4.1), is suggestive that this component originates from8Li+within the vdW gap, where the magnitude of EFGs at interstices are minimized. Resonancesof the two helicities can be combined to give an overall average lineshape (see bottom panel inFigure 4.15), whose evolution with temperature is shown in Figure 4.7 from Section 4.3.3.4.6.3 8Li+ sitesHere we consider the stopping sites in more detail. Generally, ion-implanted 8Li+ occupieshigh-symmetry crystallographic sites that locally minimize its electrostatic potential. This mayinclude metastable sites that are not the energetic minimum, but have a significant potentialbarrier to the nearest stable site. While these sites are characteristic of the isolated implantedion, they may be related to the lattice location of Li+ obtained by chemical intercalation.100quadrupolesatellitescentralLorentzian1.00.50.00.51.0AsymmetryChange (%)positivehelicitynegativehelicityT=172 KB0=6.55 T (001)41.20 41.25 41.30 41.35Frequency (MHz)6.57.0Asymmetry (%)combinedhelicitiesFigure 4.15: Typical helicity-resolved 8Li resonance spectra in BTSwith 6.55 T ∥ (001), revealingthe fine structure of the line. Four quadrupole satellites, split asymmetrically in each helicityabout a “central” Lorentzian line are evident. Note that, in contrast to conventional NMR, thesatellite amplitudes are determined primarily by the high degree of initial polarization [127].The solid coloured lines are global fits to a sum of five Lorentzians with positions given byEquations (2.4) to (2.6) and (2.8) (see Section 4.3.3 for further details). Upon combining helicities,a nearly symmetric line about the resonance centre-of-mass, shown in the bottom panel, isobtained. Adapted from [274]. Copyright © 2019, American Physical Society.BTS is structurally similar to the TMDs that consist of triatomic layers separated by a vdWgap between chalcogen planes. This spacious interstitial region accommodates many types ofextrinsic atoms and small molecules in the form of intercalation compounds [333]. Similarly, avariety of dopants have been intercalated into the tetradymite Bi chalcogenides, such as Cu [300],Ag [334], Au [335], and Zn [336]. Lithium has also been inserted into Bi chalcogenides [294, 295,337, 338], but the precise sites for Li+ in the vdW gap have not been determined [295]. Note thatin all these cases, intercalation is at the level of atomic %, so that intercalated species certainlyinteract (e.g., forming “stage” compounds [333]).Based on this, we expect the lowest energy site for implanted 8Li+ is within the vdW gap,similar to NbSe2 [286]. Here, the EFGs are likely minimized, yielding a small quadrupole101Figure 4.16: High-symmetry sites within the vdW gap of BTS, shown as coloured circles, in aplane perpendicular to the trigonal 𝑐-axis. TheWyckoff position is indicated for each site and theboundary of the unit cell is marked by solid black lines. Neighbouring 3𝑏 (blue) sites, enclosedby Te quasi-octahedra, are connected indirectly through 6𝑐 (green) sites with quasi-tetrahedralTe coordination, and directly through 9𝑑 (pink) sites of lower symmetry. The structure wasdrawn using VESTA [273]. Adapted from [274]. Copyright © 2019, American Physical Society.frequency, consistent with the unsplit component of the resonance in Figure 4.15. Withinthe vdW gap, several high-symmetry Wyckoff sites are available (see Figure 4.16): the quasi-octahedral 3𝑏 at (0, 0, 1/2); the quasi-tetrahedral 6𝑐 at (0, 0, 1/6); and the 2-fold coordinated 9𝑑 at(1/2, 0, 1/2). Here, the fractional coordinates correspond to the hexagonal unit cell in Figure 4.1from Section 4.1. Neighbouring 3𝑏 sites are connected by direct paths through the 9𝑑 sites andindirect paths (i.e., dog-leg trajectories) passing through 6𝑐 sites. The 3𝑏 site offers by far thelargest coordination volume for interstitial 8Li+ and it is reasonable that this is the preferredsite in the vdW gap. Indeed, preliminary density functional theory calculations confirm thisassignment.As indicated in Section 4.3.3, the low temperature resonances suggest a nearly 1:1 relativeoccupation of two sites with different EFGs. Noting that, in contrast to the trilayers in TMDs,the QLs in BTS account a much larger volume fraction of the crystal, which leads us to considerpossible interstitial sites therein. While interstitial sites within the QL will be characterized by102lower-symmetry and much larger EFGs, the most likely sites retain the trigonal rotation axis(e.g., 6𝑐 at (0, 0, 1/3)). A simple point-charge model of isolated 8Li+ in the BTS lattice, using ioniccharges of +0.3 for Bi and −0.2 for Se/Te (1/10 their nominal values), gives 𝜈𝑞 for these sites thatare within a factor ∼3 of the values for the 3𝑏 site in the vdW gap (∼2 kHz), consistent with thedifference required to explain the experimental spectra. Note that, while the point charge modelpredicts nearly identical |𝜈𝑞| for all sites in the vdW gap, sites on the edges of the hexagons inFigure 4.16 have the opposite sign.From the helicity-resolved resonances, the sign of the EFG does not change with temperatureand, based on the 𝑇-independent 𝜈𝑞 and satellite amplitudes, we suggest that this componentcorresponds to a fraction of implanted 8Li+ that stops at a site within the QL, where it remainsstatic over its lifetime. Similarly, we ascribe the unsplit line to 8Li+ stopped in the vdW gap, likelyin the 3𝑏 site. This component, in contrast, is dynamic above 100K, accounting for the growthin resonance amplitude and the 1/𝑇1 maxima, which we consider in more detail in Section 4.4.1.Note that we find no evidence for a site change transition up to 317K, indicating a substantialenergy barrier separates the two sites.103Chapter 5Local ionic, electronic, and magneticproperties of the doped topologicalinsulators Bi2Se3:Ca and Bi2Te3:Mn5.1 IntroductionThe bismuth chalcogenides Bi2Ch3 (Ch = S, Se, or Te) of the layered tetradymite structureare an interesting class of highly two dimensional narrow bandgap semiconductors. Strongspin-orbit coupling inverts the energy ordering of their bands, making them bulk topological in-sulators (TIs) characterized by a single Dirac cone at the Brillouin zone centre and a topologicallyprotectedmetallic surface state [259, 339]. This has augmented longstanding interest in their ther-moelectric properties with significant efforts (theoretical and experimental) to understand theirelectronic properties in detail [254, 255]. Consisting of weakly interacting Ch Bi Ch Bi Chatomic quintuple layers (QLs) (see e.g., Figure 4.1), they can also accommodate intercalantspecies such as Li+ in the van der Waals (vdW) gap between QLs [294, 295, 340], similar tothe layered transition metal dichalcogenides (TMDs) [341, 342]. Although their bandgaps are∼150meV, doping by intrinsic defects, such as Ch vacancies, yields crystals that are far frominsulating. To increase the contrast in conductivity between the bulk and themetallic topologicalsurface state (TSS), they can be compensated extrinsically. For example, Ca substitution for Bisuppresses the self-doped 𝑛-type conductivity in Bi2Se3 [258, 259]. Doping can also be used to104modulate their magnetic properties, yielding magnetic TIs where the TSS is gapped [343], forexample, by substitution of Bi with a paramagnetic transition metal [344, 345].The intriguing electronic properties of the tetradymite TIs have predominantly been inves-tigated using surface sensitive probes in real and reciprocal space (e.g., scanning tunnellingspectroscopy (STS) and angle-resolved photoemission spectroscopy (ARPES)), as well as otherbulk methods. In complement to these studies, nuclear magnetic resonance (NMR) offers theability to probe their electronic ground state and low-energy excitations through the hyperfinecoupling of the nuclear spin probe to the surrounding electrons. Such a local probe is espe-cially useful when disorder masks sharp reciprocal space features, as is the case in Bi2Ch3. Theavailability of a useful NMR nucleus, however, is usually determined by elemental compositionand natural (or enriched) isotopic abundance, as well as the specific nuclear properties suchas the gyromagnetic ratio 𝛾 and, for spin > 1/2, the nuclear electric quadrupole moment 𝑄.While the Bi2Ch3 family naturally contain several NMR nuclei [276, 279, 291], they are eitherlow-abundance or have a large 𝑄. As an alternative, here we use an ion-implanted NMR probeat ultratrace concentrations, with detection based on the asymmetric property of radioactive𝛽-decay [5].Using β-detected nuclear magnetic resonance (β-NMR), we study two single crystals ofdoped Bi2Ch3 — compensated Bi2Se3:Ca (BSC) and magnetic Bi2Te3:Mn (BTM) — each witha beam of highly polarized 8Li+. In many respects, β-NMR is closely related to muon spinrotation/relaxation/resonance (μSR), but the radioactive lifetime is much longer, making thefrequency range of dynamics it is sensitive tomore comparable to conventional NMR. In additionto purely electronic phenomena, in solids containing mobile species, NMR is also well knownfor its sensitivity to low frequency diffusive fluctuations [34, 36, 346], as are often encountered inintercalation compounds. At ion-implantation energies sufficient to probe the bulk of BSC andBTM, we find evidence for ionic mobility of 8Li+ above ∼200K, likely due to two-dimensional(2D) diffusion in the vdW gap. At low temperature, we find Korringa relaxation and a smalltemperature dependent negative Knight shift in BSC, allowing a detailed comparison with8Li in the structurally similar Bi2Te2Se (BTS) [274]. In BTM, the effects of the Mn momentspredominate, but remarkably the signal can be followed through the magnetic transition. Atlow temperature, we find a prominent critical peak in the relaxation that is suppressed in105a high applied field, and a broad, intense resonance that is strongly shifted. This detailedcharacterization of the 8Li NMR response is an important step towards using depth-resolvedβ-NMR to study the low-energy properties of the chiral TSS.5.2 ExperimentDoped TI single crystals BSC and BTM with nominal stoichiometries Bi1.99Ca0.01Se3 andBi1.9Mn0.1Te3 were grown as described in Refs. [258, 344] and magnetically characterized using aQuantumDesign magnetic property measurement system (MPMS). In the BTM, a ferromagnetictransition was identified at 𝑇𝐶 ≈ 13K, consistent with similar Mn concentrations [344, 347–349].In contrast, the susceptibility of the BSC crystal was too weak to measure accurately, but the datashow no evidence for a Curie tail at low-𝑇 that could originate from dilute paramagnetic defects.β-NMR experiments were performed at TRIUMF’s Isotope Separator and ACcelerator (ISAC)facility in Vancouver, Canada. Detailed accounts of the technique can be found in Refs. [5, 274].A low-energy highly polarized beam of 8Li+ was implanted into the samples mounted in one oftwo dedicated spectrometers [5, 122]. Prior to mounting, the crystals were cleaved in air andaffixed to sapphire plates using Ag paint (SPI Supplies, West Chester, PA). The approximate crys-tal dimensions were 7.8mm × 2.5mm × 0.5mm (BSC) and 5.3mm × 4.8mm × 0.5mm (BTM).With the crystals attached, the plates were then clamped to an aluminum holder threaded intoan ultra-high vacuum (UHV) helium coldfinger cryostat. The incident 8Li+ ion beam had atypical flux of ∼106 ions/s over a beam spot ∼2mm in diameter. At the implantation energies 𝐸used here (between 1 keV to 25 keV), 8Li+ stopping profiles were simulated for 105 ions usingthe Stopping and Range of Ions in Matter (SRIM) Monte Carlo code (see Section 5.6.1) [133].For 𝐸 > 1 keV, a negligible fraction of the 8Li+ stop near enough to the crystal surface to sensethe TSS. Most of the data is taken at 20 keV, where the implantation depth is ∼100 nm, and theresults thus reflect the bulk behaviour. A typical spin-lattice relaxation (SLR) measurement took∼20min. A single resonance spectrum typically took ∼30min to acquire.1060 5 10Time (s)0.00.20.40.60.81.0Normalized AsymmetryBeamonBeamoff20 keV 8Li+6.55 T (001)[ 0 41.27 MHz]global fit4 K100 K203 K294 K0 5 10Time (s)BeamonBeamoff20 keV 8Li+15 mT (001)[ 0 94.5 kHz]global fit4 K100 K225 K300 KBi2Se3:CaFigure 5.1: Typical 8Li SLR data in Ca doped Bi2Se3 at high (left) and low (right) magneticfield with 8Li+ implanted at 20 keV. The shaded region indicates the duration of the 8Li+ beampulse. The relaxation is strongly field dependent, increasing at lower fields, and it increasesnon-monotonically with increasing temperature. The solid black lines show fits to a stretchedexponential described in the text. The initial asymmetry 𝐴0 from the fits is used to normalizethe data which are binned by a factor of 20 for clarity.5.3 Results and analysis5.3.1 Bi2Se3:CaTypical 8Li SLR data in BSC, at both high and lowmagnetic field, are shown in Figure 5.1. Toaid comparison, 𝐴(𝑡) has been normalized by its initial value 𝐴0 determined from fits describedbelow. Clearly, the SLR is strongly temperature and field dependent. At low field, the SLR is verymuch faster, due to additional relaxation from fluctuations of the host lattice nuclear spins [139].The temperature dependence of the relaxation is non-monotonic, indicating that some of thelow frequency fluctuations at 𝜔0 are frozen out at low temperature.The relaxation is non-exponential at all temperatures and fields, so the data were fit with thephenomenological stretched exponential. This approach was also used for 8Li in BTS [274] and107in conventional NMR of related materials [276, 279, 291]. Explicitly, for a 8Li+ implanted at time𝑡′, the spin polarization at time 𝑡 > 𝑡′ follows Equation (2.13). This is the simplest model thatfits the data well with the minimal number of free parameters, for the entire Bi2Ch3 tetradymitefamily of TIs.Using Equation (2.13) convolved with the beam pulse, the SLR data in BSC, grouped bymagnetic field 𝐵0 and implantation energy 𝐸, were fit simultaneously with a shared commoninitial asymmetry 𝐴0(𝐵0, 𝐸). Note that the statistical uncertainties in the data are strongly time-dependent (see e.g., Figure 5.1), which must be accounted for in the analysis. Using customC++ code incorporating the MINUIT minimization routines [176] implemented within theROOT data analysis framework [175], we find the global least-squares fit for each dataset. Thefit quality is good ( ̃𝜒2global ≈ 1.02) and a subset of the results are shown in Figure 5.1 as solidblack lines. The large values of 𝐴0 extracted from the fits (∼10% for 𝐵0 = 6.55 T and ∼15% for𝐵0 = 15mT) are consistent with the full beam polarization, with no missing fraction. The fitparameters are plotted in Figure 5.2, showing agreement with the qualitative observations above.We now consider a model for the temperature and field dependence of 1/𝑇1. We interpret thelocal maxima in 1/𝑇1 in Figure 5.2 as Bloembergen-Purcell-Pound (BPP) peaks [34], caused by afluctuating field coupled to the 8Li nuclear spin with a characteristic rate that sweeps through 𝜔0at the peak temperature [34, 35, 225]. Potential sources of the fluctuations are discussed below.The rate peaks are superposed on a smooth background that is approximately linear, reminiscentof Korringa relaxation in metals [7, 42]. This is surprising, since BSC is a semiconductor, but itis similar to BTS [274]. We discuss this point further in Section 5.4.2.From this, we adopt the following model for the total SLR rate:1/𝑇1 = 𝑎 + 𝑏𝑇 +∑𝑖𝑐𝑖 (𝐽1,𝑖 + 4𝐽2,𝑖) . (5.1)In Equation (5.1), the first two terms account for the 𝑇-linear contribution with a finite intercept𝑎, while the remaining terms describe the 𝑖th 1/𝑇1 peak in terms of a coupling constant 𝑐𝑖(proportional to themean-squared transverse fluctuating field) and the 𝑛-quantumNMR spectraldensity functions 𝐽𝑛,𝑖 [225]. In general, 𝐽𝑛,𝑖 is frequency dependent and peaked at a temperaturewhere the fluctuation rate matches ∼ 𝑛𝜔0. While the precise form of 𝐽𝑛,𝑖 is not known a priori,10801Bi2Se3:Ca1001011/T 1 (s1 )i=1 i=215 mT (001) [ 0 94.5 kHz]1/T1 J3Dn1/T1 J2Dn20 keV 8Li+5 keV 8Li+3 keV 8Li+1 keV 8Li+0 50 100 150 200 250 300Temperature (K)10 310 21/T 1 (s1 )i=16.55 T (001) [ 0 41.27 MHz]1/T1 J3Dn1/T1 J2Dn1/T1<10 2 /20 keV 8Li+Figure 5.2: Temperature and field dependence of the 8Li SLR rate 1/𝑇1 and stretching exponent𝛽 in Ca doped Bi2Se3. 𝛽 is nearly independent of temperature and field at ∼0.6 (dotted line),except at low field around the large 1/𝑇1 peak seen in the bottom panel. The solid and dashedblack lines are global fits to Equation (5.1), consisting of a linear 𝑇-dependence with a non-zerointercept and two SLR rate peaks, labelled with index 𝑖. Independent of the choice of 𝐽𝑛 used inthe analysis, the model captures all the main features of the data.the simplest expression, obtained for isotropic three-dimensional (3D) fluctuations, has a Debye(Lorentzian) form of Equation (1.14) [34, 225]. Alternatively, when the fluctuations are 2D incharacter, as might be anticipated for such a layered crystal, 𝐽𝑛 may be described by the empiricalexpression in Equation (1.15) [35, 52]. For both Equations (1.14) and (1.15), we assume that 𝜏𝑐is thermally activated, following an Arrhenius dependence analogous to Equation (1.3). If thefluctuations are due to 8Li+ hopping, 𝜏−1𝑐 is the site-to-site hop rate 𝜏−1.Using the above expressions, we fit the 1/𝑇1 data using a global procedure wherein the kineticparameters (i.e., 𝐸𝐴,𝑖 and 𝜏−10,𝑖 ) are shared at all the different 𝜔0. This was necessary to fit the data109Table 5.1: Arrhenius parameters in Equation (1.3) obtained from the analysis of the temperaturedependence of 1/𝑇1 in Ca doped Bi2Se3 shown in Figure 5.2. The two processes giving rise tothe rate peaks are labelled with index 𝑖. Good agreement is found between the 𝐸𝐴s determinedusing the spectral density functions 𝐽𝑛 for 2D and 3D fluctuations [Equations (1.14) and (1.15)].𝑖 = 1 𝑖 = 2𝐽𝑛 𝜏−10 (1010 s−1) 𝐸𝐴 (eV) 𝜏−10 (1014 s−1) 𝐸𝐴 (eV)3D 8.4(27) 0.113(5) 7(5) 0.395(15)2D 9(3) 0.106(5) 110(90) 0.430(16)at 6.55 T where the relaxation is very slow. For comparison, we applied this procedure usingboth 𝐽3D𝑛 and 𝐽2D𝑛 and the fit results are shown in Figure 5.2 as solid (𝐽3D𝑛 ) and dashed (𝐽2D𝑛 ) lines,clearly capturing the main features of the data. The analysis distinguishes two processes, 𝑖 = 1, 2in Equation (5.1): one (𝑖 = 1) that onsets at lower temperature with a shallow Arrhenius slope of∼0.1 eV that yields the weaker peaks in 1/𝑇1 at both fields; and a higher barrier process (𝑖 = 2)with an 𝐸𝐴 of ∼0.4 eV that yields the more prominent peak in the low field relaxation, while thecorresponding high field peak must lie above the accessible temperature range. The resulting fitparameters are given in Table 5.1. We discuss the results in Section 5.4.1.We now turn to the 8Li resonances, with typical spectra shown in Figure 5.3. As anticipatedin a non-cubic crystal, the spectrum is quadrupole split, confirmed unambiguously by thehelicity-resolved spectra, which show opposite satellites in opposite helicities of polarization (seeSection 5.6.2). This splitting, on the order of a few kHz, is determined by the electric field gradient(EFG) and is a signature of the crystallographic 8Li site. Besides this, an unsplit component isalso apparent, very close to (within ∼100Hz) the centre-of-mass of the four satellites. At lowtemperature, the “central” and split components are nearly equal, but as the temperature israised, the unsplit line grows to dominate the spectrum, accompanied by a slight narrowing.The scale of the quadrupole splitting is determined by the product of the principal componentof the EFG tensor 𝑒𝑞 with the nuclear electric quadrupole moment 𝑒𝑄. We quantify this with aconventional definition of the quadrupole frequency (for 𝐼 = 2), given in Equation (2.4) [60].In high field, a first order perturbation treatment of the quadrupole interaction, given by Equa-tions (2.5), (2.6) and (2.8), is sufficient to obtain accurate satellite positions. However, at low field,where 𝜈𝑞/𝜈0 ≈ 6%, second order is necessary, defined in Equations (2.7), (2.9) and (2.10) [60,143]. Based on the change in satellite splittings by a factor 2 in going from 𝐵0 ∥ 𝑐 to 𝐵0 ⟂ 𝑐, we11060 80 100 120Frequency (kHz)050100150200250300350Temperature (K)Bi2Se3:Ca20 keV 8Li+15 mT (001)Normalized AsymmetryFigure 5.3: 8Li resonance spectra in BSC at low magnetic field. The vertical scale is the samefor all spectra; they have been normalized to account for changes in intensity due to SLR [147],with their baselines (shown as dashed grey lines) shifted to match the temperature. The spectraconsist of a small and nearly temperature-independent quadrupole split pattern, centred aboutan unsplit Lorentzian line, whose amplitude grows above ∼150K. Note the quadrupole patternof an integer spin nucleus like 8Li, has no central satellite (main line). The solid black lines arefits to a sum of this Lorentzian and 2𝐼 = 4 quadrupole satellites (see text).assume the asymmetry parameter of the EFG 𝜂 = 0 (i.e., the EFG is axially symmetric). This isreasonable based on likely interstitial sites for 8Li+ (see Section 4.6.3) [274]. Pairs of helicity-resolved spectra were fit with 𝜈0 and 𝜈𝑞 as shared free parameters, in addition to linewidthsand amplitudes. As the difference between the frequency of the unsplit line and the centerof the quadrupole split pattern was too small to measure accurately, the fits were additionallyconstrained to have the same central frequency 𝜈0. This is identical to the approach used forBTS (see Section 4.3.3) [274]. A subset of the results (after recombining the two helicities) areshown in Figure 5.3 as solid black lines.1110 50 100 150 200 250 300Temperature (K)0510152025303540Normalized Amplitude (%)Bi2Se3:Ca20 keV 8Li+15 mT (001)predicted saturation1/T1 maximumquadrupolar satellitescentral LorentzianFigure 5.4: The resonance amplitude as a function of temperature in Ca doped Bi2Se3 at15mT. While the amplitude of the satellite lines are nearly temperature independent, thecentral component increases substantially above 150K, plateauing on the high-𝑇 side of the 1/𝑇1maximum (grey band). The solid line is a guide, while the dashed line indicates the estimatedsaturation value for the Lorentizan component.The main result is the strong temperature dependence of the resonance amplitude shownin Figure 5.4. While the satellite amplitudes are nearly temperature independent, the centralcomponent increases substantially above 150K, tending to plateau above the 1/𝑇1 peak. Theother parameters are quite insensitive to temperature. Typical linewidths (i.e., full width athalf maximum) are ∼2.2 kHz for the satellites and ∼3.8 kHz for the central component. Thequadrupole frequency 𝜈𝑞 ≈ 5.5 kHz varies weakly, increasing slightly as temperature is lowered.We alsomeasured resonances at room and base temperature in high field (6.55 T) where 8Li issensitive to small magnetic shifts. From the fits, we use 𝜈0 to calculate the raw relative frequencyshift 𝛿 in parts per million (ppm) using Equation (2.11). The shifts are small: +12(2) ppm atroom temperature and −17(3) ppm at 5K, the latter considerably smaller in magnitude thanin BTS [274]. Because 8Li NMR shifts are generally so small, it is essential to account for thedemagnetization field of the sample itself. From 𝛿, the corrected shift 𝐾 is obtained by the112centimetre-gram-second system of units (CGS) expression [151]:𝐾 = 𝛿 + 4𝜋 (𝑁 −13)𝜒𝑣 (5.2)where 𝑁 is the dimensionless demagnetization factor that depends only on the shape of thesample and 𝜒𝑣 is the dimensionless (volume) susceptibility. For a thin film, 𝑁 is 1 [151], but forthe thin platelet crystals used here, we estimate 𝑁 is on the order of ∼0.8, treating them as oblateellipsoids [350]. For the susceptibility, we take the average of literature values reported for pureBi2Se3 [351–354], giving 𝜒CGS𝑣 ≈ −2.4 × 10−6 emu cm−3. Note that we have excluded severalreports [278, 355, 356] whose results disagree by an order of magnitude from those predicted byPascal’s constants [357]. Applying the correction for BSC yields𝐾s of −2(2) ppm and−31(3) ppmat room and base temperature, respectively. We discuss this below in Section 5.4.2.5.3.2 Bi2Te3:MnTypical 8Li SLR data at high and low field in the magnetically doped BTM are shown inFigure 5.5. In contrast to nonmagnetic BSC, the relaxation at high field is fast, typical of param-agnets with unpaired electron spins [154, 358, 359]. The fast high field rate produces a muchless pronounced field dependence. At low field, the SLR rate is peaked at low temperature. Therelaxation is also non-exponential and fits well using Equation (2.13), with a stretching exponentsystematically smaller than in the nonmagnetic BSC or BTS [274]. We analyzed the data withthe same global approach, obtaining good quality fits ( ̃𝜒2global ≈ 1.01) demonstrated by the solidblack lines in Figure 5.5. The shared values of 𝐴0 from the fits are large (∼10% for 𝐵0 = 6.55 Tand ∼15% for 𝐵0 = 20mT), consistent with the full beam polarization, implying that there isremarkably no magnetic wipeout from very fast relaxation [154], even at low field.The fit parameters are shown in Figure 5.6. At all temperatures, especially at high field, theSLR rate 1/𝑇1 is orders of magnitude larger than in the nonmagnetic analogs. No clear 1/𝑇1 BPPpeaks can be identified between 100–300K; however, in the low field data, a critical divergenceis evident at the magnetometric transition at about 13K. In high field, this feature is largelywashed out, with a remnant peak near 50K. Above 200K, the SLR rate increases very rapidlyand is well-described by 1/𝑇1 ∝ exp[−𝐸𝐴/(𝑘𝐵𝑇)], with 𝐸𝐴 ≈ 0.2 eV at both fields. We discuss1130 5 10Time (s)0.00.20.40.60.81.0Normalized AsymmetryBeamonBeamoff20 keV 8Li+6.55 T (001)[ 0 41.27 MHz]4 K10 K50 K148 K275 K0 1 2Time (s)BeamonBeamoff8 keV 8Li+20 mT (001)[ 0 126 kHz]9 K100 K250 K315 KBi2Te3:MnFigure 5.5: Typical 8Li SLR data in Mn doped Bi2Te3 at high (left) and low (right) magneticfield for 8Li+ implantation energies of 20 keV and 8 keV, respectively. The shaded region denotesthe duration of the 8Li+ beam pulse. The SLR is substantial and orders of magnitude faster thanin BSC (see Figure 5.1) at high field. The field dependence to the SLR is much weaker than inthe nonmagnetic tetradymites. The solid black lines are fits to a stretched exponential convolvedwith the 8Li+ beam pulse as described in the text. The initial asymmetry 𝐴0 from the fit is usedto normalize the spectra. The high and low field spectra have been binned for by factors of 20and 5.this below in Section 5.4.1.In contrast to BSC and BTS [274], the resonance in BTM consists of a single broad Lorentzianwith none of the resolved fine structure (see Figure 5.7). Surprisingly, the very broad line hassignificant intensity, dwarfing the quadrupole pattern in BSC in both width and amplitude. Inaddition, there is a large negative shift at base temperature. At room temperature, the line issomewhat narrower, and the shift is reduced in magnitude. Quantitative results from Lorentzianfits are summarized in Table 5.2.1140.00.51.0Bi2Te3:Mn101 102Temperature (K)01020304050601/T 1 (s1 )T< TC 13 K20 keV 8Li+6.55 T (001)[ 0 41.27 MHz]8 keV 8Li+20 mT (001)[ 0 126 kHz]25 keV 8Li+20 mT (001)[ 0 126 kHz]Figure 5.6: Temperature dependence of the 8Li SLR rate 1/𝑇1 in Mn doped Bi2Te3 at high andlow field. At low field, 1/𝑇1 shows a critical peak at the ferromagnetic transition at 𝑇𝐶 ≈ 13K,as the Mn spin fluctuations freeze out. Above 200K, the SLR rate increases exponentially inmanner nearly independent of applied field. The solid grey lines are drawn to guide the eye.Table 5.2: Results from the analysis of the 8Li resonance in BTM at high and low temperaturewith 𝐵0 = 6.55 T ∥ (001). The (bulk) magnetization𝑀measured with 1.0 T ∥ (001) is includedfor comparison.𝑇 (K) ̃𝐴 (%) FWHM (kHz) 𝛿 (ppm) 𝑀 (emu cm−3)294 33(5) 16.2(12) +10(9) 0.07610 14.4(8) 41.7(16) −206(12) 7.6985.4 DiscussionThe 8Li NMR properties of nonmagnetic BSC are quite similar to previous measurementsin BTS [274]. The resonance spectra show a similar splitting (𝜈𝑞 is about 25% smaller in BSC),indicating a similar site for 8Li. The resemblance of the spectra extends to the detailed tem-perature dependence, including the growth of the unsplit line approaching room temperature.Surprisingly, the BSC spectra are better resolved than BTS, implying a higher degree of order,11541.15 41.20 41.25 41.30 41.35 41.40Frequency (MHz)1.000.950.900.85Normalized Asymmetry20 keV 8Li+, 6.55 T (001)MgO | 300 Kglobal fitdemagBi2Te3:Mn | 10 KBi2Se3:Ca | 5 K3000 2000 1000 0 1000 2000 3000[Uncorrected] Shift (ppm)Figure 5.7: Typical 8Li resonances in Mn doped Bi2Te3 and Ca doped Bi2Se3 at high magneticfield. The vertical scale is the same for all spectra; they have been normalized to account forchanges in intensity and baseline [147]. The lineshape in the magnetic BTM is well-described bya broad Lorentzian (solid black line) with no quadrupolar splitting. A large negative shift is alsoapparent for BTM with respect to the reference frequency in MgO (vertical dashed line). Thedotted vertical line indicates the expected resonance position due to demagnetization, revealinga large positive hyperfine field (∼38G) at 5K in the magnetic state.despite the Ca doping. This is also evident in the SLR, with a stretching exponent 𝛽 closer tounity in BSC than in BTS. This likely reflects additional disorder in BTS from Bi/Te anti-sitedefects [262] that are much more prevalent than for Bi/Se, due to the difference in radii andelectronegativity. The sharp quadrupolar pattern indicates a well-defined crystallographic 8Li+site, and the corresponding small EFG suggests it is in the vdW gap. Density functional theory(DFT) calculations of the EFG may enable a precise site assignment. The high field SLR is alsosimilar to BTS: it is slow and near the lower limit measurable due to the finite 8Li lifetime 𝜏𝛽 andcomparable to the vdW metal NbSe2 [286], where the carrier concentration is much higher, butsignificantly slower than the TI alloy Bi1–xSbx [285]. The low field enhancement of 1/𝑇1 is alsosimilar, so it cannot be essentially related to the dilute 125Te moments that are absent in BSC,but probably determined primarily by the 100% abundant 209Bi. From such a detailed similarity,it is clear that a quantitative comparison with BTS and other vdWmaterials will be useful.116With these similarities in mind, the rest of the discussion is organized as follows: in Sec-tion 5.4.1, we consider evidence of mobility of the 8Li+ ion; in Section 5.4.2, electronic effects atlow temperature in BSC; and the magnetic properties of BTM in Section 5.4.3.5.4.1 Dynamics of the Li+ ionIn BTS,we considered if the evolution of the spectrumwith temperature (similar to Figure 5.3)was the result of a site change transition from a meta-stable quadrupolar 8Li+ site to a lowerenergy site with very small EFG at higher temperature [274], similar to elemental Nb [360].We now consider an alternative explanation; namely, dynamic averaging of the quadrupolarinteraction due to 8Li+ motion. Examples of this are found in conventional 7Li NMR, where,unlike 8Li, the 𝐼 = 3/2 quadrupole spectrum has a main line (the𝑚 = ±1/2 satellite) overlappingthe averaged resonance [361, 362]. Dynamic averaging is suggested by the onset near, but below,the SLR rate peak (see Figure 5.4). However, for hopping between equivalent interstitial sites(probably the quasi-octahedral Wyckoff 3𝑏 site in the vdW gap), one does not expect that theEFG will average to a value near zero (required to explain the unsplit line). A point chargeestimate reveals the quasi-tetrahedral (Wyckoff 6𝑐) site, thought to be the saddle point in thepotential for Li+ between adjacent 3𝑏 sites [363], has an EFG of opposite sign to the 3𝑏 site. If 6𝑐is instead a shallow minimum, the 8Li+ residence time there may be long enough that the EFGaverages to near zero. In the fast motion limit at higher temperatures, one would then expect thequadrupole splitting to re-emerge when the residence time in the 6𝑐 “transition” site becomesmuch shorter [362].We now examine the two kinetic processes causing the 1/𝑇1 peaks in BSC. It is surprising tofind two distinct thermally activated processes sweeping through the NMR frequency, especiallysince only a single process was found in BTS [274]. First, we consider the weaker feature, thelow temperature (𝑖 = 1) peaks. In layered materials, small intercalates can undergo highlylocalized motion at relatively low temperatures below the onset of free diffusion [36, 346]. Suchlocal motion may be the source of the SLR rate peak, but it is quite ineffective at narrowing theresonance, consistent with the absence of any lineshape changes in the vicinity of the 𝑖 = 1peaks. Caged local motion is usually characterized by a small activation barrier, comparable117to the ∼0.1 eV observed here. Similar phenomena have been observed at low temperature, forexample, in neutron activated 8Li β-NMR of Li intercalated graphite, LiC12 [77, 79]. It is not clearwhy such motion would be absent in BTS [274], which has a larger vdW gap than BSC (2.698Åvs. 2.568Å). Alternatively, this feature in the relaxation may have an electronic origin, perhapsrelated to the emergent low-𝑇magnetism in MoTe2 observed by μSR [364] and8Li β-NMR [365].In contrast, the SLR rate peak above 200K (𝑖 = 2) is almost certainly due to EFG fluctuationscaused by stochastic 8Li+ motion. From the data, we cannot conclude that this is long-rangediffusion, but the room temperature Li+ intercalability of Bi2Se3 [294, 295, 340] suggests it is. Itsbarrier, on the order of ∼0.4 eV, is comparable to other vdW gap layered ion conductors, but it isabout twice as high as in BTS [274], possibly a result of the Se (rather than Te) bounded vdWgap, which provides less space between neighbouring QLs.We now consider the Arrhenius law prefactors 𝜏−10 , that, for ionic diffusion, are typicallyin the range 1012–1014 s−1. For the low-𝑇 process (𝑖 = 1), independent of the form of 𝐽𝑛 (seeTable 5.1), 𝜏−10 ≈ 9 × 1010 s−1 is unusually low. In contrast, for the high-𝑇 (𝑖 = 2) process, it ismuch larger and depends strongly on 𝐽𝑛. For 3D diffusion, it is in the expected range, while the2D model yields an extremely large value, ∼1016 s−1, in the realm of prefactor anomalies [32]and opposite to the small value expected for low dimensional diffusion [50]. Similar behaviourwas observed recently in 7Li NMR of LiC6 [88], where surprisingly, 𝐽2D𝑛 was concluded to be lessappropriate than 𝐽3D𝑛 , suggesting that Li motion in the vdW gap is not as ideally 2D as might beexpected. In BSC, the anomaly may be related to local dynamics that onset at lower 𝑇, impartingsome 3D character to the motion.Given the evidence for Li+ motion in BSC and BTS [274], the absence of a relaxation peakin BTMmay seem unexpected. Both Ca2+ and Mn2+ dopants (substitutional for Bi3+) have aneffective −1 charge yielding an attractive trapping potential for the positive interstitial 8Li+, buttheMn concentration is an order of magnitude larger. The high trap density in BTMwill suppressLi+ mobility. The exponential increase in 1/𝑇1 above 200K may be the onset of a diffusive BPPpeak, but, in this case, one does not expect it to be so similar between the two very differentmagnetic fields. This may reflect a trade-off between the increase in 𝜔0 that shifts the peak tohigher temperature, slowing the relaxation on its low-𝑇 flank, and the increased polarizationof the Mn moments by the field that amplifies local magnetic inhomogeneities. A motional118origin for this increase is consistent with the apparent 𝐸𝐴 ∼ 0.2 eV, similar to8Li+ in BTS [274],which also has a Te bounded vdW gap of similar size to BTM (2.620Å). However, it may have adifferent explanation, see below in Section 5.4.3.5.4.2 Electronic effects at low temperatureBismuth chalcogenide (Bi2Ch3) crystals exhibit substantial bulk conductivity, despite a nar-row gap in the 3D band structure, making it difficult to distinguish effects of the metallic TSS.This is due to native defects (e.g., Ch vacancies) that are difficult or impossible to avoid [254].Extrinsic dopants, such as substitutional Ca/Bi, can be used to compensate the spontaneous𝑛-type doping. Brahlek et al. have argued [308] that, even for the most insulating compensatedsamples, the carrier densities far exceed the Mott criterion, making them heavily doped semi-conductors in the metallic regime. In this case, we expect metallic NMR characteristics [366],namely a magnetic Knight shift 𝐾, proportional to the carrier spin susceptibility 𝜒𝑠. In thesimplest (isotropic) case,𝐾 = 𝐴𝜒𝑠, (5.3)where 𝐴 is the hyperfine coupling constant, which is accompanied by a SLR rate following theKorringa law [7, 42],1𝑇1= 4𝜋ℏ𝐴2𝛾2𝑛 (𝜒𝑠𝑔∗𝜇𝐵)2𝑘𝐵𝑇. (5.4)Here, 𝛾𝑛 is the nuclear gyromagnetic ratio, 𝑔∗ is the carrier 𝑔-factor, and 𝜇𝐵 is the Bohr magneton.Combining Equations (5.3) and (5.4), we obtain the Korringa product, which is independent ofthe value of 𝐴,𝑇1𝑇𝐾2 =ℏ(𝑔∗𝜇𝐵)24𝜋𝑘𝐵𝛾𝑛= 𝑆(𝑔∗). (5.5)For 8Li,𝑆(𝑔∗) ≈ 1.20 × 10−5 (𝑔∗𝑔0)2s K,where, unlike inmetals, we have allowed for an effective 𝑔-factor thatmay be far from its free elec-tron value 𝑔0 ≈ 2 [367]. Indeed, recent electron paramagnetic resonance (EPR) measurementsin Bi2Se3 find 𝑔∗ ≈ 30 [368].119According to Ref. [308], BTS and BSC lie on opposite sides of the Ioffe-Regel limit, wherethe carrier mean free path is equal to its Fermi wavelength, with BSC having a higher carrierdensity and mobility. A comparative Korringa analysis could test this assertion and, to this end,using Equation (5.5) we define the dimensionless Korringa ratio as𝒦 =𝑇1𝑇𝐾2𝑆(𝑔∗). (5.6)Below the Ioffe-Regel limit, the autocorrelation function of the local hyperfine field at thenucleus, due to the carriers (that determines 𝑇1) becomes limited by the diffusive transportcorrelation time. This has been shown to enhance the Korringa rate (i.e., shortening 𝑇1) [313,314]. From this, one expects𝒦 would be smaller in BTS than in BSC.There are, however, significant difficulties in determining the experimental𝒦. First, theKorringa slope depends on magnetic field. At low fields, this is due to coupling with the hostnuclear spins, a phenomenon that is quenched in high fields where the 8Li NMR has no spectraloverlap with the NMR of host nuclei. For example, in simple metals, we find the expectedfield-independent Korringa slope at high fields in the Tesla range [5]. In contrast, in BTS, theslope decreases substantially with increasing field, even at high fields [274]. We suggested thiscould be the result of magnetic carrier freeze-out. While we do not have comparably extensivedata in BSC, we can compare the slope at the same field, 6.55 T (see Table 5.3). Here, in bothmaterials, the relaxation is extremely slow, exhibiting no curvature in the SLR during the 8Lilifetime (Figure 5.1), so the uncertainties in 1/𝑇1𝑇 are likely underestimates. The larger slopeis, however, consistent with a higher carrier density 𝑛 in BSC. The Korringa slopes shouldscale [367] as 𝑛2/3. Using 𝑛 ∼ 1 × 1019 cm−3 in BSC [258] and ∼2 × 1017 cm−3 in BTS [261], theslopes should differ by a factor of ∼14, while experimentally the ratio is ∼5.The next difficulty is accurately quantifying the shift 𝐾, which is quite small with a relativelylarge demagnetization correction. Experimentally, the zero of shift, defined by the calibrationin MgO, differs from the true zero (where 𝜒𝑠 = 0) by the difference in chemical (orbital) shiftsbetween MgO and the chalcogenide. However, because Li chemical shifts are universally verysmall, this should not be a large difference, perhaps a few ppm. The negative low temperatureshift is also somewhat surprising. The hyperfine coupling 𝐴 for Li is usually determined by a120Table 5.3: Korringa analysis of BSC and BTS [274] at 6.55 T and low temperature. To calculate𝒦, we take 𝑆(𝑔∗) in Equation (5.6) to be 2.69 × 10−3 s K, using the 𝑔∗∥ from EPR [368].1/(𝑇1𝑇) (10−6 s−1K−1) 𝐾 (ppm) 𝒦Bi2Se3:Ca 9.5(8) −31(3) 0.038(8)Bi2Te2Se 1.79(7) −115(3) 2.78(18)slight hybridization of the vacant 2𝑠 orbital with the host conduction band. As the 𝑠 orbital hasdensity at the nucleus, the resulting coupling is usually positive, with the 𝑑 band metals Pd andPt being exceptional [5]. For a positive 𝐴, the sign of 𝐾 is determined by the sign of 𝑔∗, whichhas not yet been conclusively measured in either BSC or BTS. A more serious concern is that𝐾 depends on temperature (in contrast to simple metals) and, at least in BTS, also on appliedfield [274]. To avoid ambiguity from the field dependence, we similarly restrict comparison to thesame field, 6.55 T. A similarly temperature dependent shift (for the 207Pb NMR) was found in thenarrow band semiconductor PbTe [369], where it was explained by the temperature dependenceof the Fermi level 𝐸𝐹 [370]. At low-𝑇 in the heavily 𝑝-type PbTe, 𝐸𝐹 occurs in the valence (or anearby impurity) band, but with increasing temperature, moves upward into the gap, causing areduction in |𝐾|. With this in mind, we assume the low temperature shift is the most relevantfor a Korringa comparison. Without a measured 𝑔∗ in BTS, we simply assume it is the same asBSC, and use the 𝑔∗∥ from EPR [368] to calculate the values of 𝒦 in Table 5.3.The values of 𝒦 are just opposite to the expectation of faster relaxation for diffusive BTScompared to metallic BSC [308]. Electronic correlations can, however, significantly alter theKorringa ratio to an extent that depends on disorder [315]. There should be no significantcorrelations in the broad bulk bands of the chalcogenides, but in narrow impurity bands, this iscertainly a possibility. We note that𝒦 is also less than 1 in PbTe [371], similar to BSC. At thisstage, without more data and a better understanding of the considerations mentioned above, itis premature to draw further conclusions.5.4.3 Magnetism in Bi2Te3:MnIn the Mn doped Bi2Te3 at low field, the relaxation from magnetic Mn2+ becomes faster asthe spin fluctuations slow down on cooling towards 𝑇𝐶. In particular, the low-𝑇 increase in1211/𝑇1 occurs near where correlations among the Mn spins become evident in EPR [348, 372]. Inremarkable contrast to ferromagnetic EuO [154], the signal is not wiped out in the vicinity of𝑇𝐶, but 1/𝑇1 does become very fast. This is likely a consequence of a relatively small hyperfinecoupling consistent with a Li site in the vdW gap.High applied field slows the Mn spins more continuously starting from a higher temperature,suppressing the critical peak and reducing 1/𝑇1 significantly, a well-known phenomenon inNMR and μSR at magnetic transitions (see e.g., Ref. [373]). This also explains the small criticalpeak in the 8Li SLR in the dilute magnetic semiconductor, Ga1–xMnxAs [359]. As in GaAs, Mnin Bi2Te3 is both a magnetic and electronic dopant. At this concentration, BTM is 𝑝-type with ametallic carrier density of ∼7 × 1019 cm−3 [344, 348]. However, the difference in scale of 1/𝑇1 athigh field between Figures 5.2 and 5.6 shows that the Mn spins completely dominate the carrierrelaxation.It is also remarkable that the resonance is so clear in the magnetic state, which is in contrastto Ga1–xMnxAs [359]. The difference is not the linewidth, but rather the enhanced amplitude.This may be due to slow 8Li spectral dynamics occurring on the timescale of 𝜏𝛽 that effectivelyenhance the amplitude, for example, slow fluctuations of the ordered Mn moments, not farbelow 𝑇𝐶. Similar behavior was found in rutile TiO2 at low temperature, where it was attributedto field fluctuations due to a nearby electron polaron [206]. Enhancement of the radio frequency(RF) field at nuclei in a ferromagnet [374] may also play a role.Above 200K, the activated increase in 1/𝑇1 may indicate the onset of diffusive8Li+ motion,similar to the nonmagnetic analogs, with the additional effect that the local magnetic fieldfrom the Mn spins is also modulated by 8Li hopping (not just the EFG), similar to the orderedmagnetic vdW layered CrSe2 [375]. However, it may instead mark the onset of thermal excitationof electrons across the bandgap that is narrowed by Mn doping[344]. Thermally excited carriersmay also explain the increase in 1/𝑇1 at comparable temperatures in Ga1–xMnxAs [359], a verydifferent medium for Li+ diffusion. Thermally increased carrier density would strengtheninteraction between the Mn moments, extend their effects via Ruderman–Kittel–Kasuya–Yosida(RKKY) polarization, and increase 1/𝑇1. Measurements at higher temperatures may be able todiscriminate these possibilities, but it is likely that both processes will contribute.Having established the effects of Mn magnetism in the bulk, it would be interesting to use122lower implantation energies to study how they may be altered in the surface region by couplingto the TSS.5.5 ConclusionUsing implanted 8Li β-NMR, we have studied the ionic, electronic, andmagnetic properties ofthe doped TIs BSC and BTM. From SLR measurements, we find evidence at temperatures above∼200K for site-to-site hopping of isolated Li+withArrhenius activation energy of ∼0.4 eV in BSC.At lower temperature the electronic properties dominate, giving rise to Korringa-like relaxationand negative Knight shifts, similar to isostructural BTS. A quantitative comparison revealsKorringa ratios opposite to expectations across the Ioffe-Regel limit. In BTM, the magnetismfrom dilute Mn moments dominates all other spin interactions, but the β-NMR signal remainsmeasurable through the magnetic transition at 𝑇𝐶, where a critical peak in the SLR rate isobserved. The activation energy from the high temperature increase ∼0.2 eV may be related toLi mobility or to thermal carrier excitations.With these new results, a more complete picture of the implanted 8Li NMR probe of thetetradymite Bi chalcogenides (and other vdW chalcogenides) is beginning to emerge. At hightemperatures, isolated 8Li+ has a tendency to mobilize, providing unique access to the kineticparameters governing Li+ diffusion in the ultra-dilute limit. At low temperature, 8Li is sensitiveto the local metallic and magnetic properties of the host. With the bulk NMR response nowestablished in Bi2Ch3TIs, the prospect of directly probing the chiral TSSwith the depth resolutionprovided by β-NMR remains promising.5.6 Supplemental material5.6.1 8Li+ implantation profilesThe SRIM Monte Carlo code [133] was used to predict the 8Li+ implantation profiles in BSCand BTM. At a given implantation energy, stopping events were simulated for 105 ions, whichwere histogrammed to represent the predicted implantation profile shown in Figure 5.8. From1230 100 200Depth (nm)0.000.010.020.030.040.050.060.07Stopping probability (nm1 )Bi1.99Ca0.01Se3[ 7.644 g cm 3]1 keV 8Li+3 keV 8Li+5 keV 8Li+20 keV 8Li+0 100 200Depth (nm)Bi1.9Mn0.1Te3[ 7.683 g cm 3]8 keV 8Li+20 keV 8Li+25 keV 8Li+Figure 5.8: Stopping distribution for 8Li+ implanted in the doped TIs BSC and BTM, calculatedusing the SRIMMonte Carlo code [133] using 105 ions. The implanted probe ions stop principallywithin ∼250 nm from the crystal surface, with only a minor “tail” penetrating further. Onlyat the lowest 8Li+ implantation energy 1 keV does an appreciable fraction (∼15%) stop in theregion where surface effects are expected to be important.the stopping profiles, we calculated, in the nomenclature of ion-implantation literature, the rangeand straggle (i.e., the mean and standard deviation) of the 8Li+ stopping depth. Additionally, thefraction of the simulated ions that were backscattered (i.e., did not stop in the target material) orstopped at very shallow depths below the surface, were determined for BSC (see Figure 5.9). Atthe ion beam energies used here (1 keV to 25 keV), only a minority of the implanted 8Li+ ionsstop in the top few nm where surface effects are anticipated.5.6.2 Helicity-resolved resonancesA typical helicity-resolved 8Li resonance spectrum in BSC is shown in Figure 5.10. Besides thewell-resolved fine structure, the spectra unambiguously reveal the multi-component nature ofthe line. A quadrupolar splitting, giving the anti-symmetric shape about the resonance centre-of-mass in each helicity, on the order of several kHz, can be identified from the outermost satellites.In contrast to conventional NMR, note that the satellite intensities are chiefly determined by the124050100Depth (nm)Bi1.99Ca0.01Se3 [ 7.644 g cm 3]RangeStraggle0 5 10 15 20 25 308Li+ implantation energy (keV)0.00.10.20.3Fraction of 8 Li+BackscatteredDepth 6 nmFigure 5.9: Stopping details for 8Li+ implanted in BSC calculated using the SRIM Monte Carlocode [133]. The ion stopping range and straggle (i.e., mean and standard deviation) at eachimplantation energy are shown in the top panel. The bottom panel shows the predicted fractionof 8Li+ that are backscattered and those that stop within the first ∼6 nm, where surface effectsare expected. Except at the lowest implantation energy, this latter fraction is negligible. Thesolid grey lines are drawn to guide the eye.high initial polarization, causing an increase in the relative amplitude of the outer satellites [127],but also depend on the SLR. Apart from the quadrupolar component, another contribution atthe resonance “centre”, with no resolved splitting, is discernible. This is consistent with the 8Liresonances observed in other vdW materials [274, 286, 375]. Unlike the more common caseencountered for the spin 𝐼 = 3/2 7Li nucleus, there is no𝑚±1/2 ↔ 𝑚∓1/2 main line transition thatis unshifted in first-order by the quadrupolar interaction. The absence of any peaks interlacingthe𝑚±2 ↔ 𝑚±1 transitions suggest this “central” line cannot bemulti-quantum (cf. the spectrumin Bi [127, 285]). For presentation, we combined the helicity-resolved spectra to give an overallaverage lineshape (see Figure 5.3 in Section 5.3.1).1254.55.05.5PositivehelicityBi2Se3:Ca20 keV 8Li+100 K, 15 mT (001)5.55.04.5Asymmetry (%)Negativehelicity70 80 90 100 110 120Frequency (kHz)4.55.05.5CombinedhelicitiesFigure 5.10: Typical helicity-resolved 8Li spectra in BSC, revealing the fine structure of the line.Four quadrupole satellites, split asymmetrically in each helicity about a “central” Lorentzian(marked by a vertical dashed line) are evident. Note that, in contrast to conventional NMR, thesatellite amplitudes are determined primarily by the highly polarized initial state. The solidblack lines are the fits described in the text and the vertical dashed line marks the resonancecentral frequency 𝜈0.126Chapter 6Summary & outlookWe have used ion-implanted 8Li β-detected nuclear magnetic resonance (β-NMR) to studythe dynamics of isolated Li+ in several single crystalline materials: rutile TiO2, Bi2Te2Se (BTS),Bi2Se3:Ca (BSC), and Bi2Te3:Mn (BTM).While none of these compounds natively contain lithium,their crystal structures are amenable to Li+ intercalation, thanks to a connected sublattice ofinterstitial sites in the form of a tunnel or plane. Besides the structural properties that makethem amenable to ionic diffusion, the electronic properties of the materials are also of interest,manifesting as the dominant feature of the β-NMR data at low temperature, where atomicmotion is minimal. Each of these examples are clear cases where understanding the 8Li probe’scoupling to the host’s electronic properties, along with its propensity to migrate at elevatedtemperatures, are important. The characterization of both behaviours provided here broadensthe knowledge of the capabilities of the ion-implanted β-NMR technique, as well as the scope ofmaterials it may be applied to study.Using a combination of spin-lattice relaxation (SLR) and resonance measurements, wefind evidence for motion of implanted 8Li+ below 300K, manifesting as Bloembergen-Purcell-Pound (BPP) peaks in the SLR rate 1/𝑇1 and motional narrowing. From these signatures, theArrhenius parameters governing the Li+ hop rate were identified and compared with resultsfrom theory, as well as other experiments in the literature. The rate of motion inferred inthis temperature range, often on the order of MHz, is rather exceptional for solid-state ionconductors. In several cases, anomalously large Arrhenius prefactors were found, opposite to127the usual case for spatially constrained ionic motion. Similarly, an instance of the empiricalMeyer-Neldel rule (MNR) was observed. These observations, from unique experiments onisolated Li+: suggest that the mobility of the isolated ion may be quite significant compared tomore concentrated cases; challenge conventional interpretations of Arrhenius prefactors whenlarge attempt frequencies are involved; and provide further evidence that the MNR is generic,occurring even in conceptually simple circumstances.In the one-dimensional (1D) ion conductor rutile TiO2, two sets of thermally activateddynamics were identified: one below 100K, ascribed to electron polarons (i.e., Ti3+ electronicdefects); and another at higher temperatures, associated with 8Li+ diffusion. The agreementof the diffusive activation energy with macroscopic measurements implies a single, commonmigration mechanism across all length scales. This was not clear at the outset, given therather large disagreement between theory and experiment for the ionic hopping barrier. Theresults presented here suggest that a key ingredient is missing in the calculation of the diffusionbarrier. Recent measurements using a beam of 8Li+ as an online radiotracer confirm thisassertion [253, 376], but reveal that some of the low-temperature dynamics are due to lithiumdiffusion, indicating there is further complexity to the dynamics discussed in Chapter 3.Two types of dynamics are possible for the polaron: one associatedwith its unpaired spin; andanother associated with is translational motion. Due to its charge, its motion is coupled to thenearby Li+, but since we are sensing the nuclear magnetic resonance (NMR), the magnetic effectof the spinmay dominate. The interstitial Li+maymigrate as an ion or as part of mobile Li+ Ti3+complex, which may also have its own internal motion (e.g., from binding/unbinding cycles).It has long been recognized that polaronic electron transport may be extremely important insome materials, especially mixed (ionic/electronic) conductors that are electrodes in a solid-stateionic device. Here we can study the isolated polaron-Li interaction — an ideal situation muchsimpler than the high density limit.The β-NMR measurements presented here suggest the low-temperature response is dom-inated by the proximal coupling to polarons in vicinity of the implantation site, but it is notknown what fraction of the implanted probes this affects. Key to improving the understanding ofLi+ in rutile is further investigation of these “polaronic” interactions. A detailed characterizationof the 8Li resonance below 100K, in conjunction with resonant SLR hole-burning (HB) measure-128ments (see e.g., [148]), may help further elucidate the binding/migration details of the polaron.Similarly, it may be possible to use the NMR “isotope effect” to (e.g., by comparing 8Li and 9LiSLR rates [251–253]) to identify the nature of the fluctuating field at the probe implantation site.With the results on BTS, BSC, and BTM present in Chapters 4 and 5, we have significantlyadvanced the understanding of how 8Li β-NMR may be used to study van der Waals (vdW)layered materials. The renaissance of studying materials with two-dimensional (2D) structuralcharacteristics began with graphene (2010 Nobel Prize in Physics), which instigated a renewal ofinterest in other vdW materials like transition metal dichalcogenides (TMDs) that were studiedextensively in the 1970s. These investigations on TMDs focused largely on electronic properties;however, the importance of their ability to accommodate foreign ions in the vdW gap, especiallyLi+, was also realized (2019 Nobel Prize in Chemistry). More recently, interest in tetradymitebismuth chalcogenides, structurally related to the TMDs, exploded owing to their electronicstructures manifesting a robustly conductive topological surface state (TSS). In all materialscontaining a vdW gap, developing a microscopic understanding of the mobility of foreign ions,as well as their local electronic properties, is of interest and this work establishes how implanted8Li “works” in such structural motifs.In BTS and BSC, significant thermally activated dynamics onset above ∼150K. The dra-matic changes to the 8Li resonance lineshape, similar in both materials, suggest the processis long-range diffusion of 8Li+ in the vdW gap, consistent with the SLR BPP peaks and theability to intercalate Li+ at room temperature; however, in order to be more conclusive, highertemperatures are required, where, based on the discussion in Chapter 5, we anticipate sharperresonance fine structure from (complete) motional averaging. Our estimates of the Li+ diffusivity,especially in BTS, are substantial compared to some other well-known lithium-ion conductors(e.g., LixTiS2). At lower temperatures, the electronic properties of the non-magnetic topologicalinsulators (TIs) dominate, giving rise to Korringa-like relaxation and negative Knight shifts.The metallic NMR response reflects that these materials are not really bulk insulators, withtheir behaviour dominated by (unwanted) defect properties. This feature may be unavoidablein all but exemplary circumstances (see e.g., Bi2Te3 films [377]) and is therefore important tounderstand. The observation of the strong field dependence to the SLR slope and resonanceshift in BTS was surprising, with similar behaviour found in BSC, but from a more limited129dataset. While a detailed understanding is lacking, several possible origins were discussed inChapter 4. Field dependent conventional NMR, using the (stable) host nuclei in Bi2Ch3, mayprovide a useful confirmation. In the magnetic BTM, coupling to the dopant Mn spins dominatesall other interactions, but the β-NMR remains measurable, even in the magnetic state and atlow field. This was not obvious at the outset and suggests the applicability of β-NMR to study(dilute) magnetic vdW materials, which have been scarcely explored by the technique. Thisability may be related to the weakness of 8Li’s hyperfine coupling. With the bulk NMR responsenow established in Bi2Ch3 TIs, one can consider attempting to study directly the chiral TSSwith the depth resolution provided by β-NMR; however, one first needs to carefully understandbackground signals at the very low implantation energies required [378].With the ability to study Li+ dynamics with ion-implanted β-NMRnowfirmly established, webelieve the technique will become an effective means for studying solid-state lithium diffusion,especially in situations where conventional NMR isn’t feasible. The information extracted fromsuch experiments can further be complemented by the recently developed online radiotracermethod [253, 376], which uses the same 8Li+ beam. Key to the general applicability of bothapproaches is access to higher temperatures; one really needs a couple orders of magnitudedynamic range in 𝜏−1(𝑇) to extract convincing values of 𝜏−10 and 𝐸𝐴. For β-NMR, we generallyneed 𝜏−1 in the range of 50 kHz to 50MHz, requiring that the temperature is high enough for fasthopping with residence times <1ms. Plans for upgrading the maximum operating temperaturesof the spectrometers discussed in Chapter 2 are underway.Finally, with the unique depth resolution provided by ion-implanted β-NMR, it should bepossible to study interface effects in ionic diffusion (e.g., the suppression or enhancement ofmobility near a surface or heterocontact). Note that, in practical devices, such as rechargeablebatteries, ion diffusion to and traversal across material interfaces (e.g., electrode-electrolytecontacts) is implicitly required for functionality. In particular, directly probing Li+ motion in adepth-resolved manner at space charge regions [379] (e.g., near heterocontacts) is intriguing, butdifficult to do with (ionic) conductivity measurements. 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Kuczynski, K. Tritz, M. Thoma, M. Newville, M. Kümmerer, M. Bolingbroke,M. Tartre, M. Pak, N. J. Smith, N. Nowaczyk, N. Shebanov, O. Pavlyk, P. A. Brodtkorb,P. Lee, R. T. McGibbon, R. Feldbauer, S. Lewis, S. Tygier, S. Sievert, S. Vigna, S. Peterson,S. More, T. Pudlik, T. Oshima, T. J. Pingel, T. P. Robitaille, T. Spura, T. R. Jones, T. Cera,156T. Leslie, T. Zito, T. Krauss, U. Upadhyay, Y. O. Halchenko, and Y. Vázquez-Baeza (SciPy1.0 Contributors), SciPy 1.0: fundamental algorithms for scientific computing in Python,Nat. Methods 17, 261 (2020) ∎ [cited on pp. 161, 167].[386] H. S. Carslaw and J. C. Jaeger, Conduction of heat in solids, 2nd ed. (Oxford UniversityPress, London, 1959) ∎ [cited on p. 164].[387] J. Crank, The mathematics of diffusion, 2nd ed. (Oxford University Press, London, 1975)∎ [cited on p. 164].[388] H. Mehrer, “Diffusion: introduction and case studies in metals and binary alloys”, inDiffusion in condensed matter: methods, materials, models, edited by P. Heitjans and J.Kärger, 2nd ed. (Springer, Berlin, 2005) Chap. 1, pp. 3–63 ∎ [cited on p. 164].[389] W. Meyer and H. Neldel, Über die beziehung zwischen der energiekonstanten ε und dermengenkonstanten a in der leitwerts-temperatureformel bei oxydischen halbleitern, Z. Tech.Phys. 12, 588 (1937) ∎ [cited on p. 166].[390] S. van derWalt, S. C. Colbert, and G. Varoquaux, The NumPy array: a structure for efficientnumerical computation, Comput. Sci. Eng. 13, 22 (2011) ∎ [cited on p. 167].[391] J. D. Hunter, Matplotlib: a 2D graphics environment, Comput. Sci. Eng. 9, 90 (2007) ∎[cited on p. 167].157Appendix Aβ-NMR probe ion concentrationA characteristic feature of ion-implanted β-detected nuclearmagnetic resonance (β-NMR) [5]is the ability to study the dynamical behaviour ultra-dilute probe ions (e.g., 8Li+ [89, 169, 206,274] or 31Mg+ [382]) in condensed phases. This is, in many respects, similar to the use ofmuon spin rotation/relaxation/resonance (μSR) to simulate the behaviour of hydrogen insidematerials [383]. An important consideration, however, is just how dilute are the implanted probeions? For an implanted probe X+, its concentration in a sample can be written as:[X+](𝑡) =𝑛X(𝑡)𝑉(𝑡), (A.1)where 𝑛X(𝑡) is the number of implanted ions at any given time, and 𝑉(𝑡) is the volume in thesample they occupy. Clearly, estimating [X+](𝑡) reduces to achieving good estimates for 𝑛X(𝑡)and 𝑉(𝑡). The precise answer for either quantity, however, depends largely on the conditions ofthe β-NMR experiment (e.g., beam focus, isotope production efficiency, etc. — see Chapter 2).Here we consider the quantification of [X+] and provide estimates for typical values during anexperiment.For a constant (average) beam rate 𝑅X the number of implanted ions during a beam pulse is:𝑛X(𝑡) = 𝑅X∫𝑡0exp (−𝑡′𝜏𝛽) 𝑑𝑡′ = 𝑅X𝜏𝛽 [1 − exp (−𝑡𝜏𝛽)] , (A.2)158where 𝜏𝛽 is the nuclear lifetime of X. For a sufficiently long beam pulse, we obtain the steady-statecondition (i.e., radiochemical equilibrium):̄𝑛X = lim𝑡≫𝜏𝛽𝑛X(𝑡) ≈ 𝑅X𝜏𝛽. (A.3)This is the usual situation during a continuous wave (CW) resonance measurement, but istypically not achieved during a spin-lattice relaxation (SLR) measurement using a short beampulse (see Section 2.3.1). As such, ̄𝑛X is the upper limit for the number of implanted probes atany given time. Our detected rate 𝑅X is only a fraction of the total rate 𝑅𝑡X due to finite detectorsolid angle, the steady-state number of ions can be estimated as:̄𝑛X ≈ 𝜖−1𝑅X𝜏𝛽, (A.4)where 𝜖 is the scaling factor for our detection efficiency. In an experiment using 8Li, 𝑅𝑋 ≈106 ions/s, 𝜏𝛽 = 1.21ms [18], and, assuming 𝜖 is on the order of ∼10% in the high field spectrom-eter [121–123], this puts ̄𝑛X at about ∼107 ions.Estimating 𝑉(𝑡) is somewhat trickier than estimating ̄𝑛X, since it requires some assumptionsregarding the mobility of the ions in the sample. In a liquid, the diffusivity of X+ may be highenough such that, to a reasonable approximation, the probe ions quickly homogenize overthe volume of the sample within their lifetime. By contrast, in a solid whose structure is notamenable to ionic diffusion, the implanted probes will be effectively immobile and the relevantvolume reduces to that of the implantation profile. In cases of intermediate mobility, knowledgeof the probe’s diffusion coefficient is required and the stopping profile must be evolved in time.1We forgo dealing with this more complicated situation here, in favour of exploring the other twolimiting cases. These can be used to give upper/lower bounds on acceptable values.In the limit of very slow diffusion, it is possible to estimate𝑉 using information from stoppingprofile generated from a Stopping and Range of Ions in Matter (SRIM) Monte Carlo calcula-tion [133] (see e.g., Figure A.1); From the charge-coupled device (CCD) images of typicalbeamspots at high magnetic field (e.g., from scintillating sapphire [141] — see Figure 2.9 in1Note that this also necessary when using 8Li as an online radiotracer (see e.g., [253, 376]).159x zrutile TiO24.249 g cm 38Li+A0 100 200z (nm)0.00000.00250.00500.00750.0100(z) (nm1 )B2 0 2y (mm)202x (mm)C0 100 200z (nm)202x (mm)D0200400600# of 8 Li+ ions0100200300400# of 8 Li+ ionsFigure A.1: Scaled three-dimensional (3D) 8Li+ implantation profile used to determine thetypical (peak) 8Li+ concentration in a β-NMR experiment. (A) Cartoon of the 8Li+ implantationprocess in rutile TiO2. (B) One-dimensional (1D) histogram of the stopping probability 𝜌(𝑧) as afunction of depth 𝑧 below the surface. (C) Two-dimensional (2D) histogram of the implantationprofile in the lateral directions 𝑥 and 𝑦, spread to correspond to a beamspot full width at halfmaximum (FWHM) of 2mm. (D) 2D histogram of the implantation profile in the 𝑥 and 𝑧directions, scaled to correspond to a beamspot FWHM of 2mm. This scaling corresponds to aconvex hull [384] of ∼1 × 10−7 cm3 and, assuming a typical 8Li+ beam rate, a (peak) concentrationof ∼1.8 × 1013 cm−3 (cf. 3.2 × 1022 cm−3 for Ti in TiO2). The cividis colourmap [27] is used forthe 2D histograms.Chapter 2), we know the beamspot is roughly ∼2mm in diameter. In a back-of-the-envelopeapproach, we could assume that the beamspot is perfectly circular and use the mean stoppingdepth from the SRIM calculation to compute 𝑉. This “cylindrical” approximation is simply:𝑉 ≈ 𝜋𝑟2 ̄𝑧, (A.5)where 𝑟 is the radius of the beamspot and ̄𝑧 is the mean stopping depth. Using typical values of𝑟 ∼ 1mm and ̄𝑧 ∼ 100 nm for, a 𝑉 on the order of ∼3 × 10−7 cm3 is obtained.160To improve upon this estimate, we must consider the details of the full 3D stopping profile;however, this on its own is insufficient, as the SRIM calculation assumes a point source for theincident ions. To better account for the macroscopic profile of the beam, a simple approach is tospread the incident lateral (i.e., 𝑥 and 𝑦) components of incoming ions such that the FWHM ofthe new lateral profile matches the beamspot diameter observed in the CCD images. For this, weuse a 2D Gaussian distribution of arrival positions and set the FWHM in the 𝑥 and 𝑦 directionsto 2mm. Select projections from this “scaling” procedure are shown in Figure A.1.With these modifications to the implantation profile, 𝑉 can be estimated by considering thetotal volume occupied by all simulated ions. That is, one needs to determine the minimumvolume that encloses all possible stopping positions. This can be done by computing the convexhull [384] of the 3D stopping profile (e.g., using the Python library SciPy [385]). For a typicalimplantation profile, this approach gives a volume of ∼7 × 10−6 cm3. This is somewhat larger,but still within an order of magnitude, of the much cruder “cylindrical” approximation.In the limit of very fast 3D probe diffusion, like one might expect in liquid, the appropriatevolume is simply that of the sample under investigation. For example, in 8Li or 31Mg β-NMRexperiments on ionic liquid droplets [169, 382],𝑉 is on the order of a fewµL, which is considerablylarger than either estimate in the static limit. In most solids below∼320K, however,𝐷 is unlikelyto large enough to disperse the implanted probes over its full volume.Using the assumptions outlined above, we can now make three estimates for (steady-state)probe ion concentrations. In the limit where all of the implanted ions are effectively static, the“cylindrical” approximation gives: [X] ∼ 4 × 1014 cm−3. The more robust estimate using theconvex hull [384] of the (scaled) 3D stopping profile gives: [X] ∼ 2 × 1013 cm−3. For comparison,in the limit of very fast probe diffusion (i.e., in a liquid at high temperature): [X] ∼ 6 × 1010 cm−3.This is the lower limit for the concentration in the ionic liquid experiments [169, 382]. Forcomparison, in the example shown in Figure A.1 for rutile TiO2, [Ti] = 3.2 × 1022 cm−3, clearlydwarfing the above estimates by no less than ∼108! It is surprising, however, to find that [X]differs by only three decades between the fast-diffusion and convex hull approaches. This maybe due to the fact that the hull volume is biased by outliers in the stopping profile and it doesnot account for the spatial inhomogeneity of the ions within the enclosed volume. Undoubtedly,there are regions within this volume where the concentration is higher and these regions will be161where most of the detected signal in the β-NMR experiment comes from.As a simple check, wemay consider that a smaller fraction of the hull volume (e.g.,∼0.5) mayinclude the vast majority (e.g., ∼0.875) of all implanted ions; however, this implies our estimateusing the full volume is only off by a relatively small factor, not several orders of magnitude,suggesting that the method gives a reasonable estimate.These considerations are clearly in the dilute limit and we anticipate no measurable effects ofinteractions among 8Li. It is a different question about the damage 8Li leaves behind which mayhave a significantly longer lifetime than 1.2 s. This would yield a rate dependence and evolutionof the data with overall exposure.162Appendix BContinuum diffusionIn the limit of many small steps, the theory of discrete random walks (see Section 1.1.1)crosses over to continuum diffusion, familiar for fluid phases. Even in solids, when diffusionover macroscopic length scales are probed (e.g., from ionic conductivity or tracer diffusivitymeasurements), the continuum approximation is generally good. Here, we recall its basicproperties.The phenomenological description of continuum diffusion was first expressed by Fick, whonoted that particles tend to disperse from regions of high to low concentrations. That is, the fluxof particles J at a concentration 𝑐 is related by:J = −𝐷∇𝑐, (B.1)where the proportionality constant 𝐷 is the diffusion coefficient. Generally, the diffusivity canbe anisotropic (i.e., 𝐷 is a second-rank real tensor), but often in isotropic phases (most fluids)it can be treated as a scalar quantity. Equation (B.1) is commonly referred to as Fick’s 1st law.Dimensional analysis of Equation (B.1) reveals the expected units of 𝐷; J is the number ofparticles passing per unit area per unit time (e.g., cm−2 s−1) and 𝑐 has units of particles per unitvolume (e.g., cm−3), giving 𝐷 units of area per unit time (e.g., cm2 s−1).As a consequence of the conservation of matter, time-dependent changes in concentration163Figure B.1: Sketch of a (radio)tracer diffusion measurement. Tracer atoms are first depositedon the surface of a solid, after which the material is annealed at fixed temperature for a time𝑡, facilitating the penetration of the tracers into the solid. The material is then quickly cooled,suppressing further migration, and partitioned into sections, wherein the concentration of thetracers is measured. The spatial extent of the concentration of tracer atoms after time 𝑡 isproportional to their diffusivity 𝐷. Adapted from [388]. Copyright © 2005, Springer-VerlagBerlin Heidelberg.must follow the negative of the divergence of flux (i.e., we have the continuity relation):𝜕𝑐𝜕𝑡= −∇ ⋅ J. (B.2)Hence, the full time dependence of this behaviour is captured by Fick’s 2nd law:𝜕𝑐𝜕𝑡= 𝐷∇2𝑐, (B.3)where 𝑡 is time. Equation (B.3) is an elliptic second-order homogeneous partial differentialequation with constant coefficients, sometimes called the diffusion equation. It is very similar toan equation for heat transport and it has been studied extensively [386]. Generally, 𝐷 is taken tobe a constant independent of both time and space. While solutions to Equation (B.3) depend onboth the initial and boundary conditions [387], they are generally related to a Gaussian function,equivalent to expressions derived from the microscopic theory of random walks [19]. SolvingEquation (B.3) allows one to determine 𝐷 from concentration profiles 𝑐(𝑟, 𝑡), where 𝑟 and 𝑡 arespatial and temporal coordinates. This is sketched in Figure B.1 for a (radio)tracer diffusionexperiment.164Appendix CEmpirical compensation “laws”In the study of rate processes, it is an empirical fact that the results frommany experiments arewell-described by a simple Arrhenius expression [33], analogous to that given by Equation (1.3).What is then extracted from an experiment are two parameters — an activation energy and aprefactor—which are capable of describing the full temperature dependence of the rate constant(e.g., 𝜏−1 or 𝐷). A breakthrough in explaining this Arrhenius form came in the developmentof transition state theory (TST) (see e.g., [24]) to describe how rate constants for chemicalreactions depend on temperature. A measured barrier 𝐸𝐴 is thus related to the difference inthe thermodynamic potentials between the transition and initial states of the reactive partners.Indeed, these same ideas have been extended to migration of atoms in solids [26] and, so longas no curvature is observed on an Arrhenius plot (e.g., from a strong temperature dependenceto Δ𝐺𝑚), the Arrhenius parameters are also available for comparison with results from theory;however, the conventional interpretation of these quantities can be challenging in light of theirhigh degree of correlation.With only two degrees of freedom in an Arrhenius expression (see e.g., Equation (1.3)),changes in one parameter are highly correlated with changes in the other, even for measure-ments of the same kinetic process, particularly if the measured dynamic range is relativelysmall. Empirically, this correlation results in an exponential relationship between 𝜏−10 and 𝐸𝐴,colloquially referred to as the Meyer-Neldel rule (MNR) in solid-state sciences after the work of1650.0 0.2 0.4 0.6EA (eV)101210131014101510 (s1 )A1 2 3 41000 /T (K 1)1041061081010101210141 (s1 )BArrhenius parameters:10 2×1012 s 1EA 0.05 eV10 7×1012 s 1EA 0.15 eV10 28×1012 s 1EA 0.25 eV10 106×1012 s 1EA 0.35 eV10 403×1012 s 1EA 0.45 eV10 1530×1012 s 1EA 0.55 eVFigure C.1: Illustration of the MNR for a kinetic process. (A)Meyer-Neldel plot of the expo-nential relationship between the measured Arrhenius prefactor 𝜏−10 and activation energy 𝐸𝐴for related kinetic processes [Equation (C.1)]. The Meyer-Neldel energy ΔMN can be identifiedfrom the slope of the dashed grey line in the semi-logarithmic plot. (B) Arrhenius plot of therelated rate processes following the MNR. The isokinetic temperature 𝑇iso [Equation (C.2)] canbe identified at the equivalence point (i.e., the crossing temperature) of all related rate processes,indicated by the vertical dashed grey line.Meyer and Neldel [389].1 Explicitly, it is often found that [33]:𝜏−10 ≈ 𝜏−1MN exp (𝐸𝐴ΔMN) , (C.1)where 𝜏−1MN and ΔMN are the MNR prefactor and energy, respectively. The temperature where allrelated kinetic processes are equal, known as the isokinetic point 𝑇iso, is identified as:𝑇iso =ΔMN𝑘𝐵. (C.2)These empirical relationships are illustrated in Figure C.1.1In other disciplines, the MNR is synonymously referred to as enthalpy-entropy correlations, the isokinetic rule,or the compensation law (see e.g., [33]).166ColophonThis thesis was typeset with LuaLATEX using 11 pt Scientific and Technical InformationeXchange (STIX) fonts. BibLATEX and Biber were used for the bibliography management. Thedata analysis described herein was achieved using a mix of C, C++, and Python code. Datafitting was done almost exclusively using the functionality built into ROOT [175], especiallythe implementation of the MINUIT [176] minimization algorithms. Post-processing of theresults benefited from the facilities implemented in NumPy [390] and SciPy [385], with many ofthe figures in this thesis produced using Matplotlib [391]. Crystal structures appearing in thiswork were drawn using either CrystalMaker [205] or Visualization for Electronic and STructuralAnalysis (VESTA) [273].167
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Microscopic dynamics of isolated lithium in crystalline solids revealed by nuclear magnetic relaxation… McFadden, Ryan Michael Lund 2020
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Title | Microscopic dynamics of isolated lithium in crystalline solids revealed by nuclear magnetic relaxation and resonance of ⁸Li |
Creator |
McFadden, Ryan Michael Lund |
Publisher | University of British Columbia |
Date Issued | 2020 |
Description | This thesis reports measurements on the dynamics of isolated lithium in single crystal materials using ion-implanted ⁸Li β-detected nuclear magnetic resonance. From spin-lattice relaxation and resonance measurements, we identify the kinetic parameters describing the ion’s site-to-site hop rate – the elementary process in long-range solid-state diffusion – and compare the results with theoretical work in the literature, as well as experiments at higher concentration. In addition to these “ionic” details, the nuclear magnetic resonance probe provides information on the electronic properties of the host, whose most intriguing features are also discussed. In the one-dimensional ion conductor rutile TiO₂, we find two sets of thermally activated dynamics: one below 100 K and another at higher temperatures. We suggest the low temperature process is unrelated to lithium motion, but rather a consequence of electron polarons in the vicinity of the implanted ⁸Li⁺. Above 100 K, Li⁺ undergoes diffusion as an isolated uncomplexed cation, characterized by an activation energy and prefactor that are in agreement with macroscopic diffusion measurements, but not with theory. In Bi₂Te₂Se, a topological insulator with layered tetradymite structure, implanted ⁸Li⁺ undergoes ionic diffusion above 150 K, likely in the van der Waals gap between adjacent Te planes. A comparison with structurally related materials reveals the mobility of isolated Li⁺ is exceptional. At lower temperature, we find linear Korringa-like relaxation, but with a field dependent slope and intercept, accompanied by an anomalous field dependence to the resonance shift. We suggest that these may be related to a strong contribution from orbital currents or the magnetic freezeout of charge carriers in this heavily compensated semiconductor. In the doped tetradymite topological insulators Bi₂Se₃:Ca and Bi₂Te₃:Mn, the onset of lithium dynamics is suppressed to above 200 K. At low temperatures, the nuclear magnetic resonance properties are those of a heavily doped semiconductor in the metallic limit, with Korringa relaxation and a small, negative, temperature-dependent Knight shift. From this, we make a detailed comparison with isostructural Bi₂Te₂Se. |
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Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2020-04-24 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0389972 |
URI | http://hdl.handle.net/2429/74167 |
Degree |
Doctor of Philosophy - PhD |
Program |
Chemistry |
Affiliation |
Science, Faculty of Chemistry, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2020-05 |
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UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
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