Pulsed electrically and optically detected magnetic resonance spectroscopy of disordered semiconductors

Electrically and Optically detected magnetic resonance (EDMR and ODMR) spectroscopy allows investigation of the microscopic nature of paramagnetic centers which influence the electrical or optoelectronic properties of semiconductors. Traditionally, EDMR and ODMR have been conducted as adiabatic magnetic field sweep spectroscopies under continuous wave (cw) application of electromagnetic fields. It is shown here that information about the dynamics of spin-dependent processes obtained from cwEDMR and cwODMR is determined by many electronic- and spin-relaxation parameters, which make the interpretation of experimental data quantitatively ambiguous. In contrast, it is shown that transient EDMR and ODMR experiments, so called pulsed (p)EDMR and pODMR, are significantly less ambiguous. For spin-dependent processes based on intermediate pairs of paramagnetic states, the cwEDMR and cwODMR as well as pEDMR and pODMR dynamics are derived analytically and the application of these results for the interpretation of experimental data is discussed for two examples: (i) The pEDMR study of spin-dependent recombination in silicon rich hydrogenated amorphous silicon nitride (a-SiNx:H) which showed the presence of a variety of mechanisms such as dangling bond recombination through weakly spin-coupled paramagnetic states but also recombination through band tail states which were strongly dipolar or exchange coupled. These processes had previously been observed in hydrogenated amorphous silicon (a-Si:H). However, while in a-Si:H, these processes took place solely as geminate recombination, they were of nongeminate nature in the a-SiNx:H. (ii) The pODMR study of excitonic recombination in a ?-conjugated polymer, namely, poly[2-methoxy-5-(20- ethyl-hexyloxy)-1,4-phenylene vinylene] (MEH-PPV). The presence of magnetic resonance induced spin-beat oscillations in the fluorescence intensity was confirmed. Based on the existing polaron-pair recombination model, previously pEDMR-detected beat signals seen here with pODMR in an identical manner. Two types of MEH-PPV, one fully hydrogenated and one partially deuterated were subjected to pODMR. The deuterated materials showed a different beat oscillation dependence of the driving field power pattern which was indicative of smaller hyperfine fields in the deuterated material.

PULSED ELECTRICALLY AND OPTICALLY DETECTED MAGNETIC RESONANCE SPECTROSCOPY OF DISORDERED SEMICONDUCTORS by Sang-Yun Lee A dissertation submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Department of Physics and Astronomy The University of Utah August 2011 Copyright ⃝c Sang-Yun Lee 2011 All Rights Reserved 􀀷 􀁋 􀁈 􀀃 􀀸 􀁑 􀁌 􀁙 􀁈 􀁕 􀁖 􀁌 􀁗 􀁜 􀀃 􀁒 􀁉 􀀃 􀀸 􀁗 􀁄 􀁋 􀀃 􀀪 􀁕 􀁄 􀁇 􀁘 􀁄 􀁗 􀁈 􀀃 􀀶 􀁆 􀁋 􀁒 􀁒 􀁏 􀀃 􀀶􀀷􀀤􀀷􀀨􀀰􀀨􀀱􀀷􀀃􀀲􀀩􀀃􀀧􀀬􀀶􀀶􀀨􀀵􀀷􀀤􀀷􀀬􀀲􀀱􀀃􀀤􀀳􀀳􀀵􀀲􀀹􀀤􀀯􀀃 􀀷􀁋􀁈􀀃􀁇􀁌􀁖􀁖􀁈􀁕􀁗􀁄􀁗􀁌􀁒􀁑􀀃􀁒􀁉􀀃 􀁋􀁄􀁖􀀃􀁅􀁈􀁈􀁑􀀃􀁄􀁓􀁓􀁕􀁒􀁙􀁈􀁇􀀃􀁅􀁜􀀃􀁗􀁋􀁈􀀃􀁉􀁒􀁏􀁏􀁒􀁚􀁌􀁑􀁊􀀃􀁖􀁘􀁓􀁈􀁕􀁙􀁌􀁖􀁒􀁕􀁜􀀃􀁆􀁒􀁐􀁐􀁌􀁗􀁗􀁈􀁈􀀃􀁐􀁈􀁐􀁅􀁈􀁕􀁖􀀝􀀃 􀀏􀀃􀀦􀁋􀁄􀁌􀁕􀀃 􀀧􀁄􀁗􀁈􀀃􀀤􀁓􀁓􀁕􀁒􀁙􀁈􀁇􀀃 􀀏􀀃􀀰􀁈􀁐􀁅􀁈􀁕􀀃 􀀧􀁄􀁗􀁈􀀃􀀤􀁓􀁓􀁕􀁒􀁙􀁈􀁇􀀃 􀀏􀀃􀀰􀁈􀁐􀁅􀁈􀁕􀀃 􀀧􀁄􀁗􀁈􀀃􀀤􀁓􀁓􀁕􀁒􀁙􀁈􀁇􀀃 􀀏􀀃􀀰􀁈􀁐􀁅􀁈􀁕􀀃 􀀧􀁄􀁗􀁈􀀃􀀤􀁓􀁓􀁕􀁒􀁙􀁈􀁇􀀃 􀀏􀀃􀀰􀁈􀁐􀁅􀁈􀁕􀀃 􀀧􀁄􀁗􀁈􀀃􀀤􀁓􀁓􀁕􀁒􀁙􀁈􀁇􀀃 􀁄􀁑􀁇􀀃􀁅􀁜􀀃 􀀏􀀃􀀦􀁋􀁄􀁌􀁕􀀃􀁒􀁉 􀁗􀁋􀁈􀀃􀀧􀁈􀁓􀁄􀁕􀁗􀁐􀁈􀁑􀁗􀀃􀁒􀁉􀀃 􀁄􀁑􀁇􀀃􀁅􀁜􀀃􀀦􀁋􀁄􀁕􀁏􀁈􀁖􀀃􀀤􀀑􀀃􀀺􀁌􀁊􀁋􀁗􀀏􀀃􀀧􀁈􀁄􀁑􀀃􀁒􀁉􀀃􀀷􀁋􀁈􀀃􀀪􀁕􀁄􀁇􀁘􀁄􀁗􀁈􀀃􀀶􀁆􀁋􀁒􀁒􀁏􀀑􀀃 􀀶􀁄􀁑􀁊􀀐􀀼􀁘􀁑􀀃􀀯􀁈􀁈 􀀦􀁋􀁕􀁌􀁖􀁗􀁒􀁓􀁋􀀃􀀥􀁒􀁈􀁋􀁐􀁈 􀀭􀁘􀁑􀁈􀀃􀀜􀀏􀀃􀀕􀀓􀀔􀀔 􀀽􀀑􀀃􀀹􀁄􀁏􀁜􀀃􀀹􀁄􀁕􀁇􀁈􀁑􀁜 􀀭􀁘􀁑􀁈􀀃􀀜􀀏􀀃􀀕􀀓􀀔􀀔 􀀰􀁌􀁎􀁋􀁄􀁌􀁏􀀃􀀵􀁄􀁌􀁎􀁋 􀀭􀁘􀁑􀁈􀀃􀀜􀀏􀀃􀀕􀀓􀀔􀀔 􀀶􀁗􀁈􀁓􀁋􀁄􀁑􀀃􀀯􀁈􀀥􀁒􀁋􀁈􀁆 􀀭􀁘􀁑􀁈􀀃􀀜􀀏􀀃􀀕􀀓􀀔􀀔 􀀰􀁌􀁆􀁋􀁄􀁈􀁏􀀃􀀫􀀑􀀃􀀥􀁄􀁕􀁗􀁏 􀀭􀁘􀁑􀁈􀀃􀀜􀀏􀀃􀀕􀀓􀀔􀀔 􀀧􀁄􀁙􀁌􀁇􀀃􀀮􀁌􀁈􀁇􀁄 􀀳􀁋􀁜􀁖􀁌􀁆􀁖􀀃􀁄􀁑􀁇􀀃􀀤􀁖􀁗􀁕􀁒􀁑􀁒􀁐􀁜 ABSTRACT Electrically and Optically detected magnetic resonance (EDMR and ODMR) spec-troscopy allows investigation of the microscopic nature of paramagnetic centers which influence the electrical or optoelectronic properties of semiconductors. Traditionally, EDMR and ODMR have been conducted as adiabatic magnetic field sweep spectroscopies under continuous wave (cw) application of electromagnetic fields. It is shown here that information about the dynamics of spin-dependent processes obtained from cwEDMR and cwODMR is determined by many electronic- and spin-relaxation parameters, which make the interpreta-tion of experimental data quantitatively ambiguous. In contrast, it is shown that transient EDMR and ODMR experiments, so called pulsed (p)EDMR and pODMR, are significantly less ambiguous. For spin-dependent processes based on intermediate pairs of paramagnetic states, the cwEDMR and cwODMR as well as pEDMR and pODMR dynamics are derived analytically and the application of these results for the interpretation of experimental data is discussed for two examples: (i) The pEDMR study of spin-dependent recombination in silicon rich hydrogenated amorphous silicon nitride (a-SiNx:H) which showed the presence of a variety of mechanisms such as dangling bond recombination through weakly spin-coupled paramagnetic states but also recombination through band tail states which were strongly dipolar or exchange coupled. These processes had previously been observed in hydrogenated amorphous silicon (a-Si:H). However, while in a-Si:H, these processes took place solely as geminate recombination, they were of nongeminate nature in the a-SiNx:H. (ii) The pODMR study of excitonic recombination in a π-conjugated polymer, namely, poly[2-methoxy-5-(20- ethyl-hexyloxy)-1,4-phenylene vinylene] (MEH-PPV). The presence of magnetic resonance induced spin-beat oscillations in the fluorescence intensity was confirmed. Based on the existing polaron-pair recombination model, previously pEDMR-detected beat signals seen here with pODMR in an identical manner. Two types of MEH-PPV, one fully hydrogenated and one partially deuterated were subjected to pODMR. The deuterated materials showed a different beat oscillation dependence of the driving field power pattern which was indicative of smaller hyperfine fields in the deuterated material. To my wife, Seoyoung Paik. CONTENTS ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii LIST OF TABLES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii CHAPTERS 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2. UNDERSTANDING THE MODULATION FREQUENCY DEPENDENCE OF CONTINUOUS WAVE EDMR AND ODMR SPECTROSCOPIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1 Models for the description of spin-dependent transition rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.1 Rate equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Transient behavior of cwODMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3 Modulation frequency dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.1 At low modulation frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3.2 Ambiguity of cwODMR measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3.3 Trivial case (small spin mixing rates) . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3.4 Recombination, dissociation, and flip-flop . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.5 The influence of intersystem-crossing on cwODMR experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3.6 Pair generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4 Power dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.5 Signal sign dependencies on the modulation frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.6 The interpretation of cwEDMR and cwODMR signal signs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.7 Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3. UNDERSTANDING PULSED EDMR AND ODMR SPECTROSCOPIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.1 History of development of pEDMR and pODMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2 Incoherent pEDMR and pODMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.2.1 Rate model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.2.1.1 Steady-state solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.2.1.2 Boundary conditions and solutions . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.2.2 The transient behavior of the pEDMR and pODMR observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2.2.1 Characteristics of the biexponential decay . . . . . . . . . . . . . . . . . . . 57 3.2.2.2 Modeling observable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.3 Coherent pEDMR and pODMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.3.1 Hamiltonian of spin pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.3.2 Electrically and optically detected spin Rabi oscialltion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.3.3 Effects of inhomogeneous broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.3.4 Method to analyze band-limited spin-Rabi nutations . . . . . . . . . . . . . . . 66 3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4. SPIN-DEPENDENT PROCESSES IN A SILICON-RICH AMORPHOUS SILICON-NITRIDE SOLAR CELL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.1 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.2 pEDMR transients and I-Vs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.3 Electrically detected spin Rabi nutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.3.1 Weakly coupled spin pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.3.2 Dipolar coupled spin pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.3.3 Strongly exchange coupled spin pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.5 Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5. SPIN-DEPENDENT RECOMBINATION OF POLARON PAIRS IN MEH-PPV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.1 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.2 Polaron pair recombination model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.3 Finding a lower limit on the intersystem-crossing time from a pODMR transient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.4 Polaron pair recombination dynamics in MEH-PPV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.5 Optically detected spin Rabi nutations of weakly coupled spin pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.6 Tuning hyperfine fields in organic semiconductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.7 Spin beating induced by hyperfine interaction . . . . . . . . . . . . . . . . . . . . . . . . 109 5.8 Effect of deuteration on |Bhyp| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.10 Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 LIST OF PUBLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 vi LIST OF FIGURES 2.1 Sketch of a setup of cwODMR. The basic principle of cwODMR is the same as that of conventional ESR. Square microwave modulation can be used instead of continuous B0 field modulation and a lock-in amplifier is employed to increase the signal-to-noise ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2 The intermediate pair recombination model (KSM) as relevant for cwODMR and cwEDMR. Triplet and singlet pairs are formed with two constant gen-eration rates Gt and Gs, respectively. Those pairs can dissociate into free charge carrier states with certain probabilities dt and ds (dissociation rates) or can recombine to excitonic state with recombination rates rt and rs. A spin mixing process can be introduced by ESR externally and this rate is described by α. Another spin mixing process, intersystem-crossing process is described by kISC. Note that nt and ns represent triplet and singlet pair densities, respectively. They do no necessarily correspond to eigenstate densities. . . . . . . 33 2.3 A time transient calculated from a numerical model described by a combina-tion of parameters as rs = 104, rt = 100, ds = 102, dt = 106, kISC = 10􀀀2, α = 105, ρ = 0.75, Gs = 1023, and Gt = 1020. The dash-dotted curve shows the overall response obtained from eqs. 2.34 and 2.35. The blue solid and red dashed curves are the in-phase and the out-of-phase components described by Is1 sin( 2 T t) and Ic1 cos( 2 T t), respectively. See detail in text. . . . . . . . . . . . . . . 34 2.4 Three different quantitative models result in indistinguishable frequency de-pendencies. Each quantitative model is determined by a different set of parameters. Refer to Table 2.1 for all used values. . . . . . . . . . . . . . . . . . . . . . . 35 2.5 Seven distinguishable patterns of the modulation frequency dependence of cwODMR have been found out of almost a thousand quantitative models. (b), (d), and (f) are equivalent with (a), (c), and (e), respectively, but with opposite signs. Note that the parameters used for these data are listed in Table 2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.6 Calculated transient behaviors at different modulation frequencies. Black dash-dot line is overall response and blue solid line and red dashed line are in-phase and out-of phase components of it. Parameters are the same as those in Fig. 2.4 (a). The three graphs are normalized by the same scaling factor. Thus the relative intensities among three graphs can be compared. . . . . . . . . . 37 2.7 Role of the singlet recombination rate, rs. When rs is small, no significant change in the frequency dependence pattern is found when α is increased (from (a) to (c)). But for large rs, a pattern change is observed when α is increased (from (b) to (d)). All four quantitative models have same combinations of parameters but (a) rs = 102, α = 10􀀀3, (b) rs = 107, α = 10􀀀3, (c) rs = 102, α = 108, and (d) rs = 107, α = 108. Values for the other parameters used for these data are listed in Table 2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.8 Role of the intersystem-crossing rate, kISC. At small rs, it has been observed that there appear bumps on both in-phase and out-of-phase signal at high fre-quency region when kISC becomes large (from (a) to (c)). At large rs, different pattern change also has been found. The in-phase shows local extrema and out-of-phase shows change of sign as kISC being increased (from (b) to (d)). All four quantitative models have same combinations of parameters but (a) rs = 102, kISC = 10􀀀2, (b) rs = 107, kISC = 10􀀀2, (c) rs = 102, kISC = 108, (d) rs = 107, kISC = 108. The other parameter values used for these data are listed in Table 2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.9 MW power dependence. All four quantitative models have the same com-binations of parameters but (a) f = 103, (b) f = 104, (c) f = 107. At low modulation frequencies, typical saturation curves can be found. At high modulation frequency, a nontrivial saturation behavior occurs. Refer to Table 2.1 for the values used for the other parameters. . . . . . . . . . . . . . . . . . . . . . . . . 40 2.10 Example of a modulation frequency dependence function showing a change from nonzero-crossing pattern to a zero-crossing pattern. The only difference between the two quantitative models can be found in the triplet recombination rate coefficients. (a) rt = 100, (b) rt = 106. Values for the other parameters are listed in Table 2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.11 Sign changes due to various rate coefficients. (a) In-phase intensities of the zero modulation frequency component as a function of Gt/Gs and d/rs. To distinguish positive values and negative values, different color scales are used (positive in upper left corner, and negative in lower right corner). The black dotted line describes the boundary separating positive values and negative values. (b) and (c) are two randomly chosen two dimensional subsets of the data in (a) representing a generation rate ratio slice and dissociation rate ratio dependencies. These slices are shown as white dashed lines in (a). Intensities in (a), (b), and (c) are normalized but in the same scale. (d) Changes in the numbers of singlet pairs, n0 s1 − n0s 2 as a function of the same parameters as in (a). (e) Changes in the number of triplets pairs, n0 t1 − n0 t2 as a function of the same parameters as in (a). Intensities in (d) and (e) are normalized but in the same scale. All calculations in this figure are obtained from the same condition of rs = 104, rt = 1, kISC = 1, α = 1, ρ = 0.75, Gs + Gt = 1016. . . . . 42 2.12 The sign of cwODMR signals can be negative when radiative recombination is dominant as in (a), and positive when nonradiative recombination is dominant as in (b). In contrast the signs of cwEDMR are not different, (c) and (d). Used common values for each rate parameters can be found in Table 2.1. (a) and (c) rs = 104, rs;nr = 1. (b) and (d) rs = 1, rs;nr = 104. . . . . . . . . . . . . . . . . . . . 43 3.1 Experimental setup for the pEDMR and ODMR. An optics setup for the PL detection and an electrical setup for the photocurrent detection are shown as examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.2 Plots of the steady-state pair densities, n0s and n0s as a function of kISC. Each pot shows different quantitative models: (a) ρ = 0.5, Gs = 109s􀀀1, Gt = 3×Gs, (b) ρ = 0.99, Gs = 109s􀀀1, Gt = 3×Gs, (c) ρ = 0.5, Gs = 109s􀀀1, Gt = Gs/100, (d) ρ = 0.99, Gs = 109s􀀀1, Gt = Gs/100. Parameters used identically for all simulations are rs = 2 × 104, rt = 2 × 102, ds = 4 × 104, dt = 4/3 × 104. For details see text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 viii 3.3 Plot of two cwEDMR and ce ODMR decay rates, m12 and m22, as a function of intersystem-crossing rate for two different temperatures (a) ρ = 0.5 and (b) ρ = 0.99. The parameters used identically for both simulations are rs = 2 × 104, rt = 2 × 102, ds = 4 × 104, dt = 4/3 × 104, Gs = 109, Gt = Gs/100. . . 71 3.4 kISC dependencies of the coefficients of the double exponential transient func-tions. (a) Plots of the coefficients A1p and B1p as defined by eqs. 3.11 and 3.12 as a function of kISC. (b) Plots of the coefficients I1 and I2 as defined by eq. 3.23 as a function of kISC. (c) Plot of the ratio of −I2/I1. Note that Δn > 0 is assumed for (a) and (b), but no assumption for Δn is necessary for (c). For a plots, the same parameters were used as for the data in Fig. 3.2 and ρ = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.5 Simulated pODMR transients based on the same parameters as in Fig. 3.4 as a function of the intersystem-crossing-rate kISC. As kISC becomes large, the transient becomes essentially single exponential because the fast relaxation component is not visible on this time scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.6 Calculation of a pEDMR or pODMR rate changes due to a single excitation pulse. The plot shows the Fast Fourier transform of the calculated data as a function of the excitation frequency in presence of inhomogeneous broadening. The data represent a plot of eq. 3.35 with B1=1.08 mT and FWHM of (a) 0.94 mT and (b) 1.93 mT. The oscillation components represent Rabi's frequency formula √ γ2B2 1 + (ω − ωL)2 for the case of small inhomogeneity (a). For inhomogeneities large than γB1, the hyperbolic feature vanishes and it is replaced by a broad peak (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.7 Plots of calculated Rabi oscillation data with (black) no inhomogeneous broad-ening, a short sampling time, and an infinite decay time, (red) a short sampling time but a shorter maximum pulse length, and (blue) a long sampling time (small sampling rate) and the same pulse length as red. (a) Plot of the time domain, (b) Plot of the data in (a) in the frequency domain (Fourier transformed). All data sets are calculated with the same Rabi frequency, but the different sampling rates and frequency resolutions. Black: Sinusoidal with 0.1 ns sampling time and 1.953 MHz frequency resolution. Red: Calculated according to eq. 3.35 with the same sampling rate as black curve but 3.906 MHz frequency resolution. Blue: Calculated according to eq. 3.35 with the same frequency resolution as the blue curve but a sampling rate of 4 ns. The highest frequency of the Fourier transforms of the black and red curves is 9 GHz but only data up to 200 MHz are plotted. All curves are normalized by their respective maximum intensities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.1 Transient measurements of spin-controlled currents following a τ = 172ns long excitation pulse applied at t = 0. (b) Current change ΔIph(t) as a function of time t and the excited g-value (corresponding to the static magnetic field applied to the sample). (a) and (c) are two time slices from panel (b) for g = 2.031 and g = 2.008, respectively. Panels (e),(f) and (g) are g-factor (magnetic field) slices from panel (b) for different times after the pulse. (d) Sketch of the p-i-n structure with a-SiN0:3:H bandgap of ∼ 2 eV [12] (not to scale geometrically). (h,i) I-V curves with (top) and without (bottom) illumination measured at room temperature and T = 15K. . . . . . . . . . . . . . . . 91 ix 4.2 Measurement of spin-Rabi nutation with B1 = 1.4 mT. (b) Plot of integrated photocurrent changes ΔQ after pulsed excitation as a function of the applied g-factor and pulse length τ . (a), (c) Pulse length slices from (b) for g = 2.008 and g = 2.031. (e) Fast Fourier transform of the data in (b) plotted as function of g and the Rabi frequency fRabi in units of γB1. (d), (f) Frequency slices from (e) for g = 2.008 and g = 2.031. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.3 Land´e g-factor dependence of spin-Rabi nutation with B1 = 1.4 mT. Land´e g-factor slices obtained from Fig. 4.2 at (a) fRabi = γB1 (weakly coupled spins), (b) fRabi = 1.4γB1 (predominantly dipolar coupled spins) and (c) fRabi = 2γB1 (predominantly exchange coupled spins). Green curves are final fit results. In (a), red and blue lines are two Gaussian peaks. Assignments of these peaks can be found in text. In (b), red curves are Pake doublet fit and blue curve is broad Gaussian peak representing contribution of high frequency tail of weakly coupled spin pair signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.4 The energies of four s=1/2 pair system eigenstates as a function of the mag-netic field. For simplicity, splitting due to dipolar interaction is shown only. Arrows indicate Δm = 1 transitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.5 Pake patterns for triplet-to-triplet transition (blue solid curves) and triplet-to- singlet transition (red dotted curves). Width of blue solid curve is 3D0 and width of red dotted curve is D0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.6 Electrically detected Rabi nutations at various B1 field strength. (a) Nutation spectra measured for g = 2.0047 (B0 = 347.2mT) as a function of the applied B1 field. (b) Plot of the ratio of the beat signal intensity Ibeat and nutation signal intensity Inut as a function of B1 for three different magnetic fields around g = 2.0047. The solids lines represent the expected B1-dependence of exchange coupled (constant) and weakly coupled (sloped) pairs. . . . . . . . . . . . 96 5.1 Illustration of spin-dependent excitonic charge carrier recombination in or-ganic semiconductors. Upon encounter, electrons and hole (which are usually polaronic states) form weakly spin- but strongly Coulomb-coupled intermedi-ate pairs. The pairs can exist in parallel and antiparallel configurations with pure triplet character or singlet/triplet mixtures, respectively. Triplet polaron pair will either thermally dissociate or recombine into triplet excitons. Singlet state will either dissociate at a different dissociation rate or recombine into singlet states. Changes of the precursor spin states with magnetic resonance can change netdissociation or netrecombination rates, which then influence conductivity or optical emission, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.2 Photoluminescence change of MEH-PPV as a function of the time after a short (128ns) microwave pulse and a function of the magnetic field B0. One can recognize an enhancement signal right after pulse for magnetic fields around B0=345.5mT followed by slowly relaxing weak quenching signal. . . . . . . . . . . . 116 5.3 Transient behavior of PL change at onresonance B0 fields, 345.5 mT. Double exponential function can explain enhancement-quenching behavior very well and two time constants can be extracted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 x 5.4 PODMR spectrum obtained at a time t = 40 μs after a pulsed excitation of MEH-PPV with the magnetic field expressed as g-values. A fit function consisting two Gaussian peaks, one narrow(FWHM=1.68(6) mT) and the other broad(FWHM=3.6(1) mT), is in good agreement with the data. For details see text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.5 Intensity of pODMR signals of MEH-PPV as a function of the applied B0 and spin-frequency. The displayed data are obtained from a FFT[N(τ )](B0) of a pulse length dependence measurement. The measurements were carried out at (a) at B1=0.55 mT and (b) at B1=1.54 mT. (c) FFT of opticalled detected Rabi nutations at an on resonance B0 field measured at various B1 strengths. The curves in (a) and (b) are predictions of Rabi frequencies (solid) and spin Rabi-beat frequencies (dotted) as a function of B0, based on Rabi-frequency formula (see Chapter 3). The red solid line in (c) is the linear fit of 2πfRabi vs. B1 whose slope is 1.77 × 1011 rad/sT. This is very close to the gyromagnetic ratio. The blue dashed line is a guide to the eye showing 2πfRabi = 2γB1. . . . 119 5.6 Larmor frequency distribution of noncorrelated spin pair partners expressed in g-factors. gcn and gc b are Land´e g-factors of the narrow and broad peaks, respectively, and, δgn and δgb are the FWHM of two peaks, respectively. δg is the correlation length. The red dotted curve indicates the excitation bandwidth determined by the B1 field strength. . . . . . . . . . . . . . . . . . . . . . . . . 120 5.7 Coherent spin manipulation in organic semiconductors monitored by PL-detected spin resonance. (a) Charge carrier pairs are formed in MEH-PPV by optical excitation. (b) Under spin resonance conditions, a spin flip can occur, which is recorded by a change in singlet exciton emission intensity. (c) At low microwave intensities, only one spin precesses at a time, whereas both spins precess together at high intensities (d). (e) Inset: Rabi flopping in the polymer PL is dominated by a single frequency component at low intensities as shown by the Fourier transform in the main plot (X-band 9.8 GHz radiation). (f) At high intensities, spin beating occurs, leading to a harmonic appearing in the Fourier transform. The green lines in the time domain and frequency domain plots correspond to fits of the experimental raw data and its Fourier transform, respectively. Blue lines show the fundamental contribution in the oscillation, red lines indicate the beat signal. The data analysis procedure is outlined in main text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.8 Effect of deuteration of the polymer side groups on the ODMR resonance spectrum and on spin beating [cf. Fig. 5.7 (f)]. (a), (b) Structures of the polymers studied. (c), (d) The differential PL resonance spectrum is accu-rately described by a superposition of two Gaussians, representing electron and hole resonances. (e), (f) Fourier analysis of the beating transients [see Fig. 5.7 (e), f)] allows the extraction of the spin-1 and spin- contributions to the resonance. The crossing point of the two as a function of microwave field strength B1 offers an estimate of the difference in local hyperfine fields experienced by a carrier pair. The dotted curves in (e) are the predictions. See details in text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 xi LIST OF TABLES 2.1 Parameters used for calculation of all plotted data in this chapter. All values have a unit of s􀀀1 except for ρ which is arbitrary. . . . . . . . . . . . . . . . . . . . . . . 44 3.1 Steady state pair densities at two limiting cases . . . . . . . . . . . . . . . . . . . . . . . . 76 3.2 Rates at two limiting cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.3 Coefficients at two limiting cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.4 Intensities of pODMR at two limiting cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.1 Eigenenergies of the four spin s=1/2 pair states. . . . . . . . . . . . . . . . . . . . . . . . 97 4.2 Transition energy required for triplet-to-triplet transition and triplet-to-singlet transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 ACKNOWLEDGMENTS I would like to express my gratitude to the people who have been my mentors, advisor, and colleagues during my Ph.D. studies: I especially thank Prof. Dr. Christoph Boehme who has been my advisor as well as a mentor. He has always encouraged me to overcome many obstacles that I have faced during my studies. And his deep understanding about physics, enthusiasm, and energy always have inspired me. I would like to remark that whenever I knocked on his office door for discussions, I stepped out of his office full of satisfaction. Dr. Dane McCamey, who had been a postdoctoral researcher in Christoph Boehme's group, showed me how creativity leads to successful scientific research. Discussions with him were always fun and (of course) useful. Seoyoung Paik, one of my best colleagues and my beloved wife, has always worked with me whenever I had to sit up for very long experiments. She made a most significant contri-bution to new sample holder which was essential for any pEDMR experiments. Discussions with her always revealed what I didn't know and what I had to study further. Especially she has been my faithful supporter whenever I lost my confidence. Prof. Dr. John Lupton's comprehensive knowledge has helped me a lot for organic semiconductor research. Prof. Valy Vardeny, prof. Mikhail Raikh, prof. Stephan Lebohec, and prof. Michael H. Bartl have gladly participated in my supervisory committee and provided me with advice. I also would like to thank my colleagues: Willian Baker made a contribution for building a new sample holder which has been a prerequisite for pEDMR experiments. Kipp van Schooten kindly prepared organic samples for pODMR experiments. Finally, I would like to thank my parents and brother who have provided me continuous and firm support while studying abroad. CHAPTER 1 INTRODUCTION Properties of electron spins such as the Land´e g-factor or the magnetic resonance linewidth are highly dependent on their microscopic environment and local interaction such as with neighbor electron spins, adjacent nuclear magnetic moments, and even their own motion which can result in spin-orbit coupling [1]. One of the most frequently used methods to detect electron spin properties is Electron Spin Resonance (ESR). An electron spin subjected to a static magnetic field (B0 field) can be in one of two eigenstates which are degenerate in the absence of a magnetic field but nondegenerate due to Zeeman interaction in the presence of a magnetic field. Resonance occurs when the electron absorbs or emits electromagnetic waves whose energy is equal to the energy difference between the split states. Many charge carrier recombination centers in semiconductors possess a spin, they are paramagnetic. Because of this, they are detectable by ESR and since their ESR signals are affected by surrounding environments, ESR has widely been used to study their microscopic nature [2-7]. Most electronic and optoelectronic devices use properties such as charge transport, photoconductivity, photo- and electro-luminescence, or quantum efficiency for technical applications. For many of these materials, the electron spin degree of freedom is not of significance, because of the minute energy scales of electron spin states. However, in some materials with weak spin-orbit coupling, spin properties can determine macroscopic electri-cal or optical materials properties due to spin-selection rules [8-11]. For these materials, an understanding of the properties of paramagnetic centers as well as their spin- and electronic dynamics is of profound importance for the understanding of the materials behavior. The first ESR experiment, which also happened to be the first ever magnetic resonance experiment, was reported by Zavoisky [12] in 1945. Since then, many related techniques have been developed: nuclear magnetic resonance spectroscopy (NMR) [13, 14], pulsed ESR [15], pulsed NMR [16, 17], spin echo techniques [15, 16], double magnetic resonance methods such electron-electron double resonance (ELDOR) [18], electron nuclear double resonance (ENDOR) [19] and many others. All these methods follow a common detection 2 scheme which is based on the emission or absorbtion of electromagnetic radiation from a given spin system. Due to the weak polarization of both electron and nuclear spin systems, magnetic resonance is therefore a comparatively insensitive approach: Conventional ESR is typically limited to a detection limit of 1010 spins. Moreover, when electronic processes involving paramagnetic centers are investigated using ESR, it is oftentimes hard to figure out whether an observed ESR signal actually represents the paramagnetic centers involved in the investigated processes or whether it represents other spins under which the signal of interest is buried. For instance, the ESR spectroscopy of bulk crystalline silicon without an extensive surface preparation reveals almost no information about the paramagnetic centers in these materials because any ESR spectrum obtained from a small silicon crystal will be dominated by very strong signals from crystalline silicon surface states [20]. In addition to this problem, pure ESR spectroscopy is also oftentimes not sensitive enough for semiconductor samples, especially when the investigated materials are available only as thin films. The inherent volume sensitivity of magnetic resonance spectroscopy is a great detriment for the investigation of low dimensional materials systems. The disadvantages of conventional magnetic resonance spectroscopy can be overcome by direct observation of those macroscopic observables which are influenced by spin-dependent processes. By combining conventional ESR with the detection of luminescence, absorption or electric conductivity, a vast amount of information about localized paramagnetic states and the way they influence optical and electrical properties is obtained. These methods are referred to as optically (ODMR) and electrically (EDMR) detected magnetic resonance. First ODMR experiment were carried out by Geschwind et al. in 1959 [21]. A few years thereafter, in 1965, the first EDMR experiment was carried out by Maxwell and Honig [22]. Since then, EDMR and ODMR experiments were performed on many different electronic systems in a broad range of materials systems [23-46]. Most of these studies have been carried out as continuous wave (cw) ODMR or EDMR experiments which are adiabatic field sweep experiments where the spin spectrum is obtained by a gradual sweep of a magnetic field in presence of continuously irradiated electromagnetic radiation with constant frequency and intensity. This experimental approach is simple and it allows us to obtain Land´e g-factors of paramagnetic states contributing to luminescence and conductivity. It also gives access to magnetic resonance lineshapes which contain information about disorder, spin-interactions, as well as electronic- and spin relaxation times. This broad range of experimental parameters influencing ODMR and EDMR measurement, is at the same time the origin of the limitations of these methods: Significant uncertainty typically arises for 3 most cw ODMR and EDMR spectra, because there are too many factors influencing the line-shapes and resonance positions of ESR spectra. Therefore, lineshape analysis can frequently provide ambiguous information especially when complex superpositions of lineshapes due to many overlapping spin signals with a distribution of electron- and spin-dynamics are present in a given semiconductor sample. Conventional ODMR and EDMR are traditionally performed as magnetic field or radia-tion field modulated experiments, with subsequent Lock-in detection. This approach allows for an optimized noise suppression yet it also implies, that only one particular frequency component of the investigated spin-dependent processes is detected, namely the compo-nent whose frequency is equal to the experimentally chosen modulation frequency. This aspect can be utilized to gradually scan the entire dynamics of ODMR or EDMR detected spin-dependent signals, simply by a gradual measurement of the modulation frequency dependence of an observed spin-dependent signal. Similar to the modulation frequency dependence scan for cw ODMR and EDMR exper-iments, the dynamics of spin-dependent processes can also be observed by a direct transient (broad band) measurement. Similar to conventional magnetic resonance spectrocopies, this time dependent measurement approach has evolved in recent years towards so called pulsed (p) ODMR and EDMR spectroscopies. PODMR and pEDMR are not just the time domain equivalent of cw ODMR and cw EDMR. In contrast to those techniques, pODMR and pEDMR employ very strong electromagnetic pulses in order to manipulate the investigated spin states on very short times scales, much shorter than any spin- or electronic relaxation time of the excited species. On these time scales, the spins will therefore propagate coherently, which means they will propagate deterministically in a way that depends on their Hamiltonians. The observation of coherent spin motion therefore opens up direct access to a spins' Hamiltonian and, thus, a broad range of information about its nature. This dissertation consists of four main parts, which represent, (i) a study of the exper-imental limitations of the conventional cw ODMR and cw EDMR techniques, (ii) a study of how the pODMR and pEDMR techniques can overcome these limitations as well as applications of pEDMR (iii) and pODMR (iv) to disordered materials spectroscopy which demonstrated how these modern experiments can lead to new insights into the nature of macroscopic optical and electronic properties of an amorphous inorganic semiconductor and an organic semiconductor, respectively. The results of the work presented in the following led to the first all-analytical de-scription of cw ODMR and EDMR experiments which revealed that fits of experimentally 4 observed modulation frequency dependence measurements determined by electronic and spin-relxation parameters are profoundly ambiguous and that previously made assignments based on this approach may not be accurate. In contrast, the investigation of the dynamics of pODMR and pEDMR experiments showed that due to the enhanced access to experimen-tal parameters, these methods are inherently less ambiguous. The application of pEDMR for the investigation of hydrogenated silicon rich amorphous silicon nitride then showed that a broad range of qualitatively and quantitatively different spin-dependent processes is present in this material and that the observation of these processes gives important insight into the optoelectronic properties of this material and their potential applicability for photovoltaic and photoelectrochemical device applications. Finally, the application of pODMR to the in-vestigation of spin-dependent processes in a π-conjugated polymer confirmed the previously reported [47] but at the same time disputed [44] spin-dependent exciton formation process, namely the so called polaron pair process which describes the formation of strongly exchange coupled excitonic states through initial formation of weakly spin-coupled excitonic precursor states, so called polaron pairs. 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CHAPTER 2 UNDERSTANDING THE MODULATION FREQUENCY DEPENDENCE OF CONTINUOUS WAVE EDMR AND ODMR SPECTROSCOPIES Electron spin resonance (ESR) is a useful tool for the investigation of microscopic proper-ties of paramagnetic states in a wide variety of materials. In conventional ESR experiments, the total polarization of the investigated spin ensemble is observed by the measurement of microwave absorption. In some materials, there are other observables which can be used to detect electron spin states. For instance, when electron spins control electronic transitions such as transport or recombination, macroscopic materials properties such as photoluminescence, electroluminescence or conductivity can change under spin resonance. The advantage of these electrically detected magnetic resonance (EDMR) and optically detected magnetic resonance (ODMR) spectroscopies is that they are significantly more sensitive than conventional ESR (spin polarization is usually low and single microwave pho-tons can not be detected), and these methods provide a direct insight into how paramagnetic states in semiconductors affect some of the technologically most widely used electrical and optical materials properties. ODMR has been used in a wide range of research areas since its first invention [1, 2]. ODMR and EDMR are about 8 to 9 orders more sensitive than ESR, they both are proven to have single spin sensitivity ESR [3-7], and they both can directly link a paramagnetic center to a specific luminescence center [3-5, 8]. Thanks to these advantages, ODMR can be used to deconvolute unresolved, overlapping luminescence bands in semiconductors [9]. EDMR provides information about electronically active paramagnetic centers in a similar way, again with higher sensitivity than ESR [7, 10]. In the early stage (until about the 1980s), ODMR was mainly conducted on inorganic semiconductors to identify paramagnetic recombination centers and to investigate their spin-dependent processes [3, 11]. ODMR played an important role in investigating spin-dependent processes 9 especially in amorphous silicon (a-Si) and revealed a variety of defect states which influence recombination in a-Si [4, 5, 9, 12-15]. Continuous wave ODMR and EDMR (cwODMR and cwEDMR) have been used in a wide range of research fields: They have been used to investigate spin-dependent tran-sitions involving phosphorous donors in crystalline silicon [10, 16], trapping centers and their recombination dynamics in nanocrystals [6, 17-19], transport and recombination in microcrystalline hydrogenated silicon [20], GaN [21, 22], and SiC [23], and spin-dependent recombination in nitrogen vacancy centers in diamond [24-26]. Because cwODMR and cwEDMR can be used to distinguish overlapping recombination bands and their dynamics in disordered materials, they have also been used to investigate (usually amorphous) or-ganic semiconductors: cwODMR and cwEDMR have provided information about spin-pairs dominating electronic processes and their transitions in conducting polymers [27-38], small molecules [39-41], and polymer or small molecule/fullerene blends [42, 43]. The effect of isotopic modification on magnetic field effects in organic semiconductors also has been observed by ODMR [44], and the intersystem-crossing time has been extracted from the modulation frequency dependence [45]. Experimentally, cwODMR and cwEDMR are similar to conventional ESR except that lu-minescence intensity and electric current are picked up instead of the microwave absorption. Two magnetic fields, a static field B0 and oscillating field B1, are applied to a sample with B0 ⊥ B1. The frequency of the sinusoidal B1 field is matched with the Larmor frequency of the paramagnetic center to satisfy the resonance condition. As for most ESR spectrometers, X-band (≈ 9.7GHz) is used, a frequency in the microwave (MW) range. In the case of cwODMR, to allow for optical detection, optical or electrical excitation of electronic states is necessary. Depending on the excitation method, photoluminescence detected magnetic resonance (PLDMR) or electroluminescence detected magnetic resonance (ELDMR) can be performed. In the case of PLDMR, constant optical excitation is applied using, for example, a Laser, and the resulting photoluminescence (PL) is detected. To increase the signal to noise ratio, lock-in detection is oftentimes employed. Two different modulation methods can be used. One method involves modulation of the static magnetic field, B0, as used for conventional cwESR. The other approach is based on the modulation of the MW amplitude. Experimentally, B0 modulation has been found to give weaker signals than MW amplitude modulation [3]. Square modulation of the microwaves at a fixed reference frequency is generally used. The PL intensity reflecting the varying MW amplitude is then fed into a lock-in amplifier, and both in-phase and out-of-phase signals are obtained. In 10 some studies found in the literature [20, 29, 37, 40, 46], the out-of-phase signal are ignored. However, doing so can result in the loss of important information, as will be explained later. When the optical excitation is modulated with the same reference frequency as the MW, a double modulated PLDMR (DM-PLDMR) becomes possible [37]. An experimental setup for a MW modulated ODMR experiment is shown in Fig. 2.1. For EDMR, the optical detection is replaced by a current measurement. The metallic contacts needed for this require a design that prevents the distortion of the MW field. In both cwEDMR and cwODMR, the responses of the observables to the induced magnetic resonances are determined by the underlying electronic processes. The time scales on which these processes occur depend on various experimental parameters, such as excitation density [4, 5, 12, 40, 47, 48] (or an injection current for EDMR [29, 40]), temperature [4, 9, 29, 40], and MW power (equivalently B1 field strength) [4, 5, 11, 16- 19, 25, 39, 42, 45, 49-52]. The dependencies of cwODMR and cwEDMR signals on these parameters can allow us to distinguish overlapping transitions and to understand their dynamics. For cwODMR, spectral information also can provide additional information to distinguish overlapping luminescence bands [4, 9, 25]. Another experimental parameter that can influence the observed cwODMR and cwEDMR signals is the modulation frequency, as the lock-in detected signals depend on the transient responses to the modulated MW [4, 9, 12, 19, 53]. Modulation frequency effects have often been ignored in the literature, and, as a result, studies often reported results obtained using only one (or a small number of) modulation frequencies (usually the one which maximized the obtained signal). One can, however, find a number of reports showing modulation frequency dependence. Different signals at different modulation frequencies were reported for the first time by Biegelsen et al. [50]. Other investigators have noticed that modulation frequency effects play an important role in the observed signal, which can change drastically as a function of the modulation frequency [3, 9, 12]. Qualitative reports of modulation frequencies dependencies can be found in the early ODMR and EDMR litera-ture [3, 9, 13] which were sometimes used to identify the overlap of separate spin-dependent signals [5]. Even so, very little systematic research into modulation frequency effects was undertaken before the late 1990s, when research into this question became more common [18, 19, 25, 30, 36-38, 45, 47, 48]. A number of researchers have attempted to understand modulation frequency effects by developing rate models. Dunstan and Davies were the first to develop solutions for ODMR transients [11]. Next, Street and Depinna et al. developed rate models and found transient 11 solutions [4, 12]. Lenahan et al. explained their observed modulation frequency dependence using a simple rate model described by only one time constant [14]. A number of studies based on the steady-state solutions of such rate models have been reported [16, 17, 47, 49, 51]. However, to understand the modulation frequency effects the exact solutions for the frequency dependence are necessary. There have been a number of efforts to find the solutions for modulation frequency dependence [6, 19, 30, 36-38, 42, 45, 48]. However, no closed form analytical solutions have been reported, and important aspects of modulation frequency effects remain not well understood. This has led to a number of debates regarding the underlying physical mechanisms of cwODMR and cwEDMR signals, because modulation frequency dependencies observed by different groups on similar systems have sometimes led to completely different spin-dependent transition models. For example, the source of the signal seen in organic semiconductors has been attributed to both a spin-dependent polaron model [38, 45, 54] and a triplet exciton-polaron quenching model [36, 37, 53]. Lock-in detected cwODMR and cwEDMR signals can be either positive or negative depending on the shapes of transient responses [4, 9, 12, 19, 53]. A variety of spin-dependent models have been developed based on the observed signs of cwODMR and cwEDMR signals as well as experimental parameters, like pair generation rates, temperature, MW power, and modulation frequency. Examples for such studies exist for a-si [4, 5, 7, 9, 9, 11-13, 15, 50, 55], InP nanocrystal [19], II-IV semiconductor quantum dots [18], PbI2 nanoparticles [17], and organic semiconductors [29, 30, 33, 34, 36, 38, 41, 43, 47, 48]. For instance, it has been generally accepted that radiative and nonradiative recombination result in enhancement and quenching of cwODMR signals, respectively [5, 7, 17, 56], and all recombination processes and all detrapping processes result in quenching and enhancement of cwEDMR signals, respectively [7, 56]. The qualitative explanation for signs of cwODMR signals is as follows: spin resonance induces mixing between triplet and singlet pairs, and because initial states are generally dominated by triplet pairs due to the fast recombination of singlet pairs, the number of singlet pairs is increased at resonance. Thereby, the overall transition rate increases [7]. Some studies even conclude that a certain channel is radiative or nonradiative, based on the sign of the ODMR signal [4, 9, 12, 55]. The idea here is that when a nonradiative recombination process is enhanced under spin-resonance, the competing optically detected radiative channels must be quenched. The above examples show how critical it is to understand how MW modulation affects the observed cwODMR and cwEDMR signals. In this report, we employ the widely accepted spin-dependent transition model based on weakly coupled electron-hole pairs [57], and find 12 its closed-form analytical solutions. We then use this solution to explain how a broad range of electronic transitions, including recombination, dissociation, intersystem-crossing, pair generation, and spin-flip can affect the cwODMR and cwEDMR signals. We show how serious ambiguities related to the modulation frequency dependencies can arise, which make it difficult to determine the fundamental physical processes responsible for the observed cwEDMR or cwODMR frequency dependence. For example, extensive ODMR studies have been conducted on organic semiconductors to determine their dominant recombination processes. A variety of models have been suggested based on the observations of the signs of cwODMR and cwEDMR such as the singlet exciton quenching model [34, 36, 47], the triplet-triplet annihilation model [48], the polaron-to-bipolaron decay [29, 33], and the polaron pair recombination [38]. We show that most in many cases, the modulation frequency dependence cannot be used for such assignments, since the sign of these signals can be negative or positive for both radiative or nonradiative processes. This ambiguity is one of the reasons why cwODMR and cwEDMR spectroscopies have been increasingly substituted by transient, pulsed EDMR and ODMR techniques which will be discussed in the following chapters [58-66]. 2.1 Models for the description of spin-dependent transition rates The first quantitative model explaining spin-dependent recombination was suggested by Lepin [67] who described a thermal polarization model which predicted a relative change in photoconductivity of less than 10􀀀6 at 300 K for X-band ESR. Microwave frequency and temperature dependencies were also predicted. However, it turned out that this model could neither explain the signal intensity of more than 10􀀀3 that was observed in undoped a-Si:H at R.T. [67], and the very weak dependencies on microwave frequency [68] and temperature [15, 69]. These problems were soon resolved by another model developed by Kaplan, Solomon and Mott (KSM model) [57]. In the KSM model, intermediate pair states exist prior to a spin-dependent transition and the spin pair states may recombine or dissociate. In addition, it is assumed that spin pairs in the triplet state can be annihilated only when one of pair partners is flipped by the spin-lattice relaxation process or the induced ESR otherwise pairs dissociate. Thus, the recombination of triplet pairs happens only when they experience a transition to the singlet state. In the past decades, a number of refinements were introduced to the KSM model, in which spin-spin interactions such as exchange and dipolar interactions exist within the pair, and spin-orbit coupling that is weak but not negligible is permitted such that weak triplet 13 transitions become possible [59]. Because the intermediate pairs, consisting of two spins with s=1/2, can experience spin-spin interactions, the pair eigenbasis consists in general of two parallel states (|T+⟩ and |T􀀀⟩) and two mixed states (|2⟩ and |3⟩) which change continuously from | ↑↓⟩ and | ↓↑⟩ to |S⟩ and |T0⟩, respectively, as the spin-spin interaction increases. ESR can induce transitions of |T+⟩ ↔ | ↑↓⟩, |T􀀀⟩ ↔ | ↑↓⟩, |T+⟩ ↔ | ↓↑⟩, and |T􀀀⟩ ↔ | ↓↑⟩. Thus, when the spin-spin interaction is weak, there can appear transitions among all four eigenstates, and the transition probabilities are functions of the spin-spin interaction strength. Note that transitions of | ↓↑⟩ ↔ | ↑↓⟩ are ESR forbidden but possible due to T1 relaxation, and |2⟩ ↔ |3⟩ transitions are possible via mixed relaxation processes. To understand the change of spin pair densities by ESR induced transitions, a mathematical approach will be given. Boehme and Lips have found the effective changes of spin densities by solving a Louville equations describing the propagation of a spin ensemble during an ESR excitation [59]. The density changes of each spin state are then given by, ρ1;4(τ ) = ρ0 1;4Δu(τ ), ρ2;3(τ ) = ρ0 2;3Δv(τ ) ± ρ0 2;3 J + D ~ωΔ Δw(τ ) (2.1) where indices 1 and 4 represent the states |T+⟩ and |T􀀀⟩, respectively, ρ0i is the initial density, J and D are the exchange and dipolar coupling constant, respectively, ωΔ represents the half of the frequency separation between the states |2⟩ and |3⟩. Δu(τ ), Δv(τ ), and Δw(τ ) represent the ESR duration time (τ ) dependencies. When the Larmor separation (which is the difference of the two Larmor frequencies within a pair) is larger than the applied B1 field strength, only one pair partner can be flipped. In this case the τ dependencies become, Δv(τ ) = γ2B2 1 Ω2 sin2( Ωτ 2 ) ≡ Δ(τ ), Δu(τ ) = 1 − Δ(τ ), Δw(τ ) = 0 (2.2) where Ω = 2πf represents the Rabi frequency of the flipped pair partner. Therefore, the density changes of each eigenstates become ρ1;4(τ ) = ρ0 1;4(1 − Δ(τ )), ρ2;3(τ ) = ρ0 2;3Δ(τ ). (2.3) Because either one of the states 2 or 3 is always involved in a possible transition among four eigenstates, any transition will cause a decrease or increase of ρ2 or ρ3. Density changes in state 2 and 3 are equivalent to density changes of singlet and triplet pair states. Therefore we 14 don't need to deal with four state problems. Instead two pair densities of singlet and triplet pairs are enough to describe recombination processes as long as any coherent spin motion is not of interest. Note that this is a valid statement because modulation frequency is typically not faster than the time scale of coherent spin motions so that all coherent phenomena will be averaged out. An illustration of the resulting spin pair rate model is given in Fig. 2.2. Prior to a spin pair transition to a singlet state, it is in the intermediate pair state. This pair is created with a certain rate, Gs for a singlet pair and Gt for a triplet pair. If this process is due to optical generation of electron-hole pairs and spin-orbit coupling is infinitely small, Gt can be considered to be infinitely small. In the other case, if pair generation is achieved due to electrical injection of an electron and hole, Gt/Gt becomes three, because a pair will be created with a random spin configuration. The pair can recombine to an excitonic state with a recombination rate, rs for a singlet pair and rt for a triplet pair. This pair may dissociate into two free charge carriers without recombination. This happens at a dissociation rate, ds for a singlet pair and dt for a triplet pair. Before a pair recombines or dissociates, it can change its spin configuration from singlet to triplet or vice versa. This transition is possible via two spin mixing processes. One is intersystem-crossing, which is equivalent to a longitudinal spin relaxation process which can be defined as a "radiationless transition between two electronic states having different spin multiplicities" [70]. Intersystem-crossing rate is described by kISC. The other process is ESR induced spin-mixing as can be seen from eq. (2.2) and (2.3). This ESR-induced transition rate is given by α which is proportional to the microwave power (∝ B2 1) and dependent on the spin-spin interaction controlled oscillator strength of the pair [71]. In the following section, a large number of quantitative models will be tested with analytical solutions for the observables of cwEDMR and cwODMR. Using realistic values for each transition probability, we consider experimentally relevant values for the cwODMR experiment. A wide range of transition rates have been reported. Examples include PL lifetimes in a-Si which span 11 decades from 10􀀀9 s to 102 s [72]; bound pair decay (e-h pair dissociation) life times of 5 × 10􀀀5 s in polymer-fullerene blends [73]; fluorescence life times of 2 × 10􀀀7 s and phosphorescence life times of 10􀀀4 s in conjugated polymers [74]; microsecond-millisecond time scales of recombination in nanocrystalline TiO2 thin films [75]; radiative decay rates of 106 ∼ 107 s􀀀1, nonradiative decay rates of 109 ∼ 1010 s􀀀1, and dissociation rates of 107 s􀀀1 in organic semiconductors [76], and a lower limit of intersystem-crossing time of 10􀀀5 s in organic semiconductors [77]. In the following work, we vary the electronic transition rates, including recombination, dissociation, intersystem-crossing, and 15 flip-flop, in the range between 10􀀀4 and 109s􀀀1 to cover as wide a range of experimentally observed parameters as possible. 2.1.1 Rate equations CwODMR is fundamentally similar to conventional ESR spectroscopy - the one major modification is that the observable of ODMR is not the magnetization but the changes in the number of photons induced by ESR. Generally, a lock-in detected modulation of the B0 or the B1 field is used to enhance the resulting ODMR signal. For the B1 field modulation, a square modulated microwave is continuously applied, and the responses to this excitation contain various harmonic frequency components. We will focus in the following on this kind of experiment. Based on the rate model explained in Sec. 2.1, two coupled rate equations for the singlet and triplet pair densities can be written as below, dns dt = Gs − Csns + α(nt − ns) − kISC(ns − ρns) + kISC(nt − (1 − ρ)nt), (2.4) dnt dt = Gt − Ctnt + α(ns − nt) − kISC(nt − (1 − ρ)nt) + kISC(ns − ρns), (2.5) where ρ is the Fermi-Dirac distribution function, ρ = (1 + e ΔE kT )􀀀1, which approaches zero at low temperature and 1/2 at high temperature. It should be noted that α is turned on and off for each half cycle because of the square modulated microwave with frequency of 1/T. Cs and Ct are singlet and triplet pair annihilation rate coefficients, respectively. They consist of recombination and dissociation rate coefficients, Cs;t = rs;t + ds;t. Some aspects with regard to radiative and nonradiative recombination rate coefficients should be mentioned: For radiative recombination, the spatial correlation between the electron and the hole affects the transition probability so rt and rs are dependent on separation between an electron and hole [20, 78]. Therefore, because higher generation rate causes less separation, the radiative recombination probability is also a function of generation rate. However, this effect will not be considered in this report, as we assume that the average separation is larger than the localization radii of electrons and holes. Note that this transition corresponds to the radiative tunneling in hydrogenated amorphous silicon [78]. Nonradiative recombination includes all recombination processes which are not mediated by emission of photons, but phonons and hot carriers: phonon emission, Auger processes, surface and interface recombination, and recombination through defect states [79]. Nonradiative processes quench radiation efficiency in both organic semiconductors [80] and inorganic semiconductors [79]. As treated by List et al. [47] and Dyakonov et al. [46], we consider 16 both radiative and nonradiative recombination processes, and thus Cs = (rs + rs;nr + ds) and Ct = (rt + rt;nr + dt) where the subscript nr indicates nonradiative recombination. Given the above definitions, the PL intensity and electric conductivity become I ∝ rsns + rtnt, (2.6) σ ∝ dsns + dtnt. (2.7) It should be noted that nonradiative recombination behaves as pair annihilation process as other radiative recombination and dissociation, but they do not appear as proportionality constants in eq. 2.6 and eq. 2.7. In this section, only radiative recombination will be con-sidered (rs;nr, rt;nr = 0) for simplicity and the contributions of nonradiative recombination will be discussed in Sec. 2.6. Rate equations similar to eq. (2.4) and (2.5) can be found throughout the literature. However, usually only steady state solutions were found for the consideration of cwODMR and cwEDMR experiments [16, 47, 81]. In some cases, only the time dependence was considered [11, 12, 42]. Modulation frequency dependence solutions also have been reported, but there have been no reports of closed-form analytical solutions. Some solutions reported in the literature were obtained from a simplified rate model [14, 30, 48], some solutions were based on the steady state [36, 37], some solutions based on the rate model reported here were solely reported as numerical solutions [6, 19, 38, 45, 54], or the described observable was not the number of photons or electrons but total spin densities [30, 38, 45]. One solution given by Hiromitsu et al. was based on an assumed steady state for the half cycle where the MW is off [42]. The rate equations corresponding to eq. (2.4) and (2.5) are solved for the two separated time regions where the pulse is on and off, and the closed-form solutions can be explicitly expressed as: ns1 (t) = A11e􀀀m11t + A21e􀀀m21t + n0 s1, (2.8) nt1 (t) = B11e􀀀m11t + B21e􀀀m21t + n0 t1, (2.9) ns2 (t) = A12e􀀀m12(t􀀀T 2 ) + A22e􀀀m22(t􀀀T 2 ) + n0 s2, (2.10) nt2 (t) = B12e􀀀m12(t􀀀T 2 ) + B22e􀀀m22(t􀀀T 2 ) + n0 t2, (2.11) where ns1 and nt1 are the singlet and triplet populations when the MW pulse is on, and ns2 and nt2 are the singlet and triplet populations when the MW pulse is off. Those solutions consist of double exponential functions as is often found in the literature regarding pulsed experiments [58, 59, 63, 77]. 17 The introduced constants in the above solutions are summarized below, m1j = Cs + w1j + Ct + w2j − √ (Cs + w1j − Ct − w2j)2 + 4w1jw2j 2 , (2.12) m2j = Cs + w1j + Ct + w2j + √ (Cs + w1j − Ct − w2j)2 + 4w1jw2j 2 , (2.13) n0 sj = w2jGt + (Ct + w2j)Gs (Cs + w1j) (Ct + w2j) − w1jw2j , (2.14) n0 tj = w1jGs + (Cs + w1j)Gt (Cs + w1j) (Ct + w2j) − w1jw2j , (2.15) w11 = α + kISC(1 − ρ), w21 = α + kISC · ρ, w12 = kISC(1 − ρ), w22 = kISC · ρ, (2.16) where j=1 or 2. It should be noted that the exponents, m1j and m2j , are independent on either the generation rates or the modulation frequency. It can be easily seen that m2j is decided by the fastest rate coefficient, but it is difficult to predict m1j . However, it is clear that m2j is always larger than m1j . Two constant terms, n0 sj and n0 tj , are the steady-state solutions which the system assumes for very low modulation frequency [16, 38, 45, 47, 81]. It should also be noted that the singlet and triplet pair populations will approach values at the end of each half cycle which are at the same time as initial values of the following half cycle. Therefore, the frequency dependence might be able to be explained in terms of the differences between the populations at the end of each half cycle [38, 45], ns1(T/2) − ns2(T) and nt1(T/2)−nt2(T). However, lock-in detected signals are not simply decided by these quantities. The observables are not the population changes, but the changes in the number of photons, which incoprorates both the population change and the recombination probability. 2.1.2 Boundary conditions Because the spin populations assume the steady state only for a modulation frequency f=0, the time dependent solutions must be solved to explain the transient behavior at arbitrary modulation frequencies. To find the exact solution, the expressions for the eight unknown coefficients Aij and Bij (i, j=1 or 2) in eqs. 2.8, 2.9, 2.10, and 2.11 must be derived by application of eight boundary conditions. 18 Four of the boundary conditions can be easily found from the periodicity of the solution: ns1 (0) = ns2 (T), nt1 (0) = nt2 (T), ns1 (T 2 ) = ns2 (T 2 ) , and nt1 (T 2 ) = nt2 (T 2 ) . From these boundary conditions, we obtain A1 + A2 + n0s 1 = A3e(􀀀m12T=2) + A4e(􀀀m22T=2) + n0 s2 (2.17) B1 + B2 + n0 t1 = B3e(􀀀m12T=2) + B4e(􀀀m22T=2) + n0 t2 (2.18) A1e(􀀀m11T=2) + A2e(􀀀m21T=2) + n0s 1 = A3 + A4 + n0 s2 (2.19) B1e(􀀀m11T=2) + B2e(􀀀m21T=2) + n0 t1 = B3 + B4 + n0 t2 (2.20) After each half cycle, the number of each singlet and triplet pair is decreased or increased. These changes depend on the given rate coefficients: the number of singlet or triplet pair is either decreased or increased by spin mixing and increased by pair generation, decreased by the dissociation and recombination processes. From this condition, the other four equations can be found as ns1 ( T 2 ) − ns1 (0) = Gs T 2 + ∫ T 2 0 (w21nt1 − (Cs + w11) ns1) dt, (2.21) ns2 (T) − ns2 ( T 2 ) = Gs T 2 + ∫ T T 2 (w22nt2 − (Cs + w12) ns2) dt (2.22) nt1 ( T 2 ) − nt1 (0) = Gt T 2 + ∫ T 2 0 (w11nt1 − (Ct + w21) nt1) dt, (2.23) nt2 (T) − nt2 ( T 2 ) = Gt T 2 + ∫ T T 2 (w12nt2 − (Ct + w22) nt2) dt (2.24) By plugging the eqs. 2.8, 2.10, 2.9, and 2.11 into the above eight equations, we obtain A1(e(􀀀m11T=2) − 1) + A2(e(􀀀m21T=2) − 1) = − w21B1 − (Cs + w11)A1 m11 (e(􀀀m11T=2) − 1) − w21B2 − (Cs + w11)A2 m21 (e(􀀀m21T=2) − 1) (2.25) A3(e(􀀀m12T=2) − 1) + A4(e(􀀀m22T=2) − 1) = − w22B3 − (Cs + w11)A3 m12 (e(􀀀m12T=2) − 1) − w22B4 − (Cs + w12)A4 m22 (e(􀀀m22T=2) − 1) (2.26) B1(e(􀀀m11T=2) − 1) + B2(e(􀀀m21T=2) − 1) = − w11A1 − (Ct + w21)B1 m11 (e(􀀀m11T=2) − 1) − w11A2 − (Ct + w21)B2 m21 (e(􀀀m21T=2) − 1) (2.27) 19 B3(e(􀀀m12T=2) − 1) + B4(e(􀀀m22T=2) − 1) = − w12A3 − (Ct + w22)B3 m12 (e(􀀀m12T=2) − 1) − w12A4 − (Ct + w22)B4 m22 (e(􀀀m22T=2) − 1). (2.28) Note that the terms Gs + w21n0 t1 − (Cs + w11)n0s 1 = 0, Gs + w22n0 t2 − (Cs + w12)n0s 2 = 0, Gt + w11n0 s1 − (Ct + w21)n0 t1 = 0, Gt + w12n0s 2 − (Ct + w22)n0 t2 = 0 are used here, which are obtained from eqs. 2.14 and 2.15. Solving eq. (2.17)-(2.20), (2.25)-(2.28), and by introducing the parameters βij = Cs+w1j􀀀m1j w2j , Δn0s = n0 s2 − n0 s1, Δn0t = n0 t2 − n0 t1, and γij = e􀀀mij T 2 , we realized that Bij = Aijβij and four simplified equations   1 1 −γ12 −γ22 β11 β21 −β12γ12 −β22γ22 γ11 γ21 −1 −1 β11γ11 β21γ21 −β12 −β22     A11 A21 A12 A22   =   Δn0s Δn0t Δn0s Δn0t   . (2.29) are obtained for Aij . Equation 2.29 is a fully determined system of linear equation, which can be A22 = (((β21 − β11) · (Δn0s − γ11Δn0s ) − (Δn0t − β11Δn0s ) · (γ21 − γ11)) ·((β21 − β11) · (β11γ11γ12 − β12) − (β11γ12 − β12γ12) · (β21γ21 − β11γ11)) −((β21 − β11) · (γ11γ12 − 1) − (β11γ12 − β12γ12) · (γ21 − γ11)) ·((β21 − β11) · (Δn0t − β11γ11Δn0s ) − (Δn0t − β11Δn0s ) · (β21γ21 − β11γ11))) /(((β21 − β11) · (γ11γ22 − 1) − (β11γ22 − β22γ22) · (γ21 − γ11)) ·((β21 − β11) · (β11γ11γ12 − β12) − (β11γ12 − β12γ12) · (β21γ21 − β11γ11)) −((β21 − β11) · (γ11γ12 − 1) − (β11γ12 − β12γ12) · (γ21 − γ11)) ·((β21 − β11) · (β11γ11γ22 − β22) − (β11γ22 − β22γ22) · (β21γ21 − β11γ11))), (2.30) A12 = ((β21 − β11) · (Δn0s − γ11Δn0s ) − (Δn0t − β11Δn0s ) · (γ21 − γ11) −((β21 − β11) · (γ11γ22 − 1) − (β11γ22 − β22γ22) · (γ21 − γ11)) · A22) /((β2 − β11) · (γ11γ12 − 1) − (β11γ12 − β12γ12) · (γ21 − γ11)), (2.31) A21 = ((Δn0t − β11Δn0s ) · (β21γ21 − β11γ11) −(β11γ12 − β12γ12) · (β21γ21 − β11γ11) · A12 −(β11γ22 − β22γ22) · (β21γ21 − β11γ11) · A22) /((β21 − β11) · (β21γ21 − β11γ11)), (2.32) 20 A11 = Δn0s − A21 + γ12 · A12 + γ22 · A22. (2.33) Equations 2.30, 2.31, 2.32, 2.33 represent exact and general analytical solutions for the singlet and triplet density functions during a cwODMR modulation cycle. We are thus in a position to determine the temporal evolution of the cwODMR observable. 2.2 Transient behavior of cwODMR The observable in cwODMR is the emission rate of photons, and, as described in eq. 2.6, the time dependence can be obtained by adding the contribution from the singlet and triplet pair populations multiplied by the singlet and triplet recombination rate coefficients, respectively I1 = (rsA11 + rtB11) e􀀀m11t + (rsA21 + rtB21) e􀀀m21t + rsn0s 1 + rtn0 t1, (2.34) I2 = (rsA12 + rtB12) e􀀀m12(t􀀀T 2 ) + (rsA22 + rtB22) e􀀀m22(t􀀀T 2 ) + rsn0 s2 + rtn0 t2. (2.35) Here, I1 and I2 are the photon emission rate due to recombination of both singlet and triplet pairs when the pulse is on and off, respectively. The dash-dotted curve in Fig. 2.3 is a numerical example of the time dependence. Because m1j and m2j are always positive and m2j > m1j , the first and second terms in both eq. 2.34 and eq. 2.35 determine the faster and slower decay, respectively. It is difficult to predict which response will show an enhancement or quenching behavior because the overall response depends not only on m1j and m2j but also on rsAij + rtBij . Since the coefficients of all exponential terms have very complicated dependencies on a variety of parameters (see eqs. 2.30, 2.31, 2.32, and 2.33), it is clear that sign predictions depend on the magnitudes of many parameters at the same time. Using the above solution, we have been able to reproduce a wide variety of cwODMR transients reported in the literature [4, 9, 11, 12, 19]. 2.3 Modulation frequency dependence The time dependence solutions, eqs. 2.34 and 2.35, are the collective responses to the modulated B1 field over all frequency ranges. However, in experimental implementations which utilize a lock-in technique, only the component of the transient signal which has the same frequency as the reference will be obtained. With lock-in quadrature detection, both the in- an out-of-phase components are available. While the out-of-phase components have often been ignored in the literature [20, 29, 37, 40, 46], we note that the out-of-phase components contain important information. 21 To find the in-phase and out-of-phase components at the given modulation frequency, it is better to find the Fourier series of eqs. 2.34 and 2.35, and the frequency responses will be decided from the Fourier coefficients according to the definition of the Fourier series as below, IFs (t) = I0 2 + 1Σ l=1 ( Ic cos ( 2lπ T t ) + Is sin ( 2lπ T t )) , (2.36) Ic = 2 T ∫ T 0 I (t) cos ( 2lπ T t ) dt, (2.37) Is = 2 T ∫ T 0 I (t) sin ( 2lπ T t ) dt. (2.38) Then the obtained two coefficients as well as the zero frequency component are: Ic = 2m11 T (rsA11 + rtB11) ( 1 − e􀀀m11T=2 cos (lπ) m11 2 + 4l2π2/T 2 ) + 2m21 T (rsA21 + rtB21) ( 1 − e􀀀m21T=2 cos (lπ) m21 2 + 4l2π2/T 2 ) + 2m12 T (rsA12 + rtB12) ( cos (lπ) − e􀀀m12T=2 m12 2 + 4l2π2/T 2 ) + 2m22 T (rsA22 + rtB22) ( cos (lπ) − e􀀀m22T=2 m22 2 + 4l2π2/T 2 ) , (2.39) Is = 4lπ T2 (rsA11 + rtB11) ( 1 − e􀀀m11T=2 cos (lπ) m11 2 + 4l2π2/T 2 ) + 4lπ T2 (rsA21 + rtB21) ( 1 − e􀀀m21T=2 cos (lπ) m21 2 + 4l2π2/T 2 ) + 4lπ T2 (rsA12 + rtB12) ( cos (lπ) − e􀀀m12T=2 m12 2 + 4l2π2/T 2 ) + 4lπ T2 (rsA22 + rtB22) ( cos (lπ) − e􀀀m2 22T=2 m22 2 + 4l2π2/T 2 ) + ( rsΔn0s + rtΔn0t )( cos (lπ) − 1 lπ ) , (2.40) I0 = 2 T (rsA11 + rtB11) ( 1 − e􀀀m11T=2 m11 ) + 2 T (rsA21 + rtB21) ( 1 − e􀀀m21T=2 m21 ) + 2 T (rsA12 + rtB12) ( 1 − e􀀀m12T=2 m12 ) + 2 T (rsA22 + rtB22) ( 1 − e􀀀m22T=2 m22 ) +rs(n0s 1 + n0s 2) + rt(n0 t1 + n0 t2). (2.41) 22 The Fourier series in eq. 2.36 can be simplified by introducing V0 = √ Ic 2 + Is 2 and φ = tan􀀀1 ( Ic Is ) as below, IFs (t) = I0 2 + 1Σ l=1 V0 sin (2lπft + φ) , (2.42) where f is the frequency of the square modulation, 1/T. A Lock-in amplifier multiplies the input signal by its own internal reference signals, sin(ωLt + θL) and cos(ωLt + θL), to detect in-phase and out-of-phase signals, respectively. At this moment, the in-phase Vin and out-of-phase Vout signals are Vin = I0 2 VL sin (ωLt + θL) + 1Σ l=1 VLV0 2 (cos ((2lπf − ωL) t + φ − θL) − cos ((2lπf + ωL)t + φ + θL)) (2.43) Vout = I0 2 VL sin (ωLt + θL) + 1Σ l=1 VLV0 2 (sin ((2lπf + ωL) t + φ + θL) + sin ((2lπf − ωL)t + φ − θL)) (2.44) where VL is the amplitude of the reference signals. After these signals pass through a low pass filter, only the nonAC signals will remain. And the frequency of the internal reference signal is fixed such that it has a phase which is the same as the phase of the external reference signal. Thanks to this condition, ωL ≈ 2πf, and the in-phase and out-of-phase signals become Vin = V01 2 cos (φ1) = 1 2 Is1, (2.45) Vout = V01 2 sin (φ1) = 1 2 Ic1 (2.46) where V01 = V0, Is1 = Is, Ic1 = Ic, and φ1 = φ at l = 1, and, θL is usually set to zero. Thus the in-phase and out-of-phase cwODMR signals are the Fourier coefficients of the lowest frequency sine and cosine terms of the Fourier series solution (eq. 2.36), respectively. Examples are shown in Fig. 2.3 to explain the decomposed in-phase and out-of-phase components of the time response. It should be noted that the cwEDMR solutions also can be obtained in a similar way by replacing rs and rt in front of the exponential functions in eqs. 2.34 and 2.35 with ds and dt, respectively, as shown in eq. 2.7. Similarly the solutions for B0 field modulated cwODMR and cwEDMR can be found in the same way as for microwave modulated cwODMR and cwEDMR. While the difference 23 between these two modulation techniques is that the spin resonance is modulated by a square function and a harmonic function, the lock-in detected observables are identical since the lock-in technique is sensitive to the lowest harmonic component in either case. 2.3.1 At low modulation frequency We use the low modulation frequency limit to check the solution of our model, by varifying that these solutions can explain the cwODMR response. From the solutions above, the low frequency behavior is seen to be Vin;lf = (rs + rt)(Gt + Gs)α + (rtrs + rsw22 + rtw12)(Gt + Gs) + rtdsGt + rsdtGs (Cs + Ct)α + (Cs + w12)(Ct + w22) − w12w22 · 2 π − (rtrs + rsw22 + rtw12)(Gt + Gs) + rtdsGt + rsdtGs (Cs + w12)(Ct + w22) − w12w22 · 2 π , (2.47) Vout;lf = 0. (2.48) The out-of-phase component vanishes since the transient response can easily follow the slow modulation. The in-phase component shows a typical microwave power dependence: it vanishes at small power (when α → 0) and it becomes saturated at high power (i.e., it has a nonzero constant value). MW power dependence of eqs. 2.47 and 2.48 will be explained in the section 2.4. 2.3.2 Ambiguity of cwODMR measurements To understand the modulation frequency dependence of cwODMR, we inspected a large number of quantitative models. There is an extremely large number of possible qualitative and quantitative relationships betwen the model parameters. To limit the number of cases that we inspected, we chose a number of relationships between these parameters. We considered that i) the triplet recombination coefficient is the smallest one among all the recombination and dissociation rate coefficients (rt < rs, ds, dt) (unless otherwise noted), and ii) the singlet dissociation rate coefficient is smaller than the triplet dissociation rate coefficient (ds < dt) which means that the singlet intermediate state is assumed to be energetically lower than the triplet intermediate state (unless otherwise noted). Under these assumptions, a large number of quantitative models were investigated by varying rt, rs, ds, dt, kISC, and α in the range from 10􀀀4 to 109 s􀀀1. We investigated almost a thousand different variations of the relationship between different parameters. After looking through these cases, we find that it is almost impossible to distinguish some of the quantitative models based on their modulation frequency behaviors. Fig. 2.4 illustrates this ambiguity. Figure 2.4 (a), (b), and (c) show nearly identical frequency 24 dependencies of three very different quantitative models. The frequencies at which the in-phase signals have their maximum slope and the out-of-phase signals show their local maximum values are almost identical, and their shapes are also indistinguishable. The patterns shown in Fig. 2.4 represent in fact the most common frequency dependency that we have found out by the tested quantitative models. This illustrates the difficulty in extracting correct values for the corresponding coefficients from a simple frequency dependence - one can find a wide range of values which can reproduce it. This ambiguity is the most serious disadvantage of using the cwODMR or cwEDMR frequency dependence to determine the rate of underlying physical processes and the realization of their ambiguity puts many interpretations of cwODMR data reported in the literature in questions. Of the nearly thousand models we tested, we were able to describe them all with only four different frequency dependency patterns. These are shown in Fig. 2.5. We find that those patterns are determined mostly by the recombination rate coefficients, the microwave power, the spin mixing rates, as well as the generation rates. How each parameter influences the frequency dependence will be discussed in the following sections. The most trivial cases, seen in Fig. 2.5 (a) and (c), will be discussed first. 2.3.3 Trivial case (small spin mixing rates) To understand the behavior of the response to the modulation frequency, the trivial patterns will be discussed. "Trivial" means that the spin mixing rates, both kISC and α are negligible when compared to all the other rates. In this case, only the spin pair annihilation processes determined by the recombination and dissociation rate coefficients become dominant. All the patterns in Fig. 2.4 as well as the patterns in Fig. 2.5 (a) and (c) are obtained under the assumption of insignificant spin mixing rates, kISC and α. The pattern in Fig. 2.5 (c) is identical to the one in (a), but inverted due to different ratios between Gs and Gt. We found that the sign of the lock-in detected signal depends on almost all transition processes as one can deduce from Table 2.1. The most often seen patterns in Fig. 2.5 (a) and (c) are easily described qualitatively: at low frequencies, the in-phase signal has a constant nonzero value with no out-of-phase component. This is because approach to the steady-state takes place on a time scale much faster than the modulation period, and the recorded transient response looks like the applied microwave pulse train shown in Fig. 2.6 (a). These in-phase and out-of-phase responses are not seriously changed until the modulation frequency approach the slowest time constant, m􀀀1 1j , and this can be confirmed by the low-frequency responses in Fig. 2.4. For all cases in Fig. 2.4 and in Fig. 2.6, m1j and m2j are in the ranges of 102 ∼ 104 and 25 104 ∼ 106, respectively. As the modulation frequency approaches m1j , the system begins to lag behind the applied MW modulation, and the overall response ceases to resemble the simple harmonic function. This results in a decrease of the in-phase signal and an increase of the out-of-phase signal as seen in Fig. 2.6 (b). At very high frequencies, much faster than than the fastest time constant, m􀀀1 2j ∼ 10􀀀6, both the in- and out-of-phase components tend to approach zero. This behavior is explained by the exponential decay functions which become linear with small arguments and thus, they become constants (no change) when T → 0 [14, 48]. 2.3.4 Recombination, dissociation, and flip-flop Because cwODMR measures emission rates of photons, which are usually determined by the dominant singlet recombination rate rsns, one might expect that rs should have a dominant role in determining the frequency dependence pattern. In general, this is not the case though: other rate coefficients, especially spin mixing rates, can be most significant for the behavior of an cwODMR signal. Fig. 2.7 shows one of the most frequently observed examples of the frequency dependence patterns influenced by both rs and α. When α is small, an increasing rs changes little in the observed frequency dependence (Fig. 2.7 (a) and (b)). The most significant effect is a shift of the frequencies where both the in-phase and the out-of-phase components show their maximum rate changes. This is due to the the increase of the time constants, m􀀀1 ij , from m1j ∼ 104 and m2j ∼ 106 to m1j ∼ 106 and m2j ∼ 107, due to very fast rs. It should be noted that dt is 106 in all examples in Fig. 2.7 and rs is 107 in Fig. 2.7 (b) and (d). The frequency dependence also shows little change when rs remains small and α is increased (Fig. 2.7 (c)). This corresponds to Fig. 2.5. However, when α becomes fast enough to compete with the slower time constant, m􀀀1 1j , or even faster than m􀀀1 2j , and rs becomes faster than any dissociation rate coefficients, a more complicated frequency dependence emerges. The in-phase signal does not show the simple behavior as it has a local extremum. The out-of-phase signal does not only show the local extremum (as in the simple pattern) but also a zero-crossing point, due to a sign change (Fig. 2.7 (d)). This pattern corresponds to Fig. 2.5 (b) and (d). It should be noted that the intersystem-crossing rate, kISC, has been assumed to be small to investigate the influence of α, and this pattern also appears when kISC becomes large with a small α. This aspect will be explained further in the following section. Note that for cwODMR experiments this pattern appears only when rs becomes faster than any dissociation rate coefficient and α or kISC is fast too. It can also be seen only for cwEDMR experiment when the dissociation rate coefficients and α or kISC are fast (not shown here). We can thus infer that the effect 26 of dissociation in cwEDMR is very similar to that of recombination in cwODMR. 2.3.5 The influence of intersystem-crossing on cwODMR experiments Because the intersystem-crossing rate, kISC, also represents a spin mixing processes, it acts in a similar way as α even though kISC is always on, in contrast to α which turns on and off periodically. To investigate the influence of kISC, α is assumed to be small in this section. When kISC is slow, very little change of the frequency dependence as a function of rs is seen, similar to the behavior described in the previous section. In contrast to the case of a large α and small rs, a major change in the frequency dependence pattern can be seen at fast kISC and slow rs (Fig. 2.8 (c)). A second local extremum appears in the out-of-phase component and a small bump at high frequency in the in-phase component. When both kISC and rs increase enough to compete with each other, a new pattern appears (Fig. 2.8 (d)). Note that this pattern is similar to Fig. 2.7 (d). But they become similar to the pattern in Fig. 2.5 (e) and (f) when Vin;lf → 0 at small α (eq. (2.47)). 2.3.6 Pair generation Due to spin-selection rules, optically generated electron-hole pairs are formed in singlet states and remain in this configuration unless strong spin-orbit coupling is present [82]. Thus, we can assume Gs ≫ Gt. Figure 2.5 (a) corresponds to this case in which the in-phase and the out-of-phase components are always negative and positive, respectively. This case represents the frequency dependence of photoluminescence detected ODMR (PLDMR). Used parameters are rs = 106, rt = 10􀀀2, ds = 102, dt = 104, kISC = 1, α = 10􀀀3, ρ = 0.75, Gs = 1024, Gt = 1020. In contrast to optical generation, spin configuration of electron-hole pairs formed electrically, i.e., via electrical injection, is determined by spin statistics and we can assume 3Gs ≈ Gt. All parameters in Fig. 2.5 (a) and (c) are the same except that 3Gs = Gt = 1020 in Fig. 2.5 (c). We can see from these calculations that electroluminescence detected ODMR (also called ELDMR) can show the opposite sign compared to PLDMR, for very similar underlying physical processes. It should be noted that this inversion could be found only for certain parameter sets, and this inversion can also happen when 3Gs ̸= Gt but Gs > Gt. For example, the sign of the in-phase component also becomes positive (not shown here) if every parameter remains the same except for Gs = 10×Gt. Thus, cwODMR can result in a positive in-phase and negative out-of phase signal even though Gt can be orders of magnitude but not many orders of magnitude smaller than Gs. This is because 27 the sign inversion is also determined by rate coefficients and not just the generation rates. These cases will be discussed in Section 2.6. 2.4 Power dependence The spin flip rate coefficient, α, is proportional to the applied microwave power [71]. Thus we can calculate the power dependence of cwODMR signals. Examples are shown in Fig. 2.9. For low modulation frequencies, (see Fig. 2.9 (a)), a simple saturation behavior is predicted by eqs. 2.47 and 2.48. Note that the out-of-phase is not always zero, but approaches zero at low frequencies, as expected from eq. 2.48. The saturation characteristics become more complicated as the modulation frequency increases. At 104 Hz, the in-phase component shows a local extremum before it returns to a saturation value (Fig. 2.9 (b)). Similar behavior has been reported recently for low magnetic field cwEDMR on crystalline silicon interface defects [16]. At high modulation frequency, the in-phase component shows the usual saturation behavior (even though its saturation occurs at much higher power) but the out-of-phase component shows a local extremum before it approaches a saturation value. It also has a different sign than at lower frequencies (Fig. 2.9 (c)). This shows that one can find opposite signs of in-phase and out-of-phase signals at high MW power and high MW modulation frequencies. 2.5 Signal sign dependencies on the modulation frequency Sign changes of cwEDMR and cwODMR signal have been found in InP nanoparticles [19] and organic semiconductors [38, 45]. The sign change of cwODMR response in organic semiconductor has been attributed to the imbalance between changes in the numbers of singlet and triplet pairs when the pulse is on and off, which are equivalent to ns1(T/2) − ns2(T) and nt1(T/2) − nt2(T) in our model. The zero-crossing point of the modulation frequency dependence function has also been used to estimate the intersystem-crossing time [38, 45]. According to those reports, the zero-crossing can appear at a certain frequency where the increase in the number of singlet pairs is matched with the decrease of the number of triplet pairs so that the change in the total number of pairs is zero. However, we show here that the zero-crossing can be due to not only the imbalance of changes between singlet and triplet pairs but also to other more complicated relationships betwen physical parameters. As can be seen in the solutions of the rate equations given above, the frequency de-pendence is not simply obtained from ns1(T/2) − ns2(T) and nt1(T/2) − nt2(T), but has a complicated dependence on various parameters. Among the quantitative models tested 28 here , zero-crossing behavior is rarely seen. Fig. 2.10 shows one example: no zero-crossing is observed for small rt, but when rt becomes larger and very close to rs, zero-crossing is observed (Fig. 2.10 (a), (b)). It should be noted that the origin of this zero-crossing is not obvious because of the complexity of the solutions, although we note that ns1(T/2)−ns2(T) and nt1(T/2) − nt2(T) do not meet each other at the zero-crossing point in this case, in contrast to the model described elsewhere [38, 45]. Thus the imbalance between changes in ns and nt cannot be the reason for the observed zero-crossing. We note that zero-crossing also can appear due to an overlap of two different spin-dependent recombination mechanisms whose signs are opposite (e.g., in cwODMR of a radiative and a nonradiative channel). Note however that all zero-crossing effects demonstrated here are obtained from a single recombination process. The existence of zero-crossing indicates that one can observe different signs of cwODMR and cwEDMR signals from the identical sample at different modulation frequencies. 2.6 The interpretation of cwEDMR and cwODMR signal signs The signs of the cwEDMR and cwODMR signals have long been considered important indicators for the natures of electronic transitions. For example, it has been generally accepted that radiative recombination results in positive ODMR in-phase signal [5, 7, 17]. However, the recent observations of sign changes [19, 38, 45] at certain frequencies suggest that signs may depend on complicated processes and the interpretation based soley on the sign of a modulated cwODMR or cwEDMR signal is not possible. CwEDMR and cwODMR signal signs are determined by the transient responses of optical or electrical observables to a repeated change between on- and off-resonance, as described in Section 2.3. Because the time constants and prefactors of the double expo-nential functions in eq. (2.8), (2.9), (2.10), and (2.11) are functions of all the transition rate coefficients, there are many scenarios which can produce quenching and enhancement signals for both radiative and nonradiative ODMR signals as well as for EDMR signals. Many transitions are competing with each other. For instance, recombination as well as dissociation are pair annihilation processes but only recombination causes PL while dissociation does not. Thus when a radiative recombination process is slow but dissociation is faster the resonant response may lead to quenching. This example shows that the following qualitative description of the sign of cwODMR signals is important. The study of the cwODMR signal the sign change as functions of all individual parame-ters is beyond the scope of this work. Instead, only the low modulation frequency behavior 29 will be discussed. This is a reasonable restriction because the sign does not change as long as there is no zero crossing. The solution for the in-phase cwODMR signal at low modulation frequency is given in eq. (2.47). A quantitative analysis has been done by calculating Vin;lf while changing some parameters, for an example shown in Fig. 2.11 for which it is assumed that both singlet and triplet dissociation probabilities are not distinguishable, two mixing rate coefficients, kISC and α, are slower than any other recombination and dissociation, and total generation rate, Gs +Gt is fixed to 1016, rt to 1, and ρ to 0.75. Fig. 2.11 (a) shows the zero frequency in-phase cwODMR signal, Vin;lf , as a function of the relative ratio of the triplet generation rate to the singlet generation rate, Gt/Gs, and the ratio of the dissociation rate coefficient to the singlet recombination rate coefficient which is fixed to rs = 104. Color reflects the normalized intensity of Vin;lf . It should be noted that positive and negative values are intentionally placed in different scales to make them clearly distinguishable. One can find two noticeable features. (i) The intensity tends to increase as Gt/Gs becomes larger and becomes negative at low Gt as in Fig. 2.11 (b). (ii) The intensity also depends on the dissociation rate coefficients: when d is larger or smaller than the singlet recombination rate coefficient rs, Vin;lf becomes very small, and shows an extremum and sign change. Fig. 2.11 (a), (b), and (c) show that the signs are positive at high triplet generation rates and low dissociation rates or, equivalently, high recombination rates. When dissociation is not fast, signs are positive as long as triplet generation is not too much slower than singlet generation rate. This means that changing the pair generation method between optical and electrical methods can induce a sign change in cwODMR. This behavior can be more easily understood by means of competing singlet and triplet pairs. In Fig. 2.11 (d) and (e), the differences n0s 1 − n0s 2 and n0 t1 − n0 t2, are calculated and plotted as the same parameters as (a). Note that the low-frequency solution for the in-phase cwODMR signal, Vin;lf , is proportional to rt(n0s 1 − n0s 2) + rs(n0 t1 − n0 t2). Both plots show different behavior compared to Vin;lf but the boundaries dividing positive and negative values are very similar. When the pair annihilation is dominated only by singlet recombination process, one can infer that the number of singlet pairs quickly decreases in the steady-state offresonance condition. Thus, the steady-state is dominated by triplet pairs. Consequently a resonant MW converts triplet pairs to singlet pairs, it increases the number of singlet pairs which results in an enhancement of cwODMR signal. This qualitative pictures applies to the region where n0s 1−n0s 2 is positive and n0 t1−n0 T2 is negative, in the upper left regions in Fig. 2.11 (a), (d), and (e) for example. In contrast, if 30 the triplet generation is too low (Gt < rt+dt rs+ds Gs), (lower-left corners in Fig. 2.11 (a), (d), and (e)), only a small number of triplet pairs forms during the off resonance steady-state, and the steady-state at offresonance is dominated by singlet pairs. In this case spin resonance induced changes to the number of singlet pairs can become negative. The statements above are based on the assumption of low kISC and α. When kISC becomes larger than the other rates, patterns of Vin;lf (not shown here) similar to the pattern in Fig. 2.11 are found, but the slight shifts of boundaries dividing positive and negative can exist as well. Similar shifts have been found at different ρ. Consequently, cwODMR and cwEDMR signs also depend on intersystem-crossing rate kISC and the temperature (Note that ρ is a function of temperature). We could not identify a shift of boundaries due to a change of α but this does not exclude the possible case that α can also cause sign changes. We note again that sign changes can also occur at a certain modulation frequency as already explained above. Therefore, we conclude that dissociation, recombination, relative ratio between singlet and triplet generation, intersystem-crossing, temperature, and modulation frequency are the factors which all can change the sign of cwODMR signals. Finally, we want to address the question of whether radiative and nonradiative recombi-nation results in opposite cwODMR signal signs. We have checked a number of quantitative models and two examples are shown in Fig. 2.12. In contrast to all other cases discussed above, the nonradiative singlet recombination coefficients, rs;nr is taken into account. In Fig. 2.12 (a) and (c), rs;nr is assumed to be smaller than rs to simulate the modulation frequency dependence in which radiative recombination is dominant. In Fig. 2.12 (b) and (d), rs;nr is assumed to be the larger than rs to investigate the nonradiative process. It should be mentioned again that rs;nr contributes to the pair annihilation process but it does not contribute to the radiative emission rate term as explained in Section 2.1.1. Note that Fig. 2.12 (a) shows one of the modulation frequency dependence patterns that are discussed above. The in-phase signal is negative even though rs is most dominant because Gs ≫ Gt. Fig. 2.12 (b) shows a zero-crossing behavior. Thus, the in-phase component can be positive and negative even though rs;nr is dominant. In contrast to the cwODMR cases, the signs of the cwEDMR in-phase signals are positive in both cases as shown in (c) and (d). To summarize, our results show that cwODMR signals can be negative and positive for both, radiative and nonradiative recombination processes. Any conclusion about the nature of a spin-dependent recombination process from the sign of an observed cwODMR signal is therefore prohibitive. 31 2.7 Summary and conclusion A set of rate equations based on an intermediate pair recombination model are presented and generalized analytical solutions have been obtained. These solutions have been used to calculate modulation frequency dependencies of cwEDMR and cwODMR signals. It has then been investigated how experimental parameters affect these modulation frequency dependencies which revealed that a large number of quantitatively different models show nondistinguishable modulation frequency dependence patterns. This implies that the in-terpretation of cwODMR and cwEDMR experiments can be very ambiguous. It is further shown that signs of cwODMR and cwEDMR signals depend on most rate coefficients as well as experimental parameters such as temperature and modulation frequency. Thus, there are many variables which can reverse the sign of cwEDMR and cwODMR signals and consequently, conclusions about the radiative or nonradiative nature of an observed spin-dependent transition solely based on the sign of an observed spin-dependent process or its modulation frequency dependence is not possible. 32 B0 Square modulated microwave (B1) PL MW source Microwave modulator Lock-in amplifier Transient Inphase Outphase Optical excitation sample Figure 2.1. Sketch of a setup of cwODMR. The basic principle of cwODMR is the same as that of conventional ESR. Square microwave modulation can be used instead of continuous B0 field modulation and a lock-in amplifier is employed to increase the signal-to-noise ratio. 33 rs Singlet ds rt Triplet dt energy Intermediate spin pairs free charge carrier state kISC a Spin mixing Gt Gs ns nt Figure 2.2. The intermediate pair recombination model (KSM) as relevant for cwODMR and cwEDMR. Triplet and singlet pairs are formed with two constant generation rates Gt and Gs, respectively. Those pairs can dissociate into free charge carrier states with certain probabilities dt and ds (dissociation rates) or can recombine to excitonic state with recombination rates rt and rs. A spin mixing process can be introduced by ESR externally and this rate is described by α. Another spin mixing process, intersystem-crossing process is described by kISC. Note that nt and ns represent triplet and singlet pair densities, respectively. They do no necessarily correspond to eigenstate densities. , ....... -- ........ , I , a k '*' , , ' .... _--, ~ J/f , ....... -- ........ , I, ,, ........ _--, 34 Figure 2.3. A time transient calculated from a numerical model described by a combination of parameters as rs = 104, rt = 100, ds = 102, dt = 106, kISC = 10􀀀2, α = 105, ρ = 0.75, Gs = 1023, and Gt = 1020. The dash-dotted curve shows the overall response obtained from eqs. 2.34 and 2.35. The blue solid and red dashed curves are the in-phase and the out-of-phase components described by Is1 sin( 2 T t) and Ic1 cos( 2 T t), respectively. See detail in text. -..C.-.I,) c ::::s ..c• ~ -co ..>.-., CI) .cc..u, 0 .c-modulated MW I I off I on I 44 _. - PL • • In-phase • \ -----------------------L ----\ _ -_---o-uF6f--pnase- -/-- \ ,. \ ,. o • • • • , I \ I • ..... ., • • ......-• 100 time (IlS) -' 200 35 Figure 2.4. Three different quantitative models result in indistinguishable frequency dependencies. Each quantitative model is determined by a different set of parameters. Refer to Table 2.1 for all used values. -11\ .t:: c j 0 ..r..i -nI (a) '\ I \ I , _______t ! ______ "_ .~ .. __ _I -11\ .t:: c j 0 ..r..i -nI (b) '\ ,I \ , ------ _'!_-----"- .~.. - --I -11\ .t:: c j 0 ..r..i -nI (c) '\ ,I \ , ----... - ~------ ,- ~"---I 10-1 101 103 105 107 10-1 101 103 105 107 10-1 101 103 105 107 modulation freq. (Hz) modulation freq. (Hz) modulation freq. (Hz) In-phase - - - _. Out-of-phase 36 Figure 2.5. Seven distinguishable patterns of the modulation frequency dependence of cwODMR have been found out of almost a thousand quantitative models. (b), (d), and (f) are equivalent with (a), (c), and (e), respectively, but with opposite signs. Note that the parameters used for these data are listed in Table 2.1. v..;.- a) v..;.- "2 "2 0 -... -.. -----...... -...... ~ ::J ::J .J:i 0 .J:i , ... ... ~ ~ 11\ 11\ Q) Q) ".;::: ~ "in "in s::: s::: .Q..) .Q..) s::: 10-' 10' 10' 105 107 s::: 10-' 10' 10' 105 107 Modulation freq" (Hz) Modulation freq. (Hz) v..;.- C) v...;.- "2 "2 ::J ::J .J..:.i .J..:.i , ~ ~ , 11\ 0 11\ , Q) , , Q) ".;::: , , ~ 0 _ _._ .... _ __ ..... _ ......t I!. , "in " "in \ , s::: s::: ... .Q..) .Q...) s::: 10-' 10' 10' 105 107 s::: 10-' 10' 10' 105 107 Modulation freq. (Hz) Modulation freq. (Hz) v;- " .... v...;.- (f) "2 , "2 ::J , ::J .J:i 0 ... .J..:.i ~ ~ , III III 0 Q) Q) , ~ ~ "in "in s::: s::: .Q...) .Q...) .E 10-' 10' 10' 105 107 .E 10-' 10' 103 105 107 Modulation freq. (Hz) Modulation freq. (Hz) v;- ... "~ , , s::: , , -_ ...... ::J 0 _____ f//II!_ .J..:.i ~ In-phase III Q) ~ "si:n:: - -Out-af-phase .Q..) .E 10-' 10' 10' 105 107 Modulation freq" (Hz) 37 Figure 2.6. Calculated transient behaviors at different modulation frequencies. Black dash-dot line is overall response and blue solid line and red dashed line are in-phase and out-of phase components of it. Parameters are the same as those in Fig. 2.4 (a). The three graphs are normalized by the same scaling factor. Thus the relative intensities among three graphs can be compared. 38 Figure 2.7. Role of the singlet recombination rate, rs. When rs is small, no significant change in the frequency dependence pattern is found when α is increased (from (a) to (c)). But for large rs, a pattern change is observed when α is increased (from (b) to (d)). All four quantitative models have same combinations of parameters but (a) rs = 102, α = 10􀀀3, (b) rs = 107, α = 10􀀀3, (c) rs = 102, α = 108, and (d) rs = 107, α = 108. Values for the other parameters used for these data are listed in Table 2.1. (j 0.0 C Vl rn Q) ..... U C Increasing r s '.i.i.i. ' 'c ::J ..c.. ~ .>... 'iii c: .Q...I c: ~ ,t':": 0 (a) , ," , - ---~ ... "! ~ - - - - .'.._ . ...... -------1 r =102 a=10·3 10 1 modulation freq, (Hz) I I (C) ,- .. § 0- , .. - ---~- ~ ~ ------:-~-~-------~ ..c.. ~ ~ 'iii c: .Q...I ,5 . 10 1 I modulation freq, (Hz) In-phase '.i.i.i. ' 'c 0 ::J ..c.. ~ .>... 'iii c: .Q...I ,5 ~ .'.".. 'c::J 0 ..c.. ~ ~ 'iii c: .Q...I ,5 • I (b) , .. , , ' ' .. - - --- - - - - ----"-': - - - - """';:---1 modulation freq, (Hz) (d) ,- .. - --------- -----.-, - - - -,-, - - -"-- .. , ,I " modulation freq, (Hz) Out-af-phase 39 Figure 2.8. Role of the intersystem-crossing rate, kISC. At small rs, it has been observed that there appear bumps on both in-phase and out-of-phase signal at high frequency region when kISC becomes large (from (a) to (c)). At large rs, different pattern change also has been found. The in-phase shows local extrema and out-of-phase shows change of sign as kISC being increased (from (b) to (d)). All four quantitative models have same combinations of parameters but (a) rs = 102, kISC = 10􀀀2, (b) rs = 107, kISC = 10􀀀2, (c) rs = 102, kISC = 108, (d) rs = 107, kISC = 108. The other parameter values used for these data are listed in Table 2.1. 40 Figure 2.9. MW power dependence. All four quantitative models have the same combi-nations of parameters but (a) f = 103, (b) f = 104, (c) f = 107. At low modulation frequencies, typical saturation curves can be found. At high modulation frequency, a nontrivial saturation behavior occurs. Refer to Table 2.1 for the values used for the other parameters. '.i..i.i '0 '2 :::I ..c.. ~ .>... 'iii s: .Q....J ,5 (a) ... " ... ... _-- f=1o' Hz 10 2 microwave power (arb, units) '.i..i. i'0 'c :::I ..c.. ~ .>... 'iii s: .Q....J s: Increasing frequency (b) , , , , f=lO' Hz 10 2 10° 1d 10' microwave power (arb, units) • (c) , .... --, '.i.i.i. ' '2 :::I ..c.. ~O .>... 'iii s: .Q....J s: 10' \ \ ... , ,, ------- f=107 Hz 10' 107 microwave power (arb, units) lOlD In-phase - - - - Out-af-phase 41 Figure 2.10. Example of a modulation frequency dependence function showing a change from nonzero-crossing pattern to a zero-crossing pattern. The only difference between the two quantitative models can be found in the triplet recombination rate coefficients. (a) rt = 100, (b) rt = 106. Values for the other parameters are listed in Table 2.1. -I/) .~ C => 0 ..c.. -<C > .~ I/) C CU +J C (a) I \ I \ ~ , . . _ ... _ .. ,_ .... _ .. _. -... -... ~ , .. r =106 r =100 t 102 104 106 108 Modulation freq. (Hz) Inphase -I/) .~ C => ..c.. -<C > .~ I/) C CU +J C (b) 0 ... -.-.-... -.,-, .. -... -.. ,.~ .... ~ ... , .... " I , I , , Modulation freq. (Hz) - - - - - Outphase 42 Figure 2.11. Sign changes due to various rate coefficients. (a) In-phase intensities of the zero modulation frequency component as a function of Gt/Gs and d/rs. To distinguish positive values and negative values, different color scales are used (positive in upper left corner, and negative in lower right corner). The black dotted line describes the boundary separating positive values and negative values. (b) and (c) are two randomly chosen two dimensional subsets of the data in (a) representing a generation rate ratio slice and dissociation rate ratio dependencies. These slices are shown as white dashed lines in (a). Intensities in (a), (b), and (c) are normalized but in the same scale. (d) Changes in the numbers of singlet pairs, n0s 1 − n0s 2 as a function of the same parameters as in (a). (e) Changes in the number of triplets pairs, n0t 1−n0t 2 as a function of the same parameters as in (a). Intensities in (d) and (e) are normalized but in the same scale. All calculations in this figure are obtained from the same condition of rs = 104, rt = 1, kISC = 1, α = 1, ρ = 0.75, Gs + Gt = 1016. (a) V (arb. units) in,lf ,:,:,:,:2d 10-6 +-""T""""'" 10-6 10-4 10-2 10° d/r 5 (d) nO _no (arb. units) 51 52 (e) nO _no (arb. units) t1 t2 SxlO" -3xl013 (!) '" 10-2 0 ""'... 0 (!) 10-4 10-6 -lxlO· 10-6 4xlO9 10-6 10-4 10-2 10° 102 10-6 10-4 10-2 10° 102 d/r 5 d/r 5 43 Figure 2.12. The sign of cwODMR signals can be negative when radiative recombination is dominant as in (a), and positive when nonradiative recombination is dominant as in (b). In contrast the signs of cwEDMR are not different, (c) and (d). Used common values for each rate parameters can be found in Table 2.1. (a) and (c) rs = 104, rs;nr = 1. (b) and (d) rs = 1, rs;nr = 104. r >d >d >r >r=a>k r >d >d >r >r =a>k 5 t 5 snr t SL snr t s s t SL ',ii~i" . (a.) -,I~II (b) c , ... I \ C :::::I , , \ ... .. :::::I ODMR ,g 0 --- --- - - ~-.. ... - - - -- ,.g.. (non-radiative) ~ ODMR (radiative) ~ .>... ~ 0 'iii 'iii ... \ c C \ I <II <II .... .... \ " C 1 1 C 10 2 100 102 104 106 10 2 100 1d 104 106 modulation freq, (Hz) modulation freq, (Hz) 'iii" (c) 'iii" ,~ ,~ (d) c C :::::I :::::I EDMR ,.g.. EDMR (radiative) ,.g.. (non-radiative) ~ ~ > 0 > 0 ,~ -------.. - .;: - - - - - - ,.,. ,~ -------..,.- .;: ------ .. ... , ... , III ... , III \ , C \ I C \ , .<..I.I ... ' .<..I.I " c C 10-2 100 102 104 106 10-2 100 1d 104 106 modulation freq, (Hz) modulation freq, (Hz) In-phase ------ Out-af-phase 44 Table 2.1. Parameters used for calculation of all plotted data in this chapter. All values have a unit of s􀀀1 except for ρ which is arbitrary. rs rs;nr rt ds dt kISC α ρ Gs Gt f 2.3 104 0 1 102 106 10􀀀2 105 0.75 1023 1020 - 2.4 (a) 102 0 1 104 106 10􀀀2 103 0.75 1023 1020 - (b) 104 0 10􀀀1 10 102 10􀀀2 10􀀀1 0.75 1025 1020 - (c) 104 0 1 102 106 10􀀀2 103 0.75 1020/3 1020 - 2.5 (a) 104 0 1 102 103 10􀀀2 10􀀀3 0.75 1024 1020 - (b) 106 0 1 102 104 10􀀀2 107 0.75 1022 1020 - (c) 104 0 1 102 103 10􀀀2 10􀀀3 0.75 1020/3 1020 - (d) 106 0 1 102 104 10􀀀2 107 0.75 1020/3 1020 - (e) 106 0 104 1 102 104 10􀀀3 0.75 1024 1020 - (f) 106 0 104 1 102 104 10􀀀3 0.75 1020/3 1020 - (g) 1 0 10􀀀1 102 104 106 10􀀀3 0.75 1020/3 1020 - 2.7 (a) 102 0 1 104 106 10􀀀2 10􀀀3 0.75 1022 1020 - (b) 107 0 1 104 106 10􀀀2 10􀀀3 0.75 1022 1020 - (c) 102 0 1 104 106 10􀀀2 108 0.75 1022 1020 - (d) 107 0 1 104 106 10􀀀2 108 0.75 1022 1020 - 2.8 (a) 102 0 1 104 106 102 101 0.75 1022 1020 - (b) 107 0 1 104 106 10􀀀2 101 0.75 1022 1020 - (c) 102 0 1 104 106 108 101 0.75 1022 1020 - (d) 107 0 1 104 106 108 101 0.75 1022 1020 - 2.9 (a) 106 0 1 102 104 10􀀀2 - 0.75 1022 1020 103 (b) 106 0 1 102 104 10􀀀2 - 0.75 1022 1020 104 (c) 106 0 1 102 104 10􀀀2 - 0.75 1022 1020 107 2.10 (a) 106 0 1 102 104 10􀀀2 101 0.75 1022 1020 - (b) 106 0 106 102 104 10􀀀2 101 0.75 1022 1020 - 2.11 104 0 1 - - 1 1 0.75 Gs + Gt = 1016 - 2.12 (a) 104 1 10􀀀1 10 102 10􀀀2 10􀀀1 0.75 1025 1020 - (b) 104 1 10􀀀1 10 102 10􀀀2 10􀀀1 0.75 1025 1020 - (c) 1 104 10􀀀1 10 102 10􀀀2 10􀀀1 0.75 1025 1020 - (d) 1 104 10􀀀1 10 102 10􀀀2 10􀀀1 0.75 1025 1020 - 45 2.8 References [1] S. 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