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December 5, 2017 19:13 Spin in Organics (Vol. 3) (in 4 Volumes) - 9in x 6in b2966-v3-fm page viii

viii Spin in Organics — Volume 3

2.2. Organic Magnetoresistance in small moleculedevices . . . . . . . . . . . . . . . . . . . . . . . . . 109

2.3. Universality of the OMAR effect . . . . . . . . . 1093. Simplistic Derivation of Lorentzian Lineshape . . . . 1114. The Effect of Traps . . . . . . . . . . . . . . . . . . . . . 1145. Exciplex-Based OMAR Devices . . . . . . . . . . . . . 119

5.1. Theoretical model and discussion . . . . . . . . 1286. Fringe-Field Magnetic-Field-Effect Devices . . . . . . 130

6.1. Non-volatile effects using remanent fields . . . 1327. Summary, Conclusion and Outlook . . . . . . . . . . . 134References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

4. Room-Temperature Quantum Coherence inEmission from Organic Semiconductors 143

Nicholas J. Harmon and Michael E. Flatte

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1431.1. Optical excitation and luminescence in

organic semiconductors . . . . . . . . . . . . . . . 1451.2. Exciplex excitation and luminescence

in donor-acceptor blends of organicsemiconductors . . . . . . . . . . . . . . . . . . . . 149

1.3. Spin mixing in emission from organicsemiconductors . . . . . . . . . . . . . . . . . . . . 151

2. Thermally-Activated Delayed Fluorescence(TADF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1522.1. Orbital states and TADF . . . . . . . . . . . . . 1532.2. A simple model of TADF . . . . . . . . . . . . . 1542.3. Comparison with electron spin relaxation

rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 1583. Theory of Organic Magneto-

Electroluminescence . . . . . . . . . . . . . . . . . . . . . 1583.1. The δg mechanism . . . . . . . . . . . . . . . . . . 161

3.1.1. The exciton picture . . . . . . . . . . . . 1643.1.2. The exciplex picture . . . . . . . . . . . . 165

December 5, 2017 19:13 Spin in Organics (Vol. 3) (in 4 Volumes) - 9in x 6in b2966-v3-fm page ix

Contents ix

3.2. Magnetoelectroluminescence from δg andhyperfine interactions . . . . . . . . . . . . . . . . 166

3.3. Magneto-photoluminescence . . . . . . . . . . . . 1673.4. Exciplex dynamics with separate T and

T ∗ levels . . . . . . . . . . . . . . . . . . . . . . . . 1684. Spin Decoherence . . . . . . . . . . . . . . . . . . . . . . 1715. Experimental Measurements of MEL in Exciplex

Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1736. Fringe Field Effects in TADF Materials . . . . . . . . 1777. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 182References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

5. Magnetic-Field Effects in Organic Transistors 189

Tobat P. I. Saragi

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1892. Photoinduced Magnetoresistance in Acenes-Based

Field-Effect Transistors . . . . . . . . . . . . . . . . . . . 1923. The Origin of Magnetoresistance in Unipolar

Organic Transistors . . . . . . . . . . . . . . . . . . . . . 1994. Sign Change of Magnetoresistance in Organic

Transistors . . . . . . . . . . . . . . . . . . . . . . . . . . . 2075. Magnetoresistance in Mixed System and

Ultra-Small Magnetic Field Effect . . . . . . . . . . . . 2126. Summary and Outlook . . . . . . . . . . . . . . . . . . . 216Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . 217References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

6. Photophysics of Thermally ActivatedDelayed Fluorescence in Organic Molecules 227

Fernando B. Dias, Thomas J. Penfold,Mario N. Berberan-Santos and Andrew P. Monkman

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2282. Fundamental Observations in TADF

Photophysics . . . . . . . . . . . . . . . . . . . . . . . . . 2313. Rate of Reverse Intersystem Crossing . . . . . . . . . 242

December 5, 2017 19:13 Spin in Organics (Vol. 3) (in 4 Volumes) - 9in x 6in b2966-v3-ch04 page 143

CHAPTER 4

Room-Temperature Quantum Coherencein Emission from Organic Semiconductors

Nicholas J. Harmon∗ and Michael E. Flatte†

University of Iowa, USA∗[email protected]†michael [email protected]

The role of quantum coherence in the excited states that produce lightemission from organic semiconductors is reviewed. A general theoreticalframework, applicable to emission from excitons as well as to emis-sion from charge-transfer excitations/exciplexes is described. Analyticsolutions exist for certain special cases, and numerical evaluation ispossible for general situations. Connections to recent observations ofvery large room-temperature magnetoelectroluminescence in materialscharacterized by thermally-activated delayed fluorescence is discussed.

1. Introduction

Quantum coherence plays a central role in dynamical properties, withconsequences for both the fundamental understanding of dynamicsas well as for future quantum technologies.1,2 Although frequentlythe evolution of quantum coherence follows a simple path (suchas in Fermi’s golden rule calculations of transition rates), whosesimple form allows coherent effects to be neglected or trivialized,our focus here will be on the exceptions when coherent effectspersist beyond the timescales for various other dynamical eventsand effects. Spin coherence forms a prominent subcategory withinthis area of quantum coherence,3,4 as spin coherence can persist to

143


144 Spin in Organics — Volume 3

room temperature and above, providing the mechanism for practicalspin resonance sensing technologies, including magnetic resonanceimaging. Spin resonance techniques are also applied to diagnose theorigin of small numbers of trap states in electronic devices,5–8 relyingon studies of the coherent response of various types of defects inan insulating barrier to radio-frequency electromagnetic waves. Spinmanipulation and coherence also offer prospects for reducing theenergy consumption of charge-based electrical devices9,10 through themanipulation of coherence times and spin filtering. There are evencompelling proposals for biological implications of spin coherencein the dynamics of chemical reactions including as a techniqueto sense small magnetic fields by migrating birds.11,12 Connectedthrough quantum coherence, a similar set of fundamental conceptsand theories underlies the sensitivity of transport, luminescence,and chemical reactions, at room-temperature to extremely smallmagnetic fields, whose Zeeman energies are orders of magnitudesmaller than the thermal energy.

Some general considerations motivate our focus on quantumcoherence in the optical emission from organic semiconductors,caused by spin-spin interactions. The materials considered arenonmagnetic, so the interacting spins are in general randomlyoriented and interact primarily in pairs. In order for such a pairwiseinteraction to influence phenomena at room temperature, the typicalexchange and orbital energy scales associated with these spins haveto be much larger than the thermal energy of ∼25 meV, so a typicalenergy scale for exchange and orbital energies will be ∼1 eV. In thisChapter substantial magnetic field effects (MFE’s) are found for verysmall external fields (i.e. gµBB ∼ 1µeV, where g is the Lande gfactor, µB is the Bohr magneton, and B is the applied magnetic field).Thus the energies associated with these small external fields are muchsmaller than the exchange or orbital energy scales. This ordering ofenergy scales immediately focuses our attention on systems in anincoherent hopping transport regime. For external fields to switchoff band transport, the energies associated with them would typicallyneed to exceed the energy width of the band, which itself must bemuch larger than the thermal energy. Thus significant effects of small


Room-Temperature Quantum Coherence in Emission 145

external fields on band transport should not be expected, and thefocus here is on systems exhibiting incoherent hopping transport.Many systems which have large exchange and orbital excitationenergies, have incoherent hopping transport, and are thus sensitiveto very small external fields at room temperature, are disorderedorganic and molecular materials. Other systems with similar behaviorare localized states such as defects in an insulating barrier weaklycoupled to their surroundings. Here we focus on disordered organicsemiconductors. Although the emphasis here is on the phenomenon ofluminescence, much of the analysis and theoretical dynamics can beapplied to similar phenomena in conductivity and chemical reactions.

1.1. Optical excitation and luminescence in organicsemiconductors

The field of photochemistry13 was established over two centuriesago with the Grotthuss-Draper law stating that light absorptionmay lead to chemical reactions. Photochemistry provides the micro-scopic understanding of many vital biological processes, includingphotosynthesis and vision. Chemical reactions that are not possiblethrough thermal routes alone become allowed in photochemistrybecause the much larger optical energy provided (∼eV) overcomesactivation barriers that are inaccessible at practical temperatures(thermal energies <0.1 eV). As a result the typical considerationsof equilibrium and detailed balance that apply to reactants andproducts in a thermally-activated chemical reaction may be modifiedby the near-instant departure of emitted light (luminescence) fromthe sample, leading to highly nonequilibrium reactions that areessentially not reversible.

Optical excitation in organic molecules is directly connected tospin statistics as the ground state of these organic molecules is a spinsinglet and excitation by light typically preserves the spin quantumnumbers of the excited material. Figure 1 compactly describes thebasic nature of light-matter interactions in organic molecules. Onthe left, the ground state of a molecule is shown; two electronsoccupy the highest occupied molecular orbital (HOMO). The Pauliexclusion principle requires their spins to orient antiparallel, which



ground state(singlet) S0

excited state(singlet exciton) S1

LUMO

HOMO

LUMO

HOMO

LUMO

HOMO

excited state(triplet exciton) T1

Fig. 1. Left: ground state of molecule (S0) excited by green light which promotesone electron to the LUMO (center and right). Center: if the transition does notproduce a spin flip then the excited state is a singlet exciton (S1). Right: if thereis a spin flip during the transition or later, then a triplet exciton state (T1) ispossible.

is the singlet state denoted S0. The wavy green line indicates lightabsorption, which kicks one electron to an excited state in the lowestunoccupied molecular orbital (LUMO). This excitation is named anexciton and is classified as an intra-molecular excitation — it existson a single molecule (thus a Frenkel exciton, not a Wannier exciton).Absent a substantial spin-orbit interaction, the molecule does notrespond to light with a preferential excitation of one spin directionover the other, and thus the excited spin orientation is random. Thisexciton can also be described as consisting of an electron and a hole;the “hole” indicates the absence of the electron in the HOMO. Thehole has a positive charge and a spin whose orientation is oppositeto that of the electronic state that has been emptied in the HOMO.As a result the effective spin of the hole is oriented parallel to that ofthe remaining electron residing in the HOMO. The positive chargeof the hole in the HOMO keeps the excited electron in the LUMOattracted to its home molecule. If the excited electron is removed,leaving just a HOMO with a single electron (and thus one hole), themolecule is effectively positively charged, with the positive chargeattributed to the hole.

The most likely electronic transition caused by optical absorptionis an electric-dipole allowed transition, which does not flip the spinof the excited electron, and thus the exciton is usually generatedin a singlet state, denoted S1 for the first excited singlet state.The manifolds of excited singlet and triplet energies, along with theground state, are illustrated in Fig. 2. The excited triplet state, T1,commonly has a lower energy than the excited singlet state, due to



Spatial configuration coordinate

Exchangeenergy

Triplet manifold

Singlet manifold

Ener

gy

(a)

S1

S0

T1

∆ST

(b)

Fig. 2. (a) A schematic dependence of the energy of the first excited singlet (S1)and triplet (T1) exciton states, as a function of a general configuration coordinate.For nearby electron and hole the energy splitting is large, whereas for largerseparations the energies approach each other. (b) Effective energy diagram ofground state (S0) and singlet/triplet excited states. The shaded regions aboveeach excited state energy denote combinations of the first excited exciton stateswith vibrations or variations of the configuration coordinate. ∆ST is the energygap or exchange splitting between the lowest-energy excited singlet and tripletstates.

the exchange energy, ∆ST. Exchange is a many-body effect involvingthe many-electron state, and falls off exponentially with separationbetween the relevant carriers. Here the two involved carriers arethe electron and hole, which are bound in an exciton. For excitonswithin small molecules the exchange energy often exceeds 1 eV. Theelectrons in these molecules are coupled to vibrations of the ionswithin the molecules, denoted phonons, which produces additionalstructure in Fig. 2. The thick edge shows the electronic energieswhen the molecules are in the vibrational ground state. The shadedregion above denotes excited vibrational states combined with thesame discrete electronic excitation, or excited many-body electronicexcitations that are again combined with the same discrete electronicexcitation. The excited vibrational states may also modify theelectronic wave functions to a degree, and thus they can be referredto as vibronic states. From excitation into the S1 band, even if theexcitation energy does not correspond to the lowest-energy state in



the S1 band, the emission emerges in largest yield from the lowest-energy S1 state, which corresponds to no additional vibrationalexcitations (Kasha’s rule).14

The low likelihood of generating T1 excitons is a characteristicof the optical excitation process. Different forms of excitation,however, produce different densities of singlet and triplet excitons.For example, organic electrical devices can generate excited statesby injecting electrons and holes from either side of a a device withrandom spin orientations, and if there are no other spin-dependentprocesses in the resulting dynamics, those electrons and holes endup as triplet excitons three times as often as they end up as singletexcitons, due to the different degeneracy of singlet and triplet spinstates. This effect, and the nature of optical emission from theseexcited states, has a dramatic effect on the properties of organiclight emitting diodes (OLEDs).

The two basic types of luminescence in organic semiconductors,fluorescence and phosphorescence, are intrinsically spin-dependentprocesses and thus may be modified by the effect of appliedmagnetic fields on the spins of nonequilibrium excitations in thematerials. Fluorescence is optical emission from the S1 state tothe ground-state singlet (S0). Fluorescence can occur without aspin flip, and thus is often quite rapid and efficient. Emission fromthe T1 state to S0 requires a spin flip, and thus occurs moreslowly; it is referred to as phosphorescence. The large exchangesplitting found for excitons in small molecules strongly suppressesthermally-mediated transitions from the excited triplet T1 stateto the excited singlet S1 state. Those triplet excitons thus mustrecombine nonradiatively and do not contribute substantially to theelectroluminescence from an OLED.15 This loss of triplet excitonshas been a major motivation to explore additives called phosphors,such as heavy metals, that relax spin conservation in OLEDs.These phosphors increase the OLED emission efficiency, but atan additional expense. An alternate approach is to reduce thesinglet-triplet energy splittings by using organic molecule blends,allowing triplets to convert to singlets through various mechanisms(with the help of thermal activation to overcome ∆ST) in an



approach referred to as thermally-activated delayed fluorescence(TADF).

Aside from excitons, other correlated electron complexes existand influence the optical emission from organic semiconductors.There are polaron pairs, which are electron and holes localized ondistinct and separated molecules (“sites”). Their spatial separationleads to a very small wave function overlap and, therefore, a verysmall exchange energy relative to room temperature thermal ener-gies. Thus the energetic obstacle to S ↔T singlet/triplet transitionsis readily overcome, although the spin selection rule obstacle remains.Considerable research on magnetic field effects (primarily magneto-transport or magneto-luminescence) in organic semiconductors overthe last decade focuses on ‘spin mixing’ in the polaron pair stageand how it affects exciton or bipolaron formation.16–23 The theoryof these phenomena is described in detail in other chapters inthis book. Here we focus on mixtures of molecules with eithera low ionization potential or a high electron affinity, commonlyco-evaporated together; these two types of molecules are knownas donors and acceptors respectively, and the excitations of thesemixtures are known as exciplexes.

1.2. Exciplex excitation and luminescence indonor-acceptor blends of organic semiconductors

In blends of organic molecules the energetic level of the HOMOand the LUMO can be independently controlled. For donor andacceptor molecules positioned near one another, such as during co-evaporation, electron-donor-acceptor complexes form. These com-plexes generally have lower energy than individual excited moleculesbecause the electron and hole can delocalize to an extent across thetwo molecules. When this molecule pair is excited, the complex iscalled an excited state complex or more succinctly an exciplex. Thedelocalization (or resonance) effect reducing the excitation energyoccurs even for two identical molecules, for which the excitation isknown as an excimer.

For large differences in the energies of the HOMO and LUMOsbetween the donor and acceptor molecules, an excited complex is



local exciteddonorD*A

local excitedacceptor

DA*

LUMO

HOMO

ground stateDA

charge transferstateD+A–

(a) (d)(c)(b)

Fig. 3. Four different contributions to the exciplex wave function. (a) groundstate, (b) local excited acceptor (DA*), (c) local excited donor (D*A) and chargetransfer state (D+A−).

likely to be characterized by charge transfer, in which the electronand the hole reside predominately on different molecules. This typeof exciplex is the charge transfer (CT) excitation.24,25 For a charge-transfer excitation the spatial separation between the electron andhole can be independently determined by the donor and acceptormolecular structure, and thus the singlet-triplet exchange energy ∆ST

can be substantially reduced. To visualize this effect the ground statemolecular orbital energies of donors (D) and acceptors (A) are shownside-by-side in Fig. 3(a).

In general the wave function of the exciplex can be written as

Ψexciplex = c1ψ(DA) + c2ψ(DA∗) + c3ψ(D∗A) + c4ψ(D+A−) (1)

where D∗A and DA∗ are complexes of an excited D or A state witha corresponding A or D ground state, DA is the complex with bothD and A in their respective ground states, and D+A− is the chargetransfer excitation. These four different contributions are shown inFig. 3. The ground state is shown in Fig. 3(a) and the excited donorand acceptor states are shown in Fig. 3(bc). Figure 3(d) shows the CTorbital diagram; it is this configuration that gives rise to the magneticfield effects that are highlighted in this chapter. Although the termexciplex applies to Fig. 3(bcd) the lowest-energy exciplex (and theone with large magnetic field effects) is Fig. 3(d), and thus the termexciplex is often used synonymously with the CT complex.26 Justas with excitons, the CT excitations can be either singlet or triplet



and an equivalent energy diagram to that of Fig. 2 can be drawnwith the exception that ∆ST can now be <100 meV for appropriatelychosen D and A constituents. Emission from the CT state of Fig. 3occurs when the extra electron from the A− molecules recombineswith the positive hole on the D+ molecule thus emitting radiation.For convenience, and to be consistent with earlier literature, fromnow on in this chapter we use the term exciplex and CT stateinterchangeably.

1.3. Spin mixing in emission from organicsemiconductors

In addition to radiative transitions (such S1 → S0 and T1 → S0), anunderstanding of photochemistry also requires study of radiationless(or, nonradiative) transitions. Transitions that can occur radiativelycan also occur nonradiatively, and in addition there are new processessuch as Sn → S1, Tn → T1, and S1 ↔ T1. It is these latter transitionsthat are of particular importance when spin statistics is relevantbecause the timescales of recombination for S1 and T1 are vastlydifferent. The term “spin mixing” is generically used for these typesof transitions. More specifically, S1 → T1 is an “intersystem crossing”(ISC) and T1 → S1 is a “reverse intersystem crossing” (RISC),although both are sometimes referred to as intersystem crossings.The fluorescence from materials that allow such spin mixing oftenexhibits two timescales. In “prompt” fluorescence the initially createdsinglets recombine radiatively, whereas spin mixing allows tripletsto convert to singlets and emit via fluorescence at a later time(“delayed” fluorescence).

These intersystem crossings can occur through ordinary thermalprocesses, if the exchange splitting between T1 and S1 are comparableto or smaller than the thermal energy. The spin non-conservingprocess can occur via spin-orbit scattering, inhomogeneous hyperfineinteractions, and other sources of different electron and hole preces-sion frequencies. These different precession frequencies can occur ina uniform magnetic field through differences in the Lande g factors,which also traces back to spin-orbit coupling, or through inhomoge-neous magnetic fields (such as fringe fields from a ferromagnet with



anode

cathode

donorm

olecule

acceptor molecule

acceptor LUMO

donor HOMO

TS ISCRISC

light

Fig. 4. Schematic of OLED with an emissive region consisting of a D−A blend.Electrons are injected from the cathode into the acceptor LUMO and holes fromthe anode into the donor HOMO. Electrons and holes meet in the interior aseither singlet or triplet exciplexes. When the electrons and holes recombine (fromsinglets) light is emitted. Intersystem and reverse intersystem crossings (ISC andRISC) convert singlet and triplets into one another. Motivated by Ref. 27.

inhomogeneous magnetization, commonly coming from magneticdomains). In each of these cases the effect can be referred to asthermally-activated delayed fluorescence, and large magnetic-fieldeffects should be expected in the fluorescence efficiency.

Figure 4 sketches an OLED that is based on emission fromexciplexes. Exciplex emission is enhanced by coevaporating D andA molecules so that they are frequently adjacent to one another.Electrons and holes meet at DA interfaces as part of either thesinglet or triplet manifold. If any spin mixing occurs it does so beforerecombination or dissociation.

2. Thermally-Activated Delayed Fluorescence(TADF)

Investigations of the phenomenon of thermally-activated delayedfluorescence (TADF),24,28–30 which is delayed fluorescence from athermally-activated reverse intersystem crossing, have acceleratedin the past half decade after indications31,32 that the effect might



provide a practical avenue to improve OLED emission efficiency andspeed without requiring heavy metal phosphors. Early phenomeno-logical work on TADF suggested that the intersystem crossingrate (kISC) and reverse intersystem crossing rate (kRISC) wereproportional to their thermal equilibrium populations such that

kRISC =kISC

3e−∆ST/kBT . (2)

An effective second rate of (delayed) fluorescence, kTADF, caused bytriplet states converting to singlet and then recombining is limitedby kRISC:

kTADF =kS

3e−∆ST/kBT , (3)

where kS is the rate of fluorescence.33 The processes leading to kRISC

and kISC must depend on spin if the spin multiplicity is to be changed;the natural mechanism for the conversion process is the spin-orbitinteraction, although hyperfine interactions can also mediate thetransitions. ISC (S1 → T1) typically occurs by spin-orbit couplingto a vibronic T1 level or some other excited Tn level which then maydecay down to T1.24

2.1. Orbital states and TADF

Careful consideration of the nature of the orbital states involved inthe intersystem crossings relevant to TADF is required. For electron-hole separations that are large enough that the exchange energyis very small compared with the intramolecular excitation energies[corresponding to the right-hand side of Fig. 2(a)], then the electronand hole wave functions that enter into the triplet and singletexciplex wave functions are nearly identical. The El-Sayed rules34

dictate that spin-orbit matrix elements do not yield large transitionrates between such singlet and triplet exciplexes. As the electronand hole move together, towards the left-hand side of Fig. 2(a), thenthe exciplex energies for singlet and triplet can differ substantiallyand as a result the electron and hole wave functions are alsomodified. Thus the spin-orbit-induced coupling between the singlet



and triplet exciplexes will increase, producing more rapid intersystemcrossings.

However, as indicated in Fig. 2, there are a range of possibleexcitation characteristics for the electron and hole, as well as for thevibrational excitations of the two molecules. These often mix intovibronic states in which the reduced wave function overlap of singletsand triplets with differing vibrational character is compensated bythe increased overlap of the orbitals (due to a relative displacementof electron and hole wave functions for triplet and singlet excitationsthat breaks the assumptions of the El-Sayed rules). Such consid-erations also motivate the suggestion in some models for TADFthat the intersystem crossings occur through an intermediary localtriplet excitation (which could be on either a donor or acceptor)35

which is vibronically coupled to the triplet exciplex.33 As a muchlarger spin-orbit coupling exists between a local excitation and theintermolecular excited states,36 this provides motivation for largeTADF rates.

In our analysis it is not essential that the relevant triplet statethat spin mixes with the singlet excited state is the T1 state, oreven that the relevant states are exciplex states; the key featurewhich justifies our analysis is that the relevant energy scale forthe intersystem crossing is much smaller than for excitons (andcomparable to the thermal energy), and that TADF is demonstratedin the experimental system explored. TADF itself is only possible ifthe El Sayed rules do not apply to the relevant singlet and tripletexcitations which enter into the intersystem crossings. We note thatthe El Sayed rules do not apply to some other mechanisms ofconverting between singlet and triplet excitations, such as hyperfineinteractions or differences in the Lande g factor, and thus magnetic-field effects on TADF may use different singlet-triplet conversionchannels than zero-magnetic-field TADF.

2.2. A simple model of TADF

We now present a simple model of TADF which will illustratethe main features and identify some subtleties, but without themultitude of additional states that can be required to describe



some experimental situations quantitatively. This analysis also fore-shadows the formalism to be used in later sections. We considerpopulations of singlet and triplet charge transfer states or exciplexeswhich are represented by a spin density matrix ρ described by thefollowing equation

∂ρ

∂t= −1

2{kSPS + kT PT , ρ} + LISCρL

⊤ISC − 1

2{L⊤

ISCLISC, ρ}

+ LRISCρL⊤RISC − 1

2{L⊤

RISCLRISC, ρ}, (4)

where LISC = LS→T =√

kISC|T ⟩⟨S| and LRISC = LT→S =√kRISC|S⟩⟨T | are Lindblad operators that describe the (reverse)

intersystem crossings between singlets and triplets. The curly brack-ets denote anticommutation. The average population of singlets(triplets) are then determined by nS = TrPSρ (nT = TrPT ρ) wherePS and PT are singlet and triplet projection operators, respectively.Using these relations the temporal dynamics appears in a morefamiliar rate equation form:37

d

dt

(nS

nT

)=

(−(kS + kISC)nS + kRISCnT

−(kT + kRISC)nT + kISCnS

). (5)

Studies of TADF frequently use time-dependent measurements ofphotoluminescence; the initial condition for optical excitation is

ρ(0) =

(1 00 0

)

, (6)

(i.e. all initial states are spin singlet). Although the differentialequations above can be solved analytically, the full expressions arenot very transparent and in any case desirable TADF propertiesrequire the parameters to reside in the regime kS ≫ kISC, kRISC, kT,which reduces the complexity of the solutions. The solutions for thetwo populations are then

nS(t) = nS(0)e−(kS+kISC)t +kRISCn∗

T

kSe−(kT+kRISC)t, (7)

nT(t) = n∗Te−(kT+kRISC)t − n∗

Te−(kS+kISC)t, (8)



where n∗T = kISCnS(0)/kS is the population of triplets after the

prompt fluorescence has finished. At sufficiently long times

nS(t > 1/kS) =kRISCn∗

T

kSe−(kT+kRISC)t,

nT(t > 1/kS) = n∗Te−(kT+kRISC)t, (9)

which is the regime of delayed fluorescence. Figure 5 shows thedistinct time dependence for the prompt and delayed fluorescentcomponents for excitation at t = 0.

The fluorescence in the steady state is found by solving

∂ρ

∂t= 0 = −1

2{kSPS + kTPT, ρ} + LISCρL

⊤ISC − 1

2{L⊤

ISCLISC, ρ}

+ LRISCρL⊤RISC − 1

2{L⊤

RISCLRISC, ρ} + G (10)

where G is a generation term which, for photoluminescence, appliesonly to the singlet population. For continuous-wave (cw) excitation,

G =

(GS 00 0

). (11)

0.1 1 10 100 100010–9

10–7

10–5

0.001

0.100

t ( s)

n S(t)

/nS(0

)

µ

Fig. 5. Fluorescence decay in time showing the prompt decay and the delayeddecay using kISC = 0.01 µs−1, kRISC = kISC/10, kS = 0.32 µs−1, and kT =0.027 µs−1. The sharper decay after the delay is due to nonradiative transitionsvia kT . Phosphorescence is not included in the model for emission (only emissionfrom singlets is shown here). The black curve is the exact analytic calculationand the orange dashed curve is an approximate calculation of Eq. (7).



For general rates

nS(t → ∞) = GSkRISC + kT

kS(kRISC + kT) + kTkISC, (12)

nT(t → ∞) = GSkISC

kS(kRISC + kT) + kTkISC. (13)

If kRISC is the smallest rate (kRISC ≪ kT), which is the lowtemperature regime [Eq. (2)], then

nS(t → ∞) =GS

kS + kISC. (14)

We are chiefly concerned with the singlet population, becausethe intensity of light emission depends centrally on it. At highenough temperatures (kRISC ≫ kT) the singlet population becomesindependent of temperature (nS ∼ GS/kS).

The density matrix formalism provides a convenient way to focuson transitions between different spin states in an environment withinwhich transitions between different orbital and vibrational states arevery rapid compared to spin mixing. The reduction of the densitymatrix of the entire system to a two-spin-channel model is justifiedwhen the transitions between the spin channels are much slower thanany transitions within the channels. Thus the triplet channel shouldnot be considered to be one specific triplet eigenstate, and the singletchannel should not be considered to be one specific singlet eigenstate.Various scattering events will efficiently and rapidly mix individualsinglet eigenstates with other singlet eigenstates, but this does notnecessarily need to be tracked within this density matrix formalism.Thus it is also not necessary to specifically identify which tripletstates are converted into singlet states via the Lindblad operator; it issufficient to know that there is some coupling, and that this couplingis weak compared with the scattering rates within the singlet channeland within the triplet channel, in order to formulate the densitymatrix expression in this fashion.

For example, recent in-depth experiments35,38 have led to newunderstandings of the processes and specific states that lead to TADFat zero magnetic field. The pivotal conclusion is that singlet-triplet



spin mixing of charge transfer or exciplex states occurs not directlybetween those singlet and triplet levels but instead via intermediatelocal excited states (e.g. either 3LED or 3LEA) which are vibronicallycoupled to the CT/exciplex states. Although these most recentadvancements in understanding zero-field TADF have yet to beincorporated into existing models of the magnetic-field effects onTADF, they do not invalidate the general density matrix approachas the nature of the states involved are still specifically tripletor singlet, and the processes involved still involve intersystem andreverse intersystem crossings.

2.3. Comparison with electron spin relaxation rates

A similar two-component density matrix treatment is frequently donefor electron spin decoherence in conduction states in III-V semicon-ductors and quantum wells, through an analysis known as D’yakonov-Perel’ spin relaxation.4,39–41 Here the density matrix is written forthe conduction electron spin-up and conduction electron spin-downchannels, which are coherently mixed by spin-orbit interaction ratesthat are very much smaller than the orbital scattering rates orthe range of thermally-occupied energies the conduction electronsexplore. One difference between the singlet-triplet dynamics and thespin-relaxation dynamics in III-V semiconductors is the role of thesinglet-triplet energy splitting, which quenches any transitions fortriplets to singlets for energies below the singlet energy. A similar rolewould be played by a Zeeman energy splitting in spin relaxation, butthe consequences of such an effect are usually dominated by orbitalmagnetic field effects.40,41 For the zero-field case considered abovethe effect of this splitting is simply to reduce the effective RISC rateby the thermal occupation factor, as only those states above thesinglet energy threshold can produce RISC. The effect on magnetic-field-induced dynamics will be examined later.

3. Theory of Organic Magneto-Electroluminescence

The emission from either excitonic or charge transfer states isdetermined by solving for the spin density matrix ρ which evolves



according to the stochastic Liouville equation (SLE):42–47

∂ρ

∂t= − i

! [H , ρ] − 12{kSPS + kT PT , ρ}− kDρ+ G (15)

where PS and PT are

PS =

⎛

⎜⎜⎜⎜⎝

1 0 0 00 0 0 00 0 0 00 0 0 0

⎞

⎟⎟⎟⎟⎠, PT =

⎛

⎜⎜⎜⎜⎝

0 0 0 00 1 0 00 0 1 00 0 0 1

⎞

⎟⎟⎟⎟⎠.

kD is a spin-independent dissociation rate and G is the pair-generation matrix. The density matrix is 4x4 with basis states ofsinglet and triplet polaron pairs for excitonic emission and of singletand triplet exciplexes for exciplex emission.

For the exciton model, the spin mixing is occurring betweenpolaron pairs, and thus it is assumed that the generation matrixis diagonal with elements GS = GT , as there is no singlet-triplet gapfor polaron pairs. For exciplexes the generation matrix is:

G =

⎛

⎜⎜⎜⎜⎝

GS 0 0 00 GT ∗ 0 00 0 GT ∗ 00 0 0 GT ∗

⎞

⎟⎟⎟⎟⎠,

where GT ∗ is the number of triplet exciplexes generated in the tripletchannel at energies that exceed ∆ST. Exciplexes are generated inthe entire triplet channel as well, with a rate GT , but only thosewhose energies exceed ∆ST participate in the spin mixing. Thusa temperature dependence characteristic of TADF enters throughGT ∗ = GT e−∆ST/kBT . If the exciplexes are formed from polaron pairsthen GT and GS from electrical injection should be comparable. Allrates introduced are identified on Fig. 6, which shows a Jablonskienergy diagram for exciplexes.



exciton exciplex

spin mixing state polaron pair exciplexemissive state exciton exciplex

kS rate of S excitonformation

rate of fluorescence

kT rate of T excitonformation

rate of phosphorescence

kD dissociation rateof polaron pair

dissociation rate ofexciplex

generation rates generation rates

GS for S polaron pair for S exciplexGT for T polaron pair for T exciplexGT ∗ — for T ∗ exciplex

singlet (S) triplet (T)

ground state (S0)

spin mixing

kT

kS

kDG GT*

exciplexes ∆ST

NO spin mixing

kD

free polarons/polaron pairs

T*

T

Fig. 6. Exciplex Jablonski diagram. The Jablonski diagram for excitons looksidentical except ∆ST is much larger for exciplexes, which precludes any thermally-occupied T ∗ states at typical temperatures.



The Hamiltonian of the polaron pair or exciplex is H = HZ +HHF + HFF + HHF,δg + HFF,δg where

HZ =g1 + g2

2µBB0 · (S1 + S2)

+g1 − g2

2µBB0 · (S1 − S2), (16)

HHF =g1 + g2

2µB

(BHF(r1) · S1 + BHF(r2) · S2

), (17)

HFF =g1 + g2

2µB

BFF(r1) + BFF(r2)2

· (S1 + S2)

+g1 + g2

2µB

BFF(r1) − BFF(r2)2

· (S1 − S2); (18)

HHF,δg =δg

2µB

(BHF(r1) · S1 − BHF(r2) · S2

), (19)

HFF,δg =δg

2µB

BFF(r1) + BFF(r2)2

· (S1 − S2)

+δg

2µB

BFF(r1) − BFF(r2)2

· (S1 + S2); (20)

where BHF is the hyperfine field, r1,2 (S1,2) are the positions (spins)of the two constituents, and δg = g1−g2. The fringe field Hamiltonian(HFF) and its associated δg factor counterpart (HFF,δg) are onlyrelevant when inhomogeneously magnetized magnets are in proximityto the active area.48–50 We neglect HHF,δg and HFF,δg as of higherorder compared to HZ , HHF , or when appropriate, HFF . Figure 7(a)shows a physical representation of an exciplex on adjacent donorand acceptor molecules. Figures 7(b), (c) are schematics of the δgand hyperfine interactions. Further description of the δg mechanismprovided in the next section while other Chapters cover the hyperfineinteraction in detail. We now describe the solutions to these equationsfor various models appropriate to different material regimes.

3.1. The δg mechanism

The formalism thus presented can be overwhelming given the numberof terms in the SLE, and the number of possible spin interac-tion Hamiltonians. In general numerical solutions to the SLE are



(a)

(b) (c)

Fig. 7. (a) Schematic of an exciplex. (b) Spin mixing due to variations in theLande g-factor. (c) Spin mixing due to the hyperfine interaction. Taken fromRef. 47.

necessary. However if all interactions except for HZ are ignored, ana-lytic solutions are possible. Aside from being merely tractable, sucha situation appears to arise in certain donor-acceptor blends.47,51,52

A very simple analysis will first be presented, followed by theapplications to specific cases with some additional rates included.

The central physics can be understood by focusing on two states,the S and T0 states, which are coupled through the different Landeg factors of the electron and hole that comprise them:

Hδg =δg

2µBBz · (Se − Sh). (21)

When this term is commuted with the two-channel density matrix,

ρ =

(ρSS ρST0

ρT0S ρT0T0

)

(22)



to determine an equation of motion, we obtain

! ∂dt

(ρSS ρST0

ρT0S ρT0T0

)

= iBδgµB

((ρST0− ρT0S) (ρSS − ρT0T0)−(ρSS − ρT0T0) −(ρST0− ρT0S)

)(23)

The dynamics expressed by this equation can be succinctly summa-rized by noting that the rate of change of the off-diagonal terms(e.g. ρST0) is related to the difference of the diagonal terms (whichare the populations of singlets and triplets), and the rate of changeof the diagonal terms (e.g. ρSS) is related to the difference of theoff-diagonal terms. Thus the enhancement with magnetic field of theluminescence from the singlet population, kSρSS, will be proportionalto the magnetic field squared. If the initial condition correspondedto a generated population on the diagonal of the density matrix ata specific time (t = 0) then for subsequent times the off-diagonalterm grows linearly in time and magnetic field. Similarly one thenexpects magnetic-field-dependent changes in the diagonal terms thatare linear in the time and magnetic field, multiplying the off-diagonalterms; this yields a magnetic field effect on the diagonal that is thesquare of the magnetic field and the square of the time. Physicallythis can be viewed as the consequence of two initially parallel spins(thus in a triplet configuration) which precess at a different rateand thus end up in a linear superposition of parallel and antiparallelconfigurations as shown in Fig. 7(b). The steady-state solution forlow-field MEL, when generation and other rates are included, isreported in the sections below. It can be useful to consider thephysical nature of this two-spin precession. The eigenstates of theδg Hamiltonian are | ↑↓> and | ↓↑>, and the time evolution of a T0

state in the presence of that Hamiltonian is

|Ψ > = eiδgµBBt/2| ↑↓> /√

2 + e−iδgµBBt/2| ↓↑> /√

2 (24)

= eiδgµBBt/2(|T0 > +|S >)/2 + e−iδgµBBt/2(|T0 > −|S >)/2(25)



We now return to our consideration of the effect of the singlet-triplet splitting, ∆ST on the MEL. Two scenarios are consistentwith survival of the MEL in the presence of a substantial singlet-triplet splitting. In the first, the timescale for energy changes inthe triplet state is long compared with the precession time betweensinglet and triplet. As the precession times are of order microseconds,and the vibrational scattering times are likely much shorter thanpicoseconds, this regime appears implausible. An alternate versionin which the MEL occurs while the triplet state is in a stable excitedconfiguration, such as a local triplet exciton, might be possible. Morelikely, however, is that the triplet/singlet coherent state is largelyunaffected by the temporary difference in energy between triplet andsinglet, as the two precess. Relative excursions in the singlet andtriplet energies may produce dynamical phase differences betweentriplet and singlet in Eq. (25), which appear as an additional time-dependent phase

|Ψ > = eiδgµBBt/2(|T0 > +eiθ(t)|S >)/2

+ e−iδgµBBt/2(|T0 > −eiθ(t)|S >)/2. (26)

If the T0 and S manifolds that are tracked in the density matrixtreatment (i.e. those states within the manifolds that can be mixedby the spin-orbit interaction) are selected so that they have thesame average energy (discarding the low-energy triplet states thatcannot be mixed with the singlet states), then θ(t) will averageto zero, and the magnetic field effects on the relative populationsof T0 and S will be unaffected. This phenomenon is very differentfrom the role of scattering in, for example, spin relaxation inconduction electrons in III-V semiconductors.40 There the dephasingof different spins relative to the overall population, followed byscattering that scrambles the orbital information, is central to themotional narrowing regime for electron spin relaxation.

3.1.1. The exciton picture

We first examine situations where the eventual emissive recombina-tion stems from excitons (intramolecular pairs) instead of exciplexes



(intermolecular pairs). The difference between the two is that theS-T splitting, ∆ST, is much larger for excitons (∼ 1 eV ≫∼ 50 meV).Such a large splitting precludes any spin mixing with the exciton spinstates (so for all intents and purposes δg = 0). Moreover the negativeand positive polarons are tightly bound (binding energy EB ≫ kBT )which suppresses any exciton dissocation into polaron pairs (kD = 0).

The SLE of Eq. (15) is then solved in the steady state. Byconsidering only fluorescence (emission from singlets excitons), thequantity of interest is TrρPS and

MEL =TrρPS(B0) − TrρPS(B0 → 0)

TrρPS(B0 → 0)(27)

which for this case yields

MELo =[

kS − kT

2kD + kS + kT

](δgµBB/!)2

(kD + kT )(kD + kS) + (δgµBB/!)2.

(28)

Note that the width of the Lorentzian function scales as√(kD + kS)(kD + kT )/δg ≈

√kSkT /δg. The bracketed expression

gives the high-field MEL.

3.1.2. The exciplex picture

If the emissive precursors are exciplexes instead of excitons, the muchsmaller exchange energy (∆ST ∼ kBT ) allows spin mixing within theexciplex manifold. A consideration of that spin mixing47 provides

MEL =[GT ∗(kS + kD) − GS(kT + kD)

GS(2kD + kS + kT )

]

× (δgµBB/!)2

(kD + kT)(kD + kS) + (δgµBB/!)2(29)

which has the same form as the excitonic MEL in Sec. 3.1.1 if GT ∗ =GS . An activated (Arrhenius) temperature dependence for the MELfollows from Eq. (29) for GT ∗ = GT e−∆ST/kBT and GT = GS , so longas (kS + kD)e−∆ST/kBT > kT + kD. Thus mathematically there is nodifference between the exciton and exciplex results, however the rates



–100 –50 50 100–20

0

20

40

60

80

100

0B (mT)

% M

EL

∆ST

kBT= 0

0.5

1

Fig. 8. Percent MEL versus applied magnetic field for exciplexes (black, blue,red), using Eq. (29), and excitons (dashed green), using Eq. (28). The exciplexcurves are labeled by their ratio of exchange splitting to temperature. Thefollowing values were used; exciplex: kS = 1×10−4ns−1, kT = 1×10−5ns−1, kD =1× 10−6 ns−1, δg = 1× 10−4; exciton: kS = 1× 10−1 ns−1, kT = 2× 10−1ns−1,kD = 0, δg = 1× 10−4.

ki differ in value and in physical process described, as summarizedin Table I.

Figure 8 displays the MEL (in %) as a function of appliedmagnetic field. For excitons (green dashed curve), kS ≈kT and alsothese rates are much larger than what is found for exciplexes. Notemperature dependence is visible as excitons are formed after spinmixing occurs between polaron pairs, and there is negligible spinsplitting between singlet and triplet polaron pairs. The size of kS

and kT lead to much larger widths in the magnetic-field dependenceof the MEL than what is found for the exciplexes (black, blue, andred curves). Exciplex line shapes are shown for different ratios ofspin splitting to temperature; the large temperature dependence ofthe MEL is clearly present.

3.2. Magnetoelectroluminescence from δg andhyperfine interactions

When analyzing MEL curves in realistic materials the influence ofboth the δg mechanism and the hyperfine mechanism should beconsidered. Figure 9 shows the high-field MEL percentage of theδg mechanism (a) and the hyperfine mechanism (b). The hyperfine



g-mechanism effect on MEL

kD/kS = 0GT/GS = 1

0.750.5

0.5 1.0 1.5 2.0kT kS

50

50

100

HF-mechanism effect on MEL

kD/kS = 0

B0hf = 0.1kS

B0hf = 10kS

0.25

0.5 1.0 1.5 2.0kT kS

40

20

20

40

MEL (%) MEL (%)

(a) (b)

Fig. 9. (a) Theoretical calculations for the MEL at large field with the δgmechanism (B0hf = 0). (b) Theoretical calculations for the MEL from hyperfinespin mixing at large field (δg = 0). Increasing either GT /GS or B0hf causes theMEL to be more negative. Orange lines are GT = GS = 0.25 and black lines areGT = GS = 1. From Ref. 47.

curves are calculated assuming that semiclassical hyperfine fields areGaussian distributed with width B0hf . These two plots apply equallyto excitons and exciplexes. For excitons the regime of validity is nearkT /kS ≈1 where the MEL is small and the MEL sign hinges on therelative size of the two formation rates. For exciplexes the regime ofvalidity is near kT /kS ≪ 1 where the MEL is large in either case (δgor hyperfine) but the sign is dependent on the dominant mechanism,which greatly assists assigning the correct mechanism to the MEL ofexciplex recombination.

3.3. Magneto-photoluminescence

Magneto-photoluminescence (MPL) can be covered within thistheory as well, by taking GT = 0 and GS finite due to opticalexcitations. It is evident from Eq. (29) that MPL < 0; at leastone experiment has viewed negative MPL46 but others51,53 haveobserved MPL > 0. m-MTDATA/3TPYMB (1:1) has shown bothsigns.46,53 The model of Eq. (29) assumes that the δg mechanismis solely responsible for spin mixing. This cannot be the completepicture, for TADF is observed in the absence of an applied mag-netic field and the δg mechanism is ineffective without an appliedmagnetic field. The theory is readily generalizable to include field-independent spin mixing rates (kISC and kRISC) between S1 andT1 which are due to the spin-orbit interaction. The SLE then



becomes

∂ρ

∂t= − i

! [H , ρ] − 12{kSPS + kT PT , ρ} − kDρ+ G

+6∑

i=1

(LiρL

Ti + LiρL

Ti − 1

2{LT

i Li + LTi Li, ρ}

)

where Li is the Lindblad operator that sends state i to anotherstate j = i. For instance L1 = LS→T ∗

0=

√kISC |T ∗

0 ⟩⟨S|. Thesolution for the MPL is long but it can be shown that MPL > 0if kISC > kD + kT + kRISC . ISC and RISC also have effects onMEL, and the MEL can surpass 100% if kISC becomes comparableto and then larger than kS . The reason for this enhancement is thatthe fluorescence from initial singlets is reduced and only regainedthrough δg spin mixing (assuming that kRISC is very small comparedto kISC).

Approximate values of kISC are 1− 10×10−4 ns−1. The analysisof Hontz et al. for a 1:1 blend of m-MTDATA:3TPYMB indicateskISC = 140×10−6 ns−1, which in our model yields an MEL ≈3.5%.An intersystem crossing rate one-tenth that yields an MEL ≈26%.

3.4. Exciplex dynamics with separate T and T ∗ levels

A more sophisticated model is produced when excitation and de-excitation between the T and T ∗ levels is included explicitly. Someimmediate consequences of this theory will be described here, as anexample of the behavior of this system in the presence of additionalrelated states. The Lindblad operators describing transitions betweenthese states appear in the stochastic Liouville equation in thefollowing way:

∂ρ

∂t= − i

! [H , ρ] − 12{kSPS + kT PT , ρ}− kDρ+ G

+∑

i=T0,T+,T−

(Li(u)ρLi(u)T + Li(d)ρLi(d)T

−12{Li(u)T Li(u) + L3(d)T Li(d), ρ}

)



where Li(u) is the Lindblad operator that sends an electron up inenergy from Ti to a vibrational state T ∗

i and Li(d) sends an electrondown in energy from the vibronic level T ∗

i to Ti. The Lindbladoperators, Li, describe different rates of population shuffling withinthe system. For instance LT0(u) = LT0→T ∗

0=

√kup|T ∗

0 ⟩⟨T0|,LT+(d) = LT ∗

+→T+ =√

kdown|T+⟩⟨T ∗+|. There are six possible

incoherent transitions encompassed in the Lindblad terms. Theserates, kup and kdown are assumed to be spin independent and relatedby a Boltzmann factor such that kup = kdowne−∆/kBT , where ∆ is theenergy separation between T1 and a specific vibronic level. Our chiefconsideration is that the vibronic triplet levels T ∗ that are resonantwith S are added to the model, so ∆ = ∆ST. The generation matrix,basis vector, and projection operators are now

G =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

GS 0 0 0 0 0 00 0 0 0 0 0 00 0 GT 0 0 0 00 0 0 0 0 0 00 0 0 0 GT 0 00 0 0 0 0 0 00 0 0 0 0 0 GT

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

|S⟩|T ∗

0 ⟩|T0⟩|T ∗

+⟩|T+⟩|T ∗

−⟩|T−⟩

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

PS =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, PT =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 00 0 0 0 0 0 00 0 1 0 0 0 00 0 0 0 0 0 00 0 0 0 1 0 00 0 0 0 0 0 00 0 0 0 0 0 1

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

A physical picture is shown in Figure 10. Triplets are generatedat the rate GT in T1 and then fluctuate between T1 and the vibronic



singlet (S) triplet (T)

ground state (S0)

spin mixing

kT

kS

kDG GT

exciplexes ∆ST

NO spin mixing

kD

free polarons/polaron pairs

T*

Tkupkdown

Fig. 10. An exciplex model that explicitly includes T ↔ T ∗ transitions.

levels of T ∗ (with energy ET ∗ = ES) with the rates kup and kdown.When T ∗ is occupied, coherent spin evolution into the S state ispossible. Singlets are electrically generated as well, with a rate GS =GT . S states may spin mix to unoccupied T ∗ state in which case theycan fall down to T states at the rate kdown. There will be vibronicsinglet levels as well, though for simplicity only S1 is included in themodel.

The MEL can be calculated analytically though the expression iscomplex. In the limit of dissociation kD smaller than all other ratesthe maximum MEL is

MELmax =kSkup − kT kdown

kT kdown + kS(kT + kup), (30)

where it is evident that the T ∗ states play an important role; eitherkup and kdown must be finite if any MEL is to appear. For orbitaland vibrational scattering rates that are very large compared to theother rates of the problem (kup and kdown much larger than all otherrates) and kT ≪ kS , the thermal equilibrium within the excitedtriplet manifold will require kup/kdown ∼ e−∆ST/kBT , and the limit ofEq. (30) is MELmax = 1. Noteworthy is that the maximum MEL is100%, which indicates that even though most triplets cannot spin mix



(a) (b)

–300 –200 –100 0 100 200 300

0

20

40

60

80

100

B (mT)

ME

L (%

)

ME

L (%

)

0 2 4 6 8 10–100

–50

0

50

100

∆ST

kB T

∆ST /kBT = 0

1

2 10−4

kdown = 10−5 ns−1

10−3

Fig. 11. (a) MEL calculated from the more sophisticated model with kdown =10−3ns−1 and kup = kdownexp(−∆ST /kBT ) for three different temperatures.(b) MEL at high magnetic field calculated as a function of ∆ST /kBT for threedifferent kdown and kup = kdownexp(−∆ST /kBT ). Other values used are kS =1×10−4ns−1, kT = 1×10−5ns−1, kD = 0, δg = 1×10−4. No hyperfine interactionwas used in these calculations.

at any given time, they will effectively be spin mixed into the singletchannel over time, so that all the T0 states produce luminescencethrough TADF.

Theoretical results from this model are shown in Fig. 11;Figure 11(a) is the MEL for three different ratios of ∆ST /kBT .As the ratio decreases (e.g. temperature increases), the MELincreases in agreement with the several experiments with TADFmaterials.27,47,51,52 Figure 11(b) shows that for low enough tempera-ture, the model predicts the MEL to change sign since the excitationof triplets T → T ∗ is too weak to compete with the spin mixing ofS → T ∗. As a result S becomes less populated as the applied fieldincreases and the MEL turns negative.

4. Spin Decoherence

Transitions included to this point include those between states likeS1 and S0, with rate kS , or between S1 and T1, with rates ofkISC or kRISC ; the coherent evolution of spin from the Zeemaninteraction and hyperfine interactions also appears. So far neitherspin relaxation nor spin decoherence of the individual spins thatmake up either a polaron pair or an exciplex have been included.



Although spin relaxation (with rate 1/T1) is typically very weakin organic semiconductors, the spin decoherence rates (1/T2) areon the order of microseconds and may contribute to the physicsof these systems. Although these spin decoherence times are muchslower than the dominant transition times in the excitonic model,the spin decoherence times are comparable to 1/kS in the exciplexmodel. The previous model can be generalized to account fora finite T2 for the two spins forming either a polaron pair orexciplex.

We define decoherence Lindblad operators in the Zeeman basisfor each of the two spins, L3(1) and L3(2):

L3(1) = 1 ⊗ L3, L3(2) = L3 ⊗ 1 (31)

where

L3 =√

12T2

σz.

Additional operators L1 and L2 could be included if spin relaxationwere strong. To express these operators in the S − T basis, thefollowing transformation matrix is applied:

T =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 1 0

− 1√2

1√2

0 0

1√2

1√2

0 0

0 0 0 1

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎠

(32)

which gives new operators in the preferred basis as T T Li(j)T . TheLindblad operators enter the stochastic Liouvile equation as

∂ρ

∂t= − i

! [H , ρ] − 12{kSPS + kT PT , ρ}− kDρ+

14G

+ L3(1)ρL3(1)T + L3(2)ρL3(2)T

− 12{L3(1)T L3(1) + L3(2)T L3(2), ρ}. (33)



The effect of decoherence on the simple exciplex model of theMEL from the δg mechanism explored here increases the MEL widthaccording to

∆B =√

(2kD + kS + 4/T2)(kDkS + (2kD + kS)/T2)δg√

2kD + kS(34)

with kD ≪ kS ∼ 1/T2 and GS = GT = G. kT = 0 here but the resultcan be modified slightly to account for finite kT . The MEL at largefield, (the saturated MEL), is now

MELsat. =(kS − kT )(kD + kT )

(2kD + kS + kT )(kD + kT + 2/T2)(35)

or

MELsat. =kD + kT

2/T2(36)

when kD, kT ≪ kS , 1/T2. In either case the spin decoherence reducesthe overall MEL.

5. Experimental Measurements of MEL in ExciplexMaterials

Early work on exciplex materials54–56 proposed hyperfine interactionsto explain the MEL, and the values of MEL were not large and hadvarious signs. For example, in a study of magnetic field effects inexciplexes using m-MTDATA/Bphen,56 the EL spectra confirmedexciplex emission at the donor-acceptor interface, although thedonor and acceptor molecules were not coevaporated to maximizeexciplex efficiency. In Ref. 56, the maximum measured MEL averagedover the emission spectrum was near 6% and was attributed tohyperfine interactions between polaron pairs; spin mixing betweenexciplex constituents was assumed to be energetically suppressed.The hyperfine picture is bolstered by spectrally resolved MELmeasurements which showed that low energy, “cold” polarons tendto have wider MEL widths. This observation is consistent with largerhyperfine fields as Bhf ∝ 1/

√volume. The widths measured extended



from 10 mT for higher energy polaron pairs/exciplexes to 25 mTfor lower energy polaron pairs/exciplexes. These widths, however,appear also to be consistent with later measurements that do notattribute the MEL to nuclear moments.

δg spin mixing among exciplexes and hyperfine spin mixing ofpolaron pairs was proposed as the mechanism for subsequent work onmagneto-photoluminescence and MEL in exciplex materials.51 Therole of hyperfine spin mixing in contributing to the observed signalswas not detailed. A strong temperature dependence in both the MPLand MEL was identified, and the MEL width appeared too large tosupport a hyperfine mechanism alone. At 300 K, the MEL surpassed30% (38% at 317 K), making the effect one of the largest observed atthe time although the values were not qualitatively larger than thelargest MEL’s seen in exciton-emissive materials. The first MFEsexceeding 100% at room temperature appeared shortly thereafter.27

A key finding was that the blend ratio was an indicator of the size ofthe MFE. In this case27 using 75% 3TPYMB to 25% m-MTDATAwas found to yield the largest responses. The authors noted thattheir results pointed to a magnetic-field modulated reverse inter-system crossing (RISC) between the triplet and singlet exciplexesbut no model was provided except to say that the magnitudeof the effect was inconsistent with the hyperfine mechanisms’sexpectations.44

The first detailed account of the exceptionally large experimentalMEL and MC with a corresponding theory47 used a coevaporateddonor-acceptor blend that had been earlier used in investigations ofTADF at zero field.32 Device conditioning has been known to increaseMFEs for some time57–62 but this work was the first to use the proce-dure on TADF materials. Unconditioned devices (pristine) displayedin Fig. 12(b) a MEL of about 20% at room temperature (in constantcurrent mode) in a coevaporated m-MTDATA/3TPYMB(1:3) blendwhich is smaller than found by Basel et al. in a different materialcombination.51 Under the particular conditioning procedure whichcan be found in Ref. 47, the MEL increased to 40% [Fig. 12(d)]. Thesame samples under constant voltage operation display dramaticallylarger responses: 60% and 1200% for pristine and conditioned devices,



Fig. 12. MEL measurements in a coevaporated m-MTDATA/3TPYMB(1:3)blend. (a), (b) pristine devices. (c), (d) conditioned devices. Experiments wereperformed in either the constant voltage (a), (c) or constant current (b), (d)modes. Taken from Ref. 47.

respectively. The large amplification is understood to be the resultof the large nonlinearity of the I-V and EL-V curves,47 which isindicative of the increase in traps present from the conditioningprocess.63,64 An implication is that large MFEs in a constant voltagesetting are much more sensitive to trap creation. Even so-calledpristine devices still exhibited highly nonlinear EL-V curves whichsuggests that traps are unavoidable at present. Moreover reports oflarge MFEs may be misconstrued in terms of the underlying spinmixing mechanism instead of as a result of a highly nonlinear I-V orEL-V curve.



A quantitative model, based on the stochastic Liouville equation,was put forth that considered the spin mixing to be within theexciplex levels and to be dominated by difference in the g-factor.From this analysis the hyperfine interaction was deduced to yieldobservations contrary to their measurements (e.g. opposite sign ofMEL and narrower widths). The model, as described in Section 3.1.2,produces an MEL width proportional to

√kDkS/δg where kD is the

dissociation rate of the exciplexes and kS is the rate of fluorescenceand kS ≫ kD ≫ kT where kT is the rate of phosphorescence.The model implicitly assumes that exciplexes are created fromfree polarons and not polaron pairs. Temperature dependence isaccounted for by an exciplex generation matrix that preferentiallyplaces triplets in excited states resonant with S1 as temperaturesincrease.

Shortly after Ref. 47, previous experimental work27 was extendedin Ref. 52. The maximum MEL increased to near 160%, and the MELand linewidth versus temperature were measured. A model of the δgmechanism was suggested in which

MEL ∼ 11 + (B/∆B)2

, (37)

where ∆B = !/(2µBδgτ) is the MEL width; τ is the lifetime of theelectron-hole pair. The authors concluded that the large MEL widthsruled out hyperfine interactions and instead scaled with the lifetimeτ in their model. Thus these results are consistent with Ref. 47.

In addition a comparison of different donor-acceptor combina-tions was conducted.52 With a single donor type, m-MTDATA, theacceptor was varied among the following options: 3TPYMB, Alq3,PPT, TPBi, and PBD. The assumption was that different donor-acceptor combiniations would have different singlet-triplet energydifferences which would alter the magnetic-field modulated RISC.By measuring temperature dependence of the MEL, they extractedthe activation energy, Ea, believed to correspond to the singlet-tripletenergy gap. Ref. 52 noted that although δg could only spin mix Sand T0 states, spin lattice relaxation could convert T± to T0 and socould therefore indirectly contribute to the RISC.



6. Fringe Field Effects in TADF Materials

As seen above, spin mixing driven by the δg mechanism onlyeffectively mixes one of the three triplet eigenstates (T0) with thesinglet, leading to an anticipated efficiency improvement47 of 100%.The remaining two triplet eigenstates, T+ and T−, are not efficientlyharvested into the singlet channel by the applied magnetic field.This limitation can be overcome by using the fringe magnetic fieldfrom a nearby ferromagnetic film.49 Harvesting all three tripleteigenstates would enhance OLED emission by 300%. A magneticfield gradient can be generated by the remanent magnetization ofpatterned ferromagnetic films which will mix in the other two tripleteigenstates, T+ and T−. Ref. 49 predicts that even OLED activeregions that are 100 nm away from the ferromagnetic film can achievethe maximal (factor of 4) enhancement of light emission.

Shown in Fig. 13(a) is a schematic diagram of the recombinationof polaron pairs into excitons. As described previously in the theoryof magnetoelectroluminescence (Sec. 3), the spin character of theprecursors (singlet or triplet) influences the rate of exciton formation.For exciplexes [Fig. 13(b)], the spin-mixing of singlet/triplet exciplexpopulations changes the emission efficiency. Figures 13(c–e) showspatterned domain stripes of magnetic material, whose magnetizationcan be reoriented by applied magnetic fields. These magnetic domainsthen remain configured (through the remanent magnetization) whenthe applied magnetic fields are turned off. Through this reconfig-uration of the magnetic domains, the spin mixing via fringe fieldscan be switched on and off without any remaining applied externalmagnetic field. The form of the fringe field interaction produces verylarge changes in the electroluminescence ∆EL/EL = (EL(ON) −EL(OFF))/EL(OFF) = 300%.

Two options for striped patterns are shown in Figs. 13(c,d)for a thin magnetic film, with thickness t, with striped domainsthat generate fringe fields in the organic layer. The stripes repeatevery distance a along the x-axis and the film extends far out intothe y-direction to ±c. The domains can be fixed perpendicular tothe stripes [Fig. 13(c)] with sufficiently large a and by applying



Fig. 13. (a) Diagram of exciton energies and recombination pathway.(b) Diagram of exciplex energies and recombination pathway. (c) and (d) depict,respectively, the domain configurations that either produce fringe fields andenhanced luminescence (ON) or do not produce fringe fields (OFF) (e) In-planemagnetic configuration (c) with fringe fields directions (arrows only depictdirections and not magnitudes) in the x − z plane. Parameters used: t = 20nm,Ms = 8 × 105A/m, and a = 160 nm. The calculation assumes imax ≫ 1. FromRef. 49.

a magnetic field to set the domains perpendicular to the stripes.Strong fringe fields appear in this orientation, so this configurationis defined to be ON. Figure 13(d) is the most energetically favorableconfiguration, and corresponds to negligible fringe fields if edgeeffects are unimportant. This magnetic state is defined OFF, anda switching figure of merit is thus ∆EL/EL.

The magnetic scalar potential from a ferromagnet with magneti-zation M , volume V , and surface S, [Fig. 13(c)] is65

Φ(R) =∑

i

14π

Ms,i

∫

Si

ni · M i(ri)|R − ri|

dAi, (38)



where each domain is assumed to be uniformly magnetized and isdenoted by an index i. R = (X,Y,Z) is the position outside themagnet, r = (x, y, z) the position within the magnet, and the Ai

are area elements of the magnet’s surface. H = B/µ0 outside amagnetized volume, so B = −µ0∇Φ(R) where µ0 = 4π×10−7 N/A2.Ms,i and the length and width of Si are assumed to be constant forall i in a given configuration of the magnet.

The magnetized domains are separated spatially by non-magneticstripes [Fig. 13(e)]. Magnetic surface charge densities form on eachdomain wall and alternate between positive and negative magneticcharge. These surfaces are located at each xi and have an area 2c(zt−zb) = 2ct, where zt and zb are the top and bottom positions of themagnet. x = 0 is halfway in between two such oppositely “charged”plates (the negative plate lies at x = +a/2 and the positive plate atx = −a/2). Each plate (or domain wall) is indexed by xi = (i + 1

2)awith −imax < i < imax − 1. The x-edge length is thus Lx = (2imax −1)a. An approximation for infinite stripes (i.e. c → ∞) that t ≪ a,Z,allows for simplified expressions for the fringe fields by expandingeach term in small t:

BFF(R)

= −µ0

4πMs

4πt

a

{− cos(

πXa

)cosh

(πZa

), 0, sin

(πXa

)sinh

(πZa

)}

cos(

2πXa

)+ cosh

(2πZ

a

) .

(39)

Figure 13(e) shows the direction of the fringe fields above themagnetic domains in the x − z plane.

For the calculation of a figure of merit, Eq. (15) is solved insteady state with BFF(r1) calculated in the organic layer from Eq.(39). r1 is selected randomly within a box of height 30 nm, positionedat Z = Zmin above the magnet. The lateral size of the box is equalto the magnetic film’s lateral size. The current path is along the zdirection, so the average carrier hop occurs in the z-direction andBFF(r2) = BFF({x1, y1, z1 + d}), where d = 1nm is the hoppinglength. After determining the density matrix ρ, the fluorescent EL isdetermined from EL ∝ kSTrPSρ.



Fig. 14. Change in EL versus average fringe field gradient and height abovemagnets in the absence of hyperfine fields. For the exciplex curves: kS = 3 ×10−3ns−1, kT = 0, and each curve is labeled by the choice of kD (in ns−1). Theexciton curves are labeled by their values of kS (in ns−1); other values used arekT = 2kS . We find kD has negligible effect on exciton responses and so it is takento be zero. We specify the patterned FM to be permalloy with Ms = 8×105A/m,t = 20 nm, and a = 160 nm. From Ref. 49.

Figure 14 shows the figure of merit ∆EL/EL in a simplified modelincluding only the fringe field (HHF = 0) within both the excitonand exciplex pictures. Beginning far above the magnet, and thus farfrom the fringe fields, the lack of a substantial fringe field gradientproduces no spin mixing or change in electroluminescence. As thefringe field gradient is increased by reducing the height above themagnets, Zmin, the figure of merit plateaus at 100%. The reasonfor this plateau is reminiscent of the δg mechanism for which all T0

states may upconvert to S states, doubling the electroluminescence.47

The spin mixing between T± and S states is much slower thanthe spin mixing between T0 and S states. For kD large enough toprohibit further mixing the effect saturates. Thus these curves shiftleft as kD is increased. As the gradient increases further, and forsufficiently small dissociation, the figure of merit approaches 300%.This indicates that now all triplet exciplexes are spin mixing tosinglet exciplexes and are recombining as singlets. Spin-selective ratesfor excitons vary little, contrary to what occurs with exciplexes,so the figure of merit is much smaller. We assume kS ! kT .66



Fig. 15. Change in EL versus average fringe field gradient and height abovemagnets in the presence of hyperfine fields. Exciplexes (red): kS = 3× 10−3ns−1,kT = 0, kD = 10−6 ns−1. Excitons (blue): kS = 6 × 10−1ns−1, kT = 12 ×10−1ns−1, kD = 0.66 ∆hf is the width of the gaussian distribution from which thefields BHF are sampled when calculating the average figure of merit. The thicknessof the organic layer is taken to be 10 nm. Calculations for shorter heights are notdisplayed as the condition t << Zmin is not met for Eq. (39) and thus the resultswould not be valid. From Ref. 49.

If kS " kT , the excitonic ∆EL/EL changes sign but still does notreach the magnitude seen for exciplexes.

When hyperfine interactions are included as well the figure ofmerit is modified from the results in Fig. 14. The changes are shownin Fig. 15. High above the magnets, where the fringe field is small, thefigure of merit is zero. The fringe field in Fig. 15 acts like an externalmagnetic field influencing the usual (hyperfine-based) magnetoelec-troluminescence for spin pairs. This picture is supported by the signof the figures of merit which are opposite to those in Fig. 14.44,47 Eventhough hyperfine mixing is dominating the response, so plateaus inthe figure of merit are not apparent and the magnitude is smaller,we see that the fringe fields still act as an ON/OFF switch.

The influence of the fringe fields within the Hamiltonian descrip-tion is similar to the influence of the δg mechanism of Sec. 3.1, butbecause the fringe field gradient is not necessarily aligned with thefringe field, the precessional mechanism is more efficient in convertingtriplet to singlets, and thereby increasing radiation more than in



Sec. 3.1 (to 300% instead of 100%). Conditioned devices governedby the δg mechanism yield an MEL up to 4000% under constant-voltage operation, but only 40% for constant current (which is themode assumed in our theory).47 Thus the larger constant-currentMEL with fringe fields will lead to a corresponding larger than 4000%constant-voltage MEL in conditioned devices.

7. Conclusion

Chemists have explored photo-magnetic field effects on reactions forsome time now; their most spectacular manifestation may be therole played in avian navigation mechanisms.11,12,67 Perhaps it is notsurprising that such effects would prove to be influential in organicdevices that emit light, however their importance in materials anddevices that are of high efficiency, and their promise to potentiallyimprove the light emitting efficiency of OLEDs, is gratifying to thoselooking for substantial quantum coherent effects at room temperaturein electronic materials.

The discovery of very large magnetic field effects in exciplexemission provides an elegant example of a scenario in which themathematical description of a new physical situation (exciplexemission, as indicated in our Table I) is considerably simpler,and with much more dramatic magnetic field effects, than in theoriginal materials and excitations in which magnetic field effects werediscovered (exciton emission). Analytic solutions of the magneto-electroluminescence are possible for the δg mechanism for exciplexemission, and the maximum magnetic field effect is 100% in constant-current measurements. When the presence of fringe fields frominhomogeneous magnetic materials is included, which could providean avenue to integrate magnetic memories with organic deviceswithout electrical contacts between them, the maximum magneticfield effects are 300%!

Little discussion has been presented here regarding electricaltransport to and from the donor-acceptor pair of relevance. Similarfeatures should emerge in transport and in other forms of chemicalreaction under the right conditions. In addition there is littlefundamentally limiting the size of the regions that emit and respond



to these magnetic fields, since the dynamics considered here isessentially entirely within the donor-acceptor molecule complex.

This material is based on work supported by the U.S. Departmentof Energy, Office of Science, Office of Basic Energy Sciences, underAward No. DE-SC0014336.

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December 5, 2017 19:21 Spin in Organics (Vol. 3) (in 4 Volumes) - 9in x 6in b2966-v3-index page 423

Index

Alq3, 109acceptor, 71–79activation energy (Eact), 377, 382active control, 410ambipolar OLEDs, 304ambipolar transistors, 217Anderson, 47anthracene, 47, 63Arrhenius plot, 390

bimolecular, 198biological compass, 43biological imaging, 256bipolar, 209bipolar device, 62bipolaron, 42, 48, 51–53, 55–62, 77bipolaron mechanism, 42, 53, 59,

61–63, 97bipolaron model, 203blends module with DREIDING

force-field, 384Bohr magneton, 51branching parameter, 60, 61

C-13 isotope effect, 238central-limit theorem, 48chapters, 267, 351charge blocking, 77, 80charge transfer, 150, 155, 158, 205,

230, 232

charge-transfer exciton, 44, 71compass, 1, 28–31, 33–35conditioning, 40constant current, 123constant voltage, 123copolymer, 72, 81, 82copolymerization, 81

∆ES1−T1 , 231, 232∆EST, 316∆B mechanism, 102∆g effect, 64∆g mechanism, 1, 2, 11, 12, 14, 16,

17, 20, 44, 102, 127, 128, 354, 377,393, 394

D-A exciplex, 377decay rate, 6, 11, 16, 18, 24–26, 28delayed fluorescence (DF), 43, 228,

234delayed fluorescence lifetime, 240density functional theory (DFT),

382density matrix, 45density of traps, 203density operator, 45, 52, 53, 64, 65,

268deuterated organic semiconductors,

110deuterated polymer devices, 97deuteration, 42, 70

423



deuterium, 42deuteron, 70device conditioning, 118, 123, 125Dexter energy transfer (DET), 381diffusion constant, 77, 79, 80disorder, 115donor, 71–74, 76–80donor-acceptor polymer, 80, 81donor/acceptor, 213DOO-PPV, 42, 69doping, 205doublet, 63duty cycle, 313dynamic equation, 297

E-type delayed fluorescence, 227,228

earth’s magnetic field, 43effective magnetic field, 48, 51, 53, 55,

56, 70, 74, 75EHP model, 293Einstein’s relation, 77El-Sayed rules, 153electrically detected magnetic

resonance, 69electrochemically doped organic

light-emitting diodes, 40electroluminescence, 40, 98, 228electroluminescence quantum

efficiency, 116electron donor–acceptor exciplex

chromophores, 377electron-electron interaction, 77electron-hole pair, 44, 206electron-hole pair model, 98electron-hole polaron-pair

mechanism, 42, 53energetic disorder, 206Ern, 63exchange, 144, 147–151, 153, 165exchange coupling, 80exchange energy J, 232exchange interaction, 50exchange splitting, 99, 276

exciplex (EX), 40, 44, 99, 119, 128,149, 150, 153–155, 158, 160, 161,165, 173, 180, 264–266, 268, 271,385

exciplex devices, 119exciplex OLED, 71exciplex spectrum, 127exciplexes, 379exciton, 95, 146, 148, 149, 154,

158–160, 164, 177, 180, 229, 264,266

exciton binding energy, 57exciton mechanism, 100excitonic models, 190excitons, 21excitons kinetics, 294exoergic, 199extrinsic MEL, 407

F4-TCNQ, 82F6-TNAP, 199Forster resonance energy transfer

(FRET), 379fast-hopping limit, 67Fermi contact interaction, 49Fermi energy, 75, 76ferromagnetic thin film, 130field effect transistor, 1, 2, 8field-induced circular polarization,

355floats, 279fluorescence, 148, 151, 165Franck-Condon factors, 242fringe fields, 151, 178fringe-field effects, 132fringe-field magnetic-field-effect,

130fringe-field OMAR, 132fringe-field-induced

magnetoresistance, 101fullerenes, 238fusion, 198

g-factor, 44, 51, 65, 102, 151, 154,268, 395


Index 425

gating effect, 62gyromagnetic ratio, 70, 76

HAT-CN, 72, 82, 212high-field effects, 43highest occupied molecular orbital, 73HMDS, 199HOMO, 73, 78, 82, 94hopping transport, 144hot-exciton rISC, 229huge magnetoresistance, 80hybrid spintronic exciplex-OLED, 407hyperfine, 7, 114hyperfine coupling (HFC), 245hyperfine field, 48, 110hyperfine frequency, 52hyperfine Hamiltonian, 111hyperfine interaction (HFI), 1, 2, 7,

30, 35, 42, 96, 151, 154, 166, 264,266, 316, 391

hyperfine magnetic field, 101

interaction, 7, 146interchain hopping, 80intermolecular radical pair, 214intersystem crossing (ISC), 151, 294intra-molecular charge-transfer, 318intrachain hopping, 80ionization, 73, 75

Jablonski, 159Johnson, 47

kinetic effect, 59Kubo, 47

level crossing, 70Lindblad operators, 155, 172lineshape, 68, 278, 279

Lorentzian, 40non-Lorentzian, 40

Liouville equation, 45–47, 51Liouville-von Neuman equation, 46Lorentzian, 40, 41, 43, 60, 61, 68Lorentzian function, 110

Lorentzian line shape, 195Lorentzian lineshape, 111lowest unoccupied molecular orbital,

73luminescence, 264luminescent efficiency, 43LUMO, 73, 78, 82, 94

m-MDATA, 119m-MTDATA, 99MAPbI3, 349magnetic color spin-XOLED display,

413magnetic dipolar interaction, 47, 49magnetic dipole-dipole interaction,

43, 50magnetic field effect (MFEs), 40, 95,

100, 106, 144, 149, 164, 286, 340,377

magnetic field effect onelectroluminescence, 105

magnetic moment, 48magnetic resonance, 42magnetic tunnel junction (MTJ), 407magneto-conductance (MC), 1magneto-electroluminescence (MEL),

40, 64, 67, 285, 377magneto-luminescence, 264magneto-optic Kerr effect (MOKE),

132magneto-photoinduced absorption

(MPA), 3, 24magneto-photoluminescence (MPL),

3, 24, 167, 351, 377magneto-RISC (M-RISC), 382, 387magnetoconductance, 40magnetodiffusion, 79magnetoelectroluminescence, 93magnetoresistance, 40majority carrier, 62master equation, 270MC, 7, 129, 131McConnell, 49McConnell rule, 50MEH-PPV, 42



MEL, 93, 131, 348, 367Merrifield, 47, 63migratory birds, 43Miller-Abrahams hopping rates, 61,

71minority carrier, 62mixed state, 46mixing, 2mobility, 107, 109Monte Carlo, 77Monte Carlo simulation, 61, 62, 77, 79MPC, 365MPC(B), 13

9,10-Bis[N,N-di-(p-tolyl)-amino]anthracene (TTPA),382

N ,N ,N ′ ,N ′-Tetrakis(4-methoxyphenyl)benzidine(MeO-TPD), 381

non-Lorentzian, 40, 41, 43, 60, 61, 63non-Lorentzian line shape, 195non-radiative decay, 58non-volatile effect in organic MFE

devices, 132nonvolatile information storage, 130nuclear magnetic moment, 48nuclear spin, 48

1,4,5,8,9,11-hexaazatriphenylenehexacarbonitrile, 72

OFET, 72, 74, 192OMAR, 93, 95, 101, 104, 107, 109,

114, 116, 119, 190optical thermometry, 242optoelectronic-spintronic (O-S)

devices, 363organic compass, 43organic field-effect transistor, 72organic light emitting diode (OLED),

1, 3, 15, 40, 43, 44, 58, 59, 63, 64,69, 82, 94, 99, 148, 189, 227, 228,256, 378

organic magnetoresistance, 93, 109organic photovoltaics, 44

organometal trihalide perovskites, 343overshoot, 328oxygen sensing, 242

π-electron, 49π-orbitals, 49P3HT, 81paramagnetic species, 198Pauli exclusion principle, 97, 145Pauli principle, 60Pauli spin blockade, 60, 97PEDOT:PSS, 81pentacene, 109, 192percolation, 62perpendicular magnetic anisotropy,

130PFO, 103phenyl spacer, 80, 81Phonon, 47phonon-assisted tunneling, 44phosphorescence, 98, 148, 176phosphorescence lifetime, 240phosphors, 148photocurrent, 40photoinduced, 192polaron, 46polaron hopping, 53polaron pair (PP), 3, 42, 45, 47, 50,

51–53, 56, 62, 64, 66, 70, 100, 149,160, 161, 341

polaron-pair mechanism, 42, 53poly(p-phenylene vinylene, 40poly(3,4-ethylenedioxythiophene)-

poly(styrenesulfonate),81

poly(3-hexylthiophene-2,5-diyl), 81poly[2,5-dioctyloxy-1,4-p-phenylene

vinylene, 42poly[2-methoxy-5-(2′-ethylhexyloxy)-

p-phenylene vinylene],42

polyfluorene, 40, 103, 106polymer-fullerene blend, 43, 44PPV, 40, 50, 69prompt fluorescence (PF), 233


Index 427

Pt-rich polymer, 345pulsed voltage, 289pure state, 46

quantum dynamics, 251quantum-statistical ratio, 67quartet, 63quasi-equilibrium, 80

radical anion, 213radical cation, 213radical-pair mechanism, 42recombination, 73, 75, 79regio-random polythiophene, 106regio-regular polythiophene, 106resistor model, 79resistor network, 73, 75, 76reverse intersystem crossing (RISC),

151, 228, 316, 377rISC, 249–251, 253–255

σ-π exchange coupling, 50σ-electron, 49Schulten, 48semiclassical approximation, 45, 48sensing, 256sign change, 207sign flip, 411simulation, 300singlet, 4, 99singlet exciton, 3, 22, 24, 95, 100, 196singlet exciton fission, 98singlet fission, 196singlet-triplet energy splitting, ∆EST,

378singlet-triplet inter-conversion, 323slow hopping, 52slow-hopping limit, 52, 55, 67–69slow-hopping regime, 70small molecule, 109SOC, 228space-charge-limited current (SCLC),

130, 410spacer unit, 80spin, 2

spin blocking, 73, 77, 79, 80spin chemistry, 49spin coherence, 143spin configuration, 4–6, 11spin decoherence, 171spin diffusion, 55spin dipole-dipole interaction, 196spin filter, 60spin Hamiltonian, 50, 53, 55spin mixing, 2, 4, 9–12, 26, 28, 42, 52,

68, 70, 73, 149, 151, 157, 159, 318,342

spin orbit coupling, 228spin pairs, 10spin polarized injection, 274spin projection factor, 55, 56, 68, 75spin relaxation, 14, 16, 55, 172, 267,

270, 273spin splitting, 59spin subspace, 51spin-dependent reaction, 43, 47, 51spin-mixing, 101spin-mixing channels, 395spin-orbit, 146, 153, 167spin-orbit coupling, 44, 55, 266, 342spin-orbit interaction, 41, 264, 316spin-spin interaction, 197spin-XOLED, 407Spiro-DPPFPy, 209Spiro-TAD, 199Spiro-TTB, 72, 82, 212state, 7steady, 7steady state, 7, 11, 16, 17, 24, 25stochastic interactions, 47stochastic Liouville equation, 44, 45,

47, 50, 64, 128, 159super yellow, 50super-fluorescent-OLEDs

(SF-OLEDs), 379

2,2,7,7-tetrakis-(N,N-di-p-methylphenylamino)-9,9-spirobifluorene,72



3TPYMB, 99, 119t-Bu-PBD, 99TADF, 99, 119, 227, 228, 256TADF kinetic mechanism, 233TADF-assisted fluorescent OLEDs,

379template, 265, 344tetracene, 192tetrafluorotetracyanoquinodimethane,

82tetraphenyldibenzoperiflanthene

(DBP), 382thermal polarization, 16, 17thermal spin polarization, 1, 2, 14,

15, 20, 35thermally activated delayed

fluorescence (TADF), 99, 127, 149,152, 227, 228, 230, 316, 378

thermally stimulated current, 117time-correlation function, 269time-resolve MELs, 293TIPS-Pentacene, 192transient EL, 287transient method, 286trap, 64, 114, 117trion, 63, 204triplet, 1–6, 10, 11, 14, 18, 19, 21, 22,

26–28, 35, 95, 99, 100, 196triplet exciton, 1, 2, 4, 21, 22triplet exciton-polaron interaction,

63, 64triplet exciton-polaron model, 98triplet exciton-polaron reaction

mechanism, 43

triplet formation yield, ΦISC, 235triplet fusion, 229triplet harvesting, 230triplet triplet pair (TT pair), 35triplet-polaron interaction, 191triplet-triplet annihilation, 43, 47, 64,

199, 229tris-[3-(3-pyridyl)mesityl] borane

(3TPYMB), 381TSC, 117TTA, 229TTA model, 293

ultra-small, 215ultra-small magnetic field effect, 43ultra-small MFE, 43ultralarge MFEs, 119unipolar, 203unipolar device, 62, 63, 304Universality of the OMAR effect, 109USMFE, see ultra-small MFE

vibronic coupling, 253–255

Wolynes, 48

X-ray bremsstrahlung, 115X-ray irradiation, 116

Zeeman energy, 197Zeeman interaction, 5, 267, 268zero field splitting (ZFS), 1, 2, 21, 28,

43, 56, 63zinc phthalocyanine (ZnPc), 82

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