Exciton transport in amorphous polymers and the role of morphology and thermalisation
EExciton transport in amorphous polymers and the role of morphology and thermalisation
Francesco Campaioli ∗ and Jared H. Cole † Chemical and Quantum Physics, and ARC Centre of Excellence in Exciton Science,School of Science, RMIT University, Melbourne 3000, Australia (Dated: February 15, 2021)Understanding the transport mechanism of electronic excitations in conjugated polymers is key to advancingorganic optoelectronic applications, such as solar cells, OLEDs and flexible electronics. While crystalline poly-mers can be studied using solid-state techniques based on lattice periodicity, the characterisation of amorphouspolymers is hindered by an intermediate regime of disorder and the associated lack of symmetries. To over-come these hurdles we use a reduced state quantum master equation approach based on the Merrifield excitonformalism. Using this model we study exciton transport in conjugated polymers and its dependence on mor-phology and temperature. Exciton dynamics consists of a thermalisation process, whose features depend on therelative strength of thermal energy, electronic couplings and disorder, resulting in remarkably different trans-port regimes. By applying this method to representative systems based on poly(p-phenylene vinylene) (PPV)we obtain insight into the role of temperature and disorder on localisation, charge separation, non-equilibriumdynamics, and experimental accessibility of thermal equilibrium states of excitons in amorphous polymers.
I. INTRODUCTION
Organic semiconductors (OSCs) are at the forefront of the cur-rent research efforts for the development and improvementof optoelectronic technology, such as solar cells [1–3], or-ganic light-emitting diodes (OLEDs) [4, 5], thin-film transis-tors [6–9], sensors and flexible electronics [10–13]. In partic-ular, conjugated polymers are a key class of OSCs for pho-tovoltaic applications, due to their ability to transport elec-tronic excitations, i.e., excitons , over tens of nanometers, to-gether with their low productions cost, ease of fabrication andflexibility [14, 15]. Depending on their chemical compositionand fabrication conditions, polymeric semiconductors can befound in a variety of different morphologies characterised byspecific exciton transport properties [16].While the fundamental features of energy and charge trans-port across crystalline, semicrystalline and amorphous poly-mers are qualitatively known, the dependence of exciton dy-namics on temperature and morphology is still an importantarea of investigation [17–20]. For example, the introduc-tion of amorphous polymers characterised by high electronmobility has challenged the idea that crystalline ones wouldbe optimal for charge transport [21–26]. The study of ex-citon transport in amorphous polymers is however hinderedby the complexity that arises from their disordered nature,which prevents the use of solid-state techniques based onlattice periodicity. In this intermediate transport regime be-tween band conduction and incoherent hopping, semiclassi-cal techniques such as Marcus theory often fail to reproducethe coherent quantum attributes of the exciton dynamics [27–30], while first-principles calculations like multiconfigurationtime-dependent Hartree method (MCTDH) become compu-tationally intractable due to the large size of the systems ofinterest [19, 20, 31–34].An effective alternative to study exciton transport in dis-ordered OSCs is given by reduced state quantum master ∗ [email protected] † [email protected] equations [15]. These allow for the efficient description ofthe interaction between excitons and nuclear vibrations, i.e., phonons , and account for both coherent and incoherent dy-namics. Quantum master equations have therefore been ap-plied to a variety of exciton transport problems in OSCs, suchas natural photosynthetic complexes [35–38], molecular ag-gregates [39–42], and disordered systems [34, 43]. A key in-sight from this body of literature is that exciton-phonon inter-actions have an essential role for exciton transport in OSCs.On one hand, decoherence induced by local and uncorrelatedphonon modes is known to improve transport efficiency in dis-ordered systems [36, 44, 45], counteracting weak and stronglocalisation phenomena, such as Anderson localisation [46–49], that otherwise prevail when dynamics is primarily coher-ent. On the other hand, strong spatial correlations betweenbath modes can facilitate the emergence of decoherence-freesubspaces, improving transport of certain states in orderedsystems [50].Recently, quantum master equations have also been usedto study exciton transport in conjugated polymers, albeit onlyfor the case of crystalline morphologies. In Ref. [19], Lyskov et al. bridge first-principles calculations to a Lindblad masterequation [51–55], and obtain a phenomenological model fortriplet exciton dynamics in crystalline poly(p-phenylene viny-lene) (PPV). Their results show that, conversely to the case ofdisordered OSCs, decoherence slows down exciton dynamicsin crystalline PPV, inducing a rapid transition from ballistic(coherent) to diffusive (incoherent) transport. However, themodel used in Ref. [19] cannot be directly applied to amor-phous polymers, whose difficult theoretical characterisationpresents several questions [33, 56–67]. How do temperatureand morphology affect exciton transport? What are the fea-tures of the dynamical transition between localised and delo-calised states? Are thermal equilibrium states experimentallyaccessible? How does disorder affect charge-separation dy-namics?Here we answer these questions generalising the masterequation approach used in Ref. [19] to conjugated polymerswith arbitrary morphology. Firstly, since excited electronicstates of amorphous polymers are known to display charge a r X i v : . [ c ond - m a t . m t r l - s c i ] F e b separation over a few monomers [24, 26], we formulate a mas-ter equation for the dynamics of Merrifield excitons [31, 68],i.e., a strongly coupled electron-hole pair. Accordingly, ourapproach can be seen as a direct extension of the master equa-tions used in Refs. [19, 36–38] for the dynamics of charge-neutral Frenkel excitons [69], characteristic of OSCs with lowdielectric constant.Furthermore, our model, introduced in Sec. II, provides arigorous thermodynamic description of exciton-phonon inter-actions, overcoming the limitations of the phenomenologicalapproach used in Ref. [19]. There, the authors fit the masterequation parameters to MCTDH calculations carried out onsmall two-monomer subsystems over short time scales. Suchapproach, affected by artificially short recurrence times, mayfail to correctly reproduce the dynamics for long time scalesand returns decoherence rates that do not necessarily obeythermodynamics in the long-time limit. Here, instead, weuse the Bloch-Redfield formalism to calculate temperature-dependent decoherence rates from the phonons’ correlationfunctions [53, 70], to then express the dynamics of the exci-ton’s reduced state using a master equation of the Lindbladform [35, 40–42, 54]. This approach has the additional ad-vantage of providing ensemble average transport propertieswithout calculating many individual stochastic realisations ofsystem-environment dynamics.In Sec. III, we use the model to study the exciton transportproperties of some representative oligo and poly(p-phenylenevinylene) (OPV and PPV) systems, across different mor-phologies, from crystalline to highly disordered. Instead of re-lying on material-specific modelling of polymers, we expressour calculations in terms of electronic couplings and thermalenergy. This allows us to explore the impact that disorder andtemperature have on localisation, charge-separation dynam-ics and equilibration time scales. Exciton transport is there-fore interpreted as a thermalisation process [53, 54, 70–73],whose features vary with the relative strength between ther-mal energy, electronic couplings and disorder. We show howsuch differences in electronic coupling determine whether thephonon bath cools or heats the system, with dramatic effectson exciton transport properties. We conclude discussing ourresults in Sec. IV, where we compare and contrast our find-ings with those of other works, and outlining the use of ourmethod to exciton transport problems of great importance foroptoelectronic applications. II. METHODOLOGY
The exciton transport model that we introduce in this article isbased on the following two premises. First, the characteristictimes and energies associated with exciton transport in OSCsat room temperature and natural illumination imply that ex-citons do not interact with each other, as they are sufficientlyspatially and temporally separated [15]. Under these circum-stances, it is a standard approach to focus on the transport ofa single electronic excitation, formally defined by the single-exciton manifold [15, 19, 20, 31, 36–38]. Second, we considerpolymeric materials whose structural changes are far slower
FIG. 1. Polymeric materials are here modelled as a network of N nodes, each of which represents a monomer. Monomers positions (cid:126)r k and conformational coordinates Q k define the morphology ofthe medium and the structure of the exciton’s Hamiltonian. Excitontransport on such network is mediated by through-bond couplingsbetween conjugated monomers that are part of the same polymer, orby through-space couplings ( gray line ) between pairs of monomersin proximity of each other. than the characteristic femtosecond to nanosecond time scalesof exciton dynamics, such as for the paradigmatic case of PPVand its amorphous derivative MEH-PPV [24, 26]. This allowsus to adopt a snapshot approach for a medium’s morphology,which is assumed to be time-independent. Additionally, weassume that the interaction between excitons and rapidly os-cillating conformational coordinates like bond-length alterna-tions, here modelled as a bath of independent harmonic os-cillators, are well described in the weak coupling approxima-tion [15, 35].Under these assumptions, the transport of a single elec-tronic excitation in OSCs is typically modelled as a quantumstochastic walk (QSW) of a charge-neutral Frenkel excitonover a network of N nodes [74, 75], each of which repre-sents a monomer or a chromophore [15, 19, 35, 37]. How-ever, motivated by recent results on the electronic excitationsof amorphous polymers [24, 26], we here generalise this stan-dard approach and extend the dynamics to charge-separatedexciton states using the Merrifield model [31]. In this formal-ism the states of the single-exciton manifold are here givenby the product of the localised single-electron and single-holebases, B := {| jk (cid:105)} Nj,h =1 , (1)where | jk (cid:105) := | j (cid:105) e ⊗ | k (cid:105) h , and where | j (cid:105) e ( | k (cid:105) h ) repre-sents the state of the electron (hole) localised on monomer j ( k ) of a polymeric material. Every monomer k is also as-sociated with its centre’s position (cid:126)r k := ( x k , y k , z k ) and aset of local conformational coordinates Q k , such as torsionalangles and bond-length alternations, as schematically repre-sented in Fig. 1. These, together with the bonds betweenpairs of monomers, define the morphology of the medium anddetermine the structure of the exciton’s Hamiltonian, as de-scribed in the next section. A. Exciton Hamiltonian
The exciton Hamiltonian H ex = H + H F + H CT is given bythe Coulomb term H , which defines the energetic landscapeassociated with electron-hole configurations on the polymericmaterial, and the Frenkel H F and charge H CT transfer terms,which model the coherent transport of energy and charges, re-spectively. In the electron-hole basis of Eq. (1), the Coulombterm H reads (cid:104) eh | H | e (cid:48) h (cid:48) (cid:105) = δ ee (cid:48) δ hh (cid:48) (cid:18) E − π(cid:15) (cid:15) r d eh (cid:19) , (2)where E − = (4 π(cid:15) (cid:15) r r ) is the exciton binding energyof electron-hole pairs that are localised within the samemonomers, i.e., Frenkel excitons. The electron-hole distance, d eh = r + (cid:107) (cid:126)r e − (cid:126)r h (cid:107) , is obtained from the distance betweentwo monomers offset by the intrinsic electron-hole distance r for Frenkel excitons in the considered polymer [31]. TheCoulomb term is diagonal in the electron-hole basis, and van-ishes for Frenkel excitons. The morphology of the mediumintroduces static disorder in this term via the inter-monomerdistance d eh .The Frenkel term H F models the coherent transport ofFrenkel excitons and is given by (cid:104) eh | H F | e (cid:48) h (cid:48) (cid:105) = δ eh δ e (cid:48) h (cid:48) V F ( (cid:126)r e , Q e ; (cid:126)r e (cid:48) , Q e (cid:48) ) , (3)where V F ( (cid:126)r j , Q j ; (cid:126)r k , Q k ) is the strength of the Frenkel cou-pling between a pair of monomers j, k , as a function ofthe morphology of the medium. Frenkel transfer terms pre-serve charge separation, and can only mediate the transportof Frenkel exciton states, which form a N dimensional subsetof the full N dimensional Hilbert space associated with thesingle-exciton manifold of the Merrifield model.Similarly, the charge transfer (CT) term H CT models thecoherent transport of individual charges and is given by (cid:104) eh | H CT | e (cid:48) h (cid:48) (cid:105) = δ ee (cid:48) V CT ( (cid:126)r h , Q h ; (cid:126)r h (cid:48) , Q h (cid:48) )++ δ hh (cid:48) V CT ( (cid:126)r e , Q e ; (cid:126)r e (cid:48) , Q e (cid:48) ) , (4)where V CT ( (cid:126)r j , Q j ; (cid:126)r k , Q k ) represents the strength of the CTcoupling between a pair j, k of monomers, as a function of themorphology of the medium. The CT term can map Frenkelstates into charge-separated states and vice versa, and its ad-dition to the Hamiltonian allows exciton states to explore theentirety of the single-exciton manifold.The dependence of the coupling strengths V F and V CT on themorphology of the medium is described in terms of the rela-tive arrangement of pairs of monomers that are conjugated,or close enough to each other to allow for weaker through-space couplings. Without loss of generality, V F and V CT areassumed to be symmetric in the pairs’ indices and to vanish formonomers that are not connected by a bond, or out of range forthrough-space interactions. An example of the dependence of V F and V CT on the morphology for PPV-like conjugated poly-mers is given by Eqs. (14) and (15) in Sec. III. B. Exciton-phonon interaction
Exciton-phonon interactions induce decoherence and relax theexciton states until a steady state is reached. The composite dynamics of exciton and phonons is governed by the system-environment Hamiltonian H = H ex + H ex-ph + H ph . The vi-brational modes involved in this process, such as bond-lengthalternations and ring-breathing modes [19, 24, 26], are mod-elled as a bath of local and independent harmonic oscilla-tors [15, 19]. Let us write the exciton-phonon interactionHamiltonian H ex-ph as H ex-ph = (cid:88) l A l ⊗ B l , (5)such that each coupling operator A l acting on the excitonspace is associated with a bath operator B l = (cid:88) m g l,m (cid:16) b † l,m + b l,m (cid:17) (6)given by a sum over the displacement operators associatedwith the modes with frequencies ω l,m , written in terms of cre-ation and annihilation operators. The tensor g l,m representsthe coupling strengths between the modes ω l,m and the ex-citon coupling operators A l . These are assumed to be weak ,with respect to the characteristic energies of H ex and H ph . Inthis notation the phonon Hamiltonian reads H ph = (cid:88) l,m (cid:126) ω l,m (cid:18) b † l,m b l,m + (cid:19) . (7)The exciton coupling operators A l model the exchange ofenergy between the exciton and the vibrational modes. Thesecan either involve a single monomer, or a bond between twomonomers. For example, the interaction between a Frenkelstate | kk (cid:105)(cid:104) kk | and ring-breathing modes concerns an individ-ual monomer, here labelled by k . Conversely, the interac-tion between Frenkel transfer terms | jj (cid:105)(cid:104) kk | + h.c. and bond-length alternation modes is associated with the bond betweena pair j, k of monomers. The set of exciton coupling operators A l considered in this model is given in Table I. C. Master equation
The dynamics of the reduced exciton state ρ t is governedby a Lindblad master equation [76] that is obtained fromthe system-environment Hamiltonian H by tracing over thephonon environment [53] ˙ ρ t = − i (cid:126) [ H ex , ρ t ] + (cid:88) ω,l γ l ( ω ) (cid:18) A l ( ω ) ρ t A † l ( ω )++ 12 (cid:110) A † l ( ω ) A l ( ω ) , ρ t (cid:111)(cid:19) . (8)The Lindblad collapse operators A l ( ω ) are here given by thespectral representation of the coupling operators A l , A l ( ω ) = (cid:88) E λ (cid:48) − E λ = ω | E λ (cid:105)(cid:104) E λ | A l | E λ (cid:48) (cid:105)(cid:104) E λ (cid:48) | , (9)where | E λ (cid:105) are the eigenstates of the exciton Hamiltonian, H ex | E λ (cid:105) = E λ | E λ (cid:105) , such that ω matches the Bohr fre-quencies E λ (cid:48) − E λ . The relaxation rates γ l ( ω ) are obtainedfrom the Fourier transform of the bath correlation functions (cid:104) B l ( t ) B l (cid:48) ( t (cid:48) ) (cid:105) = Tr[ B l ( t ) B l (cid:48) ( t (cid:48) ) ρ ph ] [53, 54]. Here, ρ ph =exp( − βH ph ) Z − is the steady state of the phonon bath, inthermal equilibrium at inverse temperature β = ( k B T ) − ,with partition function Z = Tr[exp( − βH ph )] [54].This general approach, also known as the Bloch-Redfieldformalism, allows to obtain relaxation rates from the energyand temperature dependent correlation functions of the bath.The rates obtained this way satisfy thermodynamics and de-tailed balance condition, leading to the correct thermal equi-librium distribution for long time scales [70]. Here, we obtaina prescription for the dynamics of exciton states in the Marko-vian approximation, under which condition the master equa-tion can be equivalently expressed in either Bloch-Redfield orLindblad formalism [70].In the Markovian limit that characterises Eq. (8), the bathcorrelation functions only depend on the time interval τ be-tween two time steps t − t (cid:48) = τ [53, 54]. Moreover, in virtueof the mutual independence of the vibrational modes for eachpair of coupling operators A l , A l (cid:48) , the bath correlation func-tions reduce to (cid:104) B l ( τ ) B l (cid:48) (0) (cid:105) = δ ll (cid:48) (cid:88) m g l,m (cid:18)(cid:16) ν β (cid:0) ω l,m (cid:1)(cid:17) e − iω l,m τ ++ ν β (cid:0) ω l,m (cid:1) e iω l,m τ (cid:19) , (10)where ν β ( ω ) = (exp( (cid:126) βω ) − − is the bosonic distributionfunction at inverse temperature β [53, 54].To calculate the rates γ l ( ω ) , the sum over the couplings g l,m of Eq. (10) is replaced with an integral in the frequency do-main, (cid:80) m g l,m → (cid:82) J l ( ω ) dω , where J l ( ω ) is a spectral den-sity that can be used to sample the coupling strengths g l,m . Todo so we use an Ohmic spectral density J l ( ω ) J l ( ω ) = Λ l (cid:126) ω Ω l exp (cid:18) − ω Ω l (cid:19) , (11)where Λ l = (cid:126) (cid:82) ∞ dωJ l ( ω ) /ω is the reorganisation energyassociated with coupling operator A l [35, 53]. In this way therelaxation rates read γ l ( ω ) = 2 π (cid:18) J l ( ω )(1 + ν β ( ω )) + J l ( − ω ) ν β ( − ω ) (cid:19) . (12)Exciton-phonon interactions also introduce further disor-der, which is captured by energy fluctuations δE l in the ex-citon Hamiltonian, H ex → H ex + (cid:80) l δE l A l [34, 43]. Theseare sampled from a normal distribution with zero average andstandard deviation σ l ( β ) , given by σ l ( β ) = (cid:90) ∞ dω (cid:126) J l ( ω ) coth (cid:18) (cid:126) βω (cid:19) , (13)as done in Ref. [42, 77]. The addition of such energy fluctua-tions affects site energies and coherent transfer terms, leading Exciton coupling operators ( A l ) Site k Bonds j, k
Frenkel | kk (cid:105)(cid:104) kk | | jj (cid:105)(cid:104) kk | + h.c. Electron | k (cid:105)(cid:104) k | e ⊗ h ( | j (cid:105)(cid:104) k | e + h.c. ) ⊗ h Hole e ⊗ | k (cid:105)(cid:104) k | h e ⊗ ( | j (cid:105)(cid:104) k | e + h.c. ) TABLE I. Exciton coupling operators A l classified by their action onFrenkel, electron, and hole states and transfer terms. to asymmetry in the charge separation landscape and promot-ing localisation in the centre of mass dyamics [49, 78, 79].In the next section we use Eq. (8) to study exciton transportin conjugated polymers across different morphologies. Beforediscussing the results, let us make a few remarks on the ap-proximations that characterise Eq. (8). First of all, it is worthpointing out that, in Eq. (8) we have neglected the Lamb-ShiftHamiltonian term. This term accounts for the contributionsof the coupling operators A l to the coherent dynamics of theexciton. Lamb-Shift corrections have often been neglectedin exciton transport problems, especially for the case of localand uncorrelated phonon baths [35, 37, 38]. When applyingEq. (8) in Sec. III we will assume the exciton Hamiltonian toalready include energy shift arising from the weak couplingwith the phonon environment. There, Lamb-Shift correctionsare expected to be well below the precision limit of a qualita-tive study focused on order-of-magnitude differences in exci-ton energies and electronic couplings.Another important observation is that we have assumed thevibrational modes to be uncoupled with each other, whichis the reason why Eq. (8) does not feature combinations of A l ( ω ) and A l (cid:48) ( ω ) Lindblad operators. It is reasonable toimagine that this approximation would not be justified forsome combinations of coupling operators, such as for thosevibrational modes coupled with the same monomer. Whenspatial correlations become important, e.g., when phononmodes are delocalised over several monomers, they can besimply accounted for and included into Eq. (8) via the ad-dition of terms in A l and A l (cid:48) , and the associated rates aris-ing from the spatial dependence of the correlation functions (cid:104) B l ( t ) , B l (cid:48) (0) (cid:105) [80]. The presence of strong and long-rangespatial correlations allows for the formation of decoherencefree subspaces and gives rise to super and subradiance phe-nomena [81], which have been studied for biological light-harvesting complexes [70], but remain rather unexplored forthe case of organic polymers.Finally, the weak coupling approximation for the exciton-phonon interaction, necessary to obtain Eq. (8), is arguablythe main source of deviations from a rigorous description ofthe exciton transport. While some vibrational modes, such asthose associated with torsional angles and foldings of a poly-mer chain, are very likely to be well described by a weak cou-pling approximation, others modes, such as bond-length alter-nations, can couple strongly with the excitons. In such cases itis possible to overcome the limitations imposed by Eq. (8) byperforming a polaron-transformation [82–85], which modelsthe dynamics of an exciton followed by nuclear deformations, Exciton Hamiltonian parameters — OPV, PPV
Parameter Hartree AU
Frenkel electron-hole distance r . Exciton biding energy E . × − Frenkel transfer parameters j . × − j . × − j . × − Charge transfer parameter t . × − Relative permittivity (cid:15) TABLE II. Exciton Hamiltonian parameters used in Eqs. (2), (14),and (15) for the OPV and PPV-like materials considered inSecs. III A and III B, expressed in Hartree atomic units [31]. Frenkeltransfer parameters have been scaled in Sec. III C in order to studythe dependence of transport properties on the strength of the elec-tronic couplings. i.e., a polaron (or dressed-exciton ), using master equationssimilar to Eq. (8), such as the secular polaron-transformedRedfield equation (sPTRE) [33, 34, 43, 86–91].
III. RESULTS
We now use Eq. (8) to study room-temperature exciton trans-port in some representative conjugated oligomers and poly-mers, exploring the transition from ordered to disordered mor-phologies. To compare our findings with previous results, webase our calculation upon the electronic properties of OPVand PPV, used in Refs. [19, 31]. In particular, we focus on thefollowing systems. First, we consider a short OPV hexamer toexplore centre of mass (CoM) and charge separation (CS) dy-namics with a full Merrifield exciton approach, in Sec. III A.Then, in Sec. III B, we model a long PPV polymer with 51repeated unit to explore intrachain Frenkel exciton dynamicsacross different morphological regimes. Finally, we examinethe role of intrachain and interchain transport and its depen-dence on electronic coupling strength in Sec. III C, by lookingat Frenkel exciton transport in OPV oligomers.The different regimes of disorder, from crystalline to amor-phous , are here achieved sampling torsional angles and bondlength-alternations from a normal distribution with zero av-erage and tunable standard deviation. The functional de-pendence of Frenkel and CT terms on the relative arrange-ment between two conjugated monomers, given by V F ( d, θ ) = f ( d )Θ F ( θ ) and V CT ( d, θ ) = f ( d )Θ CT ( θ ) , respectively, is cal-culated upon their distance d and the relative angle θ betweenthe two monomers’ planes. The angular dependence Θ F ( θ ) and Θ CT ( θ ) is modelled as in Ref. [31], Θ F ( θ ) = ( j − j ) cos θ + ( j − j ) cos θ − j , (14) Θ CT ( θ ) = t cos θ, (15)while the dependence on the intra-monomer distance isheuristically modelled with a Gaussian factor f ( d ) =exp[( d − r ) / r ] . FIG. 2. Exciton transport for different instances of OPV hexamermorphologies at room temperature T = 300 K . For each case theinitial state ρ is a Frenkel exciton localised on the first monomer.Expectation value (solid lines) and standard deviation (shaded areas)of CoM (blue) and CS (red) are expressed in the sites basis. Theenergy required for charge dissociation (dotted-dashed magenta line)is shown in eV (on the same y -axis as CS). The histograms insetsschematically represent the populations (not in scale) of the partialstates of electron (black bars) and hole (light gray bars) on each siteat a given time. The parameters used in Eqs. (2), (14), and (15) are givenin Tab. II and expressed in Hartree atomic units, while therelative permittivity of the considered materials is assumedto be (cid:15) r = 1 for simplicity. The characteristic electroniccouplings associated with Frenkel and charge-transfer termscan be as high as 2 eV, and, thus, much larger than the 26meV associated with the thermal energy of the phonon bath atroom temperature of 300K. When required, Frenkel through-space couplings are included for non-conjugated monomersthat are separated by ≈ r , with characteristic energies around10 meV [19, 92].Reorganisation energies Λ l and cut-off frequencies Ω l as-sociated with vibrational modes define relaxation rates γ l ( ω ) and energy fluctuations σ l ( β ) , as prescribed by Eq. (11) [53,54]. For the considered systems we have chosen reorgani-sation energies of 500 meV and 50 meV for monomer-localand bond-local vibrational modes, respectively. The chosencut-off frequencies are 1500 cm − for modes that couple withFrenkel states, 1000 cm − for modes that couple with indi-vidual charges and Frankel transfer terms, and 500 cm − formodes that couple with CT terms [19, 32, 93, 94].To propagate an initial exciton state ρ we exactly solve thesystem of linear differential equations associated with Eq. (8)written in Liouville space [95]. In this way we obtain a map Λ t for the dynamics of the state ρ t = Λ t [ ρ ] . Exciton trans-port properties are studied evaluating time-dependent expec-tation value and standard deviation of CoM and CS opera-tors. These can be expressed in terms of monomer indices FIG. 3. (
Left ) Exciton transport for crystalline ( top-left ) and amorphous ( bottom-left ) PPV polymers with 51 monomers, at room temperature T = 300 K . For each case the initial state ρ is a Frenkel exciton localised on the central monomer ( k = 26) . Expectation value (solid lines)and standard deviation (shaded areas) of CoM (blue) are expressed in the sites basis. Delocalisation is monitored by purity (dashed light bluelines) and average IPR in the sites basis (dashed-dotted red line). The histograms insets schematically represent the populations (not in scale)of the exciton on each site at a given time. (
Right ) The dependence of steady state delocalisation on the morphology is studied varying thestandard deviation σ θ of random torsional angles, and sampling 100 PPV polymers with 51 repeated units for each value of σ θ . (cid:104) eh | X CoM | e (cid:48) h (cid:48) (cid:105) = δ ee (cid:48) δ hh (cid:48) ( e + h ) / , and (cid:104) eh | X CS | e (cid:48) h (cid:48) (cid:105) = δ ee (cid:48) δ hh (cid:48) ( e − h ) , as done in Ref. [31]. Alternatively, they can beexpressed in terms of monomer coordinates (cid:126)r k = ( x k , y k , z k ) ,by replacing e ( h ) with a given Cartesian coordinate, such as x e ( x h ).Localisation properties of Frenkel states are inferred from purity , P [ ρ t ] = Tr[ ρ t ] , and average inverse participation ra-tio ( IPR ) [29, 34, 66, 94, 96]. Let ρ = (cid:80) k p k | r k (cid:105)(cid:104) r k | beexpressed in its eigenbasis {| r k (cid:105)} , and let {| n (cid:105)} be the sitesbasis then IPR( ρ ) = (cid:88) k p k (cid:18) (cid:88) n |(cid:104) n | r k (cid:105)| (cid:19) − , (16)i.e., the weighted sum of the IPR = 1 / (cid:80) n |(cid:104) n | ψ (cid:105)| for eachpure state | r k (cid:105) making up the density operator ρ [34, 97].From the IPR , delocalisation length can be estimated using l = (IPR( ρ )) /d for a d -dimensional system [34]. Since theconsidered systems are either one-dimensional or ensemblesof coupled one-dimensional systems we estimate the delocal-isation length simply using IPR( ρ ) . A. OPV hexamer
This system consists of six conjugated monomers and it isstudied with a full Merrifield exciton approach, i.e., includingall the possible CS states. For each morphology, we initialisethe exciton in a Frenkel state localised on the first monomer.For comparison, a similar system was studied in Ref. [31] witha MCTDH method, requiring around configurations for itsnumerical implementation, as opposed to the 36 required forthe solution of Eq. (8) [98]The crystalline morphology, shown in Fig. 2 ( top ), is ob-tained for a perfectly planar arrangement of the monomers with constant bond-length alternation and no static disorder.It is characterised by trivial CS dynamics, which remains con-stant and equal to zero for all times. The exciton undergoesa initial ultrafast ballistic transport regime (1-10 fs), followedby a diffusive transport regime that lasts for the first 100 fs.During the evolution the partial states of electron and holeare perfectly symmetric, as shown in the histograms insets ofFig. 2 representing the population (not in scale) of the partialstates of electron (black bars) and hole (light gray bars) oneach site at a given time.The semicrystalline morphology, an instance of which isshown in Fig. 2 ( middle ), is obtained for torsional angles nor-mally distributed with standard deviation of 0.5 rad. The pres-ence of disorder allows for a non trivial CS dynamics, thusreducing the energy barrier required for charge dissociation(CD) of the exciton. Despite the CoM dynamics has similarfeatures to those of the crystalline morphology, asymmetry inthe partial states of electron and hole is generally present bothduring the evolution and in the steady states.The amorphous phase, an instance of which is shown inFig. 2 ( bottom ), is obtained for torsional disorder with stan-dard deviation of 1 rad. It is characterised by noticeableCS dynamics, with electron-hole separation over more than2 monomers, lasting for hundreds of femtoseconds. SuchCS dynamics can considerably reduce the CD energy for rel-atively long time spans of hundreds of femtoseconds. Re-markably, high-disorder allows for the formation of long-livednon-equilibrium states, before thermal equilibrium is reachedaround 100 ps. Exciton lifetimes may be shorter than the timerequired to reach thermal equilibrium in such systems. Thisimplies that thermal equilibrium states of the excitons are gen-erally not experimentally accessible. FIG. 4. (
Left ) Exciton transport for an arrangement of three semicrystalline OPV octamers with weak (10 – 100 meV) ( top-left ) and strong(0.1 – 1 eV) ( bottom-left ) electronic couplings at room temperature T = 300 K . For each case the initial state ρ is a Frenkel exciton localisedon first monomer of the first octamer. Expectation value (solid lines) and standard deviation (shaded areas) of CoM (blue) are expressed inthe sites basis. Delocalisation is monitored by purity (dashed light blue lines) and average IPR in the sites basis (dashed-dotted red line). Theinsets schematically represent the populations of the exciton on each site at a given time. (
Right ) The dependence of exciton delocalisation onthe electronic couplings is studied varying the characteristic strength of Frenkel couplings from 10 meV to 1 eV. For each coupling regime wesampled 100 OPV oligomers with 10 repeated units and semicrystalline morphology ( σ θ = 0 . rad). Delocalisation is studied by evaluating IPR , purity and standard deviation of CoM for steady states and for states at 100 fs, with ρ initialised in a Frenkel state localised on the firstmonomer. B. PPV polymer
We now consider a PPV polymer with 51 repeated units. Werestrict the dynamics to the Frenkel manifold to study thetransport regimes of the CoM while limiting the computa-tional cost. Excitons are here initialised in a Frenkel statelocalised on the central monomer ( k = 26 ) of each consid-ered polymer.The crystalline morphology, shown in Fig. 3 ( top-left ), ischaracterised by a transition from ballistic to diffusive excitontransport during the first 100 fs, in analogy with the resultsof Ref. [19]. The remaining part of the dynamics consists ina relaxation process that guides the exciton towards thermalequilibrium. The absence of disorder in the sites energies andelectronic couplings allows for the formation of highly delo-calised steady states, with IPR > .However, in contrast with the results of Ref. [19], steadystate populations are not evenly spread across the polymer,and thus only partially mixed. This is because the decoher-ence model used in Ref. [19] does not account for the spectralresponse between exciton coupling operators and the phononbath. The results of Ref. [19] can be qualitatively reproducedusing Eq. (8) by replacing the Lindblad operators A l ( ω ) withthe coupling operators A l , and the associated relaxation rates γ l ( ω ) with frequency independent rates γ l . In such simpli-fied limit, steady states become diagonal in the sites’ basis.Here, instead, we use a Bloch-Redfield approach for rates anddissipators and obtain the correct thermal equilibrium states,which are diagonal in the exciton Hamiltonian basis. As disorder is increased, localisation becomes more evi-dent, undermining the ultrafast ballistic transport regime thatwould otherwise characterise a crystalline morphology. Theamorphous morphology, an instance of which is shown inFig. 3 ( bottom-left ), is characterised by sub-diffusive excitontransport within the first 100 fs, followed by thermal relax-ation. States are remarkably less delocalised both during thedynamics and at equilibrium, with IPR ≈ and CoM stan-dard deviations much lower than for the crystalline morphol-ogy.To further examine the relation between morphology andexciton delocalisation we study the IPR of steady states atroom temperature for different morphological regimes. To doso we vary the standard deviation σ θ for the random torsionalangles between 0 (crystalline) and 1 rad (amorphous), sam-pling 100 different PPV polymers with 51 monomers for eachmorphology. The results, presented in Fig. 3 ( right ), show thehigh sensitivity of exciton delocalisation to conformationaldefects. Delocalisation rapidly drops from IPR ≈ for crys-talline polymers to IPR ≈ for amorphous ones. C. Dependence on electronic couplings
We now study the dynamics of Frenkel excitons for fixedmorphology while varying the strength of the electronic cou-plings. This allows us to explore the different regimes ofexciton transport and thermal equilibrium that are associatedwith weak (10 – 100 meV) or strong (0.1 – 1 eV) electroniccouplings. First, we illustrate such difference by looking atFrenkel exciton dynamics for a system given by three closelyarranged OPV octamers that interact via through-space cou-plings, shown in Fig. 4 ( left ).Systems characterised by weak electronic couplings rapidlylose their coherence, which vanishes within the first 10 – 20 fs.As shown in Fig. 4 ( top-left ), exciton transport is incoherentboth during an initial intrachain transport over one oligomerand during the slower interchain transport across the differentoligomers. The thermal energy is sufficient to populate severaleigenstates of the exciton Hamiltonian. Thermal equilibriumis therefore characterised by low-purity and large CoM stan-dard deviation.Strong electronic couplings ( ≈ eV) also display ultrafastintrachain dynamics, followed by a slower interchain trans-port. However, the thermal energy is not high enough to pop-ulate several states of the exciton Hamiltonian, therefore thesteady states are rather pure and localised around the mostenergetically favourable clusters of conjugated monomers, asshown in Fig. 4 ( bottom-left ).To systematically explore the dependence of exciton de-localisation on the strength of the electronic couplings (andthus on the thermal energy) we study IPR , purity and CoMstandard deviation of transient (100 fs) and steady statesof semicrystalline oligomers. The characteristic strength ofFrenkel couplings is varied from 10 meV to 1 eV. For eachcoupling regime we sample 100 OPV oligomers with 10 re-peated units and fixed semicrystalline morphology ( σ θ = 0 . rad). As shown in Fig. 4 ( right ), the average IPR decreasesonly slightly ( ≈ ) over two orders of magnitude ofelectronic couplings strength. However, transport propertieschange dramatically, with CoM standard deviation and purityvarying remarkably for both transient and steady states. D. Final remarks
Using the model introduced in this article the dynamics ofexcitons in conjugated polymers is understood as a quantumthermalisation process, whose features strongly depend on theamount of disorder (i.e., morphology) and on the relative mag-nitude of thermal energy and electronic couplings (i.e., tem-perature). The non-equilibrium dynamics is characterised byan ultrafast and ballistic transport transient, followed by an in-termediate diffusive or sub-diffusive process that occurs boththrough-bond and through-space. The dynamics culminateswith thermal relaxation. Increasing disorder enables localisa-tion, thus decreasing delocalisation lengths and
IPR and hin-dering the efficacy of exciton transport. The amorphous phaseis characterised by the presence of long-lived non-equilibriumstates and prominent charge-separation dynamics, which how-ever is limited to a few monomers, as previously reported inRef. [24].In polymers characterised by weak electronic couplings,of the order 10 – 100 meV, the environment heats the sys- tem, driving the excitons low-purity thermal states, with non-negligible populations across the whole spectrum of energyeigenstates of the exciton Hamiltonian. Even though the in-trinsic localisation lengths are low due to the presence of dis-order, the thermal energy of the phonon bath (around 26 meVat room temperature) is sufficient to populate monomers thatare far apart from the initial exciton location. In contrast, inmaterials characterised by strong Frenkel and charge-transfercouplings ( ≈ eV), the phonon environment cools the sys-tem, leading the excitons to high-purity thermal state, withlow-energy eigenstates of the exciton Hamiltonian being theonly few populated ones. For this reason, despite the slowerintrachain transport, systems with weak electronic couplingscan display a more efficient transport mechanism over the pi-cosecond time-scale at room temperature. This becomes par-ticularly important for triplet excitons, which typically havelower mobility and longer lifetime than singlet excitons. IV. CONCLUSIONS
In this article we have introduced a general master equationto study the dynamics of Merrifield excitons in conjugatedpolymers as a function of temperature and morphology of themedium. Using this method we have explored the generalfeatures of energy and charge transport in some representa-tive systems based on PPV’s electronic properties, confirm-ing well known paradigms of quantum transport in disorderedsystems, and revealing non-equilibrium features such as long-lived states and charge-separation dynamics.Beyond the qualitative understanding of exciton dynamicsin amorphous polymers, we expect our method to be appli-cable for the quantitative study of energy and charge trans-port properties of specific materials. This can be done us-ing a multi-scale approach based on molecular dynamics andfirst-principle calculations, as demonstrated in Refs. [19, 41,42]. We also anticipate that this approach could be used tostudy disorder-dependent effects in OSCs, such as trapped-charge induced photoluminescence peak displacement [99,100] and individual charge-carrier pathways in amorphouspolymers [101].
ACKNOWLEDGMENTS
This research was supported by the Australian ResearchCouncil under grant number CE170100026. FC thanks IgorLyskov and Ivan Kassal for insightful discussions. This re-search was undertaken with the assistance of resources fromthe National Computational Infrastructure (NCI), which issupported by the Australian Government. The authors ac-knowledge the people of the Woi wurrung and Boon wurrunglanguage groups of the eastern Kulin Nations on whose un-ceded lands we work. We respectfully acknowledge the Tra-ditional Custodians of the lands and waters across Australiaand their Elders: past, present, and emerging. [1] H. Hoppe and N. S. Sariciftci, Polymer solar cells, Advancesin Polymer Science , 1–86 (2008).[2] S. Zhang, Y. Qin, J. Zhu, and J. Hou, Over 14% Efficiency inPolymer Solar Cells Enabled by a Chlorinated Polymer Donor,Advanced Materials (2018).[3] G. Wang, F. S. Melkonyan, A. Facchetti, and T. J. 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