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Entanglement entropy in organic semiconductors

Yao Yao1∗1Department of Physics and State Key Laboratory of Luminescent Materials and Devices,

South China University of Technology, Guangzhou 510640, China

(Dated: September 17, 2019)

Different from traditional semiconductors, the organic semiconductors normally possess moder-ate many-body interactions with respect to charge, exciton, spin and phonons. In particular, thediagonal electron-phonon couplings give rise to the spatial localization and the off-diagonal cou-plings refer to the delocalization. With the competition between them, the electrons are dispersivein a finite extent and unfavorable towards thermal equilibrium. In this context, the quantitiesfrom the statistical mechanics such as the entropy have to be reexamined. In order to bridge thelocalization-delocalization duality and the device performance in organic semiconductors, the quan-tum heat engine model is employed to describe the charge, exciton and spin dynamics. We adoptthe adaptive time-dependent density matrix renormalization group algorithm to calculate the timeevolution of the out-of-time-ordered correlator (OTOC), a quantum dynamic measurement of theentanglement entropy, in three models with two kinds of competing many-body interactions: two-bath lattice model with a single electron, Frenkel-charge transfer mixed model, and the Merrifieldmodel for singlet fission. We respectively investigate the parameter regime that the system is in themany-body localization (MBL) phase indicated by the behavior of OTOC. It is recognized that thenovel effects of coherent electron hopping, the ultrafast charge separation and the dissociation oftriplet pairs are closely related to the MBL effect. Our investigation unifies the intrinsic mechanismscorrelating to charge, exciton and spin into a single framework of quantum entanglement entropy,which may help clarify the complicated and diverse phenomena in organic semiconductors.

I. INTRODUCTION

Organic semiconductors are increasingly attractive inthe field of photoelectricity, as the organic light emit-ting diodes have been successfully industrialized and theorganic photovoltaics and spintronics are of high poten-tial in applications [1]. Thanks mainly to the cost con-tainment, the single crystal phase of organic materials israrely present in practice and the utilization of molecu-lar aggregates are always in fashion up to date [2]. Theubiquitous disorders induced by the local thermal vibra-tions of molecules (phonons) intuitively give rise to thelocalization of every eigen-states of electrons with the lo-calization length being one molecule [3, 4]. In presenceof the strong nonlocal electron-phonon (vibronic) cou-plings or the so-called off-diagonal dynamic disorders, thecharge transport can be bandlike implying the electronis in some sense delocalized [5, 6]. The delocalizationis also regarded to be essential in the charge separationin photovoltaics [7–10]. The localization and delocaliza-tion duality of electrons in organic semiconductors long-termly serves as the major puzzle for the community [11].On the experimental side, in order to distinguish the twophases, one can measure the mobility-temperature rela-tionship in a macroscopic manner and assign the posi-tive and negative dependency to localized and delocalizedphase, respectively [12, 13].

In a normal insulator, the Anderson localization playsthe essential role in the electron transport [14]. With

∗Electronic address: yaoyao2016@scut.edu.cn

relatively weak localization, Mott’s variable-range hop-ping (VRH) mechanism matters [15]. In terms of delo-calized Bloch waves in traditional semiconductors, theband theory achieved great successes. All these theo-ries are more or less correlated to the statistical mechan-ics of (near) equilibriums. The picture of free electrongas is normally applicable, and a single term such asthe static disorder or the single electron hopping in thelattice dominates the transport mechanisms. The tra-ditional techniques of transport measurement were thusestablished on the basis of these widely-accepted theo-ries. In presence of strong many-body interactions suchas the electron-phonon couplings, however, the thermalequilibrium is not easy to be achieved, and the dynamicalprocesses out of equilibration become important. More-over, there is not an apparent small parameters in or-ganic semiconductors to make the perturbation theoryavailable, and the competition among different terms re-sults in complicated many-body effects [11]. As a result,we have to develop a quantum dynamics theory for or-ganic semiconductors to bridge the microscopic mecha-nisms and the traditional device measurements.

Theoretically, the phonons are modeled with a bosonicenvironment which may diagonally and/or off-diagonallycouple to the electrons. Accordingly, the most sim-plest but highly nontrivial model is the so-called spin-boson model [16, 17], that is extensively utilized tothe researches of localization and delocalization of elec-trons. In the study of two-bath spin-boson model, anew phase named critical phase was uncovered, whichis distinguished from both the localized and delocalizedphases [18]. When the diagonal and off-diagonal electron-phonon couplings are equal, it can be proven that theground state of the system is highly degenerate due to the

http://arxiv.org/abs/1909.06742v1yaoyao2016@scut.edu.cn

2

FIG. 1: Schematic for nuclear potential surfaces in localized(LP), delocalized (DP) and critical (CP) phases. The general-ized coordinate is set to z direction for the diagonal couplingand x direction for the off-diagonal coupling. |1〉 and |2〉 arethe local states on molecules labeled as 1 and 2. The min-imums (min.) of the potential surface are plotted on z − xplane as well.

parity symmetry. Fig. 1 shows a scheme of the nuclearpotential surfaces in three phases. We take two moleculeslabeled as 1 and 2 as instance and set the generalized co-ordinate on z direction for diagonal couplings and x di-rection for off-diagonal couplings. In the localized phase,the minimums of the potential surface appear on the zaxis and entangle with the local states on molecule 1 and2, respectively. In the delocalized phase, they appearon the x axis and entangle with two delocalized states,namely the bonding state |1〉+ |2〉 and the anti-bondingstate |1〉 − |2〉. On the lattice of more molecules, the de-localized states become Bloch waves. Of more interestsis the critical phase with equal diagonal and off-diagonalcouplings. An infinite number of minimums form a circleon the z−x plane and the system can shift between local-ized and delocalized without any energy barriers, whichmay lead to exotic phenomena. In realistic materials,these two couplings can not be explicitly the same. Butso long as they are close to each other which is normallythe case, there will be numerous local minimums in thehigh-dimensional potential surfaces resulting in the sim-ilar effects with that in the critical phase [19].

With the novel findings of ultrafast spectroscopy inbloom, researches on the dynamics of quantum coher-ence in organic semiconductors emerged to be prevalentin recent years [7–10, 20–23]. By definition, the quan-tum coherence reflects the correlation between differentquantum states, which strongly depends on the choice ofbasis [24]. For example, in the localized phase, when wetalk about the electron transport in the real space, the

incoherent hopping between nearest molecules (differentfrom VRH) is always mentioned, but in the energy spacethe electron is persistently localized in a single eigen-state other than ergodic suggesting the wavefunction ofelectron is coherent. The mobility measurement com-monly refers to the real space while the ultrafast spec-tra account for the excitations and emissions in the en-ergy space, making the assignment of quantum coherencecomplicated. Moreover, nonspecialists are always easy tobe confused with the electron coherence, the exciton co-herence, the spin coherence and the vibronic coherence.In this context, a quantity that is basis-free should be re-markably useful for getting rid of the confusion, and theentropy turns out to be a good candidate [25–37]. How-ever, the state-of-the-art experiments mostly focus on theentropy in a statistical manner, that is, the number of di-abatic states or molecular frontier orbitals, irrelevant tothe quantum coherence [35]. The quantum-mechanicalentropy, namely the quantum entanglement, character-izes the correlation among substances, and using theentanglement entropy as a fingerprint in organic semi-conductors can largely help comprehend the complicatedand diverse microscopic processes related to the charge,exciton, spin and phonon dynamics. Furthermore, peo-ple employ the ultrafast spectroscopy to reveal the mi-croscopic processes with the timescale being femtosec-ond to picosecond, while the devices work in a macro-scopic circumstance. The entanglement entropy shouldalso play essential roles in understanding the bridge be-tween microscopic mechanisms and device performance,which serves as an essential motivation of the presentpaper.The paper is organized as follows. In the next Section,

a generic scenario of entanglement entropy in terms ofquantum heat engine model is described, in which thedefinition of entanglement entropy is introduced. In thefollowing three sections, three microscopic models areinvestigated, all of which take two kinds of competingmany-body interactions into account. Section III is as-signed to the charge transport which mainly refers to thecompetition between diagonal and off-diagonal vibroniccouplings, and the time evolution of the two-bath latticemodel with a single electron is shown. Section IV is forthe exciton and charge transfer (CT) state dynamics andthe regime of many-body localization (MBL) stemmingfrom the competition between the Coulomb interactionbetween charges and nonlocal vibronic coupling is deter-mined. Section V is for the singlet fission and the compe-tition of spin-spin interaction and the local bosonic envi-ronment is explored. The last Section is for the summaryand outlook.

II. FORMALISM OF ENTANGLEMENT

ENTROPY

In order to proceed with the entanglement entropy,let us first describe the intrinsic mechanisms in organic

3

semiconductors with the language of quantum heat en-gine model. The first law of thermodynamics tells us that[38]

dU = d̄Q+ d̄W, (1)

where U is the internal energy, Q is the heat and W isthe work. For a system with the Hamiltonian being Ĥ,the internal energy can be written as

U =∑

n

pnεn, (2)

where εn is the energy of n-th eigenstate of Ĥ and pn isthe relevant population. Taking the differential of U , weobtain

dU =∑

n

(εndpn + pndεn) . (3)

If the system is in a canonical ensemble, pn =exp(−εn/kBT )/Z with Z being the partition functionand T being the temperature. The first term on the righthand side of Eq. (3) turns out to be

∑

n

εndpn = −kBT∑

i

d(pn ln pn). (4)

With the definition of entropy in the canonical ensem-ble being S = −kB

∑

n pn ln pn, this term is nothing butd̄Q = TdS. As a result, the differential of work is thengiven by d̄W =

∑

n pndεn, implying the change of theeigen-energies refers to the work performed by a gen-eralized force conjugated to the generalized coordinates[39–43]. Therefore, the thermodynamic quantities in thestatistical mechanics, the heat and the work, are relatedto the quantum dynamic quantities, the change of pop-ulation and eigen-energy. We are thus able to analyzethe device physics of organic semiconductors in the mi-croscopic scenario of the quantum heat engine [44–48].In the microscopic scope, the change of the eigen-

energy spectrum of the Hamiltonian stems from the ex-plicit temporal dependence of the Hamiltonian. Consid-ering the Berry phase is not detectable in a macroscopicmanner in disordered molecular materials [49], when theHamiltonian changes with time sufficiently slowly, thepopulation distributions {pn} on the eigen-energy spec-trum do not change so we can call this process as adia-batic or coherent (d̄Q = 0) [42]. The instantaneous eigen-states of the Hamiltonian can be named as the adiabaticstates. Subsequently, the heat exchange is regarded to beoriginated from the nonadiabatic and incoherent hoppingamong the adiabatic states (d̄W = 0). In organic semi-conductors, as discussed, what makes the Hamiltonianof electrons time dependent is the vibronic coupling tonuclei which normally move slower than electrons. Thediagonal (intramolecular) vibronic coupling serves as thelocal energy disorders for localization. In terms of theadiabatic approximation, the diagonal couplings merelychange the spatial coherence but do not drive the elec-trons move, so the local disorders can be regarded as

“static” in some sense. The off-diagonal (intermolecular)vibronic coupling and thus the dynamic nonlocal disor-der enables the nonadiabatic hopping (i.e. the conicalintersection) between nuclear potential surfaces and alsochanges the heat and entropy [50, 51]. From the view-point of quantum heat engine, this is why we have toconsider more details of the potential surfaces in differ-ent phases as discussed above.

Now let us describe the basic quantum dynamicsof electrons in organic semiconductors in the followinggeneric way [52]. Initially there is an electron completelylocalized on one molecule, and then it expands to sev-eral molecules enabled by the off-diagonal vibronic cou-plings or the nonadiabatic hopping. After the quantumcoherence among molecules is quenched by the diagonalvibronic couplings, the electron returns to be localizedon another molecule and one step of incoherent hoppingin the real space finishes. The electron spin resonance(ESR) experiment has shown that the spatial extent ofelectrons in pentacene is roughly ten molecules corrob-orating this localization and decoherence scenario [53].One may notice that, the electron is not fully delocalizedto all molecules but just dispersive in a finite extent, sothat it is not precise to say the system is in a delocalizedphase. It is probably in a critical phase. To be specific,we may call the critical phase as the “dispersive phase” todistinguish from the delocalized phase. In our previousstudies, we found this dispersive effect appears in bothcharge transport and charge transfer processes [23, 54].Furthermore, in nonfullerene solar cells with the energyoffset between donor and acceptor being negligibly small,we think that the ultrafast charge transfer is also origi-nated from the similar mechanism [55].

In presence of both disorders and many-body interac-tions, there may emerge a novel effect which is so-calledMBL [56–58]. In the MBL phase, the electronic eigen-states are localized, but the quantum information canbe transferred in a logarithmic manner since the entropyis persistently and slowly improved [59, 60]. As a re-sult, the electrons are still mobile in a quantum mannerwithout assistance of heat. This is because the numberof many-body states is overwhelmingly larger than thatof single-body states and the many-body interaction en-ables the transition between localized many-body eigen-states. In organic semiconductors, the on-site disordersare naturally present and the vibronic couplings play therole of many-body interactions which are always on thesame order with other parameters [11]. If the MBL effectis actually present, the novel phenomena relating to thequantum coherence and insensitive to the thermal energy,such as the coherent hole transfer in nonfullerene cells,can be well understood [55]. Therefore, it is rational toexamine if the critical and dispersive phase is equivalentto the MBL phase at least in an approximate manner.

Recently, following the rapid developing progress ofexperiments [61, 62], a quantum dynamic quantity formeasuring the entanglement entropy originally from thecosmology was reactivated to characterize the entropy-

4

increasing process in MBL phase which is the so-calledout-of-time-ordered correlator (OTOC) defined as [63,64],

F (t) = 〈Ŵ †(t)V̂ †(0)Ŵ (t)V̂ (0)〉. (5)

Herein, the expectation value is averaging overthe eigen-state of the Hamiltonian and Ŵ (t) =

exp(iĤt/~)Ŵ exp(−iĤt/~). Ŵ and V̂ are two local op-erators which commute with each other at time 0, andŴ 2 = V̂ 2 = 1. For a spin-half lattice, e.g., Ŵ and V̂ arenothing but the Pauli operators at different sites. Physi-cally, the OTOC quantifies both the spatial and temporalcorrelation of two initially local operators [65–68].To let the nonspecialists easily understand and make

sense, Fig. 2 sketches a typical instance to interpretthe physical meaning of OTOC. At time 0 we excite amolecule labeled as 0 by a laser pulse and let the systemevolve. After time t we excite another molecule labeled asx and let the system evolve back to time 0 (echo dynam-ics). Then we measure the emission signal of molecule0 and let the system again evolve to time t and finallywe measure the emission signal of molecule x. If the sys-tem is fully localized, the excitations of molecule 0 andx are independent with each other so that the OTOCis definitely 1, namely the emission signals of these twomolecules are always completely detectable. On the otherhand, if the system is fully delocalized, the exciton atmolecule 0 can be quickly transferred to molecule x be-fore time t, so that the molecule x can not be repeatedlyexcited and the OTOC in this case decays exponentiallyfrom 1 to 0. The MBL phase is in between. Due to thefinite dispersion, there is a certain probability that themolecule x is occupied by the excitons initially generatedon molecule 0, so the OTOC normally oscillates between0 to 1 showing an uncertainty feature [65, 66]. It hasbeen demonstrated that the OTOC is closely related tothe second Rényi entropy [67].There are many theoretical measures for the entangle-

ment entropy. One may ask why we do not use the sim-pler quantities such as the von Neumann local entropyand the inverse participation ratio (IPR). The advan-tage of using OTOC as an indicator is that, the OTOCas an explicit function of real time is specifically devel-oped for characterizing the quantum dynamics of spa-tial localization and MBL. Especially, in MBL phase thevibrational modes are essential in a quantum dynamicmanner while in localized phase they are just a mediumfor heat exchange, and only the OTOC is able to make aclear distinction between them. Furthermore, comparedwith the local entropy which is defined in the equilibra-tion, OTOC is a purely dynamic quantity that can beeasily measured by the commonly-used ultrafast spec-troscopy as indicated by Fig. 2 [62]. On the other hand,in presence of translational symmetry, the IPR can beirrationally large in the localized phase as discussed be-low, making it difficult to distinguish it from the MBLphase. Whereas, they are quite easy to be distinguishedby OTOC which is symmetry insensitive. Therefore, it

FIG. 2: Schematic for the physical scenario of OTOC in lo-calized and delocalized phases. At time 0, the molecule 0is excited by a laser pulse. During the following time evo-lution, the wavepacket of exciton will be kept unchanged inlocalized phase while expanded to molecule x in delocalizedphase. At time t, the molecule x is excited in localized phasewhile it is not able to absorb the light as the exciton has beentransferred to it before the laser pulse comes. Afterward, thesystem evolves to time −t via an echo dynamics. In experi-ments, this can be realized through reversing the sign of allparameters in the Hamiltonian. In the echo duration, the lo-calized states do not change, but the expanded wavepacket isshrunk to its original form in the delocalized phase. Subse-quently, in the following emission events at time 0 and t, oneobtains different emission signals in the two phases.

is remarkable to calculate the OTOC as the measure ofentanglement entropy in organic semiconductors to de-termine which phase they are residing in and what is thedynamics of entanglement entropy in these systems. Itis also worth mentioning that, the quantitative connec-tion between the entanglement entropy and the thermalentropy in the statistical mechanism is on the currenttheoretical stage not straightforward except in some spe-cific cases, so the more explicit quantification for the realdevices is beyond our present scope.

5

FIG. 3: Schematic for the mobility-temperature relationshipand the category of the three phases.

III. ENTANGLEMENT ENTROPY IN CHARGE

TRANSPORT

In organic semiconductors, the mechanism of chargetransport shifts between bandlike and hopping [69, 70].As sketched in Fig. 3, there are three regimes of themobility-temperature relationship [71–73]. When thetemperature is lower than T1 or higher than T2, namelyat low or high temperature regime, the mobility increaseswith different slopes following temperature increasing. Inthe intermediate regime, the relationship becomes weaklydependent and negative indicating a completely differ-ent mechanism dominates. In field-effect transistors, T1is roughly 200K and T2 is 250K [72]. This novel phe-nomenon thus inspires the discussion of the bandlike-hopping duality. The bandlike transport normally refersto the coherent motion of electron wavepackets. Thethermal fluctuation breaks the quantum coherence sothat the mobility-temperature relationship could be neg-ative [6]. The suffix “like” indicates it is not fully bandtransport as that in the inorganic crystals, since theorganic semiconductors are usually amorphous and theelectrons are not fully delocalized as stated. The hop-ping mechanism suggests the electron moves as a classi-cal particle which stochastically hops among the near-est molecules. The thermal fluctuation facilitates thehopping mechanism by either providing energy to crossthe potential barriers or eventually pulling the moleculescloser [52].In the language of entropy, we may have a more

sophisticated scenario to describe the novel mobility-temperature relationship as depicted in Fig. 3. Below thetemperature of T1, the phonon modes around 100cm

−1

are activated which are from intermolecular vibrations.These off-diagonal vibronic couplings act as nonlocal ran-dom forces to drive the electron move, and the higher thetemperature is, the stronger the force is. As a result, atlow temperature regime, it is a nonadiabatic entropy-

increasing process, and the phonons play a mediate rolein the heat exchange. Above the temperature of T2, thephonon modes larger than 500cm−1 are activated whichare from intramolecular vibrations. These diagonal vi-bronic couplings induce the localization of electrons, andthe correlation between two molecules is quenched by thehigh-frequency vibrations such that the entropy decreases[52]. Afterward, the normal diffusion of electrons via theincoherent hopping among molecules takes place. In to-tal, the entropy decreases in the process of localizationand increases in the diffusion, so it is roughly an adia-batic and isentropic process at high temperature regime.Of the most interests is the abnormal regime betweenT1 and T2, in which both the intra- and inter-molecularvibrations are active and comparable with each other.This is the regime that the dispersive phase and the MBLmay emerge. The eigen-states are dispersive and the en-tropy increases very slowly. In this phase, the dispersiveelectron moves among different eigen-states via quantumtunnelings driven by the off-diagonal many-body vibroniccouplings. It is dominated by a quantum-mechanicalprobability, and the temperature does nearly not matterin this process. In a previous work, we call this process as“coherent hopping”, in comparison with the incoherenthopping in the diffusive transport [54].In the charge transport, the competition between diag-

onal and off-diagonal vibronic couplings serves as the crit-ical issue. To this end, we write down a one-dimensionalHamiltonian solely taking both couplings into account,i.e.,

H =∑

i,µ

λi,µxi,µ|i〉〈i|+∑

i,ν

δν x̃i,ν(|i〉〈i + 1|+ h.c.)

+1

2

∑

i

mi

[

∑

µ

(ẋ2i,µ + ω2µx

2i,µ) +

∑

ν

( ˙̃x2i,ν + ω̃2ν x̃

2i,ν)

]

,

(6)

where |i〉 is a local state of electron on i-th moleculewith the local (diagonal) vibronic coupling being λi,µand the nonlocal (off-diagonal) coupling being δν ; xi,µ(x̃i,ν) is the displacement of i-th molecule in µ-th lo-cal (ν-th nonlocal) vibrational modes with mi being themass of the i-th molecule and ωµ (ω̃µ) being the rele-vant frequency. In order to induce the localization ofelectrons by diagonal couplings, λi,µ is set to λµ on theodd sites and vanishing on even sites, so the system re-serves the translational symmetry. The phonon spectraldensities for both couplings are chosen to be sub-Ohmicas usual, that is J(ω) = 2πα(β)ω1−sc ω

se−ω/ωc with α(β)being the dimensionless diagonal (off-diagonal) coupling,s being the exponent, and ωc being the cut-off frequency.We do not consider the electronic interactions, so the off-diagonal coupling drives the electrons move whose cou-pling strength β is fixed to be 0.1. The total number ofsites is set to 20, and s equals to 0.5 for both couplings.The evolution of electron population on the lattice is

computed by the adaptive time-dependent density matrix

6

5 10 15 200

25

50

75

100(c) =0.3

Site index5 10 15 20

(d) =1.0

(b) =0.1

0.0

0.1

0.2

25

50

75

100

ct(a) =0.01

FIG. 4: Evolution of population distribution on the latticewith two baths and one electron for α = (a) 0.01, (b) 0.1, (c)0.3 and (d) 1.0.

renormalization group algorithm [22, 23], as displayed inFig. 4. The initial state is that obtained via applyingthe operator V̂ ≡ exp(iπn̂x) onto the ground state, withn̂x being the number operator of electrons at moleculex. Here, we choose x as the central molecule (“c”) in thelattice. When α is small, the off-diagonal coupling dom-inates so that the electron freely expands on the lattice.Only finite number of sites are considered so the electronsbounded against the ends and are not able to be fullydelocalized. Following α increasing, there emerge visi-ble spatially dispersive states as indicated by the spots,and when α = 1.0 the electrons are almost localized nearthe initial site. The competition between diagonal andoff-diagonal couplings gives rise to the transition of thecharge transport.

We then calculate the time evolution of the OTOCwith the operators defined as V̂ ≡ exp(iπn̂c) and Ŵ ≡exp(iπn̂c+x). Fig. 5 shows the results for x = 1, andresults for other x’s are quite similar due to the transla-tional symmetry so they are not shown. It is clear thatfor α = 0.01 the OTOC oscillates and decays, and withincreasing α the OTOC increases. All these behaviorsexhibit the remarkable MBL features of electrons show-ing very similarity of dispersive and MBL phases. Forα = 1, it keeps at a value close to 1 indicating the elec-trons are localized. It is worth noting that, the value ofIPR for α = 1 can be remarkably larger than one as thesites other than the initial one are also populated, so itfails to reflect the localization length.

Before ending this Section, let us briefly discuss themeasurements of mobility. In the traditional semicon-ductor physics, the mobility and the mean velocity ofcharge carriers are dependent on the scattering rate ofimpurities and phonons [74]. The former gives rise to apositive temperature coefficient while the latter refers to

0 20 40 60 80 1000.7

0.8

0.9

1.0

OTO

C

ct

=0.01 =0.1 =0.3 =1.0

FIG. 5: Time evolution of OTOC on the lattice with twobaths and one electron for x = 1 as a function of α.

negative coefficient. Most techniques of mobility mea-surements such as the time-of-flight (TOF) approach arebased upon the random walk of injected carriers [75].If the semiconductor is dominated by purely incoherenthopping of localized electrons, the Einstein’s relation be-tween mobility and diffusion coefficient is available andthe TOF still properly works. The Einstein’s relationis applicable in the ergodic phase, and the dispersive orMBL phase do not explicitly fit for TOF [76]. Consid-ering the mixture of coherent and incoherent hoppingmechanisms, one can also divide the mobility into coher-ent and incoherent components, respectively [77]. In ouropinion, therefore, the mobility measurements should becombined with the time-resolved microwave or terahertzconductivity to determine the component from coherenthopping induced by MBL effect as well as the ESR toquantify the dispersion of the charge carriers.

IV. ENTANGLEMENT ENTROPY IN EXCITON

AND CHARGE TRANSFER STATE DYNAMICS

Upon photoexcitation, a local exciton is generated fol-lowed by a charge separation process to produce pho-tocurrent. In organic photovoltaic devices, the chargeseparation comprises two steps: The first is the chargetransfer between donor and acceptor and the second isthe further dissociation of the CT state. Borrowing thelanguage of traditional semiconductors, the CT state canbe regarded as a tight-binding electron-hole pair withthe binding energy mainly from the Coulomb attraction.Considering the relatively small dielectric constant, how-ever, a puzzle arises: what serves as the driving force tomake the CT state dissociate. As stated, recent experi-ments [7–10, 20, 21] highlight the quantum coherence andthe emergence of the long-range CT states which pos-sess much smaller binding energy than the short-rangeones. In this context, it is thus unjustified to define theCT state by the Coulomb interaction. In our opinion,

7

it makes more sense to define the CT state as a coher-ent electron-hole pair, and once the coherence betweenelectron and hole is lost, the CT state is regarded to bedissociated.

It is rather convenient to discuss the entanglement en-tropy in the photophysics of organic photovoltaic deviceswith a quantum heat engine model. In thermodynamics,an ideal cycle comprises four processes: isothermal ex-pansion (heat injection Q1), adiabatic expansion (powerexport), isothermal compression (heat rejection Q2), adi-abatic compression (internal energy recover). The effi-ciency is defined as η = 1 − Q2/Q1. In the generationof local exciton, a single excited state (S1 state) is pop-ulated and the entropy increases. This is obviously theheat injection process, and Q1 could be simply regardedas the energy of the absorbed photons. As discussed inthe last Section, the charge transport and extraction areapproximately isentropic and adiabatic processes whichcould be regarded as the process of power export. Dur-ing the process that the local exciton is conversed to theCT state the entanglement entropy increases which willbe quenched in the following CT-state dissociation andcharge separation process. Combined with the energy re-laxation from higher-energy excitation to the lower one,these two processes serve as the major channel of heatrejection and determine the value of Q2. The subsequentrebalance of the electrostatic potential with the externalcircuit can be regarded as the final adiabatic process ofthe cycle of quantum heat engine.

To rationalize the efficiency of the heat engine, it is es-sential to quantifyQ2, so the entropy change must be wellmeasured. The transition from the local exciton to theCT state turns out to be essential. Let us now assume thecoherent size of CT state is N molecules, and the electronand hole are dispersive in this region. Both of them arecompletely entangled to form an entangled state. As thelocal exciton is singlet, the (coherent) CT state must besinglet as well. If we further assume the singlet exciton,the electron and the hole are respectively distributed onN molecules with equal probabilities, the entropy of a sin-glet exciton SS should equal to kB lnN and the entropy ofa CT state SCT should be twice of it. The entanglemententropy that is quenched during the CT-state dissocia-tion thus equals to kB lnN . Imagine a coherent spherefor the excitons with the radius being 10 molecules [53],so N should roughly equal to 4200. At room tempera-ture, we thus obtain Q2 = T∆S ≃ 217meV, which couldbe regarded as the heat rejection in the photo-electricconversion process. In our previous work, we have calcu-lated the ultrafast long-range charge transfer process tak-ing the intermolecular vibrations into account. Therein,the entanglement entropy in a three-dimensional case isestimated to be 210meV, very similar with the presentestimation [23]. In addition, we believe the ultrahighopen-circuit voltage loss in organic photovoltaic devicesmay stem from this effect [78, 79].

The above arguments rely on two prerequisites.Firstly, the excitons, electrons and holes must be quan-

tum matters. There is no way the classical particles pre-fer to be confined in a relatively small region rather thanto diffuse to much larger extent and produce unrealis-tically large entropy increase. The dispersive feature ofthe wavepackets induces the spatial confinement as wellas the coherent size. Secondly, the dispersive electronsand holes are driven by many-body interactions, suchas the off-diagonal vibronic couplings [23] or electronicinteractions [80]. This MBL effect ensures the entan-glement entropy increases in a logarithmic manner, suchthat the entropy does not significantly increase during thesubsequent charge extraction process after charge sepa-ration. This picture is available in both fullerene andnon-fullerene solar cells.Since there are always two kinds of charge carriers in

the issues of exciton and CT state, the Coulomb interac-tion between them apparently plays the essential role.The Coulomb interaction induces localization and thenonlocal vibronic couplings give rise to nonlocal disorderand delocalization, and they compete with each other tolet novel many-body effects emerge. In order to demon-strate the ultrafast long-range charge transfer process isfrom the MBL effect, we thus calculate the OTOC in anFrenkel-CT mixed model with the Hamiltonian being [23]

H = He +Hp +Hep. (7)

The first term He is written as,

He = Eex| − 1〉〈−1|+ VDA| − 1〉〈−1, 1| −∑

i,j

C

ri,j|i, j〉〈i, j|

+∑

i

∑

{j,j′}

VA|i, j〉〈i, j′| −

∑

j

∑

{i,i′}

VD|i, j〉〈i′, j|, (8)

where |i〉 represents the state of the local Frenkel excitonand |i, j〉 is the CT state with the electron on site i andhole on site j, Eex is the on-site energy of the Frenkel ex-citon, VD/A/DA denotes the electronic interactions, C isthe prefactor of the Coulomb attraction between electronand hole with ri,j being the distance which will induce thelocalization of electron and hole. Without lose of general-ity, we set the parameters as Eex=0.25eV, VDA=0.08eV,C=0.48eV, and VD = VA=0.06eV. The phonon Hamilto-nian Hp is expressed as (~ = 1)

Hp =∑

ν

ω±,ν b̂†±,ν b̂±,ν, (9)

where b̂†±,ν (b̂±,ν) denotes the creation (annihilation) op-erator of ν-th phonon mode on the donor (−) or accep-tor (+) molecules, and ω±,ν is the relevant frequency.For the vibronic couplings, we consider the off-diagonalcouplings:

Hep =∑

i,j,ν

γ−,ν

[

|i, j〉〈i− 1, j|b̂†−,ν + |i− 1, j〉〈i, j|b̂−,ν

]

+∑

i,j,ν

γ+,ν

[

|i, j〉〈i, j + 1|b̂†+,ν + |i, j + 1〉〈i, j|b̂+,ν

]

,(10)

8

0 50 100 150 2000.7

0.8

0.9

1.0

time (fs)

(b) x=8

=0.02 =0.04 =0.1 =0.06 =0.12 =0.08 =0.14

0.6

0.8

1.0

OTO

C

x=4 x=6 x=8 x=10 x=12

(a) =0.08

FIG. 6: Time evolution of OTOC in the Frenkel-CT mixedmodel as a function of (a) x for β = 0.08 and (b) β for x = 8.

with γ±,ν being the coupling strength for ν-th mode, de-pending on the spectral density J(ω) = π

∑

ν γ2±,νδ(ω −

ω±,ν). A continuous spectral density is set for thephonons with the same spectral density function withthat in the last Section.

In the previous work [23], we have calculated the ultra-fast dynamics of the long-range CT state, showing thatfor an intermediate vibronic coupling (β = 0.06–0.1) thecharge transfer is the most efficient. The Coulomb attrac-tion between electron and hole induces the localizationand the off-diagonal coupling gives rise to quantum tun-neling. Moreover, we found that during the whole chargeseparation process, both the electron and hole modestlyhold the shape of dispersive wavepackets rather than fur-ther delocalize, suggesting that the ultrafast long-rangeCT process is related to an MBL effect. To examine thisidea, we calculate the OTOC dynamics with the site in-dex for the operator V̂ (≡ exp(iπn̂1)) being the acceptormolecule 1. Fig. 6 shows the results for various x whichis the site index for the operator Ŵ (≡ exp(iπn̂x)) andthe off-diagonal vibronic coupling β. It is found that,the change from x=4 to x = 12 exhibits the path alongwhich the charges move. Except the case x = 4, the val-ues of OTOC for other four cases are similar implyingthe long-range feature of the CT state. For the depen-dency of β, we can find for 0.02 and 0.04, the OTOCdeviates from 1 slightly implying the localization dom-inates. From β = 0.06 to 0.1 the OTOC oscillates be-

tween 0.7 and 1 rather than quickly decays to 0 indi-cating the MBL effect functions. We do not calculatethe longer time duration because of the numerical pre-cision loss, but we expect the OTOC will finally decayto 0 in a very slow manner [67]. The minimum appearsat β = 0.08, which is the case that the charge transferprocess is efficient. Combined with the dynamics of pop-ulation as shown in Ref. [23], β = 0.06 to 0.1 is the regimethat the long-range CT process takes place manifestingits similarity with MBL. In addition, for β = 0.12 and0.14, the case that charge transfer does not occur, theOTOC keeps nearly constant above 0.9. The coupling isso strong that the phonon energy exceeds the electronicband, so the electron is again localized. We can thusconclude here that the ultrafast long-range CT processis originated from the MBL effect, and neither weak norstrong vibronic couplings are friendly to the charge sep-aration — the intermediate ones are best.

V. ENTANGLEMENT ENTROPY IN SINGLET

FISSION

In organic semiconductors, both charge carriers andtriplet excitons carry non-zero spin. The singlet fissionthat a singlet exciton is split into two triplet excitonsturns out to be an appealing effect to facilitate the ef-ficiency of photovoltaics. However, a usual doubt onthe statement that the singlet fission can fairly enhancethe power conversion efficiency is, although one singletis split into two triplets and subsequently two electron-hole pairs might be formed to increase the short-circuitcurrent, the energy of these electron-hole pairs are lostin the singlet fission and thus the open-circuit voltageshould be decreased. As a result, the efficiency can notbe significantly enhanced after all.In order to comprehend this puzzle, it is again useful

to employ the framework of quantum heat engine model.Let us assume the dissociation of each triplet exciton con-sumes the same energy with that discussed in the lastSection, giving the fact that the entanglement entropywhich is quenched in the exciton dissociation is insensi-tive to the energy of excitons. Since the relaxation pro-cesses are almost absent in singlet fission materials, thesinglet energy (Q1) must be doubled to keep the quantumefficiency η unchanged. We notice that, the singlet fissionmaterials exhibit higher external quantum efficiency forlarger photon energy [81], so they are indeed more suit-able for the ultraviolet regimes. In the practical applica-tion of photovoltaics, one should then add additional ab-sorbers into the device to absorb the low-energy photons.In ideal cases, the maximum efficiency will increase from45.7% to 47.7% by inserting an additional layer [82, 83].On the other hand, the transition from triplet pair state(TT state) to the uncorrelated two triplets (T+T state)is also an entropy-increasing process which may furtherlower the efficiency. Therefore, it is necessary to decreasethe energy dissipation in TT state dissociation as much

9

as possible, and we have to carefully investigate the dis-sociation process of TT state.A big question now arises: What makes the TT state

dissociate? In absence of the dissipated environment,the two triplet excitons are consistently entangled witheach other, no matter how long they separate in the realspace. Therefore, the dissociation means the loss of co-herence between two triplet excitons, analogous to theCT state dissociation. The triplet exciton is of chargeneutrality and large binding energy so in terms of chargedegree of freedom the exciton-phonon couplings shouldbe relatively weak. Meanwhile, the organic materials arebasically composed of light elements with negligible spin-orbital couplings, so the spin coherence is also of longlifetime. Fortunately, there are still many factors relatedto the spin degree of freedom, such as the hyperfine inter-action of hydrogen atoms [84], the hybridization of σ andπ orbitals due to out-of-plane vibrations [85], the othertriplet excitons [86], the kinetically blocked radicals [87]and also the rotation with different chirality of polarizedmolecules. All these factors can be grouped into a lo-cal spin-related bath for the electrons, holes and tripletexcitons, which may influence the decoherence and local-ization of TT state.The intermolecular distance may also serve as a key

factor in the TT dissociation. The distance must be suf-ficiently short such that the electron transfer integral islarge enough to induce the conversion of local singlet ex-citon and the TT state. The short distance will how-ever enhance the spin-spin exchange interaction betweentriplets and make the TT dissociation unfavorable. Thisconflict heavily limits the practical application of singletfission in photovoltaics. Therefore, the spin-spin interac-tion which is the main many-body interaction for delo-calization of spin should be carefully investigated as well.To this end, let us write down a Hamiltonian for two

pairs of electron and hole taking the diagonal bosonicenvironment into account. That is

H = −2J Ŝ1Ŝ2 +D∑

i=1,2

Ŝ2i,z + E∑

i=1,2

(Ŝ2i,x − Ŝ2i,y)

+∑

i,µ

λi,µŜi,z(b̂†i,ν + b̂i,ν) +

∑

i,ν

ωi,ν b̂†i,ν b̂i,ν . (11)

where Ŝi is the spin operator for the i-th triplet, J is thespin-spin interaction between two triplets, D and E de-note the interaction strengths with respect to the molec-ular field, and the environmental part takes the same for-mula with those in the last two Section. Herein, the firstline is nothing but the Merrifield model considering thedipole-dipole interaction [88–90]. The parameters are setas usual cases, namely D = 0.15cm−1 and E = 0.1cm−1.Different from charge degree of freedom which can off-

diagonally couple to the intermolecular vibrations, thespin-related environment is merely diagonal such thatthe spin-environment coupling itself does not change thepopulation on each spin state [91]. There are two param-eters influencing the change of local populations: J and

E. As we are focusing on the dissociation of triplet pairs,the influence of J is mainly considered which is sensitiveto the distance between triplets as stated. The effect ofthe spin-spin interaction is the conversion of the states|+1,−1〉, |−1,+1〉 and the state |0, 0〉, with ±1 and 0 be-ing the common notations of the relevant state for the 1stand 2nd triplet in order. Now, let us assume the singletstate is initially conversed to the TT states |+1,−1〉 and|−1,+1〉, and they entangle with relevant environmentalstates. The quantum state can be expressed as

a|+ 1,−1, E+−〉+ b| − 1,+1, E−+〉, (12)

where E denotes the relevant environmental states, anda, b the combination coefficients. In order to make theTT state dissociate, each triplet must turn to entanglewith their own local environmental state. That is, thequantum state have to change to

|σ1, E1〉 ⊗ |σ2, E2〉, (13)

where σi is a linearly combined spin state locally for i-th triplet. The larger change of environmental state is,the easier the decoherence takes place, so the entropychange heavily depends on the change of the environ-mental states. Interestingly, however, the environmentalstates are hardly changed and thus both | + 1,−1〉 and|− 1,+1〉 are not easy to be dissociated. This is because,during the time evolution both | + 1,−1〉 and | − 1,+1〉are firstly conversed to the state |0, 0〉, and |0, 0〉 does notcouple to the environment in terms of the present Hamil-tonian. As a result, if we want to increase the yield offree triplet excitons, it is essential to decrease the yieldof | + 1,−1〉 and | − 1,+1〉 states during the singlet-TTconversion. This can be done by, e.g., exerting an exter-nal magnetic field to open energy gaps among +1, 0 and-1 states [90].The |0, 0〉 state is easier to be dissociated since it firstly

converses to either | + 1,−1〉 or | − 1,+1〉 state duringthe time evolution. We then calculate the time evolu-tion with the initial state being that, on each moleculethe electron and hole form a state |↑↓〉, namely the com-bination of the singlet and the 0 component of triplet.Both the OTOC and von Neumann entropy SVN are cal-culated for the two electrons of the excitons on the twomolecules, as displayed in Fig. 7. It is found that, withJ = 0.02meV and α ≤ 0.02 the system is delocalized,i.e. the correlation between two triplets is strong. In thissituation the TT state is hard to be dissociated. WhenJ is between 0.005 and 0.01meV, the behavior of OTOCexhibits the MBL feature which may make the dissoci-ation much easier. For J = 0.002meV it becomes lo-calized. Therefore, J =0.01meV (or equivalently 90mT)and α = 0.02 emerge to be critical values for efficient dis-sociation of TT state. We also calculate SVN for the localelectrons. The basic feature of SVN is similar with that ofOTOC, but some of the details such as the oscillations be-come weaker exhibiting the advantages of using OTOC.Interestingly, we find very similar behavior of SVN with

10

(b) =0.02 J=0.02meV

J=0.01meV

J=0.005meV

J=0.002meV

100 200 300 400

(d) =0.02

J=0.02meV J=0.01meV

J=0.005meV

J=0.002meV

0 100 200 300 4000.0

0.2

0.4

0.6

0.8

1.0

(c) J=0.01meV

=0.01=0.02

=0.03

=0.04

=0.05

=0.06

S VN

time (ps)

0.2

0.4

0.6

0.8

1.0

=0.06

=0.05=0.04

=0.03

=0.02

OTO

C

=0.01(a) J=0.01meV

FIG. 7: Time evolution of OTOC in the Merrifield model forsinglet fission as a function of (a) α for J = 0.01meV and(b) J for α = 0.02. The relevant evolution of von Neumannentropy SVN is shown in (c) and (d).

that in the many-body model with electron-electron in-teractions [59], that is, the SVN firstly saturates and thenincreases slowly. This phenomenon clearly manifests theMBL characteristics of the triplet-environment couplingsystem.

In the usual study of Shockley-Queisser’s detailed bal-ance model for the singlet fission, the spin degree of free-dom is not taken into account [83]. People mainly fo-cused on the energy profile of singlet, triplet and CTstate and dedicated to synthesize exergonic singlet fis-sion materials. Whereas, the TT state must be firstlydissociated into free triplets before they are able to pro-vide two pairs of free electron and hole. As calculated,the entropy increase and thus the free energy decrease inthe TT dissociation are considerable. Different from thatfor charge degree of freedom, the energy scale for the spinsystems is normally 2-4 orders smaller than the thermalfluctuation energy at room temperature. It is thus notable to compensate the energy dissipation in TT dissoci-ation through designing the appropriate energy profilesof different spin states. Fortunately, the singlet fission isan issue of dynamics, not only energetics. Properly deal-ing with the quantum coherence and entanglement willgreatly facilitate the improvement of efficiency. To bemore specific, due to the difference of energy scales, thespin dynamics is much slower than the charge dynamics,so the quench of entanglement at proper time point maylargely change the yield of free triplets. Consequently, arational design of high-efficiency singlet fission materialsneeds very detailed dynamics simulations in addition tothe energetic computations.

VI. SUMMARY AND OUTLOOK

In summary, the localization, delocalization and dis-persion of electrons in organic semiconductors are inves-tigated with the theoretical framework of quantum heatengine model and entanglement entropy, and the pro-cesses of charge transport, photoelectric conversion andsinglet fission are analyzed as well. For each issue, weconsider two kinds of competing many-body interactions— one refers to the localization and the other to delocal-ization. Two-bath lattice model with a single electron,Frenkel-CT mixed model, and the Merrifield model forsinglet fission are studied by the adaptive time-dependentdensity matrix renormalization group algorithm and thetime evolution of entanglement entropy quantified byOTOC is calculated. The parameter regimes that theMBL effect functions are accordingly determined. Thethree models we studied involve many-body charge, spinand vibronic interactions, covering the majority cases ofthe theories in the field. Therefore, our research makesa step forward to establish a unified quantum dynamictheory relating to charge, exciton and spin in organicsemiconductors.

The statistical mechanics relies on the basic postulateof the ergodic hypothesis, which straightforwardly leadsto the entropy increase and free energy decrease principle.As discussed, however, the organic semiconductors arerarely in the ergodic phase, and the disorders and many-body interactions give rise to localization-delocalizationduality and MBL effect. It is therefore insufficient to ana-lyze the issues in organic semiconductors simply with theargument from the energetics, such as the electron nat-urally hop from the higher-energy level to the lower one.The dynamic information turns out to be essential. Es-pecially, the time of decoherence and disentanglement inthe dynamics, which is dependent of the electron-phononcouplings, plays a significant role in affecting the bandlikeand hopping mechanism, the ultrafast long-range chargetransfer, the triplet pair dissociation, and other relevantprocesses. As a consequence, the proper computationsof entanglement entropy may tell us the explicit valueof dissipated energy and thus the efficiency of the realdevices.

Throughout this work, we quantitatively determinedparameter regime of MBL phase but merely investigatedthe factors that influence the dynamics of entanglemententropy in a qualitative manner. One would intuitivelyconsider that how the change of entropy in real devicescan be explicitly quantified. Commonly speaking, boththe von Neumann entropy and OTOC are defined be-tween 0 and 1. We have to properly calibrate them to fitfor the standard statistical mechanics and the relevantexperiments. This is not easy in a general manner butwe can first try to bridge the entanglement entropy andthe entropy in organic solar cells which will be the nextsubject.

11

Acknowledgment

The author gratefully acknowledges support from theNational Natural Science Foundation of China (Grant

Nos. 91833305, 11974118, 11574052). We thank Prof.Haibo Ma for fruitful discussions.

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