Effects of Charge Transfer State and Exciton Migration on Singlet

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Effects of Charge Transfer State and Exciton Migration on Singlet Fission Dynamics in Organic Aggregates Hang Zang, Yaling Ke, Yi Zhao, and WanZhen Liang J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b02943 • Publication Date (Web): 09 Jun 2016 Downloaded from http://pubs.acs.org on June 14, 2016

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The Journal of Physical Chemistry

Effects of Charge Transfer State and Exciton Migration on Singlet Fission Dynamics in Organic Aggregates Hang Zang,†,‡ Yaling Ke,‡ Yi Zhao,∗,‡ and WanZhen Liang∗,‡ †Department of Chemical Physics, University of Science and Technology of China, Hefei, Anhui 230026, China ‡State Key Laboratory of Physical Chemistry of Solid Surfaces, Collaborative Innovation Center of Chemistry for Energy Materials, Fujian Provincial Key Laboratory of Theoretical and Computational Chemistry and Department of Chemistry, College of Chemistry and Chemical Engineering, Xiamen University, Xiamen, Fujian 361005, China E-mail: [email protected]; [email protected] Phone: +86-592-2189197 (Y.Z.); +86-592-2184300 (W.L.)

Abstract A time-dependent wavepacket diffusion method is used to investigate the effects of charge transfer (CT) states, singlet exciton and multiexciton migrations on singlet fission (SF) dynamics in organic aggregates. The results reveal that the incorporation of CT states can result in a different SF dynamics from the direct interaction between singlet exciton and multiexciton, and an obvious SF interference is also observed between the direct channel and the indirect channel mediated by CT states. In the case of direct interaction, although the fast population transfer of singlet exciton in monomers,

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caused by the increase of exciton-exciton interaction, can accelerate the SF process, the spatial coherence alternatively has a counter-productive effect, and their competition leads to an optimal exciton-exciton interaction at which SF has a maximal rate. This trade-off relationship in SF dynamics is further analysed from different perspectives, specifically in spatial and energy representations, is also confirmed through the indication that static energy disorders can speed up SF process by destructing the coherence. Meanwhile, it is found that the couplings among multiexciton states decrease SF rates by the multiexciton coherence and backward conversion from multiexciton to singlet exciton states.

1

Introduction

Singlet fission (SF) in an organic dimer is referred as the conversion of a singlet excited state (S1 ) on a monomer into two triplet excited states (2T1 ) on the dimer. 1,2 Although its intrinsic conversion mechanism is still under debate, it is recognized that there exists a multiexciton (ME) state as a dark intermediate state to bridge S1 and 2T1 . 3,4 The ME state represents a doubly excited pair of spin-correlated triplet exciton (TT) state which is an overall singlet state 5–7 , and it has been experimentally demonstrated recently 8–10 . As the SF process potentially allows single-junction photovoltaic devices to surmount the Shockley-Queisser limit in terms of power conversion efficiency, 11,12 it has attracted renewed attentions. 13–21 Numerous investigations have revealed that the interaction between the initial singlet exciton (SE) state and TT state, which plays an important role in SF dynamics, may include a direct statestate interaction and an indirect superexchange interaction mediated by a charge-transfer (CT) state. 3,4,22–32 This interaction in organic crystals can be strong enough to induce the quantum coherence between SE and TT states, 5,7,33,34 resulting in an ultrafast timescale of SF. Recent works have further exhibited that the SF with a high efficiency can be per2


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formed by optimizing the morphologies of organic aggregates. 35–43 In this case, a largely unexplored question is the relationship between SF process and SE migration. Several experiments 37,38,40 have observed that the SE delocalization can drive a rapid SF process in well-arranged molecular crystals. However, other researches 41,42 indicate that the SF process is slower in single-crystal samples than that in films. Such contradictory observations require an alternative theoretical scenario that differs from the dimer model, which only considers two proximately coupled molecules, to fully understand the effect of SE dynamics on SF process. 44–48 In this paper, we concentrate on this specific problem and theoretically exhibit how the SF process is affected by CT states and SE migration in organic aggregates. For simplicity, we use a one-dimensional model to describe the dominant SF dynamics because the exciton-exciton interactions in organic crystals are commonly anisotropic and they are obviously larger in a direction than in other directions 39,49 . According to the Frenkel exciton model, SE in a molecular aggregate can spread over many monomers 33,50 and whether its migration exhibits a coherent band-like or incoherent hopping-type behaviour is determined by the ratio of exciton-exciton coupling strength and exciton-phonon interaction strength. 51,52 In our previous works, 53–57 a time-dependent wavepacket diffusion (TDWPD) method has been proposed to uniformly describe the SE motion from coherent to diffusive regimes in a system with thousands of units. In SF process, however, the conversion channels from SE states to CT and TT states open during the SE migration. We thus extend the TDWPD method to describe SF dynamics with incorporation of SE, CT and TT states and theoretically unearth the impacts of monomer number, CT states, and exciton delocalization on the SF process. Some interesting results are demonstrated, for instance, the SF process via direct and indirect superexchange interactions has obviously different time-dependence, and there exists an optimal exciton-exciton coupling strength at which the SF rate has a maximal value. These findings could make an important step towards understanding the underlying mechanism of the experimentally debated observations.

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The paper is arranged as follows. In section 2, we briefly outline the TDWPD method and the aggregate model. Section 3 shows results and corresponding discussion. The concluding remarks are given in Section 4.

2

Aggregate Model and TDWPD Method 2(N-1) CT states JCT VSE-CT

VTT-CT

N SE states JSE

VSE-TT

N-1 TT states JTT

Figure 1: Schematic picture of SE, CT and TT states and the interactions among them. For a one-dimensional organic aggregate with N monomers, the electronic states in SF process involves N SE states, 2N-2 CT states and N-1 TT states. These states and their interactions are schematically shown in Figure 1, and the corresponding electronic Hamiltonian can be expressed as follows HE = H SE + H CT + H T T + H int .

(1)

Here, the electronic Hamiltonians H SE , H CT , H T T are given by H SE =

N X n=1

H

CT

=

N −1 ³ X

EnSE Bn† Bn +

N −1 ³ X

´

SE Jn,n+1 Bn† Bn+1 + h.c. ,

n=1

´

† † CT CT Cn+1,n Cn+1,n + Cn,n+1 Cn,n+1 + En+1,n En,n+1

n=1

4


(2)

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N −1 ³ X

´

† CT Jn,n+1 Cn,n+1 Cn+1,n + h.c. +

n=1

H

TT

=

N −2 ³ X

´

† CT Jn+1,n Cn+1,n Cn+1,n+2 + h.c. ,

(3)

n=1

N −1 X

† TT En,n+1 Dn,n+1 Dn,n+1

+

n=1

N −2 ³ X

´

† TT Dn,n+1 Dn+1,n+2 + h.c. , Jn,n+1

(4)

n=1

where the creation operator Bn† generates a SE state at the n-th monomer, the creation † operator Cn,n+1 creates an electron and a hole residing at the n-th and (n + 1)-th monomer, † respectively, the creation operator Dn,n+1 spans two triplet states over the n-th and (n+1)-th

monomer, and Bn , Cn,n+1 and Dn,n+1 are the corresponding annihilation operators. In the above model, the only exciton coupling between two adjacent states are considered, which is typical for dynamic simulations in organic aggregates. Generally, the CT-CT coupling J CT is negligible because it requires the exchange of two particles. Compared to SE-SE couplings J SE , TT-TT couplings J T T are quite small because multistep processes are involved even in the nearest neighbor TT energy transfer. The Hamiltonian H int in Eq. (1) represents the interactions among different-type exciton states H int =

N −1 ³ X n N −1 ³ X n N −1 ³ X

´

SE−CT † SE−CT † Vn,n+1 Bn Cn,n+1 + Vn+1,n Bn+1 Cn+1,n + h.c. +

´

T T −CT † T T −CT † Vn,n+1 Dn,n+1 Cn,n+1 + Vn+1,n Dn,n+1 Cn+1,n + h.c. +

´

SE−T T † SE−T T † Vn,n+1 Bn Dn,n+1 + Vn+1,n Bn+1 Dn,n+1 + h.c. ,

(5)

n=1 SE−T T SE−CT where Vn,n+1 represent direct interactions between the SE and TT states, and Vn,n+1 and T T −CT Vn,n+1 are the indirect superexchange interactions between SE and CT states, and TT and

CT states, respectively. In organic aggregates, carrier dynamics is greatly affected by the molecular vibrational motions especially in the case of strong carrier-phonon interaction strength. We model these vibrational motions as a collection of harmonic oscillators, and the total Hamiltonian for SF

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thus reads H = HE + Hph + HE−ph .

(6)

Here, Hph and HE−ph are molecular vibrational Hamiltonian and exciton-phonon interactions, respectively. With use of the creation and annihilation operators of vibrational modes, they are expressed as follows (Hereinafter, the mass scaled coordinates and the unit with h ¯ = 1 are used), Nn

Hph =

ph XX

1 ωnj (a†nj anj + ), 2 j=1

n

(7)

Nn

HE−ph =

ph XX

Cnj (a†nj + anj )G†n Gn .

(8)

n=1 j=1

Here, the sum of n indicates that SE, CT and TT states are all covered and Gn are corresponding operators of electronic states, i.e., it runs over Bn , Cn,n±1 and Dn,n+1 states. Nnph is the number of vibrational modes on the n-th state and a†nj and anj are the creation and annihilation operators of the j-th vibrational mode with frequency ωnj , respectively. Cnj represents the carrier-phonon interaction strength and it is proportional to the displacements ∆Qnj from equilibrium configurations of ground state to that of exciton state, and 2 ∆Q2j /2. the corresponding mode-specific reorganization energy is ωnj

Following our previous works, 53–58 the carrier dynamics can be obtained by TDWPD method. To proceed, we rewrite total Hamiltonian in the interaction representation with respect to the phonon Hamiltonian H(t) = eiHph t (HE + HE−ph )e−iHph t ph

= HE +

N N n X X n

Cnj (a†nj eiωj t + anj e−iωj t )G†n Gn .

(9)

j

The wavefunction for the entire system satisfies the following time-evolving equation

i

∂|Ψ(t)i = H(t)|Ψ(t)i, ∂t 6


(10)

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with a formal solution |Ψ(t)i = U (t)|Ψ(0)i, where |Ψ(0)i is the initial wavefunction and U (t) = eiHph t e−i(HE +Hph +HE−ph )t is the unitary propagator. The reduced density operator of carrier dynamics is obtained by ρ(t) = Trph (|Ψ(t)ihΨ(t)|) = Trph (U (t)ρtot (0)U † (t)).

(11)

Assuming that the molecular vibrational motions with harmonic oscillators initially remain in the thermal equilibrium state and the initial density operator is factorizable, ρtot (0) = |Ψ(0)ihΨ(0)| = |ψ(0)ihψ(0)|ρTph , where |ψ(0)i is the initial wavefunction of carrier and ρTph represents initial vibrational density operator. In the unnormalized Bargmann coherent state representation of vibrational motions, Eq. (11) becomes Z

ρ(t) =

dα2 −|α|2 Z d2 β − n¯ +1 |β|2 α∗ β+β ∗ α e e n¯ e |ψα,β (t)ihψα,β (t)|, π π¯ n

(12)

with ∗

|ψα,β (t)i = e−α β hα|U (t)|βi|ψ(0)i.

(13)

Here, |βi and |αi represent coherent states of vibrational motions at time 0 and t. n ¯ (ω) = 1/(eω/kB T − 1) is the thermally averaged occupation number of vibrational modes at temperature T . The quantity ρα,β (t) = |ψα,β (t)ihψα,β (t)| can be interpreted as the carrier density operator along the vibrational quantum trajectories starting from |βi coherent state to |αi coherent state. The wavefunction |ψα,β (t)i can be obtained from the following differential equation 53–58 Ã

!

Z t X ∂|ψα,β (t)i 0 0 = HE + Fα,β (t) − i i G†n Gn dτ 0 α(τ 0 )e−iHE τ G†n Gn e−HE τ |ψα,β (t)i, (14) ∂t 0 n

where α(t) =

P j

2 −iωnj t e in the correlation function of carrier-phonon interaction at zero Cnj

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temperature, and Fα,β (t) is given by X

Fα,β (t) =

n

Fn G†n Gn =

X

∗ iωnj t Cnj (αnj e + βnj e−iωnj t )G†n Gn .

(15)

nj

It should be mentioned that the time integral term in Eq. (14) is obtained upon the Markovian approximation and the limitation of weak carrier-phonon interaction strength. These approximations have been demonstrated to be reasonable because Eq. (14) can correctly predict exact carrier dynamics for most of realistic organic aggregates. 53–58 Fα,β (t) is the so-called stochastic force in numerical simulations as we take α and β as random variables, it is generated by the following expression Nph

Fn (t) =

X

s

k=1

· ¸ q Jn (ωk )∆ω q A(ωk ) cos(ωk t + φk ) + i B(ωk ) sin(ωk t + φk ) , π

(16)

where A(ωk ) = coth(ωk /2kB T ) + csch(ωk /2kB T ), B(ωk ) = coth(ωk /2kB T ) − csch(ωk /2kB T ), and {φk } is a series of random variables that are uniformly distributed in [0, 2π]. It is easily proved that the time-correlation function of Fn (t) is consistent with the correlation function of carrier-phonon interaction Cn (t) = hFn∗ (t)Fn (0)iα,β Z

=

0

∞

Ã

!

ω/kB T Jn (ω) coth cos(ωt) − i sin(ωt) , dω π 2

with the spectral density Jn (ω) = π

P j

(17)

C nj 2 δ(ω − ωnj ).

Finally, the time dependent density matrix of carrier dynamics is obtained through the stochastic average. To obviously calculate the carrier dynamics, we expand the wavefunction in the basis sets composed of SE, CT and TT states

|ψ(t)i =

N X n=1

cSE n |SEn i +

N −1 X

cCT n,n±1 |CTn,n±1 i +

n=1

N −1 X n=1

8


T cTn,n+1 |T Tn,n+1 i.

(18)

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The density matrix ρ(t) that describes both exciton population and coherence is thus straightCT TT forwardly obtained in terms of cSE n , cn,n±1 and cn,n+1 , for example, the population of SE at

D

E

SE∗ SE the n-th site is given by ρSE nn (t) = cn (t)cn (t) with a stochastic average.

To quantitatively describe the carrier population transfer and coherence, we use the population participation ratio (PPR) and coherence participation ratio (CPR) 59–63 , given by

PPR(t) = 1/

X n

CPR(t) =

³X n

´ X

ρ2nn (t) (

ρ2nn (t),

(19)

³X

|ρnn0 (t)|)2 /

nn0

´

|ρnn0 (t)|2 .

(20)

nn0

For an aggregate with N monomers, the PPR of SE ranges from 1 to N, serving as the measurement of the number of sites at which SE is likely to be found and corresponding to the spatial distribution of population of SE state. The CPR measures the spatially coherent length and is determined by the off-diagonal elements of the density matrix. The PPR and CPR can be individually applied to the carrier dynamics on SE, CT and TT states. However, the populations among each type of states are varying as the time goes on, and they are not constant. To avoid this artificial effect in PPR and CPR, we thus individually rescale the population with respect to each type of states at every time step.

3

Results and Discussion

In numerical simulations, the basic parameters in the Hamiltonian Eq. (6) are assigned to ³

´

³

´

³

CT TT model a pentacene crystal. 4,22,34,47 The energies of SE EnSE , CT En,n±1 and TT En,n+1

´

states are chosen as 200 meV, 400 meV and 0 meV respectively. The SE couplings J SE , CT couplings J CT and TT couplings J T T are 20 meV, 0 meV, and 0 meV, respectively. The interactions among different-type exciton states, V SE−CT , V T T −CT and V SE−T T , are 50 meV, 50 meV and 5 meV, respectively. Spectral density Jn (ω) is chosen as a Debye-Drude form Jn (ω) =

2λωc ω ω 2 +ωc2

, where λ = 50 meV is the reorganization energy and ωc = 1450 cm−1 is the

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characteristic frequency of vibrational motions. Temperature T is set to be 300K. Based on these initial values of parameters, we will further change the values of some specific parameters to investigate the dependence of SF dynamics on SE coupling strength J SE , TT coupling strength J T T and the energies of CT states E CT .

3.1

SF Dynamics With and Without CT States

We start from a one-dimensional aggregate composed of 10 monomers as a model to study SF dynamics, and the initial SE population is at the central monomer. This aggregate length is found to be long enough because most of exciton wavefunction is located within the aggregate and is less likely to reach the edges of chain during SF process. Figures 2 (a) and (b) show the temporal and spatial resolved evolutions of exciton population dynamics with and without the CT states incorporated. It is found that SF dynamics is obviously different between the two cases. As the CT states are incorporated, a fraction of population can transiently occupy CT states initially, then quickly converts into TT states, and the appearance of TT population is obviously earlier than that from considering only direct interaction. Furthermore, SE and TT populations are also more localized. The different SF dynamics can be further exhibited by the total population evolutions on SE, CT and TT states, shown in Figure 3. As the CT states are incorporated, it is found that both the populations on SE and TT states evolve exponentially, for instance, the population evolution of TT states can be fitted by an exponential function P (t) = P0 + Ae−t/τ (the dots in Figure 3 (a)), and the fitted coefficients are P0 =0.95, A=−0.93 and τ =298.35 fs. In this case, the SF rate can be naturally taken as the inverse of τ , i.e., k = 1/τ =3.35 ps−1 . The exponential increase of TT population might be explained by an instantaneous conversion of CT population into TT states. Indeed, a quite large amount of CT state population (30%) occurs at a very short initial time and it rapidly converts into TT states. In spite of obvious CT population, the effect of CT states can be still thought as a superexchange interaction 62 . For the direct SF process, an obviously different population dynamics is observed, espe10


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(a)

0.6

SE

0

8

200 Time (fs)400 600

2

0 10

(b)

6 4 State 0.06

0 200 Time (fs)400

0 200 Time (fs)400 600

2

14 10 6 State

0 18

600

0 8

200 Time (fs)400 0

600

2

0.3

0

TT

200 Time (fs)400

2

8 6 4 State

0 10

TT

0.4

0

0.6

SE

CT

600

2

6 4 State

8 6 4 State

Figure 2: Spatially-temporally resolved exciton population dynamics with CT states (a) incorporated and (b) not incorporated, in an organic aggregate chain with 10 monomers.

0.8

(a) SE CT TT fit

0.6 0.4 Population

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.2 0.8

(b)

0.6

SE TT fit

0.4 0.2 0

200

400 Time (fs)

600

800

Figure 3: Total population evolutions with CT states (a) incorporated and (b) not incorporated.

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cially, the TT state population has an S-shape evolution with respect to time. Together with the spatial SE distribution displayed in Figure 2(b), the S-shape dynamics can be explained as follows. Initially, the SE migration may dominate the dynamics by the SE-SE coupling. As the population of SE states spatially transfer into enough monomers, the SF process can be sped up because of the many conversion channels on the occupied SE monomers, later on, a small fraction of remaining SE population leaks into the TT states and the decay process of the population should be exponential. The S-shape TT state population can be well fitted with an approximation of Heaviside step function P (t) = a/(1 + e−w(t−t0 ) ), as shown by dots in Figure 3 (b). Here, t0 is the time at which the TT population reaches the half of plateau value a. It is noted that the SE population also decays to the half at t0 , thus, the inverse of t0 (k = 1/t0 ) can represent SE population decay rate. w corresponds to the width of S-shape curve and representing the population rising rate, and it can serve as a measure of SF rate for the direct process. The fitted values of a, k and w from Figure 3 (b) as 1.00, 3.67 ps−1 and 13.10 ps−1 , respectively. It is noted that both the exponential 10 and S-shape 8,33 evolutions of TT population have been observed experimentally, and that may be taken as a simple judgement to discern whether CT states are obviously got involved in SF processes or not. It should be addressed that even if the CT states are incorporated, the S-shape TT dynamics may be also observed when the CT-state energies are high or the SE-CT and CTTT interactions are small enough. In this case, the effective indirect interaction via CT states should become quite small. To show this property, we calculate the population evolutions in terms of CT state energies (E CT ), and the results are exhibited in Figure 4. As expected, at low E CT (200 and 400 meV), the TT state populations have exponential shapes and SF rates become larger with lower CT energies. As the CT energies are higher than 800 meV, the exponential shapes of TT population are switched into S-shapes. In these cases, the CT states act as virtual states since they are nearly empty during all the time. Interestingly, although the TT population has an S-shape increase at the CT energy of 800 meV, the corresponding TT rising rate is obviously smaller than that from the direct SF

12


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Population

0.9

(a)

(b)

0.7 0.5 E

0.3

CT

=200meV

E

CT

=400meV

CT

=2000meV

0.1 0.9 Population

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(d)

(c)

0.7 0.5 E

0.3

CT

=800meV

E

0.1 100

300 500 Time (fs)

700

100

300 500 Time (fs)

700

Figure 4: Total population evolutions on TT (green lines) and CT (red lines) states at different CT-state energies. The solid lines come from the direct interaction, whereas dashed and dashed-dotted lines correspond to the CT-mediated interactions with the SE-TT interactions of 5 meV and −5 meV, respectively. process. It is known that the sign of effective interaction via CT states is negative 24,30 whereas the sign of direct interaction is positive in the present model. One may expect that the magnitude of indirect interaction is similar to that of direct interaction, and the interference between direct and indirect conversion channels leads to a small SF rate. This behaviour may cause a destructive interference 28,30 for SF conversion. To further demonstrate it, we change the direct interaction V SE−T T to −5 meV and the obtained TT state generation rates, shown in Figure 4, become larger than that from the direction interaction. For the CT states with higher energies, TT population evolutions both from positive and negative SE-TT interactions V SE−T T converge into that from direct interaction, manifesting that the indirect interaction becomes small and the direct interaction dominates the SF conversion.

3.2

Effects of SE Migration on SF Dynamics

We now concentrate on the effect of SE migration on SF process. Without loss of generality, we consider the SF process with the direction interaction, thus the disturbance via CT states can be avoided. The results in the above subsection already show that the SF rate 13



is controlled by the timescales of initial SE migration and SF conversion. One might thus expect that the SF could become faster with stronger SE-SE couplings because a stronger coupling can lead to a faster spatial SE population transfer, subsequently, more conversion channels. Indeed, the mechanism has been used to explain an experimental observation. 40 To reveal quantitative dependence of SF process on the SE migration process, we thus calculate the SF population dynamics in terms of SE-SE coupling and aggregate length while the other parameters are kept unchanged. In a broad regime of SE-SE couplings, it is found that the evolutions of TT population always have an S-shape property, and can be fitted by the approximate Heaviside step function. Figures 5 (a) and (b) exhibit the SE decay rate k and the SF rate w fitted from S-shape dynamics of TT population in terms of aggregate length and SE-SE coupling.

k (ps-1)

3.5 (a) 2.5 1.5

w (ps-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 31

12

(b)

8

2monomers 4monomers 10monomers 20monomers

4 0

20

40 JSE (meV)

60

80

Figure 5: (a) SE decay rates k and (b) SF rates w with respect to the SE-SE coupling and the number of monomers. For a given SE-SE coupling strength, Figure 5 clearly shows that both k and w increase with the increase of aggregate length. This feature manifests that the broadly spatial distribution of SE population indeed benefits SF. However, the rates become converged as the number of monomers is greater than 10 with different SE-SE couplings. This independence of aggregate length may result from the competition between SE migration and SF conversion 14


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processes. As the SE population is distributed on more than 10 monomers, the number of channels from SE to TT states may be enough to make the SF process faster than the SE migration, leading to little SE population remained. Therefore, the SF rates go to convergence with respect to aggregate length. For a given aggregate length, surprisingly, the SE decay rate k and SF rate w each has a maximal value with respect to SE-SE couplings no matter whether the aggregate length is short or long. In the weak regime of SE-SE couplings, the increase of SF rate with the increase of SE-SE coupling strength is easily explained by faster SE migration process caused by a larger coupling strength. In the strong coupling regime, however, the SF rates decrease as the SE-SE couplings increase. This feature is contradicted with the above concept and the corresponding mechanism is unclear. To understand this intriguing property, we reveal detailed SE energy relaxation dynamics, which is important to understand SE migration and is also straightforwardly related to ultrafast experimental observations. We diagonalize the Hamiltonian matrix HE (S † HE S = E), and transform the population distribution from the spatial representation into the energy representation by S matrix. The initial SE wavepacket in the energy representation is assumed to be a normalized Gaussian wavepacket f (E) =

1 −(E−E 0 )2 /2∆E e , Z

where E 0 and ∆E are the averaged energy and

the magnitude of energy distribution, respectively. It is noted that this initial condition represents the distribution over multiple discrete nearly degenerate SE states in the spatial representation. In ultrafast experiments, E 0 corresponds to a laser central frequency and ∆E is determined by laser pulse duration. We use E 0 = 200 meV (resonant excitation) and ∆E = 20 meV (pulse duration in femtoseconds) in the calculations. Figures 6 (a)-(c) exhibit the energy relaxation dynamics at weak (10 meV), moderate (25 meV) and strong (60 meV) SE-SE couplings. In the weak coupling regime, SE needs time to relax into low energy SE states, and as SE is on low energy SE states, it quickly converts into TT states. In this case, the energy relaxation process is a rate-determining step. With the increase of SE-SE coupling, the energy relaxation is sped up, leading to a large SF rate. In the strong

15



210 Energy (meV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(a)

195

190

175

180 0.0 -0.5 0

200 400 Time (fs)

600

0.5

215

200

Page 16 of 31

(b)

(c)

225

0.4

180

0.3

135

155

90

0.0 -0.5

0.0 -0.5 0

200 400 Time (fs)

600

0.2 0.1 0

200 400 Time (fs)

600

0

Figure 6: Energy relaxation dynamics in a 10-monomer chain at the SE-SE coupling strengths of (a) 10 meV, (b) 25 meV and (c) 60 meV. coupling regime, interestingly, although the energy relaxation time is very short, the relaxed SE on the low-energy states has a long lifetime and does not immediately convert into TT states. The SE-TT conversion alternatively becomes rate-controlling step. To explain this slow conversion on the low energy states, one may consider the concept of symmetry forbidden on the low energy states which has been proposed by several works 64,65 . By doing block diagonalization for SE states, we find that the SE-TT interactions on low energy SE states are indeed smaller than those on high energy states. It can be intuitively expected that the SE relaxation into the low energy states is too fast to convert into TT states from relative high energy states, leading to slow SF rates. To further demonstrate this mechanism, we use negative SE-SE couplings (J-type aggregate), and again calculate the SE-TT interactions in the energy representation. In this case, the SE-TT interaction become the largest on the lowest energy states. However, similar energy relaxation dynamics is observed in the strong SE-SE coupling regime. Therefore, the symmetry-forbidden is not a dominant factor to determine the slow SF rate at low energy SE states, and it cannot be used to explain the slow SF rates in the strong SE-SE coupling regime. Figure 5 already shows that even for two monomers there exists a maximal SF rate with respect to SE-SE couplings. In this case, SE dynamics is simplified as a spin-boson model and a strong SE-SE coupling can lead to a strong quantum coherence between two SE states. In organic semiconductors, it is also known that a strong SE-SE coupling strength causes

16


Page 17 of 31

a coherent band-like exciton dynamics. 52 It is thus expected that the long lived low-energy exciton states should be related to the spatial quantum coherence of SE. Qualitatively, the strong spatial SE coherence would prefer to make SE wavefunction as a wavepacket. It is well known the wavepacket is commonly bounded and moves together, therefore, the SE wavepacket performed by a strong SE-SE coupling can help SE migration but make it hard to disperse for the SF conversion. As SE-SE couplings are strong enough, although the fast SE population transfer can accelerate SF rates, the slow conversional rates caused by strong SE coherence may dominate SF process. Thus, the joint effects of population transfer and coherence of SE state could lead to a maximal SF rate. To confirm the mechanism, one should demonstrate that the spatial SE coherence is maintained at the low energy states. Normally, this coherence is easily destructed by exciton-phonon interactions. On the other hand, if the SE coherence is artificially destructed, for instance, by the static disorders from the morphologies of organic aggregates, the SF rates should be enhanced.

PPR

10 (a) 7

JSE=10meV SE J =15meV JSE=25meV JSE=40meV JSE=60meV

4 1

10 (b) CPR

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


7 4 1 0

200 400 Time (fs)

600

Figure 7: (a) PPR and (b) CPR of SE states at several SE-SE coupling strengths. We first investigate whether the SE coherence on the low energy SE states is really maintained or not. Figures 7 (a) and (b) display the PPR and CPR of SE dynamics in the aggregate with 10 monomers at several different SE-SE coupling strengths. As expected, the 17



PPR is positively correlated to the SE-SE coupling strength. When the coupling strength is greater than 25 meV, the values of PPR rapidly reach 10 in the first 50 fs duration, i.e., the SE population spatially transfer among the whole aggregate chain. Alternatively, the SE wavefunctions are nearly incoherent in the weak SE-SE coupling regime because the CPR is quite small. In the strong SE-SE coupling regime, however, the SE wavefunction can transiently be fully coherent (CPR = 9), and more importantly, the SE coherence can be still maintained at long time limit (on the low energy states), for instance, the wavefunction becomes coherent among 6 monomers at the coupling of 60 meV. These results indeed manifest that the coherence of SE wavefunction on the low energy states becomes stronger with

k (ps-1)

a larger SE-SE coupling strength. 6.5 (a) 5.5 4.5 3.5 2.5 24 w (ps-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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(b)

18 12

σ=0meV σ=50meV σ=100meV σ=200meV

6 10

30 50 JSE (meV)

70

Figure 8: (a) SE decay rates k and (b) SF rates w with respect to the SE-SE coupling at several SE energy disorders. We now introduce the static disorders in SE energies to see how the SF rates change as the SE coherence is destructed. The static disorders ²n are assumed to have a Gaussian distributions

√ 1 2πσn

2

n i) exp(− (²n −h² ), where h²n i is the expectation value and it is taken as 2σ 2 n

zero, and σn represents the width of fluctuation energies, i.e., the fluctuation strength. In the simulations, these fluctuation energies are added in the original SE energies. Figure 8 exhibits the corresponding SF rates. Obviously, static disorders accelerate the SF rates 18


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PPR

10 (a) 7 σ=0meV σ=50meV σ=100meV σ=200meV

4 1

10 (b) CPR

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


7 4 1 0

200 400 Time (fs)

600

Figure 9: (a) PPR and (b) CPR of SE states at several exciton-energy disorders. by destructing the SE coherence, and the optimal SE-SE coupling strength with respect to the maximal SF rates are shifted into larger values with the increase of the static disorder strengths. It should be addressed that the static disorder should not be too strong, otherwise, the Anderson localization may occur and thus block SF process. To find the detailed dependence of SE migration on static disorder, Figures 9 (a) and (b) display the PPR and CPR with the SE-SE coupling of 60 meV at several disorder strengths. At weak disorders, the values of PPR have only slight changes, however, the CPR decrease with the increase of disorder strengths, manifesting that the disorders do not significantly change the spatial SE population transfer but dramatically affect SE coherence. In the case of σ=200 meV, according to the Gaussian distribution, there exists a chance that about 16% of the SE states to be less energetic than TT states, however the SF rates from moderate to strong SE-SE couplings are still larger than that with smaller energy disorders, indicating that the coherence destruction effect plays a major role on SF process. These results are consistent with our qualitative analysis from Figure 8. It is noted that at very strong disorder, both PPR and CPR decrease, and the disorder-induced Anderson localization begins to play a role.

19



3.3

Effect of TT-TT Coupling on SF Rate

Commonly, the TT-TT coupling strength is quite small compared to a SE-SE coupling strength. In our above calculations, this coupling strength is thus assumed to be zero. For non-zero TT-TT couplings, the TT state population may also migrate in aggregates. In order to reveal the effect of TT migration, we calculate the SF dynamics with several different TT-TT coupling strengths, and the results are shown in Figure 10. Interestingly, the TT-TT coupling essentially hinders SF process, and the rates decrease as the TT-TT coupling strengths increase. It is known that the TT-TT couplings can lead to the TT population distribution along the aggregate chain, and the occupied TT states has more channels than the localized one to convert TT to SE backwardly. This behaviour may lead to a slow SF rate. Meanwhile, the cooperative transport of SE and TT may occurs, and a similar phenomenon has been observed experimentally 21,50 . 1.0 TT state population

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Page 20 of 31

0.8

JTT=0meV JTT=5meV JTT=10meV JTT=20meV

0.6 0.4 0.2 0.0 0

200

400 600 Time (fs)

800

1000

Figure 10: TT population dynamics in terms of TT-TT coupling. The dashed line is the result when both SE-SE and TT-TT couplings are zeros. To quantitatively demonstrate the TT migration property with respect to the TT-TT coupling strength, Figures 11 (a) and (b) display the corresponding PPR and CPR. Similar to the case of SE, these values are positively correlated to TT-TT couplings. As expected, the TT states have an obviously coherent property as their coupling strength is close to that of SE-SE coupling. This coherence may have a similar effect on the SF process with that 20


Page 21 of 31

10 PPR

8

(a)

6 4 2 5

CPR

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


TT

J =0meV JTT=5meV TT J =10meV TT J =20meV

(b)

3 1 0

200 400 Time (fs)

600

Figure 11: (a) PPR and (b) CPR of TT states in terms of TT-TT couplings. The dashed line is the result for zero SE-SE and TT-TT couplings. from SE-SE coherence. Indeed, the corresponding rate is even smaller than that for the localized SF process, shown by the dashed line in Figure 11 (a), where both the SE-SE and TT-TT coupling strengths are zeros.

4

Concluding Remarks

In summary, we have applied the TDWPD method to investigate the effects of CT states, SE and TT migrations on SF dynamics, and found many interesting properties. First of all, the TT population increases exponentially as the superexchange interaction is obviously incorporated via CT states whereas it exhibits an S-shape dynamics for direct SE-TT interactions. This feature could possibly serve as a simple tool to judge whether the CT states are involved or not in perfect organic crystals. Second, an obvious interference is observed between two conversion channels from direct and indirect superexchange interactions, and whether interference is constructive or destructive is determined by the signs of these interactions. As the CT-state energies are high enough, the SF dynamics is converged into that from the direct interaction. Third, the SF rates increase with increase of aggregate length

21



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Page 22 of 31

and can reach a convergent value as the aggregate length is long enough. Fourth, in the case of direct coupling, the population transfer and spatial coherence of SE have opposite effects on SF process, resulting in an optimal SE-SE coupling strength at which the SF rate is maximal. The mechanism has been confirmed by the effect of static exciton-energy disorders on SF rates. Finally, it is found that the couplings among TT states decrease the SF rates by increasing TT coherence as well as promoting backward conversion. The present finding should provide a physical insight for understanding experimentally inconsistent observations with respect to crystalline morphologies, and designing concepts for realizing a high efficiency SF process in organic aggregates.

5

AUTHOR INFORMATION

Corresponding Authors Yi Zhao, E-mail [email protected]; phone +86-592-2189197 (Y.Z.). WanZhen Liang, E-mail [email protected]; phone +86-592-2184300 (W.L.). Notes The authors declare no competing financial interest.

6

ACKNOWLEDGMENTS

This work is partially supported by NSFC (Grant Nos: 21290193, 21373163, 21573177). YZ acknowledges financial purports from the NSFC (Grant Nos. 21573175,91333101,21133007).

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Effects of Charge Transfer State and Exciton Migration on Singlet

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