Polariton-assisted Singlet Fission in Acene Aggregates

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Surfaces, Interfaces, and Catalysis; Physical Properties of Nanomaterials and Materials

Polariton-Assisted Singlet Fission in Acene Aggregates Luis Angel Martínez-Martínez, Matthew Du, Raphael Florentino Ribeiro, Stéphane Kéna-Cohen, and Joel Yuen-Zhou J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.8b00008 • Publication Date (Web): 18 Mar 2018 Downloaded from http://pubs.acs.org on March 26, 2018

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The Journal of Physical Chemistry Letters

Polariton-assisted Singlet Fission in Acene Aggregates Luis A. Martínez-Martínez,† Matthew Du,† Raphael F. Ribeiro,† Stéphane Kéna-Cohen,‡ and Joel Yuen-Zhou∗,† †Department of Chemistry and Biochemistry, University of California San Diego, La Jolla, California 92093, United States ‡Department of Engineering Physics, École Polytechnique de Montréal, Montréal H3C 3A7, QC, Canada E-mail: [email protected]

Abstract Singlet fission is an important candidate to increase energy conversion efficiency in organic photovoltaics by providing a pathway to increase the quantum yield of excitons per photon absorbed in select materials. We investigate the dependence of exciton quantum yield for acenes in the strong light-matter interaction (polariton) regime, where the materials are embedded in optical microcavities. Starting from an open-quantum-systems approach, we build a kinetic model for time-evolution of species of interest in the presence of singlet quenchers and show that polaritons can decrease or increase exciton quantum yields compared to the cavity-free case. In particular, we find that hexacene, under the conditions of our model, can feature a higher yield than cavity-free pentacene when assisted by polaritonic effects. Similarly, we show that pentacene yield can be increased when assisted by polariton states. Finally, we address how various relaxation processes between bright and dark states in lossy microcavities affect polariton photochemistry. Our results also provide insights on how to choose microcavities to enhance similarly related chemical processes. ACS Paragon Plus Environment

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TOC Graphic

Singlet fission (SF) is a spin-allowed process undergone by select materials that permits the conversion of a singlet exciton into a triplet-triplet (TT) state with an overall singlet character, which later decoheres and forms two triplet excitons. This process has been used to enhance the external quantum efficiency of organic solar cells 1,2 by allowing a single absorbed photon to produce more than one exciton. In this work we explore the influence of strong light-matter coupling (SC) on the TT yield of acenes. This regime can be achieved at room-temperature, for example, in optical microcavities enclosing densely packed organic dyes 3 . Under these conditions, the energy of interaction between the microcavity photonic modes and the molecular degrees of freedom of the material is larger than their respective linewidths. The hybrid states that arise from this interaction are called polaritons. The latter have previously been exploited to tune the properties and functionality of organic materials at the molecular level. For instance, there have been experimental and theoretical efforts to explore the potential applications of SC in photochemistry, where the electrodynamic vacuum can significantly alter molecular processes 3–11 . Previous studies have also explored SC in the context of exciton harvesting and transport 12,13 , Raman scattering 14,15 and photoluminescence spectroscopy 16,17 , Bose-Einstein condensation 18–20 , and topologically-protected states 21 , just to mention a few examples. By developing a microscopic model for the relevant processes, we address the effects of SC on the TT yield in aggregates of acene dyes (tetracene, pentacene and hexacene) and determine the important molecular parameters that rule this yield. Our starting point is a kinetic model based on a Pauli master equation formalism that describes the population dynamics of the states that ACS Paragon Plus Environment

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take part in SF 22 . We then use this model to elucidate the circumstances under which polaritons can enhance SF under realistic dissipative conditions. Theoretical model.— We consider a simplified one-dimensional acene aggregate comprised of N identical molecules embedded in a microcavity and strongly interacting with a single electromagnetic mode supported by the latter. The Hamiltonian of the model is given by H

=

HS + HB + HS−B + Hp + Hp−S + HT T + HT T −B + HT T −S , (1)

where HS (HT T ) is the electronic singlet (TT) Hamiltonian of the aggregate given by (~ = 1)

X

HS =

n N −1 X

HT T =

ω e |nihn|,

(2a)

ω T T |Tn Tn+1 ihTn Tn+1 |,

(2b)

n=0

where |ni is a localized singlet (Frenkel) exciton 23 at the nth site (molecule), and |Tn Tn+1 i denotes P P a TT state delocalized over sites n and n+1. Here ω e = ωe + i ωi λ2S,i (ω T T = ωT T +2 i ωi λ2T,i ) is the vertical singlet (TT) excitation frequency, where ωe (ωT T ) and λS,i (λT,i ) are the 0-0 excitation frequency and the square root of the Huang-Rhys factor 24 for the ith vibrational mode coupled to the transition |Gi → |ni (|Gi → |Tn Tn+1 i), respectively, and |Gi is the state corresponding P to all molecules in the electronic ground state. HB = n,i ωi b†n,i bn,i accounts for the vibrational degrees of freedom of the ensemble, where b†n,i (bn,i ) is the creation (annihilation) operator of the i-th harmonic vibrational degree of freedom with frequency ωi on site n. The singlet (TT) vibronic couplings are encoded in HS−B (HT T −B ), given by

HS−B =

X

|nihn|ωi λS,i (bn,i + h.c.),

(3a)

n,i

HT T −B =

N −1 X

|Tn Tn+1 ihTn Tn+1 |

n=0

×

X

ωi λT,i (bn,i + bn+1,i + h.c.).

i

ACS Paragon Plus Environment

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(3b)


The singlet-TT electronic coupling is (assuming periodic boundary conditions |T−1 T0 i = |TN −1 T0 i): 25 N −1 i VT T −S X h (|Tn Tn+1 i + |Tn−1 Tn i) hn| + h.c. . (4) HT T −S = 2 n=0 Finally, the photonic degree of freedom is included in Hp = ωph a† a where a† (a) is the creation (annihilation) operator of the cavity photonic mode. Its interaction with the singlet excitons is described by the light-matter Hamiltonian X Hp−S = g a† |Gihn| + h.c. n

√

N g a† |Gihk = 0| + h.c. Ω † = a |Gihk = 0| + h.c. 2 =

(5)

where in the second line we have introduced a delocalized Fourier basis for the singlet excitons √ P , m = 0, 1, 2, . . . , N − 1. The N g term in (5) is the collective |ki = √1N n eikn |ni k = 2πm N light-matter coupling and Ω is the so-called Rabi splitting. Importantly, the singlet excitons are optically bright, in contrast to the dark TT states 25 . Because of this property, our model does not feature a TT term analogous to Eq. (5).

a)

b)

kSF~15ps-1 Singlet TT

~10-4ps-1

Energy

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c)

γ~5ps-1 γ~5ps-1 α

γ~5ps-1

TT UP

Dark states LP

α/N

Singlet

κ|c-ph|2~20ps-1

TT

κ|c+ph|2~20ps-1

κrad~10-4 ps-1

Q

Figure 1: a) Bare triplet-triplet (red) and singlet (blue) vibronic energies of the different molecules considered. From left to right: tetracene, pentacene and hexacene. For clarity only the vibrational mode with highest frequency is shown. The SF dynamics is schematically shown for bare case (b) and the SC scenario (c). In (c) the fastest decay constant α ≈ 100 ps−1 is due to vibrational 2 − 2 relaxation, γ is the dressed SF rate, κ is the cavity-photon leakage rate and |c+ ex | (|cex | ) is the exciton fraction in the upper (lower) polariton. kf is the singlet fluorescence decay rate. Continuous arrows denote radiative decay. Thicker lines indicate larger density of states. Approximate SF and relaxation rates for pentacene interacting with a resonant photonic mode are included. The decay timescale for the triplet-triplet state is significantly longer than the timescales considered here. ACS Paragon Plus Environment

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In our approach we consider the reduced population dynamics of the manifold of TT states ({|Tn Tn+1 i}), the two polariton states (|±i) and the so-called dark states ({|di = |k 6= 0i}) that emerge from SC (see Fig. 1 and the Supporting Information (SI) for additional details of the method), where

+ |+i = c+ ph |Gi ⊗ |1ph i + cex |k = 0i ⊗ |0ph i,

(6a)

− |−i = c− ph |Gi ⊗ |1ph i + cex |k = 0i ⊗ |0ph i.

(6b)

with (zeroth-order) eigenenergies given by ω e + ωph ± ω± = 2

s

ω e − ωph 2

2

√ +

2 Ng ,

(7)

In Eq. (6), |nph i is the state with n photons in the photonic space and |+i (|−i) is the upper ± (lower) polariton state. Meanwhile, c± ph and cex are the Hopfield coefficients for the photon and

exciton components, respectively, of the polariton states 26 . Importantly, the validity of Eq. 6 as the starting point of our (perturbative) approach (see SI for more details) relies on the consideration of large Rabi splittings compared to the singlet reorganization energy 27–29 . In our model, the latter is assumed to be 50 meV, a value which even though is well below experimental observations, it provides an accurate description of the dynamics for pentacene 30,31 . The reduced dynamics under SC is described by means of a Pauli master equation derived using the Redfield formalism under the secular and Markov approximations 22 (see SI), in analogy to previous theoretical descriptions of SF 30,31 . The kinetic model can be summarized by the following equations,

α(ω±D ) 2 α(ωD± ) total (N − 1)P± + |c± PD ex | N N 2 γ(ωT T,± ) 2 γ(ω±,T T ) total − |c± (N )P± + |c± PT T ex | ex | N N ± 2 2 − |c± ex | kc (ω± ) + |cph | kphot P± ,

2 ∂t P± (t) = −|c± ex |

α(ω+D ) 2 α(ωD+ ) total (N − 1) P+ − |c+ PD ex | N N 2 α(ω−D ) 2 α(ωD− ) total + |c− (N − 1) P− − |c− PD ex | ex | N N

2 ∂t PDtotal (t) = |c+ ex |

− γ(ωT T,D )PDtotal + γ(ωD,T T )PTtotal T ACS Paragon Plus Environment

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(8a)

(8b)


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− kc (ωD )PDtotal , + 2 + 2 ∂t PTtotal T (t) = |cex | γ(ωT T,+ )P+ − |cex |

γ(ω+,T T ) total PT T N

(8c)

total − γ(ωD,T T )PTtotal T + γ(ωT T,D )PD 2 − 2 + |c− ex | γ(ωT T,− )P− − |cex |

γ(ω−,T T ) total PT T , N

where P+ (t) (P− (t)) is the population in the |+i (|−i) state and PDtotal (PTtotal T (t)) is the total population in the dark (TT) state manifold. The photon and exciton content of the polaritonic 2 ± 2 states are given by |c± ph | and |cex | , while ωD = ω e and ωab = ωa − ωb . In Eqs. (8a) and (8b),

we phenomenologically introduce the rate constant kc (ω) to account for the contribution to the decay rate of the dressed states due to their singlet exciton fraction. kc (ω) can account for e.g., radiative or non-radiative relaxation of the singlet to the ground electronic state, or its conversion into charges at the interface with charge acceptors, which is the case in donor-acceptor blends used for organic solar cells. The various cases are treated in more detail below. The α(ω) transfer rates appearing in Eqs. (8a) and (8b) are calculated in terms of a bath spectral density and thermal populations at frequency ω. The γ(ω) rates are computed with a Bixon-Jortner-like 32 equation adapted to the SC regime. For sake of simplicity, in the calculation of γ(ω) the vibrational bath is treated by using an effective high (low) frequency ω h (ω l ) that satisfies ω h 1/β (ω l 1/β), to which we associate a so-called inner (outer) sphere reorganization energy 33 . We refer the reader to the SI for details of the derivation and other relevant parameters employed in the calculation of α(ω) and γ(ω). The parameters ∆G = ωT T − ωe , and VT T −S /2 take part in these calculations and are treated as material-dependent; they are taken from Ref. 2 and are summarized in Table 1. Table 1: Summary of bare material-dependent parameters. The values for VT T2−S and ∆G = ωT T − ωe were taken from Ref. 2 . Based on these, we calculate the bare SF rates kSF and bare TT yields T T . Importantly, T T is calculated assuming that SF competes with the fast singlet decay process (charge production) kCT = 17 ps−1 , as discussed in the main text. Molecule VT T −S /2 (meV) Tetracene 41.5 Pentacene 42 Hexacene 22

∆G (meV) 150 -110 -630

kSF (ps−1 ) 0.01 17 6


6

T T (%) 0.01 100 48

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Discussion of results.— We first consider the population dynamics of bare (cavity-free) systems containing one of the acene molecules,

dPS (t) = − kSF + kc (ωD ))PS (t) + kT F PT T (t) dt dPT T (t) = kSF PS (t) − kT F PT T (t) dt

(9a) (9b)

where PS (t) (PT T (t)) is the population of the singlet (TT) electronic state of a given acene, kSF is the bare SF rate, and kT F is the bare triplet fusion rate, which corresponds to the reverse process to SF. kSF and kT F are calculated by means of the Bixon-Jortner equation 32 , with the parameters in Table 1. For all cases we compute PS (t) and PT T (t) assuming PS (0) = 1, PT T (0) = 0 (see Fig. 2). Under these conditions, we define the TT yield

T T (t∗ 0) = 200% × PT T (t∗ )

(10)

as the relevant figure of merit for our subsequent analysis where t∗ was chosen to reach a stationary PT T (t) value for pentacene and hexacene. We notice that when kc (ω) = kf = O(10−4 ) ps−1 (singlet fluorescence rate 22 ) pentacene and hexacene are expected to exhibit a 200% TT yield in view of kSF kf and kT F = eβ∆G kSF kSF (detailed balance, where β is the inverse temperature). This contrasts with tetracene, in view of its higher TT energy compared to the singlet, (ωT T > ωe ) so that kSF kT F and the TT population decays to zero for long t. The experimental TT yield of tetracene is well above zero 34,35 , which is in contrast with our findings (see Fig. 2). However, the mechanism to explain this observation is still under debate. 36,37 To the best of our knowledge, a coherently assisted pathway is the most recent proposal to explain this unexpected high yield. 38 Regardless, we have opted to analyze the results that follow from our model, as they should be valid for any system with similar singlet and TT energetic arrangement in the absence of the aforementioned coherent mechanism. In organic solar cells TT yield is typically below 200% because the processes of singlet migration and charge separation are fast enough to compete with kSF 1 .TT yields closer to 200% in the presence of the fast charge separation process are desirable since they translate into a greater


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number of charges collected per photon absorbed by the organic molecules. For instance, a 200% (100%) TT yield equals to 2 (1.5) electrons collected per photon absorbed (assuming that all singlet and TT states undergo charge separation, which is reasonable in view of the long lived singlet and TT states compared with the fast charge separation time scale.) In our model, to consider an organic-solar-cell-like scenario we assume that the singlet state quickly decays to a charge-transfer state. This would correspond to the incorporation of a charge-acceptor next to each of the acene molecules of the chain. For simplicity and for the purpose of showing the possibilities of control of TT yield by polaritonic means, we assume kc (ω) = δωD, ω kCT (δi,j being the Kronecker delta function), where kCT is equal to the bare pentacene rate kSF = 17 ps−1 . The form introduced for kc (ω) is approximately correct as long as the spectral density describing the singlet-chargetransfer state is peaked around ωD and decays quickly with ω. The aforementioned kCT value is experimentally reasonable as it has been observed in solar cells with a thin slab of SF material 1 . We use Eq. (10) to compute the T T values summarized in Table 1 in the presence of kCT . Our definition of yield is thus different from that obtained in steady-state, but follows the spirit of

Population

many experiments that measure SF using time-domain spectroscopy techniques 39–41 .

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

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Time (ps)

Time (ps)

Figure 2: Time evolution of populations in the singlet (blue) and TT electronic states (red) of the molecules considered in this work in the bare case. We consider the initial conditions that follow from pumping the singlet state at t = 0 (PS (t = 0) = 1, PT T (t = 0) = 0), for all the molecules in question. Inset: time evolution of PT T (t) for tetracene. Turning now to the polariton-assisted SF case, the non-trivial dynamics that emerge are due to differences in the density of states (DOS) between the polariton and exciton manifolds, as well as 2 to the photonic character of each polariton state. These traits are encoded in the prefactors |c± ph | , 2 total |c± ex | , N and 1/N in Eqs. (8a)-(8c). Notably, in the N 1 limit, the transfer rates from PD


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b)

1.0

0.16 0.14 0.12 0.10 0.08 0.06

0.0 - 0.5 - 1.0

0

Triplet Yield (%)

Δ ωh

Δ ωh

0.5

Triplet Yield (%) 100 90 80 70 60 50

-1

Out[6]=

-2 -3

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

Ω ωh

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

Ω ωh

4

6

4

5

Triplet Yield (%) 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01

2

]=

1

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

Ω ωh

3

Δ ωh

3

0

c)

1

Triplet Yield (%) 140 120 100 80 60 40 20

2

Out[36]=

1 0 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

Ω ωh

Triplet Yield (%)

4

Δ ωh

a)

Δ ωh

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


160 140 120 100 80 60 40 20

3 2

Out[23]=

1 0 -1

2

3

4

5

6

Ω ωh

Figure 3: TT yield as a function of ∆/ω h = (ωph − ωe )/ω h and Ω/ω h featured by a) tetracene, b) pentacene and c) hexacene. For each acene, the upper (lower) plot considers initial conditions with upper (lower) polariton pumping, i.e. P+ (0) = 1,Pa6=+ (0) = 0 (P− (0) = 1, Pa6=− (0) = 0) . to the polariton manifold are largely suppressed. This is a consequence of the large and PTtotal T DOS of the former (which act like a population sink) and the small DOS of the latter (which is spectrally isolated). The reverse transfers are fast as they have single-molecule relaxation scalings and correspond to going into the population sink. Similar findings are reported in Ref. 15, in the context of the dynamics of molecular vibrations under the SC regime and in 42,43 for exciton polaritons. Therefore, once population reaches the dark and TT states, it is no longer transferred back to the polariton manifolds, and the subsequent dynamics is determined by transfer rates between dark and TT states. We stress, however, that such asymmetry is approximate, as we are ignoring the polariton bandwidth that emerges from the many photonic modes hosted by the microcavity, which yields non-zero transfer rates between to the aforementioned manifolds 43,44 . We performed numerical simulations of the dynamics of the polariton-assisted scenario by assuming two different initial conditions: pumping of the upper polariton (UP) (P+ (0) = 1, Pa6=+ (0)) and of the lower polariton (LP) (P− (0) = 1, Pa6=0 (0)) for mentioned acenes. We denote ˜T T as the polariton-assisted TT yields. They were calculated using the same criteria as in Eq. (10). From a comparison of T T (Table 1) and the values ˜T T (∆, Ω) (where ∆ = ωph − ωD is the detuning ACS Paragon Plus Environment

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between the cavity photon and the singlet) in Fig. 3 we notice an enhancement in the TT yield for hexacene, especially when the LP is pumped. Under these conditions and high ∆ values, the state 2 |−i is almost purely excitonic (|c− ex | ≈ 1), and the rate of the channel associated to photon leakage

is suppressed. Moreover, ω− becomes closer to resonance with the third vibrational state of the TT manifold (with frequency ω h ) for a given range of Ω. Thus population transfer |−i → {|Tn Tn+1 i} is faster than the bare SF, as the (outer sphere) energetic barrier is lower in the former. Finally, there is no competition between the previous transfer process and the decay channel associated to kc , as kc (ω− ) = 0, under the assumptions of our model. The enhancement considering pumping of the UP for the same molecule (Fig. 3c, upper) is weaker since the fast rate of the transfer |+i → {|di} competes with the rate of |+i → {|Tn Tn+1 i}. Pentacene shows a similar behavior: when ∆ ≈ −ω h and the UP is pumped (Fig. 3b, upper), then |+i is mainly excitonic and population of the TT states is mainly determined by transfer from the dark state manifold, since α(ω+D ) γ(ωT T,+ ), i.e. the population from the UP is quickly transferred to dark states before transfer to TT states is carried out. Hence, noting that γ(ωT T,D ) ≈ kSF , we recover (the bare) pentacene yield T T . Notice however that for large detunings ˜T T (∆ ≈ −3ω h , Ω ≈ 1.6ω h ) > T T because a phonon blockade prevents fast UP decay into dark states and additionally the charge-transfer decay channel is suppressed (kc (ω+ ) = 0). On the other hand, ˜T T values are higher for pumping of the LP (Fig. 3b, lower) as a result of a reduced decay rate α(ω−D ) to the dark states such that the dominant transfer process is from |−i to TT states. Tetracene shows a distinct behavior in view of the bare energetic arrangements of its singlet 2 + 2 and TT states (see Fig. 1). More concretely we have |c− ex | γ(ωT T,− ), |cex | γ(ωT T,+ ), γ(ωT T,D )

γ(ωD,T T ) for most of the explored (∆, Ω) values. This translates into a rapid depletion of the population of the TT manifold during the considered timescale, which is a consequence of the energy of the TT states lying above the dark state energy, such that the rate of population depletion of TT states towards the dark states outcompetes the rate of the inverse process (in view of detailed balance) as well as rates from the polariton manifold to the TT states. The largest ˜T T values are reached when the UP is pumped (Fig. 3a, upper) for parameters (∆,Ω) which yield a predominantly excitonic character to |+i, and with a rate for the |+i → {|Tn Tn+1 i} process comparable with the


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rates associated to |+i → {|di} and {|Tn Tn+1 i} → {|di}. Considering the pumping of the LP (Fig. 3a, lower), the maximal ˜T T values are lower in view of ωT T − ω− > 0, which greatly diminishes the rate of the transfer |−i → {|Tn Tn+1 i}. 1.0 0.8

Triplet Yield (%)

0.6

Δ ωh

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


0.4 0.2 0.0 - 0.2 2.0 2.2 2.4 2.6 2.8 3.0

70 60 50 40 30 20 10

Ω ωh Figure 4: TT yield as a function of ∆/ω h = (ωph − ωe )/ω h and Ω/ω h featured by a model poor SF material with VS−T T = 44 meV , ∆G = −630 meV, and an outer reorganization energy of 5 meV. We consider initial conditions with LP population (P− (0) = 1, Pa6=− (0) = 0). In this case the only competing decay channel with SF is fluorescence. We confirmed that the order of magnitude of the rates calculated in our model are qualitatively similar to the ones obtained in experiments. For instance, the rate of decay from the UP to dark states for hexacene (Ω = 1.6ω h ,∆ = 0) is approximately 100 ps−1 , while a 20 ps−1 has been observed in J aggregates which feature vibronic couplings of 37 meV. 42,45 In addition, even though the rotating wave-approximation assumed in our model breaks down for very large Rabi splittings, we noticed that the yields reported here do not change significantly (in fact, they experience a modest increase of about 10% for the particular case of the dynamics from the LP in hexacene) when we incorporate ultrastrong coupling corrections (see SI for a detailed explanation). We also point out that for high detunings (more specifically ∆ ≥ 3ω h ), the photonic cavity mode is close in resonance to the high-energy α-band of the acenes considered 46,47 . However the oscillator strength of the latter is almost one order of magnitude smaller than the singlet electronic state considered in this work (the so-called p-band) 47 . In view of the big energy gap between the p- and α− bands, the corrections to the LP energy level are perturbative and do not qualitatively change the results presented in Fig. 3, lower panels. The TT yields that follow from pumping the UP ACS Paragon Plus Environment

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(Fig. 3, upper panels) also remain the same, since the UP (even at the maximum ∆ considered) is sufficiently off-resonant with the α band (See SI for further details on these arguments.) Until now, we have only studied SF materials that already feature high TT yields in the bare case if fast singlet quenching mechanisms like charge transfer are absent. We wonder if similar results hold for poor SF molecules, where the SF rate is low enough to compete with the singlet spontaneous emission rate. To address this, we consider a SF material with a singlet fluorescence decay rate kc (ω) = kf = 2.5 × 10−4 ps−1 and an outer sphere reorganization energy of 5 meV, while keeping the rest of the parameters as for hexacene. This situation could correspond to the latter in a solvent that significantly increases the outer sphere SF energy barriers (see Eq. (S14c) of Supplementary). Under these assumptions, the (cavity-free) SF rate occurs on the nanosecond time scale, and T T = 18%. Upon introducing a strongly coupled photonic mode, pumping the LP (Fig. 4) leads to ˜T T > T T for high ∆ and modest Ω. One of the reasons of this behavior is that the competing role of decay due to photonic leakage is minor in view of the dominant exciton character of the LP. Furthermore, the outer-sphere-mediated transfer to the TT manifold from the LP competes with the nonradiative relaxation to the dark states, but it does so less efficiently than in the (low-outer-sphere activation energy) hexacene case (see Fig. 3c, lower and Fig. 5), hence we observe lower TT yields for the poor SF molecule. The proposed system could give a straightforward verification of polariton assisted SF in the absence of a fast singlet quenching process. To summarize, in this letter, we have shown that when SF materials are subjected to SC with a microcavity mode, the photonic leakage of the resulting polariton states constitutes an important decay channel that can decrease TT production, when compared to the bare case. However, the rates associated with this competing decay channel can be tuned by modifying the ratio ∆/Ω (see Fig. 3), in such a way that the dynamics are dictated by the energy differences and the DOS of the dressed states involved in SF. Given the large DOS of the dark state manifold and TT states, the latter must lie lower in energy with respect to the former so as to avoid population leakage towards the dark state manifold and enhance TT yield. Remarkably, while hexacene exhibits a low TT yield in the bare case when singlet quenchers are present, it is the material which features


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Figure 5: Left: potential energy surfaces (PESs) that describe the outer-sphere-mediated transfer of population from the singlet (blue) to the TT state (red) in a poor SF molecule. When Rl ∆G (Marcus inverted regime) high activation energies for SF emerge. Notice the existence of one TT PES per energy level of the high-frequency (inner sphere) vibrational mode with frequency ω h (for simplicity, we only show two TT PESs). Right: under SC the outer sphere activation energy for transfer of population from the LP (purple) to the TT (red) PES can be tuned as a function of detuning (∆) and Rabi splitting (Ω). When the LP is mainly excitonic, the nonradiative relaxation (jagged arrow) to the dark states (blue) is the main competing channel with transfer of population to the TT manifold. Thicker lines indicate larger DOS. the highest enhancement under the proposed polaritonic approach and can even outcompete the bare pentacene TT yield under the conditions considered in this work. Similarly, we notice an increase of the pentacene TT yield, although the improvements are modest in comparison with those obtained for hexacene. Finally, we have also considered the putative scenario of a SF material with low TT yield, where the SF rate competes with fluorescence. In this case, our model predicts (for acene-like molecules) that for molecules with ∆G 0 substantial enhancement of TT yield can be achieved by polariton methods.

Autor Information Corresponding author *E-mail: [email protected]

Acknowledgments L.A.M.M is grateful for the support of the UC-Mexus CONACyT scholarship for doctoral studies and with Jorge Campos-González-Angulo for useful discussions. R.F.R., M.D. and J.Y.Z. acknowledge support from the NSF CAREER award CHE-1654732. S.K.C. acknowledges support from the ACS Paragon Plus Environment

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Canada Research Chairs program and NSERC RGPIN-2014-06129. L.A.M.M., R.F.R., M.D. and J.Y.Z. are thankful with UCSD for generous startup funds. L.A.M.M. and J. Y. Z. acknowledge Ming Lee Tang, Michael Tauber, Shane R. Yost and Troy Van Voorhis for helpful comments. Supporting Information Available: the derivation of the main equations shown in this work is presented in the Supporting Information.

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Polariton-assisted Singlet Fission in Acene Aggregates

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