Wenbin Lu(蘆文斌), Yongcong Chen(陳永聰),?, Xuyun Yang(楊旭云), and Ping Ao(敖平)

1Shanghai Center for Quantitative Life Sciences&Physics Department,Shanghai University,Shanghai 200444,China

2College of Biomedical Engineering,Sichuan University,Sichuan 610064,China

Keywords: organic nanocrystals,phosphorescent emission,resonance energy transfer,exciton-polariton

1.Introduction

Artificial light-harvesting systems of organic selfassembled crystals have attracted considerable research interests due to their potential applications in solar energy and micro electronics.[1-3]Inspired by the ultra-high efficient energy transfer(over 95%)in natural plants and photosynthetic bacteria, the light-harvesting nanocrystal systems, including those assembled by transition metal with different ligands, have been extensively studied.[4-6]Vast numbers of light-absorbing antennas (donors) and rapid transfer of excitations to a small number of acceptors are the key features observed in these systems.High degree of spatial organization and appropriate energy gradient appear conducive to the efficient transfer.[7]Specific cases of energy transfer between two fluorescent molecules[8,9]or between fluorescent donor and phosphorescent acceptor have been frequently explored.[10-12]Nevertheless, some fundamental questions regarding the energy transfer between two phosphorescent molecules remain elusive.[13]

We focus in this work on the reported energy transfer between two transition-metal complexes in nanocrystals.(i)Light-harvesting nanorods are from coassembly of iridium(Ir) and ruthenium (Ru) metallophosphors.[7]The crystals of[Ir(ppy)2(pzpy)](PF6)(Ir donor)have smooth surface and uniform size.The emission spectrum is predominated by doped[Ru(bpy)3](PF6)2(Ru acceptor)when the doping ratio reaches 1.5%, resulting in 90% quenching of the Ir-donor’s emission.(ii) Thermal-responsive phosphorescent nanotubes are from coassembly of two Ir complexes[Ir(dmppy)3](Ir-D)and[Ir(piq)3](Ir-A).[14]Again,when the concentration of Ir-A saturates at 0.5%,85%of the energy is transferred.

Neither Dexter electron exchange nor F¨orster resonance energy transfer(FRET)mechanism is able to fully account for the observed transfer rates in the experiments.A spherical model that mimics the crystal packing of the Ir donor complexes around the Ru center was used in Ref.[7].Their analysis concluded that the short-range of the T1-T1Dexter mechanism alone could not account for the high efficiency when theA/Dratio＞1/100.Furthermore,the T1-S1FRET rate from donor to acceptor, which is characterized by the Coulombic dipole-dipole interaction and can be related to the radiative decay rate of the molecules, would only drop by roughly a factor of 2,while the rate constant obtained from the lifetime decay measurement dropped more than ten folds in the second experiment from 298 K to 118 K.[14]

We propose an alternative energy transfer pathway that is mediated by resonance-confined polaritons in this work.Note that the natural boundaries of a nanocrystal provide a microcavity of resonance for the lights emitted from the donors.They can be confined within the cavity as excitonpolaritons (EPs) to facilitate the energy transfer.A previous study showed that delocalized EPs could lead to ultra-efficient(＞90%) S1-S1energy transfer at less than 1/1000 of doping ratio.[15,16]In the current context, the presence of transition metal elements such as Ir or Ru makes the T1state of the organic complexes the lowest visible excitation via intersystem crossing.There arises complex inter-plays and population dynamics among the S1,T1and the EP states.The energy transfer can then occur via the route(donor)T1→S1→EP states→(acceptor)S1→T1→emission.Our analysis is able to account quantitatively, with no fitting parameters, for the observed dependence of the donor’s phosphorescent lifetime with temperature and doping ratio.

This work is organized as follows.In Section 2, we describe the core process of the proposed mechanism, starting with the resonance inside the micron-scale crystal and quantization of the electromagnetic(EM)fields.We show explicitly how EPs as composite entities mediate the energy transfer between the S1states of donors and acceptors.Then in Section 3,a population dynamics engaging the T1-excitons is formulated for a quantitative study of the donor’s decay process.In Section 4 we turn to the primary subject of this study,applying the theory to both the Ir-Ru metallophosphor binary and the Ir-Ir donor-acceptor nanocrystals.The final section (Section 5)summarises the main findings with a perspective on further exploration of EPs in the context of resonance energy transfer(RET).

2.Polariton-mediated RET process

2.1.Resonance inside the micron-scale crystal

We first present a brief review of a previous study[16]in which the standing EM waves are sustained by the natural boundaries of a nanocrystal.The micron-scale crystal dimensions are crucial for the confinement of EPs formed as a result of strong coupling between the EM field and localized S1exictons of the molecules.The resonant modes become captive under total internal reflections.To exemplify the concept,we take for simplicity and with no loss of generality that the nanocrystal is a rectangular isotropic dielectric medium of dimensionsL1×L2×L3.Let

where the electric field inside the cavity under background relative permittivityεbthat excludes specifically the contribution from the excitons.Following Ref.[16],Ukα(r)is a superposition of plane waves taking the form of[17-19]

whereukαare the two polarization vectors and the wave vectork=(kx,ky,kz)is limited to the first octant

A wave is trapped inside when the tangent component of the wave vector

holds on all the surfaces,more explicitlyk2‖=(k2x+k2y),(k2y+k2z),(k2z+k2x).In the above,εsis the dielectric constant of the surrounding medium andEkis the lower-branch energy of EP to be shown below.

2.2.Quantization of the EM wave and S1-exciton

For quantum mechanical modeling of EM interaction with the S1excitons, one can quantize the macroscopic Maxwell equations using above eigenmodes.On a givenk,the photon Hamiltonian and the electric field operator inside the cavity are given by(see Ref.[20])

As for the exciton,we can approximate it by a harmonic oscillator that is restricted to its zeroth and first levels,namely the one excitation manifold.The position of thej-th molecule isRj, the creation and annihilation operators of the oscillator are ?b?jand ?bj, respectively, and the dipole matrix element between the ground and the excited states isd=〈g|?r|e〉, we have the exciton-photon interaction Hamiltonian

Note that in a self-assembled narocrystal,Rjform a crystalline lattice ofNmolecules.We can further take all molecules with the identical matrix element, i.e., ignoring the orientation difference between the different groups of molecules in the crystal.

Similar to Ref.[21], we can construct a standing-wave(SW) superposition of the identical S1states from all the molecules

Two more SW states|φxk〉 and|φyk〉 can be likewise defined respectively for thexandycomponents.The three SW states are mutually orthogonal for a givenk(for largeN).More details can be found in part A of the supplementary information(SI)[19]of this work.

2.3.Polariton as a S1-exciton-photon composite

The total Hamiltonian under Eq.(8), which includes the photon Hamiltonian,the exciton Hamiltonian and the interactive Hamiltonian between them,reads

where the photon-exciton frequency detuning isˉhΩk=ˉhωk-ˉhωe.

Knowledge ofEkcan be used in Eq.(4)to determine the modes that are confined by the natural boundaries of the crystal.Details relevant to the two experiments considered in this work are presented in part C of SI.[19]The results show that the majority of EPs emitted from the donor’s S1state are indeed trapped in the cavity.Such a characteristic is assumed throughout the rest of this work.

2.4.Transition rate to acceptor S1 state

The transition from a trapped EP state to the doped acceptors is enabled by a small photonic componentAph≤20%in the EP entity.We can calculate the transition rate via the Fermi golden based on the component.For further details,cf.part C of SI.[19]For an EP inkαmode,the rate reads

whereNa/Nis the doping ratio of acceptors evenly distributed in the crystal with a spectral bandwidth Δωroughlyωe/Δω ≈10.It turns outΓ0≈ωe.

3.Population of T1,S1 and EP states

We next study the interplay between the T1, S1and the EP states that are engaged in the energy transfer process.Figure 1(a)presents a Jablonski diagram of these states,also depicts the virtual levels of Stokes shift (ˉhωs) associated with radiation.They appear in the vertical transition picture for absorption and emission.The first absorption peak corresponds to the donor’s energy of S1+ˉhωs.The phosphorescent emission occurs at that of T1-ˉhωs.We can infer the intrinsic energy difference between T1and S1states ΔETSfrom the measured difference between the two peaks, with the subtraction of 2ˉhωs.In addition, the fluorescent emission of S1exciton into an EP state is accompanied by an energy drop ofˉhωs.We will later identify the energy difference between T1and EP as the activation energyEathat dictates the temperature dependence of the energy transfer from donor to acceptor.We obtain ΔETS= ˉhωs+Eaas shown in the figure.Finally,ΔEkin Eq.(11)is actually≈2ˉhωs,as ˉhωeshould be raised by ˉhωsitself.[16]

Fig.1.(a)Jablonski diagram for the states of S1,T1 and EPs along with Stokes shift of±ˉhωs for absorption/emission process.ΔETS ≈ˉhωs+Ea with Ea ≈0 for the Ir-Ru and Ea ≈0.04 eV for the thermal-responsive phosphorescent Ir-Ir system.(b) Transitions between different energy levels and typical values for the rates.Their population dynamics sets the transfer rate from T1 to T1 of donor to acceptor.

To proceed further, let us denote the time-dependent ensemble population of T1excitons asA(t),that of EPs asB(t),and S1asC(t).The possible transition paths between them are shown in Fig.1(b).Their time evolution can be expressed as

In the above, the parameters are associated with the following physical processes.kST≈1011-1012s-1stands for the fast inter-system crossing from S1to T1.WhilekTS≈109s-1is for the reverse, which is temperature dependent and suppressed by the ΔETSenergy barrier.kP≈106s-1is the total decay rate of T1that includes phosphorescence.kF≈109s-1is the fluorescence emission rate from S1to EP.The later can either return to S1or go down to an acceptor(completing energy transfer).Γre≈1011-1012s-1indicates the rate for the reabsorption process.It can be higher thankPdue to the vast number of donor molecules that compete against the relative minor energy drop.Finally,Γacis given by Eq.(12)scales with the doping ratio.

One can deduct an overall process of exciton migration based on the rate magnitudes as shown in Fig.1(b).An exciton created by UV radiation would quickly reach the T1state of a donor from S1via inter-system crossing,assisted by spinorbit coupling in the metallophosphor.It would do roughlykTS/kP～103round-trips to the S1during its life time.But there is a small probabilitykF/kST～10-2that would detour to an EP state, about～10 times, setting a saturation limit of～90%efficiency.While at the later, it would have either returned to S1or be captured by an acceptor, depending on the relative ratio ofΓrevs.Γac.Equation (12) forΓacwould then give rise to significant portion of transfer at relative low doping ratio of～10-4.In the absence of doping, there would sustain a quasi steady distribution with a Boltzmann-like energy factor relatingAandB,as bothA →CandB →Cwould rely on thermo-activation over energy deficit.The Boltzmann factor would correspond to the observed temperature dependence observed on the Ir-Ir system.More detailed analyses below confirm this gross picture.

4.Experiments and analysis

4.1.Nanorods of Ir-Ru metallophosphors

The first system comprises a donor-acceptor arrangement achieved via the coassembly of two polypyridyl metallophosphors,namely Ir and Ru,which function as the antenna chromophore and the energy acceptor respectively.The nanorods exhibit multicolor phosphorescence due to efficient S1to T1inter-system crossing in the presence of Ir element(transition metal).These rods have a uniform size and smooth surface.The Ir donor possesses a T1energy of 2.61 eV(by emission),which is lower than the S1energy of 2.82 eV(by absorption).Notably,the brighter emission at the two ends of nanorods displays color-tunable optical waveguide characteristics, which supports the core point of EP confinement used in this work.

The exciton-photon interaction depends on the crystallographic and spectroscopic characteristics of the S1state.The dipole moment of the Ir(ppy)2(pzpy) donor molecule can be estimated to be approximately 6.4 D (Debyes), which corresponds to|d|≈1.3 °A based on the integral of extinction coefficient over the absorption frequencies (see Refs.[22,23]and part B of SI[19]).A unit cell of the crystal contains 8 donor molecules with a volume of 5636 °A3(Table S1 in Ref.[7]).Part C of SI[19]has the details regarding the formation of EP states facilitated by the resonance-confined EM waves.At the resonance level ˉhωe= ˉhωk ≈2.82 eV, the coupling constantgk ≈0.56 eV.

Fig.2.The dependence of the lifetime of the Ir donor on Na/N of the doped [Ru(bpy)3](PF6)2 in [Ir(ppy)2(pzpy)](PF6) crystal.Open cycles(blue) are the theoretical curve with no fitting parameter, cf.part C of SI.[19] Solid triangles (green) are the experimental data taken from Table S4 of SI in Ref.[7].

The lifetime of the Ir donors descends from 800 ns to 108 ns as the doping of the Ru acceptors rises from 0 to 1.5%.Numerical results of Eq.(13), using parameters derived from the experimental data(cf.Table S4 of Ref.[7])as summarized in Table S2 of SI,[19]are shown in Fig.2 alongside the experimental ones.The overall match is excellent.At zero doping,the lifetime of T1exciton is 798 ns vs.800 ns.The agreement goes all the way to the saturated doping of 1.5% where the theoretical value is 110 ns vs.the observed lifetime of 108 ns.

4.2.Thermal-responsive Ir-Ir tubes

The second set of experiment is on thermal phosphorescent [Ir(dmppy)3] nanotubes which were fabricated with the ability to display color-tunable phosphorescence.The doping of [Ir(piq)3] is adjusted from 0 to 0.5% and the temperature is within the range 298 K to 77 K.Upon exposure to UV radiation,the tips of these nanotubes exhibit brighter emissions compared to the body section, indicating again the presence of an active optical waveguide.Although the tubes have a smooth hexagonal outer shape in general, we use a similar size of the rectangular waveguide and assume the polarization of the molecules aligned to a fixed orientation (see part C of SI[19])to simplify the modeling work.

The high extinction coefficient on the tubes yields a dipole moment of 8 D(see part B of SI[19]).The crystal contains 6 molecules in a unit cell with a size of 4491 °A3,and S1state of the metallophosphor has ˉhωe=2.72 eV (by absorption).For completeness, the T1state emits at 2.43 eV.The confined EP modes are studied in part C in Ref.[19].We obtain the coupling constantgk ≈0.61 eV(at ˉhωk=ˉhωe).

Figure 3(a)shows the dependence of the phosphorescent lifetime of[Ir(dmppy)3]on the doping ratioNa/Nof[Ir(piq)3]spanned from 0 to 0.15%.The comparison substantiates a high degree of agreement between the theoretical results and the experimental data obtained from Table S3 of the Supporting Information in Ref.[14].The required parameters for the calculation are summarized in Table S2 of SI[19]and Refs.[24-29].The slight discrepancy at the larger doping end might be attributed to the inadequate etching by poor solvents that caused non-penetrated morphology of hollow structures.This could add extra decays not included in our theoretical model.Furthermore,the resonant confinement of EPs under a closer shape, such as cylindrical symmetry should be further explored.

Fig.3.(a)Room-temperature lifetime of[Ir(dmppy)3]metallophosphor as a function of doped ratio Na/N of[Ir(piq)3].Theoretical results with no adjustable parameters are in open circles(red).They are plotted against the experimental ones in the solid triangle (green), taken from Table S3 of support information in Ref.[14].(b) The temperature dependence of lifetime,with an activation energy of 40 meV as a result of the difference between the T1 and the EP levels.Theoretical predictions in open circles(grey)are drawn alongside the experimental data at Na/N=0.06%.

Unlike the first experiment, the lifetime has a significant dependence on temperature.Arrhenius fitting of experimental data reveals an activation energy ofEa=40 meV.In the population dynamics outlined above, such barrier can be naturally attributed to the difference between the T1and the EP states.Namely, it is the balancing result between the reverse inter-system crossing(cf.kTS)and the reabsorption(Γre)to the donor’s S1state at the top.These parameters are given in Table S1 of SI[19]and Refs.[27,30].Figure 3(b)shows that the theoretical calculations are in excellent agreement with the experimental results.Explicitly,it predicts a lifetime of roughly 550 ns even at 118 K(with 0.06%doping),which matches the experimental observation of 570 ns.

5.Discussion

We have assumed that there are adequate number of EP modes near the S1emission spectrum.But in reality, the assumption can become an issue as it would depend on the geometry and dimensions of the nanocrystal in question.It would be interesting for such dependence to be taken into account in both theoretical and experimental works in the future.There are also minor mismatches between the theory and experiment in Fig.3.While the discrepancy at the higher doping levels might be caused by material quality control with experiment, the deviation at the low-temperature end is likely due to some overestimate of the activation energyEafrom the Arrhenius equation, cf.Fig.1.In the latter case, the activation rate(from T1to S1)is more sensitive to the energy difference as the temperature decreases.

To summarize, we present an alternative explanation for the efficient transfer of metallophosphor (T1) excitons in organic nanocrystals observed in Refs.[7,14].The proposed mechanism focuses on the intermediate states of excitonpolaritons.We first show that these EPs can be trapped by the natural boundaries of the crystal.[16]Then a population dynamics involving T1, S1and the confined EPs is formulated with various transition parameters taken or deducted entirely from known experimental data.The results on the phosphorescent lifetime of the metallophosphors are in excellent agreement with that from experiments.Theoretical predictions match experimental curves over a wide range of doping concentrations and temperatures.In particular, the temperature dependence is shown to arise from the energy difference between the T1and the EP states,which is much smaller than that between the T1and S1states.The success of the theory may be of significance to the applications of these materials in miniaturized photonic devices and solar energy harvesting.[31,32]Finally, we hope more experiments with regard to the mechanism can be conducted in future studies.

Acknowledgments

Project supported by the National Natural Science Foundation of China (Grant No.16Z103060007) (PA).One of us(YCC) thanks Prof.Y W Zhong for visiting Shanghai University and for an insightful discussion on the experimental works.Thanks Shanghai Nanobubble Technology Co., Ltd.for supporting this work.