Signatures of Strong Vibronic Coupling Mediating Coherent Charge Transfer in Two-Dimensional Electronic Spectroscopy

2.1 Model Dimer of a Donor–Acceptor Interface

We consider an electronic dimer system as the conceptually simplest conceivable model for a donor–acceptor interface. We define |0j⟩ and |1j⟩, with j = D, A, as the ground and excited electronic states of donor (j = D) and acceptor (j = A), respectively. For simplicity, we take the ground state energy ℏω0j=0 eV for both donor and acceptor. The model Hamiltonian of such a dimer is essentially that of a two-particle system and reads

where ℏωD and ℏωA are the electronic excitation energy of the donor and acceptor, and J is the electronic coupling matrix element between their excited states. Interaction with a series of resonant optical pulses induces optical transitions between |0j⟩ and |1j⟩ and is described in dipole approximation [63] as H^I(t)=μ^∑kEk(t), with μ^ denoting the dipole operator. Each optical field Ek(t), k = 1, 2, 3, interacting with the system is modelled as a Gaussian pulse of the form Aexp[−2ln2(t−t0ktF)2]cos(ω0(t−t0k)+φk). Here, A is the field amplitude, and t0k is the time delay of the k − th pulse with respect to the third pulse, centred at t₃ = 0 fs (see next sections and Fig. 1). The duration of each pulse, t_F, is defined as the full-width-at-half-maximum of the intensity profile; ω₀, the carrier frequency; and φ_k, a possible phase shift. For the simulations presented in this article, we take ω0=2 eV/ℏ and tF=5 fs. For each moiety, j = D, A, the light–matter interaction of the system is defined as

Figure 1:

Scheme of the pulse sequence used in a 2DES experiment. Three ultrashort pulses (red) interact with the sample inducing a coherent nonlinear polarisation P⁽³⁾. The corresponding field E_s reemitted by the sample (blue) after interaction with the third pulse is measured as a function of the coherence and waiting times τ and T.

where the transition dipole moment matrix element μj=⟨0j|μ^|1j⟩ is a constant, and |1j⟩⟨0j| and |0j⟩⟨1j| describe absorption and stimulated emission (SE) processes, respectively. This model (1–2) describes a purely electronic dimer. Unless otherwise specified in the text, we assume that the donor is optically bright, whereas the acceptor is initially dark, i.e. we take μD≠0 and μA=0, respectively.

To model vibronic coupling, we assume that the electronic system is coupled to a single vibrational mode with vibrational energy E_vib along the nuclear coordinate Q for all electronic states. The equilibrium geometry of the ground state is chosen as the origin of the coordinate system Q = 0, and each potential energy surface is described by the same normal mode coordinate. The energy of the ground state then becomes that of a one-dimensional harmonic oscillator, i.e. h0=Evib2(∂2∂Q2+Q2), with the minimum energy of the electronic ground state potential equal to zero as in (1). We assume that the excited state potentials of donor and acceptor are coupled to the vibrational mode with a coupling strength κ_j, resulting in excited state energies of hj=h0+ℏωj+κjQ, with j = D, A for donor and acceptor states, respectively. These vibronic couplings result in finite displacements of the excited state harmonic oscillator potentials along Q with respect to that of the ground state. Specifically, κ_j induces a (dimensionless) displacement Δj=−κjEvib, j = D, A, of the equilibrium position of the excited potential energy surface with respect to the one of ground state (which is the reference, Q = 0) along the vibrational coordinate. Depending on the specific coupling parameters, κ_j may modify the energetic detuning between donor and acceptor states. Moreover, κ_j define the position of the Franck-Condon region on the excited state potentials along Q. Thus, upon optical excitation, they provide the initial direction of propagation for the excited state wave packet motion. As we assume that the acceptor is initially dark, the optical excitation launches wave packet motion exclusively on the donor side of the excited state potential in a region defined by κ_D (Franck-Condon region of the donor state).

For the numerical calculations, we choose to work in the basis of displaced harmonic oscillator states [1]. This basis transformation is obtained introducing the unitary displacement operator for the vibrational mode, D(Δj)=eΔj(a†−a), where a†,a are the creation and annihilation operators, and Δ_j denotes the dimensionless displacement along Q defined above [1], [63]. In the displaced basis, the wave functions associated with the excited states in our model become displaced harmonic oscillator wave functions. This means that they are centred around different equilibrium positions [37], defined by the vibronic couplings via the dimensionless displacements Δ_j.

The optical excitation of the vibronic system is described by a similar light–matter interaction Hamiltonian as introduced above (2). Here we make use of the Condon approximation [1], in which the expectation value of the dipole operator between ground state with vibrational level n and the excited state with vibrational level m_j, j = D, A, is given by μe,jnm=⟨0j,n|μ^|1j,mj⟩. It can be separated into an electronic dipole moment contribution μe,j=⟨0j|μ^|1j⟩, which is independent of the vibrational coordinate, and a Franck–Condon overlap integral matrix element μFC,jnm=⟨n|mj⟩, such that μe,jnm=μe,jμFC,jnm. Taking the harmonic oscillator wave functions Ψn0(Q) for the vibrational level n in the (nondisplaced) ground state and the displaced harmonic wave functions Ψm1j(Q) for the m level of either the donor or acceptor excited state, the Franck–Condon overlap integral is given by μFC,jnm=⟨Ψn0(Q)|Ψm1j(Q)⟩. In the displaced basis, the Franck–Condon overlap integral matrix is equivalent to the expectation value of the displacement operator. Hence, the operator describing, e.g. absorption transitions between the ground and excited donor state in the displaced basis becomes D(Δj)|1j⟩⟨0j|D†(0)=D(Δj)|1j⟩⟨0j|, where we have used the fact that for the ground state Δ = 0, and hence the expectation value of the displacement operator is the identity matrix.

2.2 Master Equation and Numerical Calculation of the 2DES Signals

For sufficiently weak optical pulses, 2DES is a third-order nonlinear optical spectroscopy technique. The measured signal in 2DES is proportional to the third-order nonlinear polarisation, ES∝iP(3)(τ,T,t), induced in the system by a sequence of three time-delayed optical pulses interacting with the sample (Fig. 1) and reemitted in a specific phase-matched direction [23]. The delay between the first and second pulses is called the coherence time τ. The delay between the second and third pulses is termed waiting time T, and t is the detection time (here the simulation time). In our simulations, we thus start by calculating the nonlinear polarisation in the time domain as P=N⟨μ^⟩=NTr(μ^ρ), with N the particle density and ρ denoting the density matrix of the system. Thus, the expectation value of the dipole operator ⟨μ^⟩ contains all relevant information about the optical response of the system upon interaction with the pulse sequence [38]. For each pair of time delays τ, T, we calculate the time evolution of the density matrix along the simulation time t, by numerically solving the master equation in the Lindblad form [64]

The total Hamiltonian H=H0+HI describes the free evolution of the system, H₀, and its interaction with the optical pulses, H_I. Dissipation processes, such as dephasing and relaxation, are accounted for phenomenologically within the Lindblad formalism [64]. Specifically, they are described by superoperators ℒn(ρ) of the form ℒn(ρ)=γn(VnρVn†−12{Vn†Vn,ρ}), with γ_n denoting the damping rate, and Vn† and V_n being the creation and annihilation operators for the n-th damping process. We note that, with the exception of the system Hamiltonian H₀, all other operators in the master equation (3) depend on the time delays (τ, T, t).

For our model system, we define creation and annihilation operators describing electronic dephasing as Vdeph†=Vdeph=|1j⟩⟨1j|, whereas the ones for population relaxation to the ground state are taken as Vrel†=|1j⟩⟨0j| and Vrel=|0j⟩⟨1j|, respectively. Vibrational relaxation is also accounted for by a Lindblad term with the creation and annihilation operators coinciding with the raising and lowering (ladder) operators a†,a of the harmonic oscillator [63]. For the simulations presented in this article, the electronic dephasing time is taken as T₂ = 30 fs, and the population relaxation time to the ground state is set to T₁ = 5 ps, consistent with values estimated from our previous experiments on conjugated polymers [35], [37]. The vibrational relaxation time is set to T1,vib=500 fs, in agreement with experimental values for C=C stretching modes in conjugated polymers observed by us and other groups [37], [65].

The master equation (3) is solved numerically by a nonperturbative approach, which yields the total polarisation, including all allowed phase-matched signals as well as all linear and third-order nonlinear contributions arising from the interaction with the pulse sequence [66]. In the nonperturbative scheme, the optical pulses are included in the light–matter interaction part of the Hamiltonian, and no assumptions are done on the timings between the three pulses [66]. This has the advantage that the calculated signals include directly the effect of pulse overlaps, which may result, e.g. in perturbed free induction decays or other coherent interferences during pulse overlap [67]. The nonlinear signal in a specific phase-matching condition is subsequently isolated from the total polarisation. Experiments probing absorptive signals are usually most interesting for 2DES of molecular systems. Therefore, in our simulations, we will calculate absorptive pathways by applying a phase-cycling algorithm. To this aim, at each time delay τ, T, we calculate the total polarisation for four different phases φ=nπ/2, with n = 1 − 4, between the phase-locked excitation pulse pair (E₁ and E₂ in Fig. 1) and the third pulse (E₃ in Fig. 1) and average them. Subsequently, we subtract the linear contributions. This phase-cycling scheme yields the directional dependence of the nonlinear signal as in a partially collinear experimental configuration, and it is analogous to the one previously reported for the calculation of pump-probe signals by other authors [68]. Similar phase-cycling procedures for the computation of 2DES signals with arbitrary phase-matching conditions have been reported in the literature by different groups [66], [69], [70]. To obtain the 2DES signals, we take the real part of the Fourier transform along τ and t for each waiting time T. This results in absorptive 2DES energy–energy maps for excitation E_X and detection E_D energy, respectively (see Section 3 for more details on 2DES).

2.3 Wave Packet Dynamics

To discuss the effect of possible ground state contributions to spectra, we calculate wave packet dynamics in the ground and excited donor state of such a dimer model coupled to a high-frequency harmonic oscillator mode for different time profiles of the excitation pulses. Such wave packet dynamics are calculated by projecting the density matrix ρ(t) on the diabatic harmonic potential energy surface of the excited and ground states according to [1]:

where the wave functions are that defined above, and ρkl0,1j(t) is the part of the density matrix that describes the ground (0) or excited (1_j, j = D, A) states, respectively.

3.1 Electronic Coherence and Charge Transfer

To discuss the signatures of strong couplings and coherent charge transfer in 2DES, we first consider the case of electronic coherence without contribution of vibrations. For this we use a dimer model mimicking a donor–acceptor interface and assume that the effect of vibrations merely results in decoherence of the optically driven coherent polarisations, accounted for by a finite decoherence time of the optical excitations. This is equivalent to the case of weak coupling to all molecular vibrations in the system, which is, however, quite unlikely in functional organic materials [50], but helps us to introduce the typical spectral signatures of coherent coupling in 2DES.

For simplicity, we consider only the electronic ground and first (singly) excited state of each moiety and neglect the effect of possible higher-lying electronic states. Specifically, we do not include the possible excited state absorption (ESA) signals that may arise from transitions to these states. This simplification is motivated by the fact that for P3HT ESA of singlet excitons is strongly red-shifted with respect to the excitonic resonances centred around 600 nm [35], [80], [81], [82]. The ESA peaks around 1200 nm, and therefore, such transitions are not excited even when using extremely short optical pulses in the 2DES experiments. Quite generally, such ESA transitions will lead to negative peaks in absorptive 2DES spectra. These negative peaks may easily mask the 2DES signatures of the transitions between the ground and singly excited electronic excited state, which can lead to both ground state bleaching (GSB) or SE. In contrast to ESA, both GSB and SE enhance the transmission through the sample and thus result in positive peaks in absorptive 2DES maps. It will be important for the later discussion that, in OPV materials, negative peaks in 2DES associated with ESA may also arise from other types of quasiparticle excitation, e.g. from polaronic excitations, charge transfer excitons, or also from triplet excitons [81]. Signatures of polaronic excitations in experimental 2DES studies of P3HT thin films and of its blend with fullerene acceptors have indeed been observed in recent experiments [37], [39]. In the case of P3HT, these ESA features reflect photoinduced absorption of delocalised polaron excitons in the polymer aggregate and overlap with the positive SE cross-peaks expected from singlet excitons [37]. This complicates the interpretation of the spectra. In our dimer model, we neglect for simplicity higher excited states that could give rise to ESA transitions, even though they may be relevant for the interpretation of certain experiments.

In this electronic dimer model, we describe the interaction between donor and acceptor by an electronic coupling strength J connecting their excited states. This electronic coupling controls the transfer of energy from one moiety to the other. When this coupling exceeds the damping, i.e. for J>ℏ/T2, the electronic coupling results in strong mixing of donor and acceptor states and hence in the formation of new, hybrid states with energies ℏω1,2=ℏωD+ℏωA2±124J2+(ℏωD−ℏωA)2. Here, ℏωD and ℏωA denote the electronic excitation energies of the uncoupled donor and acceptor, respectively [83]. In this strong coupling regime, the hybrid eigenstates are split apart of a quantity ΔJ=±124J2+(ℏωD−ℏωA)2 with respect to the mean energy of the two uncoupled ones. This splitting exceeds the detuning between donor and acceptor excited states ΔEDA=ℏωD−ℏωA.

In a typical configuration in OPV materials, the donor is the main absorber with a broad absorption covering much of the visible spectral range [43]. The acceptor, on the other hand, is only weakly absorbing and essentially transparent in this range. A specific example for such donor–acceptor systems is a conjugated polymer mixed with a high concentration of fullerene derivatives as acceptors [43]. In the following, we therefore assume μA≈0. Hence, the optical pumping initially mainly excites the donor moiety. For J>ℏ/T2, the excitation results in coherent population oscillations between donor and acceptor with a period 2πℏ/4J2+(ℏωD−ℏωA)2. If the uncoupled states are in resonance (ΔEDA=0) and decoherence is sufficiently weak, nearly all the excited state population is coherently transferred from the donor to the acceptor and back. However, disorder inherent in organic materials results in a certain amount of energetic detuning ΔEDA at the donor–acceptor interface. This reduces the amplitude and increases the frequency of the coherent oscillations. Additionally, dissipation processes damp the oscillations on a timescale defined by T₂.

To illustrate the effect of electronic coupling on 2DES spectra, we assume a resonant case with ℏωD=ℏωA=2 eV, an electronic coupling strength J = 50 meV, and a dephasing time of T₂ = 30 fs. The level scheme showing the excited states before (dashed) and after (solid) the coupling is depicted in Figure 3a, where we have neglected the ground state for simplicity.

Figure 3:

Electronic dimer in the strong coupling limit. (a) Scheme depicting the excited states of the electronically coupled dimer in the absence (dashed) and presence (solid) of electronic coupling. We have assumed that the (uncoupled) donor and acceptor states are initially resonant and the coupling strength J = 50 meV exceeds the damping ℏγ=ℏ/T2=22 meV. (b–c, e–f, h–i) Absorptive 2DES maps of the electronic dimer at selected waiting times show two diagonal peaks at excitation and detection energies corresponding to the coupled (eigen)states and their cross-peaks suggesting coherent coupling. Strong amplitude oscillations are seen during the dephasing time T₂ = 30 fs. (d, g) Waiting time dynamics of selected (d) diagonal and (g) cross-peaks show oscillations with a period πℏ/J defined by the electronic coupling strength, which rapidly decay with a time constant according to the electronic dephasing time T₂.

Upon excitation at T = 0 fs, the corresponding absorptive 2DES map shows diagonal peaks, at excitation and detection energies corresponding to the hybrid eigenstates, and their cross-peaks suggesting their interaction (Fig. 3b). The peaks are arranged in a regular square pattern with the splittings defined by the coupling strength and thus reflecting the structure of the excited states (Fig. 3a). For the resonant case in Figure 3, the splitting is 2J = 100 meV. The waiting time dynamics (Fig. 3c–i) show amplitude oscillations of the peaks with a period defined by the energetic splitting and decaying with a time constant determined by T₂ (Fig. 3d, g). These dynamics probe the coherent transfer of population between the coupled states of an electronic donor–acceptor dimer. The oscillatory peaks are signature of electronic coherence during the transfer. The appearance of these oscillations implies that electronic charge is transferred from the donor to acceptor, i.e. over a distance of approximately 0.5–1.0 nm, on a timescale of πℏ/2J. For typical coupling strengths of J ≈ 100 meV [57], [58], this implies an average speed of ballistic electron wave packets of ∼0.1 nm/fs or 10⁵ m/s. Due to the short characteristic electronic dephasing times of few tens of fs, we expect this initial ballistic transport in purely electronically coupled donor–acceptor systems to occur over distances of at most a few nanometres, too short to be of significant functional relevance for device applications.

For coupling strengths smaller than the damping, J<ℏ/T2, donor and acceptor are in the weak coupling regime, and the electronic properties of the interacting system differ only slightly from those of the uncoupled states. In this case, dissipation dominates, and the no coherent population oscillations between the states can occur. Population transfer from the donor to the acceptor slows down, and it approaches a classical incoherent relaxation, consistent with predictions by Marcus’ theory of electron transfer [3]. Upon excitation, the 2DES map will therefore mainly show a single diagonal peak around the donor and essentially no diagonal peak around the acceptor (μA≈0). Cross-peaks would be so extremely weak that they can be hardly recognised even in the simulations. The waiting time dynamics would essentially follow that of incoherent population decay from the donor to the acceptor.

3.2 Vibronic Coupling and Charge Transfer

To discuss the effect of vibrations on the 2DES spectra and dynamics, we include coupling of the electronic states to a high-frequency, underdamped vibrational mode in the electronic dimer model. In OPV materials, C=C stretching and ring breathing vibrations in the range of 1400–1500 cm⁻¹ (∼170–180 meV) are usually strongly coupled to electronic excitations with large Huang–Rhys factors of about unity [45], [84] and result in pronounced vibrational progression in the linear optical spectra. In nonlinear experiments of donor conjugated polymers, long-lived vibrational oscillations up to about 1 ps, arising from intramolecular C=C stretching, have been observed [35], [37], [65], [85]. This suggests long vibrational relaxation times of the order of 500–600 fs. For fullerene acceptors widely used in OPV, the dominant breathing mode is found in the same frequency range (1470 cm⁻¹) [84]. Therefore, in our model, we assume that donor and acceptor strongly couple to the same high-frequency vibrational mode with vibrational energy Evib=180 meV and vibrational relaxation time of T1,vib=500 fs, in agreement with experimental results. We further assume that all other molecular vibrations are weakly coupled such that their effect can be accounted for by the dissipation terms in the model (Section 2.1). For the other parameters, we set the electronic coupling J = 50 meV and electronic dephasing time T₂ = 30 fs as in the electronic dimer discussed in Section 3.1 and additionally introduce an energetic detuning of ΔEDA=−200 meV between donor and acceptor electronic states. This detuning accounts for the energy difference between the excited states in, e.g. polymer-fullerene heterojunctions and also for an average disorder, typical in organic materials [58].

In light of experimental observations [35], [36], [37], these assumptions seem a reasonable approximation to explain the signatures of vibronic coupling in 2DES in a dimer model mimicking a prototypical donor–acceptor system with parameters relevant for OPV materials. As for the case of the electronic dimer (Section 3.1), we neglect the effect of possible higher excited states that could result in ESA transitions. In the next two sections, we discuss the signatures of vibronic coupling in this dimer model in 2DES for the limiting cases of weak and strong vibronic coupling.

3.2.1 Weak Vibronic Coupling Between Donor and Acceptor

We first consider the case that vibronic coupling does not influence the charge transfer dynamics between an optically bright donor and an optically dark acceptor. For that, we assume an electronic coupling J = 50 meV and the same vibronic coupling strength κD=κA=180 meV for both donor and acceptor, which are detuned by ΔEDA=−200 meV. This results in the same displacement ΔD=ΔA=1 of the excited state potentials of donor and acceptor with respect to the ground state [47]. In the absence of a relative displacement between donor and acceptor, the vibrational quantum number in the excited states remains unaffected upon charge transfer. Thus, the charge transfer dynamics and their effects on the 2DES spectra are quite similar to the case of purely electronic coupling.

As the vibrational mode frequency exceeds by far thermal fluctuations at room temperature, the population is initially in the vibronic ground state |0j,n=0⟩. Optical excitation launches large-amplitude oscillatory vibrational wave packet motion on the donor excited state potential with an oscillation period given by the vibrational mode frequency (Fig. 4). For weak excitation pulses, as those chosen in experiment, variation of the pulse duration t_F has two important consequences (Fig. 4d–f): first, the probability to excite the system in the excited state of the donor scales with the square of the pulse area A=1ℏ∫μ→D⋅E→(t)dt (E→(t): time profile of the electric pulse in the rotating frame) and thus with the square of the pulse duration; second, the maximum amplitude of the nuclear wave packet decreases with increasing t_F. For longer pulse durations, the nuclear wave packet on the excited state surface has time to evolve during the pulse, and this washes out the coherent wave packet motion and reduces its maximum amplitude. The amplitude of all wave packets shows an exponential decay reflecting the timescale for the charge transfer between donor and acceptor. Due to the finite detuning ΔEDA, the two states are now weakly coupled, even though the strength of the electronic coupling J = 50 meV exceeds ℏ/T₂.

Figure 4:

Wave packet dynamics in the (a–c) ground and (d–f) excited state potentials of the donor moiety along the nuclear coordinate Q for increasing values (left to right) of the excitation pulse duration of (a, d) 1 fs, (b, e) 5 fs, and (c, f) 10 fs. Note that, for the ground state wave packets (d–f), the amplitude square of the ground state vibrational wave function |Ψn=00|2 has been subtracted to better display the oscillatory components of the wave packet. The data are multiplied by a constant factor of (a) 50, (b) 5, and (c) 2.5 for the ground state wave packets and of (d) 100, (e) 10, and (f) 5 for the excited state ones, respectively. Vertical dashed lines indicate the equilibrium geometry of each potential energy surface along Q.

Impulsive stimulated Raman scattering [86], [87] also drives coherent vibrational wave packet motion in the ground state. Here, the shape and dynamics of the wave packet launched in the ground state depend critically on the duration of the excitation pulses.

This can be seen in Figure 4a–c, showing ground state vibrational wave packets for different values of the pulse duration t_F. To display the oscillatory part of the ground state wave packet more clearly, we have subtracted the amplitude square of the first vibrational wave function |Ψn=00|2 from the ground state wave packets in Figure 4a–c.

For the shortest pulses (t_F = 1 fs) in our model, the wave packet displays a characteristic oscillation with a period of ∼11.5 fs, half the period of vibrational mode (∼23 fs). For such short pulses, the excited state wave packet will not evolve substantially before being driven back to the ground state. In this case, two weak, counter-propagating wave packets, phase shifted by 180°, are launched in the ground state (Fig. 4a). For longer pulse durations (Fig. 4b, c), the excited state wave packet can evolve during the duration of the pulse. Hence, for the chosen displacement Δ_D, the ground state wave packet is preferentially launched near the outer turning point (Q > 0), and the wave packet motion becomes more similar, except for a 180° phase shift, to that in the excited state.

In the 2DES, such coherent vibrational wave packet motion in the excited donor state results in a series of diagonal and cross-peaks. These peaks reflect optical transitions between vibronic ground states |0D,n⟩ and excited states |1D,m⟩ of the donor and appear at energies ℏωD±(n−m)Evib along either the excitation or detection axis. The number of diagonal peaks is determined mainly by the displacement of the donor potential Δ_D and by the spectral overlap of the vibrational transitions with the pulse spectra. In Figure 5, for example, the choice of the vibronic coupling, determining the displacement, together with the finite pulse spectrum extending between ∼1.6 and ∼2.4 eV (c.f. Fig. 2b, e), results in mainly the first two vibrational quanta to be clearly visible in the 2DES map. Excitation of the vibrational wave packet results mostly in the appearance of cross-peaks for excitation and detection energies corresponding to the diagonal peaks at 2.0 and 2.18 eV (Fig. 5b). Cross-peaks at lower detection energy around ED=1.8 eV (Fig. 5b) arise from transitions between |0D,m+1⟩ and |1D,m⟩. We observe that the peaks are arranged in a regular pattern with the spacing determined by the vibrational mode energy of the donor E_vib. Very similar patterns have been experimentally observed upon excitation of vibrational wave packets in 2DES studies of P3HT thin films [65] and polymer-fullerene nanoparticles in solution [36].

Figure 5:

Vibronic dimer in the weak vibronic coupling limit. (a) Scheme of the harmonic oscillator potential energy surfaces of donor and acceptor. (b–c, e–f, h–i) Absorptive 2DES maps show diagonal and cross-peaks arranged in a checkerboard pattern suggesting excitation of a vibrational wave packet mainly in the donor moiety. Peaks for excitation and detection energies between 2.0 and 2.2 eV reflect bleaching transitions of the donor. Cross-peaks at detection energies E_D ∼ 1.8 eV arise from SE of the excited donor to the ground state. The peak splittings along detection and excitation energy axes are determined by the frequency of the coupled vibrational mode Evib=180 meV. (d, g) Waiting time dynamics of selected (d) diagonal and (g) cross-peaks show persistent beatings with a period corresponding to the vibrational mode 2πℏ/Evib.

The coherent motion of the wave packets in the donor potentials modulates the amplitude of the peaks along the waiting time (Fig. 5b–i), resulting in coherent vibrational beatings with a period of 2πℏEvib. While the contribution of the impulsively launched ground state wave packet to these peak oscillations remains time-invariant on the timescale given by the vibrational relaxation time, the excited state contribution decays much more quickly, with a rate given by the charge transfer rate. The analysis of the dynamics of the coherent peak oscillations thus provides direct insight into the charge transfer dynamics. In the present case of a weakly coupled system, a single-exponential relaxation dynamics is expected. Therefore, the 2DES patterns and their dynamics show no significant signatures of coherent dynamics between donor and acceptor, but they are dominated by vibrational wave packet motion in the donor moiety.

In this vibronic dimer model, population transfer between donor and acceptor is not affected by the vibronic coupling but it is determined only by the coupling strength between electronic states J and by the energetic detuning between them ΔEDA. As the chosen parameters result in donor and acceptor being in a weak coupling regime, the population transfer approaches an incoherent hopping from the donor to the acceptor. Hence, in this weak coupling case, our model yields essentially the same results as those obtained by Marcus theory. Also, the dynamics probe vibrational wave packet motion in the donor, but do not report signatures of coherent transfer to the acceptor.

This is partly due to the relative displacement between donor and acceptor ΔDA=ΔD−ΔA=0, which results in an overlap integral matrix that supports only donor–acceptor transitions that do not alter the vibrational quantum number. This implies that the transfer dynamics are determined only by the electronic coupling and the energetics of the donor–acceptor interface, as in the case of an electronic dimer. Vibronic coupling of the electronic states to the high-frequency mode may launch vibrational wave packet motion on the potentials of the single moieties (if both are bright), but these do not influence the population transfer dynamics between the electronic states.

With the exception of the SE peaks in Figure 5, the pattern generated by the excitation of a vibrational wave packet in the donor is very similar to the one resulting from strong coherent electronic transfer between donor and acceptor in Figure 3. Importantly, despite the fact that the physics of the underlying system is quite different, both models result in oscillatory dynamics. While in Figure 3 they reflect electronic coherence between donor and acceptor, the beatings in Figure 5 arise from vibrational coherence in only one of the moiety. It is now evident that the interpretation of persistent beatings of the 2DES peaks is not straightforward and may easily lead to misinterpretations of the underlying physics [6], [7]. Hence, the presence of cross-peaks and long-lived beatings in experimental 2DES data cannot be taken as an unambiguous signature of electronic coherence [21], [25], [61], [62]. Simulations of 2DES spectra and their dynamics based on model Hamiltonians, as presented in the present work, or more advanced ab initio methods are therefore indispensable for a substantiated analysis of experimental data.

3.2.2 Strong Vibronic Coupling Mediating Coherent Transfer

We now consider different strengths of the vibronic couplings, i.e. different displacements, for the excited donor and acceptor states in the dimer (Fig. 6a). When the vibronic couplings of donor and acceptor are significantly different, they may modulate the electronic coupling between the states and hence may take an active role in the charge transfer dynamics. In this case, both the 2DES spectral pattern and dynamics may differ significantly from those seen in the case of pure electronic or vibrational coherence. In the dimer model coupled to a high-frequency vibrational mode introduced in Section 3.2, the excited states are electronically coupled to each other and mutually strongly coupled to a single vibrational mode. For a sufficiently strong electronic coupling, this results in the formation of new eigenstates of the system that are mixed states between the electronic donor and acceptor states and the vibrational mode ladder. The coupled potential energy surfaces show an avoided crossing along the vibrational coordinate Q where the uncoupled potentials of each component would instead cross (Fig. 6a, dashed black). The strong mixing of electronic and vibrational states induces delocalisation of the wave function between donor and acceptor. Excitation by the pulse sequence in 2DES results in a coherent superposition of these mixed donor–acceptor vibronic states. The optically excited wave packet is initially launched on the donor side of the coupled potential energy surface. This wave packet is now no more a vibrational wave packet oscillating only within the donor moiety, as in the case discussed in Section 3.2.1, but it is a coherent superposition of vibronic wave functions delocalised across the donor and acceptor. Hence, a complex and completely new energy level structure forms in the excited state. In the 2DES pattern, this results in a splitting of the vibrational wave packet structure defined by the vibrational mode into a subpeak structure, with many peaks (Fig. 6a) corresponding to optical transitions in the ground and the complex mixed excited state.

Figure 6:

Strong vibronic coupling mediating coherent charge transfer dynamics between donor and acceptor. (a) Scheme showing the coupled donor–acceptor excited state potential energy surface. Strong mixing of electronic and vibrational degrees of freedom results in vibronic states (solid) delocalised across the donor and acceptor. The dashed lines show the position of vibrational states of donor and acceptor in the absence of coupling. (b–c, e–f, h–i) Absorptive 2DES maps show a clear substructure of the checkerboard pattern on top of the one observed for excitation of a vibrational wave packet in the donor (c.f. Fig. 5). Multiple peak splittings reflect the more complex structure of the excited state and probe the strongly coupled vibronic eigenstates. (d, g) The waiting time dynamics of selected (d) diagonal and (g) cross-peaks of this subpeak structure show clear and persistent beatings, suggesting multiple oscillation frequencies, which reflect the different splittings. Note that, although the value of electronic dephasing time and electronic coupling are the same as in Figure 3, the coherent vibronic oscillations are not damped within the electronic dephasing time T₂, but persist on a longer timescale set by the vibrational relaxation time T1,vib. This is a result of the coupling to the underdamped vibrational mode.

The waiting time dynamics of diagonal and cross-peaks (Fig. 6d, g) show beatings suggesting multiple oscillation frequencies that reflect both the coherent vibrational motion and also the coherent population oscillations between donor and acceptor induced by the electronic coupling. Such beatings persist on a timescale that is much longer than the electronic dephasing time. In fact, although T₂ and J are the same as in case discussed in Sections 3.1 and 3.2.1 (c.f. Figs. 3 and 5d, g), here the coherent oscillations do not decay within T₂, but persist on a much longer timescale, which is set by the vibrational relaxation time T1,vib. This is a result of the interplay between electronic coupling and strong vibronic coupling to an underdamped vibrational mode. In this case, the eigenstates of the system are mixed electronic and vibrational states with different damping. As such, the dephasing time of such vibronic states might be significantly different from those of the uncoupled donor and acceptor states.

Due to the strong vibronic coupling, the optical excitation generates a coherent wave packet of strongly mixed donor–acceptor electronic and vibrational states. As long as vibronic coherence survives, the transfer can be regarded as essentially barrierless across the interface. As such, energetic detuning ΔEDA, induced by electronic energy level offsets and disorder in the sample, does not significantly affect coherent charge transfer as long as it does not exceed the strength of the electronic coupling by far. Moreover, our simulations suggest that, in the limit of sufficiently large electronic coupling, the initial transfer rate is mainly controlled by the vibrational mode frequency. For C=C stretching modes (∼180 meV), as assumed in our model, this means a timescale of ∼11 fs, i.e. half a vibrational period, for moving the electron from the donor to the acceptor. In contrast, in the weakly coupled dimer, this timescale is greatly influenced by ΔEDA, and this may result in substantially slower transfer dynamics in the limit of large energetic detuning. In this case, even moderate detunings may completely suppress coherent dynamics [38].

We note that, while vibronic coupling strengths govern the timescale for equilibration of the population between donor and acceptor, the relaxation time of the coupled vibrational mode controls the coherent population beatings between the donor and acceptor during transfer. Hence, for shorter T1,vib than assumed here, these coherent oscillations will decay on a faster timescale. In the limit of T1,vib shorter than the first surface crossing of the wave packet, vibronic coupling will clearly have no significant effect on charge transfer. In this case, the coherent beatings are suppressed, and charge transfer falls back into the incoherent regime, as discussed in Marcus theory. In the 2DES maps, the faster vibrational dissipation will wash out the subpeak structure, and hence the maps will become more similar to the ones in Figure 5. It appears that not only vibronic coupling strengths, but also vibrational dissipation, play a crucial role for the coherent charge transfer dynamics.

Our results show that the observation of oscillating cross-peaks in 2DES does not unequivocally indicate coherent transfer. Instead, additional features in the 2DES spectra, in particular spectral peak splittings, seem to be more reliable signatures of coherent transfer mediated by a strongly coupled vibration (Fig. 6). Recently, we have indeed observed such peak splittings in ultrafast 2DES experiments of the prototypical polymer P3HT, reflecting strong vibronic couplings between localised excitons and more delocalised polaron excitons within the polymer moiety [37]. In P3HT, spectral subpeaks are detected for both excitonic GSB transitions and for ESA cross-peaks probing transitions within the polaron manifold [37]. An even more complex peak pattern is seen at early waiting times in experimental 2DES maps of P3HT blended with the fullerene acceptor PCBM [39]. These experimental features could reasonably well be explained by strong mixing of the electronic states (excitons and polaron excitons) to the underdamped intramolecular C=C stretching mode, giving rise to delocalised vibronic states [37]. This explanation, based on a relatively simple model Hamiltonian similar to the one used here, has been recently fully confirmed by advanced first-principles quantum theory [88]. The resulting peak structures and dynamics in 2DES are similar to the ones reported in Figure 6. In other 2DES experiments on P3HT, such peak splittings have not been directly resolved [65]. The dynamics show clear peak oscillations with the period of the high-frequency vibrational mode but little other dynamic signatures of strong vibronic coupling. These 2DES spectra can be also understood within the vibronically coupled dimer model presented here, but they reflect vibrational wave packet motion in a weakly coupled dimer, as discussed in Section 3.2.1 in this article. It appears that, despite its simplicity, this dimer model can explain surprisingly well the essential experimental features and thus aid in the interpretation of 2DES experiments of complex OPV materials.