Theory of ultrafast photoinduced electron transfer from ...

Theory of ultrafast photoinduced electron transfer from a bulk semiconductor to a quantum dot Andrew M. Rasmussen, S. Ramakrishna, Emily A. Weiss, and Tamar Seideman Citation: The Journal of Chemical Physics 140, 144102 (2014); doi: 10.1063/1.4870335 View online: http://dx.doi.org/10.1063/1.4870335 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/14?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Gain dynamics of an InAs/InGaAsP quantum dot semiconductor optical amplifier operating at 1.5 m Appl. Phys. Lett. 98, 011107 (2011); 10.1063/1.3533365 Time-resolved amplified spontaneous emission in quantum dots Appl. Phys. Lett. 97, 251106 (2010); 10.1063/1.3529447 Density-dependent carrier dynamics in a quantum dots-in-a-well heterostructure Appl. Phys. Lett. 96, 031110 (2010); 10.1063/1.3294309 Ultrafast relaxation and optical saturation of intraband absorption of GaN/AlN quantum dots Appl. Phys. Lett. 94, 132104 (2009); 10.1063/1.3114424 Spin-preserving ultrafast carrier capture and relaxation in InGaAs quantum dots Appl. Phys. Lett. 87, 153113 (2005); 10.1063/1.2103399

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THE JOURNAL OF CHEMICAL PHYSICS 140, 144102 (2014)

Theory of ultrafast photoinduced electron transfer from a bulk semiconductor to a quantum dot Andrew M. Rasmussen,a) S. Ramakrishna, Emily A. Weiss, and Tamar Seidemanb) Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3113, USA

(Received 30 January 2014; accepted 24 March 2014; published online 8 April 2014) This paper describes analytical and numerical results from a model Hamiltonian method applied to electron transfer (ET) from a quasicontinuum (QC) of states to a set of discrete states, with and without a mediating bridge. Analysis of the factors that determine ET dynamics yields guidelines for achieving high-yield electron transfer in these systems, desired for instance for applications in heterogeneous catalysis. These include the choice of parameters of the laser pulse that excites the initial state into a continuum electronic wavepacket and the design of the coupling between the bridge molecule and the donor and acceptor. The vibrational mode on a bridging molecule between donor and acceptor has an influence on the yield of electron transfer via Franck-Condon factors, even in cases where excited vibrational states are only transiently populated. Laser-induced coherence of the initial state as well as energetic overlap is crucial in determining the ET yield from a QC to a discrete state, whereas the ET time is influenced by competing factors from the coupling strength and the coherence properties of the electronic wavepacket. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4870335] I. INTRODUCTION

This paper describes the relationship between the electronic and vibrational structure of an electron donor-bridgeacceptor system, where the donor is a quasicontinuum of energy levels and the bridge and acceptor are discrete sets of energy levels, and the rate and yield of electron transfer in this system. Quantum dots (QDs) are appealing for use as electron donors and acceptors due to their size-tunable electronic structure and their ability to be readily incorporated into a variety of systems, for example, by direct growth on a substrate or deposition from solution to a surface.1, 2 The electronic structures of QDs are intermediate between that of a molecular system, characterized by a discrete set of sharp eigenvalues, and a bulk semiconductor with a continuous density of states. States close to the band-edges of a QD are wellseparated in energy and can be spectroscopically resolved at low temperatures. At energies far above the bandgap the density of states increases and is essentially bulk-like.3, 4 The band structure is degenerate according to the symmetry of the QD material’s Brillouin zone; hence the QD can accommodate multiple excitons or extra electrons at roughly the same energy. It is this last feature which motivates our study of electron transfer from a bulk to a discrete system. To optimize a catalytic reaction requiring multiple electrons arriving in fast sequence, it is critical to understand first how to move single electrons coherently from a bulk semiconductor (an energetic quasicontinuum, QC) to the reactive site (a set of energetically well-separated states). In general, an electron at constant energy will always tend to move toward a higher density of states. This tendency is a barrier that must be overcome in ora) Electronic mail: [email protected] b) Electronic mail: [email protected]

0021-9606/2014/140(14)/144102/13/$30.00

der to create useful devices that exploit high concentrations of charge in a sparse electronic structure. To this end we develop, in the present work, a timedependent, fully analytical model, and a numerical Hamiltonian approach to propagate an electronic wavepacket in time, and apply them to explore the relationship between the electronic and vibrational structure of an electron donorbridge-acceptor system and the rate and yield of electron transfer in such a system, where the donor is a QC of energy levels and the bridge and acceptor are discrete sets of energy levels. The model Hamiltonian approach based on a few parameters does not permit accurate modeling of a specific molecular system, but has the merit of generality; this generality allows for investigation of parameters expected to play a role in the reaction outcome, such as the density of states, band alignment, and coupling.5–7 Model Hamiltonians have been used extensively to investigate discrete-stateto-discrete-state coherent electron transfer,8–12 discrete-stateto-continuum coherent electron transfer,13–18 and transport between junctions composed of quasicontinua,19, 20 whereas limited work has been done modeling continuum-to-discrete state electron transfer (ET).21, 22 We remark here that the case of ET from a high density of states to a low one has been studied intensively experimentally and is relevant to a variety of applications including heterogeneous photocatalysis and photoinduced desorption.23–29 Our model and several of our conclusions apply to these processes. Similar time-dependent models of ET from an electrode to a molecular state have previously been employed for the study of thermal ET from an electrode to an adsorbed molecule, and of substrate-mediated photodesorption.21, 28, 30–33 In the study of electrochemical ET, an initial thermal distribution of electronic population in the substrate is assumed.22 Photodesorption studies

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typically make either a similar assumption,21 or the assumption that population is instantaneously localized on a molecular state.31, 33 The focus in these studies is on vibronic dynamics that are relevant to the desorption process, and not on the electron transfer dynamics. In contrast, this work examines in detail the effect of a coherent initial state in the substrate and briefly a that of a thermal initial state on the yield and timescale of ET to discrete electronic states. Our study focuses on fully coherent processes, relevant to experiments where a laser pulse prepares a wavepacket of states of the bulk QC and the electron transfer occurs on a time-scale rapid compared to thermalization. The inverse limit, that of rapid thermalization, is accommodated within our analytical and numerical models and is noted only briefly. We discuss the bulk-to-discrete ET dynamics in this paper in terms of three processes occurring on different time scales: (i) forward ET from the quasicontinuum to the discrete system, (ii) back ET from the discrete system to the quasicontinuum, and (iii) electronic beating. These processes are illustrated for the cases where the discrete system models (i) a molecular system with a single accessible electronic state, (ii) a molecular/nanostructured system with multiple electronic states, e.g., a QD, and (iii) a molecular/nanostructured system linked to the bulk material by a bridging electronic state with an associated vibrational mode. The latter example is discussed in both the elastic limit and the case of vibrational excitation. Our model allows us four insights into the dynamics of the quasicontinuum-to-discrete ET process: (i) In the case of an electron that starts as a coherent wavepacket in a quasicontinuum, as would result from the excitation of an electron from the valence to the conduction band of a semiconductor with a coherent laser pulse, the width of the wavepacket controls the yield and the time scales of ET to a discrete system. This width is a parameter which does not have a counterpart for the discrete-to-quasicontinuum ET case, since in that latter case the donor is a single state. Although discrete-tocontinuum ET can be very fast,15, 34 we find that continuumto-discrete ET can occur on an even shorter time scale with a sufficiently broad initial wavepacket. (ii) The electronic coupling between electron-donating and electron-accepting states plays a role complementary to that of the wavepacket’s shape in quasicontinuum-to-single-state ET; the coupling influences the yield of ET more than it does the rate. This observation is in contrast to the single-state-to-bulk process, where the coupling, along with the density of states in the quasicontinuum, determines the rate but not the yield (which is unity). (iii) The addition of a bridging electronic state does not change the time scale on which forward ET occurs, but it reduces the amount of back ET. In general for discreteto-continuum ET or discrete-to-discrete ET, the addition of a bridging state or sequential bridging states increases the time scale of ET.15, 35, 36 (iv) A vibrational mode on a bridging molecule is excited during the ET if the starting energy of the electron is on resonance with, or higher in energy than, that of the bridging electronic state, and if the discrete system does not have electronic states on-resonance with the wavepacket. Excitation of the vibration may enhance or suppress the yield of bridge-mediated ET, with maximum enhancement occur-

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ring when the gap between the electron’s initial energy and the energy of the bridging electronic state is equal to the reorganization energy of the vibrational mode. Observations (i) through (iii) illustrate the contrast between discrete-to-bulk and bulk-to-discrete ET. This article begins, in Sec. II, with a derivation of our analytical and numerical models and an analysis of the general insights furnished by the analytical formulation. Section III applies our model to the three model systems described above. Regarding these systems we discuss the effects of resonance, wavepacket shape, and coupling strengths on the time scales and yield of quasicontinuum-to-discrete-state electron transfer. With this minimal set of input parameters we are able to observe and gain an understanding of the basic features of this transfer process, which underlies the electron transfer dynamics of more complex models and experiments. II. THEORY

We consider a single electron moving through the conduction bands/LUMOs of the system. The present model does not explicitly include pathways for energetic relaxation of the electron, since our interest is in the short-time dynamics. When a bridging state is present in the system, our model allows for a vibrational mode on the bridge, and for electronvibrational energy exchange. Energy and phase are thus not conserved in the electronic subspace, but are conserved in the complete electronic-vibrational space. Our approach in (i) modeling the bulk states as a quasicontinuum of electronic states and (ii) computing the time-dependent Schrodinger wave equation of the system in a basis of vibronic states which takes into account the combined electronic-vibrational dynamics is similar to the previous studies of electron injection from a molecular donor to an electrode.13, 14, 17 A. Hamiltonian

The model Hamiltonian consists of an electronic Hamiltonian Hˆ 0 and an electronic coupling term Vˆ , Hˆ = Hˆ 0 + Vˆ ,

(1)

Hˆ 0 = Hˆ l + Hˆ b + Hˆ r ,

(2)

where

the subscript l denotes the quasicontinuum (QC), b denotes the bridge, and r, the accepting sub-system, comprises a single state (SS), multiple states (MS), or a QD electronic structure (QD). Each of the subsystem operators Hˆ I , I = l, b, r, takes the form Hˆ I = εI ν |χν ; ϕI ϕI ; χν |, (3) I,ν

where the vibronic states |χ ν ; ϕ I are products of vibrational eigenstates (|χ ν ) and nonstationary electronic states (|ϕ I ). The electronic states |ϕ I are the eigenstates of the uncoupled subsystems of the left (I = l) manifold, right (I = r) manifold, and bridge (I = b). The wavefunctions |χ ν denote vibrational eigenstates of the bridging molecule of a single generalized


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normal mode, where ν denotes the vibrational quantum number. The model can be readily extended to account for multiple normal modes covering high and low frequencies. In order to account for the vibrational modes of a bridging molecule between the electron donor and acceptor each electronic subsystem I is associated with a displaced harmonic potential surface along the dimensionless coordinate Q, 2 Q ¯ω (Q − QI )2 = ¯ω + gI , (4) 4 2 where QI denotes the equilibrium configuration. The second, equivalent form of Eq. (4), which includes the gI = −QI /2 term, is for convenience in calculating the vibrational FranckCondon overlap factors χν | χν¯ via the recursive method of Kühn et al.37 For simplicity we assume that the frequency ω is the same in the neutral and charged states. The equilibrium distance is shifted in the charged relative to the neutral state, which is typically the case in realistic systems and which corresponds to the reorganization energy of the system in the language of the Marcus theory. The Franck-Condon factors determine the probability of a vibrational transition concomitant with a change in the electronic state. Hence energy exchange between the electronic and vibrational modes is included exactly, thereby accounting for processes which do not conserve the electronic energy (but conserve total energy). The reorganization energy λ corresponding to an electron transition from subsystem I to subsystem I is

λ = ¯ω(g I − g I )2 .

(5)

The coupling operator Vˆ , which quantifies the degree of electronic orbital overlap, couples states of physically adjacent subsystems, Vˆ = Vˆl,b + Vˆb,b + Vˆb,r + Vˆl,r , where the VÎ,I are VÎ,I = VI,I

χν | χν |χν ; ϕI ϕI ; χν | + h.c.

(6)

(7)

ν,ν

In general, the model can include an arbitrary number of bridging sites with interactions Vb,b coupling them, although in this work we consider a single bridging site in Model C. Numerically it is convenient to scale the couplings with quasicontinuum states (Vl,r and Vl,b ) by the inverse of the energy gap between energy levels in the bulk quasicontinuum,14 Vl,I = Vl,I . (8) This scaling eliminates the dependence of the electron population dynamics before the Poincaré recurrence time38, 39 on the number of states in the quasicontinuum.40 Our discussion of the results below is thus in terms of the scaled coupling Vl,I . For simplicity we will refer to Vl,I as just V when there is only one coupling value in a system. B. Model systems and parameters

We consider the three model systems mentioned in the Introduction, characterized by features of their electronic struc-

FIG. 1. Electronic structures of model systems discussed in this paper (not exactly to scale). V is the electronic coupling, E denotes the position of the single site of Model A relative to the QC band edge, EF is the Fermi level, Vl,b denotes the bulk-bridge coupling, and Vb,r is the bridge-QD coupling.

ture, Fig. 1. Model A is a quasicontinuum coupled to a single site. It is characterized by the position of the single site relative to the lower QC band edge (E), and by the electronic coupling V , which quantifies orbital overlaps. E may be positive or negative. Model B is the same as A except for the inclusion of multiple discrete electron-accepting sites. The energies of the electronic states in Model B are literature values for the unoccupied orbitals in a bare lead selenide QD with a radius of 30.6 Å, scaled to span 1.0 eV.41 Model B thus corresponds to a bare QD attached directly to a semiconducting surface. To investigate the case of a QD bound to a surface via a molecule, Model C includes a bridging site at energy Eb , measured relative to the lower band edge of the QC. We ignore effects of the passivating ligands on the QD electronic structure, such that Models B and C have the same QD electronic structure. The couplings Vl,b and Vb,r (bulk-bridge and bridge-QD couplings, respectively) vary independently, and in the case of Model C, there is only nearest-neighbor coupling (Vl,r = 0). To avoid effects due to the artificial cutoff of the QD band at high energies, we restrict Eb to below the highest-energy QD state. Unless otherwise specified, the parameters for the system under consideration are as depicted schematically in Fig. 1: the QC consists of 251 states spanning 2 eV, and the bridging state representing the ligand LUMO is at 0.41 eV or 0.82 eV above the bottom of the QC band. We assume that states far off-resonance do not make a significant contribution to the electron population dynamics, hence the QD conduction band edge is aligned with the QC conduction band edge. The exception is Model A, wherein the QD energy is at the center of the bulk CB. The harmonic vibrational mode in the bridging molecule has an energy of 0.1 eV, and we include the first 11 vibrational eigenstates (ν = 0. . . 10). The default couplings are 8.71 meV, 3.81 meV, and 27.2 meV in Models A, B, and C, respectively (by default Vl,b and Vb,r have the same value in Model C). C. Time propagation

The system propagates in time subject to the timedependent Schrödinger equation, the initial condition being a Gaussian wavepacket of quasicontinuum eigenstates, reflecting in shape and parameters the excitation pulse. This starting condition corresponds to excitation of the bulk semiconductor


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time-evolving acceptor amplitude, cr (t) =

FIG. 2. (a) An energy level schematic for Model A, where the initial state is a coherent superposition of QC states. The electronic wavepacket has a width 2σ E and its central energy, μE , coincides with the single accepting state energy in Panel (a). (b) A comparison of numerical (red curve) and analytical (Eq. (12), blue dashed curve) results for the electron population on the single accepting state of Model A (Pr (t)). In the wideband limit (bandwidth > σE , V ), the numerical and analytical Pr (t) match almost perfectly until after the recurrence time. The arrows indicate the time constants of FET and BET. The black dotted line is an exponential decay fit to the numerical data between 0.3 τ r and 0.5 τ r , with time constant 2κ and amplitude allowed to float. The green dashed-dotted line is Pr (t) with a weighted-average starting condition, √ 2 2 corresponding to Eq. (B3) with wli = 1/( 2π σE ) × e−(Ei −μE ) /2σE .

V e−iElr t − e−κt , cl ¯ l Elr + iκ

(11)

where the cl are the (real arithmetic) wavefunction amplitudes of the QC band states at t = 0, Elr are the (frequency) gaps between energy eigenvalues, Elr = (εl − εr )/¯, and κ is defined 2 as πV , where is the QC energy level spacing. This lat¯ ter ratio is readily recognized as the Fermi Golden Rule rate of transition from a single discrete state into a continuum of equally spaced states.42–45 The corresponding time-evolving population on acceptor state r is thus, |cr (t)|2 =

cl∗ cl V2 ¯2 l,l Elr2 + κ 2 El2 r + κ 2 × Elr El r + κ 2 cos (Ell t) − e−κt (cos (Elr t) + cos (El r t)) + e−2κt

+κEl l (sin (Ell t) − e−κt (sin (Elr t) − sin (El r t))) . (12)

by a short Gaussian pulse (short with respect to the inverse of the bulk energy level spacing), |ψ(t = 0) =

1 −(Ej −μE )2 /4σE2 e |ϕj ; χ0 , N j

(9)

where N is a normalization constant, the index j enumerates the quasicontinuum energy levels, μE is the center of the wavepacket in energy space, and σ E is the energy breadth of the wavepacket. The vibrational degree of freedom is in its ground state |χ0 (see Fig. 2(a)). The dynamical evolution discussed below is largely determined by the relative magnitudes of the system energy parameters, V , , and σ E . The primary observables of relevance for addressing the questions raised in Sec. I are the electron populations in the subsystems I = l, b, and r, denoted PI (t), |ψ(t) | χν ; ϕI |2 = |cI,ν (t)|2 . (10) PI (t) = I,ν

Interesting insights of relevance to the questions posed in Sec. I can be gleaned from the simple model of Eq. (12). In particular, if the initial wavepacket is symmetric about the accepting state energy, then at long times the population of that state, |cr (t)|2 , vanishes. In this limit, the exponential factors decay to zero and the remaining sine and cosine terms in Eq. (12) interfere destructively, such that their sum cancels. A similar result applies in the case of a wavepacket which is offset from the accepting state in energy but is sufficiently broad to be essentially symmetric. By contrast, if the initially prepared QC wavepacket is not broad and not symmetric about Er , Eq. (12) predicts retention of charge on the electron-accepting state at long times. It is instructive to compare Eq. (12) to the decay of the electron population from a single electronic state into a continuum, Pr (t) = e−2κt .

I,ν

1. Analytical limits

A single accepting level: Although numerical propagation of the time-dependent wavepacket for our simple model is straightforward, analytical solutions are useful as a route to generating physical insight. Here we begin by developing an analytical solution of the time dynamics for Model A, applicable to the case where the quasicontinuum band is resonant with the acceptor state, and its band width is broad relative to the coupling strength V (wideband limit). Appendix A contains details of the derivation, and Appendix B contains an application to a thermalized initial condition. The derivation is related to that done in previous work, where the boundary condition is that the wavefunction is localized on the state r at t = 0.17 With the constraint that the wavefunction amplitude on the acceptor state r is zero at t = 0, we find for the

(13)

A term of the form of Eq. (13) appears in Eq. (12), and describes the contribution of the same discrete-to-continuum ET process. In the (academic but pedagogically useful) limit of a narrow-band laser excitation of the QC, where the laser energy band width is smaller than the QC energy level spacing, the population is initially in a single quasicontinuum state li . This condition places the constraint cl∗ cl = δlli δl li on Eq. (12), which leads to the expression, |cr (t)|2 =

V 2 1 − 2 cos (Eli r t)e−κt + e−2κt . ¯2 El2i r + κ 2

(14)

Equation (14) is equivalent to expressions previously derived for molecule-to-bulk electron injection,17, 22 the reverse process from that discussed here. This intriguing symmetry is anticipated from the microscopic reversibility of the Schrödinger equation. At long times Eq. (14) simplifies


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further to |cr |2 =

2

1 V , ¯2 El2i r + κ 2

(15)

from which we draw the conclusion that, for this initial condition, there will be charge retention in the acceptor state in the long-time limit. In addition there is no coherent oscillation of |cr |2 at long times, in contrast to Eq. (12) where oscillations persist indefinitely if the initial state is a superposition of QC states. Maximum charge retention is expected (and also explicitly seen through Eq. (15)) when Eli r = 0, that is, when the initial and final states are on resonance. A manifold of accepting levels: Our analytical derivation (Appendix A) is readily extended to multiple electronaccepting states. In addition to the wideband assumption made in the one-level case, a perturbative approximation must be made when recursively solving the differential equation for cr (t), as discussed in Appendix A. Including up to the thirdorder perturbative term in V , the expression for cr (t) takes the form, iErl t e − e−κt V iV κ cl cr (t) = + ¯ l Elr + iκ ¯ eiErl t −e−κt e−(κ−iErr )t −e−κt . × − cl (Elr +iκ)(Erl −iκ) (Elr +iκ)Err l r = r

(16) The coefficient cr (t) is dominated by the first term of Eq. (16) in this perturbative limit. This term accounts for the first-order transition of population prepared in initial states l to state r. Additional terms in the expansion, the first of which is in Eq. (16), originate from higher-order transitions. For example, the third-order term accounts for transition of the electron from state l to r to l to r.40 It is through this third-order transition that the energy gap between QD states, Err , enters into cr (t). This expression accurately describes the evolution of the electron density on the accepting sites in the wideband, low-coupling regime, where the ratio V /Err < 1. The same set of assumptions leading to Eq. (15), namely, cl∗ cl = δlli δl li and t → ∞, gives the expression, |cr |2 =

V2 1 × 2 2 ¯ Elr + κ 2 ⎧ ⎡ ⎫⎤ ⎨ 2 1 Elr Elr +κ ⎬⎦ 2 1−2 2 × ⎣1−κ 2 ⎩ Elr +κ 2 Elr +κ 2 ⎭ r =r r V ∗ domain arises from an increasing contribution from high-frequency terms in the wavefunction, seen in Eq. (As a simpler example, the maxi (12). 1 sin(nt) occurs at smaller t as N inmum of the sum N n=1 n creases). At short times there is a coherent oscillation of elec-

FIG. 3. (a) Electron population in the accepting state in Model A (Pr (t)) transferred from a coherent wavepacket in the bulk, as a function of the energy bandwidth σ E of the QC wavepacket given in units of V ∗ . (b) Normalized data from A, for direct comparison of time constants. (c) Energy profiles of the QC wavepacket for the bandwidths studied in Panel (a) compared with the Lorentzian lineshape of the accepting level (colored curves corresponding to colors in (a)). The transfer yield and time-scale are maximum for equal energetic widths of the QC wavepacket and accepting level, σE ∼ V ∗ .

tron population from the QC to the single state and back, and coherent dephasing damps further oscillations. At long times three of the five terms in Eq. (12) approach zero exponentially, at a rate κ, and coherent dephasing among the remaining (sine and cosine) terms causes |cr (t)|2 to tend to zero. The population decays exponentially from the accepting state back to the QC with a time constant, 2κ = 2V ∗ /¯, determined by Eq. (18) (see Fig. 2(b)). On one hand, our results suggest σ E as a useful design tool for experiments (where this parameter is directly controlled by the excitation pulse energy bandwidth in the coherent limit). On the other they suggest that an experimental determination of the optimal σ E would furnish useful information about the Golden Rule rate κ = π V 2 /¯, which could be contrasted with the results of numerical fits. While ET can be ultrafast in the incoherent limit (see Fig. 2(b)), tuning σ E allows for control of ET yield and timescale over a wide range of values. The tendency toward negligible electron population in the accepting state at long times, illustrated in panel (a) of Fig. 3, is a unique feature of the initial-wavepacket electron dynamics, reminiscent of the behavior of an electron in a single state that couples to a continuum. When the electron is initially prepared in a single QC state, by contrast (see Eq. (B3)), it decays exponentially onto the accepting state, and eventually reaches a steady state with a nonzero population on the discrete state. In the case of initial excitation of a QC wavepacket, laser experiments will only be able to make use of the electron on the discrete state at times shorter than


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FIG. 4. (a) Electron population on the accepting state of Model A, taken to be in resonance with the center of the bulk QC wavepacket vs the coupling V between the bulk QC and the single accepting state. Except for σ E all system parameters are as detailed for Model A in Sec. II B. (b) The same data as in (a), normalized by the maximum of acceptor population |cr (t)|2 for each value of V . (c) The accepting resonance lineshape vs the coupling parameter V . Here we use the same color scheme as in panels (a) and (b), with the addition of the discrete state’s broadened energy profile (gray curve). Also shown are the energetic positions of the QC energy levels (vertical black lines). The coupling V varies such that the acceptor resonance width ranges from below to above the QC energy level spacing.

the decay back to the QC (t < 1/κ), e.g., observing a change in the transient absorption signal from the electron arriving on the single state. In the V ∗ < regime, where the resonance width is smaller than the QC spacing, τ f is dependent only on σ E , as shown by comparison of the population evolution corresponding to different values of V ∗ and fixed σ E and in Fig. 4(a). In particular, the temporal location of the maximum is independent of V ∗ . Conversely, in this low-V ∗ regime the BET time τ b is dependent only on V ∗ , and in the small V ∗ limit τ b is essentially infinite on the time scale of the simulations shown here (Fig. 4(b)). Although τ f decreases monotonically with increasing V , its V -dependence is not as strong as its σ E -dependence. In the small- to medium-V regime with respect to the quasicontinuum spacing (black dotted curve through green dashed-dotted curve in Fig. 4(c)), the amplitude of ET is approximately proportional to the square of the coupling, as predicted by the prefactor V 2 in the analytical model derived in Sec. II (Eq. (12)). For large V there is a turnover in the amplitude of ET, which occurs, as discussed above, when σE ≈ V ∗ . For V ∗ > σE , back electron transfer happens on a faster time scale than forward electron transfer, and hence the maximum ET yield drops. This observation substantiates the design principle proposed above: the choice of an optimal energy bandwidth for the excitation pulse so as to satisfy σE ≈ V ∗ produces a FET time scale that is on the order of the BET time.

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FIG. 5. (a) Dependence of Pr (t) on σ E and on the overlap of the initial wavepacket with the accepting states in Model B where μE is 408 meV above the bulk band edge. (b) Normalized data from A. (c) Energy profiles of the QD wavepacket with different widths, compared with the spectrum of QD states with a Lorentzian broadening applied.

In Model B, the electron transfer yield also depends on the overlap of the accepting states with the initial wavepacket (Fig. 5). When σ E is smaller than the linewidth of a broadened QD state, the yield increases as σ E grows until σE > V ∗ , when the overlap begins to decrease. However, the presence of multiple accepting QD states means that as σ E increases further, the overlap with multiple states improves. This progression is visible in Fig. 5, where a high yield with small σ E (black dotted curve) gives way to a poorer yield at intermediate σ E (blue dashed curve), and eventually the yield recovers as σ E allows overlap with multiple states. For σE < V ∗ , the time at which Pr (t) reaches its maximum is strongly dependent on σ E (compare the black dotted and blue dashed curve in Fig. 5). For σ E greater than the linewidth of a single QD state, there is little dependence of the initial maximum of Pr (t) on σ E (blue dashed, red solid, green dashed-dotted curves). It is a consistent feature that some amount of back ET follows an initial rise in population on the QD states; the peak in population is a common phenomenon, namely, at short t, Pr (t) is transiently well above its long-time limit. This observation, evident also from inspection of Eq. (12), suggests on one hand that coherent “pump-dump” experiments, where a second pulse, time-delayed by roughly the maximum temporal location τ f , could serve to enhance the transfer yield dramatically. On the other hand, our result suggests that choice of the system parameters such that decoherence (possibly accompanied by population relaxation) of the QD levels, which occurs on time-scales comparable to the BET, may serve to enhance the acceptor population (provided that incoherent relaxation to the ground bulk state is sufficiently slow or can be suppressed). These observations invite the application of


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FIG. 6. The addition of a bridging state does not change the time constant of forward ET for a given σ E , but the amount of back ET is less (compare red and blue dashed curves). The vertical black line marks the end of the forward ET process, which has nearly the same time constant independent of the presence of a bridge and of coupling values.

optimal control theory to optimize only the laser parameters or both the optical and material parameters. B. Role of a bridge

Our analytical and numerical studies of a discrete level, or a manifold of discrete levels, directly coupled to a QC furnish very general insights and, as discussed above, are relevant to a variety of experimental scenarios. In many experiments, however, a molecular bridge is between the donating (e.g., bulk semiconductor QC) and accepting (e.g., semiconducting QD) manifolds. Inclusion of a bridging electronic state in the model is necessary to capture the observed enhancement of ET rate by an increased driving force (energy difference between the final and initial states), characteristic of molecule- or QD-to-bulk ET processes.46 Without a bridge the rate of discrete-to-bulk ET is solely dependent on the coupling and the density of states in the quasicontinuum (see Eq. (13)); there is no dependence on the driving force. When a bridging electronic state is between the discrete system and the continuum, the ET rate becomes a function of the central energy of the initially prepared electronic wavepacket relative to the bridge energy level. This factor can contribute to an increased ET rate with increased driving force observed in experiments.46 Model C serves as a minimal model to investigate the effects of a bridge. Figure 6 compares the features of quasicontinuum-todiscrete ET with and without a bridging state. Specifically, we illustrate the time evolution of the QD population for Models B and C for the same values of μE and σ E . The couplings Vl,b and Vb,r are smaller in Model C (9.54 × 10−2 V ∗ ) than the coupling in Model B (1.67 × 10−1 V ∗ ). In Model C, the bridge energy Eb (816 meV) is above μE . The vertical black line in Fig. 6 is at the time τ f at which ET from the quasicontinuum to the discrete states reaches its maximum yield. This time is the same for Models B and C, despite the significant difference in electronic coupling between the systems. The parameter controlling τ f is the width of the wavepacket, σ E . This ties in with recent studies47 which show evidence for subpicosecond coherent ET in molecular systems where ET would be expected to occur on a ns time scale. The amount of back ET is less in Model C relative to Model B for a range of μE , σ E , and Eb (data not shown). We

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find that BET occurs in general when Vl,b > Vb,r , and not in the opposite case. When Vl,b ≈ Vb,r it is ambiguous whether BET occurs. For very small Vl,b the amount of FET is small enough that it is not very meaningful to discuss BET (not shown in figure). In the Vl,b < Vb,r case (green dashed-dotted line), the strong coupling between the bridging state and the QD states causes a rapid oscillation from the bridge to the QD states and back; the electron simply decays back into the QC after residing briefly on the bridge. When the coupling to the QD states Vb,r is smaller than Vl,b (black dotted line), however, BET is greatly reduced or eliminated. In this case, the electron moves preferentially toward the QD from the bridge. This result gives us an important insight into designing systems for unidirectional electron transfer. The presence of BET creates a significant difference between Pr,max , defined as the maximum of Pr (t) over the course of the simulation, and Pr (t → ∞) for large Vb,r . For fixed σE = 27.2 meV, μE = 408 meV, and Eb = 816 meV in Model C the largest Pr,max is about 0.4, but the largest longtime Pr (t) is about 0.2 (data not shown). Electronic resonance is the primary factor determining the long-time yield of ET if there is no mechanism for dissipation of energy. If the initial wavepacket is not on resonance with any states on the electron-accepting QD, any electron population that arrives on the QD at short times will reside only for a short time. Similarly, if the initial wavepacket is far from resonance with the bridging state, the ET yield is less because the transient population on the bridge is smaller. The more important of these two resonance effects is that of resonance with the electron-accepting states. For a given position of the bridging state, the greatest amplitude of Pr (t) occurs when μE is on resonance with the highest density of QD states. Thus, for efficient ET to occur it is preferable to excite the electron such that it is on resonance with the final accepting states. Resonance of the excited electron with the bridging state is less important for the efficiency of the ET than is resonance with the accepting states. C. Electron-vibration coupling and inelastic events

Perhaps the most interesting property of the bridging molecule is its ability to couple the passage of an electron to vibrational modes of the molecule. These vibrations can provide a pathway for inelastic tunneling, which increases the number of channels by which the electron can cross the bridge.15, 48 As it tunnels, the electron charges the bridge transiently. We take the bridging state to be associated with a single vibrational mode of energy 0.1 eV. The vibrational eigenstate populations in the transiently charged and neutral states of the bridge, Pionic,ν and Pneutral,ν , respectively, are |cb,ν |2 , Pionic,ν = b

Pneutral,ν =

l

|cl,ν |2 +

|cr,ν |2 ,

r

where ν is the vibrational quantum number, and b, l, and r are indices running over the bridge, bulk, and QD electronic states, respectively. Figure 7 shows the vibrational


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FIG. 7. Vibrational eigenstate (ν = 0. . . 10) populations for the transient ionic (Pionic,ν (t)) and neutral (Pneutral,ν (t)) states of the bridge molecule for Model C in two cases differing in the relative energies of initial electron wavepacket central energy μE and the bridge state energy level. In Panels (a) and (b) μE is below the bridging energy level and in Panels (c) and (d) μE is above the bridge level (details in text). Excited vibrational eigenstates (ν > 0) are always populated transiently, as the electronic population transfers to the bridge (first few fs in Panels (a) and (c)). As the electron moves off of the bridge (t > 5 fs), higher vibrational eigenstates are mostly de-excited.

populations for two examples of Model C. The QC wavepacket energy bandwidth, σ E , the coupling parameters Vl,b and Vb,r and the driving force λ are equal in the two examples; they differ only in the bridging state energy level compared to the center of the QC electronic wavepacket μE . In panels (a) and (b), the bridge energy is above μE by 0.41 eV, whereas in panels (c) and (d) it is below μE by the same amount. The first case is in the superexchange regime of ET as described by previous work, while the second case is in the sequential ET regime.19 In both cases the transient occupation of the bridge state accompanies substantial population of highly excited vibrational states at short time scales. However, inelastic tunneling, indicated by excited vibrational populations at long times, only occurs when μE exceeds the bridge energy level, panels (a) and (b) in Fig. 7.

1. Vibrational dynamics

The mechanism of vibrational excitation is responsible for current-driven dynamics in molecular conduction junctions—see Refs. 49–51 for reviews and Ref. 52 for a collection of experimental and numerical studies—and is very general. Equilibrium displacement between the neutral and ionic states of the bridge molecule gives rise to the population of a nonstationary vibrational wavepacket of the ionic state vibrational levels. Rapid neutralization of the charged state (which corresponds to transfer of the charge to either the initial or the final manifold) will lead to a vibrationally excited neutral state, hence an inelastic event, however, only if the vibrational system has evolved non-negligibly from the neutral state equilibrium while residing in the charged state. To zero order, the extent of inelastic ET events is thus largely determined by the equilibrium displacement of the neutral and charged states and the lifetime of the charged state. The analogous process in the present system (Fig. 7) results in essentially complete vibrational relaxation upon neutralization of the bridge as the electron population transfers either to the QD or back to the bulk QC in the case where

the bridge energy level exceeds μE by 0.41 eV. In the case where μE exceeds the bridge energy by 0.41 eV, there is non-negligible excitation of the second vibrational eigenstate (ν = 1, blue dashed curve, Fig. 7(d)). Inelastic tunneling dominates when there are no QD states on resonance with μE . When the initial electronic energy is higher than that of all QD states, the tunneling has mostly inelastic character and higher (ν > 1) vibrational eigenstates retain population at long times (data not shown). The higher the density of QD states that are on resonance with the initial wavepacket center, the greater the probability of elastic tunneling taking place.

2. Electronic dynamics

As mentioned in Sec. III B, resonance of the initial wavepacket with the bridging electronic state correlates with a greater yield of ET due to a larger population residing transiently on the bridge. This holds for vibronic states as well as electronic states. The Franck-Condon factors in Eq. (7) determine which vibronic states on the bridge will have the best overlap with the initial state of the electron in the bulk. A high reorganization energy means that the vibronic states with the largest overlap with an initial ν = 0 vibrational state will be high in energy (the vibrational quantum number on the bridge ν f > ν i , the vibrational quantum number on the initial state). In the case where Eb (0.41 eV) < μE (0.82 eV), a nonzero reorganization energy causes the vibronic states with the best overlap with the initial state to be further off-resonance relative to the λ = 0 case. As expected, with increasing detuning from resonance the ET yield decreases (Fig. 8(a)). In the opposite (Eb (0.82 eV) > μE (0.41 eV)) case, nonzero Franck-Condon factors allow energetic pathways across the bridge that are closer in energy to the initial wavepacket than the electronic state without reorganization. With nonzero λ the electron therefore has a greater probability of transferring to the bridge, from which it transfers to the QD with a greater yield than in the zero-λ case (Fig. 8(b)).


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FIG. 8. A comparison of electronic population dynamics on the QD in Model C for two relative positions of the initial wavepacket and the bridging electronic state. The small schematics portray the configuration of Eb above μE (Panel (a)) or below μE (Panels (b)–(d)). (a) The initial wavepacket is below the bridge. The yield of ET is larger with zero reorganization (solid red curve) than it is with finite reorganization. (b) The initial wavepacket is above the bridge. The yield of ET increases when reorganization is nonzero (dashed and dotted curves), and is maximum when the reorganization energy matches the 0.4 eV gap between the initial wavepacket and the bridge (blue dashed curve). (c) Normalized Pr (t) from (b) shows no change of the FET time. (d) Varying σ E causes a change in the time scale of FET in Model C when the initial wavepacket is above the bridge. The value of the reorganization energy is 0.4 eV; the same value as for the blue curves from Panels (a)–(c).

A comparison of the two cases in Figs. 8(a) and 8(b) demonstrates that, although the yield of ET may be similar for the pure electronic case (red curves), the presence of a vibrational mode has a dramatic effect on the ET yield. Specifically, the yield is several times larger in Fig. 8(b) than in Fig. 8(a) for moderate to large values of λ. The vibrational mode has very little effect on the ET times; however, the main determinant of the FET time is σ E . Fig. 8(c) shows that λ has no effect on the FET time, and very little effect on the BET dynamics. The inverse relationship of σ E in and the FET time in Model C is shown in Fig. 8(d). In Panel (d), λ = 0.4 eV, matching λ for the blue lines in Panels (a)–(c). IV. CONCLUSIONS

In this work, we considered the problem of electron transfer from a dense QC to a discrete state (or a manifold thereof). We are motivated by the unusual and rarely-addressed physics of transfer from a high into a low density of states as well as by potential applications, e.g., in catalysis. We have focused on two aspects of ET dynamics in a system comprising an electronic QC coupled to electron-accepting state(s), namely, (i) the factors determining the time scale of the ET process and (ii) the yield of the transferred population on the accepting states. In order to provide qualitative insights that also assist the interpretation of the numerical and future experimental results, we began with an analytical solution of a simplified model comprising the semiconductor electrode coupled to a single electronic level. The analytical solution Eq. (12) reveals that there are three (sets of) factors that determine the ET time from the QC to discrete states. These factors are the discrete-to-bulk rate constant κ, the coherences between electron donor and acceptor levels with electronic energy gap Elr ,

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and coherences between donor levels with energy gap Ell . For an ET process that starts from a thermalized initial state only the Elr terms are present. These terms alone can lead to ultrafast ET but the resulting Pr,max is low compared to that resulting from a coherent initial state. In general for Models A, B, and C we find that a coherent initial state gives a higher ET yield (Pr,max ) than a thermally averaged initial state. While it is well known that the decay rate of population from a single state population to an electrode is proportional to the square of the electronic coupling and to the electrode density of states,43, 44 the result that the electrode to single state ET has ultrafast components that arises from the phase properties of the initial state has not been discussed in the literature. Such fast ET also takes place for the population decay from a single state into an electronic QC provided one observes the population rise in a narrow window of electrode states.17 The latter study suggested energy-resolved spectroscopy as a tool to observe ultrafast oscillations of populations at different energies within the bulk. In the reverse case of bulk-to-discrete ET in the systems considered in this work, coherent oscillations of the electron population on the accepting state reflecting the coherence properties of the initial wavepacket would be directly visible by a transient absorption experiment on the accepting system. The ultrafast components that appear in the ET dynamics in either direction arise in the analytic expression (Eq. (12); Eq. (4) in Ref. 17) from sinusoidal functions. These functions are indicative of coherences between QC states that are either introduced upon short pulse excitation into the electronic wavepacket or are mediated via the single state. The coherences lead to oscillatory ET which the ultrafast components reflect. We remark that the fast ET components persist when a manifold of final states replaces the single state mimicking the QD, as well as when a molecular bridging state is introduced. The maxiumum yield of electron population over time in the accepting state(s) is related to how well the initial coherent distribution in the electrode can be resonantly mapped energywise onto the accepting state(s) that have been broadened due to interaction with the electrode states. In other words, when the entire initial distribution can undergo isoenergetic ET to the final broadened electronic state one finds the maximum yield. The fact that ET yield is so strongly linked to electronic resonance implies that to have high-yield ET from a high to a low density of states, the tendency of a resonantly transferred electron to return to the higher density of states must also be overcome. In the case of single final electronic state on resonance with the initial wavepacket, the back ET reduces the ET yield to zero. When there are multiple accepting states interacting with the electronic QC, the back ET is never complete and the accepting states retain a finite population at long times. We observe that in our Model C, the tendency of the electron to move to the higher density of states in the QC can be most effectively avoided with a bridge-acceptor coupling that is larger than the donor-bridge coupling. With the addition of a molecular bridge with vibrations, reorganization energy participates in determining the ET yield. A finite reorganization energy can increase or


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decrease the yield of ET by decreasing or increasing (respectively) the detuning of the initial superposition state with the final set of vibronic states. Another potentially important feature of donor-bridge-acceptor systems, vibrational excitation leading to inelastic tunneling, is revealed by our model to be largely unimportant when the bridge electronic energy is sufficiently high. Although inelastic tunneling may not occur, the reorganization energy maintains its role in determining the ET yield. The relationship between the ultrafast laser-driven dynamics and the ET properties of a material in an actual device can be understood using an extension of the current model to include an incoherent source of excited electrons. The present study invites several applications and extensions of our model. First, the model Hamiltonian approach allows us to describe different chemical and physical scenarios qualitatively but yet in a way that generates new insights. Problems other than the one considered here that would be interesting to explore within this framework include substratemediated photochemistry on metal surfaces, molecular conduction junctions where resonant tunneling may give rise to inelastic processes, and hot electron generation. Second, for the problem at hand and a specific system of experimental or technological relevance it is possible to compute ab initio parameters using cluster approximations.16, 18 The results are no longer general but they can serve as a basis for a quantitative description of a specific system. Finally, this work sets the stage for studies of the effects of processes such as exchange of phase and energy of the electron with a surrounding bath on the yield of charge transfer to the discrete system. It will be interesting to learn whether an electron can receive a coherent “push” to a discrete system from an initial wavepacket in a quasicontinuum, and once in the discrete system be prevented from returning to the quasicontinuum. This and related challenges are the topics of ongoing research in our groups.

ACKNOWLEDGMENTS

T.S. is grateful to the NSF (Award No. CHE-1012207) and the NSF IGERT: Quantum Coherent Optical and Matter Systems program (Award No. 0801685) for support. This work was supported in part by a grant from the Air Force Office of Scientific Research (AFOSR) (FA9550-10-1-0220) through a Young Investigator Award to E.A.W.

APPENDIX A: DERIVATION OF TIME PROPAGATION

In this Appendix, we derive an analytical expression for the electron population dynamics in an acceptor state (denoted by a subscript r) for the case where the initial state is a coherent wavepacket in an electronic QC coupled with the accepting state. We start with the Hamiltonian Hˆ =

r

εr |rr| +

l

εl |ll| +

Vlr |lr| + h.c.,

l,r

(A1) where r is an index over discrete states, and l is an index over quasicontinuum states. The equations of motion for the wave

function’s coefficients (in atomic units) are i Vrl cl (t)e−iElr t , c˙r (t) = − ¯ m c˙l (t) = −

i Vlr cr (t)e−iErl t . ¯ r

Elr = (εl − εr )/¯. Integrating Eq. (A3) gives t i Vlr dt cr (t )e−iErl t . cl (t) = cl (t = 0) − ¯ r 0

(A2)

(A3)

(A4)

We will henceforth refer to cl (t = 0) as just cl , and Vlr and Vrl as just V . Substituting (A4) into (A2) we have V2 t c˙r (t) = − 2 dt cr (t )e−iElr (t−t ) ¯ l 0 V2 t − 2 dt cr (t )e−iEr l t e−iElr t ¯ 0 l r = r

−

iV −iElr t cl e . ¯ l

(A5)

The r = r term is written separately from the other terms in the sum over r , and the couplings Vrl are set as a constant V . Introducing the Poisson summation formula written in the form, ∞

e−2πinx/P = P

n=−∞

∞

δ(−x − kP ),

(A6)

k=−∞

we rewrite the summations over l in the form, V2 t c˙r (t) = − 2 dt cr (t )e−2πil(t−t )/τr ¯ l 0 t V2 − 2 eiEr t dt cr (t )e−iEr t e−2πil(t−t )/τr ¯ 0 l r = r

−

iV −iElr t cl e , ¯ l

(A7)

where τr = 2π is the recurrence time. This assumes that εr,r = εm for some l, and that the quasicontinuum band extends well above and below this resonance. With this result we have t dt cr (t )δ(t − (t − lτr )) c˙r (t) = − κ l

− κeiEr t

0

l r = r

−

t

dt cr (t )e−iEr t δ(t − (t − lτr ))

0

iV −iElr t cl e . ¯ l

(A8)

At times t before the first recurrence time, τ r , the argument of the delta function is nonzero for all nonzero values of l, and


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is equal to zero only for l = 0 and t = t. Given that − a)f (x) = 12 f (a), one finds cr (t)e−iEr t c˙r (t) = − κcr (t) − κeiEr t

a 0

dxδ(x

r =r

iV −iElr t − cl e . ¯ l

(A9)

A solution to the ordinary linear differential equation 2 (A9) is obtained by defining a rate constant κ as V¯π and in−κt troducing an integrating factor e , obtaining iV e−iElr t − e−κt cl cr (t) = − ¯ l κ − iElr t − κe−κt dt e(κ+iErr )t cr (t ). r =r

PTr (t) =

APPENDIX C: CODE

l r = r

e−(κ−iErr )t − e−κt (Elr + iκ)Err

(A11)

.

The fourth-order term has the form t t 2 −2κt iErr t κ e dt e dt e(κ+iEr r )t cr (t ). r = r r = r

0

0

(A12)

APPENDIX B: THERMALLY AVERAGED INITIAL STATES

The discussion in Sec. II C 1 focuses on the case where the initial QC state is a pure state. This case corresponds to excitation of a single eigenstate li into an electronic wavepacket of QC states by means of a fully coherent source. In order to account for a thermal distribution of electron population in the initial state within the Schrödinger wavepacket formalism, all observables Oli (t) obtained for a pure initial state li are averaged according to a Fermi-Dirac distribution at temperature T, wli (T )Oli (t), (B1) OT (t) = li

wli =

V2 1 − 2 cos (Elr t)e−κt + e−2κt w (T ) . (B3) l ¯2 l κ 2 + Elr2

(A10)

0

The first term in Eq. (A10) is the expression for cr (t) in a level-band system with one discrete level. Substituting Eq. (A10) into itself recursively constructs a perturbative series in V . Up to the third order this series is iErl t e − e−κt V cl cr (t) = ¯ l Elr + iκ iV eiErl t − e−κt κ + cl ¯ Elr + iκ (Erl − iκ)

−

where Eli is the energy eigenvalue of state li and kB is Boltzmann’s constant. In particular, thepopulation of accepting levels at temperature T is PTr (t) = li wli (T )Plri (t). Relevant to experiments is the case where relaxation of the initially excited coherent wavepacket to the band edge occurs on a faster time scale than the electron transfer dynamics. In this limit it is useful to consider the thermalized photoexcited bulk as the initial state. This situation allows us to focus on the electron transfer physics, rather than on the nature of the laser excitation. Using Eqs. (14) and (B1), we have for the population of the accepting level in this limit

1 (Eli −EF )/kB T

1+e

,

(B2)

The raw data used in this publication, along with the source code used to generate the data, are available online at https://github.com/andyras/pub-jcp-2014/. 1 M.

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