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Functional Mode Hot Electron Transfer Theory Justin E. Elenewski, Jesse Yu Cai, Wei Jiang, and Hanning Chen J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b00099 • Publication Date (Web): 23 Feb 2016 Downloaded from http://pubs.acs.org on February 28, 2016

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The Journal of Physical Chemistry

Functional Mode Hot Electron Transfer Theory Justin E. Elenewski,† Jesse Y. Cai,‡ Wei Jiang§ and Hanning Chen*† †

: Department of Chemistry, the George Washington University, Washington, DC 20052, the United States of America ‡

: Thomas Jefferson High School for Science and Technology, Alexandria, VA 23312, the United States of America

§

: Argonne Leadership Computing Facility, Argonne National Laboratory, Argonne, IL 60439, the Untied States of America *: corresponding author 800 22nd Street, NW, Washington, DC 20052 Fax: 202-994-5873 Phone: 202-992-4492 Email: [email protected]

ACS Paragon Plus Environment


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Abstract Charge carriers that have been driven out of thermal equilibrium by their excessive vibrational energies are term hot-carriers. A theory has been developed to model the injection of these vibrationally excited electrons by explicitly accounting for the time-dependent thermal relaxation of the electron-transfer driving vibrational mode as ascertained using functional mode analysis. Specifically, the thermal relaxation rate of the driving mode is first determined through the so-called frozen-phonon approach before the energy-dependent line shape function is revisited to include memory effects for the vibrational quanta within the framework of Fermi’s golden rule. As shown by the numerical simulations of a 6-methyl-azulene-2-carboxylic acid dye molecule bound to an anatase TiO 2 [101] surface, our new theory not only yields persistently faster electron injection rates with higher incident photon energy, but also exhibits a sharp increase when the vibrational quanta of the photo-excited dye molecule changes from 2 to 3, in excellent agreement with a recent femtosecond pump-probe spectroscopy study. These methods comprise a practical first-principles simulation protocol to model vibrationally resolved electron injection by accommodating the subtle coupling between molecular vibration, thermal relaxation and electron transfer under arbitrary thermalization conditions. Moreover, only trivial extensions are needed to enable the application of our new theory to vibrationally controlled electron transfer reactions in a wide range of chemical and biological systems, particularly those engineered using time-delayed laser pulses. Keywords: electron transfer, vibronic coupling, functional mode analysis, timedependent density functional theory, constrained density functional theory, phonon scattering

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1. Introduction: Electron transfer is one of the most important dynamical processes in chemical and biological systems. For instance, the synthesis of adenosine triphosphate (ATP) is aided by a series of sequential electron transfer events1 across several eukaryotic protein complexes in mitochondrion, each with increasing standard reduction potentials.2 Thanks to the natural evolution, the structural optimization of the mitochondrial redox carriers has allowed the adaptation of the electron transfer process to specific ecological entities,3 making their energy conversion efficiency much higher than any artificial photosynthetic system that has been made so far.4-5 The strong interplay between molecular structure and electron transfer has also been demonstrated in deoxyribonucleic acid (DNA) strands6-7 whose electric conductivities can vary by an order of magnitude through single base pair mutations.8 A further example is given by the intra-molecular electron transfer rate within a dyad molecule; a process that may be tightly regulated by its solvation environment. When a betaine-30 (B30) dyad is solvated by the small and thus mobile acetonitrile molecules, its charge recombination rate after photoexcitation is 2.0 ps-1,9 consistent with the thermal relaxation rate of acetonitrile at room temperature. By contrast, if the bulky and thus sluggish glycerol triacetate (GTA) is selected as the solvent, the charge recombination of B30 notably outpaces the thermal relaxation of GTA by at least ten times across a wide range of temperatures,9 suggesting the predominance of intramolecular vibrations in the electron transfer mechanism. The delicate interplay between molecular vibration, thermal relaxation and electron transfer has attracted more and more attention since 2009 when the extraction of vibrationally excited electrons (the so-called hot electrons) was first reported in a core/shell CdSe/ZnSe colloidal quantum dot.10

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Shortly thereafter, hot electron transfer from a PbSe thin film to its TiO 2 substrate was directly observed using the time-resolved optical second harmonic generation spectroscopy,11 which reveals a coherent excitation of TiO 2 transverse surface phonons accompanied by interfacial electron injection from PbSe. In both experiments, the excess vibrational energy possessed by a hot electron carrier was successfully harvested before being dissipated through heat loss. As such, the hot electron transfer holds great promise to circumvent the long-standing Shockley-Queisser thermodynamic limit12 by thermally insulating the electron-hole pairs when their photo-activation energies exceed the band gap.13 According to the Shockley-Queisser detailed balance analysis, the incident photon conversion efficiency (IPCE) can be raised from 33% to 66% when there is no thermal contact between a charge carrier and its environment.13 This improvement has been ascribed to a lower threshold incident energy, a reduced blackbody radiation loss, and a higher Carnot limit.13 Since the thermalization of a photo-excited species typically occurs through multi-phonon scattering in the condensed phase, the thermal insulation of the carrier would become most efficacious when its vibrational density of states (DOS) is distinct from that of its substrate. More specifically, if the slowest vibrational mode of a charge carrier is n times as fast as the fastest vibrational model of its substrate, its thermalization must involve the collective scattering of at least n+1 phonons. Because the phonon scattering intensity usually decays rapidly when increasing the number of involved phonons,14 only the scattering processes with the least number of participating phonons are considered in practice.15 In a dye-sensitized solar cell (DSSC), most vibrational normal modes of the organic sensitizer are substantially faster than the

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phonon modes of the semiconducting metal oxides, making the sensitizer/semiconductor interface a promising venue for hot electron extraction. As an example, vibrationally resolved electron transfer has been observed on the carboxyazulene/TiO 2 interface,16 which exhibits a 9-fold increase in quantum yield when the vibrational quanta of the photo-excited chromophore is raised from 0 to 3. The substantially slower rate of electron injection commensurate with thermal relaxation is attributed to the sparse population of surface trap states when compared to the densely populated and higher-energy conduction band. As a consequence, engineering band-structure alignment is an appealing approach to modulate interfacial hot electron transfer. The first vibrationally resolve electron transfer investigation was conducted on the ion pair,17 where the charge recombination rate after photoexcitation is tripled when the vibrational quanta in the C=O stretching mode of the species increases from 0 to 1. More importantly, this picosecond transient infrared absorption spectroscopy study suggested that the electron transfer rate could be regulated by simultaneously stimulating electronic transitions and molecular vibrations. Supporting this notion, a femtosecond optical-pulse shaping experiment18 demonstrated that the photo-induced spin separation efficiency can be improved by 20% if the coherent vibrational oscillations at 123 cm-1, 130 cm-1 and 200 cm-1 are activated alongside an electronic excitation at 2.34 eV. More recently, the charge separation rates of a library of donor-bridge-acceptor (DBA) assemblies were found to be sensitive to the selected vibrational modes of their bridge moieties by using the Ultraviolet-Infrared-Infrared (UVIR-IR) pulses,19 further confirming the significance of vibronic coupling in electron transfer. Despite this rapid progress in vibrationally controlled electron transfer, the

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identification of vibrational models that are required to drive a given electron-transfer process remains theoretically daunting. In a combined femtosecond stimulated Raman spectroscopy (FSRS) and density functional theory (DFT) study,20 six vibrational modes ranging between 1000 cm-1 and 1700 cm-1 were proposed to be responsible for the charge recombination in a perylene-xylene-perylene-3,4:9,10-bis(dicarboximide) (Per-Xy-PDI) dyad. However, the relative contributions of these essential vibrational modes to charge recombination remain obscure, making the quantitative regulation of vibronic interference and thermal relaxation a rather challenging task. In one of our recent studies, the functional mode electron transfer (FMET) theory21 was conceived to quantitatively correlate molecular vibration and electron transfer through statistical analysis. In FMET, the electron-transfer driving mode is first ascertained by maximizing the Pearson correlation coefficient between the atomic displacement and the diabatic energy gap. Then, the revised Jortner formula22 or the energy gap law22 is chosen to calculate the electron transfer rate for a thermalized system. In the present study, this FMET theory is extended to treat electron transfer under arbitrary thermalization conditions, paving the way to the systematic design of ultraefficient photovoltaic devices and photosynthetic apparatuses on the basis of hot electron extraction. The remainder of this paper is organized as follows. In section 2, the FMET theory is briefly reviewed before introducing the so-called frozen-phonon approach14 for calculation of the thermal relaxation rate. The time-dependent generating function for nuclear quantum dephasing is then revisited to account for the initial population of vibrational states and their thermal relaxation. Using these results, the revised Jortner formula is applied to calculate the electron injection rate from a photo-selected molecular

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vibronic state to a surface Bloch state within the conduction band of a semiconductor. Thereafter the overall hot electron transfer rate is ultimately determined by summing the density-weighted state-resolved injection rates. In section 3, our functional mode hot electron transfer theory (FMHET) is justified by numerical simulations of a 6-methylazulene-2-carboxylic acid (MeAz) dye molecule that is bound to an anatase TiO 2 [101] surface. Through a direct comparison with experimental data,16 a detailed mechanism for non-equilibrium electron injection at the molecule-semiconductor interface will be obtained. Finally, we summarize our results and succinctly discuss possible applications of our new theory in section 4.

2. Theoretical Methods: As detailed in Ref. 21, the driving vibrational mode, transfer process between donor,

, and acceptor,

, for a given electron

, diabatic states can be formulated

as a linear combination of all vibrational normal modes: (1) where the expansion coefficient, vibrational mode,

, indicates the relative contribution of the

th

, to the degeneracy of vibronic energy that is demanded by a

radiationless transition. In the present study, the vibrational modes were ascertained by diagonalizing the mass-weighted cross-correlation matrix of atomic displacements in a molecular dynamics trajectory. Inspired by the functional mode analysis method,23 is identified through the maximization of the Pearson’s correlation coefficient, is defined as:

6


, which


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(2)

is the projection of instantaneous nuclear configuration

where on

,

is the instantaneous diabatic energy gap, is the covariance function between and

deviations are denoted as value of

and

, and their standard

, respectively. The search for the maximum

reduces to solving the following coupled linear equations: (3)

Since the greatest change of along

is warranted when the system’s geometry is displaced

, it can be also regarded as the electron transfer reaction coordinate. Once

is ascertained, we are in a good position to evaluate its thermal

relaxation rate. According to phonon scattering theory,24 the decay rate of a parent phonon,

into two new ones of

and

is given by

(4)

where

are the corresponding angular frequencies,

is the

is the third-order derivative of energy

Bose-Einstein distribution function, and

with respect to all participating phonons. For the completeness of the phonon scattering scheme, all thermal relaxation channels that conserve energy (i.e.,

7


) should be

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considered. However, for our MeAz/TiO 2 model system, its four-phonon scattering intensity is about two orders of magnitude smaller than its three-phonon counterpart. Therefore, only the three-phonon scattering is assessed in the present study. With regard to the quasi-independence of the molecular vibrational modes, the overall thermal relaxation rate of

reads as: (5)

Under this non-thermalized condition, the radiationless transition rate,

, is given by

Fermi’s golden rule25 so that (6) where

is the electronic coupling strength, and

is the time-dependent generating

function,22 which can be expressed as (7) where quanta,

is the electron transfer driving force,

is the time-dependent vibrational

is the effective angular frequency of

angular frequency and Huang-Rhys parameter of the As defined in Ref. 21, the dimensionless quantity,

,

are the

th vibrational mode, respectively. , is given by

.

Under the short time approximation with strong vibronic coupling (i.e., ), Eq. 7 simplifies to (8)

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after expanding and

to second-order. In this expression,

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is the reorganization energy,

is the functional-weighted vibrational frequency defined as (9)

Since the thermal relaxation of a single phonon follows first-order kinetics,

is a

vertically shifted exponential function: (10) where

is the thermally averaged vibrational quanta as given by the Bose-Einstein

distribution function: (11) If

is substantially faster than the thermal fluctuation, i.e.,

, then

and Eq.8 can be rewritten as

(12) and its Fourier transform yields (13) Thus,

reduces to

(14) for instantaneous thermalization with

, whereas it is given by

(15)

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for ideal insulation with

. By comparing the two preceding equations, it is easy

to conclude that thermal insulation not only shifts the electron transfer resonance peak to

from

, but also broadens the peak by a factor of

. As a consequence, hot electron transfer may be fully controlled by selectively populating an initial set of vibrational states,

, as will be demonstrated at a molecule-

semiconductor interface in the next section. For a band of electron acceptor Bloch states with a DOS of

, the overall hot electron injection rate,

, is simply (16)

where

is the Heaviside step function to ensure thermodynamic irreversibility,

Fermi energy,

and

is the

are the energies of the donor and acceptor states, respectively.

Since a Bloch acceptor state,

, is normalized by its unit cell volume, i.e.,

, the calculated

by Eq. 16 is invariant upon laterally adding

(or removing) unit cells in our simulation box. Provided that the TiO 2 surface is large enough to diminish the artificial interactions between the periodic images of the sensitizer, the linear increase (or decrease) in decrease (or increase) in

is exactly negated by the linear and thus in

.

3. Simulation Results: Azulene dye and its functional derivatives have been long employed as model systems to study vibrationally dependent interfacial electron transfer26 due to their well-

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separated S 1 and S 2 states27-28 that impede an otherwise very fast S 2 →S 1 internal conversion. Another appealing feature of theses molecules is their accessible redox potentials, placing their S 1 states near the conduction band edge of TiO 2 , the most widely used semiconductor in dye-sensitizer solar cells.29-30 Among the three naturally occurring polymorphs of TiO 2 , anatase is not only more photochemically active than rutile,31 but also is more thermodynamically stable than brookite.32 Thus, the well-characterized anatase [101] surface33 is chosen as our semiconducting substrate to which a 6-methylazulene-2-carboxylic acid (MeAz) dye molecule is anchored through its carboxyl group. Specifically, the hydroxyl hydrogen atom and carbonyl oxygen atom of MeAz bind to a twofold-coordinated oxygen atom and a fivefold-coordinated titanium atom on the anatase [101] surface, respectively. In our simulation box (Fig. 1), the bottom half of the anatase slab is frozen to mimic the experimental bulk phase crystal structure,34 while the other atoms are allowed to undergo surface reconstruction. Unless otherwise specified, all simulations were performed using the CP2K software35 with the Goedecker-Teter-Hutter

(GTH) pseudopotential,36 Perdew-Burke-Ernzehof hybrid

(PBE0) exchange-correlation functional,37 and polarized-valence-double-ζ (PVDZ) basis set.38 In addition, a wavelet-based Poisson solver39 was applied to treat periodic boundary conditions in the anatase [101] plane with no periodicity assumed normal to the surface. The success of our FMHET theory requires an accurate determination of the collective vibrational mode that drives electron transfer between two diabatic states. For our MeAz/TiO 2 adduct, a pair of diabatic states were constructed using constrained density functional theory (CDFT)40, and were quantified through the net density derived atomic point charges41 of the dye molecule,

11

, defined as +1 before and 0 after charge


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recombination (Fig. 2). Since the structural reorganization associated with the S 1 excited state relaxation is usually much smaller than that originating from interfacial charge recombination, the vibrational mode that drives S 1 excited state electron injection can be assumed to propel S 0 ground state charge recombination as well. As expected, this approximation is only valid when the excited state thermal relaxation energy is much less than the energy released during ground state charge recombination, as will be justified in our later discussion. Under this approximation, a total of 1,000 atomistic snapshots were randomly selected from a 50-ps ab initio molecular dynamics (AIMD) trajectory at 300K before their diabatic energy gaps were evaluated by CDFT.40 Functional mode analysis was then applied to ascertain the

that maximizes the correlation between electron

transfer and molecular vibration. As shown in Fig. 3, the primary constituents of the are the atomic displacements of the MeAz dye, however, the contribution from the dyebinding domain of the anatase [101] surface is also considerable. Collectively, the tends to translocate the MeAz dye’s surface-binding carboxyl group with respect to its 6methyl-azulene ring. Interestingly, out of a large vibrational mode ensemble, only five modes have

and hence are essential to the electron transfer. Moreover, all of

them are notably faster than the thermal fluctuations at room temperature (i.e., ). This underscores the significance of vibrational tunneling effects, which are also signified by noting that statistical quality of

. As a prudent measure to ensure the

, its coefficients of

were projected onto another 1,000

snapshots that were extracted from a separate 50-ps AIMD trajectory. As shown in Figure 4, the cross-validated Pearson’s coefficient,

12

, is 0.77, which is only slightly smaller



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than the training coefficient

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of 0.81. The small deviation of

from

clearly

indicates that our sampling is sufficient for functional mode analysis. Calculations for the thermal relaxation rate of

were initiated from an

optimized geometry of the MeAz/TiO 2 adduct and a library of displaced nuclear configurations was generated by imposing linear and nonlinear atomic displacements onto the optimized geometry according to the frozen-phonon approach.14 Then, the energy perturbations arising from atomic displacements were evaluated to provide input into ALAMODE,15 a phonon simulation package that uses a linear least-square fitting algorithm to iteratively determine harmonic and anharmonic force constants by fully accounting for system symmetry and particle permutation. In a subsequent step, Eq.4 was employed to calculate the thermal relaxation rate for each constituent mode of

. As

shown in Fig. 5, the profile of the coefficient–weighted thermal relaxation rate, , is featured by two groups of peaks, one centered at ~150 cm-1 and the other centered at ~1400 cm-1. Since the higher-energy group lies beyond the upper boundary of the phonon DOS of a pristine anatase crystal,42 it can be ascribed to the selfdissipation of the MeAz dye molecule. By contrast, the lower-energy group is associated with phonon-phonon scattering within the dye-binding domain of the anatase slab. It is 0.85 ps-1, which is well in

, of

turns out the overall thermal relaxation rate, line with the experimental vibrational lifetime of

ps in the S 1 excited state of an

azulene molecule solvated in benzene.43 As shown in Fig. 2, the

vertical and adiabatic excitation energies

calculated by time-dependent density functional theory44 (TDDFT) are given by

13


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and

, respectively. The associated thermal relaxation energy,

, of 0.30 eV is thus nontrivial for photovoltaic applications. With regard to

, a vertical

excitation will most likely populate the third

vibrationally excited states of S 1 (i.e.,

) if the geometry of the MeAz dye is

optimized at S 0 . Moreover, a dye molecule’s optical absorption spectrum is usually broadened due to vibronic coupling and orbital hybridization, enabling the selective population of the initial vibrational states of S 1 by varying the incident photon energy, : (17) This equation also implies that the excessive vibrational energy of S 1 is distributed to all constituent modes of

according to their relative importance to electron transfer. If

the dye molecule is under monochromatic light irradiation,

reflects the extent of

vibrational heating upon vertical excitation. According to the nonadiabatic transition theory,45 an appropriate alignment between the dye’s S 1 state and the semiconductor’s surface conduction band (SCB) is essential for interfacial electron injection. In the anatase [101] slab, only the atoms of its outermost thinnest repeating layer are significant constituents of

(Fig. 2). As a consequence, we assume that the active electron-

accepting surface layer is made solely of those atoms. Then, the corresponding surface DOS,

, can be calculated by projecting the total DOS onto the active surface layer.

As shown in Fig. 6, the calculated

is rather flat when

vibrational energy of S 1 whereas it persistently increases when

14


is below the ground changes from 0 to 4.


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The step-function like profile of

with onset at

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suggests that more electron-

accepting surface states become available when more excess energy is deposited into the photo-excited dye’s molecular vibrations. Charge recombination competes with the current-carrying diffusion of injected electrons into the bulk phase of the semiconductor. The energy change associated with the charge recombination,

, in the MeAz/TiO 2 adduct is given by: (18)

where

indicates the site of the electron being transferred and

geometry of the adduct for a given value of

. It was found that

is the optimized is 1.75 eV, which

is substantially greater than the dye’s S 1 thermal relaxation energy of 0.30 eV. The distinct difference between these quantities is sufficient to justify our approximation of a single reaction coordinate for both ground state charge recombination and excited state electron injection. Aiming to further support this supposition, we also evaluated the ratio and

between

, which are the changes in electron injection energy and charge

recombination energy, respectively, when the system’s geometry is perturbed along . Under an optimal scenario, this ratio is 1 and change along

, which by definition also maximizes

adduct, the calculated ratio of

along

by

should experience the greatest . For our MeAz/TiO 2

is 1.2 when its optimized structure is perturbed

. Therefore, our approximation is a physically sensible

choice in light of the burdensome computational cost of TDDFT using our chosen hybrid

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exchange-correlation functional. Under the same approximation, the reorganization energy of both processes,

, calculated by (19)

turns out to be 0.31 eV, placing the charge recombination reaction into the Marcus ) for a potentially long-lived surface-trapped polaron.46

inverted region (i.e.,

Due to the large number of Bloch surface states in our system, it is computationally cumbersome to evaluate the electronic coupling strength,

, between the

dye’s S 1 state and each individual Bloch state. Nevertheless, beyond the wide-band approximation limit47 that assumes a constant DOS near the Fermi surface,

can be

more accurately determined from the width of the broadened S 1 state due to the molecule-semiconductor coupling. As shown by a typical projection of the unoccupied DOS onto the MeAz dye (Fig. 7), the line shape is described very well by a Lorentzian function that is centered at

with a full width at half maximum of

. Interestingly, an

analysis of 100 atomistic configurations reveals that the state-averaged width defined as is nearly invariant at 0.0023 eV upon the shift of

Condon approximation of a constant

, justifying the

value of 0.0023 eV.

Now with all needed parameters in hand, we can apply Eq. 16 to calculate the electron injection rate,

, as a function of the incident phonon energy,

thermalized and thermalized conditions. As shown in Fig. 8, 1

at

to 0.45 ps-1 at

increases from 0.079 psof 0.85 ps-1. By

with a thermal relaxation rate

contrast, if instantaneous thermalization is assumed (i.e.,

16


, under non-

),

is a constant of


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0.079 ps-1 and thus irrelevant to the photon-induced vibrational heating. This indicates that the partial thermal insulation of the MeAz dye can drastically improve the interfacial electron injection efficiency. Very encouragingly, the sharp upward turn in occurs when

that

increases from 2 to 3 is in excellent agreement with the experimental

dependence of electron injection quantum yield on the incident photon energy.16 Moreover, the average energy of injected electrons in the conduction band, given

, for a

, can be defined as:

(20)

As shown in Fig. 9,

increases almost linearly with

under non-thermalized

conditions, suggesting that a much more efficient harvesting of solar energy can be achieved when thermal relaxation is outpaced by electron injection at excited states.

4. Discussion and Conclusions: The concept of hot-carrier solar cell48 belongs to the third-generation photovoltaic technologies49, which seek to overcome the Shockley-Queisser thermodynamic limit also through multiple-exciton generation,50 singlet fission,51 and multiple-junction tandem structures.52 In a hot-carrier solar cell, the cooling of excited charge carriers must be impeded to ensure an efficient extraction of their energies in excess of the band gap. Consequently, a key ingredient in the conceptual hot-carrier dye-sensitized solar cells is the ultrafast injection of the vibrationally excited electrons from a dye molecule to its semiconducting substrate. In a recent femotosecond pump-probe spectroscopy study under ambient conditions,16 the electron injection rate from a high-lying vibrational state

17


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of an excited dye molecule was found to be 9-fold faster than that from a low-lying one, opening new avenues for mitigating wasteful heat dissipation and enhancing photocurrent generation in cost-effective organic photovoltaic devices. Despite encouraging achievements on the experimental side, computational modeling of the hot electron injection remains a challenge task due to the delicate interplay between molecular vibration, thermal relaxation and electron transfer. To tackle this challenge from a theoretical perspective, we have developed a non-equilibrium electron transfer theory that accounts for the time-dependent thermal relaxation of the driving vibration mode upon optical excitation. As shown by our numerical simulation of the MeAz/TiO 2 adduct, the photo-induced electron injection rate can be readily modulated by selectively populating the initial vibrational states within the excited dye molecule, thus achieving photophysical properties that are in excellent agreement with experimental data.16 Of broader significance, our novel functional mode hot electron transfer theory can be easily extended to allow for arbitrary time-dependent vibrational perturbations, enabling the simulation of vibrationally controlled electron transfer reaction using multiple ultraviolet and infrared laser pulses.19

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Acknowledgements The research was supported by a start-up grant and the Columbian College Facilitating Fund of the George Washington University. Computational resources utilized in this research were provided by the Argonne Leadership Computing Facility (ALCF) at Argonne National Laboratory under Department of Energy contract DE-AC0206CH11357 and by the Extreme Science and Engineering Discovery Environment (XSEDE) at Texas Advanced Computing Center under National Science Foundation contract TG-CHE130008.

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Figures:

Figure 1. Molecular structure of 6-methyl-azulene-2-carboxylic acid bound to an anatase [101] surface. The titanium, oxygen, carbon and hydrogen atoms are colored pink, red, cyan and white, respectively. All atoms below the blue dashed line are frozen to mimic the bulk phase crystal structure, while those above the green dashed line are considered to form an active electron-accepting surface layer.

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Figure 2. Energy diagram of photo-induced hot electron injection from 6-methyl-azulene2-carboxylic acid to the anatase TiO 2 [101] surface. The vertical and adiabatic excitation energies are denoted as and , respectively, while their difference is the excited-state thermal relaxation energy of . The thermal relaxation rate of the electron-transfer driving mode, , is given by , and is the electron injection rate. is the energy associated with the charge recombination, which corresponds to a , from +1 to 0. change of the dye’s net charge,

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Figure 3. Profile of MCM coefficients, , for the vibrational modes of the MeAz/TiO 2 adduct. The five essential driving modes with are colored red, while the others are colored black. The is also depicted in the inset.

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Figure 4. Scatter plot for cross-validation of MCM coefficients, energy gap, .

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, on the diabatic

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Figure 5. Profile of the MCM coefficient-weighted thermal relaxation rate, vibrational modes of the MeAz/TiO 2 adduct.

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, for the


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Figure 6. Density of states within the active electron-accepting surface layer, where is the ground state energy of the MeAz dye and vibrational displacement upon vertical excitation.

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is the quanta of the dye’s

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Figure 7. Unoccupied density of states projected onto the MeAz dye. The curve is fitted to a Lorentzian function in the form of

.

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Figure 8. Electron injection rate, under non-thermalized (

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, as a function of the incident phonon energy, ) and thermalized (

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) conditions.

,

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Figure 9. Dependence of the average energy for electrons injected into the conduction , on the incident photon energy, , under the non-thermalized conditions. band,

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TOC

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Functional Mode Hot Electron Transfer Theory - The Journal of

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