68
J. Phys. Chem. B 2000, 104, 68-77
Pump-Probe Spectroscopy of Ultrafast Electron Injection from the Excited State of an Anchored Chromophore to a Semiconductor Surface in UHV: A Theoretical Model S. Ramakrishna† Fritz-Haber-Institut der MPG, Faradayweg 4-6, 14195 Berlin, Germany
F. Willig* Hahn-Meitner-Institut, Glienickerstrasse 100, 14109 Berlin, Germany ReceiVed: April 29, 1999
Decay of excited state absorption, owing to charge injection into the conduction band continuum of semiconductor states, is obtained via a qualitative theoretical model. The density matrix formalism is utilized to obtain an expression for the sequential pump-probe signal in terms of the nonlinear third-order polarization of the molecular system. Electron transfer slower than the pump and probe pulses is assumed and reorganization of the molecule upon charge injection is ignored while obtaining the final expressions. The lifetime of the excited state decouples from nuclear motion as a consequence. Decay of the excited state into a continuum of electronic states is examined for various energy positions of the injecting state and for different bandwidths of the continuum. The decay can be fitted by exponential functions for the majority of the cases considering different dimensionalities of the semiconductor continuum. Model calculations are performed for the snapshot limit of the pump-probe signal. Under the assumed conditions one obtains oscillations due to vibronic coherences that are superimposed on temperature-independent irreversible charge transfer decays, as is reported in recent experiments.
1. Introduction Transfer of an electron from an electronic excited localized state to the continuum of electronic states in the conduction band of a semiconductor is the simplest type of interfacial electron transfer reaction involving a large molecular reactant.1 It is a prototype reaction for certain electrochemical and photoelectrochemical interfacial processes. Recently, it has been utilized as the charge-separating step in a new type of electrochemical solar cell.2 The process of light-induced electron injection is essentially as follows: A molecular donor anchored to the semiconductor (typically TiO2) surface has its electronic ground state located energetically in the band gap of the semiconductor. Photoexcitation of the molecule promotes an electron to an excited molecular state that is resonant with the empty conduction band levels; thereafter the electron is injected into the semiconductor on an ultrafast (femtosecond) time scale.1 One of the techniques employed, to directly determine the charge-injection process, has been pump-probe spectroscopy, wherein the pump prepares the excited state of the molecule and the probe which follows it at a variable time delay can measure for instance, the excited state absorption.3 The subject of this article is the theoretical modelling of such a pumpprobe signal which explicitly incorporates the charge-injection into the continuum of semiconductor levels that gives rise to the observed decay in the excited state absorption. The pumpprobe signal can be obtained via the time-dependent density † Present address: Hahn-Meitner-Institut, Glienickerstrasse 100, 14109 Berlin, Germany.
matrix and this is usually achieved by considering the interaction between the molecular system and the laser fields perturbatively to obtain a third-order optical polarization of the molecule.4 Such an expression contains all the microscopic information necessary for calculating the signal. In order to achieve this, one needs to specify a molecular model Hamiltonian appropriate to the given problem. In the present case, one needs the relevant molecular states with the excited state being coupled to the semiconductor continuum. Previous work on photoinduced charge transfer from a donor to a semiconductor substrate has been worked out for (i) much slower time scales, as an activated process in a solvent environment,5 or (ii) has ignored the optical preparation and the accompanying vibronic coherences that have to be taken into account for an ultrafast charge-injection process.6,7 For similar reasons, the present work is different from the theory developed for adiabatic electron transfer processes for semiconductor8,9 as well as for metal electrodes,10 although all orders of the electronic coupling that mediates charge transfer will be included here, as in an adiabatic electron transfer theory.8-10 Unlike in most of the above cases, in the present theory, charge transfer does not take place from a thermalized donor but rather from an initial coherent superposition of molecular vibronic states and, consequently, will be shown to take into account the presence of vibronic coherences. Moreover, the objective here is to model the charge transfer as a signature appearing in a pump-probe signal. The presence of the anchor group increases the chromophore-substrate distance and thus weakens the electronic
10.1021/jp991428r CCC: $19.00 © 2000 American Chemical Society Published on Web 12/09/1999
Pump-Probe Spectroscopy of Ultrafast Electron Injection coupling between the excited state and the substrate levels.3 This results in line widths for the excited state to be in the range of millielectronvolts,3 whereas in the case of chemisorption the line widths are usually in the range of 0.5 to several electronvolts.11 The work of Persson and Avouris12 deals with photoexcitation of electrons at metal surfaces and contains more ingredients in the molecular model Hamiltonian than what will be considered here. Taking into account electron correlations, it has been shown by these authors that photoinduced electron injection can in principle be accompanied by excitations of several electron-hole pairs in the substrate, but such processes contribute negligibly to the lifetime of the excited state when the magnitude of the electronic coupling is already down to a few tenths of an electronvolt.12 Moreover, as the excited state is far above the filled levels of the semiconductor, its lifetime is essentially determined by its electronic coupling to the substrate and not by intramolecular electron correlations or molecular-plasmon (substrate) couplings.12 In recent times, a number of articles on the theoretical description of the complex dynamics in ultrafast charge transfer arising as a result of vibronic and/or quantum coherences that decay due to vibrational relaxation have appeared.13,14 However, the problem of photoinduced charge transfer from a donor to an electronic continuum of a surface at ultrafast time scales in the context of vibrational coherence and relaxation remains to be addressed. The need for developing such a theory arises in the context of recent experiments establishing certain features of this reaction some of which were not known earlier, such as (1) the ultrafast nature of the charge-injection process from chromophore to the semiconductor surface in the time scales of femtoseconds;15-18 (2) the temperature independence of the injection process from 22 K to room temperature;15 spacer control of charge-injection time by varying the distance between the chromophore and the surface;3 and the presence of vibronic coherences in the observed signal.19 Moreover, Willig and co-workers have performed experiments on the excited state decay for the perylene-TiO2 system and have verified that the decay is indeed due to charge transfer (and not energy transfer) by measuring the rise of the signal due to the formation of a molecular cation arising with an identical time behavior as the decay of the excited state.15 Subsequently, the optical absorption of the injected electrons in the semiconductor has been probed and found to show a similar rise time.17,18 Thus, it is meaningful and timely to formulate a theory for the excited state decay due to transfer of charge to a surface, as observed in a pump-probe signal. The UHV environment simplifies the requirements for a theoretical model since the influence of a solvent bath can be ignored. The intramolecular modes can be assumed to oscillate undamped during the electron-transfer process (30-100 fs) and consequently relaxation of vibronic coherences can be ignored.19 In large molecules like perylene, the presence of several vibrational modes eventually leads to the redistribution of the initially prepared vibrational excitation to other vibrational modes but this happens at much slower time scales of subpicoseconds and picoseconds.20 The next section starts with the basic definitions and essentials required to obtain a general form for the differential probe absorption signal. It is ended by formulating a specific qualitative model for the molecular Hamiltonian to describe the chargeinjection process. Following this, one obtains a general expression for a sequential pump-probe signal valid for a two-color, three-level system under conditions of pulses much shorter than charge-injection times and negligible reorganization energies.
J. Phys. Chem. B, Vol. 104, No. 1, 2000 69 The expression thus obtained shows decoupling of the lifetime from molecular vibrations. Thus, the next section is an exploration of the lifetime of the excited state owing to its coupling to an electronic continuum. The energy of the injecting excited state as well as the bandwidth of the substrate is varied to ascertain if the numerical decays can be fitted with exponential fits. Since the problem is sensitive to the dimensionality of the substrate levels, all three dimensions are studied. Finally, model calculations are performed for the snapshot limit of the pumpprobe signal which displays vibronic coherences arising in the timescale of charge injection. 2. Differential Probe Absorption of the Excited State 2.1. General Expression for the Signal. The complete Hamiltonian of a molecular system, interacting with an external electric field E(t), can be simplified in the dipole approximation as21
H ) Hs - VE(t)
(1)
where Hs represents the molecular system coupled to the semiconductor states and V is the molecular dipole operator. In order to keep the notation simple it has been assumed that the molecule is located at the origin of the spatial coordinate.21 The external electric field consists of the pump pulse centered around the frequency ω1, and the probe pulse around ω2, and whose temporal profiles are E1(t) and E2(t), respectively. The probe difference absorption S, the relevant quantity which needs to be calculated for pump-probe measurements, defined as the total probe absorption in the presence of the pump minus the probe absorption in the absence of the pump, can be expressed as21,22
S ) 2ω2 Im
∫-∞∞ E/2(t) P(t) dt
(2)
where P(t) is the polarization induced in the molecular system due to the pump and probe pulses. Now
P(t) ) Tr{V F(t)}
(3)
and the time-dependent density matrix F(t) obeys the Liouville equation
i i dF ) - [Hs, F] - [Hint, F] ≡ (Ls + Lint) F dt p p
(4)
where Hint is the interaction between the molecule and the laser fields of the pump and probe pulses.
Hint ) - VE(t)
(5)
The Liouville operators Ls and Lint defined in the above equation are the commutators with Hs and Hint, respectively. The timedependent density matrix and thereby the polarization P(t) is obtained by solving the Liouville equation perturbatively in Lint. It has been shown that pump-probe spectroscopy is obtained from a third-order expansion in powers of E. A third-order representation of the polarization P(t) is obtained from such an expansion of the density matrix using the Liouville equation and may be expressed as21,22
P(3) (t) )
∫0∞ dt3 ∫0∞ dt2 ∫0∞ dt1 R(3)(t3,t2,t1) E(t - t3)
E(t - t3 - t2) E(t - t3 - t2 - t1) (6)
70 J. Phys. Chem. B, Vol. 104, No. 1, 2000
Ramakrishna and Willig
and R(3) is the nonlinear response function of the third order.
R(3) ) (i/p)3 Tr{VG(t3) VG(t2) VG(t1) VF(-∞)}
(7)
The Liouville space dipole operator is defined by its action on an arbitrary operator A, VA ≡ [V, A]. Also G(t), the time evolution operator or the Green function of the Liouville equation is defined by its action on an operator A, in the absence of the perturbing external field
G(t)A ≡ exp
(pi L t)A ) exp(- pi H t) A exp(pi H t) s
s
s
(8)
The nonlinear response function R(3) is the essential quantity which needs to be obtained to arrive at the pump-probe signal S. The three time arguments t1, t2, and t3 found in it represent the time intervals between successive interactions with the electric field as follows:22 initially the system is in thermal equilibrium given by the density matrix F(-∞) upon which the first V acts. Thereafter, the system freely propagates for a period t1 after which there is a second interaction with V, followed by another free propagation during the period t2. This is followed by yet another interaction with the electric field and again another free propagation during the period t3. Finally, the dipole operator V acts from the left and one performs the trace. R(3) contains all the microscopic information concerning the molecular system and its interactions with other subsystems. Therefore, to proceed further one needs to explicitly specify the Hamiltonian Hs which characterizes the molecular system. Finally, by employing P(3) in the expression for the signal S, one obtains a general expression for the probe absorption as a third-order perturbative expansion in the external field. 2.2. A Model Molecular Hamiltonian. The model for a molecular Hamiltonian will be constructed from a diabatic basis set. Members of this electronic set considered relevant for the pump-probe signal consists of (i) three bound molecular electronic levels belonging to the neutral molecular species, namely the ground state |φ1〉, the first excited state |φ2〉, and a higher excited state |φ3〉 with corresponding energy values 1, 2, and 3, respectively, and (ii) a continuum of electronic states |φk〉 labeled k, representing the conduction band levels of the semiconductor wherein the electron is injected. The three bound electronic states are assumed to be linearly coupled to a set of molecular vibrational modes of frequencies ων in the harmonic approximation via the dimensionless displacement quantity gνi. Furthermore, the excited electronic state |φ2〉 is coupled to the set of conduction band levels of the semiconductor by the various hopping matrix elements denoted as V2k. An electron transfer from the excited state |φ2〉 to the semiconductor results in an ionized molecule. The ion core of the molecule shifts to a new equilibrium position when the electron is in the continuum of the conduction band levels. This can be characterized by coupling each of the electronic states |φk〉 to the molecular vibrational modes by the same dimensionless displacement factor i.e., gνk ≡ gK.23 It can be noted that the present characterization of the molecular system is similar to autoionization models.23 The conduction band continuum of levels is virtually empty as the Fermi level, which identifies the acceptor levels as belonging to a semiconductor, plays little role as it is situated in the band gap. Such a qualitative model Hamiltonian can be denoted as
3
Hs )
i|φi〉〈φi| + ∑ pωνb†νbν + ∑ pωνgνi (bν + ∑ i)1 ν ν,i † bν)|φi〉〈φi| + ∑ k|φk〉〈φk| + ∑ V2k|φ2〉〈φk| + k k / pων (bν + b†ν)|φk〉〈φk| ∑k V2k|φk〉〈φ2| + gK ∑ ν,k
(9)
The interaction Hamiltonian can be simplified by incorporating both the rotating wave approximation (whereby highly oscillatory off-resonant terms are omitted) as well as by assuming optical selectivity of the pump and probe frequencies. The latter condition implies that the pump frequency is tuned such that it is near resonant only to the ground to first excited state transition, and that the probe frequency, being very different from the pump frequency, is similarly tuned to a transition from the first excited to a higher excited state. This leads to
Hint ) -V12|φ1〉〈φ2|E1(t + τ) eiω1t V23|φ2〉〈φ3|E2(t) eiω2t + h.c. (10) where V12 and V23 are the molecular dipole operators in the Condon approximation. The peaks of the pump and the probe pulses are separated by a time interval τ. 3. Sequential Pump-Probe Signal for a Three-Level System A formal expression for the difference absorption of the probe signal relating to a particular molecular model Hamiltonian (specified in the previous section) can now be given. However, such an expression would in general contain terms pertaining to all possible time orderings of the excitation pulses interacting with the system. A sequential process is one in which the molecular system initially interacts only with the pump pulse and after a well-separated delay subsequently interacts with the probe pulse. The other combinations in which the pulses overlap in time and lead to “coherent artifacts” will not be considered. After making changes in the integration variables, namely, (i) setting t to t + τ, (ii) changing t2 to t′ ) t + τ - t2, and (iii) extending the upper limit of of the t′ integration from τ + t to ∞ since the integrand E1(t′) vanishes for t′ g τ, one obtains the expression for the sequential pump-probe signal as21,22
Sseq )
2ω2 3
Re
∫-∞∞ dt ∫0∞ dt3 ∫-∞∞ dt′ ∫0∞ dt1 [exp(-iω2t3 -
p iω1t1) E/2 (t + t3) E2(t)E/1(t′) E1(t′ - t1) R1(t3, t1, t, t′, τ)] + [exp(-iω2t3 + iω1t1) E/2(t + t3) E2(t) E1(t′)
E/1(t′ - t1) R2(t3, t1, t, t′, τ)] (11) where R1 and R2 are terms representing Liouville space pathways given as22
R1 ) Tr{V32 G(t3) V23 G(t) G(τ) G†(t′) V21 G(t1) V12 F(-∞)} (12) R2 ) Tr{V32 G(t3) V23 G(t) G(τ) G†(t′) V12 G(t1) V21 F(-∞)} (13) 3.1. A Unitary Transformation. Evaluation of the trace in the above equations would lead to the necessary expression for the sequential signal and to accomplish this one can perform a
Pump-Probe Spectroscopy of Ultrafast Electron Injection unitary transformation of the two Liouville space pathways (the trace is invariant to a unitary transformation). The unitary transformation helps to remove the linear coupling terms between the electronic eigenstates and the nuclear displacement operators of the molecular vibrational modes. However, because of the transformation, all electronic transition operators are now accompanied by exponentiated momentum operators of the vibrational modes with appropriate dimensionless displacement cofactors gνi. The unitary transformation of the trace is, in effect, essentially a transformation of the total Hamiltonian. Such a transformed Hamiltonian can be given as
HTT ) S-1(Hs + H ˜ int)S ≡ H ˜ 0 + HI
(14)
where H ˜ int is Hint but without the pulse envelope and frequency terms. The unitary operator S is defined as
S ) exp[-
gνa(b†ν - bν)|φa〉〈φa|] ∑ aν
J. Phys. Chem. B, Vol. 104, No. 1, 2000 71
∑ν gν23(b†ν - bν)] exp[-(i/p)Hbt3] × exp[-(i/p)Hb(t + τ - t′)] - exp[∑gν12(b†ν - bν)] × ν † exp[-(i/p)Hbt1] exp[∑ gν12(bν - bν)] exp[(i/p)Hbt1] × ν exp[(i/p)Hb(t + τ - t′)] exp[∑ gν23(b†ν - bν)] × ν
R1 ≈ Tr{exp[-
[exp(i/p)Hbt3]} exp[(i/p)(˜ 3 - ˜ 2)t3] × exp[(i/p)(˜ 2 - ˜ 1)t1]| 〈2|exp[-(i/p)H0τ]|2〉|2 (18)
{
∑ν gν23(b†ν - bν)]exp[- i/pHbt3] ×
R2 ≈ Tr exp[-
[ () ]
exp[- i/pHb(t + τ - t′)] exp -
i
Hbt1 exp[-
p
(15)
where a ) i, k. The transformed molecular and the interaction Hamiltonians can be respectively given as
bν)] exp[(i/p)Hbt1] exp[ τ - t′)] exp[
∑ν gν12(b†ν -
∑ν gν12(b†ν - bν)] exp[(i/p)Hb(t +
}
∑ν gν23(b†ν - bν)][exp(i/p)Hbt3]
exp[i/p(˜ 3 -
˜ 2)t3] exp[i/p(˜ 1 - ˜ 2)t1]| 〈2|exp[-(i/p)H0τ]|2〉|2 (19)
3
H ˜0 )
˜ i|φi〉〈φi| + ∑ pωνb†νbν + ∑ ˜ k|φk〉〈φk| + ∑ i)1 ν k (∑ V2k exp[∑ gν2K (b†ν - bν)] |φ2〉〈φk| + h.c.) k ν
where Hb ) ∑ν pωνb†νbν and
(16)
and
HI ) -V12|φ1〉〈φ2| exp[
∑ν gν12(b†ν - bν)] - V23|φ2〉〈φ3| × exp[∑ gν23(b†ν - bν) ] + h.c. (17) ν
It should be noted that ˜ a ) a - ∑ν pωνgνa2 and that gνab ) gνa - gνb. The above transformed Hamiltonians would enable one to evaluate the boson (vibrational) operator averages easily. 3.2. Evaluation of the Liouville Space Pathway Expression. Before we start the evaluation of R1 and R2, two physical approximations will be considered. The first is to assume that the electron transfer process or the formation of the ionic state is a much slower process compared to the time scales of interaction of the molecular system both with the pump as well as with the probe pulses. In other words, electron transfer will be considered only during the longest time scale in the problem i.e., during the time interval between the pulses, namely τ, and the electronic hopping terms associated with the V2k will thus be ignored in the Hamiltonian for all other time variables excepting τ. Thus, the electron transfer is akin to a small parameter in a perturbative expansion and excepting for the time interval between the pulses, only the zeroth-order terms in the Hamiltonian will be considered. Second, it shall be assumed that the equilibrium position of nuclear modes changes very little upon ionization of the molecule. This enables one to set V2k exp[∑ν gν2K(b†ν - bν)] ≈ V2K since gν2K , 1. This amounts to the assumption that the reorganization energy for the electron transfer process is negligible and the consequences of this assumption will be dealt with later in the discussion section. Incorporating the above approximations, R1 and R2 can be expressed as
H0 ) ˜ i|φi〉〈φi| +
∑k ˜ k|φk〉〈φk| + (∑k V2k|φ2〉〈φk| + h.c.)
(20)
Thus in the above expressions for R1 and R2, the fermion (electronic) and boson (vibrational) averages have been factored out. The last term, which is the modulus square of the timedependent overlap of the excited state with itself, is nothing but the excited state lifetime. The Hamiltonian H0 describes a single state interacting with a continuum of levels, well-known as the Fano-Anderson Hamiltonian.24 The excited state has a finite lifetime as a result of its being coupled to a continuum of states. The time-dependent probability of occupany Pe(t) of the excited state can be recast in terms of a spectral function A(ω ˜) as follows:
|〈 | ( ( ) )| 〉| | ∫
Pe(τ) ) 2 exp -
i Hτ 2 p 0
-iθ(τ)
2
∞
-∞
≡
( ( ) )|
dω ˜ i A(ω ˜ )exp - ω ˜τ 2πp p
2
(21)
and the spectral function can be expressed in terms of the selfenergy ∑(ω ˜ ) as
A(ω ˜))
-2 Im Σ(ω ˜) [ω ˜ - ˜ 2 - Re Σ(ω ˜ )]2 + [Im Σ(ω ˜ )]2
(22)
where
Σ(ω ˜))
|V2k|2
∑k ω˜ - ˜
(23) k
At this stage, evaluating the boson averages, one arrives at an expression for the sequential pump-probe signal for a threelevel system with the excited state being coupled to the continuum of levels in the semiconductor.
72 J. Phys. Chem. B, Vol. 104, No. 1, 2000
2ω2 ∞ ∞ ∞ Re -∞dt 0 dt3 -∞dt′ p3 ˜ 2)/p]t3}E/2(t + t3)E2(t) exp[-
∫ ∫ ∫
Sseq )
Ramakrishna and Willig
∫0∞dt1 exp{-i[ω2 - (˜ 3 -
∑ν (g2ν12 + g2ν23)(1 + 2njν)]
∑ν g2ν23C*(t3)]{exp[∑ν g2ν12C*(t1)]exp{-i[ω1 - (˜ 2 ˜ 1)/p]t1}E/1(t′)E1(t′ - t1) exp[-∑gν12gν23{C(t1 + t + τ ν
exp[
t′) + C*(t3 + t + τ - t′) - C*(t1 + t3 + t + τ - t′) C(t + τ - t′)}] + exp{i[ω1 - (˜ 2 - ˜ 1)/p]t1}E1(t′)E/1(t′ t1) exp[-
∑ν gν12gν23{-C(t1 + t + τ - t′) - C*(t3 + t +
τ - t′) + C*(t1+ t3 + t + τ - t′) + C(t + τ ˜ ∞ dω t′)}] exp[ g2ν12C(t1)]}|-iθ(τ) -∞ A(ω ˜ )e-(i/p)ω˜ τ|2 (24) 2πp ν
∑
∫
where
C(m) ) (1 + 2njν) cos ωνm + i sin ωνm (1 + 2njν) ) coth
( ) pων 2κT
(25) (26)
In the above, κ and T are the Boltzmann constant and the temperature, respectively. The sequential pump-probe signal is essentially a function of τ, the time delay between the pump and the probe pulses, since all the other time variables are integrated over. It is also a function of the pump and the probe frequencies, namely ω1 and ω2, since the two laser pulses in principle can be a few tens of femtoseconds duration and hence usually have a frequency spread equivalent to over several levels of vibronic energy spacings. The general expression for the sequential pump-probe also reveals a clear factorization between the electron-transfer component, which is the excited state lifetime P2, and the pump-probe spectroscopic components. The neglect of the electron-transfer reorganization energy, λ ) ∑νpων(gν2 - gνK)2, has resulted in the electron transfer being a purely electornic process. The oscillatory components of the signal which arise from nuclear dynamics have no contribution from the FranckCondon overlap factors of electron transfer (curve crossing). Such oscillations, which arise from nuclear motion, are due to the creation of wavepackets which are nothing other than coherent superposition of vibronic levels, created from suitable pulses of laser excitation. 4. Decay of the Excited State into a Continuum of Levels A description of the solid, to which there is an electron transfer from the first excited state, is now necessary, and also the exact nature of the electronic overlap factors between the solid surface and the molecule. This will enable one to obtain the precise functional forms for the self-energy Σ(ω ˜ ) and thereby the spectral function A(ω ˜ ). In the case of the perylene-TiO2 system, recent experimental evidence suggests the excited state is about 1 eV above the conduction band edge.25 The electronic states of the solid which are most likely to interact with the molecular level are those of the bulk three-dimensional conduction band. The anchor group determines the orientation of the adsorbate at the surface and also the chromophore-substrate distance. The empty electronic levels of the anchor group lie much higher than the π -orbital of the perylene donor chro-
mophore. The presence of the anchor group decreases the strength of the electronic interaction of the donor molecule with the conduction band levels. A good description of the electronic overlap between the relevant conduction band states and the molecular electronic states requires a microscopic modeling of the interface, obtaining the precise eigenstates of the two subsystems, their orientational geometry, and finally even the potential energy surface for the entire interface.8,9 To obtain precise results on the detailed microscopic aspects of the electronic coupling implies solving a complex problem in electronic structure.26 Calculations of the structure and energetics of water adsorption on TiO2 surfaces have recently appeared.27 Corresponding calculations for the binding of the dye are currently not available. For the present problem a calculation of the electronic interaction between the excited dye orbital with the conduction band levels of the TiO2 semiconductor would be required. However, since the present work focuses on dynamics, it would be sufficient and relevant to initially adopt standard and idealized descriptions of the interface and explore the qualitative dynamics it entails. A nearest-neighbor tight-binding Hamiltonian would be the simplest description for a solid. The eigenstates of such a Hamiltionian can be denoted as k. These had been designated earlier as the continuum electronic states of the model molecular Hamiltonian and subsequently due to the canonical (unitary) transformation shifted in energy and are presently denoted as ˜ k. Recalling the expression for the self-energy
Σ(ω ˜))
|V2k|2
∑k ω˜ - ˜
k
it may be noted that one has to model the the electronic coupling |V2k|2. As a first approximation one usually assumes that the coupling be a constant quantity, independent of k.28 In other words
Σ(ω ˜ ) ≈ Cd
1
∑k ω˜ - ˜
(27) k
where Cd is a constant and the subscript denotes the dimensionality of the solid. The above approximation for the selfenergy makes it identical to the Green function of a tight-binding solid except for the constant factor. The Green function that is usually employed in such cases is that of the bulk,26 also known as the Hubbard Green function. This is actually the local density of states on the first atom of a semiinfinite chain in a tightbinding model.26 For illustrative purposes one- and twodimensional Green functions will also be considered in the calculations. Making use of the functional forms for the real and imaginary parts of the Green function available in literature,29 one obtains for a continuum of k states
1. a 1D solid Im Σ(ω ˜)) 2. a 2D solid
C1
; Re Σ(ω ˜))0 (w - ω ˜ 2)1/2 2
(x(
))
(28)
2 C2 C2 ω ˜ +w ; Re Σ(ω ˜)) ln (29) 2w 2w ω ˜ -w 3. a 3D solid (Hubbard Green function) 2C3ω ˜ 2C3 ˜ 2)1/2; Re Σ(ω ˜)) (30) Im Σ(ω ˜ ) ) 2 (w2 - ω 2 w w
Im Σ(ω ˜))
Pump-Probe Spectroscopy of Ultrafast Electron Injection
J. Phys. Chem. B, Vol. 104, No. 1, 2000 73 pertaining to different dimensionalities of the solid substrate. For a given dimension and bandwidth of the solid substrate, the energy position of the excited state has been varied in Figures 1 and 2, starting from the middle of the band and then proceeding toward the lower edge. For the case of 1D and 3D the decay rate varies with position whereas for the 2D case it is independent of position. Also, the decay rate increases as the position of the excited state is moved closer towards the band edge for the 1D case, whereas the opposite trend is observed for the 3D case. However, what is interesting in Figures 1 and 2 is that the various decays for the dimensionality of the three cases can be fitted quite accurately by exponential fits given as follows:
(
exp(-τ/τ1) ) exp -
2C1
(
exp(-τ/τ2) ) exp -
(
exp(-τ/τ3) ) exp -
Figure 1. Probability of excited state occupancy versus time in femtoseconds for a bandwidth (2w) of 2 eV: (a) 1-D substrate, (b) 2-D substrate, and (c) 3-D substrate. The energy of the excited state is varied from the middle of the band toward the lower edge of the band for values m ) 0, 0.2, 0.4, 0.6, and 0.8 eV. In (a) the slowest decay curve (uppermost one) is for m ) 0 and for increasing values of m one obtains the respective curves of faster decay. In (c) the opposite trend is observed. The fastest decay curve (lowermost one) is obtained for m ) 0 and for increasing values of m one obtains the respective curves of slower decay. The solid lines are from the calculations and the dashed lines are the respective exponential fits.
where 2w is the bandwidth of the solid. It has to be mentioned that the above expressions are valid only for |ω ˜ | e w and that Im Σ(ω ˜ ) ) 0 for |ω ˜ | g w. As a result, the limits of integration for the excited state lifetime reads
|-iθ(τ)
dω ˜ A(ω ˜ ) exp(-(i/p)ω ˜ τ)|2 ∫-ww 2πp
(31)
As one of the main objectives of this work, the excited state lifetime is numerically calculated from the above expression (1) for various energy positions of the excited state in the conduction band, (2) for various widths of the conduction band of the semiconductor substrate, and (3) for various dimensionalities of the semiconductor substrate. Furthermore, it would be of utmost interest to fit the exact numerical results with pure exponential fits. Accordingly, we obtain in Figures 1-3 the decay of the excited state for three different bandwidths, namely, 2, 1, and 0.5 eV, respectively. Each figure has three parts
4C3 pw
2
C2 τ pw
)
τ
pxw2 - m2
)
(32)
(33)
)
xw2 - m2τ
(34)
m indicates the energy position of the excited state within the conduction bandwidth (for example, m ) 0 implies that the excited state is resonant with the mid-position of the bandwidth). The value of the coupling parameter Ci, where i ) 1, 2, has been set to 0.004, whereas C3 ) 0.002 in all calculations. The three lifetimes τ1, τ2, and τ3 which provide excellent fits have been defined such that the rate of decay of the excited state is proportional to the density of conduction band states at that energetic position. This is to be expected since from the value of the parameters namely, Ci , (2w)2 one is in the weakcoupling limit of chemisorption.26 However, it should be noted, from the curves, that exponential decays are not so accurate when the excited state is positioned close to the band edge or when the bandwidth becomes narrow as in Figure 3. The decay starts to deviate appreciably from exponential behaviour and shows mild oscillatory trends especially in the case of the 1D substrate. The decay of the excited state is seen from the calculations as an irreversible one, since it is well-known that the large continuum of final states available for the electron act as a kind of an “electronic bath”, making the electron transfer akin to a relaxation process.7 The rate of decay as seen from the exponential fits can be related to three parameters, namely, the coupling strength, the half-width of the band w, and the relative position of the excited state Vis-a-Vis the middle of the band. For a 2D substrate however, it is clear from the expressions (and the curves shown in Figures 1 and 2) that the decay is independent of position and only two parameters are needed to specify the decay rate. If the excited state is not situated close to the band edge, for a sufficiently large bandwidth (g2 eV) one can ignore the position of the excited state i.e., the parameter m. In such situations, the strength of the electronic coupling can be ascertained from the decay since the information regarding the bandwidth of the substrate is usually known. Excited state decay for three different dimensionalites have been analyzed. This is to clarify the role played by them and the distinct signatures which arise thereof that could be discerned from the decay. These trends, however, may not be sufficiently realistic since for actual systems the electronic density of states seen by the molecule at the surface will not be a smooth function (as in the above models) but rather display a lot of structure in
74 J. Phys. Chem. B, Vol. 104, No. 1, 2000
Figure 2. Probability of excited state occupancy versus time in femtoseconds for a bandwidth (2w) of 1 eV: (a) 1-D substrate, (b) 2-D substrate, and (c) 3-D substrate. The energy of the excited state is varied from the middle of the band toward the lower edge of the band for values m ) 0, 0.1, 0.2, 0.3, and 0.4 eV. In (a) the slowest decay curve (uppermost one) is for m ) 0 and for increasing values of m one obtains the respective curves of faster decay. In (c) the opposite trend is observed. The fastest decay curve (lowermost one) is obtained for m ) 0 and for increasing values of m one obtains the respective curves of slower decay. The solid lines are from the calculations and the dashed lines are the respective exponential fits.
general. Although these models delineate the essential physics behind the process of decay into a continuum, one has to exercise caution in seeking precise agreement with experimental data based merely on them. 5. Sequential Pump-Probe Signal in the Snapshot Limit The sequential pump-probe signal obtained in the earlier section has four time integrations to be performed. A very useful limiting case of the signal is the snapshot limit which is obtained by considering two simplifying approximations which require only two integrations to be considered.21,22 The first is the ultrafast dephasing limit, which amounts to pure dephasing time scales being much shorter than pulse durations. During the time periods t1 and t3 the system is in an optical coherence (|φ1〉〈φ2| and |φ2〉〈φ3|, respectively) whose typical time scales are determined by either pure dephasing processes or pulse dura-
Ramakrishna and Willig
Figure 3. Probability of excited state occupancy versus time in femtoseconds for a bandwidth (2w) of 0.5 eV: (a) 1-D substrate, (b) 2-D substrate, and (c) 3-D substrate. The energy of the excited state is resonant with the middle of the band (m ) 0). The solid lines are from the calculations and the dashed lines are the respective exponential fits.
tions. If the inverse absorption line width of the corresponding optical transition is shorter than the pulse duration then the ultrafast dephasing limit holds. As a result, one can neglect the variation of the external pulses on the t1 and t3 time scales, leading to
E/1(t′) E1(t′ - t1) ≈ |E1(t′)|2
(35)
E/2(t + t3) E2(t) ≈ |E2(t′)|2
(36)
Even though the pulse durations are longer than the pure dephasing time scale, it can still be much shorter than nuclear dynamics. Making this second approximation, one neglects nuclear motions during the time periods t′ and t. This amounts to setting the time arguments in the function C from (t + τ t′) to simply τ. This decouples the t′ and t integrations from the dynamics of the pump-probe signal and results in a snapshot spectrum which does not depend on pulse shapes but rather on their frequencies ω1 and ω2. Such a snapshot spectrum Sss can be expressed as
Pump-Probe Spectroscopy of Ultrafast Electron Injection
Sss ∝ Re
J. Phys. Chem. B, Vol. 104, No. 1, 2000 75
∫0∞ dt1 ∫0∞ dt3 exp{-i[ω2 - (˜ 3 - ˜ 2)/p]t3} exp[-(g212 + g223)(1 + 2nj0)] exp[g223C*(t3)]
{exp[g212C*(t1)] exp{-i[ω1 - (˜ 2 - ˜ 1)/p]t1} exp[-g12g23{C(t1 + τ) + C*(t3 + τ) - C*(t1 + t3 + τ) C(τ)}] + exp{i[ω1 - (˜ 2 - ˜ 1)/p]t1} exp[-g12g23{-C(t1 + τ) - C*(t3 + τ) + C*(t1 + t3 + τ) + C(τ)}] ∞ dω ˜ exp[g212C(t1)]}|-iθ(τ) -∞ A(ω ˜ )e-(i/p)ω˜ τ|2 (37) 2πp
∫
The snapshot spectrum depends only on the intrinsic properties of the molecular and substrate subsystems and is therefore of importance to understand the dynamics of electron transfer. A single vibrational mode of frequency ω0 in the zero temperature limit (pω . κT) has been considered in the above equation for the snapshot signal for the sake of simplicity in numerical calculations. In Figure 4 a snapshot signal is plotted in the absence of any coupling to the conduction band levels. The plot shows undamped vibrational coherences of frequency ω0. The oscillations arise from the formation of wavepackets which are in-built into the snapshot limit (the pulses are assumed to be much shorter than the time period of nuclear oscillations). The calculation also includes electronic dephasing factors of the order of 10 fs for numerical convergence. The inclusion of electronic coupling between the excited state and the conduction band continuum results in an oscillatory decay which is shown in Figure 5. The pump-probe signal is thus a superposition of an exponential decay component and an oscillatory component each arising from the electronic coupling to the semiconductor substrate and vibrational coherences, respectively. The different panels of Figure 5 indicate that for the observance of vibrational coherences, in a three-level system, it is important that the product g12g23 ≡ (g1 - g2) (g2 - g3) should be large, say g0.3, rather than one of them being large, whereas their product itself being quite small. This is a reflection of the well-known fact that for creation of wavepackets of significant amplitude one needs sufficient displacements between the concerned potential energy surfaces.13 Thus the above result serves to illustrate how, for instance, one would be able to observe vibronic coherences in a stimulated emission signal (a two-level system) whereas it may not be observable in an excited state absorption. This would imply that the displacement between ground and excited is sufficiently large whereas, the same between the excited and the higher excited state is insufficient for creating observable wavepackets. The snapshot limit may not always be realizable under experimental conditions but it is a useful limit since the essential dynamics is revealed in the signal without complications from convolutions involving external fields. Even if one of the above two conditions for the snapshot limit is satisfied, the pumpprobe signal can be given as an appropriate convolution of the external fields (either temporal or spectral, as the case may be) with the snapshot spectrum.22 Otherwise one has to use the general expression derived for the pump-probe in the earlier section. 6. Discussion and Conclusions An expression for the sequential pump-probe signal, probing the charge transfer from an excited molecule to a continuum of conduction band levels of a semiconductor, has been obtained. This has been achieved by (i) utilizing the existing theoretical
Figure 4. Pump-probe signal (snapshot limit) in arbitrary units plotted as function of the delay time between the pump and probe pulses demonstrating vibronic coherences in the absence of electron transfer. The harmonic oscillator energy pω is set to 0.1 eV.
Figure 5. Pump-probe signal (snapshot limit) versus the delay time between the pump and probe pulses in arbitrary units when the excited state (at m ) 0) is coupled to a continuum of electronic energy levels of a 1-D substrate of bandwidth 2.0 eV: (a) g12 ) g23 ) 1; (b) g12 ) g23 ) 0.7; (c) g12 ) 1, g23 ) 0.1. For all cases C1 ) 0.004.
machinery available for a sequential pump-probe signal in a three-level system22 and (ii) formulating a specific model
76 J. Phys. Chem. B, Vol. 104, No. 1, 2000 molecular Hamiltonian (similar to the one used in autoionization problems23), so as to embed it in the general expression for the pump-probe signal. The final expression has been cast in a diabatic representation consisting of the bound and continuum electronic states of the molecular model Hamiltonian. Such a representation is convenient for considering the continuum levels of the semiconductor substrate. Nuclear motion performs free undamped oscillations in the above model. It is a straightforward task to incorporate in the above expressions a more general multimode Brownian oscillator model which interpolates between coherent nuclear motion considered here and damped nuclear oscillations.22 However, this may be necessary when the injecting molecule is in a solvent environment rather than in ultra-high-vacuum (which is the focus of this work) where one has observed that the intramolecular modes are essentially undamped during the initial 100 fs or so.19 The expression for the signal obtained in this work are, however, constrained by the two physical approximations. The first one, namely that the electron transfer times should be much longer than the pulse durations, sets a relative time limitation on its applicability to fast electron transfer processes. The case addressed by our theory has recently also been realized in an experiment.19 Recently, Willig and co-workers19 have utilized pulses of less than 20 fs duration and have examined cases of electron injection in the range of 60-80 fs. The expressions derived in this work can be used to match the signals coming from similar electron injection times and pulse durations. Interestingly, the probe signals in the above-mentioned experimental work showed clearly the presence of vibronic coherences. The second approximation arises from the assumption that reorganization energy of the electron transfer process can be ignored on the basis that the equilibrium position of the nuclear modes of the ionized molecule is negligibly different from that of the excited state i.e., (gν2 - gνK) , 1. It is not clear whether such a condition may be common for large molecules injecting into semiconductor substrates. However, it should be approximately correct for large aromatic molecules like perylene with delocalized π-electrons. What is being ignored in this approximation, as stated earlier, are the Franck-Condon factors which are responsible for vibronic transitions between the excited manifold and the multiple manifolds which characterize the vibrational states of the ionized molecule with the electron being located in the continuum of conduction levels. The inclusion of these factors could in general modulate the oscillations but in some cases influence the lifetime of the excited state also. In order to clarify the role of these neglected terms it is pertinent to examine the theoretical work of Cinni et al.30 and in particular that of Hewson and Newns.31 From their work it can be seen that the lifetime of an impurity state coupled to a single mode of vibration as well as to a continuum of states is in general a difficult many-body problem. However, their study shows that when the reorganization energy of electron transfer λ is much less than the bandwidth (namely, 2w) of the continuum of levels i.e., λ e 2w/5, the lifetime of the impurity state becomes independent of boson coupling and can be given purely in terms of the electronic coupling, exactly as has been obtained in this work. In other words, this implies that the excited state lifetime derived in this work will be valid and be dictated by purely electronic factors, ably fitted by the exponential expressions, as long as the reorganization of electron transfer is much smaller than the bandwidth of the substrate. Of course, the excited state is assumed to be away from the band edges, at least by a few reorganization energy widths. For
Ramakrishna and Willig most substrates typical values for bandwidth are usually of the order of 2 eV or more whereas for λ the values are usually not expected to exceed 0.1 eV for large aromatic molecules in vacuum.32 The general expression for the pump-probe signal indicates that the decay will not be temperature dependent, although the line shape of the oscillations will be. The occupancy of the empty semiconductor states has been taken as zero since they are far above the Fermi level and hence no temperature dependence is likely to arise from the semiconductor substrate. It is well-known that the Franck-Condon factors for electron transfer lead to a temperature-dependent rate (nonadiabatic) in the case of a two-level electronic system.33 However, if there is an electronic continuum of acceptor states as in an heterogeneous electron transfer, it has been clarified that, in the case of a wide band, the electron-tranfer rate (nonadiabatic) becomes temperature-independent.1 The presence of a wide continuum makes all the Franck-Condon factors realizable and hence they can be independently summed to unity, leaving the expression for the rate of electron transfer to be independent of temperature. However, the situation is different when the width of the continuum of the acceptor levels is sufficiently narrow (compared to the reorganizaion of electron transfer) as the lifetime (adiabatic) in such a case has been shown to depend on the Franck-Condon factors,31 and thus on temperature. In fact, when the band is sufficiently narrow such that 2w , pω0, the imaginary part of the self-energy can be rescaled as Im Σ(ω ˜) 2 (1 + 2nj0)). This is the well-known large f Im Σ(ω ˜ ) exp(-g2K polaron behavior of the Fro¨hlich type wherein one uses the diagonol transition case with no change in phonon numbers during electron transfer (or hop).34 In this extreme narrow band limit one clearly sees how the temperature dependence is built into the lifetime of the excited state. Thus the temperature independence of electron transfer can be understood to arise essentially from the large continuum (wide band limit) of semiconductor levels. However, if there is appreciable vibrational relaxation on the same time scale of electron transfer this will also lead to temperature-dependent decays even in the wideband limit. This corresponds to the overdamped limit of the Brownian oscillator model.22 The fact that no temperature dependence of electron transfer was observed experimentally15 testifies to (i) the applicability of the wide-band limit as well as (ii) the absence of significant vibrational relaxation during the electron injection process. In order to obtain the excitation energy required to promote an electron between the occupied and unoccupied levels in the molecule, the intramolecular electron correlation is necessary.12 On the other hand, one can redefine the ground and the excited states by appropriate shifts in energy that take into account Coulombic repulsion energies between electrons in the ground state and between excited and ground state and appropriately define the molecular Hamiltonian, as has been done in this work. Importantly, intramolecular electron correlations can lead to changes in the lifetime of the excited state if its Lorentzianlike line shape overlaps with filled levels of the substrate.12 This will not be appropriate for the case considered here since the ground state as well as the excited state of the molecule is situated far above the filled states of the semiconductor.1 However, more significant effects, arising from intramolecular cum substrate-molecular electron correlations, are energetic shifts in the absorption spectra of the adsorbed/anchored molecule vis-a-vis its free state.12 This effect has been ignored here since the focus is to obtain dynamical features rather than energetics.
Pump-Probe Spectroscopy of Ultrafast Electron Injection To summarize, this work has made the first attempt to obtain qualitative expressions for a pump-probe signal investigating electron injection into a continuum of substrate levels. Charge injection has been considered sufficiently slow to be neglected during the time scale of the pump and probe pulses. Also, the reorganization energy for the charge transfer has been ignored in the final expressions. As a result, the lifetime of the excited state turns out to be independent of the dynamics created by the pump and probe pulses. The lifetime under these conditions is shown to be determined solely by the density of electronic states of the substrate that the excited state samples from its specific energetic location. The approach has been sufficiently general to explore different dimensionalities for the substrate, the different dynamics that ensue from each of them, and useful exponential expressions that admirably fit the actual decay. This suggests the interesting possiblity of estimating the electronic coupling between the injecting molecular orbital and the substrate levels from actual experimental decays. The expressions obtained are realistic in that they indicate temperatureindependent irreversible decays, at time scales that have been observed experimentally. Moreover, the theory incorporates nuclear motion on the time scale of electron transfer and examines conditions for observing vibronic coherences in a three-level system. Acknowledgment. The authors thank Dr. B. Burfeindt for useful discussions and the Joule III and the Volkswagen programs for financial support. References and Notes (1) Miller, R. D.; McLendon, G. L.; Nozik, A. J.; Schmickler, W.; Willig, F. In Surface Electron Transfer Processes; VCH: New York, 1995; Chapter 5. (2) (a) O’Regan, B.; Gra¨tzel, M. Nature 1991, 353, 737. (b) Hagfeldt, A.; Gra¨tzel, M. Chem. ReV. 1995, 95, 49. (3) Burfeindt, B.; Ramakrishna, S.; Meissner, B.; Hannappel, T.; Storck, W.; Mahrt, J.; Willig, F. In Ultrafast Phenomena XI; Springer-Verlag: Berlin, 1998; p 636. (4) Mukamel, S. Annu. ReV. Phys. Chem. 1990, 41, 647. (5) Guo, L. H.; Mukamel, S.; McLendon, G. J. Am. Chem. Soc. 1995, 117, 546. (6) Suzuki, M.; Nasu, K. J. Chem. Phys. 1990, 92, 4576. (7) (a) Lanzafame, J. M.; Miller, R. J. D.; Muenter, A. A.; Parkinson, B. A. J. Phys. Chem. 1992, 96, 2820. (b) Lanzafame, J. M.; Palese, S.; Wang, D.; Miller, R. J. D.; Muenter, A. A. J. Phys. Chem. 1994, 98, 11020. (8) Boroda, Y. G.; Voth, G. A. J. Chem. Phys. 1996, 104, 6168.
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