Optical Switching of Charge Transmission through a Single Molecule

Feb 15, 2010 - Shane M. Parker , Manuel Smeu , Ignacio Franco , Mark A. Ratner , and Tamar Seideman. Nano Letters 2014 14 (8), 4587-4591. Abstract | F...
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J. Phys. Chem. C 2010, 114, 4179–4185

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Optical Switching of Charge Transmission through a Single Molecule: Effects of Contact Excitations and Molecule Heating Luxia Wang† and Volkhard May*,‡ Department of Physics, UniVersity of Science and Technology Beijing, Xueyuan road 30, 100083 Beijing, China, and Institut fu¨r Physik, Humboldt-UniVersita¨t zu Berlin, Newtonstraβe 15, D-12489 Berlin, Germany ReceiVed: NoVember 9, 2009; ReVised Manuscript ReceiVed: January 26, 2010

Photoinduced changes of the current voltage (IV) characteristics of a single molecule attached to a left and a right electrode are revisited including the inevitable photon absorption by the leads due to electron-hole pair generation. To determine the related nonequilibrium electron distribution, a method is utilized based on the introduction of an effective electron temperature Tel in the Fermi distribution. By varying Tel, possible excitation conditions in the experiments are modeled. Heating of the molecule as well as the effect of intramolecular vibrational energy redistribution (IVR) is also accounted for. An intermediate excitation regime can be identified where the optical current switch is dominated by molecular excitations and less by an electron-hole pair generation in the leads. Vibrational distributions in the molecule induced due to either current formation or optical excitation are found to strongly deviate from thermal equilibrium. To observe the current switch, it is essential that these distributions are only slightly affected by IVR. I. Introduction A considerable number of schemes has been suggested to manipulate a single molecular junction by external laser fields. Either the fields are supposed to induce molecular conformational transitions1-4 or directly produce excited electronic states of the molecule. For the latter case, mechanisms were studied based on laser pulses from the infrared5-8 or from the visible region (see refs 9-19). However, only a restricted number of experimental tests of all of these suggestions have been reported so far.20-24 In any case, such a disturbance essentially influences the leads contacting the molecule. Accompanying thermal expansion may destroy the whole junction and an excitation of electrons in the leads far above the Fermi sea shall result in an additional and uncontrollable electron injection into the molecule. The latter mechanism is discussed for many years in the field dealing with photodesorption of molecules attached to metal surfaces. Here, one standardly distinguishes between desorption induced by electronic transitions (DIET) working in the low flux cw range and desorption induced by multiple electronic transitions (DIMET), where intensive but short (some hundred femtoseconds long) laser pulses are used (see ref 25 for a recent overview). For the DIMET mechanism, the excited state of the metal surface is well described by an effective electron gas and an effective lattice temperature changing in the range of some thousands of Kelvin but on a time interval of a few picoseconds. To shine light mainly on the molecular part of the junction, one may utilize the techniques of nanooptics.24 Nevertheless, the very restricted geometry would result in an unavoidable excitation of the leads, which may alter the IV characteristics. By including this obstacle of lead excitation when discussing photoinduced changes of the IV characteristics of a single molecule, we will put our recent discussion of an all optical current switch17-19 into a more realistic frame. * To whom correspondence should be addressed. E-mail: may@ physik.hu-berlin.de. † University of Science and Technology Beijing. ‡ Humboldt-Universita¨t zu Berlin.

Figure 1. Molecular energy level scheme for a sequential net leftlead to right-lead charge transmission (A f B f C or A′ f B′ f C′). B and B′ label the singly negatively charged state with energy E1. Otherwise, the energy E0 of the neutral state is concerned. The electronic ground and first excited state are indicated by bold horizontal lines and are both extended by the sets of reaction coordinate vibrational levels (blue and red). EF is the Fermi energy (at zero voltage), identical for both leads, and µL (µL′ ) and µR (µR′ ) fix the finite voltage position of the chemical potentials of the left and right leads, respectively. (To have an easy view on energetically allowed charge transmission processes, a fictitious combination has been introduced of the neutral molecule’s energy levels E0 with available energies of lead electrons Eel and E′el.) Possible charging and discharge is visualized by horizontal arrows. Vertical arrows indicate optical excitation (full line) and IVR (dotted line). While the A f B f C scheme corresponds to current formation mainly via electronic ground-state transitions, the A′ f B′ f C′ scheme involves the first excited electronic state of the neutral molecule.

The suggested optical current switch shall work if photoinduced transitions into excited electronic states of the neutral and charged molecule become possible (see Figure 1). If the charging energy of the molecule is positioned above the lead’s Fermi energy, current formation is suppressed in a low voltage regime. This suppression may be removed if photoexcitation

10.1021/jp910667h  2010 American Chemical Society Published on Web 02/15/2010

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populates an excited electronic state. Now, one may define a new charging energy which is reduced by this amount of excitation energy, and charging of the molecule becomes possible (cf. Figure 1). We will demonstrate in the following that the current switch may survive the presence of electrons high above the Fermi energy originated by the simultaneous optical excitation of the leads. Because of the rare number of related experiments, it seems reasonable to vary characteristic parameters of the molecular junction instead of focusing on a particular example. In the subsequent section, we introduce our parametrized model and explain the way to compute steady state IV characteristics (our choice of parameters is motivated in section IIA). The results are explained in section III. The paper ends with some brief concluding remarks in section IV. II. Model and Computational Approach The basic physical picture utilized in the following is shown in Figure 1. It is based on the generic model of a single molecule attached to a left and a right lead. Charge transmission is restricted to the sequential type assuming weak molecule-lead coupling. The overall Hamiltonian takes the form

H(t) ) Hmol(t) + Hmol-lead + Hlead

(1)

with its different components explained in the following. The molecular part

Hmol(t) ) Hel+vib + HIVR + Hsc + Hfield(t)

(2)

covers all molecular degrees of freedom which are of interest here and the coupling to the external laser field (remember that optical excitation of the leads is only indirectly accounted for). Besides the neutral molecule (charging number N ) 0), just the singly charged state (N ) 1) is considered (the cationic state N ) -1 is assumed to be energetically well separated and thus negligible). Both charging states are further characterized by their electronic ground state with energy ENg and their first excited state with energy ENe. Related electronic wave functions are denoted as φNa (a ) g, e; the importance of a full molecular electronic state description has been recently emphasized in ref 15 but already suggested in ref 26 and used in refs 18, 19, and 27). The chosen electronic level configuration should be of such a type that the quantity

∆E10 ) E1g - E0g - µ0

(3)

becomes positive and large enough to clearly demonstrate optical current switching (µ0 is the equilibrium chemical potential identical for both leads). Moreover, to stay simple enough, we will also assume that the excitation energies E0e - E0g and E1e - E1g coincide. If the considered sequential charge transmission regime is connected with a residence time of the injected electron in the molecule comparable with or somewhat less than typical times of IVR, nonequilibrium vibrational distributions in the molecule become essential (see, e.g., refs 28 and 29). To account for this in a tractable way, the whole set of intramolecular vibrations is separated into a small set of coordinates which couple most strongly to all involved electronic transitions (the primary or reaction coordinates), and a remaining set of coordinates which coupling to the electronic transitions is neglected. In order to

describe IVR, however, a coupling to the reaction coordinates has to be assumed and is accounted for in HIVR. (The coupling potential to the reservoir coordinates contained in HIVR will be specified later when introducing respective transition rates.) These secondary (reservoir) coordinates with Hamiltonian Hsc shall form a heat bath and do not enter the description explicitly. Therefore, only the reaction coordinate states χNaµ (with vibrational quantum numbers µ and vibrational energy pωNaµ) have to be taken into consideration. Introducing the overall molecular state quantum number R ) Naµ, we arrive at the following notation of the molecular electron-vibrational Hamiltonian

Hel+vib )

∑ pεR|ΨR〉〈ΨR|

(4)

R

The |ΨR〉 ) |χNaµφNa〉 are the electron vibrational states and the pεR ) ENa + pωNaµ are the respective energies. The electronic ground state and first excited state together with the quasicontinuum of the primary vibrational states are displayed in Figure 1 for the neutral molecular state (combined with the energy of a lead electron) and the singly charged state. Of course, such a representation of the Hamiltonian is only meaningful if the reaction coordinate states are explicitly known. To consider vibrational dynamics and electron-vibrational coupling in the molecule, the shifted oscillator model provides a reasonable description. It is characterized by the potential )2/4 defined by energy surfaces (PES) of type ∑j pωj(Qj - Q(Na) j some reaction coordinates Qj having frequency ωj (for simplicity, we assume the vibrational frequencies to be state independent). The coupling to a charge transfer step or an optical . While all the transition is quantified by the PES shifts Q(Na) j following rate expressions are valid for an arbitrary number of reaction coordinates, concrete computations focus on a single one. Charge transfer between the neutral (N ) 0) and the singly charged state (N ) 1) is considered by the molecule-lead coupling part Hmol-lead of eq 1 and contains the various transfer couplings VX(0a, 1b; k) (cf. ref 18). The leads with Hamiltonian Hlead are counted by X () L, R) and the respective electrons by Bloch-vectors k. Those are only of symbolic meaning here, since it is more appropriate to replace the k-dependence of the transfer coupling by a continuous frequency dependence and to introduce the lead DOS NX(Ω). Be aware that the coupling notices the actual electronic state of the molecule but neglects any dependence on the vibrational coordinates. Therefore, when changing to a representation in the electron vibrational states, VX(0a, 1b; Ω), for example, has to be multiplied by the vibrational overlap expression 〈χ0aµ|χ1bν〉. Optical excitation and deexcitation of the molecule is described by Hfield(t) of eq 1 (see also ref 18) and shall be characterized by the common Rabi energy ER ) degE, where deg denotes the transition dipole moment which is assumed to be equal for the neutral and the charged state (E is the electric field strength of the cw excitation). Concerning the optical excitation of the leads, we have to stay in a less concrete description. Obviously, the actual electron distribution originated by optical excitation depends on the concrete atomistic structure of the nanocontact as well as on the form of the local radiation field. The study of these peculiarities would require the simulation of nonequilibrium dynamics in a many electron system of a metal with restricted geometry. While this would be of general interest, here we circumvent such computations and consider the effect of lead excitation in an implicit way.

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Therefore, lead electrons are further described by a Fermi distribution but defined with an effective temperature Tel strongly exceeding that of the surroundings. This is a common procedure used, for example, when discussing photodesorption of molecules at a metal surface.25 Accompanying lattice heating depends on the type of electron-phonon interaction in the lead. The concrete coupling of the molecule to the leads would be responsible for the degree of molecular heating. It will be considered by changing the temperature Tmol which characterizes the state of the secondary vibrational coordinates introduced earlier. Since a determination of Tel, Tmol, and their dependence on the actual strength of molecular excitation ER is outside the scope of the present approach, both temperatures are considered as independent parameters to be varied appropriately. To compute steady state IV characteristics, we use the master equation approach of refs 17-19, 26, and 27. In a stationary situation, one may deduce the time-independent electron-vibrafrom the following balance equation: tional level population P(stat) R

P(stat) R

∑ kRfβ ) ∑ Pβ(stat)kβfR β

(5)

β

The transition rates kRfβ include contributions due to the molecule-lead coupling, IVR, and optical excitation. The assumed weakness of the molecule-lead coupling allows us to restrict the considerations to second-order rate expressions. The resulting rate of molecular charging takes the standard form18,19 (mol-lead) k0aµf1bν )

∑

4π |〈χ0aµ |χ1bν〉|2 × p2

NX(ε1bν,0aµ)|VX(0a, 1b;ε1bν,0aµ)|2fF(pε1bν,0aµ - µX)

X)L,R

(6) where ε1bν,0aµ ) ε1bν - ε0aµ are transition frequencies and fF denotes the Fermi distribution (depending on the temperature (mol-lead) of discharge follows from the Tel). The respective rate k1bνf0aµ charging rate in replacing fF by 1 - fF. Since the approach does not distinguish between up and down spin states of the electron injected into the molecule, the prefactor 2 enters the rates. When carrying out concrete computations, we do not work with transfer couplings which depend on the different electronic states but use a common quantity. We also apply the wide-band approximation by replacing 2πNX(ε1bν,0aµ)|VX(0a, 1b; ε1bν,0aµ)|2/p2 by the common conj |2/p2 (N denotes the mean lead DOS and V j stant Γ ) 2πN |V the mean transfer coupling). Γ represents a measure for the charge injection efficiency as well as for the molecular level broadening due to the molecule-lead coupling. Moreover, the model of a symmetrically applied voltage V is used (the chemical potentials entering the rates are for the left lead µL ) µ0 + |e|V/2 and for the right lead µR ) µ0 - |e|V/2). The overall steady state current can be written as18,19

I)

∑ (I0af1b + I1bf0a)

(7)

a,b

(mol-X) (mol-X) The part k0aµf1bν (as well as k1bνf0aµ ) of eq 6, referring to a particular lead X together with the populations, determine the partial currents due to molecular charging

I0af1b ) |e|

(stat) (mol-X) k0aµf1bν ∑ P0aµ

(8)

µ,ν

and due to discharge

I1bf0a ) -|e|

(mol-X) ∑ P(stat) 1bν k1bνf0aµ

(9)

µ,ν

Rates of IVR, also entering the balance eqs (5), are based on a bilinear coupling between the reaction coordinates and all other molecular vibrations treated as a heat bath (with temperature Tmol). This type of coupling represents the most basic form and results in the following model for IVR rates (be aware of the additional restriction to a single reaction coordinate oscillating with frequency ωvib) (IVR) kMaµfNbν ) δMa,Nb2πJMa(ωvib)(δµ+1,ν(µ + 1)n(ωvib) + δµ-1,νµ[1 + n(ωvib)]) (10)

The function n is the Bose-Einstein distribution depending on the temperature Tmol and JMa denotes the spectral density of the coupling to the secondary vibrational coordinates. Since the reaction coordinate is harmonic, the spectral densities are only needed at the respective vibrational frequency ωvib. The neglect of a possible electronic state dependence, then, allows the actual strength of IVR to be characterized by the single value J. The rates of optical excitation and deexcitation of the molecule (for a given charging state and a * b) read (opt) kNaµfNbν ) 2ER2 |〈χNaµ |χNbν〉|2

γNaµ,Nbν

+ (ω0 - εNaµ,Nbν)2 + γNaµ,Nbν2 (ω0 f -ω0) (11)

ω0 is the frequency of the external laser pulse, and the broadening of the transitions among the different electron-vibra(IVR) (IVR) + kNbνfNbκ ) + γpd is tional states γNaµ,Nbν ) 1/2 × ∑κ (kNaµfNaκ determined by the IVR rates (the pure dephasing rate γpd ensures broadening of zero-zero reaction coordinate quanta transitions; note that a more involved theory may also account for a broadening due to molecule-lead coupling30). Field-induced optical transitions may be balanced by recombination due to spontaneous photon emission as well as nonradiative transitions (molecular de-excitation due to electron-hole pair generation in the lead, cf. ref 10). As demonstrated in ref 19, this type of molecular deexcitation is of less importance when calculating IV characteristics (it does not introduce any type of vibrational relaxation) and will be neglected here. A. Choice of Parameters. The parameters used are listed in Table 1. The quantity pΓ determining the molecular level broadening due to the molecule-lead coupling has been used to scale all energies. A reference current I0 ) |e|Γ and a reference voltage V0 ) pΓ/|e| can be introduced and are used to display the IV characteristics which follow. Although being of interest, a lead and electronic state dependent variation of the moleculelead coupling will not be considered here (an asymmetry of the IV characteristics for negative and positive voltages and a suppression or enhancement of the excited state molecule-lead coupling may result). However, we will demonstrate the effect of changing the various reorganization energies (nuclear rearrangements) upon electronic transitions. This is of general importance, since these energies regulate the influence of

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TABLE 1: Parameter Sets Used in the Computations (First Numbers Give the Reference Values; for Further Explanation, See Text) ∆E10/pΓ (E0e - E0g)/pΓ ) (E1e - E1g)/pΓ ωvib/Γ Q0g Q0e Q1g Q1e J/pΓ γpd/pΓ kBTel/pΓ, kBTmol/pΓ

80 300 10 0 4...1 6...3 2...5 0...1 1 0.5...50

vibrational overlap. However, such a variation is also necessary, since concrete data for charged molecules are not available. The computations start with those values already used in our earlier of the PES studies of refs 17 and 19. There, the shifts Q(Na) j have been chosen in such a way to arrive at reorganization energies upon charging or discharge (in the electronic ground state) which are larger than those related to photoinduced transitions (see first values in Table 1). As already indicated, the charging energy E10 ) E1g - E0g and, thus, the quantity ∆E10, eq 3, will be taken large enough to visualize the switching mechanism (∆E10/pΓ ) 80 guarantees current suppression in the low temperature and symmetrically applied voltage case up to V ) 160V0). Computations of molecular charging energies are rather rare in the literature. Some estimates on distryl-benzene, distryl-paracyclophane, and magnesium porphine can be found in refs 31 and 32 where a HF-CIS level of theory has been used but any coupling to the leads ignored (the effect of possible polarizations of the lead’s electron gas are outside of these computations). Since we are concerned with sequential charge transmission, 1 meV would be a reasonable choice for pΓ (resulting in V0 ) 1 mV and I0 ≈ 250 nA). Then, the related lifetime of the charged molecular state amounts to half of a picosecond. Consequently, steady state conditions for the current require (as a lower bound) optical excitation in the range of a few picoseconds. This excludes a DIMET type of surface excitation but justifies at the same time the lower effective electron gas temperature (here assumed to be time independent). Moreover, we get ∆E10 ) 0.08 eV, resulting in a current suppression up to a voltage of 0.16 V. Effective temperatures which correspond to the chosen numbers of 0.5, 5, 10, 20, and 50 for kBTel/pΓ are about 5.8, 58, 116, 232, and 580 K, respectively. Finally, the taken ratio ER/pΓ ) 1 corresponds to an electric field strength of about 105 V/cm if the transition dipole moment takes a few debyes. III. Current Voltage Characteristics Photoinduced changes of the IV characteristics are discussed first in concentrating on the excitation of the leads while neglecting optical excitation of the molecule. The simultaneous excitation of the molecule and the leads is considered afterward. Finally, IVR is included and the effect of possible heating of the molecule is addressed. A. Absence of Molecular Photoexcitation. In assuming the absence of any molecular excitation (the external laser field is off-resonant with respect to the electronic transition in the molecule), Figure 2 shows the change of the IV characteristics if the degree of lead excitation is increased (mimicked by an increase of the effective temperature of the electron gas). At the absence of any optical excitation (case kBTel ) 0.5pΓ, where Tel has to be understood as the temperature arranged in the experiment), the current is suppressed for voltages below 2∆E10/

Figure 2. Steady state current (upper panel) and mean number of excited vibrational quanta (lower panel) versus applied voltage and at off-resonant photoexcitation of the molecule (absence of IVR, kBTmol ) 0.5pΓ, other parameters see text). Optical excitation of the leads is varied by changing Tel. Solid line (black), kBTel ) 0.5pΓ; dashed line (red), kBTel ) 5pΓ; chain dotted line (green), kBTel ) 10pΓ; dotted line (blue), kBTel ) 20pΓ; thin solid line (black), kBTel ) 50pΓ.

|e| (V/V0 < 160; this is easily deduced from the energy expression pε1gν,0gµ - µL ≡ ∆E10 + pωvib(ν - µ) - |e|V/2, entering the Fermi function in eq 6). According to the small vibrational wave function overlap 〈χ0g0|χ1g0〉 (the respective reorganization energy is rather large), the current stays small for somewhat higher voltages (Franck-Condon blockade, see ref 33). The following stepwise increase of the current is related to the population of excited vibrational states (the IV characteristics display this simple stepwise form, since we assumed that the vibrational energy in the neutral molecule coincides with that in the charged state). The IV characteristics are smoothed out if Tel increases to end up with a linear dependence of I on V. The appearance of this linear dependence, which can also be reproduced analytically in the large Tel limit, is considered here as an upper limit for the used values of kBTel. Thus, it implicitly fixes the case of the strongest optical excitations of the leads. Comparing the kBTel ) 50pΓ line with that for kBTel ) 0.5pΓ, it already becomes obvious that such a strong lead excitation offers a mechanism of current switching without the incorporation of molecular excitations. j vib Respective mean numbers of excited vibrational quanta N ) ∑N,a,µ µP(stat) Naµ are also shown in Figure 2. If any photoexcitation is absent, vibrational excitation appears at the voltage of 2∆E10/ |e| where a current starts to flow. As in the IV characteristics, j vib (drawn versus V) follow the excited vibrational the jumps of N j vib reaches states involved in the charge transmission. Finally, N a value somewhat larger than 6 (at V/V0 ) 280). This indicates the formation of a pronounced nonequilibrium vibrational distribution, i.e., a heating of the molecule, caused by charge transmission through it (note that in the present case no heat dissipation appears, since IVR has been neglected). For kBTel ) 10pΓ and larger, vibrational excitation is already present at V ) 0. The molecule is heated up due to equilibrium charging and discharge (case of vanishing net current). This effect becomes stronger if V is increased and if the overall current reaches finite values.

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Figure 3. IV characteristics for a simultaneous photoexcitation of the molecule and the leads (absence of IVR, kBTmol ) 0.5pΓ). The optical excitation of the molecule is according to ER ) pΓ, and the lead excitation has been changed as in Figure 2.

B. Presence of Molecular Photoexcitation. Figure 3 displays the change of the IV characteristics if an optical excitation of the molecule is also considered. The case Tel ) Tmol (kBTmol ) 0.5pΓ) where any excitation of the leads is neglected confirms the result of our recent studies.17,18 The current suppression in the low voltage range due to the chosen value of the charging energy ∆E10 (cf. Figure 2) is overcome by optical excitation moving the molecule in its excited electronic state (cf. the transition scheme A′ f B′ f C′ in Figure 1). This behavior has been interpreted as an optical current switch. If we allow for a certain degree of lead excitation by increasing Tel, the current is somewhat increased in the low voltage range. Arriving at kBTel ) 50pΓ, the IV characteristics practically coincide with those obtained in the absence of any optical excitation of the molecule. The chosen high value of Tel results in a strongly softened Fermi edge, which enables arbitrary transitions between the neutral and singly charged molecule. The initial A (A′) and final configurations C (C′) of Figure 1 can be placed far apart from µL (µL′ ) and µR (µR′ ), respectively. Comparing Figure 2 with Figure 3, we can draw three main conclusions. First, possible electron-hole pair generation in the leads accompanying photoexcitation of the molecule does not remove the current switch. This becomes obvious in a comparison of the thick solid line of Figure 2, upper panel (case of no optical excitation), with the thin solid line of the same figure (absence of molecular excitation) as well as with the thin solid line of Figure 3 (inclusion of molecular excitation). Second, if the simultaneous optical excitation of the molecule and the leads results in a high effective electron temperature, the current switch is dominated by electron-hole pair generation in the leads. Third, an intermediate range of lead excitations with kBTel ) 10pΓ exists where the current switch is dominated by photoinduced transitions in the molecule. This observation is highlighted by a comparison of the chain dotted line in Figure 2, upper panel, with the chain dotted line in Figure 3. (Here, the identification of an intermediate range of lead excitations does not refer to possible laser pulse intensities of the optical excitation process but to the relation of the actual kBTel value to the upper value used where I starts to depend linearly on V.) To assess the influence of nuclear rearrangement upon charging and electronic transitions, Figures 4 and 5 again display the curves of Figures 2 and 3 but for differently chosen reorganization energies. The case of smaller reorganization energies due to transitions into excited electronic states is shown in Figure 4 (the values are reduced from 4pωvib to pωvib/4). Only marginal changes of the IV characteristics are obtained. If all reorganization energies are reduced, however, the IV characteristics change drastically. Figure 5 displays a situation where the reorganization energy of the unexcited molecule upon

Figure 4. Total current versus applied voltage with off-resonant photoexcitation of the molecule (upper panel) and with resonant photoexcitation (lower panel). In contrast to Figures 2 and 4, the reorganization energies upon electronic excitation have been reduced (Q0e ) 1, Q1e ) 5; optical excitation of the molecule is according to ER ) pΓ; the lead excitation has been changed as in Figure 2).

Figure 5. Total current versus applied voltage with off-resonant photoexcitation of the molecule (upper panel) and with resonant photoexcitation (lower panel). In contrast to Figures 2 and 3, all reorganization energies have been reduced (Q0e ) 2, Q1g ) 3, Q1e ) 1; optical excitation of the molecule is according to ER ) pΓ; the lead excitation has been changed as in Figure 2).

charging is reduced from 9pωvib to 9pωvib/4. If any optical excitation is absent, the current starts with a big jump which is caused by the large vibrational overlap 〈χ0g0|χ1g0〉. The presence of a lead excitation, as in the foregoing cases, smears out this behavior and sets up an efficient current switch. This is not the case if we concentrate on an exclusive molecular excitation (thick solid curve of the lower panel of Figure 5). The related changes of the IV curves reach an acceptable magnitude only if the lead excitation corresponds to kBTel ) 20pΓ. Thus, a sufficient large reorganization energy upon charging is necessary

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Figure 6. Optical switching of IV characteristics at the presence of IVR (kBTmol ) 0.5pΓ). Upper panel, absence of photoexcitation (Tel ) Tmol); lower panel, presence of photoexcitation with ER ) pΓ and according to kBTel ) 10pΓ. Solid line (black), J ) 0; dashed line (red), J ) 0.1Γ (thin dashed line: kBTmol ) 10pΓ); chain dotted line (green), J ) Γ (thin chain dotted line: kBTmol ) 10pΓ).

Figure 7. Mean number of excited vibrational quanta versus applied voltage and for various strengths of IVR. Absence of optical excitation (kBTel ) kBTmol ) 0.5pΓ): solid line (black), J ) 0; dashed line (red), J ) 0.1pΓ; chain dotted line (green), J ) pΓ. Presence of optical excitation and molecular heating (ER ) pΓ, kBTel ) kBTmol ) 10pΓ): thin solid line (black), J ) 0; dotted line (blue), J ) 0.1pΓ (the case J ) pΓ nearly corresponds to the chain dotted line).

to get a current switch which is mainly based on molecular electronic transitions. C. Effect of IVR. So far, any effect of vibrational relaxation, i.e., IVR, has been neglected. Figure 6 addresses this important issue for the absence as well as the presence of photoexcitation. In all cases, IVR reduces the net current. Such a behavior has already been reported in refs 27 and 33 but for a situation without any optical excitation. Obviously, the current reduction is due to the partial or total removal of the nonequilibrium character of the vibrational distribution (for J ) pΓ, it only remains a population of the vibrational ground state, see Figure 7). To understand this in more detail, we consider the relation ∆E10 + pωvib(ν - µ) - |e|V/2 characterizing the charging process (cf. eq 6). If IVR operates the various transition channels related to vibrational quantum numbers, µ > 0 are suppressed. In contrast, they are present if IVR is absent. Just these additional µ > 0 contributions (due to excited vibrational state (stat) ) let the current become larger. contributions in P0gµ To combine photoexcitation of the junction with IVR, we considered the intermediate case where the excitation of the

Wang and May molecule is characterized by ER ) pΓ and that of the leads by an effective electron temperature according to kBTel ) 10pΓ. As it becomes obvious from the lower panel of Figure 6, the suggested current switch is diminished for an increasing strength of IVR. This behavior is only slightly changed if heating of the molecule is taken into account (characterized by kBTmol ) 10pΓ). The related mean vibrational numbers are displayed in Figure 7, indicating that the current induced nonequilibrium vibrational distributions are removed by strong IVR. D. Current Switching On and Off. We use the obtained results for the steady state current to present a brief discussion on the behavior of the current when the optical excitation is switched on and off. In section IIA, we emphasized that pΓ ) 1 meV would be a reasonable value for the studied scheme of sequential charge transmission. Since 1/Γ is an estimate for the residence time of an injected electron in the molecule, we can conclude that the response of the current to a change of the laser field excitation lies in the picosecond range. Let us assume that the intensity change of the radiation field takes place in this time region or even faster with a final value corresponding in the present study to the choice ER ) pΓ. Then, the current is switched on if the applied voltage is chosen in the respective range (V < 2∆E10/|e|). Since in this range of radiation field intensity the switching due to an exclusive molecular excitation works, a strong excitation of the leads (corresponding to kBTel ) 50pΓ in the present studies) would only amplify the switch while a smaller lead excitation (BTel < 10pΓ) does not hinder it. As already indicated, the strong nonequilibrium character of the lead’s electron distribution is lost within some picoseconds and a heating of the lead’s lattice follows.25 If this results in a modest heating of the molecule only, a stable current should be formed within the time interval of the laser pulse action (a possible thermal expansion of the leads may be absorbed by flexible linkers connecting the active molecular part with the leads). The off-state of the current is again reached in a time interval comparable to 1/Γ if the light field is switch off as fast as it was switched on. In the contrary case of a switching on and off slow compared to 1/Γ, we may use the steady state current curves drawn versus a varying strength of the radiation field, i.e., I(ER) (not shown here, but see refs 17 and 19). This would offer the response of the current when ER becomes time dependent. A simultaneous excitation of the leads would not suppress the current switch. However, the presence of excited electron-hole pairs in the leads during the decrease of the optical excitation possibly would disturb the switch off of the current. Obviously, the time region between the two switching scenarios discussed here needs a more detailed study. IV. Conclusions Photoinduced changes of the IV characteristics of a single molecule attached to a left and a right electrode have been discussed by considering optical excitation of the molecule as well as of the leads. The first is accounted for by determining the steady state solution of the master equations governing the molecular electron-vibrational populations. Besides rates of charging and discharge, there are also rates describing photoinduced electronic transitions of the molecule. Optical excitation of the leads is taken into consideration by introducing into the lead’s electron Fermi distribution an effective electron temperature Tel. Heating of the molecule as well as IVR are also accounted for. To correspond to the specialty of lead excitation which is determined by their atomic structure as well as by the concrete form of the local exciting field, Tel is varied indepen-

Optical Switching of Charge Transmission dently of the actual strength of the applied laser field. Molecular heating is accounted for in a similar way by changing the molecular temperature. A current switch due to photoexcitation may be realized by an exclusive excitation of the leads. If the lead excitations are less pronounced, the current switch also requires an optical excitation of the molecule. Vibrational distributions in the molecule induced either due to a current formation or an optical excitation are found to strongly deviate from thermal equilibrium. It is essential for the switching mechanism that these nonequilibrium distributions are not completely removed by strong IVR. While the present studies address steady state properties of a molecular junction, it would be of particular interest for future work to focus on transient phenomena connected with an excitation of finite duration. Respective computations are under progress. Acknowledgment. Financial support by the Deutsche Forschungsgemeinschaft through Sonderforschungsbereich 450 and by the National Science Foundation of China (Grant No. 10604004) is gratefully acknowledged. References and Notes (1) Derosa, P. A.; Seminario, J. M. J. Phys. Chem. B 2001, 105, 471. (2) Emberly, E. G.; Kirczenow, G. Phys. ReV. Lett. 2003, 91, 188301. (3) Zhang, C.; Du, M.-H.; Cheng, H.-P.; Zhang, X.-G.; Roitberg, A. E.; Krause, J. L. Phys. ReV. Lett. 2004, 92, 158301. (4) Pati, R.; Karna, S. P. Phys. ReV. B 2004, 69, 155419. (5) Ha¨nggi, P.; Kohler, S.; Lehmann, J.; Strass, M. In Introducing Molecular Electronics; Cuniberti, G., Fagas, G., Richter, K., Eds.; Lect. Notes Phys., Vol. 680; Springer-Verlag: Berlin, Heidelberg, New York, 2005; p 55. (6) Kohler, S.; Lehmann, J.; Ha¨nggi, P. Phys. Rep. 2005, 406, 379. (7) Welack, S.; Schreiber, M.; Kleinekatho¨fer, U. J. Chem. Phys. 2006, 124, 044712. (8) Li, G.; Welack, S.; Schreiber, M.; Kleinekatho¨fer, U. Phys. ReV. B 2008, 77, 075321.

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