Laser pulse induced transient currents in a molecular junction: Effects of plasmon excitations of the leads.

THE JOURNAL OF CHEMICAL PHYSICS 142, 224701 (2015)

Laser pulse induced transient currents in a molecular junction: Effects of plasmon excitations of the leads Yaroslav Zelinskyy1,2,a) and Volkhard May1,b) 1

Institute für Physik, Humboldt-Universität zu Berlin, Newtonstraße 15, D-12489 Berlin, Germany Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, Metrologichna st., 14-b, UA-03360 Kiev, Ukraine 2

(Received 27 December 2014; accepted 22 May 2015; published online 8 June 2015) The transient response of a molecular junction excited by a single laser pulse or by a sequence of pulses is studied theoretically utilizing a density matrix description. The approach focuses on the sequential transmission regime and accounts for intramolecular vibrations and vibrational relaxation. Besides the optical excitation of the molecule, also the laser pulse action on the leads is considered. It is accounted for by collective plasmon excitations which also couple to the molecular transitions. Transient currents are calculated as well as averaged dc resulting from a huge sequence of laser pulses. While the transient currents give some insight into the dynamics of the junctions, the averaged dc is ready to be measured in the experiment. Different enhancement effects due to resonant lead-plasmon excitations are highlighted. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4922072]

I. INTRODUCTION

One of the most promising substitutes for today’s microelectronic circuits are organic molecules attached to metallic leads forming arrangements of molecular junctions.1–3 Further improvement of experimental techniques in combination with the latest achievements in organic chemistry allows to create a huge variety of single-molecule systems.4–12 So far, research mainly concentrated on an exploration of stationary transport properties of these single-molecule nanostructures. However, transient phenomena induced by time-dependent external fields offer new insight into the dynamical characteristics. For example, to understand the interplay between the external coherent control of charge transmission through a molecular junction and the accompanying dissipative processes is of extreme importance for the fabrication of novel types of high speed electronic devices. Those are expected to operate in the terahertz regime since the electronic and vibronic response time of a single molecule typically lies in the picosecond time region. Accordingly, the operation speed can be significantly increased compared to solid state devices. This all strongly motivates the extent of steady state single-molecule transport measurements to the investigation of transient phenomena in molecular junctions. Recently, we have suggested a theoretical model which is ready to describe the transient response of a molecular junction to either a single voltage pulse or a sequence of pulses acting in the ps time-region.13 Here, we consider laser pulse induced transient currents through a single molecule. Exciting a single-molecule junction optically, excitations of the lead are unavoidable. Their consideration becomes less sophisticated if one chooses spherical leads characterized by simple multipole plasmon excitations. In this spirit, we a)Electronic address: [email protected] b)Electronic address: [email protected]

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continue our previous work on laser pulse induced transient currents14 and profit from our studies on effects of the leads plasmon excitations in a molecular junction.15–20 Such work is in line with similar investigations by other groups (for a recent overview, see Ref. 21). The sensitivity of the optical response of a plasmonic cavity to the conductance of the junction has been illustrated in Ref. 22 using a semiclassical model. The transient behavior of a molecular junction perturbed by a laser pulse was considered in Ref. 23 in the framework of Keldysh’s Nonequilibrium Green’s Function formalism (NEGF). There, the influence of localized surface plasmon-polaritons on the transient current could be studied. The transient dynamics in a donor-bridge-acceptor molecular junction were investigated in Ref. 24 also utilizing the NEGF formalism. Based on an equation of motion method, a theory of laser pulse induced currents through a molecular junction could be suggested in Ref. 25. The description accounts for excitation energy transfer between the molecule and the electron-hole pairs of the leads. Reference 26 suggested a single molecule electron pump. An asymmetric donor-bridge-acceptor molecule attached to two metallic leads has been considered and charge pumping caused by external light irradiation even at zero applied bias was suggested. A nearly complete suppression of the transient current through a molecular wire by a short laser pulse has been predicted in Ref. 27. Further on, it was shown that transient molecular dynamics reveal not only time-dependent currents but also time-integrated dc currents.28 And finally, transient currents in a single-molecule photo-diode system were studied in Refs. 29 and 30. In the following, we continue our studies on transient dynamics in molecular junction exposed to laser pulses. The work focuses on a molecular junction formed by two spherical leads and a single molecule sandwiched between both (cf. Fig. 1). The spherical leads are given by two identical gold metal nanoparticles (MNPs). To account for their plasmon

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transient and averaged currents are also presented in this section. Simulation results on transient dynamics of a molecular junction driven by a single laser pulse or a sequence of pulses are shown in Secs. IV and V. Concluding remarks can be found in Sec. VI. II. MODEL OF THE MOLECULAR JUNCTION

FIG. 1. A single molecule attached to two spherical gold nanoparticles forming a left (L) and a right (R) lead. The leads are subject to an applied voltage. Optical excitation of the molecular junction by a single laser pulse or by a sequence of pulses with controllable delay τ del may result in a transient current I (t). Upper right corner: excitation energy scheme of the molecular + and E − are the hybrid plasmon levels of the coupled spherical junction. E pl pl leads displaying strong life-time broadening. Molecular excitation energies E eg are positioned in between the hybrid levels.

excitations, we utilize our recent work on coupled moleculeMNP systems.15–20 There, the molecule and the MNP are treated as a uniform quantum system. The inter-lead and molecule-lead Coulomb-couplings appear as an excitation energy exchange processes between the MNPs and the molecule. It results in an oscillator strength transfer from the leadplasmon excitations to the molecule causing, in particular, an increase of its absorption. Thus, it may strongly affect the conductance of the molecular junction especially if the molecular excitation energy is in resonance to the hybrid plasmon level formed as a result of the plasmon excitation exchange between the two leads. In order to describe the processes of charge transfer in such a molecular junction, we restrict ourselves to the case of weak molecule-lead coupling. This allows us to stay in the regime of sequential electron transmission when a voltage is applied to the leads. The influence of the lead’s plasmon excitations on the transient current has been already studied in a nonequilibrium Green’s-function approach in Refs. 23 and 31. However, intramolecular vibrations as well as effects of intra-molecular vibrational energy redistribution (IVR) were not addressed in these studies. Here, we also account for nonequilibrium vibrational distributions in the molecule and how they affect the transient currents as well as the time-integrated dc in the molecular junction. In order to tackle this problem, we apply our density matrix theory of electron-vibrational dynamics which is particularly suited for a description of transient phenomena in molecular junctions. The molecular electronvibrational dynamics is governed by rates of charging and discharge. Effects of IVR are accounted for by separate rates. Furthermore, the dynamics is also governed by time-dependent laser pulses as well as by processes of excitation energy exchange between the molecule and the leads. The paper is organized as follows. In Sec. II, we briefly describe the model of the molecular junction. The used density matrix theory is explained in Sec. III. The expressions for

The junction model used in the following has been described at several places.14,17,18,20 Here, we only briefly recapitulate the main ingredients. Fig. 2 shows the energy level scheme of the considered molecular junction (the model supposes a symmetrically applied voltage). The total electronvibrational energies are denoted by E N a µ , with N counting the number of excess electrons. We restrict to N = 0, 1, i.e., besides the neutral molecule, only the singly negatively charged state is considered. a = g labels the electronic ground-state, a = e the excited state, and µ denotes the set of vibrational quantum numbers. The quantities E1a,0b = E1a µ=0 − E0bν=0 define the energies necessary to charge the neutral molecule (without vibrational excitations). The inequality E1g,0g > µ X indicates that charging (without electronic excitations) is impossible and that the molecule remains neutral (µ X is the chemical potential of the left/right lead X = L/R). However, the relation E1g,0e = E1g,0g − (E0e µ=0 − E0gν=0) < µ X might be fulfilled indicating possible charging if an excited electronic state of the neutral molecule has been populated. It results a current switch caused by photoexcitation of the molecule.

FIG. 2. Energy level scheme of the junction at two different applied voltages. The electron-vibrational energy E 0a µ of the neutral molecule is combined with the energy E el of an electron of the left lead’s (L) Fermi-sea (blue region). The molecule may be in its electronic ground-state (a = g ) or in its excited state (a = e) and with µ vibrations excited. The energies of the neutral molecule are also combined with the energy Fel of an electron above the right lead’s (R) Fermi-sea (light grey region). The central part M shows the energy E 1aν of the singly charged molecule. The scheme displays charge transmission from L to R as a horizontal transition with molecular charging via L and discharge via R. Charge transmission from R to L is obtained by an interchange of E el and Fel. Panel (a): case of small applied voltage. Charging is only possible from E 0e µ to the energy levels E 1g ν . Discharge may proceed into E 0g µ . Panel (b): case of increased voltage. Charging is possible from E 0g µ and discharge may proceed also to E 0e µ .


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The total Hamiltonian of the molecular junction is given by H(t) = HS (t) + H R + HS−R , where HS is the system Hamiltonian, H R describes different types of a reservoir, and the respective system-reservoir coupling is denoted by HS−R . The system Hamiltonian separates in HS (t) = Hmol + Hpl + Hmol−pl + Hfield(t).

(1)

Hmol describes the electron-vibrational levels of the molecule at different charging states. The coupled plasmon excitations of the two spherical leads are considered by Hpl. Excitation energy exchange between the molecule and the leads is described by Hmol−pl, and Hfield(t) is the coupling to the laser field. The molecular part of the system Hamiltonian reads  Hmol = E N a µ |ΨN a µ ⟩⟨ΨN a µ |. (2) N,a, µ

The molecular wave functions ΨN a µ factorize into the electronic part φ N a and the vibrational part χ N a µ . Often, highfrequency vibrational modes dominate the coupling to the molecular charging and discharge processes. We consider a single mode with dimensionless coordinate Q and with potential energy surface UN a (Q) = UN(0)a + ~ωvib(Q − Q N a )2/4. The Q N a are the electronic state dependent equilibrium values of Q, and UN(0)a is the energy at the equilibrium configuration. The energy E N a is introduced as UN(0)a plus the vibrational zero-point energy ~ωvib/2. So, the overall energies take the form E N a µ = E N a + µ~ωvib Reorganization energies follow as λ N a, M b = ~ωvib/4 × (Q N a − Q M b )2. According to Ref. 17, the plasmon Hamiltonian referring to dipole plasmons of the two spherical leads X = L, R and their interaction is written as   Hpl = ~Ω0|X0⟩⟨X0| + ~Ω I |X I⟩⟨X I| X

+

X, I



VL I, R I ′|LI, R0⟩⟨RI ′, L0| + H.c. .

(3)

I, I ′

The MNP ground-state energies are ~Ω0 and the dipole plasmon energies ~Ω I (identical for both leads). The latter are threefold degenerated with I = x, y, z counting the possible excitations in the three spatial directions. We may set Ω I − Ω0 = ωpl, where ωpl is the dipole plasmon frequency. The interlead energy exchange coupling VL I, R I ′ is taken in a form of a standard dipole-dipole coupling. The transition dipole moments of the spherical leads are d X I = d ple I , where e I are unit vectors of a Cartesian coordinate system. When the two spherical leads are placed along the x-axis particular elements of VL I, R I ′ remain finite,17,18 VL x, R x = −2Vpl and VL y, R y = VLz, Rz 2 3 = Vpl (note Vpl = d pl /Xlead , where Xlead is the distance between the centers of mass of the two spherical leads). The strong coupling between the two leads results in the formation of hybrid dipole plasmon states with the hybrid plasmon energies: E xφ=± = Epl ± 2Vpl and E yφ=± = Ezφ=± = Epl ± Vpl. This restriction to dipole plasmons has been relaxed in the Appendix. Respective computations indicate that the lowenergy part of the hybrid level spectrum is not filled with additional levels but only shifted. Such an energetic shift can be compensated by a slight change of the lead–lead distance or a change of the molecular excitation energy. The newly appearing levels in the higher part of the spectrum might be

only of some importance for higher lying vibrational levels. So we do not expect a qualitative change of the enhancement effect described in the following. The restriction to dipole plasmons already covers the essential physics. The molecule lead-plasmon coupling reads  Hmol−pl = VN, X I |ϕ N e ⟩⟨ϕ N g | ⊗ |X0⟩⟨X I| + H.c. (4) N X, I

The molecule-lead energy exchange couplings VN, X I are also taken in a standard dipole-dipole coupling form.18,20 The remaining coupling to an external laser field is written as ˆ Hfield(t) = − µE(t).

(5)

The dipole operator includes a molecular and a MNP contribution   dmol|ϕ N e ⟩⟨ϕ N g | + d X I |X I⟩⟨X0| + H.c. (6) µˆ = N

X, I

The laser pulse field-strength is written as E(t) = ep E(t)e−iω0t + c.c.

(7)

Here, e p is the unit vector of field polarization, E(t) is the pulse envelope, and ω0 is the carrier-frequency. There are three types of reservoirs involved in our model. The first type is given by the Fermi-sea of the left and the right lead electrons. The related system-reservoir coupling is responsible for molecular charging and discharge  Hmol−lead = VX (1a, 0b, k)a X k s |ϕ1a ⟩⟨ϕ0b | X,k, s

+ VX (0b, 1a, k)a+X k s |ϕ0b ⟩⟨ϕ1a | .

(8)

Lead electron energies are ε X k and electron creation and annihilation is considered by a+X k s and a X k s , respectively. The operators refer to electrons in the lead X = L, R with quantum number k (wave vector k in the bulk case) and with spin s. The matrix element VX (1a, 0b, k) describes charging of the neutral molecule via a coupling to lead X, and VX (0b, 1a, k) is responsible for the reverse process. The second type of reservoir is formed by lead electronhole pair excitations which originate the fast plasmon decay (cf. also Refs. 32–34). The process is characterized by the nonradiative decay rate 2γpl of the dipole plasmons as well as by the dephasing rate γpl. Finally, IVR is accounted for by the third type of reservoir. It is constituted by secondary vibrations which coupling to the high-frequency vibrational coordinate causes finite life times of their excited states (for details, see also Refs. 35–37).

III. DENSITY MATRIX APPROACH

As in our earlier work,14,18,20,38 we will use the density matrix theory to describe the transient behavior of the molecular junction. We utilize the so-called secular approximation and arrive at equations of motion where the dissipative part does not couple diagonal and off-diagonal elements of the density matrix


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i i  ∂ ρα β (t) = − ε˜α β ρα β (t) − Vαγ (t)ργ β (t) ∂t ~ ~ γ  − Vγ β (t)ραγ (t) − δα, β k α→ γ ραα (t) γ

− kγ→ α ργγ (t) .

(9)

The quantum numbers α, β, γ, . . . abbreviate the product states of the molecule and the two MNPs. Since we only consider a low excitation regime, we can restrict the description to singly excited states of the system. Those might be either a molecular excitation or a single plasmon excitation of one of the MNPs. Of course, a superposition state of such states is also allowed. But higher excited states, for example, a double or triple excitation of MNP dipole plasmons should be very unlikely. Consequently, we consider product states of the molecule (two) MNP system of the type |ΨN e µ ⟩|X = 1, 0⟩|X = 2, 0⟩. Here, the excitation is localized at the molecule (both MNPs remain un-excited). The other type of excited states reads |ΨN gν ⟩|X = 1, I⟩|X = 2, 0⟩ or |ΨN gν ⟩|X = 1, 0⟩|X = 2, I⟩. The excitation is localized at one of the two MNPs. The state of the un-excited system (however including possible vibrational excitations) is |ΨN gν ⟩|X = 1, 0⟩|X = 2, 0⟩. The related energies ε α , ε β , . . . determine the complex transition energies ε˜α β = ε α β − i(1 − δα, β )~γα β , where ε α β = ε α − ε β . The imaginary part of the complex transition energies accounts for dephasing 1 k α→ γ − kγ→ α + γα(pd) (10) γα β = β . 2 γ The dephasing rates include transition rates and pure dephasing rates with molecular contributions γmol. All couplings among the states introduced beforehand are comprised by the Vα β . They include energy exchange couplings but in the timedependent variant also excitation and de-excitation processes due to the applied laser pulse. The influence of the latter is accounted for in the so-called rotating wave approximation (see, for example, Ref. 20). A. Transition rates

The equation of motion for the reduced density matrix, Eq. (9), contains three types of transition rates, related to the three types of reservoirs. The rates describing the nonradiative decay of the various plasmon states are 2γpl. The rates which are responsible for the molecular charging take the form  (leads) (X ) k0a = k 0a µ→ 1bν µ→ 1bν =

X 

(X ) Γ0a,1b (E1bν,0a µ )|⟨ χ0a µ | χ1bν ⟩|2

X

× f F(E1bν,0a µ − µ X )

(11)

and those responsible for discharge read  (leads) (X ) k1bν→ = k 1bν→ 0a µ 0a µ =

X 

In the so-called wide-band limit, it becomes energy indepen(X ) dent, i.e., we take Γ0a,1b . The above introduced rates stay valid if plasmons are excited in the leads. Finally, the IVR-rate reads14,35,38 2π JN a (ωvib)[δν, µ+1(µ + 1)n(ωvib) k (IVR) N a µ→ N aν = ~ + δν, µ−1 µ(1 + n(ωvib))]. (14) n(ωvib) denotes the Bose-Einstein distribution and JN a (ωvib) is the spectral density for the coupling of the single highfrequency vibration to the reservoir modes. It is taken as a constant independent on the charging state and the electronic level. B. Tunneling and displacement currents

The total current I X(tot)(t) flowing from lead X into the molecule consists of two parts: the tunneling current I X (t) and the displacement (screening) current I X(dis)(t). While the tunneling current is determined by quantum mechanical sequential electron transfer, the displacement current is caused by the influence of screening charges at the leads. To specify it in somewhat more detail, we follow a reasoning taken in mesoscopic physics39,40 and note that the screening charges are induced by temporal charge accumulation at the molecule during electron transmission. Such a charge accumulation results in a change of the electrostatic potential between the molecule and the adjacent lead X, characterized by the capacitance CX . Then, the displacement current is determined as the time derivative of the screening charges I X(dis)(t) = Q˙ (dis) X (t) accumulated at the respective capacitance CX , whereas the charge accumu lated at the molecule is Q˙ mol(t) = X I X (t). The total charge in the molecular junction is conserved at any time: Qmol(t)  + X Q(dis) X (t) = 0. The introduction of displacement currents guarantees the electrostatic equilibrium of the capacitors after any tunneling event. It also guarantees that the total current  through the molecular junction is conserved, i.e., X I X(tot)(t) (tot) (tot) = 0 or I L (t) = −I R (t). However, the sequential currents I X (t) do not satisfy such a relation since excess charge at the molecule Qmol(t) changes in time. Turning to the total current, it reads39 CX  I X ′(t), (15) I X(tot)(t) = I X (t) − C X′ where C = CL + CR is the total capacitance of the molecular junction. The tunneling current flowing from lead X into the molecule reads35,38  (X ) (X ) I X (t) = I0a→ (t) + I1b→ (t) . (16) 1b 0a a,b

(X ) Γ0a,1b (E1bν,0a µ )|⟨ χ1bν | χ0a µ ⟩|2

X

× [1 − f F(E1bν,0a µ − µ X )].

The expressions cover Franck-Condon factors, the Fermifunction f F(E), and the molecule-lead coupling parameter 2π  (X ) Γ0a,1b (E) = |VX (0a, 1b; k s)|2δ(E − ε X k ). (13) ~ k, s

(12)

It includes the partial current of molecular charging  (X ) (X ) I0a→ (t) = |e| k0a P (t) 1b µ→ 1bν 0a µ

(17)

µ,ν


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and the partial current due to molecular discharge  (X ) (X ) I1b→ (t) = −|e| k1bν→ P (t). 0a 0a µ 1bν

J. Chem. Phys. 142, 224701 (2015) TABLE I. Used parameters (for explanation, see text).

(18)

µ,ν

The lead specific rates have been defined in Eqs. (11) and (12). Note also the introduction of the molecular state populations  P0a µ(t) and P1bν (t). Finally, we quote the relation dt I L (t) = − dt I R (t). The displacement currents do not contribute to the time average of the total current because their time averages are zero. So, they also do not contribute to the dc but only affect the time-dependent ac. C. Mean dc

While it is possible to initiate by a laser pulse transient currents which change on a picosecond or subpicosecond timescale, it is impossible to directly measure these transient currents. Producing, however, such transient currents by identical and repeatedly applied laser pulses, one can measure the total amount of charge transferred by the whole sequence of pulses.28,41 The resulting mean current can be used to characterize the effect of the laser pulse. This scheme has been recently used by us in Ref. 13 to characterize the effect of voltage pulses. If in the present situation a single laser pulse is applied to the junction, a certain (net) number Nel of electrons are transferred from the left lead to the right one,   Nel = dt I L (t)/|e| = − dt I R (t)/|e|. (19) Since the displacement currents average to zero, there is no need to compute them when focusing on Nel. But if the transient current is superimposed by a steady-state current due to the applied voltage, I L (t) and I R (t) have to be properly corrected. We indicate that for the present case, the relation  dt I L (t) = − dt I R (t) replaces the relation I L = −I R valid for thesteady state situation. Accordingly, we may also write Nel = dt I L (t) − I R (t) /2|e|. Let us consider the dc Idc resulting from a huge number np of laser pulses.13 We introduce the time ∆Tp between two laser pulses. Accordingly, the total time of the pulse sequence is (np − 1)∆Tp (or np∆Tp if the relaxation of the system following the last pulse is considered). Then, the dc follows as Idc =

np|e|Nel ≈ |e|Nel/∆Tp. (np − 1)∆Tp

(20)

However, in the subsequent discussion, we exclusively focus on the quantity Nel which can be considered alternatively to Idc. D. Used parameters

Concerning the formation of laser pulse induced transient currents, we are not in the position to compare our data with experimental results. Consequently, our studies do not focus on a particular molecule. Instead, we use a parameterized model which offers the flexibility to describe different scenarios of photoinduced transient current formation (see Table I). As

R MNP ∆x L R E pl γ pl d pl − E pl

10 nm 5 nm 2.615 eV 28.5 meV 2912 D 1.927 eV

k BT d mol γ mol Γ ∆E 10 ~ω vib Q 0g Q 0e Q 1g Q 1e J E0 τp

3 meV 5D 5 meV 0.1, . . . , 1 meV 0.5, . . . , 1.68 eV 62.5 meV 0 4 6 2 0, . . . , 5 meV 105 V/m 0.5, . . . , 50 ps

the MNPs, we take two identical Au spheres with a diameter of 20 nm and with a dipole plasmon excitation energy around Epl = 2.6 eV (for the respective dipole moment d pl and the plasmon damping γpl, see also Table I). To stay in the regime of dominant dipole-dipole coupling either for the inter-lead as well as for the molecule-lead energy exchange coupling, we chose the lead surface-to-surface distance ∆x L R = 5 nm. Optical excitation is always taken to be resonant to the molecular transition. A Gaussian shaped pulse is taken with E(t) = E0 exp(−4(t − t p)2/τp2) (pulse maximum at t p and pulse length τp). Considering a pair of laser pulses, we take identical pulses but separated by the delay time τdel. In order to stay in the weak excitation limit even if plasmon enhancement becomes dominant, we set E0 = 105 V/m. The transition dipole moment of the molecule shall point in the x-direction (cf. Fig. 1) and shall take the typical value d mol = 5 D. The molecular transition couples to the lower MNP hybrid state with energy E x− ≡ Epl− . For the chosen lead surface-to-surface distance, Epl− amounts to 1.927 eV. The relative charging energy ∆E10 = E1g µ=0 − E0gν=0 − µ0 is set equal to 0.5 eV. We further assume that the excitation energy of the neutral molecule E0e µ=0 − E0gν=0 ≡ E0eg coincides with one of the charged molecule E1e µ=0 − E1gν=0. Moreover, we consider the molecule to be symmetrically coupled to the leads with the molecule-lead electron trans(L) (R) fer coupling parameter Γ0a,1b = Γ0a,1b ≡ Γ. According to the considered sequential charge transmission regime, Γ has been chosen in the range from 0.1 to 1 meV. The corresponding molecular charging (discharge) time τch can be estimated as ~/Γ ≈ 0.6–6 ps. Concerning the molecular vibrations, we proceed as in our earlier study of Ref. 14 and take the values also quoted in Table I. The strength of IVR is characterized by the parameter J. It shall be independent on the electronic state of the molecule and has been chosen between 0 (absence of IVR) and J = 5 meV (strong IVR). The characteristic time of IVR follows as τIVR = ~/2π J and varies between τIVR = ∞ and τIVR = 20 fs (very fast IVR).


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IV. TRANSIENT CURRENTS DUE TO A SINGLE LASER PULSE

We start our discussion with the consideration of transient currents induced by a single laser pulse (for the junction and laser pulse parameters, see Table I). Fig. 3 displays respective results for the transient currents I L (t) and −I R (t) induced by laser pulses with different durations. The total currents I L(tot)(t) and I R(tot)(t) are presented in Fig. 4. Respective tunneling currents in the absence of a plasmon enhancement can be found in Fig. 5. All simulations have been performed at different strengths of IVR, for the single value Γ = 1 meV

(tot)

(tot)

FIG. 4. Total currents I L (t) (thick lines) and I R (t) (thin lines) versus time for a simultaneous excitation of the molecule and the leads. Upper panel: τ p = 0.5 ps, middle panel: τ p = 5 ps, lower panel: τ p = 50 ps. Variation of the IVR strength, solid line (black): J = 0, dashed line (red): J = 0.1 meV, chain-dotted line (blue): J = 1 meV (Γ = 1 meV, E 0eg = E −pl , other parameters according to Table I).

FIG. 3. Laser pulse induced transient currents I L (t) (thick lines) and −I R (t) (thin lines) versus time for a simultaneous excitation of the molecule and the leads. Upper panel: τ p = 0.5 ps, middle panel: τ p = 5 ps, lower panel: τ p = 50 ps. Variation of the IVR strength, solid line (black): J = 0, dashed line (red): J = 0.1 meV, chain-dotted line (blue): J = 1 meV (Γ = 1 meV, E 0eg = E −pl , other parameters according to Table I).

(corresponding residence time of the charge in the molecule is τres ≈ 0.6 ps), and at E0eg = Epl− . The chosen values of the charging energy and the applied voltage guarantee the absence of any current if no laser pulse excitation is present. Note that the temporal behavior of the currents shown in Fig. 5 is identical with the results of Ref. 14, Fig. 8. However, the absolute values of the currents presented here are much smaller since E0 is weaker. Let us consider the case of ultra-short laser pulse excitation with τp = 0.5 ps. Comparing Figs. 3 and 5, we notice the strong plasmon-induced enhancement of the transient currents I L (t) and −I R (t) which amounts about five orders of


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rents follow more directly the temporal profile of the laser pulse and the maximum values of the transient currents increase with increasing pulse duration. The delay between the maximum of I L (t) and of −I R (t) becomes less pronounced indicating that charge injection from the left lead into the molecule coincides with charge outflow into the right lead. A pronounced nonequilibrium behavior of the current can be observed for the absence of IVR (τIVR = ∞). The transient current undergoes a self-stabilization, i.e., the nonequilibrium state of the junction persists although the optical excitation is switched off.14 The conservation of total currents, i.e., I L(tot)(t) = −I R(tot)(t) is illustrated in Fig. 4. Again, at τIVR = ∞, a self-stabilization of the current appears. At finite strength of IVR, the changes in the curves are less pronounced. The case of a somewhat weaker molecule-lead coupling Γ = 0.1 meV (residence time τres = 6 ps) is considered in Fig. 6 where the tunneling currents in the presence of a

FIG. 5. I L (t) (thick lines) and −I R (t) (thin lines) versus time as in Fig. 3 but ignoring lead-plasmon excitations.

magnitude. These huge differences also appear for the case of longer optical excitation (middle and lower panels of Figs. 3 and 5). The strong increase of molecular absorption caused by the resonant coupling to the lead’s hybrid plasmon excitation results in a strong current enhancement. We further underline that the inequality I L (t) > −I R (t) indicates transient molecular charging, whereas I L (t) < −I R (t) is a signature of discharge. In the first 2 ps time-interval of the 0.5 ps laser pulse excitation, we have I L (t) > −I R (t), and I R (t) > 0, what is the result of charge injection from the left lead and also from the right lead (upper panels of Figs. 3 and 5). Then, it follows −I R (t) > 0 indicating that charge outflow from the molecule to the right lead takes place. At t > 2 ps, both currents decrease, and for the case of finite IVR strength they vanish. Optical excitation by the two longer laser pulses with τp = 5 ps and τp = 50 ps is characterized by the inequality τp > τres (middle and lower panels in Figs. 3 and 5). Now, the cur-

FIG. 6. Transient currents as in Fig. 3 but with Γ = 0.1 meV.


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J. Chem. Phys. 142, 224701 (2015)

FIG. 8. Transient currents (upper panel) with I L (t) (thick lines) and with −I R (t) (thin lines) and transient total currents (lower panel) versus time. Change of the molecule hybrid plasmon detuning. Solid line (black): E 0eg − , dashed line (red): E − = E pl 0eg = E pl + 0.1 eV, chain-dotted line (blue): E 0eg − + 0.25 eV (∆E = E pl 1g ,0g = 0.5 eV, τ p = 0.5 ps, ~ω 0 = E 0eg , J = 1 meV, other parameters according to Table I).

FIG. 7. Transient total currents as in Fig. 4 but with Γ = 0.1 meV.

molecule lead-plasmon coupling are drawn (compare with Fig. 3). Now, the slower discharge induces a time interval where I L (t) > −I R (t), and I R (t) > 0 (simultaneous charge injection from both leads). Fig. 7 displays respective total currents. As above, a strong plasmon induced enhancement of the transient current is obtained what becomes obvious if one compares with data obtained without the consideration of lead excitations (not shown). The influence of a detuning between the molecular excitation energy E0eg and the lower hybrid plasmon level Epl− is shown in Fig. 8. As expected, an increasing detuning reduces the magnitude of the tunnel currents and the total currents. However, perfect resonance is not necessary to observe an effect. A. Number of transferred electrons

As already described in Sec. III C, a reliable scheme to get access to phenomena related to picosecond transient currents

is to measure the dc induced by a continuous sequence of laser pulses. The dc is proportional to the quantity Nel, Eq. (19), which is the net number of electrons transferred from the left lead to the right lead due to the action of a single laser-pulse. Computations of Nel will be presented for the relative charging energy ∆E10 = 1.68 eV. Such a large value has been chosen to avoid the formation of a current in the absence of laser pulse excitation. For the taken value of ∆E10, charge transmission only starts at the critical voltage of Vc = 2∆E10/e = 3.36 V. Thus, for the low values of the applied voltage, only optical excitation can induce currents flowing through the junction. Fig. 9 displays Nel versus the applied voltage V what can be understood as a dynamical IV-characteristic. (If the laser pulses are properly arranged, overheating of the junction can be circumvented.) The step-like behavior of Nel reflects the consideration of a single high-frequency molecular vibration. According to the energy level scheme in Fig. 2, molecular charging after photoexcitation is possible if Eel + E0e µ = E1gν . Here, we combined the energy of occupied electron levels of the left lead Eel with the energy of the excited electronic state of the neutral molecule E0e µ . Discharge takes place if E1gν = E0g µ + Fel, where Fel denotes energies of unoccupied levels of the right lead. Consequently, higher vibrational states of the molecule can be populated and a nonequilibrium distribution across the vibrational levels may be established. At the same time, discharge of the molecule is also possible via transitions from the electronically excited charged molecule to the excited


224701-9


FIG. 9. Number Nel of transferred electrons caused by a single laser pulse −, of 0.5 ps duration versus applied voltage V (∆E 10 = 1.68 eV, E 0eg = E pl other parameters according to Table I). Variation of the IVR strength. Blue chain-dotted line: J = 0.1 meV, red dashed line: J = 1 meV, black solid line: J = 5 meV.

neutral molecule: E1eν = E0e µ + Fel. But the population of P1e µ stays small and the transition does not contribute. Thus, only different excited vibrational states of the charged molecule’s electronic ground state become populated leading to the steps in Nel versus V . Noting the different curves in Fig. 9 for the different values of the IVR-strength, one realizes that Nel is approximately independent on IVR at small applied voltages (V < 0.4 V). There is only a small enlargement of Nel at V < 0.4 V in the case of a slow IVR (J = 0.1 meV) in comparison to intermediate and fast IVR (J = 1 or 5 meV). It can be explained by the fact that for slow IVR, higher excited vibrational states of charged molecule’s states can be populated and contribute to Nel. At larger applied voltages, the effect of IVR on Nel becomes more pronounced. First, we underline that the height of the step near the critical bias Vc = 2(E0eg − ∆E10)/e = 0.5 V is the largest one. This is due to an additional discharge from the charged molecule being in its ground-state with energy E1gν . Charge transition proceeds to the right lead (see right diagram of Fig. 2), but the subsequent neutral molecule remains in

J. Chem. Phys. 142, 224701 (2015)

FIG. 11. Number Nel of transferred electrons as in Fig. 9 but for different laser pulse amplitudes E 0. Black solid line: E 0 = 106 V/m, red dashed line: E 0 = 5 × 105 V/m, blue chain-dotted line: E 0 = 105 V/m (∆E 10 = 1.68 eV, E 0eg = E −pl , J = 1 meV).

its excited electronic state. Accordingly the relation E1gν = Fel + E0e µ has to be fulfilled. This type of discharge process overcomes other possible processes where the neutral molecule stays in its in electronic ground-state. By increasing V , the transition into higher electronvibrational states of the electronically excited molecule becomes possible. These transitions may be connected with smaller Franck-Condon factors. Consequently, the mean number of transferred electrons is reduced. In contrast, at strong IVR, discharge starts from E1gν=0 into lower excited vibrational states (ideally into the vibrational ground-state of the excited molecule). These transitions are more favorable because they are characterized by larger Franck-Condon factors. So, fast relaxation processes at large applied voltage increase the number of transferred electrons and, thus, the respective dc. Interestingly, such a current increase with increasing strength of IVR has been already described in Ref. 42 for steady-state IV characteristics. Fig. 10 illustrates the effect of a detuning of the molecular excitation with respect to the lower hybrid level of the lead plasmons. The behavior of Nel versus V at different values of the laser pulse amplitude E0 is displayed in Fig. 11. Decreasing the detuning or increasing E0 leads to an increase of Nel.

V. TRANSIENT CURRENTS DUE TO A PAIR OF LASER PULSES

FIG. 10. Number Nel of transferred electrons as in Fig. 9 but for different detunings of the molecular excitation energy with respect to the lower plasmon hybrid level. Black solid line: E 0eg = E −pl , red dashed line: E 0eg = E −pl + 0.1 eV, blue chain-dotted line: E 0eg = E −pl + 0.25 eV (J = 1 meV, ∆E 10 has been chosen as 1.68, 1.78, and 1.92 eV, respectively).

Correlations among excitation processes can be considered if a first pulse is followed by a second one and if the temporal delay is varied. This has been suggested recently in Ref. 28 and is reminiscent of pump-probe experiments common in laser spectroscopy. Each of the laser pulses shall have the same amplitude of E0 = 105 V/m and the same duration of τp = 0.5 ps. The delay time between both is given by τdel. Figs. 12 and 13 display I L (t) and −I R (t) for different τdel and for a molecule-lead coupling Γ = 1 meV and Γ = 0.1 meV, respectively. The simulations have been performed for the simultaneous excitation of the molecule and the leads and at E0eg = Epl− .


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FIG. 12. Transient currents I L (t) (upper panel) and −I R (t) (lower panel) due to a pair of laser pulses versus time (E 0 = 105 V/m, τ p = 0.5 ps, V = 0.75 V, − , ∆E = 0.5 eV, J = 1 meV, Γ = 1 meV, remaining parameters E 0eg = E pl 10 due to Table I). Variation of the delay time τ del. From lower to upper panel: τ del = 0, 1, 3, 6 ps.

For the large delay time, τdel = 6 ps I L (t) and −I R (t) display substructures which behave independently from each other and are similar to those of a single pulse excitation (upper panel of Fig. 3, chain-dotted blue curves). Changing to smaller τdel, the current structures start to interfere. At τdel = 0, we arrive at a current form like in the case of a single pulse excitation but with an enlarged pulse amplitude. A. Number of transferred electrons

Nel versus τdel is shown in Fig. 14. The computation has been performed for the case of a simultaneous excitation of

J. Chem. Phys. 142, 224701 (2015)

FIG. 13. Transient currents I L (t) (upper panel) and −I R (t) (lower panel) versus time as in Fig. 12 but for a molecule-lead coupling Γ = 0.1 meV.

the molecule and the leads, choosing E0eg = Epl− and different strengths of IVR. The independence of Nel on τdel for large values of τdel reflects the fact that the currents induced by each of the pulses contribute independently to Nel. The values of Nel at τdel > 1 ps are approximately twice as large as those calculated for the case of a single pulse, Fig. 9. If τdel decreases, laser pulse overlap increases Nel. Finally, let us briefly comment on the relation between the delay time and the response time of the molecular junction. The latter depends on the resistance of the molecule Rmol and its capacitance C as τRC = RmolC. A simple calculation utilizing experimental data on the single molecule resistance43 and capacitance44 allows us to compute the RC time. Consequently, it lies in the range of 0.1, . . . , 1 ps.13 On the other hand, a response time of molecular junction is the inverse to


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J. Chem. Phys. 142, 224701 (2015)

APPENDIX: CONSIDERATION OF MULTIPOLE PLASMONS

We change to a consideration of higher lead multipole plasmons to judge their influence on the current enhancement effect (their importance has been already addressed in Ref. 46). The introduction of multipole plasmons is based on our earlier work of Ref. 16 where a scheme has been presented to establish a multipole plasmon Hamiltonian for a spherical MNP. The generalization of the dipole plasmon Hamiltonian, Eq. (3), referring to a MNP dimer, to an inclusion of all multipole plasmon excitations and their mutual coupling reads  Hpl = El |X, lm⟩⟨X, lm| X,l m

FIG. 14. Number Nel of transferred electrons, Eq. (19), caused by a pair of laser pulses versus delay time (∆E 10 = 1.68 eV, other parameters as in Fig. 12 and according to Table I). Variation of the IVR strength. Blue chain-dotted line: J = 0.1 meV, red dashed line: J = 1 meV, black solid line: J = 5 meV.

the tunneling rates, i.e., τRC ∼ 1/Γ (see, for example, Ref. 45). In the simplest consideration, one can argue that if τdel ≫ τRC, τs, the molecular junction, can always enter its equilibrium state before the next pulse is switched on. It allows us to consider each of the pulses independently and to interpret Nel as the average number of transferred electrons from one lead to another.41

VI. CONCLUSIONS

The transient response of a molecular junction excited by a single laser pulse or by a sequence of pulses was analyzed. The sequential transmission regime, intramolecular vibrations, and vibrational relaxation could be considered by utilizing a density matrix description. Optical excitations of the leads were included by introducing collective plasmon excitations which also couple to the molecular excitations. Transient currents as well as averaged dc resulting from a huge sequence of laser pulses were calculated. While the transient currents give some insight into the dynamics of the junctions, the averaged dc is ready to be measured in the experiment. Our computations demonstrate strong current enhancement due to the coupling to the lead plasmon excitations. A vibrational progression becomes visible if the dc caused by a huge sequence of laser pulses (or the mean number of transferred electrons) is drawn versus the applied voltage. This can be considered as a dynamically determined IV-characteristic. IVR has a distinct influence on all considered quantities. If the dc caused by a huge sequence of pairs of laser pulses is drawn versus delay time, its absolute value increases with increasing strength of IVR. At the same time, the differences of the dc at zero and very large delay time increase too. Ongoing work tries to find out if this difference can be used as a signature for the importance of IVR. ACKNOWLEDGMENTS

Financial support by the GIF research Grant No. 114673.14/2011 is gratefully acknowledged.

+

 ( ) Vl m,l ′m′|L, lm⟩⟨R, l ′m ′| + H.c. . (A1) l, m,l ′m ′

The ground state energy has been set equal to zero and the El are multipole plasmon excitation energies following the Mie formula (3l/(2l + 1))1/2 El=1 (El=1 is the dipole plasmon energy taken from experiment). Moreover, the |X, lm⟩ describe a state where in the MNP X = L, R, a multipole plasmon labeled by the multipole indices lm is excited. The transfer of these multipole excitations from the right to the left MNP is accounted for by Vl m,l ′m′ (the multipole indices lm belong to the left sphere and the l ′m ′ to the right one). The matrix elements Vl m,l ′m′ result from a two-center multipole expansion (see, for example, Ref. 47) of the general excitation energy transfer coupling,  enk0(x)en∗k ′0(x′) . (A2) Vk,k ′ = d 3x d 3x′ |X + x − x′| The quantum numbers k and k ′ label an arbitrary electronic excitation of the left and the right MNP, respectively. These excitations are represented by the electronic transition densities  nk0(x) = Nel dx δ(x − x1)ψk∗ (x)ψ0(x). (A3) Here, x = {x1, . . . , x Nel} labels the whole set of MNP electronic coordinates and their total number is given by Nel. The excited state wave function is ψk and that of the ground-state ψ0. In order to deduce an expression for Vl m,l ′m′, we note the following expansion (cf., e.g., Ref. 48):  1 = (−1)m Il,−m (X)ql m (x − x′). (A4) |X + x − x′| l, m The Il m = ql m/x 2l+1 are defined by the single-particle multipole operator  4π l ql m (x) = x Yl m (ϑ, ϕ). (A5) 2l + 1 According to Ref. 47, a further expansion of ql m (x − x′) results in 1/2

′  1 l ′+m+m ′*2l + 2l + = (−1) |X + x − x′| l, m l ′, m′ , 2l -

× Il+l ′,−m−m′(X)ql m (x)ql ′m′(x′)

× ⟨l, m; l ′, m ′|l + l ′, m + m ′⟩,

(A6)


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J. Chem. Phys. 142, 224701 (2015)

TABLE II. Multipole excitation energy transfer coupling Vl m, l ′m ′ (in meV) according to Eq. (A8) (two spherical Au leads with 10 nm radius and a surface-to-surface distance of ∆x L R = 5 nm). l=1 m = −1

l=2

m=0

m=1

m = −2

m = −1

m=0

m=1

m=2

l=1

m = −1 m=0 m=1

−502 0 167

0 334 0

167 0 −502

303 0 −61

0 −171 0

−149 0 149

0 171 0

61 0 −303

l=2

m = −2 m = −1 m=0 m=1 m=2

−303 0 149 0 −61

0 171 0 −171 0

61 0 −149 0 303

256 0 −90 0 22

0 −147 0 88 0

−90 0 132 0 −90

0 88 0 −147 0

22 0 −90 0 256

where the ⟨l, m; l ′, m ′|l + l ′, m + m ′⟩ are Clebsch–Gordan coefficients.48 Inserting this expansion into Eq. (A2), we arrive at the MNP transition multipole moments,16  Q l m (k0) =

d 3x ql m (x)nk0(x).

(A7)

Applying the so-called plasmon resonance approximation,16 the Q l m (k0) deviate from zero only if the general MNP excitation with quantum number k coincides with a multipole excitation with multipole indices lm. The resulting Q l m can  be approximated by a2l+1 El /2, where a is the MNP radius. So, we arrive at

Vl m,l ′m′ =



(−1)

l, m l ′, m ′

l ′+m+m ′*2l

+ 2l ′+ 2l -

1/2

, × Il+l ′,−m−m′(X)⟨l, m; l ′, m ′|l + l ′, m + m ′⟩ × Q l mQ l ′m′.

(A8)

FIG. 15. Formation of hybrid plasmon energy levels originated by a pair of spherical MNPs (Au, radius of 10 nm, surface-to-surface distance ∆x L R = 5 nm). Upper panel: hybrid levels caused by a consideration of dipole plasmons only, lower panel: inclusion of quadrupole plasmons. TABLE III. Molecular dipole MNP multipole coupling matrix elements Vl m (in meV) according to Eq. (A9) (spherical Au lead with 10 nm radius and a molecule-to-MNP-surface distance of 2.5 nm). l=1

Table II displays the magnitude of the V up to a quadrupole–quadrupole coupling and for a MNP (lead) configuration as used in the main text (cf. Table I). The data of Table II indicate that the dipole–quadrupole and quadrupole–quadrupole coupling have a similar strength as the dipole–dipole coupling. To characterize the influence of higher multipoles (here quadrupole moments) on the MNP hybrid state formation, we diagonalized the plasmon Hamiltonian, Eq. (A1) (see also the earlier work in Ref. 49). Fig. 15 compares the result with that one which is obtained if a restriction to dipole plasmons has been taken. Therefore, the energy splitting of dipole plasmons is compared with the splitting due to an inclusion of quadrupole plasmons. As it has to be expected, more hybrid levels appear if quadrupole plasmons are included. However, the low energy part of the hybrid state spectrum is only slightly affected (energy level shift of 230 meV, cf. also Ref. 49). So we cannot expect a fundamental change of the current enhancement effects if we include higher multipoles provided that the molecular excitation is in resonance with the lowenergy part of the plasmon hybrid spectrum. To be complete, we quote the values of the molecule multipole plasmon energy transfer coupling (it generalizes the molecule dipole plasmon coupling of Eq. (4)). According to l m,l ′m ′

m = −1 −6.5

l=2

m=0

m=1

m = −2

m = −1

m=0

m=1

m=2

0

6.5

−7.0

0

5.7

0

−7.0

Ref. 16, we get 

4π Yml (ϑ, ϕ) Q∗ [dmol∇X] . (A9) 2l + 1 l m X l+1 Respective data of Table III indicate the importance also of the molecular-dipole MNP-quadrupole coupling. However, it does not affect the current enhancement in a noticeable way since the quadrupole plasmon is out of resonance to the molecular excitation (cf. Fig. 15). Vl m =

1E.

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