A simple model for laser-electrode interaction and its role in photo-assisted electron transport processes in molecular interfaces

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A simple model for laser-electrode interaction and its role in photo-assisted electron transport

processes in molecular interfaces

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2005 J. Phys. B: At. Mol. Opt. Phys. 38 3779

(http://iopscience.iop.org/0953-4075/38/21/001)

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INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS

J. Phys. B: At. Mol. Opt. Phys. 38 (2005) 3779–3794 doi:10.1088/0953-4075/38/21/001

A simple model for laser-electrode interaction and itsrole in photo-assisted electron transport processes inmolecular interfaces

I Urdaneta1,2, A Keller1, O Atabek1 and V Mujica2

1 Laboratoire de Photophysique Moleculaire du CNRS, Bat. 210, Universite Paris-SUD,Campus d’Orsay, 91405 Orsay Cedex, France2 Universidad Central de Venezuela, Facultad de Ciencias, Escuela de Quımica, Apartado 47012,Caracas 1020A, Venezuela

E-mail: [email protected]

Received 12 May 2005, in final form 26 July 2005Published 10 October 2005Online at stacks.iop.org/JPhysB/38/3779

AbstractA simple model to assess the relative importance in photo-assisted conductanceof laser-induced excitation processes in metal electrodes is presented. Weconsider both one and two electrodes subject to the combined effect of adc bias and a time-dependent field using a model for the photo-induceddynamic electron distribution function. Floquet formalism is used to analysethe multiphoton processes inducing electron tunnelling through the surface.Under the assumption of low laser intensity, the relative importance of the directcurrent and of the one resulting from photon absorption are carefully examinedby varying the characteristics of the external fields (applied voltage for the staticfield, intensity and frequency for the laser). For the one-electrode case, ourapproach renders results that are consistent with other available treatments forphotofield emission current. Under certain conditions, that may be reproducedin current experiments, we observe a great sensitivity of the single-photoncurrent with respect to the laser intensity and frequency, advocating for the useof a laser modified electron distribution in the metal instead of the commonlyused equilibrium Fermi–Dirac distribution. However, for the case of a vacuumjunction with two metal electrodes, we find that these transient effects canbe safely neglected for the transport problem, under current experimentallimitations. This seems to justify commonly used models that include onlythe molecule–laser interaction in the calculation of the conductance but leavethe possibility open for measurable consequences in situations where there iscompetition between through-space and through-molecule transport.

0953-4075/05/213779+16$30.00 © 2005 IOP Publishing Ltd Printed in the UK 3779

3780 I Urdaneta et al

1. Introduction

Individual molecule bridged nano-contacts are one of the main subjects of interest for thedevelopment of molecular electronic devices [1–4]. Controlled laser irradiation results intoconsiderable enhancement (up to several orders of magnitude) of the conductance of suchsystems, opening the way to the nanoscale switch design. Up to now, theoretical approacheshave emphasized the role of the molecule–laser interaction in modifying the electron tunnellingcurrent between two electrodes through a molecular wire [5–9]. Within the assumption of lowinter-electrode bias voltage and low laser intensity, not only the through-space current fromone electrode to the other has been neglected, but also the whole laser-electrode excitationand relaxation processes have not been taken into account. Photon–leads interactions maybe classified according to the laser frequency. For high frequencies (one-photon processes)or high intensities (multiphoton processes) a photoelectric [10] or photoemission [11] currentPEC is generated from the metal surface that may dominate over other currents if the electrodesare close enough. For frequencies below the electrode’s work function, a photofield emissioncurrent PFEC [12–17] is expected as a result of electron tunnelling from the metal surface andcan modify the characteristics of the total current. References [10, 11] deal with the first familyof processes (PEC). The aim of this work is to construct a simple, but general enough modelto accommodate processes belonging to the second family (PFEC) and estimate their relativeimportance in the description of electronic transport in metal–molecule–metal nanojunctions.This is an important step towards a comprehensive model for laser-assisted conduction inthese systems, because in most models, as those employed by us and other authors [5–9], thephoto-induced processes in the electrodes are neglected.

The most interesting case for nanojunctions, and the one we consider here, is that of apulsed laser. This is because of the severe limitations imposed by the need to preserve thethermal stability of the junction, which translate into important constraints in our theoreticalmodel where lattice heating, which would occur even for moderate increases of the temperature,is neglected. Use of a pulsed laser also means that, to make a meaningful comparison betweentransient and stationary currents, one has to take into account the time scale of the photo-induced current as compared to the relevant experimental time scales.

Section 2 is devoted to the theoretical model. This includes the calculation of thewavefunction of the normal motion of the electrons from the metal surface, described bythe Sommerfeld model, to the vacuum under the effect of the two external fields described bya vector potential (for the laser) and a bias potential (for the dc field). Floquet formalism is usedfor the multiphoton excitation process within a one-dimensional spatial description. Next, wepresent the calculation of the PFEC current for the one-electrode case and the total current forthe two-electrodes situation. Different statistics concerning the stationary electron distributionfunction are considered. The equilibrium Fermi distribution around the metal Fermi energyis used as a lowest order approximation, and the deviation from equilibrium induced by laserexcitation, which takes into account the electron–electron and electron–photon collision terms,is introduced following the derivation of [27–29, 31], as a first-order correction term assuminglow laser intensity. This correction measures the response of the medium resulting from thepenetration of the electromagnetic wave in the electrode region. The results are presentedin section 3 by analysing the sensitivity of the current to different parameters: bias voltage,electromagnetic field intensity and frequency and interelectrode distance. In both cases, themain result is that the through-space current can be strongly influenced by the laser in a timescale that depends on the duration of the pulse. The experimental possibility of detectingsuch effects will depend on the repetition rate of the laser and the response time of theelectronics. Despite this apparent limitation, our results bring information in situations where

Laser-electrode interaction and its role in photo-assisted electron transport processes 3781

the through-space component of the current can compete or interfere with the through-moleculecurrent.

2. Theoretical model

Most of the quantum formalism is common to the one- and two-electrode cases. For simplicity,we present the details for the single-electrode situation and generalize them accordingly whenneeded.

2.1. The wavefunction

The normal motion of the electron is described within a one-dimensional spatial model labelledby the coordinate x. The metal occupies the half space x < 0, which is referred to as region I.In the Sommerfeld model, neglecting the radiation field penetration effects, the target potentialis

V (x) = −V0�(−x), (1)

where �(−x) is the Heaviside stepfunction and V0 is the surface potential confining a freeelectron gas given as the sum of the work function φ and the Fermi energy EF of the metal:

V0 = φ + EF. (2)

The external electric field acts in vacuum region II, where x > 0. The normal component ofthe vector potential of the electromagnetic field is written as

A(x, t) = A0(x) cos(ωt), (3)

ω being its frequency. Since the electromagnetic field penetrating inside the metal representsless than 0.1% of the incident initial field intensity, the number of electrons whose motion isaffected by the absorbed field is not considered and

A0(x) = A0�(x). (4)

The static electric field potential results from the application of a bias voltage W = Fx inregion II:

W(x) = −eFx�(x), (5)

−e being the electric charge of the electron. Finally, the time-dependent total Hamiltonian, inthe velocity gauge, is

H(x, ωt) = 1

2m

(ih

∂

∂x− eA(x, t)

)2

+ V (x) + W(x), (6)

m being the mass of the electron. The different potential functions involved in equation (6) areschematically illustrated in figure 1. Among different formalisms to treat this problem is theBloch–Nordsiech representation which operates a coordinate transformation that eliminatesthe field (see for instance [37]). In particular, such an approach has been used to obtainthe exact solution of a charged particle submitted to a linear potential and a time-dependentexternal field [38]. Here we are rather referring to a Floquet approach which clearly identifiesthe relative contributions of the multiphoton processes.

The time-dependent Schrodinger equation TDSE to be solved is

ih∂

∂t�(x, t) = H(x, ωt)�(x, t). (7)

3782 I Urdaneta et al

φ

−E

−Vx<0 x=0 x > 0

I II

Ef

Ef=Ei+hω

Ef=Ei

PFEC−F x

Ei

ωh

METAL

n = 0

n = 1

0

F

0

Figure 1. Schematic view of the theoretical model (potential in region I (x < 0) and II (x > 0))and of the effect of the tunnelling on the photofield current PFEC as a function of the electron finalenergy Ef . All the notations are those of the main text.

The periodic time behaviour of H,

H(x, ωt + 2π) = H(x, t), (8)

allows us to use a Floquet approach where the scattering process is considered as timeindependent, but in an enlarged Hilbert space [8, 33, 35]. The Floquet Hamiltonian

K(x, θ) = H(x, θ) − ihω∂

∂θ, (9)

acting in the extended Hilbert space is time independent, with θ a new degree of freedom. Theimportant point, with respect to the periodic structure of the θ -dependence, is that if �λ(x, θ)

is a so-called Floquet state, i.e. an eigenfunction of K with λ as eigenvalue:

K(x, θ)�λ(x, θ) = λ�λ(x, θ), (10)

then �λ+nhω(x, θ) e−inθ , n ∈ Z, is also an eigenfunction of K corresponding to an eigenvalue(λ + nhω). In region I (x < 0), the eigenfunctions of the Floquet Hamiltonian (9) are planewaves:

�±E,n(x, θ) = e±iknx einθ , (11)

where E is the initial energy of the electron prior to the photon interaction and n is the numberof absorbed (n < 0) or emitted (n > 0) photons. The wavenumber kn for channel n is givenby

kn =[

2m

h2 (E + nhω)

]1/2

. (12)

The general solution of the TDSE for region I is a linear superposition of the eigenfuctionsequation (11) with appropriate coefficients:

�IE(x, θ) =

∑n

[t In(E) eiknx + r I

n(E) e−iknx]

e−inθ , (13)

where t In(E) and r I

n(E), to be determined later, are the transmission and reflection amplitudesfor each photon channel n at the energy E, respectively. In region II (x > 0), the analytic formof the Floquet eigenfunctions can be derived in terms of the Airy–Gordon–Volkov (AGV)waves [35, 38].

�±E,n(x, θ) = e−iF(θ) e−iG(θ)Ai±(Yn(x, θ)) e−inθ , (14)

Laser-electrode interaction and its role in photo-assisted electron transport processes 3783

where Ai±(z) = Ai(z) ∓ iBi(z) is the appropriate outgoing or incoming combination of Airyfunctions. The argument of the Airy functions Yn(x, θ) is

Yn(x, θ) = −K0

(x +

eA0 sin θ

mω+

E − e2A20

/4m + nhω

eF

), (15)

with K0 = (2meF/h2)1/3, a wavenumber. The arguments of the exponentials of equation (14)are given by

F(θ) = e2FA0 cos(θ)

hmω2, (16)

and

G(θ) = e2A20 sin(2θ)

8hmω. (17)

It is worthwhile noting that these solutions cannot be continuously extrapolated to F = 0(no static field), due to the diverging behaviour of the imaginary parts of the Airy functionsdescribing the electronic motion in the classically forbidden regions of large spatial extension.The general solution of the TDSE in region II is then derived as a linear combination of theFloquet states with their periodic time structure:

�IIE(x, θ) = e−i[F(θ)+G(θ)]

∑n

t IIn Ai+(Yn(x, θ)) e−inθ + r II

n Ai−(Yn(x, θ)) e−inθ , (18)

t IIn (E) and r II

n (E) being now the transmission and reflection amplitudes in region II,respectively.

The complete determination of the wavefunctions �E(x, θ) in each sector necessitates thecalculation of the unknown transmission and reflection coefficients. This is done by ensuringcontinuity of the wavefunction and of the current defined as (h/m) Im(�∗∂x� − ieA|�|2),at x = 0, for all values of θ (‘Im’ stands for the imaginary part). For practical purposes thesummation over the photon number index n is limited to the range −N < n < N , whereN is the maximum number of photons effectively absorbed or emitted by the electrons inthe metal surface. This number depends on the intensity of the laser. Taking into accountthe θ dependance, additional matching has to be done, either by identification of the Fouriercomponents, or by identification of the functions at a set of (2N + 1) phase points [36]. Thenumber of these phase points in the Floquet formalism

θ = 2π(i − 1)

(2N + 1), i = 1, 2, . . . , 2N + 1, (19)

is chosen such that the amplitudes of the waves on each sector could be related through asquare matrix. The total number of amplitudes and of equations is 2(2N + 1) since bothfunctions and their derivatives must match. The system of equations has a dimension of2(2N + 1) × 2(2N + 1) which can be written in a compact form as

M Iτ I = M IIτ II, (20)

where I and II refer to each region, and M I(II) is the scattering matrix, containing the sequenceof matching operations. The amplitudes of the right- and left-hand-side transfer tn andreflection rn coefficients are represented as the τ vectors for each side. The scattering boundaryconditions are applied at this stage: t I

0(E) = 1 and r IIn (E) = 0 for all n.

3784 I Urdaneta et al

2.2. The photofield emission current (PFEC)

The wavefunctions for both sectors being completely determined we can now proceed to thecalculation of the current density. Starting from the definitions of the current densities andafter integration over the optical cycle one has, for sector I:

j I(E) = limx→−∞

eh

2πm

∫ 2π

0Im

(�I∗

E

∂�IE

∂x

)dθ, (21)

which yields, inside the metal:

j I(E) = eh

m

(k0 −

∑n

kn

∣∣r In(E)

∣∣2

), (22)

and for sector II:

j II(E) = limx→∞

eh

2πm

∫ 2π

0Im

(�II∗

E

[∂

∂x− ieA(θ)

]�II

E

)dθ, (23)

j II can be recast in the following form:

j II(E) =∑

n

j IIn (E), (24)

where

j IIn (E) = ehK0

πm

∣∣t IIn (E)

∣∣2. (25)

Since we are interested in the stationary current, integration over a laser period yields thesimple expressions given by equations (22) and (24), and reduces the transport problemto a connection between scattering matrix coefficients. The Schrodinger equation with thematching conditions imposing continuity of the current implies charge conservation which inturn yields current conservation j I(E) + j II(E) = 0.

Specifying by indices i and f initial and final observables, the quantity of physical interest,j IIn (ki) dkiδ(Ei − (Ef ∓ nhω)) dEf , represents the charge per unit time with an initial energy

Ei corresponding to an intial wavenumber ki taken within the range (ki ± dki) and detectedat a final energy Ef ± dEf following the absorption (n > 0) or emission (n < 0) of n photonsof frequency ω. The photofield emission is obtained considering the number Nki

of electronswith normal wavenumber kx = ki given by

Nki=

∑kz,ky

f (E(ki)), (26)

where f (E) is the statistical electron distribution in the metal. The PFEC is then obtained byintegrating the current density probability over the initial kinetic momentum ki of electronsand over all the electron states in the metal taking into account the photon exchanges with thelaser:

J (Ef) dEf dS = dEf dS∑

n

∫dkiNki

j IIn (ki)δ(Ei − (Ef ∓ nhω)), (27)

where dS is the surface of the electrode irradiated by the laser and J (Ef)dEf represents thetotal charge per unit time and surface at the final energy Ef .

Laser-electrode interaction and its role in photo-assisted electron transport processes 3785

2.3. Total current for a two-electrode junction

Including a second electrode is straightforward: a third region (III), x > L,L being the inter-electrode distance, has to be introduced in the description of the potential in the Schrodingerequation. It represents an electrode characterized by the same parameters as the first one, butwith energy levels shifted by an amount corresponding to the bias voltage, W = FL. Thematching conditions have to be adapted (equation (20)) to the existence of a second boundaryto obtain the transmission amplitudes. We include at this stage some notations. In orderto compute the current, we take into account the forward current J + from region I to III,calculated with the electron density of region I (left electrode), and the backward current J−

from region III to I, calculated with the electron distribution of region III (right electrode),at the shifted energy already mentioned above. For the forward current one has to define thecurrent density in region III. We also have to compute the backward current in terms of thecurrent density in region I, taking the proper matching conditions:

j I(III)(E) = eh

m

(∑n

kI(III)n

∣∣t I(III)n (E)

∣∣2

), (28)

where kIn is given by equation (12), whereas kIII

n by the same equation but with an energyappropriately shifted by the bias. The transmission coefficients t I(III)

n are calculated withthe boundary conditions corresponding to each current direction: for t III

n , t I0(E) = 1 and

r IIIn (E) = 0; while for t I

n, tIII0 (E) = 1 and r I

n(E) = 0.The forward and backward currents are then given by the following expressions:

J +(ω) =∫

dkN Ikj

III(k), (29)

J−(ω) =∫

dkN IIIk j I(k), (30)

where the ± sign in the subscript of J indicates the direction of the current and wherethe statistics N I and N III including the laser interaction with the left and right electrodes isconsidered in the next section. The total current is simply taken as the difference of the forwardand backward contributions:

J (ω) = J +(ω) − J−(ω), (31)

2.4. Electron distribution function

At the lowest order approximation f (E) is taken as f 0(E), the equilibrium Fermi–Diracdistribution function:

f 0(E) = 1

1 + exp((E − EF)/kBT ), (32)

where T is the temperature and kB is the Boltzmann constant. However, when theelectromagnetic field is illuminating the metal, the electron distribution function changesfrom its equilibrium value, as a result of the penetration of the laser. This is accounted for,following the treatment of [31], using the Boltzmann rate equation. The photon-inducedexcitation of the electron is described through a parameter ξ (the probability of an electron toabsorb one photon per unit time). The electron–electron relaxation is modelled by a singleenergy independent time τee. As for the electron–phonon interaction, it is neglected as aconsequence of its much slower time scale as compared to τee.

3786 I Urdaneta et al

(E)fo

EF E

hω

Figure 2. Schematic view of the electron distribution function f0 as a function of energy. Thedashed lines corresponds to the MDF term, whereas the solid line represents the Fermi distributionfunction

We consider a typical pulse duration of 10−12 s [20–22]; at this time scale one mayhave noticeable changes in the electron distribution function without modifying the latticetemperature [23–26], since the pulse duration is lower than the electron–phonon relaxationtime. At the first-order approximation (low laser intensity) and if the electron gas is degenerate,the distribution function is finally given by

f0 = f 0 + sign(E − EF)(ξτee)[1+g]. (33)

ξ = (1 − R)I0/(nldphω), where (1 − R)I0 is the absorbed laser intensity, R is the opticalreflection coefficient, I0 is the intensity of the field, nl is the effective electron densityparticipating in the absorption process:

nl = n

(1 −

[(EF − hω)

EF

]3/2)

, (34)

and dp is the depth penetration of the laser radiation dp = λ/4π Im(η), with η the complexrefractive index. As for [1 + g], it represents the integer part of (1 + g), with g = (|E−EF|)/hω.

The photofield emission current can now be calculated by using either the equilibriumFermi–Dirac distribution function, or its modified version (MDF), which includes acontribution referred to as thermic, because it depends basically on the relaxation time ofthe electron–electron interaction. Physically this effect would correspond to an increase ofthe electron temperature where, as a first-order approximation, it is the result of one-photonprocesses. Figure 2 shows the behaviour of this modified distribution function. The model canbe interpreted as follows: a compensation scheme whereby electrons are extracted from theinitial distribution and replaced at energies larger than EF in such a way that the total numberof electrons is conserved. The distribution function is affected the most in the neighbourhoodof the Fermi energy.

Substitution of equation (32) by (33) in expression (24) yields for the one-electrode case:

J II(Ef, ω) =√

m

2

eK0

π2h

∑n

(kBT log(1 + exp

(EF−(Ef−nhω)

kBT

))+ γ hω(ξτee)

[1+g])√

(Ef − nhω + V0)

×∣∣t IIn (Ef − nhω)

∣∣2, (35)

Laser-electrode interaction and its role in photo-assisted electron transport processes 3787

where γ = 0 for the equilibrium electron distribution and γ = 1 for the electron distributionincluding the thermic contribution. Each channel n in the summation contributes as a partialcurrent j II

n (E) as expression (24) shows.Similarily, for the two-electrode case, expressions (29) and (30) should also include this

substitution; the number of electrons depends on the total distribution function followingequation (26):

N I(E) = Sm

πh2

(kBT log

(1 + exp

(EF − E

kBT

))+ hω(ξτee)

[1+g]

), (36)

N III(E) = N I(E + eW), (37)

S being the surface of the electrode.

3. Results and discussion

Most of the more relevant results are already present in the single-electrode case. In addition,PFEC currents are routinely measured regardless the limitations in laser pulse duration due tothe use of devices that can discriminate the final kinetic energy of electrons. This is importantas a consistency check of our model before we proceed to analyse more complex situations.

A caveat about the use of the Floquet formalism is in place here because, as was mentionedin the introduction, we are forced to consider a pulsed laser due to junctions thermal stabilityconsiderations. Two crucial points are to be taken into account in applying the Floquet theory;first, whether a reasonable number of optical oscillations are included within the pulse duration,and, second, whether the time scale of the electronic relaxation dynamics is much shorter thanthe pulse duration. Both requirements are satisfied for sub-picosecond visible–UV lasers[27, 30]. It is worthwhile noting that the whole model, so far presented on continuous wavelasers, can also be generalized to properly take into account short pulsed fields, using, forinstance, the (t, t ′) method of Peskin and Moiseyev [33, 34].

3.1. PFEC current

A simple view of the multiphoton excitation mechanisms involved in the total current canbe seen in figure 1. As expressed by equation (35), the effect of the laser on the current istwofold. It provides energy for vertical excitation of the electrons and simultaneously modifiesthe electronic distribution function.

In the contribution to the current corresponding to zero-photon absorption, an electronwith an initial kinetic energy lower than the Fermi limit (Ei < Ef) may be extracted by theeffect of the applied bias voltage F, through a tunnelling process. Obviously, the higher thenumber of electrons with Ei , the larger will be the current, due to the smaller tunnelling barrierto overcome. For Ei > EF a very low number of electrons are found. A direct current, reachedat an energy Ef = Ei = EF, is obtained displaying an asymmetrical shape that reflects theasymmetry of the Fermi–Dirac distribution at finite temperature around E = EF. The Ef < EF

wing results from the variations of∣∣t II

0 (Ef)∣∣2

describing the tunnelling at different energies,whereas Ef > EF abruptly falling off wing is a consequence of the electron distribution abovethe metal Fermi level. In addition to this current, which we shall refer to as direct current,the laser field induces photon absorption and emission processes. For photon absorption, asingle-photon absorption brings the initial kinetic energy to Ei + hω, for which the electrontunnels through a lower barrier leading to an additional peak at a final energy Ef = Ei + hω.This peak is made up mainly by the contribution from the channel n = 1 (all other multiphoton

3788 I Urdaneta et al

Table 1. Values of the parameters taken from Girardeau et al [27] for gold. EF = 5.53 eV,φ = 4.65 eV. η designates the complex refractive index, R is the optical reflection coefficient, τeeis the electron–electron relaxation time and dp is the depth penetration of the laser.

λ (nm) hω (eV) η R τee (s) dp (nm)

496 2.50 0.82 + i1.59 0.735 3 × 10−14 24.8625 1.98 0.13 + i3.36 0.957 3 × 10−14 14.8

processes are not represented in the figure), which contributes with∣∣t II

1 (Ei � EF � Ef −hω)∣∣2

.This probability is weighted by the electron distribution statistics at energies close to EF. Forthe sake of completeness we have to mention that the n = −1 channel also contributes to themain peak at EF, but only up to a negligible amount, since photon emission from an initialenergy Ei < EF will slow down the electron by hω yielding to less tunnelling (no current isobserved at a final energy Ef = Ei − hω).

The thermic contributions (which are not explicited in the same figure) can reverse such atendency, as we will see in the following, even at room temperature, owing to the modificationof the electron distribution following laser induced excitation–relaxation processes in the metal.Since the thermic processes accounted for are derived from one-photon processes, at the sameEf = Ei + hω there are electrons coming not only from the excitation process mentionedabove, but also from the tail corresponding to those electrons that have acquired this energyvia all the processes that lead to the new distribution function. This results in a contribution tothe current that belongs to a n = 0 case, weighted by a factor

∣∣t II0 (Ef = EF + hω)

∣∣2, although it

is detected at the same energy where the coherent one-photon absorption contribution is beingmeasured. Hence, at each final energy corresponding to the channels, we can also encounterthe thermic contributions.

As a practical application, the results are hereafter presented in terms of the partialphotofield currents Jn(Ef) (given in A/eV/µm2 units) for different photon channels, includingthe thermic contribution to the electron distribution (γ = 1 in equation (35), the equilibriumdistribution being evaluated at room temperature T = 293 K. Gold is taken as an illustrativeexample for the calculations. All relevant metal and laser parameters are collected in table 1as a function of the laser wavelength λ (i.e. in the vacuum).

3.2. The role of the static field

The laser frequency is fixed at hω = 2.5 eV, whereas its intensity is taken as I = 107 W cm−2.Using the parameters of table 1 for these laser characteristics, one gets the value ξ =2.63 × 10−11 s−1, which measures the efficiency of the thermic contribution. The typicalexperimental applied voltages are of the order of a few volts per nanometer. We are consideringtwo such voltages, namely F = 4×109 V m−1 and F = 3×109 V m−1, with the correspondingPFEC as a function of the final electron kinetic energy Ef as displayed in figure 3. Threeobservations are in order:

(i) The overall spectrum of the PFEC is severely attenuated (i.e. by two orders of magnitude),when decreasing F only by a factor of 3/4. This results from a much larger barrier forelectron tunnelling. We initially note that figure 3(a) corresponds to parameters leadingto the commonly expected direct current peak at Ef = EF, much higher than the single-photon peak at Ef = EF + hω.

(ii) At low applied voltage F = 3 × 109 V m−1 (figure 3(b)) the contribution of then = 1 excitation peak at (EF + hω) is much higher than the direct current one. This

Laser-electrode interaction and its role in photo-assisted electron transport processes 3789

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3

0

0.5

1

1.5

Ef − E

F (eV)

JII n (A

/eV

/µm

2 )×10

−6

(a)

−1 0 1 2 3 4

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Ef − E

F (eV)

JII n (A

/eV

/µm

2 )×10

−8

(b)

Figure 3. Partial photofield emission currents (in A/eV/µm−2) as a function of the electron finalenergy Ef (in eV). The solid line belongs to the n = 0 channel, the dotted line to the n = 1 channel(i.e. the n = 0 or n = 1 contribution to the sum on the right-hand side of equation (35)). Theapplied voltages are F = 4 × 109 V m−1 (panel a) and F = 3 × 109 V m−1 (panel b). The lasercharacteristics are: frequency hω = 2.5 eV, intensity I = 107MW cm−2. All other parametersare taken from table 1.

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3

0

0.5

1

1.5

Ef − E

F (eV)

JII n (A

/eV

/µm

2 )×10

−6

(a)

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

Ef − E

F (eV)

JII n (A

/eV

/µm

2 )×10

−9

(b)

Figure 4. Same as for figure 3 but with the laser intensity I = 105 W cm−2.

somehow surprising effect is again related to the very different tunnelling behaviour, atlow bias, of electron motions prior to and after photon absorption, in such a way that∣∣t II

1 (EF)∣∣2

>∣∣t II

0 (EF)∣∣2

, even though the laser intensity is low.

(iii) Thermal contribution of the n = 0 channel, i.e.∣∣t II

0 (EF + hω)∣∣2

, to the single-photon peakat (EF + hω), overpasses the n = 1 photon contribution. It is important to note that suchan effect is a direct consequence of including the laser modified electron distribution inthe metal that brings enough electron population at the energy EF + hω to increase (as aweighting factor) the transition probability

∣∣t II0 (EF + hω)

∣∣2.

3.3. The role of the laser intensity

If the laser intensity is now taken two orders of magnitude weaker, i.e. I = 105 W cm−2,which leads to a ξ = 2.63 × 10−13 s−1 for the same frequency hω = 2.5 eV, the results for theprevious two applied bias voltages F are displayed in figures 4(a) and (b).

3790 I Urdaneta et al

−2 −1 0 1 2 3

0.5

1

1.5

2

Ef − E

F (eV)

JII n (A

/eV

/µm

2 )×10

−6

(a)

−2 −1 0 1 2 3 4

2

4

6

8

10

12

14

Ef − E

F (eV)

JII n (A

/eV

/µm

2 )×10

−9

(b)

Figure 5. Same as for figure 3 but with a laser frequency hω = 1.98 eV and intensityI = 108 W cm−2.

(i) The direct current, accounting for by∣∣t II

0 (EF)∣∣2

, is not affected by the laser field strengthincrease, and this for both bias voltages.

(ii) The n = 1 single-photon contribution peak at EF + hω, accounted for by∣∣t II

1 (EF)∣∣2

, showsa linear behaviour of the transition probability as a function of the laser intensity sinceit decreases by a factor 102. This, of course, is an expected behaviour in low intensity(perturbative radiative coupling) regime. Hence, at this very low intensity, we find thatthis direct contribution is the leading one at both static fields with respect to the n = 0thermic and n = 1 contributions appearing at EF + hω.

(iii) The n = 0 thermal contribution to the single-photon peak at EF + hω, accounted for by∣∣t II0 (EF + hω)

∣∣2weighted by the laser modified distribution at EF + hω, is also affected

by a factor 102. The reason for such a linear laser field intensity dependence is now tobe looked for in the behaviour of the electron distribution function (and not in t0, whichdoes not depend upon intensity). This linear behaviour with respect to intensity merelyresults from the transition probability per unit time ξ .

3.4. The role of the laser frequency

To study the effect of the laser frequency, we take a different value hω = 1.98 eV, consideredin [27]. All other parameters are changing within the same ranges, i.e. F from 4 × 109 V m−1

to 3 × 109 V m−1 and field strenghts of I = 108 W cm−2 (see figure 5).Here again we observe the following:

(i) The direct current is, as expected, not affected by the frequency change, simply becauseit does not depend over the laser field.

(ii) The single-photon peak at EF + hω is much decreased due to the unfavorable barriertunnelling for a lower photon energy. This behaviour is observed both for the n = 0and n = 1 thermal and excitation contributions to the single-photon peak. The generalobservation is that the secondary peaks at EF + hω remain negligible as compared to thedirect current peaks at EF, at least for the high bias voltage F = 4 × 109 V m−1.

(iii) As opposite to all previous calculations with the high frequency (hω = 2.5 eV), thelow frequency (hω = 1.98 eV) laser induces, in the single-photon peaks at EF + hω, amore important n = 1 photon contribution as compared to the n = 0 thermal one. Thisis rendered even more spectacular with high intensity I = 108 W cm−2 (Figure 5(b)).

Laser-electrode interaction and its role in photo-assisted electron transport processes 3791

10 15 20 25 30 350

2

4

6

8

10

12

14

16

18

Electrode distace (Amstrong)

J (A

/µm

2 )×10

−4

Figure 6. Total photofield emission current (in A µm−2, dotted line) as a function of the electrodedistance (in A). The applied voltage is F = 4.25 × 109 V m−1, for a photon energy hω = 2.5 eV.The laser intensity is I = 108 W cm−2. The single-electrode current is indicated by the solid line.

The interpretation is again to be looked for in the frequency behaviour of ξ which inducesan important decrease in the electron distribution function weighting

∣∣t II0 (EF +hω)

∣∣2, such

that the n = 1 coherent contribution∣∣t II

1 (EF)∣∣2

dominates.

3.5. Two-electrode junction current

The consideration of a second electrode introduces the inter-electrode distance, L, as a newparameter. Although the inclusion of this additional electrode in the model brings differentmatching conditions at x = L, one would expect that the two problems merge into the samefor large L (i.e. L EF/eF ), as seen in figure 6, where it is shown how the two-electrode casereaches the single-electrode one. The existence of a second electrode induces oscillations dueto resonances (or multiple reflections inside the cavity). Taking into account the asymptoticbehaviour of the Airy functions, we have done an analysis in terms of a coherent superpositionof reflected and transmitted partial waves and found a quasi-periodic behaviour with a localperiod Tl � 4πK

−3/20 L−1/2, which amounts to Tl = 3 A for L = 40 A, close to the result

displayed in figure 6. More interestingly, due to the nature of the tunnelling phenomenaconsidered here, the second electrode plays a similar role to the voltage insofar it modifiesthe exponent that dominates the physics of tunnelling conductance. As a consequence, itis possible to reverse the relative magnitude of the zero-photon and one-photon currents byplaying with either the voltage, the distance or a combination of both (see figures 3 and 7).

Figure 7 shows the total current as a function of photon energy as given by equation (31).The basal level (J � 0.15 × 10−5 A µm−2) is determined by the frequency-independent zero-photon component. The subsequent raise of the current is purely determined by the laser and itis of the same order of magnitude as the one-photon component for the parameters consideredhere. A word of caution is needed here because although the absolute magnitudes of thelaser-induced currents are perfectly comparable to stationary currents observed in molecularjunctions, they correspond to transient phenomena whose detection may be challenging,because they have to be measured within the duration of the pulse. To clarify this point weestimate that for a typical response time of 1 ns for a measuring electronic device, we would

3792 I Urdaneta et al

1.4 1.6 1.8 2 2.2 2.4 2.60

1

2

3

4

Photon energy (eV)

J (A

/µm

2 )× 1

0−5

Figure 7. Photofield emission currents (in A µm−2) as a function of the photon energy (in eV). Thesolid line shows the total current and the dotted line the current without the thermic contribution.The applied voltage is F = 4.25 × 109 V m−1, for an inter-electrode distance L = 15 A. The laserintensity is I = 108 W cm−2.

need 1 THz repetition rate for a 1 ps pulsed laser to obtain comparable values of the one-photonand zero-photon currents.

Despite the measuring difficulties alluded to in the previous paragraph, laser-inducedcurrents may be of importance if through-space current can compete with through-bondcurrent in molecular junctions or in junctions with a metal and a semiconductor electrode.One may also speculate that in the presence of a molecule there may occur excitation andenergy transfer processes, due to the presence of resonances, that can completely change thetime scale where the laser-induced currents are of importance.

4. Conclusions

The main conclusions, which are basically transposable to other metal–laser systems withinthe assumptions of the model, concern the relative importance of three peaks (direct, single-photon contributions and the thermal contribution) in either the photofield emission currentspectrum as a function of the final electron kinetic energy (for the one-electrode case) or thetotal current for a vacuum junction as a function of the laser frequency. When the appliedvoltage of the static external field is high enough for obtaining an important amount of direct(zero-photon) current, the effect of the laser field may be safely neglected. The situation iscompletely different for lower values of the bias voltage, where the laser field characteristics(intensity and frequency) play a major part when considering the relative importance ofthe direct current observed in the vicinity of EF and the single-photon current peaked atEF + hω. For relatively low laser intensities, basically including single-photon processes,two contributions are involved when analysing the single-photon current. One results from asingle-photon absorption process, measured by a probability

∣∣t II1 (EF)

∣∣2which behaves linearly

as a function of the intensity (at least for the intensity regime studied). This probability ofthe so-called n = 1 channel is weighted by an electron distribution function which behaves asa Fermi–Dirac equilibrium distribution around EF. To this contribution, we have now addedthe one of the n = 0 channel referred to as the thermal contribution, given by the transition

Laser-electrode interaction and its role in photo-assisted electron transport processes 3793

probability∣∣t II

0 (EF +hω)∣∣2

weighted again by the electron distribution function. Obviously, theequilibrium Fermi–Dirac distribution would lead to a negligible population at EF + hω. It isprecisely the laser modified electron distribution, leading to a population proportional to ξτee

at EF + hω, that can enhance the probability of this n = 0 channel. The resulting incoherentthermic current is in turn proportional to the laser intensity, not through the probability |t0|2(field independent) but through the linear intensity dependence of the electron distributionfunction ξ .

With respect to the laser frequency, a great sensitivity of the thermic electron distributionfunction is evidenced such that for some parameters regimes the single-photon peak at EF +hω

may be much enhanced and dominate over the direct current peak at EF. This inversion mayhave important consequences under conditions such that the through-space contribution canbecome of importance and cause interference with the through-bond current. We are currentlyexamining some models that include both molecular resonances in the interelectrode region andthe possibility of combining two laser fields with different frequencies, where current-inducednoise and laser control of the current can be studied.

In the two-electrode case the total current is much enhanced by the laser irradiation,not through the thermal effect but by the single-photon process. Although this is a transientphenomena, it can be experimentally evidenced by using pulsed sources of 1 THz repetitionrate. Multiphoton processes can also be analysed with our model, even though in our presentwork we have restricted ourselves to one-photon processes.

Acknowledgments

This work has been supported by Programme Alban, the European Union Program of HighLevel Scholarships for Latin America, scholarship no. E04D045094VE. We acknowledgepartial financial support from France Venezuela ECOS cooperation (projet no. V00P01), andFONACIT (through the PPI program).

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