Photon-Assisted Tunneling through Molecular Conduction Junctions with Graphene Electrodes.

# Photon-Assisted Tunneling through Molecular Conduction Junctions with Graphene Electrodes.

Boris D. Fainberg Faculty of Science, Holon Institute of Technology, 5810201 Holon, Israel School of Chemistry, Tel-Aviv University, 69978 Tel-Aviv, Israel
July 12, 2019
###### Abstract

Graphene electrodes provide a suitable alternative to metal contacts in molecular conduction nanojunctions. Here, we propose to use graphene electrodes as a platform for effective photon assisted tunneling through molecular conduction nanojunctions. We predict dramatic increasing currents evaluated at side-band energies ( is a whole number) related to the modification of graphene gapless spectrum under the action of external electromagnetic field of frequency . A side benifit of using doped graphene electrodes is the polarization control of photocurrent related to the processes occurring either in the graphene electrodes or in the molecular bridge. The latter processes are accompanied by surface plasmon excitation in the graphene sheet that makes them more efficient. Our results illustrate the potential of graphene contacts in coherent control of photocurrent in molecular electronics, supporting the possibility of single-molecule devices.

###### pacs:
73.23.b, 73.63.Rt, 78.67.Wj, 42.50.Hz

## I Introduction

The field of molecular-scale electronics has been rapidly advancing over the past two decades, both in terms of experimental and numerical technology and in terms of the discovery of new physical phenomena and realization of new applications (for recent reviews please see Refs.Kohler05 (); Chen09 (); Heath09 ()). In particular, the optical response of nanoscale molecular junctions has been the topic of growing experimental and theoretical interest in recent years Park11PRB (); Wang11PCCP (); Fainberg_Galperin11PRB (); Haertle10JCP (); Reuter08PRL (); Li08SSP (); Thanopulos08Nanotech (); Prociuk08PRB (); Li08NJP (); Galperin08JCP (); Li07EPL (); Fai07PRB (), fueled in part by the rapid advance of the experimental technology and in part by the premise for long range applications in optoelectronics.

A way of the control of the current through molecular conduction nanojunctions is the well-known photon-assisted tunneling (PAT) Platero04 (); Kohler05 () that was studied already in the early 1960’s experimentally by Dayem and Martin Dayem_Martin62PRL () and theoretically by Tien and Gordon using a simple theory which captures already the main physics of PAT Tien_Gordon63 (). The main idea is that an external field periodic in time with frequency can induce inelastic tunneling events when the electrons exchange energy quanta with the external field. PAT may be related either to the potential difference modulation between the contacts of the nanojunction when electric field is parallel to the axis of a junction Tien_Gordon63 (); Gri98 (); Platero04 (); Kleinekathofer06EPL (); Li07EPL (), or to the electromagnetic (EM) excitation of electrons in the metallic contacts when electric field is parallel to the film surface of contacts Tien_Gordon63 (). According to the Tien-Gordon model Tien_Gordon63 (); Platero04 (); Li07EPL () for monochromatic external fields that set up a potential difference , the rectified dc currents through ac-driven molecular junctions are determined as Platero04 (); Li07EPL ()

 ITG=∞∑n=−∞J2n(eV0ℏω)I0dc(eV0+nℏω)=∞∑n=−∞In (1)

where the current in the driven system is expressed by a sum over contributions of the current in the undriven case but evaluated at side-band energies shifted by integer multiples of the photon quantum and weighted with squares of Bessel functions. A formula similar to Eq.(1) can be obtained also for EM excitation of electrons in the metallic contacts Tien_Gordon63 (). Note that the partial currents contain contributions from . The term denotes the -th-order Bessel function of the first kind. The photon absorption () and emission () processes can be viewed as creating effective electron densities at energies with probability . These probabilities strongly diminish with number when that severely sidelines the control of the current for not strong EM fields ( Kohler05 ()).

In the last time graphene, a single layer of graphite, with unusual two-dimensional Dirac-like electronic excitations, has attracted considerable attention due to its exceptional electronic properties (ballistic in-plane electron transport etc.) Novoselov09RMP (); Trauzettel07PRB (); Efetov08PRB (). Quite recently they have shown interest to a new kind of graphene-molecule-graphene (GMG) junctions that may exhibit unique physical properties, including a large conductance (achieving conductance quantum), and are potentially useful as electronic and optoelectronic devices Yang_graphene_junctions10JCP (). The junction consists of a conjugated molecule connecting two parallel graphene sheets. In this relation it would be interesting to investigate PAT in such a junction to control the current through it. The PAT in GMG junctions under EM excitation of electrons and holes in the graphene contacts may be rather different from that for usual metallic contacts. It was shown that the massless energy spectrum of electrons and holes in graphene led to the strongly non-linear EM response of this system, which could work as a frequency multiplier Mikhailov07EPL (). The predicted efficiency of the frequency up-conversion was rather high: the amplitudes of the higher-harmonics of the ac electric current fell down slowly (as ) with harmonics index . Sure, the strongly non-linear EM response should also lead to a slow falling down currents evaluated at side-band energies (see Eq.(1)) with harmonics index in comparison to nanojunctions with metallic (or semiconductor Fainberg13CPL ()) leads (see below). This makes controlling charge transfer essentially more effective than that for molecular nanojunctions with metallic contacts. Additional factors that may enhance currents evaluated at side-band energies in nanojunctions with graphene electrodes are linear dependence of the density of states on energy in graphene Novoselov09RMP (), and the gapless spectrum of graphene that can change under the action of external EM field (see below).

Here we propose and explore theoretically a new approach to coherent control of electric transport via molecular junctions, using either both graphene electrodes or one graphene and another one - a metal electrode (that may be an STM tip). Our approach is based on the excitation of dressed states of the doped graphene electrode with electric field that is parallel to its surface, having used unique properties of graphene mentioned above. As a first step, we calculate a semiclassical wave function of a doped graphene under the action of EM excitation. Then we obtain Heisenberg equations for the second quantization operators of graphene and calculate current through a molecular junction with graphene electrodes using non-equilibrium Green functions (GF). We address different cases, which are analytically soluble, hence providing useful insights. We show that using graphene electrodes can essentially enhance currents evaluated at side-band energies in molecular nanojunctions.

## Ii Model Hamiltonian

Consider a spinless model for a molecular wire that comprises one site of energy , positioned between either both graphene electrodes (leads) (Fig.1) or one graphene and another one - a metal electrode (Fig.2).

The leads are represented by electron reservoirs and , characterized by the electronic chemical potentials , , and by the ambient temperature . The corresponding Fermi distributions are in the absense of external EM field, and the difference is the imposed voltage bias between the electrodes. External EM field acting on electrode , , changes the corresponding Fermi distribution (see below). The Fermi energy of the graphene electrode may be controlled via electrical or chemical modification of the charge carrier density Mak08PRL (); Chen8Nature (); Abajo11NL (); Chen12Nature (); Fei12Nature (). We consider that steady-state current through a nanojunction does not influence on the Fermi energy, since such current does not change a charge of the graphene electrode.

The corresponding Hamiltonian is

where the wire Hamiltonian is , () are creation (annihilation) operators for electrons at the molecular wire. The molecule-leads interaction describes electron transfer between the molecular bridge and the right () and left () leads that gives rise to net current in the biased junction

 ^V=∑+,−∑σ,p∈{L,R}(Vp±,σ;m^a†p±,σ^cm+H.c.) (3)

Here denotes Hermitian conjugate, are creation operators for graphene electrodes (see below). The corresponding contribution to from a metal electrode does not contain summation with respect to positive and negative energies () and quasispin index .

## Iii Calculation of Semiclassical Wave Function

The states of electrons in graphene are conveniently described by the four-component wave functions, defined on two sublattices and two valleys. Electron motion in the time-dependent EM field is described by the 2D Dirac equation Novoselov09RMP (); Efetov08PRB ()

 (4)

written for a single valley and for a certain direction of spin. Here is the momentum of the quasiparticle, - the Fermi velocity ( m/s), - the vector of the Pauli matrices in the sublattice space (“pseudospin” space), and are vector and scalar potentials of an EM field, respectively. Suppose a graphene film is excited by a linearly polarized monochromatic electric field that is parallel to its plane (). Then , . Eq.(4) can be brought to more symmetric form introducing matrices and , where

 ^P=−iℏ3∑k=1γk∂∂xk, ^A=3∑k=1γkAxk, (5)

, , and . To obtain a semiclassical solution of Eq.(4), we shall use a method of Ref. Pauli32 () (see also akhiezer-berestetskii69 ()). Let us put . Then one can obtain the following equation for

 [iℏe2c3∑k,l=1γkγl(1−δkl)Fxlxk−3∑k=1 (ℏ∂∂xk−iecAxk)2]Φ=0 (6)

where is the field tensor. Let us seek a solution of Eq.(6) as an expansion in power series in

 Φ=exp(iS/ℏ)w=exp(iS/ℏ)(w0+ℏw1+ℏ2w2+...) (7)

where is a scalar and is a slowly varying spinor berestetskii-lifshitz99 (). Substituting series, Eq.(7), into Eq.(6) and collecting coefficients at the equal exponents of , we get that is the action obeying the Hamilton-Jacobi equation where is the classical Hamilton function of a particle:

 exp(iℏS)=exp[−iℏ(v∫t0√¯p2x+¯p2ydt′+e∫t0φpotdt′)], (8)

and the equation for spinor

 3∑k=1 {[∂∂xk(∂S∂xk−ecAxk)]w0+2(∂S∂xk−ecAxk)∂w0∂xk−e2c3∑l=1γkγl(1−δkl)Fxlxkw0}=0 (9)

In Eq.(8), is the normal momentum that obeys the classical equations of motion for a particle with charge , according to which ; where is the generalized momentum. If one takes only the first term in series, Eq.(7), into account, it can be shown that wave packets behave like particles moving along classical trajectories.

Let us solve Eq.(9) for spinor . We shall introduce a linear combination of the components of the Hermitian conjugated wave function by akhiezer-berestetskii69 (). Then using equation and Eqs.(5), one can show that electronic flux obeys the continuity equation

 3∑k=1∂∂xksk=0 (10)

Put

 w0=√ξφ0 (11)

where we denoted

 ξ=−i2¯w0^πw0 (12)

and , . Then in our approximation the electronic flux is reduced to that gives, bearing in mind Eq.(10),

 3∑k=1∂∂xk(πkξ)=0 (13)

Here quantities can be written as and with the aid of the Hamilton-Jacobi equation and . Here signs plus and minus are related to positive and negative energies, respectively. Eq.(13) can be write over as

 3∑k=1 (∂πk∂xkξ+πk∂ξ∂xk)=0 (14)

Using Hamilton’s equations the time derivative can be written as

 ˙xk=±v¯pk¯p=ivπkπ3, k=1,2 (15)

This enables us to write down the second term on the right-hand side of Eq.(14) in the form

 3∑k=1 πk∂ξ∂xk=−iπ3v[2∑k=1 ∂ξ∂xkdxkdt+∂ξ∂t]=−iπ3vdξdt

and Eq.(14) becomes

 dξdt=−1¯p∂¯p∂tξ (16)

since for . Integrating Eq.(16), one gets

 ξ(t)=ξ(0)¯p(0)¯p(t) (17)

where . Furthermore, substituting Eq.(11) into Eq.(9), we obtain equation for spinor : , the solution of which may be written as

 φ01,2 =12√p¯p(1+cosφ)(1+cos¯φ){φ01,2(0)[¯p(1+cos¯φ)+ +p(1+cosφ)]±φ02,1(0)[¯p(1+cos¯φ)−p(1+cosφ)]} (18)

The quantities and in Eqs.(17) and (18) are chosen in such a way that the wave function should be normalized and coincide with the wave function of unperturbated graphene in the absence of external EM field Novoselov09RMP (). After combersome calculations we get the wave function normalized for the graphene sheet area :

 ψ =1√sexp(ipxx/ℏ+ipyy/ℏ)exp[iℏ(∓v∫t0¯pdt′− −e∫t0φpotdt′)]¯up± (19)

where slowly varying spinors are equal to

 ¯up±=1√2(exp(−i¯φ/2)±exp(i¯φ/2)), (20)

, , , , .

Eqs.(19) and (20) show remarkable and very simple result, according to which the time-dependent part of the semiclassical wave function is defined by the same formula as that for the unperturbated system with the only difference that the generalized momentum should be replaced by the usual momentum . The space-dependent part of the wave function remains unchanged.

### iii.1 Heisenberg Equations for the Second Quantization Operators of Graphene

The wave function of the graphene sheet interacting with molecular bridge may be represented as the superposition of wave functions, Eqs.(19) and (20). Passing to the second quantization, we get

 Ψ =1√s∑+,−∑p^ap±exp[iℏpr+iℏ(∓v∫t0¯pdt′ −e∫t0φpotdt′)]¯up± (21)

where are annihilation operators. To obtain the Hamiltonian in the second quantization representation, consider an average energy of a particle with wave function that is given by . Replacing wave functions for operators and integrate with respect to , we get

 ^H=∫Ψ†^HΨdr=∑pσ∑+,−^a†p±,σ^ap±,σ[±v¯p(t)+eφpot(t)] (22)

where is the ”quasispin” index. In deriving Eq.(22), we have taken into account that the main contribution to in the semiclassical approximation is given by the exponential term on the right-hand side of Eq.(21) (see Ref.landau-lifshitz.65 (), chapter II). In addition, we beared in mind that the summation over can be substituted by the integration over phase space

 ∑p→∫dΓ(2πℏ)2=s(2πℏ)2∫dp (23)

Using Hamiltonian, Eq.(22), we obtain the Heisenberg equations of motion

 d^ap±,σ(t)dt=iℏ[^H,^ap±,σ]≃iℏ[∓v¯p(t)−eφpot(t)]^ap±,σ(t) (24)

## Iv Formula for the Current

The current from the lead () can be obtained by the generalization of Eq.(12.11) of Ref.haug-jauho.96 ()

 IK=−2ϰeℏRe∑+,−∑σ,p∈KVp±,σ;mG

where for the metal electrode, and for the graphene electrode that accounts for the valley degeneracies of the quasiparticle spectrum in graphene. denotes the lesser GF that is given by

 G

where and are the retarded and lesser wire GFs, respectively; and are the lesser and advanced lead GFs, respectively; is the unit function. Using Eq.(24), we get

 g

and

 gap±,σ(t1,t′) =iθ(t′−t1)exp{iℏ[−eφpot,K(t1−t′) ∓v∫t1t′dt′′¯p(t′′)]} (28)

where is the Fermi function and - the chemical potential of lead . Substituting Eqs.(26), (27) and (28) into Eq.(25), and converting the momentum summations to energy integration, Eq.(23), we get

 IK =4eℏ∫t−∞dt1∑+,−Im∫∞0d(vp)2πexp[±iℏeφpot,K(t−t1)]× ×ΓKmm(±vp,t1,t)[Grmm(t,t1)fK(±vp)+G

where

 ΓKmm(±vp,t1,t) =2πℏ(s2π2ℏv2)∑σ∈K∫π0dθvpVp±,σ;m(t)× ×V∗p±,σ;m(t1)exp[±iℏv∫tt1dt′¯p(t′)] (30)

is the level-width function.

To proceed, we shall make the time expansion of into the Fourier series, and then use the Markovian approximation, considering time as very short. This will also enable us to use the non-interacting resonant-level model haug-jauho.96 () for finding the time dependence of and as functions of and where is the population of molecular state .

According to the Floquet theorem Kohler05 (), the general solution of the Schrödinger equation for an electron subjected to a periodic perturbation, takes the form , where is a periodic function having the same period as the perturbation, and is called quasienergy. Then the expansion of function on the right-hand side of Eq.(19) into the Fourier series will be as following

 exp[iℏv∫t0dt′¯p(t′)]=exp[iℏε(p,θ)t]∞∑l=−∞cl(p,θ)exp(iltω) (31)

where

 cl(p,θ)=ω2π∫π/ω−π/ωexp[iℏv∫t0dt′¯p(t′)−iℏε(p,θ)t−ilωt]dt (32)

Using expansion, Eq.(31), into Eq.(30) and neglecting fast oscillating with time terms, we get

 ΓKmm(±vp,τ) =2πℏ(s2π2ℏv2)∑σ∈K∫π0dθvp|Vp±,σ;m|2× ×∞∑n=−∞|cn(p,θ)|2exp{±i[ε(p,θ)ℏ+nω]τ} (33)

Then going to the integration with respect to in Eq.(29) and bearing in mind Eq.(33), we get

 IK =4e∑σ∈K∫π0dθ∞∑n=−∞[nm(t)−fK(vpn±)]× ×|cn(pn±,θ)|2¯γ(n)±GKσ,m (34)

where we denoted

 ¯γ(n)±GKσ,m =s2πℏ3v2∫∞0vpd(vp)|Vp±,σ;m|2× ×δ[±(ε(p,θ)+nℏω)+eφpot,K−εm] (35)

is the spectral function for the -th photonic replication, is the Dirac delta, arguments are defined by equation

 ε±(p,θ)=±(εm−eφpot,K)−nℏω (36)

and should be positive. Below we shall consider not dependent on and quasispin .

## V Molecular Bridge between Graphene and Metal Electrodes

Consider a specific case when the molecular bridge is found between graphene and metal (tip) electrodes (Fig.2). In that case one can use Eq.(34) for :

 IL =4e∑σ∈K∞∑n=−∞[nm(t)−fL(vpn±)]× ×∫π0dθ|cn(pn±,θ)|2¯γ(n)±GLσ,m (37)

If represents the metal electrode, then

 IR=2eγRm[nm(t)−fRp] (38)

where is the charge transfer rate between the molecular bridge and the metallic lead. In the case under consideration the equation for becomes

 dnmdt=−IL/e−IR/e (39)

that is written as the continuity equation. Inserting Eqs.(37) and (38) into Eq.(39), solving the latter for the steady-state regime and substituting the solution into Eq. (38) for current , we get

 IR=2eγRm∑σ∞∑n=−∞¯γ(n)±GLσ,m∫π0dθ|cn(pn±,θ)|2[fL(vpn±)−fRp]∑σ∞∑n=−∞¯γ(n)±GLσ,m∫π0dθ|cn(pn±,θ)|2+γRm/2 (40)

For a special case

 γRm/2>>∑σ∞∑n=−∞¯γ(n)±GLσ,m∫π0dθ|cn(pn±,θ)|2

we obtain

 IR=4e∑σ∞∑n=−∞∫π0dθ|cn(pn±,θ)|2¯γ(n)±GLσ,m[fL(vpn±)−fRp] (41)

Eq.(41) seems similar to that of Tien and Gordon, Eq.(1), and generalizes it. To calculate current, we shall use a variety of approaches.

### v.1 Calculations using Cumulant Expansions

Function may be written in the dimensionless form as

 exp(iαb∫y0dx√1+2bcosθsinx+b2sin2x)

where and represent the work done by the electric field during one fourth of period weighted per unperturbated energy and photon energy , respectively; , and we assume . If , one can use the cumulant expansion, and we get

 exp[iαb∫y0dx√1+2bcosθsinx+b2sin2x] =exp[G1(y)+G2(y)] (42)

where correct to fourth order with respect to ,

 G1(y) =iαcosθ(1−b23sin2θ)+iαb[1+b24sin2θ− −3b464sin2θ(1−5cos2θ)]y, (43)
 G2(τ)=iz1cosy+iz2sin2y+iz3cos3y+iz4sin4y (44)

Here parameters are defined by and .

As a matter of fact, the second term on the right-hand side of Eq.(43) that is proportional to describes the quasienergy weight per photon energy

 ε(p,θ)/(ℏω)=αb[1+b24sin2θ−3b464sin2θ(1−5cos2θ)] (45)

that is anisotropic: when the momentum is parallel to electric field ( or ), and is most different from when the momentum is perpendicular to the electric field (). The term can be expanded in terms of the Bessel functions as Abr64 ()

 exp(iz2nsin2ny) =∞∑s=−∞Js(z2n)exp(i2sny), exp[iz2n−1cos((2n−1)y)] =∞∑s=−∞Js(z2n−1)× (46) ×exp[isπ2+is(2n−1)y]

where . This gives expansion

 |cl(p,θ)|2 =[∑s2s3s4Jl−2s2−3s3−4s4(z1)J−s2(z2)J−s3(z3)Js4(z4)]2 (47)

for quantities , Eq.(32), that converge fast.

For a linear case (weak fields) , , , and we get from Eq.(36): . In that case quantities , Eq.(35), become

 ¯γ(n)±GLσ,m=γ0π[±(εm−eφpot,L)ℏω−n] (48)

where , and the expression in the square brackets is proportional to the DOS for graphene that is proportional to energy Novoselov09RMP (). The current, Eq.(41), calculated in the linear regime using Eq.(48), as a function of applied voltage bias is shown in Fig.3. In our calculations temperature , and the leads chemical potentials in the biased junction were taken to align symmetrically with respect to the energy level Li_Fai12Nano_Let (), i.e., for the left lead, and for the right lead (, ) where for both leads. Both curves of Fig.3 show photon assisted current -

the steps when the potential of the graphene electrode achieves the value corresponding to the photon energy. The steps are found on the background that decreases linearly for a n-doped graphene electrode and increases linearly for a p-doped electrode when increases. This is related to the linear dependence of DOS on energy. Fig.2 shows our model together with the photonic replica of the graphene electrodes and elucidates the behavior observed in Fig.3.

When the interaction with external field is not small, , the linear consideration does not apply. In case of large momenta (far from the Dirac point), , Eq.(48) applies, and we get from Eq.(47) . The current, Eq.(41), calculated for large momenta when , as a function of applied voltage bias is shown in Fig.4. The number of steps and their heights increase in comparison with the linear case.

### v.2 Calculations of Current including Small Momenta

To calculate coefficients , Eq.(32), in general case, we need to know quasienergy . The latter may be found as zero harmonic of the Fourier cosine series of normal momentum on the left-hand side of Eq.(32). Consider first limiting points when the momentum is parallel to the electric field. Then the quasienergy weighted per the work done by the electric field during one fourth of period is equal to . If , like above. When ,

 ¯ε(p;θ=0,π)=2πb[arcsin(1b)+√1−1b2] (49)

that gives for

 ε(p;θ=0,π)=1π[2αℏω+(vp)2evE0/ω] (50)

- a quadratic dependence of on for small