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Chemical Physics 319 (2005) 368–379

Tunneling and resonant conductance in one-dimensionalmolecular structures

M.A. Kozhushner a,*, V.S. Posvyanskii a, I.I. Oleynik b

a Institute of Chemical Physics RAS, Moscow, Russiab Department of Physics, University of South Florida, Tampa FL, USA

Received 13 December 2004; accepted 15 June 2005Available online 8 August 2005

Abstract

We present a theory of tunneling and resonant transitions in one-dimensional molecular systems which is based on Green�s func-tion theory of electron sub-barrier scattering off the structural units (or functional groups) of a molecular chain. We show that themany-electron effects are of paramount importance in electron transport and they are effectively treated using a formalism of sub-barrier scattering operators. The method which calculates the total scattering amplitude of the bridge molecule not only predicts theenhancement of the amplitude of tunneling transitions in course of tunneling electron transfer through onedimensional molecularstructures but also allows us to interpret conductance mechanisms by calculating the bound energy spectrum of the tunneling elec-tron, the energies being obtained as poles of the total scattering amplitude of the bridge molecule. We found that the resonant tun-neling via bound states of the tunneling electron is the major mechanism of electron conductivity in relatively long organicmolecules. The sub-barrier scattering technique naturally includes a description of tunneling in applied electric fields which allowsus to calculate I–V curves at finite bias. The developed theory is applied to explain experimental findings such as bridge effect due totunneling through organic molecules, and threshold versus Ohmic behavior of the conductance due to resonant electron transfer.� 2005 Published by Elsevier B.V.

Keywords: Molecular electronics; Electron tunneling; Resonant electron transfer; Sub-barrier scattering

1. Introduction

In the recent years, substantial research efforts weredirected towards understanding the mechanisms of elec-tron transport in one-dimensional structures such asmetallic nanowires, carbon nanotubes, and single organ-ic molecules [1,2]. This interest is motivated by activeexperimental investigations of one-dimensional nano-structures as perspective electronic device elements thatwould allow the fundamental scaling limitations ofsilicon-based electronics to be overcome [3]. Molecularelectronics based on organic polymers, oligomers, andsmall organic molecules exploits intriguing electric

0301-0104/$ - see front matter � 2005 Published by Elsevier B.V.

doi:10.1016/j.chemphys.2005.06.023

* Corresponding author.E-mail address: [email protected] (M.A. Kozhushner).

properties of single molecules with the aim to reachthe ultimate limit of miniaturization by producing singlemolecule diodes, transistors, and switches [4–7]. In orderto fully utilize the unique properties of these one-dimen-sional molecular nanostructures, a fundamental under-standing of conduction mechanisms in such systemshas to be achieved.

The transport properties of single molecular devicesare remarkably different from the electrical behavior oftraditional solid-state devices. In general, most organicmaterials including single molecules do not conductelectricity in the usual way as it occurs in metals, i.e.,by moving free electrons at applied bias (band conduc-tivity). Due to the discrete nature of the electron energyspectrum and the presence of a substantial gap betweenoccupied and unoccupied molecular energy levels,

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M.A. Kozhushner et al. / Chemical Physics 319 (2005) 368–379 369

transport of electrons in the molecule attached to themetallic electrodes usually occurs via electron tunneling.

In many experiments, strong amplification of the tun-neling current through organic molecules was observed.Without a bridging molecule, vacuum tunnelingbetween metallic electrodes would exhibit exponentialdependence of the tunneling current as a function ofthe distance between the electrodes, the tunneling expo-nent 2(2meW/�h2)1/2 � 2.2 A�1 being determined by thework function of the electrodes W � 4–5 eV. When amolecule is inserted between the electrodes, tunnelingcurrent dependence on the length of the bridging mole-cule exhibits exponential dependence with a much smal-ler tunneling exponent �(0.4–1.2) A�1 [8]. Thisphenomenon is known in chemistry as the bridge effectand is primarily responsible for electron transfer reac-tions in biological systems [9] as well as electron tunnel-ing through self-assembled monolayers in STMexperiments [8,10,11].

Another important experimental discovery was madeby several groups who observed extremely high values ofthe electric conductance through very long (tens andhundreds of nanometers) segments of DNA [12–15] withweek dependence on the segment length. DNA is theorganic molecule with a large HOMO–LUMO gap(�7–9 eV). Even taking into account the bridge effect,the tunneling current through the molecule would bepractically zero already at lengths of several nanometers.But typical values of the conductance measured in manyexperiments are comparable with that of a metallic wireof the same thickness and length.

More surprisingly, different experiments demon-strated differing characteristics of I–V dependence. Inparticular, the I–V curve in experiment on bundles ofDNA strands [12,14,15] showed an Ohmic behavior,i.e., the conductance was approximately constant forsmall biases. However, in another experiment [13], theelectrical transport through a 10 nm long single doublehelix DNA segment had shown a pronounced thresholdas a function of applied bias. Other groups lately repro-duced both the Ohmic and threshold I–V features ofDNA conductance. In addition, the threshold in I–V

curves was observed in other organic systems such as re-cently discovered di-block oligomers [16,17] that showeddiode type I–V behavior.

The wide spectrum of the electrical behavior inducedspeculations in the scientific community about either the‘‘insulating’’, ‘‘semiconducting’’, or ‘‘metallic’’ characterof electron transport in organic molecules. However, itis important to make a distinction between electrontransport in traditional bulk solid-state materials suchas semiconductors or metals and a pure organic medium(we exclude the case of heavily doped conducting organicpolymers). These two types of materials arefundamentally different because organic molecules areinsulating in a sense that they do not possess free

carriers as opposed to the case of semiconductors andmetals where electrical conductance is due to the move-ment of free carriers such as electrons and holes. Obvi-ously, the absence of free carriers in an organicmedium does not allow direct transfer of transportmechanisms operational in bulk inorganic materials toexplain the electrical conductance in single organic mol-ecules and requires development of new theoretical con-cepts to explain the fundamental mechanisms of electrontransport in molecular systems.

In this paper, we present a theory of tunneling andresonant transitions that is able to rationalize andexplain the fundamental features of electron transportin molecular systems within a conceptually simple andunified framework. Our approach is based on Green�sfunction theory of electron sub-barrier scattering [18–20] off the structural units (or functional groups) of amolecular chain. The concept of sub-barrier scatteringallows us to treat effectively the many-body effects inelectron tunneling through organic molecules using aformalism of sub-barrier scattering amplitudes [18–20].The method which calculates the total scattering ampli-tude of the bridge molecule not only predicts theenhancement of the amplitude of tunneling transitionsin course of tunneling electron transfer through one-dimensional molecular structures but also allows us tointerpret conductance mechanisms by calculating thebound energy spectrum of the tunneling electron, theenergies being obtained as poles of the total scatteringamplitude of the bridge molecule. In particular, wefound that the resonant tunneling via bound states ofthe tunneling electron is the major mechanism of elec-tron conductivity in relatively long organic molecules.

We show that the many-electron effects are of para-mount importance in electron transport and its inclusionis critical for attaining a quantitative description of elec-tron transport in one-dimensional molecular nanostruc-tures. The sub-barrier scattering technique naturallyincludes a description of tunneling in applied electricfields which allows us to calculate I–V curves at finitebias. The developed theory is applied to explain experi-mental findings such as bridge effect due to tunnelingthrough organic molecules, threshold versus Ohmicbehavior of the conductance due to resonant electrontransfer, and temperature effects in electron transportthrough organic molecules.

2. Amplitude of the tunneling transition

Almost all observable electron tunneling transitionsare the transitions between the states of a continuousspectrum. These are the continuum of electronic statesin metallic electrodes in the case of a molecule attachedto metallic electrodes or the quasi-continuous vibrationspectra of donor and acceptor in the case of donor–

370 M.A. Kozhushner et al. / Chemical Physics 319 (2005) 368–379

acceptor tunneling transitions. In most of these cases theFermi ‘‘golden rule’’ is an excellent approximation for cal-culation of the probabilityW klkrðV Þ of the electron tunnel-ing transition from the state kl of the left electrode to thestate kr of the right electrode under applied bias V

W klkrðV Þ ¼ 2pjAklkr j2d EðklÞ � EðkrÞð Þ; ð1Þ

where Aklkr is the amplitude of the transition. In this pa-per, we use the atomic system of units �h = e = me = 1.The Fermi ‘‘golden rule’’ works because the transitionamplitude Aklkr is much smaller than the characteristicenergy scale of the substantial change of the density ofstates of metallic electrodes.

It is important to understand that the tunnelingtransitions through organic molecules are essentiallyquasi-equilibrium phenomena. Three main mechanismsmight result in a non-equilibrium situation: (1) non-equilibrium kinetic phenomena in left and right elec-tron subsystems due to removal of a tunneling electronfrom the left electrode and its addition to the rightelectrode in the course of tunneling; (2) inelastic inter-actions of tunneling electrons with vibrational degreesof freedom of the molecule; and (3) non-equilibriumoccupations of electronic levels inside the molecule inthe course of tunneling. All three factors are of minorimportance in the case of relatively small (�10�10–10�8

A) currents flowing through the molecule. For exam-ple, the time interval between individual electrontunneling transitions s � 10�11–10�9 s is much largerthat the characteristic time for establishment of ther-mal equilibrium in the electron gas of the metallicelectrodes se � 10�16–10�15 s and we can consider theelectronic subsystem to be unperturbed by occasionalelectron tunneling transitions from the left to the rightelectrodes. In addition, the probability of vibrationalexcitations of the molecule in the course of tunnelingtransitions is small, therefore, we consider tunnelingtransitions as purely elastic. It is also worth mentioningthat highly sophisticated non-equilibrium Green�s func-tion techniques widely used to describe transport inmolecules [21,22] is an overcomplication of essentiallyquasi-equilibrium tunneling phenomena in one-dimen-sional molecular nanostructures.

Using equilibrium statistics of electrons in the rightand left electrodes, we can write an expression for thetunneling current through the molecule as

IðV Þ ¼ 2pZ

dkl dkrjAklkr j2½f ðeðklÞ; T Þ � f ðeðkrÞ � V ; T Þ�;

ð2Þ

where f(e(kl),T) and f(e(kr) � V,T), are the Fermi distri-bution functions for electrons in the left and right elec-trodes, and V is applied bias. The tunneling amplitudeAklkr is expressed via the one-electron Green�s functionof the molecule G(r 0, r; e) and the electron wave func-

tions, wr,l and electron potentials, Ur,l of right and leftelectrodes, respectively

Aklkr ¼Z

dr

Zdr0wlðr0; klÞU lðr0ÞGðr0; r; eÞU rðrÞwrðr; krÞ;

ð3Þwhere G(r 0, r; e) also includes the electric field applied be-tween the electrodes. It is also worth noting that the Fer-mi golden rules (1) and (2) combined with expression (3)for the amplitude of electron transition is formallyequivalent to Bardeen�s transfer Hamiltonian theory[23] which is widely used to describe tunneling phenom-ena in condensed matter systems.

Our approach for calculating the current through themolecule is similar to the Landauer–Buttiker formalismor its generalization for the case of finite biases, non-equilibrium Green�s function (NEGF) technique [24].In both cases the current is expressed via a single-elec-tron Green�s function of a molecular system and theself-energies (or interface potentials, Ur,l as in our case)that take into account electronic interactions of the mol-ecule with the electrodes, see (2) and (3). However, incontrast to standard NEGF theory where the Green�sfunction of a neutral molecule is evaluated, we workwith the Green�s function of a molecule plus an extraelectron (or negative ion) which corresponds to thephysical situation of an extra electron interacting withthe molecule during the course of an electron transitionfrom one electrode to another. At negative electronenergies close to the Fermi energies of the electrodes,the Green�s function of the negative molecular ion ismostly determined by the electronic states of a continu-ous spectrum, i.e., by the states of the electron scatteredoff the molecule. Therefore, the developed theory of sub-barrier scattering takes this contribution into accountnaturally within its remit by expressing Green�s functionvia operators of sub-barrier scattering. In standardNEGF approaches based on the inversion of the Ham-iltonian matrix of a neutral molecule, the continuousspectrum is completely ignored.

Let us first consider the case of small applied biases.In the case of tunneling through vacuum (no moleculepresent between electrodes) the one-electron Green�sfunction is

G0ðr; r0; eÞ ¼ � 1

2pjr� r0j expð�jjr� r0jÞ; ð4Þ

where j = (2|e|)1/2 is the tunneling exponent in vacuum(the energies of the tunneling electron are negative in re-spect to the vacuum energy level which is chosen as zeroenergy). The Green�s function of a single molecularstructural unit placed between electrodes is [18–20]

Gðr; r0; eÞ ¼ G0ðr; r0; eÞ þ G0ðr;R1; eÞ� ð�2paðe; #ÞÞG0ðR1; r

0; eÞ; ð5Þ


where a(e,#) is the electron sub-barrier scattering ampli-tude off this structural unit or center located at R1 and #is the scattering angle between vectors R1 � r andr 0 � R1. The expression (5) has a clear physical meaning:the electron tunnels free plus scatters off the centerlocated at R1. The amplitude of potential scatteringcan be obtained as an analytic continuation of theamplitude at positive electron energies e > 0 or real mo-menta to the region of negative energies or imaginarymomenta k ¼ ij ¼ i

ffiffiffiffiffiffiffi2jej

p.

We have developed a variational asymptotic methodfor calculation of tunneling scattering amplitudes [18–20]. Within this approach the exponential tail of thewave function of the system of nc + 1 electrons (nc elec-trons of the center plus one tunneling electron) is deter-mined by varying the total energy functional. Therefore,the exchange interaction between the tunneling electronand the electrons of the scattering center is explicitlytaken into account. The general form of sub-barrierscattering amplitude a(e,#) is [25,20]

aðe; #Þ ¼ apoleð#Þe� e0

þ apotðe; #Þ; ð6Þ

where the energy e0 in the pole term is the energy of thevirtual bound state of the tunneling electron which isformed in course of scattering off the center. The secondterm in (6) is the potential part of the scattering ampli-tude which is an analytic and smooth function of energy.The energy and angular dependencies of the sub-barrierscattering amplitude off the atoms H (singlet and tripletscattering), He, Ne, Be [20] are shown in Fig. 1. Thehydrogen and beryllium atoms are the examples of openshell systems that exhibit strong scattering, whereasclosed shell systems such as helium and neon exhibitan order of magnitude weaker scattering. In all casesthe scattering amplitude a(e,#) shows similar energy

Fig. 1. The energy dependence of the amplitude of sub-barrierscattering a(e,#) at two scattering angles # = 0 and # = p for (a)singlet and triplet scattering off the hydrogen atom, beryllium, and (b)helium and neon.

and angular behavior. The potential part apot(e,#)exhibits a rapid monotonic increase for systems witheffective repulsion when apot(e,#) > 0, (triplet H, Be,Ne, He) and a monotonic decrease for systems with aneffective attraction when apot(e,#), (singlet scatteringoff the hydrogen atom). For example, in case of tripletelectron scattering off the hydrogen atom, apot(e,#) in-creases from �30 to �2.4 when varies from 0.18a.u. = �5 to 0 eV. The minimum and subsequent in-crease of the total scattering amplitude is a result ofapproaching the energy of the bound state e0 (see (6))when the pole term starts to contribute to a(e,#). Inaddition, a(e,#) strongly depends on the scattering angle# if the energy e is within the tunneling energy interval,but this dependence disappears when |e|! 0.

The expression for a tunneling Green�s function forthe general case of a molecular chain consisting of sev-eral scattering centers is

G ¼ G0 þ G0bT G0; ð7Þ

where bT is the total scattering operator which is relatedto the total scattering amplitude as bT ¼ �2pA. The scat-tering operator depends on coordinates, r, r 0 via scatter-ing angles for the electron coming from point r,scattered off the centers {R1, . . .,RN} and arriving atpoint r 0. The total scattering operator bT is determinedvia solution of the system of N linear equations [20]

bT nðeÞ ¼ tnðeÞ þXNk ¼ 1

k 6¼ n

tnðeÞG0ðRn;Rk; eÞbT � k; n¼ 1; . . . ;N

ð8Þ

as

bT ðeÞ ¼ XNn¼1

bT nðeÞ; ð9Þ

where tn � �2panðe; #ÞdðR� RnÞ is the scattering opera-tor off the individual nth center, and the nth componentTn of the solution vector {T1, . . .,TN} is the partial scat-tering operator that gives a subset of all multiple scatter-ing events that start from center n. In deriving (8), weassumed that the distance between any two centers issufficiently large so that the effective short-range poten-tials of the centers weekly overlap. The system (8) can begeneralized for the case of non-zero overlap of thecenters.

The tunneling Green�s function of the system of N

centers contains all the possible multiple scatteringevents

Gðr; r0; eÞ ¼ G0ðr; r0; eÞ

þX

G0ðr;Ri; eÞCðRi;RkÞG0ðRk; r0; eÞ; ð10Þ

i;k

Fig. 2. Graphs contributing to the tunneling Green�s function for asystem of two centers. Top panel corresponds to partial scatteringoperator T1, bottom panel corresponds to T2.


CðRi;Rk; eÞ ¼

1D ftiG0ðRiRa; eÞtaG0ðRa;RbÞtb� � � tvG0ðRvRk; eÞtkg; i 6¼ k;

1D ti; i ¼ k;

8><>:

ð11Þwhere D is the determinant of the system (8), and greekindices, a,b, . . .,m enumerate all the intermediate centersother than endpoints i and k. Each term in the sum (10)can be easily visualized using graphical diagrams: eachgraph connects space points r and r 0 by all the possiblepaths running through the centers {R1,R2, . . .,RN}.The vertex of each graph at the center Ri is representedby the center�s scattering operator ti(e,#i) which dependson the scattering angle #i, and the segment connectingany two centers centers i and k is represented by the vac-uum Green�s function G0(|Ri � Rk|; e). In Fig. 2, we showall the diagrams contributing to the tunneling Green�sfunction of the system consisting of two scatteringcenters.

The poles of the total scattering operator constitutethe spectrum of the bound states of the tunneling elec-tron. They are easily calculated as roots of the determi-nant D of the system (8). The graphs contributing to Dare all possible self-returning paths starting from eachscattering center. Based on a knowledge of the energyspectrum, specifically, the position of the energy spec-trum in respect of the Fermi energies of the left and rightelectrodes, we can identify different mechanisms oftransport through the molecule such as ordinary tunnel-ing and resonant electron transfer mechanisms.

Fig. 3. Exponent b of the bridge enhancement factor as a function ofelectron tunneling energy (bold curve). Dotted curve is for the casewhen only the pole term apole(#)/(e � e1) is taken into account.

3. Ordinary tunneling

When the bound energy spectrum of the tunnelingelectrons is higher than the Fermi energies of both the

left and right electrodes, the ordinary tunneling is themajor mechanism of electron transport through the mol-ecule. This regime is characterized by the exponentialdependence of the tunneling current along the lengthof the molecule I / exp(�2jL). The physical conse-quence of the interaction of the tunneling electron withthe molecule is the substantial reduction of the tunnelingexponent j compared to that for tunneling in vacuumj0 ¼

ffiffiffiffiffiffiffiffiffiffi2jeFj

p’

ffiffiffiffiffiffiffi2W

p� 1.1 A

�1where W � 4�5 eV is

the work function of the metallic electrodes.The total tunneling amplitude of this transition is ob-

tained by substituting the expression for the tunnelingGreen�s function (10) in the expression for the tunnelingamplitude (3). It is possible to show that the generalexpression for tunneling amplitude Alr can be written as

Alr ¼ eAlrG0ðjRl � Rrj; eÞf1þ BðjRl � Rrj; eÞg; ð12Þwhere the prefactor eAlr is

eAlrðe; kl; krÞ ¼Z

drwlðr; klÞU lðrÞG0ðjRl � rj; eÞ� �

�Z

dr0wrðr0;krÞU rðrÞG0ðjRr � rj; eÞ� �

;

ð13Þ

and the bridge enhancement factor has exponentialdependence on the length of the molecule

BðjRl � Rrj; eÞ ¼ expðþbðeÞjRl � RrjÞ. ð14ÞThe expression (14) was derived for the molecules thatconsist of more than five centers, and is not valid forvery short molecules.

The dependence of the bridge enhancement exponentb(e) on e for a model system consisting of the chain ofhydrogen atoms separated by the distance d = 6 a.u. forthe case of singlet scattering as(e,#) > 0 is shown inFig. 3. For this particular case, the lowest energy level


of the bound spectrum is close to vacuum level (zero en-ergy). Therefore, the pole term of the scattering ampli-tude of each individual center (6) gives a smallcontribution at e � eF and the physics of the bridge effect(amplification of the tunneling current) is mainly deter-mined by the potential term apot(e,#). In order to demon-strate this, the bridge enhancement exponent wascalculated for the case when only the pole term apole(#)/(e � e0) was included and as is seen from Fig. 3 (dottedcurve), the pole contribution is very small. However, itscontribution increases as the tunneling energy ap-proaches to the lowest energy level of the bound spec-trum, e � �0.025 a.u., see Fig. 3.

The qualitative behavior of the bridge enhancementexponent can be understood by considering the specificcase that allows us to obtain an analytic solution forB(|Rl � Rr|; e).

In particular, if the energy of the tunneling electron ismuch lower than the bound energy spectrum and the scat-tering amplitude of individual centers is positive (effectiveattraction), then the following condition is satisfied:

aðe; #ÞG0ðRi;Riþ1; eÞ ¼aðe; #Þ

dexpð�jdÞ � 1; ð15Þ

and the bridge enhancement factor is

BðRlr; eÞ ¼ C 1þ aðe; #Þd

� �Nþ1

; ð16Þ

whereN is the number of centers comprising themolecule,and the numerical coefficient C � 1. Within this model,the bridge enhancement exponent b obtained from (16) is

bðeÞ ¼ ln 1þ aðe; #Þd

� �. ð17Þ

The expression (17) shows that the energy dependence ofthe bridge enhancement exponent in Fig. 3 is determinedentirely by the energy dependence of the scatteringamplitude of the individual scattering center a(e,#). Inparticular, the minimum in b(e) shown in Fig. 3 is dueto the minimum of singlet scattering amplitude a(e,#)of the hydrogen atom, see Fig. 1(a).

The expression for the tunneling current in the case ofa small applied bias V � |eF| = W is obtained by substi-tuting (12) and (14) in (2)

IðV Þ ¼ V�qlðeFÞqrðeFÞ

2peA2

lrðeFÞexpð�2j0jRl � RrjÞ

jRl � Rrj2

� 1þ exp bðeFÞjRl � Rrjð Þ½ �2�; ð18Þ

where eF = �W, W is the work function of the left andright electrodes, ql(eF) and qr(eF) are their densities ofstates at the Fermi level, j0 ¼

ffiffiffiffiffiffiffiffi2jeF

pand eAlrðeFÞ is the

prefactor eAlrðe; kl; krÞ averaged over the energy surfacese(kl) = e(kr) = eF. The expression in curly brackets is thetunneling conductance G = I/V = dI/dV at V = 0.

4. Effect of an electric field

If a finite bias is applied, then the problem of sub-bar-rier scattering is solved in the presence of an electricfield. We must address the issue of possible charge redis-tribution within the molecule due the applied electricfield and a tunneling current passing through themolecule.

An important reference is the magnitude of themicroscopic electric field inside the molecule,Emol � e

a20

¼ 1 a.u. � 1012 V=m. If several volts is appliedacross a molecule several nanometers long, then themagnitude of the external electric field is 1 V/1 nm =109 V/m which is at least three orders of magnitudesmaller than the internal microscopic electric field.Therefore, the change of the scattering operator due tothe modification of the local electronic structure by theexternal electric field is negligible. In addition, the exter-nal electric field due to an applied voltage is essentiallyan electrostatic electric field that would exist in a systemof two bare electrodes without a molecule because: (1)there are no external charges between electrodes present,(2) the molecular polarizability effects are of minorimportance, and (3) tunneling current passing throughthe molecule is small. Then, in the case of flat electrodesand a linear molecule the external electrostatic potentialis distributed linearly along the molecule and there is noneed to solve self-consistently the Poisson equation.More over, the change of the total energy of the isolatedmolecule upon application of external electrostatic fieldis very small. We found that total energy of eight-ringthiophene oligomer changed only by 0.08 eV when themolecule was placed in the electric field correspondingto bias 2 V applied across the length of the molecule.

Due to the above arguments, the effect of an electricfield can be easily included into our formalism of sub-barrier scattering by parametric referencing of the localvacuum levels of the scattering centers by the local elec-trostatic potential and an additional modification of thevacuum Green�s function to include explicitly the elec-trostatic potential. In particular, we assume that thepolarity of applied bias is such that the electrons tunnelfrom the left electrode to the right electrode and the en-ergy of ith scattering center is shifted as ei ! ei + Vi,where Vi is the value of the local electrostatic potentialat the center,V i ¼ �V Ri

Rlr< 0, see Fig. 4.

The vacuum Green�s functions G0(Rn,Rk; e) connect-ing individual scattering centers Rn and Rk in (8) are re-placed by quasi-classical Green�s functions for theelectron in a homogeneous electric field E = V/Rlr

GVðRn;Rk; e;EÞ ¼ � 1

2pjRn � Rkj� expð�SVðRn;RkÞÞ;

ð19Þ

where the action SV(Rn,Rk) is

Fig. 5. Current–voltage curves for the case of vacuum tunneling(dotted line with squares, right scale) and for the tunneling through themolecule (left scale).

Fig. 4. Energy diagram for the left-electrode molecule right-electrodejunction. Zero energy is the energy of the vacuum level.


SVðRn;RkÞ¼Z Rk

Rn

dzffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðjej�EzÞ

p

¼2ffiffiffi2

pjRl�Rrj3V

ðjej�jV njÞ3=2�ðjej�jV kjÞ3=2n o

.

ð20Þ

Here, we assumed that the scattering centers are along astraight line but it is easy to generalize the quasi-classicalexpression (19) to include a general three-dimensionalconfiguration of the molecule.

The system of linear equations for the total scatteringoperator is modified accordingly

T nðe;V Þ ¼ tnðe� V nÞ þXk 6¼n

tnðe� V nÞGVðRn;Rk; eÞT kðe;V Þ;

ð21Þ

where the total scattering operator is

T ðe; V Þ ¼XNn¼1

T nðe; V Þ. ð22Þ

It is easy to modify the expressions (10) and (11) for thetotal Green�s function and the expression for the tunnel-ing amplitude to include explicitly the electric field.Then, assuming that |V| < |eF|, we can derive the expres-sion for the tunneling current at finite bias V:

IðV Þ ¼Z eF

eF�VdeqlðeÞqrðe� V ÞeA2

lrðeÞ

� expð�2SVðe; jRl � RrjÞÞ2pjRl � Rrj

� ½1þ expðbðe; V ÞjRl � RrjÞ�2. ð23Þ

The I–V curve for the same model system, a molecu-lar wire of hydrogen atoms separated by a distanced = 6 a.u., is shown in Fig. 5. We can interpret gen-eral features by examining (23) in the case of smallbiases (|V| � |eF|). The differential conductance forthe case of vacuum tunneling (two electrodes withoutmolecule) is

dðln IÞdV

¼ �jjRl � RrjV

2jeFj; ð24Þ

that is the tunneling current increases exponentially withbias V. For the case of tunneling through the molecule(bridge enhanced current) the differential conductance is

dðln I tÞdV

� �jjRl � RrjV

2jeFj� 2jRl � Rrj

dbðe; V ÞdjV j . ð25Þ

Because the energy dependence of the bridge enhance-ment exponent b(e) is determined entirely by the energydependence of the scattering amplitude a(e,#), of an indi-vidual center, see (17), we can make the conclusion thatthe derivative db(e,V)/d|V| is always negative based onthe fact of the general monotonic behavior of the scatter-ing amplitude for both cases of attractive a(e,#) > 0, andrepulsive a(e,#) < 0 interactions, see Fig. 1.

Therefore, the bridge enhanced current increases withbias much slower as compared to vacuum tunnelingexcluding the interval of very small bias, see Fig. 5. Atthe same time, the tunneling current through the mole-cule is enhanced by a factor 1015 (bridge enhancement)compared to vacuum tunneling.

The tunneling current in Fig. 5 was calculated for 20A long molecule and its value is on the order of picoam-peres. In most experiments, however, the measured tun-neling current is on the order of nanoamperes fororganic molecules of such length, which indicates thepresence of another mechanism of electron transfer thatsubstantially enhances the current through the molecule.This mechanism is the resonant electron transferthrough the bound energy levels of the tunneling elec-tron that we are going to consider in the next section.

5. Resonant electron transfer

If the bound energy spectrum of the tunneling elec-tron is close to the Fermi energies of the electrodes, a

s

Fig. 6. Resonant tunneling transfer occurs when some energy levels ofthe bound spectrum are within energy interval eFr 6 es 6 eFl , whereeFr ¼ eFl � jV j.


new transport mechanism comes into play, see Fig. 6.The sub-barrier scattering formalism developed so farhas to be modified to include resonant electron transfer.One of the critical pieces of information is the bound en-ergy spectrum that is required to assess the states con-tributing to resonant electron transfer.

As was already mentioned, the bound energy spec-trum is automatically obtained in course of solving sys-tem (21), which is solved in order to find the totalscattering operator T(e,V). The energies of the boundstates es in the electric field E are found as poles of thetotal scattering operator, i.e., as the roots of the determi-nant of system (21)

det jdik � tiðe� V i; #ÞGVðRi;Rk; e; V Þj ¼ 0. ð26ÞThe bound state wave-function ws corresponding to s-throot of the secular equation (26) is expressed via the nor-malized solution ~T ðsÞ ¼ fT 1ðsÞ; T 2ðsÞ; . . . ; T N ðsÞg ofhomogeneous system (21) for sth scattering vector

wsðr; esÞ ¼Xl

T lðsÞ~uðr� Rl; esÞ; ð27Þ

where ~uðr� RlÞ are additional contributions to theexponential tail of the electron wave function of the tun-neling electron due to its interaction with scattering cen-ter l. These wave functions are determined in the courseof a variational minimization procedure that was devel-oped with a specific focus to resolve exponentially smallcontributions.

There is a clear analogy between functions ~uðr� RlÞthat form the wave function of the tunneling electron(27) and the atomic wave functions that form the inde-pendent electron, tight-binding wave function (29) inLCAO method. The components Tl(s) of the scatteringvector ~T are similar to the coefficients Cl of the LCAOexpansion of the tight-binding wave functions. TheLCAO energies and wave-functions for a one-dimen-sional chain are written as

ek ¼ e0 � 2h cosðkdÞ; ð28Þ

wTBk ¼

ffiffiffiffi2

N

r XNl¼1

Cluðr� RlÞ; ð29Þ

where the one-dimensional wave vector of a molecularwire is k = pn/(N + 1)d, n = 1, . . .,N, u(r � Rl) is theatomic wave function centered at atom l that has on-siteenergy e0, and h is the hopping integral between nearestneighbor atoms. The LCAO expansion coefficients Cl

are determined via the solution of the tight-binding sec-ular equation that includes explicitly the effect of theelectric field by referencing the on-site energies of eachcenter by the corresponding local electrostatic potential:e0 ! e0 � |Vi|. In the case of zero applied biasCl = sin (kRl), i.e., these are the usual Bloch wave phasefactors for the wave function of the system with 1-dperiodicity.

It is worth discussing the connection of many-elec-tron sub-barrier scattering theory and the essentiallyone-electron tight-binding approach that gives thetight-binding energy spectrum (28) and the tight-bindingwave functions (29). The tight-binding method is a sim-plified version of the standard density-functional LCAOmethod for an electronic structure widely used for thedescription of electronic transport in molecules. It ispossible to show that if the distance between scatteringcenters is large, d > 10 a.u., the solution for the eigen-spectrum and wave-functions of the tunneling electronobtained within the sub-barrier scattering approachcan be cast into the tight-binding form (28) and (29),if the hopping integral h is related to the parameters apoleand e0 of the pole term in (6) via expression

h ¼ apole=d � exp �ffiffiffiffiffiffiffiffiffiffiffiffi2je0jd

p� �. ð30Þ

However, at smaller distances between the scatteringcenters the many electron effects substantially modifythe physics of the resonant tunneling and the two ap-proaches give drastically different results. In particular,within the tight-binding approach the probability of res-onant tunneling is close to zero. In order to see this, it isnecessary to write down the general form of the resonanttunneling amplitude Ares(e).

Let us consider one of the bound energy states s thatis in the resonance condition, i.e., eF � |V| < es < eF, seeFig. 6. In contrast to the case of ordinary tunneling,we do not need to use the previous expression for thetotal Green�s function (10) that explicitly takes intoaccount the contribution of the entire energy spectrumof the tunneling electron including continuous states.Instead, only resonant state es makes a dominant contri-bution to the total Green�s function which can be explic-itly written down using spectral representation as

GresðR;R0; eÞ ¼ wsðR; esÞwsðR0; esÞe� es þ ic

; ð31Þ

Fig. 7. Sub-barrier scattering vector~T ðsÞ ¼ fT 1; T 2; . . . ; T Ng and tight-binding LCAO coefficients Cl(s) along molecular wire.


where ws(R; es) is the wave function of the bound state sof the tunneling electron that is given by expression (27)and cs is the imaginary part of the energy of resonantstate s associated with the finite lifetime of this energylevel. Substituting (31) into the expression for the tran-sition amplitude (3), we obtain the expression for theresonant amplitude Ares(e, es)

Aresðe; esÞ ¼ ~AlðesÞ~ArðesÞT 1ðsÞT NðsÞe� es þ ics

; ð32Þ

where amplitudes

~Alðes; klÞ ¼Z

drwlðr; klÞU lðrÞ~uðr� R1; esÞ;

~Arðes; krÞ ¼Z

drwrðr; krÞU rðrÞ~uðr� RN ; esÞ;ð33Þ

describe the coupling of the electronic states kl and krof the left and right electrodes with the resonant wavefunction ~uðr� R1; esÞ of the tunneling electron. Obvi-ously, the amplitude of the resonant tunneling transi-tion (32) depends on partial scattering operators T1(s)and TN off the first and last centers of the molecularwire that are obtained as components of the solutionof the system (21) in the case of sub-barrier scatteringor as the first C1 and last CN LCAO coefficients of thetight-binding solution (29). We would like to examinethe spatial behavior of both partial scattering opera-tors Tl(s) and the LCAO coefficients Cl(s) in orderto see the difference in the two approaches in describ-ing resonant tunneling.

In general, the tight-binding amplitude of resonanttunneling Ares is negligible because the LCAO coefficientC1 closest to the left electrode is exponentially smallcompared to the coefficient CN at the right electrode.In order to demonstrate this let us consider the specificexample of a molecular chain of N = 30 centers eachhaving on-site energy e0 = 1 eV and hopping integralh = 1 eV, the values being chosen to capture character-istic valence electronic interactions in organic molecules.We also assume that the Fermi energy of metallicelectrodes at V = 0 is eF = �4 eV which correspondsto a typical work function of metals. Under these condi-tions the lowest energy level of the tight-binding bandcorresponding to the state with k1 = p/(N + 1)d is 1 eVabove the Fermi energy at zero bias. The resonant con-dition within the tight-binding model is satisfied if theapplied bias is equal to the threshold value Vth = 1.25eV, i.e., when the lowest energy level of the tight-bindingband aligns with the Fermi level. The tight-bindingLCAO coefficients Cl(s) as a function of center index l

are shown in Fig. 7 (curve with dots). As is seen fromFig. 7, the tight-binding solution decays extremely fastwhen going from the right to the left electrodes, i.e.,C1/C30 � 10�8. As we will see later, the amplitude ofthe resonant transition is proportional to the coefficientof the wave function at the first center close to the left

electrode, i.e., T1 in the case of sub-barrier scatteringor C1 in the case of tight-binding. Therefore, even ifthe resonant condition is attained in the course of a biasincrease, the amplitude of the transition is negligible.

This numerical result has a simple physical explana-tion that is valid for any electronic structure methodthat uses a single electron approximation in describingelectron-electron interactions within a potential frame-work, including DFT. The external electron passingthrough the molecule is essentially a weakly interacting,free-electron like particle. When the bias is applied, theelectron in the lowest bound state with e � eF becomeslocalized near the right electrode and penetrates to theleft under a triangular potential barrier U = �Vz/Rlr

with the effective mass m* = �h2/(d2e/dk2) = 1/2hd2 whichgives an exponentially small value of the tight-bindingwave function near the left electrode.

In contrast, the sub-barrier scattering approach thattakes into account the many-electron interactions givesa correct picture of resonant electron transfer. Let usconsider the same molecular wire used for the tight-binding exercise above. The parametrization of the scat-tering amplitude off the individual center is

aðeÞ ¼ 1.0

eþ 0.037þ ð�10þ 350eÞ; ð34Þ

where the parameters of the pole part in (34) were cho-sen based on the tight-binding parameters used above(see relationship (30)). The potential part was approxi-mated based on our previous calculations of the scatter-ing amplitudes for different systems. For simplicity, wealso neglected the angular dependence of the scatteringamplitude in (34).

The numerical solution of the secular determinant(26) gives the energy of the lowest state in the boundspectrum to be in resonance with the Fermi level of


the left electrode at applied bias V = 1 eV. The scatter-ing vector ~T corresponding to this energy has apprecia-ble component T1 near the left electrode, and the ratioT1/T30 � 2. · 10�5 is much larger than that of thetight-binding solution, see Fig. 7. In contrast to singleelectron potential methods the sub-barrier scatteringapproach takes into account many-electron interac-tions and, as a result, the resonant wave functiondecays much more slowly towards the left electrode.Therefore, 21 resonant tunneling transfer does havean appreciable amplitude and we might expect a newmode of the electron transport that greatly assists thetransfer of electrons in the case of relatively longmolecules.

The reason for the slower decay of scattering vectorcomponents Tl towards the left electrode can be tracedback to mutually compensating energy dependences ofthe scattering operator off the individual center t(e)and the Green�s function GV(Ri,Ri±1; e) connectingnearest neighbor centers. This results in a weak energydependence of the matrix elements of the secular matrix(26). Because spatial dependence is coupled to energydependence as a result of referencing of local vacuumlevels by electrostatic potential as we go along the wire,we obtain a weak spatial dependence of the solution, i.e.,scattering vector ~T .

Our calculations of resonant electron transitions inan external electric field are based on the fact that nei-ther the external electrostatic potential nor the currentproduce substantial changes in the electronic densitycompared to the state of the molecule in the absenceof the electric field. This is because the occupationof the charged resonant states contributing to the cur-rent is very low. However, recent calculations basedon NEGF formalism revealed substantial changes inthe charge density and the corresponding electrostaticpotential due to an applied electric field and thecurrent passing through the molecule [26]. The statescontributing to the current in NEGF theory are thehole states, i.e., occupied states of the neutral mole-cule that start to participate in transport when anapplied bias raises them above the Fermi energy. Itis not surprising that these ‘‘charged states’’ (i.e., thestates that correspond to the Hamiltonian of the pos-itively charged hole) are very sensitive to the appliedelectric field and their wave functions are substantiallydeformed in the electric field (e.g., see the tight-bind-ing wave-function in Fig. 7). However, the occupa-tions of the hole states (on the order of T1/TN) arevery low due to the rapid emptying of the levels thatlie in the energy interval eF � |V| < es < eF. Therefore,it is not completely clear why the states with a verysmall occupation probability are contributing in a sub-stantial way to the charge density and electrostaticpotential of the molecule in NEGF calculations ofthe transport through the molecule.

6. Resonant tunneling current

The resonant tunneling current is determined by thetotal resonant tunneling amplitude

Areslr ðeÞ ¼

Xs

Aðe; esÞ; ð35Þ

which includes contributions from each resonant state swithin the energy interval elF � jV j 6 es 6 elF. For a givens only a narrow energy interval around es, es � cs < e< es + cs, contributes to the resonant tunneling transi-tions because each partial amplitude A(e, es) contains adominant pole factor (e � es + ics)

�1. Beyond this en-ergy interval only ordinary tunneling takes place, butbecause of its exponentially small values for sufficientlylong molecules, we can neglect its contribution and con-sider only transitions via resonant energy level es closestto a given energy e.

By substituting total resonant tunneling amplitude(35) and (32) in the general expression for the tunnelingcurrent (2), the total resonant tunneling current is ob-tained as a sum of partial currents from each boundstate within an energy interval elF � jV j 6 es 6 elF andis given by

I reslr ðV Þ ¼X

elF�jV j6es6el

F

I resðesÞ; ð36Þ

where

I resðesÞ ¼2p2

cseA2

leA2

rT21ðesÞT 2

N ðesÞqlðesÞqrðesÞ. ð37Þ

In deriving (36) and (37), we assumed that the width cs issmaller than the separation between neighboring reso-nant energy levels {es} which allowed us to replace theLorentz function by the d-function: ððe� esÞ2 þ c2s Þ

�1

! p=csdðe� esÞ.The expression (36) for the resonant tunneling cur-

rent contains the resonant width cs that must be deter-mined in a self-consistent manner. The inverse lifetimecs is determined by the probability of the transition ofthe electron from state es to all other states. The Fermigolden rule gives the following expression for this prob-ability cs

cs ¼ 2pfeA2

l ðesÞT 21ðesÞ þ eA2

r ðsÞT 2N ðesÞg; ð38Þ

where the first and second terms in (38) give the proba-bilities of transitions from state s to the left and to theright electrodes, respectively. We have already learnedthat the left component of the scattering vector T1 ismuch smaller than the right component TN, thereforethe first term can be dropped in (38). Then, we obtainthe final expression for the partial resonant tunnelingcurrent due to resonant state es

Fig. 9. Ohmic I–V curve that was calculated for a model system thatmimic the transport in DNA bundles.


I resðesÞ ¼ peA2

l ðesÞT 21ðesÞqlðesÞ. ð39Þ

The apparent asymmetry of the expression (39) is due tothe presence of a rate limiting step in the resonant trans-fer which is a transfer of the electron from one of thestates of the left electrode to the resonant bound states. Once the electron reaches the virtual bound state s,then this level is quickly emptied by a fast transfer toone of the states of the right electrode.

The current–voltage curve in resonant tunneling re-gime would exhibit a step-like structure as the bias is in-creased because more and more resonant states from thebound state spectrum are included. However, due to tem-perature effects, the step-like structure is smeared out as aresult of the fluctuations in the resonant energy levelsaround zero-temperature values e0s . This effect is presenteven at low temperatures because positions of the energylevels depend exponentially on the distance between thenearest-neighbor centers d, see (26) and the effect of smallmolecular vibrations to be greatly amplified.

We calculated the resonant I–V curve for the samemodel system that we considered in Section 4 usingparametrization (34) for the scattering amplitude a(e).The calculated I–V curve shown in Fig. 8 has a distinctthreshold Vth � 1 eV, the currents being on the order ofnA which is in quantitative agreement with experimenton single strand DNA [13]. In another experiment [12]bundles of DNA were used in measurements and nothreshold was observed. Instead, I–V curves showedOhmic behavior, i.e., a linear increase of the currentwith the applied bias. In order to take into accountthe conditions of experiment, we assumed that the weakinter-strand interaction in the bundle will only slightlymodify the scattering operator. Therefore, we increaseda linear slope in the potential part of the scatteringamplitude by �10%

Fig. 8. Resonant I–V curve for a model of single strand of DNA, thatshows clear threshold behavior, was calculated using the parameterscorresponding to experiment [12].

aðeÞ ¼ 1

eþ 0.037þ ð�10þ 400eÞ. ð40Þ

As in experiment, we were specifically interested in therange of small applied biases 0 6 V 6 0.1 V. A calcu-lated I–V curve for this case is shown in Fig. 9. In con-trast to the previous case, we did not observe a thresholdin the I–V curve. Moreover, the conductance dI/dV isconstant, i.e., the I–V curve is linear and the regime isindeed Ohmic. Also, the values of the current at suchsmall bias are much higher as compared to the case ofsingle strand DNA.

The resonant mode of transport is usually character-ized by a weak length dependence of the tunneling cur-rent on the distance between the left and rightelectrodes. Therefore, we investigated the length depen-dence of the conductance. The I–V curve was calculatedfor a molecule consisting of N = 60 and compared to thecase N = 30. We did not observe the length dependenceof the conductance, i.e., the I–V curves are very similar.

We explained the substantial conductivity of rela-tively long DNA molecules with the dominant contribu-tion of the resonant electron transfer to electrontransport. Recently, a new mechanism of charge trans-port in DNA molecules due to hopping of the holeshas been proposed by Jortner and co-workers [27,28].The variable-range hoping is based on the assumptionthat there are electron (or hole) traps in the mediumdue to electronic defects. In the case of DNA the elec-tron energies of these defect levels with respect to vac-uum are comparable to the ionization potential ofDNA, i.e., they lie several eV below the Fermi energiesof the electrodes. Therefore, this mechanism is not oper-ational for the case of electron transport in metal–mol-ecule–metal systems. An additional experimentalconfirmation of this statement is a weak temperature


dependence of the DAN conductance [29]. If variable-range hopping were important it would exhibit a strongtemperature dependence due to its activation nature.

7. Conclusions

In this paper, we presented a new approach for inves-tigating electron transport through organic molecules.This theory is drastically different from the standard,one-electron potential description of the electron struc-ture widely used to model electron transport throughsingle molecules. We found that the many-electroneffects play an important role in electron transport andin order to address them, we have developed a theoryof sub-barrier scattering that includes exchange interac-tions naturally within its remit. Our approach predictedtwo important mechanisms of electron transport: ordin-ary tunneling and resonant tunneling. In the first case,sub-barrier-scattering theory predicts a substantialamplification of the tunneling current by the molecule(bridge) compared to vacuum tunneling, the amplifiedtunneling exponents being in good agreement withexperiment. The physics of resonant tunneling is deter-mined by the bound energy spectrum of the tunnelingelectron. Based on the position of the lowest level ofthe band in respect to the Fermi energy of one of theelectrodes, we predicted threshold and Ohmic modesof transport. Although we illustrated the features oftransport mechanisms by performing model calcula-tions, we are confident that several aspects of electrontransport are fundamental phenomena that will also bepresent in more elaborate calculations that we plan todo in the future.

Acknowledgments

M.A.K. and V.S.P. thank the Russian Foundationfor Basic Research for financial support under Grant05-03-32102. I.I.O. thanks the National ScienceFoundation for financial support under Grant CCF-0432121.

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