yongqiang xue and mark a. ratner- microscopic study of electrical transport through individual...

8/3/2019 Yongqiang Xue and Mark A. Ratner- Microscopic study of electrical transport through individual molecules with metallic contacts. I. Band lineup, voltage drop, an

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Microscopic study of electrical transport through individual molecules with metallic contacts.

I. Band lineup, voltage drop, and high-field transport

Yongqiang Xue* and Mark A. RatnerDepartment of Chemistry and Materials Research Center, Northwestern University, Evanston, Ilinois 60208, USA

Received 4 March 2003; revised manuscript received 27 June 2003; published 11 September 2003

We present the first in a series of microscopic studies of electrical transport through individual molecules

with metallic contacts. We view the molecules as heterostructures composed of chemically well-definedatomic groups, and analyze the device characteristics in terms of the charge and potential response of theseatomic groups to the perturbation induced by the metal-molecule coupling and the applied bias voltage, whichare modeled using a first-principles based self-consistent matrix Greens function SCMGF method. As thefirst example, we examine the devices formed by attaching two benzene-based molecular radicalsphenyldithiol PDT and biphenyl dithiol BPDsymmetrically onto two semi-infinite gold electrodes through theend sulfur atoms. We find that both molecules acquire a fractional number of electrons with similar magnitudeand spatial distribution upon contact with the electrodes. The charge transfer creates a potential barrier at themetal-molecule interface that modifies significantly the frontier molecular states depending on the correspond-ing electron density distribution. For both molecules, the metal Fermi level is found to lie closer to thehighest-occupied-molecular-orbital HOMO than to the lowest-unoccupied-molecular-orbital LUMO. Trans-mission in the HOMO-LUMO gap for both molecules is due to the metal-induced gap states arising from thehybridization of the metal surface states with the occupied molecular states. Applying a finite bias voltage leads

to only minor net charge injection due to the symmetric device structure assumed in this work. But as currentflows, the electrons within the molecular junction redistribute substantially, with resistivity dipoles developingin the vicinity of potential barriers. Only the delocalized electrons in the benzene ring can effectively screenthe applied electric field. For the PDT molecule, the majority of the bias voltage drops at the metal-moleculeinterface. But for the BPD molecule, a significant amount of the voltage also drops in the molecule core. Thefield-induced modification of the molecular states the static Stark effect becomes significant as the biasvoltage increases beyond the linear-transport region. A bias-induced reduction of the HOMO-LUMO gap isobserved for both molecules at large bias. The Stark effect is found to be stronger for the BPD molecule thanthe PDT molecule despite the longer length of the former. For both molecules, the peaks in the conductance aredue to electron transmission through the occupied rather than the unoccupied molecular states. The calculationis done at room temperature, and we find that the thermionic-emission contribution to the current-voltagecharacteristics of both molecules is negligible.

DOI: 10.1103/PhysRevB.68.115406 PACS numbers: 85.65.h, 73.63.b, 73.40.c

I. INTRODUCTION

Exploring the use of individual molecules as active com-ponents in electronic devices1 has been at the forefront ofnanoelectronics research in recent years due to the potentialadvantage of ultrahigh density/speed and low-cost devicefabrication through self-assembly/self-organization processesthat such molecular-scale devices may bring.2,3 Numeroususeful device characteristics including molecular rectifyingdiodes, negative differential resistance and field-effect tran-

sistors have been demonstrated using molecular-scale struc-tures including small conjugated molecules,415 single- andmultiple-wall carbon nanotubes16 and macromolecules suchas DNA.17 A quantitative understanding of the physicalmechanisms underlying the operation of such diversemolecular-scale devices has been and remains a major chal-lenge in nanoelectronics research.

The electrical characteristics of molecular devices areusually measured by sandwiching the molecules betweentwo metallic electrodes.315 The measured transport charac-teristics reflects both the nature of the metal-molecule inter-face and the properties of the molecular layer.1829 There are

basically three electronic processes of interest in such metal-molecule-metal junctions: charge transfer between the metalsand the molecules, the change of the electrostatic potential,and the modification of the molecular geometry and elec-tronic states. The nature of these processes under both zeroand nonzero bias determines the electrical characteristics ofthe molecular junction. Here it is natural to separate theproblem into device at equilibrium and device out of equi-librium since 1 the linear dc-transport property of the mo-lecular device probes the equilibrium charge distribution of

the molecular junction established by adsorption onto theelectrodes and 2 the nonlinear transport probes the chargeresponse of the molecular junction to the applied electricalfield established through a finite bias voltage. A microscopictheory of molecular electronics will therefore need to 1determine the appropriate geometry of the molecular junc-tion, 2 determine the self-consistent charge transfer and theresulting lineup of the molecular levels relative to the metalFermi level and the modification of the molecular states uponformation of the metal-molecule-metal junction, 3 deter-mine the molecular screening of the applied electric field, thefield-induced modification of molecular states the static

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Stark effect, and the nonequilibrium electron distributionwhen current is flowing, and 4 determine the current/conductance-voltage characteristics and their correlationwith the molecular and device structures. None of these is-sues has been satisfactorily solved such that theory can beused in conjunction with experiment for an unambiguousidentification of the device operation mechanism. The pur-pose of our work is to elucidate the above device electronicprocesses and their dependence on the given molecular anddevice structures under both equilibrium and nonequilibriumconditions within a well-defined theoretical model. Specifi-cally, we will investigate the different aspects of moleculartransport due to the interplay between the electronic pro-cesses at the metal-molecule interface, the electronic pro-cesses in the molecule core and the molecular response to theapplied electrical field when current is flowing. Correspond-ingly, we focus our attention on the electron dynamics andcalculate the device characteristics within the coherent trans-port regime. Our emphasis is on the conceptual understand-ing and the chemical trends obtained from detailed micro-scopic calculation of simple but representative molecular and

device structures. Our work can therefore be considered as aminimal microscopic model of single-molecule electron-ics.

The ionic dynamics of the metal-molecule-metal junctioncan also be affected by the above electronic processes both atequilibrium and out of equilibrium, i.e., the adsorption andbias/current-induced conformation change. Neither aspectwill be treated here. By confining ourselves to the coherenttransport regime, we also neglect the effect of electron-vibrational coupling on transport.30 Solving the adsorption-induced conformation change requires an accurate knowl-edge of the surface lattice structure at the atomic scale fordevices under applied voltage. This is almost never known in

typical molecular transport measurement. Since one goal ofsuch a calculation is to provide the input nuclear configura-tion for transport calculation, our purpose can be servedequally well by examining the effect that different adsorptionand molecular geometries may have on the device character-istics, which will be treated in subsequent papers of the se-ries. A rigorous solution of the bias/current-induced confor-mational change and the general solution of inelasticcontribution to nonlinear current due to electron-vibrationalcoupling is a challenging topics which requires solving thenonequilibrium dynamics of the coupled electron-nucleisystems31,32 since the molecular potential energy surface isaffected by the nonequilibrium electron dynamics and elec-tron energy is dissipated in moving the atoms. By focusingon the electron dynamics in the molecular junction with fixedgeometry, the present work will provide a reference forevaluating the importance of such additional complicationsand provide the necessary input for further investigation insituations where they are indeed critical.

In typical molecular junction measurement, conjugatedmolecules are attached to the metallic electrodes through ap-propriate end groups. In this paper we will consider currenttransport through two benzene-based molecular radicalsphenylene dithiol PDT and biphenylene dithiol BPDadsorbed symmetrically onto two semi-infinite gold 111

electrodes through the end sulfur atoms. These structureschosen are among the simplest possible but are still repre-sentative of current experimental works. For the device atequilibrium, we obtain the self-consistent charge transfer, theadsorption-induced change in the electrostatic potential, thelineup of the molecular level relative to the metal Fermilevel, and the modification of the individual molecular statesdue to the metal-molecule coupling. For the device out ofequilibrium, we obtain the charge injection/redistributionwithin the metal-molecule-metal junction, the molecularscreening of the applied field and the resulting voltage drop,the modification of the molecular states by the applied field,the nonequilibrium occupation of molecular orbitals, and thecurrent/conductance-voltage characteristics for the givencontact geometry.

The calculation we present is performed using a recentlydeveloped self-consistent matrix Greens function SCMGFmethod18,19 which is based on the nonequilibrium generali-zation of the quasiparticle Greens function theory3337

NEGF and uses a finite local-orbital basis set. By replacing

the quasiparticle exchange-correlation self-energy38

with theexchange-correlation potential within the density functionaltheory,39,40 the well-established technique of self-consistentfield theory of molecular electronic structure can then beutilized for transport calculations.19 The real-space formula-tion of our approach allows us to provide an intuitive andcoherent physical picture of the molecular transport by ana-lyzing the device electronic processes both at the atomicscale and on the basis of individual molecular orbitals. Weview the molecules as comprised of chemically well-definedatomic groups and interpret their electrical characteristics interms of the response of these atomic groups to the pertur-bation induced by the metal-molecule coupling and the ap-

plied bias voltage. We emphasize the insight obtained withsuch atomic-scale analysis and show the important effect ofatomic-scale charge and potential inhomogeneity on devicecharacteristics at the molecular scale, which may otherwisebe obscured by the molecular-level analysis treating the mol-ecules as a whole. In particular, two important conclusionscome out of this work: 1 the adsorption-induced modifica-tion of molecular states is larger than the field-induced effectunless we go to large bias, so accurate modeling of the elec-tronic processes at equilibrium is critical for determining thelow-bias transport characteristics and 2 the effect of boththe metal-molecule coupling and the applied electric field inturning the individual molecular orbitals into effective con-

duction channels depends on the detailed charge and poten-tial distribution across the metal-molecule-metal junctionand may be different for different molecular orbitals. Engi-neering the charge and potential inhomogeneity as com-monly done in quantum semiconductor heterostructures willbe equally important at the molecular scale. Both aspectshave been largely neglected in the past. We therefore hopethe results obtained here will provide an useful guide regard-ing the nature of electron transport at the ultimate limit ofdevice scaling and the prospect of device engineeringthrough molecular design.

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II. THEORETICAL MODEL

A. The self-consistent matrix Greens function theory

The details of the self-consistent matrix Greens functiontheory have been described elsewhere,18,19 so we only give abrief summary here to clarify the assumptions and approxi-mations and to show how the physical observables are com-puted. Similar but different methods based on the NEGF

approach have been developed independently by other re-search groups.20,23,24 We feel that this approach provides anatural link between quantum transport, first-principles elec-tronic structure theory, and qualitative molecular orbitaltheory.41,42 A comparison between the present approach andother related experimental and theoretical works will begiven at the end of the paper.

Due to metallic screening in the electrodes, the charge andpotential perturbations induced by molecular adsorption ex-tend only over a finite region into the metal surface.43,44 Wedefine an extended molecule which includes both the mol-ecule and the surface atoms perturbed by molecularadsorption.45,46 The surface atoms also form an adiabatic re-

flectionless contact with the rest of the electrodes, which canthen be modeled as infinite electron reservoirs commonlyassumed for mesoscopic transport problems.19,47 The size ofthe extended molecule is chosen such that charge neutral-ity is maintained approximately under both zero and finitebiases. The central quantities in the NEGF theory are theretarded and correlation Greens function G r and G.37,36

We expand both the wave functions and the Greensfunctions using a finite set of local orbital functions i ,

G r; r,r;Ei,j

G i jr;Ei rj* r, 1

G; r

,r;Ei ,j G i j

;Ei r

j* r, 2

where is the spin index. We obtain the retarded and corre-lation matrix Greens function by solving the Keldysh-Kadanoff-Baym KKB equation in the matrix form19

GMMr;ESMMHMM

VMM

ext;L

r;ERr;E1,

3

G;Ei G r;ELEG a;EfEL

i G r;EREG a;EfER, 4

where the effect of the contact is incorporated as the self-

energy operators L(R);r

,

Lr;EESMLHML ;GLL

0r;ESLMHLM ,

Rr;EESMRHMR

GRR0r;ESRMHRM

,

L(R)i L(R)

r;r;L(R)

. 5

Here GLL (RR )0r;

(ESLL (RR )HLL (RR ) )1 are the surface

Greens functions of the left L and right R contacts theelectrodes with the perturbed surface atoms removed whichcan be obtained from that of the semi-infinite surface. The

self-energy operators L(R)r; characterizes the interaction be-

tween the molecule and the left right contacts, the real partof which leads to an effective shift of the molecular levelwhile the imaginary part L(R)

describes an effective energylevel broadening due to the finite probability of electronsescaping into the respective contact for details see Ref. 19.HM M

represents the part of the Fock matrix contributed bythe charge distributions both nuclei and electron in the ex-tended molecule only, while Vext;(r) represents the long-range coulomb potential due to the equilibrium charge ionicand electronic distribution in the contact region, which in-cludes the linear voltage drop due to the applied bias. The Sare overlap matrices. The applied potential Vext; changes thecharge and potential in the extended molecule, requiring aself-consistent solution.

Given the electron density or the density matrix in theextended molecule, the calculation of the Fock matrixHM M

is the same as that of standard molecular electronicstructure calculation, which greatly facilitates the implemen-tation of our procedure using standard molecular electronicstructure codes.19 The self-consistent calculation proceeds bycomputing the input density matrix to the next iteration fromthe correlation matrix Greens function computed in the cur-rent iteration

i j dE2i G i j;E. 6

The density matrix is simply the energy integration of the

matrix correlation function. The integration over energy canbe performed conveniently in the complex energy plane.19,48

Once the self-consistent calculation converges, we can cal-culate all the physical observables from the matrix retarded

Greens function and the density matrix.

B. Analyzing molecular transport

We calculate the charge density using

ri j ;

i ji rj* r, 7

from which we can obtain the electrostatic potential in themolecular junction from the Poisson equation. We can alsocalculate charges associated with each atom using Beckesatomic-partition scheme49

N dr ri drWi r r

iNi . 8

Here the atomic weight function Wi(r), which satisfies iWi(r)1 everywhere in space, is determined such that itis centered on the atom i and is non-negligible only in aregion close to its atomic center.49

The local density of states LDOS gives the energy-resolved charge density distribution

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n r,E1

lim

0

i jImG i j

r;Eii rj* r.

9

The spatial integration of LDOS gives the density of states

nE drn r,E 1

lim

0

TrIm G r;EiS.

10

To identify the contribution of the individual molecular or-bitals to the total density of states, we project it onto thebasis of molecular orbitals. This is done by transforming G r

into the basis of the molecular orbitals obtained by diagonal-izing the self-consistent Fock matrix corresponding to themolecule. The imaginary part of the diagonal element of thetransformed G r gives the projected density of states PDOSof the corresponding molecular orbitals. The peak position ofthe PDOS characterizes the perturbed molecular energylevel, while the broadness reflects the coupling strength ofthe molecular orbitals with the metallic surface states.

A critical question in molecular transport is how the elec-tron occupation of the molecular orbitals changes as the mol-ecule is driven out of equilibrium by a finite bias voltages.This information is contained in the nonequilibrium densitymatrix i j

Eq. 6. Transforming to the basis of molecularorbitals, the diagonal elements of the density matrix give thenonequilibrium electron occupation of the correspondingmolecular levels.

A general formula for the current through a mesoscopicsystem with arbitrary interaction in contact with two nonin-teracting electrodes is

Ie

h dE

Tr LR

E,ViG;E,V

fEL LE,VfER R

E,V

AE,V , 11

where Ai(G r;G a;) is the spectral function. Since theonly scattering mechanism in the coherent transport regimeis that by the contacts introducing additional scatteringmechanisms within the molecule will lead to additionalterms in self-energy and current, we arrive at the familiarLandauer-type current formula

Ie

h dE

TE,VfELfER , 12

where the transmission probability T through themolecule19,5052 is obtained from

TE,VTr TE,VTr tt

TrL

E,VG r;E,VR

E,VG r;E,V .

13

Note that equations 12, 13 hold for both orthogonal andnonorthogonal basis functions.19 To identify the contributionof individual molecular orbitals to the transmission, we

transform the transmission matrix T(E,V) into the basis ofthe molecular orbitals whose diagonal elements give thetransmission probability through the corresponding molecu-lar orbitals.

Combined with the spatially resolved LDOS and chargedensity, the total DOS, the transmission coefficient and theirprojection onto individual molecular orbitals provide a set ofuseful qualitative analysis tools to establish the connectionbetween the molecular electronic structure and the transportcharacteristics of the metal-molecule-metal junction, similarto the use of qualitative molecular orbital theory in quantumchemistry.42 A caveat of projecting the physical observableonto the individual molecular orbitals is that phase interfer-ence information is lost. Only the summation over all mo-lecular orbitals conserved by the matrix transformation, notthe individual components, corresponds to the quantum me-chanical physical observable. The situation is similar to whatwe met when we try to decompose the electron density of themolecule into the contribution of individual atoms through aMulliken-type population analysis or other charge partitionschemes as we used here. Only the total density but not the

atom-partitioned density corresponds to a physical observ-able.

Within the coherent transport model, the only temperatureeffect is from the two electrode Fermi-Dirac distributionsassociated with the electronic temperature in the electrodes.We can separate the current into two components, the tun-neling component Itun and the thermionic emission com-ponent Ith as follows:

IItunIthe

h LR

L

R

dE TE,VfEL fER. 14

C. Device model

A convenient way of characterizing the band lineupproblem at any interface is to find a reference level to put allmaterials forming the interface on a common absolute energyscale. For the metal-molecule-metal junction consideredhere, the occupation of the electron states is determined bythe electrochemical potential of the two metallic electrodes,which are treated as large electron reservoirs. So it is naturalto choose the reference level being determined by the prop-erty of empty bimetallic junction alone, which is not af-fected by the choice of the molecules or the nature of metal-molecule interaction. Therefore we choose the electrostaticpotential energy in the middle of the empty bimetallic junc-tion as the energy reference, then the equilibrium Fermi levelEf is fixed at the negative of the metal work function5.31 eV for single-crystal gold 111 electrodes. The elec-trochemical potential of the two electrodes is fixed by theapplied bias voltage V at LEfeV /2 and REfeV/2. Here the bias polarity is chosen such that for posi-tive bias the electron is injected from the right electrode.Note the Fermi-level positions are fixed by the bimetallic

junction alone without the molecular insertion. The symmet-ric shift of the electrochemical potential of the two electrodes


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from the equilibrium Fermi level Ef is independent of theproperty of the molecule as well as its interaction with theelectrodes, and has nothing to do with the common assump-tion of symmetric shift of molecular levels with respect tothe metal Fermi levels at nonzero bias which is an approxi-mation that is not correct in general. Instead, the symmetricshift of the electrochemical potential of the two electrodesdetermines the electrostatic boundary condition from whichthe self-consistent molecular charge and potential responseto the applied bias voltages can be deduced, which, in gen-eral, depends on both the property of the molecule and themetal-molecule coupling, as will be detailed in this series ofwork.

In this work, the electronic structure of the extendedmolecule is described using the Becke-Perdew-WangBPW91 Refs. 53,54 parameterization of the spin-polarized density-functional theory SDFT Refs. 39,55within the generalized-gradient approximation GGA.54 Wealso replace the atomic core by an appropriatepseudopotential56 with the corresponding optimized Gauss-ian basis sets.57,58 The geometry of the adsorbed molecule istaken to be the same as the singlet geometry of the freemolecule optimized at the BPW91/631G* level we as-sume the bare molecule to be the molecular biradical withthe end H atoms removed. The adsorption geometry is cho-sen such that the end atoms sit in front of the center of thetriangular pad of the three gold atoms on the Au111 sur-face the end sulfur-surface distance is 1.9 . Six nearest-neighbor gold atoms on each metal surface are included intothe extended molecule. Within the range of second-nearest-neighbor coupling, there are 12 metal atoms in thefirst surface layer and 14 metal atoms on the second surfacelayers on each side coupled to the extended molecule.Only the blocks of the surface Greens function correspond-

ing to these atoms are needed, which in turn can be calcu-lated from the surface Greens function of the semi-infinitemetal using tight-binding method parametrized by fitting ac-curate bulk band structure calculations.18,19,59 The results andconclusions given in the following are not affected signifi-cantly by small changes in the molecular and adsorption ge-ometry. The structures of the molecule junction are shown inFigs. 1 and 2. The calculation is performed using a modifiedversion ofGAUSSIAN98.18,19,60

III. DEVICE AT EQUILIBRIUM

The problem for device at equilibrium is a generalizationof the familiar chemisorption problem in surface sciencesince two metallic surfaces are involved. For a molecularorbital to be turned into an effective conduction channel ofthe metal-molecule-metal junction within the coherenttransport regime, it needs to couple to both electrode statesand its energy must lie within the energy window determinedby the bias voltage and the thermal broadening through theelectrodes Fermi distribution. Both aspects are affected bythe molecular adsorption, which may well be more signifi-cant than the subsequent application of the bias voltage, asdetailed in subsequent sections.

A. The electronic structure of the molecules and the

identification of chemical groups

The transport property of the molecular junction is deter-mined mainly by the hybridization of the surface metal stateswith the frontier molecular orbitals, i.e., the molecular stateswhose energies lie closest to the metal Fermi level. So westart by illustrating in Figs. 3 and 4 the charge distribution ofthe four frontier states of the two isolated molecular biradi-cals with the end H removed. The states correspond to theHOMO-1, HOMO, LUMO, and LUMO1 of the isolatedmolecules with the end H atoms and the HOMO, LUMO,LUMO1, and LUMO2 of the molecular biradicals withthe end H removed.18 Since the former characterization cor-responds to the actual occupation of the molecular states

within the metal-molecule-metal junction, we adopt this no-tation here to avoid unnecessary confusion.

For both molecules, the HOMO-1 states are localized onthe end sulfur atoms, the HOMO states are delocalized overentire conjugated backbone while the LUMO and theLUMO1 states are benzene based. The LUMO state ofBPD also has a finite weight on the end sulfur atoms. Both

FIG. 1. Color online Atomic geometry of the gold-PDT-gold junction. Six gold atoms closest to the end sulfur atoms on eachelectrode are included into the extended molecule.

FIG. 2. Color online Atomic geometry of the gold-BPD-gold junction. Six gold atoms closest to the end sulfur atoms on eachelectrode are included into the extended molecule.


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molecules can be viewed as heterostructures composed ofthe end sulfur atoms and the benzene rings. Due to the non-coplanar geometry of the two benzenes in BPD moleculethe torsion angle is 37), the orbital overlap between thecorresponding electrons is weak. Our analysis of the elec-trical transport through the two molecules will be based onthe electrical response of these chemical units.

B. Charge transfer and electrostatic potential change in the

molecular junction

The adsorption-induced charge transfer across the metal-

molecule interface is the central quantity for the device atequilibrium since the linear-response transport probes theequilibrium charge distribution of the molecular junction.The perturbation introduced by the metal-molecule coupling

is largely a localized interaction. The corresponding charge-transfer process is determined by the interfacial chemistryand can be understood qualitatively from the local bondinganalysis across the metal-molecule interface as in the chemi-sorption problem.18,41

Due to the identical end group/bonding configurationacross the metal-molecule interface and similar charge distri-bution of the frontier molecular states, both the magnitude

and the spatial distribution of the transferred charge for thetwo molecules are quite similar. This is illustrated in Figs. 5,6, where we plotted the difference between the self-consistent charge distribution in the gold-molecule-gold

junction and the charge distribution in the isolated moleculeplus the charge distribution of the isolated bimetallic junc-tion. For clarity and to aid visualization, we have plottedboth the regions where charge accumulation and depletionoccur and the spatial distribution of the transferred charge asa function of position in the X-Y plane defined by the leftbenzene ring. The charge transfer process involves mainlythe end sulfur atoms and neighbor carbon atoms and decaysrapidly as we move away from the metal-molecule interface.This is clearly seen for the BPD molecule where the chargeperturbation induced by the two metal-molecule contacts de-cays into the interior of the molecule without effectively in-terfering with each other Fig. 6. Note this is not necessarilydue to the noncoplanar geometry. The decay of charge per-turbation is already obvious for the PDT while for BPD itbecomes negligible before reaching the interbenzene bond-ing region. The number of electrons increases in the sulfur-gold bonding region due to the rehybridization of the sulfurPz orbital in forming gold-sulfur bond and the transfer ofcharge from gold atom to sulfur atom due to electronegativ-ity difference. Electron density also increases in the neighbor

FIG. 3. Color online Orbital shape of the HOMO-1, HOMO,LUMO, and LUMO1 states of PDT.

FIG. 4. Orbital shape of the HOMO-1, HOMO, LUMO, andLUMO1 states of BPD.

FIG. 5. Charge transfer upon the formation of the gold-PDT-gold contact. The top and middle figures show the isosurface plot ofthe region where electron increases and decreases, respectively. The

isodensity values of the isosurfaces are one-fourth of the corre-sponding maximum electron density change. The bottom figureshows the spatial distribution of the transferred electrons as a func-tion of position in the xy plane after integrating over the z axis.


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carbon Pz orbital since the HOMO level has large weights onboth the sulfur and carbon Pz orbitals. The electron densitydecreases in the sulfur-carbon -bonding region, the sulfurPx orbitals and also the gold s orbitals. This is because theformation of surface gold-sulfur bond involves charges origi-nally residing in the sulfur Px and gold s orbitals thus weak-ening the bonding between sulfur and carbon. The direction

of charge transfer is from the gold electrodes to the mol-ecules.Accompanying the charge transfer across the gold-

molecule interface, the electrostatic potential also changes.The difference of the electrostatic potential in the junctionand the electrostatic potential in the isolated molecule plusthat in the isolated bimetallic junction gives the potentialperturbation due to the formation of the contact. A cross-sectional view of the change in the electrostatic potential inthe xy plane as defined by the bezene ring the left benzenering for the BPD molecule is plotted in Figs. 7 and 8. Thenet transfer of electrons into the molecule increases the elec-trostatic potential inside the molecule creating a potentialbarrier between the metal surface and the end sulfur atom.There is also a potential well between the sulfur atom and thebenzene ring because of the decrease of electron density inthe sulfur-carbon bonding region. Although the pattern ofcharge transfer at the metal-molecule interface is similar forthe two molecules, the long-range electrostatic potential per-turbation is quite different. For the BPD molecule, the elec-trostatic potential profile inside the two benzene rings be-comes quite complicated which creates additional barrier forthe electron motion within the molecule core. Due to thenoncoplanar geometry, the barrier for injection from the rightmetal into the right half of the molecule is larger than that

from the left metal to the left half the right benzene ringgives a slightly smaller orbital overlap with the right elec-trode. Since the probability of electron tunneling through apotential barrier is sensitive to its shape, the charge-transferinduced potential change will have a profound influence onthe electron transmission through the metal-molecule-metal

junction. The presence of the potential barrier at the interfaceand also in the molecule core for BPD will affect thecharge redistribution within the molecule when an additionalfield is applied, since it impedes the flow of electrons across

the metal-molecule-metal junction. This will be discussed inthe next section.

FIG. 6. Charge transfer upon the formation of the gold-BPD-gold contact. The top and the middle figure show the isosurface plotof the region where electron increases and decreases, respectively.

The isodensity values of the isosurfaces are one-fourth of the cor-responding maximum electron density change. The bottom figureshows the spatial distribution of the transferred electrons as a func-tion of position in the xy plane after integrating over the z axis.

FIG. 7. Color online Cross-sectional view of electrostatic po-tential change upon the formation of the gold-PDT-gold contact as afunction of position in the xy plane (Z0). Also shown is the

projection of the molecule onto the xy plane.

FIG. 8. Color online Cross-sectional view of electrostatic po-tential change upon the formation of the gold-BPD-gold contact asa function of position in the xy-plane (Z0). Also shown is theprojection of the molecule onto the xy plane.


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C. Contact-induced modification of molecular states,

band lineup, and conductance of the metal-molecule-metal

junction

The coupling between the gold electrodes and the mol-ecule reflected in the self-energy operator of the electrodesand the induced potential change in the molecule reflectedin the self-consistent molecular Fock matrix modify boththe charge distribution and the energy of the molecular statesrelative to the metal Fermi level. They also broaden the dis-crete molecular electronic spectrum into a quasicontinuousone. These effects can be analyzed through the local densityof states, the transmission coefficient and their projectiononto the individual molecular orbitals.

The transmission versus energy (T-E) and the projectedDOS PDOS corresponding to the five frontier molecularstates of the two molecules LUMO1, LUMO, HOMO,

HOMO-1, and HOMO-2

closest to the Fermi-level forspin-up electrons are shown in Figs. 9 and 10. The moleculesstudied here contain an even number of electrons and thegold electrode is non-magnetic, so the spin-up and spin-down channels show identical characteristics. For other mol-ecules in contact with ferromagnetic electrodes, the spin-resolved analysis is essential for understanding the transportcharacteristics.

From both the transmission versus energy curve and theprojected DOS curve, we find the Fermi level lies closer tothe HOMO state than to the LUMO state for both molecules.Comparing the peak position in the PDOS plot with the en-

ergy levels of the isolated molecule, we find that the contactwith the metallic electrodes significantly shifts the energylevels of the frontier molecular states. The change is found to

be larger for the occupied states than for the unoccupiedstates since the unoccupied states are benzene based whichalso leads to sharper peaks in the PDOS plot.

The contact with the electrodes also modifies the chargedistribution of the molecular states. This is illustrated by ex-amining the LDOS at energies corresponding to the peakpositions of the PDOS of the HOMO and LUMO for bothmolecules in Figs. 11 and 12, where we show both the shapeand the spatial distribution of the surface-perturbed molecu-lar states. The spatial distribution is obtained by integratingthe three-dimensional 3D LDOS with respect to the z axisand plotted as a function of position in the xy plane. Theperturbation of the molecular states due to the coupling tothe electrodes is different for the two molecules. In general,the surface-induced change in the charge distribution associ-ated with the molecular states correlates closely with thechange in the electrostatic potential. For the PDT molecule,the large increase of the electrostatic potential in the middleof the benzene rings pushes the electrons located on the car-bon atoms to the peripheral hydrogen atoms for both theHOMO and the LUMO states Fig. 11. The LUMO levelremains highly localized giving rise to sharp peak in bothPDOS and T-E plots. For the BPD molecule, electrons alsomove from the right benzene to the left benzene for both theHOMO and LUMO states due to the difference in potential

FIG. 9. Band lineup at the gold-PDT-gold contact. The curvescorresponding to the spin-up and spin-down electrons are virtuallyidentical. The left figure plots the transmission versus energy, while

the right figure plots the projected density of states onto the fivefrontier molecular states which can be identified from their peakpositions. For PDT, these are 2.25 eV LUMO1, 2.45 eVLUMO, 6.5 eV HOMO, 7.25 eV HOMO-1, and7.7 eV HOMO-2. The horizontal line corresponds to the energylevels of four frontier molecular orbitals as plotted in Fig. 3. Notethat the sharp peaks in the PDOS plot have been truncated herebecause showing them in full would have made the peaks corre-sponding to HOMO and LUMO less visible.

FIG. 10. Band lineup at the gold-BPD-gold contact. The curvescorresponding to the spin-up and spin-down electrons are virtuallyidentical. The left figure plots the transmission versus energy, whilethe right figure plots the projected density of states onto the fivefrontier molecular states which can be identified from their peakpositions. For BPD, these are 2.5 eV LUMO1, 2.85 eVLUMO, 6.35 eV HOMO, 6.95 eV HOMO-1, and7.45 eV HOMO-2. The horizontal line corresponds to the en-ergy levels of four frontier molecular orbitals as plotted in Fig. 4.Note that the sharp peaks in the PDOS plot have been truncatedhere because showing them in full would have made the peakscorresponding to HOMO and LUMO less visible.


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arising from the difference in local geometries. Similaranalysis applies to other molecular states.

The peak positions in the PDOS plots of the HOMO andLUMO states align closely with the two peak positions in theT-E plot around the Fermi level, corresponding to the onsetof resonant transmission. By projecting the total transmissioncoefficient onto the individual molecular orbitals, we find thefirst transmission peak above EF at E2.45 eV] arisesmainly from the nearly degenerate LUMO and LUMO1states. The first transmission peak below EF at E6.5 eV] instead arises mainly from the HOMO state. Inaddition to the transmission peak at the HOMO and LUMOstates, there is also a transmission peak in the HOMO-

LUMO gap of the PDT molecule at energy E3.6 eVFig. 9. This peak corresponds to transmission throughmetal-induced gap states MIGS Refs. 6164 due to thehybridization of metal surface states with the occupied mo-

lecular states, which is significant for such a short moleculeas PDT. The projected transmission analysis shows contribu-tion coming mainly from the HOMO and also other occupiedmolecular states. The nature of the MIGS state is mostclearly seen in the corresponding LDOS plot Fig. 13, whichshows similar charge distribution to the HOMO state. This isthe case for both the LDOS and the transmission coefficientthroughout the HOMO-LUMO gap region including that atthe Fermi level. For such a short molecule as PDT, the trans-mission through the metal-induced gap states in the HOMO-LUMO gap is significant. A similar situation occurs also forthe BPD molecule except that the transmission through theHOMO-LUMO gap of the BPD molecule is much reducedFig. 10. The molecule is longer and the orbital overlap withthe right electrode states is weaker, so the transmission prob-ability in the gap is much smaller.

The magnitude of the transmission coefficient at theFermi level determines the zero-bias conductance at lowtemperature. We find a conductance of 4.8 and 1.4 S forthe PDT and BPD molecules. Since the length of the mol-ecules are 6.4 and 10.7 , respectively, the resistance is notproportional to the conjugation length, as expected for tun-neling transport.

IV. DEVICE OUT OF EQUILIBRIUM

A. Current-voltage and conductance-voltage characteristics

The current-voltage (I-V) and differential conductance-voltage (dI/dV-V or G-V) characteristics are calculated forthe two molecules using the method described in Sec. II inthe bias range from 4 to 4 V and plotted in Figs. 14 and15. Due to the symmetric device structure, the current-voltage and conductance-voltage characteristics for bothmolecules are nearly symmetric with respect to bias polarity.In the low bias regime, current changes approximately lin-early with the bias voltage for both molecules, correspondingto tunneling transport in the HOMO-LUMO gap. Both tun-

FIG. 11. Color online Characteristics of the surface perturbedHOMO and LUMO molecular states at the gold-PDT-gold contact.

FIG. 12. Color online Characteristics of the surface perturbedHOMO and LUMO molecular states at the gold-BPD-gold contact.

FIG. 13. Color online Characteristics of the metal induced gapstates. Left figure: gold-PDT-gold contact. Right figure: gold-BPD-gold contact.


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neling and thermionic emission contribution to the total cur-rent are shown. As expected, for tunneling transport througha large and thin barrier, the thermionic emission contributionto the current is negligible even at low bias shown in theinsets. Since the variation of the thermionic emission con-tribution with bias voltage is small in the coherent transportregime, this further reduces its contribution at high bias.Since this is the only temperature dependence in the coherenttransport model, we expect the temperature dependence ofthe coherent current transport through the two molecules tobe weak ignoring any disorder effects. Within the coherent

transport model, the thermionic emission and, correspond-ingly, the temperature dependence of the current transportcan be important only if the barrier for electron transmissionis small and/or thick.

For comparison, we have also plotted the conductance-voltage characteristics obtained using the equilibrium trans-mission coefficient, i.e., replacing T(E,V) in Eq. 12 withT(E,V0). Since the gold surface density of states are ap-proximately symmetric with respect to the Fermi level thegold surface band around the Fermi level is due primarily tothe sp electrons, the difference between the G-V character-istics thus obtained and the self-consistent G-V characteris-tics reflects the effect of the bias-induced modification ofmolecular states.

For both molecules, only the low-bias conductance-voltage characteristics before reaching the first conductancepeak can be reasonably well reproduced by the equilibriumtransmission characteristics. The deviation in both the mag-nitude and the peak position of the conductance becomessignificant at large bias, there are also more conductancepeaks than would be obtained from the equilibrium transmis-sion characteristics. So any attempt to predicting the nonlin-ear transport characteristics from the equilibrium transmis-sion characteristics combined with an assumption about thevoltage drop will lead to significant error. The effect of thebias-induced modification of molecular states is also obviousfrom looking at the shift of the frontier molecular levels bythe applied voltage, as shown in Fig. 16. The molecular lev-els plotted are obtained by diagonalizing the molecular partof the self-consistent Fock matrices at each bias voltage, i.e.,the submatrix of HM M containing only the molecule itselfsince the molecular levels thus obtained do not include theeffect of self-energy induced shift, their values dont matchthe peaks in the PDOS or T(E,V) plot shown later. Themolecular levels are nearly constant at low voltages, but be-gin to shift before reaching the first conductance peak. Thevoltage at which the shift occurs corresponds to the voltagewhere major deviation in the G-V characteristics occurs.

FIG. 14. I-V upper figureand G-V lower figure characteris-tics of the gold-PDT-gold device. The inset in the I-V plot gives themagnified view at low bias. The dotted line in the G-V plot is

obtained assuming the transmission-energy relation to be bias inde-pendent.

FIG. 15. I-V upper figure and G-V lower figure characteris-tics of the gold-BPD-gold device as in Fig. 14.

FIG. 16. Bias-induced modification of molecular levels at gold-PDT-gold junction left figure and gold-BPD-gold junction rightfigure. We have also shown the position of the equilibrium Fermi

level EF and the electrochemical potential of the two electrodeL(R) in the plot.


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An important question in current transport through mol-ecules is which molecular states are responsible for the ob-served conductance peak. From the position of the plottedmolecular levels and also from the T(E,V) plots shownlater, we expect that for both molecules, the peaks in theconductance are due to the occupied molecular levels, indisagreement with previous calculation where a jelliummodel of the electrode is used.21,22 We believe this is because

the jellium model underestimates the metal work functionwhich effectively pushes the HOMO level down relative tothe metal Fermi level.18 The shifts in the individual molecu-lar levels are not identical in either direction or magnitude,so a rigid shift in energy levels as occurring in a planarmetal-semiconductor contact does not apply here. Instead theshift of the individual molecular states will depend on thedetailed charge and potential distributions. For the two mol-ecules considered here, there is an effective reduction in theHOMO-LUMO gap as bias increases Fig. 16.

B. Charge response and voltage drop

The charge response of the molecule to the applied elec-trical field is reflected in both the net charge injection intothe molecule and the charge redistribution inside the mol-ecule. For PDT molecule, we find there is only slight addi-tional charge injection into the molecule as we apply the biasvoltage. For BPD molecules, the net charge injection be-comes important only when we go to high bias (2.5 eV).The charge injection under nonzero bias voltage is deter-mined by the balance of charge injection and extraction atthe source-molecule and drain-molecule contacts. Since forthe PDT molecule the contact geometries are identical andthe gold surface density of states are approximately symmet-ric with respect to the Fermi levels, little net charge accumu-

lation will be induced by the applied bias voltages. For theBPD molecule, the two rings are noncoplanar, but due to thelargely localized nature of the charge injection process, thedifference in the two contact becomes non-negligible only athigh bias voltages. In addition to the total net charge in themolecule, we can also calculate the nonequilibrium occupa-tion of the individual orbitals from the nonequilibrium den-sity matrix Eq. 6 as a function of the bias voltage Fig.17. We find the change in the electron occupation is frac-tional and small, with the maximum being 0.2. The elec-tron occupation of the molecular orbital is fractional due tothe broadening of the molecular level by the coupling to thecontact. The difference in the voltage dependence of the oc-cupation of the four molecular states is due to the differencein the voltage dependence of their energies and their differentbroadening. Note the change in the electron occupation ofthe molecular states is rather gradual, far less than the unitchange implied by electron oxidization or reduction.

Although the net charge injection due to applied voltage isnegligible, there can be significant charge redistributionwithin the molecule. This calls for closer investigation intothe spatial distribution of the charge and potential distribu-tion within the molecule as a function of applied voltage. 65

Samples of the spatial distributions of the transferred chargeand the electrostatic potential drop for both molecules at bias

voltage of 3.0 V are shown in Fig. 18. Similar charge andpotential distributions are obtained for all bias voltages. Thespatial distribution of charge transfer is obtained by integrat-ing the difference in electron density at finite and zero biasesalong the z axis and plotted as a function of position in the xyplane defined by the left benzene ring. The potential drop isobtained by evaluating the difference between the electro-static potential at finite and zero biases, which obeys theboundary condition of approaching V/2 (V/2) at the leftright electrode. The molecules exhibit different conduc-tance at different voltages, but the calculated charge and po-

tential distributions show similar patterns and do not dependon whether the molecules are in low or high conductancestate. The reason is as follows: The charge and potential

FIG. 17. Nonequilibrium occupation of molecular orbitals as afunction of voltage for the PDT left figure and BPD right figuremolecules in the molecular junction. Here we show the electron

occupation of the HOMO-1, HOMO, LUMO, and LUMO1 states.

FIG. 18. Color online Spatial distribution of charge transferand potential drop at the gold-PDT-gold and gold-BPD-gold con-tacts for bias voltage of 3.0 V.


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response of the molecular junction is determined by the totalelectron population. But to be in high conductance state, onemolecular orbital needs to align with one of the metal Fermilevels. As it moves away from alignment, the change in theelectron population is gradual Fig. 17, in contrast with thechange in conductance. Given the net charge injected into themolecule, the determination of the electrostatic charge andpotential response of the molecule in contact with the twoelectrodes will be equivalent to that of an isolated chargedmolecule under the same boundary conditions of charge andpotential.

The nature of the charge redistribution can be understoodreadily by partitioning the molecules into the atomic groups,i.e., the benzene rings and the end sulfur atoms. As biasvoltage increases, electrons move from the source-side car-bon and sulfur to the drain-side carbon and sulfur. This is dueto fact that the electrons in the carbon Pz orbitals canmove freely under the applied electric field. This flow ofelectrons is impeded at the molecule-drain contact becausethe presence of potential barrier there, which inhibits thecharge flow into the drain electrode. Similar considerations

apply to the molecule-source contact. This leads to the cre-ation of two resistivity dipoles at the metal-molecule inter-face, i.e., the accumulation of electrons on the injecting sideand the depletion of electrons on the extracting side of thepotential barrier. For the BPD molecule, where the two ben-zene rings are noncoplanar, the charge response of the twobenzene rings show identical behavior and are similar to thebenzene ring in the PDT molecule. The weak orbital overlapbetween the two benzene rings disrupts the flow of elec-trons across the interbenzene bonding region, which createsanother potential barrier with corresponding resistivity di-pole and partially insulates the two benzene rings from eachother.

The spatial variation of the current-induced electron redis-tribution could also have a significant effect on the structuralstability of the molecule under high bias. As shown in Fig.18, applying bias voltage increases the electron density in thesulfur-carbon bond at the source-molecule contact but de-creases the electron density in the sulfur-carbon bond at thedrain-molecule contact. According to Hellman-Feynmantheorem,66 this would lead to stronger attractive forces andconsequently shorter sulfur-carbon bond at the source-molecule contact but weaker forces and longer sulfur-carbonbond at the drain-molecule contact. For the BPD molecule,there is also a shift of electron density in the carbon-carbonbond connecting the two benzene rings which may affect theinter-ring spacing. Since the magnitude of the electron redis-tribution increases with the bias voltage, we expect this bias-induced modification of structural change will have strongereffect at high bias. The problem of current-induced confor-mational change is the molecular analog of the more familiarelectromigration problem in metallic systems.67 Althoughthis problem has been treated recently by several groups,22,68

we feel that much needs to be done before a satisfactorytheory emerges.

The resistivity dipole is a well known concept in mesos-copic electron transport47 and is common in inhomogeneoustransport media where their presence in the vicinity of local

scattering center helps to overcome the barrier for transportand ensure current continuity. This can lead to a nonlineartransport effect due to the strong spatial variations in localcarrier density and transport field. Such nonlinear transportcharacteristics induced by the device charge and potentialinhomogeneity are a hallmark of band engineeringthrough mesoscopic semiconductor heterostructures.69 Mo-lecular junctions are an analog of the mesoscopic hetero-

structures, since the choice of the component functional andstructural groups allows the possibility of engineeringcharge and potential inhomogeneity within a single mol-ecule, suggesting the possibility of device engineeringthrough molecular design and permitting the extension ofdevice concepts to the molecular scale. This effect can befurther elucidated by examining the potential response of thetwo molecules.

The mobile electrons of the benzene ring effectivelyscreen the applied field, leading to rather flat electrostaticpotential drop across the benzene rings. The electrostatic po-tential drops rapidly in the molecule-electrode contact re-gion, and becomes flat again as it approaches the boundary

of the electrode where strong screening of the applied elec-tric field occurs. For the PDT molecule, the majority of thevoltage therefore drops across the molecule-electrode contactregion.20,65 Since the unoccupied molecular states of the PDTmolecule are located mainly on the benzene ring where thescreened electric field is small, their energy levels do notchange much as bias voltage increases. The HOMO-1 stateshows stronger voltage dependence than the HOMO statebecause it is end-sulfur based where the potential variation isthe strongest. For the BPD molecule, although the elec-trons of each benzene ring can screen the applied field effec-tively, the potential barrier between the two benzene ringsprevents the electron from flowing freely across, leading tosignificant voltage drop across the molecule core. As a result,there exists strong spatial variation of the electrostatic poten-tial across the molecule core in addition to the metal-molecule contact. Since the four frontier molecular states ofthe BPD molecule are either sulfur based or have chargedistributed across the entire molecule core, they all showclear voltage dependence. An empirical model of rigid shiftof the molecular level relative to the Fermi levels of the twoelectrodes has often been assumed,6 with the proportion ofthe voltage drop being associated with the contact geometry.It is clear that this model can only be utilized as a crudecheck of the device characteristics, since it neglects the pos-sible voltage drop across the molecules and the different ef-fects this may have on different molecular states.

C. Bias-induced modification of molecular states and

transmission characteristics

The molecular charge and potential response to the ap-plied electrical field will affect the transport characteristics intwo ways: 1 it shifts the molecular levels relative to theFermi levels of the two electrodes and, 2 it modifies thecharge distribution of the molecular states and therefore theircapability for carrying current. The bias-induced modifica-tion of molecular states can be analyzed by examining the


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local density of states and its projection onto the individualmolecular orbitals at finite bias voltages.

A snapshot of the bias dependence of the transmissioncharacteristics and the projected DOS corresponding to thefive frontier molecular orbitals is shown for the PDT mol-ecule at bias voltages of 1.6, 3.0, and 4.0 V Fig. 19 and forthe BPD molecule at bias voltages of 1.4, 3.0, and 4.0 VFig. 21 to illustrate their voltage dependence. We have alsoshown for PDT molecule the LDOS at energies correspond-ing to the peak position in the PDOS plots of the HOMO and

LUMO states at bias voltage of 3.8 V Fig. 20.

For the PDT molecule changing from zero bias Fig. 9 toV1.6 V Fig. 19, both the peak positions in the PDOS andthe electron distribution associated with the LUMO states donot change much. The HOMO energy level also does notchange much, but there is a shift in charge distribution fromthe right end sulfur atom to the right carbon atom not shownhere. The energy of the HOMO-1 state increases with bias,leading to a decrease in the energy spacing between theHOMO and HOMO-1 states. At 1.6 V, the transmission ver-sus energy of the PDT molecules reaches a peak at energycorresponding to the drain Fermi level at EFeV/2, giving

rise to the first peak in the conductance. As we increase thebias voltage further from V1.6 to 3.0 V Fig. 19, the en-ergy levels of the HOMO and HOMO-1 states shift graduallytoward the equilibrium Fermi level EF, increasing the trans-mission coefficient there. As we further increase the bias toV3.8 V, the charge distribution the LDOS at energiescorresponding to the peak position of the LUMO state alsochanges, developing large weight on the end sulfur PZ andthe peripheral hydrogen atoms Fig. 20. Because the HOMOand HOMO-1 states move up to the metal Fermi level as thebias increases, the current increases more gradually thanwould be obtained if the bias-induced modification of mo-lecular states is neglected Fig. 14.

For the BPD molecule, the first peak in conductance isreached at V1.4 V Figs. 15 and 21. From zero bias to 1.4V, the energies of the LUMO and HOMO-1 states decreaseand there is a shift of charge from the right benzene to theleft benzene ring. The energy of the HOMO states increases,and the shift of charge distribution is from the left benzenering to the right benzene ring not shown here. Compared tothe PDT molecule, the bias-induced modification is strongerdespite the longer molecule length, due to the stronger spa-tial variation of the potential profile. As we increase the biasto V3.0 V Fig. 21, the energy of the LUMO continues todecrease, but the energies of both the HOMO and HOMO-1

FIG. 20. Color online Characteristics of field-induced modifi-cation of molecular states at V3.8 V for the gold-PDT-gold junc-tion. The left figure right figure shows the LDOS at energy corre-sponding to the peak position in the projected DOS of the LUMOHOMO states.

FIG. 21. Bias-induced modification of molecular states andtransmission coefficient at voltages of 1.4, 3.0, and 4.0 V for thegold-BPD-gold contact. The sharp peaks in the PDOS plot are notshown in full here.

FIG. 19. Bias-induced modification of molecular states andtransmission coefficient at voltages of 1.6, 3.0, and 3.8 V for thegold-PDT-gold contact. The sharp peaks in the PDOS plot are notshown in full here.


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states increase toward the metal Fermi level EF. The trans-mission coefficient at both energies decreases significantlysince the electrons are now more localized on the right ring.A notable change is that a second peak in the PDOS of theHOMO state develops at E7.0 eV. Examining the cor-responding LDOS shows similar charge distribution to thatof the HOMO-2 state with energy of7.15 eV which hascharge distribution on both benzene rings not shown here

and correspondingly large transmission probability. Increas-ing the bias to 4.0 V Fig. 21 further decreases the energy ofthe LUMO state and increase the energies of the HOMO andHOMO-1 states. At V3.8 V, the electrons shift from theright benzene ring back to the left for both the HOMO andHOMO-1 states not shown increasing the transmission co-efficients there. Since the electrochemical potential of the leftright electrode approaches alignment with the HOMO-2and also the LUMO state, there is a rapid increase in theconductance as we increases the bias further toward V4.0 V Fig. 15.

V. DISCUSSION AND CONCLUSION

A. Tunneling or single-electron tunneling?

In this work we have considered electron-electron inter-action in the molecule within a self-consistent-field frame-work through the use of density-functional theory. Tunnelingthrough the molecular junction is therefore a single-step co-herent process equivalent to that of a noninteracting particletunneling through an effective potential barrier determinedthrough the self-consistent procedure detailed in Sec. II,which can be understood qualitatively through the energymismatch between the injected electrons from the electrodesand the molecular eigenstates. This is in contrast to the phe-nomena of single-electron tunneling through metallic or

semiconducting quantum dots separated from the electrodesby two insulating gaps,70 where the tunneling is a correlatedtwo-step process mediated by the electron transfer onto thequantum dots. Such quantum dots are often characterized asartificial molecules. The difference between these artifi-cial molecules and the molecules considered in this worklies in their different coupling with the electrodes and thedegree of delocalization of the electron states within themetal-molecule-metal junctions. For the devices consideredhere, the molecule couples strongly to at least one of theelectrodes through the end group and the metallic states pen-etrate sufficiently deep inside the molecule MIGS. TheCoulomb blockade of tunneling is therefore washed out byquantum fluctuations of the charge on the electrodes, similarto the situation of electron tunneling through ultrasmall junc-tions with low-impedance electromagnetic environments.71,72

This is no longer true if the molecule is physisorbed ontoboth electrodes and the electron states of the molecule can beconsidered as localized, where Coulomb Blockade phenom-ena has indeed been observed.73,74

A different system that combines tunneling through mo-lecular junction with single-electron tunneling phenomena isrealized using molecular-assembled nanoparticles, where thenanoparticle is connected to the electrodes through molecularbridges.75,76 Here the tunnel barrier in the conventional

single-electron devices is replaced by the molecular tunneljunction, and the theory of electron transport through mo-lecular junction described in this work can be combined withthe theory of charging-dynamics on the nanoparticle to pro-vide a microscopic understanding of transport through suchmetal-molecule composite system.7779 Interestingly, thescaling limit of such molecular-assembled nanoparticle sys-tem will correspond to a metal complex where a single metal

atom is embedded within the molecule and connected to theelectrodes through the molecular bridges. Such systems havebeen realized recently using transition-metal complex, whereboth Coulomb blockade and Kondo effect have been suc-cessfully demonstrated.80,81 It will be of great interest to ex-tend the theory presented here to incorporate both the strongelectron-electron interaction and the discrete electron stateson the metal atoms82 such that a single theory can be used todescribe all the transport regimes involved in single-molecule devices. This is a challenging topic currently underinvestigation.

B. Electron transport or hole transport?

A common practice in the literature on molecular elec-tronics is to characterize the molecular transport as electrontransport if the conduction is mediated by tunneling throughthe unoccupied molecular states, or as hole transport if theconduction is mediated by tunneling through the occupiedmolecular states, following the terminology commonly usedin semiconductor/organic device and electron transfer re-search. Here it is important to recognize their differences.

For bulk semiconductor devices, the concepts of elec-tron or hole are associated with introducing shallow-impurity dopant atoms into the ideal lattice structure, whichintroduces electron into the unoccupied conduction band or

removes electrons from the filled valence band. The elec-tron or hole thus introduced are mobile charge carrierswith delocalized wave functions. The system as a whole re-mains charge neutral, since the charges associated with thesecarriers are compensated by the impurity ions left behind.Therefore, the type of charge transport depends only on thetype and amount of dopant atoms introduced, which is anequilibrium property. By contrast, for organic devices, thesystem is often not doped. Charge transport occurs by inject-ing charge carriers into the system from the electrodes.83 Thetype of transport depends on the charging state of the mol-ecule and the nonequilibrium injection of charge carriers intothe neutral system. For both cases, the type of transport isdetermined by the change in the occupation of electron statesthrough doping or contact injection, independent of the na-ture of charge transport itselfband transport or polaron hop-ping.

The situation for coherent molecular transport is quite dif-ferent. For the symmetric molecular devices considered here,both the total number of electrons the charging state of themolecule and the occupation of the individual molecularorbitals change little with the applied bias. Although a frac-tional amount of charge is transferred from the gold to themolecule upon electrode contact, the transferred charge islocalized in the interfacial region and characterizes the


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changes in the interfacial bond. The change in the occupationof molecular states is fractional and gradual, either upon con-tact to the electrodes at equilibrium or upon application of afinite bias voltage out of equilibrium Fig. 17. Due to thequantum mechanical nature of the coherent tunneling trans-port, we cannot characterize the tunneling electron as beingphysically injected into the molecule and subsequently ex-tracted out. Therefore, it is more appropriate to characterize

molecular transport through the relevant resonant molecularstates without associating it with specific transport types.

C. Single molecule versus molecular monolayer

The focus of the present series of work is on current trans-port through a single molecule in contact with two metallicelectrodes. This corresponds closely to the molecular trans-port measurements using atomic-size break junctions,4,12,14,15

where either one or several molecules are probed. However,many molecular transport experiments are performed on mo-lecular monolayers where thousands or tens of thousands ofmolecules are probed by the contacts to the electrodes.3,8,10,11

Such experiments with monolayer configuration present aquite different situation from the single molecule configura-tion considered here. In addition to the additional complexityof intermolecular interactions within the monolayer, the mostimportant difference lies in the interface electrostatics. Notethat the same boundary condition for the electrostatic poten-tial across the molecular junction applies to both the singlemolecule configuration and the monolayer configuration:deep inside the electrodes they approach the bulk value,which can be shifted rigidly with respect to each other by theapplied bias voltage. For the single molecule configuration,the transfer of charge across the metal-molecule interface isconfined in a small region, whose contribution to the electro-

static potential decays to zero in regions far away from themolecule. But for the monolayer configuration, the electro-static potential in regions far away from the molecule is thesuperposition of contributions from the transferred charge ona large number of molecules. To satisfy the same boundarycondition of the electrostatic potential, the charge transferper molecule in a molecular monolayer can be orders ofmagnitude smaller than that in the single molecule consid-ered here.84,85 This is the situation often met in organic elec-tronics, where a monolayer of self-assembled molecules isused to modify the work function of the metalliccontacts.86,87 Similar problems have also been considered inthe electron transport through metal-carbon nanotube

interfaces.88,89

Here it is important to recognize the differentphysics of the metal-molecule interface in single-moleculedevices and monolayer devices, since it may have profoundeffect on the device functionalities achievable using molecu-lar materials.

D. Comparison with experiments

Although the present work focuses on the conceptual un-derstanding of fundamental electronic processes involved inthe operation of any single-molecule device rather than onexplaining specific experiment, it is appropriate here to com-

ment on the relation between the present work and the cur-rent experimental activities in molecular electronics. Perhapsthe single most important discrepancy between the presentcalculation and experiment is the magnitude of current andconductance. This is especially true for the PDT molecule,for which the present calculation gives result two orders ofmagnitude larger than experiment.4 Other theoretical calcu-lations using similar formalism as the present work20,90 aswell as the plane-wave based work of Di Ventra, Pantelides,and Lang21 have given similar results in terms of the magni-tude of current and conductance. The situation is more opti-mistic for longer phenylene thiolate molecules PDT mol-ecule is the smallest of such molecules, where bothexperiment12,15 and theory91 give currents in the A range.

Since the existing experiment on PDT molecule4 givessmaller conductance than the longer phenylene thiolatemolecules,12,14 in contradiction to the expectation of decreas-ing molecular conductance with molecular length, we con-sider the following two factors as the likely cause of thediscrepancy between theory and experiment: 1 atomic ge-

ometry of the metal-molecule interface and 2 the use ofdensity-functional theory in computing molecular energylevels. Within the current theoretical model, the magnitudeof the current/conductance is determined by the lineup ofmolecular level relative to the metal Fermi level as well asthe broadening of the frontier molecular states, both of whichare sensitive to the details of the metal-molecule interaction,especially for the small PDT molecule. The use of density-functional theory in describing the molecular electronicstructure, on the other hand, is an approximation to the morerigorous quasi-particle theory and its non-equilibriumextensions,19,38 whose error generally increases with decreas-ing molecule size. This is reflected in the relatively large

difference in the molecular levels of PDT 0.5 eV or largercomputed using different parameterization scheme withinDFT. Although other possibilities including inter-molecularinteraction exist, it is likely that much better agreement withexperiment can be achieved using carefully characterized in-terface models and improved electronic structure theories.

E. Conclusion

We have presented a first-principles-based microscopicstudy of current transport through individual molecules. Thereal-space formulation allows us to establish a clear connec-tion between the transport characteristics and the molecularelectronic structure perturbed by the metal-molecule cou-pling and the applied bias voltage. By separating the probleminto equilibrium and nonequilibrium situations, we identifythe critical electronic processes for understanding the linearand nonlinear transport characteristics. At equilibrium, thecritical problem is the charge transfer process upon forma-tion of the metal-molecule-metal contact, which modifies themolecular states and determines the energy level lineup rela-tive to the metal Fermi level. This is mainly an interface-related process and can be controlled by controlling the con-tact. Out of equilibrium, the central problem is the molecular


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charge response and the consequent molecular screening ofthe applied electric field, which depends on both the mol-ecule core and the nature of the metal-molecule contact andcan be understood by viewing the molecules as heterostruc-tures of chemically well-defined local groups and analyzingtheir electrical response to the applied bias voltage.92

ACKNOWLEDGMENTS

We thank A. Nitzan, S. Datta, V. Mujica, and H. Basch foruseful discussions. This work was supported by the DARPAMolectronics program, the DoD-DURINT program, and theNSF Nanotechnology Initiative.

*Author to whom correspondence should be addressed. Email ad-dress: [email protected]

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yongqiang xue and mark a. ratner- microscopic study of electrical transport through individual...

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