Michael Galperin, Abraham Nitzan and Mark A. Ratner- Molecular Transport Junctions: Current from Electronic Excitations in the Leads

8/3/2019 Michael Galperin, Abraham Nitzan and Mark A. Ratner- Molecular Transport Junctions: Current from Electronic Excita…

http://slidepdf.com/reader/full/michael-galperin-abraham-nitzan-and-mark-a-ratner-molecular-transport-junctions 1/4

Molecular Transport Junctions: Current from Electronic Excitations in the Leads

Michael Galperin,1 Abraham Nitzan,2 and Mark A. Ratner1

1 Department of Chemistry and Nanotechnology Center, Northwestern University, Evanston, Illinois 60208, USA2School of Chemistry, The Sackler Faculty of Science, Tel Aviv University, Tel Aviv 69978, Israel

(Received 17 November 2005; published 25 April 2006)

Using a model comprising a two-level bridge connecting free electron reservoirs we show that coupling

of a molecular bridge to electron-hole excitations in the leads can markedly effect the source-drain currentthrough a molecular junction. In some cases, e.g., molecules that exhibit strong charge transfer transitions,the contribution from electron-hole excitations can exceed the Landauer elastic current and dominate theobserved conduction.

DOI: 10.1103/PhysRevLett.96.166803 PACS numbers: 73.23.ÿb, 73.50.Lw, 85.65.+h, 85.80.ÿb

Introduction.—Electron transport in molecular tunnel junctions has been the focus of intense recent research[1–4]. Theoretical modeling of tunnel conduction [5,6]starts from Hamiltonians that contain electron transfer(tunneling) interactions between molecule and leads as

essential elements for current transport in such junctions.At the same time, energy-transfer interactions— excitation(deexcitation) of the molecule accompanied by electron-hole pair annihilation (creation) in the metal—are knownto strongly affect the lifetime of excited molecules nearmetal surfaces [7]. An essential difference between theseinteractions is that electron transfer is a tunneling processthat depends exponentially on the molecule-metal dis-tance, while energy transfer is associated with dipolarcoupling that scales like the inverse cube of this distance,and can therefore dominate at larger distances.

How will such dipolar interactions affect the conductionproperties of molecular junctions? Here we address this

question by using the nonequilibrium Green function(NEGF) formalism to derive an expression for the conduc-tion in a junction model that contains both electron andenergy-transfer interactions, then analyze several exampleswith reasonable parameters. We conclude that currentcaused by electron-hole excitations in the leads may besignificant, sometimes even dominant, in situations whenstrong asymmetry in the molecule-lead coupling, of aparticular type explained below, is present.

As an extreme example consider the case where thehighest occupied molecular orbital (HOMO) is coupledonly to one lead, while the lowest unoccupied molecularorbital (LUMO) is coupled only to the other. Such a

junction cannot pass current (in the absence of electroniccorrelations) if the dipolar interaction is absent. However,mixing of the HOMO and LUMO orbitals by dipolarcoupling to the leads makes conduction possible.Realistic situations will not be that extreme; still, wheneverthe HOMO-LUMO transition is associated with electroniccharge transfer along the molecular bridge, we expectsome degree of such asymmetry. For example, the dipolemoment of DMEANS (4-Dimethylamino-4’-nitrostilbene)

is 7 D in the ground state and 31 D in the first excitedsinglet state [8], and 40 A CdSe nanocrystals change theirdipole from 0 to 32 D upon transition from their ground tothe first excited state [9]. We have recently shown [10] thatsuch situations may give rise to light induced current under

zero voltage. We show below that another manifestation of this charge transfer property is the increasing importanceof dipolar energy-transfer interactions between moleculeand leads in determining the junction conductionproperties.

Model and method.—We consider a tunneling junction(Fig. 1) consisting of a molecule positioned between twometal contacts (L and R). The molecule is represented byits HOMO, j1i, and LUMO, j2i, with energies "1 and "2

and gap "21 "2 ÿ "1. The contacts are assumed to befree electron reservoirs, each at its own equilibrium, char-acterized by electronic chemical potentials L and R,where the difference L ÿ R e is the imposed volt-

age. The corresponding Hamiltonian is

H H 0 V M V N ; (1)

FIG. 1. The junction model. The dark and light gray areasdenote occupied and unoccupied leads states that are connectedvia the molecular bridge. The latter is represented by itsHOMO (1) and LUMO (2) levels. The solid-line double arrowsrepresent the electron transfer interactions, the dashed doublearrows, the energy-transfer interactions, and the vertical openarrows, the electron excitation/deexcitation processes in themolecule and the metals (see text for notations).

PRL 96, 166803 (2006)P H Y S I C A L R E V I E W L E T T E R S week ending

28 APRIL 2006

0031-9007=06=96(16)=166803(4)$23.00 166803-1 © 2006 The American Physical Society

http://dx.doi.org/10.1103/PhysRevLett.96.166803

http://dx.doi.org/10.1103/PhysRevLett.96.166803



H 0 X

m1 ;2

"mcymcm

Xk2fL;Rg

"kcyk ck ; (2)

V M X

K L;R

Xm1 ;2;k2K

V MK km cy

k cm H:c: ; (3)

V N XK L;R XkÞk02K

V NK kk0 cyk ck0 cy

2 c1 H:c: ; (4)

where H.c. denotes the Hermitian conjugate. Here the

operators cm and cym (m 1 ;2) are annihilation and crea-

tion operators of electrons in the bridge, while ck and cyk

are annihilation and creation operators of electrons in the

leads. The Hamiltonian H 0 is a sum of terms that corre-spond to the isolated molecule (represented by its HOMO-LUMO levels in our model) and free electron reservoirs

representing the leads. V M describes the electron transfer

(tunneling) process between these subsystems. H 0 V M isthe standard Hamiltonian used in simple models of mo-lecular conduction that yield the Landauer conduction

expression [Eq. (11) below]. The additional term V N rep-resents coupling of the molecular HOMO-LUMO transi-tion to electron-hole excitations in the leads which is oftenused in models of energy transfer between the moleculeand the metals. The effect of this interaction on the junctiontransport properties is the subject of our discussion. Effectsof electron-hole recombination on the bridge were previ-ously considered [11] at the level of master equationapproach to transport.

In the Keldysh NEGF formalism [12] the steady-statecurrent through the junction is given in terms of the lesser

and greater Green functions (GFs), Gh ;i, and the corre-sponding self-energies (SEs) associated with the electrontransfer process, <;>

MK , by [13]

I sd e

@

Z dE2

Tr<MKEG>E ÿ>

MKEG<E

(5)

calculated at the left (K L with ‘‘’’ sign) or right (K R with ‘‘ÿ’’ sign) contact, where the direction from left to

right is chosen to be positive. The GFs Gh ;i needed in (5)can be obtained from the Keldysh equation

G h ;iE GrEh ;iEGaE (6)

where the retarded and advanced GFs, Gr;a

, are given bythe Dyson equation

G rE E ÿHm0

ÿrEÿ1 ; GaE GrEy:

(7)

HereHm0

is a matrix that corresponds to the molecular part(first term on the right) of the Hamiltonian (2) and rE isthe retarded self-energy matrix due to both e transfer anddipolar coupling to the leads. These are 2 2 matrices inthe representation of molecular states j1i and j2i.

The SEs needed in Eqs. (5)–(7) are obtained within theusual diagrammatic technique on the Keldysh contour. Inthe noncrossing approximation [14] they contain additivecontributions associated with the electron and energy-transfer processes at the two leads (L and R):

ML MR NL NR: (8)

The evaluation of these self-energies and their corre-

sponding retarded, advanced, lesser, and greater projec-tions on the real time axis is described in detail in Ref. [15].The additive structure [Eq. (8)] translates into additiveexpressions for the corresponding projections and makesit possible to separate the lesser and greater Green func-tions [Eq. (6)], and consequently also the source-draincurrent [Eq. (5)], into contributions due to direct electrontransfer to the leads, I et

sd, and coupling to the electron-holeexcitations, I eh

sd .

I sd I etsd I eh

sd : (9)

In particular, the self-energies associated with the electron

transfer processes are obtained in the wide band approxi-mation in the familiar forms [16,17]

MK rmm0 MK a

m0m ÿimm0 ÿMK m =2 ; (10a)

MK <mm0 imm0 f K EÿMK

m ; (10b)

MK >mm0 ÿimm0 1 ÿ f K EÿMK

m ; (10c)

ÿMK m 2

Xk2K

jV MK km j2E ÿ "k ; (10d)

f K E expfE ÿ K =kBT g 1ÿ1 (10e)

(note that the matrix ÿMK was assumed diagonal, disre-garding possible level mixing due to coupling to the leads),

where K , K L;R, are the chemical potentials of the leftand right leads, respectively. This results in the standardLandauer expression for I et

sd:

I etsd

e

@

Z 1

ÿ1

dE

2

X

m1 ;2

ÿMLm Gr

mmEÿMRm Ga

mmE f LE ÿ f RE

(11)

The evaluation of I ehsd is more involved. The SEs asso-

ciated with the energy-transfer processes can be expressedin terms of the GFs [see Eqs. (19), (22), and (36) of

Ref. [15] ], and the self-consistent solutions of the resultingcoupled equations are found by iterations. We use the levelpopulations, nm ÿi

RdE2 G

<mmE (m 1 ; 2), as a test for

convergence, which is declared when the population valuesat subsequent iteration steps do not change within a pre-defined tolerance, chosen below as 10ÿ6.

The results shown in Figs. 2 and 3 were obtained fromthis procedure. A simple analytical expression for I eh

sd can

be obtained when ÿm ÿLm ÿR

m "21 (where ÿK m

ÿMK m ÿNK

m with ÿNK m ÿ2ImNK r

mm ) [18] and in the


28 APRIL 2006

166803-2



strong bias limit. ‘‘Strong bias’’ implies, e.g., for nega-tively biased left electrode, that L "2 and R "1 sothat f L 1 and f R 0 in the relevant energy range.Under these conditions the results for the Landauer currentI et

sd and for the electron-hole excitations induced current I ehsd

are obtained [15] in the forms

I etsd

e

@Xm1 ;2

ÿMLm ÿMR

m

ÿm

sgnL ÿ R ; (12)

I ehsd

e

@B

ÿML

2 ÿMR1

ÿ1ÿ2

L ÿ R

ÿÿML

1 ÿMR2

ÿ1ÿ2

R ÿ L

; (13)

where x 1 for x > 0 and 0 for x < 0, and where B

BL BR is the function

BK !;K 2 Z dE

XkÞk02K

jV NK kk0 j2E ÿ "k

E ! ÿ "k0 f K E1 ÿ f K E !

2 jV NK j2ehK ! (14)

evaluated at ! "2 ÿ "1. Here ehK ! is density of

electron-hole excitations in the lead K . Equations (12)and (13) show that the magnitude of I et

sd depends on the

product ÿMLm ÿMR

m m 1 ; 2, while I ehsd is determined by

the mixed products ÿML2 ÿMR

1 or ÿML1 ÿMR

2 . The relativecontribution of the latter current component will be sig-nificant in asymmetric coupling situations, e.g., for L >

R when ÿML2 > ÿMR

2 and/or ÿMR1 > ÿML

1 . Note that,

since ÿML2 ÿMR1 can be different from ÿML1 ÿMR2 , jI ehsd jcan be asymmetric to bias reversal (see also Fig. 2). Belowwe compare the magnitude of the two contributions to thecurrent for different junction parameters.

Numerical results.—In the calculations reported belowwe used the following ‘‘standard’’ choice of parameters:

T 300 K, "1 0 eV, "2 2 eV, and ÿM 1 ÿM

2 0:2 eV. Values of other parameters are indicated in thefigures. The Fermi energy is taken at the midpoint of theHOMO-LUMO gap and the electrochemical potentials inthe left and right leads are assumed to shift with the voltagebias symmetrically relative to this point. Numerical inte-gration was done on the energy grid spanning range fromÿ3 to 5 eV using integration step 10ÿ3 eV.

Figure 2 depicts the current-voltage characteristic of the junction for the cases of symmetric and asymmetric cou-pling between the leads and the molecular LUMO. Shownare the total current and its two components. The followingpoints are noteworthy: (1) In the symmetric case the cur-rent is dominated by the usual electron and/or hole trans-port through the LUMO and/or HOMO, respectively, and issymmetric with respect to voltage reversal. (2) The asym-metric case shows a significant contribution of the current

associated with electron-hole excitations when the LUMOis coupled more strongly to the negatively biased electrode.Indeed, I eh

sd is expected to be pronounced when the LUMO

is populated and the HOMO is empty, which happens atsuch bias. (3) The asymmetric LUMO coupling leads toasymmetry in I eh

sd and consequently in the total current with

respect to bias reversal.Figure 3 shows the results of a model study of the

dependence of the source-drain current on the molecule-

lead distance R. These results are obtained using ÿMK m

AMK m expÿMK

m R, which accounts for the tunneling

nature of electron transfer, and taking B

K

K

=R

3

,which reflects a dipolar distance dependence of the energy

transfer to the leads [7]. The parameters used are AML1

AMR1 0:27 eV, AML

2 0:52 eV, AMR2 0:027 eV,

MK m 1 Aÿ1, and K 0:01 eVA3 (K L;R and

m 1 ; 2). The choice of AMK m corresponds to a total life-

time broadening for electron transfer into the electrodes of 0.2 eV at a distance (from each electrode) of 1 A. The

choice of K corresponds to taking BK 0:01 eV at thisdistance. Both choices reflect the orders of lifetimes ob-

-2

-1

0

1

( 10-5

)

I s d

( A )

-6 -4 -2 0 2 4 6e (eV)

-1

0

1

( 10-5

)

I s d

( A )

(a)

(b)

FIG. 2 (color online). The source-drain current I sd vs appliedvoltage . Shown are the total current I sd (solid red line) and itscomponents due to direct electron transfer I et

sd (dashed green line)

and electron-hole excitations I ehsd (dotted blue line), for the sym-

metric ÿML1 ;2 ÿMR

1 ;2 0:1 eV (a) and asymmetric ÿML=R1

0:1 eV, ÿML2 0:19 eV, and ÿMR

2 0:01 eV (b) cases.


28 APRIL 2006

166803-3



served for the corresponding processes for molecules ad-sorbed on metal surfaces.

The results shown in Fig. 3 demonstrate an importantaspect of the distance dependence of the two contributionsto the total current. Obviously, both I et

sd and I ehsd involve

electron transfer between molecule and leads, thereforeboth drop exponentially with increasing tunneling length.Their relative importance, however, depends on the detailsof the molecule-leads couplings. It is easily seen fromEqs. (12) and (13) that I eh

sd=I etsd B=ÿ. For small and

intermediate distances, where ÿ is dominated by the elec-tron tunneling process, this ratio increases with increasing

distances. (Note that this limiting behavior is obtained onlywhen both left and right molecule-metal couplings de-crease together. Experimentally, one of these distancescan be controlled by moving a tip, while the other can bechanged by adding insulating layers between molecule andsubstrate [19,20].) Detailed calculation shows that thisratio can in fact become substantially larger than 1 at largemolecule-lead distances, as seen in Fig. 3. In this case thecurrent is dominated by the contribution associated withthe electron-hole excitation process.

Conclusion.—We have studied, within a simple model,the effect of dipolar energy-transfer interaction betweenmolecule and leads on molecular conduction. We found

that such interaction, which leads to electron-hole excita-tions in the contacts, can affect the current-voltage char-acteristic of the junction in a substantial way and cannot ingeneral be disregarded. The contribution of this interactioncan dominate the overall conduction for particular asym-

metric coupling where the molecular LUMO and/orHOMO are coupled differently to different leads. In addi-tion, because of the different dependence of electron andenergy transfer on the molecule-leads distance, the relativeimportance of I et

sd and I ehsd depends on this distance, and can,

in some cases, result in strong dominance of I ehsd at large

molecule-lead separations.We thank the NSF-NNI program, the DARPA MolApps

initiative, and the Durint/MURI program of the DOD forsupport. A. N. thanks the Israel Science Foundation and theU.S.-Israel Binational Science Foundation for support.

[1] Molecular Electronics II , edited by A. Aviram, M. Ratner,and V. Mujica (The New York Academy of Sciences,New York, NY, 2002), Vol. 960; Molecular Electronics

III , edited by J.R. Reimers, C.A. Picconatto, J.C.Ellenbogen, and R. Shashidhar (The New York Academy of Sciences, New York, NY, 2003), Vol. 1006.

[2] A. Nitzan, Annu. Rev. Phys. Chem. 52, 681 (2001).[3] J. R. Heath and M. A. Ratner, Phys. Today 56, No. 3, 43

(2003).[4] Molecular Nanoelectronics, edited by M. A. Reed and

T. Lee (American Scientific Publishers, StevensonRanch, CA, 2003).

[5] A. Nitzan and M. A. Ratner, Science 300, 1384 (2003).[6] S. Datta, Quantum Transport: Atom to Transistor

(Cambridge University Press, Cambridge, England, 2005).[7] R. Chance, A. Prock, and R. Silbey, Adv. Chem. Phys. 31,

1 (1978).[8] S. N. Smirnov and C. L. Braun, Rev. Sci. Instrum. 69, 2875

(1998).[9] V. L. Colvin and A. P. Alivisatos, J. Chem. Phys. 97, 730

(1992).[10] M. Galperin and A. Nitzan, Phys. Rev. Lett. 95, 206802

(2005).[11] Y. M. Niquet et al., Phys. Rev. B 65, 165334 (2002).[12] L. V. Keldysh, Sov. Phys. JETP 20, 1018 (1965).[13] Y. Meir and N. S. Wingreen, Phys. Rev. Lett. 68, 2512

(1992).[14] N. E. Bickers, Rev. Mod. Phys. 59, 845 (1987).[15] M. Galperin and A. Nitzan, cond-mat/0602217.[16] G. D. Mahan, Many-Particle Physics (Kluwer Academic

and Plenum, New York, 2000), 3rd ed.[17] H. Haug and A.-P. Jauho, Quantum Kinetics in Transport

and Optics of Semiconductors (Springer, Berlin, 1996).[18] Explicit expressions for ÿN

m are obtained in this limit interms of the level populations nm (m 1 ; 2) [see Eq. (37)of Ref. [15] ], which in turn are obtained from the self-consistent calculations.

[19] N. A. Pradhan, N. Liu, and W. Ho, J. Phys. Chem. B 109,8513 (2005).

[20] J. Repp, G. Meyer, S.M. Stojkovic, A. Gourdon, andC. Joachim, Phys. Rev. Lett. 94, 026803 (2005).

2 4 6 8 10 12 14

R (Angstrom)

10-11

10-10

10-9

10-8

10-7

10-6

10-5

I s d ( A )

FIG. 3 (color online). The source-drain current I sd at voltage 3 V vs molecule-leads distance R. See text for the choice of coupling parameters. Shown are the total current I sd (solid redline) as well as its I et

sd (dashed green line) and I ehsd (dotted blue

line) components.


28 APRIL 2006

166803-4

Documents

Michael Galperin, Abraham Nitzan and Mark A. Ratner- Molecular Transport Junctions: Current from Electronic Excitations in the Leads