Coherent charge transport through molecular wires: "Exciton blocking" and current from electronic excitations in the wire

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by  GuangQi Li

Coherent charge transport through molecular wires: “Exciton blocking”and current
from electronic excitations in the wire
GuangQi Li,1 Boris D. Fainberg,1,2, Abraham Nitzan,1 Sigmund Kohler,3 and Peter Ha¨nggi4
∗
1Raymond and Beverly Sackler Faculty of Exact Sciences,
School of Chemistry, Tel-Aviv University, Tel-Aviv 69978, Israel
2Faculty of Sciences, Holon Institute of Technology, 52 Golomb St., Holon 58102, Israel
3Instituto de Ciencia de Materiales de Madrid (CSIC), Cantoblanco, 28049 Madrid, Spain
4Institute for Physics, University of Augsburg, Augsburg, D-86135, Germany
(Dated: January 27, 2010)
0
We consider exciton effects on current in molecular nanojunctions, using a model comprising a
1
twotwo-levelsitesbridgeconnectingfreeelectronreservoirs. Expandingthedensityoperatorinthe
0
many-electron eigenstates of the uncoupled sites, we obtain a 16×16 density matrix in the bridge
2
subspace whose dynamics is governed byLiuoville equation that takes into account interactions on

n the bridge as well as electron injection and damping to and from the leads. Our consideration can
a beconsiderablysimplifiedbyusingthepseudospindescriptionbasedonthesymmetrypropertiesof
J LiegroupSU(2). Westudytheinfluenceofthebiasvoltage,theCoulombrepulsionandtheenergy-

6 transferinteractionsonthesteady-statecurrentandinparticularfocusontheeffectoftheexcitonic
2 interaction between bridge sites. Our calculations show that in case of non-interacting electrons
this interaction leads to reduction in the current at high voltage for a homodimer bridge. In other

] words, we predict the effect of “exciton”blocking. The effect of “exciton”blocking is modified for
l
e a heterodimer bridge, and disappears for strong Coulomb repulsion at sites. In the latter case the
- excitontypeinteractionscanopennewchannelsforelectronicconduction. Inparticular,inthecase
r
t ofstrongCoulombrepulsion,conductionexistsevenwhentheelectronicconnectivitydoesnotexist.
s
.
t PACSnumbers: 73.63.Rt,73.23.Hk,73.22.Lp
a
m
- I. INTRODUCTION interactionscanaccompanythechargetransferprocesses
d
aswasshownfor charge-transferexcitons[17]in (quasi-)
n
o Electron transport through molecular wires has been one-dimensionalstructures[18,19],includingpolysilanes
c under intense theoretical(see e.g. [1, 2]) and experimen- [20–22]. The latter show a weak coupling between the
[ tal(seee.g. [3,4])studyinthelastfewyears. Theoretical Frenkel exciton with the admixture of charge transfer

4 studies usually fall into two categories. The first focuses states and nuclear motions [21, 22].
v onthe ab-initio computationsofthe orbitalsrelevantfor In molecular bridges energy-transfer interactions can
2 the motion of excess charges through the molecular wire also sometimes have important effects on charge trans-
7 [5–9], while the other [10, 11] employs generic models to fer dynamics. Charge and energy transfer in a lin-
9 gain qualitative understanding of the transport process. ear 2,2’:6’,2”-terpyridine-based trinuclear Ru-(II)-Os(II)
4 At the simplest level[10, 11] the wire Hamiltonianis de- nanometer-sized array [23], and one-dimensional en-
.
0 scribedbyatight-bindingchaincomposedofN siteswith ergy/electron transfer of amylose-encapsulated chain
1 nearest-neighborcoupling(Huckelmodel)thatrepresents chromophores [24] are examples. In addition, it seems
9 theelectrontransfer(tunneling)interactionsbetweenad- likely that energy transfer takes place in chemically re-
0 jacent sites. This model has been generalized to include sponsive molecular transistors based on a dimer of ter-
:
v Coulombinteractionsbetweenelectronsonthe samesite pyridyl molecules combined with ion Co2+ [25].
Xi [12] (Hubbard model) and/or electron-phonon interac- It should be noted that electron transfer is a tunnel-
tions [13]. In the present paper we investigate another ing process that depends exponentially on the site-site
r
a extension of this model, in which we take into account distance,while energytransferis associatedwith dipolar
energy transfer interactions between adjacent molecular couplingthatscalesliketheinversecubeofthisdistance,
sites. and can therefore dominate at larger distances. The im-
Energy-transferinteractions-excitation(deexcitation) portance of the latter stems also from geometric issues,
of a site accompanied by deexcitation(excitation) of an- whicharerelatedtothedipole-dipoleinteractionbetween
other are well-known in the exciton theory [14–16]. In different sites occurring in the vicinity of metal particles
particular, Frenkel excitons - neutral excited states in in molecular nanojunctions. Really, Gersten and Nitzan
which anelectron anda hole areplaced on the same site [26,27] predictedaccelerateddipole-dipole energytrans-
are readily transferred between sites, and such intersite fernearasolidparticle(see also[28,29]), andinthe last
time a number of works devoted to the exciton-plasmon
interactionshavebeen published[30–33]thatarerelated
tophysicaleffectsduetothelocalfieldenhancement[34–
∗Electronicaddress: [email protected] 39].
2
How will such dipolar interactions affect the conduc-
tion properties of molecular junctions? This question
was addressed by Galperin, Nitzan and Ratner by the
exampleofajunction composedofone-site-wireandtwo
metal leads [40], where they predicted the existence of
non-Landauer current induced by the electron-hole exci-
tations in the leads. To the best of our knowledge, there
were no analog treatment of simultaneous electron and
energy transfer (excitons) in multisite bridges. Here we
addressthisproblembyusingtheLiouville-vonNeumann
equation(LNE)forthetotaldensityoperatortoderivean
FIG.1: Amodelforenergy-transferinducedeffectsinmolec-
expression for the conduction of a molecular wire model ular conduction. The right (R=|{r}i ) and left (L=|{l}i)
that contains both electron and energy-transfer interac- manifoldsrepresent twometalleadscharacterized byelectro-
tions. While not a central issue of the present work, we chemical potentials µR and µL respectively, each coupled to
note that energytransfer is closelyrelatedto heattrans- its nearest molecular site. The molecular bridge is a dimer,
fer through the molecular nanojunction - an issue of im- where each site is represented by its ground, |1gi and |2gi,
portantconsequencesforjunctionstabilityandintegrity. and excited, |1ei and|2ei, states.
Treatedseparately,the simplestmodelsofexcitonand
electron transport may be represented by tight-binding
transport models, albeit in different representations. In-
deed, in the wire Hamiltonian (see Eq.(3) below), both
Hˆ =Hˆ +Hˆ +Hˆ (1)
the electron- and energy-transfer terms are binary in wire leads contacts
termsofthe annihilationandcreationoperatorsforelec-
trons and excitons, respectively. Their simultaneous
Hˆ = ε cˆ+cˆ (2)
treatment, however, constitutes a rather complex non- leads k k k
linear problem. In this work we combine a tight-binding k XL,R
∈{ }
model for electron transport [10, 11] with that of one-
dimensional Frenkel excitons [14–16] to investigate the
Hˆ = ε cˆ+ cˆ ∆ (cˆ+ cˆ +cˆ+ cˆ )+
effectofenergytransferinteractiononelectrontransport wire mf mf mf − f 2f 1f 1f 2f
in one-dimensional nanowires. The outline of the pa- mX=1,2 fX=g,e
f=g,e
per is as follows. In Sec.II we introduce our model and
in Sec.III we derive a master equation in the eigenbasis +~J(b+1b2+b+2b1)+ UmNm(Nm−1) (3)
of many-electron wire Hamiltonian. Sec.IV is devoted mX=1,2
to the analytical solution of the problem where we con-
sider both non-interacting electrons at a site and strong
Hˆ =Vˆ +Wˆ (4)
Coulomb repulsion at sites. In Sec.V we show that the contacts
excitontypeinteractionscanopennewchannelsforelec-
tronicconduction. InSec.VIwecarryoutnumericalsim-
ulations, compare them with the analytical theory and Vˆ = Vˆ = V(mf)cˆ+cˆ +H.c., (5)
show the existence of the “exciton blocking” effect. We mf k k mf
summarizeourresultsinSec.VII. InAppendix Awecal- Xmf mf,Xk∈Km
culate the eigenbasis of many-electron wire Hamiltonian
fWorignnoenr-itnrtaenrsafcotrimngateiloenct[r4o1n]s. aItnaAspitpee,nudsiixngBthweeJporredsaennt- Wˆ = Wˆm = Wˆk(km′)bk′kb+n +H.c., (6)
auxiliary calculations. Xm m,k6=Xk′∈Km
wherecˆ+ (cˆ )(m=1,2,f =g,e)arecreation(annihi-
mf mf
lation) operators for electrons in the different site states
of energies ε , while cˆ+ (cˆ ) (k L,R) are creation
II. MODEL (annihilation)mfoperatorskfor fkree ele∈ctrons (energies ε )
k
in the leads L and R. nˆ = cˆ+ cˆ are the occupa-
mf mf mf
We consideraspinlessmodelforamolecularwirethat tion operators for the different site states, and site oc-
comprises two interacting sites, each represented by its cupation operators are given by N = nˆ +nˆ . The
m mg me
ground, g , and excited, e , states positioned between operators b+ = cˆ+ cˆ and b = cˆ+ cˆ are excitonic
| i | i m me mg m mg me
twoleadsrepresentedbyfreeelectronreservoirsLandR (creation and annihilation) operators on the molecular
(Fig.1). The electronreservoirs(leads) are characterized sites m = 1,2, while bk′k = cˆ+kcˆk′ = b+kk′ (k,k′ ∈ L or R
bytheir electronicchemicalpotentials µ andµ ,where ) corresponds to electron-hole pairs in the leads. In the
L R
thedifferenceµ µ =eV istheimposedvoltagebias. wire Hamiltonian, Eq. (3), the ∆ terms represent elec-
L R bs f
−
The corresponding Hamiltonian is tron hoping between site states of similar energies (i.e.
3
between g and between e states of adjacent molecu-
| i | i
lar sites), the J terms representexciton hopping (energy
i
transfer)betweenmolecularsites andthe U terms corre- Oˆint( x)=exp[ (Hˆ +Hˆ )x]
− −~ wire leads
spond to on-site Coulomb interactions. The molecular-
plehaydssicianlteprraoccteisosness:HˆVˆcodnteascctrsibaerseetlaekcetnrotnotraacncosfuenrtbfeotrwteweno Oˆexp[~i(Hˆwire+Hˆleads)x] (9)
the molecular bridge and the leads that gives rise to net
and where Tr =Tr Tr .
currentin the biasedjunction, while Wˆ describes energy K L R
Considerfirsttermswith the electrontransferinterac-
transferbetweenthebridgeandelectron-holeexcitations tionsVˆ. WritingthecouplingHamiltoniansVˆ (Eq.(5))
nf
in the leads. In (5) and(6) K is the leadcloser the the
m as
molecular site m (K = L, K = R) and H.c. denotes
1 2
Hermitian conjugate. In what follows it will be useful Vˆ =cˆ Λ+ +cˆ+ Λ (10)
nf nf nf nf nf
also to define the population operators
where Λ = V(nf)cˆ , we have Vˆint( x) =
λf =nˆ2f +nˆ1f (7) cˆ+int( xn)Λfint( Px)k∈Kn+ k cˆintk( x)Λ+int( xn)f − with
in the manifolds of ground (f =g ) and excited ( f =e) nf − nf − nf − nf −
Λint( x)= V(nf)cˆ exp(iε x).
sitWe leevceolns.sider electronic transport through the molecu- nSfim−ilarly,Pwrki∈tiKnng thke coukpling~Hakmiltonianforenergy
transfer Wˆ = Wˆ as
lar wire where the leads K = L,R are taken to be each n n
in its own equilibrium characterized by its temperature P
Wˆ =b+Θ +b Θ+ (11)
T (here taken equal for the two leads) and electronic n n n n n
electrochemical potential µ . Therefore, the lead elec-
trons are described by theKequilibrium Fermi functions where Θn = k6=k′∈KnWk(kn′)bk′k, then
f (ε )=[exp((ε µ )/k T)+1] 1. Consequentlyex- P
K k k K B −
−
pectation values for lead operators can be traced back
Wˆint( x)=b+int( x)Θint( x)+bint( x)Θ+int( x)
to the expression hcˆ+kcˆk′i= fK(εk)δkk′ where δkk′ is the n − n − n − n − n −(12)
Kronecker delta. The excitonic operators are equal to
where
b+ =cˆ+ cˆ . Theeffectofthecorrespondinginteraction
m me mg
in the bridge (=~Jb+b +H.c.) on the charge transport i
properties is the subj1ec2t of our discussion. Θinnt(−x)= Wk(kn′)bk′kexp[~(εk′ −εk)x] (13)
k6=Xk′∈Kn
Bearing in mind that ρ(t) = σ(t)ρ where σ(t) =
III. MASTER EQUATION K
Tr ρ(t) and Eqs.(10), (11) and (12), we get for the sec-
K
ond term on the RHS of Eq.(8)
Our analysis is based on the LNE, or the generalized
master equation for the reduced density matrix of the 1 Tr ∞dx[Vˆ,[Vˆint( x),ρ(t)]] =
molecular subsystem, obtained using a standard proce- − ~2 K{Z − }
dure [10, 11, 42] based on taking Hˆ as a per- 0
turbation. Briefly, one starts with thecoLntNacEtsfor the to- 1 ∞dx Tr [VˆVˆint( x)ρ ]σ(t)
− ~2 Z { K − K
tal density operator and use the projectors of the type 0
P ρ(t) = ρ Tr ρ(t) in order to derive an equation Tr [Vˆρ σ(t)Vˆint( x)] Tr [Vˆint( x)ρ σ(t)Vˆ]
K K K K K K K
− − − −
for the time evolution of the reduced density matrix
+Tr [ρ σ(t)Vˆint( x)Vˆ] (14)
σ =TrRTrLρ. The calculation is facilitated by invoking K K − }
the so called non-crossing approximation that assumes
In evaluating the RHS of Eq.(14) we encounter reservoir
thattheeffectsofdifferentreservoirs(hereL,R)anddif-
correlationfunctionsthatreflectthereservoirequilibrium
ferent relaxationprocesses (here Vˆ, Wˆ) are independent propertiesaswellasthenatureofitsinteractionwiththe
and additive. This leads to wire. For example,
C ( x)=Tr [Λ Λ+ ( x)ρ ]
dσ(t) = i[Hˆ ,σ(t)] nf − K nf nf − Kn
dt −~ wire − = V(nf) 2[1 f (ε )]exp( iε x) (15)
− ~12TrKZ ∞dx[Vˆ,[Vˆint(−x),ρ(t)]] kX∈Kn| k | − Kn k −~ k
0
Turning to the energy transfer contribution, third term
− ~12TrKZ ∞dx[Wˆ,[Wˆint(−x),ρ(t)]] (8) on the RHS of Eq.(8), we obtain an expression of the
0 form(14) with the energy transfer interactionWˆ replac-
where for any operator Oˆ, Oˆint is the corresponding in- ing Vˆ. Using the Wick’s theorem, we obtain correlation
teraction representation functions of the type
4
Define also the pseudospin (Bloch) vector in the second
quantization picture
D ( x)=Tr [Θ Θ+( x)ρ ]
n − K n n − Kn
2 i
=k6=Xk′∈Kn(cid:12)(cid:12)(cid:12)Wk(kn′)(cid:12)(cid:12)(cid:12) fKn(εk)[1−fKn(εk′)]exp[~(εk−(1ε6k′))x] rrr123fff = inˆ(b2b+fff−+−nˆbbf+f1f); f =g,e (21)
Below we get a Markovian master equation in the wide- Its components have the following properties: (a) They
bandlimit. Thefullmasterequationobtainedinthisway satisfy the same commutation rules as Pauli matrices
constitutes a set of 256 coupled equation for the 16x16 σˆ [44–46]; (b) the operators λ = nˆ + nˆ =
1,2,3 f 2f 1f
elements of the wire density matrix,which canbe solved cˆ+ cˆ , f = e,g (cf. Eq.(7)) and rf commute:
numerically by diagonalizing the corresponding Liouvil- m=1,2 mf mf i
P[rf,λ ] = 0 (i = 1,2,3); (c) any linear operator of the
lianmatrix. Inparticularwe areinterestedinthe steady i f
”donor acceptor” system can be written as linear super-
state solution, σ , which is given by the eigenvector of
zeroeigenvalue. SOSnceσSS hasbeenfound,thecurrentis position of the operators{rif} and λf. In particular,the
obtained from wire Hamiltonian can be written as
1 1
hIi=Tr(IˆσSS) (17) Hˆwire = 2λe(ε1e+ε2e)+ [2r3f(ε2f −ε1f)−∆fr1f]
fX=g,e
where the current operator (defined, e.g., as the rate of ~J
change of electron population on the left of the dashed − 2 (r1er1g +r2er2g)+ UmNm(Nm−1) (22)
line in Fig. 1) is given by mX=1,2
InEq.(22)wehaveput,withoutlossofgenerality,(ε +
1g
Iˆ=e dNˆ = ie[Hˆ,Nˆ] (18) ε2g)/2 = 0. Because the operators λf and rif commute,
dt ~ λ is conservedunder unitary transformationsrelatedto
f
the diagonalization of Hˆ . Therefore, a total 24 24
wire
×
Nˆ = cˆ+cˆ +nˆ +nˆ (19) spacecanbe partitionedinto nine smallersubspaces,i.e.
k k 1g 1e
kXL theLiouvillianmatrixintherequiredbasisisblockdiag-
∈ onal with blocks, according to the values of λ = 0,1,2
f
InsectionVIweshowsomeresultsofsuchnumericalcal- (see Fig.2): four one-dimensionalsubspaces for λ =0,2
f
culations. To gain better insight of the transport prop- for either f = e,g (type I); four two-dimensional sub-
erties of this model, analytical simplifications in some spaces for λf = 1 and λf′ = 0,2 where f = f′ (type
limits are useful. These are discussed next. II); and one four-dimensional subspace for λ6 = λ = 1
e g
(type III). The type I submatrix is diagonal, while four
state pairs with each pair coupled by the charge trans-
IV. ANALYTICAL EVALUATION fer interaction are associated with the four 2 2 blocks
×
of the type II subspace. The remaining four states are
It is known [11] that for the evaluation of Eqs. (8) coupledby both the chargetransfer and excitontransfer
and (14) it is essential to work in the representation of interaction and constitute the 4 4 block of subspace
×
the eigenstates of the Hamiltonian Hˆ +Hˆ that III.Eachofthesesubspacesischaracterizedbyassigning
wire leads
definesthe zeroth-ordertime evolution. Theuse ofother thevalues(λe,λg)oftotalpopulationsinthegroundand
representations bears the danger of generating artifacts, excited states of the two bridge sites.
which,forinstance,mayleadtoaviolationoffundamen- Using the identity
talequilibriumproperties[43]. Wethusfacetheproblem
of diagonalizing a matrix of order 256. This procedure
0 for λ =0,2
may be facilitated by using the pseudospin description (r1f)2 =(r2f)2 =(r3f)2 =λf−2nˆ2fnˆ1f =(cid:26) 1 for λff =1 (cid:27),
basedonthesymmetrypropertiesofLiegroupSU(2)as-
(23)
sociated with the two state problem (1f,2f); f = e,g.
the wire Hamiltonian (22) can be written in the form
Sucha“donoracceptor”systemmaybedescribedbythe
“chargetransfer” operatorsb+ =cˆ+ cˆ and b =cˆ+ cˆ 1
f 2f 1f f 1f 2f Hˆ = λ (ε +ε )+ U N (N 1)+
that describe intersite charge transfer 1 2 and 2 1, wire 2 e 1e 2e m m m−
respectively, in upper and lower states o→f the molec→ular mX=1,2
dimer. The non-diagonalpartofHˆ ,Eq.(3),canthen +0 For subspaces I
wire
be written in terms of operators b only +1rf(ε ε ) ∆ rf For subspaces II
f 2 3 2f − 1f − f 1
+[1 rf(ε ε ) ∆ rf
Hˆw(niroendiag) =− ∆f(b+f +bf)−~J(b+ebg+b+gbe) (20) −~22J(Pr1efr=1gg+,er32er2g2)f] − 1f −FPorfs=ugb,espafce1I−II(24)
fX=g,e
5
that the current in channel f exists only for the case of
one of states f is occupied and another one of f is
{ } { }
unoccupied.
Further simplification is made below, when we con-
sider two specific limiting cases. The first limit,
U = 0, describes noninteracting electrons at each
m
sites. In the opposite limit with strong on-site Coulomb
repulsion U (m = 1,2) is much larger than any
m
other energy scale of the problem. In the latter
case we disregards states with more than one elec-
tron on any of the molecular site 1 and 2 so we
need to consider only 9 bridge states: 0 ,0 ,0 ,0 ,
1g 2g 1e 2e
| i
0 ,0 ,1 ,1 and 1 ,1 ,0 ,0 in subspaces
FIG.2: Aschematicdisplayoftheblockstructureofthewire | 1g 2g 1e 2ei | 1g 2g 1e 2ei
I; 0 ,0 ,0 ,1 , 0 ,0 ,1 ,0 , 0 ,1 ,0 ,0
Hamiltonian. | 1g 2g 1e 2ei | 1g 2g 1e 2ei | 1g 2g 1e 2ei
and 1 ,0 ,0 ,0 in subspaces II; 1 ,0 ,0 ,1
1g 2g 1e 2e 1g 2g 1e 2e
| i | i
and 0 ,1 ,1 ,0 in subspace III.
1g 2g 1e 2e
| i
This prediagonalizationprovides an important simpli- The diagonalization procedure yields the transforma-
fication of our problem. From Eqs. (17), (18), (19) the tion between the eigenstates of the wire Hamiltonian
current is given by and the states of the non-interaction molecular wire,
n ,n ,n ,n displayed in Fig.2. Denoting the col-
1g 2g 1e 2e
| i
Iˆ= i~e ∆f(bf −b+f)= ~e ∆fr2f (25) utimvenly,veacntdortsheoftrtahnessfeorsmtaatteisonbybe{twΦe}enantdhem{χ}b,yrYeˆs,pie.ce-.
fX=g,e fX=g,e χ =Yˆ Φ , we can characterized each eigenstate Φ by
{ } { }
Using Eq.(23), this yields the corresponding subspace (λe,λg). In this basis, the
fermionicinteractionpictureoperators(seeEq.(9))read,
Iˆ= e ∆ rf(λ =1) (26) for example
~ f 2 f
fX=g,e
Obviously λ = 1 in Eq.(26) is another way of saying
f
αcˆint( x)β =[Yˆ+(λ (α),λ (α))χ˜+(λ (α),λ (α))cˆ χ˜(λ (β),λ (β))Yˆ(λ (β),λ (β))]
h | nf − | i e g e g nf e g e g αβ
i
exp[ (E (λ (β),λ (β)) E (λ (α),λ (α)))x]
× ~ β e g − α e g
where (λ (α),λ (α)) denotes the subspace associated if f = g. χ˜ denotes the transpose matrix χˆ. The relax-
e g
with the eigenstate α and points to the corresponding ation terms in the master equation (8) take in this basis
valuesofλ andλ ,andwhere(λ (β),λ (β))=(λ (α)+ the forms
e g e g e
1,λ (α))iff =eand(λ (β),λ (β))=(λ (α),λ (α)+1)
g e g e g
− ~12TrKZ ∞dx[Vˆ,[Vˆint(−x),ρ(t)]]αβ = 21 Γnf{cˆnf,αα′σα′β′cˆ+nf,β′β[2−fKn(Eβ′ −Eβ)
0 nfXα′β′
−fKn(Eα′ −Eα)]+cˆ+nf,αα′σα′β′cˆnf,β′β[fKn(Eβ −Eβ′)+fKn(Eα−Eα′)]−{cˆnf,αα′cˆ+nf,α′β′fKn(Eα′ −Eβ′)
+cˆ+nf,αα′cˆnf,α′β′[1−fKn(Eβ′ −Eα′)]}σβ′β −σαα′{cˆnf,α′β′cˆ+nf,β′βfKn(Eβ′ −Eα′)+cˆ+nf,α′β′cˆnf,β′β[1−fKn(Eα′ −Eβ′)]}}
(27)
where
2π
Γ = V(nf) 2δ(ε ε ) (28)
nf ~ | k | k− nf
kX∈Kn
6
and
1 Tr ∞dx[Wˆ,[Wˆint( x),ρ(t)]]
− ~2 KZ − αβ
0
1
= 2 {−BKn[Eβ′(λe+1,λg)−Eα′(λe,λg+1),µKn][b+n,αα′bn,α′β′σβ′β(t)+σαβ′(t)b+n,β′α′bn,α′β]
nXα′β′
−BKn[Eα′(λe,λg+1)−Eβ′(λe+1,λg),µKn][bn,αβ′b+n,β′α′σα′β(t)+σαα′(t)bn,α′β′b+n,β′β]
+b+n,αα′σα′β′(t)bn,β′βBKn[Eβ′(λe,λg+1)−Eβ(λe+1,λg),µKn]
+bn,αα′σα′β′(t)b+n,β′βBKn[Eβ′(λe+1,λg)−Eβ(λe,λg+1),µKn]
+b+n,αα′σα′β′(t)bn,β′βBKn[Eα′(λe,λg +1)−Eα(λe+1,λg),µKn]
+bn,αα′σα′β′(t)b+n,β′βBKn[Eα′(λe+1,λg)−Eα(λe,λg+1),µKn]} (29)
where
2π 2
BKn(Eα−Eβ,µKn)= ~ k6=Xk′∈Kn(cid:12)(cid:12)(cid:12)Wk(kn′)(cid:12)(cid:12)(cid:12) δ(εk−εk′ +Eα−Eβ)fKn(εk)[1−fKn(εk′)] (30)
Inevaluatingtheseformswehavetakenthewideband A. Zero on site coupling
limit for the electrodes spectral densities.
Next consider the diagonalization procedure itself. In The case of zero on site coupling is discussed in Ap-
subspacesIthe unitarytransformationYˆ(λe,λg)isobvi- pendix A. We find the eigenfunctions andenergiesofthe
ously the unity matrix. The diagonalizationof the block 2-site bridge summarized in Table I
matrices insubspaces II andIII is carryout inthe limit-
ing cases of zero and infinite on-site interactions.
λ =0 λ =1 λ =2
g g g
Φ(0,0)= Φ (0,1)= 0 ,1 ,0 ,0
λe =0 =|01g,E02=g,001e,02ei Φ−+(0,1)=E||111=gg,0022gg,011ee,022eeii Φ(0,2)=E|1g=,102g,0e,02ei
Φˆ(1,1)=Y+(1,1)χˆ(1,1)
Φ (1,0)= Φ (1,2)=
= √12(|01±g,02g,01e,12ei∓ E = 121(ε1e4+∆2ε2+e)J±2~212J~± = √12(|1±1g,1g,01e,1ei∓
λe =1 χˆ(1,0)∓=|0E1(cid:18)g=,||000211εggg2,,,e100±122egg∆,,,001e211eeei,,)1022eeii(cid:19) χˆ(1,1)=±2p||||11001111gggge,,,,00112222gggg,,,,10101111eeee,,,,01012222eeeeiiii χˆ(1,2∓)|=1E1g(cid:18)=,1||112εggg2,,,e111±122egg∆,,,010e2eee,,i01)22eeii (cid:19)
Φ (2,1)= 0 ,1 ,1 ,1
λe =2 Φ(2,0)E==|0ε11ge,0+2gε,21e1e,12ei Φ−+(2,1E)==||1ε11gg,+022εgg,111ee,122eeii Φ(2,2)E==|1ε1g1e,1+2gε,21e1e,12ei
1e 2e
where and where τ is given by
sinτ cosτ cosτ sinτ J~ 2∆
Y+(1,1)= √12 ccsoionssτττ −csoisnisnτττ −−ssiicnnosτττ −cocsoisnsτττ , cos2τ = p4∆−2e+J2~2 and sin2τ = p4∆2e+eJ2(~322)
 − − 
(31)
7
The current in this case is found to be Yˆ+(1,0)andYˆ+(0,1)andthe correspondingeigenstates
and eigenvalues are defined by the same Eqs. (46), (47)
and (48), respectively, as before (see Appendix A). The
2e
I = ∆ Im [σ (1,1)+σ (1,1)]cos2τ operator Yˆ+(1,1) is reduced to (see Appendix B)
h i − ~ e { 32 41
+[σ (1,1) σ (1,1)]sin2τ σ (1,λ )
31 − 42 −λgX=0,2 −+ g } Yˆ+(1,1)= √12(cid:18) 11 11(cid:19) (35)
(33) −
Yˆ+(1,1) is used to obtain the corresponding eigenstates
Indices ”+” and ”−” in Eq.(33) correspond to the func- Φˆ(1,1)=Y+(1,1)χˆ(1,1)andeigenvaluesE1,2 =εe J~.
tions Φ (1,λ ) and Φ (1,λ ), respectively, in Table I. ∓
+ g g Substituting Eq.(35) into Eq.(49) of Appendix A for
Indices 1,2,3 and 4 lab−el the the eigenstates of the wire
the current, we get for ∆ =0
g
Hamiltonian in subspace III. The corresponding ener-
gies are given by formulas E E , E E , 2e
E3 E+, , E4 E+,+ where 1 ≡ −,+ 2 ≡ −,− hIi= ~ ∆eImσ−+(1,0) (36)
≡ − ≡
1 1
E =ε J~ 4∆2+J2~2 (34)
±,± e± 2 ± 2 e V. CURRENT FROM THE ENERGY
p TRANSFER INTERACTION IN THE WIRE
B. Rotating-wave approximation In a recent paper [40] Galperin, Nitzan and Rat-
ner have predicted the existence of non-Landauer cur-
The calculation of the non-diagonal elements of the rent induced by energy transfer interactions between
density matrix σ (1,λ ) in Eq.(33) for the current is a bridge molecule and electron-hole excitations in the
αβ g
essentially simplified for very weak wire–lead coupling leads. Here we show that a similar non-Landauer cur-
whenthe coherenttime-evolutiondominatesthe dynam- rent arises from the exciton type interaction J in the
ics of the wire electrons. This means that the largest wire itself. For simplicity we limit ourselves to elec-
time-scaleofthecoherentevolution,givenbythesmallest trontransfer interactionbetween the wire and the metal
energy difference, and the dissipative time-scale, deter- leads, Eq.(27), and disregard the corresponding excita-
minedbytheelectronandenergytransferrates,Γ and tion transfer, Eq.(29). Also for simplicity we consider
nf
B (E E ,µ ),respectively,arewellseparated,i.e., a large bias limit in the Coulomb blockade case when
bα~yΓK6=nntfhβ,e.~αBfiT−rKhstnen(tβEefrαomrK−λnoEen=βt,hµ1eKaRnn)dH≪Sα.|6=CEoαβn,−sEeqEqu.β(e|8n)ftolirys,λdσeoαm=β(i1n1a,aλtnegdd) sµleiLtvieol>nseodεferbaaotnhthderlµefRaadrs<(≫soεgtk,hBaaTtn,df~L|tJ(hε|e,)|∆s=tea|t)1esfarnoεmde,εftgRh(eaεr)Fee=rpmo0-i
can be calculated in the first order of ~Γ /(E E ) can be taken on the RHS of Eq.(27). Finally, we dis-
nf α β
and ~B (E E ,µ )/(E E ). This const−itutes regard electron transfer interaction in the ”g” channel,
theessenKcneofαa−rotaβtingK-nwaveaαp−proxβimation(RWA)[11]. i.e. we take ∆g = 0. Landauer type current would be
Within it,one canprovideaclosedexpressionforthe re- realized in channel ”e” when it is isolated from channel
duceddensitymatrixelementsσαβ andforthestationary ”g”, i.e. when J = 0, Γ1g = Γ2g = 0 and λg = 0. Solv-
current. We shall use the RWA below in Sec.V and Ap- ingEqs. (8),(27)intheRWAapproximationunderthese
pendix B. conditionsandsubstitutingthesteady-statesolutioninto
Eq.(36), we get, using also the normalization condition
Trσ(λ ,0)=1,
λe=0,1,2 e
C. Strong Coulomb repulsion at sites P
Γ Γ
1e 2e
I = e (37)
RWA
h i − Γ +Γ
In the limit of strong Coulomb repulsion, U 1e 2e
m
is assumed to be so large that at most one ex- Eq.(37) describes the Landauer current and coincides
cess electron resides on each site. Thus, the avail- with Eq.(21) of Ref.[11] (excluding the sign).
able Hilbert space for uncoupled sites is reduced to In fact, the current vanishes for Γ1e = Γ2e = 0 even
t0hre,e0 st,a1tes,1χˆ(0,a0n)d=χˆ(0|,021)g,0=2g,011e,,012ei,,0χˆ,(02,0) fo=r wtivheencoΓu1pgl,inΓg2gto6=th0e, lseinadcesc∆ogul=db0e(osebetaFinige.d3)f.orStuhcehbsreidlegce-
1g 2g 1e 2e 1g 2g 1e 2e
| i | 0 ,0 ,0 ,1i madeofaquadruplequantumdotwherethe lateralones
subspaces I; two states χˆ(1,0) = (cid:18)|011gg,022gg,111ee,022eei(cid:19) are strongly coupled to the leads [47, 48].
| i Considernowthe casewhenΓ =Γ =0;Γ ,Γ =
and χˆ(0,1) = |01g,12g,01e,02ei for subspaces II; 0;∆ =0andJ =0. Forthisca1seeEqs.2(e8),(27)1tgoge2tghe6 r
(cid:18) 11g,02g,01e,02e (cid:19) g 6
| i with Eq.(36) lead to
1 ,0 ,0 ,1
and the state χˆ(1,1) = | 1g 2g 1e 2ei for the
now 2-dimensional subspa(cid:18)ce|0II1Ig.,1T2hg,e1u1en,i0ta2eriy(cid:19)operators hIi=−4eΓgJ2∆2e∆21Γ−2[σ+−1−6(∆0,21J)2++σ~(22Γ,0)2]J2 (38)
e g e g
8
system in state 1 ,0 ,0 ,1 , electron transmission
1g 2g 1e 2e
| i
1
takes place along route such as 1 ,0 ,0 ,1
1g 2g 1e 2e
| i −→
2 3
0 ,1 ,1 ,0 0 ,0 ,1 ,0
1g 2g 1e 2e 1g 2g 1e 2e
| i −→ | i −→
4
0 ,0 ,0 ,1 1 ,0 ,0 ,1 . Step 1 is
1g 2g 1e 2e 1g 2g 1e 2e
| i −→ | i
anenergytransferprocess,steps 2 and 3 relyonΓ =0
2g
6
and ∆ =0, respectively, and step 4 closes the circle via
e
6
the Γ process.
1g
Eq.(38) clearly shows that the current exists only for
J = 0 and ∆ = 0. For small J, I J2. For large J
e
6 6 h i ∼
we obtain
FIG. 3: A possible physical realization of the selective tun-
neling configuration, where only |gi levels are coupled to the
1 [σ (0,1)+σ(2,0)]
leads. hIi≃−4eΓg∆2e − 1−6−∆2+~2Γ 2 (39)
e g
which does not depend on J. In the limit ~ J , ∆
e
| | | | ≫
Γ , Eq.(38) yields for σ (0,1)=σ(2,0)=0
g
−−
e Γ Γ
2g 1g
I = (40)
RWA
h i −2Γ +Γ
2g 1g
In deriving Eq.(40) we have not put Γ = Γ . This
1g 2g
limit corresponds to the range of validity of the RWA.
Indeed, it can be shown that Eq. (40) can be obtained
for this model in the RWA (see Appendix C).
Ifσ (0,1),σ(2,0)=0,the non-Landauercurrentde-
crease−s,−since the popu6 lations of states 1 ,0 ,0 ,0
1g 2g 1e 2e
| i
(σ (0,1)) and 0 ,0 ,1 ,1 (σ(2,0)) suppress cur-
1g 2g 1e 2e
ren−t−. Two latter|states are also siteady-states in the case
under consideration (Coulomb blocking, Γ = Γ =
1e 2e
∆ = 0) along with the states described by Fig.4. The
g
existenceofseveralsteady-statescorrespondstothepres-
ence of the respective zero eigenvalues of the relaxation
matrix. Our numerical calculations give three such zero
eigenvalues corresponding to three above steady-states.
If Γ ,Γ = 0, state 1 ,0 ,0 ,0 is only steady-
1e 2e 1g 2g 1e 2e
6 | i
statethat“locks”the currentdue to Coulomb blocking,
since ∆ = 0. Numerical simulations of other situations
g
when ∆ = 0 and non-interacting electrons at a site are
g
6
carried out in the next section.
VI. NUMERICAL RESULTS
The results presented in this section are based on di-
rect numerical solution of Eq. (8), and are in com-
FIG. 4: Different stages of the energy-transfer induced cur-
plete agreement with the analytical solutions when ap-
rent. a) energy transfer, σ(1,1) 6= 0. b) the charge transfer
totheright lead. c) theintersitecharge transfer, σ(1,0)6=0; plied to the special cases treated in Sections IV and V.
thecharge transfer from theleft lead. The numerical solution was carried using the the basis
of eigenstates of the Hamiltonian Hˆ , Eq.(3). Once
wire
σ(t) is obtained from Eq. (8), the expectation value
where for simplicity we put Γ = Γ Γ . Eq.(38) of the current is calculated as I = Tr(Iˆσ(t)) where
1g 2g g
≡ h i
describesanon-Landauercurrentcausedby transportin the current operator was defined by Eq.(26). In this
different channels: the intersite transfer occurs in chan- calculation we have limited ourselves to the case where
nel ”e”, and the charge transfer between the molecular the wire-leads energy transfer coupling Wˆ is disregarded
bridge and the leads occurs in channel ”g”. The inter- and, unless otherwise specified, have used the following
channel mixing is induced by the energy-transfer term parameters: ε = ε = 0.0eV, ε = ε = 2.0eV,
1g 2g 1e 2e
J (see Fig.4). For example, starting with the molecular ∆ = ∆ = 0.01eV, Γ = Γ = 0.02eV for f = g,e
g e 1f 2f
9
1 1 1
0.8 0.8
V = 4.0eV
Vbs = 2.0eV 0.8
bs
_∆<I> [e/h ]e00..46 Vbs =1.96eV _∆<I> [e/h ]e0.6 PPPP1212ggee 00..46Population
0.2
0.2
0.4
0
-0.05 -0.04 -0.03 -0.02 -0.01 -0.05 -0.04 -0.03 -0.02 -0.01 0
0 J [eV] J [eV]
-0.05 -0.04 -0.03 -0.02 -0.01 0
J [eV]
FIG. 6: The current hIi as a function of the parameter J in
FIG.5: ThecurrenthIidisplayedasafunctionoftheexciton the case of non- interacting electrons for V = 4.0eV. The
bs
coupling parameter J . Vbs = 1.96eV (dashed line); Vbs = current hIi is shown in left panel, and the populations P1g,
2.0eV (dotted line); Vbs=4.0eV (solid line) P1e, P2g and P2e are shown in theright panel.
1
(belowweuseΓtodenotetheorderofmagnitudeofthese 0.5
widths) and T = 100K. The Fermi levels were taken to
0.8
alignsymmetrically with respect to the energy levels ε
1g
Waunneditaεol1sfeo,cuiu.resre.edµntLthh=eIiv.(aεl1uge+oεf1ee∆+eV/b~s)=/22a.n45d•µR10=−6µALa−sVtbhse. _∆<I> [e/h ]e0.4 PPPP1212ggee 00..46Population
Consider first non-interacting electrons. Figs.5, 6 and 0.3
7 show the expectation value of the current I and one-
particle populations P = Tr(cˆ+ cˆ σ) ashfuinctions of 0.2
nf nf nf
the exciton interaction parameter J. One can see that 0.2
0
if the imposed voltage bias Vbs is larger than εe εg, -0.05 -0.04 -0.J0 3[eV-0].02 -0.01 -0.05 -0.04 -0.J03 [eV-0].02 -0.01 0
−
the expectation value of the currentdiminishes when J
| |
increases (we have used J < 0 which is typical to J-
aggregates, however the trend is similar with J > 0). FIG. 7: Same as Fig.6 for the parameters ∆g = 0, ∆e =
Suchabehaviorcanbeunderstood,usingEq.(34)forthe 0.01eV and Vbs=4.0eV.
energies in subspaces (III) and Eq.(33) for the current.
Thelatterequationshowstwodirectcontributionstothe
valuesofenergyE =E =ε +∆ andE =E =ε
current. The firstone has its originin states ofsubspace 1 4 e e 2 3 e
−
∆ ,wherethesplittingisofthesameorderofmagnitude
(III),theenergiesofwhichdependonboth∆ andJ (the e
e
as the hopping matrix element ∆ . We obtain
first and the second terms on the RHS of Eq.(33)). The e
second contribution arises from states of subspaces (II),
2e
theenergiesofwhichdependon∆ only(thethirdterms ∆ Im[σ (1,1) σ (1,1)] eΓ (42)
e ~ e 31 − 42 ∼
on the RHS of Eq.(33)). Using Eqs.(8) with Wˆ =0 and
(27), the non-diagonalelements ofthe density matrix on This contribution is of the same order of magnitude as
the RHS of Eq.(33) for the steady-state condition can that from the states of subspaces (II).
Ebe evaEluat=ed a2s∆σα,βE∼q.E(4−α8−i~)E,Γβwe(sgeeetaflosor tSheec.IcVonBtr)i.bSuitniocne useInJop<po0si-taescianseJ,-a~g|gJr|eg>a>tes∆) ea,ndcossi2nτ2τ≈ 10(.agIaninthwies
of−th−e th+ird t−ermeon the RHS of Eq.(33) caseonlythefirsttermin(33)contributesto≈thecurrent
from the states of subspace (III). For this case we get
E E ε and E ε J~. This leads to
2e 2 ≈ 4 ≈ e 1,3 ≈ e∓
∆ Im σ (1,λ ) 2eΓ (41)
~ e λgX=0,2 −+ g ∼ 2~e∆eIm[σ32(1,1)+σ41(1,1)]∼eΓ∆J~e (43)
Thecontributionofthefirstandthesecondtermsonthe
RHS of Eq.(33) depends on the relation between J and This contribution is much smaller than that of Eqs.(41)
∆ . and (42), since the hopping matrix element ∆ is much
e e
When ~ J <<∆ , Eq.(32)yields cos2τ 0, sin2τ smaller than the splitting between states 3 and 2, and
e
| | ≈ ≈
1,andonlythesecondtermin(33)givesacontributionto states 4 and 1 due to the exciton interaction (indices
the current from the states of subspace (III). Under this 1,2,3 and 4 label the the eigenstates of the wire Hamil-
condition one gets from Eq.(34) two doubly-degenerated tonian in subspace III). This can cause the value of the
10
1 1 0.1
Vbs = 2 eV Vbs = 2 eV 0.6
0.8 VVbbss == 48 eeVV VVbbss == 48 eeVV 0.8
0.5 0.08
_∆/h ]e0.6 0.6_∆/h ]e _∆/h ]e _∆/h ]e
<I> [e0.4 0.4<I> [e <I> [e0.4 0.06<I> [e
0.3
0.2 0.2
0.04
0.2
00 1 2 3 4 5 0 1 2 3 4 5 60 -0.1 -0.08 -0.06 -0.04 -0.02 -0.1 -0.08 -0.06 -0.04 -0.02 0
U [eV] U [eV] J [eV] J [eV]
1 1
FIG. 8: The current hIi as a function of the Coulomb in- FIG.10: ThecurrenthIiplottedagainsttheexcitoncoupling
teraction parameter U1 = U2. Vbs = 2.0eV (solid line); parameter for bias Vbs = 4.0eV for different energies in the
Vbs = 4.0eV (dashed line); Vbs = 8eV (dotted+dashed line); e-channel: ε1e = 1.95eV and ε2e = 2.05eV. ∆g = 0.01eV
J =0.0eV (left panel) and J =−0.05eV (right panel). (left panel), ∆g =0 (right panel).
1 1
0.8 JJJ === -0- 00.0..00025 eeeVVV JJJ === --000...000025 eeeVVV 0.8 00..65 JJJ === -0- 00.0..00025 eeeVVV JJJ === --000...000025 eeeVVV 00..56
_∆<I> [e/h ]e00..46 00..46_∆<I> [e/h ]e _∆<I> [e/h ]e00..43 00..34_∆<I> [e/h ]e
0.2 0.2
0.2 0.2 0.1 0.1
00 1 Vbs 2[eV] 3 0 1 Vbs 2[eV] 3 40 00 1 Vbs 2[eV] 3 0 1 Vbs 2[eV] 3 40
FIG. 9: The current hIi as a function of the bias voltage FIG. 11: Same as Fig. 9 except that ε1e =1.95eV and ε2e =
V shown for different values of the exciton coupling pa- 2.05eV.
bs
rameter: J = 0.0 (solid line), J = −0.02eV (dotted line),
J =−0.05eV (dash-dotted line). Left panel - noninteracting
electrons. Right panel U1,U2 =∞. which are acted upon exciton interaction (see Fig.2).
Moreover, in the case of Coulomb blocking, the effect
of exciton-induced current exists (Sec.V).
totalcurrentto decrease. Inother words,the transitions The effect of “exciton blocking” depends on the en-
3 2 and 4 1 do not participate in electron transfer ergydetuning ε ε inchannel”f”foraheterodimer
2f 1f
du→e to their l→arge splitting for ~ J >>∆ . This is in a bridge. Figs.10an−d11showthecurrent I asafunction
e
| | h i
sense”excitonblocking”ofelectrontransmissionthrough ofJ forε ε =0.1eV. I isseentoincreaseforsmall
2e 1e
− h i
the bridge. J , then to decrease as J becomes larger. This can be
| | | |
Next we turn to situations where electron-electron in- relatedto the modificationof resonanceconditions when
teraction is taken into account. Fig.8 shows the current ε ε =0.
2e 1e
− 6
I as a function of the Coulomb interaction parameter Finally, figures 12, 13, 14 and 15 show more of the
h i
U = U . Fig.9 depicts the current I as a function system behavior for the model with U ,U = . Fig.12
1 2 1 2
h i ∞
of the bias voltage V for different values of the exci- showsthe currentas a function of J for different values
bs
| |
ton coupling J for the case of non-interacting electrons of the imposed voltage bias V . If V is large com-
bs bs
as well as for the case of infinite on-site interaction be- pared to the energy difference between the excited and
tween electrons. The “exciton blocking” effect seen for ground site energies, the current behaves in accordance
non-interacting electrons (smaller current for larger J ) withEq.(38). IfV is close to this energydifference, the
bs
| |
disappears in the case of Coulomb blocking. currentincreasesinitially with J ,andthendecreasesto
| |
This is supported by Eq.(36) that does not show zero. Furthermore, in accordance with Eq.(38), the left
a direct contribution of the states of subspace (III) panelofFig.13showsthatthesteady-statecurrentiszero
to the current, and the above evaluation of the term for the initial condition σ (0,1) = 1. The steady-state
2e∆ Imσ (1,0). The point is that in the case of in- current is zero also for th−e−initial condition σ(0,0) = 1,
~ e +
teracting−electrons, subspace (III) includes only states, sincethelatterstaterelaxestoσ (0,1)=1. Figures 14
−−