Compensation of Coulomb blocking and energy transfer in the current voltage characteristic of molecular conduction junctions.

Abstract

We have studied the influence of both exciton effects and Coulomb repulsion on current in molecular nanojunctions. We show that dipolar energy-transfer interactions between the sites in the wire can at high voltage compensate Coulomb blocking for particular relationships between their values. Tuning this relationship may be achieved by using the effect of plasmonic nanostructure on dipolar energy-transfer interactions.

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Compensation of Coulomb Blocking and Energy Transfer in the
Current Voltage Characteristic of Molecular Conduction Junctions
Guangqi Li,*,†Manmohan S. Shishodia,‡,§Boris D. Fainberg,‡,§Boris Apter,∥Michal Oren,§
Abraham Nitzan,§and Mark A. Ratner†,#
†Non-Equilibrium Energy Research Center (NERC), Northwestern University, Evanston, Illinois 60208, United States
‡Faculty of Science, Holon Institute of Technology, 58102 Holon, Israel
§School of Chemistry, Tel-Aviv University, 69978 Tel-Aviv, Israel
∥Engineering Faculty, Holon Institute of Technology, 58102 Holon, Israel
#Department of Chemistry, Northwestern University, Evanston, Illinois 60208, United States
*
S Supporting Information
ABSTRACT: We have studied the influence of both exciton
effects and Coulomb repulsion on current in molecular
nanojunctions. We show that dipolar energy-transfer inter-
actions between the sites in the wire can at high voltage
compensate Coulomb blocking for particular relationships
between their values. Tuning this relationship may be achieved
by using the effect of plasmonic nanostructure on dipolar
energy-transfer interactions.
KEYWORDS: Molecular conduction nanojunctions, exciton effects, energy transfer, Coulomb blocking, plasmonic effects
E
transfer interactions can sometimes have important effects on
the dynamics of such processes. Charge and energy transfer in a
linear 2,2′:6′,2″-terpyridine-based trinuclear Ru-(II)-Os(II)
nanoscale array9and one-dimensional energy/electron transfer
along amylose-encapsulated chain chromophores10are exam-
ples. In addition, it seems likely that energy transfer takes place
in chemically responsive molecular transistors based on a dimer
of terpyridyl molecules chelating with Co2+.11We have recently
developed a theory of electron transport through molecular
wires in the presence of intersite dipolar energy-transfer
interactions in the wire, using a model comprising a bridge
with two two-level sites connecting free electron reservoirs.12,13
Our calculations show that for noninteracting electrons, this
structure leads to reduction in the current at high voltage for a
homodimer bridge. This effect, called “exciton blocking”,
disappears for strong on-site Coulomb repulsions.13
Although in free exciton systems, dipolar interactions (≲ 0.01−
0.1 eV)14are considerably smaller than on-site Coulomb
interaction U (characteristically U ∼ 1 eV);15the former may
still have strong effects under some circumstances, e.g., in the
vicinity of metallic structures in or near the nanojunctions. In such
cases dipolar interactions may be enhanced, while Coulomb
interaction is screened.16−19The question “how do such dipolar
lectron transport through molecular wires has been under
intense theoretical1−5and experimental6−8study. Energy-
interactions affect the junction transport properties when they are
of the same order of magnitude as on-site Coulomb repulsion U?”
can become pertinent.
In this letter we consider the simultaneous effects of energy
(exciton) transfer and Coulomb repulsion on the conduction
properties of molecular nanojunctions. We evaluate the
enhancement of dipolar energy-transfer near plasmonic
nanostructures and show that the magnitude of the enhanced
interaction can approach U. We then show that this interaction,
J, can have a substantial effect on the transport and can in fact
substantively reduce the high-voltage Coulomb blocking effect
for particular values of J and U.
We consider a spinless molecular wire model that comprises
two interacting sites (each represented by its ground, |g⟩, and
excited, |e⟩, states) positioned between two leads. The latter are
represented by free electron reservoirs L and R (Figure 1),
characterized by the electronic chemical potentials μK, K = L, R,
and by the ambient temperature T. The corresponding Fermi
distributions are fK(εk) = [exp((εk− μK)/kBT) + 1]−1, and the
Received:
Revised:
Published: March 30, 2012
November 23, 2011
March 14, 2012
Letter
pubs.acs.org/NanoLett
© 2012 American Chemical Society
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difference μL− μR= eVbsis the imposed voltage bias. The
corresponding Hamiltonian is
=
HHHH
wireleadsint
∑
∈
k
{L,R}
∑
=
=
f
g,e
++≡+
HH
0 int
(1)
=
†
HE c c
k k k
leads
(2)
∑
=
f
∑
=
m
=−Δ+
+ℏ++−
††
2 1
†
1
†
1 2
H E c c
mf mf mf
c cc c
Jb b
U
2
N N
(
m
()
(H.c.)1)
m
fffff
m
wire
1,2
g,e
2
1,2
(3)
∑
∈
,
=+
†
k mf
HV c c
H.c.
mf k K
†(cmf) (m = 1, 2; f = g, e, see Figure 1) are creation
(annihilation) operators for electrons in the different site states
of energies Emf, while ck
(annihilation) operators for free electrons (energies Ek) in the
leads L and R. The occupation operators are nmf= cmf
different site states, and site occupation operators are given by
Nm= nmg+ nme. The operators bm
exciton creation and annihilation operators on the molecular
sites m = 1, 2. In eq 3, Δfrepresents electron tunneling between
site states of similar energies (i.e., between |g⟩ levels of sites 1
and 2 and between |e⟩ levels on these sites), the J terms
represent exciton hopping (energy transfer) between molecular
sites, and the U terms correspond to on-site Coulomb
interactions. The molecule−leads interaction Hintdescribes
electron transfer between the molecular bridge and the leads
that gives rise to net current in the biased junction. In eq 4, Km
is the lead closer the molecular site m (K1= L, K2= R), and
H.c. denotes a Hermitian conjugate. Below we will also use the
population operators λf= n1f+ n2fin the manifolds of ground
(f = g) and excited (f = e) site levels.
Our analysis is based on the generalized master equation for
the reduced density matrix of the molecular subsystem,12,13
obtained using a standard procedure20−22based on taking Hint
as a perturbation. Briefly, one starts with the equation for the
total density operator and uses the projectors PKof the type
PKρ(t) = ρKTrKρ(t) with ρKbeing the density matrix of the
leads in their equilibrium state, in order to derive an equa-
tion for the time evolution of the reduced density matrix
σ = TrRTrLρ. This leads to13
k
mf
(
int
)
m
(4)
where cmf
†(ck) (k ∈ L, R) are creation
†cmffor the
†= cme
†cmgand bm= cmg
†cmeare
∫
σ
d
= −ℏ
−ℏ
σ
τ−τρ
∞
t
t
i
Ht
HHt
d ( )
[ , ( )]
1Tr
2
d [, [(), ( )]]
K
wire
0
intint
int
(5)
where TrK= TrRTrLand Hint
exp(iH0τ/ℏ) is the interaction representation of Hint. The second
term on the right-hand side of eq 5 can be evaluated in the
Markovian and wide-band limits and is in terms of the rate constant
of charge transfer from state f of site m to the corresponding lead:
∑
ℏ
∈
k K
n
int(−τ) = exp(−iH0τ/ℏ)Hint
Γ=
π
|| δ−
VEE
2
()
mf
k
mf
(
kmf
) 2
(6)
The resulting rate equation can be solved numerically.
The electric current I is defined as the current going into the
system on the left side, using the electron number operator of
the left lead NL= ∑k∈Lck
⎧
⎨
⎩
t
d
k L
†ck,22,23
∑
∈
=| |ρ
†
k k
⎪⎪
⎭
⎪⎪
⎬
⎫
I t
( )
c ct
Tr ed( )
(7)
that at steady state is equal to the current going out of the
system on the right.
In this calculation we use the following parameters: E1g= E2g≡
Eg= 0, E1e= E2e≡ Ee= 2 eV, Δg= Δe= 0.01 eV, Γ1f= Γ2f= Γ =
0.02 eV for f = g, e (below we use Γ to denote the order of
magnitude of these widths) and T = 100 K. The leads chemical
potentials in the biased junction were taken to align symmet-
rically with respect to the energy levels E1gand E1e, i.e., μL=
(E1g+ E1e+ Vbs)/2 and μR= μL− Vbs. We also used the value
of I0= e × 0.01 eV/ℏ = 2.45 × 10−6A as the unit of current I.
Figures 2 and 3 show the current as a function of the
parameters of Coulomb interaction U and the dipole−dipole
interaction J between sites and as a function of the imposed voltage
bias for different values of J and U, respectively. For J = 0, Figures 2
and 3 show the effect of Coulomb blocking: current has a
maximum at U = 0 and decreases with increasing U. For U = 0,
the figures show the effect of “exciton blocking” (as predicted in
ref 13): current has a maximum at J = 0 and decreases with
increasing the absolute value of J. The most remarkable observation
is however the mutual compensation of exciton and Coulomb
blocking seen in Figures 2 and 3, where the current in Figure 2
goes through a maximum at U = −ℏJ with a U-independent peak
value I/I0= 1. Remarkably, if either Δgor Δeare taken to zero
(Figure 4), we observe peaks for both U = −ℏJ and U = ℏJ.
Figure 1. A model for energy-transfer induced effects in molecular
conduction. The molecular bridge is a dimer, where each site is
represented by its ground, |1g⟩ and |2g⟩, and excited, |1e⟩ and |2e⟩,
levels with the nearest-neighbor site coupling Δgand Δe. The two
metal leads characterized by electrochemical potentials μRand μLare
coupled to their nearest molecular site with the transfer rate Γ2e, Γ2g,
Γ1e, Γ1g. J represents exciton hopping.
Figure 2. Current I displayed as function of the energy-transfer
coupling J and on-site interaction U. The current shows a maximum
at U = −ℏJ with the same peak value I/I0= 1. The bias voltage
Vbs= 4.0 eV, and Δg= Δe= 0.01 eV.
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To understand this behavior, we extend the analytical
evaluation of ref 13 to include finite on-site Coulomb repulsion
between charge carriers. Details can be found in ref 13 and in
the Supporting Information. Since the total molecular
populations described by operators λfare conserved under
the unitary transformations associated with the diagonalization
of Hwire,13the total 24× 24occupation state-space can be
partitioned into nine smaller subspaces (see Figure 2 of ref 13):
four one-dimensional subspaces for [λf] = 0, 2 for either f = e, g
(type I); four two-dimensional subspaces for [λf] = 1 and
[λf’] = 0, 2, where f ≠ f′ (type II); and one four-dimensional
subspace for [λe] = [λg] = 1 (type III). Here we use [λf] to
denote the eigenvalues of matrix operator λf. The type I
submatrix is diagonal, while four pairs of states with each pair
coupled by the charge-transfer interaction are associated with
the four 2 × 2 blocks of the type II subspace. The four type III
states are coupled by both the charge- and exciton-transfer
interaction and constitute the 4 × 4 block of subspace III.
This partitioning can be exploited to diagonalize the kinetic
matrix and evaluate the current through the junction. For J < 0
(J aggregates) the current can be shown to be given by
2e
Im{[ (III)
e32 41
∑
λ =
[ ] 0,2
g
= −ℏΔ
− σ
σ+ σϕ + σ
ϕ −σ λ =
e
λ
−+
I
(III)]cos[(III)
31
(III)]sin (II; [ ]1, [ ])}
42g
(8)
with
ϕ =
−ℏ + Δ
Δ ℏ
J
4
e
UJ
1
2arccot
4
2 2 2
e
2
(9)
and the reduced density matrix σ shown in eq 5. Equations 8
and 9 are the generalizations of eqs 32 and 33 of ref 13 to finite
on-site Coulomb repulsion. The matrix−element indices “+”
and “−” in eq 8 label the eigenstates of the wire Hamiltonian in
subspaces II, and indices 1−4 are the corresponding labels in
subspace III. The nondiagonal elements of the density matrix
on the right-hand side of eq 8 can be evaluated using eq D2 of
ref 13: Imσαβ∼ −(Eα− Eβ)ℏΓ/(Eα− Eβ)2+ 4ℏ2Γ2, where Eα
and Eβare eigenstate energies of Hwire, and Γ is of the order of
the relaxation parameter, eq 6. In particular, we get the same
contribution from the third summation term on the right-hand
side of eq 8, as in case of U = 0 (eq 41 of ref 13):
∑
+ ℏ Γ Δ
λ =
0,2
g
ℏΔ
σ∼
Γ
−+
2e
Im
2e
1/
e
22
e
2
(10)
Let us evaluate the contributions of the first and the second
terms on the right-hand side of eq 8 at U2= J2ℏ2, corresponding
to maxima shown in Figure 4. The corresponding energy
differences between the states of subspace III are given by
−= Δ − ℏ −
e
Δ +
e
ℏ
− = −Δ − ℏ +
e
Δ +
e
ℏ
−= Δ − ℏ +
e
Δ +
e
ℏ
−= Δ + ℏ +
e
Δ +
e
ℏ
EEJJ
EEJJ
EEJJ
EEJJ
14
2 2 2
23
22 2
13
22 2
42
22 2
(11)
and
ϕϕ =±
Δ
Δ +
e
ℏ
J
sin, cos
1
2
1
e
22 2
(12)
When ℏ|J| ≪ Δe, eq 12 yields cos φ ≈ 0, sin φ ≈ 1, and only
the second term in eq 8 gives a contribution to the current from
the states of subspace (III). Under this condition one gets using
eq 11
2e
Im[]
1
ℏΔ
σ− σ∼
Γ
2
+ ℏ Γ Δ
e
/
e3142
2
e
2
(13)
This contribution is of the same order of magnitude as that
from the states of subspaces (II), eq 10.
In the opposite case, ℏ|J| ≫ Δe, cos φ ≈ sin φ ≈ 1/√2, and
both first and second terms in eq 8 contribute to the current.
For this case using eq 11 we get
ℏ
Δσ + σ + σ
41
− σ∼
Γ
2
+ ℏ Γ Δ
4
2e
2
Im[]
2 2e
1/
e323142
2
e
2
This contribution is of same order of magnitude as those of eqs
10 and 13. In other words, in the case under consideration, a
simultaneous change in the values of J and U for U2= J2ℏ2does
not affect the current. This stands in contrast to the separate
effects of exciton coupling and Coulomb repulsion.13In a sense
this is an “exciton compensation” of the Coulomb blocking
(ECCB) effect on electron transmission through the bridge.
Anticipating the discussion below on the possibility for
controlling the value of J by manipulating the electromagnetic
Figure 3. Current I displayed as a function of bias voltage Vbsfor
different J and U. Since we assume a symmetric bias voltage, the
energy level will be in the open window when the bias voltage achieves
2 eV, since we took 2 eV as Ee− Eg, where we see a jump shoulder of
the current. The current shows its maximum value at U = −ℏJ, and
Δg= Δe= 0.01 eV.
Figure 4. Current I displayed as function of the energy-transfer cou-
pling J and on-site interaction U for Δg= 0 and Δe= 0.01 eV. The current
goes through a maximum at U = ±ℏJ. The bias voltage Vbs= 4.0 eV.
The same results are obtained when Δg= 0.01 eV and Δe= 0.
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environment of the junction using plasmonic response of the
metal contact, we address the possibility of complex-valued
J due to coupling to the plasmonic resonances involving decay
processes. The latter, as is known, is described by the imaginary
part of interaction. Figure 5 shows that the ECCB phenomenon
is conserved for complex-valued J with maximum current
obtained at U = ± ℏ|J|, depicting additional maxima with
respect to the case of real J. These figures also demonstrate that
the relevant magnitude for complex-valued J is its modulus, |J|.
It is interesting to examine possible experimental setups in
which ECCB may be realized. In free space, U is considerably
larger than J. On the other hand, dipolar energy transfer is
known to be effected by proximity to plasmonic metal
structures. Coulomb repulsion is affected by metal screening,
but for our geometry, the effect is relatively small. We assume
that the site (point dipole) is found at the distance of 1 nm
from the nearest metal lead (if the distance is less, the classical
description of the plasmonic effects is incorrect). If an extra
electron is placed on the site, the distance between the site’s
electrons is about 0.1 nm (the atom size). The (positive) image
of the extra electron of the site is in the metal lead at the
distance of 1 nm from the metal surface. This means that the
distance between the extra electron of the site and its image is 2
nm, much larger than the inter-electron distances between the
site’s electrons. Therefore, the metal screening of the Coulomb
repulsion is small, for the distances considered here. In contrast
to U, the energy-transfer coupling J can be controlled by the
electromagnetic environment. This can in principle be achieved
via the plasmonic response of the metallic contacts, that can
greatly alter the effective dipole−dipole interactions in the
molecular bridge.16−19To demonstrate this, we have calculated
the energy-transfer coupling in the gap between two metal
nanospheres by the finite-difference time-domain (FDTD)
method. We used commercially available FDTD Solutions
software by Lumerical.28This is shown in Figure 6. For
comparison, we also show the results calculated within the
quasistatic approximation17,18,24,25for a single nanosphere (the
right sphere in the inset to Figure 6 is absent) (the calculational
details can be found in ref 27). The metal particles are
represented by a Drude dielectric model, ε = ε0− ωp
with parameters ε0= 3.57, ωp= 9.1 eV, and γ = 0.052 eV
corresponding to silver; the host medium was taken as
2/[ω(ω + iγ)]
transparent with εd= 2. Figure 6 shows enhancement of
dipole−dipole interactions (J) both in the gap of a dimer of
silver spheres and near the single sphere as a function of the
transition frequency |e⟩ → |g⟩.
It appears that proximity to plasmonic structures can indeed
be used to enhance energy transfer to the level needed to
observe ECCB, however to complete this assessment, we also
need to consider metal-induced damping of excitation energy.
To this end, we evaluated the rate of the energy transfer to
metal γmfor a single sphere, using the approaches from refs 17
and 18, which is shown in Figure 6 together with |J|. One can
see that due to different frequency dependence of |J| and γm,
rather large values of |J| can be induced at frequencies for which
γm≪ |J|.
In conclusion, the coexistence of electron and energy-transfer
interactions in molecular junctions may result in a new effect:
ECCB of electron transmission at high voltage. ECCB can
permit efficient electron transport even in systems with strong
Coulomb repulsion for carriers and can be realized by
controlling the plasmonic response of metallic contacts. Here
the enhancement of the dipole−dipole interaction calculated
using FDTD simulation for the dimer of silver spheres, and
within the quasistatic approximation for a single sphere,
reached the value of 0.13 eV for nanosphere-shaped metallic
contacts. It is conceivable that more realistic geometries of the
Figure 5. Current I displayed as a function of |J| for complex-valued J
when U = 0.07 eV. All the lines have their peaks at the position ℏ|J| =
U, while Δg= 0.01 eV and Δe= 0.01 eV (upper two panels), and Δg=
0.01 eV and Δe= 0.0 (bottom two panels). The bias voltage is Vbs=
4.0 eV.
Figure 6. The dipole−dipole interaction |J| in the gap of the dimer of
silver spheres of radius of R = 50 nm as a function of energy difference
between the excited and ground molecular states (curve a). The
metallic nanosystem is described through a frequency-dependent
dielectric function εm(ω) and assumed to be embedded in a host
medium characterized by its dielectric permittivity εd. Two point
dipoles D1and D2with dipole moments of 10 D are situated in the
dimer axis and directed along it in the same direction (see lower inset).
The dipoles are placed at a distance of 1 nm from each other, and the
distance between a sphere surface and the nearest dipole is 1 nm. The
upper inset shows a 2 D distribution of electric field intensity, induced
by the left dipole at 3.8 eV. The field enhancement due to plasmon
excitation inside the gap can be clearly seen. We also show the
calculation results for a single nanosphere (the right sphere in the
lower inset is absent): |J| (curve b), ReJind(curve c), −ImJ (curve d),
and the energy-transfer rate from the nearest dipole to metal γm(curve e).
Here Jinddenotes the part of J induced by a nanosphere. Curve ReJ
should be shifted below with respect ReJindon the value of the dipole−
dipole interaction in free space (−0.06125 eV). The Fano-like peaks at
3.8 eV mirror the plasmonic resonance.
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Page 5

contacts, like the bowtie antenna,26etc., will give a larger
enhancement. This issue will be studied elsewhere.
■ASSOCIATED CONTENT
*
This material is available free of charge via the Internet at
http://pubs.acs.org.
■AUTHOR INFORMATION
Corresponding Author
*E-mail: guangqili@northwestern.edu
Notes
The authors declare no competing financial interest.
■ACKNOWLEDGMENTS
Authors are supported by the Non-Equilibrium Energy
Research Center (NERC) which is an Energy Frontier
Research Center funded by the U.S. Department of Energy,
Office of Science, Office of Basic Energy Sciences under award
number DE-SC0000989 (G.L. and M.R.), the Israel Science
Foundation grant no. 1646/08, the Germany−Israel Founda-
tion, the European Research Council under the European
Union’s Seventh Framework Program (FP7/2007-2013; ERC
grant agreement no. 226628) (A.N.), the Israel−U.S. Binational
Science Foundation (A.N. and B.F.), and the Russia−Israel
Scientific Research Cooperation (B.F.).
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