Gross R., Sidorenko A., Tagirov L. Nanoscale Devices

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NOVEL IDEAS AND PRINCIPLES OF DEVICES

Negative U Molecular Quantum Dot

A. S. Alexandrov

Department of Physics, Loughborough University, Loughborough, United

Kingdom

Abstract: While the correlated transport in mesoscopic systems with

repulsive electron-electron correlations received considerable

attention in the past, and continues to be the focus of intense

investigations, much less has been known about a role of

attractive correlations in molecular nanowires and quantum

dots. Here a negative

−

Hubbard model of a d -fold

degenerate quantum dot is reviewed. The attractive electron

correlations caused by a strong electron-vibron interaction

and/or by valence fluctuations in molecules provide a molecular

switching effect, when the current-voltage (I-V) characteristics

show two branches with high and low current for the same

voltage, if the degeneracy of the dot is larger than two.

Keywords: attractive correlations, molecular junctions, switching, nano-

wires, quantum dots

Introduction

Molecular electronic junctions are structures in which single molecules or

small groups of molecules conduct electrical current between two

electrodes [1]. Combine unique I-V characteristics of molecules with the

endless possibilities of synthesis offered by chemistry, molecular devices

may revolutionize the future of the electronic industry. There is a real

progress

on that front. A number of laboratories are now testing new types

137

R. Gross et al. (eds.), Nanoscale Devices - Fundamentals and Applications, 137–151.

138 A. S. Alexandrov

of reversible switches, as well as fabricating nanowires needed to connect

the circuit elements together. Once one has the appropriate switches and

wires, the next significant step will be integrating them together into more

complex structures that perform useful functions. Hewlett-Packard

Research Labs have already constructed a 64-bit molecular memory chip

that fits inside a square micron. More recently they fabricated in

collaboration with UCLA one kilobit cross-bar molecular memory circuits

by nano-imprint lithography [2]. Consequently the actual mechanisms of

molecular switching and transport in molecular nanowires are of the highest

current experimental and theoretical value [3, 4]. A few experimental

studies [5, 6] provide evidence for various molecular switching effects,

where the current-voltage (I-V) characteristics show two branches with

high and low current for the same voltage below the conventional

threshold. This phenomenon can result from a conformational

transformation of certain molecules containing a "moving part" like a

bypyridinium ring, which changes its position if the voltage is sufficiently

high, or from the interaction of metallic leads with the molecules. Such

transformations necessarily involve a large displacement of many atoms so

this "ionic" switching is rather slow, perhaps operating on a millisecond

scale. Remarkably, reversible switching was also observed in simple

molecules (organic molecular films [6]). Molecular devices that exhibit

intrinsic switching could be the basis of future active elements of molecular

electronics. Thus further progress will depend upon finding reversible

molecules and understanding intrinsic mechanisms of their switching from

low to high current state.

Solid-state mesoscopic systems with repulsive electron correlations have

received considerable interest in the past and continue to be the focus of

intense experimental and theoretical investigations [4]. The Coulomb

repulsion suppresses tunneling for certain range of applied voltages, leading

to what is commonly known as the Coulomb blockade. Our [7, 8] and other

[9] recent studies led to a new insight into a possible bistable current state

of molecular circuits due to attractive electron-electron correlations. This

mechanism of current-controlled electronic molecular switching requires

many-particle attractive correlations, which could appear as a result of

strong electron-vibron interactions and/or valence fluctuations in molecular

quantum dots (MQDs). Here I describe the switching behaviour of MQD

caused by attractive correlations in the framework of a simple

ative U−

Hubbard model [7].

Negative U Molecular Quantum Dot 139

One mechanism that can produce a negative attractive potential in

molecular systems is a strong electron-vibron interaction. It is well known

that the lattice deformation strongly affect the interaction between electrons

in solids. At large distances electrons repel each other in ionic crystals, but

their Coulomb repulsion is substantially reduced due to the ion polarization.

However, when a short-range deformation potential and molecular-type

(e.g. Jahn-Teller) vibrations are taken into account together with the long-

range Fröhlich electron-phonon interaction, they overcome the Coulomb

repulsion in complex lattice structures [10]. The resulting interaction

becomes attractive at a short distance of about the lattice constant.

Strong electron-vibron interactions and attractive correlations are quite

feasible also in molecular nanowires and quantum dots (MQD) used as the

“transmission lines” and active elements in molecular-scale electronics.

There is a wide range of bulk molecular conductors with polaronic carriers

formed by the strong electron-phonon interaction. Since the formation of

polarons in polyacetylene (PA) was theoretically discussed [11], they were

detected optically in PA [12], in conjugated polymers such as

polyphenylene, polypyrrole, polythiophene, polyphenylene sulfide [13], Cs-

doped biphenyl [14], n-doped bithiophene [15], organic-based light

emitting diodes [16], and other molecular systems. More recently it has

been experimentally demonstrated that the low-bias conductance of

molecules is dominated by resonant tunneling through coupled electronic

and vibronic levels [17]. Conductance peaks due to electron-vibron

interactions has been seen in C

[18].

Fig. 1. Localized electrons shift the equilibrium position of the ion (

3 ). As a result

two electrons on neighboring sites

1 and 2 can attract each other.

Attractive Correlations

140 A. S. Alexandrov

Let us elaborate the physical origin of attractive correlations in solids and

molecules caused by lattice deformations. We consider a toy model of two

electrons localized on the neighboring sites 1,2 and interacting with a single

ion vibrating near other site 3 as a three-dimensional harmonic oscillator,

Fig. 1. The vibration part of the Hamiltonian in our the model is

∂



=− + ,



∂



(1)

where

is the ion mass, k is the spring constant, and u is the ion

displacement (here and further we use

h ). Electron potential energies

due to the Coulomb interaction with the ion are approximately

12 0 12

(1 )VV a

−⋅ /,ue

(2)

where

VZea

−/ is the Coulomb energy in a rigid lattice (an analog of

the crystal field potential),

a is the average distance between electron sites

and the ion, and

12,

e are units vectors connecting sites 12

and site 3 ,

respectively. Hence the Hamiltonian of the model is given by

12 1 2

()( )

ˆˆ ˆ ˆ

nn n n

∂



++⋅ + − + ,



∂



uf f

(3)

where

is the atomic level at site 1 and 2 in the rigid lattice, which

includes the crystal field,

12 12

Ze a

/fe is the Coulomb force, and

†

12 12

= are occupation number operators at every site expressed in

terms of the electron annihilation

and creation

†

operators. This

Hamiltonian is readily diagonalized by a displacement transformation of the

vibronic coordinate

u as

(

)

ˆˆ

=− + /.uv f f

(4)

The transformed Hamiltonian has no electron-phonon coupling,

12 12

()()

ˆˆ ˆˆ

ap ph

HEE V

nn nn

∂



−++ − +,



∂



(5)

Negative U Molecular Quantum Dot 141

where we used

12 12

ˆˆ

= because of the Fermi statistics. It describes two

small polarons at the atomic level shifted by the polaron level shift

=/, which are entirely decoupled from ion vibrations. The ion

vibrates near a new (shifted) equilibrium with the "old" frequency

=/. As a result of the local ion deformation, the total energy of

each polaron decreases by

since a decrease of the electron energy by

− overruns an increase of the deformation energy by

. Another

important consequence of the lattice deformation caused by two electrons is

the appearance of an effective interaction between carriers,

V , which

should be added to their Coulomb repulsion,

V ,

12ph

−⋅/.ff

(6)

When

V is negative and large compared with positive

V the full

interaction becomes negative. Our toy model allows us to elaborate its

physical origin. If a carrier (electron or hole) acts on an ion with a force

f ,

it displaces the ion by some vector

/xf. The total energy of the carrier-

ion pair is

(2 )k−/f . In case of two carriers interacting with the same ion

(see Fig.1) the ion displacement is

()k

+/xff and the energy is

12 12

(2 ) (2 ) ( )kk k−/ − / − ⋅ /ff ff. The last term depends on the scalar product

f and

f and consequently on the relative positions of the carriers with

respect to the ion. If the ion is an isotropic harmonic oscillator, as we

assume here, then the following simple rule applies. If the angle

between

f and

f is less than 2

/ the polaron-polaron interaction will be attractive,

if otherwise it will be repulsive. In general, some ions will provide

attraction, and some repulsion between polarons.

Applying the displacement transformation, Eq. (4), to a generic

“Fröhlich-Coulomb” Hamiltonian, allows us explicitly calculate the

effective interaction of small polarons in complex lattices and molecules

[19]. An estimate of the short-range attraction (called here the negative

Hubbard-

U ) yields its value c.a. 0.1 eV or larger in ionic solids. We expect

the negative

U of the same order of magnitude in carbon-based molecules.

142 A. S. Alexandrov

characteristics of quantum dots [20]. However, they cannot cause a

switching. We show here that a negative Hubbard

U of any origin provides

an intrinsic non-retarded current switching of MQD. One mechanism that

can provide the negative

U in molecular systems is the strong electron-

vibron interaction as discussed in the previous section. Even if bipolarons

are not formed in molecular quantum dots because of a short life-time of

carriers on the molecule, the attractive correlations between them still

remain. Moreover, attractive short-range correlations are feasible even

without electron-phonon interactions. They might be of a pure “chemical”

origin, as in the mixed valence compounds [21].

The exact solution of MQD with strong electron-vibron interactions

involves not only attractive correlations but also the vibron side-bands [8].

Since main physical conclusions with respect to the switching are

qualitatively similar to the results of a much simpler negative Hubbard-

model [7] we limit our consideration to this model.

Steady current

We employ the Landauer-type expression for the steady current through a

region of interacting electrons, derived by Meir and Wingreen [22] as

[]

() () ()ImTr () ()

IV d f f G

ωω ω ω ω

∞

−∞





−−Γ,





∫

)

(7)

where

{}

1(2)

( ) exp[( 2) ] 1feVT

ωω

−

=+∆//+,m T is the temperature, ∆ is

the position of the lowest unoccupied molecular orbital (LUMO) with

respect to the chemical potential.

()

Γ depends on the density of states

(DOS) in the leads and on the hopping integrals connecting one-particle

states in the left (1) and the right (2) leads with the states in MQD, Fig. 2.

Negative Hubbard- U Model of MQD

Repulsive electron correlations cause the “Coulomb blockade” in the I-V

Negative U Molecular Quantum Dot 143

Fig. 2. Schematic of energy levels of the molecular quantum dot under bias voltage

V . MQD is assumed to be (quasi)4-fold degenerate (4)d = . Switching occurs in

the voltage range

VVV<< (

2( )eV U= ∆− | | and

2eV = ∆ ) due to a

lowering of the HOMO-LUMO gap by the attractive electron-electron potential U

in the current state.

Formula (7) includes, by means of the Fourier transform of the the

molecular retarded Green’s function (GF),

()

, the electron-vibron and

Coulomb interactions inside MQD and coupling to the leads. Since the

leads are metallic, electron-electron and electron-phonon interactions in the

leads, and interactions of electrons in the leads with electrons and vibrons

in MQD can be neglected. We are interested in the tunneling near the

effective attractive potential

The attractive energy is the difference of two large interactions, the

Coulomb repulsion and the phonon mediated attraction, of the order of 1 eV

each. Hence,

U|| is of the order of a few tens of one eV. We neglect the

energy dependence of

()

Γ≈Γ on this scale, and assume that the coupling

to the leads is weak,

Γ U

|. In this case

()

does not depend on the

leads. Moreover we assume that there is a complete set of one-particle

molecular states

, where

()

is diagonal. With these assumptions

we can reduce Eq. (7) to

conventional threshold,

eV= 2∆ , Fig. 2, within a voltage range about an

144 A. S. Alexandrov

[]

012

() () () ()IV I d f f

ωωρω

∞

−∞

=−,

∫

(8)

allowing for a transparent analysis of essential physics of the switching

phenomenon. Here

Ie=Γ and the molecular DOS, ()

is given by

() ()

ρ ωω

−,

∑

(9)

where

()

is the Fourier transform of

†

() () ()

titctc

µµ







−,,

{

}

,LL is the anticommutator, ()

iHt iHt

ct ece

µµ

−

, () 1t

for 0t > and

zero otherwise.

MQD Green’s functions

The negative Hubbard-

U Hamiltonian includes the attractive electron-

electron interaction

U in the molecular eigenstate

coupled with the le

and

ht leads by the hopping integrals

†

ˆˆˆ

kkk

HU aa

nnn

µµ

ααα

εξ

′

≠

=+ + +

∑∑∑

†

()

tac Hc

αµ α µ

µα

...

∑

(10)

Here

and c

are the annihilation operators in the left ( 1)

and right

(

= leads, and in the molecule, respectively,

†

= ,

is the

energy dispersion in the leads, and

. Similar Hamiltonian was applied

to glassy semiconductors [23, 24], high

T superconductors [10, 25]

including doped fullerenes [26], and mixed valence compounds [21].

We use the perturbation theory with respect to the hopping integrals,

neglecting any contribution to the current other than

, but keeping all

orders of the negative Hubbard

Terms of higher orders in

cannot

change the gross I-V features for any voltage except the narrow transition

regime from one branch to another. Applying the equations of motion for

Negative U Molecular Quantum Dot 145

the Heisenberg operators () ()

ct t

and ()

we obtain a set of coupled

equations for the molecular GFs as

()

() (0)

dG t

itn

µµ µ

−

≠ ≠...

∑

∏

() ( 1)

[(1)]() ()

NUGtUG t

µµ

+− + ,

(11)

where

†

() () ()nt ctct

µµµ

= is the expectation number of electrons on the

molecular level, and the

N -particle retarded Green’s function is defined as

() †

() () () ()

Gt it ct tc

µµµ

µµ µ

−













≠ ≠...



=−Θ ,

∑

∏

(12)

for

2N <∞ and

(1)

() ()

Gt t

≡.

For the sake of analytical transparency we solve this system for a

molecule having one

d -fold degenerate energy level with

In this

case the set is finite, and it is solved using the Fourier transform. The one-

particle GF is found as

(1)

()

rU i

−

−+

∑

(13)

where

=+ , (0)nn

=, and

(1)

() (1 )

(1)

rdr

Zn n n

rd r

−

=−.

!−−!

This is an exact solution with respect to correlations which satisfies all sum

rules.

The electron density

()nt

obeys the rate equation, which is obtained by

using the equations of motion as

(1)

()

2()

dn t

tImA t

αµ αµ

∑

(14)

where

Gross R., Sidorenko A., Tagirov L. Nanoscale Devices - Fundamentals and Applications

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