Roi Baer and Daniel Neuhauser- Many-body scattering formalism of quantum molecular conductance

8/3/2019 Roi Baer and Daniel Neuhauser- Many-body scattering formalism of quantum molecular conductance

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Many-body scattering formalism of quantum

molecular conductance

Roi Baer a,*, Daniel Neuhauser b

a Institute of Chemistry and Lise Meitner Center for Quantum Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israelb Department of Chemistry and Biochemistry, University of California at Los Angeles, Los Angeles, CA 90095-1569, USA

Received 5 April 2003

Abstract

A general formalism is developed for the theory of electronic current through a molecule attached to macroscopic

electrodes in a given bias. The current is calculated in a non-perturbative many-body formalism. We obtain an intu-

itively appealing formula. We identify two distinct limits: the Kubo linear response formula valid in the small bias

regime and the Landauer formula valid for non-interacting electrons. Finally, we develop a new correlation function

formula, which is more amenable to numerical computations and discuss the application of absorbing potentials in the

electrodes as a possible means of calculation.

2003 Elsevier Science B.V. All rights reserved.

Passing electric currents through molecules

connected to electrodes is the essence of single

molecule electronics [1]. This is a relatively new

basic science with the potential of considerable

technological impact, due to new effects [26]. It is

apparent, following the recent literature, that while

experimentalists in this discipline are still strug-

gling to define reproducible procedures and stan-dards of measurement, theory is also lacking

rigorous results and benchmarks.

Perhaps the most widely accepted theory of

molecular conductance relies on the fundamental

work of scientists from the field of mesoscopic

conductors [7]. However, at the molecular level the

situation seems quite different. Many effects that

can be ignored in mesoscopic physics may become

dominant on the molecular scale. It has been es-

tablished that the charge distribution [8,9] and

therefore molecular structure is exceedingly im-

portant. The role of electron correlation is im-

portant in molecular conductors because of the

strong inhomogeneities in the external potential.Such effects can modify the conductance consid-

erably [10]. Also, these effects may lead to new

phenomena such as finite size effects and disorder

[11].

The purpose of this Letter is to take a step

forward towards establishing a theory of molecu-

lar scale electronics that serves as a foundation for

developing new approximations and models. We

present a theory of conductance, encompassing

electronelectron interaction and correlation, as

Chemical Physics Letters 374 (2003) 459463

www.elsevier.com/locate/cplett

* Corresponding author. Fax: +972-2-651-3742.

E-mail address: [email protected] (R. Baer).

0009-2614/03/$ - see front matter 2003 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0009-2614(03)00709-7
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well as electronnuclear interaction. Furthermore,

our theory is formulated for a general setup of

electrode voltages so it is not limited to small

voltages.Our approach combines several concepts from

different fields. First, we adopt Landauers point of

view that conductance is quantum mechanical

transmission [12]. Next, we are motivated by re-

active scattering theory concepts and techniques,

in particular the currentcurrent correlation ap-

proach of Miller [1315] and the negative imagi-

nary potential approach of Neuhauser and Baer

[1618]. Taking these elements via a rigorous

route, we arrive at formulae which are general and

taking full account of electron correlation. In the

non-interacting electron limit we obtain Landauers

conductance formula [12]. In the limit of zero bias

we obtain Kubos currentcurrent correlation

function formula for conductance [19]. The for-

malism presented is formally equivalent to a

Greens function description of the conductivity of

Meir et al. [20] in which the unknown is the full

Greens function (see Fig. 1).

We treat the electrodes as black-body cavities

with respect to their ability to absorb and dissipate

electrons. The small amount of charge that escapes

in the form of electric current does not affect thisequilibrium. The state of each metallic electrode is

characterized by a given chemical potential ll and

a temperature (for simplicity, we assume the same

temperature for all electrodes, so bl b). We ne-glect any currentcurrent interactions and limit

our discussion to cases where magnetic fields are

negligible.

The electric current entering electrode l is de-

fined by

^

IIl qe_

NN^

NNl qei

h ^

HH;^

NNl: 1

Here, qe is the electron charge, ^HH TT UUVE VW is the total Hamiltonian of the system,composed of kinetic energy TT, electronelectron

repulsion UU and the interaction of the electrons

with the nuclei of the electrodes VVE and the nuclei

of the wire VVW. We first assume fixed nuclei, al-

though we mention later that the formulation ex-

tends to include nuclear vibrations. We will also

refer to the electrodes Hamiltonian HE ^HHVVW. The electrodes Hamiltonian has an important

property, in that it does not allow the electrons to

move from one electrode to another, because of a

barrier (in the absence of the wire) through

which they only undergo negligible amounts of

tunneling.

Next, we assume that the electrons do not in-

teract strongly in the electrodes. More precisely,

we assume that deep within the electrodes, the

electrons are not strongly interacting, so that one

can define asymptotic states associated with elec-

tron scattering.

The number of electrons in electrode l is

^NNl Xs

Zhlrw

ysrwsr d

3r; 2

where hlr is defined as [14]

hlr 1 r is in electrode l;0 otherwise

&3

and we sum over both spins. It is clear that all the^NNl commute: Nl;Nl0 0 for any l and l0. How-

ever, the electrodes do not cover space and there-fore we may not assume thatP

l Nl is the total

number of electrons in the system. In particular,

when currents start to flow, there can be electrons

within the wire NW.

The zero-order Hamiltonian HE does not let

electrons tunnel appreciably from one electrode to

the next. It therefore conserves the number of

electrons in each electrode HE;Nl 0. Thisproperty would be important in the derivations

below.

Fig. 1. Schematics of the molecular wire. L and R are the

electrodes, composed of atomic cores in their equilibrium po-

sition. The vertical dotted line is the surface defining the current

IR. The dotted curve describes the atomic cores of the wire. The

region of the wire must be large enough so that the electrodes

are decoupled when there is no wire.

460 R. Baer, D. Neuhauser / Chemical Physics Letters 374 (2003) 459463


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Other preliminaries are in order now. First, we

assume is the electrodes are in some sense infinite.

Thus the Mller operator [21]

X limt!1

eiHt=heiHEt=h 4

is well defined. It operates on many-electron

wavepackets as follows: first, a wavepacket at

t 0 is transformed into its primitive parent bypropagating it back in time to t! 1 using theuncoupled Hamiltonian ^HHE; next, the primitive

parent is propagated back forward to t 0 by thecoupled Hamiltonian ^HH. This combination trans-

forms any initial scattering eigenstate ofHE into a

scattering state of ^HH with well defined asymptotic

currents. The following intertwining relations holdthen [22]:

HX XHE; ^NNX XXl

^NNl: 5

Actually, the second term in Eq. (5) is not a

strict equality; there are contributions of very high

energy states of HE associated with electrons lo-

calized on the wire. However, the contribution of

these states is negligibly small due to the Boltz-

mann averaging at finite temperature.

We can now write the expression for the cur-rent entering electrode R, IR, which is a thermal

weighted average of the expectation value of ÎIR at

t! 1. In other words, we average over all scat-tering eigenstates ofHE (where for each scattering

state there are several electrons that come from

one electrode, several from another, etc.). This

yields a trace under the assumption that the

scattering states are complete, i.e., that there are

no bound states in the wire. The corresponding

procedure is summarized in the following equa-

tion:

IR Z1Tr eb

^HHE eb~ll~NNNNXy

ÎIRX

h i; 6

where ZQ

l Zl Treb ^HHE eb~ll

~NNNN is the elec-

trode grand canonical partition function, and

~ll ~NN P

l llNl. In brief, the expression says that

we should thermally average over all states, asso-

ciate an electrode-dependent chemical potential

with each electron, and calculate the eventual flux

from such a state.

One important feature about this formula is

that since the scattering states span the full Hilbert

space of the problem, we are free to include a term

lWNW in the ~ll ~NN sum, for any value oflW. Thisfollows from two facts: first, NW commutes with

Nl, and second, it yields zero for any of the scat-

tering states of HE. The freedom in the choice of

the chemical potential of the wire will prove con-

venient in the discussion below, for example when

we consider the case of near-zero bias when all the

chemical potentials are equal.

Using the intertwining relation equation (5)

in the expression, we obtain: IR Z1Treb~ll

~NNNNXy

ÎIR eb ^HHX. The basic expression for the currentinto electrode R then becomes

IR Z1 lim

t!1Tr ebHeb~ll

~NNNNÎIRt

h i; 7

where ÎIRt eiHt=hÎIRe

iHt=h. This is the basic formula

for quantum conductance. The interpretation is

simple and intuitively appealing: the steady-state

current into an electrode R is the expectation value

of the corresponding current operator at t! 1,thermally and chemically averaged over all possible

scattering states.

Note that we could have used the Hermitian

conjugate intertwining relation HEX

y

X

y

H andhave the eb ^HH trade places with e~ll

~NNNN in Eq. (7).

Thus, we see that the order of placement of op-

erators in the formula does not affect the final re-

sult. A similar observation was made by Miller [14]

in the context of chemical reaction rates.

Eq. (7) is the general equation for describing the

electronic current in molecular devices. It formally

takes into account all electronelectron correla-

tions.

Let us study some limiting cases. First, consider

non-interacting electrons. The trace formula, Eq.

(7), immediately simplifies in the standard Fermi

Dirac way

IR 2Xl

limt!1

Tr1 1h

ebHll1 ^NNlÎIRti; 8

where the trace Tr1 is over single electron states.

Here ^NNl simply acts as a projection operator on

electrode l and the factor of 2 accounts for spin

degeneracy. Note that Eq. (8) is equivalent to the

Landauer formula [7,23]

R. Baer, D. Neuhauser / Chemical Physics Letters 374 (2003) 459463 461


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IR 2qe

h

Xl

ZFDbE liTlRE dE; 9

where h is Plancks constant, FDx 1 e

x

1

and TlRE is the cumulative transmission proba-bility, given by Millers formula [24] TlRE h=qe limt!1 Tr1dE ^HH ^NNlÎIRt.

Next, consider a different limiting case, of in-

teracting electrons at zero bias ll l for all l, andwe also include lW. In that case the chemical po-

tential term simplifies to ~ll ~NN lN. This imme-diately implies that the total current should be

zero. Indeed, since ^HH and ^NN commute, Eq. (7)

simplifies to

IR Z1 limt!1

Tr eb^HHl

^NNÎIRt

h i: 10

But ÎIR is a commutation relation involving H (Eq.

(1)) so the trace evaluates to zero as it should.

Next, consider the conductance GlR oIR=oll atzero bias. From Eq. (7)

Gll0 llR bZ

1 limt!1

Tr eb^HHlN ^NNl

h tÎIR

i: 11

This can be simplified. Following Miller et al [13],

we write the t! 1 limit as an integral:

limt!1 ct R1

0_cct dt c0, obtaining the well-

known Kubo result, derived from linear-response

theory, that the conductance is the currentcurrent

correlation function

Gll0 llR bZqe

1

Z10

Tr ebHlNIlIRt

dt: 12

Once again, it is seen that our general formula

Eq. (7) is compatible with known special limits.

Going back to the general formalism, we now

discuss how the formalism can be used in an actual

calculation. While in the one-particle case theGreens function formalism is useful, in the general

case, we propose complex absorbing potentials for

limiting the grid size [15,18]. Before introducing

these, let us convert Eq. (7) into a correlation

function, as this will facilitate the calculations. We

transform the limit of t! 1 to a definite integral

IR qeZ1

Z10

Tr eb^HHeb~ll

~NNNN=2 ^JJeb~ll

~NNNN=2ÎIRt

h idt;

13

showing that the current expectation value is a

correlation function between current entering the

electrode R and a weighted current

^JJ 2Xl

sin hbll=2ÎIl: 14

In the course of deriving Eq. (13), we used the

following relation, to be proved in a more detailed

account, applicable for Fermions:

ea^NNlÎIle

a ^NNl coshaÎIl sinhaÎIl; ^NNl: 15

Now that the current formula is a correlation

function, we add a negative absorbing potential to

each electrode. The role of this potential is to ab-

sorb any out-going flux from scattering events.

The imaginary potential should be placed deepenough inside the electrodes so that it absorbs only

electrons that have a negligible chance to be re-

flected back into the molecular wire. Thus the

non-Hermitian Hamiltonian ^HHc ^HH iP

l CCl is

formed. This Hamiltonian induces the following

non-reversible Heisenberg evolution of any ob-

servable ÂA:

ÂAt eiHc t ÂAeiHct: 16

The electrons in each electrode are eventually

consumed, soR1

0

_

NN^NNt dt

^NNl0, thusZ1

0

ÎIRt dt 2

Z10

^NNRCCRt dt ^NNR0: 17

Using this, it is possible to obtain the following

expression:

IR qeZ1

Z10

Tr ebHeb~ll~NNNN=2 ^JJeb~ll

~NNNN=2 ^NNRCCR

h it

h idt:

18

Practical calculation of these expressions or of

Eq. (12) could proceed in several directions.

Quantum Monte Carlo methods, such as Auxil-

iary-Field method [25] or semiclassical approaches

can be used, in principle. Alternately, usual

Quantum Chemical approaches, such as configu-

ration-interaction methods, can be deployed.

These methods would be quite limited in the

number of electrons they can simulate if one

wanted a fully correlated description; however,

462 R. Baer, D. Neuhauser / Chemical Physics Letters 374 (2003) 459463


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with proper construction of a HartreeFock or

density functional-like ground-state made of scat-

tering states [10,26], it is possible to design a

multiple-excitation approach that can in principlehandle a large number of electrons, as will be

shown in a future work.

Finally, we remark that the same formalism is

applicable even when we relax the clamped-nuclei

assumption and include nuclear motion. The only

difference in formulation is that the trace has to be

extended to include also all possible states of the

nuclei.

Acknowledgements

We thank M. Ratner, E. Rabani and R. Kosloff

for useful discussions. This work was supported by

the Israel Science Foundation founded by the Israel

Academy of Sciences and Humanities, the USA

National Science Foundation and the Petroleum

Research Fund.

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Roi Baer and Daniel Neuhauser- Many-body scattering formalism of quantum molecular conductance