Tamar Seideman- Current-driven dynamics in quantum electronics

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Current-driven dynamics in quantum electronics

TAMAR SEIDEMAN

Department of Chemistry, Northwestern University

2145 Sheridan Road, Evanston, Illinois 60208-3113, USA;

e-mail: [email protected]

(Received 12 March 2003; revision received 22 April 2003)

Abstract. We suggest that inelastic electron tunnelling via molecular-scaleelectronics can induce a variety of fascinating dynamic processes in the

molecular moiety. These include vibration, rotation, intermode energy flowand reaction. Potential applications of current-induced dynamics in molecular-scale devices range from new forms of molecular machines and approaches toenhancing the conductivity of molecular wires, to new directions in nanochem-istry and nanolithography. Understanding the molecular properties thatencourage current-triggered dynamics is relevant also to the design of devicesthat would be guaranteed to remain stable under current. We discuss thequalitative physics underlying current-driven dynamics, briefly sketch a theorydeveloped to explore these dynamics, describe a few examples from recent andongoing numerical work and note avenues for future research.

1. Introduction

Considerable theoretical effort has been devoted in recent years to the problem

of molecular-scale electronics. This research [1–17] has been fuelled by the

fascinating physics of conductance at the molecular level, by the anticipation for

technological applications and by the development of experimental techniques to

produce and characterize molecular-scale electronics [18–36]. Reviews of the

theoretical and experimental aspects are given in [17] and [21] respectively.

A recent special issue on transport in molecular wires [37] provides a useful

up-to-date overview of this exciting field. The combination of theoretical and

numerical work in this area has yielded substantial insight into the response of the

conductivity to several general attributes [2–8], as well as quantitative predictions

of the conductance of a variety of specific systems [10–15].

A problem of significant fundamental interest, which may also carry practical

benefit and which is only starting to be explored, is the effect of the current on the

molecule. Recent theoretical work [38], supported by indirect experimental

evidence, suggests that inelastic tunnelling via molecular-scale devices can induce

a variety of dynamic processes in molecular heterojunctions. These include

vibration, rotation, intermode energy flow and reaction.

The mechanism through which a tunnelling current at the mild conditionstypical of molecular electronics research can trigger large-amplitude motion of a

massive molecular backbone is simple and general. For applications such as

journal of modern optics

2003, vol. 50, no. 15–17, 2393–2410



and at most weakly distance dependent. By contrast, direct tunnelling falls

exponentially with increasing distance. Resonance conductance is often inelastic.

Provided that the initial and resonant electronic states are displaced in equi-

librium, the system evolves during the resonance lifetime. Upon electronic

relaxation the molecule carries internal excitation which may suffice to triggerchemical dynamics before relaxing to the initial vibrational state [38].

Current-triggered dynamics in nanosystems carries potential implications and

applications in several fields.

(i) New forms of molecular machines. The area of artificial molecular

machines [39] has been evolving rapidly for nearly two decades, leading

to laboratory-made turnstiles, shuttles; switches and rotors, driven by

light, chemistry or electrochemistry. On the one hand, this work points

to a host of potential technological applications [39]. On the other

hand, it reveals the need for new forms of molecular machines, whichcould be addressed individually, thus eliminating the ensemble average

that is inherent to the solution phase. Current-triggered dynamics in

molecular heterojunctions could make a route to that end. We envisage

using resonant inelastic current to deposit energy selectively into a

given mode and hence to induce directed motion of the molecular

component.

(ii) Dynamic modification of the conductivity of molecular wires. Often in

molecular wires the conductivity depends sensitively on the structure of

the molecule [22]. An example is the biphenyl family, where the

conductivity changes by over an order of magnitude as the torsion anglebetween the rings varies between zero and p/2. In these systems, torsion

may be expected to ensue resonance tunnelling since the ionic states of

biphenyl differ substantially from the ground neutral state in equilibrium

configuration, specifically along the torsion angle. Current-triggered

large-amplitude vibration may likewise modify the conductivity of wires.

An experimental study of electron transfer from a metal surface to

a molecule [40] found the electron transfer probability to be greatly

enhanced by excitation of the molecule to high vibrational levels. Clearly,

one can envisage systems where molecular motion inhibits the conduc-

tivity, because the equilibrium is ideally configured for electron transferto take place.

(iii) Nanochemistry. An intriguing possibility is that of using a scanning

tunnelling microscope tip to induce surface reactions that do not occur

spontaneously and to study them on a single-molecule, site-specific level.

One class of systems of interest is the reactions of the common gas sources

used for thin-film growth on, and doping of, Si by chemical vapour

deposition (NH3 for N, PH3 for P, and AsH3 for As, for instance). The

thermal reactions currently used by advanced growth and doping tech-

nologies are not ideal to that end, as is evident from the broad disparity

between observations of different laboratories [41] and from the variety of

undesired effects reported [42]. The current-triggered approach may

2394 T. Seideman



The design of current-immune devices relies on understanding the

molecular properties that encourage current-triggered dynamics.

(v) Opportunities in molecule-scale lithography. Nanolithography based on

current-induced desorption of H atoms from Si(100) is an established field

with a variety of applications [43–47]. Organic molecules may offeradvantages. In the adsorbed state conjugated organics support low-energy

resonances (lying close to the Fermi level), corresponding to much lower

voltages than those used for H desorption from silicon; hence smaller tip–

surface separations and improved resolution. Further, as discussed in [48],

the properties of organic adsorbates can be tuned via functional group

substitution, providing a means of control over the process.

In the next section, we outline a recently developed theory of current-driven

dynamics in molecular devices. Section 3 describes several applications and the

final section concludes with an outlook to future research.

2. Theory and interpretation

We consider nuclear dynamics that are triggered by an inelastic resonance-

mediated tunnelling event. In the tip–adsorbate–substrate environment, this

would be a surface reaction induced by electron transfer from the tip to the

surface or vice versa via an adsorbate-derived resonance. In the molecular wire

environment, we envisage directed motion of the molecular moiety, induced by

tunnelling from one electrode to the other via a resonance of the molecule þ

electrodes Hamiltonian, as modified by the bias voltage. Formally the two prob-lems are equivalent and can be studied within the same theoretical framework.

Numerically, however, they differ substantially, as outlined below.

The complete Hamiltonian is written as

H ¼ H e þ H N þ H eÀN, ð1Þ

where H e describes the electronic dynamics and allows for hopping of electrons

between the two electrodes via the adsorbate orbital, H N is a system-specific

nuclear Hamiltonian and H e–N is the non-adiabatic interaction coupling the

electronic and vibrational motions. It is convenient to expand the Hamiltonian

in eigenstates of H e, such that coupling between the extended electronic states of the two electrodes and the discrete state localized on the adsorbate is taken into

account non-perturbatively. In this representation,

H e ¼X

cy c þ

X

cyc, ð2Þ

where ji and ji are stationary electronic states on the two electrodes and and

, are the corresponding energies. The coupling is written in the H e representa-

tion as,

H eÀN ¼ hNðQÞX;

ðhjr i hr ji cyc À hnriÞ, ð3Þ

Current-driven dynamics in quantum electronics 2395



where hN (Q) is a function of the nuclear coordinates Q, jri denotes the resonant

orbital and hnri is its equilibrium occupancy.

Assuming that the non-adiabatic interaction can be described within first-order

perturbation theory, we express the reaction rate w as

w ¼ 22p

~

X,

Xv

f i ðÞ½1 À f f ðÞjhv, jH eÀNjv0, ij2ð À À Þ, ð4Þ

where jv0, i ¼ jv0iji, jv, i ¼ jviji, jv0i is the initial vibrational state, a bound

eigenstate of the nuclear Hamiltonian, and jvi is a final, bound or free eigenstate of

the same Hamiltonian. We denote by the energy transferred from electronic into

vibrational modes, ¼ À , and the subscript to v serves as a reminder that

the energy of the final state is restricted through energy conservation to above

the initial state energy. In equation (4), v0 denotes collectively the set of quantum

indices specifying the parent state. In the case of a bound–bound process in the

nuclear mode (such as intermode energy flow, rotational or vibrational excitation

or a surface exchange reaction), v has a similar significance. In the case of a

bound–free reaction in the nuclear mode (such as a desorption or a dissociation), vdenotes the scattering energy and the quantum indices specifying the internal state

of the free product. In the latter case, the v summation in equation (4) implies

summation over the discrete indices and integration over the energy. The f i ( f )(E )

in equation (4) are Fermi–Dirac distribution functions.

The assumption of Q-independence of H e allows us to reduce equation (4) to a

form that provides better insight and is substantially easier to handle numerically.

To that end we first separate the vibronic matrix element in equation (4) intointerdependent electronic and nuclear factors and introduce a complete set of

eigenstates of the ground nuclear Hamiltonian

I ¼X

v

jvi hvj þX

n

ð d jnÀihnÀj, ð5Þ

where jvi are bound and jnÀi are incoming wave scattering states of H N. The

collective index v denotes the bound-state vibrational indices and n specifies

the internal quantum numbers of the free state. With equations (3) and (5), the

transition matrix element in equation (4) assumes the form

hv,jH eÀNjv0, i ¼ hjr ihr jihvjX

v

jvihvj þX

n

ð d jnÀihnÀj

!hNðQÞjv0i

hjr ihr jihvjCi, ð6Þ

where we denote by jCi a superposition of eigenstates of H N:

jCi ¼X

v

Avjvi þX

n

ð d AðnÞjnÀi, ð7Þ

with expansion coefficients Av ¼ hvjhNðQÞjv0i and AðnÞ ¼ hn

À

jhNðQÞjv0i, deter-mined by the current-triggered excitation event.

The physical picture suggested by equations (6) and (7) is simple and general.

2396 T. Seideman



vibration, rotation, intermode energy transfer or a surface exchange reaction.

The scattering projection of jCi translates into bound–free dynamics, such as

desorption or dissociation.

Considering first the case of bound–free dynamics, we relate the expansion

coefficients in equation (7) to observable probabilities by introducing a set of eigenstates jni of an asymptotic Hamiltonian H N À V N through

jnÀi ¼ jni þ GNV Njni: ð8Þ

In equation (8), GN is the Green operator corresponding to the Hamiltonian

H N and V N is the potential energy operator in that Hamiltonian. In terms of the

asymptotically observed states, equation (7) is written as

jCi ¼X

v

Avjvi þX

n

ð dBðnÞjni, ð9Þ

where BðnÞ ¼P

n0Ð

d0

hnj0

n0À

iAð0

n0

Þ is the amplitude to observe the state jni atasymptotically long time. Formally, the B(n) are readily expressed in terms of the

scattering matrix corresponding to H N. Numerically, however, it is more con-

venient to solve for the nuclear dynamics in the time domain. As shown for

example in [50],

jCðtÞi ¼X

v

AvðtÞjvi þX

n

ð d AðntÞjnÀi

¼

Xv

AvðtÞjvi þ

Xn

ð d BðntÞjni, ð10Þ

with the initial condition jCðt ¼ 0Þi ¼ jCi: The information of interest,

BðnÞ ¼ limt!1 ½eiEt=~ BðntÞ ¼ limt!1 hnjCðtÞi, is determined by propagating

the wave packet on the neutral ground surface to a long time and projecting it

on to the jni. The initial condition jCi is determined by the resonance scattering

event and contains the information about the resonance state structure and

lifetime.

Substituting equations (6) and (9) in equation (4), we express the bound–free

reaction rate as

wðV b; Þ ¼

ð d P reacð

; ÞW excð, V b

; Þ, ð11Þ

where

P reacð; Þ ¼X

n

jBðnÞj2 ¼X

n

limt!1

hnjCðtÞi

2

, ð12Þ

W excð, V b; Þ ¼ 22p

~

X,

f i ðÞ½1 À f f ðÞjhjr i hr jij2ð À À Þ, ð13Þ

and we indicated explicitly the dependence of the rate on the bias voltage V b and

its parametric dependence on the resonance lifetime . Equations (11)–(13)formulate the reaction rate in terms of the current that drives the reaction;

Wexc(, Vb; ) is seen to be the resonant component of the transmission rate




factor, P reac(; ), is a normalized probability, describing the reaction probability

per resonance excitation. Both conceptually and numerically, it is convenient

to express equation (13) in terms of projected densities of states as

W excð, V b; Þ ¼ 2

2p

~ ð

d f i ðÞ½1 À f f ð À Þi ðÞ f ð À Þ, ð14Þ

where

i ðE Þ ¼X

jhjr ij2ðE À Þ ð15Þ

is the density of electronic states (DOS) in the initial state, projected on to the

resonance orbital, and an analogous expression holds for f (E ).

The problem of bound–bound dynamics in the nuclear modes is addressed

along similar lines, the rate taking a discrete form (section 3.2). Here, however,

internal relaxation needs to be taken into account, except in cases where thedriving rate can be assumed fast compared with the relaxation rate.

3. Applications

In this section, we describe several applications of the theory outlined in

section 2. We start in section 3.1 with a discussion of scanning tunnelling

microscopy (STM)-triggered reactions. In section 3.2, we turn to the problem

of current-induced dynamics in single-molecule transistors.

3.1. Scanning tunnelling microscopy-triggered nanochemistry

Two experimental curves of yield versus voltage are shown as symbols in figure

1. Figure 1 (a) shows the results from [51] for desorption of H atoms from H-

passivated Si(100) whereas figure 1 (b) gives the data from [52] for desorption of

C6H6 from Si(100). The two reactions differ substantially in detail but share

several features. Both experiments [51, 52] provide evidence for an electronic

mechanism; the large tip–surface distance precludes STM manipulation mechan-

isms that rely on the electric field and field gradient at the surface or on chemical

contact. The temperature dependence precludes thermal desorption. The sigmoi-

dal shape of the voltage dependence in both figure 1 (a) and figure 1 (b) should be

noted. These features are common also to the rate versus voltage curves of [53–55],

pertaining to the rotation of DCCD on Cu(100) [53], the rotation of CCD on

Cu(100) [54], and the displacement of Si adatoms on Si(111)(7 Â 7) [55]. An

interesting question in connection with figure 1 (a) is the lifetime of the underlying

resonance, this problem being relevant to H-atom nanolithography with a rich

variety of proven and projected applications [43–46]. Two remarkable features of

figure 1 (b) are the sharp voltage dependence and the large yield observed under

mild conditions; under comparable conditions rates about five orders of magnitude

lower were reported for the STM-triggered resonance-mediated desorption of CO

from a Cu surface [56]. The former question is addressed in section 3.1.1 and thelatter in section 3.1.2.

We begin this section by exploring the qualitative physics responsible for the

2398 T. Seideman



be treated as a perturbations compared with its strong interaction with thesubstrate continuum. Under these conditions, the projected DOSs in equation

(14) can be replaced by those of the uncoupled tip and substrate–adsorbate

complex. Subject to the assumption that the resonance is isolated and the DOSs

of the tip and the bare substrate are almost constant on the energy scale of the

resonance width, the product of DOSs in equation (14) reduces to a Lorentzian

function . In the low-temperature limit the Fermi distribution functions are

approximated by step functions, with which the integral over is analytically

soluble. One finds that

W excð, V b; Þ / tanÀ1 E F þ eV b À À r

À=2

À tanÀ1 E F À r

À=2

, V b > 0,

tanÀ1 E F À r

G=2

À tanÀ1 E F þ eV b þ À r

G=2

, V b < 0,

ð16Þ

where E F is the Fermi energy, r is the resonance energy and G ¼ ~ = is the

resonance width. The closed-form approximation of equation (16) makes explicit

the physical content of the electronic function in equation (11). By contrast with

an electron beam, a scanning tunnelling microscopy is a broad-band source of electrons, with an energy band of the order of e|V b| À . The excitation function

Wexc(, Vb; ) measures the fraction of the current that is transferred via the

Figure 1. STM-triggered resonance-mediated desorption. (a) Desorption of H atomsfrom Si(100)2 Â 1:H: (.), (m), experimental data of [51]; (—), (– – –), theoreticalresults of section 3.1.1. (b) Desorption of C6H6 from Si(100): (.), experimental data of [52]; (—) theoretical results of section 3.1.2. The error bars are omitted from both setsof experimental data.




it suggests the possibility of extracting resonance lifetimes through fit of equa-

tion (11) to experimental data. This invitation is taken up in sections 3.1.1 and

3.1.2 below.

3.1.1. H-atom desorption from Si(100)2 Â 1:H. Electron-energy-loss spectros-

copy of Si(100)2 Â 1:H showed a peak with a maximum at 8 eV and an onset at

6 eV [57]. Calculations [58] identified this feature as the ! * transition of the

SiÀH bond, and direct optical excitation experiments with 7.9 eV light [59]

confirmed this assignment. The experimental results of [51] shown as full codes

and full triangles in figure 1 (a) provide clear evidence for resonance-mediated

desorption via the * state of the H–Si complex.

Potential energy curves for motion perpendicular to the surface in the ground

and the excited * states of H/Si(100) were constructed in [47] based on ab initio

electronic structure calculations. These curves are used to compute the energy-resolved probability P reac(, ) of equation (12) using the method discussed in [60].

Figure 2 shows the probabilities obtained as a function of the electronic-to-

vibrational energy transfer and the resonance lifetime . P reac(, ) is seen to be

2400 T. Seideman



a narrowly distributed function of the scattering energy À E B (E B being the

binding energy), peaked at about zero. As the lifetime increases, the distribution

broadens but throughout the range of lifetimes relevant for the H/Si(100) system

(vide infra) the desorbate energy distribution remains sharply peaked with width

below 0.3 eV. The implications of figure 2 are discussed below.Figure 3 illustrates schematically the energetics of the W-tip–Si:H-surface

junction. Since the desorption experiments [43, 45, 47, 51] considered here are

performed on H-terminated surfaces, the dangling bonds, which significantly affect

the DOS of the clean Si(100) surface, do not play a role and 0s is better

approximated as the density of unoccupied states of bulk Si than as that of the

clean surface. This function, as computed within the density functional theory

(DFT), is superimposed on the energy scheme of figure 3. Our results agree with a

previous calculation of the bulk Si DOS [61]. The tip DOS is approximated as

energy-independent (see [60] for a discussion of the range of validity of this

approximation) and the lifetime of the SiÀH * state is determined by incorporating

the dynamic simulation in a least-squares fit loop to the data of [51]. The solid curve

of figure 1(a) gives the numerical desorption yield thus obtained. The lifetime

determined is % 1.2 fs, essentially independent of the substrate temperature.

The H-atom desorption problem is an example of an effectively one-

dimensional reaction; excited-state motion is along the ground-state reaction

coordinate, the transient excitation event deposits energy into the reaction mode,

and intermode energy transfer does not take place. More interesting are multi-

dimensional processes, where energy is initially deposited in an internal mode and is

subsequently transfered to the reaction mode, owing to efficient mode coupling.The latter case is exemplified by the C6H6/Si desorption problem, discussed in the

next section.

Figure 3. A schematic illustration of the energetics of a W-tip–Si(100):H-surface junction. Superimposed is the electronic DOS of bulk Si. The position and width of the s ! s

Ã resonance, as obtained from a fit of the theoretical results to experimentaldata (see the text for details), are indicated by the broken curve. We note that thesample work function is 4.91 eV. (Reproduced with permission from [60].)




of electronic states of C6H6/Si(100), indicating that a localized state is accessed in

the STM-triggered tunnelling event. The peak corresponds to the binding energy

of a feature observed around 2.3 eV in photoemission studies of benzene/Si(100)

and attributed to the highest occupied molecular orbital of the adsorption complex

[62]. These observations suggest that the desorption process summarized in figure1 (b) is mediated by a positive-ion resonance that is predominantly localized on the

adsorbate.

In [52, 63] the physical considerations involved in the construction of an

orthogonal coordinate system and the calculation of ab initio potential energy

surfaces for the neutral and ionic states of C6H6/Si are discussed. Below we

provide only a qualitative discussion, assisted by the contour plots of figure 4. Here

Z and X are dimensionless coordinates whose mathematical definition [38] is not

essential for the present discussion. In terms of physical content, Z takes the role of

a desorption coordinate whereas X is a collective mode that accounts for internal

2402 T. Seideman



motion; predominantly into-plane motion of the carbon backbone accompanied by

relaxation of the Si dimer that supports the ring.The potential energy surfaces of figure 4 are employed to study the nuclear

dynamics triggered by the resonance tunnelling process. Figure 5 describes the

evolution in the neutral state subsequent to the resonance tunnelling event through

plots of the expectation values of X and Z in the wave packet. The initial condition

C is determined through 25 fs evolution in the resonant state. Desorption is

initiated by large-amplitude vibrational motion across the chemisorption well.

Energy initially deposited in the X mode shuttles periodically between the coupled

modes for a brief period, after which the oscillation amplitude of hX i decays and

desorption is under way, as indicated by a rapid increase in hZ i. The inset of figure

5 shows a snapshot of the wave packet at 750 fs, illustrating vibrationally mixedmotion in the chemisorption well, some probability density in a weak physisorp-

tion site [63], located just beyond the transition state, and a portion of the wave

packet propagating along the exit valley.

With much shorter residence times (5–20 fs) in the resonance state, the

tunnelling event just barely vibrationally excites the system. It undergoes low-

amplitude vibrations in the well but remains bound. With much longer times in

the charged state (30–40 fs) the system explores highly repulsive regions of the

neutral potential energy surface, the amplitude of the short-time vibrational

motion in the ring mode grows and desorption is faster [52, 64].

The time-evolving wave packet is employed to extract the energy-resolved

desorption probability using the approach of [63]. As was found for the H/Si(100)

Figure 5. Nuclear dynamics leading to desorption of benzene molecules from a Si surface:expectation values of X (—) and Z (– – –) in the evolving wave packet, subsequent to25 fs evolution in the ionic state. The inset shows a snapshot of the wave packet 750 fsafter relaxation, illustrating highly vibrationally mixed motion in the chemisorptionwell, some probability density in the physisorption site and a portion of the wavepacket propagating along the exit valley. (Reproduced with permission from [64].)




resonance energy of the threshold in the yield versus voltage curve predicted by

equation (16) is thus roughly 0.3 eV, consistent with observation (figure 1 (b)).

The solid curve in figure 1 (b) gives the results of a numerical fit of equation

(11) to the observed yield. Since our fit is based on a small number of data points,

and since the experimental error bars are of the order of the data, determination of a definite lifetime is not appropriate but rough upper and lower bounds can be

estimated, 7 fs9 917fs: Together with figure 4, the relatively long lifetime of the

adsorbate-derived C6H6 resonance rationalizes the large yield and the sharp

voltage dependence observed for this system.

We remark that the ability of a two-dimensional model to capture the dynamics

of the many-mode problem is due to features that are specific to the C6H6/Si(100)

system, in particular the near conservation of C2v symmetry along the reaction

pathway. Nonetheless, the rapid nature of the resonance-driven processes envi-

saged here is expected more generally to restrict the number of active modes to

relatively few. Evolution in the resonant state competes with electronic relaxationand is thus of femto second time scale. Reaction in the neutral state competes with

internal relaxation and is thus of a time scale of sub- to a few picoseconds. On such

short time scales, a small portion of the resonance-state potential energy surface is

explored and, subsequent to electronic relaxation, energy is distributed within a

restricted subset of modes.

3.2. Current-triggered dynamics in single-molecule transistors

In this section, we apply the theory of section 2 to a particularly simple

problem, namely current-induced vibrational excitation of a fullerene adsorbedbetween two Au contacts. Our choice of a simplest-case-senario model has several

motivations. Firstly, an experiment by Park et al . [65] recorded the signature of

current-induced vibrational motion in single Au–C60 –Au transistors. We are not

aware of other experimental demonstrations of current-induced dynamics in a

molecular device, although we believe that the concept is general1. Secondly, the

Au–C60 –Au system offers rich and relatively well-studied [66] electronic dynamics.

The ability to control the electronic dynamics, already demonstrated for this

system [65, 66] provides potentially a route of manipulating also the nuclear

dynamics (vide infra). Thirdly, the nuclear dynamics allow for an analytical

approximation which provides useful insight [66].

The molecular device in mind is depicted in figure 6. The Au electrodes extend

to electron reservoirs at Z ¼ Æ1, where the bias voltage V b is applied and electric

current is collected. A metal gate with a gate voltage V g is placed near the C60

molecule, providing a third probe capacitatively coupled to the C60.

In figure 7 we describe the electronic dynamics of the device of figure 6, as

computed within a DFT-based non-equilibrium Green function approach [66].

The inset shows the computed transmission T (E ,V b,V g) versus the electron energy

E at zero bias and gate voltage. The energy scale is set such that E F ¼ 0.

T (E ,V b

,V g

) is sharply peaked about E % 0.15 eV, with a line shape close to the

Breit–Wigner form, indicating an isolated resonance and essentially energy-

independent direct transmission. This resonance is responsible for the nuclear

2404 T. Seideman



lifetime of about 26 fs. At finite bias voltages V b 6¼ 0, the eigenlevels, and with them

the transmission peaks, shift significantly but the overall shape of T (E ,V b,V g) is

only slightly modified. The effect of a finite gate voltage on the transport properties

is rather more substantial. By changing the charge distribution in the molecular

region and shifting the energy levels, a non-zero gate voltage alters qualitatively

both the line centre and the shape of the transmission curve [66].

Figure 7. Electronic dynamics in the Au–C60 –Au device of figure 6 as obtained within a

DFT-based non-equilibrium Green function approach. The main diagram shows theexcitation function W exc of equation (14) versus the electronic-to-vibrational energytransfer and the bias voltage V b. The inset gives the transmission coefficientT ðE , V b, V gÞ versus energy E for zero bias and gate voltages. The transmission peakarises from the resonant orbital that mediates the current-triggered dynamics of figure 8.

Figure 6. Current-triggered dynamics in single-molecule transistors. Schematic diagramof the Au–C60 –Au molecular device. The Au electrodes extend to Z ¼ Æ1 where the

bias voltage V b is applied. A gate voltage V g may be applied in the scattering regionof the device.




biased molecular transistor follows nicely the form of the analytical approximation

of equation (16), namely a sigmoidal function of voltage and energy. This intuitive

structure is due to the near-Breit–Wigner shape of the resonance and no longer

obtains in devices and energy regimes where resonances overlap.

The nuclear dynamics triggered by the electronic transmission of figure 7 aresimilar in concept to the STM-triggered dynamics of section 3.1. The tunnelling

event transiently places the nuclear system in a negative ion state of the Au–C60 –

Au system. Owing to the equilibrium mismatch between the neutral and ionic

states, a non-stationary superposition of vibrational eigenstates is formed, which

travels towards the ionic state equilibrium while continuously relaxing to the

neutral state. Upon electronic relaxation, the population has been redistributed

between the vibrational levels of the neutral surface.

This qualitative picture is quantified in figure 8. The inset shows the expecta-

tion value of z in the neutral state wave packet. We find that the C60 centre of mass

oscillates between the contacts at the fundamental frequency of the neutral surfaceand an amplitude approximately equal to the distance travelled in the ionic state.

Slow damping of the oscillations is due to the anharmonicity of the potential.

The main diagram in figure 8 shows the vibrational excitation rate of the

Au–C60 –Au system given by

wvðV b, Þ ¼ P vð ÞW excðv, V b; Þ ð17Þ

(see equations (11) and (12)), as a function of the applied voltage V b and the

equilibrium displacement Z eq between the neutral and ion states for v ¼ 1. Our

Figure 8. Rate of vibrational excitation of the Au–C60 –Au device versus bias voltage (seeequation (17)). The equilibrium displacement dZ eq between the neutral and ionic

2406 T. Seideman



results correspond to the unmodified molecular junction, V g ¼ 0, and are given for

Z eq values covering the physically relevant range; for smaller Z eq the vibrational

excitation vanishes whereas for larger values we find a finite desorption

probability. The bias-voltage dependence of the wv follows the V b dependence

of W exc in equation (17) (see figure 7) and is well approximated by the analyticallimit of equation (16), as a result of the isolated nature of the resonance mediating

the dynamics. The excitation rates of vibrational levels v > 1 take a similar shape

but respond differently to the magnitude of the equilibrium displacement.

We find that the essential dynamics are contained in the combination of the

equilibrium displacement and the resonance lifetime. The former parameter

determines the outcome through the exponential dependence of P reac on Z eqThe latter parameter determines the rate through the balance between the

dependences of the electronic and nuclear dynamics. Whereas W exc decreases

sharply with increasing (see equation (16)), P reac increases (roughly exponen-

tially) with this parameter. The ability to manipulate the conductance by varyingthe distance between the electrodes and the gate voltage thereby translates into

control of the current-induced dynamics.

4. Summary and outlook

In the previous sections we reviewed recent theoretical work on the problem of

singe-molecule dynamics induced by electric current, a young and exciting field

that shares common aspects with several established areas of research, including

transport through molecular wires, photochemistry on conducting surfaces, STM

manipulation and electron–molecule scattering.Subsequent to a discussion of the qualitative physics underlying current-

triggered dynamics in section 1, we outlined a theoretical framework for study

of these dynamics (section 2). By introducing an approximation in the form of the

electronic Hamiltonian in equation (1), we converted the solution of correlated

reactions in the electronic and nuclear modes (equation (4)) into a form that

provides better insight and is substantially easier to deal with numerically

(equation (11)). In the latter form, the reaction rate is expressed in terms of the

resonant current j ¼ W exc/e that drives the reaction and a normalized reaction

probability P reac. The parameters determining the dynamics are spelled out and

the computational effort is reduced to that required for a conductance calculation

and a reaction dynamics calculation in the nuclear subspace. An analytical

approximation of W exc rationalizes a common feature of several experiments,

namely the sigmoidal shape of the observed rate versus voltage curves. Application

was made to study STM-triggered surface reactions (section 3.1) and current-

triggered dynamics in a single-molecule transistor (section 3.2).

Clearly, much remains to be done in the area of current-driven dynamics in

nanoscale devices. Exploring several of the potential applications proposed in

section 1 is one of the topics of ongoing research in our group. In particular, we

employ the concepts discussed in section 1.1 to devise a coherently drivenunidirectional molecular rotor and a nanoscale rattle, to explore the modification

of the conductivity induced by current-triggered torsion in a biphenyl device




Of considerable interest is the development of improved methods for comput-

ing the electronic dynamics. Here one seeks approaches that would go beyond the

current implementation of DFT, techniques that would economize the calculation

and, most importantly, time-domain approaches that would provide better insight

into the electronic dynamics. Another exciting avenue that we hope to address isthe development of a method that goes beyond the approximation leading to

equation (11) by solving simultaneously for the electronic and nuclear dynamics.

Note added in proofs

In a recent conference, we became aware of an unpublished experiment by D.

Ralph and coworkers, where the signature of current-driven dynamics in mol-

ecular transistors was recorded.

Acknowledgments

The author is grateful to several postdoctoral fellows and colleagues who

contributed to the work reviewed in this article, in particular Dr Saman Alavi,

Dr Roger Rousseau and Professor Hong Guo. This work was partially supported

by the Natural Science and Engineering Research Council of Canada.

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Tamar Seideman- Current-driven dynamics in quantum electronics