Chemical Physics 281 (2002) 293–303
www.elsevier.com/locate/chemphys
Current-triggered vibrational excitation in
single-molecule transistors
Saman Alavi a,1, Brian Larade b, Jeremy Taylor b,2, Hong Guo b, Tamar Seideman a,*
b
a
Steacie Institute for Molecular Sciences, National Research Council of Canada, Ottawa, Ont., Canada K1A 0R6
Center for the Physics of Materials & Department of Physics, McGill University, Montreal, Que., Canada H3A 2T8
Received 6 November 2001
Abstract
We investigate the possibility of inducing nuclear dynamics in single-molecule devices via inelastic, resonance-mediated tunneling current. Our method is based on the combination of a theory of current-triggered dynamics in molecular heterojunctions and a nonequilibrium Green’s function approach of computing electron transport properties.
The scheme is applied to study current-induced dynamics in single-molecule Au–C60 –Au transistors. Ó 2002 Elsevier
Science B.V. All rights reserved.
1. Introduction
In recent years, theoretical work in the area of
molecular-scale electronics [1] has provided substantial insight into the effects of various molecular
properties on the conductance of molecular-wire
heterojunctions. [2–14] In addition, quantitative
predictions of the conductance have been reported
for a variety of systems. A problem of fundamental (perhaps potential practical) interest, which
to our knowledge has not been addressed as yet, is
the effect of the current on the molecule.
*
Corresponding author. Tel.: +613-990-0945; fax: +613-9472838.
E-mail address: [email protected] (T. Seideman).
1
Present address: Department of Chemistry, Oklahoma State
University, Stillwater, OK 74078-3071, USA.
2
Present address: Mikroelektronik Centret (MIC), Technical
University of Denmark, East DK-2800 Kgs. Lyngby, Denmark.
In Ref. [15] two of us pointed out that the
electron tunneling event can induce a variety of
dynamical processes in single-molecular devices,
including intermode energy transfer, rotation, vibration and reaction. Several potential applications
of
current-triggered,
single-molecule
dynamics have been proposed [15]. Among other
opportunities we mention the application of these
dynamics to make new forms of molecular machines [16] and their potential as a means of enhancing the conductivity of molecular wires.
Equally interesting is the possibility of using current-triggered dynamics in single-molecule devices
as a laboratory for study of electron-induced reactions in complex media. An example is the radiation damage of DNA, ascribed in a recent
experimental study [17] to the effect of low energy
secondary electrons. Further applications will
likely be found in the areas of STM-controlled
nanolithography and nanochemistry [18].
0301-0104/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved.
PII: S 0 3 0 1 - 0 1 0 4 ( 0 2 ) 0 0 5 6 7 - 0
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S. Alavi et al. / Chemical Physics 281 (2002) 293–303
The mechanism through which low-energy
electrons can trigger large amplitude motion of the
heavy nuclei is simple and general. Favorable
candidates for molecular wires are molecules that
have resonances in energetically accessible regimes; resonance tunneling is efficient and at most
weakly distance dependent [2]. Direct conductance, by contrast, is slow and falls exponentially
with distance. Resonance-mediated conductance is
often inelastic. Provided that the resonance state
(typically a charged or partially charged state of
the molecule+contacts Hamiltonian) is displaced
in equilibrium with respect to the neutral state, the
nuclear system evolves during the resonance lifetime. Upon electronic relaxation the molecule is
internally excited and interesting dynamics may
ensue before vibrational relaxation has taken place
[15,19–21].
In Ref. [15] we developed a theory for study of
the problem of current-induced, single-molecule
dynamics in molecular heterojunctions. By introducing an approximation in the form of the electronic Hamiltonian, we reformulated the exact rate
for transition from an initial to a final state of the
coupled vibronic Hamiltonian, wðVb Þ, as an energy
integral over a product of an electronic and a
nuclear function,
Z
wðVb Þ ¼ d Wexc ðVb ; ; sÞPreac ð; sÞ;
ð1Þ
where Vb is the bias voltage and is the energy
transferred from electronic into vibrational. The
function Wexc is the rate of resonance tunneling,
related to the resonant component of the current
by j ¼ eWexc , e being the electron charge,
Z
2p
dm fi ðm Þ
Wexc ðVb ; ; sÞ ¼ 2
h
½1 ff ðm Þqi ðm Þqf ðm Þ:
ð2Þ
In Eq. (2) fiðfÞ ðEÞ is a Fermi–Dirac distribution
function for electronic states of energy E, i(f) denotes the electrode from (to) which electrons are
transfered and qiðfÞ ðEÞ is a projected density of
electronic states,
X
2
jhmjrij dðE m Þ;
ð3Þ
qiðfÞ ðEÞ ¼
m
where j r i is the resonant orbital. Preac in Eq. (1)
contains the details of the nuclear Hamiltonian
and the reaction dynamics. We have indicated
explicitly the parametric dependence of both electronic and nuclear functions on the resonance
lifetime, s, as a reminder of the crucial role played
by this parameter in determining both the nuclear
and the electronic dynamics; the longer the lifetime
the smaller the fraction of the current that is resonance mediated but the higher the degree of
vibrational excitation per resonance event. The
derivation of Eqs. (1)–(3) from the formally exact
rate and the approximation involved are detailed
in Ref. [15] and are not reproduced here.
With the reaction rate expressed in terms of the
current that drives the reaction, the calculation
proceeds by first computing the electronic dynamics, using it to determine the current, the
function Wexc and the resonance lifetime, calculating potential energy surfaces for the neutral and
ionic states and propagating the nuclear dynamics
subject to these potentials and to a decay rate
C ¼ h=s. Finally, the nuclear and electronic functions are re-coupled through the energy integration in Eq. (1) [15].
Previous work [15,19–21] applied the formalism
to several simple processes in adsorbates subject to
the tunneling current of a STM. The tip-adsorbatesubstrate environment is formally equivalent to the
molecular wire environment and the theory [15] can
be applied to the latter environment unaltered.
Numerically, however, the STM experiment allows
for drastic simplification of the electronic dynamics
which does not apply to the molecular wire. In the
former environment, at the large tip-adsorbate
distances of relevance, the STM field and tip-adsorbate chemical interactions do not modify the
eigenstates of the substrate-adsorbate Hamiltonian
to a noticeable extent. In this situation, the calculation of Wexc reduces within a good approximation
to calculation of the density of electronic states
(DOS) of the uncoupled tip and substrate–adsorbate systems, the former within an approximate
model of the tip structure. Furthermore, in the
limit of an isolated resonance and low temperature
the function Wexc can be expressed as a 2-parameter
analytical function [15,19]. Given experimental rate
vs voltage curves (see Refs. [20,22,23]) the param-
S. Alavi et al. / Chemical Physics 281 (2002) 293–303
Fig. 1. Schematic diagram of the Au–C60 –Au molecular device.
The Au electrodes extend to z ¼ 1 where bias voltage Vb is
applied. A gate voltage Vg may be applied at the scattering region of the device.
eters can be extracted from fit of the theoretical rate
vs voltage curve to the data [15]. In order to apply
the theory to a molecular wire it is essential to account for the dependence of the device eigenpairs
on the bias voltage, allow for their modification by
a gate voltage and incorporate the proper open
system boundary conditions.
In the present work we describe the quantum
transport within the first principles approach detailed in Ref. [11], which combines the Keldysh
nonequilibrium Green’s function (NEGF) with
density functional theory (DFT). This approach
allows us to compute transport properties as well as
the function Wexc without introducing phenomenological parameters. The NEGF–DFT technique
has so far been applied to the prediction of elastic
transport through molecular devices, including
molecular tunnel junctions, [10a] carbon nanotubes, [11] doped metallofullerenes [10b] and atomic
[10c] and molecular [10d] wires. In the present work
we extend it to the inelastic case using Eq. (2).
In Section 2 we briefly describe our method of
computing the electronic dynamics and in Section
3 we apply Eq. (1) to model current-induced dynamics in a molecular device, specifically a fullerene adsorbed between gold electrodes, see Fig. 1.
Section 4 summarizes our conclusions with an
outlook to future research.
2. Theory
In order to calculate electron transport properties of molecular scale conductors such as the
295
Au–C60 –Au device of Fig. 1, we have developed a
computational framework that combines first
principles density functional theory analysis with
the Keldysh nonequilibrium Green’s function
formalism. The details of this technique have been
presented elsewhere, [11] and in this section we
outline only the essentials.
In the molecular device shown in Fig. 1, the Au
electrodes extend to electron reservoirs at z ¼ 1,
where bias voltage Vb is applied and electric current is collected. A metal gate with gate voltage Vg
is placed near the C60 molecule, providing a third
probe capacitively coupled to the C60 . The effect of
Vg is to shift the molecular levels, thus controlling
the electron transport. In the electron transport
calculation we fix the atomic positions of both the
electrodes and the C60 . Each electrode is represented by a slab of Au atoms oriented along the
(1 0 0) plane with 18 atoms per unit cell repeated to
z ¼ 1. The C60 is placed a distance d ¼ 2:3 A
from the electrode planes.
Due to the external fields at nonzero bias
voltage, the electrochemical potentials li=f of the
initial (i) and final (f) leads are not equal, and the
molecular device is in a nonequilibrium steady
state. The electronic density, therefore, must be
constructed under nonequilibrium conditions. We
note that the device in Fig. 1 is an open system
due to the presence of the infinitely long leads. In
contrast, typical situations analyzed within DFT
involve either finite (as in quantum chemistry) or
periodic (as in solid state physics) systems. In
order to deal with the open structure and the
nonequilibrium situation from first principles, we
exploit the fact that the Kohn–Sham potential
Veff ðrÞ deep inside the electrodes is very close to
the corresponding bulk Kohn–Sham potential,
since the influence of C60 is rapidly attenuated
with distance from the contact. The bulk potential, being a periodic system, can be calculated
with standard DFT techniques and stored in a
database to provide a boundary condition for the
device analysis. The device is thus naturally divided into three sections: the left and right electrodes, and the central cell, which forms our DFT
simulation box (see Fig. 1). The central cell
contains a portion of the electrodes and all the
atoms inside the central cell are included in our
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S. Alavi et al. / Chemical Physics 281 (2002) 293–303
DFT self-consistent iterations. During the iterations, we require the potential Veff ðrÞ to match the
bulk Kohn–Sham potential of the perfect electrode at the boundary of the central cell. The
external bias potential is then readily applied as
the boundary condition for the Hartree potential,
which we solve in real space using a multi-grid
numerical technique [11].
Proceeding to calculate the charge distribution,
qðrÞ, under nonequilibrium conditions, we note
that in conventional DFT qðrÞ is constructed by
summing over the Kohn–Sham eigenstates. For
an open system such as that of Fig. 1, while determination of the scattering states is straightforward, it is rather difficult to determine all the
bound states which exist inside the molecular
region [11]. The reason for which the Hamiltonian may support bound states is that the bandwidth of the molecule can be larger than that of
the electrodes, hence behaving as a potential well.
To bypass this difficulty, and indeed, to deal with
the nonequilibrium condition due to external
bias, we construct the density matrix q^ via the
Keldysh nonequilibrium Green’s function [24,25]
G< ,
Z
i
dE G< ðEÞ;
ð4Þ
q^ ¼ 2p
where G< is calculated using the Keldysh equation
G< ¼ GR R< GA :
R=A
ð5Þ
is the retarded/advanced Green’s funcHere G
tion of the device and R< is the lesser self-energy,
representing injection of charge from the electrodes [24–26]. The latter function is obtained
from the self-energies Rii;i and Rff;f , arising from
coupling to the initial and final electrodes, respectively, and the Fermi distribution functions of
the electrodes [11]. The charge density constructed
in this manner accounts for both scattering and
bound states and incorporates the proper boundary condition for open systems under external
bias.
We use a s, p, d real space Fireball LCAO basis
set [11,27–29] in the DFT analysis, where atomic
cores are defined by the standard nonlocal norm
conserving pseudopotential [30]. In this way the
Hamiltonian of the device is organized into a tri-
diagonal form and the Green’s functions GR , GA
are obtained by direct matrix inversion in the orbital space [11]. Finally, we calculate the self-energies Rii;i and Rff;f by extending [11] a technique
discussed in Ref. [31]. With qðrÞ determined, we
evaluate all other terms in the effective potential
Veff ðrÞ, including the exchange-correlation and the
core contributions. The self-consistent Kohn–
Sham equations are iterated to numerical convergence, chosen in the present study as 103 eV
accuracy in the band-structure energy, and the
physical observables are collected. In particular,
the elastic electric current is obtained through the
Landauer formula,
Z
2e lmax
I¼
dE ðfi ff ÞT ðE; Vb Þ;
ð6Þ
h lmin
where lmin (lmax ) is the smaller (larger) of the
chemical potentials li and lf and T ðE; Vb Þ is the
transmission coefficient at energy E and bias potential Vb [24,26],
f
A
T ðE; Vb Þ ¼ 4 Tr½ ImðRii;i ÞGR
i;f ImðRf;f ÞGf;i :
ð7Þ
It is emphasized that, since the current is calculated from a self-consistent analysis, the functions
inside the trace in Eq. (7) are all functions of bias
potential Vb .
Once the Kohn–Sham effective potential Veff ðrÞ
is iterated to self-consistency for a given set of bias
and gate voltages, we determine the self-consistent
Hamiltonian of the complete Au–C60 –Au device.
This allows us to calculate the projected density of
states from Eq. (3) and hence the excitation function Wexc of Ref. [15], as defined in Eq. (2). To that
end we first determine the resonant orbital jri of
Eq. (3) by diagonalizing the submatrix of the
Hamiltonian that is associated with the atomic
orbitals in the C60 molecule. Owing to the selfconsistent nature of Veff ðrÞ, the levels obtained in
this way are derived from a molecule interacting
with the electrodes under bias and gate potentials.
The properties of these fully interacting states have
been discussed in Ref. [10b]. Having obtained the
resonant orbital jri, we project onto it the scattering states jmi, obtained through our NEGF
analysis, to determine the projected density states
of Eq. (3). Finally, Wexc is calculated according to
Eq. (2).
S. Alavi et al. / Chemical Physics 281 (2002) 293–303
3. Results
In this section we apply the method outlined in
Sections 1 and 2 to a particularly simple problem,
namely, current-induced vibrational excitation of a
C60 adsorbed between two gold contacts. While
our ultimate goal is to apply the formalism to
chemically more complex processes (see Section 4),
our choice of a simplest-case-scenario model for
this first application has interesting motivations.
First is a recent experiment by Park et al. which
records the signature of current-induced vibrational motion in single Au–C60 –Au transistors [32].
We are not aware of previous experimental demonstrations of current-induced dynamics in a molecular device, although we believe that the
concept is general. Second, the Au–C60 –Au system
offers rich electronic dynamics (see Section 3.1).
Third, the nuclear dynamics allows for an analytical approximation which provides useful insight
(see Section 3.2). A rather different approach to
the same model system is discussed in Ref. [33].
For the simple problem at hand, Eq. (1) reduces
to
wv ðVb ; Vg ; sÞ ¼ Wexc ðv ; Vb ; Vg ÞPv ðsÞ;
ð8Þ
where wv is the rate of excitation of vibrational
level v and v , the energy transfer, is now quantized, being the energy level difference between the
ground and the vth vibrational levels of the neutral
Au–C60 –Au system. The dependence of our observable on Vg , the gate voltage, arises from the
strong effect of the gate voltage on the electronic
eigenenergies of the device (vide infra).
Section 3.1 discusses the electronic dynamics
and in Section 3.2 we describe the nuclear dynamics. In Section 3.3 we couple the electronic and
nuclear dynamics to determine the excitation rate
as a function of the applied voltage.
3.1. Electronic dynamics
The linear and nonlinear transport through a
C60 molecular tunnel junction have been the subject of both experimental [32,34] and theoretical
[10] interest. An isolated pristine C60 molecule has
a filled HOMO state (highest occupied molecular
orbital) and an empty sixfold (including spin) de-
297
generate LUMO (lowest unoccupied molecular
orbital). The HOMO–LUMO gap is approximately 1.8 eV and hence one expects C60 to be a
poor conductor. Interestingly, experiments found
single C60 molecular junctions to conduct efficiently [34,32]. In previous work, [10] where an Al–
C60 –Al device was investigated, we found that
charge transfer doping of the C60 by the Al electrodes half-fills the LUMO which thus aligns with
the Fermi level of the electrodes and substantially
improves the equilibrium conductance of the device. For the Au–C60 –Au device studied here, we
find a similar physical picture, namely charge
transfer doping from the Au electrodes to the C60
partially occupies the LUMO, thereby aligning it
to EF of the Au electrodes. In the Au–C60 –Au case,
the charge transfer is about 0:7e, and the equilibrium conductance of the tunnel junction is about
G ¼ 0:94G0 , where G0 2e2 =h is the conductance
quantum. Thus, the scattering of electrons by the
C60 junction somewhat reduces the value of G
from the ideal value G0 but the conductance remains good due to the efficient doping. We note
that the amount of charge transferred, and hence
the conductance, depends on the electrode material as well as on the junction distance d.
The inset of Fig. 2 shows a typical I–V curve for
the Au–C60 –Au device, where a metallic behavior
is observed. The current is in the lA range, compatible with STM observations of conductance
through C60 [34]. We omit a detailed discussion of
the I–V curves as our primary purpose is to explore the nuclear dynamics that is triggered by the
current. The main panel of Fig. 2 shows the
transmission coefficient, T ðE; Vb Þ, vs the electron
energy E for three bias voltages at zero gate voltage. The energy scale is set such that EF ¼ 0. At
zero bias voltage, T ðE; Vb Þ is sharply peaked about
E 0:15 eV, with a lineshape close to the Breit–
Wigner form, indicating an isolated resonance and
essentially energy-independent direct transmission.
This resonance is responsible for the nuclear dynamics. For reference below, we note that its width
corresponds to a resonance lifetime of ca. 26 fs.
The overall shape of T ðE; Vb Þ is not modified at a
finite Vb , but its center shifts. The magnitude of
this energy shift depends on how the bias is
dropped across the two electrode-molecule con-
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S. Alavi et al. / Chemical Physics 281 (2002) 293–303
Fig. 2. The transmission coefficient T ðE; Vb Þ vs energy E for
three bias voltage values and zero gate voltage. The circles indicate the energies of the molecular orbitals that mediate the
transmission and are responsible for the dynamics. The inset
gives a current–voltage curve of the device, showing a metallic
behavior.
tacts [10b]. The three circles superimposed on the
curves in Fig. 2 mark the real part of the energy
level corresponding to the molecular orbitals jri
discussed in the previous section. The level positions align very well with the peaks of the T ðE; Vb Þ,
clearly confirming that transport is mediated by
these orbitals. We recall an early remark due to
Breit and Wigner [35], that if there is an elastic
resonant transmission of the Breit–Wigner form,
such as shown by Fig. 2, then the weakly coupled
inelastic channels (e.g., inelastic transmission of
electrons due to nuclear scattering) are also characterized by a resonant transmission [35].
In the presence of a finite gate voltage Vg , the
transport through the molecular device becomes
more complicated. Vg has two main effects: it
changes the charge distribution in the molecular
region and shifts the energy levels of the molecule+contacts Hamiltonian. Fig. 3 shows T ðE; Vb Þ
and the scattering density of states DOSðEÞ vs E at
Vg ¼ 0:25 a.u. and three different bias voltage
values. At this value of Vg , sharp transmission
features can be pushed across the Fermi energy EF
(compare Figs. 3a–c, the inverted triangle indicates
the position of EF ) by the applied bias voltage.
This indicates a desirable character of this molec-
Fig. 3. The transmission coefficient T ðE; Vb Þ for a gate voltage
of Vg ¼ 0:25 a.u. and three bias voltage values. The lower panels
show the corresponding scattering density of states divided by a
factor 2000. The peaks of T ðE; Vb Þ are well aligned with those of
the DOS, identifying the corresponding orbitals as the resonances mediating the transmission at nonzero gate potential.
The inverted triangles indicate the Fermi energy.
ular device, namely, resonance transmission features can be controlled by a combined effect of Vg
and Vb . Furthermore, the peaks of DOSðEÞ are
aligned with the transmission peaks, clearly
marking the molecular level mediating the resonance transmission at a finite Vg . It is interesting to
note that the applied voltage modifies the resonance lineshape in addition to shifting its position.
Finally, the most relevant quantity for the present work is the excitation function Wexc ð; Vb Þ of
Eq. (2). This function is plotted in Fig. 4 vs the
Fig. 4. The excitation function Wexc of Eq. (2) vs the electronicto-vibrational energy transfer, , and the bias voltage Vb . Note
that Wexc ¼ 0 for < Vb .
S. Alavi et al. / Chemical Physics 281 (2002) 293–303
electronic-to-vibrational energy transfer, , and the
bias voltage, Vb . The projected density of states
required for this calculation has been obtained
using the procedure described in Section 2 for the
orbitals indicated by circles in Fig. 2. Wexc follows
nicely the shape of the analytical approximation
derived in Ref. [15]. This is expected given the
nearly pure Breit–Wigner form of the density of
states.
3.2. Nuclear dynamics
The electron tunneling event described in the
previous section transiently places the nuclear
system in a negative ion state of the Au–C60 –Au
system. Due to the equilibrium mismatch between
the neutral and ionic states, a nonstationary superposition of vibrational eigenstates is formed,
which travels toward 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 – the fullerene bounces between
the gold surface until vibrational relaxation returns the system to the ground vibrational state.
In the harmonic limit these dynamics are readily
solved for analytically. We expand the nonstationary superposition evolving on the ionic surface
as,
X
wðt; zÞ ¼ eixt=2
Av uv ðz dzeq Þeivxt ;
ð9Þ
v
where z denotes distance from the surface, measured with respect to the neutral state equilibrium
configuration, dzeq is the equilibrium displacement
of the ionic with respect to the neutral state, x is
the vibrational frequency, assumed equal in the
two states, uv are harmonic oscillator functions
and
Av ¼ hu0 ðzÞ j uv ðz dzeq Þi
fveq expðf2eq =4Þ
pffiffiffiffiffiffiffiffi
¼
;
2v v!
feq pffiffiffiffiffiffiffi
lxdzeq ;
ð10Þ
l being the mass. (The second equality of Eq. (10)
can be derived in one of several ways. See, e.g., Eq.
(27) of Ref. [36].) Substituting Eq. (10) in Eq. (9)
we have that the probability density in the reso-
299
nance state oscillates without change of shape
about the ionic state equilibrium configuration
with amplitude dzeq and frequency x,
rffiffiffiffiffiffiffi
pffiffiffiffiffiffiffi
lx ðffeq cos xtÞ2
2
jwðt; zÞj ¼
; f lxz:
ð11Þ
e
p
In the harmonic limit the probability of excitation
of the vth vibrational level of the neutral state
upon electronic relaxation is
2
X
0
Pv ðsR Þ ¼ Av0 huv ðzÞ j uv0 ðz dzeq Þieiv xsR ;
v0
ð12Þ
where sR is the residence time in the ionic
state. Closed-form expressions for the huv ðzÞ j
uv0 ðz dzeq Þi are given in Ref. [36]. In particular,
the probability of capture into the ground vibrational level is
2
X f2v expðf2 =2Þ
eq
eq
ivxsR e
P0 ðsR Þ ¼ v
v
2 v!
2
¼ efeq ½1cosðxsR Þ ;
ð13Þ
where we used Eq. (1.211) of Ref. [37] to perform
the v P
summation. As expected, vibrational excitation, v6¼0 Pv , vanishes in the limit of small equilibrium displacement; P0 ðsR ; dzeq ! 0Þ ! 1, and in
the limit of short residence time in the ionic state;
P0 ðsR ! 0; dzeq Þ ! 1.
Considering next the vibrational dynamics of
the Au–C60 –Au device, we adopt the potential
energy surfaces given in Ref. [32] based on electronic structure calculation, Ref. [38], of the
C60 =Auð1 1 0Þ system. The equilibrium displacement between the neutral and ionic states is left as
a variable, however, since this parameter is only
approximately known [38] while the dynamics depends crucially on its precise value [see Eq. (13)].
Our results, taken together with the observations
of Park et al. [32] place upper and lower bounds on
the physical equilibrium displacement (vide infra).
Schr€
odinger’s equation for the vibrational motion is propagated using the split operator technique and the probability of capture into different
vibrational states vs the residence lifetime, Pv ðsR Þ,
is computed by projecting the ionic state wave
300
S. Alavi et al. / Chemical Physics 281 (2002) 293–303
packet at different times during the propagation
onto the eigenstates of the neutral state Hamiltonian. An equivalent, time-dependent, observable is
the damped oscillation of the C60 center-of-mass
against the gold electrodes, described through the
expectation value of z in the wavepacket subsequent to electronic relaxation, hziðt; sR Þ ¼ hwðt; sR Þ
jzjwðt; sR Þi. To account for the continuous nature
of the relaxation both observables are averaged
over sR with an exponentially decaying weight
function [39],
Z
1
dsR Pv ðsR Þ expðsR =sÞ;
Pv ðsÞ ¼ s
ð14Þ
hziðt; sÞ ¼ s1
Z
dsR hwðt; sR Þjzjwðt; sR Þi
expðsR =sÞ:
ð15Þ
The details of the numerical procedure are essentially the same as in the work discussed in Refs.
[15,19–21].
The insets of Fig. 5 show the probability of
capture into the lowest five vibrational levels as a
function of sR , the residence time in the ionic
state. The equilibrium displacement is taken to be
dzeq ¼ 4 and 20 pm in Figs. 5(a) and (b), respectively. The smaller value is the equilibrium
displacement computed in Ref. [38]. The larger
(a)
one falls in the middle of the physically relevant
range. The long sR -behavior of the Pv ðsR Þ is given
for pedagogical completeness; only the small sR
edge of the figure is of physical relevance since,
for the lifetimes determined in Section 3.1, sR
values beyond 450 fs do not contribute to the
lifetime averaged result of Eq. (14). P0 ðsR Þ (solid
curve) follows closely the structure predicted by
Eq. (13) although, due to the anharmonicity of
the potential, the periodicity of Eq. (13) is lost
and the amplitude of subsequent recurrences decreases slowly with sR . As v increases, Pv ðsR Þ
broadens and shifts to larger values of sR (modulo 2p=x), reflecting the spatial location and
breadth of the vibrational eigenfunctions of the
neutral state Hamiltonian. With increasing equilibrium displacement dzeq (see the inset of Fig.
5b), the Pv ðsR Þ become better localized in time,
decaying exponentially to zero between recurrences, as predicted by Eq. (13). As dzeq decreases, the amplitude of oscillations of P0 ðsR Þ
below unity decreases and the higher-v vibrational excitation probabilities diminish. We find
that for an equilibrium displacement of 4 pm the
dynamics is captured by the five lowest vibrational states, whereas many more states are required to describe the post-relaxation dynamics
for the dzeq ¼ 20 pm.
(b)
Fig. 5. The probability of excitation of the Au–C60 –Au device into different vibrational states in the course of resonance transmission.
The equilibrium displacement between the neutral and ionic states is dzeq ¼ 4 pm in panel (a) and 20 pm in panel (b). Solid curves:
v ¼ 0, dotted curves: v ¼ 1, dashed curves: v ¼ 2, long-dashed curves: v ¼ 3, dot-dashed curves: v ¼ 4. The insets give the corresponding unaveraged probabilities, Pv ðsR Þ vs the residence time sR , see Eq. (14).
S. Alavi et al. / Chemical Physics 281 (2002) 293–303
The physical vibrational excitation probabilities, Pv ðsÞ of Eq. (14), are plotted vs s in the main
frames of Figs. 5a and b for v ¼ 0; . . . ; 4 and
equilibrium displacements of dzeq ¼ 4 and 20 pm,
respectively. P0 ðsÞ decays to a nonzero asymptotic
value that decreases with increasing dzeq . For small
equilibrium displacements (Fig. 5a), the higher-v
probabilities increase monotonically from zero
and saturate on a dzeq -dependent value. As dzeq
increases (Fig. 5b), the vibrational excitation
probabilities of progressively larger levels reach a
maximum before decaying to the asymptotic plateau. This behavior follows from Eq. (14) and the
sR -dependence of the Pv ðsR Þ, which becomes
smoother with increasing v and with decreasing
dzeq .
The solid curve of Fig. 6 shows the expectation
value of z [Eq. (15)] in the neutral state wavepacket
for an equilibrium displacement of 10 pm and a
lifetime of 26 fs, corresponding to the width of the
resonance shown in Fig. 2 (i.e., C ¼ h=s 0:025
eV). For comparison, the dotted and dashed
curves of Fig. 6 give the unaveraged expectation
values, hziðt; sR Þ, for several values of the residence
time sR . We find that the C60 center-of-mass oscillates between the contacts at the fundamental
Fig. 6. The expectation value of z in the wave packet subsequent to the resonance tunneling event. The bold solid curve
gives h z iðt; sÞ for s ¼ 26 fs, corresponding to the width of the
resonance feature shown in Fig. 2. The dotted and dashed
curves give hziðt; sR Þ for sR ¼ 0 (dotted curve), sR ¼ 10 fs
(dashed curve), sR ¼ 50 fs (long-dashed curve), and sR ¼ 100 fs
(dot-dashed curve).
301
frequency of the neutral surface and an amplitude
approximately equal to the distance traveled in the
ionic state. Slow damping of the oscillations is due
to the anharmonicity of the potential.
3.3. Vibrational excitation rates
In Fig. 7 we show the vibrational excitation rate
of the Au–C60 –Au system, wv ðVb Þ of Eq (8), as a
function of the applied voltage Vb for v ¼ 1; . . . ; 4.
Our results correspond to the unmodified molecular junction, Vg ¼ 0 in Eq. (8), and are given for
dzeq values covering the physically relevant range;
for smaller dzeq the vibrational excitation vanishes
[see Eq. (13) and the discussion below]. For larger
values we find (within the model potential energy
surfaces used [32,38]) a finite desorption probability. The bias-voltage-dependence of the wv follows the Vb -dependence of the Wexc in Eq. (8), see
Fig. 4, and is well approximated by the analytical
expression of Ref. [15], as a result of the nearly
pure Breit–Wigner form of the resonance mediating the dynamics.
(a)
(b)
(c)
(d)
Fig. 7. The vibrational excitation rate, Eq. (8), as a function of
the bias voltage. The equilibrium displacement between the
neutral and ionic states is dzeq ¼ 1:24 pm (circles), 4 pm (solid
curves), 10 pm (dotted curves), 20 pm (dashed curves), 25 pm
(long-dashed curves) and 31 pm (dot-dashed curves). (a) v ¼ 1,
(b) v ¼ 2, (c) v ¼ 3, (d) v ¼ 4. To allow display on the same
scale the dzeq ¼ 1:24 pm results in panels (a) and (b) are multiplied by factors 5 and 100, respectively, and the 4 pm results in
panels (c) and (d) are multiplied by factors 5 and 10, respectively.
302
S. Alavi et al. / Chemical Physics 281 (2002) 293–303
The equilibrium distance between the neutral
and ionic states is seen to play a crucial role; the
vibrational excitation increases by orders of magnitude with a small increase in dzeq . A similarly
important role is played by the resonance lifetime,
which determines both the nuclear and the electronic functions in Eq. (8). We find that both the
conductance and the current can be controlled by
the combination of bias and gate voltages and
chemical substitution.
4. Conclusions
Our goal in the work described above has been
to develop and apply a method of describing
current-driven chemical dynamics in molecularscale devices. To that end we combined a theory
of current-induced dynamics [15] with a DFTbased approach of computing electron transport
through molecular heterojunctions [10,11] and
time-dependent simulations of the nuclear dynamics.
As a first application of our scheme we considered the simplest consequence of inelastic
current through a molecular device, namely centerof-mass motion of a fullerene between the two
gold contacts of single-molecule Au–C60 –Au heterojunctions. Our choice of model was motivated
by recent experimental studies [32] of the conductance of Au–C60 –Au transistors. We found substantial probabilities for excitation of high
vibrational levels of the neutral state Au–C60 –Au
Hamiltonian, corresponding to oscillatory motion
of the fullerene center-of-mass between the contacts. The vibrational excitation is mediated by a
long-lived negative-ion resonance located ca. 0.15
eV above the Fermi level.
We feel that the possibility of current-inducing dynamics in molecular devices opens a
variety of new opportunities in the field of single-molecule science. One of our goals in ongoing work in this area is to devise a molecular
rotor consisting of a 9,10-bis(cyanoethynyl)
anthracene molecule attached between gold
contacts. A second subject of ongoing study is
current-triggered intermode energy flow in molecular heterojunctions.
Acknowledgements
We gratefully acknowledge financial support
from the Natural Science and Engineering Research Council of Canada and le Fonds pour la
Formation de Chercheurs et l’Aide a la Recherche
de la Province du Quebec.
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