ABSTRACT
we report a theoretical analysis of a nano electromechanical shuttle
based on a multi scale model that combines microscopic electronic structure
data with macroscopic dynamics. The microscopic part utilizes a (static)
density functional description to obtain the energy levels and orbitals of the
shuttling particle together with the forces acting on the particle. The
macroscopic part combines stochastic charge dynamics that incorporates the
microscopically evaluated tunneling rates with a Newtonian dynamics. We
explained how to apply the multi scale model to describe the shuttling of a
single copper atom between two gold-like jellied electrodes. We find that
energy spectrum and particle surface interaction greatly influence shuttling
dynamics; in the specific example that we studied the shuttling is found to
involve only charge states Q = 0 and Q = + e. The system
is found to exhibit two quasi-stable shuttling modes, a fundamental one and an
excited one with larger amplitude of mechanical motion, with random transitions
between them.
CONTENTS
Ø INTRODUCTION
Ø METHOD
·
SET UP
·
DYNAMICS
o
DISSIPATION
Ø RESULTS
Ø DISCUSSION
INTRODUCTION
Nano electromechanical
systems that combine electrical and mechanical functionalities on the nanometer
scale have in recent years attracted a great deal of theoretical and
experimental interest . The nano electromechanical shuttle is a structure
that resembles a single electron transistor but incorporates mechanical motion
of the central island. Previous theoretical studies on the shuttle have shown
that in the presence of a dc applied bias the charge and velocity of the
central island are correlated, which implies that the shuttle absorbs energy from the dc field
and converts it into mechanical motion. The shuttle motion facilitates charge
transfer through the system, and signatures of mechanical motion can be seen
both in the current–voltage characteristics and in the noise properties of the
device .
Several theoretical studies have been carried
out for different set-ups of the shuttle since the first description of this
phenomenon. The
theoretical studies cover different size regimes of the
shuttle, featuring coherent or sequential tunneling and quantum
mechanically or classically described mechanical motion. The studies
have shown that the shuttle instability strongly depends on the bias voltage
and the system set-up. This sensitivity also renders the shuttle behavior
dependent on the precise description of the problem. Experimental evidence of
coupling between vibrational degrees of freedom and electron transfer has been
found for both microscopic and macroscopic systems. In particular,
the experiment by Park et al , using a C60 molecule between
gold electrodes, has demonstrated the type of coupling that has been considered
by many theoretical studies and has increased the interest for a molecular
shuttle . In the shuttle geometry the mechanical motion is on a nearly
macroscopic time scale, typically from picoseconds for small molecules to
nanoseconds for large molecules such as carbon nano tubes. The motion is
excited due to tunneling events between the mobile object and the stationary
electrodes, which have a typical time scale of femto seconds that is determined
by the electronic structures of the mobile molecule and Hence, a theoretical
description of the shuttle system naturally calls for multi scale methods that combine
the fast electronic time scales with the slower mechanical ones. Thus far
research has concentrated on the slow degrees of freedom (long length scales)
while dealing with the fast ones (short length scales) in a phenomenological
approximation.
In this study, we examine
some of the issues that are important for a molecular shuttle system at zero
temperature. We focus on the thus far unexplored effects that emerge in the
shuttling of quantum mechanical objects with realistic electronic structure comprising
many orbitals with different symmetries. A key feature of the system is the
interaction between the shuttling object and the stationary electrodes which
includes a purely energetic part associated with the potential energy surfaces
of the surface-shuttle system, and a charge transfer part describing electron
transfer between the different subsystems (shuttling object and electrodes).
The latter issue is problematic from an electron structure point of view since
the charge equilibration rate strongly depends on the separation between the
shuttling particle and the electrode, and at large separations the subsystems
are independent of each other as far as their electronic structures are
concerned while at near proximity the electronic structures must be treated
jointly. In this study, we concentrate on the situation when the hybridization
between the shuttling object and the surfaces is small and charge transfer can
be described by stochastic tunneling events between unhybridized electronic
states—systems in which chemisorption plays an important role are hence
excluded from this study. Since the impact of the small shuttling object on the
electronic structure of the electrodes is quite small in the absence of
chemisorption, we have opted for modeling the electrodes as two semi-infinite
jellied slabs. This allows us to restricted the numerical electronic structure
calculations to the shuttling object alone and incorporate the effects of the
electrodes as external surface potentials. This decoupling also eliminates
spurious effects such as hybridization between two degenerate orbitals that are
separated by a large distance.
A priori, it is unclear if
the predominantly attractive surface forces are so strong that shuttling of a
nano scale object in general is prohibited, or if the adhesive forces can be
overridden by reasonable electric fields. Addressing this issue is one of the
main motivations of the present study. The surface forces are included as
external potentials in the static density functional description of the
shuttling object, which provides information on the energy spectrum of the
island as well as structure of the relevant orbitals.
The atomistic description of
the central island is hence coupled to an effective description of the
electrode surfaces. The mechanical motion of the shuttle is described using
classical dynamics that allows us to treat macroscopic time scales. The
electronic and mechanical descriptions are connected by the forces that are
determined by the atomistic model and by stochastic tunneling events that
describe charge transfer between the electrodes and the shuttle. The mechanical
model also incorporates a dissipative term which prevents catastrophic runaway
by transferring mechanical energy from the shuttle to lattice vibrations in the
electrodes.A more accurate treatment of
this problem in a future study would be to incorporate a time-dependent density
functional theory (TD-DFT) module that describes both the island and the
electrode during the crucial parts of the shuttle cycle; at present, however,
that type of description is prohibitively expensive from a computational point
of view.
METHOD
SET UP:
For computational
efficiency, we have chosen to focus on the simplest possible system where the
central island comprises just one atom. However, the methods and qualitative
results should be applicable also to more complex systems. The system we
consider includes two electrodes 15 Å apart, described as semi-infinite
jellied slabs. The central island is a copper atom that can move in a direction
normal to the electrode surfaces. For the electrodes, the Wigner–Seitz radius
is set to 3 au and the electrode work function W is set to 3.5 eV.
The Fermi energy, εF, is calculated from the Wigner–Seitz radius
while other material specific electrode parameters are taken from gold. A bias
voltage of 3 V is applied over the gap, the potential dropping from left
to right. The region described by the DFT module consists of the region between
the two electrodes plus a buffer region of 2.5 Å inside each of the
electrodes as shown in figure 1. The buffer regions are needed so that the
plane-wave-based code can better describe orbitals that are localized in the
inter-electrode gap.
Figure1. In the system, a copper atom is placed in between
two jellied surfaces 15 Å apart. A buffer with the width of 2.5 Å is
added on both sides of the gap for the DFT calculations in order to localize
the electrons. A small smoothing area between the gap and barrier is needed to
speed up calculations. The effective surface of the material is denoted z0
while zb is the edge of the positive jellied
DYNAMICS
A
dynamic Monte Carlo approach is used to calculate the shuttle dynamics, in the
spirit of the method proposed by Tully .Input parameters to this module
consist of forces and transition rates as functions of the island position. The
motion of the central island is described classically by
M(x)=F(ext){x.Q}+F(dissip){x.Q} à (1)
where
Fext are the core forces given by the previous calculations and Fdissip
is a dissipation term. The island position x(t) is calculated by numerically
integrating the equation of motion, while the island charge Q(t) is allowed to
change stochastically using the tunnel rates determined above. This results in
a coupled stochastic dynamics for the mechanical and electrical degrees of
freedom.
DISSIPATION
Since
the shuttle absorbs energy from the bias voltage, a dissipation term is
essential for the stability of the island motion .Earlier theoretical
study has mainly used viscous damping, -nX . While viscous damping is often used in macroscopic
systems, its justification is less clear on the nanoscale where the damping
arises from the interactions between the small subsystem (the shuttling island)
and the large system that is assumed to stay in a near thermal equilibrium (the
electrodes). The damping rate is therefore largest when the interaction between
the subsystems is strongest, i.e., when the shuttling island is close to the
electrodes.
The
model we employ is based on mechanical damping motivated by phonon emission
into the electrodes. An additional electronic dissipation mechanism is due to
the creation of electron–hole pairs in the electrodes. The relative importance
of these mechanisms scales roughly as (ms/m)(πh/mvd) where ms is the mass of a substrate atom, v the
impact velocity and d the distance that the shuttle impinges into the
surface .For our case, m 
0.5 ms and the electron–hole pair creation contribution
is only a few percent. Upon impact, the shuttling island exerts a force on the
electrode surface which sets a part of the surface in motion (phonon emission).
The size of the surface area that is appreciably affected by interactions with
the shuttle changes with the island–surface separation. We incorporate these
physical effects in a simple phenomenological dissipation model where the
surface is described by a single degree of freedom X(t). The mass M(x–X)
associated with this degree of freedom is determined so that the acceleration
of the single surface degree of freedom subjected to the force felt by the entire electrode
surface equals the acceleration of the surface atom that experiences the
maximum force (i.e. the surface atom closest to the shuttle island). The resulting
equation of motion for the island (x, m) and the surface (X, M) degrees of
freedom reads.
M(x)=F(tot){x}
– XF’(tot){x} – (XF’(tot){x}x’)
à (2)
where K
= kM(X)/m is the effective elastic spring constant for the surface
degree of freedom and the Θ-function restricts the flow of energy so that
island motion is dissipated. The reverse energy flow, from an oscillating
electrode surface to the shuttling island, would correspond to the island
absorbing phonons from the electrode. Since we assume that the electrode
surfaces equilibrate rapidly to their ground states (T = 0), the
absorption processes are excluded. The atomic spring constant k is given by k
= 8mvs2/a2 where a is the lattice constant and vs is
the sound velocity. All parameters for the electrode surface atoms have been
chosen to correspond to those of gold.
The choice
of the dissipation term has rather a small influence on the behavior of the
system, and it mostly changes details such as the threshold voltage for onset
of shuttling. It also renders the threshold dependent on the initial
conditions. Note that for separations beyond the range of surface forces our
dissipation model implies free propagation of the shuttle, while at very small
separations the total force is dominated by the maximum force so that M ~m and K ~k. At intermediate separations the
damping coefficient goes smoothly to zero with increasing separation.
Figure 2. The Kohn–Sham one-electron
effective potential comprises several parts rendering diverse behaviors for
different positions and charge states. The above figures are for Q = 0 and
island position Z = 10.17 Å. The z-axis is the direction of
island motion; r is parallel to the electrodes. In (a) and (b) z encompasses
the gap while (c) includes the entire DFT cell with the 2.5 Å buffer
regions on both sides of the gap. (a) Equation (1) and the bias voltage. (b) One electron
interaction with induced charge due to other system charges. (c) The effective
potential as used by DaCapo. Close to the electrodes (1) forms deep wells.
RESULTS
The
different parts of the one-electron potential are depicted in figure 2. The small widths of the form function (σ~
0.24–0.27 Å for an unperturbed pseudo potential, Q = 1··· –1 and ~
0.30–0.35 Å for the double junction potential) imply that sufficiently far
from the surface the point charge approximation would be quite accurate: for
distances ≥ 2 Å from the surface, the spatial distribution of charge has
little effect on the potential. For island positions close to the electrode
surfaces, the spread in the valence electron distribution and the rapid
saturation effectively bare the core image making the effective potential
strongly repellent.
The resulting Kohn–Sham Eigen values, depicted in figure 3, are used as energy spectra for the central
island. Comparison between the Eigen values and the electrode chemical
potentials gives the possible transitions. Full relaxation into the N/2 lowest
bands is assumed to be instantaneous, where N is the number of valence
electrons (11 for the used Cu generalized gradient approximation pseudo
potential, Q = 0). Higher bands are treated as excitations. The
temperature is taken to be zero in the treatment of tunneling events; however,
in the DFT calculation a finite temperature is used to improve convergence.
Figure 3. The lowest eight
non-spin polarized Kohn–Sham Eigen values for nine positions of the central
island. Levels indicated by filled circles are fully occupied, empty circles
half-occupied and crosses unoccupied. The upper and lower solid lines are μR
and μL respectively. (a) Q = 0. The lower lying
excitations (core positions in the left half of the gap) have been calculated
including the surface potential well near the electrode. The higher lying
excitations (core positions near the right electrode) have been calculated
without the surface potential well near the left electrode as transitions
directly from the right electrode to the left surface well have very small
tunneling rates. For Z = 7.5 Å and Z =
10.17 Å results are depicted for both approaches (solid and dashed lines).
For the positions closest to the electrodes, the width of the surface potential
wells causes a substantial drop in Eigen energies. (b) Q = 1.
A small correction is needed for some calculations in order to use
the equilibrium DFT calculations within a dynamic picture. For some island
positions and Q ≤ 0, it is energetically favorable to place some of the extra
charge in the surface potential well outside the positive electrode surface
instead of on the central island. However, the time scale for this direct
equilibration between leads is very long. In order to find the energy spectra
and orbitals that are relevant for the dynamic evolution, the surface well near
the left electrode surface is manually suppressed for core positions near the
right electrode; due to the polarity of the applied bias, similar problems do
not arise for core positions near the left electrode. The possibility of
transitions directly between the electrodes is kept, but the transition rates
are small enough to be of no importance for the results.For the tunneling rate
calculation the Kohn–Sham eigen values are regarded as electron energies, which
is known to be rigorously correct for the ionization potential involving the
highest occupied molecular orbital level, and believed to be reasonably
accurate for the other levels as well . We assume that tunneling rates are
sufficiently low so that the island fully relaxes between each tunneling event
to the configuration determined by time-independent DFT.
The time scale for this relaxation is typically in the femto
second range which is fast compared to the tunneling rates except for core
positions very close to the electrode surfaces; however, since energetics
severely limits the possible tunneling processes, the instantaneous relaxation
approximation is reasonably well justified for all core positions. For the
chosen bias voltage, the possible charge transitions for the island are 1 → 0
and 0 → 1.
It is interesting to notice the asymmetry of attainable charge
states that arises from the asymmetry of the energy spectrum of the Cu atom:
the dynamical evolution only involves charge states Q = 0 and Q
= 1 but not Q = –1. The symmetric expression for
charging energy E = Q2/2C used for larger metallic
grains is only justified if the level spacing on the island is small enough so
that the electrostatic energy scales dominate. This asymmetry implies that the
shuttling is asymmetric also in the sense that energy is absorbed from the DC
field only during half a period which makes the system more sensitive to
dissipative mechanisms. The occupied Kohn–Sham Eigen functions are identified
as d- and s-orbitals in accordance with the expected electron configuration of
Cu, 3d104 s1. Close to the electrode surfaces, the
orbitals deform against the repulsion wall. For all but the closest position to
the electrodes the 4s-orbital gives the widest electron distribution and the
largest contribution to the transition rates.
The core forces for the central positions are strongly dependent
on the delocalization of the electron distribution of the central island
(figure 4). For the positive ion with Q = 1 the
dominant force is the electrical bias while the potential of Q = –1 is a
nearly symmetric image charge potential. For Q = 0, the sign depends on
the description of the surface interactions, and with the interaction model we
have chosen the neutral atom feels a slight net force towards the negatively
charged right electrode. The repulsion from the surface is dominant for the two
outermost positions on either side giving a physisorption minimum between 3 and
5 Å from the electrode surface.
Figure 4. Total forces on the grain core
for a range of central positions. Lines between data points correspond to the
used interpolation scheme. The inset shows the forces for a wider range of
island positions
The transition rates are much less sensitive to the surface
description than the forces (figure 5). Their distance dependence is approximately
exponential as assumed by effective theories with slight saturation for
core positions nearest to the electrodes with a tunneling length that is
approximately 0.4 Å with some variation for the different allowed
transitions. Near the electrodes the energetics considerations inhibit
tunneling, as seen in figure 3, which can be viewed as a molecular equivalent
of a Coulomb blockade.
Figure 5. Transition rates: electron
current from right to left. (a) Squares: transitions from negative right lead
to the island. Close to the negative lead, current mediating transfer is
blocked by energetics (Coulomb blockade). Instead, an electron can transfer
against the bias back to the negative lead, (O). (b) Electron transitions from the
island to positive left lead.
In the dynamics simulations, the calculated forces and transition
rates are joined. The result is indeed a stable shuttling regime where Q(r)Z(t)= 0. We have performed dynamical
simulations starting from a variety of initial states, and seen that for most
starting conditions the results are quite similar: as a rule, the model does
shuttle electrons (figure 6). However, for some initial configurations such
as Z(t = 0)~ 10 Å and Q(t = 0)
= 0 the applied bias of 3 V is not sufficient to initiate
shuttling.
Figure 6. Shuttle regime for initial
conditions Z(0) = 0, Q(0) = 0 and v(0) = 0.
(a) Island position. (b) Charge state as a function of time. The shuttle
carries one electron per period. (c) Island velocity. The main acceleration and
deceleration are close to the electrodes
One of the differences between our results and previous studies is
the complexity of the forces acting on the shuttle, particularly near the
electrode surfaces. The results are quite sensitive to the details of the
surface interaction. A slightly different potential may render the forces on the
neutral atom positive over a larger range of positions, and the range of
initial conditions that would result in shuttling would be smaller, implying
that the threshold voltage for shuttling depends sensitively on the model for
surface-island interactions and on the initial conditions. The detailed
structure of the forces of the middle positions is less important after
shuttling is well established. The main forces become the close-range
exponential forces of the electrodes and the applied electric field. For the
asymmetric shuttle, energy is absorbed by the charged shuttle from the field
during half a cycle while during the other half-cycle, after an elastic
collision with the electrode surface, a neutral shuttle moves nearly freely in
the opposite direction. The energy loss during the shuttle–electrode collision
cannot exceed the energy absorbed from the field if a stable periodic motion is
to be established.
The shuttle reaches a stable shuttling
motion quickly with a current of ~0.19 μA and an amplitude of
~11.1 Å, (see figure 8). The random character of the transition
processes influences the turning points very little. There is, however, a
possibility for the system to undergo semi-stable excitations due to randomness
of transfer events and the position dependence of the energy spectrum near the
right lead (figure 9). Very close to the negative (right) lead,
there is a possibility of a process in which an electron first tunnels from the
electrode to an initially positively charged (Q = 1) shuttle that
continues to move towards the right lead, followed by tunnelling against the
bias back into the lead, and finally a new tunnelling event after the shuttle
has changed its direction of motion.
During the time that the charged shuttle
spends near the electrode surface after the second tunnelling event, it
experiences a larger force than a neutral shuttle would, which allows it to
absorb more energy from the potential and results in an enhanced shuttling
amplitude. The increase in amplitude enhances the possibility for this sequence
of three tunnelling events to take place also in the next period. The
excitation lasts until a transfer without reverse tunnelling takes place near
the negative lead. For the system we have studied, the amplitude of this
excited cycle is about 0.3 Å larger than that of the simple cycle, and the
current level is increased by approximately 20% to 0.23 μA.
DISCUSSION
The energy spectra and the (Kohn–Sham) orbitals of small molecules
near metal surfaces exhibit a rich structure and vary substantially as a
function of the molecule–metal surface separation. The transition rates are
largely exponential functions of the tunneling distance as assumed in
phenomenological theories, but the allowed transitions are determined by the
energy spectrum, and in particular near the surfaces certain transitions are
forbidden by energy considerations. This results in an asymmetry in possible
charge states and in asymmetric shuttling where energy is absorbed only during
one half cycle of the periodic motion.
The forces in the system are highly sensitive to the description
of the electrode–molecule interaction and to the electronic structure of the
shuttling object. For the particular system that we studied we found that the
attractive surface forces can be overcome by reasonable electric fields and
shuttling of a nano scale object can be established. In the stable shuttling
regime the island velocity is large enough to overcome physisorption allowing
the island to bounce between the electrodes. The details of the forces near the
middle of the gap are less important than the balance between dissipation and
short range surface forces.
The shuttle excitations depicted in figure 9 are an example of effects that arise due to the
details of the energy spectrum of a small system. For a more complicated
spectrum and larger bias voltage more phenomena of the same type can be
expected; for a slightly different model of interactions between the shuttle
and the electrodes we have even observed that the regular shuttling motion may
pass into a more chaotic behavior. Therefore, it is likely that a microscopic
picture of both forces and energy levels is paramount for both quantitative and
qualitative predictions of molecule-sized shuttles.
REFERENCES
WEB SITES:
BOOKS:
- A book on Nano Technology
- Electro Mechanical Systems
- Dictionary of Nano technology
