Real time simulation of transport and
current driven dynamics in nanoscale devices
This problem has numerous attractive features as an area for
fundamental research. At its most basic level it involves the
non-equilibrium exchange of energy between current-carrying electrons
and the atoms in the nanoscale device. From a practical point of view
it has major ramifications in understanding the functioning of next
generation electronic devices. In addition, the breadth of scope of
this problem facilitates contact with the numerous experiments in the
field of conduction in atomic scale systems.
Over the last several years, we have developed a number of theoretical
and computational techniques at the ASC which has allowed us to make
major advances in the understanding of the fundamental processes
affecting the functioning of these devices. These techniques include
The use of the Wigner function for the numerical simulation of
correlated electron transport in molecules
The calculation of current noise to investigate molecular
rotation
The development of a correlated electron-ion dynamics
formalism for describing the proper exchange of energy between
electrons and ions in molecular-scale electronic devices.
In addition, these methods, together with associated parallel computer
codes developed in the ASC, have allowed us to open up new avenues of
research. One example that illustrates all of the above is the problem
- of great current interest both in our group and in other groups,
from experiment and theory - of non conservative current-induced
forces and their capacity to drive atomic-scale motors. We have shown
conclusively (mathematically and computationally) that current-induced
forces are fundamentally non-conservative in nature. Since its
publication, less than a year ago, this work has been echoed in two
New and Views articles in the Nature journals, and has sparked off
further work in Leiden (experiment) and in Copenhagen (theory).
Current-driven atomic waterwheel
An Ehrenfest dynamical simulation of a bent atomic wire was performed. The dynamics of the
corner atom in the 2D plane was investigated using a nearest-neighbour single-orbital
orthogonal tight-binding model. For an applied bias of 1V, a bend of 70o and
specific for the onsite energies of the corner atoms and its nearest neighbours, we observe
the atom spiralling outwards with its kinetic energy increasing. This is a signature of the
non-conservative nature of current-induced forces.
Heating in atomic wires
Remember these?
Experimentalists can now make the thinnest wires possible - one-atom thick, like this:
(For real-life images see Nature 395 (1998) 780, Phys. Rev. B 63 (2001) 073405.)
It is possible to pass currents through them, and large ones too: the current
densities in atomic wires can exceed those in their ancestors above by many
orders of magnitude. So if those got hot, how hot do these get?
We, together with our collaborators, work on understanding and modelling this problem,
moving from steady-state [1-3] to dynamical quantum-mechanical methods [4-10].
The simplest case is just one dynamical atom, like the swinging atom in the sketch,
heating up in the electron current. (One can work out this example by hand, so it is
a useful test case.) Here is a simulation of its vibrational energy (top) and the
current (bottom) in the 1d wire as a function of bias and time.
This 1-atom problem displays key characteristics of inelastic quantum transport.
First look at the ''big picture''. The initial energy of the atom is the quantum zero-point energy.
As current starts to flow the atom starts to heat up, eventually settling at an elevated energy.
So what is this bias-dependent steady vibrational energy? The answer depends on many factors:
the electronic structure of the nanoconductor, whether - and how - vibrations in the
atomic wire couple to phonons in the leads (ignored in this example), etc. But in certain limits (like this one)
there is an analytical upper bound on the vibrational energy per degree of freedom:
Emax∼ eV ⁄ 2 where V is the bias.
Now notice how the current drops slightly in time as the atom is heating up. This is the 1-atom analogue
of the phonon resistivity of metals: the ''hotter'' the atom the more it scatters the electrons, creating a
tiny hot resistor.
Now look closely at low bias. There is a critical bias below which there is no heating. This bias
is eV ∼ ℏ ω: the well-known threshold for the onset of inelastic
electron-phonon scattering. It results from a combination of spontaneous phonon emission
by electrons and Fermi statistics. You can see its signature also in the I-V: the arrow
is pointing at a slight kink. It corresponds to the small increase in resistance due to the
onset of inelastic scattering above the critical bias.
So with ∼ 200 lines of MATLAB one can heat up atoms one by one on a laptop
(using a simple method we are experimenting with [10]).
T. N. Todorov, Phil. Mag. B 77 (1998) 965
T. N. Todorov, J. Hoekstra and A. P. Sutton, Phys. Rev. Lett. 86 (2001) 3606
J. M. Montgomery, T. N. Todorov and A. P. Sutton, J. Phys.: Condens. Matter 14 (2002) 5377
A. P. Horsfield, D. R. Bowler, A. J. Fisher, T. N. Todorov and M. J. Montgomery, J. Phys.: Condens. Matter 16 (2004) 3609
A. P. Horsfield, D. R. Bowler, A. J. Fisher, T. N. Todorov and C. G. Sanchez, J. Phys.: Condens. Matter 17 (2005) 4793
E. J. McEniry, D. R. Bowler, D. Dundas, A. P. Horsfield, C. G. Sanchez and T. N. Todorov,
J. Phys.: Condens. Matter 19 (2007) 196201
E. J. McEniry, T. Frederiksen, T. N. Todorov, D. Dundas and A. P. Horsfield,
Phys. Rev. B 78 (2008) 035446
E. J. McEniry, T. N. Todorov and D. Dundas, J. Phys.: Condens. Matter 21 (2009) 195304
J.-T. Lu, M. Brandbyge, P. Hedegaard, T. N. Todorov and D. Dundas, arXiv:1205.0745 (2012)
T. N. Todorov and J. Kohanoff, work in progress
Bond currents in azulene
These cartoons show the development of bond currents: corresponding to those bonds
highlighted. These currents flow as the time dependent Schroedinger equation is solved
starting with coherent mixtures of one of the three states illustrated, which are
highest occupied (|0>), and lowest (|1>) and next lowest (|2>) unoccupied molecular
orbitals in azulene. In this case, all three states are made up exclusively from
electrons in the \pi-system.
The coherent mixtures are |0> + |1> (first 6 frames); |0> + |1> (next 6 frames) and
|0> + |1> (final 6 frames).
The Wigner function
The picture shows the electronic Wigner function for an atomic wire with a resonant
device, a few tens of femtoseconds after the onset of current flow. We are in the Landauer
picture of transport, in which two 'boxes' of electrons, with an initial population
imbalance between them, are allowed to 'discharge' through a nanoscale 'sample'.
The Wigner function is as close as quantum mechanics gets to classical statistical
mechanics: it is a phase-space density, except that i. it is not constrained to be
non-negative and ii. there are constraints on the allowed initial conditions. But for
large numbers of particles and/or high temperature it can look convincingly quasi-classical.
In the picture the x-axis is position along the atomic wire, and the y-axis is electron
crystal momentum. The resonant device filters out special energies/momenta, resulting in the
transmitted narrow 'beams' (bottom left for electrons, top right for holes).
Barry Dillon, Paul Delaney, Tchavdar Todorov
In terms of its place in the work of ASC as a whole, this subject
combines several threads of research in our Centre:
time-dependent tight binding theory
time-dependent density functional theory
the theory of transport
the use of molecular dynamics (in this case, under
non-equilibrium conditions with electronic open
boundaries)
to help understand the behaviour of individual atoms and groups of
atoms, in the problem at hand.
