Real-time simulation of nanoscale electronic devices

What is the thinnest wire possible? What would happen if we drove a current through it? Does
it make sense to speak of single-atom heaters, refrigerators, and watermills?

These questions form a subject often called Molecular Electronics: the study of how electrons
and heat flow in atomic-scale conductors, and what happens as a result. The subject has evolved over
decades through a series of experimental and theoretical breakthroughs, eventually enabling people to
make, investigate and simulate nature's ultimate wires. It is a rich area that appeals to experimentalists,
theorists and application-driven thinkers alike.

We work on the real-time simulation of current flow in these systems, and of the dynamics of the atoms
driven by the huge current densities possible in atomic wires. Our work uses in-house parallel computer
packages based on our own theoretical methods, and combines several strategic directions in the ASC:

time-dependent tight binding

time-dependent density-functional theory

transport theory

non-adiabatic electron-nuclear dynamics

A recent breakthrough was to prove theoretically that the forces on atoms that current flow exerts are
non-conservative, and to simulate the resultant operation of a one-atom 'waterwheel'. This work was featured
in two News and Views articles in the Nature Journals and sparked off experimental and theoretical interest
internationally. See below for this and other examples of these problems.

Atomic waterwheels

An open-boundary non-adiabatic molecular dynamics simulation of the corner atom in a bent
atomic wire. The current in the wire is in the region of 70 μ A. The atom is driven in
an expanding orbit by the non-conservative current-induced force on it. Its kinetic energy
grows exponentially in time, till other factors kick in to slow it down.

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:
E_{max}∼ ^{ 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).

Ring currents in azulene
A.T. Paxton, T.N. Todorov amd A.M. Elena Chemical Physics Letters, 483, 154-158 (2009).

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).