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. 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.

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.

A cornerstone 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.

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.

Current-driven atomic waterwheels
D. Dundas, E.J. McEniry and T.N. Todorov
Nature Nanotechnology, 4, 99-102 (2009).

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 modelling this problem, moving from steady-state to dynamical quantum-mechanical methods. 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.

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, in which two ‘boxes’ of electrons, with an initial population imbalance, 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

Staff involved

Tchavdar Todorov
Daniel Dundas