Andrew P Horsfield

Beyond Ehrenfest: correlated non-adiabatic molecular dynamics

A method for introducing correlations between electrons and ions that is computationally affordable is described. The central assumption is that the ionic wavefunctions are narrow, which makes possible a moment expansion for the full density matrix. To make the problem tractable we reduce the remaining many-electron problem to a single-electron problem by performing a trace over all electronic degrees of freedom except one. This introduces both one- and two-electron quantities into the equations of motion. Quantities depending on more than one electron are removed by making a Hartree–Fock approximation. Using the first-moment approximation, we perform a number of tight binding simulations of the effect of an electric current on a mobile atom. The classical contribution to the ionic kinetic energy exhibits cooling and is independent of the bias. The quantum contribution exhibits strong heating, with the heating rate proportional to the bias. However, increased scattering of electrons with increasing ionic kinetic energy is not observed. This effect requires the introduction of the second moment.

The transfer of energy between electrons and ions in solids

In this review we consider those processes in condensed matter that involve the irreversible flow of energy between electrons and nuclei that follows from a system being taken out of equilibrium. We survey some of the more important experimental phenomena associated with these processes, followed by a number of theoretical techniques for studying them. The techniques considered are those that can be applied to systems containing many non-equivalent atoms. They include both perturbative approaches (Fermis Golden Rule and non-equilibrium Greens functions) and molecular dynamics based (the Ehrenfest approximation, surface hopping, semi-classical Gaussian wavefunction methods and correlated electron–ion dynamics). These methods are described and characterized, with indications of their relative merits.

Correlated electron-ion dynamics: the excitation of atomic motion by energetic electrons

Correlated electron-ion dynamics (CEID) is an extension of molecular dynamics that allows us to introduce in a correct manner the exchange of energy between electrons and ions. The formalism is based on a systematic approximation: small amplitude moment expansion. This formalism is extended here to include the explicit quantum spread of the ions and a generalization of the Hartree-Fock approximation for incoherent sums of Slater determinants. We demonstrate that the resultant dynamical equations reproduce analytically the selection rules for inelastic electron-phonon scattering from perturbation theory, which control the mutually driven excitations of the two interacting subsystems. We then use CEID to make direct numerical simulations of inelastic current-voltage spectroscopy in atomic wires, and to exhibit the crossover from ionic cooling to heating as a function of the relative degree of excitation of the electronic and ionic subsystems.

Molecular conduction: Do time-dependent simulations tell you more than the Landauer approach?

A dynamical method for simulating steady-state conduction in atomic and molecular wires is presented which is both computationally and conceptually simple. The method is tested by calculating the current-voltage spectrum of a simple diatomic molecular junction, for which the static Landauer approach produces multiple steady-state solutions. The dynamical method quantitatively reproduces the static results and provides information on the stability of the different solutions.

Power dissipation in nanoscale conductors: classical, semi-classical and quantum dynamics

Modelling Joule heating is a difficult problem because of the need to introduce correct correlations between the motions of the ions and the electrons. In this paper we analyse three different models of current induced heating (a purely classical model, a fully quantum model and a hybrid model in which the electrons are treated quantum mechanically and the atoms are treated classically). We find that all three models allow for both heating and cooling processes in the presence of a current, and furthermore the purely classical and purely quantum models show remarkable agreement in the limit of high biases. However, the hybrid model in the Ehrenfest approximation tends to suppress heating. Analysis of the equations of motion reveals that this is a consequence of two things: the electrons are being treated as a continuous fluid and the atoms cannot undergo quantum fluctuations. A means for correcting this is suggested.

Open-boundary Ehrenfest molecular dynamics: towards a model of current induced heating in nanowires

We present a time-dependent method based on the single-particle electron density matrix that allows the electronic and ionic degrees of freedom to be modelled within the Ehrenfest approximation in the presence of open boundaries. We describe a practical implementation using tight binding, and use it to investigate steady-state conduction through a single-atom device and to perform molecular dynamics. We find that in the Ehrenfest approximation an electric current allows both ionic heating and cooling to take place, depending on the bias.

Correlated electron-ion dynamics with open boundaries: formalism

We extend a new formalism, which allows correlated electron-ion dynamics to be applied to the problem of open boundary conditions. We implement this at the first moment level (allowing heating of ions by electrons) and observe the expected cooling in the classical part of the ionic kinetic energy and current-induced heating in the quantum contribution. The formalism for open boundaries should be easily extended to higher moments of the correlated electron-ion fluctuations.

Density functional calculations of surface free energies

We propose a general method of thermodynamic integration to find the free energy of a surface, where our integration parameter is taken to be the strain on the unit cell of the system (which in the example presented in this paper is simply the extension of the unit cell along the normal to the surface), and the integration is performed over the thermal average stress from a molecular dynamics run. In order to open up a vacuum gap in a continuous and reversible manner, an additional control interaction has been introduced. We also use temperature integration to find a linear relation for the temperature dependence of the free surface energy. These methods have been applied to the titanium dioxide (110) surface, using first principles density functional theory. A proof of principle calculation for zero temperature shows excellent agreement between the integral calculation and the difference in energy calculated by the DFT program. Calculations that have been performed at 295 and 1000 K give excellent agreement between the two integration methods.

An Open-Boundary, Time-Dependent Technique for Calculating Currents in Nanowires

In this paper we briefly review the current state of models for computing electrical conduction in nanoscale devices, highlighting the progress made, but also some limitations still present. We then summarise our recent novel theory that allows the simultaneous evolution of the electronic and ionic degrees of freedom to be modelled within the Ehrenfest approximation in the presence of open boundaries. We describe our practical implementation using tight binding and use this theory to investigate steady-state conduction through an atomic scale device. We then use the model to investigate two systems not accessible with other contemporary techniques: the response of a nano-device to a rapidly varying external field, and non-adiabatic molecular dynamics in the presence of a current.