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.

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.

Electron-phonon interaction in atomic-scale conductors: Einstein oscillators versus full phonon modes

Two extreme pictures of electron-phonon interactions in nanoscale conductors are compared: one in which the vibrations are treated as independent Einstein atomic oscillators, and one in which electrons are allowed to couple to the full, extended phonon modes of the conductor. It is shown that, under a broad range of conditions, the full-mode picture and the Einstein picture produce essentially the same net power at any given atom in the nanojunction. The two pictures begin to differ significantly in the limit of low lattice temperature and low applied voltages, where electron-phonon scattering is controlled by the detailed phonon energy spectrum. As an illustration of the behaviour in this limit, we study the competition between trapped vibrational modes and extended modes in shaping the inelastic current-voltage characteristics of one-dimensional atomic wires.

Inelastic current-voltage spectroscopy of atomic wires

A tight-binding model is developed to describe the electron-phonon coupling in atomic wires under an applied voltage and to model, their inelastic current-voltage spectroscopy. Particular longitudinal phonons are found to have greatly enhanced coupling to the electronic states of the system. This leads to a large drop in differential conductance at threshold energies associated with these phonons. It is found that with increasing tension these energies decrease, while the size of the conductance drops increases, in agreement with experiment.

Reply to comment on 'Counterbalancing forces in electromigration'

Reply to comment by K-H W Chu.

Power dissipation in nanoscale conductors

A previous tight-binding model of power dissipation in a nanoscale conductor under an applied bias is extended to take account of the local atomic topology and the local electronic structure. The method is used to calculate the power dissipated at every atom in model nanoconductor geometries: a nanoscale constriction, a one-dimensional atomic chain between two electrodes with a resonant double barrier, and an irregular nanowire with sharp corners. The local power is compared with the local current density and the local density of states. A simple relation is found between the local power and the current density in quasiballistic geometries. A large enhancement in the power at special atoms is found in cases of resonant and anti-resonant transmission. Such systems may be expected to be particularly unstable against current-induced modifications.

Tight-binding simulation of current-carrying nanostructures

The tight-binding (TB) approach to the modelling of electrical conduction in small structures is introduced. Different equivalent forms of the TB expression for the electrical current in a nanoscale junction are derived. The use of the formalism to calculate the current density and local potential is illustrated by model examples. A first-principles time-dependent TB formalism for calculating current-induced forces and the dynamical response of atoms is presented. An earlier expression for current-induced forces under steady-state conditions is generalized beyond local charge neutrality and beyond orthogonal TB. Future directions in the modelling of power dissipation and local heating in nanoscale conductors are discussed.

Counterbalancing forces in electromigration

In electromigration (EM) experiments on metallic wires, a flux of atoms can lead to motion of the centre of mass (COM) of the wire. Hence, it may be tempting to assume that the flow of current produces a net force on the wire as a whole. We point out, on the basis of known momentum-balance arguments, that the net force on a metallic wire due to a passing steady-state current is zero. This is possible, because in addition to EM driving forces, acting on scattering centres, there are counterbalancing forces, acting on the rest of the system. Drift of the COM in EM experiments occurs inevitably because the substrate keeps the crystal lattice of the wire fixed, while allowing diffusion of defects in the bulk of the wire. This drift is not evidence for a net force on the wire.

Time-dependent tight binding

Starting from a Lagrangian mean-field theory, a set of time-dependent tight-binding equations is derived to describe dynamically and self-consistently an interacting system of quantum electrons and classical nuclei. These equations conserve norm, total energy and total momentum. A comparison with other tight-binding models is made. A previous tight-binding result for forces on atoms in the presence of electrical current flow is generalized to the time-dependent domain and is taken beyond the limit of local charge neutrality.

A simple model of atomic interactions in noble metals based explicitly on electronic structure

A total energy tight-binding model with a basis of just one s state per atom is introduced. It is argued that this simplest of all tight-binding models provides a surprisingly good description of the structural stability and elastic constants of noble metals. By assuming inverse power scaling laws for the hopping integrals and the repulsive pair potential, it is shown that the density matrix in a perfect primitive crystal is independent of volume, and structural energy differences and equations of state are then derived analytically. The model is most likely to be of use when one wishes to consider explicitly and self-consistently the electronic and atomic structures of a generic metallic system, with the minium of computation expense. The relationship to the free-electron jellium model is described. The applicability of the model to other metals is also considered briefly.

Non-linear conductance of disordered quantum wires

The self-consistent electron potential in a current-carrying disordered quantum wire is spatially inhomogeneous due to the formation of resistivity dipoles across scattering centres. In this paper it is argued that these inhomogeneities in the potential result in a suppression of the differential conductance of such a wire at finite applied voltage. A semi-classical argument allows this suppression, quadratic in the voltage, to be related directly to the amount of intrinsic defect scattering in the wire. This result is then tested against numerical calculations.

Electromigration of vacancies in copper

The total current-induced force on atoms in a Cu wire containing a vacancy are calculated using the self consistent one-electron density matrix in the presence of an electric current, without separation into electron-wind and direct forces. By integrating the total current-induced force, the change in vacancy migration energy due to the current is calculated. We use the change in migration energy with current to infer an effective electromigration driving force F-e. Finally, we calculate the proportionality constant rho* between F-e and the current density in the wire.

Magnetomechanical interplay in spin-polarized point contacts

We investigate the interplay between magnetic and structural dynamics in ferromagnetic atomic point contacts. In particular, we look at the effect of the atomic relaxation on the energy barrier for magnetic domain wall migration and, reversely, at the effect of the magnetic state on the mechanical forces and structural relaxation. We observe changes of the barrier height due to the atomic relaxation up to 200%, suggesting a very strong coupling between the structural and the magnetic degrees of freedom. The reverse interplay is weak; i.e., the magnetic state has little effect on the structural relaxation at equilibrium or under nonequilibrium, current-carrying conditions.