Recent Publications

Tchavdar Todorov

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  1. Title: Electron-phonon thermalization in a scalable method for real-time quantum dynamics

    Author(s): Rizzi V., Todorov T.N., Kohanoff J.J., Correa A.A.,

    Physical Review B, 93, No. 2 (27 January 2016)

    doi: 10.1103/PhysRevB.93.024306

    Full Text

    We present a quantum simulation method that follows the dynamics of out-of-equilibrium many-body systems of electrons and oscillators in real time. Its cost is linear in the number of oscillators and it can probe time scales from attoseconds to hundreds of picoseconds. Contrary to Ehrenfest dynamics, it can thermalize starting from a variety of initial conditions, including electronic population inversion. While an electronic temperature can be defined in terms of a nonequilibrium entropy, a Fermi-Dirac distribution in general emerges only after thermalization. These results can be used to construct a kinetic model of electron-phonon equilibration based on the explicit quantum dynamics.

  2. Title: Efficient simulations with electronic open boundaries

    Author(s): Horsfield A.P., Boleininger M., D'Agosta R., Iyer V., Thong A., Todorov T.N., White C.

    Physical Review B, 94, pp. 075118- (10 August 2016)

    doi: 10.1103/PhysRevB.94.075118

    We present a reformulation of the hairy-probe method for introducing electronic open boundaries that is appropriate for steady-state calculations involving nonorthogonal atomic basis sets. As a check on the correctness of the method we investigate a perfect atomic wire of Cu atoms and a perfect nonorthogonal chain of H atoms. For both atom chains we find that the conductance has a value of exactly one quantum unit and that this is rather insensitive to the strength of coupling of the probes to the system, provided values of the coupling are of the same order as the mean interlevel spacing of the system without probes. For the Cu atom chain we find in addition that away from the regions with probes attached, the potential in the wire is uniform, while within them it follows a predicted exponential variation with position. We then apply the method to an initial investigation of the suitability of graphene as a contact material for molecular electronics. We perform calculations on a carbon nanoribbon to determine the correct coupling strength of the probes to the graphene and obtain a conductance of about two quantum units corresponding to two bands crossing the Fermi surface. We then compute the current through a benzene molecule attached to two graphene contacts and find only a very weak current because of the disruption of the π conjugation by the covalent bond between the benzene and the graphene. In all cases we find that very strong or weak probe couplings suppress the current.

  3. Title: Length Matters: Keeping Atomic Wires in Check

    Author(s): Cunningham B., Todorov T.N., Dundas D.

    MRS Proceedings, 1753 (2015)

    doi: 10.1557/opl.2015.197

    Dynamical effects of non-conservative forces in long, defect free atomic wires are investigated. Current flow through these wires is simulated and we find that during the initial transient, the kinetic energies of the ions are contained in a small number of phonon modes, closely clustered in frequency. These phonon modes correspond to the waterwheel modes determined from preliminary static calculations. The static calculations allow one to predict the appearance of non-conservative effects in advance of the more expensive real-time simulations. The ion kinetic energy redistributes across the band as non-conservative forces reach a steady state with electronic frictional forces. The typical ion kinetic energy is found to decrease with system length, increase with atomic mass, and its dependence on bias, mass and length is supported with a pen and paper model. This paper highlights the importance of non-conservative forces in current carrying devices and provides criteria for the design of stable atomic wires.

  4. Title: Nonconservative current-driven dynamics: beyond the nanoscale

    Author(s): Cunningham B., Todorov T.N., Dundas D.

    Beilstein Journal of Nanotechnology, 6, pp. 2140-2147 (13 November 2015)

    doi: 10.3762/bjnano.6.219

    Full Text

    Long metallic nanowires combine crucial factors for nonconservative current-driven atomic motion. These systems have degenerate vibrational frequencies, clustered about a Kohn anomaly in the dispersion relation, that can couple under current to form nonequilibrium modes of motion growing exponentially in time. Such motion is made possible by nonconservative current-induced forces on atoms, and we refer to it generically as the waterwheel effect. Here the connection between the waterwheel effect and the stimulated directional emission of phonons propagating along the electron flow is discussed in an intuitive manner. Nonadiabatic molecular dynamics show that waterwheel modes self-regulate by reducing the current and by populating modes in nearby frequency, leading to a dynamical steady state in which nonconservative forces are counter-balanced by the electronic friction. The waterwheel effect can be described by an appropriate effective nonequilibrium dynamical response matrix. We show that the current-induced parts of this matrix in metallic systems are long-ranged, especially at low bias. This nonlocality is essential for the characterisation of nonconservative atomic dynamics under current beyond the nanoscale.

  5. Title: Nonconservative dynamics in long atomic wires

    Author(s): Cunningham B., Todorov T.N., Dundas D.

    Physical Review B, 90, pp. 115430 - (24 September 2014)

    doi: 10.1103/PhysRevB.90.115430

    The effect of nonconservative current-induced forces on the ions in a defect-free metallic nanowire is investigated using both steady-state calculations and dynamical simulations. Nonconservative forces were found to have a major influence on the ion dynamics in these systems, but their role in increasing the kinetic energy of the ions decreases with increasing system length. The results illustrate the importance of nonconservative effects in short nanowires and the scaling of these effects with system size. The dependence on bias and ion mass can be understood with the help of a simple pen and paper model. This material highlights the benefit of simple preliminary steady-state calculations in anticipating aspects of brute-force dynamical simulations, and provides rule of thumb criteria for the design of stable quantum wires.

  6. Title: Current-induced forces: a simple derivation

    Author(s): Todorov T.N., Dundas D., Lü J., Brandbyge M., Hedegård P.

    European Journal of Physics, 35, No. 6, pp. 065004- (02 September 2014)

    doi: 10.1088/0143-0807/35/6/065004

    We revisit the problem of forces on atoms under current in nanoscale conductors. We derive and discuss the five principal kinds of force under steady-state conditions from a simple standpoint that—with the help of background literature—should be accessible to physics undergraduates. The discussion aims at combining methodology with an emphasis on the underlying physics through examples. We discuss and compare two forces present only under current—the non-conservative electron wind force and a Lorentz-like velocity-dependent force. It is shown that in metallic nanowires both display significant features at the wire surface, making it a candidate for the nucleation of current-driven structural transformations and failure. Finally we discuss the problem of force noise and the limitations of Ehrenfest dynamics

  7. Title: Current-induced atomic dynamics, instabilities, and Raman signals: Quasiclassical Langevin equation approach

    Author(s): Lü J.T., Brandbyge M., Hedegård P., Todorov T.N., Dundas D.,

    Physical Review B, 85, pp. 245444- (25 June 2012)

    doi: 10.1103/PhysRevB.85.245444

    We derive and employ a semiclassical Langevin equation obtained from path integrals to describe the ionic dynamics of a molecular junction in the presence of electrical current. The electronic environment serves as an effective nonequilibrium bath. The bath results in random forces describing Joule heating, current-induced forces including the nonconservative wind force, dissipative frictional forces, and an effective Lorentz-type force due to the Berry phase of the nonequilibrium electrons. Using a generic two-level molecular model, we highlight the importance of both current-induced forces and Joule heating for the stability of the system. We compare the impact of the different forces, and the wide-band approximation for the electronic structure on our result. We examine the current-induced instabilities (excitation of runaway “waterwheel” modes) and investigate the signature of these in the Raman signals.

  8. Title: An ignition key for atomic-scale engines

    Author(s): Dundas D., Cunningham B., Buchanan C., Terasawa A., Anthony T Paxton A.T., Todorov T.N.

    Journal of Physics: Condensed Matter, 24, pp. 402203-1-402203-6 (2012)

    doi: 10.1088/0953-8984/24/40/402203

    A current-carrying resonant nanoscale device, simulated by non-adiabatic molecular dynamics, exhibits sharp activation of non-conservative current-induced forces with bias. The result, above the critical bias, is generalized rotational atomic motion with a large gain in kinetic energy. The activation exploits sharp features in the electronic structure, and constitutes, in effect, an ignition key for atomic-scale motors. A controlling factor for the effect is the non-equilibrium dynamical response matrix for small-amplitude atomic motion under current. This matrix can be found from the steady-state electronic structure by a simpler static calculation, providing a way to detect the likely appearance, or otherwise, of non-conservative dynamics, in advance of real-time modelling.

  9. Title: Nonconservative current-induced forces: A physical interpretation

    Author(s): Todorov T.N., Dundas D., Paxton A.T., Horsfield A.P.

    Beilstein Journal of Nanotechnology, 2, pp. 727-733 (2011)

    doi: 10.3762/bjnano.2.79

    Full Text

    We give a physical interpretation of the recently demonstrated nonconservative nature of interatomic forces in current-carrying nanostructures. We start from the analytical expression for the curl of these forces, and evaluate it for a point defect in a current-carrying system. We obtain a general definition of the capacity of electrical current flow to exert a nonconservative force, and thus do net work around closed paths, by a formal noninvasive test procedure. Second, we show that the gain in atomic kinetic energy over time, generated by nonconservative current-induced forces, is equivalent to the uncompensated stimulated emission of directional phonons. This connection with electron–phonon interactions quantifies explicitly the intuitive notion that nonconservative forces work by angular momentum transfer.

  10. Title: Nonconservative generalized current-induced forces

    Author(s): Todorov T.N., Dundas D., McEniry E.J.

    Physical Review B, 81, No. 7, Art. No. 075416 (2010)

    doi: 10.1103/PhysRevB.81.075416

    A recent result for the curl of forces on ions under steady-state current in atomic wires with noninteracting electrons is extended to generalized forces on classical degrees of freedom in the presence of mean-field electron-electron screening. Current is described within a generic multiterminal picture, forces within the Ehrenfest approximation, and screening within an adiabatic, but not necessarily spatially local, mean-field picture.

  11. Title: Density-potential mapping in time-dependent density-functional theory

    Author(s): Maitra N.T., Todorov T.N., Woodward C., Burke K.,

    Physical Review A, 81, No. 4 (2010)

    doi: 10.1103/PhysRevA.81.042525

    The key questions of uniqueness and existence in time-dependent density-functional theory are usually formulated only for potentials and densities that are analytic in time. Simple examples, standard in quantum mechanics, lead, however, to nonanalyticities. We reformulate these questions in terms of a nonlinear Schrodinger equation with a potential that depends nonlocally on the wave function.

  12. Title: Modelling non-adiabatic processes using correlated electron-ion dynamics

    Author(s): McEniry E.J., Wang Y., Dundas D., Todorov T.N., Stella L., Miranda R.P., Fisher A.J., Horsfield A.P., Race C.P., Mason D.R., Foulkes W.M.C., Sutton A.P.

    European Physical Journal B , 77, No. 3, pp. 305-329 (October 2010)

    doi: 10.1140/epjb/e2010-00280-5

    Here we survey the theory and applications of a family of methods (correlated electron-ion dynamics, or CEID) that can be applied to a diverse range of problems involving the non-adiabatic exchange of energy between electrons and nuclei. The simplest method, which is a paradigm for the others, is Ehrenfest Dynamics. This is applied to radiation damage in metals and the evolution of excited states in conjugated polymers. It is unable to reproduce the correct heating of nuclei by current carrying electrons, so we introduce a moment expansion that allows us to restore the spontaneous emission of phonons. Because of the widespread use of Non-Equilibrium Green's Functions for computing electric currents in nanoscale systems, we present a comparison of this formalism with that of CEID with open boundaries. When there is strong coupling between electrons and nuclei, the moment expansion does not converge. We thus conclude with a reworking of the CEID formalism that converges systematically and in a stable manner.

  13. Title: Ring currents in azulene

    Author(s): Paxton A.T., Todorov T.N., Elena A.M.

    Chemical Physics Letters, 483, No. 1-3, pp. 154-158 (2009)

    doi: 10.1016/j.cplett.2009.10.041

    Full Text

    We propose a self consistent polarisable ion tight binding theory for the study of push–pull processes in aromatic molecules. We find that the method quantitatively reproduces ab initio calculations of dipole moments and polarisability. We apply the scheme in a simulation which solves the time dependent Schrödinger equation to follow the relaxation of azulene from the second excited to the ground states. We observe rather spectacular oscillating ring currents which we explain in terms of interference between the HOMO and LUMO states.

  14. Title: Current-driven atomic waterwheels

    Author(s): Dundas D., McEniry E.J., Todorov T.N.

    Nature Nanotechnology, 4, No. 2, pp. 99-102 (2009)

    doi: 10.1038/nnano.2008.411

    A current induces forces on atoms inside the conductor that carries it. It is now possible to compute these forces from scratch, and to perform dynamical simulations of the atomic motion under current. One reason for this interest is that current can be a destructive force—it can cause atoms to migrate, resulting in damage and in the eventual failure of the conductor. But one can also ask, can current be made to do useful work on atoms? In particular, can an atomic-scale motor be driven by electrical current, as it can be by other mechanisms? For this to be possible, the current-induced forces on a suitable rotor must be non-conservative, so that net work can be done per revolution. Here we show that current-induced forces in atomic wires are not conservative and that they can be used, in principle, to drive an atomic-scale waterwheel.

  15. Title: Current-assisted cooling in atomic wires

    Author(s): McEniry E.J., Todorov T.N., Dundas D.,

    Journal of Physics: Condensed Matter, 21, No. 19 (2009)

    doi: 10.1088/0953-8984/21/19/195304

    The effects of inelastic interactions between current-carrying electrons and vibrational modes of a nanoscale junction are a major limiting factor on the stability of such devices. A method for dynamical simulation of inelastic electron-ion interactions in nanoscale conductors is applied to a model system consisting of an adatom bonded to an atomic wire. It is found that the vibrational energy of such a system may decrease under bias, and furthermore that, as the bias is increased, the rate of cooling, within certain limits, will increase. This phenomenon can be understood qualitatively through low-order perturbation theory, and is due to the presence of an anti-resonance in the transmission function of the system at the Fermi level. Such current-assisted cooling may act as a stabilization mechanism, and may form the basis for a nanoscale cooling 'fan'.

  16. Title: Newtonian origin of the spin motive force in ferromagnetic atomic wires

    Author(s): Stamenova M., Todorov T.N., Sanvito S.,

    Physical Review B, 77, No. 5 (2008)

    doi: 10.1103/PhysRevB.77.054439

    We demonstrate numerically the existence of a spin-motive force acting on spin carriers when moving in a time and space dependent internal field. This is the case for electrons in a one-dimensional wire with a precessing domain wall. The effect can be explained solely by adiabatic dynamics and is shown to exist for both classical and quantum systems.

  17. Title: Inelastic quantum transport in nanostructures: The self-consistent Born approximation and correlated electron-ion dynamics

    Author(s): McEniry E.J., Frederiksen T., Todorov T.N., Dundas D., Horsfield A.P.,

    Physical Review B, 78, No. 3 (2008)

    doi: 10.1103/PhysRevB.78.035446

    A dynamical method for inelastic transport simulations in nanostructures is compared to a steady-state method based on nonequilibrium Green's functions. A simplified form of the dynamical method produces, in the steady state in the weak-coupling limit, effective self-energies analogous to those in the Born approximation due to electron-phonon coupling. The two methods are then compared numerically on a resonant system consisting of a linear trimer weakly embedded between metal electrodes. This system exhibits an enhanced heating at high biases and long phonon equilibration times. Despite the differences in their formulation, the static and dynamical methods capture local current-induced heating and inelastic corrections to the current with good agreement over a wide range of conditions, except in the limit of very high vibrational excitations where differences begin to emerge.

  18. Title: Correlated electron-ion dynamics in metallic systems

    Author(s): Horsfield A.P., Finnis M., Foulkes M., LePage J., Mason D., Race C., Sutton A.P., Bowler D.R., Fisher A.J., Miranda R., Stella L., Stoneham A.M., Dundas D., McEniry E., Todorov T.N., Sanchez C.G.,

    Computational Materials Science, 44, No. 1, pp. 16-20 (November 2008)

    doi: 10.1016/j.commatsci.2008.01.055

    In this paper we briefly discuss the problem of simulating non-adiabatic processes in systems that are usefully modelled using molecular dynamics. In particular we address the problems associated with metals, and describe two methods that can be applied: the Ehrenfest approximation and correlated electron-ion dynamics (CEID). The Ehrenfest approximation is used to successfully describe the friction force experienced by an energetic particle passing through a crystal, but is unable to describe the heating of a wire by an electric current. CEID restores the proper heating. (C) 2008 Elsevier B.V. All rights reserved.

  19. Title: Structure-related effects on the domain wall migration in atomic point contacts

    Author(s): Stamenova M., Sahoo S., Todorov T.N., Sanvito S.

    Journal Of Magnetism And Magnetic Materials, 316, No. 2, pp. E934-E936 (SEP 2007)

    doi: 10.1016/j.jmmm.2007.03.147

    We investigate the interplay between magnetic and structural dynamics in magnetic point contacts under current-carrying conditions. In particular we address the dependence of the results upon the specific parameterization of the elastic properties of the material. Our analysis shows a strong influence of the structural relaxation on the energy barrier for domain wall migration, but a negligible dependence of the mechanical properties on the magnetic state. These results are stable against the various material specific choices of parameters describing different transition metals. (c) 2007 Elsevier B.V. All rights reserved.

  20. Title: Dynamical simulation of inelastic quantum transport

    Author(s): McEniry E.J., Bowler D.R., Dundas D., Horsfield A.P., Sanchez C.G., Todorov T.N.

    Journal Of Physics-Condensed Matter, 19, No. 19, Art. No. 196201 (MAY 16 2007)

    doi: 10.1088/0953-8984/19/19/196201

    A method for correlated quantum electron-ion dynamics is combined with a method for electronic open boundaries to simulate in real time the heating, and eventual equilibration at an elevated vibrational energy, of a quantum ion under current flow in an atomic wire, together with the response of the current to the ionic heating. The method can also be used to extract inelastic current voltage corrections under steady-state conditions. However, in its present form the open-boundary method contains an approximation that limits the resolution of current-voltage features. The results of the simulations are tested against analytical results from scattering theory. Directions for the improvement of the method are summarized at the end.

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

    Author(s): Sanchez C.G., Stamenova M., Sanvito S., Bowler D.R., Horsfield A.P., Todorov T.N.

    Journal Of Chemical Physics, 124, No. 21, Art. No. 214708 (JUN 7 2006)

    doi: 10.1063/1.2202329

    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. (c) 2006 American Institute of Physics.

  22. Title: The transfer of energy between electrons and ions in solids

    Author(s): Horsfield A.P., Bowler D.R., Ness H., Sanchez C.G., Todorov T.N., Fisher A.J.

    Reports On Progress In Physics, 69, No. 4, pp. 1195-1234 (APR 2006)

    doi: 10.1088/0034-4885/69/4/R05

    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 nonequivalent atoms. They include both perturbative approaches (Fermi's Golden Rule and non-equilibrium Green's 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.

  23. Title: Magnetomechanical interplay in spin-polarized point contacts

    Author(s): Stamenova M., Sahoo S., Sanchez C.G., Todorov T.N., Sanvito S.

    Physical Review B, 73, No. 9, Art. No. 094439 (MAR 2006)

    doi: 10.1103/PhysRevB.73.094439

    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.

  24. Title: Current-driven magnetic rearrangements in spin-polarized point contacts

    Author(s): Stamenova M., Sanvito S., Todorov T.N.

    Physical Review B, 72, No. 13, Art. No. 134407 (OCT 2005)

    doi: 10.1103/PhysRevB.72.134407

    A method for investigating the dynamics of atomic magnetic moments in current-carrying magnetic point contacts under bias is presented. This combines the nonequilibrium Green's function (NEGF) method for evaluating the current and the charge density with a description of the dynamics of the magnetization in terms of quasistatic thermally activated transitions between stationary configurations. This method is then implemented in a tight-binding (TB) model with parameters chosen to simulate the main features of the electronic structures of magnetic transition metals. We investigate the domain wall (DW) migration in magnetic monoatomic chains sandwiched between magnetic leads, and for realistic parameters find that collinear arrangement of the magnetic moments of the chain is always favorable. Several stationary magnetic configurations are identified, corresponding to a different number of Bloch walls in the chain and to a different current. The relative stability of these configurations depends on the geometrical details of the junction and on the bias; however, we predict transitions between different configurations with activation barriers of the order of a few tens of meV. Since different magnetic configurations are associated with different resistances, this suggests an intrinsic random telegraph noise at microwave frequencies in the I-V curves of magnetic atomic point contacts at room temperature. Finally, we investigate whether or not current-induced torques are conservative.

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

    Author(s): Horsfield A.P., Bowler D.R., Fisher A.J., Todorov T.N., Sanchez C.G.

    Journal Of Physics-Condensed Matter, 17, No. 30, pp. 4793-4812 (AUG 3 2005)

    doi: 10.1088/0953-8984/17/30/006

    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.

  26. Title: Correlated electron-ion dynamics with open boundaries: formalism

    Author(s): Bowler D.R., Horsfield A.P., Sanchez C.G., Todorov T.N.

    Journal Of Physics-Condensed Matter, 17, No. 25, pp. 3985-3995 (JUN 29 2005)

    doi: 10.1088/0953-8984/17/25/024

    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.

  27. Title: Transport in nanoscale systems: the microcanonical versus grand-canonical picture

    Author(s): Di Ventra M., Todorov T.N.

    Journal Of Physics-Condensed Matter, 16, No. 45, pp. 8025-8034 (NOV 17 2004)

    doi: 10.1088/0953-8984/16/45/024

    We analyse a picture of transport in which two large but finite charged electrodes discharge across a nanoscale junction. We identify a functional whose minimization, within the space of all bound many-body wavefunctions, defines an instantaneous steady state. We also discuss factors that favour the onset of steady-state conduction in such systems, make a connection with the notion of entropy, and suggest a novel source of steady-state noise. Finally, we prove that the true many-body total current in this closed system is given exactly by the one-electron total current, obtained from time-dependent density-functional theory.

  28. Title: Beyond Ehrenfest: correlated non-adiabatic molecular dynamics

    Author(s): Horsfield A.P., Bowler D.R., Fisher A.J., Todorov T.N., Sanchez C.G.

    Journal Of Physics-Condensed Matter, 16, No. 46, pp. 8251-8266 (NOV 24 2004)

    doi: 10.1088/0953-8984/16/46/012

    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.

  29. Title: A Maxwell relation for current-induced forces

    Author(s): Sutton A.P., Todorov T.N.

    Molecular Physics, 102, No. 9-10, pp. 919-925 (MAY 10 2004)

    doi: 10.1080/00268970410001703354

    A Maxwell relation is presented involving current-induced forces. It provides a new physical picture of the origin of current-induced forces and in the small-voltage limit it enables the identification of a simple thermodynamic potential which drives electromigration. The question of whether current-induced forces are conservative or non-conservative is discussed briefly in the light of these insights.

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

    Author(s): Horsfield A.P., Bowler D.R., Fisher A.J., Todorov T.N., Montgomery M.J.

    Journal Of Physics-Condensed Matter, 16, No. 21, pp. 3609-3622 (JUN 2 2004)

    doi: 10.1088/0953-8984/16/21/010

    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.

  31. Title: Are current-induced forces conservative?

    Author(s): Di Ventra M., Chen Y.C., Todorov T.N.

    Physical Review Letters, 92, No. 17, Art. No. 176803 (APR 30 2004)

    doi: 10.1103/PhysRevLett.92.176803

    The expression for the force on an ion in the presence of current can be derived from first principles without any assumption about its conservative character. However, energy functionals have been constructed that indicate that this force can be written as the derivative of a potential. On the other hand, there exist specific arguments that strongly suggest the contrary. We propose physical mechanisms that invalidate such arguments and demonstrate their existence with first-principles calculations. While our results do not constitute a formal resolution to the fundamental question of whether current-induced forces are conservative, they represent a substantial step forward in this direction.

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

    Author(s): Montgomery M.J., Todorov T.N.

    Journal Of Physics-Condensed Matter, 15, No. 50, pp. 8781-8795 (DEC 24 2003)

    doi: 10.1088/0953-8984/15/50/011

    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.

  33. Title: Inelastic current-voltage spectroscopy of atomic wires

    Author(s): Montgomery M.J., Hoekstra J., Todorov T.N., Sutton A.P.

    Journal Of Physics-Condensed Matter, 15, No. 4, pp. 731-742 (FEB 5 2003)

    doi: 10.1088/0953-8984/15/4/312

    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.

  34. Title: Reply to comment on 'Counterbalancing forces in electromigration'

    Author(s): Hoekstra J., Sutton A.P., Todorov T.N.

    Journal Of Physics-Condensed Matter, 14, No. 25, pp. 6603-6604 (JUL 1 2002)

    doi: 10.1088/0953-8984/14/25/327

    Reply to comment by K-H W Chu.

  35. Title: Power dissipation in nanoscale conductors

    Author(s): Montgomery M.J., Todorov T.N., Sutton A.P.

    Journal Of Physics-Condensed Matter, 14, No. 21, pp. 5377-5389 (JUN 3 2002)

    doi: 10.1088/0953-8984/14/21/312

    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.

  36. Title: Tight-binding simulation of current-carrying nanostructures

    Author(s): Todorov T.N.

    Journal Of Physics-Condensed Matter, 14, No. 11, pp. 3049-3084 (MAR 25 2002)

    doi: 10.1088/0953-8984/14/11/314

    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.

  37. Title: Counterbalancing forces in electromigration

    Author(s): Hoekstra J., Sutton A.P., Todorov T.N.

    Journal Of Physics-Condensed Matter, 14, No. 6, pp. L137-L140 (FEB 18 2002)

    doi: 10.1088/0953-8984/14/6/101

    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.

  38. Title: Time-dependent tight binding

    Author(s): Todorov T.N.

    Journal Of Physics-Condensed Matter, 13, No. 45, pp. 10125-10148 (NOV 12 2001)

    doi: 10.1088/0953-8984/13/45/302

    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.

  39. Title: A simple model of atomic interactions in noble metals based explicitly on electronic structure

    Author(s): Sutton A.P., Todorov T.N., Cawkwell M.J., Hoekstra J.

    Philosophical Magazine A-Physics Of Condensed Matter Structure Defects And Mechanical Properties, 81, No. 7, pp. 1833-1848 (JUL 2001)

    doi: 10.1080/01418610108216639

    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.

  40. Title: Quantum electronics - Nanotubes go ballistic

    Author(s): White C.T., Todorov T.N.

    Nature, 411, No. 6838, pp. 649-651 (JUN 2001)

    doi: 10.1038/35079720

  41. Title: Current-induced embrittlement of atomic wires

    Author(s): Todorov T.N., Hoekstra J., Sutton A.P.

    Physical Review Letters, 86, No. 16, pp. 3606-3609 (APR 16 2001)

    doi: 10.1103/PhysRevLett.86.3606

    Recent experiments suggest that gold single-atom contacts and atomic chains break at applied voltages of 1 to 2 V. In order to understand why current flow affects these defect-free conductors, we have calculated the current-induced forces on atoms in a Au chain between two Au electrodes. These forces are not by themselves sufficient to rupture the chain. However, the current reduces the work to break the chain, which results in a dramatic increase in the probability of thermally activated spontaneous fracture of the chain. This current-induced embrittlement poses a fundamental limit to the current-carrying capacity of atomic wires.

  42. Title: Non-linear conductance of disordered quantum wires

    Author(s): Todorov T.N.

    Journal Of Physics-Condensed Matter, 12, No. 42, pp. 8995-9006 (OCT 23 2000)

    doi: 10.1088/0953-8984/12/42/306

    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.

  43. Title: Electromigration of vacancies in copper

    Author(s): Hoekstra J., Sutton A.P., Todorov T.N., Horsfield A.P.

    Physical Review B, 62, No. 13, pp. 8568-8571 (OCT 1 2000)

    doi: 10.1103/PhysRevB.62.8568

    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.

  44. Title: Current-induced forces in atomic-scale conductors

    Author(s): Todorov T.N., Hoekstra J., Sutton A.P.

    Philosophical Magazine B-Physics Of Condensed Matter Statistical Mechanics Electronic Optical And Magnetic Properties, 80, No. 3, pp. 421-455 (MAR 2000)

    doi: 10.1080/13642810008208601

    We present a self-consistent tight-binding formalism to calculate the forces on individual atoms due to the flow of electrical current in atomic-scale conductors. Simultaneously with the forces, the method yields the local current density and the local potential in the presence of current flow, allowing a direct comparison between these quantities. The method is applicable to structures of arbitrary atomic geometry and can be used to model current-induced mechanical effects in realistic nanoscale junctions and wires. The formalism is implemented within a simple Is tight-binding model and is applied to two model structures; atomic chains and a nanoscale wire containing a vacancy.