Dr. Tchavdar Todorov

Reader in Applied Mathematics and Theoretical Physics


+44 (0) 28 9097 6030

Atomistic Simulation Centre School of Mathematics and Physics Queen's University Belfast University Road Belfast BT7 1NN Northern Ireland

Degrees, Awards and Honours

  • 2003 | Maxwell Medal and Prize (Institute of Physics, 2003)


  • Transport in nanoscale conductors
  • Current-induced forces and current-driven atomic motion
  • Time-dependent tight binding
  • Electron-nuclear dynamics

Most Recent Publications

  1. Electron-phonon thermalization in a scalable method for real-time quantum dynamics, Physical Review B, 2016, 93, No. 2
    doi: 10.1103/PhysRevB.93.024306 Abstract 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. Efficient simulations with electronic open boundarieshttp://dx.doi.org/10.1103/PhysRevB.93.024306, Physical Review B, 2016, 94, pp. 075118
    doi: 10.1103/PhysRevB.94.075118 Abstract
    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. Length Matters: Keeping Atomic Wires in Checkhttp://dx.doi.org/10.1103/PhysRevB.94.075118, MRS Proceedings, 2015, 1753
    doi: 10.1557/opl.2015.197 Abstract
    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.

All of Tchavdar's publications




"If we take the view that quantisation of energy levels, tunnelling and interference are where quantum mechanics departs most violently from classical notions, we may ask where do the two come closest? Nowhere is this proximity more tangible than in the realm of particle interactions." Undated, Anon


This we can put to a simple test. In a random array of barriers, a combination of quantisation, tunnelling and interference generates
trapped quantum states (left); but couple this to another set of degrees of freedom, and the picture washes out (right):



Without electron-phonon interactions life would crash. For starters we’d lose our sense of smell (PRL 98 (2007) 038101) (not always a bad thing). Electrical and electronic equipment would seize up. Even if we could keep appliances going, we wouldn't be getting the energy in the forms and places needed. E-phonon scattering is what turns the laptop armies at meetings into one of today's biggest energy wasters. E-phonon scattering is how electrons and atomic vibrations exchange energy and momentum. This controls charge and energy transport in molecular and condensed matter systems, and enforces thermal equilibration. Despite their fundamental role e-phonon interactions remain a challenge. Stepping beyond the Born-Oppenheimer approximation (the basis for the vast majority of MD simulations) opens a many-body problem (even ignoring e-e interactions). The hardest case are problems involving the simultaneous coupled dynamics of the two subsystems. We work on this problem in molecular electronics and systems under irradiation. Something these two very different contexts have in common is that in each case it is possible to drive the electrons and the ionic motion very far out of equilibrium, resulting in exceptionally violent momentum and energy exchange. We've developed a scalable method for real-time quantum e-phonon dynamics that can reach significant size/time scales [eceid] . Take say a 2d quantum dot on a substrate, following electronic excitation. We can then track the subsequent relaxation back towards equilibrium, as monitored say via the evolving dot state occupancies. Notice the interplay between transitions below. A vital process here is spontaneous phonon emission. It's often the hardest process to capture, and is where mean-field electron-nuclear dynamics (aka Ehrenfest dynamics) usually breaks down.



[eceid] Valerio Rizzi, Tchavdar Todorov, Jorge Kohanoff and Alfredo Correa, PRB 93 (2016) 024306


Quantum correlated e-phonon simulations are never easy though. A short-cut which throws out some physics but makes the problem a lot easier is to keep just the 'diagonal part' of the dynamics, resulting in a set of coupled kinetic equations for the electronic and phonon occupancies [eceid] . As an example, consider a narrow-gap doped semiconducting system. Initially a population of electrons from the valence band have been excited into impurity levels just below the conduction band. Below we track the e-phonon relaxation from there. The electronic occupancies eventually relax into a thermal distribution but notice how the process is split between two very different time-scales: fast initial thermalization with the nearby conduction band, followed by much slower overall thermalization involving the far valence band.