1. EIPAM, 19 Apr 2005 Coupled electron-ion dynamics: Introduction to CEID David Bowler [1,2], Andrew Fisher [1], Andrew Horsfield [1], Tchavdar Todorov [3], Christian Sanchez [3] [1] University College London [2] International Center for Young Scientists, NIMS, Tsukuba [3] Queen’s University Belfast

2. EIPAM, 19 Apr 2005 Those who did the work... Hervé Ness (now CEA Saclay) David Bowler (UCL and ICYS/NIMS) Thanks to EPSRC, IRC in Nanotechnology, Royal Society for funding Andrew Horsfield (UCL) Tchavdar Todorov (Belfast) Christian Sanchez (Belfast)

3. EIPAM, 19 Apr 2005 Electron-ion dynamics: context Interactions between electronic and atomic degrees of freedom important in many places in physics, chemistry and nanoscience: –Local heating in nanostructures; –Local current-voltage spectroscopy and STM-induced surface chemistry; –Decoherence of electronic processes used for quantum information processing; –Molecular electronics. C. Durkan, M. A. Schneider, and M. E. Welland, J. App. Phys. 86, 1280 (1999)

4. EIPAM, 19 Apr 2005 Models for atomic-scale electronics Rigid-molecule (elastic) transport Need to worry about: Fluctuations (e.g. ring torsions) Feedback of electrons on geometrical structure (breakdown of Born-Oppenheimer approximation) Local heating (diffusion, electromigration) X Molecular and electronic motions strongly coupled Exceptions: nanotubes, small-molecule STM (mostly) Bloch-like states

5. EIPAM, 19 Apr 2005 Overview An example of a “conventional” approach: solution of the time- independent coupled electron-lattice Schrödinger equation The CEID approach: –Aim: a Car-Parrinello-like revolution for coupled electron-ion dynamics –Analysis of the local heating problem –The Ehrenfest approximation –Going beyond Ehrenfest –First results from the DINAMO code Survey of future plans

6. EIPAM, 19 Apr 2005 Conducting polymers: the simplest model + - + - + - + - + - + - Π-electron tight-binding model linearly coupled to atomic displacements (Su, Schrieffer and Heeger, 1980)

7. EIPAM, 19 Apr 2005 The method Many ‘copies’ of electronic system with different states of vibrational excitation ‘Transitions’ mediated by annihilation/creation operators. (Bonca and Trugman, 1995)

8. EIPAM, 19 Apr 2005 The basis set Reference system: ‘Neutral’ chain (N atoms, N  electrons) Add single carrier (electron or hole) in one of N/2 states Include lowest N max states of M chosen oscillators

9. EIPAM, 19 Apr 2005 The numerics Solve for ‘embedded’ molecule Green function in presence of leads using sparse-matrix techniques: Use the Green function to ‘propagate’ incident electron wave: Use single-particle eigenstates of reference system as electronic basis (makes correct incorporation of Exclusion Principle easier) Careful identification of most important phonon modes (long- wavelength optical modes) is essential Embedding potential

10. EIPAM, 19 Apr 2005 Coherent transport with polarons Polaron moves coherently through the chain as a coupled entity Tunnelling occurs through introduction of ‘virtual polarons’ into the molecule Correlation between electron position and dimerisation

11. EIPAM, 19 Apr 2005 Polarons affect conductance Increases tunnel conductance, because carrier has to ‘borrow’ less energy to tunnel through the molecule Polaron-assisted Elastic (neutral chain) β-factor (attenuation) depends strongly on inelastic terms Elastic (charged chain)

12. EIPAM, 19 Apr 2005 Heating These large effects on current also involve a small probability of energy loss (corresponding to excitations remaining within the molecule). Dominant processes are “virtual” ones where lattice vibrations are produced and then re-emitted. Nevertheless corresponds to substantial heating rate: One phonon emitted Polaron-dominated conductance (even chain):

13. EIPAM, 19 Apr 2005 A new approach to heating: time-dependent quantum mechanics in an open system Couple time-dependent treatment of one-electron density matrix with classical dynamics for atoms in device region under instantaneous (Ehrenfest) forces Device DEnvironment E Device: Device-environment coupling: Numerical integration Analytic integration Environment: Finite-range in space and time

14. EIPAM, 19 Apr 2005 Towards CEID: time-dependent conduction model Model current as the discharge of a capacitor through a resistor. Enables incorporation of other time-dependent effects due to ions/atoms. Want a method that works for a general (possibly large) R having many almost classical degrees of freedom.

15. EIPAM, 19 Apr 2005 The Ehrenfest Approximation Simplest approach to coupled quantum-classical dynamics: Ehrenfest approximation Approximation: represent distributions of ionic positions and momenta by a single average value: True distribution of ionic positions at time t: R

16. EIPAM, 19 Apr 2005 First results (Ehrenfest approximation) Implemented in tight-binding (non-self-consistent so far): V bias =0.1V (cooling) V bias =1.0V (heating) Dynamic atoms (T initial =300K) V gate V bias =0 V bias =1.0V Static atoms: Landauer value

17. EIPAM, 19 Apr 2005 Is Ehrenfest good enough? In an exact calculation, would decompose general electron- ion Hamiltonian as Lowest eigenvalue of H e, gives Born- Oppenheimer potential surface Expand H I and H eI about reference ionic positions R 0 : Full ionic heating rate is then

18. EIPAM, 19 Apr 2005 Is Ehrenfest good enough? (2) In Ehrenfest approximation: expand around instantaneous average values R(t) and P(t) of ionic position and momentum: Ionic heating rate is now Lose correlations between electrons and ions; heating may contain large errors (or even be wrong sign) Average force from electrons This term usually neglected Ionic motion Electronic evolution

19. EIPAM, 19 Apr 2005 Is Ehrenfest good enough? (3) Calculate heating/cooling of a single Einstein oscillator, forming a 1eV potential barrier between two reservoirs and heated by electrons of different biases. Shows ionic cooling (and heating of electrons) even for biases (~1eV) much larger than initial ionic K.E. Ehrenfest approximation does not give correct physics

20. EIPAM, 19 Apr 2005 Going Beyond Ehrenfest Must keep the terms we formerly neglected. Do this by making a systematic moments expansion about the average ionic trajectory, keeping correlations between electrons and ions. First moment approximation. First moments of X, P R ρ e varies X

21. EIPAM, 19 Apr 2005 Going Beyond Ehrenfest (2) As a starting point, neglect electronic correlations, use Hartree-Fock approximation. Define Then work entirely in terms of one-particle quantities by using the extended Hartree-Fock ansatz

22. EIPAM, 19 Apr 2005 Beyond Ehrenfest – results (1) Local Heating Ionic energy change now contains original (classical) part plus new quantum part: Classical (cools ions) Quantum (heats ions) DINAMO code (Sanchez et al) Increasing bias

23. EIPAM, 19 Apr 2005 Beyond Ehrenfest – results (2) Inelastic Spectroscopy CEID (at first moment level) already contains enough information to describe IETS Sanchez, Todorov, Horsfield Expected position of inelastic peak: 0.26 V

24. EIPAM, 19 Apr 2005 Our plans We plan a three-pronged development programme for CEID over the next four years, focussing on –Implementing the second moment approximation –Local heating and vibrational spectroscopy in nanostructures –Electron-lattice coupling and degradation in conducting polymer films –Electron-ion energy transfer during radiation damage in solids We will also be working on –STM-IETS (with Geoff Thornton, Werner Hofer) –Charge transport and oxidative damage in biomolecules (with Sarah Harris, William Barford) –Decoherence induced by electron-lattice coupling in other quantum systems (e.g. dopant spins in semiconductors, quantum dots)

25. EIPAM, 19 Apr 2005 To read more: Open-boundary Ehrenfest molecular dynamics: towards a model of current induced heating in nanowires. A.P. Horsfield, D.R. Bowler and A.J. Fisher. J. Phys.: Conden. Matt. 16 L65 (2004). Power dissipation in nanoscale conductors: classical, semi-classical and quantum dynamics. A.P. Horsfield, D.R. Bowler, A.J. Fisher, T.N. Todorov and M.J. Montgomery. J. Phys.: Conden. Matt. 16 3609- 3622 (2004). Beyond Ehrenfest: correlated non-adiabatic Molecular Dynamics. A.P. Horsfield, D.R. Bowler, A.J. Fisher, T.N. Todorov, and C. Sanchez. J. Phys.: Conden. Matt. 16 8251-8266 (2004). Thank you for your attention!