Coupled electron-ion dynamics: Introduction to CEID

1. 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 EIPAM, 19 Apr 2005

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

3. 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) EIPAM, 19 Apr 2005

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

5. 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 EIPAM, 19 Apr 2005

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

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

8. 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 Nmax states of M chosen oscillators EIPAM, 19 Apr 2005

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

10. 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: Polaron-dominated conductance (even chain): One phonon emitted EIPAM, 19 Apr 2005

11. 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. EIPAM, 19 Apr 2005

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

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

14. Is Ehrenfest good enough? In an exact calculation, would decompose general electron-ion Hamiltonian as Lowest eigenvalue of He, gives Born-Oppenheimer potential surface Expand HI and HeI about reference ionic positions R0: Full ionic heating rate is then EIPAM, 19 Apr 2005

15. Is Ehrenfest good enough? (2) In Ehrenfest approximation: expand around instantaneous average values R(t) and P(t) of ionic position and momentum: Ionic motion This term usually neglected Electronic evolution 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 EIPAM, 19 Apr 2005

16. 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 EIPAM, 19 Apr 2005

17. R Going Beyond Ehrenfest ρe varies 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. X First moments of X, P First moment approximation. EIPAM, 19 Apr 2005

18. 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 EIPAM, 19 Apr 2005

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

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

21. 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) EIPAM, 19 Apr 2005

22. 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! EIPAM, 19 Apr 2005