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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 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 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; C. Durkan, M. A. Schneider, and M. E. Welland, J. App. Phys. 86, 1280 (1999) – Molecular electronics. EIPAM, 19 Apr 2005 Models for atomic-scale electronics Bloch-like states X Rigid-molecule (elastic) transport Molecular and electronic motions strongly coupled Need to worry about: •Fluctuations (e.g. ring torsions) •Feedback of electrons on geometrical structure (breakdown of Born-Oppenheimer approximation) •Local heating (diffusion, electromigration) Exceptions: nanotubes, small-molecule STM (mostly) EIPAM, 19 Apr 2005 Overview • An example of a “conventional” approach: solution of the timeindependent 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 Conducting polymers: the simplest model + + + + + - - + - Displacement ui Π-electron tight-binding model linearly coupled to atomic displacements (Su, Schrieffer and Heeger, 1980) K M Hˆ [t0 (ui 1 ui )](cˆi†cˆi 1 h.c.) (ui 1 ui ) 2 u 2 2 i 2 i i EIPAM, 19 Apr 2005 The method Product Hilbert space H el H vib Many ‘copies’ of electronic system with different states of vibrational excitation nq 3 nq 2 nq 1 nq 0 ‘Transitions’ mediated by annihilation/creation operators. (Bonca and Trugman, 1995) 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 Nmax states of M chosen oscillators Hilbert space of overall dimension N el ( N max 1) M 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 (charged chain) Elastic (neutral chain) β-factor (attenuation) depends strongly on inelastic terms Elastic (neutral chain) 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: Polaron-dominated conductance (even chain): IPinelastic phonon e 1 nA 15pW One phonon emitted W 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. EIPAM, 19 Apr 2005 The Ehrenfest Approximation Simplest approach to coupled quantum-classical dynamics: Ehrenfest approximation True distribution of ionic positions at time t: R Tre ˆ (t ) R Approximation: represent distributions of ionic positions and momenta by a single average value: R R (t ) Tr Rˆ (t ) EIPAM, 19 Apr 2005 First results (Ehrenfest approximation) Implemented in tight-binding (non-self-consistent so far): Static atoms: Landauer value Vgate Dynamic atoms (Tinitial=300K) Vbias=0.1V (cooling) Vbias=1.0V (heating) Vbias=0 Vbias=1.0V EIPAM, 19 Apr 2005 Is Ehrenfest good enough? In an exact calculation, would decompose general electronion Hamiltonian as Lowest eigenvalue of He, gives BornHˆ TˆI Hˆ e ( R) Oppenheimer potential surface I TI E0 ( R) E0 ( R0 ) Hˆ e ( R0 ) Hˆ e ( R) E0 ( R) Hˆ e ( R0 ) E0 ( R0 ) I Hˆ H ( R ) Hˆ I e 0 eI Expand HI and HeI about reference ionic positions R0: 1 Hˆ I Hˆ eI TˆI Xˆ I K IJ ( R ) Xˆ J Xˆ I Fel ( R ) where Xˆ I Rˆ I RI (t ) 2 I ,J I I Full ionic heating rate is then 1 Hˆ I , Hˆ eI w i PˆI ˆ I FI M I 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: R (t ) Tr Rˆ ˆ (t ) ; P (t ) Tr Pˆ ˆ (t ) Ionic motion Electronic evolution dRI P dP I ; I Tre ˆ e Fˆel ( R ) dt M I dt ˆ e i Hˆ e ( R ), ˆ e Fel ( R ), ˆ I t I with ˆ e TrI ˆ (t ) and ˆ I TrI Xˆ I ˆ Ionic heating rate is now Average force from electrons PˆI ˆ FI MI I This term usually neglected Lose correlations between electrons and ions; heating may contain large errors (or even be wrong sign) 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 EIPAM, 19 Apr 2005 Going Beyond Ehrenfest R Tre ˆ (t ) R 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 F Tre ˆ e Fˆel ( R ) Tre Kˆ IJ ˆ J with ˆ e TrI ˆ (t ) and J ˆ I TrI Xˆ I ˆ ˆ e i Hˆ e ( R ), ˆ e Fel ( R ), ˆ I t I ˆ dˆ I ˆ i H e R , ˆ I i I dt M ˆI TrI PˆI ˆ FˆI FˆI F I dˆI ˆ i i i H e R , ˆI Fˆ , ˆ e dt 2 2 ρe varies Kˆ J IJ , ˆ J First moment approximation. EIPAM, 19 Apr 2005 2 Hˆ K IJ RI RJ R R First moments of X, P Going Beyond Ehrenfest (2) As a starting point, neglect electronic correlations, use Hartree-Fock approximation. (1) NTr2,3...N [ ˆ e ] Define (2) N ( N 1)Tr3...N [ ˆ e ] etc. Then work entirely in terms of one-particle quantities by using the extended Hartree-Fock ansatz (2, HF ) (1, 2;1', 2 ') (1) (1,1') (1) (2, 2 ') (1) (1, 2 ') (1) (2,1') I(2, HF ) (1, 2;1', 2 ') I (1) (1,1') (1) (2, 2 ') (1) (1,1') I (1) (2, 2 ') I (1) (1, 2 ') (1) (2,1') (1) (1, 2 ') I (1) (2,1') I(2, HF ) (1, 2;1', 2 ') I (1) (1,1') (1) (2, 2 ') (1) (1,1')I (1) (2, 2 ') I (1) (1, 2 ') (1) (2,1') (1) (1, 2 ')I (1) (2,1') EIPAM, 19 Apr 2005 Beyond Ehrenfest – results (1) Local Heating Ionic energy change now contains original (classical) part plus new quantum part: w I 1 ( FI PI Tre FˆI ˆI ) MI Quantum (heats ions) DINAMO code (Sanchez et al) Classical (cools ions) EIPAM, 19 Apr 2005 Increasing bias 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 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 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