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Current-driven runaway instabilities in molecular bridges – DFT/NEGF studies. Jing Tao Lü 1 , Mads Brandbyge 1 , Per Hedegård 2. Dept . of Micro and Nanotechnology, DTU-Nanotech, Technical University of Denmark (DTU) Niels Bohr Institute, University of Copenhagen
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Current-driven runaway instabilities in molecular bridges– DFT/NEGF studies Jing Tao Lü1, Mads Brandbyge1, Per Hedegård2 • Dept. of Micro and Nanotechnology, DTU-Nanotech, Technical University of Denmark (DTU) • Niels Bohr Institute, University of Copenhagen • Acknowledgement: Prof. Jian-Sheng Wang • Prof. Baowen Li The Third International Workshop onTransmission of Information and Energy in Nonlinear and Complex Systems (TIENCS), Singapore, 5-9 July, 2010 • TexPoint fonts used in EMF. • Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAAAA
Outline • Molecular dynamics and electrical current: Motivation • Non-conservative (”Water-wheel”) forces • Our approach: Semi-classical Langevin equation - Current-induced forces, run-away modes • DFT-NEGF calculations on atomic chains • Conclusion
Motivation • How much will the current excite vibrations? Stability! • Can the current be used to drive functions? Anharmonic coupling lower frequency vibrations ”High frequency” vibrations Harmonic coupling to Electrode phonons Electrons Molecular Dynamics in the presence of current
Current-induced ”decomposition” of C60 Cu(110) Before high current (micro amps) After high current (micro amps)
Graphene: Current-induced edge modification Scenario based on DFT calculations: Heating of localized edge-vibrations M. Engelund et al., Phys. Rev. Lett. 104, 036807 (2010) e Jia et al., Science, 2009, 323, 1701 Estimated mode heating using LOE-NEGF-DFT: Frederiksen et al., Phys Rev. B 75, 205413 (2007) Brandbyge et al., Phys Rev. B 65, 165401 (2002)
Joule heating from Raman Nano Letters, 8, 919 (2008) NATURE NANOTECH., 3, 727, 2008
Outline • Molecular dynamics and electrical current: Motivation • Non-conservative (”Water-wheel”) forces • Our approach: Semi-classical Langevin equation - Current-induced forces • Example: DFT-NEGF calculations on atomic chains • Conclusion
Populate Depopulate Current-induced forces Energy mL EF Anti-Bonding mR Bonding • Conservative? • - No(Sorbello, Sol. State Phys. 51, 159, 1998) • - Maybe (Di Ventra, Chen, Todorov, Phys Rev Lett 92, 176803, 2004) • - No(Dundas, McEniry, Todorov, NATURE NANOTECH 4, 99, 2009) Bond in the contact L Electrode R Electrode M. Brandbygeet al., PRB 67, 193104 (2003)
Tuning a: W1> W2 W1= W2 y x Current-driven atomic water-wheels Dundas, McEniry, Todorov, NATURE NANOTECH 4, 99, 2009 y a 2D model x current Energy-transfer mechanism from current to atomic motion different from Joule heating - important for actual systems??
Outline • Molecular dynamics and electrical current: Motivation • Non-conservative (”Water-wheel”) forces • Our approach: Semi-classical Langevin equation - Current-induced forces • Example: DFT-NEGF calculations on atomic chains • Conclusion
Classical Langevin Molecular Dynamics (MD) Stochastic equation of motion Mimic reservoir: Friction Fluctuating force
System and Reservoir • Heavy ions = System: semi-classical description, adiabatic expansion (fast electrons) • Only want influence of Reservoirs(many degrees of freedom) on system Reservoir: Electrons (non-equilibrium) System Reservoir: Phonons in electrodes (Equilibrium temperature T, harmonic)
Dynamics of System only: Propagator of Classical Adiabatic Action Corrections to Adiabaticup to 2nd order Semi-classical dynamics of the system: Stationary action Langevin equation in Q ”All possible paths” – Stochastic Noise From quantum equations to semi-classical MD • Schmit, J. Low Temp. Phys. 49, 609 (1982); A. O. Caldeira, A. J. Legget, e.g. Physica 121A, 587 (1983); • Feynman and Vernon, Ann. Phys. (N. Y.) 24, 118 (1963) System only: Influence functional from phonon baths
Model and approximations System System Electron reservoir System-reservoir coupling • 2nd order expansion of semi-classical action in M • Wide-band approximation T. Frederiksen, M. Paulsson, M. Brandbyge, A.-P. Jauho, Phys. Rev. B 75, 205413 (2007).
Equilibrium eV=0 Langevin equation for the vibrational modes (Q): Electronic Friction force Fluctuating stochastic force M. Head-Gordon, J. C. Tully, MOLECULAR-DYNAMICS WITH ELECTRONIC FRICTIONS, J. CHEM. PHYS., 103, 10137 (1995)
Non-equilibrium eV>0 Langevin equation for the vibrational modes (Q): Non-conservative ”water-wheel” forces ConservativeLorentz-like forces (”Berry phase” of electrons) Non-equilibrimfluctuating forces J.-T. Lü, M. Brandbyge, P. Hedegård, Nano Lett. 10, 1657(2010) M. Brandbyge, P. Hedegård, Phys. Rev. Lett. 72, 2919 (1994)
Adiabatic approximation: Berry phase Total Hamiltonian: Adiabatic ansatz: electrons ions Average over electrons Review by Resta, J. Phys.: Condens. Matter 12 (2000) R107–R143
Mode-mode coupling mediated by electrons I J electrons ”Water-wheel” (NC) ”Berry”
Simple 2D model: modes Non-conservative Berry May obtaingrowing solutions: ”runaway” Q-factor<0
2 1 -3 -2 -1 1 2 3 Average fluctuations Non-equilibrium fluctuating forces N Effective number of vibrational Excitations, N Effective mode temperature kT Threshold
Outline • Molecular dynamics and electrical current: Motivation • Non-conservative (”Water-wheel”) forces • Our approach: Semi-classical Langevin equation - Current-induced forces, run-away modes • DFT-NEGF calculations on atomic chains • Conclusion
Without With NC With NC+BP Runaway Atomic gold chain: Runaway modes J.-T. Lü, M. Brandbyge, P. Hedegård, Nano Lett. 10, 1657(2010) Gold Damping Acceleration (100) Normal modes Runaway mode
Without With NC With NC+BP Runaway Runaway Excitation by the fluctuating force Au Pt
Coupling to electrode phonons: Calculated Q-factors M. Engelund, M. Brandbyge, A.-P. Jauho, Phys. Rev. B, 80, 045427 (2009) Bulk Band edge • Strong, non-trivial dependence on • Chain length/strain
Summary • Semi-classical Langevin equation for MD with current • Joule heating by non-equilibrium fluctuating forces • Non-conserving (”Water-wheel”) • Conserving Lorentz-like (”Berry”) forces makes NC forces more robust • DFT-NEGF: The non-conserving forces important in realistic systems • Next step: Molecular dynamics with the Langevin equation • Experimental verification of ”runaway”??
Tacoma bridge • μL • current • μR • Tacoma bridge (Nov. 7, 1940) • A runaway mode, T= 5 s, driven by • the wind.