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Current-driven runaway instabilities in molecular bridges – DFT/NEGF studies

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

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  1. 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

  2. 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

  3. 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

  4. Current-induced ”decomposition” of C60 Cu(110) Before high current (micro amps) After high current (micro amps)

  5. 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)

  6. Joule heating from Raman Nano Letters, 8, 919 (2008) NATURE NANOTECH., 3, 727,  2008

  7. 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

  8. 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)

  9. 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??

  10. 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

  11. Classical Langevin Molecular Dynamics (MD) Stochastic equation of motion Mimic reservoir: Friction Fluctuating force

  12. 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)

  13. 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

  14. 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).

  15. 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)

  16. 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)

  17. Adiabatic approximation: Berry phase Total Hamiltonian: Adiabatic ansatz: electrons ions Average over electrons Review by Resta, J. Phys.: Condens. Matter 12 (2000) R107–R143

  18. Mode-mode coupling mediated by electrons I J electrons ”Water-wheel” (NC) ”Berry”

  19. Simple 2D model: modes Non-conservative Berry May obtaingrowing solutions: ”runaway” Q-factor<0

  20. 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

  21. 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

  22. 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

  23. Without With NC With NC+BP Runaway Runaway Excitation by the fluctuating force Au Pt

  24. 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

  25. 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”??

  26. Tacoma bridge • μL • current • μR • Tacoma bridge (Nov. 7, 1940) • A runaway mode, T= 5 s, driven by • the wind.

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