Large Fluctuations, Classical Activation, Quantum Tunneling, and Phase Transitions

1. Large Fluctuations, Classical Activation, Quantum Tunneling, and Phase Transitions Daniel Stein Departments of Physics and Mathematics New York University Conference on Large Deviations: Theory and Applications University of Michigan June 4-8, 2007 Collaborators: Jerome Bürki (Physics, Arizona), Andy Kent (Physics, NYU), Robert Maier (Math, Arizona),Kirsten Martens (Physics, Heidelberg), Charles Stafford (Physics, Arizona) Reference: DLS, Braz. J. Phys. 35, 242—252 (2005). Partially supported by US National Science Foundation Grants PHY009484, PHY0351964, and PHY0601179

2. Outline of Talk • Decay of monovalent metallic nanowires • Classical Activationin Stochastic Field Theories • Magnetization Reversal in Quasi-2D Nanomagnets • Experimental Evidence for the Phase Transition?

3. Why nanowires? Moore’s law International Technology Roadmap for Semiconductors: 1999 Extrapolates to 1nm technology by 2020

4. C.-H. Zhang et al., PRB 68, 165414 (2003). Theoretical stability diagram But: why should these wires exist at all? Rayleigh instability: cylindrical column of fluid held together by pairwise interactions is unstable to breakup by surface waves

5. Electron Shell Potential J. Bürki, R.E. Goldstein and C.A. Stafford, Phys. Rev. Lett. 91, 254501 (2003). J. Bürki, R.E. Goldstein and C.A. Stafford, Phys. Rev. Lett. 91, 254501 (2003).

6. Because the zero-noise dynamics are gradient, where Consider an extended system with gradient dynamics perturbed by weak spatiotemporal white noise, for example the stochastic Ginzburg-Landau equation:

7. The infinite line case (Langer `69, Callan-Coleman ’77) Let Then the stable, unstable, and saddle states are time-independent solutions of the zero-noise GL equation: That is, the states that determine the transition rates are extrema of the action.

8. Need to study extrema of the action, which are solutions of the nonlinear ODE Uniform solutions: Nonuniform (bounce*) solutions: * Or critical droplet

9. But: how do we know which is the true saddle configuration? Ans: the saddle is the lowest energy configuration with a single unstable direction. Kramers rate: G ~ G0 exp [ -DE / kBT ]

10. To compute the prefactor G0 , must examine fluctuations about the optimal escape (classical) path. This is essentially the same procedure as computing quantum corrections about the classical path in the Feynman path-integral approach to quantum mechanics.

11. _ Treat radius fluctuations as a classical field φ(z,t) on [-L/2,L/2]: R(z,t) = R0 + φ(z,t) _ Fluctuations occur in potential _ Escape rate calculable from Kramers theory: G = G exp[-DE/T] Model of nanowire `decay’ rate • Will use a continuum approach • Thermal fluctuations responsible for nucleating changes in radius 0 J. Bürki, C. Stafford, and DLS,inNoise in Complex Systems and Stochastic Dynamics II(SPIE Proceedings Series 5471, 2004), pp. 367 – 379; Phys. Rev. Lett.95, 090601-1—090601-4 (2005).

12. Putting everything together, we find:

13. Conductance histograms for Na and Au Counts (a.u.) Counts A.I. Yanson et al., Nature400, 144 (1999) E. Medina et al., Phys. Rev. Lett.91, 026802 (2003)

14. related to L via boundary conditions What about finite L? Boundary conditions: periodic, antiperiodic, Dirichlet, Neumann … In all cases, find a phase transition (asymptotically sharp second order or first order, depending on the potential). For symmetric quartic: R.S. Maier and DLS, Phys. Rev. Lett. 87, 270601 (2001). R.S. Maier and DLS, in Noise in Complex Systems and Stochastic Dynamics, (SPIE Proceedings Series 5114, 2003), pp. 67 - 78 DLS, J. Stat. Phys. 114, 1537 (2004).

15. Consider action difference (energy barrier) first:

16. Now compute the prefactor:

17. R~200 nm R~5-10 nm Magnetization reversal in nanoscale ferromagnets The stochastic dynamics are now governed by magnetization fluctuations in the Landau-Lifschitz-Gilbert equation: K. Martens, DLS, and A.D. Kent, in Noise in Complex Systems and Stochastic Dynamics III, L.B. Kish et al.,eds., (SPIE Proceedings Series, v. 5845, 2005), pp. 1-11; and Phys. Rev. B 73, 054413 (2006).

18. where and with The nonlocal magnetostatic term simplifies to a local shape anisotropy because of the quasi-2D geometry: R. Kohn and V. Slastikov, Arch. Rat. Mech. Anal.178, 227 (2005).

19. Extrema of the action satisfy

20. Instanton saddle `uniform’ saddle What determines the crossover? Permalloy ring of mean radius 200 nm at a) 52.5 mT and b) 72.5 mT

21. Has the classical activation `phase transition’ been seen?

22. J. Bürki, C.A. Stafford, and DLS, Appl. Phys. Lett. 88, 166101 (2006)

23. Conclusions • The study of the effects of small amounts of noise on fundamental processes in physical systems still contains surprises --- and many applications. • In certain classical field theories perturbed by spatiotemporal noise, an asymptotically sharp phase transition exists and is experimentally observable. • A formally similar transition exists in the classical activation → quantum tunneling transition for a number of systems. • But in this case the finiteness of ħ removes the divergence of the rate prefactor at the transition point. • Similarity between two cases may lead to increased understanding of the classical → quantum transition through experiments on magnetization reversal, nanowire decay, …

24. Goldanskii 1959: Crossover at Transition from Thermal Activation to Quantum Tunneling Affleck; Wolynes; Caldeira and Leggett; Grabert and Weiss; Larkin and Ovchinnikov; Riseborough, Hanggi, and Freidkin; Chudnovsky; Kuznetsov and Tinyakov; Kleinert and Chernyakov; Frost and Yaffe …

25. T>>T0 : Thermal activation over barrier T<<T0 : Quantum tunneling through barrier T~T0 : Crossover Solutions:

26. so …

27. So there exists amapping from the activation of classical fields to the quantum ↔ classical crossover problem: Classical Field Quantum ↔ Classical Small parameter T ħ External `control’ variable L,H,… T Periodic in L βħ But there’s also an important physical difference: T can be varied, and ħ cannot! Why does this matter? It affects the `nature’ of the `second order phase transition’.

32. Linearize the zero-noise evolution about a stationary state > 0 Stable mode < 0 Unstable mode A (meta)stable state has all > 0 ; a saddle has one < 0 . Finally, To compute the prefactor G0 , must examine fluctuations about the optimal escape (classical) path.

34. related to L via boundary conditions As an example, consider Neumann BC’s: What about finite L? Boundary conditions: periodic, antiperiodic, Dirichlet, Neumann … In all cases, find a phase transition (asymptotically sharp second order or first order, depending on the potential). For symmetric quartic: Period of sn function is 4K(m). R.S. Maier and DLS, Phys. Rev. Lett. 87, 270601 (2001). R.S. Maier and DLS, in Noise in Complex Systems and Stochastic Dynamics, (SPIE Proceedings Series 5114, 2003), pp. 67 - 78 DLS, J. Stat. Phys. 114, 1537 (2004).

35. is a classical field defined on the interval [-L/2,L/2] It is subject to a potential like or With specified boundary conditions (periodic, antiperiodic, Dirichlet, Neumann, …) classical (thermal) Now add noise … or quantum mechanical

36. Pervasive in physics (and many other fields) – controls dynamical phenomena in a wide variety of processes • Classical: Micromagnetic domain reversal, pattern nucleation, dislocation motion, nanowire instabilities, … • Quantum: Decay of the `false vacuum’, anomalous particle production, … The Kramers escape rate (Kramers, 1940): when kBT << DE, then G ~ G0 exp [ -DE / kBT ]. Also called the Arrhenius rate law when G0 is independent of T. • DE is the energy difference between the `saddle’ state and (meta)stable state • G0 governed by fluctuations about the `optimal escape path’ M.I. Freidlin and A.D. Wentzell, Random Perturbations of Dynamical Systems (Springer, 1984); W.G. Faris and G. Jona-Lasinio, J. Phys. A15, 3025 (1982).