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

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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, 242252
(2005).
Partially supported by US National Science
Foundation Grants PHY009484, PHY0351964, and
PHY0601179
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Outline of Talk

• Decay of monovalent metallic nanowires

• Classical Activation in Stochastic Field
  Theories

• Magnetization Reversal in Quasi-2D Nanomagnets

• Experimental Evidence for the Phase Transition?

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Why nanowires? Moores law
International Technology Roadmap for
Semiconductors 1999
Extrapolates to 1nm technology by 2020
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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
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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).
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Consider an extended system with gradient
dynamics perturbed by weak spatiotemporal white
noise, for example the stochastic Ginzburg-Landau
equation
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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.
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Need to study extrema of the action, which are
solutions of the nonlinear ODE
Uniform solutions
Nonuniform (bounce) solutions
Or critical droplet
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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
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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.
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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, in Noise in
Complex Systems and Stochastic Dynamics II (SPIE
Proceedings Series 5471, 2004), pp. 367 379
Phys. Rev. Lett. 95, 090601-1090601-4 (2005).
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Putting everything together, we find
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Conductance histograms for Na and Au
Counts (a.u.)
Counts
A.I. Yanson et al., Nature 400, 144 (1999)
E. Medina et al., Phys. Rev. Lett. 91, 026802
(2003)
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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).
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Consider action difference (energy barrier) first
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Now compute the prefactor
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• 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).
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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).
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Extrema of the action satisfy
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What determines the crossover?
Permalloy ring of mean radius 200 nm at a) 52.5
mT and b) 72.5 mT
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Has the classical activation phase transition
been seen?
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J. Bürki, C.A. Stafford, and DLS, Appl. Phys.
Lett. 88, 166101 (2006)
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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 h 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,

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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
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TgtgtT0 Thermal activation over barrier
TltltT0 Quantum tunneling through barrier
TT0 Crossover
Solutions
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so
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So there exists a mapping from the activation of
classical fields to the quantum ? classical
crossover problem
Classical
Field Quantum ? Classical Small
parameter T
h External control
variable L,H,
T Periodic in
L ßh
But theres also an important physical
difference T can be varied, and h cannot!
Why does this matter? It affects the nature of
the second order phase transition.
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To compute the prefactor G0 , must examine
fluctuations about the optimal escape
(classical) path.
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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).
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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
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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 ltlt 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. A
15, 3025 (1982).