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

Partially supported by US National Science Foundation Grants PHY009484, PHY0351964, and PHY0601179

Reference: DLS, Braz. J. Phys. 35, 242—252 (2005).

Collaborators: Jerome Bürki (Physics, Arizona), Andy Kent (Physics, NYU),

Robert Maier (Math, Arizona), Kirsten Martens (Physics, Heidelberg),

Charles Stafford (Physics, Arizona)

Outline of TalkOutline of Talk

• Decay of monovalent metallic nanowiresDecay of monovalent metallic nanowires

• Classical ActivationClassical Activation in Stochastic Field Theoriesin Stochastic Field Theories

• Magnetization Reversal in Quasi-2D NanomagnetsMagnetization Reversal in Quasi-2D Nanomagnets

• Experimental Evidence for the Phase Transition?Experimental Evidence for the Phase Transition?

Why nanowires? Moore’s law

Extrapolates to 1nm technology by 2020International Technology Roadmap for Semiconductors: 1999

Theoretical stability diagramTheoretical stability diagram

C.-H. Zhang et al., PRB C.-H. Zhang et al., PRB 6868, 165414 (2003)., 165414 (2003).

But: why should these wires exist at all?But: why should these wires exist at all?

Rayleigh instability: cylindrical column of fluid held Rayleigh instability: cylindrical column of fluid held together by pairwise interactions is unstable to together by pairwise interactions is unstable to

breakup by surface wavesbreakup by surface waves

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

Electron Shell Potential

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

Because the zero-noise dynamics are gradient,

Vdzz z

L

L

2)()][ 2[2/

2/

Η

where

Consider an extended system with gradient dynamics perturbed by weak spatiotemporal white noise, for example the stochastic Ginzburg-Landau equation:

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:

30 Η

That is, the states that determine the transition rates are extrema of the action.

Need to study extrema of the action, which are solutions of the nonlinear ODE

Uniform solutions:

Nonuniform (bounce*) solutions:

* Or critical droplet

Vdzz z

L

L

2)()][ 2[2/

2/

Η

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: ~ 0 exp [ -E / kBT ]

To compute the prefactor 0 , must examine fluctuations about

the optimal escape (classical) path.

This is essentially the same procedure as computing quantum This is essentially the same procedure as computing quantum corrections about the classical path in the Feynman path-corrections about the classical path in the Feynman path-

integral approach to quantum mechanics.integral approach to quantum mechanics.

Model of nanowire `decay’ rate

• Will use a continuum approach

• Thermal fluctuations responsible for nucleating changes in radius

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: = exp[-E/T]_0

J. BJ. Bürki, C. Stafford, and DLS,rki, C. Stafford, and DLS, inin Noise in Complex Systems and Stochastic Dynamics IINoise in Complex Systems and Stochastic Dynamics II (SPIE Proceedings Series 5471, 2004), pp. 367 – 379; Phys. Rev. Lett. 95, 090601-1—090601-4 (2005).

Putting everything together, we find:Putting everything together, we find:

A.I. Yanson et al., Nature 400, 144 (1999)

Conductance histograms for Na and Au

E. Medina et al., Phys. Rev. Lett. 91, 026802 (2003)

Cou

nts

(a.u

.)C

ou

nts

Co

unt

s

0/GG

What about finite L?

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).

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:

related to L via boundary conditions

Consider action difference (energy barrier) first:

Now compute the prefactor:

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).

R~200 nm

R~5-10 nm

MH

The stochastic dynamics are now governed by magnetization The stochastic dynamics are now governed by magnetization fluctuations in the Landau-Lifschitz-Gilbert equation:fluctuations in the Landau-Lifschitz-Gilbert equation:

][)/(][ 0 MMHMM eff tt M

Magnetization reversal in nanoscale ferromagnets

where where

MHMxM extR

xdUxdxdE 3232323

)]([

andand

withwith

0)( MU

The nonlocal magnetostatic term simplifies to a local shape anisotropy because The nonlocal magnetostatic term simplifies to a local shape anisotropy because of the quasi-2D geometry: R. Kohn and V. Slastikov, of the quasi-2D geometry: R. Kohn and V. Slastikov, Arch. Rat. Mech. Anal.Arch. Rat. Mech. Anal.

178178, 227 (2005)., 227 (2005).

Extrema of the action satisfyExtrema of the action satisfy

sincossin/ 22 hdsd

.ˆMsolution metastable a and

ˆMsolution stable a is there

,ˆ and ,10When

0

0

ext

M

M

Hh

])|([m)|cn(

m)|sn(s-sdncot2),(:tNonconstan

)(cos :solutions saddleConstant

01

-1

RR

mms

h

What determines the crossover?What determines the crossover?

21/2 cc hl

h

l

Instanton saddleInstanton saddle

`uniform’ saddle`uniform’ saddle

Permalloy ring of mean radius 200 nm at a) 52.5 mT and b) 72.5 mTPermalloy ring of mean radius 200 nm at a) 52.5 mT and b) 72.5 mT

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

J. BJ. Bürki, C.A. Stafford, and DLS, Appl. Phys. Lett. ürki, C.A. Stafford, and DLS, Appl. Phys. Lett. 8888, , 166101166101 (2006) (2006)

Conclusions

• 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, …

• 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.

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 …

Goldanskii 1959: Crossover at

]32

[)( 322

0 qu

qMqV

T>>T0 : Thermal activation over barrier

T<<T0 : Quantum tunneling through barrier

T~T0 : Crossover

. /2)q(/2)q(- conditionsboundary periodic with

0)()()(q- limit, friction-low thein So 220

qq

Solutions:

(stable) 0)( q

)TT when(saddle /)( 020, cq

)TT when(saddle)()2(13

)|2

)((dn)(

2

3),( 0

2002

20

, ][ mmmm

mmqc

0

4

12 (m)(m)4

and )1()( where

K

mmm

so …

So there exists a mapping 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’.

To compute the prefactor 0 , must examine fluctuations about

the optimal escape (classical) path.

Linearize the zero-noise evolution about a stationary state

> 0

< 0

Stable mode

Unstable mode

A (meta)stable state has all > 0 ; a saddle has one < 0 .

Finally,

What about finite L?

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).

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:

related to L via boundary conditions

Period of sn function is 4K(m).

As an example, consider Neumann BC’s:

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, …)

Now add noise …classical (thermal)

or quantum mechanical

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 << E, then ~ 0 exp [ -/ kBT ].

E is the energy difference between the `saddle’ state and (meta)stable state

• 0 governed by fluctuations about the `optimal escape path’

Also called the Arrhenius rate law when 0 is independent of T.

M.I. Freidlin and A.D. Wentzell, M.I. Freidlin and A.D. Wentzell, Random Perturbations of Dynamical SystemsRandom Perturbations of Dynamical Systems (Springer, 1984); W.G. Faris and G. Jona-Lasinio, (Springer, 1984); W.G. Faris and G. Jona-Lasinio, J. Phys. AJ. Phys. A 1515, 3025 , 3025

(1982).(1982).