[PDF] - Computational Studies of Dynamical Phenomena in Nanoscale PDF Document

SLIDE 1

Computational Studies of Dynamical Phenomena in Nanoscale Ferromagnets

PI: Mark A. Novotny

Dept. of Physics and Astronomy

Mississippi State University co-PI: Per Arne Rikvold

Dept. of Physics, MARTECH, and CSIT

Florida State University Supported in part by NSF CARM-95 (DMR,DMS,ASC,OMA) DMR9520325 DMR9871455 DMR9971001 (Interdisciplinary Workshop) DMR0120310

http://www.msstate.edu/dept/physics/profs/novotny.html http://www.physics.fsu.edu/users/rikvold/info/rikvold.htm

SLIDE 2

Motivation: Nanoscale Magnets Dynamics for Magnetic Recording

Bits on single-domain particles
Thermal effects important

(now, not in 1995)

Superparamagnetic limit important

(now, not in 1995)

Nanoscale ferromagnets also in MRAM & MEMS

SLIDE 3

Motivation: Experimental Nanoscale Magnets Dynamics

New methods for forming nanoscale magnets
New methods for measuring nanoscale ferromagnets

AFM (a) and MFM (b) images of Fe nanopillars. Courtesy of D.D. Awschalom.

µ 2.5 m 5.0

(b)

2.5 5.0

SLIDE 4

Magnetization Switching Span Disparate Timescales

t < 0 t = 0 t ≫ 0

Thermally Activated Metastable Escape

m Free Energy Free Energy m

stable metastable saddle point

Metastable Lifetime τ is first passage time to m=0

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Time (sec)

Inverse phonon frequency CPU clock cycle Magnetic disk access time second minute Age of universe/earth/life year Human/Nation lifetime last earth mag. field reversal Gregorian calendar zero last ice age

SLIDE 5

Monte Carlo Dynamics (Ising model, s=±1)

Randomly choose a lattice point
Calculate energy change ∆E if spin si changes
Calculate transition probability

Wsi→−si = [1 + exp(∆E/kBT)]−1 fermion: Martin ’77 Wsi→−si =

∆E [1 − exp(∆E/kBT)]−1
phonon: Park ’01
Calculate a random number r
Flip spin si→−si if r≤W
Repeat ∼ 1030 times!!!!!!!

Free Energy m

stable metastable saddle point

Ising Model

Start with all si=1
Applied magnetic field H<0
Measure τ, average first time when m= 1

N

i si=0

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Time (sec)

SLIDE 6

Our Long-Time Simulation Algorithms Simple Models → Realistic Models Novel Simulation Algorithms

Monte Carlo with Absorbing Markov Chains

(MCAMC) Absorbing Markov Chains + Monte Carlo

Projective Dynamics

lumpability of absorbing Markov chain

Constrained Transfer Matrix Method

analogy with stationary ergodic Markov information source

Rejection free for continuous spin systems

related to MCAMC for discrete spin systems

Projective Dynamics (+‘String Method’)

being worked on for finite T micromagnetic simulations

Non-Trivial n-fold way Parallelization

parallel discrete event simulations (Korniss ITR)

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SLIDE 7

0.0 10.0 20.0 30.0 40.0 1/H

2

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mean lifetime 0.0 1.0 2.0 3.0 4.0 1/H

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mean lifetime

The right panel is a close-up of the lower-left corner of the left

panel. The age of the universe is about 1033 femtoseconds.

Extreme Long-time Simulations Projective Dynamics with Moving Constraint 3D Ising Model at 0.6Tc

SLIDE 8

Model Interface Dynamics: Projective Dynamics

Fe sesquilayers [between 1 and 2 monolayers] on W(110) Uniaxial in-plane ferromagnets: H = −J

ij sisj − H i si

Digitalization of STM pictures of real sesquilayers published in: H. Bethge et al., Surf. Sci. 331-333, 878 (1995). Domain-wall motion driven by field Monitor probabilities g(n) and s(n) of growing or shrinking stable phase unstable phase

14000 15000 16000 # of spins in stable phase 0.990 1.000 1.010 shrinkage/growth ratio

H=0.03J H=0.04J

(b) VA VB

(n ≈ domain-wall position)

SLIDE 9

Micromagnetic Simulations: Projective Dynamics

Single pillar: finite T: Langevin: (Fast Multipole Method)

0.6 0.7 0.8 0.9 1

Mz

0.01 0.02 0.03 0.04 0.05

P

Pshrink Pgrow

T = 20 K T = 50 K T = 100 K

20 40 60 80 100 120

T (K)

0.7 0.75 0.8 0.85 0.9

Mz

SLIDE 10

¿What Have We Learned About Dynamics of Nanomagnets from Model Simulations? simple models → realistic models → experiments

Field Reversal
Different decay regimes for L, T, H
Peak in Hswitching vs. L even for single-domain
Functional forms for Pnot(t) different
Thermally activated Domain Wall motion
Change in Barkhausen volumes with H and T
Change in Activation volumes with H and T
Dependence of coercive field on frequency
Hysteresis: (for single-domain)
Stochastic Hysteresis
Area of hysteresis loop, A, on L, T, H, ω
Stochastic Resonance
Dynamic (non-stationary) Phase Transition (fss)

−2000 −1000 1000 2000

H (Oe)

−2000 −1000 1000 2000

Mz (emu/cm

3)

SLIDE 11

d=2: Different Decay Regimes

0.00 0.25 0.50 0.75 1.00 1.25 1.50

1/|Hz|

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<τ> [MCSS]

Multidroplet L=64, H=1.0 Single Droplet L=64, H=0.75

SF MD SD

Homogeneous nucleation & growth:

different decay regimes

Four length scales: a, Rcrit, R0, L
τ different dependences on H and L
‘Metastable phase diagram’: experimentally relevant

0.5 1 1.5 2

kBT/J

1 2 3 4

|H|/J

L= 20 L=200 MFSp L= 20 L=200 µm square soccer field

SF SD MD

SLIDE 12

Switching Fields and Switching Times

Ising: Maximum in Switching Field
No dipole-dipole interactions
Finite Temperature micromagnetics (LLG)

H = ±ˆ zH, reverses at t=0

Fe single-domain nanopillar (aspect ratio ≈17)

0.0005 0.001 0.0015 0.002

1/H0 (Oe

−1)

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tsw (ns)

100K, <tsw> 100K, σt 20K, <tsw> 20K, σt

SLIDE 13

Pnot vs time — Ising and LLG

Square Lattice Ising

47.5 50 52.5 55 57.5 60 62.5 65 Time HMCSSL 0.2 0.4 0.6 0.8 1

Finite Temperature LLG: T = 100 K
Fe single-domain nanopillar (aspect ratio ≈17)

0.0 10.0 20.0 30.0 40.0 50.0

t (ns)

0.0 0.2 0.4 0.6 0.8 1.0

Pnot(t)

simulation error function two exponential

b)

SLIDE 14

Hysteresis Loop Area: A

1

R=ωτ/2π

Thin film nn Ising

6 5 4 3 2 1 log101R 1.2 1 0.8 0.6 0.4 0.2 log10 A

4H0

L 64 MC asymptote scaled SD linear

num. integration
Finite Temperature LLG
Fe single-domain nanopillars (aspect ratio 17)

−2000 −1000 1000 2000

H (Oe)

−2000 −1000 1000 2000

Mz (emu/cm

3)

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1/R = ω<τ(H0,T)>/2π

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<A>/(4H0)

T = 100 K T = 20 K

SLIDE 15

Hysteresis: Dynamic Phase Transition

MD regime:

Θ = Period

2τ

Thin film square-lattice nn Ising

0.0 200.0 400.0 600.0 800.0 1000.0

periods

−1.0 −0.5 0.0 0.5 1.0

Q

Θ=0.27 Θ=0.98 Θ=2.7

Order Parameter: Q =

1 Period

m(t)dt
Use finite-size scaling

0.50 0.75 1.00 1.25 1.50 Θ 0.0 0.2 0.4 0.6 0.8 1.0 <|Q|> L=64 L=90 L=128 L=256 L=512 0.70 0.80 0.90 1.00 1.10 1.20 Θ 2000 4000 6000 <(∆|Q|)

2>L 2

L=64 L=90 L=128 L=256 L=512

SLIDE 16

Funded Personel (collaborators)

Postdoctoral fellows
Alice Kolakowska
Kyungwha Park
Gregory Brown, Oak Ridge & Florida State U.
Jos´

e Mu˜ noz, U. Nacional de Colombia

Gy¨
rgy Korniss, Rensellear Polytechnic Inst.
Miroslav Kolesik, U. Arizona
Hans Evertz, Technical U. Graz
Raphael Ramos, U. Puerto Rico, Mayaguez
Graduate Students
Steven Mitchell, physics, Ph.D. 2001,

Eindhoven U. Technology

Daniel Valdez-Balderas, physics, M.S. 2001, Ohio State
Xuekun Kou, EE M.E. 1998, industry
Scott Sides, physics Ph.D. 1998, U.C.S.B.
H.L. Richards, physics Ph.D. 1996,

Texas A&M, Commerce

S. Weaver, physics M.S. 1995, industry

SLIDE 17

Funded Personel (collaborators)

Undergraduate Students
Ashley Frye, physics, 2002
Shannon Wheeler, biology, 2002
Daniel Roberts, physics, 2002, U.I.U.C.
Christina White Oberlin, physics, 2002, U. Wisc.,

Madison

Dean Townsley, physics/math/ME, 1998, U.C.S.B.
Jarvis A. Addison, EE, 1997, industry
Steven Duval, EE, 1997, industry
D’Angelo Hall, EE, 1997, industry
Adam Hutton, EE, 1997, industry
Frederick M. Jenkins, EE, 1996, industry
Sabbatical faculty
Gloria Buend´

ıa, U. Sim´

n Bol´

ıvar, Caracas, Venezuela

Underrepresented Groups
1 physically challenged
8 minorities
6 women

SLIDE 18

Greater Good to Society (Dissemination of Results)

about 55 articles published
Physical Review A & B & E & Letters
J. Magnetism and Magnetic Materials
Computer Physics Communications
J. Non-Crystalline Solids
IEEE Trans. Magn.
Annual Reviews in Computational Physics
Many papers with undergraduate co-authors
Presentations in: physics, chemistry, materials science,

applied mathematics, engineering, computer science

About 19 invited conference presentations
Multidisciplinary workshop (biology, chemistry, physics)
Advanced algorithms: applicable to other areas in

science & engineering & technology

Web-based dissemination of papers & simulations

(general public and K-12 education)

Patent applications (2 – different stages)

SLIDE 19

Conclusions

(& outlook)

Understanding dynamics of nanoscale magnets
Simple model simulations → realistic model simulations
Simple models have complicated behavior:

more realistic models · · ·

Different regimes for metastable decay
Pnot different in different regimes
Statistical interpretation of hysteresis & loop area
Dynamic Phase Transition in hysteresis
Advanced algorithms to bridge disparate time scales
Monte Carlo with Absorbing Markov Chains
Projective Dynamics
Thermal Micromagnetics (fast multipole method)
Constrained Transfer Matrix
Thermal effects important for nanoscale ferromagnets
Interdisciplinary projects: Ideal for education &

Societal ‘greater good’ More advances in algorithms, simulations, understanding, education, & applications

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SLIDE 20

Proposed Work, 2002-04

Specific systems and models

– Micromagnetics simulations of nanomagnets of more complicated shapes, and of arrays of nanomagnets. – Search for theoretical foundation of the friction constant in the micromagnetic Landau-Lifschitz-Gilbert equation. – Nucleation in driven, pinned domain walls. – Magnetization switching in systems with surfaces and bulk defects. – Hysteresis at the nanoscale.

Algorithm development

– Development of accelerated simulation algorithms for systems with continuum spins. – Applications of time-bridging algorithms to Langevin simulations.