Computational Studies of

Dynamical Phenomena in

Nanoscale Ferromagnets

PI: Mark A. NovotnyDept. of Physics and Astronomy

Mississippi State University

co-PI: Per Arne RikvoldDept. of Physics, MARTECH, and CSIT

Florida State University

Supported in part by NSFCARM-95 (DMR,DMS,ASC,OMA) DMR9520325

DMR9871455DMR9971001 (Interdisciplinary Workshop)

DMR0120310

http://www.msstate.edu/dept/physics/profs/novotny.html

http://www.physics.fsu.edu/users/rikvold/info/rikvold.htm

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

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.

µ0 2.5m

5.0

(b)

0

2.5

5.0

Magnetization Switching

Span Disparate Timescales

t < 0 t = 0 t � 0Thermally Activated Metastable Escape

m

Free

Ene

rgy

Free

Ene

rgy

m

stable

metastable

saddle point

Metastable Lifetime

τ is first passage time to m=0

10−15

10−10

10−5

100

105

1010

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Tim

e (s

ec)

Inverse phonon frequency

CPU clock cycle

Magnetic disk access time

secondminute

Age of universe/earth/life

yearHuman/Nation lifetime

last earth mag. field reversal

Gregorian calendar zerolast ice age

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

• Randomly choose a lattice point

• Calculate energy change ∆E if spin si changes

• Calculate transition probabilityWsi→−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

Ene

rgy

m

stable

metastable

saddle point

Ising Model

• Start with all si=1

• Applied magnetic field H<0

• Measure 〈τ〉, average first time when m= 1N

∑i si=0

10−15

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10−5

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105

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Tim

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se

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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 Dynamicslumpability of absorbing Markov chain

◦ Constrained Transfer Matrix Methodanalogy with stationary ergodic Markov informationsource

◦ Rejection free for continuous spin systemsrelated to MCAMC for discrete spin systems

◦ Projective Dynamics (+‘String Method’)being worked on for finite T micromagnetic simulations

◦ Non-Trivial n-fold way Parallelizationparallel discrete event simulations (Korniss ITR) 10

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100

105

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Tim

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0.0 10.0 20.0 30.0 40.0

1/H2

100

1012

1024

1036

1048

1060

me

an

life

tim

e

0.0 1.0 2.0 3.0 4.0

1/H2

100

102

104

106

108

me

an

life

tim

e

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

Model Interface Dynamics: Projective Dynamics

Fe sesquilayers [between 1 and 2 monolayers] on W(110)

Uniaxial in-plane ferromagnets: H = −J∑

〈ij〉 sisj − H∑

isi

Digitalization of STM pictures of real sesquilayers publishedin: H. Bethge et al., Surf. Sci. 331-333, 878 (1995).

Domain-wall motion driven by fieldMonitor 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

shrin

kage

/grow

th rat

io

H=0.03JH=0.04J

(b)

VA

VB

(n ≈ domain-wall position)

Micromagnetic Simulations: Projective Dynamics

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

0.6 0.7 0.8 0.9 1Mz

0

0.01

0.02

0.03

0.04

0.05

P

Pshrink

Pgrow

T = 20 K

T = 50 K

T = 100 K

0 20 40 60 80 100 120T (K)

0.7

0.75

0.8

0.85

0.9

Mz

¿What Have We Learned AboutDynamics 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 0 1000 2000H (Oe)

−2000

−1000

0

1000

2000

Mz (

emu/

cm3 )

d=2: Different Decay Regimes

0.00 0.25 0.50 0.75 1.00 1.25 1.501/|Hz|

100

101

102

103

104

105

<τ>

[MC

SS

]

MultidropletL=64, H=1.0

Single DropletL=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 0.5 1 1.5 2kBT/J

0

1

2

3

4

|H|/J

L= 20L=200MFSpL= 20L=200µm squaresoccer field

SF

SD

MD

Switching Fields and Switching Times

• Ising: Maximum in Switching Field

• No dipole-dipole interactions

• Finite Temperature micromagnetics (LLG)

• ~H = ±zH, reverses at t=0

• Fe single-domain nanopillar (aspect ratio ≈17)

0 0.0005 0.001 0.0015 0.0021/H0 (Oe

−1)

10−1

100

101

102

103

104

t sw (

ns)

100K, <tsw>100K, σt

20K, <t sw> 20K, σt

Pnot vs time — Ising and LLG

• Square Lattice Ising

47.5 50 52.5 55 57.5 60 62.5 65Time HMCSS L

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 50.0t (ns)

0.0

0.2

0.4

0.6

0.8

1.0

Pno

t(t)

simulationerror functiontwo exponential

b)

Hysteresis Loop Area: 〈A〉

• 1R=ω〈τ〉/2π

• Thin film nn Ising

-6 -5 -4 -3 -2 -1log10H1�RL

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

log 1

0<

A>

����������

����������

��4

H0

L = 64 MC

asymptote

scaled SD

linear

num. integration

• Finite Temperature LLG

• Fe single-domain nanopillars (aspect ratio 17)

−2000 −1000 0 1000 2000H (Oe)

−2000

−1000

0

1000

2000

Mz (

emu/

cm3 )

10−3

10−2

10−1

100

1/R = ω<τ(H0,T)>/2π10

−1

100

<A

>/(

4H0)

T = 100 KT = 20 K

Hysteresis: Dynamic Phase Transition

• MD regime: Θ = Period2〈τ〉

• Thin film square-lattice nn Ising

0.0 200.0 400.0 600.0 800.0 1000.0periods

−1.0

−0.5

0.0

0.5

1.0

Q

Θ=0.27Θ=0.98Θ=2.7

• Order Parameter: Q = 1Period

∮m(t)dt

• Use finite-size scaling

0.50 0.75 1.00 1.25 1.50Θ

0.0

0.2

0.4

0.6

0.8

1.0

<|Q

|>

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

0.70 0.80 0.90 1.00 1.10 1.20Θ

0

2000

4000

6000

<(∆

|Q|)

2 >L

2

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

Funded Personel (collaborators)

• Postdoctoral fellows

◦ Alice Kolakowska

◦ Kyungwha Park

◦ Gregory Brown, Oak Ridge & Florida State U.

◦ Jose Munoz, U. Nacional de Colombia

◦ Gyorgy 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

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 Buendıa, U. Simon Bolıvar, Caracas, Venezuela

• Underrepresented Groups

◦ 1 physically challenged

◦ 8 minorities

◦ 6 women

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 inscience & engineering & technology

• Web-based dissemination of papers & simulations(general public and K-12 education)

• Patent applications (2 – different stages)

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 10

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Proposed Work, 2002-04

• Specific systems and models

– Micromagnetics simulations of nanomagnets of morecomplicated shapes, and of arrays of nanomagnets.

– Search for theoretical foundation of the frictionconstant in the micromagneticLandau-Lifschitz-Gilbert equation.

– Nucleation in driven, pinned domain walls.

– Magnetization switching in systems with surfacesand bulk defects.

– Hysteresis at the nanoscale.

• Algorithm development

– Development of accelerated simulation algorithmsfor systems with continuum spins.

– Applications of time-bridging algorithms toLangevin simulations.