Computational Studies of Dynamical Phenomena in Nanoscale ... · Computational Studies of Dynamical...
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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
1015
1020
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
10−10
10−5
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105
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Tim
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se
c)
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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10−5
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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.