Computational Studies of Dynamical Phenomena in Nanoscale - PDF document

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

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. 5.0 (b) 2.5 0 0 2.5 5.0 µ m

Magnetization Switching Span Disparate Timescales t < 0 t ≫ 0 t = 0 Thermally Activated Metastable Escape Free Energy Free Energy saddle point metastable stable m m 20 10 Age of universe/earth/life 15 10 last earth mag. field reversal Metastable Lifetime last ice age Gregorian calendar zero 10 10 Human/Nation lifetime τ is first passage time to m =0 year Time (sec) 5 10 minute 0 10 second −5 10 Magnetic disk access time −10 10 CPU clock cycle Inverse phonon frequency −15 10

Monte Carlo Dynamics (Ising model, s = ± 1 ) • Randomly choose a lattice point • Calculate energy change ∆ E if spin s i changes • Calculate transition probability W s i →− s i = [1 + exp(∆ E/k B T )] − 1 fermion: Martin ’77 � ∆ E [1 − exp(∆ E/k B T )] − 1 � � W s i →− s i = phonon: Park ’01 � • Calculate a random number r 20 10 • Flip spin s i →− s i if r ≤ W • Repeat ∼ 10 30 times!!!!!!! 15 10 10 Free Energy 10 saddle point Time (sec) 5 10 metastable stable 0 10 m Ising Model • Start with all s i =1 −5 10 • Applied magnetic field H< 0 −10 10 • Measure � τ � , average first time when m = 1 � i s i =0 N −15 10

Our Long-Time Simulation Algorithms Simple Models → Realistic Models Novel Simulation Algorithms 20 10 ◦ Monte Carlo with Absorbing Markov Chains (MCAMC) 15 10 Absorbing Markov Chains + Monte Carlo ◦ Projective Dynamics 10 10 lumpability of absorbing Markov chain Time (sec) 5 ◦ Constrained Transfer Matrix Method 10 analogy with stationary ergodic Markov information source 0 10 ◦ Rejection free for continuous spin systems −5 related to MCAMC for discrete spin systems 10 ◦ Projective Dynamics (+‘String Method’) −10 10 being worked on for finite T micromagnetic simulations ◦ Non-Trivial n -fold way Parallelization −15 10 parallel discrete event simulations (Korniss ITR)

Extreme Long-time Simulations Projective Dynamics with Moving Constraint 3D Ising Model at 0.6 T c 60 8 10 10 48 10 6 10 mean lifetime mean lifetime 36 10 4 10 24 10 2 10 12 10 0 0 10 10 0.0 10.0 20.0 30.0 40.0 0.0 1.0 2.0 3.0 4.0 2 2 1/H 1/H The right panel is a close-up of the lower-left corner of the left panel. The age of the universe is about 10 33 femtoseconds.

Model Interface Dynamics: Projective Dynamics Fe sesquilayers [between 1 and 2 monolayers] on W(110) Uniaxial in-plane ferromagnets: H = − J � � ij � s i s j − H � i s i 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 1.010 (b) V B shrinkage/growth ratio 1.000 H=0.03J V A H=0.04J 0.990 14000 15000 16000 # of spins in stable phase ( n ≈ domain-wall position)

Micromagnetic Simulations: Projective Dynamics Single pillar: finite T : Langevin: (Fast Multipole Method) 0.9 0.05 T = 100 K 0.85 0.04 0.03 T = 50 K M z P 0.8 0.02 T = 20 K 0.75 P shrink 0.01 P grow 0 0.7 0.6 0.7 0.8 0.9 1 0 20 40 60 80 100 120 M z T (K)

¿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 H switching vs. L even for single-domain ◦ Functional forms for P not ( 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 3 ) M z (emu/cm 0 −1000 −2000 −2000 −1000 0 1000 2000 H (Oe)

d =2 : Different Decay Regimes 5 10 Multidroplet L=64, H=1.0 4 10 SD < τ > [MCSS] 3 10 MD 2 10 1 SF 10 Single Droplet L=64, H=0.75 0 10 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1/|H z | • Homogeneous nucleation & growth: different decay regimes • Four length scales: a , R crit , R 0 , L • � τ � different dependences on H and L • ‘Metastable phase diagram’: experimentally relevant 4 L= 20 L=200 MFSp 3 L= 20 L=200 µ m square soccer field |H|/J SF 2 MD 1 SD 0 0 0.5 1 1.5 2 k B T/J

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) 4 10 100K, <t sw > 3 100K, σ t 10 20K, <t sw > 20K, σ t 2 10 t sw (ns) 1 10 0 10 −1 10 0 0.0005 0.001 0.0015 0.002 −1 ) 1/H 0 (Oe

P not vs time — Ising and LLG • Square Lattice Ising 1 0.8 0.6 0.4 0.2 47.5 50 52.5 55 57.5 60 62.5 65 Time H MCSS L • Finite Temperature LLG: T = 100 K • Fe single-domain nanopillar (aspect ratio ≈ 17) 1.0 simulation b) error function 0.8 two exponential 0.6 P not (t) 0.4 0.2 0.0 0.0 10.0 20.0 30.0 40.0 50.0 t (ns)

Hysteresis Loop Area: � A � num. integration 0 1 • R = ω � τ � / 2 π linear scaled SD � 0.2 asymptote • Thin film nn Ising L � 64 MC � 0.4 ���������������������� � A � 4 � H 0 � 0.6 2000 log 10 � 0.8 1000 3 ) M z (emu/cm 0 � 1 −1000 � 1.2 � 6 � 5 � 4 � 3 � 2 � 1 log 10 � 1 � R � −2000 −2000 −1000 0 1000 2000 H (Oe) • Finite Temperature LLG • Fe single-domain nanopillars (aspect ratio 17) 0 10 <A>/(4H 0 ) T = 100 K T = 20 K −1 10 −3 −2 −1 0 10 10 10 10 1/R = ω < τ (H 0 ,T)>/2 π

Hysteresis: Dynamic Phase Transition Θ = Period • MD regime: 2 � τ � • Thin film square-lattice nn Ising 1.0 Θ =0.27 Θ =0.98 Θ =2.7 0.5 Q 0.0 −0.5 −1.0 0.0 200.0 400.0 600.0 800.0 1000.0 periods 1 � • Order Parameter: Q = m ( t ) dt Period • Use finite-size scaling 1.0 L=64 L=64 6000 L=90 L=90 0.8 L=128 L=128 L=256 L=256 L=512 L=512 0.6 2 4000 2 >L <|Q|> <( ∆ |Q|) 0.4 2000 0.2 0.0 0 0.50 0.75 1.00 1.25 1.50 0.70 0.80 0.90 1.00 1.10 1.20 Θ Θ

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¨ orgy 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 Buend´ ıa, U. Sim´ on Bol´ ıvar, 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 in science & engineering & technology • Web-based dissemination of papers & simulations (general public and K-12 education) • Patent applications (2 – different stages)