Magnetisation dynamics at different timescales: dissipation and - PowerPoint PPT Presentation

Magnetisation dynamics at different timescales: dissipation and thermal processes part II. O.Chubykalo-Fesenko Instituto de Ciencia de Materiales de Madrid, Spain

Different timescales: Electron-spin All-optical Langevin dynamics 10 -14 s fs relaxation laser-pulsed on atomistic level processes. experiments 10 -11 s ps Langevin dynamics Magnetisation Fast-Kerr on micromagnetic precession. measurements 10 -9 s ns level dynamics 10 -6 s  s acceleration techniques Hysteresis Conventional 10 -3 s ms magnetometers measurements. (VSM, SQUID) 10 -0 s s Magnetic viscosity 10 3 s hs kinetic Monte experiments . Carlo with energy 10 6 s month barriers calculations Long-time thermal stability 10 9 s years for magnetic recording.

Outline of the talk  Long-time (>  s) magnetisation dynamics  The concept of energy barriers and switching for an individual nanoparticle  Calculation of energy barriers for more complicated systems  Energy barriers for systems of interacting nanoparticles  Kinetic Monte Carlo for thermal decay evaluation in a completely interacting system. • Ultra-short timescale magnetisation dynamics (fs-ps)  The ultra-fast pump-probe experiment  Modelling  Introduction of the correlated noise approach.

Long timescale (up to years)

The Stoner-Wolfarth particle  M Easy   2    E KV cos MH cos axis H Energy barrier:   2    E KV cos MHV cos 2    H    E KV 1 , H 2 K / M   K s H   K 6 10 -13 s T relax 10 5 The Arrhenius-Neél law 10    1    4 f exp( E / kT ) 10 = f 1,2  0 3 10 Asymptote    / dm dt f m f m integration of the - Master equation 1 1 1 2 2 2 10 Landau-Lifshitz equation     ( ) exp( / ) f f m t t LLL-MC 1 10 2 1 0 10 0 2 4 6 8 10  E barr /k B T

Brown’s asymptote: For an independent small particle with applied field parallel to anisotropy The reversal probability is obtained from the solution of the Fokker-Planck equation    1 2 K KV      2 2 ( 1 h )( 1 h ) exp KV ( 1 h ) / k T B     2 ( ) k T M 1 B s •In general, f 0 =1/  0 =F(H,  but in most of cases the approximation f 0 =10 9 -10 11 s -1 is sufficient The general problem does not have solution.

Relaxation in complex systems       1            f exp( E / kT ) ( ) exp / ( ) ( ) m t t E E d E  0            E  , E E E const If in some interval     ( ) log( ) m t E M S t -widely observed behavior 0 •Based on the assumption of Homogeneous energy barrier distribution H=const which is constant in time T=const M(t) S  dM / d (log t ) -magnetic viscosity log (t/t 0 ) V act S   /( ) V k T HM Activation volume H c act B 0 s H

Superparamagnetism  The relaxation time of a grain is given by the Arrhenius-Neel law    1    f exp( E / kT )  0  where f 0 = 10 9 s -1 . and  E is the energy barrier  This leads to a critical energy barrier for superparamagnetic (SPM) behaviour  E   KV k Tln t f ( ) c c B m 0  where t m is the ‘measurement time’  Nanoparticles with  E <  E c exhibit thermal equilibrium (SPM) behaviour - no hysteresis KV>25k B T – for stability at room temperature, KV>60k B T – for magnetic recording

Slow processes: B T <<  E (Energy barrier) k   t t exp( E / k T ) i 0 B  Energy barrier calculation is essential part for determination of long-time thermal stability and slow thermal relaxation  This is important from the point of view of magnetic recording applications.  Evalulation of energy barriers should be done in a multidimensional space and is a difficult problem  in an interacting system energy barriers are dynamical and should be constantly recalculated.

Slow processes: B k T <<  E (Energy barrier) Energy barriers should satisfy conditions: grad E =0  2 E Only one (lowest) eigenvalue of the Hessian matrix   m i m j   

Ridge optimisation Constrained (Lagrangian method multiplier) method: (for simple cases only)  m       Ridge optimization   ( ), F H N i i 0  m method Fe dot i i Altura D=2r -34.815 -34.820 -34.825 -34.830 The obtained point is checked: -34.835 -34.840 •To have a unique negative eigenvalue 0.0 0.2 •To separate the basins of attractions of 0.4 1.0  the two minima from which one is initial 0.6 0.8 /  0.6 0.8 0.4   / 0.2 1.0 Similar method –elastic band

Energy barriers in a single FePt grain One atomistic FePt grain 440 400 Lagrangian method 360 KV 320 E B / (k B T Room ) 280 240 200 160 120 80 4 6 8 10 12 14 16 18 20 22 24 26 L(nm) Varying the length-> different saddle point configurations corresponding to different reversal mechanism

saddle point configuration Energy barriers as a function of Js: soft/hard grain For Js>0.1J energy barriers are collective and larger than individual Collective energy barrier of the hard material energy barrier 180 is defined by the 2 1/2 4S(A (K+  M Hard )) 160 hard layer energy of the domain 1/2 4S(A K) 140 follows wall in the hard layer 120 100 E(k B T) K hard V hard / 300 k B Individual 80 3 M Hard =1100 emu/cm energy barriers 60 3 M Soft =1270 emu/cm soft 40 J s /J layer goes first 20 0.0 0.2 0.4 0.6 0.8 saddle point configuration

Energy barriers for systems of nanoparticles

Angle between easy axes solid symbols- approxim (Pfeiffer). 1.1 empty symbols - exact 1.0 Angles between anistoropy axes 0 0.9  /4 The Pfeiffer approximation  /2 0.8 0.7  E/KV H.Pfeiffer Phys Status Solidi A 118, 295 (1990): 0.6 0.5  0.4  k ( )     E KV [ 1 h / g ( ) Pf int 0.3  0.2  3 / 2 2 / 3 2 / 3      g ( ) [cos ( ) sin ( ) 0.1     k ( ) 0 . 86 1 . 14 g ( ) 0.0 0.0 0.1 0.2 0.3 0.4 0.5   h ex /h K   E   ( M , h ) 0 Co particles Pf int   E    B 2 1 exact    E E E   ( M , h ) 0  Pfeiffer Pf min min int 0.15 frequency 0.10 0.05  is the angle between anisotropy and interaction field 0.00 0.2 0.3 0.4 0.5 0.6 0.7 0.8  E B /2KV The Pfeiffer approximation is slightly displaced to smaller values

Multidimensional energy barrier distribution for Co and FePt particles (only magnetostatic interactions) annealed structure 0,3 regular array Co FePt 0.15 annealed regular 0,2 frequency frequency 0.10 0,1 0.05 0,0 0.00 0,48 0,50 0,52 0,54 0.2 0.4 0.6 0.8 1.0 1.2  E/(2KV)  E B /(2KV) 0.16 all barriers 0.14 Multidimensional energy barrier FePt structure1 0.12 distribution H ex =0 0.10 frequency H ex /H K =0.2 for annealed FePt particles array 0.08 0.06 in the presence of exchange 0.04 0.02 0.00 0.4 0.6 0.8  E/(2KV)

Multidimensional energy barrier distributions evaluated at the remanence Magnetostatic interactions: • Broaden distributions • Displace the center to c=0.2 c=0.36 c=0.56 larger values 0.35 Volume c=0.05 0.30 c=0.2 c=0.36 0.25 c=0.56 This is consistent with probability 0.20 experimental observations that with the increase of the 0.15 strength of interactions: 0.10 0.05 • Magnetisation decay starts earlier 0.00 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 • The blocking temperature E B /(2K<V>) increases

Kinetic Monte Carlo  Evaluate all energy barriers in multidimensional space  Evaluate all transition rates, according to the Arrhenius law      f f exp( E / k T ) f f i i 0 B  Choose a particle (cluster) with the probability proportional to its transition rate and invert it  Approximate the waiting time from the exponential distribution   D ( t ) dt f exp( ft ) dt  Recalculate all the energy barriers in practice is possible for only small interaction  Only plausible, if a good initial guess for all the clusters is known  Energy barrier distribution is a dynamical property and requires a large computational effort. For initial guess, we use the Metropolis MC with simulated annealing

Thermal decay for an emsemble of 2D Co particles: ( starting from the remanent state, in-plane field 2D easy random easy axes distribution) Initial energy barrier distributions 0.35 0.9 Volume c=0.05 0.30 c=0.2 0.8 c=0.36 0.25 c=0.56 0.7 magnetisation 0.20 probability 0.6 c=0.05 0.15 c=0.2 0.5 c=0.56 0.10 0.4 0.05 0.00 0.3 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1E-11 1E-10 1E-9 1E-8 1E-7 1E-6 E B /(2K<V>) time