Simulation Of Spin Wave Switching In Perpendicular Media P. B.Visscher Department of Physics and Astronomy The University of Alabama
Abstract We propose to build on our understanding of spin wave switching in longitudinally pulsed media to study the interaction of precessional motions with spin wave instabilities in perpendicular media: In perpendicular media the write field undergoes rapid changes of direction as well as of magnitude. We propose to simulate a medium subjected to a field history similar to that it would encounter in passing under a quasi-monopolar longitudinal write head. Within the simulation, we will calculate the Lyapunov exponents and gain factors for spin wave growth, to estimate switching time. We will study the energy cascade to shorter wavelengths, which dissipates the Zeeman energy.
Spin-wave switching (Safonov &Bertram 1999) Magnetization vs time for a 4 4 4 system (finite temperature initial condition) (see also Poster #21) Hext spin wave energy
d M dt Basic equation for precession = γ ext K M H = H + H + H H K = exch = 2K 2 M J S nbrs M H Z Hext M' exch zˆ H - external magnetic field; Landau-Lifshitz equation without damping (not important in early stages of fast switching) thermal (Langevin) noise (small, k B T<<Zeeman energy) - effective anisotropy field due to intrinsic crystalline anisotropy or to sample shape - effective exchange field;
Fourier-transform to get spin wave amplitudes: Each M(r) can be written as the sum of its Fourier components: Visualizing spin waves M k) = 1 N ( 3 M ( r) = M ( r) k k r M( r) e where the Fourier component is ( ik r M ( r) Re M( k)e ) Rather than display the complex vector M(k) at each point in k-space, we display all the individual components M k (r) for all cells r (N different values, 4 here) they lie on an ellipse. In k-space we show red polygon (approx. to ellipse) k ik r
Simple case: 1 spin wave k=(0,0,1) k=(0,0,0)
Superimposed spin waves k=(0,0,1) k=(0,0,0) k=(1,0,0) k=(0,1,0)
Spin-wave switching Initial condition: Slight tilt from vertical, finite temperature. Suddenly apply downward H. Stoner-Wohlfarth orbit would get close to the equator, but not switch. Allowing incoherence & exchange interactions allows spin waves: (movie at http://bama.ua.edu/~visscher/mumag/switch1.mov)
Growth of spin-wave amplitudes (n x,n y,n z ) = wave vector in integer form: k = 2π ( n n ) x y nz,, L x L y L z
ω 2 Why is it unstable? Calculating (FMR) precession frequency General formula for spin-wave frequency (=resonance frequency for FMR, if k=0) = γ 2 [ + ( D D ) M ][ H + ( D D ) M ] H M t1 s M t 2 where D is the demag+anisotropy factor along M and D M t1 and D t2 are demag+anisotropy factors transverse to M In the simple case of M along the easy (z) axis, D M =2K/M s and D t1 = D t2 = 0, so ω 2 > 0 and ω is real: we get circular precession. If M is along a hard axis, D t1 = 2K/M s and one factor can be negative, making ω imaginary (exponentially growing instability). This is the origin of the switching instability. s
Application to perpendicular recording N Perpendicular-recording head medium grain H soft underlayer
Summary We have identified the instability responsible for spin wave switching. We will extend this from simple field pulses to longitudinal-head field histories. Other plans include: Add magnetostatic interactions (Fast Multipole Method) Determine which wavelengths are most unstable Model subsequent dissipation of spin-wave energy