Josh Deutsch. University of California. Santa Cruz

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 1/6 Nonequilibrium symmetry breaking and pattern formation in magnetic films Josh Deutsch University of California Santa Cruz

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 2/6 Collaborators Michael S. Pierce, Conor R. Buechler, Larry B. Sorensen, University of Washington Eduardo A. Jagla, The Abdus Salam International Centre for Theoretical Physics Trieu Mai, Onuttom Narayan, University of California, Santa Cruz Joshua J. Turner, Steve D. Kevan, University of Oregon, Eugene Karine M. Chesnel, Jeff B. Kortright, Lawrence Berkeley National Laboratory Olav Hellwig, Eric E. Fullerton, Hitachi Global Storage Technologies, San Jose, Joseph E. Davies, Kai Liu, University of California, Davis, Jonathan Hunter Dunn, MAX Laboratory, Lund

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 3/6 Outline Introduction Non-equilibrium statistical mechanics Return point memory Co/Pt multilayer films Experiments Correlations between states Lack of configurational symmetry Theoretical explanation Simulations Predictions Conclusions

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 4/6 Equilibrium vs non-equilibrium Simple example:water-vapor: In equilibrium, we can calculate many things very precisely, like velocity distribution function, specific heat. If we cool down below saturation, we may start nucleating water droplets. Their formation is an example of non-equilibrium statistical mechanics. Critical nucleus of a crystal of hard colloidal spheres. (Daan Frenkel and Stefan Auer)

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 5/6 Harder examples The equilibrium of a spin glass (e.g. CuMn) below the spin glass transition temperature. The ground state is hard to find. This can be proved to be NP complete, or take exponentially long to compute. (Michael Jünger)

Harder examples The equilibrium of a spin glass (e.g. CuMn) below the spin glass transition temperature. The ground state is hard to find. This can be proved to be NP complete, or take exponentially long to compute. The nonequilibrium behavior of a turbulent fluid. This is related to computational complexity. Simplified lattice models are capable of showing universal computation. Many questions related to such systems are provably undecidable. (Michael Jünger) Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 5/6

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 6/6 Irreversibility and Avalanches Zoom in on part of a hysteresis loop: M H

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 6/6 Irreversibility and Avalanches M H

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 6/6 Irreversibility and Avalanches M Irreversible H There is very fast motion during the avalanche, independent of the speed at which the field is varied (within reason).

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 7/6 Applications of Ferromagnetism Doodle Pads

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 7/6 Applications of Ferromagnetism Doodle Pads Refrigerator Magnets

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 8/6 Return Point Memory Magnetic systems containing only ferromagnetic couplings often display "Return Point Memory": a system can return to the same state after an excursion back to a previous magnetic field. (Perkovic, Dahmen and Sethna PRL (1995).

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 9/6 Nested Branches Nested sub-loops also return to the same state M A (1) (3) E C H min B D (2) H max H

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 10/6 RFIM They proved this for the random field Ising Model (RFIM): H = i,j J i,j S i S j i h i S i H i S i. S i = ±1.

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 10/6 RFIM They proved this for the random field Ising Model (RFIM): H = i,j J i,j S i S j i h i S i H i S i. S i = ±1. Arbitrary positive couplings J i,j. Arbitrary fields h i. RPM does not hold in general when some of the couplings become antiferromagnetic (negative). Also effects of finite temperature destroy the exact nature of this result.

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 11/6 Question: To what extent does a real system return to its initial state after an excursion away from it?

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 11/6 Question: To what extent does a real system return to its initial state after an excursion away from it? A realistic Hamiltonian is expected to have a different form, for example H = i,j J i,j s i s j B i s 2 z,i H i s z,i. The couplings J i,j can have a short range ferromagnetic component but a long range antiferromagnetic component, and are not always positive.

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 12/6 Hamiltonian Symmetry For a real system, we expect the Hamiltonian is invariant under s i s i, H H because all terms at bilinear in the variables s,h.

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 13/6 Finite Temperature In addition a RPM is not expected to hold for finite temperature. How do we characterize this case?

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 14/6 Co/Pt Multilayers Experiments were performed at the Advanced Light Source (LBL). The pinhole creates light coherent over a length of 40µm. By exploiting resonant x-ray magnetic scattering, the scattering can be made to have a strong magnetic component, producing a speckle pattern.

Speckle Pattern Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 15/6

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 16/6 3 mtorr Sample [Nb(3.5 nm)/si(3.0 nm)] 40 XTEM image using pseudocolor. (Cross section transmission electron microscopy). (E. E. Fullerton et al, PRB 48 17433 (1993))

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 17/6 15 mtorr Sample (E. E. Fullerton et al)

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 18/6 Hysteresis loops Pressures of 3, 7, 8.5, 10, and 12 mtorr

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 19/6 Correlations of states on loop Look at correlations between states for different realizations, but the same field. M H

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 20/6 Definition of correlation What we would like is to compare patterns in real space: A state i with spins s i (r) with a state j with spins s j (r) The un-normalized covariance between two spin configurations is defined as: cov(i,j) = s i (r) s j (r) r s i (r) r s i (r) r The normalized covariance is ρ = cov(i,j)/ cov(i,i)cov(j,j)

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 21/6 Experimental RPM data This correlation coefficient has been referred to as "RPM". Data for the 8.5 mtorr sample.

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 22/6 Compare complementary states Look at correlations between complementary states: H H. M H

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 23/6 Definition of CPM The normalized covariance is ρ = cov(i,j)/ cov(i,i)cov(j,j)

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 23/6 Definition of CPM The normalized covariance is ρ = cov(i,j)/ cov(i,i)cov(j,j) Consider spin configurations at field H on leg i of a hysteresis loop. The RPM normalized covariance is ρ(h,i;h,j), where i and j are both legs going in the same direction.

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 24/6 Experimental CPM data Data for the 8.5 mtorr sample. Starting off at negative fields, going to high fields.

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 25/6 CPM < RPM Why?

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 26/6 Unexpected Behavior For a real physical system, one expects complete microscopic symmetry because the Hamiltonian is invariant under s i s i, H H

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 27/6 RPM vs disorder RPM and CPM both increase with disorder. That is disorder stabilizes patterns.

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 28/6 Possible explanation There is a symmetry breaking term in the Hamiltonian, like h i s z,i. i That is some elusive random fields.

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 28/6 Possible explanation There is a symmetry breaking term in the Hamiltonian, like h i s z,i. That is some elusive random fields. i But what is the source of these random fields? Other magnetic impurities with very high coercivity that aren t completely saturated.

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 29/6 Possible explanation But the loops look pretty saturated

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 30/6 More detailed analysis The Landau-Lifshitz-Gilbert (LLG) equation describes micromagnetic dynamics. It contains a reactive term and a dissipative term: ds dt = γ 1s B γ 2 s (s B), s is a microscopic spin, B is the local effective field, γ 1 is a precession coefficient, and

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 31/6 Dynamics Break Symmetry If s i s i, H H then the effective field B B. Now consider the LLG eqn:

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 31/6 Dynamics Break Symmetry If s i s i, H H then the effective field B B. Now consider the LLG eqn: ds dt = γ 1s B γ 2 s (s B), Therefore the dynamics do not preserve spin inversion symmetry. More fundamentally, this can also be seen from the fact that although the Hamiltonian has spin inversion symmetry, the spin commutation relations (e.g. [S x,s y ] = i S z ), change sign under spin inversion.

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 32/6 What went wrong? If instead of using precessional dynamics (LLG eqns), we left out the precessional term (relaxational dynamics), the states would be complementary. MISTAKE: Leaving out the precessional nature of the dynamics. Confusing Hamiltonian symmetry with configurational symmetry.

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 33/6 Counterargument If the external field is varied adiabatically, the detailed fast time scale precessional motion must be irrelevant. Rebuttal: It is the avalanches that give rise to hysteresis in the first place. The motion during one is fast and the outcome is greatly effected by the details of the precessional dynamics.

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 34/6 Next Step We describe a simulation using the LLG equations to find out how well this agrees with experiment and make predictions.

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 35/6 Simulation of Co/Pt films We ll simulate the LLG equations with the following ingredients in the Hamiltonian: Assume the films are disordered but strongly anisotropic. The easy axis is randomly oriented but strongly biased perpendicular to the film. Assume a long range dipolar interaction between points. Assume a short range ferromagnetic coupling J + δ i, where δ i is a random variable whose strength and statistics can be adjusted.

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 36/6 Precessional Term α γ 2 /γ 1 measures how much damping there is compared to precession.

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 36/6 Precessional Term α γ 2 /γ 1 measures how much damping there is compared to precession. NeFe thin films, α.01. CoCr/Pt multilayer films, α 1. Co/Pt multilayer films, α.37.

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 37/6 Hysteresis Loops Hysteresis loops for systems with different amounts of disorder; the vertical axes are the magnetizations and the horizontal axes are the external fields. The cliff in the hysteresis curve vanishes as disorder is increased.

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 38/6 Low disorder simulations Here is a movie of the low disorder case. Domain growth for a 256 2 system with low disorder, λ = 1000 and w = 0.15. The temperature is 10 4.

Cliff Region Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 39/6

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 40/6 Low disorder images O. Hellwig and E. E. Fullerton

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 41/6 Domains vs Disorder Configurations near the coercive field. (T = 10 4 ).

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 42/6 Experimental Patterns (O. Hellwig and E. E. Fullerton)

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 43/6 Corresponding Loops Hysteresis loops for systems with different amounts of disorder; the vertical axes are the magnetizations and the horizontal axes are the external fields. The cliff in the hysteresis curve vanishes as disorder is increased.

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 44/6 Sensitivity To Noise Look at domain evolution with different realizations of thermal noise: 1 2 high disorder: low disorder: The low disorder system is much more sensitive to thermal effects. This implies that RPM will be much weaker in that case.

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 45/6 RPM CPM plots 1 0.8 0.6 0.4 0.2 0 1 0.8 0.6 0.4 0.2 0-0.6-0.4-0.2 0 0.2 0.4 0.6-0.6-0.4-0.2 0 0.2 0.4 0.6 1 0.8 0.6 0.4 0.2 0 1 0.8 0.6 0.4 0.2 0-0.6-0.4-0.2 0 0.2 0.4 0.6-0.6-0.4-0.2 0 0.2 0.4 0.6 Blue RPM, Red CPM. As disorder increases, both RPM and CPM increase. (T = 10 4 ).

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 46/6 RPM CPM vs Disorder 1 0.8 RPM, CPM 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 Disorder RPM CPM

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 47/6 Temperature dependence (predicted) 1 0.9 0.8 RPM 0.7 0.6 0.5 0.4 T = 0.0 0.3 T = 0.00001 T = 0.0001 0.2 T = 0.001-0.4-0.2 0 0.2 0.4 0.6 B 1 0.9 0.8 CPM 0.7 0.6 0.5 0.4 T = 0.0 0.3 T = 0.00001 T = 0.0001 0.2 T = 0.001-0.4-0.2 0 0.2 0.4 0.6 B

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 48/6 Low Disorder Prediction 1 0.8 0.6 0.4 0.2 0-0.6-0.4-0.2 0 0.2 0.4 0.6 Initial growth sites should be the same at sufficiently low temperatures. (O. Hellwig and E. E. Fullerton)

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 49/6 Conclusions Patterns in magnets are not symmetric between the upper and lower branches of magnetic systems.

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 49/6 Conclusions Patterns in magnets are not symmetric between the upper and lower branches of magnetic systems. This can be explained by the presence of both damping and precessional motion. Simulations incorporating this, ferromagnetism, disorder, and dipolar interactions explains the experiments fairly well.

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 50/6 Multicycle spin dynamics Consider the 3d Edwards and spin glass Hamiltonian H = i,j J i,j S i S j h i S i. The couplings J i,j are uniform random numbers between ±1. The spins S i = ± 1. Free boundary conditions. Use single spin-flip dynamics. At any step, we search for the next value of h where a spin flip occurs. Once that happens we let any subsequent avalanches occur before changing h again.

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 51/6 Multicycle spin dynamics Now we periodically cycle the field between h min and h max.

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 51/6 Multicycle spin dynamics Now we periodically cycle the field between h min and h max. In steady state, the hysteresis loop takes more than one cycle to close on itself. 0.6 0.4 0.2 M 0-0.2-0.4-0.6-1.5-1 -0.5 0 0.5 1 1.5 h

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 52/6 Power spectrum The magnetization as a function of time in steady state shows subharmonics. The power spectrum of M(t) has peaks at fractions of the driving frequency. (Driving frequency = 1 below). 20 15 3 2 T=0 T=0.2 10 3 I 10 1 5 0 0 0.2 0.25 0.3 0.35 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 f

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 53/6 Nanomagnetic pillar arrays Ni nanomagnets on silicon 1 Magnetic Force Microscopy [1] Courtesy of Holger Schmidt. Fabricated by T.Savas MIT They can be fabricated to have a wide variety of shapes and sizes

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 54/6 Modeling nanomagnetic pillars These are single domain nanomagnets where the crystalline orientation is random in each pillar.

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 55/6 Hamiltonian The Hamiltonian is the addition of four pieces due to: The external field: i hs z,i

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 55/6 Hamiltonian The Hamiltonian is the addition of four pieces due to: The external field: i hs z,i Crystalline anisotropy: [ K 1 2 (α4 x,i + αy,i 4 + αz,i) 4 + K 2 αx,iα 2 y,iα 2 z,i] 2 i α s are direction cosines relative to the crystalline axes.

Hamiltonian The Hamiltonian is the addition of four pieces due to: The external field: i hs z,i Crystalline anisotropy: [ K 1 2 (α4 x,i + αy,i 4 + αz,i) 4 + K 2 αx,iα 2 y,iα 2 z,i] 2 i α s are direction cosines relative to the crystalline axes. Dipolar self energy, that is shape anisotropy: i d z s 2 z,i Dipolar interactions between pillars: s i A(r ij ) s j j i Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 55/6

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 56/6 Results We found that to get multi-cycles it was best to use a triangular lattice. Here is a movie of a system showing two cycles. Here part of the movie is in slow motion. showing details of avalanches. We tried a range of pillar radii, heights and separations. The probability of observing a multicycle is as high as.6 This provides a viable system for observing multicycle behavior.

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 57/6 Avalanches and Precession with Andreas Berger, Hitachi Global Storage Technologies San Jose In many magnetic materials spin dynamics are dominated for short times by precessional motion as damping is relatively small. Values of α = γ 2 /γ 1 range from.01 to 1.

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 58/6 Dynamics with no damping An avalanche can transition to an ergodic phase where the state is equivalent to one at finite temperature. The temperature is often above the ferromagnetic ordering temperature. This is a movie of a 32 32 system. This is a 128 128 system.

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 59/6 Effect of finite damping α = 0.9 α = 0.8 α = 0 final configuration during the avalanche

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 59/6 Effect of finite damping α = 0.9 α = 0.8 α = 0 final configuration during the avalanche Decreasing damping increases the size of avalanches

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 60/6 "Nucleation" with no damping Model the system as being subdivided into metastable and ergodic regions.

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 60/6 "Nucleation" with no damping Model the system as being subdivided into metastable and ergodic regions. ergodic T A growing ergodic region metastable

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 61/6 Model Results 35 30 25 w 0 20 15 10 5 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 T 0 Simulation results and analytical prediction of the critical line separating growing and static avalanches. The initial avalanche of size w 0 to grow. T 0 is the energy released by a site in transitioning to an ergodic region.

Further Questions Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 62/6

Nonequilibrium symmetry breaking and pattern formation in magnetic films. p. 62/6 Further Questions If the total s z is only weakly broken, by dipolar and anisotropy terms, then what effect does this have on the dynamics? How does precessional motion effect the size of the critical field region of avalanche dynamics?