FERROMAGNETIC RESONANCE MEASUREMENTS AND SIMULATIONS ON PERIODIC HOLE AND DISC ARRAYS J. Skelenar, S. Chernyashevskyy, J. B. Ketterson; Northwestern University V. Bhat, L. Delong; University of Kentucky K. Rivkin; Seagate Technologies O. Heinonen; Argonne National Laboratory C-C. Tsai; Chang Jung Christian University, Taiwan MISM Conference August, 2011
Simulation: the discrete dipole approximation Replace the continuum of some object by an array of dipoles H 0 H 1 microwave field external field each dipole represents a cluster of atomic dipoles
Landau-Lifshitz equation Equation of motion for i th dipole: dm i dt Contributing fields: (torque) =!"m i # h i! $" m M i # m i # h i s h i = h i external +hi internal (damping) ( ) h i internal = hi exchange + hi dipole + hi anisotropy h i exchange = J! m i NN h i dipole = ) j!i # 3r ij (m j! r ij ) % 5 $ % r ij " m j r ij 3 & ( '( h i anisotropy = K(m i! Ĥ A ) M s 2 Ĥ A
Finding the equilibrium magnetization: Magnetostatics Method 1: i) Start with dipoles either along the external field or arbitrary axes; ii) calculate the total field acting on each dipole; iii) move the orientation of each dipole to align with the local field; iv) continue until the orientations stabilize. Method 2: i) Start with arbitrarily oriented dipoles; ii) integrate the Landau-Lifshitz equation with damping until system relaxes to an equilibrium. Potential problem: metastable states.
Example: the static magnetization of a Py disk in a vortex state: z - component Magnetization points out of plane in the vortex center
Equilibrium magnetization of Py disks and rings: x-component applied field H 0 700nm
Resonant Modes of Magnetic Nanostructures in the Discrete Implementations: Dipole Approximation (DDA) Codes available free at: http://www.rkmag.com/ Direct numerical integration in time of DDA (e.g., the NIST OOMMF code) Advantage: works for nonlinear cases. Disadvantage; very slow. Eignvalue method: fast; non - linear response limited to perturbation theory. Alternative: Finite element methods.
The eigenvalue method Consider a system of N spins denoted by i with magnetization m i each obeying the Landau-Lifshitz equation: dm i dt =!" m i # h i! $ " m M i # ( m i # h i ) s We linearize this equation by writing: m i = m i (0) + m i (1) (t) h i = h i (0) + h i (1) (t) + h i (rf ) (t) For the case h i (rf) = 0 this yields a first order homogeneous linear differential equation! "1 dm (1) i dt (1) (0) (0) (1) $ (0) % (1) (0) (0) (1) + m i # hi +mi # hi + m M i # mi # hi + mi #hi s & ' Assume m i (1) (t)! e "i#t ; dm i (1) dt = "i#m i (1) ( ) * = 0 Task: find the N eigenvalues, ω (j), and eigenvectors, V i (j), of this equation.
Eigenmodes m i (0) = equilibrium (or initial) orientation m i (1) = dynamic shift in orientation m i (0 z m i (1) (t) m i (t) y m i (1) (t) = Vi ( j) e!i" ( j) t + Vi ( j)* e i" ( j)* t x V i ( j) = j th eigenvector! ( j) = j th eigenfrequency
Driven inhomogeneous equation dm i (1) dt +! # (0) (0) (1) (0) (0) (1) m i " hi + mi " hi + mi " hi $% +m i (1) " hi (1) + mi (0) " hi (rf ) + mi (1) " hi (rf ) & '( + (damping) = 0 Various terms: (0) (0) m i! hi (0) (rf ) m i! hi (1) (1) m i! hi (1) (rf ) m i!h i torque due to non-equilibrium initial configuration excitation due to the external r.f. field non-linear term mode mixing, SHG, etc. non-linear term second harmonic generation etc.
General solution to linear case m i (1) (t) = (transient)+(steady state) Formally we have a set of coupled 1 st order ODE's for which has the solution (1) ( j) ( m i (t) = Vi e!i"( j) t c j + e!i"( j) t e i"( j) t ( j)% + * ' $ V i# L &g i # (t)dt-; )* i#,- here g i (t) =!. ( (0) (0) (0) (rf ) m i / hi + mi / hi (t) + )*,- and V ( j) i# L = left eigenvector Energy absorbed from a sinusoidal r.f. field follows directly from eigenvalues AND eigenvectors as: * de dt =!" Re,, # i, $ +, % V ( j)& (0) (rf ) il ' mi ( hi i,j! #! $ j ( ) ( ) j + i)! $$ ( ) ( j) V i$ ' h $ i (rf )& - / /. / For details see Phys. Rev. B 75, 174408 (2007)
Application: absorption spectra for in-plane magnetic field
Excited Modes
Extension to Periodic Systems Apply Bloch s theorem ( j) ( j) V i! Vkn (i)e ik"r j = j th eigenvector/mode i = individual spin in unit cell k = Bloch wavevector n = band index (total of N) R = n a a + n b b = real - space lattice vectors Complication: requires dipole lattice sums for each k-vector Dynamic magnetic response of infinite arrays of ferromagnetic particles Phys. Rev. B 75, 174408 2007
Summary of Simulation Techniques Discussed 1. Mode spectra of arbitrary shaped object (eigenvalue method) 2. Absorption spectra of arbitrary shaped object (eigenvalue method). 3. Generalization to periodic arrays Topics not discussed (use RK-integration of LL equation) High power FMR spectra Suppressing the Suhl instability Dynamic magnetization reversal Spin valves and oscillators
Applications 1. FMR. Computes resonant modes and absorption. 2. Spin torque memory. Magnetization reversal is dominated by a few modes. Transforming L-L equation from coordinate to eigenvalue space results in a simple linear equation with N variables and no stiffness, unlike NlogN operations needed in coordinate space. 3. Magnetic reader thermal noise. Thermal excitation of resonant modes results in GHz frequency noise in magnetic readers. Neil Smith (Hitachi), JMMM 321, 6 (2009).
Read Head Modeling Free layer in GMR, TMR element changes its magnetization as a linear response to external fields. Ideal application for the method small object, with rich dynamics dominated by a few modes. Static configuration Modes responsible for the noise:
Ferro-magnetic resonance measurements on magnetic nanostructures: Dot and anti-dot arrays.
Experimental set up Hewlett - Packard!!!!Synthesizer 10MHz - 20GHz Microwave!Detector Lock - in amplifier Magnet Coils & Pole Pieces!!!Field Modulation!!!!Coils!Power Amplifier!!Bipolar Power Supply!!Computer : Field sweeping, data aquisition, trace averaging Meanderline!!!!sample!!!!!!!cell
The Sample Holder Rigid Coax Hand wound broadband meanderline is pressed up against the surface of the sample. Microwave generator supplies frequencies from 10 MHz to 20 GHz. Wire Field lines Meanderline Meanderline creates microwave magneac fields in- plane and perpendicular to sample.
An e - beam patterned square hole array
Broadband FMR in square antidot arrays: a snapshot
Resonances observed with magnetic field aligned along the principal axis - Resonances occurring at the same field regardless of sweep direcaon suggest we are in a saturated state. - Split resonance modes observed before by Yu, Pechan, and Mankey.
Resonances observed with magnetic field aligned in 45 o position - The two strongest lines have merged together. - There are resonances len in the wake of the high field resonance. One of them doesn t move and is likely the uniform FMR mode.
Simulated magneazaaon phase profile for the mode propagaang parallel to the field with B along a principal axis (the high field resonance) SimulaAon: the different colors depict the variaaon of phase of the in- plane. MagneAzaAon. (Calculated via FFT spaaal and temporal integraaon of LL equaaon in DDA)
Simulated magnetization phase profile for the mode propagaang perpendicular to the field with the B along a principle axis (the low field resonance)
Simulated magnetization phase profile: B at 45 o where the two resonances merge
Angular Dependence: RK simulation vs. experiment (9.75 GHz)
Interpretation 1. The microwave field is uniform on the scale of the array 2. The array sets up dynamic fields with the periodicity of the lattice 3. These fields excite spin-waves having wave vectors equal to those of the reciprocal lattice. 4. All modes are at k = 0 when folded to the first Brillouin zone. 5. The allowed group symmetries correspond to Γ (singly degenerate; and E (two fold degenerate) represntations. 6. Γ - modes are four fold symmetric; E - modes are two fold symmetric
Schematic of zone-folded coupling to spin waves!(k)! d 2! d k
Damon Eshbach Modes! H = H 4"M ;!!!!!! = # 4"$M
Combining Damon-Eshbach with exchange for Permalloy Films Kreisel et al derive an approximate analytic formula for the dispersion of spin waves in YIG propagating in-plane with for varying propagation angles relative to an inplane field. The model incorporates both magnetostatic and exchange effects. The approximate expression for the dispersion relation works best for propagation angles less than 45 degrees away from applied field.
- Based on simulaaons showing phase variaaons of spin waves in (1,0) and (0,1) direcaons with periodicity of the la[ce the two strongest lines were tracked and fit to the analyac model using a wavevector with magnitude corresponding to the la[ce constant. - We are not in exchange limit so exchange constant was set to zero. Angular Dependence for Strongest Resonances to Analytic Model Lattice Constant Saturation Magnetiza -tion Effective Thickness (1,0) Spin Waves (0,1) Spin Waves 1008 nm 948nm 780 emu 780 emu 22 nm 22nm
Work in Progress: Overview of all Observed Resonances! - mode E - mode
Graduate students who do the work Joe Sklenar Vinayak Bhat
Thank you for your attention
Micromagnetic simulations show, for the largest absorption lines, periodic variation in the phase of the magnetization suggesting spin waves are excited. The RF field produced by the meander line is essentially uniform over many lattice constants. This uniform r.f. field is diffracted by antidot array causing an effective excitation field with spatial variations having the periodicities of the reciprocal lattice. Theory of spin waves propagating in-plane with varying inplane angle designed for YIG seems to agree. The theory uses a Heisenberg like Hamiltonian that incorporates both exchange and dipolar effects.
How Should we Characterize These Absorption Lines? Micromagnetic simulations show, for the largest absorption lines, periodic variation in the phase of the magnetization suggesting spin waves. The RF field produced by the meander line is essentially uniform over many lattice constants. This uniform field is diffracted by antidot array causing an effective excitation field with spatial variations having the different periodicities of the lattice. Theory of spin waves propagating inplane with varying in-plane angle designed for YIG seems to agree. The theory uses a Heisenberg like Hamiltonian that incorporates both exchange and dipolar effects. R. W. Damon, J. R. Esbach (1960); B.A. Kalinikos, A.N. Slavin (1986); A. Kreisel, F. Sauli, L. Bartosch, and P. Kopietz (2009)
Outline Brief description of measurement apparatus. Short overview of broadband resonances in a 1 micron permalloy antidot array with 500nm antidots. Experimental results of angular dependence of resonances at 9.75 GHz. Results of simulation leading into a description of spin wave modes being the most dominantly activated mode, with wavelengths related to the underlying antidot lattice. Results of applying a microscopic analytic theory of spin wave modes in YIG to the antidot resonances.