Simplified Fast Multipole Methods for Micromagnetic Modeling

Transcription

1 implified Fast Multipole Methods for Micromagnetic Modeling D. M. Apalkov and P. B.Visscher Department of Phsics and Astronom and MIT Center The Universit of Alabama This project was supported b F grants # EC and DMR , and DOE grant # DE-FG0-98ER4574 an F Materials cience and Engineering Center

2 Overview Contet: Fast multipole method (FMM), calculates magnetostatic fields hierarchicall Previous improvements: Cartesian (simpler than spherical) harmonics recursive calculation of kernels using self-similarit (PBV & DMA, J. Appl. Phs. 93, 5 Ma 003. ) Improvements described here: imple wa to treat periodic sstems (MRAM, thin films) Compute fields from surface charges on discrete cells, not from uniform magnetiation inside an F Materials cience and Engineering Center

3 Basic idea of FMM Rationale -- MRAM eample: ubdivide rectangular elements into grid (~0 5 cells) H~M/r 3 +Quad/r 4 M quad, oct, O( ) interactions FMM lumps cells together hierarchicall: + = then + = Quad. (add multipole moments) + an F Materials cience and Engineering Center

4 Basic FMM operations A FMM (fast multipole method) needs functions to Compute Talor epansion of the potential of a point charge at an arbitrar point V (r) = n! n! n Compute multipole moment of a charge or M distribution Q n n n n ) n n = ρ (,, d d d Convolute the multipole moment with the field of a point charge to get the field of the multipole hift the origin of a Talor epansion! n, n, n V n n n n This gives an O() algorithm for computing the magnetostatic field. n n an F Materials cience and Engineering Center

5 an F Materials cience and Engineering Center Application to periodic sstem eed potential of cored arra of multipoles like But... it s enough to know potential of cored arra of point charges! Just convolute this with the multipole moment of periodic cell Want Talor epansion of potential about center of one cell:

6 Conclusion: to do FMM efficientl in a periodic sstem, we need onl calculate the potential of a cored arra of point charges: Traditional Methods: Ewald sum (disadvantage: order ) FFT (disadvantage: requires uniform periodic grid) Instead, use iterative method that uses functions alread part of an FMM code. an F Materials cience and Engineering Center

7 an F Materials cience and Engineering Center Conclusion: to do FMM efficientl in a periodic sstem, we need onl calculate the potential of a cored arra of point charges: R R R Convolute with weight arra w(r) = till missing: shell from R to ~R add b hand

8 Iterative algorithm to calculate the potential of a cored arra of point charges tart with an estimate of V cored (r) Rescale to coarse grid (V of black s) Convolute with w(r) = Add V shell (r) Repeat R R Formall, V cored, new (r) = ½V cored, old (r/) * w + V shell (r) Converges ver fast. an F Materials cience and Engineering Center

9 an F Materials cience and Engineering Center umerical result for cored potential V cored (,,0) ( ) ). ( ! ! ),, ( cored O r V = Agrees numericall with brute-force summation.

10 Charge-based calculation of magnetostatic field Magnetiation of a cubical cell can be treated as () Uniform magnetiation M (dipole moment per unit volume) over cell, OR () Uniform surface charge (densit = normal component of M) on each face of cell These are mathematicall equivalent, but ma not be equivalent in an given approimation Trivial eample where charge-based method is more efficient: uniforml magnetied infinite thin film. Magnetiation method gives poorl convergent sum of dipole fields urface charge method gives ero (correct answer) immediatel there are no charges anwhere Man real problems are close enough to this case that the charge method is significantl more efficient. an F Materials cience and Engineering Center

11 Implementing charge-based field calculation in FMM contet FMM requires hierarchical description of sstem cubical cells are the lowest objects on the hierarchical tree, in usual M-based scheme In charge-based scheme, charged faces of cells are treated as children of the cells tree is etended at the bottom Whole sstem or root cell cells in the bulk own 3 faces boundar cell ma own up to 6 faces an F Materials cience and Engineering Center

12 Advantages of charge-based field calculation in FMM contet Micromagnetic kernel can be calculated recursivel (D. M. Apalkov and P. B. Visscher, imple iterative calculation of micromagnetic kernels, J. Appl. Phs. 93, 5 Ma Kernel (used to calculate interactions between nearb objects) describes average field of charged square (rather than magnetied cube) over another cube. Uniforml-magnetied regions are treated efficientl. an F Materials cience and Engineering Center

13 Charge concentration at top corners Test case: MRAM element 500 cells, in C state Magnetiation vector shown (red); charge not shown Charge concentration an F Materials cience and Engineering Center

14 Accurac of charge-based vs magnetiation-based FMM calculation Detail of upper left corner of MRAM, showing both M (red) and charge (black) Field error in a 500-cell MRAM element (averaged over element, relative to average field) Relative Error Magnetiation-based FMM Charge-based FMM Tin charges at faces, if M nearl uniform Big charges at eternal faces Multipole order an F Materials cience and Engineering Center

15 Conclusions ew method for iterative calculation of the magnetostatic field of an infinite arra of periodic images Well adapted to Fast Multipole Method Eas to implement, uses functions that alread eist in FMM code Charge (rather than magnetiation) based calculation of magnetostatic field is much more accurate in man cases, especiall where there is flu closure (MRAM, thin film) an F Materials cience and Engineering Center

16 pin-polaried Current Induced witching. P. B.Visscher and D. M. Apalkov Department of Phsics and Astronom The Universit of Alabama This project was supported b F grant # EC and DMR grant # an F Materials cience and Engineering Center

17 Motivation stud spin-polaried current induced switching in thin film multilaered Co/Cu/Co structures; include the influence of the reflected current induced torque in the thick Co-laer. an F Materials cience and Engineering Center

18 Cornell eperiment [Katine et al, Phs. Rev. Let. 84 (000) 349; F. J. Albert et al, Appl. Phs. Lett. 77 (000) 3809] device dimensions ~ 60 nm 30 nm; laer thickness: Cu (80 nm) / Co (40 nm) / Cu (6 nm) / Co (.5 nm) / Cu (5 nm)/ Au(60 nm); effect of the current induced magnetic field is assumed to be small compared to the effect of the spin-polaried current on the spin moments. an F Materials cience and Engineering Center

19 Micromagnetic Calculation Modified Landau-Lifshit (LL) equation: dm dt γ = αγ = γ M H 7.6 ( KOe ns) M M ( M H) + Jβ M is the gromagnetic ratio; M pinpartners ( M M ) P pin torque (loncewski model) α = 0.07 is the LL damping coefficient; H is the total magnetic field; M is the saturation magnetiation; J is the current densit; β is a constant, which characteries the magnitude of the current-induced torque. an F Materials cience and Engineering Center

20 Treatment of the Current Induced Torque Co Co These cells affect the magnitude of spin torque in the blue cell. The are called pin Partners for the blue cell. The radius of the clinder is 6 nm (the discretiation sie is 3 nm). dm dt current = Jβ M M pinpartners ( M M ) P an F Materials cience and Engineering Center

21 Magnetic Field rand H K H + J M( r + δ) + ( M( r) eˆ )eˆ et stat H( r) = H + H + e M δ et H is the eternal (applied) field; J e ê = 0.4 is the echange integral (dimensionless), corresponding to - echange constant ; A = µ 0 JM a = 0 ( 4π = 8. KOe) M = 440 emu M cc is the saturation magnetiation (Co); is a unit vector along the crstallographic eas ais ( - ais); H K stat H = 0.5 KOe is the effective anisotrop field; is the magnetostatic field (includes both demag field and the magnetostatic field due to the other laer of Co). J m rand H is the effective random field describing Langevin noise. an F Materials cience and Engineering Center

22 nm H et stem Geometr 3 nm free to move laers strongl pinned laers nm 60 nm Relaed to thermal equilibrium (T = 4 K) sstem is shown. an F Materials cience and Engineering Center

23 witching Process (T=300 K) - pulse is on βj = 0.3 ns ; T = 300 K; H et =.0 Koe; switching time ~ 0.4 ns; pulse duration is.7 ns. an F Materials cience and Engineering Center

24 M Z M M-J Hsteresis Loops βj (ns ) H et =. koe H et = koe weep rate is 0.5 ns - For one-domain model: + / αγ β J crit = µ = 0.68, [ ( πm + H ) + H ] 0.ns H et = koe, sweep rate is 0.5 ns - P an F Materials cience and Engineering Center - K et

25 Conclusions One-domain assumption is wrong in some cases, complete micromagnetic treatment is required switching is achieved b domain wall motion thermal fluctuations are important to nucleate the reversed domain Future work tud the probabilit of reversed domain nucleation versus temperature; compare the results with eperimental data and adjust the model; calculate the current induced torque from electron wave functions (W. Butler and J.Velev) an F Materials cience and Engineering Center