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Dynamic critical curve of a synthetic antiferromagnetHuy Pham, Dorin Cimpoesu,Andrei-Valentin Plamad,Alexandru Stancu, and Leonard SpinuCitation:Appl. Phys. Lett. 95, 222513 (2009); doi: 10.1063/1.3265739View online: http://dx.doi.org/10.1063/1.3265739View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v95/i22Published by theAIP Publishing LLC.Additional information on Appl. Phys. Lett.Journal Homepage: http://apl.aip.org/Journal Information: http://apl.aip.org/about/about_the_journalTop downloads: http://apl.aip.org/features/most_downloadedInformation for Authors: http://apl.aip.org/authors

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Dynamic critical curve of a synthetic antiferromagnet

Huy Pham,1 Dorin Cimpoesu,2,a Andrei-Valentin Plamad,2 Alexandru Stancu,2 andLeonard Spinu1,b1Department of Physics and Advanced Materials Research Institute (AMRI), University of New Orleans,

New Orleans, Louisiana 70148, USA2Department of Physics, Al. I. Cuza University, Iasi 700506, Romania

Received 30 July 2009; accepted 21 October 2009; published online 3 December 2009

In this letter, a dynamic generalization of static critical curves sCCs for synthetic antiferromagnet

SAF structures is presented, analyzing the magnetization switching of SAF elements subjected to

pulsed magnetic fields. The dependence of dynamic critical curves dCCs on field pulses shape and

length, on damping, and on magnetostatic coupling is investigated. Comparing sCCs, which are

currently used for studying the switching in toggle magnetic random access memories, with dCCs,

it is shown that a consistent switching can be achieved only under specific conditions that take into

account the dynamics of the systems. The study relies on the LandauLifshitzGilbert equation.

2009 American Institute of Physics. doi:10.1063/1.3265739

The magnetization reversal in synthetic antiferromagnet

SAF structures has been extensively studied due to their

applications as hard layers of exchange coupled composite

media,1

soft underlayers for perpendicular recording,2

pinnedand free layers for magnetic random access memory

MRAM ,3

hard disk reading heads or magnetic sensors.4

The performance of the devices using SAF structures relies

on their switching characteristics. The magnetization switch-

ing can be described using the concept of critical curve CC

developed initially for uncoupled magnetic systems5

and

then for coupled films,6,7

CC being the locus of in-plane

fields at which the irreversible magnetization reversal occurs

but not the locus of all free energys critical points, as the

name may suggests . Subsequently, CCs of SAF have been

extensively studied due to their technological importance,

especially for toggle MRAM.

815

However, these descrip-tions are restricted to quasistatic regime, where the magneti-

zation dynamics and precessional effects are neglected. The

devices using SAF structures require a short access time and

the magnetization is forced by pulsed magnetic fields to

switch at nano and subnanosecond time scales for which the

static CC sCC approach is not anymore adequate. For un-

coupled systems it was shown that a pulsed magnetic field

can provide a high-speed switching16

below the static limit

predicted by StonerWohlfarth SW model,17

and subse-

quently the dynamic CC of a SW particle was given.18

Later,

dynamic and temperature effects on toggle MRAM operating

field19

and switching diagrams20

were presented.

In this letter a dynamic generalization of sCCs for

coupled magnetic systems is presented, analyzing the mag-

netization switching of SAF elements subject to pulsed

magnetic fields while thermal effects are neglected. As in

Ref. 18 for a SW particle, the boundary between switching/

nonswitching regions represents the generalization of sCC,

namely the dynamic CC dCC . Comparing sCCs with dCCs

it will be shown that a consistent switching can be achieved

only under specific conditions that take into account the dy-

namics of the magnetic moments. Using dCCs we can also

better understand the toggle switching diagrams from

Ref. 20.

The model is based on LandauLifshitzGilbert LLG

equation21

assuming that the magnetization in each layer isuniform. We consider that the two ferromagnetic layers are

identical except for the thickness. As ferromagnetic material,

permalloy with saturation magnetization Ms =10.8

106 /4 A /m was used. The magnetic layers are assumed

to be in the shape of ellipsoids making the demagnetizing

field uniform across the layer. The ellipsoids principal axes

are taken along x, y, and z: 2a =120 nm along Ox and

2b =100 nm along Oy. The thickness of bottom layer is

t1 =5 nm, leading to demagnetizing factors Nx, Ny,Nz 0.029, 0.039, 0.932 and to in-plane uniaxial shape an-

isotropy field 0Hsh,1 =0 Ny Nx Ms =9.82 mT. Hereby

two contributions to anisotropy are taken into account: an

easy plane and an easy axis EA directed in this plane. In astatic regime the easy plane anisotropy can be neglected. The

thickness t2 of top layer is varied so that t= t2 / t11; a thick-

ness imbalance also involves a shape anisotropy imbalance.

The effective fields consist of applied field, demagnetiz-

ing field, phenomenological antiferromagnetic exchange

coupling, and magnetostatic coupling. The last two are de-

scribed by hJ= Wex / 2Ksh,1St1 and hmag= Wmag / 2Ksh,1V1 ,

respectively, where Wex and Wmag are the exchange and mag-

netostatic energy when both layers are aligned along EA, S

=ab is the layers area, and V1 is first layers volume. The

magnetostatic interaction field was calculated using the

method presented in Ref. 22. All magnetic fields presentedthroughout the paper are normalized by Hsh,1.

Because an instantaneous change of the applied field

from zero to some value is not realistic, sinusoidal time de-

pendence for the field pulse rise and fall are assumed. The

rise/fall time is a function of the pulses amplitude, so that

the field sweep rate H, defined as the ratio between the

amplitude and pulses rise/fall time, is constant. The pulse

width TH is the amount of time the pulse takes to go from

zero to high and back to zero again. The final state is taken

after a time long enough to reach the equilibrium after the

termination of applied pulse.

In Fig. 1 we present sCCs for t=0.8 and t= 1, obtained

by making the determinant of free energys Hessian equal to

aElectronic mail: [email protected].

b Electronic mail: [email protected].

APPLIED PHYSICS LETTERS 95, 222513 2009

0003-6951/2009/95 22 /222513/3/$25.00 2009 American Institute of Physics95, 222513-1

http://dx.doi.org/10.1063/1.3265739http://dx.doi.org/10.1063/1.3265739http://dx.doi.org/10.1063/1.3265739http://dx.doi.org/10.1063/1.3265739


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0.6

For t=0.8 Fig. 1 a the interior curve consists of a heart-

like part heart for short and an astroidlike part, the height ofthe last being around 0.05, so that it appears almost like a

point in our figures. The free energy has one minimum

Mheart= 1 ,2 inside the heart, where 1, 2 denote the mag-

netization angle of the two layers with respect to EA, two

minima Me,0 and Me,1 between hearts and the exterior curve

Me,0 = 0 , and Me,1 = , 0 for no applied field , and one

minimum Msat outside the exterior CC.9

For hx0 on the

heart Me,0Mheart, and on the exterior CC Me,0Msat, whilethe state Me,1 loses stability as it crosses both CCs. For t

= 1 Fig. 1 b the interior CC consists only of an astroidlikepart, two pairs of degenerate minima existing inside it, one

minimum outside the exterior CC and one pair of degenerate

minima otherwise. In a static regime a discontinuous transi-tion occurs when an increasing field starting from zero

crosses the right side of the astroidlike part. Decreasing the

field back to zero, a discontinuous transition occurs when the

field crosses the left side of the astroid, bringing the system

back to the original state.12

A strategy to point out the CCs is to use both a bias and

a pulse field Fig. 1 : for t=0.8 we have to start from , 0 ,

to apply a bias field hbias perpendicular to EA, and then for

each hbias value, field pulses with different amplitudes along

EA we denote with B1 this fields configuration . Comparingthe system state before the pulse application with the state

after completion of pulse, we identify two transitions: at the

left side of the heart and at the exterior CC. In order to detectthe right side of the heart we have to apply a L-shaped bias

field a bias field perpendicular to EA and then a bias fieldparallel to EA, so as to not induce any irreversible transition

and field pulses perpendicular to EA BL configuration . Be-cause for t=1 the B1 configuration cannot reveal the astroid-

like CC, we decrease hbias along the EA, starting from posi-

tive saturation, and then for each hbias value, field pulses are

applied perpendicular to EA B2 configuration . This con-figuration can reveal the left sides of both astroidlike CCs,

corresponding to hx0 and hx0, respectively.

The sCCs are restricted to the static regime and does not

take into account the dynamics of the magnetization, i.e., the

SAF is subject to a slowly varying field so that the systemstays within one energy well and irreversible switches occur

only when the state occupied by the system loses stability. If

the pulse field has a rise time shorter than the relaxation

time, then the precessional term from LLG equation cancarry the magnetization over a large range of motion before

energy is dissipated precessional regime . Also, there is anout of plane component of magnetization.

In order to determine the parameters values for reliable

operation of a MRAM cell, we have studied the switching

properties as a function of pulses shape and length, damp-

ing, and magnetostatic coupling.

The switching behavior of an asymmetric SAF is pre-

sented in Fig. 2. From Figs. 2 a and 2 b , we can see that for

a damping coefficient =0.008, typical for Permalloy,23

the

final state is sensitive to the sweep rate H. As H increases

the nonswitching region corresponding to outermost dCC

shrinks and an instability region, with switching/nonswitching fringes, grows up because a significant ringing

of the magnetic moments still exists during the field pulse

and the final state is determined by the positions of the mo-

ments at the end of the pulse. The region corresponding to

interior dCC, where switches can occur, is enlarged com-

pared to sCC, and what is important to observe is that only a

digit or word field applied at 45 with respect to EA intoggle MRAM, can switch the magnetization. Consequently,

using sCCs instead of dCCs can lead to inadvertent switch-

ing of half-selected memory cells. Also, we observe that ir-

reversible switches appear when the field crosses the upper

part of right part of the heart, in contrast with the static case.

In order to minimize dynamical effects, rare-earth dopantswere used in Ref. 24 to increase describing the energy

dissipation and magnetizations relaxation into the magnetic

FIG. 1. Color online sCCs for an asymmetric SAF a and for a symmetricone b . A strategy to point out CCs for an asymmetric SAF is to use anincreasing bias field perpendicular to EA, starting from , 0 state, and thena pulse field parallel to EA, combined with a L-shaped bias field and a pulse

field perpendicular to EA. For the symmetric case we apply a decreasing

field along EA, starting from positive saturation, and then a pulse field

perpendicular to EA.

FIG. 2. Color online Switching diagram of an asymmetric SAF: sweeprate dependence for =0.008 a , b , a , and b , and pulse lengthdependence for = 1 c , d , c , and d . The B1 configuration was usedfor a , b , c , and d , and correspondingly the BL configuration for a , b , c , and d . Each point hx, hy is the coordinate of the total vectorfield hbias +hpulse when hpulse reached its peak value.

222513-2 Pham et al. Appl. Phys. Lett. 95, 222513 2009



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fields direction . As increases the instability fades away

Figs. 2 c and 2 d , and by systematic simulations it wasobserved that for TH= 3 ns the interior dCC shrinks toward

sCC and the exterior dCC expands to sCC, so that for =0.05 dCCs pretty much concur with sCCs. Further, for

0.05 the magnetization cannot follow the applied field and

dCCs shift toward higher fields. Instead, using longer pulses

dCCs move back toward sCCs.

Similarly, in the case of symmetric SAF, as H increases

the nonswitching region corresponding to outermost dCCshrinks and an instability region grows up Figs. 3 a and3 b . The regions corresponding to interior CC, shrink alittle bit, and are almost independent on H. As increases

the instability fades away and the interior dCCs shift toward

higher fields Fig. 3 c . However, B2 configuration cannotreveal the entire exterior dCC, and in Figs. 3 d and 3 e we

combine this method with B1 configuration: the system

switches when the increasing field crosses the right branch of

the interior CC and reverses back when the decreasing field

crosses the left branch, so that the interior CC is not re-

vealed, but the switch corresponding to the exterior CC can

be detected.

Until now we have considered that the magnetostaticcoupling is absorbed into the exchange constant hJ. How-

ever, due to the elliptical shape of the layers the magneto-

static fields not only depend on the angle between the mag-

netization of the two layers, like the exchange coupling, but

also on each moment direction. From Figs. 2 a , 2 b , and 4

we can see the differences between exchange and magneto-

static coupling, the exterior sCC expands toward higher

fields, while the interior CC shifts toward origin, increasing

the possibility of undesirable switching of half-selected

memory cells.

In summary, based on the magnetization vectors dynam-

ics, the dCCs and switching properties of a SAF element

have been presented. We have shown that usage of sCCs

instead of dCCs can lead to inadvertent switching of half-

selected memory cells. In dealing with the problem of im-proving the MRAM speed and reliability, one need to pay

attention to the parameters describing the pulses shape,

damping parameter, and the magnetostatic coupling.

Work was partially supported by NSF under Grant No.

ECCS-0902086, and by Romanian PNII-RP3 under Grant

Nos. 9/1.07.2009 and PNII 12-093 HIFI.

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FIG. 3. Color online Switching diagrams of a symmetric SAF using B2configuration a c and B1 configuration d and e , respectively.

FIG. 4. Color online Switching diagrams of an asymmetric SAF, takinginto account both exchange and magnetostatic coupling.

222513-3 Pham et al. Appl. Phys. Lett. 95, 222513 2009

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