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IEEE TRANSACTIONS ON MAGNETICS, VOL. 45, NO. 12, DECEMBER 2009

Saturation Curve for a Synthetic Antiferromagnetic System Cristina Stefania Olariu and Alexandru Stancu Department of Physics, Alexandru Ioan Cuza University, Iasi 700506, Romania For toggle magnetic random access memory, the saturation field represents the exterior limit of the work region. Exterior magnetic fields higher than the saturation field lead to an irremediable loss of the stored information because the antiferromagnetic state of the synthetic antiferromagnetic (SAF) structure becomes a ferromagnetic state and data stored in the magnetic memory cell are altered. At the saturation point, the two ferromagnetic-layer magnetizations became parallel oriented, in a direction close to that of the applied field. In this paper, we present a method to find an analytical formula for the magnetic saturation field that depends on the angle between the applied magnetic field related to the easy anisotropy axis, on the amplitude of the antiferromagnetic coupling field between the ferromagnetic layers, on the geometrical characteristics of the SAF structure, and on the second term of the series expansion of the anisotropy energy of the system. This method allows us to obtain an analytical formula for saturation field that depends only on the controllable parameters of the SAF structure. This analytically obtained critical saturation curve perfectly matches with the critical saturation curve reported in previous papers. Index Terms—MRAM, SAF, saturation curve, saturation field, toggle mode.

I. INTRODUCTION

YNTHETIC antiferromagnetic (SAF) structures are used in magnetic read-head construction, to build magnetic memory cells and to stabilize the magnetizations of the internal layers in sandwich structures. In 2003, Savtcenko and his co-workers proposed a new magnetic memory cell—toggle magnetic random access memory (MRAM)—and a new writing method [1]. In contrast with classical magnetic memory cells, toggle mode cells have the free and the fixed magnetic layers replaced with two free and fixed SAF systems. Memory cells have the easy axis direction oriented at 45 from the bit and the word lines. To switch a bit, it is necessary to switch the whole free SAF structure. For that, two successive pulses are necessary to be applied, two pulses diphase with half of a period [2], [3]. The behavior of the toggle memory cell is determined by the amplitude and the period of the applied magnetic pulses [4], [5]. To correctly exploit toggle-mode magnetic memories cells, it is important to exactly establish the area of work region. If two diphase magnetic pulses with large amplitude are applied, it is possible to saturate the free SAF structure and thus to lose the antiferromagnetic state of the SAF structure [6]. The memory bit suffers irremediable damage and stored information is alliterated. In the magnetic applied field plain, the work region for toggle mode memory cell is delimited by exterior saturation curve given by saturation field. In this paper, we present a new method to determine an analytical formula for the saturation field of a SAF structure. In the previous literature papers, the component of the exterior saturation field along easy and hard axis was determined [7]–[9].

S

Manuscript received November 12, 2008; revised March 14, 2009. Current version published November 18, 2009. Corresponding author: C. S. Olariu (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2009.2023118

Fujiwara and co-workers used the constant angle contours to understand and to explain the ferromagnetic-layer magnetizations response to the exterior applied field. From equilibrium and stability conditions of the system, it was convenient to find constant and leaving angle contours trajectories, giving a constant to be a variable and vice versa ( and are angles between the layer magnetizations and the easy axis). Exterior saturation curve was numerically obtained like an exterior envelope of this trajectories. Using a new coordinating system for ferro, magnetic-layer magnetizations orientations ( ), it was possible to calculate the components for saturation field along easy axis and hard axis [8], [9]. At the saturation point, the layer’s magnetizations are parallel oriand . The saturation field ented, then components depend on the angle between magnetizations and the easy axis and on the antiferromagnetic coupling field between magnetic layers of the SAF system . The exterior saturation curve was numerically drawn with these two components of the saturation field. For a symmetrical SAF system, our analytical method determines the saturation field value in terms of the —angle between exterior applied field related to the easy axis, and in terms between ferromagof the antiferromagnetic coupled field netic layers of the SAF system—both exterior controllable and measurable parameters. At the saturation field, direction of the layer’s magnetizations is near or the same as the exterior applied field direction. For small angle between these two directions, we . Our anaused a usual trigonometric approximation lytical results are in perfect correlation with previous numerical results for exterior critical curves. The saturation field is also influenced by the second term of series expansion of anisotropy energy of the SAF system [9]. In this case, analytical expression of the saturation field depends on the orientation of applied field related to the easy anisotropy axis of the structure , depends on the amplitude of antiferroand depends on the magnetic coupling between SAF layers anisotropy of the ferromagnetic materials through the second term . All these parameters can be controlled. In case of an asymmetrical SAF system, analytical formula give perfect correlations with numerical obtained curve [10].

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OLARIU AND STANCU: SATURATION CURVE FOR A SAF SYSTEM

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Fig. 2. Layer’s magnetizations become parallel oriented near or overlap the applied field direction in the saturation field.

and dependence between

Fig. 1. (a) Ferromagnetic layer’s magnetizations of a SAF system are antiparallel oriented along easy axis direction in absence of an exterior magnetic field. (b) In an exterior magnetic field, ferromagnetic layer’s magnetizations leave the easy axis and oriented at  and  related to applied field direction.

The saturation field depends just on the antiferromagnetic coupling field , on the angle between applied field and the easy axis , and on the relative thickness of the two ferromagnetic ( and are ferromagnetic layer’s layers , where thicknesses).

can be combined to obtain a and

(2)

In the saturation field, the system leaves the scissor state [11] and jump into a saturation state. In this state, layer magnetiza(Fig. 2). The direction tions become parallel oriented of the magnetizations is near, or overlaps to the applied field di. We used the trigonometric usual rection, thus . approximation for small angle With this trigonometric usual approximation for small angle, the equilibrium equations (2) can be written as

II. MAGNETIC SATURATION FIELDS EXPRESSION

(3)

First, we take into consideration a symmetrical SAF system formed by two ferromagnetic layers with equal thicknesses . The and are the magnetizations of ferromagnetic layers, antiparallel oriented along easy axis because of the antiferromagnetic coupled [Fig. 1(a)]. When an external field is applied to the system at angle from the easy axis, the two-layer’s magnetizations leave the easy axis and become oriand angle, related to the exterior applied field entated at [Fig. 1(b)]. Energy of the system contains the anisotropy energy of the system, the interactions energy between exterior applied field and layer’s magnetizations and the antiferromagnetic coupling energy between ferromagnetic layer

(1) is the energy density on area unity normalized to , is the anisotropy constant of the ferromagnetic layers and is the thickness of the layers. The applied field and the antiferromagnetic coupling field are normalized to , ). the anisotropy field of the layers ( Saturation field is that critical value of the external applied field that leads the SAF system from a scissor state to a parallel state of magnetizations. This minimum external field required for saturating the ferromagnetic layers is an extreme of with respect to angles and . The equilibrium equations where

that lead to the following expression for saturation field: (4) is the antiferromagnetic coupling field and the where angle between the applied field and the easy axis. Saturation field expression (4) depends on the angle between the exterior and on the amplitude field and the easy axis of the system of antiferromagnetic coupling between ferromagnetic layers . This expression of the saturation field and trigonometric approximation for small angle satisfied the equilibrium and stability conditions. Fujiwara and his co-workers express the components of the saturation field along the easy axis and the hard anisotropy axis of the SAF system [7], [8]. These two components of the saturation field depend of the angle between and depend of the layer’s magnetizations and the easy axis antiferromagnetic coupling field . Because of the anisotropy, ferromagnetic layer’s magnetizations do not orientate in the same direction with the applied saturation field, thus and is not easily exactly determined. Between exterior saturation curve given by the formula (4) and the saturation curve obtained in the previous literature papers [8], [9] it is a perfect superposition (Fig. 3). Differences between this two saturation curves are very small. The maximum relative errors are below 1.5%. In Fig. 4, there are represented the relative errors between analytical calculated

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Fig. 4. Relative errors between analytical and numerical calculated saturation field. Errors are below 1.5% and the greatest error appears for applied field direction between easy anisotropy axis and hard axis.

layer’s magnetizations are parallel oriented and closes to the direction of the applied field. Using the usual trigonometric ap, the expression of proximations for the small angle the saturation field—when the second term of the anisotropy energy series expansion is taken into consideration—can be expressed as

(6) where

Fig. 3. Exterior critical curve obtained from (4) overlaps to exterior critical curve numerical obtained in previous literature papers [8] for different values of and (b) h . antiferromagnetic coupled field. (a) h

=2

=4

saturation field (4) and numerical determined saturation field [8], [9]. For a SAF system where the second term of the series expansion for anisotropy energy is significant, the size and the shape of the saturation curve are dramatically changed [9], [10]. Energy of the SAF system contains the second terms of the anisotropy energy series expansion

(5) where is the second term of anisotropy energy normalized to first anisotropy field constant . Similar with (2), from the equilibrium equations and it is possible to be established some depenand , angles between the layer’s magnedence between tizations and the exterior applied field. At the saturation field,

and . Formula (6) expresses very well the saturation field for a SAF symmetric system where the second term of the anisotropy energy series expansion have a significant value. There is a very good overlapping between the saturation curve obtained with analytical formula (6) and the saturation curve numerically obtained in the previous literature paper [9], [10] (Fig. 5). Differences between the saturation curve obtained with analytical formula (6) and numerical obtained saturation curve is weak appear when the antiferromagnetic coupling field and the second term of the series expansion of anisotropy enhas a significant value related with the first coefficient ergy [Fig. 5(a)]. In this case, the angle between magnetizations direction and the applied field direction is not so small and usual trigonometric approximations induce some errors. For greater , there are no differences antiferromagnetic coupling field between saturation curves analytical or numerical determined, has a significant value even if the second anisotropy term [Fig. 5(b)]. In Fig. 6, relative errors between analytical saturation field and numerical saturation field are presented, for various cou) and for two different values pling field values ( of the second term of the anisotropy energy series expansion, and for .

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OLARIU AND STANCU: SATURATION CURVE FOR A SAF SYSTEM

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= 05

Fig. 5. Saturation curve obtained with formula (5) for k : and for two and h are perfect overlapping antiferromagnetic coupled field h over the numerical saturation curve [9].

=2

=4

Relative errors are below 5% for great anisotropy second term value and for small antiferromagnetic coupling field but relative errors are below 1% for greater values for [Fig. 6(a)]. For small antiferromagnetic coupled field value of the second anisotropy term relative errors between analytical and numerical saturation fields are smaller, under 0.5% for antiferromagnetic coupling field greater than [Fig. 6(b)]. For an unsymmetrical SAF system, energy of the SAF system is influenced by the relative thickness of the ferromag(where and are ferromagnetic netic layers, layers thicknesses) [7]

(7)

Fig. 6. Relative error for different values of the second term of series expansion : and (b) k : , for different values of of anisotropy energy: (a) k the antiferromagnetic coupling field between ferromagnetic layers of the SAF system.

= 05

= 02

It can be considered that the layer 1 is the thickly ferromag. Energy of the system is netic layer of the system, so that . The fields and are the values of the normalized to exterior applied field and the antiferromagnetic coupled field between ferromagnetic layers, normalized to the anisotropy field of the thick layer. Using the equilibrium equations of the system, it can be establish two relations between and —angle between the layer’s magnetizations and the easy axis, similar with relations (2) and (3) from the symmetrical SAF system case. Saturation state appears when the exterior applied field are strong enough to orien. tate in the same direction the layer’s magnetizations Angle between saturation field and layer’s magnetizations directions is small and trigonometric approximations for small . The expression of the saturation angle can be used field for an unsymmetrical SAF system depends on the relative

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consideration a SAF system with ferromagnetic layers that have a significant value of the second term of the anisotropy energy series expansion. We determine an analytical formula for the saturation field that depends only on the controllable parameters of the SAF structure: the antiferromagnetic coupling field, the angle between applied field and the easy axis, the second term anisotropy, and the geometrical parameters of the SAF structure. These expressions for the saturation fields and trigonometric approximations for small angle satisfied the equilibrium and stability conditions. Saturation curves obtained by analytical formula match the critical saturation curves obtained in the previous papers. Fig. 7. Relative errors between analytical determined and numerical calculated saturation field for different relative thicknesses of the SAF system.

thickness of the layers , depends on the antiferromagnetic couand depends on the —angle pling field between the layers between the applied field and the easy axis of the SAF system

(8) Saturation curve obtained with analytical formula (8) is perfect overlapped over the exterior saturation curve obtained through constant angle contour in previous papers [7], [10]. The relative errors between analytical and numerical saturation field are below 1.5%, and decrease with increasing the asymmetry of the SAF system (Fig. 7). III. CONCLUSION The saturation field gives the maximum value of the applied field in the direction (related to the easy axis of the SAF structure) for that bit information is still there. Over this saturation field, the structure loses the antiferromagnetic orientations of the layers and the bit information stored in the toggle mode structure is permanently lost. Minimum value for the saturation field and depends is obtained along the easy axis direction for only on the ferromagnetic layer’s anisotropy field and on the antiferromagnetic coupling field between the layers, increasing with the SAF system asymmetry. In this paper, we have presented an analytical method to determine the expression of saturation magnetic fields for a symmetrical and an unsymmetrical SAF configuration. We also take into

ACKNOWLEDGMENT This work was supported in part by the Romanian Ministry of Education and Research under Program of Research for Young Ph.D. Student TD Grant 163/2007. REFERENCES [1] L. Savchenko, B. N. Engel, N. D. Rizzo, M. F. DeHerrera, and J. A. Janesky, “Method of Writing to Scalable Magnetoresistance Random Access Memory Element,” U.S. Patent 6545906, 2003. [2] B. N. Engel, J. Akerman, B. Butcher, R. W. Dave, M. DeHerrera, M. Durlam, G. Grynkewich, J. Janesky, S. V. Pietambaram, N. D. Rizzo, J. M. Slaughter, K. Smith, J. J. Sun, and S. Tehrani, “A 4-MB toggle MRAM based on a novel bit and switching method,” IEEE Trans. Magn., vol. 41, no. 1, pp. 132–136, Jan. 2005. [3] D. C. Worledge, “Single-domain model for toggle MRAM,” IBM J. Res. Develop., vol. 50, pp. 69–79, Jan. 2006. [4] H. J. Kim, S. C. Oh, J. S. Bae, K. T. Nam, J. E. Lee, S. O. Park, H. S. Kim, N. I. Lee, U. I. Chung, J. T. Moon, and H. K. Kang, “Development of magnetic tunnel junction for toggle MRAM,” IEEE Trans. Magn., vol. 41, no. 10, pp. 2661–2663, Oct. 2005. [5] S. Wang and H. Fujiwara, “Margin comparison of Stoner-Wohlfarth MRAM and zero total anisotropy toggle mode MRAM,” IEEE Trans. Magn., vol. 42, no. 10, pp. 2727–2729, Oct. 2006. [6] S. Y. Wang, H. Fujiwara, J. Dou, Z. J. Li, and Y. M. Huai, “Dynamic simulation of toggle mode MRAM operating field margin,” IEEE Trans. Magn., vol. 43, no. 6, pp. 2337–2339, Jun. 2007. [7] H. Fujiwara, S. Y. Wang, and M. Sun, “Magnetization behavior of synthetic antiferromagnet andd toggle—Magnetoresistance random access memory,” Trans. Magn. Soc. Jpn., vol. 4, pp. 121–129, 2004. [8] H. Fujiwara, S. Y. Wang, and M. Sun, “Critical-field-curves for switching toggle mode magnetoresistance random access memory devices (invited),” J. Appl. Phys., vol. 97, May 15, 2005. [9] C. S. Olariu, L. Stoleriu, and A. Stancu, “Influence of material anisotropy in the switching toggle mode in MRAM devices,” J. Magn. Magn. Mater., vol. 316, pp. E302–E305, Sep. 2007. [10] C. S. Olariu, L. Stoleriu, and A. Stancu, “Simulation of toggle mode switching in MRAM’s,” J. Optoelectron. Adv. Mater., vol. 9, pp. 1140–1142, Apr. 2007. [11] D. C. Worledge, P. L. Trouilloud, and W. J. Gallagher, “Theory for symmetric toggle magnetic random access memory,” Appl. Phys. Lett., vol. 90, May 28, 2007.

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