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Toggle magnetoresistance random access memory based on magnetostatically coupled bilayers
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 295 (2005) 246–250 www.elsevier.com/locate/jmmm
Toggle magnetoresistance random access memory based on magnetostatically coupled bilayers Shengyuan Wanga,, Hideo Fujiwaraa, Min Sunb a
MINT Center and Department of Physics and Astronomy, University of Alabama, P.O. Box 870209, Tuscaloosa, AL 35487, USA b MINT Center and Department of Mathematics,University of Alabama, P.O. Box 870209, Tuscaloosa, AL 35487, USA Received 7 November 2004; received in revised form 8 January 2005 Available online 25 February 2005
Abstract A new scheme of toggle magnetoresistance random access memory (MRAM) based on magnetostatically coupled (MSC) bilayers without antiferromagnetic coupling is investigated. Analysis shows that the toggle-MRAM operation and necessary storage lifetime can be achieved by choosing an appropriate aspect ratio for MSC bilayers having reasonable area and thickness. The attenuation factor defined as the ratio of the effective MSC field to the demagnetizing field is found to be essential to obtain the toggle-mode operation. r 2005 Elsevier B.V. All rights reserved. PACS: 75.50.Ss; 85.70.Ay Keywords: MRAM; Magnetostatic coupling; Toggle switch
There has been a considerable interest in magnetoresistance random access memory (MRAM) because of its potential nonvolatility, high rewritability, low-power consumption and high speed [1,2]. To realize a memory density compatible with the semiconductor memory technology, the MRAM element should have an extremely small volume, which tends to decrease the thermal stability of the memory elements. Corresponding author. Tel.: +1 205 3480376;
fax: +1 205 3482346. E-mail address: [email protected] (S. Wang).
Recently, a toggle-mode MRAM device has been proposed by Savtchenko et al. [3–5], in which a synthetic antiferromagnet (SAF) with a uniaxial anisotropy is used. The operating word field (H w ) and digit field (H d ) are sequentially applied at 451 and 45 with respect to the easy axis of the magnetic anisotropy. The present authors [6,7] investigated the magnetization response of the antiferromagnetically coupled ferromagnetic bilayers to the in-plane field by means of analytic/ numeric methods and proposed a method of optimizing the magnetic parameters. There, no magnetostatic coupling (MSC) effect was taken
0304-8853/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2005.01.012
ARTICLE IN PRESS S. Wang et al. / Journal of Magnetism and Magnetic Materials 295 (2005) 246–250
into account for simplicity. Worledge [8,9] recently investigated similar systems with the magnetostatic coupling, however he assumed that the effective magnetostatic coupling field in one of the layers is equal to the demagnetizing field in the other. In this paper, a novel scheme of toggle MRAM memory elements based on shape anisotropy and magnetostatic coupling without any exchange coupling is investigated. Our results show that the ratio of the magnetostatic coupling field to the demagnetizing field plays a crucial role in the toggle-mode operation of this kind of memory element. The MRAM memory element is assumed to have two ferromagnetic (FM) layers (Layers 1 and 2) with the same oblate ellipsoidal shape and saturation magnetization M s : We assume that the anisotropy is solely from the shape anisotropy for simplicity, and that the effective magnetostatic coupling field in one of the layers is proportional to the demagnetizing field in the other. No other coupling is assumed to exist between the two layers. The total energy density W (per unit area) includes the anisotropy energy W a ; magnetostatic coupling energy W c ; and Zeeman energy W z as W ¼ Wa þ Wc þ Wz
(1)
with W a ¼ ðK u þ K sh Þtðsin2 y1 þ sin2 y2 Þ, W c ¼ 2rK sh t½p cosðy1 y2 Þ þ sin y1 sin y2 , W z ¼ H x M s tðcos y1 þ cos y2 Þ H y M s t ðsin y1 þ sin y2 Þ, K sh ¼ 0:5ðN b N a ÞM 2s , p ¼ N a =ðN b N a Þ, where y1 and y2 denote the magnetization angles of Layers 1 and 2 with respect to the x-axis, rðo1Þ is the MSC attenuation factor, H x and H y are the applied fields in the x- and y-axis, t is the thickness of each layer, K u and K sh are the uniaxial and shape anisotropy constants, respectively, and p is the relative strength of the isotropic component to the anisotropic component of the effective magnetostatic coupling field, which is inversely proportional to the shape anisotropy as seen above. Thus, the easy direction is along the x-axis for the
247
demagnetization factor N a oN b : W c is the interaction between the MSC field of Layer 1 and the magnetization of Layer 2 and vice versa. r is estimated to be 0.8 for the two FM layers having zero separation with the single layer aspect ratio of 100:50:1 in rectangle shapes and can be controlled by inserting an insulator layer with variable thickness in between. Normalization of Eq. (1) by 2K sh t gives w ¼ 0:5ðhk þ 1Þðsin2 y1 þ sin2 y2 Þ þ r½p cosðy1 y2 Þ þ sin y1 sin y2 hx ðcos y1 þ cos y2 Þ hy ðsin y1 þ sin y2 Þ, ð2Þ where hk ¼ H u =H ksh ; hx ¼ H x =H ksh and hy ¼ H y =H ksh with the shape anisotropy field H ksh ¼ 2K sh =M s and uniaxial anisotropy field H u ¼ 2K u =M s : Fig. 1(a) shows an example of the switching critical curves obtained for hk ¼ 0; p ¼ 2 and r ¼ 0:7 by applying the equilibrium conditions [10]. The outermost oval-like curve is the critical curve for the saturation. The two small astroid-like curves, which we call just ‘astroids’ hereafter, are the critical curves for switching (abrupt changes in the magnetization configuration). A part of Fig. 1(a) is magnified and shown in Fig. 1(b) with possible toggle switching field margin (the gray-filled area). Fig. 1(c) gives a toggle switch example following the vector-field (normalized by H ksh ) path. For toggle-mode switch, the word and digit field(s) are applied sequentially following the path OP1 P2 P3 O: The successful switch means that the magnetization configuration (y1 ; y2 ) switches from ð0; 180 Þ to ð180 ; 0Þ or vice versa through the sequential application of the word and digit fields. The switching evolution of magnetization configuration ðy1 ; y2 Þ for the path shown in Fig. 1(c) is plotted in Fig. 1(d). The example given here indicates a successful toggle switch using magnetostatically coupled bilayers without the exchange coupling. The word and digit fields are assumed to have the same strength, therefore hw þ hd falls onto the hx -axis. Two critical points (hcr for spin flop and hs for saturation) on the hx -axis can be obtained using Eq. (2) from the configuration stability
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maximum field tolerance, that is, the maximum operating field margin can be achieved if hw þ hd is set in the middle of hcr and hs (Fig. 1(b)). Thus, for the optimized parameters, the following relationship should hold: pffiffiffi 2hw ¼ ðhcr þ hs Þ=2. (5)
Fig. 1. (a) Critical curve includes one oval-like envelop (saturation curve) and two small astroids (switching curve). (b) Magnified portion of critical curve with the gray-filled area to the operating field margin. Saturation critical point hs and switching critical point hcr on the hx axis can be exactly solved. (c) Magnified portion of critical curve with the toggle-mode switching path OP1 P2 P3 O: (d) Evolution of the magnetization configuration ðy1 ; y2 Þ through the field trajectory in Fig. 1(c).
Therefore, r can be solved as a function of hw for certain given p and hk values as shown in Fig. 2. Therefore, for the lower the operating field, the smaller r is required. The required anisotropy of the element has to be determined from the required storage lifetime by using the Ne´el–Arrhenius formula. Assuming that the magnetization of each layer relaxes simultaneously, the energy barrier without any applied field can be deduced from Eq. (2) with the help of the energy mapping shown in Fig. 3(a), where p1 and p2 denoting the energy peaks, s1 and s2 denoting the saddle points, v1 denoting the stable configuration at ð0; 180 Þ; and v0 or v00 denoting the stable configurations at ð180 ; 0Þ and ð180 ; 0Þ; respectively. Note that v0 and v00 are identical. Because the energy at s1 is much lower, the thermal relaxation from v1 to v00 occurs more likely through s1 than through s2 : The normalized energy barrier can be solved as Dw ¼ hk þ 1 r: Taking into account the normalization factor 2K sh t and single FM layer volume V, the energy
conditions, as hcr ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 rÞ½rð2p þ 1Þ þ 1 þ ðhk þ 1Þ2 þ 2hk rp 1,
ð3Þ hs ¼ rð2p þ 1Þ ðhk þ 1Þ.
(4)
The real critical fields H cr and H s can therefore be obtained by taking into account the normalization factor H ksh : They are consistent with the results given by Worledge [9], where the attenuation factor r was taken as 1. pffiffiffi The magnitude of hw þ hd ð¼ 2hw Þ has to fall between hcr and hs in order to obtain a successful toggle-switch and avoid saturation [6,7]. The
Fig. 2. r vs. hw curves for hk ¼ 0; p ¼ 2 and 4.
ARTICLE IN PRESS S. Wang et al. / Journal of Magnetism and Magnetic Materials 295 (2005) 246–250
249
the relevant memory elements, and f 0 is the frequency factor. Thus, the shape anisotropy field has to satisfy H ksh X
kB T ln ðt0 f 0 N 0 Þ . M s V ð1 r þ hk Þ
(8)
hk ¼ 0 is specifically given as an example in the following. We assume M s ¼ 850 emu=cm3 ; T ¼ 360 K; f 0 ¼ 109 Hz; N 0 ¼ 1012 ; r ¼ 0:7; and 10year storage lifetime (t0 ¼ 3:15 108 s), which may be typical conditions for the MRAM under development. Then, from Eq. (7), we obtain DE=kB T\68: For a single FM layer volume of V ¼ 5 1016 cm3 ; we obtain H ksh 26 Oe, which gives the following demagnetization factor relation: H ksh ¼ ðN b N a ÞM s ¼ 26 Oe.
(9)
We assume each layer is an oblate ellipsoid with the thickness t ¼ 3:5 nm; the aspect ratio (in-plane axis/vertical axis) g 120; so N c ¼ 0:987 4p is obtained [11], which determines the second demagnetization factor relation ðN b þ N a ÞM s ¼ ð4p N c ÞM s 136 Oe.
Fig. 3. Energy mapping for zero field (a) and hw ¼ 1:3 (b) for hk ¼ 0; where v1 and v0 (or v00 ) refer to the stable configurations, s1 and s2 are the saddle points, p1 and p2 are the energy peaks.
barrier can be expressed by DE ¼ 2½K sh ð1 rÞ þ K u V .
(6)
It should be emphasized that r should be kept less than one to have some energy barrier if K u ¼ 0: The energy barrier DE must satisfy the following condition: DEXkB T ln ðt0 f 0 N 0 Þ,
(7)
where kB is the Boltzmann constant, T is the storage temperature, t0 is the relaxation time of each memory element, N 0 is the total number of
(10)
From Eqs. (9) and (10), in-plane demagnetization ratio N b =N a 1:5 and p 2 can be obtained. This leads to obtain the approximate aspect ratio (inplane long axis/short axis ratio) of about 1:5 with the x-axis dimension longer. Thus, the longer and shorter diameters are determined as 0.52 and 0:34 mm: Take, as an example, a practical operating field as H w ¼ 34 Oe. Then we obtain hw ¼ 1:3; leading to obtain the optimal coupling ratio r 0:7 through Fig. 2. In the MRAM operation, the loss of memory due to the half-select disturbance must be prevented. The required energy barrier DE 0 can be obtained similarly to that for the memory life as discussed above: DE 0 ¼ kB T ln ðtp f 0 N p N c Þ,
(11)
where tp is the pulse duration of either the word field or the digit field, N c is the number of halfselected cells during each switch, and N p is the number of pulse disturbances which each element must endure. Assuming tp ¼ 10 ns, N p ¼ 1015 ; 1 N c ¼ 105 which is about 10 of a 1 Mbit MRAM,
ARTICLE IN PRESS S. Wang et al. / Journal of Magnetism and Magnetic Materials 295 (2005) 246–250
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anisotropy, shape anisotropy becomes smaller, which leads to N b ! N a and causes element shape approach to an oblate. With the above demagnetization parameters, the critical fields hcr and hs can be calculated through Eqs. (3) and (4). The maximum relative operating field margin ðRmax margin Þ can be expressed as the following using Eq. (5) [6,7]:
2 (A) (B)
Nb/Na
1.75
1.5
1.25
Rmax margin ¼
hs hcr hs hcr ¼ pffiffiffi . hs þ hcr 2 2hw
(12)
1 -0.5
0
0.5
1
1.5
2
hk
(a) 0.6
Rmargin max
0.4
0.2 (A) (B) 0 -0.5
(b)
0
0.5
1
1.5
2
hk
Therefore, Rmax margin can be plotted as a function of hk also (Fig. 4(b)). About 50% and 20% margin are obtained for r ¼ 0:7 and 0.3, respectively. Thus for proper configurations, large r can provide large margin. In summary, magnetostatically coupled bilayers can be used for toggle-mode MRAMs instead of exchange coupled ones. The attenuation factor r can be controlled by inserting an insulator between the two ferromagnetic layers. The large margin can be achieved for large r, requiring the separation between the two FM layers as small as possible. However the operating field becomes large too. This research is partially supported by MRSEC Grants (DMR-0213985).
Fig. 4. (a) N b =N a vs. hk and (b) Rmax margin vs. hk for (A) r ¼ 0:7 and (B) r ¼ 0:3:
References we obtain DE 0 =kB T 48; and DE 0c 2:4 1012 ergs at T ¼ 360 K; which gives the normalized energy barrier Dwc 0:21 for the above parameters. On the other hand, the energy barrier between the two stable magnetization configurations v1 and v00 (or v0 ) can be determined from the energy mapping for the situation where hw ¼ 1:3 is applied (Fig. 3(b)). The energy barrier 0:4 at s1 and 1:1 at s2 are obtained, which are both greater than Dwc 0:21: Thus, the half-select disturbance is negligible for the given example. The demagnetization parameters for optimal geometry can thus be obtained from Eqs. (8) and (10) for given hk : N b =N a is plotted as a function of hk for r ¼ 0:7(A) and r ¼ 0:3(B) in Fig. 4(a). As seen, with the increase of the induced uniaxial
[1] J.M. Daughton, Thin Solid Film 216 (1996) 162. [2] S.S.P. Parkin, K.P. Roche, M.G. Samant, P.M. Rice, R.B. Beyers, J. Appl. Phys. 85 (1999) 5828. [3] L. Savtchenko, A.A. Korki, B.N. Engel, N.D. Rizzo, J.A. Janesky, US Patent No. 6,545,906 B1, April 2003. [4] S.V. Pietambaram, et al., GE-03, Ninth Joint MMM/ INTERMAG Conference, 2004. [5] B.N. Engel, et al., GE-05, Ninth Joint MMM/INTERMAG Conference, 2004. [6] H. Fujiwara, S.-Y. Wang, M. Sun, Trans. Magn. Soc. Japan 4 (2004) 121. [7] S.-Y. Wang, H. Fujiwara, J. Magn. Magn. Mater. 286C (2005) 27. [8] D.C. Worledge, Appl. Phys. Lett. 84 (2004) 2847. [9] D.C. Worledge, Appl. Phys. Lett. 84 (2004) 4559. [10] E. Goto, N. Hayashi, T. Miyashita, K. Nakagawa, J. Appl. Phys. 36 (1965) 2951. [11] B.D. Cullity, Introduction to Magnetic Materials, Addison-Wesley, Reading, MA, 1972, pp. 49–61.
Journal of Magnetism and Magnetic Materials 295 (2005) 246–250 www.elsevier.com/locate/jmmm
Toggle magnetoresistance random access memory based on magnetostatically coupled bilayers Shengyuan Wanga,, Hideo Fujiwaraa, Min Sunb a
MINT Center and Department of Physics and Astronomy, University of Alabama, P.O. Box 870209, Tuscaloosa, AL 35487, USA b MINT Center and Department of Mathematics,University of Alabama, P.O. Box 870209, Tuscaloosa, AL 35487, USA Received 7 November 2004; received in revised form 8 January 2005 Available online 25 February 2005
Abstract A new scheme of toggle magnetoresistance random access memory (MRAM) based on magnetostatically coupled (MSC) bilayers without antiferromagnetic coupling is investigated. Analysis shows that the toggle-MRAM operation and necessary storage lifetime can be achieved by choosing an appropriate aspect ratio for MSC bilayers having reasonable area and thickness. The attenuation factor defined as the ratio of the effective MSC field to the demagnetizing field is found to be essential to obtain the toggle-mode operation. r 2005 Elsevier B.V. All rights reserved. PACS: 75.50.Ss; 85.70.Ay Keywords: MRAM; Magnetostatic coupling; Toggle switch
There has been a considerable interest in magnetoresistance random access memory (MRAM) because of its potential nonvolatility, high rewritability, low-power consumption and high speed [1,2]. To realize a memory density compatible with the semiconductor memory technology, the MRAM element should have an extremely small volume, which tends to decrease the thermal stability of the memory elements. Corresponding author. Tel.: +1 205 3480376;
fax: +1 205 3482346. E-mail address: [email protected] (S. Wang).
Recently, a toggle-mode MRAM device has been proposed by Savtchenko et al. [3–5], in which a synthetic antiferromagnet (SAF) with a uniaxial anisotropy is used. The operating word field (H w ) and digit field (H d ) are sequentially applied at 451 and 45 with respect to the easy axis of the magnetic anisotropy. The present authors [6,7] investigated the magnetization response of the antiferromagnetically coupled ferromagnetic bilayers to the in-plane field by means of analytic/ numeric methods and proposed a method of optimizing the magnetic parameters. There, no magnetostatic coupling (MSC) effect was taken
0304-8853/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2005.01.012
ARTICLE IN PRESS S. Wang et al. / Journal of Magnetism and Magnetic Materials 295 (2005) 246–250
into account for simplicity. Worledge [8,9] recently investigated similar systems with the magnetostatic coupling, however he assumed that the effective magnetostatic coupling field in one of the layers is equal to the demagnetizing field in the other. In this paper, a novel scheme of toggle MRAM memory elements based on shape anisotropy and magnetostatic coupling without any exchange coupling is investigated. Our results show that the ratio of the magnetostatic coupling field to the demagnetizing field plays a crucial role in the toggle-mode operation of this kind of memory element. The MRAM memory element is assumed to have two ferromagnetic (FM) layers (Layers 1 and 2) with the same oblate ellipsoidal shape and saturation magnetization M s : We assume that the anisotropy is solely from the shape anisotropy for simplicity, and that the effective magnetostatic coupling field in one of the layers is proportional to the demagnetizing field in the other. No other coupling is assumed to exist between the two layers. The total energy density W (per unit area) includes the anisotropy energy W a ; magnetostatic coupling energy W c ; and Zeeman energy W z as W ¼ Wa þ Wc þ Wz
(1)
with W a ¼ ðK u þ K sh Þtðsin2 y1 þ sin2 y2 Þ, W c ¼ 2rK sh t½p cosðy1 y2 Þ þ sin y1 sin y2 , W z ¼ H x M s tðcos y1 þ cos y2 Þ H y M s t ðsin y1 þ sin y2 Þ, K sh ¼ 0:5ðN b N a ÞM 2s , p ¼ N a =ðN b N a Þ, where y1 and y2 denote the magnetization angles of Layers 1 and 2 with respect to the x-axis, rðo1Þ is the MSC attenuation factor, H x and H y are the applied fields in the x- and y-axis, t is the thickness of each layer, K u and K sh are the uniaxial and shape anisotropy constants, respectively, and p is the relative strength of the isotropic component to the anisotropic component of the effective magnetostatic coupling field, which is inversely proportional to the shape anisotropy as seen above. Thus, the easy direction is along the x-axis for the
247
demagnetization factor N a oN b : W c is the interaction between the MSC field of Layer 1 and the magnetization of Layer 2 and vice versa. r is estimated to be 0.8 for the two FM layers having zero separation with the single layer aspect ratio of 100:50:1 in rectangle shapes and can be controlled by inserting an insulator layer with variable thickness in between. Normalization of Eq. (1) by 2K sh t gives w ¼ 0:5ðhk þ 1Þðsin2 y1 þ sin2 y2 Þ þ r½p cosðy1 y2 Þ þ sin y1 sin y2 hx ðcos y1 þ cos y2 Þ hy ðsin y1 þ sin y2 Þ, ð2Þ where hk ¼ H u =H ksh ; hx ¼ H x =H ksh and hy ¼ H y =H ksh with the shape anisotropy field H ksh ¼ 2K sh =M s and uniaxial anisotropy field H u ¼ 2K u =M s : Fig. 1(a) shows an example of the switching critical curves obtained for hk ¼ 0; p ¼ 2 and r ¼ 0:7 by applying the equilibrium conditions [10]. The outermost oval-like curve is the critical curve for the saturation. The two small astroid-like curves, which we call just ‘astroids’ hereafter, are the critical curves for switching (abrupt changes in the magnetization configuration). A part of Fig. 1(a) is magnified and shown in Fig. 1(b) with possible toggle switching field margin (the gray-filled area). Fig. 1(c) gives a toggle switch example following the vector-field (normalized by H ksh ) path. For toggle-mode switch, the word and digit field(s) are applied sequentially following the path OP1 P2 P3 O: The successful switch means that the magnetization configuration (y1 ; y2 ) switches from ð0; 180 Þ to ð180 ; 0Þ or vice versa through the sequential application of the word and digit fields. The switching evolution of magnetization configuration ðy1 ; y2 Þ for the path shown in Fig. 1(c) is plotted in Fig. 1(d). The example given here indicates a successful toggle switch using magnetostatically coupled bilayers without the exchange coupling. The word and digit fields are assumed to have the same strength, therefore hw þ hd falls onto the hx -axis. Two critical points (hcr for spin flop and hs for saturation) on the hx -axis can be obtained using Eq. (2) from the configuration stability
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248
maximum field tolerance, that is, the maximum operating field margin can be achieved if hw þ hd is set in the middle of hcr and hs (Fig. 1(b)). Thus, for the optimized parameters, the following relationship should hold: pffiffiffi 2hw ¼ ðhcr þ hs Þ=2. (5)
Fig. 1. (a) Critical curve includes one oval-like envelop (saturation curve) and two small astroids (switching curve). (b) Magnified portion of critical curve with the gray-filled area to the operating field margin. Saturation critical point hs and switching critical point hcr on the hx axis can be exactly solved. (c) Magnified portion of critical curve with the toggle-mode switching path OP1 P2 P3 O: (d) Evolution of the magnetization configuration ðy1 ; y2 Þ through the field trajectory in Fig. 1(c).
Therefore, r can be solved as a function of hw for certain given p and hk values as shown in Fig. 2. Therefore, for the lower the operating field, the smaller r is required. The required anisotropy of the element has to be determined from the required storage lifetime by using the Ne´el–Arrhenius formula. Assuming that the magnetization of each layer relaxes simultaneously, the energy barrier without any applied field can be deduced from Eq. (2) with the help of the energy mapping shown in Fig. 3(a), where p1 and p2 denoting the energy peaks, s1 and s2 denoting the saddle points, v1 denoting the stable configuration at ð0; 180 Þ; and v0 or v00 denoting the stable configurations at ð180 ; 0Þ and ð180 ; 0Þ; respectively. Note that v0 and v00 are identical. Because the energy at s1 is much lower, the thermal relaxation from v1 to v00 occurs more likely through s1 than through s2 : The normalized energy barrier can be solved as Dw ¼ hk þ 1 r: Taking into account the normalization factor 2K sh t and single FM layer volume V, the energy
conditions, as hcr ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 rÞ½rð2p þ 1Þ þ 1 þ ðhk þ 1Þ2 þ 2hk rp 1,
ð3Þ hs ¼ rð2p þ 1Þ ðhk þ 1Þ.
(4)
The real critical fields H cr and H s can therefore be obtained by taking into account the normalization factor H ksh : They are consistent with the results given by Worledge [9], where the attenuation factor r was taken as 1. pffiffiffi The magnitude of hw þ hd ð¼ 2hw Þ has to fall between hcr and hs in order to obtain a successful toggle-switch and avoid saturation [6,7]. The
Fig. 2. r vs. hw curves for hk ¼ 0; p ¼ 2 and 4.
ARTICLE IN PRESS S. Wang et al. / Journal of Magnetism and Magnetic Materials 295 (2005) 246–250
249
the relevant memory elements, and f 0 is the frequency factor. Thus, the shape anisotropy field has to satisfy H ksh X
kB T ln ðt0 f 0 N 0 Þ . M s V ð1 r þ hk Þ
(8)
hk ¼ 0 is specifically given as an example in the following. We assume M s ¼ 850 emu=cm3 ; T ¼ 360 K; f 0 ¼ 109 Hz; N 0 ¼ 1012 ; r ¼ 0:7; and 10year storage lifetime (t0 ¼ 3:15 108 s), which may be typical conditions for the MRAM under development. Then, from Eq. (7), we obtain DE=kB T\68: For a single FM layer volume of V ¼ 5 1016 cm3 ; we obtain H ksh 26 Oe, which gives the following demagnetization factor relation: H ksh ¼ ðN b N a ÞM s ¼ 26 Oe.
(9)
We assume each layer is an oblate ellipsoid with the thickness t ¼ 3:5 nm; the aspect ratio (in-plane axis/vertical axis) g 120; so N c ¼ 0:987 4p is obtained [11], which determines the second demagnetization factor relation ðN b þ N a ÞM s ¼ ð4p N c ÞM s 136 Oe.
Fig. 3. Energy mapping for zero field (a) and hw ¼ 1:3 (b) for hk ¼ 0; where v1 and v0 (or v00 ) refer to the stable configurations, s1 and s2 are the saddle points, p1 and p2 are the energy peaks.
barrier can be expressed by DE ¼ 2½K sh ð1 rÞ þ K u V .
(6)
It should be emphasized that r should be kept less than one to have some energy barrier if K u ¼ 0: The energy barrier DE must satisfy the following condition: DEXkB T ln ðt0 f 0 N 0 Þ,
(7)
where kB is the Boltzmann constant, T is the storage temperature, t0 is the relaxation time of each memory element, N 0 is the total number of
(10)
From Eqs. (9) and (10), in-plane demagnetization ratio N b =N a 1:5 and p 2 can be obtained. This leads to obtain the approximate aspect ratio (inplane long axis/short axis ratio) of about 1:5 with the x-axis dimension longer. Thus, the longer and shorter diameters are determined as 0.52 and 0:34 mm: Take, as an example, a practical operating field as H w ¼ 34 Oe. Then we obtain hw ¼ 1:3; leading to obtain the optimal coupling ratio r 0:7 through Fig. 2. In the MRAM operation, the loss of memory due to the half-select disturbance must be prevented. The required energy barrier DE 0 can be obtained similarly to that for the memory life as discussed above: DE 0 ¼ kB T ln ðtp f 0 N p N c Þ,
(11)
where tp is the pulse duration of either the word field or the digit field, N c is the number of halfselected cells during each switch, and N p is the number of pulse disturbances which each element must endure. Assuming tp ¼ 10 ns, N p ¼ 1015 ; 1 N c ¼ 105 which is about 10 of a 1 Mbit MRAM,
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anisotropy, shape anisotropy becomes smaller, which leads to N b ! N a and causes element shape approach to an oblate. With the above demagnetization parameters, the critical fields hcr and hs can be calculated through Eqs. (3) and (4). The maximum relative operating field margin ðRmax margin Þ can be expressed as the following using Eq. (5) [6,7]:
2 (A) (B)
Nb/Na
1.75
1.5
1.25
Rmax margin ¼
hs hcr hs hcr ¼ pffiffiffi . hs þ hcr 2 2hw
(12)
1 -0.5
0
0.5
1
1.5
2
hk
(a) 0.6
Rmargin max
0.4
0.2 (A) (B) 0 -0.5
(b)
0
0.5
1
1.5
2
hk
Therefore, Rmax margin can be plotted as a function of hk also (Fig. 4(b)). About 50% and 20% margin are obtained for r ¼ 0:7 and 0.3, respectively. Thus for proper configurations, large r can provide large margin. In summary, magnetostatically coupled bilayers can be used for toggle-mode MRAMs instead of exchange coupled ones. The attenuation factor r can be controlled by inserting an insulator between the two ferromagnetic layers. The large margin can be achieved for large r, requiring the separation between the two FM layers as small as possible. However the operating field becomes large too. This research is partially supported by MRSEC Grants (DMR-0213985).
Fig. 4. (a) N b =N a vs. hk and (b) Rmax margin vs. hk for (A) r ¼ 0:7 and (B) r ¼ 0:3:
References we obtain DE 0 =kB T 48; and DE 0c 2:4 1012 ergs at T ¼ 360 K; which gives the normalized energy barrier Dwc 0:21 for the above parameters. On the other hand, the energy barrier between the two stable magnetization configurations v1 and v00 (or v0 ) can be determined from the energy mapping for the situation where hw ¼ 1:3 is applied (Fig. 3(b)). The energy barrier 0:4 at s1 and 1:1 at s2 are obtained, which are both greater than Dwc 0:21: Thus, the half-select disturbance is negligible for the given example. The demagnetization parameters for optimal geometry can thus be obtained from Eqs. (8) and (10) for given hk : N b =N a is plotted as a function of hk for r ¼ 0:7(A) and r ¼ 0:3(B) in Fig. 4(a). As seen, with the increase of the induced uniaxial
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