Journal of Magnetism and Magnetic Materials 258–259 (2003) 25–28
Magnetic random access memories, Brown’s paradox and hysterons Anthony S. Arrott* Virginia State University, Petersburg, VA, USA
Abstract A novel design for magnetic random access memory elements, called hysterons, is presented. It uses the interaction between two regions of magnetization. In one the process of reversal is almost uniform rotation of a magnetically hard system. In the other it is the reversible motion of a vortex that makes it a magnetically soft system. The magnetostatic interaction between the two regions is anisotropic and hysteretic. The magnetic softness permits using thicker elements that are more stable against temperature fluctations while having low switching fields. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Magnetic memory; Magnetostatic interaction
1. Introduction Among the problems that occur in the implementation of magnetic random access memory, MRAM, one is the effect of temperature fluctuations on the modes of reversal. Another is the role of edge imperfections when switching occurs by propagation of edge solitons. A novel design of an element for MRAM, presented here, mitigates against these problems. The design is called the ‘‘hysteron’’ because it contains the basic unit for reversal of magnetization by a continuous process of domain wall motion, but without nucleation. The hysteron illustrates Brown’s paradox (the concepts discussed here can be found in the book by Hubert and Schaefer [1]): to nucleate a wall requires a large field, but, once a wall is present, reversal takes place in a small field. A resolution of the paradox is not to lose the wall on reversal. This is how the hysteron works. Further the reversal takes place without difficulties from the edges and most important, the switching fields are so low that the elements can be made thick enough to be thermally stable. The hysteron is designed to be compatible with the most recent *Corresponding author. Physics Department, Simon Fraser University, Burnaby, British Columbia, Canada. Fax: +1 8045245439. E-mail address:
[email protected] (A.S. Arrott).
advances in silicon technology, using, for example, dimensions 200 100 8 nm3 and switching in fields of 50 Oe, which are easily achieved.
2. Hysterons The hysteron was an accidental discovery while preparing a pedagogical example for a public lecture. The idea was to have a single vortex that moves from the center of one circle to the center of a second circle that overlaps the first along an axis for which the length between centers was slightly less that the diameters of the circles. For a single circle the stable position of the vortex is in its center. With two connected circles it should be possible to move the vortex from one center to the other along the line joining the centers. The lecture was to demonstrate this, using micromagnetic calculations [1, pp. 148–155]. Switching occurs by applying a magnetic field in the plane of the circles perpendicular to the axis joining their centers. This would be a very simple memory element. For it to be useful for MRAM, the vortex should move more easily from one center to the other when a bias field is applied parallel to the line joining the centers. The pair of fields allows one element to be selected from a two-dimensional array. The
0304-8853/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 2 ) 0 1 0 0 0 - 4
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Fig. 1. (a) The color code is illustrated by overlaying arrows for each of the three equilibrium states in zero field. (b) The switching process is shown for a hysteron composed of two 100 nm circles partially overlapped. A switching field of 50 Oe is applied along the line joining the centers. A bias field of 50 Oe is applied upward on the page. The calculations were carried out on a grid of 2 nm cubes using the LLG micromagnetics code of M.R. Scheinfein [2] which includes a method for correcting for the discretization at the boundaries [3].
discovery of the hysteron came when the effect of the bias field was calculated. The two-circle hysteron with one vortex has three stable points for the vortex in either circle. Perhaps this may seem obvious to some in hindsight, but there is no evidence that anyone has predicted this behavior. The magnetostatic energy has three minima for equilibrium configurations where the center of the vortex is either in the center of the circle or well displaced toward the top or bottom (as viewed on the page). These three equilibrium configurations are shown in Fig. 1a. The switching of the first hysteron, as shown in Fig. 1b, illustrates a resolution of Brown’s paradox. The low switching fields result from domain wall motion without nucleation because the wall never goes away. In the absence of the field the wall is the small blue segment from the vortex to the nearby edge. On reversal it emerges from its hiding place to bring about the reversal of the other circle. Since that first discovery, many variations on size and shape have been studied for their applicability
to MRAM. Here the discussion is limited to explaining the principles behind the operation of these devices. The two displaced minima can be explained using a simplified model of two non-intersecting circles, one with and the other without a vortex. This ignores the important exchange interactions between the circles as shown in Fig. 1, but it brings out the importance of the magnetostatic interactions. The circle without the vortex is a permanent magnet that rotates without hysteresis. The circle with the vortex is a soft magnetic material where magnetization occurs with the displacement of the vortex from the center. The susceptibility of the circle with a vortex is determined by the change in self-energy of the configuration when the vortex is displaced. The magnetostatic energy increases and the exchange energy decreases, both proportional to the square of the net magnetization of the circle with a vortex. Independently both circles are isotropic and without hysteresis. It is the dipolar interaction between the two circles that is anisotropic and hysteretic.
A.S. Arrott / Journal of Magnetism and Magnetic Materials 258–259 (2003) 25–28
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3. Simplified model
With these components given in terms of y; the energy can be written as
The energy of the permanent magnet circle in a magnetic field H is given by
E=Ek ¼ ðHx =Hx;0 Þ cos y ðHy =Hy;0 Þ
Ep ¼ ðMs Hx cos y þ Ms Hy sin yÞVp ;
where Ms is the saturation magnetization and Vp the volume of the permanent magnet circle. The energy of the vortex circle is given by Ev ¼ fðMx Hx þ My Hy Þ þ ðMx2 þ My2 Þ=ð2wÞgVv ;
ð2Þ
where Mx and My are the components of the magnetization induced by the field and w reflects the net changes of the magnetostatic and exchange energy on magnetization of the circle of volume Vv : As this will be applied to permalloy, the crystalline anisotropy will be neglected in this argument. The vortex is a soft magnetic material with a susceptibility w; which can be quite large if the decrease in exchange energy compensates the increase in self-magnetostatic energy when the vortex is displaced from the center of its circle. For 100 nm diameter with 8 nm thickness, w is about 10. The dipole–dipole interaction between the two circles is expressed in the crude approximation that these are two interacting point dipoles separated by a distance R: The energy of interaction is Ed ¼ ð2Mx Ms cos y þ My Ms sin yÞVv Vp =R3 ;
ð3Þ
where the x direction is along the line joining the centers of the circles. The calculation of R; the effective separation of the dipoles, requires considerable integration. In practice the full power of micromagnetics [1, pp. 148–155] is needed to determine the configurations in each circle and the magnetostatic interactions between them. The factors –2 and +1 are just the differences between the energy of two dipoles when they are aligned parallel to the axis joining them and when they are aligned perpendicular to that axis. The physics is clear. The energy of the interaction of the induced dipole with the permanent dipole is a minimum when the magnetizations in the two circles are parallel to each other and parallel to the axis joining them. The magnetization components of the vortex circle are given by Mx ¼ wðHx þ 2Ms cos y Vp =R3 Þ
ð4Þ
and 3
My ¼ wðHy Ms sin y Vp =R Þ:
sin y cos2 y=2;
ð1Þ
ð5Þ
The vortex is meta-stable in the absence of applied fields with y ¼ 0 or p; My ¼ 0 and Mx ¼ 72wMs Vp =R3 7Mx0 : For 100 nm diameter with 8 nm thickness, Vp =R3 is about 0.03 and Mx0 is roughly 2/3 Ms :
ð6Þ
where Ek ¼ 3wMs2 Vp2 Vv =R6 ;
ð7Þ
Hx;0 ¼ 3Ms ðVp =R3 Þ=fðR3 =Vp Þ=w þ 2g;
ð8Þ
and Hy;0 ¼ 3Ms ðVp =R3 Þ=fðR3 =Vp Þ=w 1g:
ð9Þ
For the particular case considered here Ek is about 10 picoergs, Hx;0 is about 20 Oe and Hy;0 about 100 Oe. Note that an increase in w directly increases the stability by increasing the energy of interaction, but it has much less effect on increasing the switching field. At the same time it increases the response of the circle with the vortex. An energy of 1.6 picoergs is 1 eV or 40 times kB T at room temperature. Eq. (8) should be recognized as the standard Stoner– Wohlfarth picture [1, pp. 203–207] of the magnetization reversal for a particle with uniaxial anisotropy, except that now Hx;0 aHy;0 : The switching astroid is found by requiring that the first and second derivatives vanish yielding the standard answers for the fields along the astroid Hx ¼ Hx;0 cos3 y;
ð10Þ
Hy ¼ Hy;0 sin3 y
ð11Þ
and ðHx =Hx;0 Þ2=3 þ ðHy =Hy;0 Þ2=3 ¼ 1:
ð12Þ
When there is no bias field, the switching occurs for Hx ¼ Hx;0 : With a bias field of equal magnitude the switching occurs for Hx ¼ Hx;0 =f1 þ ðHx;0 =Hy;0 Þ2=3 g3=2 : The susceptibility has a strong effect on increasing the bias field needed for selectivity as it brings the first term in the denominator closer to one. This aspect of the two non-touching circles may be relieved by introducing some exchange coupling between the circles as occurs when they overlap.
4. Discussion This simplified model makes clear the physics of the hysteron. There is a permanent magnet that rotates easily as it induces an image of itself in the adjacent soft magnetic circle. The softness is the result of the presence of the vortex for which there is a balance between increase in magnetostatic energy and a decrease in exchange energy as it is displaced from the center of its circle. When the circles overlap, the situation is more
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complicated. It is also possible to understand, qualitatively, the hysteron in terms of the energy of the wall that is seen reversing the process in Fig. 1. The hysteron started out as a device for which the vortex moves from one circle to the other so that the circles necessarily must overlap. The discovery of the off-center stable states led to an analysis of why it works. This analysis indicates that the circles do not have to touch in order for it to work nor do the circles have to have the same volume. This solves the problem of how to get a vortex in one circle and not the other. A vortex can be induced in each circle by applying a strong field. If one circle has a smaller radius than the other, the vortex will appear first in one circle. The overlapping circles have the shape of a ‘‘dog bone’’ and allow for several choices of parameters to characterize the connecting section as well as the radii of the two circles. Again it should be possible with unequal circles to produce one vortex. Other geometries could be considered. For example, one could have three circles in a line with either one vortex in the center circle or two vortices in the outer circle. In any geometry one may or may not want to overlap the circles. As shown above, the hysteron behaves quite the same as a shaped bit that switches by rotation against the magnetostatic shape anisotropy. It has a similar switching astroid. One might then ask what makes the hysteron any more useful than the bit with shape anisotropy. The answer given in the introduction is that avoiding the problem of nucleation during the reversal process, the hysteron switches more easily than a bit with shape anisotropy. That conclusion is reached by
considering a bit with two circles for which the region between the two circles was filled with material. For an ellipse 200 100 8 nm3, if it were to reverse by uniform rotation, it would take more than 400 Oe. It does reverse at 400 Oe by a process with two stages, one in which the center of the bit rotates followed by the propagation of edge reversals along the two long edges. The fields are large and sensitive to edge imperfections. Another question is ‘‘Why not use almost circular bits with just a little shape anisotropy?’’ The onset of shape anisotropy with distortion from the circular shape is quite rapid with a=b ratio. This would make production quite difficult. It is also a problem for the production of hysterons using non-intersecting circles. The circles should be circles. The overlapping circles are coupled by exchange interactions as well as magnetostatic interactions. This helps by adding additional coupling that stiffens the response and makes it less sensitive to deviations from circularity. The design of hysterons for practical applications will require extensive micromagnetic calculations, but even more important will be the experiments to determine what can be achieved in practice.
References [1] A. Hubert, R. Schaefer, Magnetic Domains, Springer, Berlin, 1998, pp. 211–212. [2] http://llgmicro.home.mindspring.com [3] Z. Gumbatis, C.J. Garcia-Cervera, Weinan E, J. Comp. Phys., in press.
