A graphical technique for finding equilibrium magnetic domain walls in ...

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Journal of Magnetism and Magnetic Materials 249 (2002) 187–192

A graphical technique for ﬁnding equilibrium magnetic domain walls in multilayer nanowires P.N. Loxley* School of Physics, University of Western Australia, Crawley, WA 6009, Australia

Abstract A graphical technique for ﬁnding equilibrium magnetic conﬁgurations in exchange coupled multilayer magnetic nanowires is presented. For the case of a two layer wire this technique is used to ﬁnd two domain wall conﬁgurations localized near the interface between the layers. Both conﬁgurations are demonstrated to satisfy some important requirements for use in a calculation of the rate of magnetization reversal due to thermal activation. It is described how the graphical technique can be used for other types of multilayer nanowires. r 2002 Elsevier Science B.V. All rights reserved. PACS: 75.60.Ch; 75.60.Jk Keywords: Multilayer nanowire; Domain wall; Magnetization reversal; Thermal activation

1. Introduction Measurement of magnetization reversal in individual nanoscale ferromagnetic wires has been recently demonstrated by several groups [1–3]. Fabrication of multilayer nanowires has also been reported [4], making theoretical study of the reversal process in multilayer nanowires particularly relevant. Thermally activated reversal in nanowires involves the nucleation and expansion of a region of reversed magnetization. The rate of such a process can be measured experimentally for individual wires [2], and has been demonstrated to obey the Arrhenius rate law I ¼ I0 expðEb =kTÞ; where I is the reversal rate, I0 a rate prefactor, and Eb an energy barrier height. This expression has also been derived using a Kramers rate theory [5], *Tel.: +618-9380-7014; fax: +618-9380-1014. E-mail address: [email protected] (P.N. Loxley).

giving a theoretical prescription for the calculation of I0 and Eb : More precisely Eb is the thermal energy required to nucleate a region of reversed magnetization of critical size, while I0 contains information about the rate at which this region expands. Within the framework of the theory this nucleus is an unstable magnetic conﬁguration resulting from an energy saddle point. In a reversal process the magnetization begins in a locally stable conﬁguration and is then deformed into the unstable conﬁguration by thermal ﬂuctuations before reversal takes place. Analytic calculation of I0 and Eb using both of these equilibrium conﬁgurations has been achieved for particular nanowire dimensions [5], and Eb has been calculated for a two layer wire in Ref. [6]. The point of this paper is to present a graphical method that can be used to ﬁnd the equilibrium magnetic conﬁgurations present in an exchange coupled multilayer nanowire, and to identify

0304-8853/02/$ - 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 0 5 2 9 - 2

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which of these conﬁgurations is suitable to use in the rate theory for the calculation of reversal rates. A two layer wire will be discussed in detail. The graphical method relies on reducing a variational problem to its ﬁrst integrals and associated boundary conditions, possible conﬁgurations are then found by exploiting an analogy with particle mechanics. Topological conditions on these conﬁgurations determine suitability for use in calculating reversal rates. The energetic stability of conﬁgurations is also important; however, will not be considered here.

2. Energy for a two layer nanowire The wire geometry is shown in Fig. 1, where the uniaxial anisotropy axis is chosen to be in the direction of the x-axis, and a magnetic ﬁeld is applied along the x-axis in the positive direction. The magnetization aligns with the anisotropy axis, preferentially pointing in the direction of the applied ﬁeld. The wire is assumed to be cylindrical, uniform in diameter, and consisting of two different types of ferromagnetic material. The ﬁrst layer of the wire (extending from L=2pxo0) has exchange constant A; and anisotropy constant K; while the second layer (extending from 0oxpL=2) has constants A0 and K 0 ; with M00 EM0 : The energy per unit area for an arbitrary magnetic conﬁguration can be described using ) Z L=2 ( 0 A dM 2 K 0 2 E¼ dx 2 Mx Hext Mx M02 dx M0 0þ ( ) 2 Z 0 A dM K 2 þ dx 2 Mx Hext Mx ; M02 dx M0 L=2 ð1Þ where MðxÞ is a vector representing the magnetization, considered only to vary in orientation along the wire axis, and where Hext is the applied magnetic ﬁeld. The energy given by Eq. (1) contains the main contributions for a thin nanowire of length L; and where M is assumed to be uniform over the diameter of the wire. It has been argued [7] that this assumption is valid in the case of wire diameters smaller than the exchange

A, K

A’, K’

x

L Fig. 1. Geometry for the two layer wire. The uniaxial anisotropy is along the x-axis, and the applied ﬁeld is also along the x-axis in the positive direction.

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ length A=pM02 ; where there is a large exchange energy contribution for any deviation from uniformity across the diameter. It has also been shown in this case that the effects of the non-local demagnetizing ﬁeld can be approximated by a local uniaxial anisotropy [7].

3. Theory It is convenient to work in spherical polar coordinates, where M ¼ M0 ðsin y cos f; sin y sin f; cos yÞ: Choosing suitable length and energy scales allows the dimensionless and E% to be ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ quantities x%p 0 0 0 % deﬁned as x ¼ A =K x; % and E ¼ 2 A K 0 E: Using polar coordinates Eq. (1) becomes, in terms of the dimensionless quantities ( " 2 # Z L=2 1 dy 2 df 2 E¼ dx þsin y 2 dx dx 0þ 1 sin2 y cos2 f h sin y cos f 2 ( " 2 # Z 0 A* dy 2 df 2 þ dx þsin y 2 dx dx L=2 K* sin2 y cos2 f h sin y cos f ; ð2Þ 2 where the dimensionless ratios h ¼ Hext M0 =2K 0 ; A* ¼ A=A0 ; and K* ¼ K=K 0 have been deﬁned. The magnetization is assumed to obey the following equations of motion: M0 qf dE ¼g ; sin y qt dy 2K 0 M0 qy 1 dE ¼ g ; 0 sin y df 2K qt

ð3Þ

P.N. Loxley / Journal of Magnetism and Magnetic Materials 249 (2002) 187–192

A* 2 df 2 A* K* 2 * cos f ¼ C2 ; sin f þ Ah 2 2 dx

xo0 ð5Þ

along with the boundary conditions df df ¼ ; A* dx x¼0 dx x¼0þ df ¼ 0; dx x¼7L=2

ð6Þ

ð7Þ

where C1;2 are constants of integration to be determined. The differential equations (4) and (5) have the same form as the equation for energy conservation in one-dimensional particle mechanics [8]. The motion of each particle is described by the coordinate fðxÞ; where x now plays the role of time. One of the particles has kinetic energy 12ðdf=dxÞ2 ; a potential energy given by V ðfÞ ¼ 12sin2 f þ h cos f; and a total energy C1 ; while the other particle has kinetic * energy A* 2 =2ðdf=dxÞ2 ; potential energy VðfÞ ¼ 2 * cos f; and total energy C2 : * ðA* K=2Þsin f þ Ah Equilibrium magnetic conﬁgurations can then be considered as conservative trajectories of the analogous particle. In the following two sections the possible solution trajectories described by Eqs. (4) and (5) will be discussed.

4. Behaviour of trajectories The general shape of the potential energy curves * V ðfÞ and VðfÞ for the analogous particles is shown in Fig. 2. Three different types of bounded motion exist for the particles, depending on the values of C1 and C2 : One possible particle trajectory with Ci ¼ E1 begins at one of the global maxima of V ðfÞ; either f ¼ 0 or 2p; with zero kinetic energy, and ends at the other maximum, f ¼ 2p or 0: Another type of particle trajectory occurs if Ci ¼ E2 . Starting at the local maximum f ¼ p this trajectory proceeds towards either f ¼ 0 or 2p; encountering a turning point before reaching either of these points. This causes the particle to retrace it’s path to end back at f ¼ p: The last type of motion occurs for Ci equal to a value such as E3 : This particle travels along a path before encountering a turning point, it then retraces its path before encountering a second turning point. The particle continually alternates between the two turning points, leading to a periodic trajectory. The behaviour of a particle trajectory near any of the maxima of V ðfÞ is exponential [9], meaning that the particle only approaches or departs these points as x-7N: This behaviour leads to a localized, non-periodic trajectory, which is the situation when Ci ¼ E1 or E2 :

E1 E2

V()

where g is the gyromagnetic ratio, and E is the dimensionless energy. These equations describe the precession of a magnetic moment about an effective ﬁeld, with the magnitude of the moment M0 conserved at all times. Equilibrium magnetic conﬁgurations are conﬁgurations that satisfy qf=qt ¼ 0 and qy=qt ¼ 0: From Eq. (3) it is seen that these can be found by solving the variational problem dE=dy ¼ 0 and dE=df ¼ 0: Upon using Eq. (2) to calculate dE; the variational problem leads to two Euler– Lagrange equations for each layer and associated boundary conditions. One of the conﬁgurations that satisﬁes dE=dy ¼ 0 is y ¼ p=2: In this case dE=df ¼ 0 becomes, after integrating once 1 df 2 1 2 sin f þ h cos f ¼ C1 ; x > 0; ð4Þ 2 dx 2

189

E3

0

2

Fig. 2. The potential energy curve VðfÞ for the analogous particle described by fðxÞ; with x as time. Three different types of bounded motion between f ¼ 0 and 2p exist for the particle, corresponding to the three different particle energies E1 ; E2 ; and E3 :

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5. Equilibrium conﬁgurations for nanowire E V()

The energy of a magnetic conﬁguration given by Eq. (2) is seen to be independent on the length of a nanowire if the energy remains ﬁnite in the limit that L-N: This limit corresponds to each of the layers in the wire being much longer than the pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ characteristic length scale A0 =K 0 : The two localized trajectories previously considered both begin and end with zero kinetic energy and maximum potential energy, i.e. ðdf=dxÞ-0 and fðxÞ-0; p or 2p as x-7N: This means magnetic conﬁgurations based on these trajectories allow convergence of Eq. (2) as L-N to be achieved, in contrast to the periodic trajectory described. This length independence is important for constructing reversal mechanisms for the nanowire as will be seen in Section 6, and only localized trajectories will be considered in the following. From Eqs. (4) and (5) the behaviour of the two localized trajectories as x-7N gives C1 ¼ 7h; * and also satisﬁes the boundary condiC2 ¼ 7Ah tion given by Eq. (7). However, the most interest* The ing case occurs if C1 ¼ h and C2 ¼ Ah: * shifted potentials V ðfÞ C1 and VðfÞ C2 are shown in Fig. 3 for this case, and the curves are seen to intersect at points where V ðfÞ C1 ¼ * VðfÞ C2 : The intersection points, for example at f ¼ f1 or f2 ; are also points where the boundary condition at x ¼ 0 given by Eq. (6) is satisﬁed. To see this square both sides of Eq. (6), upon using with Eqs. (4) and (5) the boundary condition is seen to be satisﬁed anywhere that V ðfÞ C1 ¼ * VðfÞ C2 : Now consider a particle of energy E that begins at the maximum f ¼ p with zero kinetic energy and roles down the potential curve. As previously discussed with reference to trajectory E2 in Fig. 2 the particle proceeds towards the maximum at f ¼ 0; encounters a turning point before reaching this point, and retraces its path back to f ¼ p: However, if the particle is allowed to switch to the other potential curve shown in Fig. 3 before the turning point is reached, the particle will continue unhindered to the maximum at f ¼ 0: While the kinetic and potential energies of the particle may be affected by switching curves, the total energy E will remain the same. If, in addition,

0

1

2

2

* Fig. 3. The potential energy curves V ðfÞ C1 and VðfÞ C2 are plotted together, leading to the intersection points at f ¼ f1 or f2 : A particle with energy E that starts at the maximum f ¼ p with zero kinetic energy, and roles down the potential curve can switch to the second curve shown and continue rolling up to the maximum at f ¼ 0 by conservation of energy. If, in addition, the particle switches curves once at either of the intersection points then the resulting trajectory can be considered as an equilibrium conﬁguration fðxÞ of the two layer wire, satisfying all of the boundary conditions.

the particle only switches potential curves when it is at either f ¼ f1 or f2 ; then the resulting trajectory satisﬁes the boundary condition given by Eq. (6). If f is restricted to lie between 0 and p; and the particle switches potential curves only once during its motion, then from Fig. 3 there are two possible trajectories satisfying Eq. (6). Either of these trajectories can be considered as an equilibrium magnetic conﬁguration of the two layer wire representing a p domain wall localized near the interface between the two layers. The value of fðxÞ at the interface is f1 for one of the walls and f2 for the other. These values change as the potential curves in Fig. 3 change shape depending on the applied ﬁeld and material parameters considered. The graphical technique described can be extended to more complicated multilayer nanowires. For each sandwiched layer of width pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ao A0 =K 0 ; there is a conservation equation of the form of Eq. (5) as well as a boundary condition of the form of Eq. (6) applying to the interface * between each layer. The dimensionless ratios A; and K* are now ratios of the constants A; K for the material making up a layer, and the constants A0 and K 0 from a layer used to deﬁne the dimension-

P.N. Loxley / Journal of Magnetism and Magnetic Materials 249 (2002) 187–192

% The two end layers have less quantities x% and E: C1;2 determined by the convergence of Eq. (2) as L-N: If the potential curves Vi ðfÞ Ci for each layer are now plotted for an arbitrary range of the undetermined Ci2 values such that curve intersections occur and the particle is only allowed to switch potential curves at these points, then equilibrium magnetic conﬁgurations satisfying the boundary conditions can be constructed. For the sandwiched layers periodic conﬁgurations will be found, while the two end layers will have the exponential conﬁgurations previously discussed. It should be noted for completeness that the choice of the Ci2 ’s determine each sandwiched layer width a:

6. Reversal mechanism requirements Thermally activated magnetization reversal generally requires small activation energies. For wires this means the activation energy should not depend on wire length, such a situation is observed experimentally where reversal is due to nucleation and propagation of a small region of reversed magnetization. The rate theory only requires a pair of conﬁgurations in order to calculate the energy barrier height, one locally stable and the other unstable. However, for a transition to occur a conﬁguration corresponding to the fully reversed state of the wire must also exist. The graphical technique described can identify all of these requirements, excepting stability. After a pair of equilibrium conﬁgurations has been identiﬁed, the activation energy; or energy required to get from one conﬁguration to the other should be determined. Such a transition is less likely if it requires an intermediate conﬁguration that depends on wire length, this conﬁguration would require more energy as the wire length was increased. From this energy argument such a transition is only likely if Eq. (2) remains ﬁnite as L-N for each possible intermediate conﬁguration. This can be achieved if both the equilibrium conﬁgurations fA ðxÞ and fB ðxÞ - the same set of maxima of V ðfÞ as x-7N; meaning the conﬁgurations are topologically identical.

191

In the situation of an applied magnetic ﬁeld the lowest energy conﬁguration possible is one where the magnetization is uniformly aligned with the ﬁeld. In the absence of a ﬁeld the two lowest energy conﬁgurations correspond to the uniform alignment of the magnetization in either direction along the anisotropy axis of the wire. A conﬁguration that is only locally stable should be topologically distinct from one of these lowest energy conﬁgurations. The conditions for identifying a reversal process now involve ﬁnding a pair of equilibrium conﬁgurations, using the graphical technique, that are topologically equivalent to each other as well as topologically distinct from the fully reversed conﬁguration. For the two layer wire this technique yielded a pair of p domain walls, each with a positioning of either f ¼ f1 or f2 at the interface. These walls are topologically equivalent as both pass through the same set of maxima, f ¼ 0 and p: They are also topologically distinct from the fully reversed state of f ¼ 0: This means it is possible for thermal ﬂuctuations to continuously deform one wall into the other using intermediate conﬁgurations of relatively low energy, it also means that the fully reversed wire is of lower energy than either of these states.

7. Summary A graphical technique for ﬁnding equilibrium magnetic conﬁgurations in exchange coupled multilayer magnetic nanowires has been presented. By considering localized conﬁgurations, boundary conditions, and energy conservation in the analogous mechanical problem, two equilibrium conﬁgurations were found for the two layer wire. Each consists of a p domain wall localized near the interface between the two layers. These conﬁgurations were found to be topologically equivalent as well as topologically distinct from the fully reversed state, satisfying some important requirements for use in a calculation of thermally activated reversal rates for a two layer nanowire. It was also described how the graphical technique can be generalized to more complicated

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multilayer nanowires. In this case if the boundary conditions at all interfaces can be graphically satisﬁed then equilibrium conﬁgurations can be constructed.

Acknowledgements The author wishes to acknowledge Dr. Robert Stamps for informative discussions.

References [1] [2] [3] [4] [5] [6] [7] [8]

T. Ono, et al., Sci. 284 (1999) 468. W. Wernsdorfer, et al., Phys. Rev. B 55 (1997) 11552. Y. Yokoyama, et al., J. Appl. Phys. 87 (2000) 5618. A. Fert, L. Piraux, J. Magn. Magn. Mater. 200 (1999) 338. H.B. Braun, Phys. Rev. B 50 (1994) 16501. P.N. Loxley, R.L. Stamps, IEEE Trans. Magn. 37 (2001) 2098. H.B. Braun, J. Appl. Phys. 85 (1999) 6172. H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, MA, 1950. [9] P.G. Drazin, R.S. Johnson, Solitons: An Introduction, Cambridge University Press, Cambridge, 1989.

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