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IEEE TRANSACTIONS ON MAGNETICS, VOL. 37, NO. 4, JULY 2001

Theory of Domain Wall Nucleation in a Two Section Magnetic Wire P. N. Loxley and R. L. Stamps

Abstract—The energy barrier for thermally driven magnetization reversal in a two section nanowire is calculated, based on a mechanism for domain wall nucleation at the interface between sections. It is shown that this energy barrier is lower than that for end nucleation, and can be determined by choice of material for each section. Index Terms—Domain wall nucleation, magnetization reversal, metastable decay, two section nanowire. Fig. 1. Wire geometry for a wire of length L, composed of two different ferromagnetic materials given by material parameters A; K , and A ; K . The wire axis, anisotropy axis, and external field are chosen parallel to the x-axis.

I. INTRODUCTION

F

ERROMAGNETIC nanowires and particles are currently of considerable interest both experimentally and theoretically. This is primarily a result of recent advances in the preparation and characterization of nanometer scale materials, as well as potential technological applications. Magnetic nanostructures are useful as models for examining low dimensional magnetization reversal processes. Nanowires are particularly interesting for studies of reversal processes involving nucleation and propagation of domain walls. Recent experiments [1]–[4] have succeeded in observing domain wall propagation in individual wires. Even more recently, control of wall nucleation and propagation has been achieved using pads [5], junctions [6], and necks [7]. Recent theoretical progress includes quantitative calculations of magnetization reversal rates for uniform wires [8], [9], and elongated particles [8], [10]. Reversal rates for multi section wires have not yet been considered, and are interesting because of an extra degree of freedom available for control of energy barrier height. Theories of decay rates for metastable systems [8], [11], [12] , where are described by the expression is a decay rate, is an energy barrier height or activation a prefactor describing the fluctuation rate. The energy, and derivation of this expression relies upon the existence of an energy saddle point, representing an unstable configuration that the system passes through before decaying into a stable state. for nanoscale magnetic systems can be done Calculation of using a micromagnetic model and is presented here. The prefactor is calculated from consideration of the decay rate of small dynamical fluctuations away from the unstable state. The point for metastable decay in a two secof this paper is to examine tion wire.

The mechanism for thermally activated magnetization reversal in an applied magnetic field has already been examined for homogeneous nanowires [13]. The magnetization is initially in a state uniformly aligned against the applied field, this state becomes metastable and decays into a state uniformly aligned with the field. The reversal mechanism involves a saddle configuration consisting of half a domain wall pair nucleated at either free end of the wire. In the case of a two section wire it is energetically most favorable for a wall to nucleate at the end of one particular section and then to expand through the interface for this mechanism is into the other. It will be shown that to nucleate a separate wall at the free end of the lower than other section. II. ENERGY FOR A TWO SECTION NANOWIRE The wire geometry is shown in Fig. 1, where the uniaxial anisotropy axis is chosen to be in the direction of the -axis, and an external field is applied along the 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 sec) has exchange tion of the wire (extending from constant and anisotropy constant , while the second sec) has constants and , tion (extending from , and are assumed for the following where calculation. The energy per unit area of an arbitrary magnetic configuration can be described using,

Manuscript received October 13, 2000. P. N. Loxley and R. L. Stamps are with the Department of Physics, University of Western Australia, Nedlands, WA 6907, Australia. Publisher Item Identifier S 0018-9464(01)06176-3. 0018–9464/01$10.00 © 2001 IEEE

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(1)

LOXLEY AND STAMPS: THEORY OF DOMAIN WALL NUCLEATION IN A TWO SECTION MAGNETIC WIRE

where is a magnetization unit vector that is considered to vary along the wire axis only. The energy given by (1) contains the main contributions for a thin wire of length , where is considered large, and is assumed to be uniform over the diameter of the wire. It has been argued [13] that this assumption is valid in the case of small wire diameters, where there is a large exchange energy contribution for any deviation from uniformity across the diameter. An analogous argument applies to the uniform magnetization states modeled in nanoparticles [14]. It has also been shown for small diameters [13], [15] that effects of the non local demagnetizing field can be approximated is uniform by a local anisotropy. Since the external field over the length of the wire (1) is thought to be sufficient for describing a wire comprised of two sections of different material.

Fig. 2. The nonuniform maximum and minimum energy configurations of (1) given by  x . The min. energy config. is a  wall centered in the A; K material (solid line), and the max. energy config. a wall centered in the A ; K material (dashed line), the vertical line at x is the interface between the two materials. The parameter values used were A ;K = ;h : , and

()

1=

III. THEORY It is convenient to work in spherical polar coordinates, , with approximately the in (1) such that same for both materials. Varying allows equilibrium configurations to be found, and leads to a Euler-Lagrange equation and a surface term for each section. , The surface terms describe a boundary condition at (2) with a similar expression for . For the case of zero exchange coupling at the interface between the two different materials, and the conditions for a free surface, , satisfy (2). For the case , (2) reduces to matching first derivatives across the interface. All solutions to the Euler-Lagrange equations that fulfill (2) , and satisfy the finite energy conditions of (1) at at , are static magnetic configurations of maximum or minimum energy of (1). An analytic technique has been developed [16] that allows qualitative assessment of all configurations, without having to obtain explicit solutions to the problem. Results of this analysis reveal the existence of two planar domain walls described by , in addition to two uniform configurations which are aligned with and against the applied field. The non uniform configurations are valid for and , where ratios of the material parame, , and the ters have been defined as . external field has been parameterized by depends on the material parameters, The maximum field , and provides the limit for which (2) can be satisfied. Integrating the Euler-Lagrange equations gives the two nonuniform configurations explicitly in terms of the coordinate. The exact solutions consist of a wall centered material, corresponding to a minimum energy in the , , material, configuration, and a wall centered in the corresponding to a maximum energy configuration. In Fig. 2 both of these walls are plotted together, with the interface rep, and where the parameter resented by the vertical line at , and have been used. As values , the walls have no Zeeman energy and move away from . For the minimum energy wall moves

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A =K :

=~ 0 ~ = 1 = 1 40 = 0 065

partly into the , material, while the maximum energy material. This shift in the wall moves partly into the minimum energy walls’ position increases wall energy due to exchange and anisotropy, while decreasing the Zeeman energy now present. The result is a minimization of the total wall energy for a particular applied field. Similarly the maximum energy wall shift decreases wall energy due to exchange and anisotropy, while increasing the Zeeman energy. This results in a maximization of the total wall energy for a particular applied both walls become centered at the interface field. As and have the same energy. Using the notation of [15], the external field can be paramand as , aleterized by lowing the energy of either wall configuration to be expressed as;

(3) Where stants

is the cross sectional area of the wire and the conand are given by:

using to describe the minimum energy wall, and to describe the maximum energy wall. , (3) reduces to the familiar expression In the limit for the energy of a wall with no Zeeman energy,

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IEEE TRANSACTIONS ON MAGNETICS, VOL. 37, NO. 4, JULY 2001

Fig. 3. Variation of the total energy E of the minimum energy wall (solid line) and the maximum energy wall (dashed line) from Fig. 2, with change in the centering x of each wall from its equilibrium position for fixed h 0:065.

=

for a material with constants and , and for a material with constants and . In the op, (3) gives , as discussed in posite limit relation to the wall configurations. The total energy of both the minimum and maximum energy static wall configurations from Fig. 2 are plotted together in Fig. 3, as the centering of each wall is changed from the walls’ . The equilibrium posiequilibrium position for fixed tions correspond to the two points of local minimum and maximum energy shown in Fig. 3. It can be seen that the maximum energy wall is energetically unstable to perturbations about the equilibrium position, and the minimum energy wall is only a local minimum, the total wall energy will eventually decrease if material. this wall is pushed through the IV. CONCLUSION A mechanism for thermally driven magnetization reversal can now be proposed for a two section wire in an applied magnetic . Initially the magnetization is in a unifield form state opposing the applied field. A domain wall is nuclematerial, which propagates down the ated at the end of the wire until becoming pinned at the interface. Thermal activation allows this wall to become depinned and continue through the wire, switching the magnetization to a state fully aligned with the external field. The depinning process involves nucleation of material. Thermal fluctuations push an equia wall in the material into the malibrium wall centered in the terial by climbing the energy barrier. The wall either overcomes the barrier and continues through the wire, or falls back to its equilibrium position centered in the local minimum.

Calculation of the energy barrier height for this mech. For the anism can be performed using (3), and is case of vanishing Zeeman energy, , which can be compared [13] with for nucleation at the free end of the material. For the opfor both mechanisms, however posite limit, for nucleation at the infor end nucleation. terface compared with and is clearly For finite values of lower for nucleation at the interface, as opposed to nucleation material. For the situation of at the free end of the , which by (2) means that the material has no exchange coupling at the interface, the interface energy barrier and reduce to that for the free end. REFERENCES [1] T. Ono et al., “Propagation of a magnetic domain wall in a submicrometer magnetic wire,” Sci., vol. 284, pp. 468–470, Apr. 1999. [2] T. Ono, H. Miyajima, K. Shigeto, and T. Shinjo, “Magnetization reversal in submicron magnetic wire studied by using giant magnetoresistance effect,” App. Phys. Lett., vol. 72, pp. 1116–1117, Mar. 1998. [3] W. Wernsdorfer et al., “Measurements of magnetization switching in individual nickel nanowires,” Phys. Rev. B, vol. 55, pp. 11 552–11 559, May 1997. [4] W. Wernsdorfer et al., “Nucleation of magnetization reversal in individual nanosized nickel wires,” Phys. Rev. Lett., vol. 77, pp. 1873–1876, Aug. 1996. [5] K. Shigeto, T. Shinjo, and T. Ono, “Injection of a magnetic domain wall into a submicron magnetic wire,” Appl. Phys. Lett., vol. 75, pp. 2815–2817, Nov. 1999. [6] W. Y. Lee et al., “Domain nucleation processes in mesoscopic Ni Fe wire junctions,” J. Appl. Phys., vol. 87, pp. 3032–3036, Mar. 2000. [7] Y. Yokoyama et al., “Ker microscopy observations of magnetization process in microfabricated ferromagnetic wires,” J. App. Phys., vol. 87, pp. 5618–5620, May 2000. [8] H. B. Braun, “Statistical mechanics of nonuniform magnetization reversal,” Phys. Rev. B, vol. 50, pp. 16 501–16 521, 1994. [9] D. Hinzke and U. Nowak, “Magnetic relaxation in a classical spin chain,” Phys. Rev. B, vol. 61, pp. 6734–6740, Mar. 2000. [10] H. B. Braun, “Thermally activated magnetization reversal in elongated ferromagnetic particles,” Phys. Rev. Lett., vol. 71, pp. 3557–3560, Nov. 1993. [11] H. A. Kramers, “Brownian motion in a field of force and the diffusion model of chemical reactions,” Physica, vol. 7, pp. 284–304, 1940. [12] J. S. Langer, “Statistical theory of the decay of metastable states,” Ann. Phys., vol. 54, pp. 258–266, 1969. [13] H. B. Braun, “Nucleation in ferromagnetic nanowires-magnetostatics and topology,” J. App. Phys., vol. 85, pp. 6172–6174, Apr. 1999. [14] , “Kramers’s rate theory, broken symmetries, and magnetization reversal,” J. Appl. Phys., vol. 76, pp. 6310–6315, Nov. 1994. [15] , “Fluctuations and instabilities of ferromagnetic domain-wall pairs in an external magnetic field,” Phys. Rev. B, vol. 50, pp. 16 485–16 500, Dec. 1994. [16] P. N. Loxley, unpublished.

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