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Oct 5, 2001 - The former mechanism gives rise to a Stoner-Wohlfarth type of angular variation of the coercive force while ...... to zero at µH ' 63±. It indicates ...

CHINESE JOURNAL OF PHYSICS

VOL. 39, NO. 5

OCTOBER 2001

Domain Wall Structure and the Modified Kondorsky Function Hong Yuan Deng and Huei Li Huang Department of Physics, National Taiwan University, Taipei, Taiwan 106, R.O.C. (Received July 13, 2001)

Simulated analytic solutions of the dependence of the magnetization angles (µ; ') inside a domain-wall upon the coordinate normal to a pinned domain wall have been obtained as a function of the field orientation angles and the field strength and the results were used to obtain a new modified Kondorsky function. The modified Kondorsky function shows that our results better agree with the experimental data than ever before reported. Furthermore, the modified Kondorsky function varies sensitively with the transverse coordinate, field strength, quality factor and field angles (µ H ; Ã H ). Appropriately chosen values of these parameters may make the evaluation of the Kondorsky function edge closer to the experimental pinning field data, provided we deal with a Bloch wall, or a mixed Bloch-N´ eel wall, but not a pure Ne´el wall. Evidently, the domain wall structure has to be taken into account in graphing the pinning field. PACS. 75.60.Ch – Domain walls and domain structure. PACS. 75.70.Kw – Domain structure.

I. Introduction I-1. Domain wall structure Magnetization reversal in high magnetocrystalline anisotropy materials like barium ferrite is believed to take place by either coherent rotation or reverse-domain nucleation. The former mechanism gives rise to a Stoner-Wohlfarth type of angular variation of the coercive force while the latter gives rise to a Kondorsky-type behavior [1]. The question of which type of mechanism is more appropriate and applicable to a given system apparently rests with the values of the coercivity field [2, 3], particle size [4] and field orientation angle with respect to the easy axis [5]. The Kondorsky function [6] states that the domain wall (DW) pinning field (or coercivity) is proportional to the variation of the DW surface energy density with respect to the wall’s displacement [7, 8], which is related to the structure of the DW. Notwithstanding that the study of the motion and structure of the DW has attracted a great deal of attention for a long time [9-19], its effect on the Kondorsky function remains elusive. Schumacher [7, 20] and more recently Huang et al. [3, 13, 14] all attempted to study the pinning field related to the surface DW energy in recent years, but no detail DW structure has been successfully taken into account. In particular, both failed to describe the coordinate dependent azimuthal angle of the magnetization structure inside the DW. The aim of this work is to elaborate on the coordinate dependence of both the polar and azimuthal angles of the magnetization structure inside the DW. Our task is three fold. First is to 479

°c 2001 THE PHYSICAL SOCIETY OF THE REPUBLIC OF CHINA

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evaluate numerically the variation of magnetization angles (#; ') as a function of the coordinate normal to the domain wall and, whereupon to obtain simulated and the corresponding analytic forms by a series of trial-and-error methods through appropriate adjustment of the boundary conditions for (#; ') and their first derivatives. Second, the DW surface energy density is properly optimized (minimized) and, for the first time, the modified Kondorsky function incorporating the coordinate dependence of the magnetization structure inside the DW was successfully obtained. Third, comparison is made of our simulation analytical calculation results on the DW pinning field with the experimental data available in detail. The description of the one-dimensional DW structure is usually based on the assumption that the azimuthal angle ' of the magnetization vector inside the DW is independent of the coordinate normal to the DW plane [21]. However, the presence of the magnetization component Mn = M (sin# sin' ¡ ¹) normal to the DW plane, where ¹ is the normal component of magnetization far from the DW, gives rise to a magnetic charge proportional to @Mn [email protected]» where » (= y=4, where ¢ is the wall width parameter) is a dimensionless reduced coordinate along the normal to the DW plane. The appearance of the magnetostatic poles on the DW surface is expected to give rise to a demagnetization field Hd = 4¼M (sin#sin' ¡ ¹) and hence the demagnetization energy is as follows [9, 12]: 1 WD = Mn Hd = 2¼M 2 (sin # sin ' ¡ ¹)2: 2

(1)

The polar and azimuthal angle of magnetization vector inside the DW may vary along the coordinate normal to the DW due to the presence of the demagnetization energy to minimize the DW energy. Consequently, it is necessary to consider the coordinate dependence of the polar angle # and the azimuthal angle ' of the magnetization within the DW simultaneously. The parameter ¹ is equal to sin #± sin à H which is determined by the value of the normal component of the magnetization (M¢n) existing inside domains (far from the DW) in the presence of the transverse field H = H (cos à H sin µH , sin à H sin µH ; cos µH ). Numerical solutions of the magnetization angles (#; ') versus coordinate normal to the DW has been attempted and their results reported in recent years [15-19]. I-2. Kondorsky function According to reports [4, 5], the barium ferrite particles were known to have a distribution of coercivity, all considerably less than the anisotropy field, even for aligned materials. Namely, Hc is much less than 2K=M S (K: uniaxial constant, MS : saturation magnetization). Correspondingly, the reversal process for each particle is unlikely to be a coherent rotation, but a 180 ± domain-wall motion. Then the effective field is the component of the applied field along the easy axis. Thus for each particle the coercivity is a minimum along the axis (µH = 0) and increases with the field polar angle µH as 1= cos µH , as has been observed in orthoferrite single crystal [2], barium ferrite (BaO¢6 Fe2O3) [4, 5], YIG21 and SmCo5 [23, 24]. Coherent rotation obeys the Stoner-Wohlfarth theory [25] and the reverse-domain nucleation obeys the relation described by the Kondorsky function [6], The question then arises: in a typical polycrystalline specimen, what portion of the total change in M is due to wall motion and what is due to rotation? The question has no precise standard answer. It depends on the particle size and the orientations of all of the particles in the powder assembly, as can be seen in a typical magnetization curve. However, this division of the magnetization curve is a bit arbitrary to say

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the least, since wall motion and coherent rotation processes are not sharply divisible processes. In fact, at any one level of M, the domain wall motion may occur in one portion of a specimen and rotation in another. Even in certain orientations of a single-crystal specimen relative to the applied field, the domain wall motion and domain rotation may occur simultaneously in the same part of the specimen [26]. Therefore the particle assembly of any given specimen may be assumed to consist of a fraction r(µH ) of the particles which have a nucleation field Hn =cos µH for reverse-domain nucleation, and a fraction [1-r(µH )] which has the ideal SW behavior at the polar angle µH. Assuming the linearity of demagnetization curves by both mechanisms and based on the seriesconnected model, one can obtain the resultant angular variation of coercivity as follows: [4, 5] Hc (µH ) = [1 ¡ r(µH )]HSW (µH ) + r(µH )Hn=cosµH; where HSW (µH) ¸ Hn= cos µ is the SW coercivity and Hc(µH) is the coercivity of the oriented assembly of particles. When HSW (µH ) < Hn= cos µH, it is assumed that H c(µH) = HSW (µH ). By varying Hn and r(µH) a number of hypothetical curves may be obtained indicating that the reverse-domain nucleation at defect sites greatly influences the intrinsic coercivity of barium ferrite particles, even for stress-free chemically precipitated particles. For example, Brown showed in theory [26] that the field required for magnetization reversal is given by Hc = 2K=M for a crystal sample of any size, whether larger or smaller than the critical diameter Ds . (Here Hc is a particle value of the true field acting on the crystal, where H is the sum of the applied field and the demagnetizing field Hd): Thus, Hc is just the value obtained for a coherent rotation. For barium ferrite, 2 K=M = 2(3:3 £ 10 6)=380 = 17; 000 Oe. It would not be surprising to find this value corresponds with single-domain particles with a D s of about 100 Å (angstrom). But in larger particles this value Hc is only 3000 Oe, and it decreases still further as the particle size increases. This large discrepancy between theory and experiment for larger particles is known as ‘Brown’s paradox.’ The reason is that, for particles large enough to contain domain walls, the main mechanism of magnetization reversal is by way of reverse-domain nucleation, not by coherent rotation. Up until this date, although there were numerous papers [3, 7, 13, 14] offering different models to modify the Kondorsky function, there is still room for improvement since, as mentioned above, these models considered only the cases where the azimuthal angle ' of the magnetization inside the DW is coordinate-independent. Presently, we intend to propose a (simulated) model by treating ' as an function of the applied magnetic field and the coordinate normal to the DomainWall and obtain two-dimensional graphs of the pinning field HP versus the field polar angle µH at different coordinate and field azimuthal angle à H ; in addition, a three-dimensional plot of the pinning field HP versus the field polar angle µH and the coordinate under different magnetic fields and field azimuthal angle à H , and a three-dimensional plot of the pinning field HP versus the polar angle µH and field azimuthal angle à H under different magnetic fields have also been obtained for purpose of comparison. In short, we shall utilize the simulated analytic function for the structure of DW-(#(»); '(»)) to modify the DW energy density distribution and to analyze the Kondorsky function closely. The result is significantly important since it illustrates the fact that the coercivity due to the DW pinning field is dependent upon both #(») and '(»).

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II. Theory and model II-1. Domain wall structure We shall proceed to analyze the domain wall structure based on the micromagnetic model [9, 21]. Consider a domain wall (DW) in a magnetic crystal (e.g. barium ferrite) with the anisotropy axis of the uniaxial ferrimagnet oriented along the z-axis of the coordinate frame under a homogeneous magnetic field H = H(cos à H sin µH ; sin à H sin µH; cos µH), where the same axis is chosen as the polar axis for the magnetization vector. In addition, the DW plane is assumed to be pinned parallel to the x ¡ z plane and its direction of translational motion is along the y-axis. The configuration of the 180 ± domain wall (DW) under the action of an external magnetic field is shown in Fig. 1. For the magnetic domain structure, we work in the low temperature regime so that the fluctuation of spins can be completely neglected; but the Curie temperature for the transition metals or ferrites is much larger than room temperature, therefore the anticipated domain structure is applicable even at room temperature. The total energy density of the DW is due to several competing volume magnetic energy densies in the system as follows:

FIG. 1. The domain wall coordinate system without applied magnetic field (above) and with an applied magnetic field (below).

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WDW = WE + WA + WD + WZ ;

483

(2)

where WE = A[(@#[email protected])2 + (sin # @'[email protected])2 ] is the exchange energy, which favors parallel alignment of neighboring spins due to exchange interaction; it is the largest energy scale in the equation. WA = K sin2 # is the uniaxial anisotropy energy due to the crystalline fields of the underlying lattice symmetry. For a sputtered sample, a uniaxial anisotropy is commonly assumed. WD = 2¼M 2(sin ' sin #¡ u)2 is the demagnetization energy due to shape anisotropy. Though this energy is small the long range nature of the interaction makes it crucial in determining the domain structure. WZ = ¡ [MHt sin # cos(' ¡ à H ) + MHp cos #] is the Zeeman energy due to external magnetic fields, where ¹ = sin #± sin à H , with sin#± = ht = h sin µH ; Ht = H sin µH is the transverse field component perpendicular to the easy axis, and Hp = H cos µH is the longitudinal field component parallel to the easy axis. The volume energy density of a DW in a uniaxial magnetic crystal may be expressed as follows: [3, 14-17] WDW =Kf42B[#02 + sin2 #'02] + "(sin # sin ' ¡ ¹)2

+ sin2 # ¡ 2ht sin # cos(' ¡ Ã H ) ¡ 2hp cos #g:

(3)

The DW surface energy density is: Z EDW = WDW dy Z = K f42B[#02 + sin2 #'02] + "(sin # sin ' ¡ ¹)2 + sin2 #

¡ 2ht sin # cos(' ¡ Ã H ) ¡ 2hp cos #g 4B d» Z = K 4B f[#02 + sin2 #'02] + "(sin # sin ' ¡ ¹)2 + sin2 #

(4)

¡ 2ht sin # cos(' ¡ Ã H ) ¡ 2hp cos µgd» Z p = AK f[#02 + sin2 #'02 ] + "(sin # sin ' ¡ ¹)2 + sin2 # ¡ 2ht sin # cos(' ¡ Ã H ) ¡ 2hp cos #gd»:

Then the normalized DW surface energy density is: Z EDW ° =p = d»f#02 + sin2 #'02 + sin2 # AK +"(sin # sin ' ¡

¹)2

(5)

¡ 2ht sin # cos(' ¡ Ã H ) ¡ 2hp cos #g;

where K is the uniaxial anisotropy constant; A is the exchange stiffness constant; #0 ´ d#=d»; '0 ´ p d'=d»; » = y=¢ B ; ¢ B = A=K is the Bloch wall width parameter; " = Q¡ 1; the material quality factor Q = HK =4¼M (HK = 2K=M, 4¼M is the saturation magnetization); Ht = H sin µH, Hp = H cos µH ; ht = Ht =HK , hp = Hp=HK ; ¹ = sin #0 sin à H ; #0 is the polar angle of the magnetization inside domain; ht = sin #0.

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DW DW By means of the Euler equation ( ±W±# = ±W±' =0), the simultaneous differential equations describing the distribution of the magnetization inside the DW have the following form [14]: 00

# ¡ sin # cos #(1 + '02 + " sin2 ') + "¹ cos # sin '

(6)

+ht cos # cos(' ¡ Ã H ) ¡ hp sin # = 0;

(sin2 #'0)0 ¡ "(sin # sin ' ¡ ¹) sin # cos ' ¡ ht sin # sin(' ¡ Ã H ) = 0:

(7)

To obtain the solution to these equations the condition sin #0 = ht inside domains has been used if hp = 0 (i:e:; µH = ¼=2). The boundary conditions for the solutions describing the variation of the angles inside the DW have been chosen as follows: ( #0 ; » ! ¡ 1; #(») = (8) ¼ ¡ #0; » ! +1; » ! § 1;

'(») = Ã H ;

and 0

# (») ! 0;

» ! § 1;

' (») ! 0;

» ! § 1:

0

(9)

If the parameter " is considered to be small (i.e., the quality factor Q À 1), it is easy to see immediately that Eq. (7) has a solution ' = Ã H in the zeroth order with respect to this parameter " and the corresponding analytic solution of Eq. (6) may be represented in the form [18-23] sin #(») = sin #0 + hence ¡1

#(») = sin

·

cos2 #0 ; cosh u + sin #0

¸ cos2 #0 sin #0 + ; cosh u + sin #0

(10)

u = » cos #0

q 1 + " sin2 Ã H:

(11)

Equation (11), being an analytic function, describes the variation of the polar angle of the magnetization inside the DW with respect to the transverse reduced coordinate » corresponding to thepconstant azimuthal angle of the magnetization ' = Ã H and a DW width ¢ = ¢ B [cos #0 1 + " sin2 Ã H ]¡ 1 . In order to evaluate the numerical solutions of the system of Eqs. (6) and (7), subject to the boundary conditions (8) and (9) for small and arbitrary quality factors Q, we have successfully solved the system of Eqs. (6) and (7) by computer simulation using the fourth Runge-Kutta method. Meanwhile, in order to analyze the Kondorsky function [6] closely, its corresponding analytic functions was obtained as follows. Equation (12) (see below) is an empirical, simulated analytic function, obtained by a series of trial-and-error methods. The equation describes the variation of the azimuthal angle of the magnetization inside the DW as a function of the transverse coordinate y (i.e., the transverse dimensionless reduced coordinate »): µ ¶ ³ ³ d´ d´ ¡1 ¡1 '(»; µH) = Ã H + (0:6 ¡ ht ) ¢ tan »¡ ¡ tan »+ : (12) 4 4

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Here à H = ¼=2 or à H = ¼=4 corresponds to a Np e´el wall or a mixed Ne´el-Bloch wall, respectively, and ht = h sin(µH), » = y=4, 4 = 4B[cos #± 1 + " sin2 à H ]¡ 1, and d is a parameter related to the adjustable DW width which depends on à H , 4 and ht . II-2. Modified Kondorsky function: ° =  (y) ¢£ (µH). The derivation of the Kondorsky relation [6] was based on the theory of Becker and Do¨ring. Supposing that there is a pinned domain wall in an magnetically uniaxial crystal with saturation magnetization MS under an applied field H, causing a displacement dy and hence a change of in the wall energy dEH by dEH = 2¹ ± MS H cos µH S dy, where S is the DW surface area and ¹ ± is permeability of the crystal. A pinned domain wall may be detached if dEH =S ¸ d° where d° is the change of the surface DW energy density being affected by the deformation of the wall at the pinning center. Equating the two energies (i.e., dEH =S and d°), we may re-express the DW pinning field as HP (µH ) =

1 d° H P (0) = : 2¹ ± MS cos(µH ) dy cos(µH )

(13)

This is the well-known Kondorsky relation, where H P (0) may be identified as HP (0) = (2¹ ± MS )¡ 1 d°=dy. It is to be noted that, due to large field strength, the coherent rotation process for µH 6= 0 may also be involved in the magnetization process whenever the DW is deformed, where the wall is no longer a 180± wall, and the wall energy density has been changed. Thus, for an acceptable modified Kondorsky function, any change in the wall energy has to be duly taken into account. Schumacher [7] proposed that the surface wall energy density ° may be separated into two parts: (1)  (y) is a function of the coordinate y normal to the DW, which is intimately related to the switching field strength H P (0± ), and (2) £ (µH ) is a function of the polar angle of the applied magnetic field with the easy axis of crystal. Thus, ° =  (y) ¢£ (µH ). Accordingly, the Kondorsky pinning field becomes µ ¶ 1 d° HP (0 ± ) £ (µH ) HP (µH ) = = ; (14) 2¹ 0 MS cos(µH ) dy cos(µH ) £ (0± ) where HP (0± ) = (2¹ 0MS )¡ 1£ (0 ± ) d =dy and £ (µH)=£ (0 ± ) is the modified part for the Kondorsky function, which acts in part to avoid a divergence as µH ¡! ¼=2. II-3. Re-modified Kondorsky function: ° =  (y) ¢£ (µH ; à H ; y). Equation (14) properly identifies an explicit field polar angle dependence of the pinning field of the Kondorsky function, in addition to the normal transverse coordinate dependence through d =dy. However, it is to be rememberes that the DW surface energy density, °, is a sensitive function of the field azimuthal angle à H as well, through its interconnection with the coordinate dependence of the magnetization vector of (#(»); '(»)) upon the DW energy density. We hereby propose a new modified model, taking into account all the energy terms of the DW, including exchange, anisotropy, demagnetization and Zeeman energy, all combined to act to modify the Kondorsky function. The total volume energy density for the DW in an uniaxial magnetic crystal may be expressed

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as follows [14-17]: WDW =A[(@#[email protected])2 + (sin # ¢@'[email protected])2 ] + [K sin2 # + 2¼M 2(sin ' sin # ¡ ¹)2 ] ¡ [MHt sin # cos(' ¡ Ã H ) + MHp cos #]

(15)

where A, K have the usual meanings, ¹ is the limiting value of the magnetization component normal to the DW plane in the presence of the transverse field H = H(sin µH cos à H; sin µH sin à H , cos µH) when it is at a distance from the DW and sinµ± = ht = hsin µH: The polar angle µ may be approximated as # = 2 ¢tan¡ 1 e» , or [email protected]#[email protected] ¼ sin #, where 4 = 4(Q; h; µH ; à H ), and (Ht; Hp) = H (sin µH ; cos µH ), as defined above. The DW surface energy density can be obtained by integrating the volume energy density W along the direction normal to the DW with respect ' as a coordinate-independent variable as follows : °=

Z1

¡1

WDW dy =

¼Z¡ µ2 µ1

@y WDW d# = @#

¼Z¡ µ2 µ1

WDW

µ

@# @y

¶¡1

d#

n h A = [cos µ1 + cos µ2] + K 4 (cos µ1 + cos µ2) + " sin2 '(cos µ1 + cos µ2) 4 cscµ2 + cot µ2 i ¡ 2¹ sin '(¼ ¡ µ1 ¡ µ2) + u2 ln j j cscµ1 + cot µ1 h sin µ2 io ¡ 2ht cos(' ¡ ª H )(¼ ¡ µ1 ¡ µ2 ) + 2hp ln j j ; sin µ1

(16)

where µ1 = sin ¡ 1 (sin #± ¡ hp tan #± + h2p tan #± = cos #± ) and µ2 = sin¡ 1(sin #± + hp tan #± + h2p tan #± = cos #± ). Note that the boundary condition for the polar angle # is not symmetrical with respect to the easy axis in the presence of an external field with non-vanishing parallel field component, hp 6= 0. To satisfy the condition for the DW being in equilibrium we let @°[email protected] = 0, we then have ½ · ¸ cscµ +cot µ 2 2u sin '(¼ ¡ µ1 ¡ µ2) u ln j cscµ 21+cot µ21 j 2 4 =4± 1 + " sin ' ¡ + cos µ1 + cos µ2 cos µ1 + cos µ2 (17) µ2 ¾ ¡ 1=2 [2ht cos(' ¡ ª H)(¼ ¡ µ1 ¡ µ2) + 2hp ln j sin j] sin µ1 ¡ ; cos µ1 + cos µ2 p here 4± = A=K is the domain wall width in the absence of an applied field. Substituting Eq. (17) into Eq. (16), then gives ½ h p 2u(¼ ¡ µ1 ¡ µ2) sin ' ° =2 AK(cos µ1 + cos µ2 ) 1 + " sin2 ' ¡ cos µ1 + cos µ2 µ 2 ¾ 1=2 2 +cot µ 2 i u2 ln j cscµ [2ht (¼ ¡ µ1 ¡ µ2) cos(' ¡ ª H ) + 2hp ln j sin cscµ1 +cotµ 1 j sin µ 1 j] + ¡ ; cos µ1 + cos µ2 cos µ1 + cos µ2 (18)

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in which h = H=HK , " = 1=Q. For numerical purpose we obtain ' in the following form : ½ · ¸¾ (¼ ¡ µ1 ¡ µ2 ) ht ¡1 @°[email protected]' = 0 jà H =¼ =2 =)' = sin ¹+ : cos µ1 + cos µ2 " The magnetization azimuthal angle ' is an explicit function of the external field and its polar angle µH , ½ · ¸¾ (¼ ¡ µ1 ¡ µ2 ) Ht ¡1 '(H; µ H) = sin ¹+ ; (19) cos µ1 + cos µ2 4¼M where ¹ = sin #± sin à H , Ht = H sin µH . p By defining ° =  (y) ¢£ (µH ) and °± =  (y) ¢£ (µH = 0± ) = 4 AK and substituting the result into Eq. (14) we obtain the DW pinning field µ ¶ µ ¶ 1 d° HP (0 ± ) £ (µH ) HP (0± ) ° HP (µH ) = = = : (20) 2¹ 0 MS cos(µH ) dy cos(µH ) £ (0± ) cos(µH ) ° ± The method described above considers only the functional dependence of the azimuthal angle ' upon the applied field H and its polar angle µH , but not its coordinate dependence. To make up the deficiency, we shall adopt a more rigorous approach by treating both # and ' as coordinate-dependent. In place of the simple minded relation # = 2tan¡ 1 ey=4 we now adopt the polar angle #(») (see Eq. (21) below) to fit more closely to the numerical solution of the DW structure. By solving the Q-part of the DW Hamiltonian one obtains exactly the relation [14-17] sin #(») = sin #0 +

cos2 #0 : cosh w + sin #0

Thus, the polar angle of magnetization inside the DW varies as: · ¸ cos2 #0 ¡1 #(H; µH ; ») = sin sin #0 + ; cosh w + sin #0

(21)

here w = » cos #0

q

1 + " sin2 Ã H :

Equation (21) describes the variation of the polar angle of the magnetization inside the DW versus the reduced coordinate » in the so called quasi-Bloch wall with p a constant azimuthal angle of the magnetization ' = à H and the DW width ¢ = ¢ B [cos #0 1 + " sin2 à H ]¡ 1. To obtain a simulated analytic solution describing a '(y) dependent DW structure across the wall proper, we proceed to simulate the structure by a trial and error method guided by the numerical solution. We have ½ µ ¶ µ ¶¾ d d ¡1 ¡1 '(µH ; H; »; d) = à H + (0:6 ¡ ht) ¢ tan »¡ ¡ tan »+ ; (22) 4 4

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in which we take à H = ¼=2 and à H = ¼=4 which corresponds to a Ne´el wall and an intermittent Ne´el-Bloch wall, respectively, and ht = hsin(µH ); d is an adjustable parameter related to the DW width depending on à H , 4, ht and sµ 2 ¶ sin à H @#[email protected]» = 1+ (sin # ¡ sin #± ); (23) Q @'[email protected]» = 4(0:6 ¡ ht ) 4 »d=[(42 + » 2 + d2)2 ¡ 4»2d2 ]:

(24)

On substituting # and ' from Eqs. (21), (22) and their derivatives (23), (24) into (14), (15) we obtain HP (µH )=

1 d° 1 = WDW 2¹ ± M S COS(µH ) dy 2¹ ± MS COS(µH)

(25)

KW HP (0± )W = = ; 2¹ ± M S cos(µH ) cos(µH ) where

WDW = A[(@#[email protected])2 + (sin # ¢@'[email protected])2 ] + [K sin2 # + 2¼M 2(sin ' sin # ¡ u)2] ¡ [MHt sin # cos(' ¡ Ã H) + MH p cos #]

= Kf[(@#[email protected]»)2 + (sin # ¢@'[email protected]»)2] + [sin2 # + "(sin ' sin # ¡ u)2 ]

(26)

¡ [2ht sin # cos(' ¡ Ã H ) + 2 hp cos #]g = KW; with HP (0± ) = K=(2¹ ± MS ), ht = Ht =2K=MS , hp = Hp =2K=MS , 4B = W =[(@#[email protected]»)2 + (sin # ¢@'[email protected]»)2] + [sin2 # + "(sin ' sin # ¡ u)2 ] ¡ [2ht sin # cos(' ¡ Ã H ) + 2 hp cos #]:

p

A=K, and (27)

To proceed with numerical evaluation we input the following numerical values for barium ferrite [7]: K = 3.3 £ 105 (J/m3), the exchange constant J = 5(pJ/m)=5 £ 10 ¡ 12 (J/m) and the saturation magnetization MS = 500 emu/cm3 . III. Discussion Both the numerical and simulated DW structures obtained by solving the system of Eqs. (6) and (7) are shown in Figures 2 to 7. Note that the boundary conditions of the DW structures are very sensitive to the normalized applied magnetic field ht and the quality factor Q. The boundary values for the first derivative of µ 0 at à H = ¼=2 as » ! 1 can be adjusted as small as possible as the reduced field ht and/or the Q-value decreases. However, the boundary values for '0 show the opposite trend. This may have been connected with the DW instability or with chaos of the

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FIG. 2. Variation of the magnetization angle # and ' along the coordinate normal to the DW plane at à H = ¼ =2 for normalized field ht = 0.158.

FIG. 3. Variation of the magnetization angle ' along the coordinate normal to the DW plane at à H = ¼ =2 for normalized field h = 0.017, 0.158, 0.363.

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FIG. 4. Variation of the magnetization angle # and ' along the coordinate normal to the DW plane at à H = ¼ =4 for normalized field h t = 0.158.

FIG. 5. Variation of the magnetization angle ' along the coordinate normal to the DW plane at à H = ¼ =4 for normalized field h = 0.017, 0.158, 0.363.

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FIG. 6. Variation of the magnetization angle ' along the coordinate normal to the DW plane at à H = ¼ =2 for normalized field h = 0.01 ! 0.32 (from bottom to top).

FIG. 7. Variation of the magnetization angle ' along the coordinate normal to the DW plane at à H = ¼ =4 for normalized field h = 0.01 ! 0.6 (from bottom to top).

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DW motion [27, 28]. This is so since the diagram corresponding with the numerical solution becomes discontinuous and asymmetrical if inappropriate boundary values are used. Fig. 2 shows both the numerical and simulated analytic solutions of the DW at à H = ¼=2 with the transverse normalized magnetic field ht = 0.158. Fig. 3 shows the numerical structure of the azimuthal angle of the magnetization across the DW proper at à H = ¼=2 and at three different values of ht corresponding to that observed earlier [7]. It is necessary to note one distinctive feature of the solution for the dependence of the magnetization azimuthal angle upon the transverse coordinate normal to the DW. As shown in Fig. 3, the azimuthal angle '(») of the magnetization inside the DW becomes smaller than that inside the domain if the normalized field ht is smaller than a critical field. Such behavior can be understood from the point of view that, in order not to increase the total energy density of the DW, the demagnetizing field contribution to the total energy works to adjust the magnetization component normal to the DW to be as small as possible, therefore, the azimuthal angle '(») of the magnetization vector inside the DW can be adjusted as small as possible. Moreover the phase transition from a mixed Ne´el-Bloch wall to a Ne´el wall (' = à H = ¼=2) will occur when ht ¸ 0:32, as predicted [12]. Fig. 4 shows both the numerical and simulated analytic solutions of the DW at à H = ¼=4 with ht = 0.158. Fig. 5 shows the numerical structure of the azimuthal angle of the magnetization across the DW proper at à H = ¼=4 and at three different ht values. It is necessary to note one distinctive feature of the solution for the dependence of the magnetization azimuthal angle upon the transverse coordinate normal to the DW. No DW structure transition has taken place. Fig. 6 shows the variation of the magnetization angle ' along the coordinate normal to the DW plane at à H = ¼=2 for the reduced transverse field ht = 0:017 ! 0:32, which manifests the fact that the azimuthal angle '(») of magnetization inside the DW is equal to the field azimuthal angle (à H ) whenever ht ¸ 0:32, consistent with the result reported by Dimashko [12]. This means that the phase transition takes place at ht = 0:32 for Q = 2 provided à H = ¼=2. In the case where à H = ¼=4, no DW structure phase transition will ever take place even if ht becomes large. This is illustrated in Fig. 7 where ht is as large as 0.6. Fig. 8 shows that the volume energy density of DW is inversely proportional to the normalized field ht , that is, the smaller the normalized field, the larger the total DW energy density, hence, the smaller the azimuthal angle '(»). Figs. 9 and 10 show the two-dimensional graphs for the DW pinning field HP versus µH at à H = ¼=2 and ¼=4, respectively. We see that the newly obtained simulated graphs agree better with the experimental data than earlier reports [16-20]. Fig. 9 shows the peculiar behavior of the pinning field, with ' being treated as coordinate–independent at à H = ¼=2 and ht = 0:363, in which the strength of the pinning field (represented by dash lines) is shown to decrease down to zero at µH ' 63± . It indicates that the Stoner-Wohlfarth theory prevails when the grain size (of barium ferrite) is small (or down to ' 0.5 ¹m, as reported elsewhere [4]), due to the Bloch ! N¶eel wall phase transition which may have taken place (à H = ¼=2). This is consistent with the results heavily speculated by Ratnam et al. [4] and by Haneda et al. [5]. However, this peculiar feature is absent for a mixed or a pseudo Ne´el-Bloch wall (à H = ¼=4), as shown in Fig. 10. In addition, by comparing the graphs of Fig. 9 and 10, it indicates that the graph of Fig. 10 is much closer to the experimental data than that shown in Fig. 9. This is to say that the theory developed above is good for a Bloch wall, and remains applicable for a mixed Bloch-N´eel wall, but not for a pure Ne´el wall. It is significant that a similar statement of this nature has also been pointed out earlier by Schumacher [7].

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FIG. 8. Variation of volume DW energy (WDW ) vs. reduced coordinate normal to the DW for normalized field h = 0.017 ! 0.363 (from top to bottom) at à H = ¼ =2.

FIG. 9. Comparison of the calculated pinning fields vs. field polar angle µH with experimental data for barium ferrite [13] and other authors’ results [16-20]. Dash line: present result with ' as function of the normalized magnetic field h; solid line: present result with ' as function of the reduced transverse coordinate »; ‘+’ symbol: experimental data at reduced coordinate » = 0.42 and à H = ¼ =2.

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FIG. 10. Comparison of the calculated pinning fields vs. field polar angle µH with experimental data for barium ferrite [13] and other authors’ results [16-20]. Dash line: present result with ' as function of the normalized magnetic field h; solid line: present result with ' as function of the reduced transverse coordinate »; ‘+’ symbol: experimental data at reduced coordinate » = 0.2 and à H = ¼ =4.

Thus, Figs. 9 and 10 demonstrate clearly that our newly re-modified Kondorsky function fits excellently well with the measured switching field distribution unlite than earlier reports [3, 7, 13, 14], even at large field polar angle µH , when the appropriate dependence of the coordinate normal to the DW plane of the azimuthal of magnetization is fully taken into account. This statement is all the more true when a mixed Ne´el-Bloch wall configuration (for example, à H = ¼=4) is considered. Fig. 11 shows both the three-dimensional and two-dimensional graphs for the DW pinning field HP versus µH and » when à H = ¼=2, in which Figs. 11(a) and 11(b) show the threedimensional graphs for H P versus µH and » as viewed from different direction, and Figs. 11(c) and 11(d) show the two-dimensional graphs for HP versus µH and », respectively. Correspondingly, Fig. 12 shows the same pinning field variation as in Fig. 11 except for this case we now set à H = ¼=4: It is not a coincidence that the features in these two graphs are rather similar, with the exception that the HP in Fig. 11 are slightly larger. This may be ascribed to the Ne´el wall nature, as it should.

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FIG. 11. Three dimensional diagram of remodified Kondorsky function vs. field polar angle µ H and the reduced coordinate » for h = 0.158 and à H = ¼ =2.

FIG. 12. Three dimensional diagram of remodified Kondorsky function vs. field polar angle µH and the reduced coordinate » for h = 0.158 and à H = ¼ =4.

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IV. Conclusion From the discussions above the following conclusions can be reached. 0 (1) The boundary value of the first derivative of µ far away from the DW is proportional to the reduced field ht and the Q-value when the field azimuthal angle à H = ¼=2. However, the boundary values for '0 show the opposite trend. This may have been connected with the DW instability or chaos of the DW motion [27, 28]. (2) The azimuthal angle '(») of magnetization inside the DW may undergo a Bloch ! N e¶el phase transition at ht > » 0:32 whenever à H = ¼=2. The statement is no longer valid if à H 6= ¼=2. The phase transition of the azimuthal angle of magnetization inside the DW varying from a mixed Bloch-Ne´el wall (0 < '(») < ¼=2) to a N´eel wall ('(») = à H = ¼=2) is closely connected to the material Q-value, applied magnetic field H and field orientation angle (µH; à H ). (3) Presence of a transverse field may disfigure the Bloch wall structure such that its wall energy density becomes reduced with increasing applied magnetic field, hence the variation of the azimuthal angle of magnetization inside the DW, consistent with the earlier reports [12, 15, 16, 18]. (4) The simulated DW structural function for '(») is an appropriate function of the normalized transverse field ht , reduced coordinate », field orientation angles (µH ; à H ), Q-value and parameters of the DW width 4 and d. (5) The DW surface energy density ° can not be separated cleanly into two parts as in ° =  (y) ¢£ (µH ) as was assumed previously [7], since the magnetization angle (#; ') in the DW volume energy density is in fact intermixed, as shown in Eqs. (3)-(5), and the corresponding differential equations are nonlinear, as shown in Eqs. (6), (7). (6) Our newly re-modified Kondorsky function obtained from our simulated wall structure function for µ and ' show that our results agree better with the experimental data than ever before reported. Carefully note, however, that the re-modified Kondorsky function varies fairly sensitively also with the transverse coordinate », field ht ; quality factor Q-value and field orientation angle (µH ; à H ). The appropriately chosen value of these parameters will make the evaluation of the Kondorsky function edging closer to the experimental data, provided the DW is not a pure N´eel wall. (7) We obtained the three dimensional graphs of the pinning field as a function of the field polar angle µH and transverse coordinate » and as a function of the field orientation angles µH and à H . Our result shows evidence that the structure of the DW has to be taken into account in graphing the pinning field. (8) The present results are consistent also with the earlier report by Ratnam et al. [4] and by Haneda et al. [5], that the reverse-domain nucleation may greatly influence the intrinsic coercive force of barium ferrite particles, especially for low applied magnetic field, large particle with multidomain and, small field polar angle µH . Acknowledgement This research is supported by National Science Council through the grants: NSC-90-2216E-002-008, 90-2112-M-002-007.

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