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Dec 15, 2002 - CoFeSiB amorphous wire, supplied by Unitika Co, Japan. The sample was previously submitted to an in-air-current anneal of 350 mA for 30 ...
JOURNAL OF APPLIED PHYSICS

VOLUME 92, NUMBER 12

15 DECEMBER 2002

Influence of surface anisotropy on magnetoimpedance in wires L. G. C. Meloa) Groupe de Recherche en Physique et Technologie des Couches Minces, De´partment de Ge´nie Physique, E´cole Polytechnique de Montre´al, Case Postale 6079, Succursale Centre-Ville, Montre´al H3C 3A7, Canada

D. Me´nard Seagate Technology, 7801 Computer Avenue, Bloomington, Minnesota 55435

P. Ciureanu and A. Yelon Groupe de Recherche en Physique et Technologie des Couches Minces, De´partment de Ge´nie Physique, E´cole Polytechnique de Montre´al, Case Postale 6079, Succursale Centre-Ville, Montre´al H3C 3A7, Canada

共Received 8 July 2002; accepted 13 September 2002兲 The variation of the amplitude of the giant magnetoimpedance maxima for a magnetic cylindrical conductor in the range of 1⬍ f ⬍300 MHz has been investigated. Emphasis is put on the effect of the surface anisotropy, K s , which was neglected in previous studies. The calculation of the impedance of a perfect anisotropic, nonsaturated wire with a helical magnetic structure also includes exchange-conductivity effects and Landau–Lifshitz damping. The results are compared with experimental data on CoFeSiB amorphous wires. It is found that other factors must also be taken into consideration to describe the data. © 2002 American Institute of Physics. 关DOI: 10.1063/1.1519345兴

I. INTRODUCTION

volved. Theoretical results are discussed and compared with an experiment on CoFeSiB wires in Sec. V, followed by conclusions in Sec. VI.

Variation of the ac resistance and inductance of soft magnetic conductors with static magnetic field applied is known as magnetoimpedance 共MI兲. A mathematical model, based on solution of the Maxwell equations and the Landau– Lifshitz equation 共which includes the effect of exchange and conductivity兲, yields both the positions and intensities of the maxima in the MI at high frequency for saturated1 and unsaturated2 wires and unsaturated plates.3 In particular, it has been shown3 how exchange effects decrease the giant MI 共GMI兲 maxima for moderate frequencies 共below 100 MHz approximately兲. Including exchange effects in the calculation also enables us to add a phenomenological surface anisotropy through the boundary conditions on the magnetization vector at the surface of the sample. This is not possible when exchange is neglected.4 In previous MI calculations,1–3 free surface spins were assumed. At moderate frequency the calculated positions of the peaks are in agreement with experiment. However, the peak intensities 共which agree at high frequency兲 are not,2,3 as illustrated in Fig. 1. One possible explanation of this is that surface torque plays a significant role in the behavior observed. This article provides a natural extension of previous calculations,1–3 permitting such torque to be included, in the form of a surface magnetic anisotropy energy density,5 K s , along with the torque induced by exchange forces.6 In Sec. II, we develop an expression for the boundary condition on magnetization. In Sec. III, this is combined with the boundary conditions for fields to obtain a general expression for the impedance of a perfect wire. In Sec. IV, a simplified solution is found for the peak amplitude of the real part of the impedance. It yields some insights into the physics in-

II. TORQUE EQUATIONS

We use the approach adopted by Rado and Weertman5 共RW兲 for study of the ferromagnetic resonance response of magnetic metals. The Landau–Lifshitz 共LL兲 equation of motion for magnetization is

␮ 0L 1 ⳵M MÃ共 MÃHeff兲 , ⫽ ␮ 0 MÃHeff⫺ ␥ ⳵t Ms

where M s is the saturation magnetization, ␮ 0 is the permeability of free space, ␥ is the gyromagnetic ratio, L ⫽4 ␲ ␭/( ␥␮ 0 M s ) is the reduced damping parameter, and ␭ is the Landau–Lifshitz damping constant. The magnetization vector M⫽M s nM ⫹m(r,t) in Eq. 共1兲 is resolved into cylindrical coordinates, as shown in Fig. 2共a兲. The direction of static magnetization, nM , is assumed to be completely uniform, such that nM "nz ⫽cos ␪, and nM "n␸ ⫽sin ␪, and the dynamic component, m(r,t) 关Fig. 2共b兲兴 is such that m"nr ⫽m r , m"n␸ ⫽m ␸ and m"nz ⫽m z . The components of m, m r , m ␸ , and m z , are components of the cylindrical modes whose spatial dependencies are of the form m r, ␸ ⫽C r, ␸ J 1 (kr) and m z ⫽C z J 0 (kr), where k is the propagation constant, J are Bessel functions of the first kind, and C is a constant.1 The GMI measurements are performed at low microwave power 共the amplitude of the alternating current is 1 mA兲, so that we may consider 兩 m兩 ⰆM s and nM "m⬵0 关Fig. 2共b兲兴. The effective field, Heff , includes the static field applied parallel to the length of the wire, H0 ⫽H 0 nz , and the dynamic field, h(t), with its components h r ⫽ v r m r , h ␸ ⫽ v ␸ m ␸ , and h z ⫽ v z m z , where v r ⫽⫺1 and v ␸ ⫽ v z ⫽

a兲

Electronic mail: [email protected]

0021-8979/2002/92(12)/7272/9/$19.00

共1兲

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J. Appl. Phys., Vol. 92, No. 12, 15 December 2002

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FIG. 1. Comparison between theoretical and experimental values of the normalized real part of the impedance, R/R dc , as a function of the field applied, H 0 , at f ⫽1 and 10 MHz. The sample is a soft, low magnetostriction CoFeSiB wire of 125 ␮m diameter. We have used the following parameters in the calculation: ␪ ⫹ ␺ ⫽85°, H k ⫽160 A/m, A⫽5.0⫻10⫺12 J/m, ␴ ⫽0.763⫻106 S m⫺1 , ␥␮ 0 /(2 ␲ )⫽352⫻102 Hz/(A/m), ␭/(2 ␲ ) ⫽40 MHz, M s ⫽635 kA/m, and 2a⫽125 ␮ m. The free spin boundary condition was used.

⫺1/(1⫺ ␮ / ␮ 0 ) are the dynamical demagnetizing factors, and ␮ is the permeability. Heff also includes an effective anisotropy field, Hk ⫽(H k /M s ) (nk "M)nk , where Hk ⫽2K u /( ␮ 0 M s ), K u being the uniaxial anisotropy constant. nk is assumed perpendicular to nr 关Fig. 2共a兲兴. This anisotropy field accounts for average magnetoelastic and demagnetizing effects. The exchange field, which is given by 2 hex⫽d ex ⵜ 2 M,

共2兲

is produced by radial variation in the orientation of M. The exchange length is d ex⫽ 冑2A/( ␮ 0 M s2 ), where A is the exchange stiffness constant. Deviations in the direction of M arise from the skin effect, which produces inhomogeneous penetration of the electromagnetic radiation. The exchange torque density, Tex , that acts on M, determines the exchange boundary condition. Tex is found by including Eq. 共2兲 in Heff in Eq. 共1兲. The term containing the normalized relaxation parameter, L, of Eq. 共1兲 provides a negligible contribution to Tex for frequencies below 100 MHz, and will be neglected 共additional comments on this problem will be given in Sec. IV兲. Thus, Tex⫽ ␮ 0



r

0

MÃhexdr ⬘ ,

共3兲

which is produced by the radial variation of m 关Eq. 共2兲兴 at r ⬘ ⫽r. Substitution of Eq. 共2兲 in Eq. 共3兲 yields

FIG. 2. Cylindrical coordinate system for the problem. 共a兲 Unit static magnetization vector, nM , and unit anisotropy vector, nk , resolved in the ␸ – z plane. The direction of the static magnetic field applied, H 0 , and the easy axis angle, ␪ ⫹ ␺ , are also shown. 共b兲 Detail of the total magnetization vector formed by static magnetization plus the dynamic vector, m(t). We have used the small signal approximation, where 兩 m兩 ⰆM s and nM •m⬵0. In 共c兲 we show the unit surface anisotropy vector, ns , in space.

2 Tex⫽ ␮ 0 d ex MÃ





⳵m m . ⫹ ⳵r r

共4兲

Ne´el7 pointed out that the spins at the surface, which are in a different environment from those in the bulk, due to the lack of nearest-neighbor spin on one side, can produce surface anisotropy, represented by surface anisotropy energy, Es , E s ⫽K s 共 nM "ns 兲 2 ,

共5兲

where K s is the surface anisotropy energy constant, in J/m2, and ns is a unit vector, generally oriented in space as shown in Fig. 2共c兲. The anisotropy can partially or completely immobilize 共or pin兲 surface spins. Other mechanisms, e.g., the difference between surface and bulk magnetization 共induced by roughness, oxidation, the existence of a ferri- or antiferromagnetic surface layer, etc.兲, stress anisotropy, crystal anisotropy, etc., may lead to the same consequences. With the surface energy given by Eq. 共5兲, the surface torque, Ts , is

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J. Appl. Phys., Vol. 92, No. 12, 15 December 2002

⳵Es . ⳵M

Ts ⫽M⫻

共6兲

Following RW,5 Eqs. 共4兲 and 共6兲 yield the equilibrium condition for the total moment per unit area, Tex⫹Ts ⫽0. Thus, M⫻

冋 冉 2 d ex

共7兲



⳵m m 1 ⳵Es ⫹ ⫹ ⳵r r ␮0 ⳵M



r⫽a

⫽0.

共8兲

Equation 共8兲 shows that, in equilibrium, the torque is zero everywhere at r⫽a. The term in square brackets in Eq. 共8兲 can be interpreted as an effective surface field parallel to M. Let us assume that M lies in the ␸ – z plane. It may be shown8 that Eq. 共8兲 yields



⳵m␸ m␸ ⫹ ⳵r r



r⫽a



Ks 关 m ␸ cos共 ␪ ⫺ ␣ 兲 A

⫹M s cos ␪ sin共 ␪ ⫺ ␣ 兲兴 r⫽a cos ␤ ⫽0

共9兲

for the ␸ component, and



⳵mr mr ⫹ ⳵r r



r⫽a



Ks 关 m r cos共 ␪ ⫺ ␣ 兲 cos ␤ A

⫺M s sin ␤ 兴 r⫽a ⫽0

共10兲

for the r component of m. In Eqs. 共9兲 and 共10兲, the total torque is taken at the surface of a cylindrical conductor of radius a. Angles ␣ and ␤ are defined in Fig. 2共c兲 In order to proceed in the calculation of the impedance Z, we must provide an orientation for ns . It can be shown8 that only the boundary condition on m ␸ is relevant to the solution, regardless of the condition on m r . According to Eq. 共9兲, ns parallel to the surface normal ( ␤ ⫽ ␲ /2) has no effect on the condition on m␸ . Therefore, we will consider ns perpendicular to the surface normal, i.e., ␤ ⫽0 in Eqs. 共9兲 and 共10兲. According to Eq. 共5兲, if ns is taken to be in the ␸ – z plane, the energy shows a minimum for K s ⬍0. Furthermore, we choose ns parallel to the static magnetization ( ␣ ⫽ ␪ ) for simplicity. Equations 共9兲 and 共10兲 then become

冋冉



Ks ⳵ m ␸ ,r m ␸ ,r ⫺ m ␸ ,r ⫹ ⳵r r A



netic component perpendicular to the z direction. The wave propagating in the bulk is obtained from simultaneous solution of Eq. 共1兲 and Maxwell’s equations. In nonsaturated materials, the general solution consists of two separate equations, one for nonmagnetic mode and one for three magnetic modes. The magnetic modes are the solutions of a bicubic secular equation,6 K 6 ⫹c 1 K 4 ⫹c 2 K 2 ⫹c 3 ⫽0,

共12兲

in terms of the normalized wave vector, K, K⫽kd ex ,

共13兲

where k⫽(1⫺i)/ ␦ , i⫽ 冑⫺1. ␦ ⫽ 冑2/( ␻␴ ␮ ) is the magnetic skin depth, ␴ is the conductivity, and ␻ ⫽2 ␲ f . In Eq. 共12兲, the coefficients c 1 , c 2 , and c 3 are functions of the static field, frequency, and the material parameters2 关see Eq. 共27兲兴. The roots of Eq. 共12兲, K 1 , K 2 , and K 3 , define mixed spinwave and electromagnetic modes, which constitute the electromagnetic field inside the magnetic medium. These modes are sometimes referred to as the Larmor electromagnetic 共LE兲, Larmor spin 共LS兲, and anti-Larmor spin 共AS兲 waves.9 For applied fields lower than the anisotropy field, the angle ␪ will never be zero, and there will be a nonzero component of the ac magnetic field along the static magnetization. In this circumstance, the anti-Larmor electromagnetic 共AE兲 wave, which relates to K 4 , will contribute to the total signal. For a nonmagnetic wave, such as K 4 , Z is determined by satisfying the boundary condition on continuity of the tangential components of the electric and magnetic fields at the surface. In the presence of exchange effects, the secular equation 关Eq. 共12兲兴 is cubic in K 2 . Consequently, as pointed out by Ament and Rado,6 two additional conditions must be satisfied. Here, exchange and surface anisotropy torque, resulting in Eq. 共11兲, supply these. Therefore, we have the following complete set of boundary equations: 3



n⫽1

h n ␸ ⫽h 0 ␸ ,

共14兲

␰ nK nh n␸⫽ ␨ M h 0␸ ,

共15兲

u n 关 K n ␰ n ⫺ ␬ s 兴 h n ␸ ⫽0,

共16兲

v n 关 K n ␰ n ⫺ ␬ s 兴 h n ␸ ⫽0.

共17兲

r⫽a

⫽0.

共11兲

If a is very large, we may neglect the term m/r, and Eq. 共11兲 reduces to the general boundary condition for the magnetization of plates.5 The torque boundary conditions are combined with the usual boundary conditions for the electromagnetic fields, and the impedance of the wire is calculated in Sec. III. III. CALCULATION OF THE IMPEDANCE

Here, we present a calculation of the impedance through solutions of the LL equation and Maxwell equations. We consider the case in which the static magnetic field is applied in the axial z direction and the ferromagnetic wire is nonsaturated, with its magnetization in the ␸ – z plane 关Fig. 2共a兲兴. In other words, the magnetic structure is helical. The ac field of frequency f propagates in the radial direction with its mag-

3



n⫽1 3



n⫽1 3



n⫽1

The summation in Eqs. 共14兲–共17兲 is over the three magnetic solutions of the secular equation for the propagating modes, K n , n⫽1, 2, and 3. ␰ n are defined as ␰ n ⫽J 0 (K n ⌳)/J 1 (K n ⌳), where J 0 and J 1 are Bessel functions and ⌳⫽a/d ex is a nondimensional constant. h 0 ␸ is the circumferential component of the alternating magnetic field at the surface, and h n ␸ are the fields related to each magnetic

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Melo et al.

J. Appl. Phys., Vol. 92, No. 12, 15 December 2002

propagating mode. ␨ M is the normalized magnetic impedance. Equations 共14兲 and 共15兲 represent continuity of the magnetic and electric fields at the surface. They are unchanged from previous calculations.1,2 The procedure8 used to find normalized Eqs. 共16兲 and 共17兲 关derived from Eq. 共11兲兴 is similar to that discussed in Ref. 1. In Eqs. 共16兲 and 共17兲, ␬ s ⫽K s d ex /A is a dimensionless pinning parameter that represents the normalized surface anisotropy contribution to the total torque at the surface. The terms u n and v n are functions of K n , that involve combinations of material parameters. They are evaluated elsewhere.1,2 Thus, the nontrivial solution of the system of homogeneous Eqs. 共14兲–共17兲 yields an expression for ␨ M . We identify this solution as ‘‘exact’’ henceforth. This is combined10 with the normalized nonmagnetic impedance, ␨ 0 ⫽ ␰ 4 K 4 , in the impedance tensor, to form the anisotropic surface impedance, Z s . For wires, the bulk value of Z/R dc⫽a ␴ Z s /2, where R dc⫽l/( ␴␲ a 2 ) is the dc resistance of a sample of length l. For a given value of exchange, in the limiting case ␬ s ⰆK n ␰ n , Eqs. 共16兲 and 共17兲 yield the free spin condition adopted in earlier work for modeling GMI in cylinders1,2 关Eqs. 共16兲 and 共17兲 are equivalent to Eqs. 共64兲 and 共65兲 in Ref. 2兴, and plates3 共the latter without the m/r term, as discussed earlier兲. In contrast, when ␬ s ⰇK n ␰ n , Eqs. 共16兲 and 共17兲 yield 3



n⫽1

u n h n ␸ ⫽0,



v n h n ␸ ⫽0.

FIG. 3. Calculated values of R/R dc at 10 MHz as a function of normalized applied field, H 0 /H k , where H k ⫽160 A/m. Curves a, b, and c are obtained with the free spin boundary condition, and a⬘ , b⬘ , and c⬘ with full spin pinning at the surface. The easy axis direction, ␪ ⫹ ␺ , is 90° for a and a⬘ , 89° for b and b⬘ , and 85° for c and c⬘ .

共18兲

3

n⫽1

7275

共19兲

Equations 共18兲 and 共19兲 are the infinite surface pinning boundary condition, when ␬ s dominates and prevents the spins at the surface from moving. Results for the calculated normalized resistance, or real part of Z/R dc , R/R dc , for free spins as a function of the static field are compared with experiment in Fig. 1. The easy direction was taken to be ␪ ⫹ ␺ ⫽85°. The material parameters used are H k ⫽160 A/m, A⫽5.0⫻10⫺12 J/m, ␴ ⫽0.763⫻106 S m⫺1 , gyromagnetic ratio ␥␮ 0 /(2 ␲ )⫽352 ⫻102 Hz/(A/m), ␭/(2 ␲ )⫽40 MHz, M s ⫽635 kA/m, and 2a⫽125 ␮ m. These parameters should be appropriate for CoFeSiB.1 The experimental results correspond to the longitudinal GMI configuration, whose measurement is explained elsewhere.1 In Fig. 3, theoretical predictions for free 共curves a, b, and c兲 and pinned spins 共curves a⬘ , b⬘ , and c⬘ ,) are compared at 10 MHz. For curves a and a⬘ of Fig. 3, the easy axis direction was taken to be circumferential, i.e., nk •n␸ ⫽1. It is shifted 1° from n␸ for curves b and b⬘ and 5° for curves c and c⬘ . We emphasize that the model only accounts for coherent rotation of the magnetization based on Stoner– Wolfarth theory for a uniaxial anisotropy. No domain wall movement is considered. For a circumferential easy axis, as in curves a and a⬘ of Fig. 3, pinning the magnetization at the surface reduces the

impedance maximum to one half of its value without pinning. For easy directions other than circumferential, the effect is smaller 共curves b and b⬘ and c and c⬘ ). Curves a and b in Fig. 4 show the calculated values of R/R dc for free and pinned spins conditions at H 0 ⫽0 and H 0 ⫽H k , respectively, as a function of the easy axis angle, ␺ ⫹ ␪ , for f ⫽1 and 10 MHz. For H 0 ⫽0, the amplitude maxima occur at ␺ ⫹ ␪ ⫽0, that is, for an easy axis parallel to the length of the wire. The free and pinned models predict roughly the same amplitudes for every anisotropy direction. In contrast, for H 0 ⫽H k , the disparity between the free and pinned curves becomes substantial around ␺ ⫹ ␪ ⫽85° reaching a maximum at 90° for the frequencies shown. Figure 5 shows the variation of the amplitude of R/R dc at H 0 ⫽H k , f ⫽10 MHz, for a circumferential easy axis as a function of K s . The curve passing through the circles was calculated from Eqs. 共14兲–共17兲. 共The solid and dashed curves are discussed below.兲 Figure 5 demonstrates how the peak of curve a in Fig. 3 turns into the peak of a⬘ in Fig. 3 as K s varies from zero to 共minus兲 infinity. The curve starts at (R/R dc兲free⫽48.65, which is the free spin peak amplitude 共curve a in Fig. 3兲, and approaches the full pinning value of (R/R dc) pinned⫽26.39 共curve a⬘ in Fig. 3兲 asymptotically. The rapid decrease in amplitude of R/R dc shown in Fig. 5 suggests that a surface with K s larger than ⫺1.0⫻10⫺4 J/m2 共⫺0.1 erg/cm2兲 will barely be distinguishable from a fully spin pinned surface. The values of K s shown are comparable with those determined from ferromagnetic resonance 共FMR兲 experiments on monocrystalline Ni–Co plates.11 For small positive values of K s 共not shown兲, R/R dc is larger than

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FIG. 5. Amplitude of R/R dc as a function of the surface anisotropy energy at H 0 ⫽H k ⫽160 A/m and f ⫽10 MHz for a circumferential easy axis. The curves vary monotonically from the free to the pinned limit. Note that these limits correspond to the peak values of curves a and a⬘ in Fig. 3. The curve passing through the circles is the exact solution. The solid line is given by the approximate Eq. 共29兲. Note that this is only slightly different from the exact solution. The dashed curve represents Eq. 共29兲 when L⫽0, i.e., Eq. 共31兲.

In the following, we derive an analytical expression for the R/R dc maxima, the resonance peaks (R/R dc) max , through a simplified solution for the impedance. As a consequence, the role of parameters such A, K s , E, L, etc. may be easily visualized, permitting us to draw some general conclusions. IV. SIMPLIFIED SOLUTION AND RESONANCE PEAKS

FIG. 4. Variation of R/R dc as a function of the easy axis direction, ␺ ⫹ ␪ , with ␪ ⫽0 for free and pinned spins. In 共a兲 the field applied is zero. In 共b兲 the magnitude of the field applied is the anisotropy field, H k ⫽160 A/m. Calculations are shown for f ⫽1 and 10 MHz.

(R/R dc) free, which is obviously nonphysical. From Eq. 共11兲 we see that positive finite K s causes the spins to be ‘‘more free’’ than those described by this equation without the surface anisotropy term.

We have seen the effect of full spin pinning on the peak value of the real part of the impedance at a particular frequency when the field applied is swept. For the frequencies shown in Fig. 1, the peak is close to the anisotropy field of the sample. Here, we develop a simplified solution for the impedance. We examine the behavior at resonance, which allows us to comprehend the role of physical parameters, A, ␭, M s , f, etc., which is not necessarily obvious from the exact calculation in Sec. III. In what follows, we derive a simplified solution for the normalized circumferential permeability, ␮⬜ / ␮ 0 , or the impedance, Z/R dc . This solution, in good agreement with the exact solution, permits us to find an analytical expression for the peak amplitude. It has been shown2,5 that the AS wave which precesses in the direction opposite to natural uniform rotation contributes very little to the final solution. If the highest order term of Eq. 共12兲, K 6 , is neglected, we keep only the LE and LS waves. The 4⫻4 determinant of the exact solution from Eqs. 共14兲–共17兲 reduces to the 3⫻3 determinant: 2



n⫽1

h n ␸ ⫽h 0 ␸ ,

共20兲

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J. Appl. Phys., Vol. 92, No. 12, 15 December 2002 2



n⫽1

␰ n K n h n ␸ ⫽ ␨ ⬘M h 0 ␸ ,

共21兲

u n 关 K n ␰ n ⫺ ␬ s 兴 h n ␸ ⫽0,

共22兲

2



n⫽1

from which the simplified impedance, ␨ ⬘M , may be calculated, giving

␨ ⬘M ⫽

K 1 K 2 关 i 共 K 1 ⫹K 2 兲 ⫺ ␬ s 兴 共 K 21 ⫹K 22 ⫹K 1 K 2 兲 ⫹i ␬ s 共 K 1 ⫹K 2 兲

␬ s ⫹ 冑c 2 (1⫹2 冑c 1 c 3 /c 2 ) 1/2 ␬ s (1⫹2 冑c 1 c 3 /c 2 ) 1/2⫹ 冑c 2 (1⫹ 冑c 1 c 3 /c 2 )

.

共25兲

Hence, we find the approximate result for the permeability,

␮0





冉 冊 R

R dc

s

⫽Re max



⌳ 2

冑2i⍀E





2

,

共26兲

共28兲

2 ⫽(H 0 ⫹M s )(H 0 ⫺H k )/M s2 when the magnetizawhere ⍀ res tion is saturated (cos ␪⫽1). Since the amplitude of the real part of the impedance, R/R dc , is proportional to the square root of the effective permeability, Eq. 共26兲, the maximum amplitude of R/R dc , (R/R dc) max , is proportional to the square root of Eq. 共26兲 at resonance, ⍀ res⫽⍀, yielding

冑i⍀L⫹2 冑2i⍀E⫹ ␬ s i⍀L⫹ 冑2i⍀E⫹ ␬ s 冑i⍀L⫹2 冑2i⍀E

where Re兵 其 stands for the real part of the complex expression in curly brackets. Figure 6 shows (R/R dc) max as a function of frequency for the exact, ␨ M , and the simplified, ␨ ⬘M , impedance, when ␬ s ⫽0 and ␬ s →⬁. Equation 共29兲 is plotted as the solid line in Fig. 5. We note that this is indistinguishable from the exact solution of Sec. III. In Fig. 6共a兲, Landau– Lifshitz damping, which is proportional to the normalized parameter, L, is ␭/(2 ␲ )⫽40 MHz. In Fig. 6共b兲, ␭⫽0. In the range of frequency in which the discrepancy between theory

共 R/R dc兲 max⫽

for c in Eq. 共25兲. This holds under moderate field and frequency conditions. Equation 共26兲 provides a simplified expression for the transverse 共or circumferential兲 permeability, ␮⬜ . When ␬ s ⫽0, Eqs. 共25兲 and 共26兲 become, as expected, the free spin result of Ref. 2, Eqs. 共92兲 and 共93兲. If exchange fields are neglected, that is, if E⫽0 in Eq. 共26兲, the permeability becomes independent of ␬ s , ␮ / ␮ 0 ⫽1/⌶, which reduces to the typical resonant form 1 ␮ ⫽ 2 , ␮ 0 ⍀ res⫺⍀ 2 ⫹i⍀L

where K 2 ⫽⫺c 3 /c 2 , the value for the LE wave, and

冑⌶⫹2 冑2i⍀E ⌶⫹ 冑2i⍀E⫹ ␬ 冑⌶⫹2 冑2i⍀E

共27兲

c 3 ⬵2i⍀E,

共24兲

␬ s⫹

c 1 ⬵1,

共23兲

.

Z K⌳ ⫽ ␰ F s 共 c 1 ,c 2 ,c 3 兲 , R dc 2



where ⌶⫽(␩⫹1)␩⬘⫺⍀2⫹i⍀L, ␩ ⫽(H 0 cos ␪⫹Hk cos2 ␺)/ M s , and ␩ ⬘ ⫽(H 0 cos ␪⫹Hk cos 2␺)/M s . Also, ⍀⫽ ␻ / ( ␥␮ 0 M s ), E⫽(d ex / ␦ 0 ) 2 /⍀, and ␦ 0 ⫽ 冑2/( ␻␴ ␮ 0 ). We have used approximations2

c 2 ⬵⌶,

The ratio of Bessel functions, ␰ n in Eqs. 共21兲 and 共22兲, was taken to be equal to i in Eq. 共23兲. This holds when the skin depth related to each magnetic mode (K 1 and K 2 ) is small compared to the radius of the wire. By writing the roots K 1 and K 2 of the reduced, biquadratic, secular equation 关Eq. 共12兲 without the term K 6 ], in terms of coefficients c 1 , c 2 , and c 3 , Eq. 共23兲 yields

F s⫽

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冊冎

共29兲

,

and experiment is relatively large 共up to 100 MHz兲, we note that the overall effect of L is very small. Therefore, if L is set to zero in Eq. 共26兲 at resonance, one finds

冉 冊 冉冑 ␮⬜ ␮0



res

␬ s ⫹ 4冑8i⍀E 2i⍀E⫹ ␬ s 4冑8i⍀E



2

.

共30兲

Moreover, from Eq. 共29兲, the real part of (Z/R dc) max is

关 2 冑⍀E/2⫹ ␬ s2 兴 cos共 ␲ /8兲 ⫺ ␬ s 关 1⫹2 cos共 ␲ /4兲兴 ⌳4 冑⍀E/2 4 . 2 共 冑⍀E/2⫹ ␬ s 兲 2 ⫺4 ␬ s 4冑⍀E/2 cos2 共 ␲ /16兲

共31兲

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J. Appl. Phys., Vol. 92, No. 12, 15 December 2002

FIG. 7. Exact calculation of R/R dc as a function of the field. The solid line is obtained from the pinned model, with an exchange constant of A⫽5.0 ⫻10⫺12 J/m. The dotted line is obtained from the free spin model, when the exchange constant is 16A. The easy axis is circumferential.

Equation 共31兲 is plotted as the dashed lines as a function of K s at f ⫽10 MHz in Fig. 5. If ␬ s ⫽0, Eq. 共31兲 becomes free ⫽cos共 ␲ /8兲 ⌳ 4冑⍀E/2, 共 R/R dc兲 max

共32兲

which is the free spin prediction. 关Note that the factor 4 in the fourth root of Eq. 共95兲 of Ref. 2 should be replaced by the factor 2, in agreement with Eq. 共32兲兴. Alternatively, if ␬ s →⬁, Eq. 共31兲 reduces to pinned ⫽ 共 R/R dc兲 max

FIG. 6. Amplitude of the resonance peaks, (R/R dc) max as a function of the resonance frequency. The exact and simplified solutions are compared for the cases of free and pinned spins. In 共a兲 we assume LL damping, ␭/(2 ␲ ) ⫽40 MHz. In 共b兲 ␭⫽0. The horizontal lines in 共a兲 and 共b兲 are the result of LL damping alone, without exchange-conductivity effects.

cos共 ␲ /8兲 ⌳ 4 冑⍀E/2, 2

共33兲

which predicts a peak amplitude half of that predicted by Eq. 共32兲. We evaluate the validity of these approximations in what follows. We substitute for E and ⌳ in Eqs. 共32兲 and 共33兲 by the quantities which define them. The resulting equations suggest that the free result should be equivalent to the pinned result, provided an effective stiffness constant is defined, such as A pinned⫽16A free . This conclusion is used to compare the approximate solution, Eq. 共24兲, with the exact solution of Sec. III. Figure 7 shows the exact solution for R/R dc as a function of the field applied at 1 and 10 MHz. The solid line is obtained from the pinned model, with an exchange constant, A⫽5.0⫻10⫺12 J/m. The dotted line is the result for free spins, with A⫽80⫻10⫺12 J/m. The excellent agreement between the two for this range of frequencies and fields shows that an increased effective stiffness constant gives behavior equivalent to that of high surface anisotropy. In Sec. V, we give an overall discussion of the results obtained. The model will be further compared with experiment.

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Melo et al.

J. Appl. Phys., Vol. 92, No. 12, 15 December 2002

FIG. 8. Comparison between theory and experiment for the normalized real part of the impedance, R/R dc , as a function of the field applied, H 0 , at f ⫽1 and 10 MHz. The sample is a soft, low magnetostriction CoFeSiB wire of 125 ␮m diameter. We have used the following parameters in the calculation: ␪ ⫹ ␺ ⫽85°, H k ⫽160 A/m, A⫽5.0⫻10⫺12 J/m, ␴ ⫽0.763 ⫻106 S m⫺1 , ␥␮ 0 /(2 ␲ )⫽352⫻102 Hz/(A/m), ␭/(2 ␲ )⫽40 MHz, M s ⫽635 kA/m, and 2a⫽125 ␮ m. The solid line is the result for the free spin boundary condition. The dotted line is the result for pinned spins.

V. DISCUSSION

We have shown that surface spin pinning reduces the peak amplitude of R at moderate frequencies. The maximum effect, when the dynamic magnetization at the surface is zero 关Eqs. 共18兲 and 共19兲兴, decreases the peak amplitude by a factor of 2 for a circumferential easy axis, compared to a surface where the spins are free to move 共Fig. 3, curves a and a ⬘ ). This factor is lower than 2 for other easy directions 共curves b, b⬘ , c, and c⬘ ), and is negligible for easy directions at angles smaller than 85° 共Fig. 4, curve b兲. Figure 8 reproduces the curves shown in Fig. 1 for free spins, plus giving the results for complete pinning, with the same parameters, and compares it to the experimental results obtained from a 125 ␮m slightly negative magnetostriction CoFeSiB amorphous wire, supplied by Unitika Co, Japan. The sample was previously submitted to an in-air-current anneal of 350 mA for 30 min,12 in an attempt to induce an average circumferential easy axis. In Fig. 8, the easy direction was taken to be ␪ ⫹ ␺ ⫽85° for the free and pinned calculation. It is clear that the reduction of the peak amplitude obtained from the pinning boundary conditions is not sufficient to fit the experimental data. We have also measured the MI behavior of a 35 ␮m diam CoFeSiBNb amorphous wire, manufactured by MXT Inc. of Montre´al, and we observed the same discrepancy between the model and experiment. This suggests that the di-

7279

ameter of the wire 共or equivalently, the thickness of a plate兲, does not affect the disagreement. In addition to the surface anisotropy discussed in this work, anisotropy dispersion has been suggested as the physical mechanism responsible for the discrepancy between theory and experiment.3,13,14 As shown previously,3 a slight variation of the easy direction from circumferential decreases the peak by a significant factor; cf. Fig. 3, curves a and b. Dispersion of the anisotropy field direction has been modeled for the case of plates14 by treating the dispersion as a distribution in the plane of the solid. The total impedance is given by linear superposition of the distributed impedance from each volume element of the sample, thus neglecting mutual interactions. However, it is likely13 that the interactions lead to more important reductions than predicted by the noninteracting model. Figure 4, despite the fact that the easy axis angle is considered to be uniform, gives us a general idea of how such dispersions can be significant in the GMI response. We believe that a complete theoretical model must incorporate mutual interaction mechanisms. How we can do this is far from clear. The irregular local environments, which may be induced by random internal stress, fluctuations in the magnetostriction constant, ␭ s 共which can cause ␭ s to be zero only as a spatial average兲, fluctuations in local order, etc., may lead to dispersion of the anisotropy. This dispersion produces fluctuations of the magnetization, that is, magnetization ripple. Since it is very difficult to completely remove stress and to induce 共e.g., by annealing兲 a definite easy magnetization axis15 on soft magnetic materials, dispersion of the anisotropy or the magnetization can be present even in low anisotropy, low magnetostriction samples. An alternative approach to take into account the anisotropy dispersion can be pursued by following the core–shell 共C–S兲 magnetic structure of negative magnetostrictive wires.16 The C–S system can be represented by two regions, defined by two uniaxial easy directions, with their magnetizations coupled and separated by a dispersion angle. Because the magnetization of the inner core is mainly bistable,17 we can calculate, for a given coupling energy, the magnetization of the outer shell as a function of the field applied, and therefore incorporate it into the MI model. It is expected that coupling could produce a slow approach to saturation, and therefore degradation of the amplitude of the GMI maxima. After this structure-related effect has been taken into account, we can further include in the model a Bloch– Bloembergen damping term, which is effective at low frequencies.3,10,13 We intend to pursue such an approach. VI. CONCLUSION

The extension of the model shown here, which takes into account the anisotropy energy, K s , present at the surface of wires, permits us to reduce the peak amplitude of the real part of the impedance from the previous model. The general boundary conditions on magnetization set in Sec. II reduce the discrepancy between model and experiment. The solution for the impedance of the previous model is a limiting case of the solution shown here for K s ⫽0. We believe that surface

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anisotropy must be incorporated into a more complete model, even though it is not sufficient to explain the origin of the discrepancy. We believe that magnetization ripple, which results from irregular variation of the easy direction, provoked by stray, random distributed anisotropies, is the principal cause of the signal degradation observed at low frequency and field. Our model supposes an ideal Stoner– Wolfarth uniaxial, single domain wire, without any ripple. Therefore, discrepancies are expected. An alternative approach to take into account dispersion of the anisotropy in the theory is proposed. A simplified solution, in excellent agreement with the exact solution in the range of low frequency and field applied, has shown that full pinning of the spins at the surface is equivalent to a surface without pinning, providing the exchange interaction is 16 times higher, showing exchange and anisotropy produce similar effects. The value of the exchange stiffness constant must be accurately known if one wishes to determinate the surface anisotropy constant from a GMI experiment. ACKNOWLEDGMENT

One of the authors 共L.M.兲 holds a fellowship from CAPES-Brası´lia 共Brazil兲.

1

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