Single domain wall dynamics in ferromagnetic lamination with variable conductivity

Physica B 407 (2012) 3992–3995

Contents lists available at SciVerse ScienceDirect

Physica B journal homepage: www.elsevier.com/locate/physb

Single domain wall dynamics in ferromagnetic lamination with variable conductivity J. Kravcˇa´k n Department of Physics, Faculty of Electrical Engineering and Informatics, Technical University of Koˇsice, Park Komenske´ho 2, 042 00 Koˇsice, Slovakia

a r t i c l e i n f o

a b s t r a c t

Article history: Received 8 June 2012 Accepted 19 June 2012 Available online 26 June 2012

The influence of variable conductivity and thickness of two outer non-ferromagnetic layers on magnetization reversal of one central ferromagnetic layer is theoretically investigated. The model of a thin rigid 1801 domain wall moving transversely through the axially magnetized ferromagnetic layer is used to calculate induced eddy currents in lamination from which the domain wall mobility is determined. The effect of asymmetric distribution of eddy currents around moving domain wall results in acceleration of the wall near the edge of the lamination. The known domain wall mobility in ferromagnetic lamination can then be used to determine either the conductivity or the thickness of deposited outer non-ferromagnetic layers as proposed in discussion. & 2012 Elsevier B.V. All rights reserved.

Keywords: Domain wall mobility Eddy currents Ferromagnetic lamination Magnetization reversal Maxwell equations

1. Introduction The model of magnetization reversal by means of movement of a rigid 1801 domain wall with negligible thickness is frequently used due to possibility of obtaining an analytical expression for induced eddy current density [1–6]. The eddy current density for a single plane domain wall moving in a rectangular ferromagnetic bar was studied by Williams et al. [1]. Based on this work an extended model was developed consisting of a rectangular bar with ferromagnetic central layer and two outer conducting non-ferromagnetic layers (see Fig. 1) [6]. It was shown that the relative thickness of non-ferromagnetic layers with respect to the thickness of the ferromagnetic layer plays a very important role in calculations of domain wall velocity and power losses. The movement of a single domain wall has also been studied by means of Helmholtz equation (diffusion equation) for eddy current field [4,5], i.e. there is a finite time delay between the wall displacement and the establishment of eddy currents. Providing that the domain wall velocity is sufficiently small, in the present paper: 1. a quasi-static approximation is introduced and the displacement currents are neglected, 2. the influence of skin effect on eddy current density is also neglected, considering that the permeability within domains is m0 , and the set of differential equations for eddy current density is derived from Maxwell equations in Section 3. The micromagnetic calculations show the internal 1801 domain wall structure with a larger wall thickness on the rectangular sample surface than in the interior of the sample. This domain wall structure must be taken into account in the case of a thin sample (r 8 mm)

n

Tel.: þ421 55 6022818; fax: þ421 55 6330115. E-mail address: [email protected]

0921-4526/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physb.2012.06.030

with a low value of magnetic anisotropy in order to avoid stray fields produced by 1801 domain wall on the sample surface [8]. The influence of domain wall non-zero thickness on eddy current losses produced during the wall movement through the sample is very small in the case of strong axial magnetic anisotropy [7]. For this reason the approximation of the negligible domain wall thickness used in this paper is valid considering the real very long rectangular ferromagnetic sample (strip) with strong uniaxial magnetic anisotropy and thickness at least 20 mm described in Section 2. The very long sample is used for minimization of the demagnetization effect. Nevertheless the total domain wall energy increase with the sample length must be taken into account, and if some critical sample length is exceeded then head-to-head domain wall type known in thin ferromagnetic strips [9] has lower energy than the 1801 domain wall proposed in the model.

2. Description of model The following calculations are connected with the extended model shown in Fig. 1. Let us consider a rectangular sample (strip) placed in cartesian coordinates with dimensions x A ½L1 ,L2 , yA ½h=2,h=2, and very long in the z direction. Let the sample contain three conducting layers, two outer layers with the conductivity g2 are not ferromagnetic (NFL) and the central layer with the conductivity g1 is ferromagnetic (FML) with strong axial magnetic anisotropy in z direction and uniform magnetization Ms parallel with z-axis. If a constant external magnetic field of strength H is antiparallel with respect to the z-axis, magnetization reversal of FML occurs by the movement of a single rigid 1801 domain wall (DW) in the x direction from the left to the right sample edge as displayed in Fig. 1. Let the

´k / Physica B 407 (2012) 3992–3995 J. Kravcˇa

h d

NFL

γ1

z

v

x

FML

γ2

NFL L1

L2

Fig. 1. The scheme of the sample, containing three conducting layers very long in the z direction, two outer layers with the conductivity g2 is not ferromagnetic and a central layer with the conductivity g1 is ferromagnetic, where the magnetization reversal occurs by means of DW movement with velocity v. (Symbols  and  assign magnetization orientation antiparallel and parallel with z-axis.)

origin of the coordinate system be placed in the center of the moving DW (x¼0). It should be pointed out that L1 and L2 are variable distances of the DW from left and right sample edge respectively. Regardless of this variability, the sample width L ¼ L1 þ L2 is constant. The FML thickness is denoted d. Such movement of the single rigid 1801 domain wall in the FML induces eddy current flow described by ! current density vector j ðx,yÞ circulating around domain wall !! ! dM ðrot j Þz ¼ g1 m0 ¼ g1 m0 2M s vd ð1Þ dt z

for x¼0 and y A ½d=2,d=2. Maxwell’s equations produce a set of partial differential ! equations valid for j ðx,yÞ in all volumes (NFL and FML), where x a 0 or y= 2½d=2,d=2: !

!

!

r2 j ðx,yÞ ¼ 0, rot j ðx,yÞ ¼ 0, div j ðx,yÞ ¼ 0

ð2Þ

with two boundary conditions: the first one is Eq. (1), because the FML is separated into two magnetic domains by the moving DW, ! and the second one is jn ¼ 0, the normal component of j ðx,yÞ on all outer surfaces is equal to zero. The method of separation of variables yields the solution of equations (2) for the above ! described model in the form j ðx,yÞ ¼ ½jx ðx,yÞ,jy ðx,yÞ:    1 hnp i X np h ðL1;2 xÞ  cos y Dn sinh jx ðx,yÞ ¼ 2 h h n¼1 jy ðx,yÞ ¼

   hnp i np h ðL1;2 xÞ  sin y Dn cosh 2 h h n¼1 1 X

ð3Þ

where L1 is valid for x r0 and L2 is for x Z 0 [12]. Since the ! following derivative with respect to time dM =dt is non-zero only for the FML area containing the moving DW, the value of coefficients Dn in Eq. (3) is given by electrical conductivity g1 : np npd sin 4g1 m0 Ms v sin 2h 2   Dn ¼ 8 npL1;2 np cosh h for n¼ 1,3,5y, where the sign þ is only valid for jy ðx,yÞ with L1 and x r 0.

3. Calculation of eddy current power loss and domain wall mobility

Let the DW be impelled (accelerated) by a constant external field of strength H, which is antiparallel with the z-axis and damped (decelerated) by an eddy current field of strength He acting on the DW. If the inertia of DW is negligible, the equilibrium He ¼ H is established almost instantly and the eddy current loss per unit length of DW is equal to the rate P ¼ 2m0 Ms Hvd, the velocity of the DW v is directly proportional to P ¼ P 1 þP 2 . The indicated integration (4) provides the eddy current damping which has viscous character: v ¼ SH, where the constant of proportionality S is the DW mobility, and P ¼ SH2 .

4. Results and discussion 4.1. Domain wall mobility The calculated dependence of DW mobility S on the DW position L1 for ratio g2 =g1 ¼ 5:88 corresponding to copper NFL is displayed in Fig. 2. After the constant external field H is switched on, the DW starts its motion from the left edge with maximum mobility S. In the range x rd the DW rapidly decelerates until it reaches the middle part of the FML, when its mobility S is of minimum value. A wide interval of constant S is observed in the middle part of the FML. Close to the right edge, the DW rapidly accelerates to the maximum S. The theoretical explanation is that during its movement in the middle part of the FML, the DW creates a symmetrical vortex of eddy currents on both of its sides, and this results in maximal damping effect of the eddy currents [6]. Near the FML edge the part of this vortex closer to the edge is asymmetrically compressed, causing the eddy current damping effect to be weakened and acceleration of the DW takes place. Further Fig. 2 shows the decrease in S with the increasing relative thickness ðhdÞ=d of deposited conducting NFL with respect to the constant thickness d of the FML. This can be explained by the stronger eddy current damping effect due to the enlarged conducting area. Since the eddy current power loss P is directly proportional to DW mobility in the constant external field H, the above-mentioned behavior of S is the same as in the case of P. 4.2. Thickness and conductivity of deposited layer The main idea of this paper is to investigate the influence of a deposited conducting cover (NFL) on DW mobility. During the

1 0.9 0.8

S ((h−d)/d)

y γ2

3993

0.7

d=2 H=1

0.6 0.5 0.4

The eddy current power loss per unit length of DW in the z-direction P consists of two parts, P1 and P2, corresponding to FML and NFL respectively Z Z 1 L2 d=2 2 2 P1 ¼ ðjx þjy Þ dy dx

g1

P2 ¼

1

g2

L1

Z

L2 L1

d=2

"Z

d=2

h=2

!2 j dy þ

Z

h=2

d=2

# !2 j dy dx

ð4Þ

0.3 0.2

0

2

4

6

8

10 L1

12

14

16

18

20

Fig. 2. The calculated dependence of mobility S on the DW position L1 for ratio g2 =g1 ¼ 5:88 corresponds to copper NFL and variable ðhdÞ=d. The sample without copper NFL corresponds to ðhdÞ=d ¼ 0.

´k / Physica B 407 (2012) 3992–3995 J. Kravcˇa

3994

Table 1 The relative ratios g2 =g1 for commonly used conducting materials with respect to conductivity of a typical iron-based ferromagnetic material (g1  107 =O m) [10], recalculated from Ref. [11].

6.21 5.88 4.55 3.65 2.78 1.69 1.14 1.07 1.00 0.96 0.91 0.48 0.23 0.10 0.086 0.0025

A1 width

2 D2

width

A2

time

Fig. 4. The calculated shape of the Barkhausen impulse induced by accelerated DW during magnetization reversal.

recording the induced Barkhausen impulse ui(t). The calculated shape of the Barkhausen impulse in the model is characterized by its width and amplitude, see Fig. 4. The DW mobility is directly proportional to the amplitude of the induced Barkhausen impulse. The impulse width is given by the condition that the time integral of all induced impulses is constant 2m0 M s dðL1 þ L2 Þ. If the inductance of the coil winding is close to zero, the above predicted DW acceleration could be detected in the recorded impulse shape. In Fig. 4 the deceleration or acceleration in DW motion can be revealed in time intervals D1,2 and A1,2 respectively.

1 0.9 0.8 S((h−d)/d) / S(0)

1 D1

amplitude

Silver Copper Gold Aluminium Calcium Zinc Indium Lithium Iron Platinum Tin (b) Lead Titanium Mercury Bismuth Graphite

amplitude

g2 =g1

ui

Material

0.7 0.6 0.5 0.4 0.3

5. Conclusion

0.2

The presented theoretical results can be summarized as ! follows: (1) the obtained solution of j ðx,yÞ and the equilibrium condition He ¼ H produces the acceleration of the DW near the sample edge, which can be observed as sharp peaks in the recorded Barkhausen impulse on condition that the relaxation time proportional to the pickup coil inductance is smaller than the time interval of the sharp peaks in the impulse. (2) The regular decrease in DW mobility with increasing thickness of the deposited conductive layer can be used for sensitive measurement of the layer thickness. The theoretical sensitivity of this measurement is the highest in the interval 0:1 r g2 =g1 r 100, where the inflexion point of S dependence can be found. Practically at room temperature the maximum g2 =g1 ¼ 6:21 is in the case of silver NFL. The extended interval 6:21 r g2 =g1 r 100 in the model reflects the FML conductivity at lower temperatures. (3) If the ratio ðhdÞ=d is known and kept constant, then the variable conductivity g2 of the thin NFL can be measured by means of DW mobility. In many other physical models the variable conductivity g2 reflects for instance the change of temperature, chemical composition, charge injection.

0.1 0 0.1

1

10

100

1000

γ2 / γ 1 Fig. 3. The calculated dependence of mobility S on relative ratios g2 =g1 for variable ðhdÞ=d. The sample without NFL corresponds to ðhdÞ=d ¼ 0.

deposition of various conducting layers (e.g. silver, copper, tin), the relative ratio of conductivities g2 =g1 is a decisive parameter in eddy current power loss calculations (4). Table 1 gives the list of relative ratios g2 =g1 for commonly used conducting materials with respect to the conductivity of typical iron-based ferromagnetic materials (g1  107 =O m). If we know the ratio g2 =g1 , we are dealing with the possibility of estimating the thickness of the conductive layer during its deposition symmetrically on both sides of the ferromagnetic layer, Fig. 3. The task can be reversed, if the relative thickness of deposited layers ðhdÞ=d is known and kept constant (Fig. 3) during the described magnetization reversal, then the measured DW mobility is dependent on variable conductivity g2 affected for instance by temperature, chemical composition, charge injection. 4.3. Induced Barkhausen impulse From the experimental point of view, when a pickup coil winding is placed around the lamination, we are able to determine the DW mobility during magnetization reversal by

Acknowledgments The ‘‘We support research activities in Slovakia’’ project is cofinanced from EU funds. This paper was developed as part of the project named ‘‘Centre of Excellence for Integrated Research & Exploitation of Advanced Materials and Technologies in Automotive Electronics’’, ITMS 26220120055.

´k / Physica B 407 (2012) 3992–3995 J. Kravcˇa

3995

Integrals P1 and P2 (4) introduced in Section 3 have been calculated analytically from Eq. (A.1) with either m¼n or m a n.

Appendix A A.1. Eddy current power loss integration !2 The squared current density j ¼ j2x þ j2y consists of the following terms: 1 X 1 hmp i hnp i X ðL1;2 xÞ sinh ðL1;2 xÞ j2x ðx,yÞ ¼ Dm Dn  sinh h h m¼1n¼1       mp h np h cos ðA:1Þ y y cos 2 2 h h j2y ðx,yÞ ¼

hmp i hnp i ðL1;2 xÞ cosh ðL1;2 xÞ h h m¼1n¼1       mp h np h sin y y sin 2 2 h h 1 X 1 X

Dm Dn  cosh

where L1 is valid for x r 0 and L2 is for x Z0 mp np mpd npd sin sin sin K sin 2h 2h 2 2     Dm Dn ¼ mpL1;2 npL1;2 2 mnp cosh cosh h h for m,n ¼ 1; 3,5 . . . and K ¼ 16g21 m20 M 2s v2 .

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

H.J. Williams, W. Shockley, C. Kittel, Phys. Rev. 80 (1950) 1090. R.H. Pry, C.P. Bean, J. Appl. Phys. 29 (1958) 532. P.D. Agarwal, L. Rabins, J. Appl. Phys. 31 (1960) 246S. F. Colaiori, G. Durin, S. Zapperi, Phys. Rev. B 76 (2007) 224416. A. Roman, J. Magn. Magn. Mater. 74 (1988) 359. J. Kravcˇa´k, Acta Phys. Slovaca 56 (2006) 153. R.P. del Real, J. Magn. Magn. Mater. 303 (2006) 160. W. Rave, A. Hubert, J. Magn. Magn. Mater. 184 (1998) 179. R.D. McMichael, M.J. Donahue, IEEE Trans. Magn. 33 (1997) 4167. R.S. Tebble, D.J. Craik, Magnetic Materials, Wiley-Interscience, London, 1969. C. Kittel, Introduction to Solid State Physics (Translation), Academia, Praha, 1985. ˇ ´ , Acta Phys. Pol. 118 (2010) 734. J. Kravcˇa´k, V. Suhajova