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Equation of the domain wall motion in amorphous ribbon with helical magnetic anisotropy
LETTER TO THE EDITOR
Journal of Magnetism and Magnetic Materials 236 (2001) L9–L13
Letter to the Editor
Equation of the domain wall motion in amorphous ribbon with helical magnetic anisotropy D.N. Zhmetko* Faculty of Physics, Zaporizhzhe State University, 66 Zhukovsky Street, Zaporizhzhe 69063, Ukraine Received 24 January 2001; received in revised form 20 July 2001
Abstract The magnetic field of eddy currents corresponding to the Matteucci emf was included in the equation of the domainwall motion while magnetizing amorphous ribbons with helical magnetic anisotropy. The obtained equations determine the dependence of the damping coefficient br on the domain wall coordinate. A sharp maximum of br ðxÞ in the middle section of the domain wall run is discovered. r 2001 Elsevier Science B.V. All rights reserved. Keywords: Magnetic anisotropy; Sandwich structure; Matteucci EMF; Domain wall motion; Damping coefficient
1. Introduction Amorphous magnetic ribbons with helical anisotropy contain two domain walls parallel to the ribbon plane. This is called a sandwich domain structure. Magnetizing experiments in such ribbons are suitable for investigation of the domainwall motion parameters, such as the damping coefficient br : The domain walls in this structure keep a simple form during magnetization of the ribbon though eddy currents exist. Owing to this the equation of the domain wall motion under the action of a constant applied magnetic field ðHa Þ has simple analytic solutions [1]. The sandwich structure is also formed during high-frequency magnetization in an amorphous alloy ribbon with helical magnetic anisotropy induced by magnetic annealing [2]. The magneti*Tel.: +38-0612-644546; fax: +38-0612-627161. E-mail address: [email protected] (D.N. Zhmetko).
zation of the ribbon is accompanied by a Matteucci emf ðEM Þ between its ends. Matteucci emf together with emf of induction coil ðEB Þ allow to find the domain wall coordinate in the sandwich structure. The purpose of this work is to get the equation of the domain wall motion during high frequency magnetization in a ribbon with helical magnetic anisotropy taking into account the eddy current transverse magnetic field corresponding to EM : And the obtained equation is used for the determination of the damping coefficient br in the ribbon.
2. Theory The sandwich structure model accepted in the calculation is represented in Fig. 1. The eddy current 2I> between the domain walls, provided by the changing of the transverse component
0304-8853/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 0 4 8 5 - 1
LETTER TO THE EDITOR
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D.N. Zhmetko / Journal of Magnetism and Magnetic Materials 236 (2001) L9–L13
Fig. 1. Sandwich-structure model in an amorphous ribbon with helical magnetic anisotropy. The arrows on lateral surface indicate the direction of components of magnetization parallel to this surface; signs " and } indicate the direction of components of magnetization perpendicular to this surface.
magnetization, is branched out in two equal currents I> in the surface layers of the ribbon. The eddy current I8 ; due to the changing of the longitudinal component magnetization, flows only in the region between the ribbon surface and the domain wall [1]. While deriving the equation of the domain wall motion we neglect the domain wall thickness, the deflection of Ms from the easy magnetization axes near the moving domain wall and the contributions to EM and EB ; caused by the rotation of Ms : The relation for Matteucci emf and emf of induction coil are [2]: 4m Ms kL dx x sin y ; ð1Þ EM ¼ 0 d dt
where j is a current density running through the ribbon and HL is the longitudinal field of a solenoid in which the ribbon stayed during the magnetic annealing. On the other hand the Matteucci emf is the voltage drop along the surface layer of thickness ðd=2 xÞ (see Fig. 1): kL ; ð5Þ EM ¼ I> r ðd=2 xÞw
dx ; ð2Þ dt where L; w and d are the length, the width and the thickness of the ribbon, n ¼ 120 is the number of induction coil turns, x is the domain wall coordinate, y is the angle between the ribbon axis and the direction of Ms near of the domain wall, k is an empirical coefficient (see below). Eqs. (1) and (2) result in: EM Lx tan y: ð3Þ ¼ nwd EB
We find the relation for the eddy-current longitudinal field similarly [1]: I8 E8 ðd=2 xÞ EB : He:c:8 ¼ ¼ ¼ 2rwkn L Lr2w=ðd=2 xÞL ð7Þ
EB ¼ 4m0 Ms wkn cos y
If the induced magnetic anisotropy constant is sufficiently large (of order 10 J/m3), the easy magnetization axes practically coincide with the vectors H of the helical magnetic field applied during magnetic annealing [2]. Thus we accepted: jx tan y ¼ ; ð4Þ HL
where r is the resistivity. From Eq. (5) we have the relation for the eddy-current transverse magnetic field in the domain-wall plane: I> d=2 x EM : He:c:> ¼ ¼ ð6Þ rkL w
The local critical field, which characterizes the hysteresis friction, may be written in the form [3]: H08 H0 ðxÞ ¼ ; ð8Þ cos y where H08 is the critical field in the middle of the ribbon ðx ¼ 0Þ: Pressure on a domain wall caused by the hysteresis friction equals: PH0 ðxÞ ¼ 2m0 Ms H0 ðxÞcos y ¼ 2m0 Ms H08
ð9Þ
Taking into consideration the pressure on a domain wall of the external field Ha ; eddy currents fields He:c:8 and He:c:> and pressure caused by the hysteresis friction and neglect of the domain-wall
LETTER TO THE EDITOR
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mass, we obtain the equation of the domain-wall motion (see Ref. [4], p. 33) in the form: Ha 1 d x m0 M s 2 knrw 2 EB " #1=2 2 EM nwd 2 2 1þ x L EB 2m0 Ms H08 2m0 Ms d EM x EB rkL 2 EB " 2 #1=2 2 EB L 1þ x2 nwd EM " #1=2 2 br EM nwd 2 2 1þ x ¼ 0: L 4knwm0 Ms EB ð10Þ In Eq. (10) we have eliminated the derivative dx=dt and y with the help of Eqs. (2)–(4).
3. Experiments The measurements were performed on a ribbon of the amorphous alloy Co68Fe4Cr4Si13B11 (ls B107 ; Hc ¼ 1:2 A/m, m0 Ms ¼ 0:58 T, r ¼ 1:16 mO m, 22 mm thick, 6.6 mm wide, 0.81 m active length) with helical magnetic anisotropy, induced by the magnetic annealing in strong helical magnetic field (annealing regime: 2001C, 1 h, j ¼ 15:32 A/mm2, HL ¼ 85 A/m) [2]. The emf of field coil EH ðtÞ; EM ðtÞ and EB ðtÞ was measured by a broadband storage oscillograph. Ha ðtÞ and BðtÞ were found by graphical integration of EH ðtÞ and EB ðtÞ: The mutual position of the curves of EB ðtÞ and EM ðtÞ along the time axis was found by fitting the result of graphical addition of EB ðtÞ to EM ðtÞ to the measured curve ðEB þ EM ÞðtÞ [2]. And the same as far as EB ðtÞ and EH ðtÞ are concerned. Also xðtÞ for Eq. (10) was found with the help of the independent Eq. (3). The damping coefficient br ðxÞ was found from Eq. (10). The empirical coefficient kðtÞ in Eq. (1) (as well as in Eq. (2)) was determined by applying the trapezoid rule for the minimal time interval Dt ¼ 6:25 ms and a small interval Dx
corresponding to it: 1 ðEM1 þ EM2 ÞDt ¼ 2 0
1 x21
x22
4m0 MS j 1B C kL @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiþqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiADx: d 2 HL2 þ ðjx1 Þ2 HL2 þ ðjx2 Þ2 ð11Þ The values found for kðtÞ for Eqs. (1) and (2) are practically equal. We determined the critical field H08 from the condition: Z 2 d=2 H08 dx: ð12Þ Hc EhH0 ðxÞi ¼ d 0 cos y After integrating we found H08 E0:81 A/m.
4. Results and discussion The results of measurement of EB ðtÞ; EM ðtÞ; Ha ðtÞ and xðtÞ along the descending branch of the dynamic hysteresis loop (f ¼ 1 kHz, Hm E19 A/m, HcB E11 A/m) are represented in Fig. 2. In regions xo6 mm and x48 mm (domain walls are near of the middle of the ribbon and its surfaces respectively) the domain wall velocity is about 0.1 m/s. In region 6oxo8 mm, dx=dtE0:01 m/s. For the amorphous ribbons with helical magnetic anisotropy the dependence shown xðtÞ is the most typical. The dependences kðxÞ and br ðxÞ have similar form (see Fig. 3). In regions xo6 mm and x4 8 mm, kðxÞo1 and br ðxÞ is of order 102 kg/m2 s. In
Fig. 2. Emf of induction coil EB (curve a), applied field Ha (curve b), the domain wall coordinate (curve c; smoothed data) and Matteucci emf EM (curve d) versus time.
LETTER TO THE EDITOR
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Fig. 3. Damping coefficient br (curve a) and empirical coefficient k (curve b) (see the text) versus the domain wall coordinate x:
region 6oxo8 mm, kðxÞ > 1 and br ðxÞB103 kg/ m2 s. The kðxÞo1 values reflect the decrease of the effective length of the ribbon ðLeff ¼ kLÞ and of the effective number of induction coil turns ðneff ¼ knÞ in accordance with Eqs. (1) and (2). It is possible if we consider the step character of the domain wall motion during of the Barkhausen microjumps (see Fig. 4 in Ref. [4]). As a result, during the small time interval Dt the magnetizing length of the ribbon is less than its active length and the change of the magnetic flux is registered in the smaller number of induction coil turns evenly distributed along of the active length of the ribbon. kðxÞ > 1 means that EB and EM are greater than their values provided by the domain wall motion. EB and EM may be increased due to Bloch lines motion separating the domain-wall areas with opposite direction of the Ms rotation [4,5]. The motion of the Bloch lines may lead to the additional He:c:8 and He:c> increase and therefore br increase in region 6oxo8 mm. The chosen regime of magnetic annealing results in Ku B10 J/m3 [2]. Accepting an exchange constant Aexc ¼ 1 10212 J/m (this value is near to that accepted in Ref. [6] for an amorphous alloy of the similar composition), pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi we find the domain wall thickness dEp Aexc =Ku B1 mm. The polarization of the pair of domain walls of this thickness by Bloch-line motion may make a significant contribution to the magnetization of the thin ribbon. Suppose, in region 6oxo8 mm (where the domain wall velocity is small, see Fig. 2) the ribbon magnetization is realized by displacement of the domain walls as well as by their polarization.
It is possible that the direction of the small increment DB due to polarization is near to that due to a corresponding small displacement of the domain walls. This exclude appearance of high frequency oscillations (b1 kHz) of the ribbon magnetization during the change of the magnetizing mechanism and also makes it possible to determine the domain wall coordinate with the help of Eq. (3) during magnetization of the ribbon by polarization of the domain walls. Substitution of the values kðxÞ > 1 in Eqs. (6) and (7) by the value k ¼ 1 considers participation in creation of the eddy-current magnetic field by displacement of the domain walls as well as by their polarization. Estimation of this field components for x ¼ 7 mm gives: He:c:8 EHe:c:> E0:1 A/m. In the same time the applied field is Ha E10 A/m (see Fig. 2). Significant excess of Ha over He:c: ; created by domain walls displacement, leads to the strong correlation between kðxÞ and br ðxÞ (see Fig. 3). If we neglect the pressure of the fields He:c:8 ; He:c:> and H08 on the domain wall, Eq. (10) reduces to the following form: 2m0 Ms
Ha br ¼ 0: cos y EB 4knwm0 Ms cos y
ð13Þ
In region 6oxo8 mm, the greatest change is for the empirical coefficient k: The values of Ha ; EB and cos y in this region change considerably less (see Figs. 2 and 3). Therefore, in order to keep Eq. (13) valid, the k change has to correspond to the same br change. The eddy currents do not screen the applied field to any considerable extent. However additional eddy currents (corresponding to kðxÞ > 1), supposedly caused by polarization of the domain walls, have a significant influence on the mobility a ¼ 1=br (see [1]) of the latter. The domain wall low mobility region 6oxo8 mm belongs to the broad minimum of the R> ðxÞ function for the eddy current I> ; excited by emf E> : R> ¼
rlðx þ d=2Þ : 2wxðd=2 xÞ
ð14Þ
The component He:c:> ; provided by I> ; is not compensated by the applied field Ha because the latter doesn’t have a transverse component.
LETTER TO THE EDITOR
D.N. Zhmetko / Journal of Magnetism and Magnetic Materials 236 (2001) L9–L13
5. Conclusions The main equation derived by us (Eq. (10)) includes eddy-current fields appearing during the domain wall motion in a ribbon with inhomogeneous magnetic anisotropy and permits to evaluate the damping coefficient br and its dependence on the domain wall coordinate, from our experimental data. A sharp maximum of the damping coefficient br is discovered in the middle section of the domainwall run. In this section the domain wall velocity is less by an order of magnitude. In a magnetic-sandwich structure of thin amorphous ribbon at f ¼ 1 kHz the eddy currents do not considerably screen the applied field. However the domain-wall mobility is being influenced by eddy currents, supposedly caused by the domainwall polarization. The domain-wall mobility
L13
change during their run corresponds to the change of resistance of the ribbon layers in which the eddy currents, causing the transverse magnetic field, flow.
References [1] F.G. Friedlaender, C.H. Smith, SMP’95, Vienna, 1995, p. 3. [2] D.N. Zhmetko, P.V. Lemish, J. Magn. Magn. Mater. 196–197 (1999) 816. [3] S. Chikazumi, Physics of Magnetism, Wiley, New York, 1964, p. 267. [4] K. Stierstadt, Tracts Mod. Phys. 40 (1966) 2. [5] A. Hubert, Theorie der Domanenwande in Geordneten Medien, Moskau, 1977, p. 206 (in Russian). [6] M. Rodriguez, A. Garcia, M. Maicas, C. Aroca, E. Lopez, M.C. Sanchez, P. Sanchez, J. Magn. Magn. Mater. 133 (1994) 36.
Journal of Magnetism and Magnetic Materials 236 (2001) L9–L13
Letter to the Editor
Equation of the domain wall motion in amorphous ribbon with helical magnetic anisotropy D.N. Zhmetko* Faculty of Physics, Zaporizhzhe State University, 66 Zhukovsky Street, Zaporizhzhe 69063, Ukraine Received 24 January 2001; received in revised form 20 July 2001
Abstract The magnetic field of eddy currents corresponding to the Matteucci emf was included in the equation of the domainwall motion while magnetizing amorphous ribbons with helical magnetic anisotropy. The obtained equations determine the dependence of the damping coefficient br on the domain wall coordinate. A sharp maximum of br ðxÞ in the middle section of the domain wall run is discovered. r 2001 Elsevier Science B.V. All rights reserved. Keywords: Magnetic anisotropy; Sandwich structure; Matteucci EMF; Domain wall motion; Damping coefficient
1. Introduction Amorphous magnetic ribbons with helical anisotropy contain two domain walls parallel to the ribbon plane. This is called a sandwich domain structure. Magnetizing experiments in such ribbons are suitable for investigation of the domainwall motion parameters, such as the damping coefficient br : The domain walls in this structure keep a simple form during magnetization of the ribbon though eddy currents exist. Owing to this the equation of the domain wall motion under the action of a constant applied magnetic field ðHa Þ has simple analytic solutions [1]. The sandwich structure is also formed during high-frequency magnetization in an amorphous alloy ribbon with helical magnetic anisotropy induced by magnetic annealing [2]. The magneti*Tel.: +38-0612-644546; fax: +38-0612-627161. E-mail address: [email protected] (D.N. Zhmetko).
zation of the ribbon is accompanied by a Matteucci emf ðEM Þ between its ends. Matteucci emf together with emf of induction coil ðEB Þ allow to find the domain wall coordinate in the sandwich structure. The purpose of this work is to get the equation of the domain wall motion during high frequency magnetization in a ribbon with helical magnetic anisotropy taking into account the eddy current transverse magnetic field corresponding to EM : And the obtained equation is used for the determination of the damping coefficient br in the ribbon.
2. Theory The sandwich structure model accepted in the calculation is represented in Fig. 1. The eddy current 2I> between the domain walls, provided by the changing of the transverse component
0304-8853/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 0 4 8 5 - 1
LETTER TO THE EDITOR
L10
D.N. Zhmetko / Journal of Magnetism and Magnetic Materials 236 (2001) L9–L13
Fig. 1. Sandwich-structure model in an amorphous ribbon with helical magnetic anisotropy. The arrows on lateral surface indicate the direction of components of magnetization parallel to this surface; signs " and } indicate the direction of components of magnetization perpendicular to this surface.
magnetization, is branched out in two equal currents I> in the surface layers of the ribbon. The eddy current I8 ; due to the changing of the longitudinal component magnetization, flows only in the region between the ribbon surface and the domain wall [1]. While deriving the equation of the domain wall motion we neglect the domain wall thickness, the deflection of Ms from the easy magnetization axes near the moving domain wall and the contributions to EM and EB ; caused by the rotation of Ms : The relation for Matteucci emf and emf of induction coil are [2]: 4m Ms kL dx x sin y ; ð1Þ EM ¼ 0 d dt
where j is a current density running through the ribbon and HL is the longitudinal field of a solenoid in which the ribbon stayed during the magnetic annealing. On the other hand the Matteucci emf is the voltage drop along the surface layer of thickness ðd=2 xÞ (see Fig. 1): kL ; ð5Þ EM ¼ I> r ðd=2 xÞw
dx ; ð2Þ dt where L; w and d are the length, the width and the thickness of the ribbon, n ¼ 120 is the number of induction coil turns, x is the domain wall coordinate, y is the angle between the ribbon axis and the direction of Ms near of the domain wall, k is an empirical coefficient (see below). Eqs. (1) and (2) result in: EM Lx tan y: ð3Þ ¼ nwd EB
We find the relation for the eddy-current longitudinal field similarly [1]: I8 E8 ðd=2 xÞ EB : He:c:8 ¼ ¼ ¼ 2rwkn L Lr2w=ðd=2 xÞL ð7Þ
EB ¼ 4m0 Ms wkn cos y
If the induced magnetic anisotropy constant is sufficiently large (of order 10 J/m3), the easy magnetization axes practically coincide with the vectors H of the helical magnetic field applied during magnetic annealing [2]. Thus we accepted: jx tan y ¼ ; ð4Þ HL
where r is the resistivity. From Eq. (5) we have the relation for the eddy-current transverse magnetic field in the domain-wall plane: I> d=2 x EM : He:c:> ¼ ¼ ð6Þ rkL w
The local critical field, which characterizes the hysteresis friction, may be written in the form [3]: H08 H0 ðxÞ ¼ ; ð8Þ cos y where H08 is the critical field in the middle of the ribbon ðx ¼ 0Þ: Pressure on a domain wall caused by the hysteresis friction equals: PH0 ðxÞ ¼ 2m0 Ms H0 ðxÞcos y ¼ 2m0 Ms H08
ð9Þ
Taking into consideration the pressure on a domain wall of the external field Ha ; eddy currents fields He:c:8 and He:c:> and pressure caused by the hysteresis friction and neglect of the domain-wall
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mass, we obtain the equation of the domain-wall motion (see Ref. [4], p. 33) in the form: Ha 1 d x m0 M s 2 knrw 2 EB " #1=2 2 EM nwd 2 2 1þ x L EB 2m0 Ms H08 2m0 Ms d EM x EB rkL 2 EB " 2 #1=2 2 EB L 1þ x2 nwd EM " #1=2 2 br EM nwd 2 2 1þ x ¼ 0: L 4knwm0 Ms EB ð10Þ In Eq. (10) we have eliminated the derivative dx=dt and y with the help of Eqs. (2)–(4).
3. Experiments The measurements were performed on a ribbon of the amorphous alloy Co68Fe4Cr4Si13B11 (ls B107 ; Hc ¼ 1:2 A/m, m0 Ms ¼ 0:58 T, r ¼ 1:16 mO m, 22 mm thick, 6.6 mm wide, 0.81 m active length) with helical magnetic anisotropy, induced by the magnetic annealing in strong helical magnetic field (annealing regime: 2001C, 1 h, j ¼ 15:32 A/mm2, HL ¼ 85 A/m) [2]. The emf of field coil EH ðtÞ; EM ðtÞ and EB ðtÞ was measured by a broadband storage oscillograph. Ha ðtÞ and BðtÞ were found by graphical integration of EH ðtÞ and EB ðtÞ: The mutual position of the curves of EB ðtÞ and EM ðtÞ along the time axis was found by fitting the result of graphical addition of EB ðtÞ to EM ðtÞ to the measured curve ðEB þ EM ÞðtÞ [2]. And the same as far as EB ðtÞ and EH ðtÞ are concerned. Also xðtÞ for Eq. (10) was found with the help of the independent Eq. (3). The damping coefficient br ðxÞ was found from Eq. (10). The empirical coefficient kðtÞ in Eq. (1) (as well as in Eq. (2)) was determined by applying the trapezoid rule for the minimal time interval Dt ¼ 6:25 ms and a small interval Dx
corresponding to it: 1 ðEM1 þ EM2 ÞDt ¼ 2 0
1 x21
x22
4m0 MS j 1B C kL @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiþqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiADx: d 2 HL2 þ ðjx1 Þ2 HL2 þ ðjx2 Þ2 ð11Þ The values found for kðtÞ for Eqs. (1) and (2) are practically equal. We determined the critical field H08 from the condition: Z 2 d=2 H08 dx: ð12Þ Hc EhH0 ðxÞi ¼ d 0 cos y After integrating we found H08 E0:81 A/m.
4. Results and discussion The results of measurement of EB ðtÞ; EM ðtÞ; Ha ðtÞ and xðtÞ along the descending branch of the dynamic hysteresis loop (f ¼ 1 kHz, Hm E19 A/m, HcB E11 A/m) are represented in Fig. 2. In regions xo6 mm and x48 mm (domain walls are near of the middle of the ribbon and its surfaces respectively) the domain wall velocity is about 0.1 m/s. In region 6oxo8 mm, dx=dtE0:01 m/s. For the amorphous ribbons with helical magnetic anisotropy the dependence shown xðtÞ is the most typical. The dependences kðxÞ and br ðxÞ have similar form (see Fig. 3). In regions xo6 mm and x4 8 mm, kðxÞo1 and br ðxÞ is of order 102 kg/m2 s. In
Fig. 2. Emf of induction coil EB (curve a), applied field Ha (curve b), the domain wall coordinate (curve c; smoothed data) and Matteucci emf EM (curve d) versus time.
LETTER TO THE EDITOR
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Fig. 3. Damping coefficient br (curve a) and empirical coefficient k (curve b) (see the text) versus the domain wall coordinate x:
region 6oxo8 mm, kðxÞ > 1 and br ðxÞB103 kg/ m2 s. The kðxÞo1 values reflect the decrease of the effective length of the ribbon ðLeff ¼ kLÞ and of the effective number of induction coil turns ðneff ¼ knÞ in accordance with Eqs. (1) and (2). It is possible if we consider the step character of the domain wall motion during of the Barkhausen microjumps (see Fig. 4 in Ref. [4]). As a result, during the small time interval Dt the magnetizing length of the ribbon is less than its active length and the change of the magnetic flux is registered in the smaller number of induction coil turns evenly distributed along of the active length of the ribbon. kðxÞ > 1 means that EB and EM are greater than their values provided by the domain wall motion. EB and EM may be increased due to Bloch lines motion separating the domain-wall areas with opposite direction of the Ms rotation [4,5]. The motion of the Bloch lines may lead to the additional He:c:8 and He:c> increase and therefore br increase in region 6oxo8 mm. The chosen regime of magnetic annealing results in Ku B10 J/m3 [2]. Accepting an exchange constant Aexc ¼ 1 10212 J/m (this value is near to that accepted in Ref. [6] for an amorphous alloy of the similar composition), pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi we find the domain wall thickness dEp Aexc =Ku B1 mm. The polarization of the pair of domain walls of this thickness by Bloch-line motion may make a significant contribution to the magnetization of the thin ribbon. Suppose, in region 6oxo8 mm (where the domain wall velocity is small, see Fig. 2) the ribbon magnetization is realized by displacement of the domain walls as well as by their polarization.
It is possible that the direction of the small increment DB due to polarization is near to that due to a corresponding small displacement of the domain walls. This exclude appearance of high frequency oscillations (b1 kHz) of the ribbon magnetization during the change of the magnetizing mechanism and also makes it possible to determine the domain wall coordinate with the help of Eq. (3) during magnetization of the ribbon by polarization of the domain walls. Substitution of the values kðxÞ > 1 in Eqs. (6) and (7) by the value k ¼ 1 considers participation in creation of the eddy-current magnetic field by displacement of the domain walls as well as by their polarization. Estimation of this field components for x ¼ 7 mm gives: He:c:8 EHe:c:> E0:1 A/m. In the same time the applied field is Ha E10 A/m (see Fig. 2). Significant excess of Ha over He:c: ; created by domain walls displacement, leads to the strong correlation between kðxÞ and br ðxÞ (see Fig. 3). If we neglect the pressure of the fields He:c:8 ; He:c:> and H08 on the domain wall, Eq. (10) reduces to the following form: 2m0 Ms
Ha br ¼ 0: cos y EB 4knwm0 Ms cos y
ð13Þ
In region 6oxo8 mm, the greatest change is for the empirical coefficient k: The values of Ha ; EB and cos y in this region change considerably less (see Figs. 2 and 3). Therefore, in order to keep Eq. (13) valid, the k change has to correspond to the same br change. The eddy currents do not screen the applied field to any considerable extent. However additional eddy currents (corresponding to kðxÞ > 1), supposedly caused by polarization of the domain walls, have a significant influence on the mobility a ¼ 1=br (see [1]) of the latter. The domain wall low mobility region 6oxo8 mm belongs to the broad minimum of the R> ðxÞ function for the eddy current I> ; excited by emf E> : R> ¼
rlðx þ d=2Þ : 2wxðd=2 xÞ
ð14Þ
The component He:c:> ; provided by I> ; is not compensated by the applied field Ha because the latter doesn’t have a transverse component.
LETTER TO THE EDITOR
D.N. Zhmetko / Journal of Magnetism and Magnetic Materials 236 (2001) L9–L13
5. Conclusions The main equation derived by us (Eq. (10)) includes eddy-current fields appearing during the domain wall motion in a ribbon with inhomogeneous magnetic anisotropy and permits to evaluate the damping coefficient br and its dependence on the domain wall coordinate, from our experimental data. A sharp maximum of the damping coefficient br is discovered in the middle section of the domainwall run. In this section the domain wall velocity is less by an order of magnitude. In a magnetic-sandwich structure of thin amorphous ribbon at f ¼ 1 kHz the eddy currents do not considerably screen the applied field. However the domain-wall mobility is being influenced by eddy currents, supposedly caused by the domainwall polarization. The domain-wall mobility
L13
change during their run corresponds to the change of resistance of the ribbon layers in which the eddy currents, causing the transverse magnetic field, flow.
References [1] F.G. Friedlaender, C.H. Smith, SMP’95, Vienna, 1995, p. 3. [2] D.N. Zhmetko, P.V. Lemish, J. Magn. Magn. Mater. 196–197 (1999) 816. [3] S. Chikazumi, Physics of Magnetism, Wiley, New York, 1964, p. 267. [4] K. Stierstadt, Tracts Mod. Phys. 40 (1966) 2. [5] A. Hubert, Theorie der Domanenwande in Geordneten Medien, Moskau, 1977, p. 206 (in Russian). [6] M. Rodriguez, A. Garcia, M. Maicas, C. Aroca, E. Lopez, M.C. Sanchez, P. Sanchez, J. Magn. Magn. Mater. 133 (1994) 36.