- Email: [email protected]
Determination of domain wall surface energy by high-frequency wall mobility measurements in amorphous ribbons
~_~
ELSEVIER
jR
journalof magnetism , J R and magnetic
Journal of Magnetism and Magnetic Materials 133 (1994) 104-106
,~
materials
Determination of domain wall surface energy by high-frequency wall mobility measurements in amorphous ribbons A. Maraner a, C. Beatrice
b,*,
p. Mazzetti a
a Phys. Dept, Politecnico of Torino, GNSM and INFM, 1-10129 Torino, Italy h Istituto Elettrotecnico Nazionale G. Ferraris, c.so m. D'Azeglio 42, 1-10129 Torino, Italy
Abstract
The aim of this paper is to determine the 180 ° domain wall surface energy density ~w by measuring the phase shift between wall oscillation at the lamination surface and the applied magnetic field. This shift occurs when the wall bows during its motion. It can be also determined theoretically by integrating the partial differential equation which describes the wall dynamics, crw can then be determined as a best-fit parameter of the experimental data. Results are given for the amorphous ribbon Metglas 2605 SC from Allied Chemicals, which shows the effect of a mechanical tension on ~rw, A comparison of experimental and the theoretical data provides an estimate for ~rw in agreement with the well known theoretical formula.
1. Introduction
In several types of ferromagnetic laminations the magnetization dynamics is determined by a structure of longitudinal, antiparallel domains separated by 180 ° Bloch walls. When the distance between walls is large with respect to the lamination thickness the system dynamics can be described in terms of the behavior of a single domain wall (DW). In several models [1], the DW is described as an infinitely thin plane with only one degree of freedom (corresponding to a rigid translation) moving within wells and over barriers of a potential determined by the DW surface energy density crw or by the stray fields at the material imperfections and subjected to the damping force of the eddy-current field induced by the moving wall itself. If the wall velocity is high enough, the eddy-current term dominates the motion [2]. Any-
* Corresponding author. Tel: +39 (11) 348 8933; fax: +39 (11) 650 7611; e-mail: [email protected].
way, the DW surface energy density plays an important role in determining the wall mobility. It has indeed been long recognized [3,4] that under the action of a nonuniform field pressure, as in the case of eddy-current field, the wall bows and the assumed curvature depends on the magnitude of ~rw, resulting from the equilibrium between internal surface tension and field pressure. It is important to point out that under a sinusoidal external field, each point of the bowed wall oscillates with a phase that depends on its position. By integrating the partial differential equation [5] that describes the DW dynamics, it is possible to evaluate the phase shift ebbs between the wall oscillation at the lamination surface and the oscillation within the bulk, as well as the phase shift &Hs between the surface wall oscillation and the applied field. These phase shifts appear to be highly sensitive to variations in the magnitude of ~rw. Magneto-optic measurements of DW displacement were performed on ribbons of Metglas 2605 SC (Allied Chemicals), annealed for residual stress relief, and subjected to tensile stress during measurements. The aim of the present work is to determine the magnitude of ~rw in these amorphous ribbons from
0304-8853/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)00053-T
A. Maraner et al. /Journal of Magnetism and Magnetic Materials 133 (1994) 104-106 phase shift measurements and by comparing the experimental and calculated data. It will be emphasized how, in a magnetostrictive amorphous lamination, the DW surface energy density o-w turns out to be predominantly of magnetostrictive origin.
105
ture and thus on the phase of the surface wall oscillation.
3. Experimental
2. Theoretical model Let us assume an infinitely thin elastic DW separating two antiparallel 180° domains in a lamination of thickness d, with saturation magnetic induction JsMoreover, let the surface energy density o-w of the elastic wall be equal to that of a rigid DW. In the absence of pinning and of demagnetizing fields, the forces acting on the moving wall are generated by the sinusoidal applied magnetic field H(t) and by the counteracting eddy-current field Hec(x, t). Eq. (2) below describes the average balance of these fields, but the local balance of the generated forces is obtained through wall bowing in Eq. (1), where the left side represents the force arising from curvature and the right side represents the field pressure:
y"(x, t) trw
(1 + y'2(x, t)) ~/~
f ( H(t)
+ H~(x,
=2Js(H(t)+Hec(x, t)),
t))dx=O.
(1)
(2)
In Eq. (1) y(x, t) is the wall profile, where x is the distance from the middle of the lamination along an axis perpendicular to the lamination surface, and t is the time. The y-axis is parallel to the longitudinal direction of the wall motion. The double and the simple primes on y indicate partial derivatives with respect to x. Eqs. (1) and (2) can be solved numerically by an iterative technique described in Ref. [6]. The numerical solution y(x, t) is a sinusoidal function with the same frequency u of H(t) and phase depending on x, u and o-,,. It is therefore possible to calculate the phase shifts b u s and ~bas, respectively, between the surface wall oscillation and the sinusoidal applied external field H(t), and between the surface wall oscillation and the mean wall oscillation within the bulk. The curves representing these quantities versus frequency are strongly dependent on o-w in an appropriate frequency range, where the behavior of the wall due to the bowing differs significantly from the rigid wall behavior. We would like to stress that up to now the theoretical curves have been calculated by neglecting the effect of wall pinning at the lamination surface, although in fact, a slight excess of wall pinning at the lamination surface has a large effect on the wall curva-
The measurements were performed on amorphous strips of 2605 SC (length 20 cm, width 5 cm, thickness 30 ~m). The samples were annealed for 2 h at 300°C in order to release frozen-in stresses. This amorphous alloy exhibits a saturation magnetostriction As = 30-35 10 -6, and a saturation induction Js = 1.55 T. The displacement of a portion of a single wall was measured through the Kerr effect using an optical microscope ( × 100) and a PIN photodiode coupled to a preamplifier as detector. The bandwidth of the system is 300 kHz and the wall displacement resolution is 0.2 Ixm. After calibration of the transfer function of this system by means of a light source modulated by a Pockels cell, the accuracy of the phase measurements was found to be + 2°. Let us stress that in these materials annealed for simple stress relief there are no negligible contributions from rotations to the magnetization process. Therefore, since the induced signal is not univocally related to the bulk mean displacement of the DW, the phase shift ~bBs becomes barely measurable. On the other hand, the measure of 4~HS implies the determination of the effective applied field and cannot be performed by simply measuring the drive current, since the solenoid is interrupted in a small region owing to the optical measurements. This fact g!ves rise to a non-uniform applied field at high frequencies with the consequence of local demagnetizing fields in the measure region. For this reason the effective field was measured using a small magnetic tensiometer 0.1 mm thick, placed in contact with the lamination surface just behind the region where optical measurements were performed. The very low magnetic cross-section of the tensiometer allows us to make reliable measurements of the phase of the effective field only above a given frequency, which, in the case of the present sample, was 50 kHz.
4. Results and discussion
Fig. 1 presents the raw data of the phase shift ~bRs between the surface wall oscillation and the effective field versus frequency, for two values of applied stress, t r = 180 and 270 MPa. It should be noted that the possible range of variation of tr is limited by the need to obtain a relatively straight wall in the longitudinal direction (lower value) and by necessity not to exceed
106
A. Maraner et al. /Journal of Magnetism and Magnetic Materials 133 (1994) 104-106
6o'
I a~=.0010 J/m e 2 o'w=.0015 J/m e • 270 MPa o 180 MPa
1 . 5 . 1 0 3 j / m 2, for ~ = 180 a n d 270 MPa, respectively. W e now consider the expression of the energy density ~rw for a rigid wall:
0 '
~w = 4~;~( X 0 + ~ A s ~ ) ,
70 '
(Ji)HS
8o'
i'O0 frequency
1000 (Kttz)
Fig. 1. Frequency dependence of the phase shift t~SH between the surface wall oscillation and the effective field for an amorphous ribbon of Metglas 2605SC. The experimental data points correspond to an applied tension of 180 MPa (open circles) and 270 MPa (filled circles). Full lines are theoretical and represent the best fit of the experimental data by using trw as an adjustable parameter. The obtained values are ~ w = l × l 0 3 and 1.5×10 -3 J / m 2, which can be compared with the values obtained from Eq. (3): 0.9-1.1X10 -3 and 1.1-1.35 X 10 -3 J / m 2 for o-= 180 and 270 MPa, respectively.
t h e s t r e n g t h limit of the m a t e r i a l ( u p p e r value). T h e large scattering f o u n d in the actual m e a s u r e m e n t s comes from the p e r t u r b i n g effect of the local d e m a g n e tizing fields, which are r e l a t e d to a slightly irregular antiparallel d o m a i n p a t t e r n . T h e s e fields are in p h a s e with the wall m o v e m e n t a n d thus o u t of p h a s e with the applied field. T h e r e f o r e they p r o d u c e an u n c e r t a i n t y in the p h a s e of t h e effective field acting o n t h e wall. T h e r e p o r t e d d a t a are sufficient to show t h a t an effect of o n crw actually exists. F r o m a best fit of e x p e r i m e n t a l data (full lines) using crw as a n adjustable p a r a m e t e r in the theory, o n e obtains the values ~w = 1 × 10 -3 a n d
(3)
w h e r e K 0 is the uniaxial anisotropy coefficient r e l a t e d to a residual directional o r d e r i n h e r e n t in the amorp h o u s r i b b o n or p r o d u c e d by field annealing; AS is the m a g n e t o s t r i c t i o n coefficient, a n d A is the exchange 3 energy coefficient. In o u r case, since the t e r m 5As~r is of the o r d e r of 10 4 J / m 3, it is possible to neglect K0, which is at least o n e o r d e r of m a g n e t u d e lower [7]. A s s u m i n g A = 5 - 8 x 10 -12 J m i [8], we could expect for % values in the r a n g e s 0 . % 1 . 1 x 10 -3 a n d 1.11.35 × 10 3 j / m 2, for ~r = 180 a n d 270 MPa, respectively. T h e s e values are in good a g r e e m e n t with those o b t a i n e d from the e x p e r i m e n t a l results r e p o r t e d in Fig. 1. Acknowledgements. W e would like to t h a n k S. Rocco for helpful c o n t r i b u t i o n in the d e v e l o p m e n t of the e x p e r i m e n t a l part.
References [1] E. Herpin, Theorie du Magnetisms (Presses Universitaires de France, 1968) ch. 33. [2] H.J. Williams, W. Shockley and C. Kittel, J. Appl. Phys. 80 (1950) 1090. [3] K.M. Polivanov, Izvest. Akad. Nauk. SSSR (Set. Fiz.) 16 (1952) 449. [4] J.E.L., Bishop, J. Phys. D: Appl. Phys. 6 (1973) 97. [5] A. Maraner, C. Beatrice and P. Mazzetti, J. Appl. Phys. 75 (1994) to appear. [6] A. Maraner, C. Beatrice and P. Mazzetti, IEEE Trans. Magn. (1994) to appear. [7] C. Appino, F. Fiorillo and A. Maraner, IEEE Trans. Magn. (1994) to appear. [8] H. Kronmueller and B. Groeger, J. Physique 42 (1981) 83.
ELSEVIER
jR
journalof magnetism , J R and magnetic
Journal of Magnetism and Magnetic Materials 133 (1994) 104-106
,~
materials
Determination of domain wall surface energy by high-frequency wall mobility measurements in amorphous ribbons A. Maraner a, C. Beatrice
b,*,
p. Mazzetti a
a Phys. Dept, Politecnico of Torino, GNSM and INFM, 1-10129 Torino, Italy h Istituto Elettrotecnico Nazionale G. Ferraris, c.so m. D'Azeglio 42, 1-10129 Torino, Italy
Abstract
The aim of this paper is to determine the 180 ° domain wall surface energy density ~w by measuring the phase shift between wall oscillation at the lamination surface and the applied magnetic field. This shift occurs when the wall bows during its motion. It can be also determined theoretically by integrating the partial differential equation which describes the wall dynamics, crw can then be determined as a best-fit parameter of the experimental data. Results are given for the amorphous ribbon Metglas 2605 SC from Allied Chemicals, which shows the effect of a mechanical tension on ~rw, A comparison of experimental and the theoretical data provides an estimate for ~rw in agreement with the well known theoretical formula.
1. Introduction
In several types of ferromagnetic laminations the magnetization dynamics is determined by a structure of longitudinal, antiparallel domains separated by 180 ° Bloch walls. When the distance between walls is large with respect to the lamination thickness the system dynamics can be described in terms of the behavior of a single domain wall (DW). In several models [1], the DW is described as an infinitely thin plane with only one degree of freedom (corresponding to a rigid translation) moving within wells and over barriers of a potential determined by the DW surface energy density crw or by the stray fields at the material imperfections and subjected to the damping force of the eddy-current field induced by the moving wall itself. If the wall velocity is high enough, the eddy-current term dominates the motion [2]. Any-
* Corresponding author. Tel: +39 (11) 348 8933; fax: +39 (11) 650 7611; e-mail: [email protected].
way, the DW surface energy density plays an important role in determining the wall mobility. It has indeed been long recognized [3,4] that under the action of a nonuniform field pressure, as in the case of eddy-current field, the wall bows and the assumed curvature depends on the magnitude of ~rw, resulting from the equilibrium between internal surface tension and field pressure. It is important to point out that under a sinusoidal external field, each point of the bowed wall oscillates with a phase that depends on its position. By integrating the partial differential equation [5] that describes the DW dynamics, it is possible to evaluate the phase shift ebbs between the wall oscillation at the lamination surface and the oscillation within the bulk, as well as the phase shift &Hs between the surface wall oscillation and the applied field. These phase shifts appear to be highly sensitive to variations in the magnitude of ~rw. Magneto-optic measurements of DW displacement were performed on ribbons of Metglas 2605 SC (Allied Chemicals), annealed for residual stress relief, and subjected to tensile stress during measurements. The aim of the present work is to determine the magnitude of ~rw in these amorphous ribbons from
0304-8853/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)00053-T
A. Maraner et al. /Journal of Magnetism and Magnetic Materials 133 (1994) 104-106 phase shift measurements and by comparing the experimental and calculated data. It will be emphasized how, in a magnetostrictive amorphous lamination, the DW surface energy density o-w turns out to be predominantly of magnetostrictive origin.
105
ture and thus on the phase of the surface wall oscillation.
3. Experimental
2. Theoretical model Let us assume an infinitely thin elastic DW separating two antiparallel 180° domains in a lamination of thickness d, with saturation magnetic induction JsMoreover, let the surface energy density o-w of the elastic wall be equal to that of a rigid DW. In the absence of pinning and of demagnetizing fields, the forces acting on the moving wall are generated by the sinusoidal applied magnetic field H(t) and by the counteracting eddy-current field Hec(x, t). Eq. (2) below describes the average balance of these fields, but the local balance of the generated forces is obtained through wall bowing in Eq. (1), where the left side represents the force arising from curvature and the right side represents the field pressure:
y"(x, t) trw
(1 + y'2(x, t)) ~/~
f ( H(t)
+ H~(x,
=2Js(H(t)+Hec(x, t)),
t))dx=O.
(1)
(2)
In Eq. (1) y(x, t) is the wall profile, where x is the distance from the middle of the lamination along an axis perpendicular to the lamination surface, and t is the time. The y-axis is parallel to the longitudinal direction of the wall motion. The double and the simple primes on y indicate partial derivatives with respect to x. Eqs. (1) and (2) can be solved numerically by an iterative technique described in Ref. [6]. The numerical solution y(x, t) is a sinusoidal function with the same frequency u of H(t) and phase depending on x, u and o-,,. It is therefore possible to calculate the phase shifts b u s and ~bas, respectively, between the surface wall oscillation and the sinusoidal applied external field H(t), and between the surface wall oscillation and the mean wall oscillation within the bulk. The curves representing these quantities versus frequency are strongly dependent on o-w in an appropriate frequency range, where the behavior of the wall due to the bowing differs significantly from the rigid wall behavior. We would like to stress that up to now the theoretical curves have been calculated by neglecting the effect of wall pinning at the lamination surface, although in fact, a slight excess of wall pinning at the lamination surface has a large effect on the wall curva-
The measurements were performed on amorphous strips of 2605 SC (length 20 cm, width 5 cm, thickness 30 ~m). The samples were annealed for 2 h at 300°C in order to release frozen-in stresses. This amorphous alloy exhibits a saturation magnetostriction As = 30-35 10 -6, and a saturation induction Js = 1.55 T. The displacement of a portion of a single wall was measured through the Kerr effect using an optical microscope ( × 100) and a PIN photodiode coupled to a preamplifier as detector. The bandwidth of the system is 300 kHz and the wall displacement resolution is 0.2 Ixm. After calibration of the transfer function of this system by means of a light source modulated by a Pockels cell, the accuracy of the phase measurements was found to be + 2°. Let us stress that in these materials annealed for simple stress relief there are no negligible contributions from rotations to the magnetization process. Therefore, since the induced signal is not univocally related to the bulk mean displacement of the DW, the phase shift ~bBs becomes barely measurable. On the other hand, the measure of 4~HS implies the determination of the effective applied field and cannot be performed by simply measuring the drive current, since the solenoid is interrupted in a small region owing to the optical measurements. This fact g!ves rise to a non-uniform applied field at high frequencies with the consequence of local demagnetizing fields in the measure region. For this reason the effective field was measured using a small magnetic tensiometer 0.1 mm thick, placed in contact with the lamination surface just behind the region where optical measurements were performed. The very low magnetic cross-section of the tensiometer allows us to make reliable measurements of the phase of the effective field only above a given frequency, which, in the case of the present sample, was 50 kHz.
4. Results and discussion
Fig. 1 presents the raw data of the phase shift ~bRs between the surface wall oscillation and the effective field versus frequency, for two values of applied stress, t r = 180 and 270 MPa. It should be noted that the possible range of variation of tr is limited by the need to obtain a relatively straight wall in the longitudinal direction (lower value) and by necessity not to exceed
106
A. Maraner et al. /Journal of Magnetism and Magnetic Materials 133 (1994) 104-106
6o'
I a~=.0010 J/m e 2 o'w=.0015 J/m e • 270 MPa o 180 MPa
1 . 5 . 1 0 3 j / m 2, for ~ = 180 a n d 270 MPa, respectively. W e now consider the expression of the energy density ~rw for a rigid wall:
0 '
~w = 4~;~( X 0 + ~ A s ~ ) ,
70 '
(Ji)HS
8o'
i'O0 frequency
1000 (Kttz)
Fig. 1. Frequency dependence of the phase shift t~SH between the surface wall oscillation and the effective field for an amorphous ribbon of Metglas 2605SC. The experimental data points correspond to an applied tension of 180 MPa (open circles) and 270 MPa (filled circles). Full lines are theoretical and represent the best fit of the experimental data by using trw as an adjustable parameter. The obtained values are ~ w = l × l 0 3 and 1.5×10 -3 J / m 2, which can be compared with the values obtained from Eq. (3): 0.9-1.1X10 -3 and 1.1-1.35 X 10 -3 J / m 2 for o-= 180 and 270 MPa, respectively.
t h e s t r e n g t h limit of the m a t e r i a l ( u p p e r value). T h e large scattering f o u n d in the actual m e a s u r e m e n t s comes from the p e r t u r b i n g effect of the local d e m a g n e tizing fields, which are r e l a t e d to a slightly irregular antiparallel d o m a i n p a t t e r n . T h e s e fields are in p h a s e with the wall m o v e m e n t a n d thus o u t of p h a s e with the applied field. T h e r e f o r e they p r o d u c e an u n c e r t a i n t y in the p h a s e of t h e effective field acting o n t h e wall. T h e r e p o r t e d d a t a are sufficient to show t h a t an effect of o n crw actually exists. F r o m a best fit of e x p e r i m e n t a l data (full lines) using crw as a n adjustable p a r a m e t e r in the theory, o n e obtains the values ~w = 1 × 10 -3 a n d
(3)
w h e r e K 0 is the uniaxial anisotropy coefficient r e l a t e d to a residual directional o r d e r i n h e r e n t in the amorp h o u s r i b b o n or p r o d u c e d by field annealing; AS is the m a g n e t o s t r i c t i o n coefficient, a n d A is the exchange 3 energy coefficient. In o u r case, since the t e r m 5As~r is of the o r d e r of 10 4 J / m 3, it is possible to neglect K0, which is at least o n e o r d e r of m a g n e t u d e lower [7]. A s s u m i n g A = 5 - 8 x 10 -12 J m i [8], we could expect for % values in the r a n g e s 0 . % 1 . 1 x 10 -3 a n d 1.11.35 × 10 3 j / m 2, for ~r = 180 a n d 270 MPa, respectively. T h e s e values are in good a g r e e m e n t with those o b t a i n e d from the e x p e r i m e n t a l results r e p o r t e d in Fig. 1. Acknowledgements. W e would like to t h a n k S. Rocco for helpful c o n t r i b u t i o n in the d e v e l o p m e n t of the e x p e r i m e n t a l part.
References [1] E. Herpin, Theorie du Magnetisms (Presses Universitaires de France, 1968) ch. 33. [2] H.J. Williams, W. Shockley and C. Kittel, J. Appl. Phys. 80 (1950) 1090. [3] K.M. Polivanov, Izvest. Akad. Nauk. SSSR (Set. Fiz.) 16 (1952) 449. [4] J.E.L., Bishop, J. Phys. D: Appl. Phys. 6 (1973) 97. [5] A. Maraner, C. Beatrice and P. Mazzetti, J. Appl. Phys. 75 (1994) to appear. [6] A. Maraner, C. Beatrice and P. Mazzetti, IEEE Trans. Magn. (1994) to appear. [7] C. Appino, F. Fiorillo and A. Maraner, IEEE Trans. Magn. (1994) to appear. [8] H. Kronmueller and B. Groeger, J. Physique 42 (1981) 83.