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Percolations of magnetic phases and coercive fields in ultrathin ferromagnetic films
Journal of Magnetism and Magnetic Materials 113 (1992) 8-12 North-Holland
Percolations of magnetic phases and coercive fields in ultrathin ferromagnetic films W. Schmidt blstitute of Molecular Physics, Polish Academy of Sciences, uL Smoluchowskiego 17, 60-179 Poznati, Poland
The coercive force of uitrathin films with a perpendicular easy directon and a domain wall strongly pinned by defects is investigated. The free energy of the film consists of the energy of exchange interactions, perpendicular uniaxial anisotropy, Zeeman and dipole contributions. The magnetization direction within the domain wall is calculated by the spline technique. Additionally, it is shown that the Laplace capillary law can be obtained from the minimum of the free energy with respect to the shape of the wall. The Laplace law is used for estimation of the distance between defects in real films.
1. Introduction
2. The free e n e ~ ,
In ultrathin films, the hysteresis phenomenon is simplified which makes the ultrathin magnetic films appropriate objects for a basic study. The aim of this paper is a calculation of the influence of the distance between defects on magnetization processes of the hysteresis curve in ultrathin magnetic films with perpendicular anisotropy. We specify our calculation to the system Au/Co/Au. These ultrathin films were widely studied by the authors of refs. [1-3]. Fredkin and Koehler [4] have developed a computer calculation of hysteresis loops and associated magnetization distributions in small magnetic particles. An alternative method of the calculation of magnetic structures is based on the Landau-Lifshitz equation [5]. We use the splineinterpolation method to determine the magnetization direction within the domain wall and its energy density. In section 3, a minimum of the free energy with respect to the shape of the wall surface is calculated nnd the stability condition between the magnetic phases (domains) is analyzed.
We take into consideration only a small region of a ferromagnetic film which is divided into two magnetic domains A and B by a domain wall strongly pinned at points ( d / 2 , 0) and ( - d / 2 , 0), see fig. i. The strong pinning means that the wall passes by the pin points and it is not possible te detach it from them. The region of the film considered extends from the centre of the domain B to the ccntre of the domain A. ~y
(o,to '2)
i'
¢////Jf -
(- d,'z,o)
_
.
o
~
..
(a '2,o)
B !
Correspondence to: Dr. W. Schmidt, Institute of Molecular Physics, Polish Academy of Sciences, ul. Smoluchowskiego 17, 60-179 Poznafi, Poland.
-
(o,-to '2)
i
Fig. 1. C~~ordinatesystems and a region of the film used in the calculatimls.
0304-8853/92/$05.00 ~¢~ 1992 - Elsevier Science Publishers B.V. All rights reserved
W. Schmidt
Magnetic phases and coercive fields in ultrathin films
The magnetization direction within the domain wall and magnetization procc, of the hysteresis curve are calculated from n~ ~ima of the free energy of the form
--Idr
•3 . 0 0
i
2.00
. . . . . . . .
I. . . . .
9
~r~
~--rT
T--r rr
r r r
r T ~ 7 v--r
~ v--~-v--r=
\\
-,,.M c~ 1.00 ~- . . . . . . . . . .
_Ku
~
COS20 -
1 . H d ), ~_M
+ E p i n.
(1)
The state of a ferromagnetic sample is determined by the spherical angles 0 and &. The angle 0 is between the local magnetization and the normal to the film. The azimuthal angle 4) of the local magnetization is in the plane of the film. On the angle 4) depends the stray field due to the volume charge. We neglect the aftereffect and consequently the time dependence of the spherical angles is determined by a tirae dependence of the external field H(t). The last term of eq. (1) describes the interaction energy of the domain wall with defects. For strongly pinned walls, we do not need to consider the form of Ep~n. The first three terms are related to the exchange interaction, Z c e m a n and anisotropy contributions, respectively. The stray field is divided into two component~ due to the surface and volume charges: H d = f}' + H v. The field H~ of the surface charges, ! ~ the film thickness t F --, O, has the form H ~ ( z = 0, ~r) =
-4wcM~
cos 0.
-~
(2)
The coefficient c describes the influence of the film thickness on the demagnetization factor, see ref. [6]. The stray field Hv due to the volume charge of a flat wall of ultrathin film gives a small correction to the wall energy (see fig. 2) and to 0 in ultrathin films. This implies that the stray field due to the volume charge can be neglected. In this approximation, the free energy depends on the angle 0. Thus static magnetic properties of the film are determined by 0 = O(r, H(t)). In a local s.xstem (o-, ~r) connected with the domain wall, see fig. 1, 0 is approximated segmentally by a flat wall. We use interpolation of 0 by the piecewise cubic Hermite splines. This is a good
0"05 o - 6 - . . . . - b'g . 0 .
.
.
.
.
.
.
.
o.oo
o.5e
H/H,I!
1.00
Fig. 2. Energy of the N~el wall (solid line) vs. the in-plane field for stripe domains nucleated at H = Her f. The dashed line is obtained by neglecting the stray field due to the volume charge. Calculations are performed for t F = 9.5 ,~, and t o = 1460.
technique for the approximation of functions arising in calculations of differential equations [7].
3. Percolation o f m a g n e t i c phases
The aim of this section is to calculate the value of the external field at which there occurs an invasion of the magnetic phase with the lower magnetic potential into the regions with a different previous phase. This invasion represents an irreversible process and determines the coercive field. For this aim, the free energy (1) is transformed into the form E
E0
fd/2 Y dx
tF
tF
" -d/2
+_d/2 r T
1+
dx.
(3)
The form (3) of the free energy introduces the wail surface and the wail energy, Y. The wail surface is represented by the function y = Y(x) and divides the film into domains A and B with the magnetic potentials FA and F B. The constant E 0 corresponds to a volume part of the free energy with the flat wall surface which passes through the pin points. The integral at the pres-
W. Schmidt / Magnetic phases and coercit'e fieMs in ultrathin films
10
sure p represents a change of the domain volume. The last term is connected with the wall energy. The pressure acting on the domain wall is determined by the difference of magnetic potentials
p=(FA-FB) , r t = -H-M-
lem is discussed in ref. [8]. The Euler equation which must be satisfied by Y(x) in order to make E a minimum is
,)213J2 (
+
(4) ( K , - 2 w c M 2) cos20,,
I=A,B.
(5)
The angles 0A and 0 B are the asymptotic va!ues of the magnetization direction in the domains (0 A = O(tD/2) and 0 B = O(--tD/2)). The wall energy is connected with the inhomoge~:eous magnetization localized at the wall surface. A comparison of (1) and (3) leads to the wall energy density. If the stray field due to the volume charge is neglected, the wall energy can be converted into the form
(7)
Such a simple equation of the wall surface can be obtained only for the constant wall energy. Eq. (7/implies that the wall surface is a segment of a cylinder with a constant radius. This gives the Laplace capillary law for two-dimensional media ( t F << t~0 )
p=y/R.
(8)
The law says that the pressure p due to the difference of magnetic potentials is compensated by the pressure of the wall curvature (y/R). For the corresponding curvature K = - l / R , the wall surface has the form x=Rsin
In this approximation, the wall energy does not depend on the orientation of the wall with respect to the directions of the domain magnetizations and is constant along the wall. In figs. 2 and 3 is shown the influence of the external field on the wall energy. The shape of the wall, y = Y(x), is calculated by minimizing the free energy (3). A similar prob-
~--
, ....
=0.
(6)
y = 2A~x,_ d~'[
1.00
~x
,,,
, , , , , , , ,
, , , , , , , , , [ ~ ' 1 , , , , , , , ! , , , , , , , ,
~o,
y=Rces~o-Rcos~o
0,
qh)-- s i n - l ( d / 2 R ) ,
(9) (10)
where the parameter q~ is taken from the interval -q~o -< ~o < qh). The corners at the pin points (see fig. l) will be removed if we consider the general form of the exchange interaction, but in this case the domain wall shape can only be calculated numerically. For the analysis of the stability condition between magnetic domains, the free energy (3) is transformed by eq. (9) into the form E = E o + tv(-pREt#o + ( p R d / 2 ) cos qh)
0.90
+2gq~oy).
_•
(11)
0.80
The conditions for a minimum of the free enea~y with respect to the curvature radius are:
aE/OR- t v ( - p g + y)[2qh , + R(Oqh)/aR)] = 0 , N
~N
. . . .
(i2)
E
o.58.oy*'' 0.10 t,
i L, ta_l
1 •,
a tLa~t~a_ta
0.20
a~La I t,,
t La--U-Lt~-X.t~_*
0.,.30
0.40
*~a
I a . a - - L t *--t t a
0.50
0.60
H / H ~,! F i g 3. Energy of the Bloch wall vs. the external p e r p e n d i c u l a r field. T h e calculation is p e r f o r m e d for t r: = 9.5 A a n d t b = 146 a. Here, Y0 is the e n e r g y of the 180 ° Bloch wall for H = 0.
~!2E/OR:: = t r p ( t a n q~()- 2qh) ) > 0, for R =
y/p.
(13)
The stability condition 02E/OR 2 > 0 is fulfilled only for the external field H < H c. The curvature
W. Schmidt / Magnetic phases and cocrcit'e fieMs in ultrathin fihns 0.15 V [-
-r-r-'r
+-r+,'w--'rTw--r~r--r-~+'r-+,r'w-r~
+ ,
,
~
+++'r~--v+-r-v-T--r-+
,
,
r-r~-7--r-
0.10
0.05
F
0"020.!0 .......
4'.00 .......
6'.1313....... 8'.130 ...... i 0!00
Fig. 4. The distance between defects vs. the film thickness. The points are calculated based on the experimental dala [I] of the coercive force with the field acting in the easy direction. The straight line is fitted by the least squares meth3d.
11
due to the external ficId and there occurs an invasion of the magnetic pt.ase. Our paper ignores the aftereffect. The relaxation time of C o / A u films is small (see ref. [3]), particularly at H---H~ where the energy barrier vanishes. Our calculation may be applied only for experimental data with greater values of the field variation rate. For an ultrathin film, the parameters of the free energy (1) depend strongly on the film thickness. An inhomogeneous thickness of ultrathin films is an essential source of the wall pinning. The wall motion in an inhomogeneous thickness film is discussed in ref. [3].
4. Summary
radius R is a function of H and can be calculated from the Laplace law (8). The critical field is determined by the equation R ( H = He) - d / 2 . For H >_H c, the stability condition is not fulfilled, see the definition equation (10) of qhl. At the critical field He, where R = d / 2 , the magnetic phase B (F~3 < FA) leaks into the volume I,),, originally filled with the phase A. For the critical field, the l a p l a c e capillary law of the form p ( H = H c) = 2 y ( H = H , . ) / d is used for cstimation of a distance d between defects from the experimental data given in ref. [1]. For the calculations we used values of the exchange stiffness constant Acx = 3 × 10 - 6 e r g / c m , of the bulk sample, the saturation magnetization M~ = 1400 G, and the effective anisotropy field of the samples (see ref. [1]) Her f = 2 K u / M ~ - 4wcM~ = ( - 16.95 + 194.8 A / t v) kOe. Fig. 4 shows the dependence of the distance between defects vs. the film thickness for the experimental data [1]. Our formalism can be applied for an estimation of the distance between defects, if the calculated distance and the wall parameter satisfy the condition d / 6 o >> 1. For two extreme films of ref. [1] with t v = 9.5 and 3.3 A, d / 6 o = 13 and 11, respectively. We have assumed that the wall is not detached by the pressure for p < y ( H ~ ) / R ( H c ) . For H = He, the wall curvature is not able to compensate the pressure o
o
We have applied the spline technique to a calculation of the magnetization direction within domain walls. We have adopted the cubic Hermite splines to the one-dimension magnetostatic problem. The l esult of this method correspond to the solution o~ the Brown's equation problem. We have analyzed an irreversible process connected with the coercive field. At high fields and the easy direction, the sample attains uniform magnetization. In the vanishing field limit (H --- 0), regions (domains) are nucleated, thus lowering the local magnetic potential. We consider a region of a ferromagnetic film divided into two magnetic domains by a strongly pinned wall. The difference of domain magnetic potentials gives a pressure which leads to a bowing of the wall. The reversible process connected with the bowing is neglected as they contribute very little to the hysteresis curve. This bcwing is described by the Laplace law. The wall curvature is abte only to compensate the pressure due to the external field H < H c. At a critical field H = H c, there occurs an invasion of the magnetic phase with lowe:" magnetic potential into the region occupied ~ruviously by other phase, present at lower filed:. The invasion at the t:ritical field is an irreversibl • process and determines the coercive field. At H = He, the Laplace capillary law, d = T(H~)/ M~H~, is used for estimation of the distance between defects in real films (see fig. 4).
12
W. Schmidt / Magnetic phases and coercit,e fields b~ ultrathin films
Acknowledgements I would like to thank Dr. A. R. Ferchmin and Dr. B. Szymafiski for their comments.
[3]
[4]
Reference [1] C. Chappert and P. Bruno, J. Appi. Phys. 64 (1988) 5736. [2] C. Chappert, P. Beauvillain, P. Bruno, J.P. Chauyineau, M. Galtier, K. Le Dang, C. Marli~re, R. M6gy, D. Renard,
[5] [6] [7] [8]
J.P. Renard, J. Seiden, F. Trigui, P. Veillet and E. V61u, J. Magn. Magn. Mater. 93 (1991) 319. P. Bruno, G. Bayreuther, P. Beauvillain, C. Chappert, G. Lugert, D. Renard, J.P. Ranard and J. Seiden, 2. Appi. Phys. 68 (1990) 5759. D.R. Fredkin and T.R. Koehler, J. Appl. Phys. 67 (1990) 5544. Y. Nakatani, Y. Uesaka and N. Hayashi, Jpn. J. Appl. Phys. 28 (1989) 2485. Y. Yafet and E.M. Gyorgy, Phys. Rev. B 38 (1988) 9145. P.M. Prenter, Splines and Variational Methods (Wiley Interscience, New York, 1975). L.F. Maga~a, J. Magn. Magn. Mater. 60 (1986) 315.
Percolations of magnetic phases and coercive fields in ultrathin ferromagnetic films W. Schmidt blstitute of Molecular Physics, Polish Academy of Sciences, uL Smoluchowskiego 17, 60-179 Poznati, Poland
The coercive force of uitrathin films with a perpendicular easy directon and a domain wall strongly pinned by defects is investigated. The free energy of the film consists of the energy of exchange interactions, perpendicular uniaxial anisotropy, Zeeman and dipole contributions. The magnetization direction within the domain wall is calculated by the spline technique. Additionally, it is shown that the Laplace capillary law can be obtained from the minimum of the free energy with respect to the shape of the wall. The Laplace law is used for estimation of the distance between defects in real films.
1. Introduction
2. The free e n e ~ ,
In ultrathin films, the hysteresis phenomenon is simplified which makes the ultrathin magnetic films appropriate objects for a basic study. The aim of this paper is a calculation of the influence of the distance between defects on magnetization processes of the hysteresis curve in ultrathin magnetic films with perpendicular anisotropy. We specify our calculation to the system Au/Co/Au. These ultrathin films were widely studied by the authors of refs. [1-3]. Fredkin and Koehler [4] have developed a computer calculation of hysteresis loops and associated magnetization distributions in small magnetic particles. An alternative method of the calculation of magnetic structures is based on the Landau-Lifshitz equation [5]. We use the splineinterpolation method to determine the magnetization direction within the domain wall and its energy density. In section 3, a minimum of the free energy with respect to the shape of the wall surface is calculated nnd the stability condition between the magnetic phases (domains) is analyzed.
We take into consideration only a small region of a ferromagnetic film which is divided into two magnetic domains A and B by a domain wall strongly pinned at points ( d / 2 , 0) and ( - d / 2 , 0), see fig. i. The strong pinning means that the wall passes by the pin points and it is not possible te detach it from them. The region of the film considered extends from the centre of the domain B to the ccntre of the domain A. ~y
(o,to '2)
i'
¢////Jf -
(- d,'z,o)
_
.
o
~
..
(a '2,o)
B !
Correspondence to: Dr. W. Schmidt, Institute of Molecular Physics, Polish Academy of Sciences, ul. Smoluchowskiego 17, 60-179 Poznafi, Poland.
-
(o,-to '2)
i
Fig. 1. C~~ordinatesystems and a region of the film used in the calculatimls.
0304-8853/92/$05.00 ~¢~ 1992 - Elsevier Science Publishers B.V. All rights reserved
W. Schmidt
Magnetic phases and coercive fields in ultrathin films
The magnetization direction within the domain wall and magnetization procc, of the hysteresis curve are calculated from n~ ~ima of the free energy of the form
--Idr
•3 . 0 0
i
2.00
. . . . . . . .
I. . . . .
9
~r~
~--rT
T--r rr
r r r
r T ~ 7 v--r
~ v--~-v--r=
\\
-,,.M c~ 1.00 ~- . . . . . . . . . .
_Ku
~
COS20 -
1 . H d ), ~_M
+ E p i n.
(1)
The state of a ferromagnetic sample is determined by the spherical angles 0 and &. The angle 0 is between the local magnetization and the normal to the film. The azimuthal angle 4) of the local magnetization is in the plane of the film. On the angle 4) depends the stray field due to the volume charge. We neglect the aftereffect and consequently the time dependence of the spherical angles is determined by a tirae dependence of the external field H(t). The last term of eq. (1) describes the interaction energy of the domain wall with defects. For strongly pinned walls, we do not need to consider the form of Ep~n. The first three terms are related to the exchange interaction, Z c e m a n and anisotropy contributions, respectively. The stray field is divided into two component~ due to the surface and volume charges: H d = f}' + H v. The field H~ of the surface charges, ! ~ the film thickness t F --, O, has the form H ~ ( z = 0, ~r) =
-4wcM~
cos 0.
-~
(2)
The coefficient c describes the influence of the film thickness on the demagnetization factor, see ref. [6]. The stray field Hv due to the volume charge of a flat wall of ultrathin film gives a small correction to the wall energy (see fig. 2) and to 0 in ultrathin films. This implies that the stray field due to the volume charge can be neglected. In this approximation, the free energy depends on the angle 0. Thus static magnetic properties of the film are determined by 0 = O(r, H(t)). In a local s.xstem (o-, ~r) connected with the domain wall, see fig. 1, 0 is approximated segmentally by a flat wall. We use interpolation of 0 by the piecewise cubic Hermite splines. This is a good
0"05 o - 6 - . . . . - b'g . 0 .
.
.
.
.
.
.
.
o.oo
o.5e
H/H,I!
1.00
Fig. 2. Energy of the N~el wall (solid line) vs. the in-plane field for stripe domains nucleated at H = Her f. The dashed line is obtained by neglecting the stray field due to the volume charge. Calculations are performed for t F = 9.5 ,~, and t o = 1460.
technique for the approximation of functions arising in calculations of differential equations [7].
3. Percolation o f m a g n e t i c phases
The aim of this section is to calculate the value of the external field at which there occurs an invasion of the magnetic phase with the lower magnetic potential into the regions with a different previous phase. This invasion represents an irreversible process and determines the coercive field. For this aim, the free energy (1) is transformed into the form E
E0
fd/2 Y dx
tF
tF
" -d/2
+_d/2 r T
1+
dx.
(3)
The form (3) of the free energy introduces the wail surface and the wail energy, Y. The wail surface is represented by the function y = Y(x) and divides the film into domains A and B with the magnetic potentials FA and F B. The constant E 0 corresponds to a volume part of the free energy with the flat wall surface which passes through the pin points. The integral at the pres-
W. Schmidt / Magnetic phases and coercit'e fieMs in ultrathin films
10
sure p represents a change of the domain volume. The last term is connected with the wall energy. The pressure acting on the domain wall is determined by the difference of magnetic potentials
p=(FA-FB) , r t = -H-M-
lem is discussed in ref. [8]. The Euler equation which must be satisfied by Y(x) in order to make E a minimum is
,)213J2 (
+
(4) ( K , - 2 w c M 2) cos20,,
I=A,B.
(5)
The angles 0A and 0 B are the asymptotic va!ues of the magnetization direction in the domains (0 A = O(tD/2) and 0 B = O(--tD/2)). The wall energy is connected with the inhomoge~:eous magnetization localized at the wall surface. A comparison of (1) and (3) leads to the wall energy density. If the stray field due to the volume charge is neglected, the wall energy can be converted into the form
(7)
Such a simple equation of the wall surface can be obtained only for the constant wall energy. Eq. (7/implies that the wall surface is a segment of a cylinder with a constant radius. This gives the Laplace capillary law for two-dimensional media ( t F << t~0 )
p=y/R.
(8)
The law says that the pressure p due to the difference of magnetic potentials is compensated by the pressure of the wall curvature (y/R). For the corresponding curvature K = - l / R , the wall surface has the form x=Rsin
In this approximation, the wall energy does not depend on the orientation of the wall with respect to the directions of the domain magnetizations and is constant along the wall. In figs. 2 and 3 is shown the influence of the external field on the wall energy. The shape of the wall, y = Y(x), is calculated by minimizing the free energy (3). A similar prob-
~--
, ....
=0.
(6)
y = 2A~x,_ d~'[
1.00
~x
,,,
, , , , , , , ,
, , , , , , , , , [ ~ ' 1 , , , , , , , ! , , , , , , , ,
~o,
y=Rces~o-Rcos~o
0,
qh)-- s i n - l ( d / 2 R ) ,
(9) (10)
where the parameter q~ is taken from the interval -q~o -< ~o < qh). The corners at the pin points (see fig. l) will be removed if we consider the general form of the exchange interaction, but in this case the domain wall shape can only be calculated numerically. For the analysis of the stability condition between magnetic domains, the free energy (3) is transformed by eq. (9) into the form E = E o + tv(-pREt#o + ( p R d / 2 ) cos qh)
0.90
+2gq~oy).
_•
(11)
0.80
The conditions for a minimum of the free enea~y with respect to the curvature radius are:
aE/OR- t v ( - p g + y)[2qh , + R(Oqh)/aR)] = 0 , N
~N
. . . .
(i2)
E
o.58.oy*'' 0.10 t,
i L, ta_l
1 •,
a tLa~t~a_ta
0.20
a~La I t,,
t La--U-Lt~-X.t~_*
0.,.30
0.40
*~a
I a . a - - L t *--t t a
0.50
0.60
H / H ~,! F i g 3. Energy of the Bloch wall vs. the external p e r p e n d i c u l a r field. T h e calculation is p e r f o r m e d for t r: = 9.5 A a n d t b = 146 a. Here, Y0 is the e n e r g y of the 180 ° Bloch wall for H = 0.
~!2E/OR:: = t r p ( t a n q~()- 2qh) ) > 0, for R =
y/p.
(13)
The stability condition 02E/OR 2 > 0 is fulfilled only for the external field H < H c. The curvature
W. Schmidt / Magnetic phases and cocrcit'e fieMs in ultrathin fihns 0.15 V [-
-r-r-'r
+-r+,'w--'rTw--r~r--r-~+'r-+,r'w-r~
+ ,
,
~
+++'r~--v+-r-v-T--r-+
,
,
r-r~-7--r-
0.10
0.05
F
0"020.!0 .......
4'.00 .......
6'.1313....... 8'.130 ...... i 0!00
Fig. 4. The distance between defects vs. the film thickness. The points are calculated based on the experimental dala [I] of the coercive force with the field acting in the easy direction. The straight line is fitted by the least squares meth3d.
11
due to the external ficId and there occurs an invasion of the magnetic pt.ase. Our paper ignores the aftereffect. The relaxation time of C o / A u films is small (see ref. [3]), particularly at H---H~ where the energy barrier vanishes. Our calculation may be applied only for experimental data with greater values of the field variation rate. For an ultrathin film, the parameters of the free energy (1) depend strongly on the film thickness. An inhomogeneous thickness of ultrathin films is an essential source of the wall pinning. The wall motion in an inhomogeneous thickness film is discussed in ref. [3].
4. Summary
radius R is a function of H and can be calculated from the Laplace law (8). The critical field is determined by the equation R ( H = He) - d / 2 . For H >_H c, the stability condition is not fulfilled, see the definition equation (10) of qhl. At the critical field He, where R = d / 2 , the magnetic phase B (F~3 < FA) leaks into the volume I,),, originally filled with the phase A. For the critical field, the l a p l a c e capillary law of the form p ( H = H c) = 2 y ( H = H , . ) / d is used for cstimation of a distance d between defects from the experimental data given in ref. [1]. For the calculations we used values of the exchange stiffness constant Acx = 3 × 10 - 6 e r g / c m , of the bulk sample, the saturation magnetization M~ = 1400 G, and the effective anisotropy field of the samples (see ref. [1]) Her f = 2 K u / M ~ - 4wcM~ = ( - 16.95 + 194.8 A / t v) kOe. Fig. 4 shows the dependence of the distance between defects vs. the film thickness for the experimental data [1]. Our formalism can be applied for an estimation of the distance between defects, if the calculated distance and the wall parameter satisfy the condition d / 6 o >> 1. For two extreme films of ref. [1] with t v = 9.5 and 3.3 A, d / 6 o = 13 and 11, respectively. We have assumed that the wall is not detached by the pressure for p < y ( H ~ ) / R ( H c ) . For H = He, the wall curvature is not able to compensate the pressure o
o
We have applied the spline technique to a calculation of the magnetization direction within domain walls. We have adopted the cubic Hermite splines to the one-dimension magnetostatic problem. The l esult of this method correspond to the solution o~ the Brown's equation problem. We have analyzed an irreversible process connected with the coercive field. At high fields and the easy direction, the sample attains uniform magnetization. In the vanishing field limit (H --- 0), regions (domains) are nucleated, thus lowering the local magnetic potential. We consider a region of a ferromagnetic film divided into two magnetic domains by a strongly pinned wall. The difference of domain magnetic potentials gives a pressure which leads to a bowing of the wall. The reversible process connected with the bowing is neglected as they contribute very little to the hysteresis curve. This bcwing is described by the Laplace law. The wall curvature is abte only to compensate the pressure due to the external field H < H c. At a critical field H = H c, there occurs an invasion of the magnetic phase with lowe:" magnetic potential into the region occupied ~ruviously by other phase, present at lower filed:. The invasion at the t:ritical field is an irreversibl • process and determines the coercive field. At H = He, the Laplace capillary law, d = T(H~)/ M~H~, is used for estimation of the distance between defects in real films (see fig. 4).
12
W. Schmidt / Magnetic phases and coercit,e fields b~ ultrathin films
Acknowledgements I would like to thank Dr. A. R. Ferchmin and Dr. B. Szymafiski for their comments.
[3]
[4]
Reference [1] C. Chappert and P. Bruno, J. Appi. Phys. 64 (1988) 5736. [2] C. Chappert, P. Beauvillain, P. Bruno, J.P. Chauyineau, M. Galtier, K. Le Dang, C. Marli~re, R. M6gy, D. Renard,
[5] [6] [7] [8]
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