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Magnetic structures with a finite number of domain walls
170
Journal of Magnetism and Magnetic Materials 94 (1991) 170-178 North-Holland
Magnetic structures with a finite number of domain walls V.A. Ignatchenko
and E.Yu. Mironov
Kirensky Insrirute of Physics, Krasnoyarsk 660036, USSR Received 29 May 1990
The equilibrium configurations and the process of magnetization of structures with a finite number N of plane domain walls are considered. In comparison with models of a periodic domain structure new effects are found: a difference from zero of the average magnetization at H = 0 and asymmetry of a hysteresis loop for even N, as well as the possibility of the existence of metastable states. The effect of changing the domain structure period in the process of magnetization known for the periodic models is manifested in the structures with a finite N as a sequence of 1st kind phase transitions corresponding to the change in the number of domain walls.
1. Introduction
Within the framework of the phenomenological theory (see e.g. ref. [l]) finding the equilibrium configurations of the magnetization M(r) for a ferromagnet is reduced to the solution of a variation problem for the functional with the energy density F=
<(M(r), g) ++V*M(r) X
/
vt
V-M(r') Ir-r')
dr'
-
H-M(r).
(1)
The first term describes the energy of short-range interactions (exchange, anisotropy). The second term shows long-range magnetostatic interaction, and the third one denotes interaction with the external magnetic field H. The variation problem results in a system of integro-differential equations which cannot be solved in a general form. A cardinal simplification of the problem has been made by Landau and Lifshitz [2]. They have shown that a characteristic domain wall width is, as a rule, much less than the domain sizes. Consequently, the problem of domain structure may be solved in the approximation of infinite thin do0304-8853/91/$03.50
main walls whose structure and energy are calculated beforehand for a solitary domain wall. Further cardinal simplification is based on the fact that in many cases the width of the domains is much less than the corresponding specimen size. This makes it possible to study models in which the specimen size along one or two directions is assumed unlimited and the domain structure periodic. In these models for periodic structures with plane domain walls the problem becomes much simpler and is characterized either by one degree of freedom (domain width) when the equilibrium configuration is determined in a zero magnetic field [3] or by two degrees when the process of magnetization is investigated [4,5]. Besides, in the study of periodic models one can apply a powerful mathematical method such as Fourier series. The results obtained for the periodic models are applied to finite samples only in the limiting case characterized by sufficiently large number of domain walls N >> 1. In real samples this relation does not always hold. Besides, we shall show below that a number of new physical phenomena (the existence of metastable states, peculiarities of magnetization process) have to exist for domain structures with a finite N and they may be ob-
0 1991 - Elsevier Science Publishers B.V. (North-Holland)
V.A. Ignatchenko, E. Yu. Mironov / Magnetic structures with a finite number of domain walls
served even at large enough N. The aim of our work is to study equilibrium configurations and the magnetization process of the plane domain structures with a finite N. But firstly we shall briefly analyze the results obtained earlier when considering the magnetization process in the model of a periodic domain structure. We need it to compare these results and those characteristic of the structures with a finite N.
2. Periodic domain structure The behaviour of a periodic structure with the plane domain walls affected by a magnetic field was first considered independently in refs. [4,5]. It is shown that a system is described by two degrees of freedom: the period of the domain structure 20 and the relation of domain sizes having opposite orientations of magnetizations which can be characterized by q = d/D (fig. 1). At a given H the equilibrium values of q and D are determined by the equations: aF _=
aD
‘7
aF _=
where the energy density has the form: F( D, q; H) = $,M’(I +
5 -
the external field. Paper [4] deals with the case of a specimen unlimited both along the x axis and y axis, that is with domains with magnetization perpendicular to the layer of thickness c. A more general case of a specimen unlimited only in the x direction is studied in ref. [5]. It gives the possibility, in particular, to describe two limiting cases: domains magnetized normal to the plane of layer [4] (b X= c) and ones magnetized in the plane of layer (b +Z c). The general case corresponds to the following expressions for the density of multipole energy Fmp and an effective demagnetizing factor n,.
C,,(D,q)=~ccx
sin’( Tmq/2)
- q)2 + F,,(D, MH(1
-
q).
q) (3)
The first and second terms describe dipole and multipole parts of the magnetostatic energy, respectively. The third term shows the energy of domain walls (y is the surface density of this energy), and the fourth one - the interaction with
m2
m=l
x
00
sin2(bk/2)(1
- eCcp) dk 9
/0
pk2
(4)
(2)
Ov
a4
171
where p2 = k2 + ( Tm/D)2. Owing to the term F,,(D, q) the system of equations (2) becomes coupled, and a change in the magnetic field leads not only to a change of q but also to that of D, i.e. to the effect of an expansion (when H increases) or contraction (when H decreases) of the domain structure period. This effect is the main and qualitatively new result obtained in refs. [4,5]. It was believed earlier that upon the application of a field only the relation between the volumes of the oppositely magnetized domains and not the period of a domain structure must change. The dependence of D on q is determined by the first equation of system (2). For D e c -=x b (domains, magnetized normal to the layer): l/2
D=&(rJITq) i ’
o,(q)
Fig. 1. Periodic domain structure.
=
E sin2(;yl/2). WI=1
(5)
172
V.A. Ignatchenko, E. Yu. Mironov / Magnetic structures with a finite number of domain walls
At H = 0 q = 1 eq. (5) transforms to the expression [3] for the domain width in a plate with an easy axis normal to the surface. For b -=x D -=z c (domains in the plane of a magnetic film) using (4) we obtain
analytical expression for the density of the total energy in the form: F(+...,x,)
2 D
=
W&s:;(q)
(6) o22(q) =
E
+ 2h4H N+l e,x, + $N, ,-_c n=O
’ sin2(;y/2)
=;q(z-q).
m=l
At H = 0 eq. (6) transforms to the expression [6] for the domain width in the film with an easy axis of anisotropy in the plane of the film. The second equation of the system (2) describes the dependence of q on H, i.e. a magnetization curve.
(7)
where the first term is the magnetostatic energy F,, the second term describes the interaction with the external field, and the third one is the energy of the domain walls. Here “configuration charges” of the lattice sites (of the domain walls and edges of the specimen) are introduced: e, = AM( x,)/2M, where AM(x,) the point x,:
(8)
is a jump of the magnetization
at
AM(x,)=M(x,+O)-1%2(x,-O), 3. Structure with a finite N general approach Let us consider a specimen finite in all three dimensions, with the location of the plane-parallel domain walls at arbitrary points with coordinates Xl,...,XN. BY &J and xN+l we denote the coordinates of the left and right edges of the specimen (fig. 2). For certainty we consider the left edge domain magnetized positively. The system is no longer infinite and periodic on the axis x therefore the Fourier series method is inapplicable here. A method adequate to the problem is in fact that the energy be written in the form of a “lattice” sum. We have found an exact
i.e., e, = f. l/2 for n = 0, N + 1 (edges of the specimen) and e, = k 1 for n = 1, . . . , N (domain walls). The potential for the interaction of the “configuration charges” between each other is defined by a function f(x): f(x)
= $arsinh& +i(c2-x2)
(x2
+ c2y2
2 +
$
arsinh % - z
+
5
( C2 -
-
&
-(x2
[
(x2
b2)
arsinh % X
arsinh
+ b2)3’2
+ c2 + b2)3’2
-
( c2 + b2)1’2 1x13+
(x2+C2)3’2
+ ( b2 + c2)3’2
2c3]
- b3 + C2
(b2+c2)
+2b
[
-(x2
+
c2 +
+ cx arctan Fig. 2. Structure with a finite number of domains.
b
arsinh
112 + (x2
b2)“2]
+
C2)1/2
- gaminhi bx
c( x2 + c2 + b2)1’2 ’
(9)
V.A. Ignatchenko, E. Yu. Mironoo / Magnetic structures with a finite number of domain walls
The magnetostatic as lattice sums:
fields can also be described
N+l
Hi=2M
c e,,gi(x,-x, n=O
y, z),
i=x,
y, z, (10)
where 8X(X, --x9 Y, r) = -gXo(x,
- x, b/2 -y,
- Lo 6% -x,
b/2 + y, c/2 - z)
+ g&,,
- x, b/2 -Y,
c/2
+ z)
+ g,,(x,,
-x,
c/2
+ z),
&z-x, =
c/2 - z)
b/2
+Y,
Y9 z) -x,
-g,o(x,
+g&,--x, -x,
- g,o(x, +gyo(x,-x, gz(x, -x9
b/2-y,
c/2-z)
b/2+Y,
c/2-r)
b/2-y,
c/2+z)
b/2+Y,
c/2+1),
=g&,-AC,
b/2-Y,
+ g&,,
Thus, a general configuration force acting on the wall is the magnetic pressure averaged over the square of the domain wall: --=
aF
- x, b/2 +Y,
+g&,,-x,
b/2-y,
+g&/-x,
bP+y,
gxo(u,
u,
w) = arsinh
g,,(u,
u,
w) = arsinh
g,,(u,
u,
w) = arctan
c/2
-z)
c/~+z) @+z),
u
(#2 + w2)1’2 ’ u
(u2 + w2)“2 ’ uu
w( u2 + u2+ It may be shown that
w2)1’2 .
y, z>)y,Z - Fek.
(12) In a specimen with a finite number of domain walls the problem of the configuration of a domain structure is formulated in a different way than for the model of a domain structure periodic in x. Since the “configuration charges” corresponding to the edges of the specimen are rigidly fixed by the forces of nonmagnetic origin for each value of N there is such a location of the domain walls that corresponds to the minimum of the magnetostatic energy independently of the energy of domain walls. Hence: 1) For each of the fixed values of N there must be found a configuration of domain walls (the values of coordinates x,, . . . , xN) minimizing the total energy (7) and representing the solution of a system of N equations:
axI
c/2-z)
- T(H;(x,,
ax,
aF -=
YP r)
173
aF
o
-
-=
o
ax, .
03)
The equations do not depend on the energy of domain walls and express the equality to zero of all general configuration forces acting on them. 2) Substituting the obtained values of coordinates in (7) we find N corresponding to the minimum of energy for a given H. At H = 0 we obtain the ground state N = No. 3) When considering the behaviour of the system in a magnetic field (or under the influence of other forces) one should find intervals where the configuration of domain walls will change with their number N being constant, and where (and in what way) the phase transitions will occur with change of N.
4. Structure with a finite N particular cases =$“ilenfx(xn-x), PI-0 where
f,(x)
=
af/ax.
Now we consider specimens in which N is small, that is the case when the Fourier expansion does not work, and the method of lattice sums
174
V.A. Ignatchenko, E. Yu. Mironov / Magnetic structures with a finite number
ofdomainwalls
Fig. 3. Dependence of the magnetostatic field HA at I = 0 on coordinate x for the equilibrium position x1 = 0 of a domain wall at N = 1 for c/a = 0.1 (a) and c/a = 10 (b).
does not require a great many calculations yet. For simplicity we investigated the limiting case b -+ 00. In this case the finiteness in x is retained and so retained are all the qualitative peculiarities of the system caused by the finiteness of N. The center of coordinates will be in the center of the specimen: x0 = -a/2, xN+i = a/2. Let us consider the equilibrium domain structure, distribution of magnetostatic fields and peculiarities of the magnetization curve for different values of N. (a) N = I: The equilibrium position of a domain wall at H = 0 corresponds to x1 = 0. The component Hk of the magnetostatic field vanishes at x = 0. In dependence on the relation between the sizes of the specimen c and a it may reach the largest values both inside the domains (fig. 3a) and at the edges of the specimen (fig. 3b). The magnetization curve is symmetric. (b) N = 2: The problem is two-particle, and the state of the system is described by a two-dimensional energy surface F, = Fm(xl, x2) which is shown in fig. 4. However, due to the symmetry x2= -x,=D both at H=O and at H#O. Therefore the problem is confined to single-particle and described only by section OA of the energy surface Fm(xlr x2) shown in fig. 5a. Fig. 5b depicts the magnetization curve. It is easy to see that the equilibrium configuration of domain walls at H = 0 does not correspond to a demagnetized state of the specimen. The difference of Do from 0.25u, at which the average magnetization (M,) would be equal to zero, is not large but principal. A significant reason for this effect is that the minimization of the magnetostatic energy corre-
-0.5 Fig. 4. The relief of the normalized magnetostatic F,(x,, x2)/M* for N = 2 at a/c =l.
energy
sponds to the minimization of a sum of dipole and multipole parts of the energy, and the minimum of the sum does not always show that the condition for the equality to zero of the magnetic moment holds exactly, as was the case for the model of a periodic structure. The second unusual effect the asymmetry of the magnetizationcurve - is due to different ways of magnetization: in the positive field domain walls approach each other and annihilate, and in the negative field they are displaced to the edges of the specimen. Both these effects increase with the growth of the ratio u/c. Fig. 6 shows the dependence H&(z) along a domain wall when it is just in the equilibrium position (b) and when it is slightly displaced in one (a) or another (c) direction from the equilibrium
0.2249
0.5
Fig. 5. Dependence F,(D) for N = 2 at a/c =1 (a) and the magnetization curve for this case (b).
V.A. Ignatchenko, E. Yu Mironov / Magnetic structures with a finite number of domain walls
position. It is clearly seen that in the equilibrium position there are a number of alternating sections where the field is not equal to zero and of a different sign. This means that the domain wall experiences the forces tending to bend it. The model of plane domain walls corresponds to the condition of a sufficiently large energy of a domain wall in comparison with magnetostatic energy. When the condition for predominance of the energy of surface tension is violated a domain wall will bend in the z direction in accordance with the pattern of magnetic fields in fig. 6b. The domain wall is located at the point where the average value of the magnetic pressure on its surface becomes zero according to expressions (12) and (13). (c) N = 3: A three-particle problem is reduced to single-particle due to symmetry only for H = 0, when x2 = 0, x3 = -xi = D; the application of a magnetic field destroys the symmetry. The dependence of I;, on D for H = 0 is shown in fig. 7a. This case is very interesting due to the fact that the condition for the equality to zero of the total magnetic moment in the specimen holds for any D. Therefore the minimum on curve F,(D) is wholly due to the multipole part of the magnetostatic energy. The distribution of the z-component of the magnetostatic field in the plane z = 0 is shown in fig. 7b.
0.3097
175
0.9
Fig. 7. Dependence F,(D) (a) and distribution of the magnetostatic field HA on x in the plane of s = 0 (b) for N = 3, a/c=l.
The magnetization curve is shown in fig. 8. At there occurs a displacement of one of H=H,,, the domain walls to the edge of the specimen. Thus, for H3_2 < 1H 1 < H,’ the state with N = 3 cannot exist in principle. (d) N = 4: The problem is reduced to two-particle both at H=O and H#O: x3= -x2, x4= -xl. In fig. 9 a dashed curve shows the states of the system upon the application of the external field H. As well as for N = 2 this case is characterized by an asymmetric magnetization curve and the slight residual magnetization at H = 0. However both effects are less pronounced than for N = 2. In a field H = - H4_ 2 two domain walls are displaced to the edges of the specimen. Therefore in the fields -Hz < H < - H4 _ 2 there cannot be in principle the state with N = 4. Let us discuss some results. Fig. 10 shows the dependence of magnetostatic energy on the num-
CM,> M 04
Fig. 6. Distribution of the field HA in the plane of a domain wall for N = 2 (a/c = 1) for the equilibrium position of domain walls: D/a = 0.2249 (b) and displaced from the equilibrium position: D/a = 0.2558 (a), D/a = 0.25 (c).
Fig. 8. Magnetization curve for N = 3, a/c = 1.
N=I
V.A. Ignatchenko, E. Yu. Mironov / Magneric structures with a finite number of domain wails
176
0.1
0.1
Fig.
9.
0.5
Relief of the normalized magnetostatic Fm(x2. x,)/M2 for N = 4, a/c =l.
energy
ber of domain walls for the equilibrium configurations at H = 0. Solid curves in this figure indicate the dependence F,(N) for the same relations of the specimen sizes calculated according to the equation from ref. [3] which was obtained in the approximation of a domain structure periodic in x and having the domain width D = a/N (formula (4) at q = 1). It is seen that for N 1 3 + 4 the
deviations between the exact and approximate values of Fh become insignificant. However the qualitative differences in the remagnetization process calculated for the model of a periodic structure and ‘structure with a finite number of domains are still retained for the large values of N. We would like to consider these processes in detail. Fig. 11 shows schematically the dependence of the total energy of the system (7) on the field for two equilibrium states corresponding to N = 2 and N = 4. At H = 0, F4 -c F2. At H = H, the energies of both states become equal and for H > H,, F4 > F2. However the state with N = 4 is separated from that with N = 2 at H = Hf by a potential barrier: the increase in the magnetostatic energy (fig. 9) prevents a transition between the states. Therefore with the growth of H the system remains in the state N = 4 and moves along a phase trajectory corresponding to the larger energy up to the point H=H,,,, where the domain walls touch tKe edges of the specimen. At this point the system jumps to the state N = 2.
F ,a--
\
-=$:
F, / M'
I<“’-\ -.
\\
F2 \
‘\\
H2-.4
Hf
H4-2
H
N=2
1
2
3
N
Fig. 10. Dependence F,(N) for a specimen with a finite number of domains (light circles) and for the model of the periodic domain structure (solid lines) at different c/a: 0.5 (a), 1 (b), 5 (c).
Fig. 11. Schematic dependence of the total energy of the system on the field H for equilibrium states of the system with N = 2 and N = 4. ( ) is the trajectory of the system when H increases, and ( + + t ) when, H decreases.
V.A. Ignatchenko, E. Yu Mironov / Magnetic structures with a finite number
Upon decreasing the field the system remains in the state with N = 2 up to H = Hf and may be there longer if additional energy is required to generate new domain walls (nucleation energy). In the latter case the transition from N = 2 to N = 4 will occur in the field H = H2 _ 4 < Hf. Similar phase transitions of the 1st kind also take place at other points of intersection of the curves F(H) corresponding to different values of N. Fig. 12 shows the dependence of N on H for a system with the equilibrium structure corresponding at H = 0 to the value of N, = 4. In the magnetization process two phase transitions occur: between the states with N = 4 and N = 2 and ones with N = 2 and N = 0. The same figure gives a smooth dependence of N on H which corresponds to the model of a periodic domain structure (5) for N 5: a/D. Accordingly, for the system with the large value of N, one would observe more phase transitions on the curve N(H). Thus, the effect of changing the domain structure period as a function of H calculated for the periodic models in refs. [4,5] in specimens with a
) is the equilibrium Fig. 12. Dependence of N on H. (trajectory for the model of the domain structure with a fiite number of domains. ( ) is the trajectory of the system when H increases, ( + + + ) when H decreases. The dotted line corresponds to the model of the periodic domain structure.
of domainwalls
177
finite N is manifested as a sequence of phase transitions of the 1st kind corresponding to a change in the number of domain walls. In the vicinity of these transitions the structure may be in the metastable states.
5. Conclusions Expressions (3) and (7) represent two different ways of giving a reduced description of the system which is described completely by the energy density (1). System (1) possesses an infinite number of degrees of freedom, whereas system (3) has only two degrees, and system (7), N degrees of freedom, where 1 G N < co. Due to this fact the model of a domain structure with a finite number of domains more nearly describes the real physical situation than the model of a periodic domain structure. It is a further approximation to the complete phenomenological description (1) and therefore some new phenomena become apparent in it which cannot be found in the frames of the model of a periodic structure. Let us summarize these. For an even number of domain walls N, the different from zero average magnetization of .the specimen at H = 0 and an asymmetric remagnetization curve correspond to the equilibrium configuration. Both effects decrease with increase of N. For N > 1 the width of domains located at the edges of the specimen is approximately a factor of 2 smaller than the widths of all other domains. The relation holds the better the larger N is. The effect of changing the domain structure period [4,5] as a function of H which is known for the periodic models is manifested for structures with a finite N as a sequence of phase transitions of the 1st kind corresponding to an uneven change in the number of domain walls. In the vicinity of these phase transitions the structure may be in the metastable states. Their existence is caused by the necessity to overcome a potential barrier in order to change the topology of the system. When H increases, the barrier is due to the increase in the magnetostatic energy in the transition to the structure with a smaller N. When H decreases, the barrier is due to the nucleation process. A high
178
barrier of when H = state with states with
VA. Ignatchenko,E. Yu. Mironov / Magnetics!rucrureswitha fin& numberof domainwall3
nucleation may lead to the situation 0 will correspond not to the ground N = NO but to one of the metastable N c NO.
References [l] W.F. Brown, 1963) p. 63.
Micromagnetics
(Interscience,
New York,
PI L.D. Landau and E.M. Lifshitz, Phys. Z. Sovjettmion 8
(1935) 153. [31 Ch. Kittel, Rev. Mod. Phys. 21 (1949) 541. [4l C. Kooy and U. Enz, Phil. Res. Rep. 15 (1960) 7. 151 V.A. Ignatchenko, J.F. Degtyarev and Yu.V. Zakharov, Izv. Akad. Nauk SSSR, Ser. Fiz. 25 (1961) 1439. English translation in: Bull. Akad. Sci. USSR, Phys. Ser. (USA) 25 (1961) 1452. 161V.A. Ignatchenko and Yu.V. Zakharov, Zh. Eksp. Tear. Fiz. (USSR) 43 (1962) 459.
Journal of Magnetism and Magnetic Materials 94 (1991) 170-178 North-Holland
Magnetic structures with a finite number of domain walls V.A. Ignatchenko
and E.Yu. Mironov
Kirensky Insrirute of Physics, Krasnoyarsk 660036, USSR Received 29 May 1990
The equilibrium configurations and the process of magnetization of structures with a finite number N of plane domain walls are considered. In comparison with models of a periodic domain structure new effects are found: a difference from zero of the average magnetization at H = 0 and asymmetry of a hysteresis loop for even N, as well as the possibility of the existence of metastable states. The effect of changing the domain structure period in the process of magnetization known for the periodic models is manifested in the structures with a finite N as a sequence of 1st kind phase transitions corresponding to the change in the number of domain walls.
1. Introduction
Within the framework of the phenomenological theory (see e.g. ref. [l]) finding the equilibrium configurations of the magnetization M(r) for a ferromagnet is reduced to the solution of a variation problem for the functional with the energy density F=
<(M(r), g) ++V*M(r) X
/
vt
V-M(r') Ir-r')
dr'
-
H-M(r).
(1)
The first term describes the energy of short-range interactions (exchange, anisotropy). The second term shows long-range magnetostatic interaction, and the third one denotes interaction with the external magnetic field H. The variation problem results in a system of integro-differential equations which cannot be solved in a general form. A cardinal simplification of the problem has been made by Landau and Lifshitz [2]. They have shown that a characteristic domain wall width is, as a rule, much less than the domain sizes. Consequently, the problem of domain structure may be solved in the approximation of infinite thin do0304-8853/91/$03.50
main walls whose structure and energy are calculated beforehand for a solitary domain wall. Further cardinal simplification is based on the fact that in many cases the width of the domains is much less than the corresponding specimen size. This makes it possible to study models in which the specimen size along one or two directions is assumed unlimited and the domain structure periodic. In these models for periodic structures with plane domain walls the problem becomes much simpler and is characterized either by one degree of freedom (domain width) when the equilibrium configuration is determined in a zero magnetic field [3] or by two degrees when the process of magnetization is investigated [4,5]. Besides, in the study of periodic models one can apply a powerful mathematical method such as Fourier series. The results obtained for the periodic models are applied to finite samples only in the limiting case characterized by sufficiently large number of domain walls N >> 1. In real samples this relation does not always hold. Besides, we shall show below that a number of new physical phenomena (the existence of metastable states, peculiarities of magnetization process) have to exist for domain structures with a finite N and they may be ob-
0 1991 - Elsevier Science Publishers B.V. (North-Holland)
V.A. Ignatchenko, E. Yu. Mironov / Magnetic structures with a finite number of domain walls
served even at large enough N. The aim of our work is to study equilibrium configurations and the magnetization process of the plane domain structures with a finite N. But firstly we shall briefly analyze the results obtained earlier when considering the magnetization process in the model of a periodic domain structure. We need it to compare these results and those characteristic of the structures with a finite N.
2. Periodic domain structure The behaviour of a periodic structure with the plane domain walls affected by a magnetic field was first considered independently in refs. [4,5]. It is shown that a system is described by two degrees of freedom: the period of the domain structure 20 and the relation of domain sizes having opposite orientations of magnetizations which can be characterized by q = d/D (fig. 1). At a given H the equilibrium values of q and D are determined by the equations: aF _=
aD
‘7
aF _=
where the energy density has the form: F( D, q; H) = $,M’(I +
5 -
the external field. Paper [4] deals with the case of a specimen unlimited both along the x axis and y axis, that is with domains with magnetization perpendicular to the layer of thickness c. A more general case of a specimen unlimited only in the x direction is studied in ref. [5]. It gives the possibility, in particular, to describe two limiting cases: domains magnetized normal to the plane of layer [4] (b X= c) and ones magnetized in the plane of layer (b +Z c). The general case corresponds to the following expressions for the density of multipole energy Fmp and an effective demagnetizing factor n,.
C,,(D,q)=~ccx
sin’( Tmq/2)
- q)2 + F,,(D, MH(1
-
q).
q) (3)
The first and second terms describe dipole and multipole parts of the magnetostatic energy, respectively. The third term shows the energy of domain walls (y is the surface density of this energy), and the fourth one - the interaction with
m2
m=l
x
00
sin2(bk/2)(1
- eCcp) dk 9
/0
pk2
(4)
(2)
Ov
a4
171
where p2 = k2 + ( Tm/D)2. Owing to the term F,,(D, q) the system of equations (2) becomes coupled, and a change in the magnetic field leads not only to a change of q but also to that of D, i.e. to the effect of an expansion (when H increases) or contraction (when H decreases) of the domain structure period. This effect is the main and qualitatively new result obtained in refs. [4,5]. It was believed earlier that upon the application of a field only the relation between the volumes of the oppositely magnetized domains and not the period of a domain structure must change. The dependence of D on q is determined by the first equation of system (2). For D e c -=x b (domains, magnetized normal to the layer): l/2
D=&(rJITq) i ’
o,(q)
Fig. 1. Periodic domain structure.
=
E sin2(;yl/2). WI=1
(5)
172
V.A. Ignatchenko, E. Yu. Mironov / Magnetic structures with a finite number of domain walls
At H = 0 q = 1 eq. (5) transforms to the expression [3] for the domain width in a plate with an easy axis normal to the surface. For b -=x D -=z c (domains in the plane of a magnetic film) using (4) we obtain
analytical expression for the density of the total energy in the form: F(+...,x,)
2 D
=
W&s:;(q)
(6) o22(q) =
E
+ 2h4H N+l e,x, + $N, ,-_c n=O
’ sin2(;y/2)
=;q(z-q).
m=l
At H = 0 eq. (6) transforms to the expression [6] for the domain width in the film with an easy axis of anisotropy in the plane of the film. The second equation of the system (2) describes the dependence of q on H, i.e. a magnetization curve.
(7)
where the first term is the magnetostatic energy F,, the second term describes the interaction with the external field, and the third one is the energy of the domain walls. Here “configuration charges” of the lattice sites (of the domain walls and edges of the specimen) are introduced: e, = AM( x,)/2M, where AM(x,) the point x,:
(8)
is a jump of the magnetization
at
AM(x,)=M(x,+O)-1%2(x,-O), 3. Structure with a finite N general approach Let us consider a specimen finite in all three dimensions, with the location of the plane-parallel domain walls at arbitrary points with coordinates Xl,...,XN. BY &J and xN+l we denote the coordinates of the left and right edges of the specimen (fig. 2). For certainty we consider the left edge domain magnetized positively. The system is no longer infinite and periodic on the axis x therefore the Fourier series method is inapplicable here. A method adequate to the problem is in fact that the energy be written in the form of a “lattice” sum. We have found an exact
i.e., e, = f. l/2 for n = 0, N + 1 (edges of the specimen) and e, = k 1 for n = 1, . . . , N (domain walls). The potential for the interaction of the “configuration charges” between each other is defined by a function f(x): f(x)
= $arsinh& +i(c2-x2)
(x2
+ c2y2
2 +
$
arsinh % - z
+
5
( C2 -
-
&
-(x2
[
(x2
b2)
arsinh % X
arsinh
+ b2)3’2
+ c2 + b2)3’2
-
( c2 + b2)1’2 1x13+
(x2+C2)3’2
+ ( b2 + c2)3’2
2c3]
- b3 + C2
(b2+c2)
+2b
[
-(x2
+
c2 +
+ cx arctan Fig. 2. Structure with a finite number of domains.
b
arsinh
112 + (x2
b2)“2]
+
C2)1/2
- gaminhi bx
c( x2 + c2 + b2)1’2 ’
(9)
V.A. Ignatchenko, E. Yu. Mironoo / Magnetic structures with a finite number of domain walls
The magnetostatic as lattice sums:
fields can also be described
N+l
Hi=2M
c e,,gi(x,-x, n=O
y, z),
i=x,
y, z, (10)
where 8X(X, --x9 Y, r) = -gXo(x,
- x, b/2 -y,
- Lo 6% -x,
b/2 + y, c/2 - z)
+ g&,,
- x, b/2 -Y,
c/2
+ z)
+ g,,(x,,
-x,
c/2
+ z),
&z-x, =
c/2 - z)
b/2
+Y,
Y9 z) -x,
-g,o(x,
+g&,--x, -x,
- g,o(x, +gyo(x,-x, gz(x, -x9
b/2-y,
c/2-z)
b/2+Y,
c/2-r)
b/2-y,
c/2+z)
b/2+Y,
c/2+1),
=g&,-AC,
b/2-Y,
+ g&,,
Thus, a general configuration force acting on the wall is the magnetic pressure averaged over the square of the domain wall: --=
aF
- x, b/2 +Y,
+g&,,-x,
b/2-y,
+g&/-x,
bP+y,
gxo(u,
u,
w) = arsinh
g,,(u,
u,
w) = arsinh
g,,(u,
u,
w) = arctan
c/2
-z)
c/~+z) @+z),
u
(#2 + w2)1’2 ’ u
(u2 + w2)“2 ’ uu
w( u2 + u2+ It may be shown that
w2)1’2 .
y, z>)y,Z - Fek.
(12) In a specimen with a finite number of domain walls the problem of the configuration of a domain structure is formulated in a different way than for the model of a domain structure periodic in x. Since the “configuration charges” corresponding to the edges of the specimen are rigidly fixed by the forces of nonmagnetic origin for each value of N there is such a location of the domain walls that corresponds to the minimum of the magnetostatic energy independently of the energy of domain walls. Hence: 1) For each of the fixed values of N there must be found a configuration of domain walls (the values of coordinates x,, . . . , xN) minimizing the total energy (7) and representing the solution of a system of N equations:
axI
c/2-z)
- T(H;(x,,
ax,
aF -=
YP r)
173
aF
o
-
-=
o
ax, .
03)
The equations do not depend on the energy of domain walls and express the equality to zero of all general configuration forces acting on them. 2) Substituting the obtained values of coordinates in (7) we find N corresponding to the minimum of energy for a given H. At H = 0 we obtain the ground state N = No. 3) When considering the behaviour of the system in a magnetic field (or under the influence of other forces) one should find intervals where the configuration of domain walls will change with their number N being constant, and where (and in what way) the phase transitions will occur with change of N.
4. Structure with a finite N particular cases =$“ilenfx(xn-x), PI-0 where
f,(x)
=
af/ax.
Now we consider specimens in which N is small, that is the case when the Fourier expansion does not work, and the method of lattice sums
174
V.A. Ignatchenko, E. Yu. Mironov / Magnetic structures with a finite number
ofdomainwalls
Fig. 3. Dependence of the magnetostatic field HA at I = 0 on coordinate x for the equilibrium position x1 = 0 of a domain wall at N = 1 for c/a = 0.1 (a) and c/a = 10 (b).
does not require a great many calculations yet. For simplicity we investigated the limiting case b -+ 00. In this case the finiteness in x is retained and so retained are all the qualitative peculiarities of the system caused by the finiteness of N. The center of coordinates will be in the center of the specimen: x0 = -a/2, xN+i = a/2. Let us consider the equilibrium domain structure, distribution of magnetostatic fields and peculiarities of the magnetization curve for different values of N. (a) N = I: The equilibrium position of a domain wall at H = 0 corresponds to x1 = 0. The component Hk of the magnetostatic field vanishes at x = 0. In dependence on the relation between the sizes of the specimen c and a it may reach the largest values both inside the domains (fig. 3a) and at the edges of the specimen (fig. 3b). The magnetization curve is symmetric. (b) N = 2: The problem is two-particle, and the state of the system is described by a two-dimensional energy surface F, = Fm(xl, x2) which is shown in fig. 4. However, due to the symmetry x2= -x,=D both at H=O and at H#O. Therefore the problem is confined to single-particle and described only by section OA of the energy surface Fm(xlr x2) shown in fig. 5a. Fig. 5b depicts the magnetization curve. It is easy to see that the equilibrium configuration of domain walls at H = 0 does not correspond to a demagnetized state of the specimen. The difference of Do from 0.25u, at which the average magnetization (M,) would be equal to zero, is not large but principal. A significant reason for this effect is that the minimization of the magnetostatic energy corre-
-0.5 Fig. 4. The relief of the normalized magnetostatic F,(x,, x2)/M* for N = 2 at a/c =l.
energy
sponds to the minimization of a sum of dipole and multipole parts of the energy, and the minimum of the sum does not always show that the condition for the equality to zero of the magnetic moment holds exactly, as was the case for the model of a periodic structure. The second unusual effect the asymmetry of the magnetizationcurve - is due to different ways of magnetization: in the positive field domain walls approach each other and annihilate, and in the negative field they are displaced to the edges of the specimen. Both these effects increase with the growth of the ratio u/c. Fig. 6 shows the dependence H&(z) along a domain wall when it is just in the equilibrium position (b) and when it is slightly displaced in one (a) or another (c) direction from the equilibrium
0.2249
0.5
Fig. 5. Dependence F,(D) for N = 2 at a/c =1 (a) and the magnetization curve for this case (b).
V.A. Ignatchenko, E. Yu Mironov / Magnetic structures with a finite number of domain walls
position. It is clearly seen that in the equilibrium position there are a number of alternating sections where the field is not equal to zero and of a different sign. This means that the domain wall experiences the forces tending to bend it. The model of plane domain walls corresponds to the condition of a sufficiently large energy of a domain wall in comparison with magnetostatic energy. When the condition for predominance of the energy of surface tension is violated a domain wall will bend in the z direction in accordance with the pattern of magnetic fields in fig. 6b. The domain wall is located at the point where the average value of the magnetic pressure on its surface becomes zero according to expressions (12) and (13). (c) N = 3: A three-particle problem is reduced to single-particle due to symmetry only for H = 0, when x2 = 0, x3 = -xi = D; the application of a magnetic field destroys the symmetry. The dependence of I;, on D for H = 0 is shown in fig. 7a. This case is very interesting due to the fact that the condition for the equality to zero of the total magnetic moment in the specimen holds for any D. Therefore the minimum on curve F,(D) is wholly due to the multipole part of the magnetostatic energy. The distribution of the z-component of the magnetostatic field in the plane z = 0 is shown in fig. 7b.
0.3097
175
0.9
Fig. 7. Dependence F,(D) (a) and distribution of the magnetostatic field HA on x in the plane of s = 0 (b) for N = 3, a/c=l.
The magnetization curve is shown in fig. 8. At there occurs a displacement of one of H=H,,, the domain walls to the edge of the specimen. Thus, for H3_2 < 1H 1 < H,’ the state with N = 3 cannot exist in principle. (d) N = 4: The problem is reduced to two-particle both at H=O and H#O: x3= -x2, x4= -xl. In fig. 9 a dashed curve shows the states of the system upon the application of the external field H. As well as for N = 2 this case is characterized by an asymmetric magnetization curve and the slight residual magnetization at H = 0. However both effects are less pronounced than for N = 2. In a field H = - H4_ 2 two domain walls are displaced to the edges of the specimen. Therefore in the fields -Hz < H < - H4 _ 2 there cannot be in principle the state with N = 4. Let us discuss some results. Fig. 10 shows the dependence of magnetostatic energy on the num-
CM,> M 04
Fig. 6. Distribution of the field HA in the plane of a domain wall for N = 2 (a/c = 1) for the equilibrium position of domain walls: D/a = 0.2249 (b) and displaced from the equilibrium position: D/a = 0.2558 (a), D/a = 0.25 (c).
Fig. 8. Magnetization curve for N = 3, a/c = 1.
N=I
V.A. Ignatchenko, E. Yu. Mironov / Magneric structures with a finite number of domain wails
176
0.1
0.1
Fig.
9.
0.5
Relief of the normalized magnetostatic Fm(x2. x,)/M2 for N = 4, a/c =l.
energy
ber of domain walls for the equilibrium configurations at H = 0. Solid curves in this figure indicate the dependence F,(N) for the same relations of the specimen sizes calculated according to the equation from ref. [3] which was obtained in the approximation of a domain structure periodic in x and having the domain width D = a/N (formula (4) at q = 1). It is seen that for N 1 3 + 4 the
deviations between the exact and approximate values of Fh become insignificant. However the qualitative differences in the remagnetization process calculated for the model of a periodic structure and ‘structure with a finite number of domains are still retained for the large values of N. We would like to consider these processes in detail. Fig. 11 shows schematically the dependence of the total energy of the system (7) on the field for two equilibrium states corresponding to N = 2 and N = 4. At H = 0, F4 -c F2. At H = H, the energies of both states become equal and for H > H,, F4 > F2. However the state with N = 4 is separated from that with N = 2 at H = Hf by a potential barrier: the increase in the magnetostatic energy (fig. 9) prevents a transition between the states. Therefore with the growth of H the system remains in the state N = 4 and moves along a phase trajectory corresponding to the larger energy up to the point H=H,,,, where the domain walls touch tKe edges of the specimen. At this point the system jumps to the state N = 2.
F ,a--
\
-=$:
F, / M'
I<“’-\ -.
\\
F2 \
‘\\
H2-.4
Hf
H4-2
H
N=2
1
2
3
N
Fig. 10. Dependence F,(N) for a specimen with a finite number of domains (light circles) and for the model of the periodic domain structure (solid lines) at different c/a: 0.5 (a), 1 (b), 5 (c).
Fig. 11. Schematic dependence of the total energy of the system on the field H for equilibrium states of the system with N = 2 and N = 4. ( ) is the trajectory of the system when H increases, and ( + + t ) when, H decreases.
V.A. Ignatchenko, E. Yu Mironov / Magnetic structures with a finite number
Upon decreasing the field the system remains in the state with N = 2 up to H = Hf and may be there longer if additional energy is required to generate new domain walls (nucleation energy). In the latter case the transition from N = 2 to N = 4 will occur in the field H = H2 _ 4 < Hf. Similar phase transitions of the 1st kind also take place at other points of intersection of the curves F(H) corresponding to different values of N. Fig. 12 shows the dependence of N on H for a system with the equilibrium structure corresponding at H = 0 to the value of N, = 4. In the magnetization process two phase transitions occur: between the states with N = 4 and N = 2 and ones with N = 2 and N = 0. The same figure gives a smooth dependence of N on H which corresponds to the model of a periodic domain structure (5) for N 5: a/D. Accordingly, for the system with the large value of N, one would observe more phase transitions on the curve N(H). Thus, the effect of changing the domain structure period as a function of H calculated for the periodic models in refs. [4,5] in specimens with a
) is the equilibrium Fig. 12. Dependence of N on H. (trajectory for the model of the domain structure with a fiite number of domains. ( ) is the trajectory of the system when H increases, ( + + + ) when H decreases. The dotted line corresponds to the model of the periodic domain structure.
of domainwalls
177
finite N is manifested as a sequence of phase transitions of the 1st kind corresponding to a change in the number of domain walls. In the vicinity of these transitions the structure may be in the metastable states.
5. Conclusions Expressions (3) and (7) represent two different ways of giving a reduced description of the system which is described completely by the energy density (1). System (1) possesses an infinite number of degrees of freedom, whereas system (3) has only two degrees, and system (7), N degrees of freedom, where 1 G N < co. Due to this fact the model of a domain structure with a finite number of domains more nearly describes the real physical situation than the model of a periodic domain structure. It is a further approximation to the complete phenomenological description (1) and therefore some new phenomena become apparent in it which cannot be found in the frames of the model of a periodic structure. Let us summarize these. For an even number of domain walls N, the different from zero average magnetization of .the specimen at H = 0 and an asymmetric remagnetization curve correspond to the equilibrium configuration. Both effects decrease with increase of N. For N > 1 the width of domains located at the edges of the specimen is approximately a factor of 2 smaller than the widths of all other domains. The relation holds the better the larger N is. The effect of changing the domain structure period [4,5] as a function of H which is known for the periodic models is manifested for structures with a finite N as a sequence of phase transitions of the 1st kind corresponding to an uneven change in the number of domain walls. In the vicinity of these phase transitions the structure may be in the metastable states. Their existence is caused by the necessity to overcome a potential barrier in order to change the topology of the system. When H increases, the barrier is due to the increase in the magnetostatic energy in the transition to the structure with a smaller N. When H decreases, the barrier is due to the nucleation process. A high
178
barrier of when H = state with states with
VA. Ignatchenko,E. Yu. Mironov / Magnetics!rucrureswitha fin& numberof domainwall3
nucleation may lead to the situation 0 will correspond not to the ground N = NO but to one of the metastable N c NO.
References [l] W.F. Brown, 1963) p. 63.
Micromagnetics
(Interscience,
New York,
PI L.D. Landau and E.M. Lifshitz, Phys. Z. Sovjettmion 8
(1935) 153. [31 Ch. Kittel, Rev. Mod. Phys. 21 (1949) 541. [4l C. Kooy and U. Enz, Phil. Res. Rep. 15 (1960) 7. 151 V.A. Ignatchenko, J.F. Degtyarev and Yu.V. Zakharov, Izv. Akad. Nauk SSSR, Ser. Fiz. 25 (1961) 1439. English translation in: Bull. Akad. Sci. USSR, Phys. Ser. (USA) 25 (1961) 1452. 161V.A. Ignatchenko and Yu.V. Zakharov, Zh. Eksp. Tear. Fiz. (USSR) 43 (1962) 459.