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The influence of domain wall dynamics on magnetization processes and internal friction
19June
1995
PHYSICS
LETTERS
A
Physics Letters A 202 (1995) 230-232
ELSEVIER
The influence of domain wall dynamics on magnetization processes and internal friction Jakob Bohr a, Victor V. Makhro b, Alexander M. Tishin ’ a Physics Department, Technical Universiv of Denmark, Building 307, DK-2800 Lyngby, Denmark b Department of Physics, Bratsk Industrial Institute, 665728 Brat& Russian Federation ’ Faculty of Physics, Moscow State University, 1198899 Moscow, Russian Federation
Received 10 March 1995; accepted for publication 6 April 1995 Communicated by V.M. Agranovich
Abstract The results of numerical investigations of the peculiarities of domain wall dynamics and its influence on internal friction and remagnetization processes in high coercivity rare-earth compounds are presented. For the first time the appearance of the nonlinear domain wall excitations in materials with intrinsic coercivity was theoretically shown.
the increase in internal friction and the remagnetization peculiarities are considered.
1. Introduction Most heavy rare-earth metals and their compounds have a very high magnetocrystalline anisotropy energy and an unusual domain structure. The domain walls are localized in the periodical potential of the intrinsic coercivity and an external field whose value is greater than the coercive force H, is needed for the delocalization of the walls. After the delocalization the walls can move inside the crystal and the remagnetization takes place. However, a domain wall can interact with the intrinsic coercivity potential even in the case when the external field is greater than H,. Additional energy losses arise because of these processes and the internal friction can increase. The influence of this phenomenon on the internal friction in domain structures of magnets is discussed in Refs. [1,2]. In the present paper the same mechanisms of both 03759601/95/$09.50
0 1995 Elsevier Science B.V. AR rights reserved
SSDI 0375-9601(95)00320-7
2. Parametrical
excitation
of Winters vibrations
The domain wall dynamics in different magnets have been investigated in many works (see for example Ref. [3]). In Ref. [4] the domain wall dynamics in the weak ferromagnets was discussed using the soliton theory of perturbation. A description of the nonlinear dynamics of the domain walls for antiferromagnetic magnet systems was given in Ref. [S]. In Ref. [6] a microscopic description of the spin wave of narrow domain walls was developed. In that paper Egami showed that a domain wall in high coercivity materials can be described as a quasi particle which corresponds to the ground state mode in the system of localized spin waves (Winters vibrations) and extended this model for investigations of
J. Bohr et al. /Physics
the domain wall tunneling phenomena. However, the results of Ref. [6] may be useful for the analysis of translational wall motion too. Let us consider a flat domain wall which moves in the periodical potential of internal coercivity. The equation for the displacement of the wall center coordinate x in the external field H B H, is a/@( ti)
+ tir-
= 2M,H
and the linearized where m L = m,(l - _?‘/c~)-~/~, equation for the Winters vibrations is
cos(27rut/a).
t).
Using the transformation + f,
(P(t)
@(t) = (a/u)f( the temporary @ + ( Ta/u)
NI-J t=800ns 2OOa Fig. 1. The temporal dependence velocity of the domain wall in the potential relief of the intrinsic coercivity.
- r2a2/4,rr2u2,
b=2Vo/mLu2. The general solution of Eq. (3) is
=f( at/u), at/u);
$(t) = (a/u)‘fi
at/u) (4)
Q + [ ( mc2k2a2/m I u”)
CS(t)+(h+bcos2t)@(t)=0,
Fb = d/dx(2riq/a)*. of of
(5) 3. Nonlinear domain wall excitations
or, if ‘p( t) = exp( - rat/2u)
and their amplitude has a very complicated type of evolution in time. This leads to additional absorption of energy by the wall in the case when bends increase, and the velocity of the wall changes abruptly. In this case one must take into account the effective braking force which arises due to this energy absorption,
For example, we present in Fig. 1 the results numerical calculations of the temporate evolution the domain wall velocity.
part of Eq. (3) yields
+ (4,rr2Vo/m I u’) cos(2lrt)] cp= 0,
(7)
so bent vibrations propagate in the plane of the wall, (3)
Let us find the solution of (3) in the form
ut/a
ma
q(t) = qoe-rr’2e-ik’r~(~ut/a),
mLij+mLQr-mcZV,2q
+‘f(
200a
h = (mc2k2a2/r2mLu2)
+ 2M,,H, (2)
4=4oe
ma
where
m.f+mL~+miT+mL~~--mc*V:q
= - (47r*V,/a*)q
t=ZODns
(1)
[7,81. Assuming that x = ut + q, where q describes the evolution of wall bends and u is the velocity of the wall, (1) yields
ut + q)/a]
ma
200a
sin(2nx/a),
sin[2r(
t=30ns
ioa. t=60Rns
where m = m,(l - i’/~*)-~‘*, m, is the Dbring mass of the wall, u = mc*, c is the Walkers velocity, a the space period of the potential and V,, the potential amplitude. Eq. (1) has been obtained and discussed in Refs.
= (27rV,,/a)
9
t=400ns
VI uv,x
+ (2rV,,/a)
231
Letters A 202 (I 995) 230-232
@( t/29r), (6)
In the case when H > H, some interesting peculiarities of wall exitations can be observed. In this
232
J. Bohr et al. /Physics
KC
45 ‘O.-^..
0
0,5 t415.
Fig. 2. The nonlinear Winters vibration.
phase of the parametrical
case Eq. (2) cannot be linearized the system of equations R + rf
to
s
evolution
of the
and one must solve
+ Fb - l/m[2M,H
+ ( Vo/a)
sin(27rx/a)]
(1 - i2/cz)3’2
q + rQ - (47rqVo/m,a2)
F,, = d/dx(27riq/u)‘.
- i2/c2)3’2
= 0,
high coercivity rare-earth compounds. Obviously, losses of energy for remagnetization will be increased due to passing to the system of the Winters vibrations. In the nonlinear case losses are especially great. Our calculations showed that due to these losses the external remagnetization field must be increased by about lo-15% to compensate the losses. Such phenomena also influence the value of the internal friction in the region where thermal resonance deepening [2] takes place. The domain wall in this case is delocalized from the intrinsic coercivity potential relief minimum due to its resonance interaction with thermal magnons. Transferring the energy into the Winters vibrations system interrupts the process of the displacement of the walls, and the wall movement becomes “jumpy”. A detailed examination of this process will be presented in a separate paper.
= 0, (8a)
x cos(23rx/a)(1
Letters A 202 (1995) 230-232
(8b) (8c)
Numerical investigations of system (8) revealed that linear Winters vibrations can be turned into the nonlinear phase. This phenomenon is accompanied by the appearance and evolution of localized spin waves on the surface of the moving domain wall. The example of origin and evolution of a soliton-like superficial wave on the domain wall is presented in Fig. 2.
4. Discussion Let us discuss the problem of the influence of such phenomena on the remagnetization processes in
Acknowledgement We are grateful to A.K. Zvezdin for the useful discussions. The authors acknowledge the financical support by grants of the International Association for the Promotion of Cooperation with Scientists from the Independent States of the Former Soviet Union.
References I11 A.M. Tishin and O.A. Shipilov, Fiz. Tverd. Tela 34 (1992) 3554.
I21 A.M. Tishin and V.V. Makhro, Phys. Lett. A 189 (1994) 331. 131 A.A. Malozemov and 1. Slonzusky, Domennye stenki v materialakh s czilindricheskimi magnitnymi domenami (Mir, Moscow, 1982) [in Russian]. [41 A.K. Zvezdin, JETP L.ett. 29 (1979) 605. 151 M.V. Chetkin et al., Sov. Phys. JETP 67 (1988) 151. I61 T. Egami, Phys. Stat. Sol. (b) 57 (1973) 211. [71 A.K. Zvezdin and A.A. Mikhin, Sov. Phys. JETP 75 (1992) 306.
1995
PHYSICS
LETTERS
A
Physics Letters A 202 (1995) 230-232
ELSEVIER
The influence of domain wall dynamics on magnetization processes and internal friction Jakob Bohr a, Victor V. Makhro b, Alexander M. Tishin ’ a Physics Department, Technical Universiv of Denmark, Building 307, DK-2800 Lyngby, Denmark b Department of Physics, Bratsk Industrial Institute, 665728 Brat& Russian Federation ’ Faculty of Physics, Moscow State University, 1198899 Moscow, Russian Federation
Received 10 March 1995; accepted for publication 6 April 1995 Communicated by V.M. Agranovich
Abstract The results of numerical investigations of the peculiarities of domain wall dynamics and its influence on internal friction and remagnetization processes in high coercivity rare-earth compounds are presented. For the first time the appearance of the nonlinear domain wall excitations in materials with intrinsic coercivity was theoretically shown.
the increase in internal friction and the remagnetization peculiarities are considered.
1. Introduction Most heavy rare-earth metals and their compounds have a very high magnetocrystalline anisotropy energy and an unusual domain structure. The domain walls are localized in the periodical potential of the intrinsic coercivity and an external field whose value is greater than the coercive force H, is needed for the delocalization of the walls. After the delocalization the walls can move inside the crystal and the remagnetization takes place. However, a domain wall can interact with the intrinsic coercivity potential even in the case when the external field is greater than H,. Additional energy losses arise because of these processes and the internal friction can increase. The influence of this phenomenon on the internal friction in domain structures of magnets is discussed in Refs. [1,2]. In the present paper the same mechanisms of both 03759601/95/$09.50
0 1995 Elsevier Science B.V. AR rights reserved
SSDI 0375-9601(95)00320-7
2. Parametrical
excitation
of Winters vibrations
The domain wall dynamics in different magnets have been investigated in many works (see for example Ref. [3]). In Ref. [4] the domain wall dynamics in the weak ferromagnets was discussed using the soliton theory of perturbation. A description of the nonlinear dynamics of the domain walls for antiferromagnetic magnet systems was given in Ref. [S]. In Ref. [6] a microscopic description of the spin wave of narrow domain walls was developed. In that paper Egami showed that a domain wall in high coercivity materials can be described as a quasi particle which corresponds to the ground state mode in the system of localized spin waves (Winters vibrations) and extended this model for investigations of
J. Bohr et al. /Physics
the domain wall tunneling phenomena. However, the results of Ref. [6] may be useful for the analysis of translational wall motion too. Let us consider a flat domain wall which moves in the periodical potential of internal coercivity. The equation for the displacement of the wall center coordinate x in the external field H B H, is a/@( ti)
+ tir-
= 2M,H
and the linearized where m L = m,(l - _?‘/c~)-~/~, equation for the Winters vibrations is
cos(27rut/a).
t).
Using the transformation + f,
(P(t)
@(t) = (a/u)f( the temporary @ + ( Ta/u)
NI-J t=800ns 2OOa Fig. 1. The temporal dependence velocity of the domain wall in the potential relief of the intrinsic coercivity.
- r2a2/4,rr2u2,
b=2Vo/mLu2. The general solution of Eq. (3) is
=f( at/u), at/u);
$(t) = (a/u)‘fi
at/u) (4)
Q + [ ( mc2k2a2/m I u”)
CS(t)+(h+bcos2t)@(t)=0,
Fb = d/dx(2riq/a)*. of of
(5) 3. Nonlinear domain wall excitations
or, if ‘p( t) = exp( - rat/2u)
and their amplitude has a very complicated type of evolution in time. This leads to additional absorption of energy by the wall in the case when bends increase, and the velocity of the wall changes abruptly. In this case one must take into account the effective braking force which arises due to this energy absorption,
For example, we present in Fig. 1 the results numerical calculations of the temporate evolution the domain wall velocity.
part of Eq. (3) yields
+ (4,rr2Vo/m I u’) cos(2lrt)] cp= 0,
(7)
so bent vibrations propagate in the plane of the wall, (3)
Let us find the solution of (3) in the form
ut/a
ma
q(t) = qoe-rr’2e-ik’r~(~ut/a),
mLij+mLQr-mcZV,2q
+‘f(
200a
h = (mc2k2a2/r2mLu2)
+ 2M,,H, (2)
4=4oe
ma
where
m.f+mL~+miT+mL~~--mc*V:q
= - (47r*V,/a*)q
t=ZODns
(1)
[7,81. Assuming that x = ut + q, where q describes the evolution of wall bends and u is the velocity of the wall, (1) yields
ut + q)/a]
ma
200a
sin(2nx/a),
sin[2r(
t=30ns
ioa. t=60Rns
where m = m,(l - i’/~*)-~‘*, m, is the Dbring mass of the wall, u = mc*, c is the Walkers velocity, a the space period of the potential and V,, the potential amplitude. Eq. (1) has been obtained and discussed in Refs.
= (27rV,,/a)
9
t=400ns
VI uv,x
+ (2rV,,/a)
231
Letters A 202 (I 995) 230-232
@( t/29r), (6)
In the case when H > H, some interesting peculiarities of wall exitations can be observed. In this
232
J. Bohr et al. /Physics
KC
45 ‘O.-^..
0
0,5 t415.
Fig. 2. The nonlinear Winters vibration.
phase of the parametrical
case Eq. (2) cannot be linearized the system of equations R + rf
to
s
evolution
of the
and one must solve
+ Fb - l/m[2M,H
+ ( Vo/a)
sin(27rx/a)]
(1 - i2/cz)3’2
q + rQ - (47rqVo/m,a2)
F,, = d/dx(27riq/u)‘.
- i2/c2)3’2
= 0,
high coercivity rare-earth compounds. Obviously, losses of energy for remagnetization will be increased due to passing to the system of the Winters vibrations. In the nonlinear case losses are especially great. Our calculations showed that due to these losses the external remagnetization field must be increased by about lo-15% to compensate the losses. Such phenomena also influence the value of the internal friction in the region where thermal resonance deepening [2] takes place. The domain wall in this case is delocalized from the intrinsic coercivity potential relief minimum due to its resonance interaction with thermal magnons. Transferring the energy into the Winters vibrations system interrupts the process of the displacement of the walls, and the wall movement becomes “jumpy”. A detailed examination of this process will be presented in a separate paper.
= 0, (8a)
x cos(23rx/a)(1
Letters A 202 (1995) 230-232
(8b) (8c)
Numerical investigations of system (8) revealed that linear Winters vibrations can be turned into the nonlinear phase. This phenomenon is accompanied by the appearance and evolution of localized spin waves on the surface of the moving domain wall. The example of origin and evolution of a soliton-like superficial wave on the domain wall is presented in Fig. 2.
4. Discussion Let us discuss the problem of the influence of such phenomena on the remagnetization processes in
Acknowledgement We are grateful to A.K. Zvezdin for the useful discussions. The authors acknowledge the financical support by grants of the International Association for the Promotion of Cooperation with Scientists from the Independent States of the Former Soviet Union.
References I11 A.M. Tishin and O.A. Shipilov, Fiz. Tverd. Tela 34 (1992) 3554.
I21 A.M. Tishin and V.V. Makhro, Phys. Lett. A 189 (1994) 331. 131 A.A. Malozemov and 1. Slonzusky, Domennye stenki v materialakh s czilindricheskimi magnitnymi domenami (Mir, Moscow, 1982) [in Russian]. [41 A.K. Zvezdin, JETP L.ett. 29 (1979) 605. 151 M.V. Chetkin et al., Sov. Phys. JETP 67 (1988) 151. I61 T. Egami, Phys. Stat. Sol. (b) 57 (1973) 211. [71 A.K. Zvezdin and A.A. Mikhin, Sov. Phys. JETP 75 (1992) 306.