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Submitted on 1 Jan 1988
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DOMAIN FORMATION IN FERROMAGNETIC
FILMS INCLUDING MAGNETOELASTIC EFFECTS
W. Wasilewski, M. Gajdek, G. Gehring
To cite this version:
JOURNAL D E PHYSIQUE
Colloque C8, Supplkment au no 12, Tome 49, dekembre 1988
DOMAIN FORMATION IN FERROMAGNETIC FILMS INCLUDING
MAGNETOELASTIC EFFECTS
W. Wasilewski ( I ) , M. Gajdek ( 2 ) and G. A. Gehring ( 3 ) ( I ) Department of Physics, Technical University, Radom, Poland ( 2 ) Department of Physics, Technical University, Kielce, Poland
( 3 ) Department of Theoretical Physics, Oxford University, Oxford OX1 3NP, G.B.
Abstract. - Static and dynamic properties of a thin ferroelastic film near the phase transition from the state of ho-
mogeneous magnetization and deformation to the state of domain and modulated structures are considered. The phase transition is caused by a change of external magnetic field. Formulae describing magnetic and magnetoelastic susceptibility tensors are derived and discussed.
The aim of this paper is to describe static and dynamic magnetoelastic properties of a thin ferromag- netic film near the phase transition from the state of homogeneous magnetization and homogeneous defor- mation to the state with domain and modulated struc- tures. Let us consider a thin film of thickness L made of uniaxial ferromagnet with the easy magnetization axis perpendicular to the surfaces of the film and par- allel t o the zaxis of the Cartesian coordinate system. Let us assume that the anisotropy energy of the mate- rial is less than the maximum value of the demagnetiz- ing energy. In such kind of films for the film thickness L
<
LC the ground state is the state of homogeneous magnetization in the film plane, i.e. in the hard mag- netic direction. The symbol LC denotes the critical thicknessof the film for which there is the phase tran- sition [I, 21 from the homogeneous magnetization state to the domain structure which is the ground state for L>
LC. A film of thickness L>
LC undergoes this transition at a critical value of an external magnetic field lying in the film plane H,' ( L ).
The energy of the film is described [3] by the func- tional:
The symbol hd denotes the vector of reduced demagne- tizing field connected with the reduced magnetization vector m via Maxwell's equations:
Let us assume that the ground state is the state of homogeneous magnetization m: and homogeneous de- formation E ! ~ determined by:
The excitations pi ( r , t ) and EG (r, t ) are defined as fol- lows:
0 0
p ; ( r , t ) = m ; ( r I t ) - m i , & ; j ( r 1 t ) = e ; j ( r , t ) - e i j .
( 5 )
In the case of strong magnetic surface anisotropy and traction free surfaces the boundary conditions have the form:
where n; is the vector perpendicular to the film sur- faces. R o m an examination of the energy of magnetic and elastic excitations in the q representation, it fol- lows [4] that the postulated ground state is stable if
. .
+
( B I I m?+
Blarn;) e,= where qo= (go,, 0, 7rL-l) and
+
(Blarn?+
~ l l m ; ) e y y+
B 3 s d e Z z+2B44 (mym.eyz
+
mzmrexz)+
2Bssmzmyexy1 9: = 22n
+ l C l ~ (e:.
+
e:,)+
C ~ t e . . e ~ ,where The symbol
P*
denotes the renormalized anisotropy constant. The function hc ( L ) defines the phase bound-C8 - 1952 JOURNAL DE PHYSIQUE
For h = h, ( L ) and q = qo there is a phase transi- tion from the homogeneous magnetization state to the domain structure with an initial period XO = 2nqc. along the *axis.
The motion equations have forms [5]:
ami/at = Wo Eijk mjhif
-
Two EijkEkln mjmlh:f,(10)
pa2ui/at2 = Oij,j
+
yij,j, (11) where wo = gMo and(12) The damping tensor
vijal
has the same structure as the elastic stiffness tensor Cijkl. Solutions of the set of motion equations (10, 11) should fulfil the Maxwell's equations (3) and boundary conditions (6). It has been shown 141 for undamped dispersion relations, that there exists a quasi-acoustic soft mode corresponding t o the structurd phase transition if the quasi equilib- rium condition 6F/6&ij = 0 is fulfilled. In this case the magnetic susceptibility tensorX;
(q, w ) has the form:XE
=-
[(I+
r2) 6.. (q)-
irwwO1] 6,' (q, w),
x z
=-
[(I+
r2) 6xx (q)-
irww~'] 6': (q, w ),
xzz
= [(I+
r2) 6x2 (q)+
i w w ~ ~ ] '6: (q, w),
x,",
= [(I+
rZ) &Z (9)-
~WWO'] '6: (q, w) (13) where2
6,, (q) = aq
+
4 ~ 9 , 2 9 - ~+
h, 6,. (q) = 4~q,q.q-~, 6,, (q) = aq2+
-
P*
+
h. (14) The symbol 6. (q, w) is determined as follows:6s (qr W) = w2wO2
+
irwwi' (6,,+
SZz)-
(1 f r2) M ~ ~ A O (q).
(15)The function A0 (q) has been defined by the formula
(7). The magnetoelastic susceptibility tensor defined as
has the form:
Susceptibility tensors (13, 17) are singular
at
the phase transition point w = 0, h = h, ( L ) and q = q o . It means that simultaneously with the appearance of the domain structure there develops the modulated struc- ture of the displacement vector. From an examination of magnetoelastic excitations it follows that they are overdamped near the phase transition point. Excita- tions parametrized by the vector q = qo relax most slowly and become unstable a t the phase transition point h = h, (L).
It corresponds t o the nucleation of domain and modulated structures.Acknowledgment
Work has been supported in part by the University of Lodz, Poland, under contract CPBP 01.08.
[I] Hubert, A., Theorie der Domanenwande in Geordnete Medien (Springer Verlag, Berlin) 1974. [2] Gajdek, M. and Wasilewski, W., J. Magn. Magn.
Mat. 37 (1983) 246; J. Phys.
C
18 (1985) 3557. [3] Brown, W. F., Micromagnetics (John Willey &Sons, New York-London) 1963.
[4] Gehring, G. A., Waailewski, W. and Gajdek, M., J . Phys.
