8/3/2019 J.A. Tuszynski- Time-Dependent Landau-Ginzburg Magnetic Modelling of Magnetic Inhomogeneities and Spiral Doma

1/3

265

TIME-DEPENDENT LANDAU-GINZBURG MODELLING OF MAGNETIC INHOMOGENEITIES

AND SPIRAL DOMAIN STRUCTURES IN THIN FILMS

J.A. TUSZYNSKIDepartment of Physics, The University of AlbertsEdmonton, Alberts, Csnada, T6G 2Jl

ABSTRACT

A Landau-Ginzburg model for a uniaxial ferromagnetic thin film ispresented which includes inhomogeneities due to nearest neighbor exchangeinteractions. The role of external magnetic fields is studied in thedetermination of exact symmetries of the equation of state. Spiral domainstructures allowed by symmetry considerations as exact solutions at thecritical and tricritical points have recently been found experimentally inepitaxially grown single-crystal garnet ferrite films. These experimentsare discussed from the point of view of our model.

THE DYNAMICAL LANDAU-GINZBURG MODEL

The Landau-Ginzburg free energy functionalf n [ 2 4 6 1 2 ]= d x A2n + A4n + A6n -hn + 2 D(Vn) (I)

where n is spatial dimensionality, ~ = a(!-! ), A4' A6 are constants, n is

the order parameter and h is the conjugate Field, has played a prominent

role in the theory of critical phenomena [I] since it can be used to des-cribe a number of phase transitions, both field- and temperature-induced(h;O and h=O, respectively), first -as well as second-order ones (when~

8/3/2019 J.A. Tuszynski- Time-Dependent Landau-Ginzburg Magnetic Modelling of Magnetic Inhomogeneities and Spiral Doma

2/3

267

for a s sine and ~ s (Hl'O). In both cases, however, the equations are ofthe same type as those analysed in [2] and [3], namely they are firstintegrals of the cubic (K2s0) or quintic (K2#O) nonlinear Klein-Gordon

equation. A particular solution for Hl=HII=O and K2=O in the form of adomain wall cos e s -tanh rK;lD x has been known for many years. The resul tspresented in [2] and [3], however, allow a systematic analysis of excita-tions in uniaxial magnets. Many of the geometries discussed in previoussection have been commonly observed in magnets, e.g. spherical a-Fe203microparticles [4], cylindrical iron whiskers [5], multilayered structures[6], and very recently spirals [7]. The formation of solitons in magneticsystems has also been given extensive exposure [8]. In our model both sechand tanh solitary waves are obtained as limiting cases of elliptic functionsbecoming of very long wavelengths, i.e. their elliptic modulus tends to one.Moreover, exact energies of all these solutions can be directly calculated.

SPIRAL DOMAINS IN UNIAXIAL THIN FILMS

A particularly interesting application of our method can be made tomodel spiral domains of uniaxial ferromagnetic thin films. It has beenrecently reported [7] that epitaxially grown single-crystal garnet ferritefilms (YISm)3(FeGa)S012 exhibit the formation of spiral domains when subjec-ted to an alternating square magnetic field along the easy magnetizationaxis along the normal to the sample for 80 Oe < IHI < 87 Oe and at a fre-quency of 00 a 300 Hz. The spiral domains of large area (~l mm) were formedfollowing a destabilization of the typical labirynthine domain structureleading to complete randomization in the magnetic structure of the sample.The lifetimes of the created spirals were large, on the order of lOs and

they persisted for as long as the external field was applied. This tends toindicate a very peculiar nonequilibrium metastable phase. These structuresalso exhibited soliton-like qualities of stability with respect to colli-sions when pairs were created.

We believe that these observations can be explained using the Landau-Ginzburg model presented in this paper. First of all, it is not uncommon touse Landau-Ginzburg modelling for thin films [9]. Symmetry reductionanalysis for two-dimensional systems is summarized in Table 1 following [2].Secondly, at both the critical (Na3) and tricritical (N-5) points exactminimum free energy solutions in two-dimensional space have been demonstra-ted to contain spiral patterns [2,3] in addition to the less exotic planeand spherical waves. Their form is given by:

M (x,t) D p(x) f(~) (8)z --

where M denotes the magnetization component along the easy magnetizationz

axis perpendicular to the sample and

2 2 2 2 112p D [4B Ilcl(B +l)(x +y )]

which is a damping factor;

-2B [ y 1 2 2 ]a BZ+I arc tan(x) + 2 B log(x +y )

which is the so-called symmetry variable in the case of the critical point,

and

p- [4B2/Idl(B2+1)(X2+y2)]1/..

with the same ~ for the tricritical point

equation:f" + f' + B2+1 f + e:fN -o

4iL

The function f(~) satisfies the

(9)

8/3/2019 J.A. Tuszynski- Time-Dependent Landau-Ginzburg Magnetic Modelling of Magnetic Inhomogeneities and Spiral Doma

3/3

269

lead to a finite width w. It is easy to demonstrate that the energy ~Equi red to form a spiral is in the vicinity of the critical point

IU;;I -41TvN~E = w(A m2 + ~) m2 -!t-- (1 -e 0) (II)4 2 2v

where m = [- M 1]-1/2, N is the number of turns, and V = -2B>0 for theo o

spiral to unwind. In the vicinity of the tricritical point, however

4 D 2 IU;;I -21TVNo~E = w(A6m + 8) m --v-- (1 -e ) (12)

where m = [- M I )-1/4 in this case. Thus, the energy increases rathero

slowly with the size of the spiral. This could explain why it was so easyto form very large spiral structures. In [7) the pitch of the spirals wastypically 20-30 jIm which wuld correspond to our 41TB provided B is smallenough to linearize the exponential. The lifetime of a spiral ~T N

exp(-B~E) should depend rather sensitively on temperature.We believe that the alternating magnetic field is necessary first to

gradually disallow more and more low energy periodic magnetic structures(starting from domain walls) and distort the form of the high energy ones,as can be seen analyzing the solutions of eq. (2). Even infinitesimallysmall fields lead to the abolition of tanh-kinks which are replaced bybumps. In fact, at the saturation field

1 2A 3/2H = -= (- ~) (13)

s IA 34

(here H N 91.5 Oe) no propagating solutions would be stable exce p t fors 8

singular ones. It has been shown that spirals can be exact solutions onlyat the critical or tricritical points [2,3). However, with an alternatingfield present, the effective free energy for solutions with relaxation times~T greater than the period of magnetic field oscillation w-1 would be:

F = fd2x [A 2 (M )2 + A 4 (M )4 + IH I IM I + ~2 (VM )2] .(14)Z z z Z Z

Effectively then, when IHzi ; -AziM I, (note that A2