Physica B 177 (1992) 207-210 North-Holland PHYSICA iI
Dipolar field effect on magnetic
H. Szymczak and R. Szymczak Institute of Physics, Polish Academy of Sciences, al. Lotnikow 32146, 02-668 Warszawa, Poland
The concept of the intermediate state arising due to the demagnetizing field has been used in order to describe magnetic field-induced first-order reorientation transitions. Special attention is devoted to the nucleation mechanisms and their effect on the magnetization curve.
The magnetic field-induced phase transitions have now become one of the most powerful tools in the investigation of highly anisotropic mag- netic materials (see ref. [l] for examples and references). The manner in which magnetic ma- terials are magnetized by an external field is rich in examples of first- and second-order phase transitions. Particularly interesting are first-order phase transitions observed in rare-earth transi- tion metal compounds such as Nd,Fe,,B. For example, when high fields are applied along the hard direction, a gradual rotation of magnetiza- tion from the easy axis toward the field direction occurs. Then, at a critical field the magnetic moments suddenly align parallel to the field. As always in phase transitions of the first order, metastable states may occur and therefore mag- netic field-induced transitions ought to be accom- panied by hysteresis loops. But, in many cases, magnetization jumps are not sharp, but rather smooth, and no appreciable hysteresis is ob- served during reorientation of magnetization. In general, such behaviour is characteristic of mag- netic systems with nonuniform magnetization. Nonuniform states near the first-order phase transition points are formed in real, finite-size samples, mainly due to the demagnetizing field effects. Usually, in a theoretical analysis of the magnetization reorientation transitions, the pos- sibility of the appearance of nonuniform phases is not taken into account. This is the source of
some discrepancies between experimental obser- vations and their theoretical interpretations.
In this paper a phenomenological description will be given for the magnetic reorientation tran- sition from the low magnetization phase (with magnetization M,) to the high magnetization phase (with magnetization M2).
2. Intermediate state
The first-order magnetic phase transition oc- curs between two nonequivalent minima of the free energy of the crystal, corresponding to par- ticular directions of M (M, and M2). The exter- nal magnetic field H can change their energy by a different amount, inducing a first-order transi- tion from one phase to another. In real magnetic materials the magnetization is changed continu- ously and it is difficult to reveal the hysteresis associated with the phase transition. The reason is probably the formation of a thermodynamical- ly stable state in which two phases coexist. This intermediate state is analogous to the inter- mediate state of type-1 superconductors. Its for- mation is due to the demagnetizing field (dipolar) effects. Theory of the phase coexist- ence has been developed  for uniaxial anti- ferromagnets in the case of spin-flop transitions. Visual observations of the coherent intermediate state have been performed for DyFeO, single crystals (, see also ref.  for further refer- ences).
0921-4526/92/$05.00 0 1992 - Elsevier Science Publishers B.V. All rights reserved
208 H. Szymczak, R. Szymczak I Magnetic field-induced phase transitions
In order to calculate the magnetization curve in the intermediate state, one should minimize the free energy of the system:
L(Oj, cp,) is a function describing the anisotropy energy in i-type domains (domains with mag- netization iVZi), 0, and pi are the polar and azimuthal angles of the magnetization vector M;, respectively, N,, NY and N, are the components of the demagnetizing tensor,
n, = Vi/V ,
V = c V, (vi is the volume of the I
i-type domains) ,
In the above procedure it is assumed that the transition occurs through the process of domain wall nucleation and displacement, i.e. by a change of the fractions ni. It means that the equilibrium equations are (with the condition (3) included into Er)
-0, i=l,2,.... (4)
Using eqs. (2)-(4) and determining the stability conditions for all of the phases in the system (including the intermediate state) it is possible (mostly by numerical methods) to determine n,(H) and M(H).
For reasons of simplicity we shall consider only a very special but important case of uniaxial symmetry (e.g. point group w/mm) and the ex- ternal magnetic field perpendicular to the c-axis. In this case the anisotropy energy can be ex- pressed as
f(O, p) = K, sin*8 + K2 sin48 + K3 sin% + . . . .
Two types of first-order magnetization processes (FOMP) are distinguished (according to ref. ; see also ref. (61 for further references), depend- ing on the fact whether the final state after the transition is the saturation state (type-l FOMP) or not (type-2 FOMP). We consider here only type-l FOMP (observed, for example, in Nd,Fe,,B and in many other rare-earth transi- tion metal compounds [l]). In such a system three phases, M, = Ml, M, = ML and M,, coex- ist (M: and ML describe the low magnetization phases with cos 13, > 0 and cos 0, < 0, respective- ly; for the M, phase the magnetization vector is oriented along the positive direction of the X- axis, i.e. along the direction of the applied field).
For reasons of symmetry one can put
n,=n,=n, n3 = 1 - 2n .
Then the equilibrium equation (4) can be written in the form
H = M-(S - 1))[K,(S - 1) + K,(S4 - 1)
+ K,(S6 - l)] + N,M[2n(S - 1) + l] , (6)
where S = sin 13, = sin 0, (S remains constant in the intermediate state and keeps its critical value).
The fact that the magnetization M depends linearly upon the external field H is in accord- ance with the theory of intermediate state  as well as with experimental observations (see, for example, refs. [l , 2, 71). According to the theory developed in ref. , the internal field Hi,
is constant in the intermediate state (assuming that magnetostrictive effects are small [S]) and equal to the critical field.
The anisotropy constants K, , K, determine not only the stability regions but also the type of the phase transition between the low (and high) mag-
H. Szymczak, R. Szymczak I Magnetic field-induced phase transitions 209
netization phase and the intermediate state. Generally, a discontinuity in the value of the magnetization is expected to exist at the transi- tion points.
A detailed study on the application of the proposed concept to the particular systems is in progress and will be published elsewhere in the near future.
3. Nucleation processes
It seems that the magnetization process during field-induced phase transitions has a completely different nature than the ordinary magnetization process. Particularly important differences are in the mechanisms of a new phase nucleation. In the case of the magnetic reorientation transi- tions, domain walls can act as nuclei for the formation of the new phases . This mechanism takes place simultaneously with the mechanism related to the structural defects.
The domain wall mechanism is effective only if the rotation of the magnetization inside the wall takes place in the xz plane. It means that the anisotropy in the basal plane may suppress this mechanism.
In order to consider the effect of the external field on the distribution of the magnetization inside the domain wall one should consider the Euler equation corresponding to the total free energy of a single domain wall:
E,= j-=[A(~)z+f(0)-(MH) -m
+ f(N,M,2 + N,M:, + NJ@)] dy , (8)
where A is the exchange parameter. For the high magnetic field transition the last
term in eq. (8) can be omitted and consequently the distribution of the magnetization M(y) is described by the formula given in refs. [6, 91. This means that deldy vanishes at the center of the wall and the (180 - 20,) wall splits into two (90 - &) walls (here, 0, is the value of 8 for y = co), giving rise to a new phase with the
magnetization parallel to the x-axis. The volume of this phase grows continuously with the exter- nal field and therefore the first-order reorienta- tion transition converts to a continuous tran- sition.
The magnetization curves of Nd,Fe,,B com- pound presented in ref. [l] illustrate the effec- tiveness of both nucleation mechanisms impor- tant for the intermediate state formation. It seems that the magnetization curve measured for the field up is determined by domain wall split- ting while the field down curve by a nucleation mechanism related to structural defects. The presence of both nucleation mechanisms is re- sponsible for the hysteresis observed for Nd,Fe,,B.
Recently, it has been shown [lo] that thermo- dynamically stable vortexes may exist in mag- netically ordered materials. This state is similar to the mixed state of a type-II superconductors. Therefore for some crystals the magnetization reorientation transitions will go through the magnetic mixed state instead of the magnetic intermediate state. It seems that the easiest way to identify the magnetic mixed state is to use the singular-point technique (SPD) [ 1 l] - an ex- perimental method that allows to reveal the dis- continuities in the magnetization curve. The presence of the mixed state results in two peaks in the differential susceptibility d2MldH2, in- stead of one expected for the intermediate state.
It has been shown that the concept of inter- mediate phase can be used successfully to de- scribe the magnetic field-induced first-order phase transitions.
We would like to thank Professor J.J.M. Fran- se for fruitful and stimulating discussions and the referee for interesting comments and remarks.
210 H. Szymczak, R. Szymczak I Magnetic field-induced phase transitions
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