Re-Orientation Transition in Molecular Thin Films: Potts Model with Dipolar Interaction

We show in below the case of a 2D system and a thin film where there is the competition between a dipolar interaction of strength D which makes the spins lie in the plane and a surface perpendicular anisotropy A which favors the perpendicular spin configuration [153]. The dipolar Hamiltonian is written as

where r,_y is the vector of modulus r,_y connecting the site i to the site j. One has г, у = r, — r,. In Eq. (7.24), D is a positive constant depending on the material, the sum *^s Hrnited at pairs of spins within a cut-off distance r_c, and S(

where s_a (a = x, y, z) is the a component with values ±1. The perpendicular anisotropy is introduced by the following term:

where A is a constant. Note that the dipolar interaction as applied in our Potts model is not similar to that used in the vector spin model where S(] = 1 as the unit of energy. The temperature T is expressed in the unit of ] /кв where кв is the Boltzmann constant.

Two-Dimensional Case

In the case of 2D, for a given A, the steepest-descent method gives the "critical value” D_c of D above (below) which the GS is the inplane (perpendicular) configuration. D_c depends on r_c. Let us take A = 0.5 and make vary D and r_c. The GS numerically obtained is shown in Ref. [153] for several sets of (D, r_c). For instance, when r_c = y/б ~ 2.449, we have D_c = 0.100.

The phase diagrams obtained from the histogram Monte Carlo technique (see Chapter 6) in the 2D case are shown in Fig. 7.23.

It is interesting to compare the present system using the 3-state Potts model with the same system using the Heisenberg spins [306]. In that work, the re-orientation transition line is also of first order but it tilts on the left of D_c, namely the re-orientation transition of type I occurring in a small region below D_c, unlike the re-orientation of type II found above D_c for the present Potts model. To explain the “left tilting” of the Heisenberg case, we have used the following entropy argument: The Heisenberg in-plane configuration has a spin wave entropy larger than that of the perpendicular configuration

Figure 7.23 (Color online) Phase diagram in 2D: Transition temperature T_c versus D, with A = 0.5, ] = 1 ,r_c = Уб (top) and r_c = 4 (bottom). Phase (I) is the perpendicular spin configuration, phase (II) the in-plane spin configuration and phase (P) the paramagnetic phase. See text for comments.

at finite T, so the re-orientation occurs in "favor” of the in-plane configuration, it goes from perpendicular to in-plane ordering with increasing T. Obviously, this argument for the Heisenberg case does not apply to the Potts model because we have here the inverse re-orientation transition. We think that, due to the discrete nature of the Potts spins, spin waves cannot be excited, so there is no spin wave entropy as in the Heisenberg case. The perpendicular anisotropy A is thus dominant at finite T for D slightly larger than D_c.

Experimentally, the "right tilting” of the re-orientation line from D_c in Fig. 7.23 has been observed in Fe/Gd [14]. So we emphasize that re-orientation depends on the system, it can be of type I or type II. Our simple model shows this possibility. Of course, to compare quantitatively our results with experimental data of Fe/Gd we need to take into account details of the real system such as Fe and Gd lattice structures and magnetic interactions.

There is another important point which is worth to mention: the scenario of the re-orientation transition. Arnold et al. [14] have suggested a two-step transition with an intermediate phase. Theoretically this scenario is possible: We have found in several exactly solved models [67, 86] that the transition between two ordered phases can be assisted by a very small disordered phase between them which is called "reentrant phase.” In this extremely small region, one can have a so-called "disorder line” with dimension reduction to help the system go from one symmetiy to another one. Numerically and experimentally the reentrance region is hardly detected due to its smallness. If the size of the reentrance region is large enough, we can have two second-order lines departing from D_c (see discussion on several exactly solved models in Ref. [86]). If the size is so small, these two lines look like a single line within numerical and/or experimental resolution, then the re-orientation on the "single" line has a first-order nature which is necessary to allow a transition between two different symmetries [380]. We think that this latter hypothesis corresponds to our finding of the first- order re-orientation shown in Fig. 7.24.

Thin Films

The case of thin films with a thickness L_z where L_z goes from a few to a dozen atomic layers has a very similar re-orientation transition as that shown above for the 2D case. Changing the film thickness results in changing the dipolar energy at each lattice site. Therefore, the critical value D_c will change accordingly. We note the periodic layered structures at large D and r_c for both cases. In the case L_z = 4, for r_c = Уб the critical value D_c above which the GS changes from the perpendicular to the in-plane configuration is D_c = 0.305.

Figure 7.24 Energy histogram P versus energy E at the re-orientation transition temperature Г = 0.930, for D =0.101, A = 0.5, J = l,r_c = s/6.

Figure 7.25 (Color online) Phase diagram in thin film of 4-layer thickness: Transition temperature T_c versus D, with A = 0.5, / = 1 and L = 24. The number (I) stands for the perpendicular configuration, the number (II) for the in-plane configuration (spins pointing along x ory axis), the number (1) for alternately one layer in x and one layer in у direction (periodic singlelayered structure), P is paramagnetic phase. See text for comments.

The whole phase diagram is shown in Fig. 7.25. Note that the line separating the uniform in-plane phase (II) and the periodic singlelayered phase (1) is vertical.

Again here, the line separating the perpendicular configuration (I) and the in-plane one (II) is a first-order line. However, we would like to emphasize that the same remark as that given above for the

2D case: We cannot exclude the possibility of a very small reentrance phase between two phases (I) and (II).

Effect of Surface Exchange Interaction

We have calculated the effect of J _s by taking its values far from the bulk value (J = 1) for several values of D. In general, when J_s is smaller than J the surface spins become disordered at a temperature T below the temperature where the interior layers become disordered. This case corresponds to the soft surface (or magnetically "dead” surface layer) [75]. On the other hand, when Js > J, we have the inverse situation: The interior spins become disordered at a temperature lower that of the surface disordering. We have here the case of a magnetically hard surface. We show in Fig. 7.26 an example of a hard surface in the case where J_s = 3 for D = 0.6 with L_z = 4. The same feature is observed for D = 0.4. Note that the surface and bulk transitions are seen by the respective peaks in the specific heat and the susceptibility. In the re-orientation

Figure 7.26 E, C_v, M and / of a 4-layer film versus T for D =0.6 with J_s = 3. The surface magnetization is shown by blue void circles, the bulk magnetization by red diamonds and the total curves by black solid circles.

region, the situation is very complicated as expected because the surface transition occurs in the re-orientation zone.

Let us discuss about experimental data of Fe/Gd. It has been found that the overall transition to the paramagnetic phase in the experiment on Fe/Gd, driven by the Gd substrate [14], is a first-order transition which occurs at room temperature. To simulate the Fe/Gd system, one can proceed by changing J_s and the nature of the surface spins to represent Fe atoms, one then can perform simulations in order to compare quantitatively with the experiment.

Conclusion

In this review, we have shown a number of studied cases on the frustration effects in two dimensions. We have discussed some properties of periodically frustrated Ising systems and limited the discussion to exactly solved models which possess at least a reentrant phase. Other frustrated Ising systems which used approximations are discussed in the chapter by Nagai et al. in Ref. [85] and in the book by Liebmann [212].

The main purpose of the review is to show some frustrated magnetic systems which present a number of common interesting features. These features are discovered by solving exactly some 2D Ising frustrated models. They occur near the frontier of two competing phases of different ground-state orderings. Without frustration, such frontiers do not exist. Among the striking features, one can mention the "partial disorder,” namely a number of spins stay disordered in coexistence with ordered spins at equilibrium, the "reentrance,” namely a paramagnetic phase exists between two ordered phases in a small region of temperature, and "disorder lines,” namely lines on which the system loses one dimension to allow for a symmetiy change from one side to the other. Such beautiful phenomena can only be uncovered and understood by means of exact mathematical solutions.

Let us emphasize that simple models having no bond disorder like those presented in this chapter can possess complicated phase diagrams due to the frustration generated by competing interac?tions. Many interesting physical phenomena such as successive phase transitions, disorder lines, and reentrance are found. In particular, a reentrant phase can occur in an infinite region of parameters. For a given set of interaction parameters in this region, successive phase transitions take place on the temperature scale, with one or two paramagnetic reentrant phases.

The relevance of disorder solutions for the reentrance phenomena has also been pointed out. An interesting finding is the occurrence of two disorder lines which divide the paramagnetic phase into regions of different kinds of fluctuations (see Section

7.4.1) . Therefore, care should be taken in analyzing experimental data such as correlation functions, susceptibility, etc. in the paramagnetic phase of frustrated systems.

Although the reentrance is found in the models shown above by exact calculations, there is no theoretical explanation why such a phase can occur. In other words, what is the necessaiy and sufficient condition for the occurrence of a reentrance? We have conjectured [16, 68, 77] that the necessary condition for a reentrance to take place is the existence of at least a partially disordered phase next to an ordered phase or another partially disordered phase in the ground state. The partial disorder is due to the competition between different interactions.

The existence of a partial disorder yields the occurrence of a reentrance in most of known cases [16, 62, 68, 76, 77, 239, 347], except in some particular regions of interaction parameters in the centered honeycomb lattice (Section 7.4.2): The partial disorder alone is not sufficient to make a reentrance, the finite zero-point entropy due to the partial disorder of the three ground states is the same, i.e., S₀ = log(2)/3 per spin [76], but only one case yields a reentrance. Therefore, the existence of a partial disorder is a necessary, but not sufficient, condition for the occurrence of a reentrance.

The anisotropic character of the interactions can also favor the occurrence of the reentrance. For example, the reentrant region is enlarged by anisotropic interactions as in the centered square lattice [62], and becomes infinite in the generalized Kagome model (Section

7.4.1) . But again, this alone cannot cause a reentrance as seen in the centered honeycomb case [76]: Only in one case a reentrance does occur. The presence of a reentrance may also require a coordination number at a disordered site large enough but below a limit to influence the neighboring ordered sites.

Finally, let us emphasize that when a phase transition occurs between states of different symmetries which have no special group- subgroup relation, it is generally accepted that the transition is of first order. However, the reentrance phenomenon is a symmetry breaking alternative which allows one ordered phase to change into another incompatible ordered phase by going through an intermediate reentrant phase. A question which naturally arises is, under which circumstances does a system prefer an intermediate reentrant phase to a first-order transition? In order to analyze this aspect we have generalized the centered square lattice Ising model into three dimensions [17]. This is a special bcc lattice. We have found that at low T the reentrant region observed in the centered square lattice shrinks into a first-order transition line which is ended at a multi-critical point from which two second order lines emerge forming a narrow reentrant region [17].

Let us mention that although the exactly solved systems shown in this chapter are models in statistical physics, we believe that the results obtained in this work have qualitative bearing on real frustrated magnetic systems. In view of the simplicity of these models, we believe that the results found here will have several applications in various areas of physics.

We have also studied frustrated magnetic systems close to the 2D solvable systems, namely thin magnetic thin films with Ising or Heisenberg spin models that are not exactly solvable. Guided by the insights of exactly solvable systems, we have introduced ingredients in the Hamiltonian to find some striking phenomena mentioned above: We have seen in thin films partial disorder (surface disorder coexisting with bulk order), reentrance at phase boundaries in face-centered cubic antiferromagnetic films. Thin films have their own interest such as surface spin rearrangement (helimagnetic films) and surface effects on their thermodynamic properties. Those systems will be presented in Chapters 9 and 10.

To conclude, we would like to say that investigations on the subjects discussed above continue intensively today. Note that there is an enormous number of investigations of other researchers on the above subjects and on other subjects concerning frustrated magnetic thin films. We have mentioned these works in our original papers, but we did not present them here. Also, for the same reason, we have cited only a limited number of experiments and applications in this review.