Theory and Simulation of Magnetic Materials: Physics at Phase Frontiers

The combination of theory and simulation is necessary in the investigation of properties of complex systems where each method alone cannot do the task properly. Theory needs simulation to test ideas and to check approximations. Simulation needs theory for modeling and for understanding results coming out from computers. In this review, we give recent examples to illustrate this necessary combination in a few domains of interest such as frustrated spin systems, surface magnetism, spin transport and melting. Frustrated spin systems have been intensively studied for more than 30 years. Surface effects in magnetic materials have been widely investigated also in the last three decades. These fields are closely related to each other and their spectacular development is due to numerous applications. We confine ourselves to theoretical developments and numerical simulations on these subjects with emphasis on spectacular effects occurring at frontiers of different phases.


Introduction
The physics at frontiers of different phases of the same system is a very exciting subject. In general, when there exist between the particles of a system several interactions, each of which favors a different symmetry, the system chooses the symmetry where its internal energy is minimum if the temperature T = 0 or where its free energy is minimum if T = 0. Let us take a simple example: we consider a chain of Ising spins interacting with each other via an interaction J 1 between nearest neighbors (NN) and an interaction J 2 between next NN (NNN). If J 1 is ferromagnetic (J 1 > 0) and J 2 = 0 then the ground state (GS) is ferromagnetic. Now, if J 2 is antiferromagnetic (J 2 < 0), we cannot arrange the spins in an order that fully satisfies at the same time J 1 and J 2 : if |J 2 | ≪ (≫)J 1 then the system chooses to satisfy J 1 (J 2 ) by taking the ferromagnetic (antiferromagnetic) order. There exists a critical value η c of J 2 /J 1 where the system changes from the ferromagnetic symmetry to the antiferromagnetic one. The critical value η c is the frontier between the two phases. In general, when there are more than two interactions, the determination of the frontiers between different phases is more complicated. As we see, the competition between "incompatible" interactions creates frontiers. This is not limited to physics. Near a frontier, physical behaviors are different on the two sides: due to their different symmetries, they have different laws that govern fluctuations etc. When we introduce into one phase an external perturbation such as the temperature or an applied field, the system can choose the symmetry of the other phase. Such a phenomenon occurs very often in many systems: it is called "reentrance".
In this review, we give various examples of physical phenomena which occur around the phase frontiers. These examples are taken from our recent and current works.
The first subject concerns frustrated spin systems where competing interactions cause many spectacular effects (see reviews given in Ref. [1]). This is shown in section 2. We will describe there the reentrance, the disorder lines and the partial disorder which have been found in exactly solved models [2,3,4,5]. We will show that simulations give complementary results that exact methods cannot reach. The second subject concerns theories and simulations of magnetic thin films. This is shown in section 3. One of the most important surface effects is the existence of localized spin-wave modes near a surface and interface. It has been shown a long time ago [6] that low-lying surface modes affect strongly macroscopic properties of magnetic systems giving rise for instance to a low surface magnetization or surface magnetically dead layer and to surface phase transitions at low temperatures. We discuss in particular surface effects in frustrated materials and in films with dipolar interaction [7,8]. Recent results of spin resistivity are shown and discussed in relation with surface disordering [9,10]. Finally, results on melting and surface lattice relaxation [11] are also shown. Concluding remarks are given in section 4.

Frustrated spin systems near phase frontiers
A system is said frustrated when all interaction bonds cannot be simultaneously satisfied in the GS. Well-known examples include the triangular lattice with NN antiferromagnetic interaction and the example given in the Introduction. For definiteness, let us take the case of the "centered square lattice" with Ising spins shown in Fig. 1, introduced by Vaks et al. [12] with NN and NNN interactions, J 1 and J 2 , respectively. The exact expression for the free energy, some correlation functions, and the magnetization of one sublattice were given in the original work of Vaks et al.
The GS properties of this model are as follows : for a = J 2 / | J 1 |> −1, spins of sublattice 2 orders ferromagnetically and the spins of sublattice 1 are parallel (antiparallel) to the spins of sublattice 2 if J 1 > 0 ( < 0 ); for a < −1, spins of sublattice 2 orders antiferromagnetically, leaving the centered spins free to flip. The phase diagram of this model is given by Vaks et al. [12]. Except at the "frontier" a = −1, there is always a finite critical temperature. When J 2 is antiferromagnetic (> 0) and J 2 /J 1 is in a small region near 1, namely near the frontier of the two  Figure 3. Temperature dependence of sublattice Edwards-Anderson order parameters, q 1 and q 2 (crosses and black circles, respectively) in the case a = J 2 / | J 1 |= −2, by Monte Carlo simulation. Susceptibility calculated by fluctuations of magnetization of sublattice 2 is also shown. The lattice used contains N = 2 × 60 × 60 spins with periodic boundary conditions. [13] phases, the system is successively in the paramagnetic state, an ordered state, the reentrant paramagnetic state, and another ordered state, with decreasing temperature (see Fig. 2). Though an exact critical line was obtained [12], the ordering in the antiferromagnetic (frustrated) region has not been exactly calculated. We have studied this aspect by means of Monte Carlo (MC) simulations [13] which show the coexistence between order and disorder. This behavior has been observed in three-dimensional (3D) Ising spin models [14,15] and in an exactly soluble model (the Kagomé lattice) [2] as well as in frustrated 3D quantum spin systems [16,17]. The results for the Edwards-Anderson sublattice order parameters q i and the staggered susceptibility of sublattice 2 , as functions of T , are shown in Fig. 3 in the case a = −2.
As is seen, sublattice 2 is ordered up to the transition at T c while sublattice 1 stays disordered at all T . This result shows a new example where order and disorder coexists in an equilibrium state. It noted that in the paramagnetic region, a Stephenson disorder line [18] has been found in Ref. [13] cosh The two-point correlation function at T D between spins of sublattice 2 separated by a distance r is zero for odd distance r and decay like r −1/2 [tanh(J 2 /k B T D )] r for even r [18]. However,there is no dimensional reduction on the Stephenson line given above. Usually, one defines the disorder point as the temperature where there is an effective reduction of dimensionality so that physical  quantities become simplified spectacularly [19]. In general, these two types of disorder line are equivalent, as for example , in the case of the Kagomé lattice Ising model (see below). This is not the case here. The disorder line corresponding to dimensional reduction, was given for the general 8-vertex model by Ref. [20]. When this result is applied to the centered square lattice, one finds that the disorder variety is given by where i 2 = −1. This disorder line lies on the unphysical (complex) region of the parameter space of this system. Only the Stephenson disorder line Eq. (1) is the relevant one for the reentrance phenomenon. Disorder solutions have found interesting applications, as for example in the problem of cellular automata (for a review see Ref. [21]). Moreover, they also serve to built a new kind of series expansion for lattice spin systems [19]. Another example is the Kagomé lattice shown in Fig. 4. The phase diagram near the phase border is shown in Fig. 5 with the Stephenson disorder line. Other very rich phase diagrams are found in Refs. [2,3,4,5] for honeycomb lattice, generalized Kagomé lattice and dilute centered square lattice. In some cases, up to five successive phase transitions and several disorder lines are found for a single set of interaction parameters.

Magnetic thin films near the phase frontiers
The presence of a surface perturbs the bulk properties of a crystal. The perturbation becomes important when the ratio "number of surface atoms to number of bulk atoms" becomes significative. Among numerous surface effects, we will outline here some results on surface magnetization and surface phase transition near phase frontiers. The first example which illustrates very well the necessary combination between theory and simulation is the case of a FCC thin film with a (001) frustrated surface [7]. The Hamiltonian is given by where S i is the Heisenberg spin at the lattice site i, i,j indicates the sum over the nearest neighbor spin pairs S i and S j . The last term, which will be taken to be very small, is needed to ensure that there is a phase transition at a finite temperature for the film with a finite thickness when all exchange interactions J i,j are short-ranged. Otherwise, it is known that a strictly two-dimensional system with an isotropic non-Ising spin model (XY or Heisenberg model) does not have a long-range ordering at finite temperatures [22]. Between the surface spins we take J i,j = J s , and between all other spins J i,j = J = 1 > 0 (ferromagnetic). The GS configuration depends on η = J s /J. When η is smaller than a critical value, the GS becomes non linear. We show cos α and cos β in Fig. 7 as functions of J s where α and β are the angles between spins defined in the caption. The critical value J c s where the collinear configuration becomes non collinear is found between -0.18 and -0.19. This value can be calculated analytically (see [7]).
Using the theory of Green's function [6], we have calculated the layer magnetizations shown in Fig. 8 (see details in [7]). The phase diagram near the phase boundary J c s is shown in Fig.  9. We notice that the surface transition occurs only below J c s . MC simulations give very similar results. Note however that at T = 0 quantum fluctuations causes a very strong "zero-point spin contraction" for surface spins: as a consequence, surface spins do not have the full length 1/2 as seen in Fig. 8, unlike the classical spins used in MC simulations.
The next example is a 2D layer with a dipolar interaction between the 3-state Potts model [8]. The competition is between the dipolar term of magnitude D, which favors the in-plane spin configuration, and the perpendicular anisotropy A which favors the perpendicular configuration. There is a critical value of D/A above (below) which the configuration is planar (perpendicular).   Near this frontier, we found by MC simulations an interesting phenomenon which is called the re-orientation phase transition shown in Fig. 12: for example, if we follow the vertical line on the left figure, we see that at low T the system is in the planar configuration (phase II), it crosses the phase separation line with increasing T to go to the perpendicular phase (phase I) before going out to the paramagnetic phase (phase P) at a higher T . Such a re-orientation at a finite T is spectacular: we have shown that the transition between phases I and II is of first order [8]. This is not similar to the reentrant region shown in Figs. 2 and 5 where all transition lines are of second order and a very narrow paramagnetic phase separates the two ordered phases (AF and F, or F and X). It is interesting to note that to allow a transition between two "incompatible" symmetries (the one is not a subgroup of the other), there are two ways: i) a single first-order transition with a latent heat (as the case shown in Fig. 12), ii) two second-order transitions separated by a narrow region of a reentrant paramagnetic phase with often a disorder line separating two zones of different pre-ordering fluctuations (as the case shown in Fig. 5). Note that the re-orientation transition has also been observed for the Heisenberg spin model [23]. Let us give now an example where the surface disordering affects strongly the spin resistivity in a magnetic thin film. The spin resistivity has been studied both theoretically and experimentally for more than 50 years. The reader is referred to Refs. [9,10,24] for references. Theoretically, the coupling between an itinerant spin with lattice spins affects strongly the spin resistivity ρ. The spin-spin correlation has been shown to be the main mechanism which governs ρ. When there is a magnetic phase transition, the spin resistivity undergoes an anomaly. In magnetic thin films, when there is a surface phase transition at a temperature T s different from that of the bulk one (T c ), we expect two peaks of ρ one at T s and the other at T c . We show here an example of a thin film, of FCC structure with Ising spins, composed of three sub-films: the middle film of 4 atomic layers between two surface films of 5 layers. The lattice sites are occupied by Ising spins interacting with each other via NN ferromagnetic interaction. Let us suppose the interaction between spins in the outside films be J s and that in the middle film be J. The inter-film interaction is J. In order two enhance surface effects we suppose J s << J. We show in Figs. 13 and 14 the layer magnetization and the spin resistivity for J s = 0.2J. We observe that the surface films undergo a phase transition at T s ≃ 4 far below the transition temperature of the middle film T c ≃ 8.8. As stated above, a phase transition induces an anomaly in the spin resistivity: the two phase transitions observed in Fig. 13 give rise to two peaks of ρ shown in Fig. 14. The surface peak of ρ has been also seen in a frustrated film [10]. The last example is the surface relaxation and the surface melting of a semi-infinite Ag crystal with a (111) surface [11]. The border here, contrary to the previous examples, is a physical border (not a phase border). The competition exists also for this case because surface and bulk atoms have different environments and the multi-body interactions compete with the two-body terms in the potential from the Embedded-Atom-Method (EAM) [25] we have used. In order to see the surface melting, we compute the structure factor S K as follows: where − → d j is the position vector of an atom in the layer, N l the number of atoms in a layer and − → K the reciprocal lattice vector which has the following coordinate (in reduced units): 2π 1; − √ 3; 0 . The angular brackets < ... > indicate thermal average taken over MC run time. The above "order parameter", which allows us to monitor the long-range surface order, is plotted for the surface layer in Fig. 15. As we can see, the long-range order is lost at ≃ 700 K. Note that the bulk Ag melts at ≃ 1235 K. In order to investigate in more details the surface melting, we have also computed the O 6 order parameter which describes the short-range hexatic orientational order of the surface : with where the sum runs over the NN pairs and Θ jk is the angle which the j −k bond, when projected on the xy plane, forms with the x axis. The δ parameter is taken as one-half the average interlayer spacing. The weighting function, W jk , allows us to differentiate the "non coplanar" and the "coplanar" neighbors. With a coplanar neighbor, the weighting function takes a maximum value. We have to calculate the spatial average of O 6 taken over all atoms of the surface layer and then calculate its thermal average over MC run time. We plot the averaged O 6 parameter versus temperature in Fig. 15. The short-range order is also lost at 700 K. We have calculated the distance ∆ between the topmost layer and the second layer. There is a contraction of this distance with respect to the distance between two layers in the bulk as seen in Fig. 16. Only at about 900 K, far after the surface melting, that surface atoms are "desorbed" from the crystal.

Concluding remarks
We have shown in this paper a number of phenomena occurring at frontiers between phases of different symmetries. The phase diagram near the phase frontiers is often very rich with reentrance, disorder lines and multiple phase transitions. One has seen that in most of the cases treated so far we need a combination of theory and simulation to better understand complicated physical effects resulting from competing forces around the frontiers. A frontier is determined as a compromise between these forces. As a consequence, frontiers do not have a high stability: we have seen that when a small external perturbation, such as the temperature, is introduced, the phase border moves in favor of one of the neighboring phases according to some criteria such as the entropy. Therefore, many interesting effects manifest themselves around frontiers. The physics near phase frontiers is far from being well understood in various situations.