Crossover from First- to Second-Order Transition in a Frustrated Thin Film

This section deals with the question whether or not the phase transition known in the bulk state changes its nature when the system is made as a thin film. In a recent work presented above, we have considered the case of a bulk second-order transition. We have shown that under a thin film shape, i.e., with a finite thickness, the transition shows effective critical exponents whose values are between 2D and 3D universality classes [270]. If we scale these values with a function of thickness as suggested by Capehart and Fisher [53] we should find, as long as the thickness is finite, the 2D universality class.

In this section, we study the effect of the film thickness in the case of a bulk first-order transition. The question to which we would like to answer is whether or not the first order becomes a second order when reducing the thickness. For that purpose we consider the face-centered cubic (FCC) Ising antiferromagnet. This system is fully frustrated with a very strong first-order transition.

On the one hand, effects of the frustration in spin systems have been extensively investigated during the last 30 years. In particular, by exact solutions, we have shown that frustrated spin systems have rich and interesting properties such as successive phase transitions with complicated nature, partial disorder, reentrance and disorder lines [67, 76]. Frustrated systems still challenge theoretical and experimental methods. For recent reviews, the reader is referred to Ref. [85].

We shall use the recent high-precision technique called "Wang- Landau” flat histogram Monte Carlo (MC) simulations to identify the order of the transition.

Model and Ground-State Analysis

It is known that the antiferromagnetic interaction between nearest- neighbor (NN) spins on the FCC lattice causes a very strong frustration. This is due to the fact that the FCC lattice is composed of tetrahedra each of which has four equilateral triangles. It is well- known [85] that it is impossible to fully satisfy simultaneously the three antiferromagnetic bond interactions on each triangle. In the case of the Ising model, the GS is infinitely degenerate for an infinite system size: On each tetrahedron, two spins are up and the other two are down. The FCC system is composed of edge-sharing tetrahedra. Therefore, there is an infinite number of ways to construct the infinite crystal. The minimum number of ways of such a construction is a stacking, in one direction, of uncorrelated antiferromagnetic planes. The minimum GS degeneracy of a L3 FCC-cell system (L being the number of cells in each directions), is, therefore, equal to 3 x 22i where the factor 3 is the number of choices of the stacking direction, 2 the degeneracy of the antiferromagnetic spin configuration of each plane and 2L the number of atomic planes in one direction of the FCC crystal (the total number of spins is N = 4L3). The GS degeneracy is, therefore, of the order of 2N' Note that at finite temperature, due to the so-called "order by disorder”

[146, 349], the spins will choose a long-range ordering. In the case of antiferromagnetic FCC Ising crystal, this ordering is an alternate stacking of up-spin planes and down-spin planes in one of the three direction. This has been observed also in the Heisenberg case [82] as well as in other frustrated systems [274].

The phase transition of the bulk frustrated FCC Ising antiferro- magnet has been found to be of the first order [26, 272, 278, 280, 334]. Note that for the Heisenberg model, the transition is also found to be of the first order as in the Ising case [82, 137]. Other similar frustrated antiferromagnets such as the HCP XY and Heisenberg antiferromagnets [83] and stacked triangular XY and Heisenberg antiferromagnets [250, 251] show the same behavior.

Let us consider a film of FCC lattice structure with [001] surfaces. The Hamiltonian is given by

where 07 is the Ising spin at the lattice site indicates the sum

over the NN spin pairs <7, and aj.

In the following, the interaction between two NN on the surface is supposed to be antiferromagnetic and equal to Js. All other interactions are equal to ] = - 1 for simplicity. Note that in a previous paper [249] we have studied the case of the Heisenberg model on the same FCC antiferromagnetic film as a function of J s (see Chapter 9).

For Ising spins, the GS configuration can be determined in a simple way as follows: We calculate the energy of the surface spin in the two configurations shown in Fig. 16.17 where the film surface contains spins 1 and 2 while the beneath layer spins 3 and 4. In the ordering of type I (Fig. 16.17a), the spins on the surface (xy plane) are antiparallel and in the ordering of type II (Fig. 16.17b), they are parallel. Of course, apart from the overall inversion, for type I there is a degenerate configuration by exchanging the spins 3 and 4. To see which configuration is stable, we write the energy of a surface spin for these two configurations

One sees that Я/ < EuwhenJs/J > 0.5. In the following, we study the case Js = J = -1 so that the GS configuration is of type I.


Thin Films and Criticality

The ground state spin configuration of the FCC cell at the film surface

Figure 16.17 The ground state spin configuration of the FCC cell at the film surface: (a) ordering of type I for ]s < —0.51/ |, (b) ordering of type II for Js > -0.51Л-

Monte Carlo Results

In this paragraph, we show the results obtained by MC simulations with the Hamiltonian (16.23) using the high-precision Wang- Landau flat histogram technique [351] (see Chapter 6).

Bulk energy vs T for L = N = 12

Figure 16.18 Bulk energy vs T for L = Nz = 12.

The film size used in our present work is L x L x Nz where L is the number of cells in x and у directions, while Nz is that along the z direction (film thickness). We use here L = 30, 40,, 150 and Nz = 2, 4, 8, 12. Periodic boundary conditions are used in the xy planes. Our computer program was parallelized and run on a rack of several dozens of 64-bit CPU. J = 1 is taken as unit of energy in the following.

Before showing the results let us adopt the following notations. Sublattices 1 and 2 of the first FCC cell belongs to the surface layer, while sublattices 3 and 4 of the first cell belongs to the second layer (see Fig. 16.17a). In our simulations, we used Nz FCC cells, i.e., 2Nz atomic layers, and two symmetric film surfaces.

Crossover of the phase transition

As said earlier, the bulk FCC antiferromagnet with Ising spins shows a very strong first-order transition. This is seen in MC simulation even with a small lattice size as shown in Fig. 16.19.

Bulk energy histogram for L = N = 12 with periodic boundary conditions in all three directions

Figure 16.19 Bulk energy histogram for L = Nz = 12 with periodic boundary conditions in all three directions (a) and without PBC in z direction (b). The histogram was taken at the transition temperature Tc indicated on the figure.

450 Thin Films and Criticality

Energy histogram for L = 20, 30, 40 with film thickness N = 4 (8 atomic layers) at T = 1.8218, 1.8223, 1.8227, respectively

Figure 16.20 Energy histogram for L = 20, 30, 40 with film thickness Nz = 4 (8 atomic layers) at T = 1.8218, 1.8223, 1.8227, respectively.

Our purpose here is to see whether the transition becomes second order when we decrease the film thickness. As it turns out, the transition remains of first order down to Nz = 4 as seen by the double-peak energy histogram displayed in Fig. 16.20. Note that we do not need to go to larger L, the transition is clearly of first order already at L = 40.

In Fig. 16.21, we plot the latent heat Д E as a function of thickness Nz. Data points are well fitted with the following formula:

where d = 3 is the dimension, A = 0.3370, В = 3.7068, C = —0.8817. Note that the term N%~x corresponds to the surface separating two domains of ordered and disordered phases at the transition. The second term in the brackets corresponds to a size correction. As seen in Fig. 16.21, the latent heat vanishes at a thickness between 2 and 3. This is verified by our simulations for Nz = 2. For Nz = 2, we find a transition with all second-order features: no discontinuity in energy (no double-peak structure) even when we go up to L = 150.

The latent heat AE as a function of thickness N

Figure 16.21 The latent heat AE as a function of thickness Nz.

Before showing in the following the results of Nz = 2, let us discuss the crossover. In the case of a film with finite thickness studied here, it appears that the first-order character is lost for very small Nz. A possible cause for the loss of the first-order transition is from the role of the correlation in the film. If a transition is of first order in 3D, i.e., the correlation length is finite at the transition temperature, then in thin films the thickness effect may be important: If the thickness is larger than the correlation length at the transition, than the first-order transition should remain. On the other hand, if the thickness is smaller than that correlation length, the spins then feel an "infinite" correlation length across the film thickness resulting in a second-order transition.

Film with 4 atomic layers(Nz = 2)

Let us show in Fig. 16.22 and Fig. 16.23 the energy and the magnetizations of sublattices 1 and 3 of the first two cells with L= 120 and Nz = 2.

It is interesting to note that the surface layer has larger magnetization than that of the second layer. This is not the case for non-frustrated films where the surface magnetization is always

Energy versus temperature T for L = 120 with film thickness N = 2

Figure 16.22 Energy versus temperature T for L = 120 with film thickness Nz = 2.

Sublattice magnetization for L = 120 with film thickness N = 2

Figure 16.23 Sublattice magnetization for L = 120 with film thickness Nz = 2.

smaller than the interior ones because of the effects of low-lying energy surface-localized magnon modes [75, 78]. One explanation can be advanced: Due to the lack of neighbors surface spins are less frustrated than the interior spins. As a consequence, the surface spins maintain their ordering up to a higher temperature.

Specific heat are shown for various sizes L as a function of temperature

Figure 16.24 Specific heat are shown for various sizes L as a function of temperature.

Let us discuss finite-size effects in the transitions observed in Figs. 16.24 and 16.25. This is an important question because it is known that some apparent transitions are artifacts of small system sizes. To confirm further the observed transitions, we have made a study of finite-size effects on the layer susceptibilities by using the Wang-Landau technique [351].

We observe that there are two peaks in the specific heat: The first peak at Tx ~ 1.927, corresponding to the vanishing of the sublattice magnetization 3, does not depend on the lattice size while the second peak at T2 1.959, corresponding to the vanishing of the sublattice magnetization 1, does depend on L. Both histograms taken at these temperatures and the near-by ones show a Gaussian form indicating a non-first-order transition [see Fig. 16.26).

The fact that the peak at Tx does not depend on L suggests two scenarios:

(i) The peak does not correspond to a real transition, since there exist systems where Cv shows a peak but we know that there is no transition just as in the case of ID Ising chain.


Thin Films and Criticality

Susceptibilities of sublattice 1 (a) and 3 (b) are shown for various sizes Las a function of temperature

Figure 16.25 Susceptibilities of sublattice 1 (a) and 3 (b) are shown for various sizes Las a function of temperature.

(ii) The peak corresponds to a Kosterlitz-Thouless transition. To confirm this we need to check carefully several points such as the behavior of the correlation length, etc. This is a formidable task, which is not the scope of this chapter.

Whatever the scenario for the origin of the peak at T, we know that the interior layers are disordered between Ti and T2, while

Energy histograms for L = 120 with film thickness N = 2 at the two temperatures (indicated on the figure) corresponding to the peaks observed in the specific heat. See text for comment

Figure 16.26 Energy histograms for L = 120 with film thickness Nz = 2 at the two temperatures (indicated on the figure) corresponding to the peaks observed in the specific heat. See text for comment.

the two surface layers are still ordered. Thus, the transition of the surface layers occurs while the disordered interior spins act on them as dynamical random fields. Unlike the true 2D random-field Ising model which does not allow a transition at finite temperature [160], the random fields acting on the surface layer are correlated. This explains why one has a finite-Г transition here. Note that this situation is known in some exactly solved models where partial

The maximum value of = {E) — {ME) / {M) versus Lin the In — In scale. The slope of this straight line gives l/v

Figure 16.27 The maximum value of = {E) — {ME) / {M) versus Lin the In — In scale. The slope of this straight line gives l/v.

disorder coexists with order at finite T [67, 76, 86]. However, it is not obvious to foresee what the universality class of the transition is at T2. The theoretical argument of Capehart and Fisher [53] does not apply in the present situation because one does not have a single transition here, unlike the case of simple cubic ferromagnetic films studied before [270]. So, we wish to calculate the critical exponents associated with the transition at T2.

The exponent v can be obtained as follows. We calculate as a function of T the magnetization derivative with respect to P = {kBT)-1-. Vt= ((In M)') = (E) - {ME) / {M) where E is the system energy and M the sublattice order parameter. We identify the maximum of V for each size L. From the finite-size scaling we know that F1max is proportional to L1/v [112]. We show in Fig. 16.27 the maximum of V versus In L for the first layer. We find v = 0.887 ± 0.009. Now, using the scaling law xmax a LY/V, we plot In/max versus In L in Fig. 16.28. The ratio of the critical exponents y/v is obtained by the slope of the straight line connecting the data points of each layer. From the value of v we obtain у = 1.542 ± 0.005. These values do not correspond neither to 2D nor 3D Ising models (y2D = 1.75,

Maximum sublattice susceptibility у  versus Lin the In — In scale. The slope of this straight line gives y/v

Figure 16.28 Maximum sublattice susceptibility у max versus Lin the In — In scale. The slope of this straight line gives y/v.

v2d = 1, Y3d = 1.241, v3D = 0.63). We note, however, that if we think of the weak universality where only ratios of critical exponents are concerned [335], then the ratios of these exponents 1/v = 1.128 and y/v = 1.739 are not far from the 2D ones which are 1 and 1.75, respectively.

Concluding Remarks

We have considered a simple system, namely the Ising model on a simple cubic thin film, in order to clarify the point whether or not there is a continuous deviation of the 2D exponents with varying film thickness. From results obtained by the highly accurate multiple- histogram technique shown above, we conclude that the critical exponents in thin films show a continuous deviation from their 2D values as soon as the thickness departs from 1. This deviation stems from a deep physical mechanism: Capehart and Fisher [53] have argued that if one works exactly at the critical temperature TC[L = oo, Nz) then the critical exponents should be those of 2D universality class as long as the film thickness is finite. At TC[L = oo, Nz), the correlation in the z direction § remains finite while those in the xy planes become infinite. Hence, £ is irrelevant to the criticality. This yields, therefore, the 2D behavior. However, when the system is away from TC[L = oo, Nz), as is the case in numerical simulations using finite sizes, the system may have a 3D behavior as long as £ <5C Nz. This should yield a deviation of 2D critical exponents. The results we obtained in this paper verify this picture. In addition, the prediction of Capehart and Fisher [53] for the shift of the critical temperature with the film thickness is in a perfect agreement with our simulations. Note, furthermore, that (i) the deviations of the exponents from their 2D values are very different in magnitude: while v and a vary very little over the studied range of thickness, у and specially p suffer stronger deviations, (ii) with a fixed thickness Nz Ф 1, the same "effective” exponents are observed, within errors, in simulations with and without periodic boundary condition in the z direction, (in) to obtain the 3D behavior, the finite size scaling should be applied simultaneously in the three directions, i.e., all dimensions should be allowed to go to infinity. If we do not apply the scaling in the z direction, we will not obtain 3D behavior even with a very large, but fixed, thickness and even with periodic boundary condition in the z direction, (iv) with regard to the critical behavior, thin films behave as 2D systems but with effective critical exponents whose values deviate from those of 2D universality.

For a film with a first-order transition when the thickness is thick enough, we have shown by the high-performance Wang- Landau technique that there is a crossover to a second-order phase transition when the thickness goes down to a few atomic layers.

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