# Monte Carlo Results

In this paragraph, we show the results obtained by MC simulations with the Hamiltonian (9.1). The spins are the classical Heisenberg model of magnitude 5 = 1.

The film size is *L* x *L* x *N _{z}* where

*N*is the number of FCC cells along the z direction (film thickness). Note that each cell has two atomic planes. We use here

_{z}*L*= 12, 18, 24, 30, 36 and

*N*=

_{z}4. Periodic boundaiy conditions are used in the *xy* planes. The equilibrating time is about 10^{6} MC steps per spin and the averaging time is 2 x 10^{6} MC steps per spin. *J =* 1 is taken as unit of energy in the following.

Before showing the results, let us adopt the following notations. The sublattice 1 of the first cell belongs to the surface layer, while the sublattice 3 of the first cell belongs to the second layer. The sublattices 1 and 3 of the second cell belong, respectively, to the third and fourth layers. In our simulations, we used four cells, *N _{z}* = 4, i.e., 8 atomic layers. The symmetry of the two film surfaces imposes the equivalence of the first and fourth cells and that of the second and third cells. It suffices then to show the results of the first two cells, i.e., four first layers. In addition, in each atomic layer the two sublattices are equivalent by symmetiy. Therefore, we choose to show in the following the results of the sublattices 1 and 3 of the first two cells, i.e., results of the first four layers.

Let us show in Fig. 9.3 the magnetizations and the susceptibilities of sublattices 1 and 3 of the first two cells, in the case where *J _{s}* = — 1.

Figure 9.3 Magnetizations and susceptibilities of sublattices 1 and 3 first two cells vs temperature for *] _{s}* = —1.0 with

*L*= 24 and

*D*= 0.1. L, denotes the sublattice magnetization of layer

*j.*

Figure 9.4 Magnetizations and susceptibilities of sublattices 1 and 3 of first two cells vs temperature for*] _{s}* = —0.8 with

*L*= 24 and

*D*= 0.1. L, denotes the sublattice magnetization of layer

*j.*

It is interesting to note that the surface layer has largest magnetization followed by that of the second layer, while the third and fourth layers have smaller magnetizations. This is not the case for non-frustrated films where the surface magnetization is always 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 suffer fluctuations due to the frustration less than the interior spins so they maintain their ordering up to a higher temperature. Let us decrease the *] _{s}* strength. The surface spins then have smaller local field, so thermal fluctuations will reduce their ordering to a lower temperature. Figures 9.4 and 9.5 show, respectively, the cases where

Figure 9.5 Magnetizations and susceptibilities of first two cells vs temperature for *J _{s}* = —0.5 with

*L =*24 and

*D =*0.1. L

_{;}denotes the sublattice magnetization of layer

*j.*The susceptibility of sublattice 1 of the first cell is divided by a factor 5 for presentation convenience.

*J _{s} =* -0.8 and -0.5. Near

*J*= —0.8 the crossover takes place: The surface magnetization becomes smaller than the interior ones for

_{s}*J*> —0.8. Note that the magnetizations of second, third and fourth layers undergo a discontinuity at the transition temperature for

_{s}*J*0.8 and -0.5. This suggests that the phase transitions for interior layers are of first order as it has been found for bulk FCC antiferromagnet [82].

_{s}= -For weak |/_{s}|, there is only one transition for all layers. An example is shown in Fig. 9.6 for *J _{s} = -*0.1. Note that the first-

Figure 9.6 Magnetization and susceptibility of first two cells vs temperature for *J _{s} =* -0.1 with

*L =*24 and

*D =*0.1. L

_{;}denotes the sublattice magnetization of layer

*j.*

order character disappears: There is no discontinuity of layer magnetizations at the transition temperature.

In the region -0.5 * S<* -0.45, there is an interesting reentrant phenomenon. To facilitate the description of this phenomenon, let us show the phase diagram in the space (

*J*

_{s},

*T*in Fig. 9.7. In the region —0.5

_{c})*-0.45, the GS is of type II as seen above. According to the phase diagram, we see that when the temperature increases from zero, the system goes through the phase of type II, undergoes a transition to enter the phase of type I before making a second transition to the paramagnetic phase at high temperature. This kind of behavior is termed as reentrant phenomenon which has been found by exact solutions in a number of veiy frustrated systems*S<

Figure 9.7 Critical temperature vs *J** _{5}* with L = 24 and

*D*= 0.1.

*L,*denotes data points for the maximum of the sublattice magnetization of layer

*j.*I and II denote ordering of type I and II defined in Fig. 9.2. Ill is paramagnetic phase. The discontinued vertical line is a first-order line. Errors are smaller than symbol sizes. See text for comments.

[67, 76]. For a complete review on these exactly solved systems, the reader is referred to the chapter by Diep and Giacomini [86] in Ref. [85]. We note here that the reentrance is often found near the frontier where two phases coexist in the GS [86]. This is the case at *Js=J _{s}^{c} = -* 0.5.

The discontinued vertical line at *J _{s}* = -0.5 is a first-order line separating phases I and II. The coexistence of these two phases which do not have the same symmetry explains the first-order character of this line. To show it explicitly, we have calculated at

*T*=0.15 the magnetization

*M*and the staggered magnetization

*M*of the first layer with varying

_{st}*J*across -0.5. From the GS configurations shown in Fig. 9.2, M should be zero in phase I and finite in phase II, and vice versa for

_{s}*M*This is observed at Г = 0.15 as shown in Fig. 9.8. The large discontinuity of

_{st}.*M*and

*M*at

_{st}*)*= —0.5 shows a veiy strong first-order character across the vertical line in Fig. 9.7.

_{s}Let us discuss on finite-size effects in the transitions observed in Fig. 9.3 to Fig. 9.6. This is an important question because it is known that some apparent transitions are artifacts of small system sizes. We have checked with *L* = 36. The results do not change.

Figure 9.8 The magnetization *M* and the staggered magnetization *M _{5t}* of first layer versus

*)*are shown, at

_{s}*T =*0.15, with

*L =*24 and

*D =*0.1.1 and II denote ordering of type I and II defined in Fig. 9.2. Ill is paramagnetic phase. See text for comments.

Of course, we may think that these sizes are still small to change the shape of the phase diagram, especially near *J _{s}* = —0.5, where finite-size effects may be strong because of the frontier between two phases. But the results from the Green's function method which are for infinite

*L*show, as will be seen later, the similar shape near

*J*= -0.5. So, we believe that the results in Fig. 9.7 remain for the infinite size.

_{s}To confirm further the observed transitions, we have made a study of finite-size effects on the layer susceptibilities at some chosen values of *) _{s}* by using the accurate MC multi-histogram technique [110-112] [see Chapter 6). At this point, let us recall that transitions in bulk Ising frustrated systems, unlike unfrustrated counterparts, have different natures: The antiferromagnetic FCC and HCP Ising lattices have strong first-order transition [272, 278, 334], while the stacked antiferromagnetic triangular lattice has a controversial nature (see references in Ref. [277]). The model studied here is the frustrated FCC film where surface effects can modify the strong first-order observed in its bulk counterpart.

Figure 9.9 Susceptibilities of layer 1 (left) and 2 (right) are shown for various sizes Las a function of temperature for *J _{s}* = -0.1 and

*D =*0.1.

Figure 9.10 Susceptibilities of layer 3 (left) and 4 (right) are shown for various sizes Las a function of temperature for/_{s} = -0.1 and *D =* 0.1.

Figure 9.11 Maximum sublattice susceptibility y^{max} versus *L* in the In — In scale, for *J _{s} = —* 0.1 and

*D =*0.1.

*Lj*denotes the sublattice magnetization of layer

*j.*The slopes of these lines give the ratios of exponents

*y/v.*

Our results show that transitions at *J _{s} =* — 1 and

*J*— 0.1 are real second-order transitions obeying some scaling law. Figure 9.9 shows the size effects on the maximum of the susceptibilities of the first and second layers for

_{s}=*J*0.1, while Fig. 9.10 shows that of the third and fourth layers. As seen, the maximum of the susceptibilities y

_{s}= —^{max}increases with increasing

*L.*

Using the scaling law y^{max} oc *L ^{y/v}* (see Chapter 6), we plot In y

^{max }versus In

*L*in Fig. 9.11. The ratio of the critical exponents

*y/v*is obtained by the slope of the straight line connecting the data points of each layer.

Within errors the third and fourth layers have the same value of *y/v* which is neither 2D nor 3D Ising universality classes, 1.75 and 2, respectively. The same holds for the values of the first and second layers. The exponent *v* can be obtained as follows. We calculate as a function of *T* the magnetization derivative with respect to *fi* = *[квТу ^{1}: Vi =* ((In

*Mj) = (E) - (ME)/(M)*where

*E*is the system energy and

*M*the sublattice order parameter. We identify the maximum of

*V*for each size

_{x}*L.*From the finite-size scaling we know that V

^{r}1

^{max}is proportional to L

^{1/v}[112]. We plot in Fig. 9.12 In К"’

^{3}’' as a function of In

*L*for

*J*= —0.1. The slope of each line gives

_{s}*1/v.*For the case

*J*= —0.1, we obtain

_{s}*v*

*=*0.822 ± 0.020, 0.795 ±

Figure 9.12 The maximum value of {(In *M)')* = {*E) **— **(ME)* / *(M)* versus *L *in the ln-ln scale for *J _{s}* = —0.1, where

*M*is the sublattice order parameter. The slope of each line gives 1/v. L, denotes the sublattice magnetization of layer

*j.*

0.020, 0.790 ± 0.020, 0.782 ± 0.020 for the first, second, third and fourth layers. These values are far from the 2D value (v = 1). Wededuce *у =* 1-510 ± 0.010,1.442±0.015,1.412±0.025, 1.395± 0.025. The values of *v* and *у* are decreased when one goes from the surface to the interior of the film.

We show in Fig. 9.13 and Fig. 9.14 the maximum of sublattice magnetizations and their derivatives for the first two layers in the case of *J _{s} =* -1. We find vj = 0.794 ± 0.022,

*v*= 0.834 ± 0.027,

_{2}*= 1.524 ± 0.0.040, and*

_{yi}*y*= 1.509 ± 0.022.

_{2}Let us discuss on the values of the critical exponents obtained above. These values do not correspond neither to 2D nor 3D Ising models (y_{2}o= 1.75, *v _{2D} = l,* y

_{3D}= 1.241, v

_{3D}= 0.63). There are multiple reasons for those deviations. Apart from numerical precisions and the modest sizes we used, there may be deep physical origins.

A first question which naturally arises is the effect of the frustration. The 3D version of this model, as said above, has a first-order transition, with a veiy strong character for the Ising case [272, 278, 334] and somewhat less strong for the continuous spin models [82, 83]. It has been shown that at finite temperature, the phenomenon called "order by disorder" occurs leading to a

Figure 9.13 Maximum sublattice susceptibility y^{max} versus *L* in the In — In scale, for *) _{s}* = — 1 and

*D =*0.1. L

_{;}denotes the sublattice magnetization of layer

*j.*The slopes of these lines give the ratios of exponents

*y/v.*

Figure 9.14 The maximum value of {(In M}') = {*E)* — *(ME)* / *(M)* versus *L *in the In - In scale for *J _{s} = —* 1, where

*M*is the sublattice order parameter. The slope of each line gives 1/v.

*L*denotes one sublattice magnetization of layer

_{k}Sj*j.*

reduction of degeneracy: Only collinear configurations survive by an entropy effect [82, 146, 349]. The infinite degeneracy is reduced to 6, i.e., the number of ways to put two AF spin pairs on a tetrahedron. The model is equivalent to 6-state Potts model. The first-order transition observed in the 3D case is in agreement with the Potts criterion according to which the transition in *q*-state Potts model is of first-order in 3D for *q >* 3.

In the case of a film with finite thickness studied here, it appears that the first-order character is lost.

A first possible cause is from the degeneracy. According to the results shown in the previous subsection, the GS degeneracy is 2 or 4 depending to *] _{s}.* If we compare to the Potts criterion according to which the transition is of first-order in 2D only when

*q >*4, then the transition in thin films should be of second order. That is indeed what we observed.

Another possible cause for the second-order transition observed here is from the role of the correlation in the film. For second-order transitions, some arguments, such as those from renormalization group, say that the correlation length in the direction perpendicular to the film is finite. Hence, it is irrelevant to the criticality. The film should have the 2D character. 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. As a consequence, two pictures can be thought of: (i) The whole system may be correlated and the first-order character is to become a second-order one; (ii) the correlation length is longer but still finite, and the transition remains of first order.

At this point, we would like to emphasize that in the case of simple surface conditions, i.e., no significant deviation of the surface parameters with respect to those of the bulk, the bulk behavior is observed when the thickness becomes larger than a few dozens of atomic layers [75, 78]: Surface effects are insignificant on some thermodynamic properties such as the value of the critical temperature, the mean value of magnetization at a given *T, ...* It should be, however, stressed that the criticality is very different. It depends on the correlation length compared to the thickness: For example, we have obtained in the case of simple cubic films with Ising model the critical exponents identical to those of 2D Ising universality class up to thickness of nine layers [270]. Due to the small thickness used here, we think that the 2D character should be assumed.

Now for the anisotropy, remember that in the case studied here, we do not deal with the discrete Ising model but rather an Ising-like Heisenberg model. The deviation from the 2D values may then result in part from a complex coupling between the Ising-like symmetry and the continuous nature of the classical Heisenberg spins. This deviation may be important if the anisotropy constant *D *is small as in the case studied here. From the renormalization group calculations, since anisotropy is a relevant parameter, one expects that any finite anisotropy will lead to Ising-like critical behavior, but with corrections due to the continuous nature of Heisenberg spins before one enters the linear regime around the Ising fixed point.

To conclude this paragraph, we believe, from physical arguments given above, that the critical exponents obtained above which do not belong to any known universality class may result from different physical mechanisms. This is a subject of future investigations. We will come back to the question of criticality in thin films in Chapter 16.

# Green’s Function Results

We can rewrite the full Hamiltonian (9.1) in the local framework as

where cos (0,-y) is the angle between two NN spins.

To study properties of quantum spins over a large region of temperatures, there are only a few methods which give relatively correct results. Among them, the GF method is known to recover the exact results at very low-T obtained from the spin wave theory. In addition, it is better than the spin wave theoiy at higher temperatures and can be used up to the transition temperature with of course less precision on the nature of the phase transition. We choose here this method to study quantum effects at low *T* and to obtain the phase diagram at high *T.*

The GF method can be used for non-collinear spin configurations [285]. In the case studied here, one has a collinear one because of the Ising-like anisotropy. In this case, we define two double-time GF by [340]

The equations of motion for G/y(t, *t*') and F,y(t, t') are written by

We shall neglect higher-order correlations by using the Tyablikov decoupling scheme [43] which is known to be valid for exchange terms [121]. Then, we introduce the following Fourier transforms:

where *u>* is the spin wave frequency, *k _{xy}* denotes the wave vector parallel to

*xy*planes, R, is the position of the spin at the site /,

*n*and

*ri*are, respectively, the indices of the layers where the sites

*i*and

*j*belong to. The integral over

*k*is performed in the first Brillouin zone whose surface is Д in the

_{xy}*xy*reciprocal plane.

The Fourier transforms of the retarded GF satisfy a set of equations rewritten under the following matrix form:

where M (a>is a square matrix *[2N _{Z}* x

*2N*g and u are the column matrices which are defined as follows:

_{Z}),

and

where

in which, Z = 4 is the number of in-plane NN, 0^_{±1} the angle between two NN spins of sublattice 1 and 3 belonging to the layers *n* and n±l (see Fig. 9.2), *б£1** _{±1}* the angle between two NN spins of sublattice 1 and 4,

*6*

*the angle between two in-plane NN spins in the layer*

_{n}*n,*and

Here, for compactness we have used the following notations:

- (i)
*]„*and*D*are the in-plane interactions. In the present model,_{n}*J*is equal to_{n }*]*for the two surface layers and equal to_{s}*J*for the interior layers. All*D„*are set to be*D.* - (ii)
*Jn,*_{n±}i are the interactions between a spin in the n-th layer and its neighbor in the (n± l)-th layer. Of course,*J„,n-i*= Oifn = 1,

= 0 if *n = N _{z}.*

Solving det|M| = 0, we obtain the spin wave spectrum *со* of the present system. The solution for the GF *g„, _{n}* is given by

with |M|„ being the determinant made by replacing the n-th column of |M| by **u **in (9.16). Writing now

one sees that *со,* (k_{x>},), *i* = 1,..., *N _{z},* are poles of the GF

*g*(k

_{ni}„. coj_{x/}) can be obtained by solving |M| = 0. In this case,

*g,*can be expressed as

_{hn}

where /„ Ц *(k _{xy}))* is

Next, using the spectral theorem which relates the correlation function {*S~S~I')* to the GF [383], one has

Figure 9.15 Layer magnetization of first four layers vs temperature for *J _{s}* = —1.0 and

*D =*4. L

_{;}denotes the sublattice magnetization of layer

*j.*Note that except the first layer (upper curve), all other layer magnetizations coincide in this figure scale.

where e is an infinitesimal positive constant and /J = 1 */k _{B}T, k_{B }*being the Boltzmann constant.

Using the GF presented above, we can calculate self-consistently various physical quantities as functions of temperature *T.* We start the self-consistent calculation from *T* = 0 with a small step for temperature: 5 x 10^{-3} at low *T* and 10^{-1} near *T _{c}* (in units of

*J/kg).*The convergence precision has been fixed at the fourth figure of the values obtained for the layer magnetizations. We know from the previous section that the spin configuration is collinear. Therefore in this section, we shall use a large value of Ising anisotropy

*D*in order to get a rapid numerical convergence. For numerical calculation, we will use

*D =*4 and

*J*= - 1 and a size of 80

^{2}points in the first Brillouin zone.

Figure 9.15 shows the sublattice magnetizations of the first four layers. As seen, the first-layer one is larger than the other three just as in the case of the classical spins shown in Fig. 9.3. This difference in sublattice magnetization between layers vanishes at *J _{s}* ~ -0.8 as seen in Fig. 9.16. Again here, one has a good agreement with the case of classical spins shown in Fig. 9.4.

Figure 9.16 Layer magnetizations of first four layers vs temperature for *J _{s}* = —0.8 and

*D =*4. L, denotes the sublattice magnetization of layer

*j.*Note that except the first layer at low

*T*(upper curve) all other layer magnetizations coincide in this figure scale.

Figure 9.17 Layer magnetization of first four layers vs temperature for *J _{s}* = —0.5 and

*D =*4. L

_{;}denotes the sublattice magnetization of layer

*j.*Note that the first layer makes a crossover: It is higher at low

*T*and smaller at high

*T*than all other layer magnetizations which coincide in this figure scale. See text for comments on the crossover of surface magnetization.

For *J _{s} >* —0.8, the sublattice magnetization of the first layer is larger at low

*T*and higher at high

*T*as seen in Fig. 9.17 for

*J*= -0.5. This crossover of sublattice magnetizations comes from the competition between quantum fluctuations and the strength of

_{s}Figure 9.18 Layer magnetization of first four layers vs temperature for *J _{5} = *—0.1 and

*D =*4.

*Lj*denotes the sublattice magnetization of layer

*j.*Only at low

*T*the surface magnetization is distinct (upper curve) from the other ones.

*J*_{s}: When |/_{s}| is small, quantum fluctuations of the surface layer are small yielding a small zero-point spin contraction for surface spins at Г = 0. So, surface magnetization is higher than the interior ones. At higher *T*, however, small |/_{s}| gives rise to a small local field for surface spins which in turn yields a smaller surface magnetization at high *T.* This crossover has been found earlier in antiferromagnetic superlattices and films [79, 80].

For *J _{s}* = —0.1, there is no more crossover at low

*T*as seen in Fig. 9.18. Moreover, there is only a single transition at

*T*~ 2.65 for both surface and interior layers.

_{c}We summarize in Fig. 9.19 the phase diagram for the quantum spin case obtained with the GF method. The vertical discontinued line indicates the boundary between ordered phases of types I and II. Phase III is paramagnetic. Note the following interesting points:

- (i) For
*J*—0.4 there is a surface transition distinct from that of interior layers._{s}< - (ii) For
*J*<—0.8, surface transition occurs ata temperature higher than that of interior layers._{s}

Figure 9.19 Phase diagram obtained by the Green’s function method with *D =* 4. *Lj* denotes the transition temperature of the sublattice magnetization of layer *j.* Errors are smaller than symbol sizes. See text for comments.

(iii) There is a reentrance between *J _{s}* = —0.4 and

*J*= —0.5. This is very similar to the phase diagram of the classical spins obtained by MC simulations shown in Fig. 9.7.

_{s}# Concluding Remarks

We have shown, by means of a Green’s function method and MC simulations, the results of the Heisenberg spin model with an Ising- like interaction anisotropy in thin films of stacked triangular lattices. The two surfaces of the film are frustrated. We found that surface spin configuration is non-collinear when surface antiferromagnetic interaction is smaller than a critical value //. In the non-collinear regime, the surface layer is disordered at a temperature lower than that for interior layers ("soft” surface). This can explain the so-called "magnetically dead surface” observed in some materials [36, 374]. The surface transition disappears for *J _{s}* larger than the critical value //. A phase diagram is established in the space (

*T, J*A good agreement between the Green's function method and the MC simulation is observed. This is due to the fact that at high temperatures where the transition takes place, the quantum nature of spins used in Green's function is lost so that we should find results of classical spins used in MC simulations. We have also studied by MC histogram technique the critical behavior of the phase transition using the finite-size effects. The result of the ratio of critical exponents

_{s}).*y/v*shows that the nature of the transition is complicated due to the influence of several physical mechanisms. The symmetry of the ground state alone cannot explain such a result. We have outlined a number of the most relevant mechanisms. Finally, we note that in surface magnetism the low surface magnetization experimentally observed [36, 374] has been generally attributed to the effects of the reduction of magnetic moments of surface atoms and/or the surface-localized low-lying magnon modes. The model considered in this chapter adds another origin for the low surface magnetization: surface frustration. It completes the list of possible explanations for experimental observations in thin films.