Frustrated Thin Films of Antiferromagnetic FCC Lattice
In this chapter, we show the effects of frustration in an antiferromagnetic film of face-centered cubic (FCC) lattice with Heisenberg spin model including an Ising-like anisotropy. Monte Carlo (MC) simulations have been used to study thermodynamic properties of the film. We show that the presence of the surface reduces the ground state (GS) degeneracy found in the bulk. The GS is shown to depend on the surface in-plane interaction Js with a critical value at which ordering of type I coexists with ordering of type II. Near this value, a reentrant phase is found. Various physical quantities such as layer magnetizations and layer susceptibilities are shown and discussed. We study here how physical properties vary as the surface bond strength changes at a fixed film thickness. The nature of the phase transition is also studied by the histogram technique.
We have also used the Green’s function (GF) method for the quantum counterpart model. The results at low-Г show interesting effects of quantum fluctuations. Results obtained by the GF method at high T are compared to those of MC simulations. A good agreement is observed.
Physics of Magnetic Thin Films: Theory and Simulation Hung T. Diep
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Some results shown in this chapter has been published in Ref. .
Effects of the frustration in spin systems have been extensively investigated during the past 30 years. Frustrated spin systems are shown to have unusual properties such as large ground state (GS) degeneracy, additional GS symmetries, successive phase transitions with complicated nature. Frustrated systems still challenge theoretical and experimental methods. For recent reviews, the reader is referred to Ref. .
On the other hand, during the same period physics of surfaces and objects of nanometric size have also attracted an immense interest. This is due to important applications in industry [35, 36, 72, 374]. In this field, results from laboratory research are often immediately used for industrial applications, without waiting for a full theoretical understanding. An example is the so-called giant magneto-resistance (GMR) used in data storage devices, magnetic sensors, ... [18, 22, 134, 345]. In parallel to these experimental developments, much theoretical effort has also been devoted to the search of physical mechanisms lying behind new properties found in nanometric objects such as ultrathin films, ultrafine particles, quantum dots, spintronic devices, etc. This effort aimed not only at providing explanations for experimental observations but also at predicting new effects for future experiments [248, 252].
The aim of this chapter is to show the effect of the presence of a film surface in a system which is known to be very frustrated, namely the FCC antiferromagnet. The bulk properties of this material have been largely studied as we will show below. We would like to see here in particular how the frustration effects on the nature of the phase transition in 3D are modified in thin films and how the surface conditions affect the magnetic phase diagram. We emphasize that we do not try to address how physical properties vaiy as the film thickness changes. Rather we work with only one film thickness and address the question how a change in the surface bond strength changes the physical properties. The effects obtained here certainly remain generic when the film thickness varies. We shall use in the following Monte Carlo (MC) simulations and the Green's function method for qualitative comparison.
In the following section, we describe the model and recall the properties of the 3D counterpart model in order to better appreciate properties of thin films. A determination of its GS properties is also given. In Section 9.3, we show our results obtained by MC simulations as functions of temperature T. The surface exchange interaction Js is made to vary. A phase diagram in the space (Г, Js) is shown and discussed. In general, the surface transition is found to be distinct from the transition of interior layers. An interesting reentrant region is observed in the phase diagram. We also show in this section the results on the critical exponents obtained by MC multi-histogram technique. A detailed discussion on the nature of the phase transition is given. Section 9.4 is devoted to a study of the quantum version of the same model by the use of the GF method. We find interesting effects of quantum fluctuations at low T. The phase diagram (T, Js) is established and compared to that obtained by MC simulations for the classical model.
Model and Classical Ground-State Analysis
It is known that the antiferromagnetic (AF) 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 corner-sharing tetrahedra each of which has four equilateral triangles. It is well-known  that it is impossible to fully satisfy simultaneously the three AF bond interactions on each triangle.
The analytical determination of the GS of systems of classical spins with competing interactions is a fascinating subject. For a recent review, the reader is referred to Ref. . For the bulk FCC antiferromagnet, the Heisenberg spins on a tetrahedron form a configuration characterized by two arbitrary angles . The ground state (GS) degeneracy is, therefore, infinite. This is also found in fully frustrated simple cubic lattice with classical Heisenberg spins : The GS is also characterized by two random continuous parameters. To give an idea about the GS of the bulk FCC antiferromagnet , let us imagine two planes, xz and ф, where ф intersects the xz plane along the z axis and makes an angle ф with the x axis. Two of the four spins make an angle в in the xz plane symmetric with respect to the z axis. The other two spins make also the same angle, symmetric with respect to the z axis, but in the plane ф. It has been shown  that the two angles в and ф are arbitrary between 0 and тс. Note that when в = 0 the spin configuration is collinear with two spins along the +z axis and the other two along the —z one. The phase transition of the bulk frustrated FCC Heisenberg antiferromagnet has been studied [82, 137]. In particular, the transition is found to be of the first order as in the Ising case [272, 278, 334]. Other similar frustrated antiferromagnets such as the HCP antiferromagnet show the same behavior .
Let us consider a film of FCC lattice structure with  surfaces. To avoid the absence of long-range order of isotropic non-Ising spin model at finite temperature (Г) when the film thickness is very small, i.e., quasi two-dimensional system , we add in the Hamiltonian an Ising-like uniaxial anisotropy term. The Hamiltonian is given by
where S, is the Heisenberg spin at the lattice site i, indicates the sum over the NN spin pairs S, and Sy.
In the following, the interaction between two NN surface spins is denoted by Js, while all other interactions are supposed to be antiferromagnetic and all equal to J = -1 for simplicity. Note that the case of pure Ising model on the simple cubic lattice has been studied by MC simulation with various surface conditions [34, 200].
We first determine the GS configuration by using the steepest descent method : Starting from a random spin configuration, we calculate the magnetic local field at each site and align the spin of the site in its local field. In doing so for all spins and repeat until the convergence is reached, we obtain easily the GS configuration without metastable states. The result is shown in Fig. 9.1.
We observe that there is a critical value // = - 0.5. For Js < _//, the spins in each yz plane are parallel while spins in adjacent
Figure 9.1 A ground state configuration of single plaquette (a) Sf is Sz of sublattice 1, (b) cos012, (c) cos023, (d) cos034. cos0f;- is the cosine of the angle between the two spins of sublattices i and j.
yz planes are antiparallel (Fig. 9.2a). This ordering will be called hereafter "ordering of type I”: In the x direction, the ferromagnetic planes are antiferromagnetically coupled as shown in this figure. Of course, there is a degenerate configuration where the ferromagnetic planes are antiferromagnetically ordered in the у direction. Note that the surface layer has an AF ordering for both configurations. The degeneracy of type I is, therefore, 4, including the reversal of all spins.
For Js > //, the spins in each xy plane is ferromagnetic. The adjacent xy planes have an AF ordering in the z direction perpendicular to the film surface. This will be called hereafter "ordering of type II." Note that the surface layer is then ferromagnetic (Fig. 9.2b). The degeneracy of type II is 2 due to the reversal of all spins.
Without using a general method [175, 262], let us calculate analytically the GS configuration in a simple manner for the present model.
Consider a tetrahedron with the spins numbered as in Fig. 9.2: Si, S2, S3 and S4 are the spins in the surface FCC cell (first cell). The interaction between Si and S2 is set to be equal to Js (—1 < Js < 0)
Figure 9.2 The ground state spin configuration of the FCC cell at the film surface: (a) ordering of type I for Js < — 0.5, (b) ordering of type II for Js > -0.5.
and all others are taken to be equal to ] (<0), and all Д = D for simplicity. The Hamiltonian for the cell is written as
Let us decompose each spin into two components: an xy component, which is a vector, and a z component S, = (S-, Sf). The numerical results shown above indicate that the spins have only z component. Taking advantage of this, we suppose that the xy vector components of the spins are all equal to zero. The angles 0, of S, with the z axis are then
The total energy of the cell (9.2), with S, = can be rewritten as
By a variational method, the minimum of the cell energy corresponds to
The solutions of Eq. (9.4) and Eq. (9.5) corresponding to the minimal energy are
Note that these solutions do not depend on D. The GS energy per spin is
We see that the solution (9.6) agrees with the numerical result.