# Thin Films and Criticality

In this chapter, we study the critical behavior of magnetic thin films as a function of the film thickness. We use the ferromagnetic Ising model with the high-resolution multiple histogram Monte Carlo (MC) simulation. We show that though the 2D behavior remains dominant at small thicknesses, there is a systematic continuous deviation of the critical exponents from their 2D values. We explain these deviations using the concept of "effective" exponents suggested by Capehart and Fisher [53] in a finite-size analysis. The shift of the critical temperature with the film thickness obtained here by MC simulation is in excellent agreement with their prediction.

In the second part of the chapter, we show the crossover of the phase transition from first to second order in the frustrated Ising FCC antiferromagnetic film. This crossover occurs when the film thickness *N _{z}* is smaller than a value between 2 and 4 FCC lattice cells. The results are obtained with the high-performance Wang-Landau flat histogram technique, which allows to determine the first-order transition with efficiency.

The results shown in this chapter have been published in Refs. [270,271].

*Physics of Magnetic Thin Films: Theory and Simulation *Hung T. Diep

Copyright © 2021 Jenny Stanford Publishing Pte. Ltd.

ISBN 978-981-4877-42-8 (Hardcover), 978-1-003-12110-7 (eBook) www.jennystanford.com

## Introduction

During the past 30 years, the physics of surfaces and objects of nanometric size has attracted immense interest. This is due to important applications in industiy [36, 374]. 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 [35, 72, 73] has been devoted to the search of physical mechanisms lying behind the new properties found in nanoscale 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.

The physics of two-dimensional (2D) systems is very exciting. Some of those 2D systems can be exactly solved: One famous example is the Ising model on the square lattice which has been solved by Onsager [265]. This model shows a phase transition at a finite temperature *T _{c}* given by sinh

^{2}(2///c

_{B}7Y) = 1 where

*J*is the nearest-neighbor (NN) interaction. Another interesting result is the absence of long-range ordering at finite temperatures for the continuous spin models (XY and Heisenberg models) in 2D [231]. In general, three-dimensional (3D) systems for any spin models cannot be unfortunately solved. However, several methods in the theoiy of phase transitions and critical phenomena can be used to calculate the critical behaviors of these systems [380].

This chapter deals with the question of criticality in thin films, namely systems between 2D and 3D. Many theoretical studies have been devoted to thermodynamic properties of thin films, magnetic multilayers, ... [35, 72, 79, 80, 87, 252]. In spite of this, several points are still not yet understood. It has been known since a long time ago that the presence of a surface in magnetic materials can give rise to surface spin waves which are localized in the vicinity of the surface [78]. These localized modes may be acoustic with a low- lying energy or optical with a high energy, in the spin wave spectrum. Low-lying energy modes contribute to reduce in general surface magnetization at finite temperatures. One of the consequences is the surface disordering which may occur at a temperature lower than that for interior magnetization [75]. The existence of low- lying surface modes depends on the lattice structure, the surface orientation, the surface parameters, surface conditions [impurities, roughness,..etc. There are two interesting cases: In the first case, a surface transition occurs at a temperature distinct from that of the interior spins, and in the second case, the surface transition coincides with the interior one, i.e., existence of a single transition. Theoiy of critical phenomena at surfaces [35, 72] and Monte Carlo (MC) simulations [200, 201] of critical behavior of the surface-layer magnetization at the extraordinary transition in the 3D Ising model have been carried out. These works suggested several scenarios in which the nature of the surface transition and the transition in thin films depends on many factors in particular on the symmetiy of the Hamiltonian and on surface parameters.

In Chapters 9 and 10 while studying the frustration effects in thin films, we have calculated some critical exponents in those frustrated thin films. We found that the critical exponents have values somewhere between 2D and 3D universality classes. However, due to the presence of the frustration in those cases, we did not make a firm conclusion that the deviation of the critical exponents from the 2D values results uniquely from the film thickness. The work presented in this chapter is, therefore, performed on a ferromagnetic Ising thin film without frustration.

We confine ourselves here to the case of a simple cubic film with the Ising model. For our purpose, we suppose all interactions are the same everywhere even at the surface. This case is the simplest case where there is no surface-localized spin wave modes and there is only a single phase transition at a temperature for the whole system [no separate surface phase transition) [75, 78]. Other complicated cases will be left for future investigations. However, some discussions on this point for complicated surfaces have been reported in some of our previous papers [248, 249] and presented in Chapters 9 and 10.

In the case of a simple cubic film with the Ising model, Capehart and Fisher have studied the critical behavior of the susceptibility using a finite-size scaling analysis [53]. They showed that there is a crossover from 2D to 3D behavior as the film thickness increases. The so-called "effective" exponent *у* has been shown to vary according to a scaling function depending both on the film thickness and the distance to the transition temperature. As will be seen below, the scaling suggested by Capehart and Fisher is in agreement with what we find here using extensive MC simulations.

We investigate in this chapter how the film thickness affects the critical exponents of the film as seen in simulations. Whatever the interpretation will be, the apparent deviations of the critical exponents from their 2D values are probably also seen in experiments. To carry out these purposes, we shall use MC simulations with highly accurate multiple-histogram technique presented in Chapter 6 (see original papers in Refs. [50,110, 111]).

Section 16.2 is devoted to a description of the model and method. Results are shown and discussed in Section 16.3. Section 16.4 shows the crossover of the transition from the first-order the second-order order when the film thickness decreases. Concluding remarks are given in Section 16.5.

## Model and Method

### Model

Let us consider the Ising spin model on a film made from a ferromagnetic simple cubic lattice. The size of the film is L x *L* x *N _{z}. *We apply the periodic boundary conditions (PBC) in the

*xy*planes to simulate an infinite

*xy*dimension. The

*z*direction is limited by the film thickness

*N*If

_{z}.*N*1 then one has a 2D square lattice.

_{z}=The Hamiltonian is given by

where

In the following, the interaction between two NN surface spins is denoted by *J _{s},* while all other interactions are supposed to be ferromagnetic and all equal to

*]*= 1 for simplicity. Let us note in passing that in the semi-infinite ciystal the surface phase transition occurs at the bulk transition temperature when

_/s~1.52/. This point is called "extraordinary phase transition” which is characterized by some particular critical exponents [200, 201]. In the case of thin films, i. e. the thickness *N _{z}* is finite, not equal to 1, it has been theoretically shown that when

*J*= 1 the bulk behavior is observed when the thickness becomes larger than a few dozens of atomic layers [78]: Surface effects are insignificant on thermodynamic properties such as the value of the critical temperature and the mean value of magnetization at a given

_{s}*T.*When

*J*is smaller than

_{s}*J*, surface magnetization is destroyed at a temperature lower than that for bulk spins [75] [see Chapter 8). The criticality of a film with uniform interaction, i.e.,

*]*has been studied by Capehart and Fisher as a function of the film thickness using a scaling analysis [53] and by MC simulations [55, 309]. The results by Capehart and Fisher indicate that as long as the film thickness is finite, the phase transition is strictly that of the 2D Ising universality class. However, they showed that at a temperature away from the transition temperature

_{s}= ],*T*the system can behave as a 3D one when the spin-spin correlation length £(T) is much smaller than the film thickness, i.e.,

_{C}(N_{Z}),*%[T)/N*<£ 1. As

_{Z}*T*gets very close to

*T*-> 1, the system undergoes a crossover to 2D criticality. We will return to this work for comparison with our results shown below.

_{C}(N_{Z}), %[T)/N_{z}### Multiple-Histogram Technique

The multiple-histogram technique is known to reproduce with very high accuracy the critical exponents of second order phase transitions [50,110,111]. We recall it briefly here.

The overall probability distribution [111] at temperature *T* obtained from *n* independent simulations, each with *Nj* configurations, is given by

where

The thermal average of a physical quantity *A* is then calculated by
in which

Thermal averages of physical quantities are thus calculated as continuous functions of *T,* now the results should be valid over a much wider range of temperature than for any single histogram.

In MC simulations, one calculates the averaged order parameter (M) (M: magnetization of the system), averaged total energy {*E), *specific heat *C _{v},* susceptibility y, first-order cumulant of the energy

*C*and nth-order cumulant of the order parameter

_{Ut}*V*for

_{n}*n*= 1 and 2. These quantities are defined as

Let us discuss the case where all dimensions can go to infinity. For example, consider a system of size *L ^{d}* where

*d*is the space dimension. For a finite

*L,*the pseudo "transition" temperatures can

be identified by the maxima of *C _{v}* and y,____These maxima do not

in general take place at the same temperature. Only at infinite Lthat the pseudo "transition” temperatures of these respective quantities coincide at the real transition temperature Г_{с}(оо). So when we work at the maxima of *V„, C _{v}* and у, we are in fact working at temperatures away from T

_{c}(oo). This is an important point to bear in mind for the discussion given below.

For large values of *L,* the following scaling relations are expected (see details in Ref. [50]):

and

at their respective "transition” temperatures *T _{C}(L),* and
and

where *A,* C_{0}, *C*i and *C _{A}* are constants. We estimate

*v*independently from K

_{1}

^{max}and k

_{2}

^{max}. With this value we calculate

*у*from у

^{max }and

*a*from C™

^{ax}. Note that we can estimate 7V(oo) using the last expression. Then, using T

_{c}(oo), we can calculate

*fi*from

*M*oo)- The Rushbrooke scaling law

_{Tc}(*a +*+

*у =*2 is then in principle verified.

Let us emphasize that the expressions Eqs. (16.11)-(16.16) are valid for large *L.* To be sure that Lare large enough, one has to allow for corrections to scaling of the form, for example,

where f?_{b} *B _{2}, D_{x}* and

*D*

*are constants and*

_{2}*u>*is a correction exponent [112]. Similar forms exist also for the other exponents. Usually, these corrections are extremely small if

*L*is large enough as is the case with today's large-memory computers. So, in general they do not alter the results using Eqs. (16.11)-(16.16).

### The Case of Films with Finite Thickness

In the case of a thin film of size *Lx Lx N _{z},* Capehart and Fisher [53] have showed that as long as the film thickness

*N*is not allowed to go to infinity, there is a 2D-3D crossover if one does not work at the real transition temperature

_{z}*T*oo,

_{C}[L =*N*Following Capehart and Fisher, let us define

_{z}).where *v _{3D}* is the 3D о exponent and

*T*the 3D critical temperature. When

_{C}(3D)*x*is larger than a value xo, i.e., at a temperature away from

*T*= oo,

_{C}(L*N*the system behaves as a 3D one. While, when

_{z}),*x < x*it should behave as a 2D one. This crossover was argued from a comparison of the correlation length in the

_{0}*z*direction to the film thickness. As a consequence, if we work exactly at

*T*= oo,

_{C}{L*N*

_{z}) we should observe the 2D critical exponents for finite

*N*Otherwise, we should observe the so-called "effective critical exponents" whose values are found between those of 2D and 3D cases. This point is fundamentally very important. There have been some attempts to verify it by MC simulations [309] but these results were not convincing due to their poor MC quality. In the following, we show with high-precision MC multiple-histogram technique that the prediction of Capehart and Fisher is really verified.

_{z}.