# Partial Phase Transition in Helimagnetic Thin Films in a Field

We study the phase transition in a helimagnetic film with Heisenberg spins under an applied magnetic field in the *c* direction perpendicular to the film. The helical structure is due to the antiferromagnetic interaction between next-nearest neighbors in the c direction. Helimagnetic films in zero field are known to have a strong modification of the in-plane helical angle near the film surfaces. We show that spins react to a moderate applied magnetic field by creating a particular spin configuration along the *c* axis. With increasing temperature (Г), using Monte Carlo simulations we show that the system undergoes a phase transition triggered by the destruction of the ordering of a number of layers. This partial phase transition is shown to be intimately related to the ground-state spin structure. We show why some layers undergo a phase transition while others do not. The Green’s function method for non-collinear magnets is also carried out to investigate effects of quantum fluctuations. Non-uniform zero-point spin contractions and a crossover of layer magnetizations at low *T* are shown and discussed.

The results shown in this chapter have been published in Ref. [105].

*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

Helimagnets have been subject of intensive investigations over the past four decades since the discoveiy of its ordering [350, 369]: In the bulk, a spin in a space direction turns an angle *в* with respect to the orientation of its previous nearest neighbor (see Fig. 12.1). This helical structure can take place in several directions simultaneously with different helical angles. The helical structure shown in Fig. 12.1 is due to the competition between the interaction between nearest neighbors (NN) and the antiferromagnetic interaction between next- nearest neighbors (NNN). Other helimagnetic structures have also been very early investigated [28, 29, 169]. Spin wave properties in bulk helimagnets have been investigated by spin wave theories [91, 139, 289] and Green's function method [286]. Heat capacity in bulk MnSi has been experimentally investigated [269].

We confine ourselves to the case of a Heisenberg helical film in an applied magnetic field. Helimagnets are special cases of a large family of periodic non-collinear spin structures called frustrated systems of XY and Heisenberg spins. The frustration has several

Figure 12.1 Spin configuration along the *c* direction in the bulk case where /2//1 = -1, *Hfh* = 0.

origins: (i) It can be due to the geometry of the lattice such as the triangular lattice, the face-centered cubic (FCC) and hexagonal- close-packed (HCP) lattices, with antiferromagnetic NN interaction [19, 82, 83, 224, 276]. (ii) It can be due to competing interactions between NN and NNN such as the case of helimagnets [350, 369] shown in Fig. 12.1. (iii) It can be due to the competition between the exchange interaction which favors collinear spin configurations and the Dzyaloshinskii-Moriya (DM) interaction which favors perpendicular spin arrangements.

Effects of the frustration have been extensively studied in various systems during the past 30 years. The reader is referred to recent reviews on bulk frustrated systems given in Ref. [85]. When frustration effects are coupled with surface effects, the situation is often complicated. Let us mention our previous works on a frustrated surface [248] and on a frustrated FCC antiferromagnetic film [249] where surface spin rearrangements and surface phase transitions have been found (see Chapters 9 and 10). We have also recently shown results in zero field of thin films of body-centered cubic (BCC) and simple cubic (SC) structures in the previous chapter. The helical angle along the *c* axis perpendicular to the film surface was found to strongly vaiy in the vicinity of the surface. The phase transition and quantum fluctuations have been presented.

In this chapter, we are interested in the effect of an external magnetic field applied along the *c* axis perpendicular to the film surface of a helimagnet with both classical and quantum Heisenberg spins. Note that without an applied field, the spins lie in the *xy *planes: Spins in the same plane are parallel while two NN in the adjacent planes form an angle *a* which varies with the position of the planes [102] (see the previous chapter), unlike in the bulk. As will be seen below, the applied magnetic field gives a very complex spin configuration across the film thickness. We determine this ground state (GS) by the numerical steepest descent method. We will show by Monte Carlo (MC) simulation that the phase transition in the field is due to the disordering of a number of layers inside the film. We identify the condition under which a layer becomes disordered. This partial phase transition is not usual in thin films where one observes more often the disordering of the surface layer, not an interior layer. At low temperatures, we investigate effects of quantum fluctuations using a Green’s function (GF) method for non- collinear spin configurations.

Section 12.2 is devoted to the description of the model and the determination of the classical GS. The structure of the GS spin configuration is shown as a function of the applied field. Section 12.3 is used to show the MC results at finite temperatures where a partial phase transition is observed. Effects of the magnetic field strength and the film thickness are shown. The GF method is described in Section 12.4 and its results on the layer magnetizations at low temperatures are displayed and discussed in terms of quantum fluctuations.

## Model: Determination of the Classical Ground State

We consider a thin film of SC lattice of *N _{z}* layers stacked in the

*c*direction. Each lattice site is occupied by a Heisenberg spin. For the GS determination, the spins are supposed to be classical spins in this section. The Hamiltonian is given by

where is the interaction between two spins S, and S_{y} occupying the lattice sites / and *j* and H denotes an external magnetic field applied along the *c* axis. To generate helical angles in the c direction, we suppose an antiferromagnetic interaction *J** _{2}* between NNN in the c direction in addition to the ferromagnetic interaction

*J i*between NN in all directions. For simplicity, we suppose that

*J*i is the same everywhere. For this section, we shall suppose

*J*is the same everywhere for the presentation clarity. Note that in the bulk in zero field, the helical angle along the

_{2}*c*axis is given by cos a = — ^ for a SC lattice [87] with I/2I > 0.25/i. Below this value, the ferromagnetic ordering is stable (see Section 3.4).

In this chapter, we will study physical properties as functions of *J**2**/J**1**, H/Ji* and *k _{B}T/Ji.* Hereafter, for notation simplicity we will take /1 = 1 and

*k*1. The temperature is thus in unit of

_{B}=*Ji/k*the field and the energy are in unit of

_{B},*J*1.

In a film, the angles between NN in adjacent planes are not uniform across the film: A strong variation is observed near the surfaces. An exact determination can be done by energy minimization [89] or by numerical steepest descent method [248, 249]. The latter is particularly efficient for complex situations such as the present case where the spins are no longer in the *xy* planes in an applied field: A spin in the i-th layer is determined by two parameters which are the angle with its NN in the adjacent plane, say and the azimuthal angle Д- formed with the *c* axis. Since there is no competing interaction in the *xy* planes, spins in each plane are parallel. We shall use here the steepest descent method which consists in calculating the local field at each site and aligning the spin in its local field to minimize its energy. The reader is referred to Ref. [248] for a detailed description. In so doing for all sites and repeating many times until a convergence to the lowest energy is obtained with a desired precision (usually at the sixth digit, namely at ~ 10^{-6} per cents), one obtains the GS configuration. Note that we have used several thousands of different initial conditions to check the convergence to a single GS for each set of parameters.

Figures 12.2a and 12.2b show the spin components *S ^{z}, Sу* and

*S**for all layers. The spin lengths in the

*xy*planes are shown in Fig. 12.2c. Since the spin structure in a field is complicated and plays an important role in the partial phase transition shown in the next section, let us describe it in details and explain the physical reason lying behind:

- (1) Several planes have negative
*z*spin components. This can be understood by examining the competition between the magnetic field which tends to align spins in the*c*direction, and the antiferromagnetic interaction*J*which tries to preserve the antiferromagnetic ordering. This is veiy similar to the case of collinear antiferromagnets: in a weak magnetic field, the spins remain antiparallel, and in a moderate field, the so-called "spin flop” occurs: The neighboring spins stay antiparallel with each other but turn themselves perpendicular to the field direction to reduce the field effect [87]._{2} - (2) Due to the symmetry of the two surfaces, one observes the following symmetry with respect to the middle of the film:

Figure 12.2 Spin components across the film in the case where *H =* 0.2. The horizontal axis *Z* represents plane *Z* (Z = 1 is the first plane etc.): (a) *S*^{z}; (b) *S ^{x}* (red) and

*S*(blue); (c) Modulus

^{y}*S*of the projection of the spins on the

^{xy}*xy*plane. See text for comments.

Note that while the *z* components are equal, the *x* and *у* components are antiparallel (Fig. 12.2b): The spins preserve their

Figure 12.3 Spin configuration in the case where *H* = 0.4, *J** _{2}* = — 1,

*N*12. The circles in the

_{z}=*xy*planes with radius equal to 1 are plotted to help identify the orientation of each spin. The spins when viewed along the

*c*axis are shown in Fig. 12.4d.

antiferromagnetic interaction for the transverse components. This is similar to the case of spin flop in the bulk (see p. 86 of Ref. [87]). Only at a very strong field that all spins turn into the field direction.

(3) The GS spin configuration depends on the film thickness. An example will be shown in the next section.

A full view of the "chain” of *N _{z}* spins along the

*c*axis between the two surfaces is shown in Fig. 12.3.

Note that the angle in the *xy* plane is determined by the NNN interaction *J*_{2}*■* Without field, the symmetry is about the *c* axis, so *x* and *у* spin components are equivalent (see Fig. 12.1). Under the field, due to the surface effect, the spins make different angles with the *c* axis giving rise to different *z* components for the layers across the film as shown in Fig. 12.2a. Of course, the symmetiy axis is still the *c* axis, so all *S ^{x}* and

*S*are invariant under a rotation around the

^{y}*c*axis. Figure 12.2b shows the symmetry of

*S*as that of

^{x}*S*across the film as outlined in remark (iii). Figure 12.2b is thus an instantaneous configuration between

^{y}*S*and

^{x}*S*for each layer across the film. As the simulation time is going on these components rotate about the

^{y}*c*axis but their symmetry outlined in remark (iii) is valid at any time. The

*xy*spin modulus

*S*shown in Fig. 12.2c, on the other hand, is time-invariant. The phase transition occurring for layers with large

^{xy}*S*disordering) is shown in the next section.

^{xy}(xyTheGS spin configuration depends on the field magnitude *H .If H *increases, we observe an interesting phenomenon: Fig. 12.4 shows the spin configurations projected on the *xy* plane (top view) for increasing magnetic field. We see that the spins of each chain tend progressively to lie in a same plane perpendicular to the *xy* planes (Figs. 12.4a-c). The "planar zone” observed in Fig. 12.4c occurs between *H* ~ 0.35 and 0.5. For stronger fields they are no more planar (Fig. 12.4d-f). Note that the larger the *xy* component is, the smaller the *z* component becomes: For example, in Fig. 12.4a, the

Figure 12.4 Top view of (projection of spins on *xy* plane) across the film for several values of *H:* (a) 0, (b) 0.03, (c) 0.2, (d) 0.4, (e) 0.7, (f) 1.7. The radius of the circle, equal to 1, is the spin full length: For high fields, spins are strongly aligned along the *c* axis, *S ^{xy}* is therefore much smaller than 1.

spins are in the *xy* plane without field *(H* = 0) and in Fig. 12.4f they are almost parallel to the c axis because of a high field.