# Phase Transition in Helimagnetic Thin Films

In this chapter, we show the main properties of a helimagnetic thin film with quantum Heisenberg spin model by using the Green's function (GF) method. Surface spin configuration is calculated by minimizing the spin interaction energy It is shown that the angles between spins near the surface are strongly modified with respect to the bulk configuration. Taking into account this surface spin reconstruction, we calculate self-consistently the spin wave spectrum and the layer magnetizations as functions of temperature up to the disordered phase. The spin wave spectrum shows the existence of a surface-localized branch which causes a low surface magnetization. We show that quantum fluctuations give rise to a crossover between the surface magnetization and interior-layer magnetizations at low temperatures. We calculate the transition temperature and show that it depends strongly on the helical angle. Results are in agreement with existing experimental observations on the stability of helical structure in thin films and on the insensitivity of the transition temperature with the film thickness. We also study effects of various parameters such as surface exchange and anisotropy interactions. Monte Carlo simulations for the classical spin model are also carried out for comparison with the quantum theoretical result.

*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

The results shown in this chapter are taken from Ref. [89] for the body-centered cubic (BCC) lattice, and from Ref. [102] for the simple cubic (SC) case.

## Introduction

Helimagnets were discovered a long time ago by Yoshimori [369] and Villain [350]. In the simplest model, the helimagnetic ordering is non-collinear due to a competition between nearest-neighbor (NN) and next-nearest-neighbor (NNN) interactions: For example, a spin in a chain turns an angle *в* with respect to its previous neighbor. Low-temperature properties in helimagnets such as spin waves [91, 139, 286, 289] and heat capacity [269] have been extensively investigated. Helimagnets belong to a class of frustrated vector-spin systems. In spite of their long history, the nature of the phase transition in bulk helimagnets as well as in other non- collinear magnets such as stacked triangular XY and Heisenberg antiferromagnets has been elucidated only recently [90, 187, 250, 251]. For reviews on many aspects of frustrated spin systems, the reader is referred to Ref. [85].

In this chapter, we study a helimagnetic thin film with the quantum Heisenberg spin model. Surface effects in thin films have been widely studied theoretically, experimentally and numerically, during the past three decades [36, 374]. Nevertheless, surface effects in helimagnets have only been recently studied: surface spin structures [229], Monte Carlo (MC) simulations [64], magnetic field effects on the phase diagram in Ho [291] and a few experiments [176,177]. We will compare our work to these in the conclusion.

Helical magnets present potential applications in spintronics with predictions of spin-dependent electron transport in these magnetic materials [150, 166, 357]. We shall use the GF method to study a quantum spin model on a helimagnetic thin film of body- centered cubic (BCC) lattice. The GF method has been initiated by Zubarev [383] for collinear bulk magnets (ferromagnets and antiferromagnets) and by Diep et al. for thin films of collinear spin configurations [78]. For non-collinear magnets, the GF method has also been developed for bulk helimagnets [286] and for frustrated films in Refs. [248, 249] with results presented in the previous chapters. Note that surface effects in thin films of stacked triangular antiferromagnets with non-collinear 120° spin configuration have been investigated by the method of equation of motion [230]. However, the model in these works did not possess a surface spin reconstruction and do not belong to the family of helical structures as our model described below.

In helimagnets, the presence of a surface modifies the competing forces acting on surface spins. As a consequence, as will be shown below, the angles between neighboring spins become non- uniform, making calculations harder. This explains why there was no microscopic calculation for helimagnetic films before Refs. [89,102].

Note that for illustration, we use below the BCC lattice structure, but the results shown below are valid for different lattices, not restricted to the BCC crystal, provided modifications on the coordination number and therefore on the value of the critical value (Уг/УОс *[Ji ^{:}* NNN interaction,

*J*NNN interaction). For example, the BCC case has (Уг/УОс = 1, while the simple cubic lattice has С/2/УОс = 1/4.

_{2}■In Section 11.2, the model is presented and classical ground state (GS) of the helimagnetic film is determined. In Section 11.3, the general GF method for non-uniform spin configurations is shown in details. The GF results are shown in Section 11.4 where the spin wave spectrum, the layer magnetizations and the transition temperature are shown. Effects of surface interaction parameters and the film thickness are discussed. The case of helimagnetic films with simple cubic (SC) lattice is shown in Section 11.5.

## Model and Classical Ground State

Let us recall that bulk helical structures are due to the competition of various kinds of interaction [19, 224, 276, 350, 369]. We consider hereafter the simplest model for a film: The helical ordering is along one direction, namely the c-axis perpendicular to the film surface.

We consider a thin film of BCC lattice of *N _{z}* layers, with two symmetrical surfaces perpendicular to the c-axis, for simplicity. The exchange Hamiltonian reads

where the isotropic exchange part is given by

*]i _{t}j* being the interaction between two quantum Heisenberg spins S, and S

*j*occupying the lattice sites / and

*j.*The anisotropic part is chosen as

where /,,_{y} is the anisotropic interaction along the in-plane local spin- quantization axes *z* of S, and S*j,* supposed to be positive, small compared to *J* i, and limited to NN on the c-axis. Let us mention that according to the theorem of Mermin and Wagner [231] continuous isotropic spin models such as XY and Heisenberg spins do not have long-range ordering at finite temperatures in two dimensions. Since we are dealing with the Heisenberg model in a thin film, it is useful to add an anisotropic interaction to create a long-range ordering and a phase transition at finite temperatures.

To generate a bulk helimagnetic structure, the simplest way is to take a ferromagnetic interaction between NNs, say *J* _{x} (> 0), and an antiferromagnetic interaction between NNNs, *J*_{2}* <* 0. It is obvious that if |/_{2}| is smaller than a critical value |/_{2}^{C}|, the classical GS spin configuration is ferromagnetic [91, 139, 289]. Since our purpose is to investigate the helimagnetic structure near the surface and surface effects, let us consider the case of a helimagnetic structure only in the c-direction perpendicular to the film surface. In such a case, we assume a non-zero *J** _{2}* only on the c-axis [see Section 3.4). This assumption simplifies formulas but does not change the physics of the problem since including the uniform helical angles in two other directions parallel to the surface will not introduce additional surface effects. Note that the bulk case of the above quantum spin model have been studied by the GF method [286].

Let us recall that the helical structure in the bulk is planar: Spins lie in planes perpendicular to the c-axis: the angle between two NNs in the adjacent planes is a constant and is given by cos a = *—)i/h* for а BCC lattice (see method of calculation for the bulk configuration in Section 3.4). The helical structure exists, therefore, if I/2I > /1, namely |/||(bulk) = /1 (see Fig. 11.1, top). To simplify the presentation, we take a zero anisotropy *I _{if}j* = 0. The effect of

*I*on the GS will be shown at the end of this section.

_{it}jTo calculate the classical GS surface spin configuration, we write down the expression of the energy of spins along the c-axis, starting

Figure 11.1 Top: Bulk helical structure along the c-axis, in the case *a = 2л/3,* namely *J**2**/J**1* = —2. Bottom: (color online) Cosinus of 0^=0! — 02, • • • ,<*7 = 07 — 08 across the film for *J**2**/J**1** =* —1.2, —1.4, —1.6, —1.8, —2 (from top) with *N _{z} =* 8:

*a,*stands for 0, — 0,+i and

*x*indicates the film layer / where the angle

*a,*with the layer (/ + 1) is shown. The values of the angles are given in Table 11.1: a strong rearrangement of spins near the surface is observed.

from the surface:

where Zi = 4 is the number ofNNs in a neighboring layer, 0, denotes the angle of a spin in the /-th layer made with the Cartesian *x* axis of the layer. The interaction energy between two NN spins in the two adjacent layers / and *j* depends only on the difference a, = 0,- — 0,+i. The GS configuration corresponds to the minimum of *E.* We have to solve the set of equations:

Explicitly, we have

where we have expressed the angle between two NNNs as follows: 0i — 0_{3} = *0i — **62** + **02* — #3 = *ct* + a_{2}, etc. In the bulk case, putting all angles a, in Eq. 11.7 equal to a we get cos a *= -J**1**/J**2* as expected. For the spin configuration near the surface, let us consider in the first step only three parameters a_{x} (between the surface and the second layer), a_{2} ^{an}d a_{3}. We take *a _{n} = a* from

*n*= 4 inward up to

*n*=

*N*the other half being symmetric. Solving the first two equations, we obtain

_{z}/2,

The iterative numerical procedure is as follows: (i) replacing аз by а = arccos(-/i/y_{2}) and solving (11.6) and (11.9) to obtain *a _{x}* and

*c/*

*2*

*,*(ii) replacing these values into (11.8) to calculate аз, (iii) using this value ofa

_{3}to solve again (11.6) and (11.9) to obtain new values of ai and аг, (iv) repeating steps (ii) and (iii) until the convergence is reached within a desired precision. In the second step, we use ai, a

_{2}and a

_{3}to calculate by iteration a

_{4}, assuming a bulk value for as. In the third step, we use a, (/ = 1 — 4) to calculate as and so on. The results calculated for various

*J*

*2*

*/J*

*1*are shown in Fig. 11.1 (bottom) for a film of

*N*= 8 layers. The values obtained are shown in Table 11.1. Results of

_{z}*N*= 16 will be shown later.

_{z}Some remarks are in order: (i) result shown is obtained by iteration with errors less than 10^{_4}°, (ii) strong angle variations

Table 11.1 Values of cos *0 _{n }*

*n+1*

*= a*between two adjacent layers are shown for various values of

_{n}*J*1

_{2}/J
-1.2 |
0.985 (9.79°) |
0.908 (24.73°) |
0.855 (31.15°) |
0.843 (32.54°) |
33.56° |

-1.4 |
0.955(17.07°) |
0.767 (39.92°) |
0.716 (44.28°) |
0.714 (44.41°) |
44.42° |

-1.6 |
0.924(22.52°) |
0.633 (50.73°) |
0.624 (51.38°) |
0.625 (51.30°) |
51.32= |

-1.8 |
0.894(26.66 ) |
0.514 (59.04°) |
0.564 (55.66°) |
0.552 (56.48 ) |
56.25° |

-2.0 |
0.867 (29.84 ) |
0.411 (65.76°) |
0.525 (58.31°) |
0.487 (60.85°) |
60° |

Only angles of the first half of the 8-layer film are shown: Other angles are, by sy mmetry, cos 07.« = cos 01,2, cos 06,7 = cos 02,з, cos 0s,6 = cos 0з,4. The values in parentheses are angles in degrees. The last column shows the value of the angle in the bulk case (infinite thickness). For presentation, angles are shown with two digits.

are observed near the surface with oscillation for strong *J _{2},* (iii) the angles at the film center are close to the bulk value

*a*(last column), meaning that the surface reconstruction affects just a few atomic layers, (iv) the bulk helical order is stable just a few atomic layers away from the surface even for films thicker that

*N*8 (see below). This helical stability has been experimentally observed in holmium films [206].

_{z}=Note that using the numerical steepest descent method described in Ref. [248] we find the same result.

Let us discuss now the effect of the anisotropy on the GS configuration. The form of Eq. (11.3) simplifies a lot: Since the interaction is limited to NN, it suffices to replace in Eq. (11.4) the parameter *J _{x}* by

*J*( =

*Ji*+ /i where

*1*= for any NN pairs (/,

*j).*The GS calculation is done exactly in the same manner. That is the reason why we choose the form of Eq. (11.3). The GS configuration is slightly modified but the method and the general aspects of the results described above remain valid. Of course, the calculation of the spin wave spectrum and layer magnetizations presented below take into account the GS modification at each value of

*1.*Choosing another form of anisotropy, for example, taking a standard single-ion anisotropy —/(Sf)

^{2}along the spin local axis, will add just a constant in Eq. (11.4) [because (S?)

^{2}= 1 in the GS], So, it will not affect the GS configuration.

In the following, using the spin configuration obtained at each *J г/J* l we calculate the spin wave excitation and properties of the film such as the zero-point spin contraction, the layer magnetizations and the critical temperature.