MONTE CARLO STUDY OF PHASE TRANSITIONS AND SKYRMION CRYSTAL IN MAGNETO-ANTIFERROELECTRIC HETEROSTRUCTURES WITH TRIANGULAR LATTICE

Chelyabinsk Physical and Mathematical Journal. 2020. Vol. 5, iss. 2. P. 202-210.

DOI: 10.24411/2500-0101-2020-15207

MONTE CARLO STUDY OF PHASE TRANSITIONS AND SKYRMION CRYSTAL IN MAGNETO-ANTIFERROELECTRIC HETEROSTRUCTURES WITH TRIANGULAR LATTICE

I.F. Sharafullin1", A.G. Nugumanov1, A.R. Yuldasheva1, N.M. Nugaeva1, M.Kh. Kharrasov1, H.T. Diep2

Cergy-Pontoise, France a SharafullinIF@yandex.ru

The study of formation of the skyrmion lattice with the non-collinear magnetoelectric interaction at the interface in a ferromagnetic/antiferroelectric heterostructure is carried out. The ground state spin configuration is calculated by using the steepest descent method. We found the formation perfect skyrmions structure at acceptable values of the magnetoelectric interaction between the antiferroelectric and magnetic layers, both with the triangilar lattice. Monte Carlo simulation has been used to study the phase transition occurring in the ferromagnetic/antiferroelectric heterostructure with and without an applied field. Skyrmions have been shown to be stable at finite temperatures. The ferromagnetic films undergo two transitions, one is due to the destruction of the skyrmion structure and one magnetic transition, separately. The first skyrmion transition occurs at a lower temperature than magnetic transition. Between these two critical temperatures the ferromagnetic/antiferroelectric heterostructure is partially disordered.

Introduction

Due to the achievements in technology toward miniaturization of devices into the nano-meter length scale within the last decade, the physics of multiferroics, heterostructures, interfaces, and surfaces is nowadays a central area of investigations [1-4]. One of the most important and interesting recent advancements in the field of multiferroics resulted in the progress in nanomagnetic systems is the discovery of the topological spin textures — magnetic skyrmions [5]. The magnetic skyrmions are particle-like textures in the magnetization. From the applied point of view, many skyrmion based innovative device storage and concepts have been proposed such as the skyrmion racetrack memory [6; 7], the skyrmion transistors, and the skyrmion logics [8; 9]. However, there are a lot of important issues should be fully investigated before the skyrmions can be put to use in real devices [10-12]. Note that, very recently, experimentally the magnetic skyrmions in ferrimagnet GdFeCo have been studied, where the magnetic moments from two sublattices were not

This work was supported by the grant from the Head of the Republic of Bashkortostan (Decree of the Head of the Republic of Bashkortostan dated 02/07/2020 № UG-43).

Monte Carlo study of phase transitions and skyrmion crystal in magneto-antiferroelectric... 203

completely compensated [13]. Kurumaji et al. observed the emergence of a Bloch-type skyrmion lattice phase in the centrosymmetric triangular-lattice magnetic material Gd2PdSi3 [14]. The interface-induced skyrmions were investigated in [15-17]. We note that the heterostructures naturally lead to the interaction of skyrmions on different interfaces [18; 19]. In [20; 21] the effects of Dzyaloshinskii — Moriya (DM) magnetoferroelectric interaction in a "unfrustrated"ferromagnetic/ferroelectric superlattice have been reported. In a zero external magnetic field, we showed that the ground state spin configuration is periodically non-collinear. We showed that when the magnetic field is applied in the direction perpendicular to the plane of layers, the skyrmions are arranged to form a crystalline structure at the interface. In [22] the effect of the frustration in a superlattice composed of alternating frustrated magnetic and ferroelectric films was investigated and have shown that frustration gives rise to an enhancement of skyrmions created by the DM interaction at the magnetoelectric interface in an external field. The heterostructures with magnetic and ferroelectric materials exhibit the magnetoelectric effect at the interface and this phenomenon is accompanied by the appearance of an antiferromagnetic phase as well as with the change in the critical temperature of the magnetic layer [23]. This has been observed experimentally in heterostructure Lao.87Sr0.i3MnO3/PbZr0.52Tio.48O3 [24]. In [25] by using Monte Carlo (MC) simulation for heterostructure LaSrMnO/PbZrTiO, phase transitions have been investigated and the correctly describing model has been proposed. Monte Carlo methods based on the Metropolis algorithm, as well as other algorithms have proven to be successful in describing physical properties of magnetic systems of different spatial dimensions [26-28]. We consider in this paper a heterostructure composed of alternate ferromagnetic films and antiferroelectric films on a triangular lattice.

The heterostructure that we study here is composed of Lm ferromagnetic (FM) layers and antiferroelectric (AFE) films with Lf layers sandwiched in the z direction. The spins and polarisations are on a triangular lattice. All interactions are limited to nearest neighbors (NN). Each xy plane has the dimension N x N. The system thus consists of a N x N x Lz sites where Lz = Lm + Lf. We assume the periodic boundary conditions in all directions. The full Hamiltonian of this multiferroic FM/AFE heterostructure is expressed as:

H = - Y, JSi ■ Sj - ^(H ■ Si) - ^ JjP Pj - Y Jmf ei,m Pfc ■ [Si x Sm].

i,j i i,j l,m,k

Here Jij > 0 characterizes the ferromagnetic interaction between one spin Si on the i-th site and its nearest neighbors (NN). We consider it to be the same for NN within a layer and NN in adjacent layers. H is an applied magnetic field along the z direction. jj < 0 denotes the NN antiferroelectric coupling between the polarizations Pi along the z axis at the i-th site assumed to have only two values ±1 (Ising-like model). For simplicity, we consider Jm = Jm for the magnetic film. We wih take the same Jij = Jf for all ferroelectric sites and interface coupling we consider Jmf < 0. The coefficient el,m = -em,l = 1. The ground state (GS) spin configurations in multiferroic superlattice were studied in detail in [20; 22]. The angle 7 between two neighbor spins in the layer have been shown to be proportional to the magnetoelectric coupling parameter Jmf. We have shown that when Jmf ^ 0, one has 7 ^ 0, and when Jmf ^ -to, one has 7 ^ П2 as it should be. We have determined the GS spin configurations in magnetic layers in a zero magnetic field by using the numerical minimization method called "steepest

I.F. Sharafullin, A.G. Nugumanov, A.R. Yuldasheva et al.

descent method" to obtain the GS spin configuration. We use a sample size N x N x Lz. The lattice sizes used in our simulations are N = 20, 30,..., 120, 300 and Lz = 4, 8,12,16. For most calculations, we select N = 40 and L = 8. Interaction parameters between spins and polarizations are taken as Jm = 1, Jf = —1 for the simulation. We investigated the following range of values for the interaction parameters Jmf: from Jmf = 0.0 to Jmf = —4.0. For region of values of interface magnetoelectric interaction Jmf = (—0.65, 0), the results of simulation shown that the GS spin configurations have periodic non-collinear structures. At Jmf = —0.66 we observed the beginning of the skyrmions creation at the interface. We noted that skyrmions in ferromagnetic/antiferroelectric heterosrtuctures with triangular lattice can created in region Jmf G (—1.25, —0.66) and skyrmions are not formed at any value of the magnetoelectric interaction at zero values of the applied magnetic field. It should be noted that such an effect was observed in unfrustrated ferromagnetic/ferroelectric heterostructures with a triangular lattice in region of Jmf G (—1.0, —0.75). The phenomena of skyrmions formation was not observed in films with a simple cubic lattice in both cases — frustrated and unfrustrated [20; 22]. Let us show some results for the GS configuration at different value of the interface interaction with and without an applied magnetic field in z-direction, which is perpendicular to the plane of films. With a moderate increase in the magnetoelectric interaction parameter, the stability of the skyrmion lattice with respect to the magnitude of the applied magnetic field increases. We show in Fig. 1 examples where Jm = 1.0, Jf = 1.0, Jmf = —0.4, H = 0, Jm = 1.0, Jf = 1.0, Jmf = —0.85, H = 0 (the beginning of bird skyrmions without an applied field) and a case with a perfect skyrmion structure with the applied external field at Jm = 1.0, Jf = 1.0, Jmf = —1.85, H = 0.6.

We have used the Metropolis algorithm [29] to calculate physical quantities of the system at finite temperatures T. For MC simulation we perform the cooling from the disordered phase: electrical polarizations are randomly assigned at lattice sites in the antiferroelectric layers, in the z direction. In the ferromagnetic layers spins with |S| = 1 are also randomly assigned in any direction, following in the spatial uniform distribution. At each T, new random Si and P were chosen, and the energy difference caused by this change is calculated. This change is accepted or rejected according to the Metropolis algorithm. In order to ensure the convergence of the observables, the lattice is swept 100000 times, where each time is considered as one MC step (MCS) that can be taken as the time scale of simulations. The observables of interest such as the averages of layer order parameters of the magnetic (Mm) layers, energy and susceptibility of layer order parameters. The averaged energy and the susceptibility per spin are defined by

ш = JHL x = (Mm) — (Mm)2

( ) N2Lz, X N2квT ,

indicates the thermal average. Order parameters of antiferroelectric films

Mf (n) = N (I E PGi£n

where (...) denotes the time average. The order parameter magnetic system defined as the projection of an actual spin configuration at a given T on its GS and we take the

where (...) defined as

Monte Carlo study of phase transitions and skyrmion crystal in magneto-antiferroelectric... 205

Fig. 1. GS spin configuration for different value of interface couplings: Jmf = -0.4, H = 0 (a); Jmf = -0.85, H = 0 (b); Jmf = -1.85, H = 0.6 (с)

time average. This order parameter of layer n is thus defined as

_______1

N 2(t«- to) 1 n

E si(T,t) ■ S0(T

where Si(T,t) is the i-th spin at the time t, at temperature T, and Sj(T = 0) is its state in the GS at T = 0. In Fig. 2 we show the dependence of the order parameter of the magnetic film versus temperature and their susceptibility for various values of the interface magnetoelectric coupling: in Fig. 2 (a), (b) for weak values Jmf = -1.75, H = 0, Jm = 1.0, Jf = -1.0, and in Fig. 2 (c), (d) for stronger values Jmf = -3.25,

H = 0.25, Jm = 1.0, Jf = -1.0.

The above mentioned figures show that the energy (not shown), order parameter of the magnetic film and susceptibility well behave at low T if we perform the heating from the correct GS according to the interface interaction. Note that the ferromagnetic films undergo two transitions, one is due to the destruction of the skyrmion structure and one magnetic transition (from an ordered phase to disordered), separately. The curves for the temperature dependence of the order parameter of the magnetic film and the susceptibility of the system for Jm = 1, Jf = -1, Jmf = -1.15, H = 0 present two second-order transitions at Tcsc ~ 0.85 and Tm — 1.40. The system at

I.F. Sharafullin, A.G. Nugumanov, A.R. Yuldasheva et al.

Fig. 2. The dependence of the order parameter of the magnetic film versus the temperature and the susceptibility, without and with an external magnetic field, for various values of the interface magnetoelectric interaction: (a)-(b) for weak values Jmf = -1.15, H = 0, Jm = 1.0, Jf = -1.0, and (c)-(d) for stronger values Jmf = -3.25, H = 0.25, Jm = 1.0, Jf = -1.0

strong interface coupling Jmf = -3.25 for magnetic layers also undergo two second-order transitions at Tcsc ~ 0.45 and Tcm ~ 1.60. One can see that the first transition temperature Tcsc decreases when we increase the values of \\Jmf |. On the other hand the transition temperature Tcm increases when we increase the values of \\Jmf \\. For values of Jmf = -3.25, the curves shown in Fig. 2 (c) indicate a deviation of the ferromagnetic state due to the non uniform interface ground state structure. The first skyrmion transition occurs at a lower temperature than magnetic transition. After the transition, spins align themselves in the field direction, giving a large value of the order parameter Mm ~ 0.25 as seen in Fig. 2 (c). Between these two critical temperatures the FM/AFE heterostructure is partially disordered. Such type of the partial disorder has been observed in many systems, for example the surface layer of a thin magnetic film can turn disordered at a low temperature while the bulk is still ordered [30]. One can also mention the partial phase transition in helimagnets in a field [31].

Conclusion

We have studied in this paper an interface coupling between a magnetic film and an antiferroelectric film in a heterostructure. This coupling connects the polarization and the spins at the interface.

The ground state shows uniform non collinear spin configurations in a zero field and skyrmions in a zero field and a stable skyrmion crystal at an enough large region applied magnetic field. Monte Carlo simulation has been used to study the phase transition occurring in the FM/AFE heterostructure with and without an applied field.

Monte Carlo study of phase transitions and skyrmion crystal in magneto-antiferroelectric... 207

Skyrmions have been shown to be stable at finite temperatures. The ferromagnetic films undergo two transitions, one is due to the destruction of the skyrmion structure and one magnetic transition, separately. The first skyrmion transition occurs at a lower temperature than magnetic transition. Between these two critical temperatures the FM/AFE heterostructure is partially disordered. The existence of skyrmions confined at the magneto-ferroelectric interface at a zero applied field is very interesting and it can be used in transport applications in spintronic devices.

References

Memory functions of magnetic skyrmions. Japanese Journal of Applied Physics, 2015, vol. 54, no. 5, p. 053001.

skyrmions — Overview of recent progress in an active research field. Journal of Applied Physics, 2018, vol. 124, iss. 24, p. 240901.

Complementary skyrmion racetrack memory with voltage manipulation. IEEE Electron Device Letters, 2016, vol. 37, iss. 7, pp. 924-927.

I.F. Sharafullin, A.G. Nugumanov, A.R. Yuldasheva et al.

The frustration-induced ferroelectricity of a manganite tricolor superlattice with artificially broken symmetry. Scientific Reports, 2017, vol. 7, p. 6201.

Article received 09.05.2020

Corrections received 25.05.2020

Monte Carlo study of phase transitions and skyrmion crystal in magneto-antiferroelectric... 209

Челябинский физико-математический журнал. 2020. Т. 5, вып. 2. С. 202-210.

УДК 538.977 DOI: 10.24411/2500-0101-2020-15207

МОДЕЛИРОВАНИЕ МЕТОДАМИ МОНТЕ-КАРЛО ФАЗОВЫХ ПЕРЕХОДОВ И СКИРМИОННОЙ СТРУКТУРЫ В ГЕТЕРОСТРУКТУРЕ

ФЕРРОМАГНЕТИК/АНТИФЕРРОЭЛЕКТРИК С ТРЕУГОЛЬНОЙ РЕШЁТКОЙ

И. Ф. Шарафуллин1’", A. Г. Нугуманов1, A. Р. Юлдашева1,

Н. М. Нугаева1, М. Х. Харрасов1, Х. Т. Дьеп2

Проведено исследование процесса формирования решётки скирмионов на интерфейсе антиферроэлектрического слоя и магнитного слоя с треугольной решёткой в гетероструктуре ферромагнетик/антиферроэлектрик с неколлинеарным магнитоэлектрическим взаимодействием. Основное состояния системы найдено методом наискорейшего спуска. Обнаружено формирование периодической структуры скирмионов при практически достижимых значениях магнитоэлектрического взаимодействия. Моделирование методом Монте-Карло выполнено для исследования фазовых переходов, происходящих в гетероструктуре ферромагнетик/антиферроэлектрик при воздействии внешнего магнитного поля. Показано, что скирмионы устойчивы до определённых значений температур. Обнаружено, что ферромагнитная плёнка подвержена двум последовательным фазовым переходам: один из них происходит при более низкой температуре и сопровождается разрушением скирмионной структуры и при более высокой температуре происходит фазовый переход второго рода в парамагнитную фазу.

Поступила в редакцию 09.05.2020 После переработки 25.05.2020

Сведения об авторах

Ш^арафуллин Ильдус Фанисович, кандидат физико-математических наук, доцент кафедры теоретической физики, Башкирский государственный университет, Уфа, Россия; e-mail: sharafullinif@yandex.ru.

Нугуманов Айдар Гайсович, аспирант кафедры теоретической физики, Башкирский государственный университет, Уфа, Россия; e-mail: aidar.nugumanov@gmail.com. Юлдашева Алина Рифовна, ассистент кафедры физики и технологии наноматериалов, Башкирский государственный университет, Уфа, Россия; e-mail: alina.yuldasheva.92@mail.ru.

Нугаева Нурия Мазитовна, студент физико-технического института, Башкирский государственный университет, Уфа, Россия; e-mail: nughaieva97@mail.ru.

Работа выполнена при поддержке гранта Главы Республики Башкортостан (Указ Главы Республики Башкортостан от 07.02.2020 № УГ-43).

I.F. Sharafullin, A.G. Nugumanov, A.R. Yuldasheva et al.

Харрасов Мухамет Хадисович, доктор физико-математических наук, профессор, советник ректора, Башкирский государственный университет, Уфа, Россия; e-mail: m-kharrasov@yandex.ru.

Хунг Т. Дьеп, доктор наук, профессор, Университет Сержи-Париж, Лаборатория теоретической физики и моделирования, Сержи-Понтуаз, Франция; e-mail: diep@u-cergy.fr.