1. Introduction
Magnetic skyrmions are topological configurations of spin vortices [
1,
2]. Skyrmions and vortices have drawn enormous attention due to their fundamental and practical importance [
3,
4,
5,
6,
7,
8]. Skyrmions hold great promise as a basis for future digital technologies [
9,
10,
11,
12,
13,
14,
15,
16,
17,
18]. Information flow in next-generation spintronic devices could be associated with metastable isolated skyrmions guided along magnetic stripes [
19,
20]. Ferromagnetic skyrmions formed in ferromagnet/ferroelectric or heavy-metal multilayers [
5] are generally considered as elements of skyrmionic race-track memory [
17,
21]. Skyrmions have been experimentally observed, created and manipulated in a number of material systems, including magnetic materials [
4,
6,
7,
10,
22,
23,
24,
25,
26,
27,
28,
29,
30], multiferroic materials and superlattices [
31,
32,
33]. Skyrmions have been recently observed in conducting and insulating helimagnets under an applied magnetic field [
4,
23]. Under an applied magnetic field, the spiral structure due to a Dzyaloshinskii–Moriya (DM) transforms into a skyrmion crystal [
34].
The anomalous thermodynamic properties of the paradigmatic frustrated spin-
triangular-lattice Heisenberg magnetic films has remained an open topic of research over decades, both experimentally and theoretically. Shen et al. have demonstrated that the observed unusual spin correlations and thermodynamics can be described by a transverse field Ising model on the triangular lattice with a ferro-multipolar order [
35]. Kurumaji et al. have reported the emergence of a Bloch-type skyrmion state in the frustrated centrosymmetric triangular-lattice magnet
[
36]. They have observed a giant topological Hall response, indicating a field-induced skyrmion phase. Note that the phase diagram of a spin-
Heisenberg antiferromagnet on the triangular lattice with first- and second-neighbor interactions was investigated in Ref. [
37]. It was discovered regions of
and collinear stripe order, separated by an intermediate nonmagnetic phase. Remarkably, numerical simulations performed in Ref. [
38] revealed two crossover low-temperature and high-temperature scales. It is shown that in the intermediate regime between the low-temperature scale and the higher one, the ’rotonlike’ excitations are activated with a strong chiral component.
Multiferroic materials, or materials with the simultaneously coexistence and coupling of ferroelectric and magnetic orders, have returned to the forefront of condensed matter research. Developing heterostructures and superlattices of multiferroics (two-component
superlattice with
A—magnetic film,
B—ferroelectric film,
N—number of repetitions) are attracting increasing interest and provoking extensive research activity due to the broken mirror symmetry at interface between two materials (the Dzyaloshinskii–Moriya interaction can only appear in systems that break bulk inversion symmetry or mirror symmetry). Therefore, superstructures naturally lead to the creation of skyrmions at different interfaces which have a unique dynamics compared to the spin structures created at the interface of the same elements. Enormous attention was paid to studying the non-uniform states in magnetic/ferroelectric superlattices both theoretically [
39] and experimentally [
40,
41,
42].
Recently, magnetic/ferroelectric superlattices attract much of attention as magneto–electric (ME) materials [
41,
42]. Spin waves and skyrmions in “unfrustrated” magnetoelectric superlattices with a Dzyaloshinskii–Moriya (DM) interface coupling have been studied [
39].
Note that in Ref. [
39], we investigated the effects of Dzyaloshinskii–Moriya (DM) magneto-ferroelectric interaction in an unfrustrated ferromagnetic/ferroelectric superlattice. In zero external magnetic field, we showed that the ground-state spin configuration is periodically non-collinear. Through the use of a two-time Green’s functions, we calculated the spin wave spectrum in a monolayer and in a bilayer sandwiched between two ferroelectric films. We showed that when 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 Ref. [
43] the effect of the frustration in a superlattice composed of alternating frustrated magnetic and ferroelectric films was investigated. We showed in particular that the frustration gives rise to an enhancement of skyrmions created by the DM interaction at the magneto–electric interface in an external field. In these works, we have considered the magnetic and ferroelectric films with a simple cubic lattice.
The purpose of this paper is to show the main features of our results obtained for a magneto–ferroelectric superlattice where the interface coupling is a DM interaction. The effect of the frustration due to the competing next-nearest-neighbor (NNN) interaction is emphasized. Since the interface coupling with a DM interaction creates the frustration at the interface, it is desirable to consider the case of triangular lattice because this lattice is known to be frustrated when non-collinear spin configurations are present. The purpose of this paper includes thus this so-far-missing case to see the effect of the DM interface coupling on the properties of skyrmions. As it turns out, we find several striking features among which (i) skyrmions are created even without an applied magnetic field, (ii) skyrmions are stable in a large region of magnetic field. These results are important for applications because skyrmions which are magnetic textures can be manipulated by an electric field.
The paper is organized as follows.
Section 2 is devoted to the description of our model and the determination of the ground-state spin configuration with and without applied magnetic field. Skyrmion crystal is shown with varying interface parameters.
Section 3 shows the results for spin-wave spectrum obtained by the Green’s function method. Results of Monte Carlo simulations for the phase transition in the system is shown in
Section 4. These results show that the skyrmion crystal is stable at finite temperatures below a transition temperature
. The case of the triangular lattice is shown in
Section 5 and
Section 6. A discussion on experimental interest of our model is given in
Section 7. Concluding remarks are given in
Section 8.
2. Model—Ground State
Consider a superlattice composed of alternate magnetic and ferroelectric films. Both have the structure of simple cubic lattice of the same lattice constant, for simplicity. The Hamiltonian of this multiferroic superlattice is expressed as:
where
and
are the Hamiltonians of the magnetic and ferroelectric subsystems, respectively,
denotes Hamiltonian of magneto–electric interaction at the interface of two adjacent films. We are interested in the frustrated regime. Therefore we describe the Hamiltonian of the magnetic film with the Heisenberg spin model on a simple cubic lattice as follows:
where
is the spin on the i-th site,
the external magnetic field,
the magnetic interaction between two spins at
i and
j sites. We shall take into account both the nearest neighbors (NN) interaction, denoted by
, and the next-nearest neighbor (NNN) interaction denoted by
. We consider
to be the same everywhere in the magnetic film. To introduce the frustration we shall consider
, namely antiferromagnetic interaction, the same everywhere. The external magnetic field
is applied along the
z-axis which is perpendicular to the plane of the layers. The interaction of the spins at the interface will be given below.
For the ferroelectric film, we suppose for simplicity that electric polarizations are Ising-like vectors of magnitude 1, pointing in the
direction. The Hamiltonian is given by
where
is the polarization on the i-th lattice site,
the interaction parameter between polarizations at sites
i and
j. Similar to the magnetic subsystem we will take the same
for all NN pairs, and
for NNN pairs.
Before introducing the DM interface interaction, let us emphasize that the bulk
model on the simple cubic lattice has been studied with Heisenberg spins [
44] and the Ising model [
45] where
and
are both antiferromagnetic (<0). The critical value
above which the bipartite antiferromagnetic ordering changes into a frustrated ordering where a line is with spins up, and its neighboring lines are with spins down. In the case of
(ferro), and
, it is easy to show that the critical value where the ferromagnetic becomes antiferromagnetic is
. Below this value, the antiferromagnetic ordering replaces the ferromagnetic ordering.
We know that the DM interaction is written as
where
is the spin of the i-th magnetic ion, and
is the Dzyaloshinskii–Moriya vector. The vector
is proportional to the vector product
of the vector
which specifies the displacement of the ligand (for example, oxygen) and the unit vector
along the axis connecting the magnetic ions
i and
j (see
Figure 1). One then has
We define thus
where
D is a constant,
the unit vector on the
z axis, and
.
For the magneto–electric interaction at the interface, we choose the interface Hamiltonian following Ref. [
39]:
where
is the polarization at the site
k of the ferroelectric interface layer, while
and
belong to the interface magnetic layer. In this expression
plays the role of the DM vector perpendicular to the
plane, given by Equation (
6). When summing the neighboring pairs
, attention should be paid on the signs of
and
(see example in Ref. [
39]).
Hereafter, we suppose independent of .
Since
is in the
z direction, i.e., the DM vector is in the
z direction, in the absence of an applied field the spins in the magnetic films will lie in the
plane to minimize the interface interaction energy, according to Equation (
7).
From Equation (
7), we see that the magneto–electric interaction
favors a canted (non collinear) spin structure. It competes with the exchange interactions
and
of
which favor collinear (ferro and antiferro) spin configurations. In ferroelectric films, there is just ferro- or antiferromagnetic ordering due to the Ising nature. Usually the magnetic or ferroelectric exchange interaction is the leading term in the Hamiltonian, so that in many situations the magneto–electric effect is negligible. However, in nanofilms of superlattices the magneto–electric interaction is crucial for the creation of non-collinear long-range spin order. It has been shown that Rashba spin-orbit coupling can lead to a strong DM interaction at the interface [
46,
47], where the broken inversion symmetry at the interface can change the magnetic states.
Since the polarizations are along the
z axis, the interface DM interaction is minimum when
and
lie in the
interface plane and perpendicular to each other. However the collinear exchange interactions among the spins will compete with the DM perpendicular configuration. The resulting configuration is non collinear. In the general case where a magnetic field is applied and there is also the frustration, we use the steepest descent method [
34,
39] to determine the ground state.
We show in
Figure 2 an example where
with no NNN interaction. The magnetic film has a single layer. The GS spin configurations have periodic structures. Several remarks are in order:
(i) Each spin has the same turning angle
with its NN in both
x and
y direction. The schematic zoom in
Figure 2c shows that the spins on the same diagonal (spins 1 and 2, spins 3 and 4) are parallel. This explains the structures shown in
Figure 2a,b.
(ii) The periodicity of the diagonal parallel lines depends on the value of
(comparing
Figure 2a and
Figure 2b). With a large size of
N, the periodic conditions have no significant effects.
Let us show now an example with
in
Figure 3 with a frustration due to NNN interaction.
At this field strength , if we increase the frustration, for example , then the skyrmion structure is enhanced: we can observe a clear skyrmion crystal structure not only in the interface layer but also in the interior layers (not shown).
3. Spin Waves
Here let us show theoretically spin-waves (SW) excited in the magnetic film in zero field, in some simple cases of the simple cubic lattice. The method we employ is the Green’s function technique for non collinear spin configurations which has been shown to be efficient for studying low-
T properties of quantum spin systems such as helimagnets [
48] and systems with a DM interaction [
49]. This technique is rather lengthy to describe here. The reader is referred to these papers for mathematical details.
Let us show here the resulting spin-wave energy in the case of the NN interaction only (the case including the NNN interaction which is more complicated will be subject of a future study):
where
,
and
being the wave-vector components in the
planes,
a the lattice constant,
the angle between two NN spins given by
.
is the statistical average of the
z spin-component. Other parameters
and
D have been defined in Equations (
2) and (
6).
We show in
Figure 4 the spin-wave energy
E versus the wave vector
, for weak and strong values of the DM interaction strength
D. For the weak value of
(
Figure 4a), namely weak
, the long-wave length mode energy (
) is proportional to
as in ferromagnets, while for strong
(
Figure 4b),
E is proportional to
k as in antiferromagnets. The effect of the DM interaction is thus very strong on the spin-wave behavior.
The case of a bilayer as well as the effects of other parameters on the spin-wave spectrum are shown in Ref. [
39].
4. Phase Transition
This section concerns the simple cubic lattice considered above. We have seen the skyrmion crystal in
Figure 3 at
. Shall this structure survive at finite
T? To answer this question we have performed Monte Carlo simulations using the Metropolis algorithm [
50]. We have to define an order parameter for the skyrmion crystal. In complicated spin orderings such as spin glasses, we have to follow each individual spin during the time. If it is frozen then its time average is not zero. This is called Edwards–Anderson order parameter for spin glasses [
51].
For the magnetic films, the definition of an order parameter for a skyrmion crystal is not obvious. Taking advantage of the fact that we know the GS, we define the order parameter as the projection of an actual spin configuration at a given
T on its GS and we take the time average. This order parameter of layer
n is thus defined as
where
is the i-th spin at the time
t, at temperature
T, and
is its state in the GS. The order parameter
is close to 1 at very low
T where each spin is only weakly deviated from its state in the GS.
is zero when every spin strongly fluctuates in the paramagnetic state.
For the ferroelectric films, the order parameter
of layer
n is defined as the magnetization
where
denotes the time average. The total order parameters
and
are the sum of the layer order parameters, namely
and
.
In
Figure 5 we show the magnetic energy and magnetic order parameter versus temperature in an external magnetic field, for various sets of NNN interactions. Note that the phase transition occurs at the energy curvature changes, namely at the maximum of the specific heat. The red curve in
Figure 5a is for both sets
and
. The change of curvature takes place at
. It means that the ferroelectric frustration does not affect the magnetic skyrmion transition at such a strong magnetic frustration
. For
, namely no magnetic frustration, the transition takes place at a much higher temperature
. The magnetic order parameters shown in
Figure 5b confirm the skyrmion transition temperatures seen by the curvature change of the energy in
Figure 5a.
6. Monte Carlo Results for the Case of Triangular Lattice
Using the same method of Monte Carlo simulation and the same definitions of the magnetic and ferroelectric order parameters as for the case of simple cubic lattice, we have studied the phase transition in the magneto–ferroelectric superlattice with the triangular lattice.
In
Figure 11 we display the magnetic energy and the magnetic order parameter versus temperature
T without an external magnetic field for various value of magneto–electric interaction. The red curve in
Figure 11a corresponds to the set
,
, which coincides with the result for
,
. The change of curvature takes place at
for both cases, meaning that the magneto–electric interaction does not affect the magnetic phase transition temperature at such a large region of
(−1.75, …, −1.0). In this region of
we do not have skyrmions at the interface without an external field. We note that in a case of magneto–electric superlattice with the simple cubic lattice,
strongly affects the critical temperature [
39]. The dependence of magnetic order parameters versus
T is shown in
Figure 11b (red and green dots) confirm the phase transition from the non-collinear phase to the paramagnetic phase occurring at temperatures seen by the curvature change of the energy in
Figure 11a. We display in
Figure 11a also the case where
, with a strong value of
:
(blue line). As seen, such a strong magneto–electric interaction at the interface removes the phase transition: the order parameter never vanishes, as seen in
Figure 11b (blue curve).
7. Discussion on the Applicability of the Model
Applications of skyrmions in spintronics have been largely discussed and their advantages compared to early magnetic devices such as magnetic bubbles have been pointed out in a recent review by W. Kang et al. [
52]. Among the most important applications of skyrmions, let us mention skyrmion-based racetrack memory [
53], skyrmion-based logic gates [
54,
55], skyrmion-based transistor [
56,
57,
58] and skyrmion-based artificial synapse and neuron devices [
59,
60].
The manipulations with skyrmions were first demonstrated in the diatomic PdFe layer on the iridium substrate, and the importance of this achievement for the technology of information storing is difficult to overestimate: it makes possible to write and read the individual skyrmions using a spin-polarized tunneling current [
7]. In Ref. [
61], the possibility of the nucleation of skyrmions by the electric field by means of an inhomogeneous magneto–electric effect was established.
The above mentioned applications of skyrmions have motivated the present work. Our model is interesting when the magneto–electric interaction is large, in the order of the magnetic interaction. Fortunately, this condition is realizable using the magneto–electric coupling in superlattices. Let us cite a number of works showing strong and very strong magneto–electric (ME) interactions to justify the applicability of our model. Values of ME interactions depend on the materials, but they are of the order of the ferromagnetic interaction or even larger as found in a lot of experiments among which we mention below. We note that ideal multiferroic memory cell could offer the possibility to electrically write the magnetic state, which then can be read out in a non-destructive manner. Such a multiferroic memory concept combines the high speed of existing electronics with the remnant properties of magnetism and thus joins the best aspects of existing ferroelectric and magnetic data storage memories. The realization of such devices requires an electrical control of magnetism, i.e., a strong magneto–electric coupling between the magnetic moments and the electric polarizations. Magneto–electric materials are one example of "smart" materials in which the coupling of such different physical quantities produces new phenomena, which are more than just the sum of the original ones, possessing, e.g., sensing and actuating functions in the same compound. The magnitude of the novel coupling phenomena between dielectric and magnetic properties is bounded by the product of the permittivity and the permeability in case of magneto–electric materials. Thus, compounds which easily polarize in response of an electric and magnetic field may exhibit large magneto–electric effects. This requirement is usually fulfilled by multiferroic materials (see Ref. [
62] and references cited therein). We mention a few experimental works among the numerous experiments showing large magneto–electric couplings with comments on some works [
63,
64,
65,
66,
67,
68].