Frustration Effect: J1 − J2 model
We consider in this section a superlattice composed of alternate "frustrated” magnetic films and "frustrated” ferroelectric films. The frustration due to competing interactions has been extensively investigated during the last four decades. The reader is referred to Ref.  for reviews on theories, simulations and experiments on various frustrated systems. In this section, we present the effect of the frustration in the presence of the DM interaction at the magnetoelectric interface. It turns out that the frustration gives rise to an enhancement of skyrmions created by the DM interaction in a field H applied perpendicularly to the films. Monte Carlo simulations are carried out to study the skyrmion phase transition in the superlattice as functions of the frustration strength.
The results have been shown in detail in Ref. . We recall in the following some principal points.
We use the same DM Hamiltonian for the interface coupling in Eqs. (15.10) and (15.11). For the magnetic and ferromagnetic Hamiltonians, H,„ and H/, we introduce the interactions between next-nearest neighbors (NNN) as follows:
We shall take into account both the nearest neighbors (NN) interaction, denoted by Jand the NNN interaction denoted by J2m. We consider J m > 0 to be the same everywhere in the magnetic film. To introduce the frustration we shall consider / 2m < 0, namely antiferromagnetic interaction, the same everywhere. The external magnetic field H is applied along the z-axis which is perpendicular to the plane of the layers.
For the ferroelectric film, the Hamiltonian is given by
where // the interaction parameter between polarizations at sites / and j. Similar to the magnetic subsystem we will take the same ]/.=]!> 0 for all NN pairs, and Jij = / 2f <0 for NNN pairs.
Let us emphasize that the bulk Ji — J2 magnetic model on the simple cubic lattice has been studied with Heisenberg spins  and the Ising model  where Ji and /2 are both antiferromagnetic (<0). The critical value |/2C| = 0.25|/i| above
Figure 15.20 (a) 2D view of the GS configuration of the interface for H = 0 with/"' = // = l,/2m =J2f = -0.3 ,)mf = -1.25, (b) 3D view.
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 /'" > 0 (ferro), and / 2m < 0 (antiferro), it is easy to show that the critical value where the ferromagnetic becomes antiferromagnetic is /2m = —0.5/"’. Below this value, the antiferromagnetic ordering replaces the ferromagnetic ordering.
For the interface coupling we use the same notation as in the preceding sections, namely / mf, independent of (/, j).
As before, since Pk is in the z direction, the DM vector is in the z direction. Thus, in the absence of an applied field the spins in the magnetic layers will lie in the xy plane to minimize the interface interaction energy.
We note that in the case when the magnetic film has a frustration and a thickness, the angle between NN spins in each magnetic layer is different from that of the neighboring layers. The determination of the angles is analytically difficult. It is more convenient to use the numerical minimization method called "steepest descent method” to obtain the ground state (GS) spin configuration . We use a sample size N x N x L. For most calculations, we select the lateral size N x N with N = 60 and we take L,„ = Lj = 4 where Lm [Lf ] is the thickness of magnetic (ferroelectric) layer. We use the periodic boundary conditions in the xy plane. As before, we take exchange parameters between NN spins and NN polarizations equal to 1, namely ] m = ] f = 1, for the simulations. We investigate the effects of the interaction parameters (/ 2m, J2f) and J mf. We note that the steepest descent method calculates the real GS down to the value/"^ = -1.25. For values lower than this, the DM interaction is so strong that the angle в tends to л/2. All magnetic exchange terms (scalar products) will be close to zero, the minimum total energy thus corresponds just to the DM energy.
Now we consider a case with the frustrated regime with (У2/,/2"') e (—0.4, 0), namely above the critical value -0.5 as mentioned above. The spin configuration in the case where H = 0 is shown in Fig. 15.20 for the interface magnetic layer. We observe here a stripe phase with long islands and domain walls. The inside magnetic layers have the same texture.
When H is increased, we observe the skyrmion crystal as seen in Fig. 15.21: the GS configuration of the interface and beneath magnetic layers obtained for J mf = -1.25, with J2m = J2f = —0.2 and external magnetic field H = 0.25. A zoom of a skyrmion shown in Fig. 15.21c and the z-components across a skyrmion shown in Fig. 15.21d indicate that the skyrmion is of Bloch type.
At this field strength H =0.25, if we increase the frustration, for example, J2m = )2f = - 0.3, then the skyrmion structure is enhanced: We can observe a clear 3D skyrmion crystal structure not only in the interface layer but also in the interior layers. This is shown in Fig. 15.22, where the interface (top) and the second layer (bottom) are displayed.
Figure 15.21 (a) 3D view of the GS configuration of the interface for moderate frustration J2m = J2f = — 0.2, (b) 3D view of the GS configuration of the second magnetic layers, (c) zoom of a skyrmion on the interface layer: Red denotes up spin, four spins with clear blue color are down spin, other colors correspond to spin orientations between the two. The skyrmion is of the Bloch type, (d) z-components of spins across the skyrmion shown in (c). Other parameters: ]m = J f = = —1.25 and H = 0.25.
The highest value of frustration where the skyrmion structure can be observed is when J2m = J2f = — 0.4 close to the critical value —0.5. We show this case in Fig. 15.23: The GS configuration of the interface (a) and second (interior) (b) magnetic layers are presented. Other parameters are the same as in the previous figures, namely J mf = —1.25 and H = 0.25. We can observe a clear 3D skyrmion crystal structure in the whole magnetic layers, not only near the interface layer. Unlike the case where we do not take into account the interaction between NNN (see the previous sections and ), in the present case where the frustration is very strong we see that a large number of skyrmions are distributed over the whole magnetic layers with a certain periodicity close to a perfect crystal.
Figure 15.22 3D view of the GS configuration of (a) the interface, (b) the second layer, for stronger frustration J2m = J2f = —0.3.Jm = )f = 1, У 2m = J2f = —0.3,у mf = -1.25 and Я = 0.25.
Though we take the same value for J 2m and ] 2f in the figures shown above, it is obvious that only the magnetic frustration ] 2m is important for the skyrmion structure. The ferroelectric frustration affects only the stability of the polarizations at the interface. As long as J 2f does not exceed -0.5, the skyrmions are not affected by ] 2f .
Now, if the magnetic frustration is not strong enough, the ferroelectric frustration plays an important role. Skyrmions disappear when ]2f = — 0.4 . We conclude that while magnetic frustration )2m enhances the formation of skyrmions, the ferroelectric frustration J 2f in the weak magnetic frustration tends to
Figure 15.23 Strongest frustration ]2m = ]2f = —0.4 (a) 3D view of the GS configuration of the interface, (b) 3D view of the GS configuration of the second magnetic layers. Other parameters Jm = J f = 1 ,Jmf = —1.25 and H = 0.25.
suppress skyrmions. The mechanism of these parameters when acting together seems to be very complicated.
Skyrmion Phase Transition
We use the same Metropolis algorithm [48, 199] as before with the size N x N x L with Af = 60 and thickness L = L,„ + Lf = 8 (4 magnetic layers, 4 ferroelectric layers). Simulation times are 106 Monte Carlo steps (MCS) per spin for equilibrating the system and 106 MCS/spin for averaging. We calculate the internal energy and
Figure 15.24 (a) Energy of the magnetic films versus temperature T for (J2m = )2f = — 0.4) (red), coinciding with the curve for (J2m= - 0.4, J2f = 0) (black, hidden behind the red curve). Blue curve is for (J2m = 0, J 2f = —0.4); (b) Order parameter of the magnetic films versus temperature T for (J 2m =) 2f = - 0.4) (red), (J 2m = - 0.4, J 2f = 0) (black), (J2m = 0, J2f = —0.4) (blue). Other used parameters: J mf = —1.25, H = 0.25.
the layer order parameters of the magnetic (Mm) and ferroelectric [Mf) films, defined in Eqs. (15.15) and (15.14).
In Fig.15.24, we show the magnetic energy and magnetic order parameter versus temperature in an external magnetic field, for various sets of NNN interaction. Note that the phase transition occurs at the energy curvature changes, namely at the maximum of the specific heat. The red curve in Fig. 15.24a is for both sets (J 2m = J2f = -0.4), (J2m = -QA,]2f = 0). The change of curvature takes place at Г™ ~ 0.60. It means that the ferroelectric frustration does not affect the magnetic skyrmion transition at such a strong magnetic frustration (/2m = -0.4). For J 2m = 0,J2f = -0.4), namely no magnetic frustration, the transition takes place at a much higher temperature Г™ ~ 1.25.
The magnetic order parameters shown in Fig. 15.24b confirm the skyrmion transition temperatures seen by the curvature change of the energy in Fig.15.24a.
We have studied in this chapter a new model for the interface coupling between a magnetic film and a ferroelectric film in a superlattice. This coupling has the form of a Dzyaloshinskii-Moriya (DM) interaction between a polarization and the spins at the interface.
The ground state shows uniform non-collinear spin configurations in zero field and skyrmions in an applied magnetic field. We have studied spin wave (SW) excitations in a monolayer and in a bilayer in zero field by the Green’s function method. We have shown the strong effect of the DM coupling on the SW spectrum as well as on the magnetization at low temperatures.
Monte Carlo simulation has been used to study the phase transition occurring in the superlattice with and without applied field. Skyrmions have been shown to be stable at finite temperatures. We have also shown that the nature of the phase transition can be of second or first order, depending on the DM interface coupling.
We have also taken into account the frustration due to the NNN interactions in both magnetic and ferroelectric layers. As expected, the magnetic frustration enhances the creation of skyrmions while the ferroelectric frustration when strong enough destabilizes skyrmions if there is no strong magnetic frustration to resist. Note that the magnetic frustration reduces the transition temperature considerably.
The existence of skyrmions confined at the magneto-ferroelectric interface is very interesting. We believe that it can be used in transport applications in spintronic devices. A number of applications using skyrmions has been mentioned in the Introduction.