Real-time observation of domain fluctuations in a two-dimensional magnetic model system

Domain patterns of perpendicularly magnetized ultra-thin ferromagnetic films are often determined by the competition of the short range but strong exchange interaction favouring ferromagnetic alignment of magnetic moments and the long range but weak antiferromagnetic dipolar interaction. Detailed phase diagrams of the resulting stripe domain patterns have been evaluated in recent years; however, the domain fluctuations in these pattern forming systems have not been studied in great detail so far. Here we show that domain fluctuations can be observed in ultra-thin two-dimensional ferromagnetic Fe/Ni/Cu(001) films with perpendicular magnetization in the stripe domain phase. Non-stroboscopic time-resolved threshold photoemission electron microscopy with high temporal resolution allows analysing the dynamic fingerprint of the topological excitations in the nematic domain phase. Furthermore, proliferation of domain ending defects in the vicinity of the spin reorientation transition is witnessed.

i.e. the ground state (dashed line) in the stripe phase [3,4]. Since the local deviation u(x) is a discrete function, the partial derivative of u(x, y) with respect to x has to be calculated as shown in (b). The two main deviations from the ground stripe state are undulations (c) and corrugations (d).
where Ω is the strength of the dipole interaction, Γ is the exchange energy, λ is the effective single-ion anisotropy constant, andr 12 = r 1 −r 2 |r 1 −r 2 | . Ω, Γ and λ are thickness as well as temperature dependent and the domain width w D in the system with perpendicular magnetization described by Eq. (1) also depends on these parameters. In fact, the dependence of the equilibrium domain width on these parameters leads to an exponential evolution of the domain width as shown in Fig. 1 (d) of the main manuscript and in Fig. 1 for a different Fe-wedge with 2.1 ML mm −1 ). Note that the orientation of the stripe domains is mainly dictated by the four-fold magnetocrystalline in-plane anisotropy due to the single crystallinity of the films. The effect of an in-plane anisotropy (by introducing an additional term to the domain wall energy per area) can be treated in analogy to the higher order exchange terms introduced in [1] leading to orientational anisotropy.

Supplementary Note 2 Elastic Energy of Stripe Domains
From elastic theory the corresponding energy density for meandering domain walls in a system featuring a perpendicular stripe phase is given by where the function u(x, y) represents the local deviations from an ideal stripe pattern, as shown in Fig. 2 (a). Without limiting generality, consider the domain walls to be parallel to the y axis. The first term is the compression energy density, due to a compressive deformation ∂ x u + 1 2 (∂ y u) 2 [1], either by local deviations of a single domain wall or by a variation of the domain width concerning two neighboring domain walls. The second term of Eq. (2) describes bending ((∂ 2 y u) 2 ) of the domain walls, and is therefore called bending energy density. The last energy term in Eq. (2) takes into account anisotropies regarding the domain wall orientation, and is based on symmetry breaking magnetocrystalline or -elastic terms or higher order exchange terms. Hence, it is referred to as the orientation energy density.
The individual constants within this elastic energy model can be connected to the coupling constants Ω (dipole), Γ (exchange) and λ (effective anisotropy) in the mean-field Hamiltonian as given in Eq. (1) by [1,3] K ∝ Ω w D compression constant or rigidity where t DW is the domain wall width. Note that the experimental determination of the coupling constants is extremely difficult in this kind of heterogeneous layered system (Fe/Ni/Cu(001)). Since the Fe/Ni layer grows in the fct-phase, the use of bulk exchange coupling parameters as well as the effective magnetization values of the bulk is highly inaccurate, which makes the determination of exact values of the elastic energy constants meaningless while qualitatively the theory describes the experimental observation well [6,7].
A stripe domain state with a large domain width is more likely to exhibit corrugations due to a small compression constant K, as shown in Fig. 2 (d). In contrast, a small domain width may exhibit more undulating excitations, Fig. 2 (c), rather than local changes of the domain width, i.e. corrugations, due to a large K. This is also supported by a smaller bending constant µ for a small domain width w D . Hence, in the vicinity of the SRT a more wavy domain pattern may occur with a narrower but more rigid domain width, which is known as transverse instability [2]. Eq. (5), however, implies that for a narrow domain configuration near the SRT the tendency towards orientation is enhanced. This is the reason why there are two scenarios for the domain pattern evolution beginning from a stripe domain phase through various other domain patterns until the magnetization direction eventually rotates into the plane as shown by Portmann et al. [6].

Method to determine the motion of domain walls
In order to extract fluctuations of individual domains, in the sense of tracing the motion of domain walls, the following procedure is used. At first a sequence of images is recorded. Due to vibrations in the experimental setup, the individual images of the recorded movie have to be realigned. Then we identify the domain walls by an edge detection algorithm. An example image of the result of this edge detection algorithm is shown in Fig. 3 (b), which displays solely the domain walls extracted from Fig. 3 (a). The summation of all these domain wall images, as shown in Fig. 3 (c where Θ y (r) is the director field of the domain configuration, which refers to the angle of the domain wall (DW) at site r ∈ DW with respect to the y-axis, see Fig. 2 (a). Note, that g n does not depend on the reference axis and is only a measure of the average deviation from a predominant orientation of the domain walls with respect to a n-fold symmetry. For instance, in the case of an ideal stripe pattern oriented along an arbitrary direction, one has g n = 1 for all n. Fig. 4 shows the evolution of g 2 for the domain patterns shown in Fig. 3 (a) of the main manuscript. For the larger domain widths the orientational order parameter is approximately equal while it decreases significantly for the pattern with 400 nm wide domains.

Supplementary Note 5
Fluctuations of a topological C-type defect A C-type defect, which can also be considered as being composed of two strongly bound for fluctuations of a C-type defect is presented in Fig. 6. In this case the motion of the two domain endings of the C-type defect occurs in opposite directions. Hence, the black domain between these two endings becomes larger. As a result the surrounding -initially straightdomains have to bend. As can be seen from Fig. 6 (b), all four domain walls move (jump).
The red dotted lines mark the two jumps of the left domain ending, which seems to be the trigger event for the domain transformation. Due to the rather high frame rate of 200 fps, a delayed response of the individual participants is revealed as shown in Fig. 6