Chiral switching and dynamic barrier reductions in artificial square ice

For an infinite artificial square ice the environment that influences the reversal of a nanomagnet forms a double-vertex configuration, as depicted in figure 1(b). Here, each tip of the central nanomagnet (black) is in close interaction to three other nanomagnet (gray), whose magnetisation remains largely unchanged during the reversal of the central nanomagnet. The extended square lattice then can be considered as an infinite tiling of these motifs.

We denote each magnetic equilibrium configuration with a state number #i determined by the arrangement of the surrounding nanomagnets, and the orientation of the central (switching) nanomagnet, which can point to the left (←) or to the right (→). The state number #i can be obtained from the binary representation of the relative configuration of the six surrounding nanomagnet numbered 1 to 6 according to the scheme shown in figure 1(b):

$$\begin{equation}i={{\Sigma}}_{j=1}^{6}{b}_{j}{2}^{j-1}\quad \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\quad {b}_{j}=\begin{cases}0\hfill & \text{if}\hspace{2pt}\;\text{moment}\hspace{2pt}\;\text{points}\hspace{2pt}\;\text{down}\hspace{2pt}\;\text{or}\hspace{2pt}\;\text{to}\hspace{2pt}\;\text{left}\hfill \\ 1\hfill & \text{if}\hspace{2pt}\;\text{moment}\hspace{2pt}\;\text{points}\hspace{2pt}\;\text{up}\hspace{2pt}\;\text{or}\hspace{2pt}\;\text{to}\hspace{2pt}\;\text{right}.\hfill \end{cases}\end{equation} \tag{ 4 }$$

Half of the 2⁶ = 64 environment configurations of the double vertex are depicted in figure 2. The remaining states #i can be derived by applying a time reversal operation on the configurations #(2⁶ − 1 − i).

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In the following, we discuss a perturbative approach to switching barriers in artificial square ice. Here, the single-nanomagnet barrier ΔE_sb, figure 1(d), is modified due to energy contributions arising from the interactions with the immediate magnetic environment.

To discuss a specific example, we focus on the fully-magnetised double-vertex environment #0 depicted in figure 1(b). The energy of the configuration where the central moment points (exactly) to the left, ←, is lower than when it points to the right, →, where three magnetic charges meet at each vertex point. Using simplified assumptions and symmetry arguments, one can derive a mean-field switching barrier ${\langle {\Delta}{E}_{i}\rangle }_{{\leftarrow}\;\text{to}\;\to }^{\text{dip}}$ (as the average of the barriers for clockwise and counter-clockwise rotation, see appendix A in reference [15]). Its value depends solely on the energies of the equilibrium configurations, as indicated by the gray arrow in figure 1(d):

$$\begin{equation}{\langle {\Delta}{E}_{i}\rangle }_{{\leftarrow}\;\text{to}\;\to }^{\text{dip}}={\Delta}{E}_{\text{sb}}^{\text{shape}}+\frac{1}{2}\left({E}_{i,\to }^{\text{dip}}-{E}_{i,{\leftarrow}}^{\text{dip}}\right).\end{equation} \tag{ 5 }$$

This mean-field barrier, however, is missing a crucial point, as independent relaxation pathways via clockwise and counter-clockwise rotation of the central nanomagnet need to be considered, i.e.

$$\begin{equation}{\Delta}{E}_{i,{\leftarrow},\mathrm{c}\mathrm{w}}^{\text{dip}}={\Delta}{E}_{\text{sb}}^{\text{shape}}+\left({E}_{i,{\uparrow}}^{\text{dip}}-{E}_{i,{\leftarrow}}^{\text{dip}}\right),\end{equation} \tag{ 6 }$$

and

$$\begin{equation}{\Delta}{E}_{i,{\leftarrow},\mathrm{c}\mathrm{c}\mathrm{w}}^{\text{dip}}={\Delta}{E}_{\text{sb}}^{\text{shape}}+\left({E}_{i,{\downarrow}}^{\text{dip}}-{E}_{i,{\leftarrow}}^{\text{dip}}\right).\end{equation} \tag{ 7 }$$

These barriers are not necessarily equivalent, as shown in figure 1(c): due to their staggered spatial arrangement, the central moment in a fully magnetised environment #0 will preferably align ferromagnetically with its neighbours, thus forming a head-to-tail flux-closure configuration which reduces the dipolar energy term in equation (2). Therefore, transitions via counter-clockwise rotations (blue) of the central nanomagnet will be largely favoured over those via clockwise rotations (red).

From the dipolar energies for all environmental states and orientation of the central moment, figure 3(a), one can obtain the respective clockwise and counter-clockwise switching barriers, figure 3(b). Under the assumption that ↑ and ↓ align perfectly along the short axis of the nanomagnets, the energy splitting between the configurations is symmetric around the mean-field barrier ${\langle {\Delta}{E}_{i}\rangle }_{{\leftarrow}\;\text{to}\;\to }^{\text{dip}}$ (marked by crosses), and equals the three distinct values shown in figure 3(c).

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Barrier splitting occurs for all environments which feature a perpendicular effective field ${\hat{M}}_{i,\perp }$ generated by the neighbouring nanomagnets, that acts on the central nanomagnet (indices b_j are as defined in equation (4)):

$$\begin{equation}{\hat{M}}_{i,\perp }={{\Sigma}}_{j=1}^{4}\frac{{\mathbf{m}}_{j}}{\vert {\mathbf{m}}_{j}\vert }=-{{\Sigma}}_{j=1}^{4}{\left(-1\right)}^{{b}_{j}}.\end{equation} \tag{ 8 }$$

The normalised perpendicular magnetisation ${\hat{M}}_{i,\perp }$ can take the values of 0 (black and purple in figures 2 and 3), ±2 (orange), and ±4 (red and blue) only. Thus, we can modify equation (5) to include an additional energy term:

$$\begin{equation}{\Delta}{E}_{i,{\leftarrow},\mathrm{c}\mathrm{w}/\mathrm{c}\mathrm{c}\mathrm{w}}^{\text{dip}}={\langle {\Delta}{E}_{i}\rangle }_{{\leftarrow}\;\text{to}\;\to }^{\text{dip}}\mp \frac{{J}_{\text{NN}}^{\text{dip}}}{3}{\hat{M}}_{i,\perp }.\end{equation} \tag{ 9 }$$

For a central moment initially pointing to the left (←), the second term (derived in appendix A) is subtracted for clockwise, and added for counter-clockwise reversal.

In conclusion, with a simplified point-dipole model the switching barriers are modified by the choice of clockwise vs counter-clockwise rotation, if the moment interacts with an effective perpendicular stray field generated by its environment. As shown in figure 2, 40 out of the 64 double-vertex configurations of artificial square ice feature a finite perpendicular magnetisation ${\hat{M}}_{i,\perp }\ne 0$ acting on the central nanomagnet. In particular, we expect a maximum chiral barrier splitting for the fully-magnetised environments (marked in red), which are commonly-used initial states for thermal relaxation studies of artificial spin ice [7].

The conclusions of this section are of general validity for arrays of interacting nanomagnets. The calculation of switching barriers for clockwise and counter-clockwise transitions can be easily adapted to other moment configurations. As an example, in supplementary material (https://stacks.iop.org/NJP/23/033024/mmedia) figure 1 we show that chiral barrier splitting occurs in over 60% of the moment configurations of artificial kagome spin as well.

The existence of separate chiral switching channels is by no means a curiosity, since nanomagnetic switching will occur predominantly via the more favourable pathway. We thus expect profound consequences on the switching rates and transition kinetics when taking into account the chiral splitting.

Contrary to simulations employing the Landau–Lifschitz–Gilbert equations to explicitly solve the time-dependent evolution of the nanomagnetic reversal, we determined the associated energy barrier using a time-independent string method. In this approach, starting from a coherent moment reversal, the moment configurations are iteratively optimised to a minimum-energy path through configuration space, and thus yield the lowest energy barrier associated to that reversal process. As in our previous work [15], which also discusses further simulation details, we implement here the simplified and improved string method [29] using the finite-element micromagnetic code magnum.fe [30]. We consider two artificial square ice geometries with distinct choices for M_sat and A_ex, representing different regimes. Meshes discretising the considered geometries, i.e. an individual nanomagnet and the double-vertex configurations, were created with the software gmsh [31].

First, we consider a geometry largely dominated by exchange interactions, which favour a coherent reversal, and thus may resemble the macrospin model derived in section 1: nanomagnets with dimensions l × w × t = 150 nm × 100 nm × 3 nm are placed on a square lattice with periodicity a = 240 nm. The material parameters M_sat = 790 kA m⁻¹ and K = 0 correspond to bulk permalloy (Fe_0.2Ni_0.8) values at 300 K. The exchange stiffness A_ex = 13 pJ m⁻¹ was obtained from a temperature-dependent scaling [32–35]

$$\begin{equation}{A}_{\text{ex}}={A}_{\text{ex}}\left(0\right){\left(\frac{{M}_{\text{sat}}}{{M}_{\text{sat}}\left(0\right)}\right)}^{1.7},\end{equation} \tag{ 10 }$$

where M_sat(0) = 950 kA m⁻¹ and A_ex(0) = 18 pJ m⁻¹ denote the respective permalloy bulk parameters at 0 K.

Second, we consider a system for which we expect sizable magnetostatic effects leading to non-uniform magnetic configurations during reversal: Nanomagnets with dimensions l × w × t = 470 nm × 170 nm × 3 nm are placed on a square lattice with periodicity a = 600 nm. This geometry, or choices close to it, have been used in several experimental studies [7, 13, 36, 37]. The saturation magnetisation M_sat = 350 kA m⁻¹ corresponds to the value given in [7], and from the scaling in equation (10) we obtain A_ex = 3.25 pJ m⁻¹. Although the saturation magnetisation is lowered significantly, the exchange stiffness is even more reduced, and thus we expect the energetics dominated by magnetostatic interactions. We again assume a vanishing magnetocrystalline anisotropy, K = 0.

We select representative environment states for each of the five classes introduced figure 2(b), for which we expect no (black and purple), intermediate (yellow), and high barrier splitting (red and blue), respectively. For these states, shown at the top of figure 4, we compare the chiral switching barriers obtained from a macrospin approximation (left) and micromagnetic string-method simulations (right) for the two different square ice geometries (schematics on the left are shown to scale). Clockwise and counter-clockwise barriers are marked in red and blue, respectively, and the difference (barrier splitting) is plotted as black bars.

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The macrospin model considers variation of the single-moment barrier ΔE_sb (calculated from the shape anisotropy of a uniformly-magnetised nanomagnet ${\Delta}{E}_{\text{sb}}^{\text{shape}}={K}^{\text{shape}}V$) due to point-dipole-like interactions quantified by ${J}_{\text{NN}}^{\text{dip}}\propto {\left({M}_{\text{sat}}V\right)}^{2}/{a}^{3}$, as derived in section 1.2. We also plot the mean-field barrier from equation (5) (black horizontal line).

The chiral barriers from the string-method simulations are calculated from the energy difference between the micromagnetic net energies of the metastable barrier configuration (↑, ↓) and the initial configuration (←):

$$\begin{equation}{\Delta}{E}_{i,{\leftarrow},\mathrm{c}\mathrm{w}}^{\text{mm}}={E}_{i,{\uparrow}}^{\text{mm}}-{E}_{i,{\leftarrow}}^{\text{mm}},\end{equation} \tag{ 11 }$$

$$\begin{equation}{\Delta}{E}_{i,{\leftarrow},\mathrm{c}\mathrm{c}\mathrm{w}}^{\text{mm}}={E}_{i,{\downarrow}}^{\text{mm}}-{E}_{i,{\leftarrow}}^{\text{mm}}.\end{equation} \tag{ 12 }$$

For the exchange-dominated square-ice geometry, figures 4(a)–(e), the mean-field barrier always overestimates the lower of the two micromagnetic barriers (for the chosen cases corresponding to counter-clockwise reversal marked in blue), as already discussed in our previous work [15]. Compared to the point-dipole model, micromagnetic simulations consistently give lower chiral switching barriers. The difference of the barrier energies, i.e. the chiral splitting, is enhanced, and remains proportional to the perpendicular moment $\vert {\hat{M}}_{i,\perp }\vert$ generated by the environment. The reversal process is still governed by an almost-coherent rotation of the central moment (appendix C.1). Therefore, one can obtain reasonable switching barriers from the perturbative decomposition of equation (9) by using a lower single-moment barrier ${\Delta}{E}_{\text{sb}}^{\text{string}}$ and stronger interactions ${J}_{\text{NN}}^{\text{mm}}$ (appendix C.2).

For the magnetostatic-dominated square-ice geometry, figures 4(f)–(j), the differences are even more pronounced. This is because the reversal behaviour is no longer uniform, as discussed in appendices C.1 and C.2. In particular, environments with vanishing perpendicular magnetisation ${\hat{M}}_{i,\perp }=0$ (black and purple) non-coherent reversal modes are promoted by C- or onion-like magnetic microstates in the central switching nanomagnet [25–28]. This leads to large reductions of the switching barrier, e.g. more than −50% in the case of state #22 in figure 4(f), when compared to the point-dipole model. In contrast, environments with M_⊥ ≠ 0 favour S-like micromagnetic configurations associated with quasi-coherent reversal [38–40]. Further details about the micromagnetic switching are discussed in appendix C and supplementary figure 2.

In general, the point-dipole barriers overestimate all micromagnetic barriers by a considerate margin, and underestimate the chiral barrier splitting, as is evident when comparing the black bars shown in figures 4(h)–(j). For environments with a finite perpendicular magnetisation the barrier splitting is of similar magnitude, figures 4(h)–(j), and thus does not follow the proportionality $\vert {E}_{i,{\uparrow}}-{E}_{i,{\downarrow}}\vert \propto \vert {\hat{M}}_{i,\perp }\vert$. Therefore, the simplified decomposition of switching barriers according to equation (9) is no longer valid when the moments can switch via non-coherent reversal mechanisms.

In general, by taking into account the micromagnetic nature of the moment reversal, we observe both a reduction of the switching barriers as well an enhanced separation of the chiral barriers for environments with ${\hat{M}}_{i,\perp }\ne 0$. These results hint to the fact that barrier reductions often utilised in simplified models to simulate the evolution of artificial spin ices [7, 8], and which are often attributed to extrinsic causes such as nanofabrication imperfections, could be—at least partially—associated with intrinsic barrier splitting and incoherent reversal modes.

As lower barriers are easier to overcome for thermally-induced reversal, we therefore expect significant enhancement of the kinetics of artificial square ice, which may yield different relaxation time scales and emergent correlations.

The temperature-dependent transition rate for spontaneous switching over an average energy barrier ΔE_avg can be obtained via the Arrhenius law [41, 42], with the attempt frequency ν₀ and the Boltzmann constant k_B = 8.62 × 10⁻⁵ eV K⁻¹:

$$\begin{equation}{\nu }_{\text{avg}}\left({\Delta}{E}_{\text{avg}},T\right)=2{\nu }_{0}\enspace \mathrm{exp}\left(-\frac{{\Delta}{E}_{\text{avg}}}{{k}_{\text{B}}T}\right).\end{equation} \tag{ 15 }$$

The attempt frequency ν₀ depends on the shape, size, and material of the nanomagnets. Typical values of ν₀ are in the order of 10^9...12 Hz [42, 43], and even faster time scales have been discussed [44, 45].

We need to consider the clockwise and counter-clockwise reversal as parallel and independent channels of relaxation, and the rates associated to each of the two barrier energies, i.e. ΔE_avg(1 − f) and ΔE_avg(1 + f), need to be added to obtain an effective transition rate. Therefore, for the definition of the rate in equation (15), we explicitly included a pre-factor of two, to account for degenerate clockwise and counter-clockwise relaxation channels over the average barrier.

Due to the pronounced non-linearity of the Arrhenius law, the summation of rates leads to an effective increase of the net transition rate ν_sum of thermally-activated switching when compared to the rate ν_avg associated with the average barrier (derivation in appendix B):

$$\begin{equation}{\nu }_{\text{sum}}={\nu }_{\text{avg}}\left({\Delta}{E}_{\text{avg}},T\right)\mathrm{cosh}\left(f\frac{{\Delta}{E}_{\text{avg}}}{{k}_{\text{B}}T}\right).\end{equation} \tag{ 16 }$$

The ratio of the joint rate compared to the average-barrier rate, i.e. ν_sum/ν_avg, are compared in figure 5(b) for different splitting ratios f and reduced energies ΔE_avg/k_B T. The exponential rate enhancement is particularly pronounced in the limit of rare events with ΔE_avg/k_B T ≫ 1 where a splitting of barriers can increase the (albeit low) transition rates by several orders of magnitude (purple lines). In contrast, in the limit of superparamagnetic fluctuations, i.e. ΔE_avg/k_B T → 1, barrier splitting increases the net rate only moderately (blue lines).

For the assumption that transitions occur predominantly via the lower barrier only, we have to consider the transition associated to the smaller barrier, i.e. (1 − f)ΔE_avg, see dashed lines in figure 5 giving the ratio ν_max/ν_avg. For high splitting ratios f and ΔE_a /(k_B T) ≫ 1 the rate ν_max will approach ν_sum, as transitions by the higher-lying barrier become irrelevant. The shaded area of figure 5 marks where ν_max exceeds 90% of the value ν_sum. Within this regime, transitions via the lower-lying barrier might be a good approximation of the net reversal rate. In the limit of f → 0, however, where both barriers are equal, using the rate ν_max will underestimate the net rate by a factor of two, i.e. ${\left({\nu }_{\mathrm{max}}/{\nu }_{\text{sum}}\right)}_{f\to 0}=1/2$.

In the case of artificial square ice, and depending on the kinetic regime given by the relation between the energies ΔE_sb, J_NN, and k_B T, approximating the transition rates via the mean-field barrier [7, 8, 13, 37] or the minimum barrier [46, 47] thus may significantly underestimate the speed of evolution.

To illustrate the consequences of barrier splitting, we now turn to the evolution of extended square-ice arrays. In many thermal relaxation studies of artificial spin ice, the main interest lies in the onset of phase transitions and formation of emergent correlations. In many experiments, the system evolves from a field-set fully magnetised state, for which we predict a particularly strong barrier splitting. We therefore expect that the initial demagnetisation of a magnetic-field-saturated artificial square ice array will be particularly affected by the modified transition kinetics.

To model the relaxation, kMC simulations are often employed [48–50]. The kMC algorithm provides a numerical solution to the master equation, which is a system of linear differential equations describing the evolution of the probabilities for Markov processes in systems that jump from one state to another in continuous time [51]. Using this method, both the equilibrium expectation values of populations and their dynamical evolution during a thermalization process can be retrieved.

In this work, kMC simulations are performed using a custom-written code [52], with a system of 50 × 50 moments and periodic boundary conditions. The initial configuration is uniformly-magnetised, with the net magnetisation being parallel to a diagonal direction of the array. The demagnetisation due to spontaneous moment reversals is tracked over time for 125 × 10³ kMC steps, and averaged over 20 individual simulation runs. As an illustrative example, we here consider the effect of symmetric barrier splitting only, and discount further barrier reductions associated to non-uniform reversal. Thus, we use the point-dipole energy barriers from figure 3(b) with parameters ${J}_{\text{NN}}^{\text{dip}}=$ 0.178 eV and ΔE_sb = 1.327 eV (i.e. using values for the exchange-dominated square-ice geometry), and calculate the environment-dependent transition rates ν_avg and ν_sum at a temperature of T = 300 K as input parameters for the kMC simulations.

Figure 6(a) compares the time evolution of the net magnetisation of square ice for rates from the mean-field barriers (dashed black line) to the model taking into account chiral barrier splitting (solid red line). The time is measured in multiples of the inverse attempt frequency, $\tau ={\nu }_{0}^{-1}$. We find that the onset of demagnetisation for the split-barrier model (red) happens two orders of magnitude earlier than for the average-barrier model (black). In the case of the average-barrier model, the demagnetisation involves bouts of rapid evolution interrupted by phases with little change, indicating avalanche-like dynamics [53–56]. In contrast, the split-barrier model shows a smooth demagnetisation, with a rate (solid thick lines indicate evolution from from 90% to 50%) which is about three orders of magnitude faster compared to the mean-barrier model.

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When assessing the emergent spatial correlations, we find that the evolution for the mean-field model is governed by the propagation of strings of ground-state vertices (in blue) wrapping the system (due to the periodic boundary conditions), as shown in figure 6(b). The final state has a magnetisation of about 16% of its initial value. The snapshots of spatial configuration of the split-barrier model at M = 0.9 and M = 0.5 (with M normalised to the initial field-set magnetisation) in figure 6(c) appear somewhat similar to that of the mean-field case. There are more possible transitions for the system to explore, and we would like to point out that this model recovers the separation of strings observed in experiments, but does not require to invoke external disorder e.g. via variation of the switching barriers as done in [8]. The final state of the evolution corresponds to a multi-domain state with almost vanishing magnetisation, M ≈ 0.

Thus, our kMC results show that the modified hierarchy of transition barriers due to the chiral barrier splitting may have subtle, but relevant, consequences: in certain cases, the kinetic relaxation pathways are not simply dictated by equilibrium-energy arguments. This will modify the emergence of spatial correlations, which needs to be explored in a systematic study and compared to experimental results [1, 9, 10, 53, 57–64].

The data that support the findings of this study are available upon reasonable request from the authors.

To quantify the uniformity of the magnetic reversal, figure C.1 shows the averaged moments for a single (non-interacting) nanomagnet and each of the representative double-vertex configurations presented in figure 4. Here, the average magnetisation of each nanomagnet is plotted for every step of the string-method minimum-energy path. The horizontal coordinate roughly corresponds to the rotation angle ϕ of the central moment, with end points denoting initial and final equilibrium states.

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In general, we find that in equilibrium the net moment |M| (magenta lines) is very close to one. Therefore, the static nanomagnets assume an almost saturated configuration, with the magnetisation largely aligned with the long axis of the nanomagnet and limited edge bending (M_x , red lines). Interactions with neighbouring moments, however, can induce sizeable perpendicular moment contributions (M_x ⊥ M_y , blue lines) in environments that feature a finite perpendicular magnetisation ${\hat{M}}_{i,\perp }\ne 0$, i.e. configurations #0, #2, and #16.

In the case of the geometry with small nanomagnets dominated by exchange interactions the magnitude |M| remains largely constant, figure C.1(a). The reversal thus represents a quasi-uniform rotation of the central moment. During reversal, the magnetisation components of the neighbouring nanomagnet can vary, allowing the system to evolve via the most efficient pathway.

For the geometry with large nanomagnets dominated by magnetostatic energy, shown in figure C.1(b), the switching of the non-interacting nanomagnet (left) involves a reduction of the net moment to 91%, and thus does not conform to a uniform moment rotation. For environment states #0, #16, and #2 with ${\hat{M}}_{i,\perp }\ne 0$ the reduction of net magnetisation is similar to that of an individual nanomagnet. In contrast, for environment states #6 and #22, with ${\hat{M}}_{i,\perp }=0$, we observe a pronounced reduction of magnetisation in the high-energy configuration to less than 70% of the net moment. This indicates reversal via non-coherent modes, which are favoured to occur in nanomagnets that conform to C- or onion-like magnetic microstates promoted by the order of the surrounding nanomagnets [38–40]. This leads to a reduction of the switching barrier as well, as discussed in section 2. The magnetic configuration of neighbouring nanomagnets varies less during reversal when compared to the small-island geometry. This is because the relative volume fraction of the large magnets meeting at the vertex point, where the spin structure varies the most, is smaller.

The switching barriers obtained from micromagnetic string-method simulations for the environment states #0-#31 are summarised in figure C.2 with (a)–(c) showing the results for the exchange-dominated, and (d)–(f) the magnetostatic-dominated geometry.

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We compare the clockwise and counter-clockwise barriers, large and small circles in figures C.2(a) and (d), to a modified mean-field model. The predictions are based on equations (5) and (9), but instead of energies derived from point-dipole calculations we use those obtained from micromagnetic simulations, as follows:

First, the switching barrier ΔE_sb of an isolated nanomagnet simulated with the string-method is used, as opposed to the shape anisotropy calculated for a uniformly-magnetised nanomagnet. For the exchange-dominated nanomagnet geometry we obtain ${\Delta}{E}_{\text{sb}}^{\text{shape}}=$ 1.540 eV compared to ${\Delta}{E}_{\text{sb}}^{\text{string}}=$ 1.327 eV, which is a reduction of -14%. For the magnetostatic-dominated nanomagnet geometry, the barrier reduction is even bigger, with ${\Delta}{E}_{\text{sb}}^{\text{shape}}=$ 2.153 eV and ${\Delta}{E}_{\text{sb}}^{\text{string}}=$ 1.691 eV (−22%), as the reversal is no longer coherent (see figure C.1(b)).

Second, ${E}_{i,{\leftarrow}}^{\text{mm}}$ and ${E}_{i,\to }^{\text{mm}}$ correspond to the micromagnetic equilibrium energies of the static configurations. Together with ${\Delta}{E}_{\text{sb}}^{\text{string}}$, a modified mean-field barrier can be calculated from equation (5) (crosses in figures C.2(a) and (d)).

Third, to estimate the barrier splitting, the nearest-neighbour interaction ${J}_{\text{NN}}^{\text{mm}}$ is rescaled. The rescaling is motivated by the point-dipole model, which predicts an energy difference of $\left(16-\frac{24\sqrt{5}}{125}\right){J}_{\text{NN}}^{\text{dip}}$ between the lowest-lying ground state and highest monopole state:

$$\begin{equation}{J}_{\text{NN}}^{\text{mm}}=\frac{{E}_{\mathrm{max}}^{\text{mm}}-{E}_{\mathrm{min}}^{\text{mm}}}{16-\frac{24\sqrt{5}}{125}}.\end{equation} \tag{ C.1 }$$

For the exchange-dominated small-island geometry we find that the modified mean-field barrier gives a passable estimate for the average switching barrier, as the small differences in figure C.2(c) show. The chiral barrier splitting is well-described by the rescaled energy ${J}_{NN}^{\text{mm}}$, albeit with a small reduction for fully-magnetised environments marked red and blue in figure C.2(c).

For the magnetostatic-dominated reversal in the large-island geometry, figures C.2(d)–(f), the mean-field approach fails: the barrier splitting is both overestimated for environments with $\vert {\hat{M}}_{i,\perp }\vert =4$ (red and blue) as well as underestimated in the case of $\vert {\hat{M}}_{i,\perp }\vert =2$ (yellow). In particular, the mean-field barrier predictions fails for environments with $\vert {\hat{M}}_{i,\perp }\vert =0$ (black and purple), where reversal via non-uniform modes are favoured, as discussed before. The reductions compared to the mean-field barrier seem to be particularly strong for environments which feature 'X' and 'C' configurations of the perpendicular moments, see figures C.2(f) and (g). The symmetry breaking by the environment promotes so-called onion or C- microstates in the central nanomagnet, respectively [38–40]. These states promote non-coherent reversal, as illustrated in supplementary material figure 2, and a reduction of the switching barrier, as discussed in figure C.1 and appendix C.1.

With the exception of ${\Delta}{E}_{\text{sb}}^{\text{string}}$, which is a result of the string-method simulation, the energies ${E}_{i,{\leftrightarrow}}^{\text{mm}}$ and ${J}_{\text{NN}}^{\text{mm}}$ can be obtained from static equilibrium micromagnetic simulations, e.g. using OOMMF [21] or MuMax3 [22]. This makes this approach attractive to estimate more realistic switching barriers based on a perturbative decomposition of a single-nanomagnet behaviour plus a correction term due to interactions with the neighbouring moments. This approach seems valid for relatively small nanomagnets favouring reversal via uniform modes, but fails if more complex reversal mechanisms are accessible.