# Polymerization in magnetic metamaterials

Samuel D. Slöetjes  Matías P. Grassi  Vassilios Kapaklis  Department of
Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden

###### Abstract

We numerically study a mesoscopic system consisting of magnetic nanorings in
the presence of thermal magnetization fluctuations. We find the formation of
dipolar-field-mediated “bonds” promoting the formation of annuli clusters,
where the amount of bonds between two rings varies between zero and two. This
system resembles the formation of polymers from artificial atoms, which in our
case are the annuli and where the valency of the atom is set by the ring
multipolarity. We investigate the thermodynamic properties of the resulting
structures, and find a transition associated with the formation of the bonds.
In addition, we find that the system has a tendency to form topological
structures, with a distinct critical temperature in relation to the one for
bond formation.

††preprint: APS/123-QED

Polymerization is typically related to the process of reaction of monomer
molecules towards the formation of larger chain-like or three-dimensional
molecular networks, polymers. Essential for these processes is the ability of
monomers to bond with other monomers and their steric effects, relating to the
way atoms can spatially arrange. Extrapolating these observations into the
realm of magnetism and artificial arrays of mesoscopic magnetic entities,
allows for a new look and design approach for the collective magnetic order
and dynamics of magnetic metamaterials [1, 2]. Important to this shift in
perspective, is introducing the concept of bonds between magnetic entities,
captured by the dipole coupling between them and the associated magnetostatic
charges. The formation of 1D chains in magnetic metamaterials has been
observed before, for example in square arrays of circular disks in the form of
antiferromagnetic lines [3, 4], which can be regarded as polymers with a
trivial topology. Another 1D entity in metamaterials consisting of bistable
elements are Dirac strings, which connect certain high energy vertex
configurations [5, 6]. However, structures with topologies beyond that of an
open chain remain scarcely explored.

Figure 1: (a) Magnetic rings, each with two domain walls with opposite-sign
magnetostatic charges [7]. These charges can couple across the elements,
forming bonds via stray fields (gray shaded areas). (b) Energy landscape for
two domain wall states in magnetic nanorings, as a function of the net
magnetic moment of the ring (indicated on the right).

The magnetic texture within the individual elements can be harnessed in order
to allow for increased complexity of the emergent artificial structures in
these materials. One way to achieve this, is by altering the topology of the
building blocks themselves. Tailoring of the topology in magnetic
metamaterials has so far only been realized on the lattice level, for the
realization of frustrated magnetic systems [8], and has lead for example to
the investigations of Shakti [9, 10], Saint George [11], and Tetris [12]
artificial spin ice, among others. In this work, we will consider a system
consisting of building blocks with an altered topology compared to the usual
Ising-like (elongated) or disk-shaped mesospins (for a classification scheme
see Table I by Skovdal  _et al._ [13]), namely magnetic rings. Nanomagnetic
rings have been studied previously, albeit in a different context. Early
efforts consisted of studying domain walls in single rings [14, 15, 16], which
focused on switching between micromagnetic states in the ring and subsequent
dynamics investigations [17, 18]. Studies on the thermally driven dynamics in
ring systems are scarce, and only focus on the thermally excited transition
from a vortex state to an onion state [19]. Experimentally, Laufenberg  _et
al._ [20] have reported on the observation of coupling between rings in
patterned arrays, showing that it is possible to realize such systems. More
recently, it was demonstrated that it is possible to do basic neuromorphic
computations in arrays of connected rings, utilizing the domain walls in these
arrays [21]. However, in all of these works, the temperature was not a
parameter of interest and concepts like emerging order and phase transitions
were not considered. Here, we inspect the temperature dependent magnetic order
in this system, and find the formation of clusters, where the amount of bonds
between rings can be more than one. Consequently, this system mimics the
formation of polymers coupling together a significant number of individual
magnetic textures, with the bonding valency set by the multipolarity of a
ring.

We used the micromagnetic simulation package MuMax3, which solves the Landau-
Lifshitz-Gilbert equation for a grid of cells describing magnetic moments
[22]. The cell size was $l_{x}\times l_{y}\times l_{z}=$ 2.5 nm $\times$ 2.5
nm $\times$ 4 nm, where $l_{z}$ is equal to the thickness of the rings. The
saturation magnetization and exchange stiffness are
$M_{\mathrm{S}}=1\times$106 A/m and $A_{\mathrm{ex}}=$ 1$\times$10-11 J/m, and
the damping is $\alpha=$ 0.01, effectively describing a material that
resembles Permalloy. The rings have an outer and inner radius of $r_{o}=$ 125
nm and $r_{i}=$ 75 nm, respectively. The square grid in the simulation
provides an effective fourfold anisotropy to the rings due to the corrugated
edges, which was partially compensated by a cubic anisotropy set along the
$[1,1]$ and $[1,-1]$ directions, resulting in a weak 8-fold anisotropy. The
rings were placed on a square 16 $\times$ 16 grid, with a mutual spacing of
$d=50$ nm. A finite temperature was taken into account by way of a stochastic
field which is proportional to the root of temperature, $\sqrt{T}$, and
uncorrelated both in space and time [23].

Figure 2: (a) Excerpt from the 16 $\times$ 16 array of rings at
$\widetilde{T}=0.171$, after relaxing from a random magnetization. The
colormap which is given by the dot product between the normalized radial and
magnetization vectors, and represents the magnetostatic charge of the domain
wall. The bonding field, $H_{\mathrm{b}}$, is shown in gray. (b) Average
cluster size for different reduced temperatures
$\widetilde{T}=k_{\mathrm{B}}T/E_{\mathrm{b}}$, starting from a random
configuration. The window average of the data is shown in blue, the red dots
are the raw data. (c) Time evolution of the average domain wall density, as a
function of azimuthal position with respect to the individual magnetic rings
(calculated as $\langle|\mathbf{\hat{m}}\cdot\mathbf{\hat{r}}|\rangle$), for
three different temperatures.

We set out by considering the magnetization within a single ring. Magnetic
rings have a fundamentally different magnetic texture than conventional
nanomagnet elements, as the topology is the same as that of a strip with
continuous boundary conditions. As such, their coupling to neighbouring
elements occurs via the stray field emitted from domain walls, instead of the
conventional majority part of the magnetic texture (see Fig. 1a). When exposed
to temperature, the domain walls are free to move around the magnetic ring,
resulting in a key difference compared to magnetic disks, which is the
additional freedom of the domain walls to move with respect to each other
without a high energy cost. The amount of bonds available for one ring, and
thereby the valency, depends on the amount and type of domain walls. A ring
with a vortex state has zero bonds, an onion state provides a double valency,
and an antivortex state has a quadruple valency. In most of the cases, the sum
of the magnetostatic charges is zero, i.e., in the case of the onion state the
two magnetostatic charges are $+$ and $-$. This is the case if the winding
numbers of the topological defects on the outer edges, as defined by
Tchernyshyov and Chern [7], are positive ($n=+1/2$), and the ones on the inner
edges are negative ($n=-1/2$). This configuration is most often the case,
since defects with positive topological numbers have the lowest energy on
positively curved edges, and vice versa for defects with $n=-1/2$ on
negatively curved edges. In some cases, $n=-1/2$ charges can be found on the
outer edges of the ring, which results in an uncompensated net magnetostatic
charge. In such a case, the state can decay through annihilation of a $n=+1/2$
charge.

The energy landscape of the magnetization in the ring can be mapped out as a
function of the total net magnetization in-plane (its components being $m_{x}$
and $m_{y}$), as shown in Fig. 1b for the case of two domain walls. In this
case, the energy landscape is flat in the azimuthal direction, due to the
rotational symmetry of the ring, but curved in the radial direction. The
landscape features one deep inner trough and a shallow outer rim. The deep
inner trough represents the groundstate, corresponding to the configuration in
which the domain walls are close together, due to the attractive interaction
between magnetostatic charges, but still remain apart due to the repulsive
interaction of the topological charges. This repulsive interaction leads to an
upturn in the energy landscape at the smallest net magnetization values. The
outer rim in the energy landscape corresponds to a state where the two domain
walls are on opposite sides of the ring.

When magnetic rings are organized in an array and starting from a paramagnetic
state, the emergent order is not immediately apparent from just considering
the magnetization. As mentioned previously, the rings can couple to one
another through the stray fields, which are produced at the domain walls in
the magnetic texture in the ring. In order to define a bond, we introduce a
scalar “bonding field”, $H_{\mathrm{b}}$, which has binary values, by
thresholding the demagnetizing field, $|\mathbf{H}_{\mathrm{dem}}|$:

$\displaystyle|\mathbf{H}_{\mathrm{dem}}(x,y)|>H_{\mathrm{thresh}}\rightarrow
H_{\mathrm{b}}(x,y)=1$
$\displaystyle|\mathbf{H}_{\mathrm{dem}}(x,y)|<H_{\mathrm{thresh}}\rightarrow
H_{\mathrm{b}}(x,y)=0$

We used $\mu_{0}H_{\mathrm{thresh}}$ = 0.03 T, and details on the
justification of this value and further identification of bonds can be found
in the supplementary material. If two rings are connected by $H_{\mathrm{b}}$,
this defines a bond. In order to reveal the underlying order, $H_{\mathrm{b}}$
must be included in the visualisation. The appearance of bonds is shown in
Fig. 2a, where bond coupling causes clustering into polymers, where the
individual rings are considered monomers. As such, the type of order that
emerges is of a percolative nature.

We will now inspect the behaviour of the artificial polymers upon thermal
excitation. In this analysis, the order parameter is taken to be the amount of
bonds on the lattice, $N_{\mathrm{b}}$. The energy needed to break one bond is
$E_{\mathrm{b}}=0.50$ eV 111This value is calculated by evaluating the
demagnetization energy between two rings connected by a bond at 50 nm, and
subsequently increasing the distance between the rings in steps of 5 nm until
the bond no longer exists., and henceforth we will make use of a dimensionless
temperature, scaled by this value,
$\widetilde{T}=k_{\mathrm{B}}T/E_{\mathrm{b}}$, where $k_{\mathrm{B}}$ is the
Boltzmann constant. We have simulated the magnetization for seven different
temperatures between $\widetilde{T}=0.043$ and 0.341, for a duration of 200
ns, in each case starting from the same magnetization state. This initial
state has 56 rings with vortex states, 151 rings with two charges 48 rings
with four charges, and one ring with six charges. This initial magnetization
state is relaxed from a random magnetization in the absence of temperature,
and is not the groundstate, but rather a metastable state with
$N_{\mathrm{b}}$ $=$ $96$. As time progresses, it can be seen in Fig. 2b that
the amount of bonds on the lattice increases for temperatures up to
$\widetilde{T}=$ 0.256. In the cases of $\widetilde{T}=$ 0.085 to 0.171, there
is a steep increase in $N_{\mathrm{b}}$ during the initial 15-20 ns, after
which the rate slows down. We attribute this increase to the annihilation of
charges (domain walls) with an inverted winding, thus providing more space for
the other charges to move to a bonding position (see supplementary material
for the total amount of domain walls over time). An exception is the high
temperature case of $\widetilde{T}=0.341$, where the thermal energy is high
enough to annihilate domain walls with regular winding in the timescales of
the simulation. Overall, the largest increase in $N_{\mathrm{b}}(t)$ over 200
ns is seen to occur for a temperature of $\widetilde{T}=0.171$. This behaviour
can be rationalized by the fact that, below this temperature, an increased
thermal agitation leads to a higher annealing rate, which means that it can be
expected that eventually the cases with $\widetilde{T}\leq 0.171$ relax to low
lying metastable minima, with a high $N_{\mathrm{b}}(t)$. At higher
temperatures, the entropy begins to prevail, which results in the breaking of
bonds and unpairing of charges. As such, we can establish that there exists a
ceiling temperature around $\widetilde{T}^{*}=0.171$, analogous to the ceiling
temperature in realistic polymer systems, at which the rate of polymerization
equals that of depolymerization. This behaviour signifies a soft transition,
involving the unpairing of charges, thereby bearing resemblance to a
Berezinskii-Kosterlitz-Thouless phase transition [25, 26]. At the critical
temperature, the system optimizes the mobility of the domain walls to find
bonding positions. A detailed investigation of this transition is due in
future work.

The difference in behaviour of the system when the temperature is varied is
ultimately reflected in the positions of the domain walls, whose motion is
governed by the underlying energy landscape. The domain wall positions are
tracked, and a histogram was made as a function of their position in polar
angle, for all timesteps, shown in Fig. 2c. At low temperatures, the domain
wall positions strongly reflect the 8-fold anisotropy of the individual rings.
When the temperature is increased, the system has enough energy to overcome
this anisotropy, and the domain wall position is mostly dominated by the
lattice symmetry, as seen from the four-fold anisotropy of the histogram. When
the temperature is increased further to $\widetilde{T}=0.341$, we observe that
the domain walls explore the full phase space, unimpeded by local or global
anisotropies. As such, the system becomes fully ergodic at the highest
temperature.

Figure 3: (a) Amount of loop-clusters for different temperatures, starting
from a random configuration. The blue line indicates the windowed average, the
red dots are the raw data (b) Example of a loop clusters spanning 2 and 4
rings. (c) The susceptibility is shown for different temperatures for the
amount of bonds (upper panel), the amount of loops (middle panel), and for the
amount of loops normalized by the amount of bonds (lower panel).

We observe that the tendency of the ring system to compensate the magnetic
charges leads to the formation of emergent topological structures on the next
length scale, namely loops, as can be seen in Fig. 3b. The typical size of
these loops is two and four rings. Loops play an important role in
conventional artificial spin ice systems, where they realize a topological
model system, in which trivial and non-trivial loops can be distinguished [27,
28]. The (possibly degenerate) groundstate of the ring lattice must feature
only loops, in order for all charges to be compensated. As such, we expect an
increased amount of loops at lower temperatures. This tendency to form loops
is enhanced by the intra-ring interactions, which favour small angles between
domain walls, causing the emerging polymers to bend. This can be contrasted to
a lattice of disks, which favours straight lines due to the contribution of
the exchange interaction in the interior [29, 30, 4, 3]. Moreover, rings with
four domain walls serve as pinch points in the polymers, also promoting the
formation of loops. In the following, we shall investigate the thermal
dynamics of these loops.

We start by considering the average polymer length, given by $\langle
L\rangle$. This parameter is closely related to $N_{\mathrm{b}}$. However, if
one is to relate the two quantities mathematically, the amount of loops, $l$,
must also be taken into account:

$\langle L\rangle=\frac{N}{N-N_{\mathrm{b}}+l}$ (1)

where $N$ is the total amount of rings. As such, once $\langle L\rangle$ and
$N_{\mathrm{b}}$ are found, $l$ can be determined via this relationship. The
amount of loops over time is shown for different temperatures in Fig. 3a. For
lower temperatures, initially the amount of loops drops drastically (for
$t=0$), before increasing again. The initial relative drop in loops ($\Delta
l/l(t=0)$) is much larger than that for the amount of bonds ($\Delta
N_{\mathrm{b}}/N_{\mathrm{b}}(t=0)$). This can perhaps be attributed to a
different relaxation time associated with loop formation versus bond
formation. The total amount of loops fluctuates between 0 and 18, where the
maximum is seen for $\widetilde{T}=0.128$, in contrast to the number of bonds,
which stabilizes at $\widetilde{T}=0.171$. This contrast suggests a different
critical temperature associated with bond formation versus that of loop
formation. We investigate this possibility by inspecting the susceptibility
(see Fig. 3c), which is calculated as
$\chi_{x}(T)=(1/k_{\mathrm{B}}T)\sigma_{x}^{2}$, where
$\sigma_{x}=\sqrt{\langle x^{2}\rangle-\langle x\rangle^{2}}$ is the standard
deviation of $x$, with $x=N_{\mathrm{b}}$ or $l$ [31]. A peak in the
susceptibility is associated with a maximum in the fluctuations, and could
indicate the occurrence of a phase transition. We observe peaks in the
susceptibilities for both $\chi_{N_{\mathrm{b}}}$ and $\chi_{l}$. However, the
susceptibility of the bonds peaks at $\widetilde{T}=\widetilde{T}^{*}=$ 0.171,
while the susceptibility peak for the loops occurs at $\widetilde{T}$ = 0.128.
The susceptibility for the loops was also calculated when normalized by the
amount of bonds, $\chi_{l/N_{\mathrm{b}}}$, and it was found that the peak
remained localized at $\widetilde{T}$ = 0.128. The difference in critical
temperature for these two entities is surprising, as the loops consist of
bonds, and one would therefore expect the same critical temperature. However,
there is one crucial difference: whereas bond formation only depends on inter-
ring interactions, the formation of loops depends on both inter- and intra-
ring interactions, thus setting a different energy scale. Additionally, the
entropy associated with $N_{\mathrm{b}}$ and $l$ should be considered. The
configurational entropy associated with placing a bond on a lattice is smaller
than the entropy associated with placing structures on the lattice that
consist of multiple bonds, such as loops, thus reducing the critical
temperature for $l$ with respect to $N_{\mathrm{b}}$.

In conclusion, we have investigated a novel artificial spin system, in which
the topology of the individual building blocks is altered with respect to
conventional magnetic metamaterials. We have found that bonding between
elements occurs via stray fields emitted and absorbed by domain walls, giving
rise to a magnetic order that resembles polymerization. Moreover, the
formation of bonds is associated with a thermodynamic transition. On the next
length scale, we observe an additional transition associated with the
formation of a topological structure known as a loop, with a definite
handedness. The critical temperatures associated with the transitions of these
two entities differ, which we attribute to different energy scales and
configurational entropy.

The data that support the reported findings are available upon reasonable
request.

##  I Acknowledgements

We wish to thank Prof. Björgvin Hjörvarsson for fruitful discussions. S.D.S.
and V.K. acknowledge support from the Swedish Research Council (Project No.
2019-03581). M.P.G. and V.K. also acknowledge support from the Carl Trygger
Foundation (Project No. CTS21:1219).

The authors have no conflicts of interest to disclose

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Supplemental Material: Polymerization in magnetic metamaterials

##  II Justification of the bonding field

The choice for a threshold of 30 mT was made based on the histogram of peak
heights in the demagnetizing field. To construct this histogram, we evaluated
the demagnetizing field along horizontal and vertical lines that divided the
spacing between the rings up in two equal parts, see Fig. S1a. The idea behind
this is that a bond has a high demagnetizing field in between the nanomagnets.
Figure S1b shows an example of the demagnetizing field along one of the lines.
By plotting the histogram of peak heights in the demagnetizing field along all
the lines, we observe that there is a trimodal distribution of peak heights,
i.e. we recognize three characteristic groups. The high values of dipolar
field correspond to domain wall that face each other, i.e. bonds. The second
peak under 30 mT corresponds to individual domain walls without a partner in
the adjacent ring, i.e. unsatisfied dangling bonds. Finally, the peak at low
dipolar field corresponds to the background expected for a long range
interaction.

Supplementary FIG. S1: (a) Intensity plot of the demagnetizing field,
$|\mu_{0}H_{\mathrm{d}}|$ outside the rings. The green lines indicate the
positions at which the cross-sections were evaluated. (b) Example of a cross-
section of $|\mu_{0}H_{\mathrm{d}}|$ along the line indicated with a black
arrow. The red line indicates the threshold for the classification as bond.
(c) Trimodal distribution of the peak heights in (b), for all lines.

##  III Identification of bonds

The identification of bonds was done by cutting out rectangular regions of
size 150 $\times$ 200 nm in between the rings, and inspecting the bonding
field. From each rectangle, a binary image was made, in which the regions of
the ring, and regions with $H_{b}=1$, are mapped to a value 1, and the rest to
0, see Fig. S2. Then, using the MATLAB function ‘bwlabel’, the different
connected regions where labeled. If there are only 2 different regions (0 and
1), this indicates the absence of a bond. In the case that there were 3
regions (0, 1 and 2), a single bond was identified, and in the case of four
different regions (see Fig. S2), a double bond was present. In order to
distinguish from artifacts, only regions with 10 or more connected pixels were
taken into account. This procedure was carried out in two steps; one for the
horizontal bonds, and one for the vertical bonds.

Supplementary FIG. S2: Identification of bonds. Left: the system of rings and
bonds at $\widetilde{T}=$ 0.128, right: zoom in of the region with the bonds,
after the labelling using MATLAB.

##  IV Domain wall number evolution

The domain wall population was quantified and plotted against time, see Fig.
S3b. This was done by plotting for each ring
$|\mathbf{\hat{m}}\cdot\mathbf{\hat{r}}|$ against angle, see Fig S3a, and
using the MATLAB ‘findpeaks’ function to identify the peaks, as each peak
indicates a domain wall.

Supplementary FIG. S3: (a) Waterfall plot of
$|\mathbf{\hat{m}}\cdot\mathbf{\hat{r}}|$ against the angle, $\theta$ in each
ring, for 128 rings, the peaks indicate the presence of a domain wall. The
temperature in this case is $\widetilde{T}=0.085$. (b) Total amount of domain
walls over time, for three different temperatures.