# Tension-free Dirac strings and steered magnetic charges in 3D artificial
spin ice

Sabri Koraltan<EMAIL_ADDRESS>Florian Slanovc  Florian Bruckner
Physics of Functional Materials, Faculty of Physics, University of Vienna,
Waehringer Strasse 17, 1090 Vienna, Austria  Cristiano Nisoli  Theoretical
Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA  Andrii
V. Chumak  Oleksandr V. Dobrovolskiy  Nanomagnetics and Magnonics, Faculty of
Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria  Claas
Abert  Dieter Suess  Physics of Functional Materials, Faculty of Physics,
University of Vienna, Waehringer Strasse 17, 1090 Vienna, Austria  University
of Vienna Research Platform MMM Mathematics-Magnetism-Materials, University of
Vienna, Austria

###### Abstract

3D nano-architectures present a new paradigm in modern condensed matter
physics with numerous applications in photonics, biomedicine, and spintronics.
They are promising for the realisation of 3D magnetic nano-networks for ultra-
fast and low-energy data storage. Frustration in these systems can lead to
magnetic charges or magnetic monopoles, which can function as mobile, binary
information carriers. However, Dirac strings in 2D artificial spin ices bind
magnetic charges, while 3D dipolar counterparts require cryogenic temperatures
for their stability. Here, we present a micromagnetic study of a highly-
frustrated 3D artificial spin ice harboring tension-free Dirac strings with
unbound magnetic charges at room temperature. We use micromagnetic simulations
to demonstrate that the mobility threshold for magnetic charges is by
$2\text{\,}\mathrm{e}\mathrm{V}$ lower than their unbinding energy. By
applying global magnetic fields, we steer magnetic charges in a given
direction omitting unintended switchings. The introduced system paves a way
towards 3D magnetic networks for data transport and storage.

Three-dimensional artificial spin ice, dirac strings, unbound emergent
magnetic monopoles

##  I Introduction

Data storage and transport devices ranging from hard disk drives to flash
memories, from CMOS to spintronic technologies are of crucial importance in
today’s technological world. Usually, these devices are based on 2D structures
approaching their limitations every day. 3D structures started to emerge over
the past years, leading to significant improvements in both reducing the
dimensions and increasing their efficiency, e.g. flash memories Zhu  _et al._
(2019); Dieny  _et al._ (2020). In these data storage devices, the third
dimension is used by the simple stacking of the same 2D structures. Thus, the
whole power of the additional third dimension is not utilised.

Over the past years, spin ices, a class of 3D materials, have been
investigated in detail. Spin ices are frustrated systems where the magnetic
moments are residing on the sites of a pyrochlore lattice, a lattice with
corner sharing tetrahedra, and commonly referred to as dipolar spin ices (DSI)
Harris  _et al._ (1997); Bramwell and Harris (1998); Ryzhkin (2005);
Castelnovo  _et al._ (2008); Morris  _et al._ (2009); Fennell  _et al._
(2009); Bramwell and Harris (2020). Its degenerate ground state obeys the ice-
rule, where two magnetic moments point to the center of the tetrahedra and two
away from it. Switching a magnetic moment breaks the ice rule and creates a
pair of magnetic charge in the centers of the vertices Ryzhkin (2005);
Castelnovo  _et al._ (2008). These magnetic charges, commonly referred to as
emergent magnetic monopoles, can be separated with a finite energy cost, and
propagated through the lattice. In a classical analogue to Dirac’s theory of
magnetic monopoles Dirac (1931), monopole motion in spin ice leaves a trace
called a Dirac String (DS), which is simply the chain of flipped magnetic
moments connecting the two separated positive and negative magnetic poles.
Theoretical and experimental studies have shown that the energetic ground
state is degenerate, constrained by the ice rule and that the magnetic
monopoles are connected via Dirac Strings at low temperatures Ryzhkin (2005);
Castelnovo  _et al._ (2008); Harris  _et al._ (1997); Bramwell and Harris
(1998); Morris  _et al._ (2009); Fennell  _et al._ (2009); Harris  _et al._
(1997); Matsuhira  _et al._ (2002); Ryzhkin  _et al._ (2013).

In order to study the geometrical frustrations and the magnetic charges on
more controllable platform, 2D artificial spin ices (2DASIs) have been
designed and investigated in detail Wang  _et al._ (2006); Möller and Moessner
(2006); Mól  _et al._ (2009, 2010); Nisoli  _et al._ (2013); Skjærvø  _et al._
(2020). There, lithographically patterned nanomagnets are arranged on
different lattices. The most common ASI lattices are the square Nisoli  _et
al._ (2007); Kapaklis  _et al._ (2012, 2014); Farhan  _et al._ (2013a) and
Kagome ices Farhan  _et al._ (2017); Qi  _et al._ (2008); Arnalds  _et al._
(2012); Hofhuis  _et al._ (2020).

In contrast to DSI, the reduced dimensionality and geometric frustration in
square ice lifts the degeneracy of the ice rule Nisoli  _et al._ (2013);
Skjærvø  _et al._ (2020); Chern  _et al._ (2013); Möller and Moessner (2006);
Mól  _et al._ (2009, 2010); Nisoli (2020), thus limiting monopole mobility.
Various methods have been explored to regain spin ice degeneracy, including
quasi three-dimensional lattices Mól  _et al._ (2010); Möller and Moessner
(2006); Farhan  _et al._ (2019); Chern  _et al._ (2014); Thonig  _et al._
(2014); Perrin  _et al._ (2016); Farhan  _et al._ (2020); Mistonov  _et al._
(2019) and interaction modifiers Östman  _et al._ (2018). After early designs
Chern  _et al._ (2014) and realizations Mistonov  _et al._ (2013) had
envisioned a 3D artificial spin ice, the first three dimensional frustrated
nanowire-lattice May  _et al._ (2019) was manufactured by two-photon
lithography Williams  _et al._ (2018), in which charge propagation was later
demonstrated May  _et al._ . However, in this lattice, the degeneracy of the
ground state is still lifted, as the 3D structure consists of an
interconnected nanowire-lattice. Additionally, the magnetic charges are
connected via DS’s, which store energy due to the presence of domain-walls at
the vertex centers.

In this work, we combine the advantages of both DSI and ASI and present a 3D
artificial spin ice (3DASI lattice), where numerical investigations are
performed. Our lattice is a 3D nano-magnetic network of vertices consisting of
four disconnected 3D magnetic ellipsoids with perfect Ising behavior enabling
tension-free DS’s and unbound magnetic charges by recovering the lost
degeneracy of the ice rule obeying states. In this new lattice, we conserve
the full accessibility of 2DASI and investigate numerically the propagation of
emergent magnetic monopoles. We demonstrate that the difference in energy,
required to create magnetic charges and their transport, is around
$2\text{\,}\mathrm{e}\mathrm{V}$, which enables the controlled propagation of
unbound charges. Considering the emergent magnetic monopoles as binary, mobile
information carriers, the presented lattice demonstrates the steered motion of
charge carriers in a 3D magnetic nano-network at room temperatures. The
controllability of these charges paves a way towards a 3D magnetic network for
data transport and storage.

##  II Results

Modeling. In recent years, the direct-write technique of focused electron beam
induced deposition (FEBID) has reached a high level of maturity for the 3D
nanofabrication. Many complex-shaped nano-architectures have become available,
providing access to experimental investigations of curvature-, geometry- and
topology-induced effects in various disciplines, including magnetism,
superconductivity, photonics and plasmonics Fernández-Pacheco  _et al._
(2020); Keller  _et al._ (2018); Dobrovolskiy  _et al._ (2020); Huth  _et al._
(2018); Fernández-Pacheco  _et al._ (2017).

Inspired by the recent developments in 3D optical lithography and focused
particle 3D nano-printing by FEBID, we present a three dimensional ASI (3DASI)
lattice, where magnetic rotational ellipsoids are arranged along the main axis
of a tetrahedron, resulting in an angle
$\theta=\mathrm{\arccos(-1/3)}\approx$109.5\text{\,}{}^{\circ}$$ between the
elements, reproducing the Ice $\text{I}_{h}$ crystal of the water ice Harris
_et al._ (1997); Ryzhkin (2005); Bramwell and Harris (2020). Figure 1
illustrates the designed 3DASI lattice.

Note that in Fig. 1 we illustrate only the magnetic ellipsoids forming the
lattice. In reality, the ellipsoids can be fabricated by direct-writing, i.e.
FEBID, and interconnected with a magnetic insulator, e.g. platinum- Keller
_et al._ (2018) or niobium-based Porrati  _et al._ (2019) compounds. We
provide the illustration of a rather realistic fabrication model for one
single vertex in the supplemental materials. With FEBID as a suitable
nanofabrication technique of the 3DASI lattice, maximal width of the
ellipsoids $w$ can be reduced down to few tens of nanometers, while the length
$L$ can be chosen up to a few micrometers Huth  _et al._ (2018); Fernández-
Pacheco  _et al._ (2020); Dobrovolskiy  _et al._ (2020); Keller  _et al._
(2018).

Figure 1: 3DASI Lattice.(a) 3D Illustration of the 3DASI lattice, and (b) the
top view. The length of the ellipsoids $L$ and maximal width $W$ as well as
the angle between the ellipsoids $\theta$ are depicted in (b).

We choose rotational magnetic ellipsoids as ASI elements, because the self-
demagnetizing field of the ellipsoids is homogeneous thus keeping the magnetic
elements uniformly magnetized in the direction of the longer axis with no edge
inhomogeneities Osborn (1945); Rougemaille  _et al._ (2013); Koraltan  _et
al._ (2020); Leo  _et al._ (2021). Hence, our model shows nearly perfect Ising
behavior and is suitable to separate and host magnetic charges. The maximal
average deviation from a perfect uniformly magnetization for one ellipsoid in
our model is found to be
$\Delta\phi\approx$4\text{\times}{10}^{-4}\text{\,}\mathrm{d}\mathrm{e}\mathrm{g}$$,
where the interactions with nearest and next-nearest neighbors were
encountered.

In this work, we restrict our model to three layers to conduct numerical
experiments in feasible times. However, we use the top and bottom layer rather
as boundary conditions, while the region of interest is only the middle layer.
In other words, we analyze the 3DASI lattice as a bulk system.

Dirac Strings and magnetic monopoles. Because of the symmetry of the
tetrahedra, the degeneracy of the ice rule is not lifted in 3DASI.
Consequently, the tension of the DS should vanish. To obtain an energy scale,
we approximate the ellipsoids by magnetic dipoles with the dipole moment
$\mu=M_{s}V$, where $M_{s}$ is the saturation magnetization and $V$ the volume
of the nanostructure. In this formulation, the dipolar interaction energy for
each pair of magnetic dipole moments $\mathbf{m}_{i},\mathbf{m}_{j}$ at
positions $\mathbf{r}_{i},\mathbf{r}_{j}$ in the lattice is given by the
dipolar interaction energy

$E_{\mathrm{dip}}=\frac{\mu_{0}}{4\pi\left|\mathbf{r}_{ij}\right|^{3}}\left[\mathbf{m}_{i}\cdot\mathbf{m}_{j}-3\frac{\left(\mathbf{m}_{i}\cdot\mathbf{r}_{ij}\right)\left(\mathbf{m}_{j}\cdot\mathbf{r}_{ij}\right)}{\left|\mathbf{r}_{ij}\right|^{2}}\right]$
(1)

with $\mathbf{r}_{ij}:=\mathbf{r}_{j}-\mathbf{r}_{i}$ and vacuum permeability
$\mu_{0}$. Then from Eq. (1) we obtain the energy scale

$J_{NN}=\frac{3}{2\sqrt{2}}\frac{\mu_{0}}{\pi}\frac{\mu^{2}}{a^{3}}$ (2)

as introduced in Leo  _et al._ (2021) , where $a$ is the lattice constant.

Vertex type |  $E^{2D}_{dip}\mathrm{(J_{NN})}$ |  $E^{3D}_{dip}\mathrm{(J_{NN})}$
---|---|---
Type I |  $2(\sqrt{2}/3-2)$ |  $-5/(2\sqrt{3})$
Type II |  $-2\sqrt{2}/3$
Type III |  $0$ |  $0$
Type IV |  $2(\sqrt{2}/3+2)$ |  $15/(2\sqrt{3})$
Table 1: Dipolar interaction energies in each vertex type in 2D and 3D ASI in
units of $J_{NN}$, which is $J_{NN}\approx$1.72\text{\,}\mathrm{e}\mathrm{V}$$
for the chosen material and geometry parameters.

From it we obtain the energy levels for the different vertex types, shown in
Table 1. Note the degeneracy in the ice rule obeying vertices.

Figure 2: Magnetostatic energies and minimum energy paths for propagation and
separation of charges.(a) Total energies per vertex calculated with a
micromagnetic treatment by taking into account energetic contributions of the
exchange and demagnetization fields, where every possible magnetic
configurations in a 2DsASI (circles) and in our 3DASI (squares) was
considered. (b) Minimum energy paths to separate (magenta) the two magnetic
charges by switching one ellipsoids magnetization in the ground state, and to
propagate it further through the lattice in a constant direction (green). The
difference of saddle points and the corresponding minima yields the separation
barrier $\Delta E_{s}$ and propagation barrier $\Delta E_{p}$, while energy of
the lattice increases by $2E_{q}\approx$2\text{\,}\mathrm{e}\mathrm{V}$$ due
the two additional charge defects. (c) - (g) Schematic illustrations of the
magnetization configurations for the ellipsoids in the Ds with the positive
(dark red) and negative (dark blue) charged magnetic monopoles. Snapshots of
magnetizations and animations available in the supplemental materials.

Additionally, we perform calculations via micromagnetic simulations using
magnum.feAbert  _et al._ (2013) using material properties similar to the
Cobalt-Iron($\mathrm{CoFe}$) alloys used in FEBID. Note that with this
technique, the material parameters are already tunable Dobrovolskiy  _et al._
(2020); Bunyaev  _et al._ for 2D structures, and advances are expected in the
near future. We choose a saturation magnetization
$M_{s}=$800\text{\,}\mathrm{k}\mathrm{A}\mathrm{/}\mathrm{m}$$ and exchange
stiffness constant
$A_{\mathrm{ex}}=$15.5\text{\,}\mathrm{p}\mathrm{J}\mathrm{/}\mathrm{m}$$. The
ellipsoids have the dimensions $L=$100\text{\,}\mathrm{n}\mathrm{m}$$,
$w=$20\text{\,}\mathrm{n}\mathrm{m}$$ and a vertex-to-vertex distance of
$a=$120\text{\,}\mathrm{n}\mathrm{m}$$. It is noteworthy that in our studies
we especially focused on the 3D structures of sizes that have been
demonstrated for platinum based nanofabrications. Magnetic CoFe/CoFeB
structures are larger, but the down-scaling is expected within the next years.

Figure 2a) shows the calculated energies of all the possible vertices in
2DsASI and 3DASI. The latter show clearly the 6-fold degeneracy of the ground
state Type I/II. This degeneracy allows for magnetic charges to be separated,
because a tension-free DS is created, making them truly unbound. In Fig. 2a),
we chose to plot all $2^{4}=16$ possible configurations within a vertex. In
reality, for vertices in the 3DASI lattice there exists only one unique ground
state configuration, the ice-rule configuration, up to symmetry operations,
i.e. rotations and mirroring, always obeying the 2in-2out ice rule.

A more detailed description of the simulations, geometries and additional
snapshots of magnetizations are given in the supplemental material.

According to the original definition of emergent magnetic monopoles, the
energy required to create a monopole-antimonopole pair (charge separation),
should be larger than the energy required to propagate it Dirac (1931);
Vedmedenko (2016); Castelnovo  _et al._ (2008); Mól  _et al._ (2010); Jaubert
and Holdsworth (2009, 2011). We verify these properties by applying a full-
micromagnetic model to calculate the energy barriers to separate and propagate
the magnetic chargesE  _et al._ (2007); Abert (2019); Koraltan  _et al._
(2020). The detailed description of the full micromagnetic model and its
application on 2DASIs is given in Ref. Koraltan  _et al._ (2020).

Figure 2b) illustrates the minimum energy paths (MEPs) for the switching
processes within a DS. Starting from a lattice in the ground state, as
illustrated in Fig. 2c), we separate two magnetic charges by switching the
magnetization of one ellipsoid, and thus creating two Type III defects, which
are depicted in Fig. 2d). The positive charge is then propagated to increase
the length of the DS, shown in Figs. 2e)-g). It can be seen, that the
separation MEP (magenta) yields an separation barrier $\Delta E_{s}$
approximately $2\text{\,}\mathrm{e}\mathrm{V}$ higher than the propagation
barriers $\Delta E_{p}$ (green MEPs), which are nearly constant. The reversal
mechanisms are highly non-coherent and complex, being dictated by domain wall
movements.

By separating the emergent monopoles, the energy of the system increases about
$2E_{q}=$2\text{\,}\mathrm{e}\mathrm{V}$$, which represents the energy
increase due to Type III defects. Since we propagate only one charge, while
the environment does not change, the energy of the system should stay constant
during the propagation. However, we need to consider the possible Coulomb
interactions between magnetic charges Ryzhkin (2005); Castelnovo  _et al._
(2008). In contrast to the classical DSI, this interaction can be neglected in
comparison to the energy barriers in our 3DASI lattice Castelnovo  _et al._
(2008).

Figure 3: Arbitrary field finite temperature micromagnetic analysis. a)
Dependence of the energy barrier on the applied external fields at
$T=$0\text{\,}\mathrm{K}$$ to switch the magnetizations of different
ellipsoids as depicted in (b). c) Upper panel: the switching probability P at
different temperatures for the relevant transitions from (b) calculated with
FTM-analysis, Eq. (3), where a Zeeman field with
$\mu_{0}|\mathbf{H}|=$180\text{\,}\mathrm{m}\mathrm{T}$$, attempt frequency
$f_{0}=$50\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ and pulse time
$t_{\mathrm{final}}=$0.15\text{\,}\mathrm{n}\mathrm{s}$$ were chosen. Lower
panel: mean error rate of successful propagation correlated with the
unintended separation, as given by Eq. (4)

Note that these results are obtained at $T=$0\text{\,}\mathrm{K}$$. In order
to analyze the dependence of the stability and the propagation properties of a
single monopole on both externally applied fields, and temperature, we use the
arbitrary field finite temperature micromagnetic analysis (FTM) Suess  _et
al._ (2011).

Arbitrary field finite temperature micromagnetic analysis. Starting from an
initial magnetization state with one magnetic monopole of charge
$Q=$-2\text{\,}\mathrm{q}_{\mathrm{m}}$$, we calculate the energy barriers to
switch the magnetization of different 3DASI elements under a uniform, external
magnetic field with $\mu_{0}\mathbf{H}=-\mu_{0}|\mathbf{H}|\mathbf{e_{x}}$.
Here, $\mu_{0}|\mathbf{H}|$ denotes the strength of the applied field, and
$\mathbf{e_{x}}$ is the unit vector along $x$. $\mu_{0}|\mathbf{H}|$ is varied
between $0-$200\text{\,}\mathrm{m}\mathrm{T}$$.

The initial magnetization state is the same as in Fig. 4e). We consider
different transitions, where the magnetic charge is propagated along $x$ (blue
arrow) and $y$ (orange arrow), or additional charges are separated in a
neighbored vertex, where the separation in three different directions is
considered, $x$ (green arrows), $y$ and $z$ (red), as illustrated in Fig. 3b).
By doing so, we cover all possible transitions in our lattice, where the
existing charges with $Q=$-2\text{\,}\mathrm{q}_{\mathrm{m}}$$ are either
propagated or new monopole-antimonopole pairs are separated at different
locations.

Figure 3a) shows the calculated energy barriers as a function of the applied
field strength for the transitions depicted in Fig. 3b). Our results indicate
that the propagation barrier along $X$ (blue, circles) is always lower than
any other barrier. At vanishing fields, the barriers for propagation $X$ and
$Y$ are equal, as demonstrated above. With the increasing field and
approaching the coercive field, all barriers are lowering, ultimately,
vanishing once the critical field is reached.

In the FTM analysis, described by Suess and co-workers in Ref. Suess  _et al._
(2011), a field-driven transition occurs with the switching probability

$P=1-\mathrm{exp}\left(-f_{0}t_{\mathrm{final}}e^{\left(\dfrac{-\Delta
E(H)}{k_{B}T}\right)}\right),$ (3)

where $f_{0}$ denotes the attempt frequency, $\Delta E(H)$ the energy barrier
at a given field magnitude, $k_{B}$ the Boltzmann constant,
$t_{\mathrm{final}}$ the time of applied field pulse and $T$ is the
temperature.

To analyze the stability of the propagation at room temperature, we consider
$T=$300\text{\,}\mathrm{K}$$ and calculate the associated probabilities where
we choose $\mu_{0}|\mathbf{H}|=$180\text{\,}\mathrm{m}\mathrm{T}$$,
$f_{0}=$50\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ and
$t_{\mathrm{final}}=$0.15\text{\,}\mathrm{s}$$. $f_{0}$ is chosen according to
values considered in ASI literature Farhan  _et al._ (2013b), and
$t_{\mathrm{final}}$ based on experimental expectations. Note that these two
values appear only as a common factor in Eq. (3) and therefore other values
for $f_{0}$ can always be compensated by an appropriate choice of
$t_{\mathrm{final}}$. The field strength
$\mu_{0}|\mathbf{H}|=$180\text{\,}\mathrm{m}\mathrm{T}$$ is chosen such that
the switching barrier of the propagation is below 300$k_{B}T$.

The upper panel of Fig. 3c) illustrates the relevant probabilities for the
switching possibilities depicted in Fig. 3b). Our results indicate that only
the element of interest will change its magnetization, as its switching
probability increases with the temperature, where
$P_{\mathrm{prop}}($300\text{\,}\mathrm{K}$)>0.9$. All the other probabilities
are nearly zero at temperatures around $T=$300\text{\,}\mathrm{K}$$. We chose
to depict only the switching probabilities for the propagation and separation
along $x$, as the other probabilities are lower than
$1\text{\times}{10}^{-15}$.

However, we still need to define the mean switching error rate

$\mathrm{ERR}(T)=1-P_{\mathrm{prop}}\left(1-P_{\mathrm{sep}}(T)\right)^{N},$
(4)

which describes the probability for an unsuccessful propagation, i.e. falsely
switch of one element leading to a separation (or propagation failure due to
insufficient thermal energy) in a lattice with $N$ elements, where
$P_{\mathrm{prob}}$ describes the propagation probabilities and
$P_{\mathrm{sep}}$ the separation probabilities.

In the lower panel of Fig. 3 c) we see that the error rate significantly
decreases after $T=$290\text{\,}\mathrm{K}$$, as the $X$ propagation
probability increases to nearly 1. For $T>$330\text{\,}\mathrm{K}$$ the
propagation is guaranteed, however the separation probability increases
leading to an increment in the mean error rate. This indicates that the
monopoles can be propagated in a controlled manner, in the desired direction,
and that they can be accessed at temperatures around room temperature,
improving the scalability and thermal stability of pyrochlore DSI.

Figure 4: Field-driven monopole propagation. Temporal evolution of the
magnitude of the applied field (a) and of the averaged magnetization
components (b) within the bulk layer during the x-propagation. The field is
applied along $-x$ in order to maximize the Zeeman energy acting on the
ellipsoid whose magnetization should be switched. The field is turned off
after $t=$4.8\text{\,}\mathrm{n}\mathrm{s}$$ in order to transfer the magnetic
charge to the next vertex controlling the propagation. (c)-(e) Schematic
illustration of the monopole (dark blue) propagation. (f)-(h) Snapshots of the
magnetization states from the output of the conducted micromagnetic
simulations, where f) illustrates the initial and g) the final magnetic
configurations. The magnetization evolution during the simulations is depicted
at different times (white boxes) in h). Color in f)-h) shows the $x$-component
of the magnetization, as given by the colorbar.

Field-driven controlled monopole propagation. Nevertheless, the controlled
propagation can be achieved at any temperature, including
$T=$0\text{\,}\mathrm{K}$$, if one adjusts the strength, direction and
duration of the globally applied magnetic fields. As illustrated in Fig. 3-a),
the energy barrier vanishes for large enough fields. To investigate this in a
rather direct manner, we conduct numerical experiments, where we continue with
our micromagnetic treatment and investigate the propagation of monopoles by
applying pulses of external magnetic fields at $T=$0\text{\,}\mathrm{K}$$.

We start from an initial system where the middle layer is in the ground state.
Reversing the magnetization in one boundary ellipsoid hosts exact one magnetic
charge with $Q=-2q_{m}$, see Figs. 4c) and f). We continue by applying a
magnetic field along $-x$ direction, as in the FTM analysis above. The field
profile can be seen in Fig. 4a), where the magnitude of the applied field
increases linearly over $4.8\text{\,}\mathrm{n}\mathrm{s}$ from
$160-$208\text{\,}\mathrm{m}\mathrm{T}$$, and is then turned off abruptly
after the propagation is successful. Note that we are confining the 3DASI
lattice to avoid finite size effects, and analyze only the middle bulk layer.

Figure 4b) shows the temporal evolution of the averaged magnetization
components within the middle layer. Our results indicate that the magnetic
charge is propagated to the next vertex via domain wall movements as predicted
by our string method calculations and FTM analysis. Magnetization snapshots at
chosen times from Fig. 4h) shows that until the critical field is reached,
small deviations from the ising state are observed for the ellipsoids in the
ground state, explaining the slow decrease in the average $x-$component of the
magnetization from Fig. 4b). Once the applied field is large enough, the
propagation takes place keeping the other vertices in the ground state,
avoiding additional charge separations. By turning off the external field, we
relax all the ellipsoids back to their ising state and control the propagation
by demand. One can see in Fig. 4b) that the average magnetization drops rather
abruptly when the magnetic charge is propagated one vertex further with each
step, demonstrating the free and controlled propagation of an unbound emergent
charge in the presented lattice. In the supplemental materials we also
demonstrate that the magnetic charge can be further propagated to the next
$x$-row by applying the same protocol with a magnetic field along $y$.

##  III Discussion

We investigated micromagnetically a new 3DASI lattice, which combines the
advantages of classical 2DASI and pyrochlore DSI lattices, recovering the lost
frustration and degeneracy of the ground state of the 2DsASI by enabling
tension-free Dirac Strings and thermally stable unbound magnetic monopoles.
Due to large distances compared to the pyrochlore DSI, the Coulomb
interactions between the charges are negligible. String method simulations and
finite temperature micromagnetic analysis show that the lattice allows to
create and propagate magnetic charges, which can exist freely or in pairs
being connected via tension-free Dirac strings at arbitrary temperatures. We
showed by numerical experiments, that a single unbound charge can be
propagated through the lattice, without creating additional magnetic charges.

Low-energy and ultra-fast switching in the 3DASI lattice, as well as the
stability of emergent magnetic monopoles at room temperature and above, paves
a way towards functional 3D magnetic nano-networks for data transport and
storage. The controlled propagation of the magnetic charges is of high
interest regarding the idea of magnetricity Vedmedenko (2016); Arava  _et al._
(2020), which might lead to further new devicesSkjærvø  _et al._ (2020).

Even though we focused only on the center layer and therefore regarded our
model as a bulk lattice, the limitations of the $z$ layers, might show
additional and equally interesting phenomena which will be investigated in
detail in a further work. By shaving off at a given $z$ dimension, the lattice
would have magnetic vertices, containing only three ellipsoids, which would
always host a magnetic charge. The top and bottom layers could act as charge
plasma layers, which might decharge in the middle layer, creating a magnetic
capacitor.

## Methods

String Method. For each switching process, we start from an initial state and
a final state interconnected via 19-20 magnetization states, corresponding to
a coherent rotation of the magnetization of the element of interest. In a
second step, the total energy of each image is driven a small step towards its
energetic minimum. The energy contributions are the demagnetization energy,
the exchange energy and Zeeman energy. The third step consists in cubic
interpolation of the new path, such that the magnetization states are
equidistant according to an appropriate energy norm E  _et al._ (2007); Abert
_et al._ (2013); Koraltan  _et al._ (2020). The last two steps are repeated
until convergence is reached.
Micromagnetic Simulations. By using the finite and boundary element method
based simulation code magnum.fe Abert  _et al._ (2013), we solve the Landau-
Lifshitz-Gilbert (LLG) equation Abert (2019):

$\dfrac{\partial\mathbf{m}}{\partial
t}=-\dfrac{\gamma}{1+\alpha^{2}}\mathbf{m}\times\mathbf{H^{\mathrm{eff}}}-\dfrac{\alpha\gamma}{1+\alpha^{2}}\mathbf{m}\times\left(\mathbf{m}\times\mathbf{H^{\mathrm{eff}}}\right),$
(5)

where $\alpha$ is the Gilbert damping constant, $\gamma$ the reduced
gyromagnetic ratio, $\bm{m}$ the magnetization unit vector, and
$\mathbf{H^{\mathrm{eff}}}$ is the effective field term, which includes the
considered energy contributions from demagnetizing, exchange, external and
anisotropy fields (magnetic confinement). Vertex state energies are computed
by calculating the demagnetizing and exchange energies from the relaxed
structures (high damping $\alpha=1$). For the field-driven dynamic
simulations, we also include global external fields, and solve the LLG for a
given time. Additionally, the magnetization components are averaged within the
middle (bulk) layer for each time step, as depicted in Fig. 4b).

## acknowledgements

We would like to thank Kevin Hofhuis and Johann Fischbacher for the fruitful
discussions. The computational results presented have been achieved, in part,
using the Vienna Scientific Cluster (VSC). S.K., C.A. A.V.C. and D.S.
gratefully acknowledge the Austrian Science Fund (FWF) for support through
Grant No. I 4917 (MagFunc). O.V.D. acknowledges the Austrian Science Fund
(FWF) for support through Grant No. I 4889 (CurviMag).

## Author contributions

S.K. and F.S. conceived the concept. C.A. and F.B. wrote and improved the
micromagnetic codes. S.K performed the micromagnetic simulations. S.K., F.S.,
F.B., C.N., A.V.C., O.V.D., C.A and D.S. further improved and developed the
concept and discussed the results. All authors contributed to the manuscript.

## Competing interests

The authors declare no competing interests.

## Additional information

Supplementary information is available for this paper at .

Reprints and permission information is available at .

Correspondence and requests for materials should be addressed to S.K.

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# Dispersion of Slow Magnetoacoustic Waves in the Active Region Fan Loops
Introduced by Thermal Misbalance

S. Belov  N. Molevich  D. Zavershinskii  Department of Physics, Samara
National Research University, Moscovskoe sh. 34, Samara, 443086, Russia
Department of Theoretical Physics, Lebedev Physical Institute, Novo-Sadovaya
st. 221, Samara, 443011, Russia

###### keywords:

Waves, Magnetohydrodynamic; Coronal Seismology; Oscillations, Solar

##  1 Introduction

Various magnetohydrodynamic waves are omnipresent in the solar corona (see,
e.g., Nakariakov and Kolotkov (2020) for a recent review). In particular, many
observed phenomena are associated with propagating or standing slow waves. For
example, strongly damped Doppler-shift oscillations observed with the Solar
Ultraviolet Measurements of Emitted Radiation (SUMER) spectrometer onboard the
Solar and Heliospheric Observatory (SOHO) mission and commonly known as SUMER
oscillations are attributed to standing slow waves in hot coronal loops (Wang,
2011). Sloshing oscillations, which are observed as localized enhancements of
the EUV emission intensity in hot coronal loops, bouncing back and forth
between the footpoints (Reale, 2016; Pant et al., 2017), are associated with
the evolution of slow modes in the closed magnetic configuration. Propagating
slow waves are usually detected as quasi-periodic EUV and soft X-ray intensity
perturbations. These perturbations propagate upward along field-aligned plasma
non-uniformities and are detected in many coronal regions, for example, in
active-region fan-like loops (Yuan and Nakariakov, 2012), and in plume and
inter-plume regions in polar coronal holes (Gupta et al., 2010).

Propagating slow waves can be used to probe the direction (Marsh, Walsh, and
Plunkett, 2009) and value (Jess et al., 2016) of the coronal magnetic field,
as well as the effective polytropic index of a plasma (Van Doorsselaere et
al., 2011). The main feature of observed propagating slow waves is their quick
decay with the damping lengths $\approx 10\,\mathrm{Mm}$. Using spectroscopic
and stereoscopic-imaging observations of slow-magnetoacoustic-wave propagation
within a coronal loop, Marsh, De Moortel, and Walsh (2011) investigated the
damping of the slow magnetoacoustic mode in three dimensions. According to
their analysis, the damping length is found to be of $20\,\mathrm{Mm}$. The
authors’ subsequent forward modeling suggest that the thermal conduction is
insufficient to explain this value, given the observed parameters of the
coronal-loop temperature, density, and wave period. In addition, the observed
damping lengths are shorter for shorter wave periods (Prasad, Banerjee, and
Doorsselaere, 2014) and show no obvious decrease with temperature (Krishna
Prasad, Jess, and Van Doorsselaere, 2019). Such features indicate that an
alternative or additional dispersive/dissipative mechanism should be taken
into consideration.

It was shown by Zavershinskii et al. (2019) and Kolotkov, Nakariakov, and
Zavershinskii (2019) that the effect called thermal misbalance can
significantly affect dispersion properties of slow waves in the solar corona.
Thermal misbalance is a manifestation of the feedback between compression
waves and heating/cooling processes operating in the plasma. This can lead to
the dependence of the phase speed and the rate of growth/decay on the period,
cause an increase or additional damping of waves and lead to the formation of
quasi-periodic patterns (see, e.g., Zavershinskii et al., 2019). Moreover, the
effect of thermal misbalance can be associated with the observed temperature
dependence of the polytropic index (see, e.g., Krishna Prasad et al., 2018).
The misbalance is also shown to be responsible for the phase shift between
perturbations of various plasma parameters (density, temperature, etc.) and
for the distribution of energy in and between eigenmodes, e.g. slow and
entropy waves (see Zavershinskii et al., 2021). Particularly, the role of
thermal misbalance in the estimation of the phase shifts is found to be
significant for the high-density and low-temperature loops. Prasad,
Srivastava, and Wang (2021) show that the variation of heating mechanism may
lead to around a five-fold increase in the phase difference. The previously
mentioned behaviour of damping length can be explained by the influence of
thermal misbalance on the slow-wave dynamics as well. Duckenfield, Kolotkov,
and Nakariakov (2021) found that the damping times of slow waves due to
thermal misbalance are of the order of $10$ – $100$ minutes, which coincides
with the wave periods and damping times observed.

In this article, we will focus on slow waves propagating in active region fan
loops. It is known from pioneering works by Zaitsev and Stepanov (1975, 1982)
and Edwin and Roberts (1983) that slow waves are a subject of dispersion due
to the finite size of the wave-guiding structure. As a result of such a
geometric dispersion, the phase speed of slow waves lies between the tube- and
sound speeds inside the wave guide. We aim to consider the combined influence
of the finite cross section and thermal misbalance on slow MA waves. It will
be shown that to describe the propagation of slow waves in magnetic flux tubes
of thermally active plasma, it is necessary to use a tube speed obtained with
both geometric dispersion and dispersion due to thermal misbalance taken into
account.

The organization of our article is as follows: in Section s:DispRealation, one
may find the evolutionary equation for compression modes propagating in a
magnetic-flux tube composed of thermally active plasma. Using the equation
obtained, we derive the relations describing dispersion properties of slow MA
waves. In order to apply the obtained theoretical results to the coronal
conditions, we specify the considered heat-loss model in Section ss:HLmodel.
Further, in Section ss:PhaseSpeed, we turn to the dependence of the phase
velocity on the period of the slow wave and compare our results with the
results for an ideal plasma without thermal misbalance. We show that
neglecting the impact from the heating/cooling process may be the source of
significant errors in the seismological estimation of plasma parameters (see
Section ss:Errors). However, these deviations may be used for seismological
determination of unknown coronal-heating mechanism as a solution of the
reverse problem. In addition, in Section s:RoleofAB, we analyze the role of
the cross-section scale and magnetic-field strength on dispersion properties
of slow waves. The summary and discussion of the results presented can be
found in the final section of this article.

##  2 Dispersion Relation

s:DispRealation The general approach of Zaitsev and Stepanov (1975, 1982) and
Edwin and Roberts (1983) gives extensive information about properties of MHD
waves propagating inside a magnetic-flux tube. However, it is not always
convenient for analysis. Therefore, we will use the second-order thin-flux-
tube approximation (Zhugzhda, 1996), which allows us to reduce the two-
dimensional consideration to a one-dimensional one. This assumption implies
that $R\ll\lambda$, where $R$ is the tube radius and $\lambda$ is the
characteristic length for density, velocity and other physical parameters
variations along the tube. Further, we will consider MHD waves propagating in
the magnetic-flux tubes composed of thermally active plasma. In order to take
into account the thermal misbalance effect, the system of equations from
Zhugzhda (1996) have been slightly modified by adding the heating and cooling
rates in the right-hand side of the energy transport Equation energy. Thus,
the basic set of equations is written as follows:

$\ilabel{system_{e}q_{f}irst}\frac{\partial\rho}{\partial t}+2\rho
V+\frac{\partial\rho u}{\partial z}=0,\ilabel{Cont}$ (1) $\displaystyle
p+\frac{b_{z}^{2}}{8\pi}-\frac{A}{2\pi}\left[\rho\left(\frac{\partial
V}{\partial t}+u\frac{\partial V}{\partial z}+V^{2}-\Omega^{2}\right)\right.+$
$\displaystyle+\left.\frac{1}{4\pi}\left(J^{2}-\frac{1}{4}\left(\frac{\partial
b_{z}}{\partial
z}\right)^{2}+\frac{1}{2}b_{z}\frac{\partial^{2}b_{z}}{\partial
z^{2}}\right)\right]=p_{\mathrm{ext}},$ (2) $\rho\frac{\partial u}{\partial
t}+\rho u\frac{\partial u}{\partial z}=-\frac{\partial p}{\partial
z},\ilabel{motionU}$ (3) $\rho\frac{\partial\Omega}{\partial t}+\rho
u\frac{\partial\Omega}{\partial z}+2\rho\Omega
V=\frac{1}{4\pi}\left(b_{z}\frac{\partial J}{\partial z}-J\frac{\partial
b_{z}}{\partial z}\right),\ilabel{motionOmega}$ (4) $\frac{\partial
J}{\partial t}+\frac{\partial uJ}{\partial z}+2VJ-
b_{z}\frac{\partial\Omega}{\partial z}=0\,,\ilabel{inductionJ}$ (5)
$\frac{\partial b_{z}}{\partial t}+u\frac{\partial b_{z}}{\partial
z}+2Vb_{z}=0\,,\ilabel{inductionBz}$ (6) $\frac{\partial A}{\partial
t}+u\frac{\partial A}{\partial z}-2VA=0\,,\ilabel{crossSection}$ (7)
$C_{V}\rho\left(\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial
z}\right)-\frac{\mathrm{k_{B}}T}{m}\left(\frac{\partial\rho}{\partial
t}+u\frac{\partial\rho}{\partial z}\right)=-\rho
Q\left(\rho,T\right),\ilabel{energy}$ (8)
$\ilabel{system_{e}q_{l}ast}p=\frac{\mathrm{k_{B}}}{m}\rho T.\ilabel{State}$
(9)

In Equations Cont – State, $\rho$, $T$, and $p$ are the density, temperature,
and pressure of the plasma, respectively; $u$ and $b_{z}$ are the plasma
velocity and magnetic field along the flux tube, $V$ is the radial derivative
of the radial velocity; $J$ and $\Omega$ are the current density and
vorticity, respectively; $A=\pi R^{2}$ is the flux-tube cross section, $R$ is
the tube radius; $p_{\mathrm{ext}}$ is the total (gas-dynamic and magnetic)
external pressure ; $\mathrm{k_{B}}$ is the Boltzmann constant; $C_{V}$ is the
specific heat capacity at constant volume; $m$ is the mean mass per one
particle. We use
$Q\\!\left(\rho,T\right)=L\\!\left(\rho,T\right)-H\\!\left(\rho,T\right)$ for
the heat-loss function (Parker, 1953; Field, 1965), where
$H\\!\left(\rho,T\right)$ and $L\\!\left(\rho,T\right)$ are heating and
radiation cooling rates, respectively. The heat-loss function equals zero
under steady-state condition:
$Q\\!\left(\rho_{0},T_{0}\right)=L\\!\left(\rho_{0},T_{0}\right)-H\\!\left(\rho_{0},T_{0}\right)=L_{0}-H_{0}=0$.

As the cooling and heating rates depend on density and temperature, the wave-
induced perturbations of these rates cause the misbalance between non-
adiabatic processes, which, in turn, affects waves. The impact of this
feedback varies depending on the wave frequency $\omega$. Using characteristic
timescales $\tau_{V}=C_{V}/Q_{0T}$ and
$\tau_{P}=C_{P}T_{0}/\left(Q_{0T}T_{0}-Q_{0\rho}\rho_{0}\right)$, one may
introduce ranges of weak ($\omega\left|\tau_{V,P}\right|\gg 1$) and strong
($\omega\left|\tau_{V,P}\right|\ll 1$) impacts of the thermal misbalance.
Here, $C_{P}=C_{V}+\mathrm{k_{B}}/m$ is the specific heat capacity at constant
pressure, $Q_{0T}=\left.\partial Q/\partial T\right|_{\rho_{0},T_{0}}$,
$Q_{0\rho}=\left.\partial Q/\partial\rho\right|_{\rho_{0},T_{0}}$. It has been
previously shown that the phase speed $\left[c_{\mathrm{ph}}\right]$ of slow
MA modes becomes frequency-dependent in the uniform plasma with thermal
misbalance and varies between $c_{\mathrm{S}}$ and $c_{\mathrm{S}Q}$ (see
Molevich and Oraevskii, 1988; Zavershinskii et al., 2019). The harmonics
weakly affected by the thermal misbalance ($\omega\left|\tau_{V,P}\right|\gg
1$) propagate with $c_{\mathrm{S}}=\sqrt{C_{P}\mathrm{k_{B}}\\!T_{0}/C_{V}m}$,
which is the standard value for the plasma without the thermal misbalance. In
the opposite case ($\omega\left|\tau_{V,P}\right|\ll 1$), the heating and
cooling processes fully determine the speed of sound
$c_{\mathrm{S}Q}=\sqrt{\tau_{V}C_{P}\mathrm{k_{B}}\\!T_{0}/\tau_{P}C_{V}m}$.
The maximum of dispersion effect (where
$\mathrm{d}c_{\mathrm{ph}}/\mathrm{d}\omega$ reaches maximum) is reached near
the period

$\ilabel{Per_{m}is}P_{\mathrm{m}}=2\pi(\tau_{P}\tau_{V})^{\frac{1}{2}}.$ (10)

Let us analyze waves in a untwisted ($J_{0}=0$) and non-rotating
($\Omega_{0}=0$) flux tube. The linearization procedure applied to Equations
system_eq_first – system_eq_last allows us to obtain evolutionary equations
for plasma eigenmodes. In the considered case, the Alfvén waves are not
coupled with compressional modes and, therefore, can be excluded from our
analysis. Focusing on compressional waves only and excluding all variables
except the density perturbation $\rho_{1}$, gives us the following equation:

$\ilabel{waveEq}\frac{\partial}{\partial
t}\hat{D}\rho_{1}+\frac{1}{\tau_{V}}\hat{D}_{Q}\rho_{1}=0,$ (11)

where

$\displaystyle\hat{D}=\left(\left(c_{\mathrm{A}}^{2}+c_{\mathrm{S}}^{2}\right)\left(\frac{\partial^{2}}{\partial
t^{2}}-c_{\mathrm{T}}^{2}\frac{\partial^{2}}{\partial
z^{2}}\right)+\frac{A_{0}}{4\pi}\left(\frac{\partial^{2}}{\partial
t^{2}}-c_{\mathrm{S}}^{2}\frac{\partial^{2}}{\partial
z^{2}}\right)\left(\frac{\partial^{2}}{\partial
t^{2}}-c_{\mathrm{A}}^{2}\frac{\partial^{2}}{\partial z^{2}}\right)\right),$
$\displaystyle\hat{D}_{Q}=\left(\left(c_{\mathrm{A}}^{2}+c_{\mathrm{S}Q}^{2}\right)\left(\frac{\partial^{2}}{\partial
t^{2}}-c_{\mathrm{T}Q}^{2}\frac{\partial^{2}}{\partial
z^{2}}\right)+\frac{A_{0}}{4\pi}\left(\frac{\partial^{2}}{\partial
t^{2}}-c_{\mathrm{S}Q}^{2}\frac{\partial^{2}}{\partial
z^{2}}\right)\left(\frac{\partial^{2}}{\partial
t^{2}}-c_{\mathrm{A}}^{2}\frac{\partial^{2}}{\partial z^{2}}\right)\right).$

The evolutionary Equation waveEq is the PDE of the fifth order in time $t$ and
fourth order in coordinate $z$. We use notations $A_{0}$ for the unperturbed
cross section and $c_{\mathrm{A}}^{2}=B_{0}^{2}/4\pi\rho_{0}$ for the square
of Alfvén speed, where $B_{0}$ is the unperturbed magnetic field along the
tube. It follows from Equation waveEq that for a loop with some specified
temperature, the wave dynamics will be governed by loop geometry (through
$A_{0}$), magnetic-field strength (through $c_{\mathrm{A}}^{2}$) and
heating/cooling processes (through $\tau_{P},\tau_{V}$). It can be shown that
applying the infinite field approximation ($\beta\rightarrow 0$ and
$A_{0}k^{2}\rightarrow 0$), Equation waveEq is reduced to the equation for
compressional perturbations previously analyzed by Zavershinskii et al.
(2019). In the case of the absence of thermal misbalance
$\tau_{P,V}\rightarrow\infty$, Equation waveEq transforms to the evolutionary
equation for an ideal plasma (Zhugzhda, 1996).

It can be seen that the multiplier $A_{0}$ is present in both of the operators
$\hat{D}$ and $\hat{D}_{Q}$. This fact implies that the geometric dispersion
will act not only at temporal scales weakly affected by the thermal misbalance
($\omega\left|\tau_{V,P}\right|\gg 1$, see $\hat{D}$), but also at frequencies
that are strongly affected by the thermal misbalance
($\omega\left|\tau_{V,P}\right|\ll 1$, see $\hat{D}_{Q}$). As a result, to
describe the properties of waves, one should introduce not only the well-known
tube speed
$c_{\mathrm{T}}=\sqrt{c_{\mathrm{A}}^{2}c_{\mathrm{S}}^{2}/\left(c_{\mathrm{A}}^{2}+c_{\mathrm{S}}^{2}\right)}$,
which is the consequence of single geometric dispersion, but also the modified
tube-speed value following from the combination of geometric and thermal-
misbalance dispersion effects:

$\ilabel{cT0}c_{\mathrm{T}Q}=\sqrt{\frac{c_{\mathrm{A}}^{2}c_{\mathrm{S}Q}^{2}}{c_{\mathrm{A}}^{2}+c_{\mathrm{S}Q}^{2}}}.$
(12)

As we are interested in the dispersion properties of waves, we search for the
solution for Equation waveEq of the form $\mathrm{e}^{-\mathrm{i}\omega
t+\mathrm{i}kz}$. Collecting together terms involving the same powers of $k$,
we can write the dispersion relation as:

$ak^{4}+bk^{2}-c=0,\ilabel{dispRelation}$ (13)

where

$\displaystyle
a=\frac{A_{0}}{4\pi}\left(c_{\mathrm{S}Q}^{2}-\mathrm{i}\omega\tau_{V}c_{\mathrm{S}}^{2}\right)c_{\mathrm{A}}^{2},\qquad\qquad\qquad\qquad\qquad$
$\displaystyle
b=\left(\left(c_{\mathrm{A}}^{2}+c_{\mathrm{S}Q}^{2}\right)\left(c_{\mathrm{T}Q}^{2}-\frac{A_{0}}{4\pi}\omega^{2}\right)-\mathrm{i}\omega\tau_{V}\left(c_{\mathrm{A}}^{2}+c_{\mathrm{S}}^{2}\right)\left(c_{\mathrm{T}}^{2}-\frac{A_{0}}{4\pi}\omega^{2}\right)\right),$
$\displaystyle
c=\left(\left(\left(c_{\mathrm{A}}^{2}+c_{\mathrm{S}Q}^{2}\right)-\frac{A_{0}}{4\pi}\omega^{2}\right)-\mathrm{i}\omega\tau_{V}\left(\left(c_{\mathrm{A}}^{2}+c_{\mathrm{S}}^{2}-\frac{A_{0}}{4\pi}\omega^{2}\right)\right)\right)\omega^{2}.$

The dispersion relation (Equation dispRelation) describes five compressional
modes including two fast and two slow MA waves, and also one entropy mode.
Considering the first order thin flux tube approximation
$A_{0}k^{2}\rightarrow 0$, reduced dispersion relation dispRelation coincides
with those presented in Duckenfield, Kolotkov, and Nakariakov (2021) (assuming
the weak impact of thermal conduction and no dependence of the heating rate on
magnetic field strength). The entropy mode is the non-propagating mode with
the real part of the frequency identically equal zero
($\mathrm{Re}~{}\omega=0$) independently of the heat-loss function. It should
be mentioned that in some specific regimes of thermal misbalance (specific
forms of the heat-loss function), the slow-mode harmonics can demonstrate
mixed properties and become non-propagating as well (for more details see
Zavershinskii et al., 2021). However, these regimes are not a subject of the
current study. Thus, to find roots that can be associated with propagating MA
waves, we look for wavenumbers in the form
$k=k_{\mathrm{Re}}+\mathrm{i}k_{\mathrm{Im}}$ and assume that the
corresponding frequency is some real non-zero quantity
$\mathrm{Re}~{}\omega\neq 0$. Following this assumption, we can find the exact
solution of the dispersion relation (Equation dispRelation) as:

$\displaystyle
k_{\mathrm{Re}}=\sqrt{\frac{\mathrm{Re}\left(r_{\pm}\right)+\mathrm{Abs}\left(r_{\pm}\right)}{2}},\quad
k_{\mathrm{Im}}=\frac{\mathrm{Im}\left(r_{\pm}\right)}{2k_{\mathrm{Re}}},\quad
r_{\pm}=\frac{-b\pm\sqrt{b^{2}+4ac}}{2a}.$ (14)

This solution (Equation solution) describes dispersion properties of slow
(root $r_{+}$) and fast (root $r_{-}$) waves propagating in the plasma flux
tube in the presence of thermal misbalance. Further, we will concentrate on
the solution corresponding to the propagating slow waves only, since
consideration of fast waves requires additional information about the external
medium.

As the final matter of this section, it is worth recalling some applicability
condition of thin-flux-tube approximation to determine the wave periods of
interest. According to Zhugzhda (1996), the second-order thin-flux-tube
approximation is applicable only when wave period $P$ is greater than the
mechanical equilibration time in the radial direction
$\left[\tau_{\mathrm{R}}\right]$. Thus, the subsequent analysis refers to
periods $P>\tau_{\mathrm{R}}$, where
$\tau_{\mathrm{R}}=R/\mathrm{min}\left(c_{\mathrm{A}},c_{\mathrm{S}},c_{\mathrm{S}Q}\right)$.

##  3 Slow Waves in Active Region Fan Loops

s:WavePrppagation

###  3.1 Heat-Loss Model

ss:HLmodel In what follows, we will consider the slow-wave propagation inside
active region fan loops. In order to account for the thermal-misbalance
influence on slow waves, one should determine the heat-loss function
$\left[Q\\!\left(\rho,T\right)\right]$. In this study, the following form of
the loss part due to the optically thin radiation is used

$\ilabel{loss_{f}}L\\!\left(\rho,T\right)=\frac{\rho}{4m^{2}}\,\Lambda\\!\left(T\right)\,,$
(15)

where $m=0.6\times 1.67\times 10^{-24}\,\mathrm{g}$ is the mean particle mass,
$\Lambda\\!\left(T\right)$ is the radiative-loss function determined from the
CHIANTI atomic database v. 10.0 (Dere et al., 1997; Del Zanna et al., 2021).
The heating function $H\\!\left(\rho,T\right)$ can be locally modeled as

$\ilabel{heat_{f}}H\\!\left(\rho,T\right)=h\rho^{a}T^{b}\,,$ (16)

where $h$, $a$, and $b$ are given constants. The first constant $h$ is
determined from the steady-state condition
$Q\\!\left(\rho_{0},T_{0}\right)=0$:
$h=L\\!\left(\rho_{0},T_{0}\right)/\rho_{0}^{a}T_{0}^{b}$. The power-law
indices $a$ and $b$ could be associated with some specific heating mechanism.
More frequently, the following five mechanisms are considered (Rosner, Tucker,
and Vaiana, 1978; Dahlburg and Mariska, 1988; Ibanez S. and Escalona T.,
1993): i) constant heating per unit volume ($a=-1$, $b=0$); ii) constant
heating per unit mass ($a=0$, $b=0$); iii) heating by coronal current
dissipation ($a=0$, $b=1$); iv) heating by Alfvén mode/mode conversion
($a=1/6$, $b=7/6$); v) heating by Alfvén mode/anomalous conduction dumping
($a=-1/2$, $b=-1/2$). However, it was shown by Kolotkov, Duckenfield, and
Nakariakov (2020) that these mechanisms are incompatible with the observations
of widespread coronal thermal stability and the rapid damping of slow
(acoustic) waves. In our research, we follow Kolotkov, Duckenfield, and
Nakariakov (2020) and Duckenfield, Kolotkov, and Nakariakov (2021) and use the
values of $a=1/2$, $b=-7/2$, for which both thermal stability $\tau_{V,P}>0$
and acoustic stability $\left(\tau_{P}-\tau_{V}\right)/\tau_{P}\tau_{V}>0$
conditions are always satisfied in the coronal plasma. We should mention that
the calculations which will be provided further are valid for the chosen
heating mechanism. However, the variation of the heating mechanism will lead
to the change of the calculated values.

###  3.2 Phase Speed of Slow Waves

ss:PhaseSpeed

In this subsection, we address the period dependence of the slow-mode phase
speed in the solar corona. Previously, in Section s:DispRealation, we
introduced the characteristic timescale $\left[P_{\mathrm{m}}\right]$ (see
Equation Per_mis) indicating the maximum of the dispersion effect caused by
the thermal misbalance only. It seems reasonable to introduce the analogous
characteristic timescale for the case of geometric-dispersion effect only. The
dispersion relation for the case of the thermal-misbalance absence can be
obtained from Equation dispRelation rewritten using the assumption
$\tau_{P,V}\rightarrow\infty$. Analyzing the obtained dispersion relation, one
may estimate that in this case $\mathrm{d}c_{\mathrm{ph}}/\mathrm{d}\omega$
reaches a maximum near the period $P_{\mathrm{g}}$:

$\ilabel{Per_{g}eom}P_{\mathrm{g}}=\frac{1}{c_{\mathrm{S}}+c_{\mathrm{T}}}\sqrt{\pi
A_{0}\frac{\left(c_{\mathrm{T}}+3c_{\mathrm{S}}\right)}{\left(c_{\mathrm{S}}+3c_{\mathrm{T}}\right)}\frac{\left(4c_{\mathrm{A}}^{2}-\left(c_{\mathrm{S}}+c_{\mathrm{T}}\right)^{2}\right)}{\left(c_{\mathrm{A}}^{2}+c_{\mathrm{S}}^{2}\right)}},$
(17)

which for low a $\beta$ plasma is reduced to $\approx\pi R/c_{\mathrm{S}}$.

In the general case, the timescale $P_{\mathrm{m}}$ may be either greater or
smaller than $P_{\mathrm{g}}$. The relative position of these timescales is
completely defined by the loop parameters and the acting heating/cooling
mechanisms. As a result, the form of the frequency dependence of the slow-wave
phase velocity can vary significantly depending on certain absolute values of
timescales $P_{\mathrm{m}}$ and $P_{\mathrm{g}}$, as well as their ratio. To
demonstrate the possible thermal misbalance and finite cross-section influence
on the slow-wave phase speed for the heat-loss model discussed above (see
Section ss:HLmodel), let us consider a fan loop with loop length
$L=100\,\mathrm{Mm}$ and radius $R=1\,\mathrm{Mm}$. We also assume that the
loop consists of plasma with temperature $T_{0}=0.8\,\mathrm{MK}$, number
density $n_{0}=10^{10}\,\mathrm{cm^{-3}}$, and magnetic field
$B_{0}=10\,\mathrm{G}$.

Figure 1.:  The phase speeds of slow waves in an active-region coronal loop.
The solid-orange line corresponds to the case of combined impact of thermal
misbalance and finite cross section. The dashed-blue line corresponds to the
case of ideal plasma with the single impact of geometric dispersion.
Calculations are made for the fan loop with $L=100\,\mathrm{Mm}$,
$R=1\,\mathrm{Mm}$, $T_{0}=0.8\,\mathrm{MK}$, $n_{0}=10^{10}\mathrm{cm^{-3}}$,
$B_{0}=10\,\mathrm{G}$, and the discussed heat-loss model (see Section
ss:HLmodel). The mechanical equilibration time in the radial direction
$\tau_{\mathrm{R}}\approx 10$ seconds, for periods less than
$\tau_{\mathrm{R}}$ the second-order thin-flux-tube approach is not
applicable. fig-1

In Figure fig-1, one may find the comparison between phase speeds of slow
waves calculated for the case of geometric dispersion only, and the case of
combined influence of the thermal misbalance and finite cross section. For the
considered heat-loss model (see Section ss:HLmodel), the misbalance timescale
is greater than the geometric one: $P_{\mathrm{m}}>P_{\mathrm{g}}$. This leads
to the fact that for waves with periods comparable to or less than
$P_{\mathrm{g}}$, the main impact on dispersion properties comes mostly from
the finite cross-section influence. As a result, one can see the change in the
phase speed from $c_{\mathrm{S}}$ to $c_{\mathrm{T}}$ at these periods,
similar to the case of an ideal plasma. Differences in the dependence of the
phase speed begin at periods comparable with $P_{\mathrm{m}}$, since now, in
addition to the geometric dispersion, the dispersion due to thermal misbalance
plays a role. As a consequence, the value of the phase speed at large periods
tends to the value $c_{\mathrm{T}Q}$ (see Equation cT0).

The last feature is quite important for the needs of MHD-seismology, since the
long-wavelength limit of the phase speed now is
$\lim\limits_{P\rightarrow\infty}c_{\mathrm{ph}}=c_{\mathrm{T}Q}$, which may
differ significantly from the generally assumed value $c_{\mathrm{T}}$ for an
ideal plasma. This issue is discussed in detail in the following section.

###  3.3 Deviations from Ideal Plasma Approximation

ss:Errors

In particular, the above mentioned difference in the phase-speed limit value
may be the source of errors in the seismological estimation of plasma
parameters. For example, Jess et al. (2016) measured the loop magnetic field
using the slow-wave phase speed $c_{\mathrm{ph}}$ interpreted as
$c_{\mathrm{T}}$. In this case, the magnetic field is easily determined if the
temperature and density of the plasma are known

$B_{\mathrm{0est}}=\sqrt{4\pi\rho_{0}\frac{c_{\mathrm{S}}^{2}c_{\mathrm{ph}}^{2}}{\left(c_{\mathrm{S}}^{2}-c_{\mathrm{ph}}^{2}\right)}}.\ilabel{B0est}$
(18)

On the other hand, if the magnetic field and plasma density are known, the
temperature can be estimated as follows:

$T_{\mathrm{0est}}=\frac{m}{\mathrm{k_{B}}\gamma}\frac{c_{\mathrm{A}}^{2}c_{\mathrm{ph}}^{2}}{\left(c_{\mathrm{A}}^{2}-c_{\mathrm{ph}}^{2}\right)}.\ilabel{T0est}$
(19)

Figure 2.: Relative errors in determination of magnetic field strength (solid-
blue curve) and temperature (dashed-red curve) caused by neglecting the
dispersion of phase velocity associated with thermal misbalance and implying
the use of tube speed $c_{\mathrm{T}}$ as phase speed of slow waves for
considered periods. fig-2

As we show in Figure fig-1, the measured phase speed $c_{\mathrm{ph}}$ for
periods $P>P_{\mathrm{m}}$ may differ significantly from $c_{\mathrm{T}}$ in
the thermally active plasma. Thus, errors in the determination of magnetic
field strength and plasma temperature from Equations B0est and T0est may
arise. Figure fig-2 shows relative errors in determination of magnetic field
strength and temperature by slow waves without accounting for the thermal
misbalance, i.e. when measured phase speed $c_{\mathrm{ph}}$ is interpreted as
$c_{\mathrm{T}}$. It is seen that for a wide range of wave periods, the error
can exceed $30\,\%$.

This means that neglecting thermal misbalance may be the reason for the
divergence between seismological and spectrometric estimations of plasma
parameters. On the contrary, this divergence may be a valuable source for
estimations of the coronal heating function $H\\!\left(\rho,T\right)$ and its
parameters (Equation heat_f).

##  4 Role of Cross Section and Magnetic Field

s:RoleofAB

###  4.1 Influence of Cross Section

Figure 3.: Finite-cross-section influence on the dispersion properties of slow
waves. a) Influence on the slow wave phase speed. b) Influence on the slow-
wave quality factor. c) Influence on the slow-wave damping length. Solid-red,
medium-dashed-green, and dashed-blue lines as well as characteristic
timescales (periods for which geometric dispersion has the most effect)
$P_{g1},P_{g2},$ and $P_{g3}$ correspond to the loop radii
$R=0.05\,L,0.01\,L,$ and $0.001\,L$, respectively. The equilibration times in
the radial direction $\tau_{\mathrm{R}}$ calculated for solid-red, medium-
dashed-green, and dashed-blue lines $\approx 50,10,$ and $1$ seconds,
respectively fig-3

It has been shown in the previous section that the thermal misbalance can
noticeably affect the slow-wave phase speed, and it may be crucial for the
task of plasma-parameter determination by analysis of observed slow waves.
Here, we investigate how the change of tube cross section affects dispersion
properties of slow waves in the plasma with thermal misbalance. For this
purpose, we use the fan-loop parameters from the previous section, except for
the loop radius. We use the following set of the loop radii:
$R=0.05\,L,0.01\,L$, and $0.001\,L$. It should be mentioned that various tube
radii lead to the different equilibration times in the radial direction
$\left[\tau_{\mathrm{R}}\right]$ and, as a result, to different ranges of
applicability for the thin-flux-tube approximation.

Figure fig-3a demonstrates how the change of cross section affects the phase
speed of slow waves. In the range of periods where the impact of thermal
misbalance is weak ($P<P_{\mathrm{m}}$), one can see that the growth of the
loop radius leads to the growth of the timescale $P_{\mathrm{g}}$ associated
with the maximum dispersion effect from the geometric dispersion only.
However, in the range of periods ($P>P_{\mathrm{m}}$) with the strong
influence of thermal misbalance, the change of loop radius does not lead to
any visible effect. Moreover, one can see that for wave periods greater than
300 seconds, which is even less than $P_{\mathrm{m}}$, the phase speed becomes
independent of the cross-section value.

Figures fig-3b and fig-3c show the cross section influence on the slow-wave
quality factor $\left[k_{\mathrm{Im}}\lambda\right]$ and on the damping length
measured in loop lengths ($\left[1/k_{\mathrm{Im}}L\right]$). The heat-loss
model considered in this article (see Section ss:HLmodel) implies the damping
of slow waves. All quality-factor curves have a maxima around
$P_{\mathrm{m}}\approx 640\,$seconds regardless of the cross section. It is
clearly seen that for the considered loop parameters the wave damping comes
primarily from the thermal misbalance, and the influence of geometry on
damping is negligible.

Estimated damping lengths (Figure fig-3c) increase with the wave period. A
similar behavior was obtained by Prasad, Banerjee, and Doorsselaere (2014) for
slow waves observed in on-disk loop structures. The minimal damping length
from Figure fig-3c is about $0.44\,L$, which for $L=100\,\mathrm{Mm}$ gives
the value of $44\,\mathrm{Mm}$. This value is close to the value of
$20\,\mathrm{Mm}$ measured by Marsh, De Moortel, and Walsh (2011) for slow-
mode wave propagation in an active-region loop system. However, this value was
obtained for $P=12\,$minutes. In the case considered here, the heating
mechanism gives the value of $88\,\mathrm{Mm}$. At the same time, the minimum
damping length, theoretically estimated by Marsh, De Moortel, and Walsh
(2011), was determined by the loop-area divergence and had the value of
$105\,\mathrm{Mm}$ (for parameters from Solar TErrestrial RElations
Observatory-A (STEREO-A)) or $65\,\mathrm{Mm}$ (for parameters from STEREO-B).
It means that the thermal misbalance has the potential to explain the observed
value and frequency behavior of slow waves damping lengths. On the other hand,
observations of slow-wave damping may provide useful information for
estimating the parameters of the coronal heating function.

###  4.2 Influence of Magnetic-Field Strength

Here, let us investigate the magnetic-field influence on the dispersion
properties of slow waves. As the considered heating rate (see Equation heat_f)
is assumed to be a function of density and temperature only (the case of
heating rate additionally proportional to the magnetic field has been
considered by Duckenfield, Kolotkov, and Nakariakov (2021)), then the magnetic
field will affect the dispersion properties only under the terms with the
Alfvén speed $c_{\mathrm{A}}$. For analysis, we will use the loop parameters
from Section ss:PhaseSpeed with $T_{0}=0.8\,\mathrm{MK}$, $B_{0}=5,10$, and
$30\,\mathrm{G}$.

Figure fig-4 shows how the phase speed and damping length of slow waves vary
with the magnetic-field strength. In the strongly non-adiabatic limit (i.e.
infinitely long period) the phase speed tends to the modified tube speed
$c_{TQ}$. The solid-red and dashed-green curves are far departed from the
infinite field limit (orange-dashed line) since the change of magnetic field
leads to the change of the Alfvén speed $c_{\mathrm{A}}$ which subsequently
causes the change of the modified tube speeds $c_{TQ}$. With the increase of
magnetic field, $c_{TQ}$ tends to $c_{SQ}$. It is seen that for
$B_{0}=5\,\mathrm{G}(\beta\approx 1)$ the impact of the geometric dispersion
is significant for the period dependencies of phase speed and damping length
and is comparable to the effect of the thermal misbalance. However, with the
growth of the magnetic field, the impact of geometric dispersion rapidly
decays. One may notice that for $B_{0}=30\,\mathrm{G}(\beta\approx 0.03)$, the
dispersion properties of slow waves are practically indistinguishable from
their properties for the case of the uniform plasma and infinite magnetic-
field approximation. Thus, for typical coronal loops with the magnetic field
$B_{0}\gtrsim 30\,\mathrm{G}$, the approximation of infinite magnetic field is
quite applicable for the description of slow waves.

Figure 4.: Magnetic-field influence on the dispersion properties of slow
waves. a) Influence on the slow-wave phase speed. b) Influence on the slow-
wave damping length. The solid-red, dashed-green, and dot-dashed-blue lines
correspond to $5,10,$ and $30\,\mathrm{G}$, respectively. The dotted-orange
lines indicate dispersion curves obtained applying the infinite field
approximation ($\beta\rightarrow 0$ and $A_{0}k^{2}\rightarrow 0$). fig-4

##  5 Discussion and Conclusion

s:Discussion

In the current research, the combined influence from thermal misbalance and
finite cross section of the loop on the dispersion properties of propagating
slow waves has been investigated. The analysis has been conducted with the
help of the second-order thin-flux-tube approximation. A linearising the basic
system of MHD equations allowed us to obtain evolutionary Equation waveEq for
compression modes in the plasma. Analysis of this equation reveals the fact
that the thermal misbalance widens the range of geometric dispersion’s impact.
This, in particular, leads to the requirement for the introduction of the
modified tube speed $c_{\mathrm{T}Q}$, which is a consequence of both thermal-
misbalance and finite-cross-section effects. Investigation of slow-mode
dispersion relation and analysis of the period dependence allows us to
demonstrate the features mentioned (see Figure fig-1).

Comparison of the phase-speed period dependencies calculated for cases of
presence and absence of the thermal misbalance shows that using
$c_{\mathrm{T}}$ as the long-period limit value may be the source of errors in
the seismological estimation of plasma parameters. Looking at Figure fig-2,
one can see that for a wide range of wave periods, the error in estimating the
magnetic field strength can exceed $30\,\%$ for the chosen heating mechanism.

We also analyze the role of the cross section value and the magnetic-field
strength on the dispersion properties of the slow waves for the heating
mechanism considered. It is shown, using the heat-loss model presented in
Section ss:HLmodel, that an increase of the cross section leads to the
increase of the period of the maximum geometric dispersion effect. However, it
does not affect the periods strongly affected by thermal misbalance $P\gtrsim
P_{\mathrm{m}}$ (see Figure fig-3a). Variations of the cross-section do not
significantly affect the period dependence of the quality factor and damping
length (see Figure fig-3b and fig-3c, respectively). The impact of the
magnetic-field strength is shown in Figure fig-4. One may see, that for the
magnetic-field strength implying plasma $\beta\approx 1$, the effect can be
considerable. However, in the case of low-$\beta$ plasma (e.g. in our
calculations $B_{0}>30\,\mathrm{G}$, i.e. $\beta\leq 0.03$), the effect of
geometric dispersion becomes negligible and allows the application of the
infinite-field approximation for describing the slow waves.

In conclusion, we want to emphasize that the constructed theory widens our
knowledge about the properties and evolution of slow modes in the solar
corona. This further understanding allows us the possibility to use slow waves
not only as a tool for the estimation of plasma parameters, but also as a tool
for the estimation of the non-adiabatic processes (e.g. for phenomenological
determination of unknown coronal-heating mechanisms). This is possible due to
the fact that thermal misbalance influence is sensitive to the choice of the
heating mechanism.

#### Acknowledgments

The study was supported in part by the Ministry of Education and Science of
Russia by State assignment to educational and research institutions under
Project No. FSSS-2020-0014 and No. 0023-2019-0003, and by RFBR, project number
20-32-90018. CHIANTI is a collaborative project involving George Mason
University, the University of Michigan (USA), University of Cambridge (UK),
and NASA Goddard Space Flight Center (USA).

#### Disclosure of Potential Conflicts of Interest

The authors declare that they have no conflicts of interest. [Edit as
appropriate.]

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