# Meta Reinforcement Learning for Resource Allocation in Unmanned Aerial
Vehicles with MIMO Visible Light Communication

Hosein Zarini†, Amir Mohammadi⋆, Maryam Farajzadeh Dehkordi⋆⋆,
Mohammad Robat Mili§, Bardia Safaei†, Ali Movaghar† and Merouane Debbah††
†Sharif University of Technology, Tehran, Iran
⋆Universidad Miguel Hernandez de Elche, Alicante, Spain
⋆⋆George Mason University, VA, United States of America
§Pasargad Institute for Advanced Innovative Solutions (PIAIS), Tehran, Iran
††Khalifa University of Science and Technology, Abu Dhabi, United Arab
Emirates

###### Abstract

This paper centers around a multiple-input-multiple-output (MIMO) visible
light communication (VLC) system, where an unmanned aerial vehicle (UAV)
benefits from a light emitting diode (LED) array to serve photo-diode
(PD)-equipped users for illumination and communication simultaneously.
Concerning the battery limitation of the UAV and considerable energy
consumption of the LED array, a hybrid dimming control scheme is devised at
the UAV that effectively controls the number of glared LEDs and thereby
mitigates the overall energy consumption. To assess the performance of this
system, a radio resource allocation problem is accordingly formulated for
jointly optimizing the motion trajectory, transmit beamforming and LED
selection at the UAV, assuming that channel state information (CSI) is
partially available. By reformulating the optimization problem in Markov
decision process (MDP) form, we propose a soft actor-critic (SAC) mechanism
that captures the dynamics of the problem and optimizes its parameters.
Additionally, regarding the frequent mobility of the UAV and thus remarkable
rearrangement of the system, we enhance the trained SAC model by integrating a
meta-learning strategy that enables more adaptation to system variations.
According to simulations, upgrading a single-LED UAV by an array of 10 LEDs,
exhibits 47% and 34% improvements in data rate and energy efficiency, albeit
at the expense of 8% more power consumption.

###### Index Terms:

Multiple-input-multiple-output (MIMO), visible light communication (VLC),
unmanned aerial vehicle (UAV), resource allocation, channel state information
(CSI), soft actor-critic (SAC), meta-learning.

##  I Introduction

Visible light communication (VLC) is anticipated to address the limitations of
existing radio-frequency (RF) communication systems. In VLC transmissions,
signal propagation relies on a light-emitting diode (LED) at transmitter,
while signal decoding is performed using a photo-diode (PD) at receiver [1].
This technology, in fact, offers a unique advantage by combining illumination
and communication simultaneously. Thanks to this multi-functionality, VLC
systems have experienced remarkable advancements and evolved within a few
years. Not long ago, the deployment of multiple LEDs in a LED array introduced
the concept of multiple-input-multiple-output (MIMO) in VLC systems,
accompanied by significant improvements in network data rate and coverage [2,
3, 4]. Later, the application of VLC band introduced in aerial communication
platforms by equipping unmanned aerial vehicles (UAVs) with a single-LED. This
achievement promised even better coverage and flexible maneuverability over
terrestrial VLC systems [5]. To date, a plethora of literature have studied
VLC-enabled UAV-based networks from network coverage [6], data rate [7] and
energy efficiency [8] standpoints. Nonetheless, our investigations reveal that
despite tremendous gain brought by MIMO-VLC [2, 3, 4], this technology has
never been studied for the UAV in the existing art hitherto [5, 6, 7, 8].

This paper is dedicated to filling the aforementioned research gap by
proposing a resource allocation mechanism to evaluate the performance a UAV
equipped with a LED array, as a UAV-MIMO-VLC system. This study faces two
design challenges: $i)$ LED arrays consume more power compared to a single-LED
and their deployment on a UAV with strictly battery limitation raises an
undeniable concern. $ii)$ Considerable dynamism of the UAV-MIMO-VLC system
originated from frequent motion of the UAV poses a challenge for real-time and
on-demand resource allocation. Toward addressing these challenges, this
treatise proposes the following contributions:

  * •

Unlike [5, 6, 7, 8] that investigated a single-LED UAV, our work centers
around a UAV that is equipped with a LED array. In light of the UAV battery
constraint and notable energy demand of the LED array, we adopt a hybrid
dimming control strategy [9] that jointly manages the direct current (DC)-bias
level and the number of glaring LEDs at the LED array to substantially reduce
its energy consumption.

  * •

The performance of this system is evaluated under the practical assumption of
imperfect channel state information (CSI) at the UAV. Specifically, we
formulate a radio resource management problem aimed at minimizing the UAV
power consumption based on its trajectory design, beamforming control and LED
selection, simultaneously. We present a real-time and robust resource
allocation approach using a soft actor-critic (SAC) mechanism, which involves
reformulation of the problem as a Markov decision process (MDP). This approach
effectively captures the problem’s dynamics and optimizes its parameters.

  * •

What’s more, to address the swift reconfiguration of the system, initiated by
frequent mobility of the UAV, we augment the trained SAC model with a meta-
learning strategy. This integration indeed enhances the model adaptability to
system variations and outperforms evaluation metrics compared to the similar
resource allocation method in [5].

  * •

It is numerically revealed that compared to the single-LED UAV [5, 6, 7, 8],
the proposed UAV equipped with a LED array composed of 10 LEDs achieves 47%
and 34% advancements in data rate and energy efficiency, respectively.
Nevertheless, this progress comes with a marginal trade-off, with an
additional 8% more power consumption incurred.

Figure 1: A UAV-MIMO-VLC system.

##  II System Model and Problem Formulation

In this section, we elaborate the UAV-MIMO-VLC system setting and formulate an
optimization problem for evaluating the performance of this system.

###  II-A Signal Model

Downlink transmission of a visible light communication (VLC) system is studied
as depicted in Fig. 1, within which an array of LEDs is patched to a rotary-
wing unmanned aerial vehicle (UAV) for the illumination and communication
purpose at the same time. In specific, the LED array includes a set
$\mathcal{N}=\left\\{{1,2,...,{N}}\right\\}$ of ${N}$ LEDs deployed at the UAV
to serve a set $\mathcal{K}=\left\\{{1,2,...,{K}}\right\\}$ of $K$ randomly
positioned users, each equipped with a single photo-detector (PD). The LEDs
are supposed to be independently modulated via separate drivers, yet all are
deployed at the UAV as a central controller that collects channel feedback and
performs resource management. Let
${\textbf{s}}=\left[s_{1},s_{2},...,s_{K}\right]\in\mathbb{R}^{K\times 1}$
denote the data symbol vector, intended for all user, such that
$s_{k}\in\left[-1,1\right]$ specifies the data symbol for the user $k$ and
$\mathbb{E}\\{|{{s}}_{k}|^{2}\\}$ = 1 $\forall k\in\mathcal{K}$. Also, let us
define a transmit precoding matrix
$\textbf{W}=\left[\textbf{w}_{1},\textbf{w}_{2},...,\textbf{w}_{K}\right]\in\mathbb{R}^{N\times
K}$ with
$\textbf{w}_{k}=\left[w_{1,k},w_{2,k},...,w_{N,k}\right]\in\mathbb{R}^{N\times
1}$ being the beamforming vector for the user $k$. Designing an efficient
precoding scheme significantly helps controlling the inter-user interference.
Without loss of generality, we assume a linear precoding for data symbols[5].
In optical communications, generated signals have to be real-valued and non-
negative. This requirement is covered by direct current (DC)-biased
modulation, through which the brightness of LEDs and average optical power are
also determined. Taking this fact into consideration, the transmit signal
vector of all LEDs can expressed as
$\mathbb{x}=\sum_{k=1}^{K}{\mathbb{w}}_{k}{\textbf{s}}_{k}+\textbf{I}^{\text{DC}},$
where the DC bias
${\textbf{I}^{\text{DC}}}=\left[I^{\text{DC}}_{1},I^{\text{DC}}_{2},...,I^{\text{DC}}_{N}\right]\in\mathbb{R}^{N\times
1}$ has been incorporated so as to ensure that the amplitude of the transmit
signal vector resides within the non-negative range of LEDs. Besides, for the
sake of uniformity of illumination in indoor environments, the DC bias is
assumed to be the same for all the LEDs[9]. On this basis, the LED indices can
be simply removed, which results in $I^{\text{DC}}_{n}=I^{\text{DC}}~{}\forall
n\in\mathcal{N}$ and ${\textbf{I}^{\text{DC}}}=I^{\text{DC}}$. Accordingly,
the transmit signal of a typical LED $n$, i.e., $x_{n}$ satisfies
$-\sum_{k=1}^{K}|w_{n,k}|+I^{\textrm{DC}}\leq
x_{n}\leq\sum_{k=1}^{K}|w_{n,k}|+I^{\textrm{DC}}~{}\forall n\in\mathcal{N}.$
All LEDs, however, work within their dynamic ranges to avoid signal clipping.
This consideration poses a constraint for the transmit beamforming vector as
$\sum\limits_{k=1}^{K}{\left|{{\mathbb{w}_{k}}}\right|}\leq\min({I^{\textrm{DC}}}-{I_{l}},{I_{h}}-{I^{\textrm{DC}}})~{}\forall
n\in\mathcal{N}$, with ${I_{l}}$ and ${I_{h}}$ being the lower and upper bound
of the LED driver current, respectively.

###  II-B Channel Model

For a typical VLC receiver, a line-of-sight (LoS) link constitutes about 95%
of the total received power, such that at least 7 dB difference is roughly
observed between the received power for the LoS link and that for the most
dominant non-line-of-sight (NLoS) one[5]. In light of this fact, by neglecting
the NLoS links, we only take the LoS links into account in our work. Suppose
that the UAV and the typical user $k$ are located at the coordinates of
$\mathbb{q}_{\textrm{U}}=[{x}_{\textrm{U}},{y}_{\textrm{U}},{z}_{\textrm{U}}]^{T}$
and $\mathbb{q}_{k}=[{x}_{k},{y}_{k},{z}_{k}]^{T}$, respectively. In
comparison with the distance between the UAV and ground users, the distance
between the LEDs deployed at the UAV is trivial. Hence, a unified (and not per
LED) coordinate is considered for the UAV. So, the direct distance between the
user $k$ and the UAV can be calculated as:
$d_{\textrm{U},k}=\sqrt{(x_{\textrm{U}}-{x}_{k})^{2}+(y_{\textrm{U}}-{y}_{k})^{2}+(z_{\textrm{U}}-{z}_{k})^{2}}$.
However, the channel condition my significantly differ between two adjacent
LEDs, regarding their transmission semi-angles. We denote the optical channel
gain from the LED $n$ to the user $k$ by $h_{n,k}\in\mathbb{R}$, which is
modelled as: $h_{n,k}\\!=\frac{(m+1)A_{k}}{2\pi
d_{\textrm{U},k}^{2}}G^{\textrm{VLC}}(\psi_{n,k})\\!\cos^{m}(\phi_{n,k})\\!\cos(\psi_{n,k})$
for $0\\!\leq\\!\psi_{n,k}\\!\leq\\!\Psi_{c}$, and $h_{n,k}=0,$ otherwise. In
such a channel modelling, the detection area of the PD at the user $k$ is
defined by $A_{k}$. Besides, $m=-\frac{\ln 2}{\ln(\cos\Phi_{1/2})}$ stands for
the order of Lambertian emission with $\Phi_{1/2}$ being the half-power semi-
angle of the LED. The angles of incidence and irradiance between the LED $n$
and the user $k$, are respectively represented by $\psi_{n,k}$ and
$\phi_{n,k}$, such that
$\cos(\psi_{n,k})=\cos(\phi_{n,k})=d_{\textrm{U},k}/(z_{\textrm{U}}-{z}_{k})$.
Moreover, $\Psi_{k}$ specifies the field of vision (FOV) semi-angle at the
user $k$ and $G^{\textrm{VLC}}(\psi_{n,k})$ indicates the gain of the optical
concentrator, with $q\geq 0$ introducing the internal refractive index.
However, the availability of perfect CSI is an ideal assumption, which leads
to remarkable performance degradation. Especially in VLC systems, where
optical lightweight signals are heavily prone to environmental obstacles or
reflections, the practical consideration of imperfect CSI at the transmitter
(the UAV in our scenario) would provide the system with a realistic
performance analysis [10]. We model the imperfect channel between the LED $n$
and user $k$ as $\hat{h}_{n,k}=h_{n,k}+\epsilon_{n,k},$ wherein
$\epsilon_{n,k}$, represents the estimation error within a bounded region.
This error is defined as $|\epsilon_{n,k}|\leq\delta$, where $\delta$
corresponds to the channel uncertainty radius and is assumed be a small
constant.

###  II-C Hybrid Dimming

As discussed earlier, the concept of hybrid dimming is a multi-domain dimming
approach that exploits analog domain (AD) and spatial domain (SD) at the same
time[9]. Thanks to invoking SD, only a subset of LEDs are glared, while the
remaining become inactive. Hence, compared to conventional systems with
digital dimming (DD), the power consumption of a hybrid dimming-enabled system
is significantly lower. For equipping UAVs to optical multiple-input-multiple-
output (MIMO) technology with a LED array therefore, a hybrid dimming control
scheme is indispensable regarding the strict energy limitation of the UAV. To
this purpose, let us introduce a binary LED selection matrix
$\mathbf{A}=\text{diag}(\mathbf{a})\in\\{0,1\\}^{N\times N},$ with
$\mathbf{a}=[a_{1},\dots,a_{N}]^{T}\in\\{0,1\\}^{N\times 1}$, such that
$a_{n}$=1 specifies the glared state of the LED $n$, whereas $a_{n}$=0
delineates that it is inactive. Accordingly, the number of active LEDs can be
calculated as ${N_{\textrm{a}}}=\sum\limits_{n=1}^{{N}}{{a}_{n}}$. By adopting
a hybrid dimming control mechanism, the signal benefits from both analog and
spatial domains simultaneously. So, the uniform DC-bias level (corresponded to
AD) and the number of glared LEDs (corresponded to SD) need to be jointly
adjusted. To do so, we firstly define a target dimming level denoted by $\eta$
with a predetermined value. Given $\eta$ then, the number of glared (actual
working) LEDs for SD can be simply rounded-off as $N_{a}=\big{[}\eta
N\big{]}.$ On this basis, the uniform DC-bias level $I^{\textrm{DC}}$ for AD
will be acquired as[9]: $I^{\textrm{DC}}=\dfrac{\eta
N(I_{0}-I_{l})}{N_{a}}+I_{l}$, in which $I_{0}=\dfrac{I_{l}+I_{h}}{2}$ defines
the original DC-bias for AD, when all $N$ LEDs are glared and $\eta=100\%$
dimming level is set. By determining ($N_{\textrm{a}}$) thus far, we are aware
of how many LEDs are glared in fact. However, it is required to exactly
specify which LEDs at the LED array have to be active. This will be clarified
by optimizing the binary LED selection matrix A in a network-wide optimization
problem, which will be mathematically formulated in upcoming subsections.

###  II-D Trajectory Control and Power Consumption for UAV

For a durable and efficient aerial communication, it is required to
meticulously trace the flight trajectory and power consumption of UAV.

Trajectory Control: Concerning the frequent mobility of the UAV and for the
sake of analysis, we discretize its overall flight time duration $T$ to a set
$\mathcal{L}=\left\\{{0,1,2,...,L}\right\\}$ with $L$ time slots, each
occupying a duration of ${\tau}$, such that $T=L\tau$. The duration of time
slots is adopted to be sufficiently small, within which the configuration of
the network is supposed to remain static and the optical channels follow a
quasi-static model. Within a typical time slot $l$, the UAV is assumed to fly
with the velocity of
${\mathbb{v}_{\textrm{U}}}\left(l\right)={\left[{v_{\textrm{U}}^{x}\left(l\right),v_{\textrm{U}}^{y}\left(l\right),v_{\textrm{U}}^{z}\left(l\right)}\right]^{T}}\in\mathbb{R}^{3\times
1}$, such that its initial and final positions are the same, i.e., it returns
to the beginning position at the end of the last time slot $L$ for recharging.
The flight restrictions of the UAV pose the some constraints to the system.
First,
${\mathbb{q}_{\textrm{U}}}\left({l+1}\right)={\mathbb{q}_{\textrm{U}}}\left(l\right)+{\mathbb{v}_{\textrm{U}}}\left(l\right){\tau}$
determines its next location with respect to (w.r.t.) the movement velocity;
Through
${\mathbb{q}_{\textrm{U}}}\left[0\right]={\mathbb{q}_{\textrm{U}}}\left[L\right]={\mathbb{q}_{\textrm{U}}^{\textrm{I}}},$
we guarantee the comeback of the UAV to its initial location
${\mathbb{q}_{\textrm{U}}^{\textrm{I}}}$ at the end of the last time slot $L$;
We restrict the UAV flight to the minimum and maximum boundaries
${\mathbb{q}_{\min}}$ and ${\mathbb{q}_{\max}}$ as
${\mathbb{q}_{\min}}<{\mathbb{q}_{\textrm{U}}\left(l\right)}<{\mathbb{q}_{\max}}$;
Finally,
$\left\|{{\mathbb{v}_{\textrm{U}}}\left({l+1}\right)-{\mathbb{v}_{\textrm{U}}}\left(l\right)}\right\|\leq
a_{\text{max}}{\tau}$ and
$\left\|{{\mathbb{v}_{\textrm{U}}}\left(l\right)}\right\|\leq V_{\text{max}}$
control the flight acceleration and velocity of the UAV by introducing the
maximum flight acceleration $a_{\text{max}}$ and maximum flight velocity
$V_{\text{max}}$, respectively.

Power Consumption: The consumed power related to hovering of the UAV,
categorized to blade power consumption $P_{b}$ and induced power consumption
$P_{i}$, can be modelled as [11]: ${P_{\textrm{Hov}}}=P_{b}+P_{i}$ where
$P_{b}\stackrel{{\scriptstyle\triangle}}{{=}}\frac{\rho}{8}\zeta\delta{A_{\textrm{U}}}\Omega_{\textrm{U}}^{3}R_{\textrm{U}}^{3}$
and
${P_{i}}\stackrel{{\scriptstyle\triangle}}{{=}}\left({1+\iota}\right)\frac{{W_{\textrm{U}}^{3/2}}}{{\sqrt{2\zeta{A_{\textrm{U}}}}}}$.
Specifically, $P_{\text{Hov}}$ includes the profile drag coefficient $\rho$,
the air density $\zeta$, the rotor solidity $\delta$, the blade angular
velocity $\Omega_{\textrm{U}}$ in radian per second, the rotor disk area
$A_{\textrm{U}}$ in square meters ($\textrm{m}^{2}$), the rotor radius
$R_{\textrm{U}}$ in meter, the incremental correction factor $\iota$ and the
UAV weight $W_{\textrm{U}}$ in Newton. Accordingly, the propulsion power
consumption of the UAV, encompassing the blade profile, the induced, and the
parasite power is given by: ${{\mathop{\rm
P}\nolimits}_{\text{Prop}}}=\underbrace{\left({1+\frac{{3\lVert\textbf{v}_{\textrm{U}}\rVert^{2}}}{\Omega_{\textrm{U}}^{2}R_{\textrm{U}}^{2}}}\right){P_{b}}}_{\text{blade
profile}}+\underbrace{\left({\sqrt{\sqrt{1+\frac{{\lVert\textbf{v}_{\textrm{U}}\rVert^{4}}}{{4v_{\textrm{UI}}^{4}}}}-\frac{{\lVert\textbf{v}_{\textrm{U}}\rVert^{2}}}{{2v_{\textrm{UI}}^{2}}}}}\right){P_{i}}}_{\text{induced}}\\\
+\underbrace{\frac{1}{2}{d_{\text{U}}}{\zeta\delta}{A_{\text{U}}}\lVert\textbf{v}_{\textrm{U}}\rVert^{3}}_{\text{parasite}},$
in which $v_{\textrm{UI}}$ specifies the mean rotor induced velocity in hover,
while $d_{\textrm{U}}$ denotes the fuselage drag ratio. Finally, the total
consumed power of the system can be calculated as:
$P^{\text{Tot}}=\ddot{\zeta}\sum_{n=1}^{N}\sum_{k=1}^{K}\mathbf{a}{w}_{n,k}+\varphi{N_{a}}{I^{\textrm{DC}}}+P^{\text{Cir}}+P^{\text{Prop}},$
with $\ddot{\zeta}$ being the amplifier efficiency factor, $\varphi$
determining the conversion factor, as well as $P^{\text{Cir}}$ designating the
consumed power of switch and control circuit modules at the UAV.

###  II-E Transmission Protocol

We adopt the power-domain non-orthogonal multiple access (NOMA) scheme for
enabling massive access in the system. Suppose
${{\left|\mathbb{h}^{H}_{1}{\mathbb{w}}_{1}\right|}^{2}}\leq{{\left|\mathbb{h}^{H}_{2}{\mathbb{w}}_{2}\right|}^{2}}\leq...\leq{{\left|\mathbb{h}^{H}_{K}{\mathbb{w}}_{K}\right|}^{2}}$
as the descending order of the channel coefficients. Then, the instantaneous
received data rate of the user $k$ can be expressed as:

$\displaystyle
R_{k}(\mathbf{W},\mathbf{A},\mathbf{Q})=\log_{2}\bigg{(}1+\dfrac{|\mathbb{h}^{H}_{k}\mathbb{A}{\mathbb{w}}_{k}|^{2}}{\sum_{i=k+1}^{K}|\mathbb{h}^{H}_{k}\mathbb{A}{\mathbb{w}}_{i}|^{2}+\sigma^{2}_{k}}\bigg{)},$
(1)

where
${\mathbb{h}_{k}}={\left[{{h_{k,1}},{h_{k,2}},...,{h_{k,{N}}}}\right]^{T}}$
denotes the channel vector of the user $k$.

###  II-F Problem Formulation

We aim to jointly optimize the network resources
$\big{\\{}\mathbf{W},\mathbf{Q}=\\{{\mathbb{q}_{\textrm{U}}},{{\mathbb{v}_{\textrm{U}}}}\\},\mathbb{A}\big{\\}}$,
such that the power consumption of the UAV is minimized and the UAV-MIMO-VLC
system requirements are met. Mathematically, this problem is formulated as
follows:

$\displaystyle\mathcal{P}_{1}:$
$\displaystyle\min_{\mathbf{W},\mathbf{A},\mathbb{Q}}{P^{\text{Tot}}}$
$\displaystyle\text{s.t.}~{}~{}~{}~{}\\!\text{C}_{1}:{R_{k}}(\mathbf{w},\mathbf{A},\mathbf{Q})\geq{R_{\text{min}}},~{}\forall
k\in\mathcal{K},$ (2a)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}\text{C}_{2}:P^{\text{Tot}}\leq
P_{\textrm{max}},$ (2b)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}\text{C}_{3}:\sum\limits_{k=1}^{K}{\left|{{\mathbb{{w}}_{k}}}\right|}\leq\min({I^{\textrm{DC}}}-{I_{l}},{I_{h}}-{I^{\textrm{DC}}}),\forall
n\in\mathcal{N},$ (2c)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}\text{C}_{4}:{\mathbb{q}_{\textrm{U}}}\left({l+1}\right)={\mathbb{q}_{\textrm{U}}}\left(l\right)+{\mathbb{v}_{\textrm{U}}}\left(l\right){\tau},\quad\forall
l\in{\mathcal{L}},$ (2d)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}\text{C}_{5}:{\mathbb{q}_{\min}}<{\mathbb{q}_{\textrm{U}}\left(l\right)}<{\mathbb{q}_{\max}},\quad\quad\forall
l\in{\mathcal{L}},$ (2e)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}\text{C}_{6}:\left\|{{\mathbb{v}_{\textrm{U}}}\left({l+1}\right)-{\mathbb{v}_{\textrm{U}}}\left(l\right)}\right\|\leq
a_{\text{max}}{\tau},\quad\forall l\in{\mathcal{L}},$ (2f)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}\text{C}_{7}:\left\|{{\mathbb{v}_{\textrm{U}}}\left(l\right)}\right\|\leq
V_{\text{max}},\quad\quad\forall l\in{\mathcal{L}},$ (2g)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}\text{C}_{8}:\eta=\dfrac{{N_{\textrm{a}}}(I^{\textrm{DC}}-I_{l})}{N(I_{0}-I_{l})}\times
100\%,$ (2h)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}\text{C}_{9}:a_{n}\in{\\{0,1\\}},~{}~{}\forall
n\in\mathcal{N},$ (2i)

where the quality-of-service (QoS) requirement of all users
($R_{\textrm{min}}$) is ensured in $\text{C}_{\text{1}}$; Constraint
$\text{C}_{\text{2}}$ guarantees the transmit power budget of the UAV denoteod
by ($P^{\textrm{max}}$); In $\text{C}_{\text{3}}$, the dynamic range of all
LEDs in the LED array is bounded; The flight restrictions of the UAV are
described in $\text{C}_{\text{4}}$-$\text{C}_{\text{7}}$ as discussed earlier;
The target dimming level $\eta$ is assured to be served in
$\text{C}_{\text{8}}$ and $\text{C}_{\text{9}}$ finally defines the binary
domain of the LED selection variable. By virtue of coupling variables drawn
from both concrete and discrete domains and also incorporating non-convex
constraint $C_{\text{1}}$, this problem is non-convex. This observation
renders $\mathcal{P}_{1}$ in a mixed-integer and nonlinear programming (MINLP)
category, thereby classifying it as non-deterministic polynomial-time hardness
(NP-hard). Unlike classical convex optimization-based solutions that offer
intricate mathematical transformation to solve non-convex optimization
problems, we propose a real-time and adaptive solution strategy to address
$\mathcal{P}_{1}$ in the following section.

##  III Detailed Description of the Proposed Solution Strategy

In this section, we reformulate (2) in MDP form and propose a Meta-SAC
algorithm that integrates meta-learning with SAC. Compared to existing
learning-driven resource allocation schemes mostly based on deep reinforcement
learning (DRL) [5], the integration of meta-learning in our proposed method
allows more adaptability into the system variations, thereby yielding a more
real-time and flexible solution framework.

###  III-A Reformulation of $\mathcal{P}_{1}$ in MDP form

We regard the UAV as the central controller, that roles as a DRL agent,
interacting with the considered MIMO-VLC system as the DRL environment. The
agent selects an action
$\textbf{a}(l)=\\{\textbf{W}(l),\textbf{A}(l),\textbf{Q}(l)\\}$ in time step
$l$ based on the current state $\textbf{s}(l)=\\{\textbf{h}_{k}\\},~{}\forall
k\in\mathcal{K},$ of the environment. We view
$P_{r}(\textbf{s}(l+1)|\textbf{s}(l),\textbf{a}(l))$ as the transition
probability, representing the likelihood of transitioning to a new state
$\textbf{s}(l+1)$ from the current state of the environment $\textbf{s}(l)$,
when action $\textbf{a}(l)$ is executed. Consequently, the environment
transits to the new state $\textbf{s}(l+1)$, and a reward function
$Re(\textbf{s}(l),\textbf{a}(l))$ evaluates the efficacy of the chosen action.
Formally, the reward function is defined as
$Re(\textbf{s}(l),\textbf{a}(l))=-P^{\text{Tot}}$, when
$\text{C}_{\text{1}}$-$\text{C}_{\text{9}}$ are met and
$Re(\textbf{s}(l),\textbf{a}(l))=0$, otherwise.

###  III-B Proposed Meta-SAC Algorithm

As a robust DRL algorithm tailored for ever-varying scenarios, SAC [12] is
equipped with an agent whose objective is to determine the optimal policy
$\pi^{*}$ that maximizes a trade-off between the expected cumulative reward
over time and a certain level of uncertainty or exploration in its actions,
known as entropy. This trade-off measures the policy’s randomness and can be
formulated as:
$\pi^{*}=\arg\max_{\pi}\mathbb{E}_{(\mathbf{s}(l),\mathbf{a}(l))\sim\mathcal{P}}\bigg{[}\sum_{l=0}^{\infty}\gamma^{l}Re(\mathbf{s}(l),\mathbf{a}(l))-\lambda\sum_{l=1}^{\infty}\gamma^{l}\log\left(\pi(\cdot|\mathbf{s}(l))\right)|\mathbf{s}(l)=\mathbf{s}(0),\mathbf{a}(l)=\mathbf{a}(0)\bigg{]},$
within which $\gamma\in(0,1]$ designates the discount factor and $\mathcal{P}$
stands for the transition probabilities. While effective in dynamic settings,
where the training and testing environments of the DRL agent match, the
traditional SAC, however, struggles to swiftly adjust to extremely updating
conditions such as our case with a flying UAV. To tackle this situation, we
introduce the Meta-SAC approach, by integrating the model-agnostic meta-
learning (MAML) [13] with SAC. By doing so, Meta-SAC showcases superior
generalization capabilities, enabling rapid adaptation to unseen upcoming
environments [14]. Whereas the traditional SAC employs one actor and two
critic networks, characterized by parameters $\phi$, $\psi_{1}$, and
$\psi_{2}$, respectively, meta-learning integrates one target actor
($\phi_{e}$) and two target critic networks ($\psi_{e1}$ and $\psi_{e2}$) to
facilitate convergence.

The system model outlined in Section II incorporates random placements of the
UAV and users in each instantiation. Consequently, the optimization problem
$\mathcal{P}_{1}$ can be considered as a meta task scenario. Here, each task
is defined by assuming distinct initial random locations for the UAV and
users. Following the MAML approach [13], We define a meta task set
$\mathcal{T}_{t}=\\{t=1,2,\ldots,T\\}$, in which each task
$t\in\mathcal{T}_{t}$ can be represented by the tuple
$(\textbf{s}_{t}(l),\textbf{a}_{t}(l),\textbf{s}_{t}(l+1),r_{t}(l)),~{}\forall
l\in\mathcal{L}$. Moreover, each task $t$ possesses its specific replay buffer
labeled as $D_{t}$, a support set $D_{t}^{tr}$ as well as a query set
$D_{t}^{val}$. The proposed Meta-SAC approach consists of two main stages,
namely Meta-training and Meta-adaptation.

Meta-Training Phase: In the initial stage of meta-learning, both the actor and
critic networks undergo training using a two-tier approach that encompasses
individual-level and global-level updates [14]. At the former level, for each
task $t$, the environment resets to initialize the state $\textbf{s}_{t}(0)$.
Next, by adopting and applying the action $\textbf{a}_{t}(l)$ in each time
step $l$, the agent updates the environment to the new state
$\textbf{s}_{t}(l+1)$ and the reward function $Re_{t}(l)$ evaluates the
efficacy of the chosen action. Relying on adaptive moment estimation (ADAM)
algorithm, subsequently, the actor and critic networks’ parameters for each
task $t$, denoted by $\hat{\phi}_{t}$, $\hat{\psi}_{1,t}$, and
$\hat{\psi}_{2,t}$ respectively, are updated as:
$\hat{\phi}_{t}=\arg\min_{\phi}\mathcal{L}_{t}^{A}(\phi,D_{t}^{tr})$,
$\hat{\psi}_{1,t}=\arg\min_{\psi_{1}}\mathcal{L}_{t}^{C1}(\psi_{1},D_{t}^{tr}),$
and
$\hat{\psi}_{2,t}=\arg\min_{\psi_{2}}\mathcal{L}_{t}^{C2}(\psi_{2},D_{t}^{tr})$,
with $\mathcal{L}_{t}^{A}(\phi,D_{t}^{tr})$,
$\mathcal{L}_{t}^{C1}(\psi_{1},D_{t}^{tr})$ and
$\mathcal{L}_{t}^{C2}(\psi_{2},D_{t}^{tr})$ quantifying the loss functions for
the actor and critic networks, respectively. After applying individual-level
updates of all tasks, the next step involves updating the global network
parameters at the global-level. This process entails sampling a query set
$D_{t}^{val}$ randomly from the dataset $D_{t}$. Accordingly, the global
network parameters are updated as:
$\phi=\arg\min_{\phi}\sum_{t\in\mathcal{T}_{t}}\mathcal{L}_{t}^{A}(\hat{\phi}_{t},D_{t}^{val})$,
$\psi_{1}=\arg\min_{\psi_{1}}\sum_{t\in\mathcal{T}_{t}}\mathcal{L}_{t}^{C1}(\hat{\psi}_{1,t},D_{t}^{val})$
and
$\psi_{2}=\arg\min_{\psi_{2}}\sum_{t\in\mathcal{T}_{t}}\mathcal{L}_{t}^{C2}(\hat{\psi}_{2,t},D_{t}^{val})$.

Meta-Adaptation Phase: During this stage, we insightfully initialize the
parameters of the actor and critic networks relying on the trained parameters
from the prior stage. In particular, we initialize $\Phi$, $\Psi_{1}$, and
$\Psi_{2}$, the parameters of the actor and critic networks, as
$\Phi_{0}=\phi$, $\Psi_{1,0}=\psi_{1}$, and $\Psi_{2,0}=\psi_{2}$,
respectively. This stage already exploits another replay buffer $D_{ada}$, a
batch of data $\mathbb{D}_{ada}$ are randomly drawn from. Accordingly, the
update rules for parameters of the actor and critic networks are respectively
expressed as: $\Phi=\Phi-\beta_{A}\nabla_{\Phi}\mathcal{L}_{A}(\Phi,D_{ada}),$
$\Psi_{1}=\Psi_{1}-\beta_{C1}\nabla_{\Psi_{1}}\mathcal{L}_{C1}(\Psi_{1},D_{ada}),$
and
$\Psi_{2}=\Psi_{2}-\beta_{C2}\nabla_{\Psi_{2}}\mathcal{L}_{C2}(\Psi_{2},D_{ada})$,
with $\beta_{A}$, $\beta_{C1}$, and $\beta_{C2}$ signifying the learning
rates, the actor and critic networks are respectively trained with. The
computational complexity for the Meta-SAC algorithm depends on the number of
incorporated layers at the SAC [14].

Figure 2: Convergence behavior of the proposed resource allocation scheme.
Figure 3: Total power consumption of the UAV $P^{\text{Tot}}$ versus the
number of users $K$. Figure 4: Average achievable data rate versus power
budget of the UAV $P_{\text{max}}$.

##  IV Simulation Results

This section discusses the performance of the proposed UAV-MIMO-VLC system and
the corresponding resource allocation scheme. To that end, three baselines are
considered, including a single-LED corresponded to [7], as well as the
baselines with $L$=10 and $L$=20 LEDs deployed at the UAV corresponded to our
proposed system. The performance is assessed under three criteria, including
power consumption of the UAV, as well as data rate and energy efficiency
gained by the PD-equipped users. Simulations are performed with $K$=30 users,
$R_{\text{min}}=2~{}\textrm{bits/s/Hz}$,
$V_{\text{max}}=20~{}\text{m}/\text{s}$,
$a_{\text{max}}=6~{}\text{m}/{\text{s}^{2}}$,
$\textbf{q}_{\text{max}}=(150,150,100)~{}\text{m}$ and $\tau=1~{}\text{s}$.
Other simulation parameters are summarized in Table I.

Fig. 2 showcases the convergence behaviour of the proposed resource allocation
scheme. Specifically, about 65% lower power is consumed in average by the UAV
while exploiting meta-learning. This achievement yields as the result of
better adaptation of the proposed meta-learning-enabled resource allocation
scheme to frequent variations of the UAV-MIMO-VLC system. Compared to the
greedy resource allocation scheme, as seen, this gain is even more.

TABLE I: Simulation Parameters Parameter | Value | Parameter | Value
---|---|---|---
$\Psi_{c}$ | $~{}60^{\circ}$ | $n_{R}$ | $1.5$
$\phi_{1/2}$ | $~{}60^{\circ}$ | $\text{A}_{\text{VLC}}$ | $1~{}\text{cm}^{2}$
$I_{h}$ | $10~{}\text{mA}$ | $I_{l}$ | $0~{}\text{A}$
$I_{0}$ | $5~{}\text{mA}$ | $\ddot{\zeta}$ | $1.2$
$\varphi$ | $1$ | $\zeta$ | $1.225$
$N$ | $10$ | $P_{\text{max}}$ | $20~{}\text{W}$
$\rho$ | $0.012$ | $\delta$ | $0.05$
$A_{\text{U}}$ | $0.79$ | $\Omega_{\text{U}}$ | $400$
$R_{\text{U}}$ | $0.05$ | $\iota$ | $1$
$W_{\text{U}}$ | $100$ | $v_{\text{UI}}$ | $7.2$
$d_{\text{U}}$ | $0.3$ | $R_{\text{min}}$ | $2~{}\textrm{bits/s/Hz}$
$V_{\text{max}}$ | $10~{}\text{m}/\text{s}$ | $a_{\text{max}}$ | $6~{}\text{m}/{\text{s}^{2}}$
$\textbf{q}_{\text{max}}$ | $(150,150,100)~{}\text{m}$ | $\tau$ | $1\text{s}$

In Fig. 3, we have exhibited the effect of increasing the number of PD-
equipped users on power consumption of the UAV. Evidently, by increasing the
number of users, fulfilling their data rate requirement, as dictated in C1,
induces more power consumption to the UAV. Quantitatively, a UAV with a LED
array with $N$=10 LEDs consumes 40% more power in average, compared to a UAV
employing a single-LED, when $K$=10. However, by increasing the number of
users, the gap between the baselines diminishes as all baselines gradually
approach the power budget of the UAV. For instance, when $K$=30, the gap
between the aforementioned baselines falls under 8%. This observation
underscores the effectiveness of the proposed UAV-MIMO-VLC system in dense
networks. Another insight, drawn from Fig. 3 corresponds to the limited gap
between the baselines with $N$=10 and $N$=20. This observation highlights the
influence of the adopted hybrid dimming mechanism, which effectively controls
the power consumption and promises the deployment of massive LED arrays to
gain much higher data rates.

Figure 5: Average achievable energy efficiency versus QoS requirement of users
$R_{\text{min}}$.

Fig. 4 depicts the impact of power budget of the UAV $P_{\text{max}}$, when it
is aimed to maximize the achievable data rate of the UAV-MIMO-VLC system. From
this figure, we perceive that the proposed resource allocation scheme
effectively exploits the available power budget, such that the inter-user
interference in (1) is managed well and the overall achieved data rate of the
network increases. More importantly, the single-LED baseline obtains much
lower data rate, compared to the baselines with $N$=10 and $N$=20 LEDs. In
average, around 47% gain is achieved, when comparing the baselines $N$=10 and
$N$=1. This observation justifies that in comparison with a single-LED, the
deployment of LED array at the UAV is much more cost effectiveness from the
network data rate standpoint. Fig. 4 also reveals that the gap between the
baselines $N$=10 and $N$=20 is much more than that between $N$=1 and $N$=10.
Conclusively, one can argue that deploying massive LED arrays can better
exploit the abundant capacity of the VLC band to fulfill the explosive data
rate requirements of upcoming wireless networks.

Eventually, Fig. 5, analyzes the performance of the UAV-MIMO-VLC system, aimed
at maximizing the overall energy efficiency, defined as a trade-off between
the achievable data rate and power consumption of the system. In essence,
assurance of more QoS requirement for users induces a user fairness within the
network, thereby degrades the overall achievable data rate of the network.
Conversely, the UAV has to spend more transmit power to meet higher levels of
QoS requirement. Under this argument, the more QoS requirement for users is
guaranteed, the lower energy efficiency is achieved. Comparatively, it is
clear that deploying a LED array with more number of LEDs at the UAV achieves
much more energy efficiency, which underscores the effectiveness of the
proposed UAV-MIMO-VLC system. For instance, in average, 34% better energy
efficiency is achieved by the baseline with $N$=10, compared to the single-LED
baseline $N$=1.

##  V Conclusions

We proposed a UAV-MIMO-VLC system in this paper and evaluated its performance
by an adaptive resource allocation framework. Through numerical results, we
found that incorporating a LED array rather than a single-LED at the UAV,
significantly improves the data rate and energy efficiency of the network, yet
at the cost of increased power consumption.

## References

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# Low-temperature giant coercivity in Co6.2Ga3.8-xGex ($x$=2.4 to 3.2)

Jiro Kitagawa1, Himawari Nomura1 and Terukazu Nishizaki2 1 Department of
Electrical Engineering, Faculty of Engineering, Fukuoka Institute of
Technology, 3-30-1 Wajiro-higashi, Higashi-ku, Fukuoka, 811-0295, Japan  2
Department of Electrical Engineering, Faculty of Science and Engineering,
Kyushu Sangyo University, 2-3-1 Matsukadai, Higashi-ku, Fukuoka, 813-8503,
Japan<EMAIL_ADDRESS>
###### Abstract

The observation of giant coercivity exceeding 20 kOe at low temperatures in
several transition-metal-based compounds has attracted significant attention
from a fundamental perspective. This research is also relevant to developing
rare-earth-free permanent magnets, wherein cobalt is one of the primary
elements used. To facilitate easy fabrication, rare-earth-free and Co-based
inorganic bulk magnets that exhibit giant coercivity are highly demanded but
rarely reported. Herein, we report the observation of low-temperature giant
coercivity in polycrystalline metallic Co6.2Ga3.8-xGex ($x$=2.4 to 3.2) with
the hexagonal Fe13Ge8-type structure composed of Kagome and triangular
lattices. As the Ge content $x$ decreases from 3.2, the magnetic ground state
changes from ferrimagnetism to ferromagnetism at $x$=2.6. In the ferrimagnetic
state, we observed a signature of spin frustration arising from the Kagome
and/or triangular lattices of Co atoms. The ferromagnetic ordering
temperatures for the $x$=2.6 and 2.4 samples are 46 K and 60 K, respectively.
The coercive fields rapidly increase upon cooling and reach values of 26 kOe
and 44 kOe in the $x$=2.6 and 2.4 samples, respectively, at 2 K.

keywords: giant coercivity, geometrical frustration, magnetization

##  1 Introduction

NdFeB and Sm-Co permanent magnets are commonly used in modern electric
society. However, due to the high supply risk associated with rare earth
elements such as Nd and Sm, developing rare-earth-free permanent magnets has
become a major research focus[1]. To be commercially viable, permanent magnets
require a large coercive field $H_{\mathrm{c}}$ and high saturation
magnetization. However, commercial rare-earth-free magnets such as Alnico and
ferrite exhibit low $H_{\mathrm{c}}$ values of approximately 3 kOe, much lower
than NdFeB and Sm-Co magnets with $H_{\mathrm{c}}$ values of 15-20 kOe[1].
Recently, some transition-metal-based magnets have been reported to exhibit
huge coercivity values that exceed 20 kOe, known as giant coercivity[2].
Typically, giant coercivity is observed at low temperatures but often
surpasses the $H_{\mathrm{c}}$ values of NdFeB and Sm-Co magnets. Rare-earth-
free transition-metal oxides like Mn2LiReO6 and Sr5Ru4.1O15 or some Fe-based
compounds have been reported as giant coercive compounds[3, 4, 5, 6, 7, 8, 9].
In most cases, $H_{\mathrm{c}}$ increases rapidly on cooling and reaches
values of 40-120 kOe at 2-4 K. The standard features among these giant
coercive compounds are an anisotropic crystal structure, canted
ferromagnetism, and low saturation magnetization. Despite the relatively high
saturation magnetization exhibited by Alnico and ferrite magnets,
understanding the underlying mechanism of giant coercivity would prove
advantageous as it can inform the exploration of new rare-earth-free hard
magnets.

One of the fundamental constituents of commercially-viable rare-earth-free
magnets is cobalt. For facile fabrication, a bulk sample is highly desirable;
however, reports of rare-earth-free and Co-based non-molecular inorganic bulk
magnets exhibiting giant coercivity values over 20 kOe are few and far
between. While it is true that room-temperature hard magnetic CoPt thin films,
cobalt monolayers, and Co3Sn2S2 wires have been shown to exhibit giant
coercivity values[10, 11, 12], the only known example of bulk magnets would be
CaBaCo4O7 with the Curie temperature $T_{\mathrm{C}}$ of 70 K and
$H_{\mathrm{c}}$ of 20 kOe at 5 K, and K2Co3(OH)2(SO4)3(H2O)2 with
$T_{\mathrm{C}}$ = 30 K and $H_{\mathrm{c}}$ = 50 kOe at 1.8 K[13, 14]. Thus,
exploring new Co-based non-molecular inorganic bulk ferromagnets with giant
coercivity is meaningful.

Our focus is on transition-metal-based compounds with the hexagonal
Fe13Ge8-type structure, which have not been extensively investigated. Only the
magnetic properties of Fe3Ga2-xAsx (0.21 $\leq$ $x$ $\leq$ 0.85) and
Fe3Ga0.35Ge1.65 have been reported[15, 16], and they exhibit soft
ferromagnetism at room temperature. While Co6.2Ga3.8-xGex also crystallizes
into the Fe13Ge8-type structure[17], its magnetic properties have not yet been
studied. Figure 1 illustrates the crystal structure, and Table 1 provides
details of the atomic positions. The space group is $P$63/$mmc$ (No. 194), and
there are three Wyckoff sites 2$a$, 6$g$, and 6$h$ for Co atoms. The occupancy
at the 6$h$ site is less than 1.0, which suggests the presence of vacancies.
The Co2 and Co3 atoms form Kagome networks, while the Co1 atoms form a
triangular lattice with a relatively long interatomic distance. We observe
that the Co2 atoms form the ideal Kagome lattice, and the Co3 atoms form a
slightly distorted Kagome lattice with vacancies. Ga and Ge atoms randomly
occupy the 2$c$ and another 6$h$ sites.

In this article, we have examined the magnetic and transport properties of
polycrystalline Co6.2Ga3.8-xGex with $x$ ranging from 2.4 to 3.2. Our
investigation has revealed that the material exhibits a metallic nature with
itinerant $d$-electrons of Co and displays a change from the ferromagnetic
(FM) to the ferrimagnetic ground state with an increase in $x$. This crossover
is accompanied by a shift towards a spin frustration regime dominated by
antiferromagnetic (AFM) interaction. Remarkably, we have found that the giant
coercivity values of 26 kOe and 44 kOe were achieved in Co6.2Ga1.2Ge2.6 and
Co6.2Ga1.4Ge2.4, respectively, at a temperature of 2 K. Our study has further
shown that the geometrical frustration with the itinerant $d$-electrons and
canted spin structure, together with the anisotropic crystal structure,
contribute to the emergence of the giant coercivity.

Figure 1:  Crystal structure of Co6.2Ga3.8-xGex. The solid black lines represent unit cell. Table 1: Atomic positions of Co6.2Ga3.8-xGex. Atom | Wyckoff | $x$ | $y$ | $z$ | Occupancy
---|---|---|---|---|---
Co1 | 2$a$ | 0 | 0 | 0 | 1
Co2 | 6$g$ | 0.5 | 0 | 0 | 1
Co3 | 6$h$ | 0.1614 | 0.3228 | 1/4 | 0.83
Ga+Ge | 2$c$ | 1/3 | 2/3 | 1/4 | 1
Ga+Ge | 6$h$ | 0.8088 | 0.6176 | 1/4 | 1

##  2 Materials and Methods

Polycrystalline specimens were synthesized via a homemade arc furnace
employing constituent elements Co (99.99 %), Ga (99.99 %), and Ge (99.999 %).
The elements were co-melted through an arc-melting process to yield button-
shaped samples weighing 1.5 g on a water-cooled Cu hearth. To ensure
homogeneity, each sample was subjected to multiple flips and remelting.
Subsequently, each as-cast sample was sealed in an evacuated quartz tube and
annealed at 700 ∘C for 4 days.

The X-ray diffraction (XRD) patterns of the powdered samples were collected at
room temperature using a Bragg-Brentano geometry X-ray diffractometer
(XRD-7000L, Shimadzu, Kyoto, Japan) with Cu-K$\alpha$ radiation. A field-
emission scanning electron microscope (FE-SEM; JSM-7100F, JEOL, Akishima,
Japan) was employed for the metallographic examination, and an energy-
dispersive X-ray (EDX) spectrometer, which was equipped with the FE-SEM, was
used for the atomic composition analysis.

The temperature dependence of the dc magnetic susceptibility,
$\chi_{\mathrm{dc}}$ ($T$), ranging from 2 K to 300 K, and the isothermal
magnetization curve were measured using MPMS3 (Quantum Design, San Diego, CA,
USA). In the magnetization curve measurements, except for the samples with
$x$= 2.4, 2.6, and 2.8 at 2 K, we employed demagnetization using the field
oscillation mode in MPMS3 prior to each measurement. Despite this
demagnetization procedure, a residual magnetization was observed when a high
coercive field was obtained. Consequently, we excluded the initial
magnetization process from the data display. To obtain precise initial
magnetization curves, magnetization curves at 2 K were measured for samples
with $x$ values of 2.4, 2.6, and 2.8 after subjecting them to zero-field
cooling. The temperature dependence of electrical resistivity, $\rho$ ($T$),
ranging from 3 K to room temperature, was measured using a dc four-probe
method with a homemade system in a GM refrigerator (UW404, Ulvac cryogenics,
Kyoto, Japan).

##  3 Results and Discussion

Figure 2:  (a) XRD patterns of Co6.2Ga3.8-xGex ($x$=2.4 to 3.2). In each dataset, the observed ($\circ$) and calculated (solid line) XRD patterns are shown at the top. The difference between the observed and calculated XRD patterns is shown at the bottom. The tick marks indicate the positions of Bragg reflections for Co6.2Ga3.8-xGex. (b) Ge content dependence of lattice parameters. (c) $c/a$ ratio vs. Ge content plot.  Table 2: Lattice parameters $a$ and $c$, $T_{\mathrm{C}}$, $\mu_{\mathrm{eff}}$, $\theta_{\mathrm{W}}$, $H_{\mathrm{c}}$ at 2 K, and room temperature $\rho$ value $\rho$ (RT) of Co6.2Ga3.8-xGex ($x$=2.4 to 3.2). $x$ |  $a$ (Å) |  $c$ (Å) |  $T_{\mathrm{C}}$ (K) |  $\mu_{\mathrm{eff}}$ ($\mu_{\mathrm{B}}$/Co) |  $\theta_{\mathrm{W}}$ (K) |  $H_{\mathrm{c}}$ at 2 K (kOe) |  $\rho$ (RT) ($\mu\Omega$cm)
---|---|---|---|---|---|---|---
2.4 | 7.906(3) | 4.970(1) | 60.3 | 1.78 | 50 | 44 | 185
2.6 | 7.900(3) | 4.977(1) | 46.4 | 1.77 | 41 | 26 | 186
2.8 | 7.894(4) | 4.984(2) | 13.2 | 1.97 | -83 | 9.3 | 209
3.0 | 7.875(2) | 4.985(1) | 5.8 | 1.95 | -116 | 1.4 | 126
3.2 | 7.857(4) | 4.992(2) |  $<$ 2 | 2.15 | -309 | 0.24 | 227

Figure 2(a) depicts the X-ray diffraction (XRD) patterns of all specimens,
which are well-explained by the Fe13Ge8-type structure. The hexagonal lattice
parameters $a$ and $c$ are obtained with the help of a Rietveld refinement
program[18, 19] and are summarized in Table 2. For this purpose, we employed
the RIETAN-FP program package[18]. To ensure a full parameter fitting process
of utmost precision, it is imperative for the peak intensity of XRD to exceed
10000 counts. However, our acquired data yielded a peak intensity of at most
1000 counts. Consequently, we adopted a strategy of fixed atomic positions, as
detailed in Table 1, while the fitting parameters encompassed lattice
parameters, background function, profile function, and scaling factor. The
lattice parameters are plotted as a function of the Ge content $x$ in Fig.2(b)
and reveal that the $a$-axis length reduces while the $c$-axis expands with
increasing $x$. The $c/a$ ratio is also calculated for each sample, as
displayed in Fig.2(c). The result illustrates a systematic increase of $c/a$
as the Ge atom replaces the Ga atom gradually. Figures 3(a) to (c) present
scanning electron microscopy (SEM) images for several samples ($x$=2.4, 2.8,
and 3.2). Each non-contrast image indicates a homogeneous chemical composition
with no conspicuous impurity phases. The chemical compositions, ascertained
through EDX analysis, are as follows: Co60.7(3)Ga14.6(5)Ge24.8(5) for $x$=2.4,
Co60.9(6)Ga10.4(7)Ge28.7(1) for $x$=2.8, and Co60.3(5)Ga6.8(5)Ge32.9(2) for
$x$=3.2, respectively. Each composition is almost identical to the starting
composition. Furthermore, elemental mappings of Co, Ga, and Ge are also
presented in Figs.3(a) to (c), revealing the homogeneous distribution of
constituent elements. The SEM images indicate the smoothness of the surface,
which may be advantageous for various application perspectives. For example,
the smoothness of the surface of magnetic material improves the sensitivity in
tunnel magnetoresistance sensors[20, 21].

Figure 3:  SEM images of (a) $x$=2.4, (b) $x$=2.8, and (c) $x$=3.2 for
Co6.2Ga3.8-xGex, respectively. The elemental mappings of Co, Ga, and Ge are
also shown. Figure 4:  (a)Temperature dependences of $\chi_{\mathrm{dc}}$ of
Co6.2Ga3.8-xGex ($x$=2.4 to 3.2) under ZFC and FC conditions. The external
field is 500 Oe. Both axes are on a logarithmic scale. (b) and (c) Temperature
derivative of $\chi_{\mathrm{dc}}$ under FC of Co6.2Ga3.8-xGex ($x$=2.4 to
3.2). (d) Temperature dependences of 1/$\chi_{\mathrm{dc}}$ of Co6.2Ga3.8-xGex
($x$=2.4 to 3.2). The solid lines represent the fitting results using the
Curie-Weiss law. Figure 5:  $M$-$H$ curves measured at temperatures denoted in
figure for (a) $x$=2.4, (b) $x$=2.6, (c) $x$=2.8, (d) $x$=3.0, and (e) $x$=3.2
Co6.2Ga3.8-xGex samples. $M$-$H$ loops at 2 K around (f) negative
$H_{\mathrm{c}}$ and (g) positive $H_{\mathrm{c}}$ for $x$=2.4, 2.6, and 2.8
samples.

We conducted measurements of $\chi_{\mathrm{dc}}$ ($T$) under both zero-field-
cooled (ZFC) and field-cooled (FC) conditions, employing an external magnetic
field of 500 Oe. The temperature-dependent dc magnetic susceptibility
$\chi_{\mathrm{dc}}$ of each sample exhibits an enhancement of
$\chi_{\mathrm{dc}}$ in the low-temperature range, which suggests a
ferromagnetic ordering, as shown in Fig.4(a). Except for the $x$=3.2 sample
with magnetic ordering temperature below 2 K, the observed irreversibility
between the ZFC and FC datasets is a characteristic of ferromagnetic behavior.
This phenomenon can be understood by recognizing the significant magnetic
domain pinning during ZFC conditions. The $\chi_{\mathrm{dc}}$ value at 2 K is
heavily reduced by increasing $x$, and the $T_{\mathrm{C}}$ systematically
decreases. The temperature at which the minimum temperature derivative of
$\chi_{\mathrm{dc}}$ under FC occurs defines $T_{\mathrm{C}}$, shown in Table
2 for each sample and represented in Figs.4(b) and (c). The $x$=3.2 sample
would possess $T_{\mathrm{C}}$ below 2 K. The definition of $T_{\mathrm{C}}$
employed in this study aligns with the methodology commonly applied in
numerous ferromagnetic investigations[22, 23, 24]. This approach is considered
reliable, as it has been validated in some ferromagnetic materials where the
$T_{\mathrm{C}}$ obtained through this method is consistent with
$T_{\mathrm{C}}$ values determined via other physical quantities, such as
specific heat[25, 26]. The temperature dependences of inverse
$\chi_{\mathrm{dc}}$ are demonstrated in Fig.4(d). In each sample,
1/$\chi_{\mathrm{dc}}$ follows the Curie-Weiss law expressed by
$\chi_{\mathrm{dc}}=C/(T-\theta_{\mathrm{W}})$ at high temperatures, as
indicated by the solid line. The effective magnetic moment
$\mu_{\mathrm{eff}}$ obtained from the $C$ value and the Weiss temperature
$\theta_{\mathrm{W}}$ are presented in Table 2. All $\mu_{\mathrm{eff}}$
values are smaller than that of an isolated Co2+ or Co3+ ion (3.87 or 4.90
$\mu_{\mathrm{B}}$/Co), which suggests an itinerant character of
$d$-electrons. For the $x$=2.6 or 2.4 sample, the positive
$\theta_{\mathrm{W}}$ is nearly identical to $T_{\mathrm{C}}$, indicating a
ferromagnetic ordering. However, in the $x$=2.8 $\sim$ 3.2 samples, negative
$\theta_{\mathrm{W}}$ values are present, indicating the dominance of AFM
interaction. The magnetization curves exhibit hysteresis loops, which are
characteristic of FM compound. Therefore, the samples with $x$$\geq$2.8
undergo ferrimagnetic ground states. Furthermore, it is notable that the
frustration index $f$, defined as $|\theta_{\mathrm{W}}|$/$T_{\mathrm{C}}$,
significantly increases from 6.3 to over 155 as $x$ increases from 2.8 to 3.2
in the ferrimagnetic state. This index is grounded in the concept that
$\theta_{\mathrm{W}}$ reflects the strength of magnetic interactions[27, 28].
Thus, an unfrustrated compound would attain a magnetically ordered state at
$\theta_{\mathrm{W}}$, resulting in $f$=1. Geometrical frustration, such as
triangular and Kagome lattices, often induces magnetic spin frustration,
significantly suppressing the ordering temperature. Compounds of this nature
frequently exhibit $f$ values exceeding 5, often surpassing 100[27, 28]. The
observed increase in $f$ within the Co6.2Ga3.8-xGex series, particularly when
$x$ exceeds 2.8, strongly supports the presence of spin frustration. This
finding aligns with the geometrically frustrated Kagome and triangular
lattices, as depicted in Fig.1. Hence, the notable characteristic of
Co6.2Ga3.8-xGex is its capacity for chemical manipulation of spin frustration.

The isothermal magnetization curves (where $M$ is the magnetization and $H$ is
the external field) for all samples, spanning -70 kOe to 70 kOe, are presented
in Figs.5 (a) through (e). The high field $M$ at 2 K displays a sudden drop as
$x$ increases from 2.6 to 2.8, which is indicative of spin compensation in the
ferrimagnetic samples with $x$=2.8 $\sim$ 3.2. However, even in the
ferromagnetic samples with $x$=2.4 or 2.6, the highest $M$ value is relatively
smaller than $\mu_{\mathrm{eff}}$, implying the existence of a canted
ferromagnetic structure. Notably, the $x$=2.4 and 2.6 samples exhibit a giant
coercivity of 44 kOe and 26 kOe, respectively, at 2 K. In the $x$=2.4 or 2.6
sample, $H_{\mathrm{c}}$ increases significantly as the temperature drops
below $T_{\mathrm{C}}$, and the initial magnetization curve at 2 K, exhibiting
a small slope, undergoes an abrupt jump at approximately $H_{\mathrm{c}}$.
This magnetization process is a typical characteristic of domain wall
pinning[29]. The ferrimagnetic samples also demonstrate hysteresis loops at
low temperatures, and the corresponding $H_{\mathrm{c}}$ values at 2 K are
provided in Table 2.

Figure 6(a) displays the temperature dependences of $\rho$ in all the samples.
The values of $\rho$ are normalized by the corresponding room temperature
values listed in Table 2. The metallic nature of the samples is evident from
the order of magnitude of $\rho$ in each one. The ferrimagnetic samples
exhibit a negative temperature coefficient of resistivity below around 150 K,
which could be indicative of carrier localization and/or a partial opening of
the gap at the Fermi surface. For a more comprehensive understanding of
carrier localization in the $x$=2.8-3.2 samples, we present the temperature-
dependent electrical conductivity ($\sigma$), normalized by the room
temperature $\sigma$ (designated as $\sigma$ (RT)) (=1/$\rho$ (RT)), as a
function of $T^{1/2}$ in Fig.6(b). In the localization regime for each sample,
$\sigma$ diminishes with decreasing temperature, adhering closely to a
$T^{1/2}$ dependence. This temperature response is explicable through a weak
localization model for three-dimensional systems[30], expressed as
$\sigma(T)=\sigma_{0}+aT^{1/2}$. In this expression, the initial term
represents residual conductivity, while the second term accounts for the
influence of weak localization due to electron-electron interactions, with the
proportional coefficient $a$. The solid lines in Fig.6(b) represent fits to
this model, effectively capturing the localization characteristics of
Co6.2Ga3.8-xGex ($x$=2.8-3.2). At lower temperatures, typically below 50 K,
the experimental $\sigma$ for each sample surpasses the predicted value of the
solid line. This observation could potentially be attributed to impurity
conduction, although further investigation is warranted.

Figure 6:  (a) Temperature dependences of $\rho$ for Co6.2Ga3.8-xGex ($x$=2.4
to 3.2). Each $\rho$ is normalized by the room temperature $\rho$ value $\rho$
(RT) listed in Table 2. (b) Temperature dependences of $\sigma$ for $x$=2.8,
3.0, and 3.2 samples plotted as a function of $T^{1/2}$. Each $\sigma$ is
normalized by the room temperature $\sigma$ denoted as $\sigma$ (RT). The
solid lines represent fits to the equation $\sigma_{0}+aT^{1/2}$ with the
associated parameters as follows: ($\sigma_{0}$ (($\mu\Omega$cm)-1), $a$
(($\mu\Omega$cm)-1 K-1/2))=(0.00416, 6.28$\times$10-5) for $x$=2.8, (0.00663,
1.11$\times$10-4) for $x$=3.0, and (0.00378, 4.83$\times$10-5) for $x$=3.2,
respectively.

We aim to investigate the correlation between Co bond length and magnetism.
The Bethe-Slater curve, widely used for analysing magnetism in magnetic
metals, suggests that longer and shorter interatomic distances between
magnetic atoms favour FM and AFM interactions, respectively[31]. This tendency
has been observed in various intermetallic compounds[32, 33]. The selected Co
interatomic distances are listed in Table 3, and, taking into account the
multiplicity (2 for Co1-Co1 and Co2-Co2, 6 for Co1-Co3, and 4 for Co2-Co3),
Co1-Co3 and Co2-Co3 bonding likely determine the magnetic ordering type. The
$x$-dependence of $\theta_{\mathrm{W}}$ in Co6.2Ga3.8-xGex strongly suggests
the coexistence of FM and AFM interactions. Co1-Co3 and Co2-Co3 bonds could
lead to AFM and FM interactions, respectively, although further investigation
is necessary. When $x$ exceeds 2.8, the Co1-Co3 and Co2-Co3 bond lengths
considerably decrease, consistent with the rapid predominance of AFM
interaction as reflected by $\theta_{\mathrm{W}}$. The coexistence of FM and
AFM interactions in the Co6.2Ga3.8-xGex system bears similarity to artificial
layered AFM-FM structures and compounds characterized by the simultaneous
presence of AFM and FM phases[34, 35]. In these systems, an intriguing
phenomenon known as the exchange bias effect often manifests, characterized by
a shift in the $M$-$H$ hysteresis loop along the magnetic field direction.
This effect’s signature can be discerned by examining the disparities between
the positive and negative $H_{\mathrm{c}}$ values in the $M$-$H$ loop.
Positive and negative $H_{\mathrm{c}}$ are the points where the $M$-$H$ loop
intersects the positive and negative $x$-axis. In Figs.5(f) and (g), we have
depicted the positions of the negative and positive $H_{\mathrm{c}}$ for the
$x$=2.4, 2.6, and 2.8 samples, which exhibit larger $H_{\mathrm{c}}$ values at
2 K, using open circles to denote these positions. For the $x$=2.4 and 2.6
samples, showcasing FM ordering with weak spin frustration, the positive
$H_{\mathrm{c}}$ values (43 kOe for $x$=2.4 and 23 kOe for $x$=2.6) are
marginally lower than their corresponding negative $H_{\mathrm{c}}$ values (44
kOe for $x$=2.4 and 26 kOe for $x$=2.6). This observation supports the
plausible occurrence of the exchange bias effect in the $x$=2.4 and 2.6
samples, aligning with the coexistence of FM and AFM interactions, as
mentioned earlier. Conversely, for the $x$=2.8 sample, which accompanies the
spin-frustration state, the positive $H_{\mathrm{c}}$ value of 9.5 kOe is
almost identical to the negative $H_{\mathrm{c}}$ value of 9.3 kOe. As $x$
surpasses 2.8, the rapid predominance of AFM interactions is implied by the
$x$ dependence of $\theta_{\mathrm{W}}$. Consequently, the emergence of spin
frustration due to the overwhelming AFM interactions may attenuate the
exchange bias effect.

It should be noted that the layered hexagonal perovskite Sr5Ru4.1O15, known
for its giant coercive properties[4], shares the same space group $P$63/$mmc$
(No. 194) as Co6.2Ga3.8-xGex. This metallic compound exhibits a highly
anisotropic crystal structure with $c/a$=4.106, wherein Ru acts as the
magnetic atom and partially forms a triangular lattice. The analysis through
Curie-Weiss fitting suggests the presence of itinerant $d$-electrons of Ru.
The saturation moment of 0.05 $\mu_{\mathrm{B}}$/Ru is much smaller than the
$\mu_{\mathrm{eff}}$ value of the Ru ion, implying weak ferromagnetism. The
giant coercivity of Sr5Ru4.1O15 arises from the large magnetocrystalline
anisotropy with the geometrical frustration[4]. Co6.2Ga1.4Ge2.4 and
Co6.2Ga1.2Ge2.6, both of which are also metallic, have a $c/a$ ratio much
smaller than 1.0 (as seen in Fig.2(c)), indicating an anisotropic crystal
structure. The Co atoms form Kagome and triangular lattices, and the Co
magnetic moment is itinerant. Co6.2Ga1.4Ge2.4 and Co6.2Ga1.2Ge2.6 exhibit
relatively low saturation moments, likely originating from a canted
ferromagnetic structure. Thus, the overall behaviour of Co6.2Ga1.4Ge2.4 and
Co6.2Ga1.2Ge2.6 is quite similar to the magnetic and transport properties of
Sr5Ru4.1O15. Therefore, we speculate that the giant coercivity in rare-earth-
free magnets with the space group $P$63/$mmc$ can be attributed to the
itinerant $d$-electrons in geometrically frustrated metals and canted spin
structure, in addition to the anisotropic crystal structure.

Table 3: Selected Co interatomic distances in Co6.2Ga3.8-xGex ($x$=2.4 to 3.2). The unit is Å. The multiplicities are 2 for Co1-Co1, 6 for Co1-Co3, 2 for Co2-Co2, and 4 for Co2-Co3, respectively.  $x$ | Co1-Co1 | Co1-Co3 | Co2-Co2 | Co2-Co3
---|---|---|---|---
2.4 | 2.485 | 2.536 | 2.485 | 2.632
2.6 | 2.489 | 2.535 | 2.489 | 2.630
2.8 | 2.492 | 2.534 | 2.492 | 2.630
3.0 | 2.493 | 2.530 | 2.493 | 2.625
3.2 | 2.496 | 2.526 | 2.496 | 2.621

As mentioned in the introduction, CaBaCo4O7 is the rare example of a rare-
earth-free Co-based bulk inorganic compound exhibiting a giant
$H_{\mathrm{c}}$[13]. This cobaltite also features Co-Kagome layers despite
its distorted orthorhombic crystal structure. While its low saturation moment
($\sim$ 0.7 $\mu_{\mathrm{B}}$/f.u.) is akin to that of Co6.2Ga3.8-xGex, the
unique dielectric behaviour is different from the metallic transport of
Co6.2Ga3.8-xGex ($x$=2.4 and 2.6). It is thus appropriate to classify
Co6.2Ga3.8-xGex ($x$=2.4 and 2.6) as a new category of rare-earth-free Co-
based inorganic compounds that exhibit giant coercivity.

A comparison of magnetic properties between Co6.2Ga3.8-xGex and isostructural
Fe3Ga0.35Ge1.65 would be of great significance. Fe3Ga0.35Ge1.65 exhibits FM
ordering below 341 K, and even at 50 K, no noticeable hysteresis loop is
detected[16]. At 50 K, the saturation magnetization is 80 emu/g, equivalent to
approximately 1.47 $\mu_{\mathrm{B}}$/Fe. Thus, it is likely that the
itinerant nature of $d$-electrons is weakened in Fe3Ga0.35Ge1.65, and the
itinerant magnetic moment is essential for the manifestation of giant
coercivity in transition-metal magnets with the $P$63/$mmc$ space group.

##  4 Summary

This study presents the transport and magnetic properties of polycrystalline
Co6.2Ga3.8-xGex ($x$=2.4 to 3.2) with the hexagonal Fe13Ge8-type structure.
The crystal structure of these materials is characterized by the presence of
Kagome and triangular lattices formed by Co atoms, as well as three distinct
crystallographic sites for Co atoms. The former feature is likely responsible
for spin frustration, while the latter is significant for the competition
between AFM and FM interactions. All samples are found to be metallic, with Co
$d$-electrons exhibiting itinerancy across all $x$ values. As $x$ increases,
the magnetic ground state changes from FM to ferrimagnetic ordering, resulting
in the emergence of spin frustration, as evidenced by the large frustration
index. Conversely, spin frustration appears to be suppressed in FM samples,
although the competition of FM and AFM interactions contributes to the canted
spin structure. Co6.2Ga3.8-xGex represents a rare system that allows for the
chemical tuning of spin frustration. We note that the chemical tuning of spin
frustration has not been reported in the other giant coercive materials.
Notably, Co6.2Ga1.4Ge2.4 and Co6.2Ga1.2Ge2.6, with FM ground states, exhibit
low-temperature giant coercivities comparable to the $H_{\mathrm{c}}$ value
(30 kOe) observed in CoPt thin film studied as potential room-temperature hard
magnets. This discovery of a new class of rare-earth-free Co-based compounds
exhibiting giant coercivity marks a promising step toward developing rare-
earth-free permanent magnets.

## Acknowledgments

J.K. is grateful for the support provided by the Comprehensive Research
Organization of Fukuoka Institute of Technology. T.N. is grateful for the
support from the Advanced Instruments Center of Kyushu Sangyo University

## Data availability statement

All data that support the findings of this study are included within the
article.

## Author contributions

Jiro Kitagawa: Conceptualization, Supervision, Formal analysis, Writing -
original draft, Writing - reviewing & editing. Himawari Nomura: Investigation.
Terukazu Nishizaki: Investigation, Formal analysis, Writing - reviewing &
editing.

## Conflicts of interest

The authors declare no conflict of interest.

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