# Giant topological and planar Hall effect in Cr1/3NbS2

D. A. Mayoh<EMAIL_ADDRESS>J. Bouaziz  A. E. Hall  J. B. Staunton
M. R. Lees  G. Balakrishnan<EMAIL_ADDRESS>Physics Department,
University of Warwick, Coventry, CV4 7AL, United Kingdom

###### Abstract

Cr1/3NbS2 is a transition metal dichalcogenide that has been of significant
interest due to its ability to host a magnetic chiral soliton lattice.
Conventional and planar Hall measurements provide valuable insight into the
detection of exotic spin structures in chiral magnets. We show that the
presence of a giant planar Hall effect can be attributed to a tilted soliton
lattice in Cr1/3NbS2. Our detailed angular dependent study shows the planar
Hall effect and anisotropic magnetoresistance are intrinsically linked in
complex non-coplanar magnets. From the conventional Hall signal we show the
presence of a giant unconventional, likely topological Hall component, that is
the fingerprint of non-coplanar spin textures.

##  I Introduction

The fundamental characteristics of two-dimensional (2D) magnetic materials can
be scrutinized by examining the behavior of their electrons in magnetic and
electric fields. 2D magnetic materials have been found to be highly tunable in
terms of their physical characteristics making them excellent candidates for
the study of exotic quantum phases and for spintronic devices [1, 2]. Chiral
helimagnetism has provided an exciting twist in the story of 2D and bulk
magnetic materials with various examples of self-localized spin textures
having been reported experimentally including skyrmions [3, 4, 5], anti-
skyrmions [6], bubbles [7] and vortices [8]. These magnetic states often
emerge due to the Dzyaloshinskii-Moriya interaction (DMI) [9, 10] but other
stabilization mechanisms can also be at play, such as multi-spin interactions
[11, 12, 13] or geometric frustration [14, 15].

Hall measurements have proved to be an excellent tool for probing spin
textures [16, 17, 18, 19]. Indeed, the hallmark of skyrmion-like materials is
the so called topological Hall effect (THE) [20, 21, 22]. However, other forms
of non-coplanar spin textures may also result in a finite THE. In particular,
non-coplanar phases occurring in metallic compounds can cause variations in
the magnetization which may lead to an emergent magnetic field that can
deflect the electrons in the transverse direction [23]. A giant topological
Hall coefficient is normally defined as $\rho^{\mathrm{THE}}_{\mathrm{xy}}\geq
1~{}\mu\Omega$ cm and has been reported in several compounds including
Gd2PdSi3 [22], NdMn2Ge2 [24] and thin film oxides [25].

Cr-intercalated NbS2 (Cr1/3NbS2) has attracted considerable attention due to
the presence of a chiral helimagnetic state and, in an applied field, the
observation of a chiral soliton lattice (CSL) by Lorentz transmission electron
microscopy (LTEM) [26]. Cr1/3NbS2 is a member of the intercalated transition
metal dichalcogenide family and has a chiral noncentrosymmetric hexagonal
structure (space group: $P6_{3}22$) [27, 28]. Cr1/3NbS2 consists of layers of
Nb and S with Cr intercalated between the layers as shown in Fig. 1(a). It is
known to have a complex magnetic phase diagram and five distinct magnetic
phases have been detected through a variety of experimental techniques
including LTEM, muon spin rotation/relaxation, magnetization, magnetocaloric
effects, and magneto-transport [29]. The magnetic ordering observed in
Cr1/3NbS2, is highly dependent on the experimental conditions. In zero-applied
field, a twisting array of magnetic moments in the $ab$-plane of Cr1/3NbS2
results in chiral helimagnetic (CHM) ordering [26, 30]. Applying a magnetic
field along the screw axis (the $c$-axis) results in the magnet moments being
tilted towards the direction of the field giving a chiral conical (CC) phase
[31], while increasing the magnetic field still further results in a forced
ferromagnetism (FFM). If the magnetic field is applied perpendicular to the
screw axis, a CSL is established. This lattice is formed of areas of forced
ferromagnetism broken by a twist in the magnetic field [26]. Recently, torque
magnetometry has revealed a tilted CSL phase in Cr1/3NbS2 when the magnetic
field is applied at an angle greater than 1.5 degrees from the $c$-axis [31].

The key finding of this work is the presence of a giant THE in Cr1/3NbS2 that
is the finger print of non-coplanar spin textures in this material. The
emergence of a finite PHE is attributed to the presence of the tilted CSL. The
temperature and field dependence of the PHE has been thoroughly investigated
in parallel with its longitudinal counterpart, the anisotropic
magnetoresistance (AMR). Both are found to be intimately linked, highlighting
that the connection between the PHE and AMR is still valid for complex non-
coplanar magnets. Furthermore, by disentangling the ordinary and anomalous
Hall components from the conventional Hall data, we reveal the presence of a
giant topological Hall effect. We attribute the origin of the topological Hall
effect in Cr1/3NbS2 to the complex magnetic textures present in this material.

##  II Experimental Details

Single crystals of Cr1/3NbS2 were produced by the chemical vapor transport
technique using iodine as the transport agent. Polycrystalline powders of
Cr1/3NbS2 along with 5 mg/cc of the transport agent were placed in an
evacuated and sealed quartz tube. The tube was then heated with one end held
at $950~{}^{\circ}$C and the other at $800~{}^{\circ}$C for three weeks,
before cooling to room temperature. Several single crystals in the form of
hexagonal platelets with sides $\sim 2$ mm in length were obtained in each
growth. DC magnetization, $M$, measurements were performed as a function of
temperature (1.8 to 300 K) and field (0 to 7 T) in a Quantum Design Magnetic
Property Measurement System. Magnetic hysteresis loops were acquired using an
Oxford Instruments vibrating sample magnetometer in magnetic fields up to 6 T.
Resistivity, conventional Hall and planar Hall effect measurements were
performed as a function of temperature (1.8 to 300 K) and field (0 to 9 T)
using the ac transport option in a Quantum Design Physical Property
Measurement System. Currents of between 1 and 10 mA at a frequency of 113 Hz
were used for all the measurements. Further details on the fitting procedure
used for the conventional Hall effect data can be found in Appendix A

##  III Magnetization and Longitudinal Resistivity.

A single crystal of Cr1/3NbS2 with a $T_{\mathrm{c}}$ of 111 K was chosen for
these experiments as shown in Fig. 1(b). A Laue diffraction pattern of the
crystal is shown in the Supplemental Material [32]. DC magnetization, $M$,
versus temperature curves for the single crystal of Cr1/3NbS2 in various
fields up to 7 T applied perpendicular to the platelet are shown in Fig. 1(c).
In low fields the system exhibits a sharp transition to a magnetically ordered
state at $T_{\mathrm{c}}=111$ K. As the field is increased the magnetic
transition broadens. In 7 T we observe a significant magnetization up to 250
K. Magnetization as a function of applied field, $H$, where the field was
applied nominally along the $c$ axis is shown in Fig. 1(d). At low-temperature
the magnetization increases with increasing applied field until saturation at
$M=2.9(1)~{}\mu_{\mathrm{B}}/\mathrm{Cr~{}atom}$. The saturation magnetization
of Cr1/3NbS2 is expected to be approximately
$3~{}\mu_{\mathrm{B}}/\mathrm{Cr~{}atom}$ for chromium in the trivalent
$(\mathrm{S}=3/2)$ state. The magnetization curves can be explained by the
presence of the CHM slowly being polarized along the direction of the applied
field. No significant hysteresis is visible in the $M(H)$ curves at any
temperature.

The different magnetic regimes in Cr1/3NbS2 are observed to have distinct
electronic transport properties. The temperature dependence of the zero-field
longitudinal resistivity $\rho_{\mathrm{xx}}$ is shown in Fig. 1(e). Also
shown in Fig. 1(f–i) is $\rho_{\mathrm{xx}}$ vs temperature for assorted
magnetic fields applied parallel to the $c$ axis. The temperature dependence
of the longitudinal resistivity differs from a conventional metallic
ferromagnet [33]. $\rho_{\mathrm{xx}}$ initially increases with decreasing
temperature until $\sim 150$ K. A similar increase in $\rho_{\mathrm{xx}}$ is
observed in UPd2Si2 and was attributed to the Kondo effect [34]. Below 150 K
the resistivity of Cr1/3NbS2 then falls over a wide temperature interval,
$\Delta T=100$ K, centered around $T_{\mathrm{c}}$. The drop in the
resistivity at temperatures below the magnetic transition is due to the
magnetic ordering which diminishes the spin-disorder scattering [35]. This
reduction is not monotonous but instead tends to saturate as observed in
previous work [36]. A possible origin of this behavior is a disorder of the Cr
atoms [37]. However, given the presence of a helical phase/CSL in this
temperature regime, there is no one to one correspondence between the behavior
seen here and that observed in a simple ferromagnet. The residual resistivity
at 2 K is $\rho_{\mathrm{xx},0}=0.339(5)$ m$\Omega$ cm.

The field dependence of $\rho_{\mathrm{xx}}$ in Cr1/3NbS2 for a selection of
temperatures is shown in Figs. 1(e-h). Different behaviors are observed in the
various temperature regimes dependent on the magnetic ordering. Between 2 and
20 K $\rho_{\mathrm{xx}}$ is seen to saturate in fields above 2 T. The rapid
decrease in the resistivity from 0 to 2 T can be attributed to the presence of
the CSL, and as the field is increased the moments are canted towards the
direction of field which in turn diminishes the spin-disorder scattering.
$\rho_{\mathrm{xx}}$ saturates when the spins are fully polarized along $H$.
This field dependence of the magnetoresistance is typical of helimagnetic
systems [38, 39]. A plateau in the magnetoresistance has also been attributed
to a skyrmion lattice phase in Gd2PdSi3 [40]. Between 20 and 110 K, the
distinct change in slope of $\rho_{\mathrm{xx}}\left(H\right)$ around 2 T
becomes more diffuse as the mean free path of the electrons decreases,
therefore, they experience less of the CSL phase. At temperatures above 120 K,
which is above $T_{\mathrm{c}}$, $\rho_{\mathrm{xx}}$ is seen to have
parabolic dependence, typical of ferromagnetic materials, before becoming
almost linear with $H$ at $300$ K.

##  IV Hall effect in Cr1/3NbS2

.

Hall effect measurements on Cr1/3NbS2 were performed at a variety of
temperatures, where the field was applied perpendicular to the largest face of
the crystal platelet (nominally along the $c$ axis) and the current applied
within the plane of the plate (nominally the $ab$ plane), and are shown in
Fig. 2. To understand the underlying physics in Cr1/3NbS2 we investigate the
Hall response and disentangle its different components. The Hall signal is
usually made up of several contributions:

$\rho_{xy}=\rho^{\text{OHE}}_{xy}+\rho^{\text{AHE}}_{xy}+\rho^{\text{THE}}_{xy}.$
(1)

The first term $\rho_{\mathrm{xy}}^{\mathrm{OHE}}$ is the ordinary Hall effect
(OHE) which scales linearly with the magnetic field $B$,
$\rho_{\mathrm{xy}}^{\mathrm{OHE}}=R_{\mathrm{H}}B$, where $R_{\mathrm{H}}$ is
the Hall coefficient. The second term $\rho_{\mathrm{xy}}^{\mathrm{AHE}}$ is
the anomalous Hall effect (AHE), which emerges from two mechanisms, one
intrinsic and the other extrinsic. The intrinsic contribution is related to
the presence of a non-vanishing Berry curvature leading to an internal
magnetic field which deflects the electrons in a direction perpendicular to
the applied electric field [41, 42]. The extrinsic contribution can be related
to side jump and skew scattering [42, 43, 44]. Usually, the skew scattering
contribution is the weakest component, however, it has been shown recently
that it can have a measurable impact on the Hall response in some materials
[45]. The third term, $\rho_{\mathrm{xy}}^{\mathrm{THE}}$, is the topological
Hall effect [18, 46, 20] whose dependence on the magnetization is non-trivial,
$\rho_{\mathrm{xy}}^{\mathrm{THE}}$ is proportional to the emergent field
$\vec{B}_{\text{e}}$ created by a non-coplanar magnetic texture [23, 47],
where
$\vec{B}_{\mathrm{e}}\propto\vec{m}\cdot\left(\frac{\partial\vec{m}}{\partial
x}\times\frac{\partial\vec{m}}{\partial y}\right)$.
$\rho_{\mathrm{xy}}^{\mathrm{THE}}$ has a non-linear evolution with the
magnetic field, and is only seen in the presence of non-coplanar phases [23].
This component of the Hall signal is anti-symmetric when the direction of the
applied magnetic field is reversed [25]. At this point it worth considering
whether any of the magnetic phases in Cr1/3NbS2 can generate a topological
Hall contribution. Neither the helimagnetic phase nor the chiral soliton
lattice will display a topological Hall effect since the scalar chirality must
vanish, i.e.
$\mathcal{C}_{ijk}=\vec{m}_{i}\cdot(\vec{m}_{j}\times\vec{m}_{k})$. However,
for the case of a tilted-CSL [31], a finite topological Hall effect
contribution is possible (see Appendix. C).

The presence of tilted chiral soliton lattice may also lead to the appearance
of a planar Hall effect in Cr1/3NbS2. The planar Hall effect emerges solely
from the anisotropic magnetoresistance [48, 49, 50], and occurs when the in-
plane component of the magnetization is misaligned with respect to the applied
current. For a current along the $x$ direction, $\rho^{\text{PHE}}_{xy}$ is
[50, 49, 51]:

$\rho^{\text{PHE}}_{xy}=(\rho_{\parallel}-\rho_{\perp})\,m_{x}\,m_{y},$ (2)

where $m_{x}$ and $m_{y}$ are the in-plane components of the magnetization and
$\rho_{\parallel}$ $(\rho_{\perp})$ are the longitudinal resistivity when
$\vec{m}$ is parallel (perpendicular) to the current direction.

##  V Planar Hall effect and anisotropic magnetoresistance

One of the keys to understanding the magnetic spin textures in our Hall effect
measurements is the presence of a PHE. Indeed, a planar Hall effect and its
longitudinal counterpart, an anisotropic magnetoresistance are observed in
Cr1/3NbS2. The presence of the PHE indicates a finite in-plane magnetization
in the magnetic structure. Thus, we consider the two possible scenarios
currently proposed for Cr1/3NbS2 when the field is applied close to the $c$
axis: a chiral conical phase (when the offset is less than $1.5^{\circ}$) and
a tilted chiral soliton lattice (when the offset is greater than
$1.5^{\circ}$) [31]. In the CC phase the magnetic moments provide a net
magnetic moment that points directly along the helical axis (out-of-plane) due
to symmetry. Once the field is sufficient to polarize all the moments along
the helical axis the magnetic moments continue to be out-of-plane. Hence, the
absence of any in-plane magnetization from the CC phase and eventual FFM phase
means no PHE would be observed. In the case of a tilted CSL where all the
magnetic moments are uniformly canted towards the helical axis, it is easy to
see that there is a net magnetic moment that is neither perpendicular nor
parallel to the helical axis. This then means that an in-plane magnetization
vector does exist, however, it is tilted slightly out-of-plane. It is
important to note that the PHE can persist outside of the tilted CSL phase as
the slightly out-of-plane magnetization persists once FFM is stabilized due to
any misalignment of the field and the plane. The different magnetic
configurations occurring when the external field is applied in different
directions are depicted in Fig. 3(a). For our conventional Hall effect
measurements we apply the field perpendicular to the plane of the platelet,
which is assumed to be nominally close to $c$ axis. Taking this symmetry
arrangement of current and magnetic field along with the modeled spin
structures of the magnetic textures we conclude that the PHE arises from the
magnetic moments in the tilted CSL and that they are pointing between the $c$
axis and the $ab$ plane.

For Cr1/3NbS2 the CSL has been observed via Lorentz transmission electron
microscopy (LTEM) when imaging along the $ab$ plane with a field applied
perpendicular to the $c$ axis [52]. For our magnetotransport measurements the
current is applied along the length of the platelet, which nominally in the
$ab$ plane. For the conventional Hall effect measurements we apply the field
perpendicular to the plane of the platelet, which is assumed to be nominally
along the $c$ axis. Taking this symmetry arrangement of current and magnetic
field along with the modeled spin structures of the magnetic textures
described in the previous section we conclude that the magnetic moments in the
tilted CSL are pointing between the $c$ axis and the $ab$ plane.

The PHE and AMR as function of angle in Cr1/3NbS2, is shown in Figs. 3(b,c).
The angle $\varphi$ between the current and the applied field is defined in
Fig. 3(b). A clear oscillatory signal of the form $\sin\varphi\cos\varphi$ for
the PHE, and $\sin^{2}\varphi$ for the AMR is observed and can be fit by Eq.
2. The amplitude of the oscillations gives the magnitudes of the PHE and AMR.

The observed temperature and field dependence of the PHE and AMR supports the
theory of a tilted-CSL in Cr1/3NbS2. It can be seen in Figs. 3(d,e) that in
all fields, close to the magnetic transition, the magnitudes of both the PHE
and AMR amplitudes increase. The field dependence of the PHE above and below
the magnetic transition are of particular note, as shown in Figs. 3(g) and
3(h), as there appear to be two components. Firstly, there exists a low-field
component below the magnetic transition, where there is a rapid increase in
the PHE amplitude below 100 mT. This component disappears above the transition
temperature. We can therefore attribute this to the magnetic state as the
transition from the CSL phase to the FFM phase is expected to occur around 100
mT [53]. The high-field component is the traditional component we would expect
to observe from a ferromagnetic system. Indeed, the maximum in the PHE
amplitude at fields above 1 T approximately corresponds to the Curie
temperature determined from magnetization. To obtain this PHE component the
only requirement is that the sample has some finite in-plane magnetization
which we have confirmed in our magnetization measurements (see Fig. 1(c).
Further planar Hall resistivity as a function of field data across all
measured temperatures is shown in the supplemental material [32].

A similar oscillatory signal, this time of the form $\sin^{2}\varphi$, is also
observed when measuring the longitudinal resistivity in this configuration as
shown in Fig. 3(d). This longitudinal resistivity is generally referred to as
the anisotropic magnetoresistance. From the resistivity tensor it is expected
that the amplitude of the PHE and AMR oscillations
$\left[\mathrm{i.e.}~{}\left(\rho_{\parallel}-\rho_{\perp}\right)\right]$
should be of equal magnitude [50]. However, any small misalignment of the
sample can cause the ordinary magnetoresistance to leak into the AMR signal.
The value for the PHE signal is significantly more reliable, however, as the
effects of any leaking of the ordinary Hall effect into the PHE due to a
misalignment is antisymmetric and can be removed by averaging the positive and
negative field data.

##  VI Emergence of a giant topological Hall effect

An anti-symmetrization procedure was first performed on the Hall effect data
to remove any symmetric components originating from the magnetoresistance and
PHE to simplify the extraction of the THE component. The individual
contributions to the Hall signal in Cr1/3NbS2, namely the OHE and AHE are
extracted from the remaining Hall signal following the procedure described in
Appendix A. A finite and significant component to the Hall signal remains once
the OHE and AHE are extracted, we attribute this contribution to the THE. The
individual Hall components at several temperatures are shown in Figs. 4(a-c)
[Further conventional Hall resistivity as a function of field data across all
measured temperatures are shown in the supplemental material [32].]. The
ordinary Hall coefficient $R_{\mathrm{H}}$ is positive across all temperatures
indicating hole-like conduction in Cr1/3NbS2; at 30 K
$R_{\mathrm{H}}=7.31(2)\times 10^{-9}$ m3/C gives a hole-density of
$8.53(2)\times 10^{26}$ m-3 consistent with what has previously been reported
[54]. The AHE component is seen to be always negative, diminishing in
magnitude with increasing temperature. However, an anomalous Hall component is
still noticeable well into the paramagnetic regime. This behavior has
previously been observed in MgZnO/ZnO heterostructures and explained using the
Giovannini–Kondo model where localized paramagnetic centers can generate an
anomalous Hall component from the skew scattering of electrons off these
centers [55]. The skew, side and intrinsic scattering terms are all found to
be significant below the magnetic transition temperature, gradually increasing
in magnitude as the temperature decreases. The scattering term can be a useful
measure of the quality of crystal used as it is normally only expected to be
significant in near perfect crystals [56, 42], hence the high quality of the
crystals used in this experiment is particularly notable.

Finally we discuss $\rho_{\mathrm{xy}}^{\mathrm{THE}}$ in the light of CSL for
Cr1/3NbS2. The temperature dependence of the maximum magnitude of
$\rho_{\mathrm{xy}}^{\mathrm{THE}}$ is shown in Fig. 4(d). At the transition
temperature the sign of the THE is seen to flip from positive
($T>T_{\mathrm{c}}$) to negative ($T<T_{\mathrm{c}}$). Below the magnetic
transition the topological Hall effect has a maximum magnitude of
$2.33~{}\mu\Omega$ cm at 90 K. The change in the sign of the
$\rho_{\mathrm{xy}}^{\mathrm{THE}}$ is likely indicative of a change in the
spin polarization of the charge carriers as the temperature passes through the
transition. Parallels can be drawn and discussed with the AHE and how it
behaves at the phase boundary. It has been shown that due to thermal spin
fluctuations, the skew scattering contributions to the AHE become more
important and can cause a change of sign [57]. Assuming that the topological
Hall effect is similar to the anomalous Hall effect, with the main difference
being that the THE emerges from non-coplanar spins instead of spin-orbit
coupling, one may conjecture that a similar mechanism occurs at the phase
transition. $\rho_{\mathrm{xy}}^{\mathrm{THE}}$ is significantly larger than
has been reported in other helical ground state antiferromagnets such as MnSi
($\sim-7$ n$\Omega$ cm) [20, 58] and FeGe ($\sim-136$ n$\Omega$ cm) [21]. Both
MnSi and FeGe host skyrmion lattices which are thought to only induce a small
topological Hall effect. This suggests that the mechanism for the THE in
Cr1/3NbS2 is provided through some other magnetic spin texture, namely the
tilted CSL. Similarly, if the rotation period is smaller, then the emergent
field is expected to be larger, which can lead to a larger
$\rho_{\mathrm{xy}}^{\mathrm{THE}}$. It is expect that the change of period
will lead to a non-linear change in the topological charge density once the
tilted CSL is stabilized since
$\mathcal{C}_{ijk}=\vec{m}_{i}\cdot(\vec{m}_{j}\times\vec{m}_{k})$, hence
important changes in the THE will occur. Furthermore, the topological Hall
effect is a complex phenomena and depends on other details of the electronic
structure such as the scattering rate off the magnetic atoms and the density
of the charge carriers. A contour map of $\rho_{\mathrm{xy}}^{\mathrm{THE}}$
[see Fig. 4(e)] reveals the presence of $\rho_{\mathrm{xy}}^{\mathrm{THE}}$
across most temperatures below the transition temperature before becoming
negligible at 2 K. As the temperature decreases, the fluctuations in the
magnetic spins from the chiral phase would be expected to reduce. Therefore,
as the temperature is reduced the magnitude of the THE would be expected to
reduce as well. The region of increased magnitude in
$\rho_{\mathrm{xy}}^{\mathrm{THE}}$ corresponds to the same region where the
CSL has been reported by LTEM [26, 30, 59] and ac magnetization [60]. This
strongly suggests that the $\rho_{\mathrm{xy}}^{\mathrm{THE}}$ is due to a
non-coplanar spin structure arising from a tilted CSL.

##  VII Summary

Remarkably, the chiral helimagnet Cr1/3NbS2 is found to host both a giant THE
and PHE. We show that the observation of a PHE and THE in Cr1/3NbS2 indicates
the presence of a tilted CSL. Furthermore, the close correspondence of the PHE
and AMR in this complex non-coplanar magnet validates the theory that the
connection between these two quantities still holds in these more complex
systems. The conventional Hall measurements performed in this study
demonstrate that non-coplanar spins arising from a tilted CSL can indeed
generate the emergent field necessary for the THE. Studying Cr1/3NbS2, where
the phase diagram has been extensively mapped out, has allowed a one-to-one
correspondence to be drawn between the relative magnetic phases and magneto-
transport properties observed. Notably, the THE exhibited in Cr1/3NbS2 is
several orders of magnitude larger than those of similar helimagnetic
materials where skyrmions are observed. This study marks an important step
forward in demonstrating that magneto-transport measurements, and more
specifically the THE, can be used to detect not just skyrmions but also non-
coplanar spin textures such as a tilted CSL.
Data will be made available via Warwick Research Archive Portal [61].

##  Appendix A Fitting procedure for the Hall effect

To estimate the contribution the topological Hall effect makes to the Hall
resistivity, the following fitting procedure was used. To simplify the problem
we first remove any Hall contributions from the PHE and any magnetoresistance.
To do this an antisymmetrization procedure was performed using
$\rho(+H)^{\prime}=[\rho(+H^{\mathrm{inc}})-\rho(-H^{\mathrm{dec}})]/2$ and
$\rho(-H)^{\prime}=[\rho(-H^{\mathrm{inc}})-\rho(+H^{\mathrm{dec}})]/2$ where
$\rho(+H^{\mathrm{inc}})$ is the Hall resistivity in the positive field
quadrants when the field is increasing in magnitude, $\rho(+H^{\mathrm{dec}})$
is the Hall resistivity in the positive field quadrants when the field is
decreasing in magnitude, $\rho(-H^{\mathrm{inc}})$ is the Hall resistivity in
the negative field quadrants when the field is increasing in magnitude and
$\rho(-H^{\mathrm{inc}})$ is the Hall resistivity in the negative field
quadrants when the field is decreasing in magnitude. For fields greater than
some critical field value, $\rho_{\mathrm{xy}}^{\mathrm{THE}}$ should be equal
to zero as forced ferromagnetism should have no spin chirality. The
contribution from the ordinary Hall effect $\rho_{\mathrm{xy}}^{\mathrm{OHE}}$
and the anomalous Hall effect $\rho_{\mathrm{xy}}^{\mathrm{AHE}}$ can be
calculated by linearizing the data using
$\rho_{\mathrm{xy}}/B=\alpha\rho_{\mathrm{xx}}^{2}M/H+R_{\mathrm{H}}$ (not
shown). Here $\alpha=\left(S_{\mathrm{Int}}+S_{\mathrm{Side}}\right)$, where
$S_{\mathrm{Int}}$ is the intrinsic contribution and $S_{\mathrm{Side}}$ is
the side jump contribution, only provides the $\rho_{\mathrm{xx}}^{2}$
contribution for the AHE but we now have a value for $R_{\mathrm{H}}$. To
estimate the full contribution of $\rho_{\mathrm{xy}}^{\mathrm{AHE}}$ we can
subtract $\rho_{\mathrm{xy}}^{\mathrm{OHE}}$ from the total Hall resistivity
and by using the following equality
$\left(\rho_{\mathrm{xy}}-\rho_{xy}^{\mathrm{OHE}}\right)/\rho_{\mathrm{xx}}M=\alpha\rho_{\mathrm{xx}}+S_{\mathrm{Skew}}$,
where $S_{\mathrm{Skew}}$ is the skew scattering contribution, the data can be
linearized. This allows for $\alpha$ and $S_{\mathrm{Skew}}$ to be extracted,
(not shown).

##  Appendix B Simulations of the Spin Dynamics

The magnetic state of the system is described by an extended Heisenberg
Hamiltonian:

$\mathcal{E}_{M}=-\frac{1}{2}\sum_{ij}J_{ij}\,\vec{M}_{i}\cdot\vec{M}_{j}+\frac{1}{2}\sum_{ij}\vec{D}_{ij}\cdot(\vec{M}_{i}\times\vec{M}_{j})-\sum_{i}\vec{M}_{i}\cdot\vec{B}\quad.$
(3)

The first term in the Hamiltonian designates the isotropic Heisenberg exchange
interaction, the second is the Dzyaloshinskii-Moriya interaction, and the
third is the Zeeman coupling to an external magnetic field. The following
parameters are used: a nearest neighbor exchange $J_{ij}=10.0$ meV, and
nearest neighbor DMI $D=5.0$ meV, and magnetic moment of
$M=3~{}\mu_{\mathrm{B}}$ with magnetic field of $6$ T. The field is oriented
in two directions in practice either along the $z$ or tilted at $45^{\circ}$
away from the $z$-axis in the $xz$ plane. The minimization of the energy is
performed using the Spirit code, which uses an algorithm based on the Landau-
Lifshitz-Gilbert equation [62].

##  Appendix C Topological Hall Effect from Chiral Phases

As discussed in the previous section, neither the helimagnetic phase nor the
chiral soliton lattice will display a topological Hall effect since the scalar
chirality must vanish, i.e.
$\mathcal{C}_{ijk}=\vec{m}_{i}\cdot(\vec{m}_{j}\times\vec{m}_{k})$. However,
for the case of a tilted-CSL, a finite topological Hall effect contribution is
possible. To illustrate this we consider three different cases: helimagnet
(A), CSL (B) and a tilted-CSL (C), these magnetic states are shown in Fig.
5(a). Using these configurations, we evaluate the topological charge density
between nearest neighbors:

$\mathbf{Q}_{i}=\vec{m}_{i-1}\cdot\left(\vec{m}_{i}\times\vec{m}_{i+1}\right).$
(4)

The results are shown in Fig. 5(b), where we found that $\mathbf{Q}_{i}$
systematically vanishes for the configurations (A) and (B), while a finite
contribution is present for (C). Thus, as a consequence of applying a magnetic
field in the $(xz)$ plane the magnetic state becomes non-coplanar.
Furthermore, in these metallic systems, coupling beyond nearest neighbors is
also possible, thus more contributions to the THE are present. To summarize,
both the PHE and THE are consequences of the tilted CSL. Furthermore, a vector
chirality contribution $(\vec{m}_{i}\times\vec{m}_{j})$ to the Hall effect can
be relevant in the non-centrosymmetric systems treated in this manuscript,
this was discussed in Ref. [47]. However, for the sake of simplicity, we only
interpret the experimental data in terms of the THE.

###### Acknowledgements.

We would like to acknowledge Tom Orton, Patrick Ruddy and Daisy Ashworth for
their technical support. This work was financially supported by three
Engineering and Physical Sciences Research Council grants: EP/T005963/1,
EP/M028941/1 (PRETAMAG), and the UK Skyrmion Project Grant, EP/N032128/1.

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Figure 1: (Color online) (a) Crystal structure of Cr1/3NbS2. The purple, green
and gray spheres denote the chromium, niobium and sulfur atoms, respectively.
The dashed line indicates the unit cell. (b) A single crystal of Cr1/3NbS2
grown by chemical vapor transport that was used for all the measurements
presented in this work, along with a schematic of the 5-point wire
configuration used for the longitudinal and Hall resistivity measurements. (c)
Magnetization versus temperature curves for Cr1/3NbS2 for fields of up to 7 T
applied perpendicular to the platelet. (d) Magnetization as a function of
field applied parallel to the $c$-axis at several temperatures. (e)
Temperature dependence of the longitudinal resistivity $\rho_{\mathrm{xx}}$ in
several magnetic fields applied parallel to the $c$-axis. Magnetoresistances
of Cr1/3NbS2 between -9 and 9 T at (f) 2 to 30 K, (g) 40 to 80 K, (h) 90 to
120 K and (i) 130 to 300 K. Figure 2: (Color online) Field-dependent Hall
resistivity $\rho_{\mathrm{xy}}$ at 10, 70, 110, 130, 190 and 230 K. There is
a significant change in the behavior of the Hall effect as the temperature is
lowered through $T_{\mathrm{c}}$. It is observed that the shape of the field
dependence in Hall effect changes from a smooth almost parabolic shape to a
sharp triangular shape. Figure 3: (Color online) (a) Possible magnetic ground
states as function of the magnetic field direction: CSL, chiral conical phase,
and tilted CSL. The chain contains 30 atoms, the model parameters are $J=10$
meV (exchange), $D=0.5J$ (DMI),$B=6$ T with a $45^{\circ}$ tilt with respect
to the $c$ axis. The simulations were performed using the atomistic spin
Spirit code [62]. Further details can be found in the Appendix. B.(b)
Schematic of the planar Hall effect and anisotropic magnetoresistance set up.
(c, d) PHE $\rho_{\mathrm{xy}}$ and AMR $\rho_{\mathrm{xx}}$ taken at 5 K. The
dashed lines indicate fits to the PHE and AMR (as described in the text). (e,
f) PHE and AMR amplitudes, $\left(\rho_{\parallel}-\rho_{\perp}\right)$, as a
function of temperature at various applied fields. The field dependence of the
PHE amplitudes at (g) 10 and (h) 130 K. Figure 4: (Color online) Observed
$\rho_{\mathrm{xy}}$ (green) as a function of applied magnetic field at (a)
10, (b) 70 and (c) 110 K with the calculated contributions of
$\rho_{\mathrm{xy}}^{\mathrm{OHE}}$ (red), $\rho_{\mathrm{xy}}^{\mathrm{AHE}}$
(blue) and $\rho_{\mathrm{xy}}^{\mathrm{THE}}$ (purple). (d) The maximum
$\rho_{\mathrm{xy}}^{\mathrm{THE}}$ as a function of temperature. (e) Contour
map of $\rho_{\mathrm{xy}}^{\mathrm{THE}}$ as function of field and
temperature. The purple points indicate the position of the transition from
resistivity as seen in Fig. 1(b). The phase boundaries are approximated from
the $\rho_{\mathrm{xx}}$ vs temperature (purple) and field (orange) data shown
in Fig. 1. The dashed lines are estimations of phase boundaries in Cr1/3NbS2
marking out the chiral, FFM and paramagnetic (PM) phases. Figure 5: (a) Three
spin configurations possible in Cr1/3NbS2: Helimagnet (A), chiral soliton
lattice (B), and tilted chiral soliton lattice (C). (b) Evolution of the
topological charge density for a Helimagnet (A), a chiral soliton lattice (B)
and a tilted chiral soliton lattice (C). The topological charge density for
the helimagnet state (A) is zero for all i so is not visible.


# Hybrid Relay-Reflecting Intelligent Surface-Aided Covert Communications

Jinsong Hu, Xiaoqiang Shi, Shihao Yan,
Youjia Chen, Tiesong Zhao, and Feng Shu,

###### Abstract

In this paper, we propose to use a hybrid relay-reflecting intelligent surface
(HR-RIS) to enhance the performance of a covert communication system.
Specifically, the HR-RIS consists of passive reflecting elements and active
relay elements to manipulate the wireless signals from a transmitter to a
desired receiver while ensuring the covertness of the transmission via
avoiding such signals being detected by a warden. To fully explore the
benefits offered by the HR-RIS, we first formulate the joint design of the
transmit power and relay/reflection coefficients of the HR-RIS as an
optimization problem to maximize the covert rate subject to a covertness
constraint. To tackle the solution to this optimization problem, we then
derive a closed-form expression for an upper bound on covert rate, based on
which we develop an alternate algorithm to solve the formulated optimization
problem. Our examination shows that the HR-RIS outperforms the traditional RIS
in term of achieving a higher covert rate. Interestingly, we also observe the
major part of the performance gain brought by the HR-RIS can be obtained by a
small number of active relay elements (e.g., 5) and further increasing this
number does not improve the covert communication performance.

###### Index Terms:

Covert communication, hybrid relay-RIS, reflection beamforming, transmit power
design

##  I Introduction

Covert communication, or low probability of detection (LPD) communication, has
emerged as a cutting-edge secure communication technique aiming at hiding
communications from a watchful warden [1]. Considering different practical
constraints and limitations, covert communication in different scenarios was
widely studied in the past few years. For example, covert communication in
relay networks was studied, and it was proved that the relay can stealthily
transmit its own information to the desired destination in the process of
forwarding data [2]. In [3], the authors presented the potentials of using the
recently emerged intelligent reflecting surface (IRS), also known as
reflecting intelligent surface (RIS), to improve covert communication
performance.

Following [3], several RIS-aided covert communication approaches have been
developed. Specifically, RIS is a flat surface composed of a large number of
reconfigurable and low-cost passive reflective elements, each of which is
capable of controlling the phase and amplitude of the incident signal for
optimal reflection, making the wireless channel between the transmitter and
receiver more favorable for communication [4]. For example, the authors of [5]
investigated the multiple-input-multiple-output (MIMO) covert communication
assisted by RIS, where the covert rate was maximized by jointly designing the
transmit covariance matrix and phase shift matrix. In [6], the authors
considered the design of a latency constrained covert communication system
with the assistance of RIS with global CSI and without the warden’s
instantaneous CSI, respectively.

It should be emphasized that the existing studies on RIS-assisted covert
communication adopted the fully-passive beamforming strategies. A main
limitation of the traditional RIS compared to relays is the that the passive
reflection limits the degrees of freedom in the beamforming. Thus, the
traditional RIS cannot outperform a half-duplex relay when the number of the
elements in the RIS is not sufficiently large [7]. This implies that if a few
passive elements of the RIS are replaced by active ones, the traditional RIS
becomes a hybrid relay reflecting intelligent surface (HR-RIS) [8], which may
lead to that the HR-RIS is able to significantly improve the assisted system
performance. Therefore, in this work we propose a HR-RIS-based covert
communication scheme, which applies active beamforming by adding some relaying
elements at the traditional RIS to mitigate the limitation of the passive
reflecting.

The main contributions of this paper are summarized as follows. We consider
the covert communication from a transmitter (Alice) to a receiver (Bob) with
the aid of an HR-RIS. Our goal is to jointly optimize the transmit power and
the HR-RIS reflection matrix, including its phase shifts and amplitudes, to
maximize the covert rate at Bob. In the optimization problem of interest, the
coefficient matrix of HR-RIS is optimized by an alternate optimization (AO)
method and then the optimal transmit power is obtained under the covertness
constraint in terms of the Kullback-Leibler (KL) divergence. Our study shows
that the HR-RIS can significantly improve the covert rate compared to
conventional RIS.

##  II System Model

###  II-A Considered Scenario and Adopted Assumptions

As shown in Fig. 1, we propose a covert communication transmission scheme
assisted by an HR-RIS, where a transmitter (Alice) intends to send
confidential information to a legitimate receiver (Bob) with the aid of the
HR-RIS, while a warden (Willie) attempts to detect the existence of this
transmission. Specifically, there are two links from Alice to Bob, i.e., the
direct link from Alice to Bob, and the reflection/relaying link form Alice to
the HR-RIS and then to Bob. Similarly, there are two paths from Alice to
Willie. Alice, Bob and Willie are assumed to be equipped with $N_{a}$,
$N_{b}$, and $N_{w}$ antennas, respectively. In addition, it is assumed that
the signals reflected by the HR-RIS twice or more are ignored due to the
significant path loss [9].

The HR-RIS is assumed to be equipped with $N$ elements, including $M$ passive
reflecting elements and $K$ active relaying elements (i.e., $M+K=N$). The
passive reflecting elements are implemented by phase shifter, while the active
relaying elements can tune the phase and amplitude of the incident signals. We
assume that the active elements work in the AF mode. Therefore, for $K=0$, HR-
RIS converges to a traditional RIS. For $K=N$, by contrast, it becomes an AF
relay station equipped with $N$ antennas. Hence, in this work we have $1\leq
K\leq N$. Furthermore, similar to the conventional RIS, we assume that each
(active/passive) element of HR-RIS can independently reflect the received
signals.

For the HR-RIS, $\mathbb{Q}$ represents the set of active relay elements. We
define $\mathbf{\Theta}=\mathbf{\Phi}+\mathbf{\Psi}$, where
$\mathbf{\Theta}=\mathrm{diag}\\{\theta_{1},\cdots,\theta_{N}\\}\in\mathbb{C}^{N\times
N}$ ,
$\mathbf{\Phi}=\mathrm{diag}\\{\phi_{1},\cdots,\phi_{N}\\}\in\mathbb{C}^{N\times
N}$ , and
$\mathbf{\Psi}=\mathrm{diag}\\{\psi_{1},\cdots,\psi_{N}\\}\in\mathbb{C}^{N\times
N}$ , where $\mathbf{\Phi}$ and $\mathbf{\Psi}$ denote the reflection
coefficients of passive elements and active elements, respectively. Therefore,
we have

$\displaystyle\mathbf{\theta}_{n}=\left\\{\begin{array}[]{ll}|\beta_{n}|e^{j\mu_{n}},&\mathrm{if}\quad
n\in\mathbb{Q},\\\ e^{j\mu_{n}},&{\mathrm{otherwise,}}\end{array}\right.$ (3)
$\displaystyle\phi_{n}=\left\\{\begin{array}[]{ll}0,&\mathrm{if}\quad
n\in\mathbb{Q},\\\ e^{j\mu_{n}},&{\mathrm{otherwise,}}\end{array}\right.$ (6)

and

$\displaystyle\psi_{n}=\left\\{\begin{array}[]{ll}|\beta_{n}|e^{j\mu_{n}},&\mathrm{if}\quad
n\in\mathbb{Q},\\\ 0,&{\mathrm{otherwise,}}\end{array}\right.$ (9)

where $\mu_{n}\in[0,2\pi)$ represents the phase shift. We notice that
$|\beta_{n}|=1$ for $n\notin\mathbb{Q}$, and $|\beta_{n}|$ for
$n\in\mathbb{Q}$ is determined by the total power of the active elements,
which will be discussed later.

Figure 1: HR-RIS aided covert communication

###  II-B Transmission from Alice to Bob

When Alice transmits confidential message, the received signal at Bob is given
by

$\displaystyle\mathbf{y}_{b}=$
$\displaystyle\sqrt{P_{a}}\mathbf{H}_{rb}\mathbf{\Psi}\mathbf{H}_{ar}\mathbf{x}+\sqrt{P_{a}}\mathbf{H}_{rb}\mathbf{\Phi}\mathbf{H}_{ar}\mathbf{x}$
$\displaystyle+\sqrt{P_{a}}\mathbf{H}_{ab}\mathbf{x}+\mathbf{H}_{rb}\mathbf{\Psi}\mathbf{n}_{r}+\mathbf{n}_{b}$
$\displaystyle=\sqrt{P_{a}}(\mathbf{H}_{rb}\mathbf{\Theta}\mathbf{H}_{ar}+\mathbf{H}_{ab})\mathbf{x}+\mathbf{n}_{bt},$
(10)

where
$\mathbf{n}_{bt}=\mathbf{H}_{rb}\mathbf{\Psi}\mathbf{n}_{r}+\mathbf{n}_{b}$
represents the total effective noise at the Bob. $\mathbf{H}_{ar}$ and
$\mathbf{H}_{ab}$ are the steering vectors from Alice to HR-RIS and the
steering vector from Alice to Bob, respectively, and $\mathbf{H}_{rb}$ denotes
the steering vector from HR-RIS to Bob. In addition, $\mathbf{n}_{r}$ $\sim$
$\mathcal{CN}(\mathbf{0},\sigma_{r}^{2}\mathbf{I}_{K})$ and $\mathbf{n}_{b}$
$\sim$ $\mathcal{CN}(\mathbf{0},\sigma_{b}^{2}\mathbf{I}_{N_{b}})$ are the
complex additive white Gaussian noise (AWGN) space vectors at the $K$ active
elements of the HR-RIS and at the Bob, respectively. For simplicity, we assume
that $\sigma_{r}^{2}=\sigma_{b}^{2}$ and $\mathbf{n}_{bt}$ $\sim$
$\mathcal{CN}(\mathbf{0},\sigma_{b}^{2}(\mathbf{I}_{N_{b}}+\mathbf{H}_{rb}\mathbf{\Psi}\mathbf{\Psi}^{H}\mathbf{H}_{rb}^{H}))$.
$\mathbf{x}$ is the signal vector transmitted by Alice, with each elements
following $\mathcal{CN}(0,1)$, and $P_{a}$ is Alice’s transmit power.

Following (10), the transmit power of the active elements at the HR-RIS can be
expressed as

$\displaystyle
P_{r}=\mathrm{trace}(\mathbf{\Psi}(\mathbf{H}_{rb}\mathbf{H}_{rb}^{H}P_{a}+\sigma_{b}^{2})\mathbf{\Psi}^{H}),$
(11)

which should meet the constraint condition $P_{r}\leq P_{r}^{\mathrm{max}}$,
where $P_{r}^{\mathrm{max}}$ is the maximum transmit power of all the $K$
active elements. We assume that Bob’s channel state information (CSI) is
perfectly known by himself through channel estimation [10]. Following (10),
Bob’s covert rate in the HR-RIS-aided covert communication system can be
expressed as [8]

$\displaystyle
f(\mathbf{\Theta},P_{a})=\log_{2}|\mathbf{I}_{Nb}+\mathbf{\Omega}_{b}|,$ (12)

where
$\mathbf{\Omega}_{b}={P_{a}\mathbf{U}_{b}\mathbf{R}^{-1}}/{\sigma_{b}^{2}}$,
the signal covariance matrix is given by
$\mathbf{U}_{b}=(\mathbf{H}_{rb}\mathbf{\Theta}\mathbf{H}_{ar}+\mathbf{H}_{ab})(\mathbf{H}_{rb}\mathbf{\Theta}\mathbf{H}_{ar}+\mathbf{H}_{ab})^{H}$,
and the aggregate noise covariance matrix is given by
$\mathbf{R}=(\mathbf{I}_{N_{b}}+\mathbf{H}_{rb}\mathbf{\Psi}\mathbf{\Psi}^{H}\mathbf{H}_{rb}^{H})$
$\in$ $\mathbb{C}^{N_{b}\times N_{b}}$.

###  II-C Binary Hypothesis Testing at Willie

In this work, we focus on the delay-constrained covert communication, that is,
the number of channel uses $L$ is finite. In order to detect the existence of
a transmission, Willie attempts to distinguish the following two hypotheses:

$\displaystyle\mathbf{y}_{w}=\left\\{\begin{array}[]{ll}\mathbf{n}_{w},&\hbox{$\mathcal{H}_{0}$,}\\\
\sqrt{P_{a}}(\mathbf{H}_{rw}\mathbf{\Theta}\mathbf{H}_{ar}+\mathbf{H}_{aw})\mathbf{x}+\\\
\mathbf{H}_{rw}\mathbf{\Psi}\mathbf{n}_{r}+\mathbf{n}_{w},&\hbox{$\mathcal{H}_{1}$,}\end{array}\right.$
(16)

where $\mathbf{H}_{aw}$ and $\mathbf{H}_{rw}$ are the steering vector from
Alice to Willie and the steering vector from HR-RIS to Willie, respectively,
and $\mathbf{n}_{w}$ $\sim$
$\mathcal{CN}(\mathbf{0},\sigma_{w}^{2}\mathbf{I}_{N_{w}})$ is the AWGN space
vector at Willie. $\mathcal{H}_{0}$ denotes the null hypothesis in which Alice
does not transmit, and $\mathcal{H}_{1}$ denotes the alternative hypothesis
where Alice transmits signals. Similarly, we assume that
$\sigma_{w}^{2}=\sigma_{r}^{2}$. Therefore, the total noise power at Willie is
$\mathbf{n}_{wt}$ =
($\mathbf{H}_{rw}\mathbf{\Psi}\mathbf{n}_{r}+\mathbf{n}_{w}$) $\sim$
$\mathcal{CN}(\mathbf{0},\sigma_{w}^{2}(\mathbf{I}_{N_{w}}+\mathbf{H}_{rw}\mathbf{\Psi}\mathbf{\Psi}^{H}\mathbf{H}_{rw}^{H}))$
under $\mathcal{H}_{1}$.

Considering the worst-case scenario for covert communications, we assume that
Willie knows the channels perfectly. The probability density function of
$\mathbf{y}_{w}$ under $\mathcal{H}_{0}$ and $\mathcal{H}_{1}$ can be
respectively given by

$\displaystyle f(\mathbf{y}_{w}|\mathcal{H}_{0})=$
$\displaystyle\mathcal{CN}(\mathbf{0},\sigma^{2}_{w}\mathbf{I}_{N_{w}}),$ (17)
$\displaystyle f(\mathbf{y}_{w}|\mathcal{H}_{1})=$
$\displaystyle\mathcal{CN}(\mathbf{0},P_{a}\mathbf{U}_{w}+\sigma^{2}_{w}(\mathbf{I}_{N_{w}}+\mathbf{H}_{rw}\mathbf{\Psi}\mathbf{\Psi}^{H}\mathbf{H}_{rw}^{H})).$

As such, the signal-to-interference-plus-noise ratio (SINR) at Willie under
$\mathcal{H}_{1}$ is given by

$\displaystyle\gamma_{w}=\frac{|\mathbf{U}_{w}|P_{a}}{(|\mathbf{M}|\sigma_{w}^{2})},$
(18)

where
$\mathbf{U}_{w}=(\mathbf{H}_{rw}\mathbf{\Theta}\mathbf{H}_{ar}+\mathbf{H}_{aw})(\mathbf{H}_{rw}\mathbf{\Theta}\mathbf{H}_{ar}+\mathbf{H}_{aw})^{H}$
and
$\mathbf{M}=(\mathbf{I}_{N_{w}}+\mathbf{H}_{rw}\mathbf{\Psi}\mathbf{\Psi}^{H}\mathbf{H}_{rw}^{H})$.
We note that $\mathbf{U}_{w}$ and $\mathbf{M}$ are Hermitian matrices. Then,
we perform the eigenvalue decomposition (EVD) on the above two matrices, which
can be written as $\mathbf{U}_{w}=\mathbf{G}\mathbf{\Xi}\mathbf{G}^{-1}$ and
$\mathbf{M}=\mathbf{J}\mathbf{\Lambda}\mathbf{J}^{-1}$, where $\mathbf{G}$ and
$\mathbf{J}$ are matrices of eigenvectors of $\mathbf{U}_{w}$ and
$\mathbf{M}$, respectively, and $\mathbf{G}$ and $\mathbf{J}$
$\in\mathbb{C}^{N_{w}\times N_{w}}$,
$\mathbf{\Xi}=\mathrm{diag}\left\\{\omega_{1},\omega_{2},\cdots,\omega_{N_{w}}\right\\}$,
$\omega_{n}$ is the $n$-th eigenvalue of $\mathbf{U}_{w}$,
$\mathbf{\Lambda}=\mathrm{diag}\left\\{\kappa_{1},\kappa_{2},\cdots,\kappa_{N_{w}}\right\\}$,
$\kappa_{n}$ is the $n$-th eigenvalue of $\mathbf{M}$. As such, we have
$|\mathbf{U}_{w}|=\prod_{i=1}^{N_{w}}\omega_{i}$ and
$|\mathbf{M}|=\prod_{i=1}^{N_{w}}\kappa_{i}$.

In covert communications, $\mathcal{D}_{01}\leq 2\epsilon^{2}$ is generally
adopted as the covertness constraint, where $\epsilon$ is a small value to
determine the required covertness level and the KL divergence
$\mathcal{D}_{01}$ is given by [11]

$\displaystyle\mathcal{D}_{01}=L\left[\ln(1+\gamma_{w})-\frac{\gamma_{w}}{1+\gamma_{w}}\right].$
(19)

###  II-D Problem Formulation

In this part, we jointly design the transmit power at Alice and
relay/reflection coefficients of the HR-RIS to maximize the covert rate at Bob
subject to the covertness and other constraints, of which the optimization
problem can be formulated as

$\displaystyle\mathrm{(P1)}:$
$\displaystyle\mathop{\mathrm{max}}\limits_{\mathbf{\Theta},P_{a}}f(\mathbf{\Theta},P_{a}),$
(20a) $\displaystyle\mathbf{s.t.}~{}\mathcal{D}_{01}\leq 2\epsilon^{2},$ (20b)
$\displaystyle~{}~{}~{}~{}~{}|\mathbf{\beta}_{n}|=1,~{}\mathrm{for}~{}~{}n\notin\mathbb{Q},$
(20c) $\displaystyle~{}~{}~{}~{}~{}P_{a}\leq P_{a}^{\mathrm{max}}.$ (20d)

##  III COVERT COMMUNICATION DESIGN

In this section, our goal is to maximize the covert rate at Bob by jointly
designing $P_{a}$ and $\mathbf{\Theta}$. We propose an alternate algorithm to
optimize $P_{a}$ and $\mathbf{\Theta}$. Specifically, we first optimize
$\mathbf{\Theta}$ for a given $P_{a}$ and the object function is transformed
into a form that is easy to handle. Then, we optimize $P_{a}$ for a given
$\mathbf{\Theta}$.

###  III-A Optimizing $\mathbf{\Theta}$ for a Given $P_{a}$

First, we randomly generate the coefficient of HR-RIS and use
$\mathcal{D}_{01}=2\epsilon^{2}$ to get the feasible $P_{a}$. The object
function $f(\mathbf{\Theta},P_{a})$ is non-convex with respect to
$\mathbf{\Theta}$. In addition, the feasible set of (P1) is non-convex due to
the unit-modulus constraint (20c). Therefore, (P1) is difficult to be tackled.
Thus, we approximate the objective function $f({\mathbf{\Theta}},P_{a})$ by
using its upper bound $f_{0}({\mathbf{\Theta}},P_{a})$, which can be written
as

$\displaystyle
f(\mathbf{\Theta},P_{a})=\log_{2}\left|\mathbf{I}_{N_{b}}+\frac{P_{a}\mathbf{U}_{b}\mathbf{R}^{-1}}{\sigma_{b}^{2}}\right|$
$\displaystyle=\log_{2}|\mathbf{R}+\rho\mathbf{U}_{b}|-\log_{2}|\mathbf{R}|$
$\displaystyle\overset{a}{\leq}\log_{2}|\mathbf{R}+\rho\mathbf{U}_{b}|$
$\displaystyle=\log_{2}\Big{|}\mathbf{R}+\rho(\mathbf{H}_{rb}\mathbf{\Theta}\mathbf{H}_{ar}+\mathbf{H}_{ab})(\mathbf{H}_{rb}\mathbf{\Theta}\mathbf{H}_{ar}+\mathbf{H}_{ab})^{H}\Big{|}$
$\displaystyle=f_{0}(\mathbf{\Theta},P_{a}),$ (21)

where $\rho=P_{a}/\sigma_{b}^{2}$ and $a$ is achieved by set
$\mathbb{Q}=\emptyset$. We note that this upper bound becomes tighter as
$\log_{2}|\mathbf{R}|$ decreases. Considering the case where the active
elements employed at the HR-RIS are few, we know that $\mathbf{\Psi}$ is
sparse, which means $\log_{2}|\mathbf{R}|$ is small as per its definition
given (12). Therefore, for a given $P_{a}$ the optimization problem (P1) can
be rewritten as

$\displaystyle\mathrm{(P2)}:$
$\displaystyle\mathop{\mathrm{max}}\limits_{\mathbf{\Theta}}f_{0}(\mathbf{\mathbf{\Theta}},P_{a}),$
$\displaystyle~{}\mathbf{s.t.}~{}(\mathrm{\ref{a}b}),(\mathrm{\ref{a}c}),(\mathrm{\ref{a}d}).$
(22)

Although $f_{0}(\mathbf{\Theta},P_{a})$ is still non-convex, the optimal
reflection coefficient of HR-RIS can be obtained by our proposed method
detailed in the following subsection.

###  III-B Transformation of the Objective Function

Generally, the proposed solution is a sequential procedure where in each
iteration, a specific coefficient of HR-RIS is updated when the others are
fixed. Specifically, we let $\mathbf{a}_{n}^{H}\in\mathbb{C}^{N_{b}\times 1}$
denote the $n$-th row of $\mathbf{H}_{ab}$, and
$\mathbf{b}_{n}\in\mathbb{C}^{N_{b}\times 1}$ denote the $n$-th column of
$\mathbf{H}_{rb}$, i.e.,
$\mathbf{H}_{ar}=[\mathbf{a}_{1},\mathbf{a}_{2},\cdots,\mathbf{a}_{N}]^{H}$,
and $\mathbf{H}_{rb}=[\mathbf{b}_{1},\mathbf{b}_{2},\cdots,\mathbf{b}_{N}]$.
Since $\mathbf{\Psi}$ and $\mathbf{\Theta}$ are diagonal matrices, we have
$\mathbf{H}_{rb}\mathbf{\Theta}\mathbf{H}_{ar}=\sum_{n=1}^{N}\mathbf{\theta}_{n}\mathbf{b}_{n}\mathbf{a}_{n}^{H}$
and
$\mathbf{H}_{rb}\mathbf{\Psi}=\sum_{n\in\mathbb{Q}}\mathbf{\theta}_{n}\mathbf{b}_{n}$.
Hence, we can rewrite $f_{0}({\mathbf{\Theta}},P_{a})$ as

$\displaystyle f_{0}(\mathbf{\Theta},P_{a})$
$\displaystyle=\log_{2}\left|\mathbf{R}+\rho(\mathbf{H}_{rb}\mathbf{\Theta}\mathbf{H}_{ar}+\mathbf{H}_{ab})(\mathbf{H}_{rb}\mathbf{\Theta}\mathbf{H}_{ar}+\mathbf{H}_{ab})^{H}\right|$
$\displaystyle=\log_{2}\Big{|}\mathbf{I}_{N_{b}}+\sum_{i\in
Q}\mathbf{\theta}_{i}\mathbf{b}_{i}\mathbf{\theta}_{i}^{*}\mathbf{b}_{i}^{H}+\rho\sum_{i=1}^{N}|\theta_{i}|^{2}\mathbf{b}_{i}\mathbf{a}_{i}^{H}\mathbf{a}_{i}\mathbf{b}_{i}^{H}$
$\displaystyle+\rho\mathbf{H}_{ab}\mathbf{H}_{ab}^{H}+\rho\sum_{i=1}^{N}\sum_{j=1,j\neq
i}^{N}\mathbf{\theta}_{i}\mathbf{\theta}_{j}^{*}\mathbf{b}_{i}\mathbf{a}_{i}^{H}\mathbf{a}_{j}\mathbf{b}_{j}^{H}$
$\displaystyle+\rho\sum_{i=1}^{N}(\mathbf{H}_{ab}\mathbf{\theta}_{i}^{*}\mathbf{a}_{i}\mathbf{b}_{i}^{H}+\mathbf{\theta}_{i}\mathbf{b}_{i}\mathbf{a}_{i}^{H}\mathbf{H}_{ab}^{H})\Big{|}.$
(23)

Consequently, $f_{0}({\mathbf{\Theta}},P_{a})$ can be rewritten as

$\displaystyle
f_{0}({\mathbf{\Theta}},P_{a})=\log_{2}\left|\mathbf{A}_{n}+|\theta_{n}|^{2}\mathbf{B}_{n}+\mathbf{\theta}_{n}\mathbf{C}_{n}+\mathbf{\theta}_{n}^{*}\mathbf{C}_{n}^{H}\right|.$
(24)

For $n\in\mathbb{Q}$, $\mathbf{A}_{n}$, $\mathbf{B}_{n}$ and $\mathbf{C}_{n}$
can be respectively written as

$\displaystyle\mathbf{A}_{n}=\mathbf{I}_{N_{b}}+\sum_{i\in\mathbb{Q},i\neq
n}^{N}\mathbf{\theta}_{i}\mathbf{b}_{i}\sum_{i\in\mathbb{Q},i\neq
n}^{N}\mathbf{\theta}_{i}^{*}\mathbf{b}_{i}^{H}$
$\displaystyle+\rho\left(\sum_{i=1,i\neq
n}^{N}\mathbf{\theta}_{i}\mathbf{b}_{i}\mathbf{a}_{i}^{H}+\mathbf{H}_{ab}\right)\left(\sum_{i=1,i\neq
n}^{N}\mathbf{\theta}_{i}\mathbf{b}_{i}\mathbf{a}_{i}^{H}+\mathbf{H}_{ab}\right)^{H},$
$\displaystyle\mathbf{B}_{n}=\rho\mathbf{b}_{n}\mathbf{a}_{n}^{H}\mathbf{a}_{n}\mathbf{b}_{n}^{H}+\mathbf{b}_{n}\mathbf{b}_{n}^{H},$
$\displaystyle\mathbf{C}_{n}=\mathbf{b}_{n}\sum_{i\in\mathbb{Q},i\neq
n}^{N}\mathbf{\theta}_{i}^{*}\mathbf{b}_{i}^{H}+\rho\mathbf{b}_{n}\mathbf{a}_{n}^{H}\left(\mathbf{H}_{ab}^{H}+\sum_{i=1,i\neq
n}^{N}\mathbf{a}_{i}\mathbf{b}_{i}^{H}\mathbf{\theta}_{i}^{*}\right).$

For $n\notin\mathbb{Q}$, $\mathbf{A}_{n}$, $\mathbf{B}_{n}$ and
$\mathbf{C}_{n}$ are defined as

$\displaystyle\mathbf{A}_{n}=\mathbf{I}_{N_{b}}+\rho\left(\sum_{i=1,i\neq
n}^{N}\mathbf{\theta}_{i}\mathbf{b}_{i}\mathbf{a}_{i}^{H}+\mathbf{H}_{ab}\right)\times$
$\displaystyle~{}~{}~{}~{}~{}\left(\sum_{i=1,i\neq
n}^{N}\mathbf{\theta}_{i}\mathbf{b}_{i}\mathbf{a}_{i}^{H}+\mathbf{H}_{ab}\right)^{H},$
$\displaystyle\mathbf{B}_{n}=\rho\mathbf{b}_{n}\mathbf{a}_{n}^{H}\mathbf{a}_{n}\mathbf{b}_{n}^{H},$
$\displaystyle\mathbf{C}_{n}=\rho\mathbf{b}_{n}\mathbf{a}_{n}^{H}\left(\mathbf{H}_{ab}^{H}+\sum_{i=1,i\neq
n}^{N}\mathbf{a}_{i}\mathbf{b}_{i}^{H}\mathbf{\theta}_{i}^{*}\right).$

It can be observed that the matrices $\mathbf{A}_{n}$, $\mathbf{B}_{n}$ and
$\mathbf{C}_{n}$ do not contain the variable $\mathbf{\theta}_{n}$, which
means that these matrices can be obtained if all the elements
$\\{\mathbf{\theta}_{i}\\}_{i=1,i\neq n}^{N}$ are fixed.

Similarly, the relationship between the transmit power of the relay and
$\mathbf{\Psi}$ can be determined as

$\displaystyle P_{r}$
$\displaystyle=\mathrm{trace}(\mathbf{\Psi}(\mathbf{H}_{rb}\mathbf{H}_{rb}^{H}P_{a}+\sigma_{b}^{2})\mathbf{\Psi}^{H})$
$\displaystyle=P_{a}\sum_{n\in\mathbb{Q}}|\mathbf{\psi}_{n}|^{2}||\mathbf{b}_{n}||^{2}+\sigma_{b}^{2}\sum_{n\in\mathbb{Q}}|\mathbf{\psi}_{n}|^{2}$
$\displaystyle=\sum_{n\in\mathbb{Q}}|\mathbf{\psi}_{n}|^{2}[P_{a}||\mathbf{b}_{n}||^{2}+\sigma_{b}^{2}].$
(25)

Denote $\widetilde{P}_{r}$ = $\sum_{i\in\mathbb{Q},i\neq
n}|\mathbf{\psi}_{i}|^{2}[\sigma_{b}^{2}+P_{a}||\mathbf{b}_{n}||^{2}]$, which
is a constant due to that the variables $\sum_{i\in\mathbb{Q},i\neq
n}\mathbf{\psi}_{i}$ are fixed. Therefore, (25) can be rewritten as

$\displaystyle P_{r}$
$\displaystyle=\sum_{n\in\mathbb{Q}}|\mathbf{\psi}_{n}|^{2}[P_{a}||\mathbf{b}_{n}||^{2}+\sigma_{b}^{2}]+\widetilde{P}_{r}$
$\displaystyle=\sum_{n\in\mathbb{Q}}|\mathbf{\beta}_{n}|^{2}[P_{a}||\mathbf{b}_{n}||^{2}+\sigma_{b}^{2}]+\widetilde{P}_{r}.$
(26)

Here, we notice that $|\mathbf{\psi}_{n}|^{2}=|\mathbf{\beta}_{n}|^{2}$ for
$n\in\mathbb{Q}$.

###  III-C An Efficient Algorithm to Solve $\mathrm{(P2)}$

####  III-C1 Problem of Updating $\mathbf{\Theta}$

In our proposed algorithm, since
$\left\\{\theta_{i}\right\\}_{i=1,i\notin\mathbb{Q}}^{N}$ is fixed when
optimizing $\mathbf{\theta}_{n}$ in each iteration, the objective function
$f_{0}({\mathbf{\Theta}},P_{a})$ can be rewritten as

$\displaystyle f_{0}(\mathbf{\Theta},P_{a})$
$\displaystyle=\log_{2}\Big{|}\mathbf{A}_{n}+|\theta_{n}|^{2}\mathbf{B}_{n}+\mathbf{\theta}_{n}\mathbf{C}_{n}+\mathbf{\theta}_{n}^{*}\mathbf{C}_{n}^{H}\Big{|}$
$\displaystyle=\log_{2}|\mathbf{A}_{n}|+f_{1}(\mathbf{\Theta},P_{a}),$ (27)

where $\mathbf{A}_{n}$ is an invertible matrix satisfying
$\mathrm{rank}(\mathbf{A}_{n})=N_{b}$. Moreover, $\log_{2}(|\mathbf{A}_{n}|)$
is a constant, and $f_{1}(\mathbf{\Theta},P_{a})$ is given by

$\displaystyle
f_{1}({\mathbf{\Theta}},P_{a})=\log_{2}|\mathbf{I}_{N_{b}}+|\theta_{n}|^{2}\mathbf{A}_{n}^{-1}\mathbf{B}_{n}+\mathbf{\theta}_{n}\mathbf{A}_{n}^{-1}\mathbf{C}_{n}+\mathbf{\theta}_{n}^{*}\mathbf{A}_{n}^{-1}\mathbf{C}_{n}^{H}|.$
(28)

Following the above transformation, the problem of updating $\mathbf{\Theta}$,
denoted by ($P_{3}$), is given by

$\displaystyle\mathrm{(P3)}:$
$\displaystyle\mathop{\textup{max}}\limits_{\mathbf{\Theta}}~{}f_{1}(\mathbf{\Theta},P_{a})$
$\displaystyle\mathbf{s.t.}~{}|\mathbf{\beta}_{n}|=1,$
$\displaystyle{\mathrm{for}~{}n\notin\mathbb{Q},}$
$\displaystyle~{}~{}~{}~{}~{}|\mathbf{\beta}_{n}|^{2}\leq\frac{P_{r}^{\mathrm{max}}-\widetilde{P}_{r}}{[\sigma_{b}^{2}+P_{a}||\mathbf{b}_{n}||^{2}]},$
$\displaystyle{\mathrm{for}~{}n\in\mathbb{Q}.}$ (29)

####  III-C2 Solution to $\mathrm{(P3)}$

$\mathrm{(P3)}$ in (29) admits a closed-form solution, and, thus, it is
efficient for practical implementation. In order to derive it, the objective
function $f_{1}(\mathbf{\Theta},P_{a})$ can be rewritten as

$\displaystyle f_{1}(\mathbf{\Theta},P_{a})$
$\displaystyle=\log_{2}|\mathbf{D}_{n}+\mathbf{\theta}_{n}\mathbf{A}_{n}^{-1}\mathbf{C}_{n}+\mathbf{\theta}_{n}^{*}\mathbf{A}_{n}^{-1}\mathbf{C}_{n}^{H}|$
(30)
$\displaystyle=\log_{2}|\mathbf{D}_{n}|+\textup{log}_{2}\Big{|}\mathbf{I}_{N_{b}}+\mathbf{\theta}_{n}\mathbf{E}_{n}^{-1}\mathbf{C}_{n}+\mathbf{\theta}_{n}^{*}\mathbf{E}_{n}^{-1}\mathbf{C}_{n}^{H}\Big{|},$

where $\mathbf{D}_{n}$ =
$\mathbf{I}_{N_{b}}+|\theta_{n}|^{2}\mathbf{A}_{n}^{-1}\mathbf{B}_{n}$ and
$\mathbf{E}_{n}$ = $\mathbf{A}_{n}\mathbf{D}_{n}$.

We next analyse the objective function $f_{1}(\mathbf{\Theta})$ by considering
the first term in (30), i.e., $\log_{2}|\mathbf{D}_{n}|$. Specifically, for
$|\mathbf{D}_{n}|$, we note that
$\mathrm{rank}(\mathbf{A}_{n}^{-1}\mathbf{B}_{n})$ $<$
$\mathrm{rank}(\mathbf{B}_{n})=1$. Moreover, the probability of
$\mathrm{rank}(\mathbf{A}_{n}^{-1}\mathbf{B}_{n})$ is close to zero (it only
happens when $\mathbf{A}_{n}^{-1}\mathbf{B}_{n}=0$). Thus, we have
$\mathrm{rank}(\mathbf{A}_{n}^{-1}\mathbf{B}_{n})=1$. Similarly, we find that
$\mathbf{A}_{n}^{-1}\mathbf{B}_{n}$ is not diagonalizable when
$\mathrm{rank}(\mathbf{A}_{n}^{-1}\mathbf{B}_{n})=0$, which usually rarely
happens. Based on this, we have $(\mathbf{A}_{n}^{-1}\mathbf{B}_{n})\neq 0$
with a high probability and $\mathbf{A}_{n}^{-1}\mathbf{B}_{n}$ is
diagonalizable. Hence, we can rewrite
$\mathbf{A}_{n}^{-1}\mathbf{B}_{n}=\mathbf{W}_{n}\mathbf{\Sigma}_{n}\mathbf{W}_{n}^{-1}$
based on EVD, where $\mathbf{\Sigma}_{n}$ =
diag$\left\\{\iota_{n},0,\cdots,0\right\\}$, $\iota_{n}$ is the only non-zero
eigenvalue of $(\mathbf{A}_{n}^{-1}\mathbf{B}_{n})$. Finally, since both
$\mathbf{A}_{n}$ and $\mathbf{B}_{n}$ are positive semidefinite, $\iota_{n}$
is nonnegative and real. Thus, we have

$\displaystyle\log_{2}|\mathbf{D}_{n}|$
$\displaystyle=\log_{2}|\mathbf{I}_{N_{b}}+|\theta_{n}|^{2}\mathbf{W}_{n}\mathbf{\Sigma}_{n}\mathbf{W}_{n}^{-1}|$
$\displaystyle=\log_{2}|\mathbf{W}_{n}(\mathbf{I}_{N_{b}}+|\theta_{n}|^{2}\mathbf{\Sigma}_{n})\mathbf{W}_{n}^{-1}|$
$\displaystyle=\log_{2}\Big{|}1+|\theta_{n}|^{2}\iota_{n}\Big{|}.$ (31)

We are now focusing on the second term of (30). By a similar argument for the
first term, we have $\mathbf{E}_{n}^{-1}\mathbf{C}_{n}$ diagonalizable with a
high probability as well. Thus, we have
$\mathbf{E}_{n}^{-1}\mathbf{C}_{n}=\mathbf{T}_{n}{\mathbf{\Gamma}}_{n}\mathbf{T}_{n}^{-1}$
based on the EVD, where $\mathbf{T}_{n}\in\mathbb{C}^{N_{b}\times N_{b}}$,
$\mathbf{\Gamma}_{n}=\mathrm{diag}\left\\{\lambda_{n},0,\cdots,0\right\\}$,
$\lambda_{n}$ is the sole non-zero eigenvalue of
$\mathbf{E}_{n}^{-1}\mathbf{C}_{n}$. Let
$\mathbf{V}_{n}=\mathbf{T}_{n}\mathbf{A}_{n}\mathbf{T}_{n}^{-1}$, and $v_{n}$
denote first element of the first column of $\mathbf{V}_{n}^{-1}$ and
$v_{n}^{{}^{\prime}}$ denote first element of the first row of
$\mathbf{V}_{n}$. Note that it follows that $v_{n}^{{}^{\prime}}v_{n}=1$. So,
according to the [12], we can write

$\displaystyle\log_{2}|$
$\displaystyle\mathbf{I}_{N_{b}}+\mathbf{\theta}_{n}\mathbf{E}_{n}^{-1}\mathbf{C}_{n}+\mathbf{\theta}_{n}^{*}\mathbf{E}_{n}^{-1}\mathbf{C}_{n}^{H}|=$
$\displaystyle\log_{2}(1+|\theta_{n}|^{2}|\lambda_{n}|^{2}+2\mathbf{\mathbb{R}}(\mathbf{\theta}_{n}\lambda_{n})-v_{n}^{{}^{\prime}}v_{n}|\lambda_{n}|^{2}),$
(32)

where $\mathbb{R}$ denotes the real part of a complex number. We note that the
additional coefficient $|\theta_{n}|^{2}$ is related to the active relay
elements in HR-RIS, which does not exist in traditional RIS.

In summary, based on (31) and (32), we have

$\displaystyle f_{1}($
$\displaystyle\mathbf{\Theta},P_{a})=\log_{2}(1+|\theta_{n}|^{2}\iota_{n})$
$\displaystyle+\log_{2}(1+|\theta_{n}|^{2}|\lambda_{n}|^{2}+2\mathbf{\mathbb{R}}(\mathbf{\theta}_{n}\lambda_{n})-v_{n}^{{}^{\prime}}v_{n}|\lambda_{n}|^{2}).$
(33)

Hence, according to (33) we have $\mu_{n}^{*}=\mathrm{arg}(\lambda_{n})$. So
the optimal solution of the problem (P3) is given by

$\displaystyle\mathbf{\theta}_{n}^{*}=\left\\{\begin{array}[]{ll}|\beta_{n}|e^{-j\mathrm{arg}(\lambda_{n})},&{n\in\mathbb{Q},}\\\
e^{-j\mathrm{arg}(\lambda_{n})},&{n\notin\mathbb{Q}.}\end{array}\right.$ (36)

In the HR-RIS, $\mathbb{Q}$ is available to determine
$\left\\{|\beta_{n}|\right\\}_{n\in\mathbb{Q}}$. Therefore, from (29), we
obtain

$\displaystyle|\beta_{n}|=\sqrt{\frac{P_{r}^{\mathrm{max}}-\widetilde{P}_{r}}{[\sigma_{b}^{2}+P_{a}||\mathbf{b}_{n}||^{2}]}},~{}~{}n\in\mathbb{Q}.$
(37)

As a result, the optimal solution to (P3) is given as

$\displaystyle\mathbf{\theta}_{n}^{*}=\left\\{\begin{array}[]{ll}\sqrt{\frac{P_{r}^{\mathrm{max}}-\widetilde{P}_{r}}{[\sigma_{b}^{2}+P_{a}||\mathbf{b}_{n}||^{2}]}}e^{-j\mathrm{arg}(\lambda_{n})},&{n\in\mathbb{Q},}\\\
e^{-j\mathrm{arg}(\lambda_{n})},&{n\notin\mathbb{Q}.}\end{array}\right.$ (40)

######  Remark 1.

Substituting $\widetilde{P}_{r}$=$\sum_{i\in\mathbb{Q},i\neq
n}|\mathbf{\beta}_{i}|^{2}[\sigma_{b}^{2}+P_{a}||\mathbf{b}_{n}||^{2}],n\in\mathbb{Q}$
to (37), it is observed that a larger $K$ results in a smaller $|\beta_{n}|$.
Therefore, increasing the number of active elements (i.e., $K$) does not
always guarantee the covert rate improvement of HR-RIS over traditional RIS.
In particular, with a limited power budget $P_{r}^{\mathrm{max}}$, the HR-RIS
can have $|\mathbf{\beta}_{n}|<1$, which attenuates the signal and degrades
the covert rate. In this case, HR-RIS with a smaller $K$ is more likely to
attain a covert rate gain than those with more active elements. This
conclusion will be further demonstrated numerically.

###  III-D Optimizing $P_{a}$ for a Given $\mathbf{\Theta}$

######  Corollary 1.

KL divergence $\mathcal{D}_{01}$ is a monotonically increasing function of
$P_{a}$.

###### Proof:

In order to determine the monotonicity of $\mathcal{D}_{01}$ with respect to
$P_{a}$, we derive its first derivative as

$\displaystyle\frac{\partial{\mathcal{D}_{01}}}{\partial{P_{a}}}$ (41)
$\displaystyle=L\left(\frac{P_{a}\mathbf{U}_{w}\mathbf{U}_{w}^{H}}{\left(\sigma_{w}^{2}\left(\mathbf{I}_{N_{w}}+\mathbf{H}_{rw}\mathbf{\Psi}\mathbf{\Psi}^{H}\mathbf{H}_{rw}^{H}\right)+P_{a}\mathbf{U}_{w}\right)^{2}}\right)>0.$

∎

For a given $\mathbf{\Theta}$, following (41), we can find optimal transmit
power of Alice $P_{a}^{*}$ by solving $\mathcal{D}_{01}=2\epsilon^{2}$.

Since $P_{a}$ does not affect the optimization of $\mathbf{\Theta}$,
$P_{a}^{*}$ is the global optimal solution.

##  IV Numerical Results and Discussions

In this section, the numerical results will validate the performance of our
proposed scheme. We assume that uniform linear arrays (ULAs) are deployed at
the Alice, Bob and Willie, respectively. In contrast, HR-RIS uses a unified
plane array (UPA) with $N$ elements. Furthermore, assuming that there is a
half-wavelength distance between Alice, Bob, Willie and HR-RIS arrays. We
consider a two-dimensional coordinate system, Alice, HR-RIS, Bob, and Willie
are respectively located at $(0\mathrm{m},0\mathrm{m})$,
$(51\mathrm{m},0\mathrm{m})$, $(50\mathrm{m},2\mathrm{m})$ and
$(30\mathrm{m},5\mathrm{m})$. All channel realizations are drawn from Rician
fading. The path loss of a link distance $d$ is given by [9, 12],
$\chi(d)=\chi_{0}(\frac{d}{1\mathrm{m}})^{\alpha_{ij}}$, where $\chi_{0}$ is
the path loss at the reference distance of 1 meter, and $\alpha_{ij}$ is the
path loss exponent. Specifically, the path loss exponents are set as
$\alpha_{ar}=2.2$, $\alpha_{rb}=2.8$, $\alpha_{ab}=4.2$, $\alpha_{aw}=4.2$,
and $\alpha_{rw}=2.8$.

Figure 2: Bob’s covert rate versus the number of reflecting elements at HR-RIS
for different numbers of the active elements $K$, for $\epsilon=0.01$,
$P_{r}^{\mathrm{max}}=-30$ dBm, $L=100$, $\sigma_{b}^{2}=\sigma_{w}^{2}=-80$
dBm.

In Fig. 2, we plot the covert rate of Bob versus the total reflection elements
$N$ of the HR-RIS under different numbers of the active elements $K$. In this
figure, we first observe that Bob’s covert rate increases as $N$ increases. It
can also be seen that Bob’s covert transmission rate increases as $K$
increases. As the number of active elements increases, we can see that the
increase in the covert rate becomes smaller. This is due to the power limit of
active elements, too many active elements can cause $|\mathbf{\beta}_{n}|<1$,
which is explained in Remark 1. Therefore, we can get the optimal number of
active elements from this figure. Based on which, we can conclude that a small
amount of active elements is sufficient for HR-RIS to achieve a significant
improvement in term of the covert rate when compared to conventional RIS-aided
covert communication schemes.

##  V Conclusion

In this work, we proposed a covert communication scheme with the help of the
HR-RIS, where several elements are active relay elements and the remaining
ones as passive reflecting elements. We first used KL divergence to represent
Willie’s detection capability, and proved that the KL divergence adopted in
the covertness constraint is a monotonically increasing function of the
transmit power at Alice. Based on this, we obtained the optimal value of
$P_{a}$. Then, we used an alternate optimization algorithm to identify the
optimal reflection coefficients of HR-RIS to improve the covert communication
performance. The numerical results demonstrated that the proposed scheme can
significantly outperform the conventional RIS-aided covert communication
schemes in term of covert rate by using a small number of active elements.

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