# Photothermal responsivity of van der Waals material-based nanomechanical
resonators

Myrron Albert C. Aguila<EMAIL_ADDRESS>Department of Engineering
and System Science, National Tsing Hua University, Hsinchu, 30013, Taiwan
Nano Science and Technology Program, Taiwan International Graduate Program,
National Tsing Hua University and Academia Sinica  Institute of Physics,
Academia Sinica, Nankang, Taipei, 11529, Taiwan  Joshoua C. Esmenda
<EMAIL_ADDRESS>Department of Engineering and System Science,
National Tsing Hua University, Hsinchu, 30013, Taiwan  Nano Science and
Technology Program, Taiwan International Graduate Program, National Tsing Hua
University and Academia Sinica  Institute of Physics, Academia Sinica,
Nankang, Taipei, 11529, Taiwan  Jyh-Yang Wang  Institute of Physics, Academia
Sinica, Nankang, Taipei, 11529, Taiwan  Teik-Hui Lee  Institute of Physics,
Academia Sinica, Nankang, Taipei, 11529, Taiwan  Yen-Chun Chen  Institute of
Physics, Academia Sinica, Nankang, Taipei, 11529, Taiwan  Chi-Yuan Yang
Institute of Physics, Academia Sinica, Nankang, Taipei, 11529, Taiwan  Kung-
Hsuan Lin  Institute of Physics, Academia Sinica, Nankang, Taipei, 11529,
Taiwan  Kuei-Shu Chang-Liao  Department of Engineering and System Science,
National Tsing Hua University, Hsinchu, 30013, Taiwan  Sergey Kafanov
Department of Physics, Lancaster University, Lancaster, LA1 4YB, United
Kingdom  Yuri A. Pashkin  Department of Physics, Lancaster University,
Lancaster, LA1 4YB, United Kingdom  Chii-Dong Chen
<EMAIL_ADDRESS>Institute of Physics, Academia Sinica, Nankang,
Taipei, 11529, Taiwan

###### Abstract

Nanomechanical resonators made from van der Waals materials (vdW NMRs) provide
a new tool for sensing absorbed laser power. The photothermal response of vdW
NMRs, quantified from the resonant frequency shifts induced by optical
absorption, is enhanced when incorporated in a Fabry-Perot (FP)
interferometer. Along with the enhancement comes the dependence of the
photothermal response on NMR displacement, which lacks investigation. Here, we
address the knowledge gap by studying electromotively driven niobium
diselenide drumheads fabricated on highly reflective substrates. We use a FP-
mediated absorptive heating model to explain the measured variations of the
photothermal response. The model predicts a higher magnitude and tuning range
of photothermal responses on few-layer and monolayer NbSe2 drumheads, which
outperform other clamped vdW drum-type NMRs at a laser wavelength of
$532\,$nm. Further analysis of the model shows that both the magnitude and
tuning range of NbSe2 drumheads scale with thickness, establishing a
displacement-based framework for building bolometers using FP-mediated vdW
NMRs.

van der Waals materials, nanomechanical resonators, interferometry,
photothermal response, static displacement, absorptive heating

††preprint: APS/123-QED

##  I Introduction

Nanomechanical resonators (NMRs) embedded in an optical cavity are valuable
platforms for studying weak forces due to the enhanced coupling between light
and motion[1]. Enhanced coupling improves the capability of nanomechanical
resonators to demonstrate nonlinear dynamics[2] and sense heat transport in
suspended nanostructures[3]. Resonators interacting with optical elements
enjoy additional degree of spatiality[4, 5, 6, 7, 8], extremely large
optomechanical coupling at ambient temperature[9, 10], and reduced mode volume
due to breakthrough technologies in focusing laser beams via free space
optics[11, 12, 13], fiber optics[14, 15, 16] and near-field interactions with
multiplexed on-chip optical waveguides and tapered fibre[17, 18].

Heating induced by the probe laser remains a concern in the optical readout of
NMRs[19, 20]. Photothermal effects[15, 21, 3, 22] emerge once the laser
illuminates a region of the resonator and raises the temperature of the
mechanical mode[20]. While the process hinders ground-state cooling of the
mechanical mode[23], it enables sensing of incident laser power with the aid
of on-chip Fabry-Perot (FP) cavities. NMRs fabricated with suspended van der
Waals (vdW) materials[24, 25, 26] show promising photothermal
sensitivities[27] due to reduced mass, and layer-dependent mechanical,
optical, and thermal properties. Few-layer and multilayer niobium diselenide
(NbSe2) is a candidate vdW material for low-noise, highly responsive
photodetectors at ultraviolet[28] and infrared wavelengths[29, 30].

There are several approaches to quantifying the power absorbed by vdW NMRs.
One can measure the mechanical mode temperature by resolving thermomechanical
motion[6, 31] while varying the incident laser power. Such detection requires
low mass and high quality factors of NMRs, which can be difficult to ensure
for vdW materials[24, 25, 26]. A less stringent yet popular method involves
monitoring of the resonant frequency shift of an electromotively driven vdW
NMRs[3, 27] and following its dependence on the incident laser power. In both
approaches, light absorption depends not only on the FP structure and
wavelength, but also on the resonator displacement from the initial
equilibrium position. Understanding the displacement-dependent absorptive
response may provide insights into managing FP-mediated heat flow in NMRs.
While experiments on NMRs fabricated from various types of vdW materials have
been published, photothermal sensing with NbSe2 NMRs has not yet been
reported. NbSe2 has low thermal conductivity[32], is flexible[33] and has
large fracture strain[34]. This combination of properties enables the
realization of low power, photothermal-strain-sensitive detectors.

In this paper, we explore the influence of the resonator displacement on the
sensitivity of the FP-based vdW NMR to incident light. We investigate the
photothermal response of electromotively driven NbSe2 drumheads. We propose an
FP-mediated absorptive heating model that accounts for resonator displacement
to explain the observed variations in the measured photothermal response. The
model consequently reveals a large tuning range of photothermal responsivity
as the drumhead moves towards the bottom electrodes. We extend the model to
drumheads of varying thicknesses and materials to gain insight into the
geometric and material impact of FP-mediated heating on NMRs.

Figure 1:  Measurement setup and device. (a) Optical interferometric setup
used to track the mechanical frequency of multilayered NbSe2 flake mechanical
resonators. (b) Schematic diagram of laser-induced photothermal heating of an
electromotively driven drumhead resonator. Cross-section showing the net
tension exerted by the drumhead resonators under photothermal heating. (c)
Optical micrograph of the circular (red dashed circle) and elliptical (blue
dashed ellipse) drumhead resonators under study. The white scale bar
corresponds to a length of 10 $\mu$m.

##  II Materials and Methods

The mechanical drums are set into motion by a combination of static and
oscillating voltage. This motion is detected through the optical
interferometric detection scheme at a laser wavelength $\lambda=532\,$nm, as
shown in Fig. 1(a) and (b). When a laser beam illuminates a region of the
drumhead, the drum surface reflects a fraction of the incident power $P_{in}$
for detection and absorbs some of the power accumulated in the FP cavity. The
detector receives the modulated reflected beam needed to resolve the driven
frequency response of the drumheads. The absorbed power can be described as
$P_{abs}=P_{in}\chi A_{FP}$, where $A_{FP}$ is the total absorbance of the FP
cavity and $\chi$ is the power fraction absorbed by the drumhead in the FP
stack. The absorbance of the FP cavity depends on the difference between the
spacer height $s$ and the resonator static displacement $z_{s}$. $P_{abs}$
heats the illuminated spot, and produces a temperature gradient due to the
radial heat transfer to the drum clamps, as shown in Fig. 1(b). Consequently,
the elevated temperature changes the mechanical tension exerted by the clamps
by photothermal tension $\gamma_{pth}$. As illustrated in Fig. 1(b),
$\gamma_{pth}$ is added to the existing initial mechanical tension
$\gamma_{0}$ generated by the drum displacement.

Figure 1(c) shows the optical micrograph of devices A (circular) and B
(elliptical). The drumheads are made from exfoliated NbSe2 flakes, which are
suspended on a lithographically patterned Au-Cr-SiO2-Si substrate covered with
electron-beam patterned resist CSAR-62. Device A has a radius
$a_{A}=b_{A}=3.5\,\mu$m and device B has a major radius $a_{B}=4.0\,\mu$m and
a minor radius $b_{B}=3.5\,\mu$m. Details of the fabrication and optical
detection are explained elsewhere[8]. Using the multilayer interface approach
(MIA)[35], the spacer height is determined to be $s=296\,$nm and the drumhead
thickness $h=55\,$nm. Observation of the photothermal effect on the mechanical
drums requires changing the laser power and monitoring the mechanical
frequency shift.

Figure 2:  Dependence of the fundamental resonant frequencies of circular
(device A) and elliptical (device B) resonators on the incident laser power.
Insets show the position of the laser spot (green circle in the center) at
which the driven responses of devices A and B are measured. The devices are
electromotively driven at $V_{DC}=4\,$V and $V_{AC}=0.125\,$Vpk. Data points
are measured resonant frequencies from the driven responses and solid lines
are linear fits.

##  III Results and Discussion

We consider the case of a tensioned circular drumhead where both flexural
rigidity $D_{p}$ and $\gamma_{0}$ have comparable effect on the resonant
frequency of the drumhead. The resonant frequency of both circular and
elliptical drumheads, with an effective radius $a_{eff}=\sqrt{ab}$ and
thickness $h$ can be written as[36]

${f_{0}(P_{in})=\frac{\lambda_{01}}{2\pi}\sqrt{\frac{D_{p}}{\rho
ha_{eff}^{4}}\left[\lambda_{01}^{2}+\frac{{{(\gamma_{0}+\gamma_{pth})}{a_{eff}^{2}}}}{{D_{p}}}\right]},}$
(1)

where $\lambda_{01}$ is a modal parameter that is determined numerically. At
$P_{in}=0$, the drumheads oscillate at their natural resonant frequency
(without heating) $f_{0}=f_{0}(P_{in}=0)$ corresponding to the y-intercept of
both plots for circular and elliptical drumheads in Fig. 2. Given $f_{0}$
values of 16.621 and 15.432 MHz for device A and B, respectively, and using
Eq. (1), we determine the initial tensions of $\gamma_{0,A}=5.01\,$N m-1 for
device A and $\gamma_{0,B}=5.15\,$N m-1 for device B, for the given applied DC
voltage. As $P_{in}$ ramps up, downward resonant frequency shift is observed
as shown in Fig. 2. Since the frequency shift is linear for small values of
$P_{in}$, Eq. (1) can be given by its first-order Taylor polynomial

${f_{0}(P_{in})\approx
f_{0}+\frac{1}{2}\left[\left(\frac{\lambda_{01}}{2\pi}\right)^{2}\frac{1}{\rho
ha_{eff}^{2}}\right]\frac{\gamma_{pth}(P_{in})}{f_{0}}}$ (2)

and the shift can be written as

${\Delta
f_{0}(P_{in})=\frac{1}{2}\left(\frac{\lambda_{01}}{2\pi}\right)^{2}\frac{\gamma_{pth}(P_{in})}{\rho
ha_{eff}^{2}f_{0}}.}$ (3)

The compressive tension is given by[27, 37]

${\gamma_{pth}(P_{in})=-\frac{E_{3D}h}{1-\nu}\alpha_{L}\Delta
T_{abs}(P_{in}),}$ (4)

where $E_{3D}$ is the Young’s elastic modulus, $\nu$ is the Poisson’s ratio,
$\alpha_{L}$ is the thermal expansion coefficient of NbSe2 at the bath
temperature $T_{0}$, and $\Delta T_{abs}$ is the temperature difference
between $T_{0}$ and the average temperature of the drumhead $T_{0}+\Delta
T_{abs}$. For linear changes in $T_{0}+\Delta T_{abs}$, the average
temperature difference is expressed as[38]

${\Delta T_{abs}(P_{in})=\frac{P_{abs}}{4\pi h\kappa}\eta,}$ (5)

where $\kappa$ is the in-plane thermal conductivity of NbSe2, and $\eta$ is
the average spot size factor. We estimate both $\chi$ and $A_{FP}$ through
MIA[39, 40, 41] whereas $\eta$ is evaluated by assuming absorptive spot
heating in the center of the drumheads[42].

Given these inputs, we define the photothermal responsivity $\Psi$ as the
frequency shift induced by the absorbed power. By solving Eq. (3) using Eq.
(4) and Eq. (5), and defining $\Psi$ as $\Psi=\Delta f_{0}/\Delta P_{in}$, the
photothermal responsivity is expressed as

${\Psi(z_{s})=-\frac{1}{8}\left(\frac{\lambda_{01}}{2\pi}\right)^{2}\frac{E_{3D}\alpha_{L}}{(1-\nu)mf_{0}}\frac{\chi
A_{FP}\eta}{\kappa},}$ (6)

where $m=\rho\pi a_{eff}^{2}h$ is the total mass of the drumhead. This
quantity can be extracted experimentally from the slope of the linear fits
acquired from Fig. 2.

Figure 3:  Photothermal responsivity dependence of the multilayer NbSe2
drumhead resonators on the static displacement. Data points are the slope
extracted from the slope in Fig. 2. Solid line refers to the photothermal
responsivity model. For comparison, the dependence of the simulated absorbance
of the Fabry-Perot cavity on static displacement, shown by magenta dotted line
with the magnitude referenced on the right Y-axis spine, is also shown. Figure
4: Dependence of the displacement-dependent photothermal responsivity profile
$\Psi(z_{s})$ of drum NMRs on the thickness and resonator material. (a)
Intensity plot of the calculated dependence of $\Psi(z_{s})$ of bulk NbSe2
drumheads on varying thickness. Colored dashed lines refer to representative
$\Psi(z_{s})$ profiles displayed in (b). For comparison, we use the diameter,
Fabry-Perot structure, and mechanical resonant frequency of Device A in
simulating the dependences. (c) Scatter plot of the photothermal responsivity
values obtained in Fig. 3, along with photothermal responsivities of other
NMRs extracted from literature. Square symbols refer to devices demonstrating
highly tensioned systems like monolayer (1L) graphene (Gr), monolayer
molybdenum diselenide (MoSe2), and thin black phosphorus (57L-BP) drumheads.
Circular symbols refer to devices dominated by flexural rigidity like thick
black phosphorus (160L-BP) drums, silicon nanowire (SiNW) cantilever beams,
and NbSe2 drums (A, B). Included also are $\Psi(z_{s})$ of both tension-
dominated (solid lines) and flexural rigidity-dominated (dotted lines) devices
by adopting the Fabry-Perot structure of device A.

Figure 3 shows the dependence of $\Psi$ extracted from Fig. 2 with the
corresponding $z_{s}$ and $\Psi(z_{s})$ generated with Eq. (6) using the
resonator specifications of device A. Device A has $\Psi_{A}=-2.19\,$kHz
$\mu$W-1 for $z_{s}$=13 nm. Device B has $\Psi_{B}=-2.38\,$kHz $\mu$W-1 for
$z_{s}=29\,$nm. The ratio of $z_{s}$ between device A and device B in Fig. 3
yields 2.23, which is close to the value of 2.16 obtained from the
theoretically derived expression
$\beta_{ellipse}a_{B}^{4}/\beta_{circle}a_{A}^{4}$, where $\beta$ is the
eccentricity factor from reference[43]. The results imply that the difference
in $z_{s}$ of these two drums lies with geometry[8]. Physically, the
compressive strain translates to an out-of-plane radial expansion of the
drumhead when aided with $z_{s}$. $z_{s}$ is controlled either through
electromotive driving of the drumhead[44, 45, 3] or through slack[46, 47, 8].
Given the nature of the transfer process using PDMS disks[48, 8], $z_{s}$
likely originates from slack.

According to the model prediction in Fig. 3, $\Psi=-2.10$ kHz $\mu$W-1 when
the resonator is in its equilibrium position ($z_{s}=0$). Displacing the
resonator to $z_{s}=69\,$nm would result in $\Psi\approx-3.33\,$kHz$\,\mu$W-1,
which amounts to increases in the measured responsivities of devices A and B
by 42$\%$ and 52$\%$, respectively. Also, we estimate a tuning range of
$1.4\,$kHz $\mu$W-1. As shown in Fig. 3, the behavior of $A_{FP}$ mimics the
observed dependence of $\Psi$ on $z_{s}$, albeit opposite in sign due to heat-
induced compression.

We then simulate the effect of varying $h$ on the $\Psi(z_{s})$ profiles of
circular NbSe2 drumheads using Eq. (6), as shown in Fig. 4(a,b). The simulated
$\Psi(z_{s})$ profiles are restricted to the material properties of bulk,
clean devices possessing $a_{eff}$, $f_{0}$, FP structure and $\lambda$ of
device A. With these parameters, only $\lambda_{01}$ is varied to decrease at
an increasing thickness to maintain $f_{0}$. For $h=6\,$nm, we estimate a
tuning range of roughly $12.2\,$kHz $\mu$W-1 from $\Psi(z_{s}=0)=-16.30\,$kHz
$\mu$W-1 to $\Psi(z_{s}=69\,$nm$)=-4.05\,$kHz $\mu$W-1. For $h=30\,$nm, we
obtain a tuning range of $2.78\,$kHz $\mu$W-1 from $\Psi(z_{s}=0)=-4.16\,$kHz
$\mu$W-1 to $\Psi(z_{s}=86\,$nm$)=1.39\,$kHz $\mu$W-1. Hence, thinner bulk
NbSe2 devices have larger tuning range and magnitude as compared to thicker
devices. Furthermore, the simulated $\Psi(z_{s})$ profile of monolayer NbSe2,
shown in Fig. 4(c) as black solid lines, has larger magnitude and tuning
ranges than the extrapolated bulk NbSe2 $\Psi(z_{s})$ curve (black dotted
line).

Intuitively, the change in $h$ modifies the properties of both the FP cavity
and the nanomechanical resonators, and consequently affects $\Psi(z_{s})$. In
the FP domain, $h$ modifies both $A_{FP}$ and $\chi$. In a non-transmissible
FP cavity with known refractive indices of the resonators $\hat{n}_{res}$[49,
50, 51, 52, 53], reflectance and absorbance dominate. For example, the
deflection-dependent behavior of the bulk NbSe2 $A_{FP}$ transitions from an
asymmetric, sinusoidal profile at thin layers to a Fano-peak profile at
thicker layers, as shown in Fig. S1 of the Supplementary Material. Also, the
deflection dependence of monolayer NbSe2 $A_{FP}$ maintains an asymmetric,
sinusoidal profile due to its large absorption coefficient, which is different
compared to bulk NbSe2[49, 35]. In device A, the values of $z_{s}$ predicted
to have maximum absorbance and magnitude of $\Psi$ would have near-zero
modulated reflectance, implying no detectable NMR responses by the FP cavity.
Hence the estimates for the tuning range of the drumheads are upper bounds.
Nevertheless, the values help confine a range of $z_{s}$ that give detectable
driven response and photothermal heating.

In the nanomechanical resonator domain, the effect of the drum head thickness
depends on whether the resonator operates in the low tension or high tension
regime (see Supplementary Material for the full expressions). In the regime
where rigidity dominates, $\Psi\propto h^{-2}$. In the tensioned membrane
regime in which the modelled multilayer and monolayer NbSe2 drums in Fig.
4(b-c) reside, $\Psi\propto h^{-0.5}$. Engineering a constant value of $\Psi$
for any value of $z_{s}$ requires thicker drum plates. For devices with larger
tuning ranges of $\Psi$, thin NbSe2 membranes are preferred, though stress-
relief structures offer the possibility of thin plate structures[54].

We note that the $\Psi(z_{s})$ curve of monolayer (1L) NbSe2 in Fig. 4(c)
represents an upper bound by assuming a clean device
($\rho\approx\rho_{NbSe_{2}}$). These conditions result in a greater magnitude
and larger tuning range of $\Psi(z_{s})$ of the monolayer NbSe2 than
$\Psi(z_{s})$ of other devices like 165 nm thick, clamped-free silicon
nanowire (SiNW) resonators[55] (cyan, solid line), graphene (1L-Gr) (orange,
solid line), cryogenically-cooled monolayer diselenide (1L-MoSe2) (purple,
solid line) and multilayer black phosphorus (BP, magenta solid and dotted
lines) drums. The mentioned devices are simulated using the same substrate as
Device A. Note that the $\Psi(z_{s})$ profiles for BP resonators were
simulated assuming an average, isotropic $E_{3D}$ and $\kappa$ even though
these resonators have reported anisotropic properties[40]. The indicated
materials have positive $\alpha_{L}$ which implies that the devices made of
these materials experience compression upon the illumination by the laser
beam. Only the $\Psi(z_{s})$ curve of graphene experiences tension upon spot
heating due to graphene’s negative $\alpha_{L}$ at ambient temperature.
Furthermore, the SiNW device falls under the optically-thick, bending regime,
which illustrates the near-zero magnitude and tunability of $\Psi$ with
$z_{s}$. Tables 1, 2, and 3 and Fig. S1 of the Supplementary Material contain
more details regarding Fig. 4.

We also see from Fig. 4(c) that the measured $\Psi$ of bulk NbSe2 devices is
comparable to that of 57L-BP device (magenta square) and significantly better
than both bulk SiNW (cyan circle) and 160L-BP (magenta circle) devices.
However, $\Psi$’s of these devices are ten times smaller than the $\Psi$
values of a 1L-MoSe2 drumhead (pink square), which possesses a significantly
smaller mass. We note that the values of $z_{s}$ extracted from literature are
for the devices in their equilibrium position except for graphene, which has
$1\,$nm deflection to resolve its motion[21].

The effect of the substrate and probing wavelength on photothermal heating of
the drumheads varies with the resonator material as its optical, thermal, and
mechanical properties modify their $\Psi(z_{s})$ profiles. For example, the
highly-reflective substrate of device A shows improved $\Psi(z_{s}=0)$ for the
160L-BP, 1L-Gr and 1L-MoSe2 device, comparable heating to SiNW device and
decreased photothermal heating for the 57L-BP drumhead as compared to their
extracted values in literature[21, 3, 40, 55] (colored symbols), as shown in
Fig. 4(c). The devices cited in literature make use of a combination of gap
heights, probe wavelength, and substrate for near-optimal FP-based
detection[56] and ease in device fabrication; their photothermal responses are
secondary. Nevertheless, our model demonstrates the variation of the tuning
range of $\Psi(z_{s})$ of thin van der Waals materials using our FP structure.

The values of $\Psi(z_{s})$ reported in this work, with the exception of clean
1L-NbSe2, are fractions of $\Psi$ reported for metamaterial string[31] and
graphene trampoline NMR bolometers[27] at infrared and visible wavelengths,
respectively. Apart from replacing NbSe2 with thermally insulating vdW
materials, a strategy to improve $\Psi(z_{s}=0)$ of NbSe2 involves decreasing
the thermal conductivity of the drumheads with high-aspect-ratio tethers that
resemble a trampoline geometry. The trampoline structure is reported to
increase $\Psi(z_{s}=0)$ of graphene, known for its very high thermal
conductivity, to $2.4\times 10^{3}\,$kHz$\,\mu$W-1[27]. The structure is
tricky to implement on NbSe2 since the focused ion beam etching step
introduces defects that alter the flake properties[57, 58]. On another note,
broadening the tuning range would entail engineering a partially-transparent
substrate at a probe wavelength to distinguish the resonator positions of
maximum absorbance and zero modulated reflectance[20].

##  IV Conclusions

In summary, we have designed and characterized NbSe2 drumheads with
controllable photothermal heating using NMR static displacements in FP
cavities. Our simulations show that the magnitude and tuning range of the
photothermal response of drumheads increase at decreasing flake thickness. Our
analysis shows that a monolayer NbSe2 drumhead NMR has promising photothermal
responsivities at room temperature. While our work focuses on NbSe2 devices,
the design framework applies to a family of vdW materials and conventional
resonator materials that absorb light.

###### Acknowledgements.

We acknowledge the contribution of T.-H. Hsu and W.-H. Chang in fabricating
devices and in building the experimental setup, B.-R. Guo for technical
assistance, A.F. Rigosi for sharing the measured dielectric constant spectra
of bulk and few-layer NbSe2 and R. Frisenda for sharing the real and imaginary
refractive index of single-layer MoSe2. We thank the Taiwan International
Graduate Program for the financial support. This project is funded by Academia
Sinica Grand Challenge Seed Program (AS-GC-109-08), Ministry of Science and
Technology (MOST) of Taiwan (107-2112-M-001-001-MY3), Cost Share Programme
(107-2911-I-001-511), the Royal Society International Exchanges Scheme (grant
IES$\backslash$R3$\backslash$170029), and iMATE(2391-107-3001). We extend our
gratitude for the Academia Sinica Nanocore Facility. Attributions.-M.A.C.A.
and J.C.E. contributed equally in this work. C.-D.C. conceived the devices and
supervised the project; J.C.E. fabricated the devices. M.A.C.A. and J.-Y.W.
modeled the photothermal responsivity dependences. C.-Y.Y. and K.-H.L.
designed and built the setup for optical measurements. M.A.C.A., J.C.E.,
Y.-C.C., and C.-Y.Y. performed the experiment. M.A.C.A., J.C.E., J.-Y.W.,
Y.-C.C., T.-H.L., K.-S.C.-L., S.K., Y.P. and C.-D.C. analyzed the data,
performed simulations and wrote the manuscript; all authors discussed the
results and contributed to the manuscript.

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[1]Miroslav Mitev

# Joint secure communication and sensing in 6G networks

Amitha Mayya  Arsenia Chorti

Keywords: Physical layer security, Joint communication and sensing, Secret key
generation, practical implementation.

Joint communication and sensing is expected to be one of the features
introduced by the sixth-generation (6G) wireless systems. This will enable a
huge variety of new applications, hence, it is important to find suitable
approaches to secure the exchanged information. Conventional security
mechanisms may not be able to meet the stringent delay, power, and complexity
requirements which opens the challenge of finding new lightweight security
solutions. A promising approach coming from the physical layer is the secret
key generation (SKG) from channel fading. While SKG has been investigated for
several decades, practical implementations of its full protocol are still
scarce. The aim of this chapter is to evaluate the SKG rates in real-life
setups under a set of different scenarios. We consider a typical radar
waveform and present a full implementation of the SKG protocol. Each step is
evaluated to demonstrate that generating keys from the physical layer can be a
viable solution for future networks. However, we show that there is not a
single solution that can be generalized for all cases, instead, parameters
should be chosen according to the context.

##  1 Introduction

The development of wireless communications and radar sensing technologies has
been progressing independently for many years. Their design has been fully
separated, relying on different resources (frequency, hardware, waveform,
etc.). However, as both services depend on the same principle, i.e.,
transmitting/receiving electromagnetic signals, it is envisioned that a co-
designed approach can bring a number of benefits. In fact, the idea of a joint
design is seeing a growing interest by both industry and academia and it is
expected to become a key enabling technology for variety of 6G applications
[1].

A simple approach to integrate both services in one device is by sharing the
same spectrum but transmitting separate signals. The main problem for this
case is finding a flexible interference cancellation technique that can
smoothly operate in different scenarios [2]. Another, more efficient, approach
is to use a single signal for both communication and sensing. This approach
can provide higher spectral efficiency but requires joint waveform design. In
this sense, depending on the application the waveform design can be
communication-centric (suitable for application with high data rate
requirements and low sensing precision, e.g., smart home applications) and
radar-centric (suitable for application with low data rate requirements and
high sensing precision, e.g., vehicular networks).

Communication signals have complex structure and are typically modulated in
one or multiple domains, e.g., time, spatial and frequency. This increases the
cost as advanced transmitter/receiver modules [2] are required. On the other
hand, signals used for radar have a simple structure, e.g., pulsed or
continuous waves. Such waveforms are specifically designed for sensing and do
not require the deployment of advanced processing modules. Such signals cannot
provide high throughput, however, their achievable data rates can still be
sufficient for certain sensing-oriented applications [3]. Some of the key
performance indicators for such applications are high sensing precision, low-
latency, low-complexity and power efficiency [4].

A question that naturally arises is: How can we secure the communication in
radar-centric applications? It is clear that current cryptographic algorithms
may not be able to support the requirements listed above. As an example, [5]
has reported that the verification of digital signature over $400$ MHz
processor in a vehicular to everything (V2X) network exceeds the low-latency
requirements and requires approximately $20$ ms. Furthermore, traditional
asymmetric public-key encryption (PKE) protocols do not satisfy low-energy
requirements and cannot scale well for massive deployment [6]. Finally, the
advances in quantum computing can leave current security solutions vulnerable,
unless key sizes increase significantly [7]. Therefore, proposals for new
lightweight security alternatives are sought.

Following from the above, the goal of this chapter is to provide a performance
evaluation of one lightweight and quantum-secure key distribution technique
(currently considered for 6G [8]). In particular, we investigate the physical
layer security (PLS)-based secret key generation (SKG) with a focus on
sensing-oriented networks. The SKG process (introduced in [9] and [10]) allows
two devices to extract a common random key sequence relying on the stochastic
behaviour and reciprocity of wireless channels. Some of the advantages of SKG
are that, it does not require a complex infrastructure (as in the case of
PKE), it can be employed using an arbitrary PHY waveform (if the transmitted
power spectral density (PSD) is known) and it does not require matched
receivers [11, 12]. This work focuses on an SKG approach that utilizes one of
the widely used signals for radar sensing, i.e., the compressed high
resolution pulse (chirp).

Thanks to their wide bandwidth, chirps can be used to harvest entropy from the
wireless channel. To extract channel observations we rely on a filterbank-
based approach, where keys are generated from power measurements at different
frequencies. This approach has been proposed in [11] and
analytically/numerically analyzed by some of the authors of this chapter in
[12, 13, 14]. As a continuation of these works, here, we take a step further
and evaluate the performance of the approach in an experimental setup. A
measurement campaign is performed in an indoor environment considering a set
of different scenarios. To arrive at the final key rate all steps of the SKG
protocol are implemented. First, advantage distillation is performed by
applying a filterbank over received signals. Next, information reconciliation
is done using Polar codes. To evaluate the key generation rate we pass the
output of the reconciliation through a conditional min-entropy (conditioning
on an eavesdropper knowledge) estimation.

The rest of this chapter is organized as follows: Section 2 motivates this
work, Section 3 gives an overview of our system model, Section 4 gives a
detailed overview of the SKG protocol, Section 5, presents the details of the
measurement campaign, Section 6 summarizes some of the main results of this
work and concludes this chapter.

##  2 Related work and motivation

Radar systems are already deployed in numerous verticals including, automotive
industry, ship and aircraft navigation, smart factories and many more [15]. It
is clear that the applications above can benefit from a co-designed
communication and sensing approach. Particularly, vehicles can continuously
sense the environment while sharing the obtained information with others on
the road [16]. Similarly, autonomous machinery can communicate to improve
production, while sensing for approaching people. Furthermore, in bistatic (or
multistatic) radar applications, radars can accurately detect multiple targets
and simultaneously convey location information to improve safety, e.g., in
traffic control. Generally, in many joint communication and sensing scenarios,
the data to be transmitted consists of confidential information, including
location, speed and identity. Therefore, it is important to protect the
transmission from possible information leakage and provide confidentiality.

As noted in the 3GPP report [17] a major limitation of 5G networks, especially
in the context Internet of things (IoT), is related to security. Standard
cryptographic solutions rely on computationally intensive modulo arithmetic
operations which makes them unsuitable for power constraint devices. This
issue has also been long investigated by the national institute of standards
and technology (NIST). Particularly an investigation on cryptography for
constrained environments was initiated by NIST in 2013 [18]. After a decade,
in 2023, the Ascon family of symmetric authenticated ciphers has been
announced as a winner [19]. The algorithm is designed to protect data
exchanged between IoT electronics. However, the issue of finding a lightweight
and quantum resistant secret key distribution technique has not been yet
resolved. As it stands at present, all approved mechanisms rely on the
employment of longer keys [20]. This leads to increased complexity and
contradicts to the requirements for low latency and low footprint. Therefore,
the current state of post-quantum innovations does not align well with the
expectations towards 6G networks.

In this sense, PLS-based SKG is a promising quantum-secure approach, that can
provide confidentiality at a low computational cost. This is a major advantage
compared to standardized solutions. The SKG procedure follows a sequence of
steps which allow a pair of devices to extract a shared secret key. A standard
source of randomness used to implement SKG is the wireless channel.
Reciprocity and stochastic behavior are the properties of the channel that
provide the ground SKG [6, 21, 22]. Devices can extract randomness from their
channel observations at a low cost and without the need to perform complex
modulo arithmetic in large fields as in current solutions. However, it is
important to note that, the achievable key generation rates depend on the
statistical properties of the channel, i.e., SKG rates in static channels are
significantly lower than in mobile environments. This makes SKG a good
candidate for key distribution in many joint communication and sensing
applications, where mobility is expected.

Only a few studies have yet considered the security of joint communication and
sensing applications [23, 24, 16, 25]. The work in [23] provides an
information-theoretic perspective on the information leakage between sensing
and communication in a discrete memoryless broadcast channel. The considered
scenario assumes a transmitter that aims to simultaneously perform channel
state estimation and reliable communication with a legitimate user. Next, [24]
proposes a multi-carrier chirp approach to perform joint communication and
sensing. The aim of the study is to maximize the secrecy rate of a wiretap
channel in the presence of one or multiple eavesdroppers. Through numerical
evaluation it is demonstrated that sensing can provide valuable information
about potential eavesdroppers. Finally, the authors of [16] and [25] provide
simulation-based analyses on the performance of chirp signals for key
generation. In particular [16], provides an estimate of the bit mismatch
probability between two nodes in a vehicular and a smart factory scenarios.
The work in [25] provides a different perspective and investigates the number
of bits that are leaked to eavesdroppers in the vicinity.

All studies above show promising results proving that employing PLS can be
beneficial for future networks. Based on that, we examine a filterbank-based
SKG approach, suitable for radar-centric application, where chirp signals are
employed. Such scenarios would typically require high sensing precision and
low-rate transfer of sensitive information. Multiple use cases that fit this
description can be found in the 3GPP report [26]. A particular example is
collective situational awareness, where devices need to simultaneously perform
sensing with high precision and exchange small packets at low latency. With a
successful history in radar application and the advantage of low-complexity
processing wideband chirp signals are seen as a viable waveform candidate for
such scenarios. Furthermore, chirps can provide a broad view of the wireless
channel and offer rich entropy source for SKG. From this perspective, channel
measurements (e.g., for equalization purposes) can be used towards SKG and
subsequently, secure the information exchange. The idea of the filterbank-
based SKG with chirp signals is to divide the received signal bandwidth into a
set of parallel sub-bands and measure the power at each sub-band. As compared
to RSS-SKG methods, where single value is obtained per transmission, the
filterbank approach allows to capture a series of power measurements that
represent the randomness of the channel.

##  3 System model

The system model in this work consists of three parties: two legitimate users,
namely Alice and Bob, and an eavesdropper, namely Eve. Each of the parties has
a single antenna. To measure the channel Alice and Bob exchange a series of
linear complex up-chirp signals. Eve is positioned in the vicinity of Bob and
eavesdrops their exchange. The goal of Alice and Bob is to generate a secret
key using the observed channel measurements while keeping the leakage to Eve
close to zero. A chirp signal can be represented in baseband as
$x(t)=\frac{1}{T}e^{j\pi ct^{2}}$, where $T$ denotes the duration of the
chirp, $c=B/T$ is chirp rate and $B$ denotes bandwidth. The instantaneous
frequency changes linearly with time, $f(t)=ct+f_{0}$, where $f_{0}=f_{c}-B/2$
is the starting frequency at $t=0$ and $f_{c}$ is the center frequency of the
signal.

To perform SKG, Alice and Bob send signals $x(t)$, up converted to a chosen
passband. The received signals at Alice, Bob and Eve are:

$\displaystyle y_{A}(t)=x(t)*h(t)+w_{A}(t),$ (1) $\displaystyle
y_{B}(t)=x(t)*h(t)+w_{B}(t),$ (2) $\displaystyle
y_{E}(t)=x(t)*h_{E}(t)+w_{E}(t),$ (3)

respectively. In the equations above, $h(t)$ is a complex channel response
representing the channel between Alice and Bob, $h_{E}(t)$ represents the
channel between Alice and Eve, which, depending on Eve’s position may be or
may not be correlated to $h(t)$, $w_{A}(t)\sim\mathcal{N}(0,\sigma^{2}_{A})$,
$w_{B}(t)\sim\mathcal{N}(0,\sigma^{2}_{B})$,
$w_{E}(t)\sim\mathcal{N}(0,\sigma^{2}_{E})$ are additive white Gaussian noise
(AWGN) variables experienced at Alice, Bob and Eve respectively. Please note
that this study considers an eavesdropper located close to Bob, hence,
measurements in the channel Bob-Eve are ignored. The channel impulse response
is a sum of multi-path components (MPCs),

$h(t)=\sum^{L-1}_{i=0}{\alpha_{i}\delta(t-\tau_{i,})},$ (4)

where $L$ is the number of MPCs, $\alpha_{i}$ and $\tau_{i}$ are the complex
amplitude and delay of $i$-th MPC, $\delta(t)$ is the Dirac delta function
($h_{E}(t)$ is defined similarly). Having their channel observations all
parties perform filterbank-based SKG as detailed in the next section.

##  4 SKG Protocol

The SKG protocol follows three basic steps, namely, advantage distillation,
information reconciliation and privacy amplification. A sketch of the protocol
is illustrated in Figure 1. Alice and Bob pass their channel estimation
vectors through a quantizer, obtaining binary sequences $\mathbf{r}_{A}$ and
$r_{B}$, respectively. To reconcile mismatches at the generated binary
sequences, one of the legitimate parties, e.g. Alice, sends syndrome
information $\mathbf{s}_{A}$ to Bob. Finally, to create maximal entropy secret
keys, both users employ privacy amplification over the reconciled information.
A detailed description of our implementation of the protocol is given below.

AliceBob[c]Advantage
distillation [c]Advantage
distillation [l]Channel [l]estimation$\mathbf{x}_{A}$$\mathbf{x}_{B}$[c]
Quantization [c]Quantization $\mathbf{r}_{A}$$\mathbf{r}_{B}$[c]Slepian Wolf
decoder [c]Slepian Wolf
decoder Information reconciliationInformation
reconciliation$\mathbf{s}_{A}$$\mathbf{r}_{A}$$\mathbf{r}_{A}$[c]Privacy
amplification [c]Privacy
amplification $\mathbf{k}$$\mathbf{k}$

Figure 1: Secret key generation between Alice and Bob.

### Advantage distillation:

To extract randomness from (1)-(3), each party passes its received signal
through a filterbank with $N$ filters. Each individual filter $n=1,\dots,N$
has an impulse response given as

$g_{n}(t)=g(t)e^{(j2\pi f_{n}(t))},$ (5)

where $g(t)$ is a protyping filter. Each filter has bandwidth $B/N$ and center
frequency, $f_{n}(t)$ calculated as $f_{n}=-\frac{B(N-2n+1)}{2N}$. The
randomness from the channel is extracted through power measurements at the
output of each filter. An example is illustrated in Figure 2. The figure shows
a normalized power spectrum densities (PSDs) of transmit and received signals
with $B=70$ MHz. The impulse response of the filterbank, for $N=5$, is also
illustrated along with the power measurements obtained at the output of each
filter.

Figure 2: Filterbank constructed of $N=5$ filters. The transmit signal has
bandwidth $B=70$ MHz. Power measurements at each sub-band are given as dots.

The obtained vector of power measurements at Alice can be expressed as
$\mathbf{{p}}_{A}=\left[{\hat{p}}_{{A,1}}\cdots{\hat{p}}_{{A,N}}\right].$ The
measurements at Bob and Eve can be similarly expressed. Assuming that
measurements are obtained over long enough periods, the power of the sampled
version of the filtered signal can be approximated as the average of the
continuous-time signals, i.e.,
${{\hat{p}}}_{{A,n}}=\overline{|g_{n}(t)*y_{A}(t)|^{2}}$ with
$\overline{(\cdot)}$ denoting time-averaging operator.

The probability distribution of a single power measurement, considering a
Rayleigh channel, has been evaluated in our previous work [14]. It was shown
that the probability is a sum of Gamma-distributed variables
$\sum_{m=1}^{M}\Gamma(\kappa_{m},\theta_{m})$, with shape $\kappa_{m}$ and
scale $\theta_{m}$ parameters and can be approximated as

$\displaystyle\hat{p}_{n}\sim\Gamma\\!\\!\left(\\!\kappa\\!\\!=\\!\\!\frac{(\sum_{m=1}^{M}\kappa_{m}\theta_{m}\\!+\\!2\sigma^{2})^{2}}{\sum_{m=1}^{M}\kappa_{m}\theta_{m}^{2}\\!+\\!\frac{8\sigma^{4}}{K-1}},\theta\\!\\!=\\!\\!\frac{\sum_{m=1}^{M}\kappa_{m}\theta_{m}\\!+\\!2\sigma^{2}}{\kappa}\\!\right)\\!\\!,$
(6)

where $M$ is the number of resolvable MPCs, $K$ is the number of digital
samples that fall within the filter bandwidth and $\sigma^{2}$ is AWGN
variance.

### Information reconciliation:

Power measurements at each party are quantized to bit sequences. Depending on
the type of quantizer each power measurement can generate one or multiple
bits. Using Gray codes, the number of bits per measurement is $\log_{2}(Q)$,
where the number of quantization levels (regions), $Q$, is chosen to be a
power of $2$. After performing quantization the three parties obtain binary
vectors
$\mathbf{r}_{A},\mathbf{r}_{B},\mathbf{r}_{E}\in\\{0,1\\}^{N\log_{2}(Q)}$. Due
to noise or imperfect power estimation the observations at Alice and Bob could
differ.

To correct errors, Alice (or Bob) generates and sends reconciliation
information (e.g., syndrome) $\mathbf{s}_{A}$ over a public channel. The
syndrome can be generated using distributed source coding techniques. Next,
$\mathbf{s}_{A}$ is used by Bob to correct $\mathbf{r}_{B}$ and obtain
$\mathbf{r}_{A}$. However, due to public exchange of the reconciliation
information Eve also obtains $\mathbf{s}_{A}$. She can use the syndrome and
her initial measurements to correctly guess part or the entire sequence. To
remove leakages to Eve and limit her chances of guessing the legitimate key a
final step is performed, i.e., privacy amplification.

### Privacy amplification:

Keys are the core of cryptographic mechanisms; hence, their quality is of
utmost importance for the security of modern communication systems. In this
sense, the final key $\mathbf{k}\in\mathcal{K}$ between Alice and Bob should
satisfy the following randomness inequality [27] $|\mathbf{k}|\leq
H_{\infty}(\mathbf{r}_{A})$, where

$H_{\infty}(\mathbf{r}_{A})=-\log_{2}\max_{\mathbf{r}_{A}\in\mathcal{R}_{A}}p(\mathbf{r}_{A}),$
(7)

denotes min-entropy [28]. However, the equation above does not consider any
leakage at Eve. Therefore, a more rigorous metric is required to represent the
scenario in this work. To measure the quality of generated keys in such
situations, conditional min-entropy is used. It is defined as [29]:

$H_{\infty}(\mathbf{r}_{A}|\mathbf{r}_{E},\mathbf{s}_{A})=-\log_{2}\max_{\mathbf{r}_{A}\in\mathcal{R}_{A},\mathbf{r}_{E}\in\mathcal{R}_{E},\mathbf{s}_{A}\in\mathcal{S}_{A}}p(\mathbf{r}_{A}|\mathbf{r}_{E},\mathbf{s}_{A}),$
(8)

where the shared sequence between Alice and Bob is conditioned on both,
initial measurements at Eve and the leaked syndrome. In fact, the difference
between (7) and (8) represents the total amount of leaked information [30]:

$\text{Leakage}=H_{\infty}(\mathbf{r}_{A})-H_{\infty}(\mathbf{r}_{A}|\mathbf{r}_{E},\mathbf{s}_{A})$
(9)

Following the above, the final key should be of size $|\mathbf{k}|\leq
H_{\infty}(\mathbf{r}_{A}|\mathbf{r}_{E},\mathbf{s}_{A})$.

The evaluation of these quantities requires knowledge on the underlying
marginal and conditional distributions. Finding a closed-form solution is a
complex task and falls out of the scope for this chapter. Instead, we use a
numerical estimator that can evaluate the above without prior knowledge on the
analytical expressions. We use the fast black-box leakage estimation (FBLEAU)
[31]. To evaluate the leakage and arrive at the conditional min-entropy, the
estimator relies on nearest neighbor and frequentist estimates [32]. Once the
conditional min-entropy is evaluated a standard way to compress the sequence
to the desired size is through the use of one-way collision resistant
compression functions, e.g., hash function.

##  5 Measurement setup

Figure 3: Sketch of the measurement setup created with floorplaner.com. The
distance between Alice and Bob is $4$ m. Alice is placed at $2.3$ m height.
Bob and Eve are at $0.85$ m height. Throughout the measurement campaign Eve is
placed on a linear positioner which moves her with one-wavelength steps with
respect to Bob’s position (from $1\lambda$ to $10\lambda$).

This section illustrates a real-life setup configured to execute the SKG
protocol presented in Section 4. Based on the system model from Section 3
three universal software radio peripherals (USRPs), two representing the
legitimate nodes (Alice and Bob) and a third acting as an eavesdropper (Eve),
were used. For the purpose of this study, USRP-2974 from National Instruments
are employed. A floorplan of the measurement setup is illustrated in Figure 3.
The distance between Alice and Bob is $4$ m. Alice is placed at $2.3$ m
height, Bob and Eve are placed on a desk at $0.85$ m height. The positions of
Alice and Bob are kept constant throughout the measurement campaign. To
evaluate the leakage towards Eve, ten different positions are considered for
her node. As it can be seen in the figure, she is placed on a linear
positioner next to Bob which moves her with steps of one-$\lambda$, starting
from $1\lambda$ and ending at $10\lambda$. For the considered frequency, i.e.,
$f_{c}=3.75$ GHz, the wavelength is $\lambda\approx 8$ cm.

To obtain channel measurements Alice and Bob transmit chirp signals in a time
division duplex (TDD) fashion. The USRP, configured as Eve, is always in
receiving mode and eavesdrops the exchange between the legitimate nodes. Note
that, due to her position, Eve is only interested in the signals received from
Alice. The duration of individual chirp symbol is $17.1875$ $\mu$s and the
bandwidth is $70$ MHz. At each of the $10$ positions of Eve, Alice and Bob
transmit/receive $10^{5}$ chirp signals. The process described above is
repeated considering four different scenarios.

### Scenarios:

The measurements were performed under a set of four different scenarios
including: line-of-sight (LoS) static, LoS dynamic, non-LoS (NLoS) static,
NLoS dynamic. Dynamic measurements are achieved by movement of people and
objects in the room. Static measurements are performed throughout night and
nothing moves during the process. The LoS scenario is illustrated in Figure 4.
One of the objects moved throughout the campaign, i.e., a circular metal
plate, is also displayed on the figure. The NLoS case is illustrated in Figure
5. The difference between the two setups is that, several absorbers are placed
to block the direct path between antennas. Apart from that the setup is kept
identical as in the LoS scenario.

Figure 4: Line-of-sight scenario. Dynamic and static cases differentiate based
on whether there is movement in the room. Eve is placed on a linear positioner
and is moved from $1\lambda$ up to $10\lambda$ with respect to Bob’s position.
Figure 5: Non-line-of-sight scenario achieved using a set of absorbers to
block the direct path between nodes. Dynamic and static cases differentiate
based on whether there is movement in the room. Eve is placed on a linear
positioner and is moved from $1\lambda$ up to $10\lambda$ with respect to
Bob’s position.

### Details on the SKG implementation:

Upon receiving a signal, each of the three nodes executes the steps of the SKG
protocol presented in Section 4. First, the received signals
$y_{A},y_{B},y_{E}$ are passed through filterbanks; different number of
filters are considered, i.e., $N=\\{8,16,32,64\\}$. Power measurements at the
output of each filter are used as a source for the SKG. Next, the measurements
are converted into binary sequences
$\mathbf{r}_{A},\mathbf{r}_{B},\mathbf{r}_{E}$ using linear multi-level
quantization technique. Two different number of quantization levels are
employed, i.e., $Q=\\{4,16\\}$, resulting in $2$ or $4$ bits per power
measurement, respectively. Bit sequences are reconciled using Slepian-Wolf
decoders implemented based on Polar (more details on Slepian-Wolf decoding,
including Polar codes can be found in [33]). Alice generates a syndrome
$\mathbf{s}_{A}$ and shares it with Bob (this is also observed by Eve). Bob
and Eve, use the syndrome along with their observations and aim at reproducing
Alice’s bit sequence. Depending on the number and distribution of errors the
error correction process might be successful or not. Finally, based on the
available conditional min-entropy
$H_{\infty}(\mathbf{r}_{A}|\mathbf{r}_{E},\mathbf{s}_{A})$ Alice and Bob
generate a shared secret key $\mathbf{k}$. A thorough performance comparison
with respect to the value of $N$, the value of $Q$, and the choice of
reconciliation technique is presented in the next section.

##  6 Results and discussion

This section provides a detailed overview on how the parameters at each step
of the SKG protocol affect the performance. First, we examine our analytical
results from [14], (also given by Eq. 6), and try to fit the histograms of the
observed power values with gamma distribution. As noted in the previous
section, the transmit signal has a bandwidth of $B=70$ MHz. Filters are
distributed evenly across this bandwidth, hence, adding more filters results
in smaller bandwidth per filter. This changes the number of digital samples
falling within a single filter and has an impact on the power distribution.
This is illustrated in Fig. 6.

Figure 6: PDF of the power measurement taken at different environments, all
fitted with different Gamma distributions. The measurements are for a single
filter when considering $N=64,32,16,8$.

The figure shows that when the channel is static the distribution of the power
measurements aligns with our analytical finding, i.e., it fits well with Gamma
distribution. Another observation, which was also confirmed in [14], is that
when the number of filters $N$ decreases (and the bandwidth of each filter
increases) the variance of the measurements also decreases. As a direct
consequence of Eq. 6 by increasing the number digital samples $K$ (i.e.,
increasing the bandwidth) the effect of noise $\sigma^{2}$ decreases. This can
be observed for both NLoS and LoS cases. Now, considering dynamic scenarios we
can see that only the tail of the distributions align with the Gamma fitted
curves and the peaks represent a mixture of several distributions. As expected
the variations in dynamic environment are greater from the static ones.
However, differently from above, they increase with the bandwidth. This is
because the noise is no longer the dominant cause of variation. Instead,
multipath provides more degrees of freedom and observing higher bandwidth
results in higher spread of the measurements. Overall, this shows that general
statistical models cannot fully capture real-life situations, hence, further
investigation is required.

The next step is quantization of the observed measurements. Based on the
combination of number of filters and quantization levels different number of
bits can be obtained (e.g., for $N=16$ and $Q=4$ we have $16\log_{2}(4)=32$
bits per channel use). At this step we evaluate the average mismatch
probability between nodes.

Figure 7: PDF of the power measurement taken at different environments, all
fitted with different Gamma distributions. The measurements are for a single
filter when considering $N=64,32,16,8$.

Figure 7 illustrates the average mismatch between legitimate users and the
average mismatch observed at Eve. First, as expected, if the legitimate users
increase the number of bits generated per frame (i.e., by increasing $N$ or
$Q$) the mismatch increases. This is because generated bits become prone to
noise variations which are not reciprocal. Second, it can be seen that in the
NLoS scenarios the generated bit sequences have more errors as compared to the
LoS scenarios. This is expected as the power observed at Alice and Bob
decreases when LoS is not present. Finally, an interesting result can be
observed for the mismatch at Eve. The error bars in the figure represent
standard deviation of her average mismatch probability taken over all $10$
positions. Note that, during the experiment it was observed that Eve’s
mismatch does not improve when she is closer to Bob, instead, she can have
better probability at further positions - this might be due to in-phase/out-
of-phase addition of multi-path components at certain location of the room.
This confirms the theoretical result [34], that channels de-correlate over a
distance of $\lambda/2$. The figure shows that, in all scenarios, the average
value of Eve’s mismatch is independent from $Q$ and $N$ and remains constant
close to $50\%$. However, it can be seen that the standard deviation increases
for smaller values. This shows that if less bits are produced per frame, i.e.,
$Q$ and $N$ are small, Eve has higher chances of observing similar bit
sequence to the legitimate one.

Figures 8 and 9 illustrate the results from the information reconciliation
step. Polar codes as in [33] are implemented to perform this step.

Figure 8: Frame error rate (FER) at Alice and Bob after reconciliation.

Figure 8 shows the frame error rate (FER) at Alice and Bob for different
values of $Q$ and $N$. It can be seen that the number of quantization levels
$Q$ has a higher impact on the FER as compared to the number of filters $N$.
In particular, if Alice and Bob harvest the entropy of the channel using $32$
filter and $4$ ($64$ bits per channel use) levels the FER is smaller compared
to $16$ filter and $16$ levels (also $64$ bits per channel use). Similarly to
Figure 7, it is observed that the performance at the legitimate users
decreases when a LoS path is not present. This is an expected result, as this
naturally decreases the SNR between the two parties. Another important
parameter for the success of this step is the code rate. At lower code rate
the FER is negligible, however, this corresponds to a larger syndrome
$\mathbf{s}_{A}$.

Figure 9: Frame error rate (FER) at Eve after reconciliation.

The syndrome is sent over a public channel, hence, Eve also has access to it.
Therefore, a lower code rate gives her a better chance to arrive at the
correct sequence. This is illustrated in Figure 9. The figure shows the FER at
Eve after she observes the syndrome information. It can be seen that at low
code rates, i.e., $0.1$, her probability to decode the correct sequence
becomes higher, hence, impacting the security of the system. Important
observations from this figure is that, similarly to Alice and Bob, Eve
performance becomes better if more filters are used as opposed to more
quantization levels. Therefore, while increasing the number of bits by
increasing $N$ (instead of $Q$) might be beneficial for the legitimate users,
this might increase the potential leakage to Eve. This is examined next.

We evaluate $H_{\infty}(\mathbf{r}_{A}|\mathbf{r}_{E},\mathbf{s}_{A})$ using
the FBLEAU numerical estimator [31]. The estimator takes as input the three
vectors $\mathbf{r}_{A}$, $\mathbf{r}_{E}$ and $\mathbf{s}_{A}$ and provides
an evaluation on the conditional min-entropy. Results are illustrated in
Figure 10. The figure gives the estimated values for different code rates and
different values of $Q$ and $N$. Due to computational constraints, i.e.,
evaluation over large dimensions of bit sequences, the figure shows only
combinations up to $32$ filters with $4$ quantization levels. This limit comes
from a problem known as the curse of dimensionality [35] which is an open
research area. It increases the time of convergence non-linearly with each new
dimension.

Figure 10: Conditional min entropy for different $N$, $Q$ and code rates.

The first observation is that increasing the code rate increases the
conditional min-entropy. This effect is due to the fact that a higher code
rate corresponds to a shorter syndrome, hence, less information is leaked to
Eve. Furthermore, as the randomness of the generated bits rely on the
variation of the channel we can see that the conditional min-entropy is low
for static scenarios. In contrast dynamic environments are good source of
randomness for Alice and Bob. Finally, controversially to the results for
reconciliation we see that capturing the small variations in the channel by
increasing $Q$ provides higher randomness as opposed to increasing $N$.
However, to execute the full SKG protocol successfully, both reconciliation
and conditional min-entropy must be respected.

The average key rate per channel use, $R$ per channel use is a function of
several parameters: i) frame rate $F$, equal to the number of bits generated
per frame (varies accordingly with $Q$ and $N$); ii) the FER after
reconciliation; iii) conditional min-entropy
$H_{\infty}(\mathbf{r}_{A}|\mathbf{r}_{E},\mathbf{s}_{A})$. The key rate is
denoted as:

$R=F\times(1-\text{FER})\times
H_{\infty}(\mathbf{r}_{A}|\mathbf{r}_{E},\mathbf{s}_{A}).$ (10)

Results on the rate are given in Figure 11.

Figure 11: Average key rate (per channel use) over different $N$, $Q$ and code
rate.

The trend in the figure is clear. To increase the key rate, $F$ must be
increased, preferably by increasing $Q$ as opposed to $N$ and the code rate
should be decreased. Based on the system parameters used in our measurement
campaign the values from Fig. 11 can also be expressed in $[$b/s/Hz$]$ or
$[$b/s$]$. Given the fact that a two-way transmission between Alice and Bob
takes approximately $35$ $\mu$s and $B=70$ MHz we present the results in Table
LABEL:tab:rates. The table shows only the maximum achievable rate based on
Fig. 11, i.e., considering code rate $0.1$, $N=Q=16$.

Table 1: Achievable secret key generation rates. Scenario | NLoS Dynamic | NLoS Static | LoS Dynamic | LoS Static
---|---|---|---|---
Rate [b/s/Hz] | 0.0121 | 0.0034 | 0.0102 | 0.0005
Rate [b/s] | 848220 | 240131 | 713940 | 31614

To conclude, our experimental results show that the information available for
SKG could vary greatly depending on both channel and system parameters. We
have shown that chirp signals can capture the channel variations over
different frequencies and provide a valid source for SKG. The evolution of
wireless systems and the introduction of sensing at the edge of the system can
provide valuable information regarding the environment. This provides the
required context information and makes SKG a viable technique for key
distribution. As a future work, the authors plan to further investigate the
results presented in this chapter, by considering multiple antenna scenarios
and work on adaptive SKG solutions. Furthermore, the proposed chirp-based
approach will be experimentally compared with waveforms with high spectral
efficiency waveforms (e.g., OFDM).

## Acknowledgement

This work is financed on the basis of the budget passed by the Saxon State
Parliament, and partially funded by Hexa-X-II project, receiving funding from
the Smart Networks and Services Joint Undertaking (SNS JU) under the European
Union’s Horizon Europe research and innovation programme under Grant Agreement
No 101095759. A. Chorti was supported by INEX funding (Project PHEBE and
ENIGMA). The authors would like to acknowledge the help from Dr. Mahdi
Shakiba-Herfeh who provided us with a Slepian-Wolf implementation of Polar
codes.

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