Fluid Antenna assisted Energy Harvesting over THz
band using Resonant Tunneling Diode
Triantafyllos Mavrovoltsos, Eleni Demarchou, Costantinos Psomas, and Ioannis Krikidis
IRIDA Research Centre for Communication Technologies
Department of Electrical and Computer Engineering, University of Cyprus, Cyprus
e-mail: tmavro03, edemar01, psomas, krikidis@ucy.ac.cy
Abstract—The deployment of nano-scale networks is a key
technology for 6G communication systems. Terahertz (THz)
communications are highly suitable for these networks, serving as a pivotal technology to facilitate their implementation.
Overcoming the impracticality of regular battery replacement
involves the harvesting of energy from electromagnetic radiation.
In this paper, we investigate low-power energy harvesting (EH)
over the THz band. For the integration of the EH circuit we
consider resonant tunneling diode (RTD), a diode suitable for
the THz band. Because of its negative differential resistance
regions, RTD is characterized by non-monotonic input-output
relationship. To this end, we propose the integration of the
RTD with fluid-antenna (FA) systems. Within this paradigm, the
freedom provided by the emlpoyment of FAs allow us to improve
the system’s performance by selecting the best port. Since a
closed-form expression for the current flow of the RTD is not
available, we propose a generalised linear piecewise model to fill
the existing gap in the literature. Next, based on the proposed
model, we study the system’s perfomance in terms of energy
outage probability. Finally, we propose a FA port selection scheme
and compare its performance with the conventional selection
combining (SC) scheme. Our results demonstrate that when
the number of ports is sufficiently large, the proposed scheme
outperforms the conventional SC.
Index Terms—Wireless power transfer, resonant tunneling
diode, fluid antennas, THz communications.
I. I NTRODUCTION
In the era of sixth generation (6G) communication systems,
the deployment of nano-scale networks emerges as a standout
application [1]. Within this paradigm, ultra-small, low-power
internet of things (IoT) sensors constitute an integral part of the
forthcoming networks. A nano-sensor is an integrated device
around 10-100 µm2 in size, able to do simple computational
tasks besides sensing, exploring potential applications accross
various sectors such as healthcare, environmental monitoring
and smart infrastructure [2]. A key enabling technology for
the 6G networks is considered to be terahertz (THz) communications which facilitates short-range, high-speed communications, essential for nano-scale IoT networks, enabling the
feasibility of such small-sized sensors [1]. In the THz band, a
variety of propagation conditions is observed, besides the high
path loss attenuation, such as molecular absorption and particle
scattering. For the research community, channel modeling is a
fundamental consideration towards the performance evaluation
of THz communications. To this end, the authors in [3],
develop a channel model to acquire the aggregate path loss
in the THz band, by taking into account both molecular
absorption and spreading loss. Moreover, the authors in [4],
investigate the small-scale fading effects under line-of-sight
(LOS) conditions. In their work, experimental measurements
support that Nakagami-m fading is able to capture the multipath effects in the THz band.
A significant consideration for the nano-scale networks
lies in nano-batteries’ lifetime. The impracticality of regular
replacements arises from the small size of the devices, and
in certain cases, replacement becomes virtually impossible
due to the nature of the application. A promising solution to
overcome this problem is wireless power transfer (WPT). In
this approach, devices are powered up by employing energy
harvesting (EH) circuits; rectifiers capable of converting the
input RF signal to a DC output [5]. A resonant tunneling
diode (RTD), distinguished by significantly reduced transient
times, proves to be a suitable diode for implementing the
EH circuit that operate over the THz band. Additionally, in
low-power regimes, the negative differential resistance regions
in RTDs give rise to a non-monotonic current-voltage (I-V)
characteristic under forward bias. The authors in [6], provide
experimental results which demonstrate that RTDs exhibit
significant non-linearity and non-monotonic behavior, and
design a Keysight ADS model to fit with their measurement
data. In addition, the authors in [7] propose an equivalent
monotonic EH model using a five-parameter logistic function
and formulate a problem to maximize the average harvested
power in a WPT system via THz-band signals.
Another key enabling technology for the implementation of
6G communications, is reconfigurable antennas that provide
more flexibility [1]. The rationale behind reconfigurable antennas, is to exploit spatial diversity through a single antenna.
Specifically, the authors in [8], propose fluid-antenna (FA), a
device composed by a radiating liquid enclosed in a container,
where the liquid can change its position among the predefined
locations, known as ports. Besides, the problem of mutual
coupling which is met in conventional antenna arrays, is vanished through FA systems which employ solely one radiating
element. The authors in [8], study the outage performance of
FA systems over spatially correlated Rayleigh fading channels.
They show that by selecting the best port, FAs can outperform
a maximum ratio combining (MRC) receiver if the number of
ports is sufficiently large. Moreover, the authors in [9], provide
an analytical framework for FAs over correlated Nakagami-m
fading channels, under various FA architectures.
path loss, which is captured by Friis’ transmission formula.
As such, the received power is given by
PR =
Fig. 1. Fluid antenna receiver architecture
In this work, motivated by the flexibility provided by the
FAs and their possible integration with the non-monotonic
behavior of the RTD, we investigate a WPT system operating
in the THz band with a FA receiver equipped with an RTDbased EH circuit. First, in order to characterize the harvested
DC output power as a function of the received RF input
power, we propose a general linear piecewise EH model. Next,
we provide a rigorous mathematical framework to study the
system’s performance in terms of energy outage probability.
Finally, we provide simulation results for three port selection
schemes and compare them with the conventional selection
combining (SC) scheme.
II. S YSTEM M ODEL
Consider a point-to-point system which operates in the
THz band and consists of a high directional single-antenna
transmitter with transmit power PT , and a FA receiver located
at a distance d from the transmitter. The FA encompasses N
ports, evenly distributed within the fluid container of size W λ,
where λ is the carrier’s frequency wavelength. Note that, the
radiating liquid within the container can be controlled such that
among the N ports, a single one can be selected at a time.
In addition, we assume that the receiver is equipped with a
rectifying circuit, able to convert the RF received signal to
DC output. The circuit is composed of an RTD, and a lowpass filter with capacitance C and a load resistance RL , as
shown in Fig.1.
A. Channel Model
We assume that the wireless link suffers from both small
scale fading (multi-path) and large scale fading (path loss), asscosiated with the THz band propagation effects. Since nanoscale communications correspond to short-range distances, the
shadowing effect can be neglected. Hence, the deterministic
aggregated path loss experienced by the receiver can be
expressed as a linear scale product of the specific attenuation
due to absorption and the spreading loss, i.e the free space
PT GR GT λ2 −ka (f )d
e
,
(4πd)2
(1)
where ka (f ) is the absorption coefficient, and GR and GT
denote the directionality gain of the receiver and transmitter
antenna, respectively.
In addition, we denote by hk the channel coefficient at the
kth port, accounting for Nakagami-m fading, where m ≥ 1
is an integer determining the LOS conditions. A Nakagamim random variable (RV) can be expressed as the square
root of a sum of m independent squared complex Gaussian
RVs with zero mean and variance σ 2 [9]. As such, a set of
N × m complex Gaussian RVs is necessary to represent the
Nakagami-m fading channels for a FA with N ports. In this
context, a Nakagami-m distributed RV rk , can be expressed
as
q
(2)
rk = χ21 + χ22 + . . . + χ2N
where χ1 , χ2 , . . . , χN are independent complex Gaussian RVs
with zero mean and variance σ 2 .
B. Spatial Correlation Model
As aforementioned, the position of the radiating liquid can
be controlled in order to select a singular predefined port.
Since these ports are located in close proximity to each other,
the received signals at all N ports are considered to be
correlated. To this end, we adopt a spatial correlation model
within which all ports are expressed with respect to a reference
port, and more spesifically the first port [8], [9]. Thereafter,
the spatially correlated Nakagami-m fading channel at port k,
is expressed as
q
Hkl = 1 − µ2k xkl + µk x0l
q
(3)
2
1 − µk ykl + µk y0l , l = {1, . . . , m},
+j
where xkl , ykl are independent Gaussian RVs with zero
mean and 1/2 variance, and µk are parameters chosen to
represent the correlation between the ports. We consider that
the correlation coefficient follows the Jake’s model, so that
2π∆dk,1
, for k = {1, . . . , N },
(4)
µk = J0
λ
k−1
where ∆dk,1 = N
−1 W λ, is the distance between the first
(reference) and the kth port and J0 (·) is the zero-order Bessel
function of the first kind. Finally, since Hkl is the complex
Gaussian RV of our interest, the appropriately normalized
channel coefficient at the kth port hk is given by
v
um
uX 1
hk = t
|Hkl |2 .
(5)
m
l=1
III. E NERGY H ARVESTING M ODEL
As previously mentioned, the EH circuit is able to convert
the received RF signal to DC output. We denote the received
Fig. 3. Generalised linear piecewise input-output power relationship.
Fig. 2. Linear piecewise approximation following I-V characteristic datapoints
from [4].
RF input power at the kth by
PRFk = PR h2k s2 ,
(6)
where s is the transmitted signal with E[s2 ] = 1.
For the sake of simplicity, we consider a triple-barrier RTD
with a single negative differential resistance region. We aim
in providing a general linear piecewise model which fits the
measumerent data of the I-V characteristic given in [6], [7]1 .
Since a closed-form expression for the current flow Id is not
available, we extract the measurement data by using curve
fitting tools and we approximate the I-V characteristic with a
linear piecewise approximation, as shown in Fig. 2.
Once we investigate a WPT system we shall further provide
information related to the RF-DC power conversion. Even
though the EH circuit’s analysis is out of the scope of this
paper, we will briefly provide some insights. For this purpose,
we adopt a simple circuit as presented in [10]. In particular,
the received signal is modeled as a voltage source, whilst
assuming an ideal low-pass filter in its steady-state response,
depicted in Fig. 1. Therefore, the voltage drop across the diode
can be expressed as Vd = Vin − Vout , where Vin corresponds
to the received signal’s amplitude after the voltage drop due to
the input resistance Rin . Vout = Id RL is the voltage across the
load resistance. Since we aim in acquiring a general model, the
received RF input power and the harvested DC output power
at any port, are given by
Vd2
,
PDC = Id2 RL ,
(7)
Rin + RL
where κ ≥ 1, is a coefficient that one can calibrate to fit the
parameters of a particular circuit design.
Based on this, we propose a linear piecewise function
PRF = κ
1 If the voltage applied to an RTD is negative and smaller than the
breakdown voltage we may cause damage to the device. Here, we assume
that breakdown voltage can be avoided.
ψ(PRF ) to characterize their non-monotonic

ψ1 (PRF ) = a1 PRF + b1 ,



ψ (P ) = a P + b ,
2
RF
2 RF
2
PDC ≜ ψ(PRF )=

ψ
(P
)
=
a
P
+
b
3
RF
3 RF
3,



ψ4 (PRF ) = γ3 ,
relationship, as
PRF ∈ [0, ρ1 ],
PRF ∈ [ρ1 , ρ2 ],
PRF ∈ [ρ2 , ρ3 ],
PRF ∈ [ρ3 , ∞),
(8)
where α1 , α2 , α3 and b1 , b2 , b3 are parameters that can
be adjusted to fit the measurement data, ρ1 , ρ2 and ρ3 are
threshold values that divide the range of PRF into monotonic
intervals, and γ3 corresponds to the saturation level, depicted
in Fig. 3.
if
if
if
if
IV. E NERGY O UTAGE P ROBABILITY
In order to study the sustainability of the WPT system we
consider the energy outage probability as the key performance
metric. An outage event occurs when the harvested power is
below the threshold γth , determined by the EH’s sensitivity.
We define the outage probability Pout as
Pout ≜ P (PDC < γth ) .
(9)
For the sake of analysis, we can express outage probability
with respect to the received power. To this end, we separate
the outage probability in three regimes, with regards to the predefined γth . As such, the outage probability can be expressed
as


if γth ∈ [0, γ1 ],
P(PRF < x1 ),
Pout ≜ P(PRF < x1 ) + P(x2 < PRF < x3 ), if γth ∈ [γ1 , γ2 ],


P(PRF < x3 ),
if γth ∈ [γ2 , γ3 ],
(10)
where γ1 , γ2 and γ3 correspond to the threshold values that
divide PDC into different regimes, and x1 , x2 and x3 represent
the cross-points of γth with ψ(PRF ) (see Fig.3),

(γth −b1 )

x1 = a1 , if γth ∈ [0, γ2 ],
(γth −b2 )
x = x2 = a2 , if γth ∈ [γ1 , γ2 ],
(11)


(γth −b3 )
x3 = a3 , if γth ∈ [γ1 , γ3 ].
Eventually, when the EH’s sensitivity lies either within
γth ∈ (0, γ1 ] or γth ∈ [γ2 , γ3 ], then there is only one
cross-point, x1 or x3 repsectively. This implies that in these
regimes the input-output relationship is monotonic and the
circuit’s behavior is similar to the one captured by utilizing
conventional linear diodes. In this case, the maximum received
power is equivalent to the maximum harvested power. On the
other hand, when γth ∈ [γ1 , γ2 ], there are three cross-points,
i.e x1 , x2 and x3 . Hence, in this regime, the input-output
relationship of our diode is non-monotonic, and the maximum
received power does not correspond to the maximum harvested
power. In this regime, an outage occurs when PRF ∈ (0, x1 ]
or PRF ∈ [x2 , x3 ], resulting in two mutually exclusive outage
events.
Throughout the rest of the paper we mainly focus in the case
where γth ∈ [γ1 , γ2 ]. We investigate three selection schemes
namely, the output-based selection scheme (OBS), the inputbased selection scheme (IBS) and the random selection scheme
(RS). When comparing input-based and output-based selection
schemes, our goal is to highlight the differences arising from
non-monotonicity. Finally, in order to compare the FA schemes
with the conventional case we consider L-antenna SC scheme
as a benchmark.
A. Output-Based Selection Scheme
To begin with, consider a port selection scheme able to
switch on the port that maximizes the harvested DC output
power. Assuming that we observe the circuit’s output, the OBS
scheme has a medium implementation complexity, since the
selection is a straightforward procedure. Once the FA locate its
radiating liquid at the port that maximizes PDC , the selected
output power PS is given by
PS = max{ψ(PRF1 ), ψ(PRF2 ), . . . , ψ(PRFN )}.
(12)
We now evaluate outage probability as Pout = P (PS < γth ).
This condition is satisfied when
P(PRF < x1 ) + P(x2 < PRF < x3 ),
(13)
where PRF is the set {PRF1 , PRF2 , . . . , PRFN }. The analytical
expression for the outage probability is presented in the
following theorem.
Theorem 1.
2mm
Pout (γth ) =
Γ(m)σ12m
m
+
2m
Γ(m)σ12m
q
Z
x1
PR
r12m−1 e
−
2
mr1
2
σ1
0
N
Y
(Ak + Bk ) dr1
k=2
q
Z
q
x3
PR
x2
PR
r12m−1 e
mr 2
− 21
σ1
N
Y
(Ak + Bk ) dr1 ,
k=2
(14)
where Ak and Bk are given in Appendix A and Qm is the
Marcum Q-function.
Proof. See Appendix A.
B. Input Based Selection Scheme
Now consider a port selection scheme able to switch on the
port that maximizes the received RF input power. Assuming
TABLE I
S IMULATION PARAMETERS
ρ1 = 1.7 mW, ρ2 = 7 mW, ρ3 = 20 mW
α
β
γ
α1 = 0.0350294
β1 = 0
γ1 = 3.9 × 10−6
α2 = −0.0105
β2 = 7.74 × 10−5
γ2 = 5.955 × 10−5
α3 = 0.0059
β3 = −3.74 × 10−5
γ3 = 8.06 × 10−5
that we observe the circuit’s input, the IBS scheme has the
same implementation complexity with the OBS scheme. Once
the FA locates its radiating liquid at the predefined port that
maximizes PRF , the received power of the selected is given
by
PS = max{PRF1 , PRF2 , . . . , PRFN }.
(15)
We now evaluate the outage probability for the IBS scheme
as Pout = P(PS < x1 ) + P(x2 < PS < x3 ). Hence, the outage
probability is expressed as
x3
x2
x1
+ Fmax
− Fmax
,
Pout = Fmax
PR
PR
PR
(16)
where Fmax is the joint CDF for the maximum port given
in
be derived by setting all integral bounds
[9] and can also
√ αN = 0, βN = γth in (22),
2
Z √γth
mr1
2mm
2m−1 − σ12
Fmax (γth ) =
r
e
1
Γ(m)σ12m 0
s
s
!#
"
N
Y
2mµ2k r12
2m
√
γth
dr1 .
×
1 − Qm
,
σ12 (1 − µ2k )
σk2 (1 − µ2k )
k=2
(17)
C. Random Selection Scheme
The random selection scheme is a low implementation
complexity scheme. In this scheme we randomly select a port
to switch on. For this scheme consider that Prand denotes
the received power of the randomly selected port. Outage
probability is defined as
Pout = P(Prand < x1 ) + P(x2 < Prand < x3 ),
(18)
where Prand follows a gamma distribution with the following
corresponding CDF
Z ∞
Γ(m, mγth )
Frand (γth ) = 1−
fγ(n) (y) dy = 1−
, γth ≥ 0,
Γ(m)
x
(19)
R∞
where Γ(a, z) = z ta−1 e−t dt is the upper incomplete
gamma function. Finally, the outage probability is expressed
as
x1
x3
x2
) + Frand (
) − Frand (
)
(20)
Pout = Frand (
PR
PR
PR
V. N UMERICAL R ESULTS
In order to validate theoretical analysis with simulation
results we use the following parameters. Here, GT = 10 dB,
GR = 0 dB, fc = 0.3 THz, N = 10, ka (f ) = 0.1 are
the transmitter and the receiver antenna directionality gains,
carrier frequency, number of predefined ports, and absorption
coefficient respectively. Parameters that give the linear piecewise function were obtained by curve fitting measurement
data from [6] and [7], and are given in Table I. Moreover,
in order to express the input-output power relationship we use
values for the input resistance Rin = 93.35Ω and the load
resistance RL = 4.65Ω. For the sake of simplicity, we consider
κ = 1, although coefficient κ can be adjusted to provide
more accurate results. All simulation results are obtained by
averaging over 106 channel realizations.
Fig. 5. Energy outage probability vs γth
Fig. 4. Energy outage probability vs PT
Fig. 4 shows the energy outage probability for LOS parameters m = 1, 2 with respect to the transmitted power
PT . In order to capture the non-monotonic behavior of RTD,
we choose a set of PT values. Here, EH’s sensitivity is
γth = 5 × 10−5 and the distance between the transmitter and
the receiver is d = 0.1m. As expected, for small PT , both the
OBS and the IBS schemes achieve the same performance since
the received power PR is less than x1 . As PT increases, PR
is approaching ρ1 . Here, as m increases we can achieve better
perfomance. In contrast, when PR is getting closer to ρ2 , we
observe that performance drops as m increases. Finally, when
the PR is greater than x3 , the OBS and the IBS have again
the same performance. Results demonstrate than for any PR
between values x1 and x3 the OBS scheme outperforms the
IBS scheme. Fig. 5 illustrates the energy outage probability for
a variety of EH’s sensitivity values γth ∈ [γ1 , γ2 ] with PT = 12
dB. In this case, as m increases we achieve better performance.
Although, this happens because of the specific PT value. For
example, when PT = 20 dB (see Fig.4), as m increases the
performance drops. We can see that OBS outperforms both
selection schemes in any possible case, with respect to both
sensitivity and transmitted power. Fig. 6 depicts the energy
outage probability with respect to the number of ports N with
PT = 12 dB. Here, we compare our proposed scheme with the
convientional L-antenna SC. As we can see, the OBS scheme
outperforms the conventional case when L = 4, if the number
of predefined port are more than 6. In addition, the minimum
Fig. 6. Energy outage probability vs number of ports N
number of ports N that is needed for the FA OBS scheme to
outperform the conventional 6-antenna SC is 7.
VI. C ONCLUSIONS
In this paper, we studied low power EH over the THz band
using an integrated FA receiver equipped with an RTD-based
EH circuit. We proposed a general linear piecewise model for
RTD-based EH circuits, in which parameters can be tuned to fit
measurement data. Also, in order to model THz propagation
enviroment we use Nakagami-m fading channel, which can
represent different LOS conditions. Furthermore, we investigate three selection schemes and their perfomance in terms
of energy outage probability. Moreover, we provide analytical
and simulation results which verify that the proposed selection
scheme outperforms the available benchmarks. Results show
that we can take advantage of the non-monotonic behavior
of RTD, combined with FA spatial diversity techniques, to
enchance system’s performance.
A PPENDIX
pressed as
2
Z β1
mr1
2mm
2m−1 − σ12
r
× Ak dr1 ,
e
Γ(m)σ12m α1 1
where for any values of α1 and β1 ,
s
s
"
!#
r
2mµ2k r12
2m
x1
Ak = 1 − Qm
,
.
σ12 (1 − µ2k )
σk2 (1 − µ2k ) PR
(26)
N
express
all
2
events
by
setting
both
i
h
i
h Then, we can
q
q
q
α1 = 0, β1 = PxR1 and α1 = PxR2 , β1 = PxR3 . Finally
we have
2 N
Z q Px1
mr1
Y
R
2mm
2m−1 − σ12
Pout (γth ) =
r
e
(Ak + Bk ) dr1
1
Γ(m)σ12m 0
k=2
2 N
Z q Px3
mr1
Y
R
2mm
2m−1 − σ12
e
(Ak + Bk ) dr1 .
+
q x r1
2m
2
Γ(m)σ1
PR
k=2
(27)
Pout2 (γth ) =
A. Appendix A
The joint PDF of N spatially correlated Nakagami-m RVs
is given by [9]
2
1
2r2m−1 mm − mr
2
σ1
e
f(|h1 |,...,|hN |) (r1 , . . . , rN ) = 1
Γ(m)σ12m
2 r 2 +mσ 2 r 2 µ2
mσ1
N
k
k 1 k
−
Y
2mr11−m rkm σ1m−1
2 σ 2 1−µ2
σ1
k(
k)
(21)
×
×
e
m+1
2 )µm−1
σ
(1
−
µ
k
k
k
k=2
2mµk r1 rk
× Im−1
,
σ1 σk (1 − µ2k )
where Im−1 (·) is the (m−1)th order modified Bessel function
of the first kind.
In order to find the conditional joint CDF for any outage
event, the following integral needs to be evaluated:
F(|h1 |,|h2 |,...,|hN |) (r1 , r2.,. . , rN )
Z β1
Z βN
=
...
f(|h1 |,|h2 |,...,|hN |) (r1 , r2 , . . . , rN ) dr1 . . . drN .
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σ1 (1 − µ2k )
σk2 (1 − µ2k )
resonant tunnelling diode-based eh circuits,” 2023, arXiv:2307.06036.
(24)
[Online]. Available: https://arxiv.org/abs/2307.06036
[8] K.-K. Wong, A. Shojaeifard, K.-F. Tong, and Y. Zhang, “Fluid antenna
where Qm (α, β) is the Marcum Q-function given in [7].
h
q i
q
systems,” IEEE Trans. Wireless Commun., vol. 20, no. 3, pp. 1950–1962,
Nov. 2021.
By setting αk = PxR2 , βk = PxR3 , this outage event is
[9] C. Psomas, G. M. Kraidy, K.-K. Wong, and I. Krikidis, “On the diversity
expressed as
and coded modulation design of fluid antenna systems,” IEEE Trans.
2
Z β1
Wireless Commun., pp. 1–1, Jul. 2023.
mr1
−
2mm
2
2m−1
σ1
[10] A. Hanif and M. Doroslovački, “Simultaneous terahertz imaging with
Pout1 (γth ) =
r
e
×
B
dr
,
k
1
information and power transfer (STIIPT),” IEEE J. Sel. Topics Signal
Γ(m)σ12m α1 1
Process., vol. 17, no. 4, pp. 806–818, May 2023.
where for any values of α1 and β1 ,
s
s
!
r
2mµ2k r12
2m
x2
(25)
Bk = Qm
,
σ12 (1 − µ2k )
σk2 (1 − µ2k ) PR
s
s
!
r
2mµ2k r12
2m
x3
− Qm
,
.
σ12 (1 − µ2k )
σk2 (1 − µ2k ) PR
h
q i
By setting αk = 0, βk = PxR1 , this outage event is ex-