#  Unravelling the atomic and electronic structure of nanocrystals on
superconducting Nb(110): Impact of the oxygen monolayer

Samuel Berman<EMAIL_ADDRESS>School of Physics and Centre for Research on
Adaptive Nanostructures and Nanodevices (CRANN), Trinity College Dublin, The
University of Dublin, Dublin 2, Ireland  Ainur Zhussupbekova  School of
Physics and Centre for Research on Adaptive Nanostructures and Nanodevices
(CRANN), Trinity College Dublin, The University of Dublin, Dublin 2, Ireland
School of Chemistry, Trinity College Dublin, The University of Dublin, Dublin
2, Ireland  Brian Walls  School of Physics and Centre for Research on Adaptive
Nanostructures and Nanodevices (CRANN), Trinity College Dublin, The University
of Dublin, Dublin 2, Ireland  Killian Walshe  School of Physics and Centre for
Research on Adaptive Nanostructures and Nanodevices (CRANN), Trinity College
Dublin, The University of Dublin, Dublin 2, Ireland  Sergei I. Bozhko
Institute of Solid State Physics, Russian Academy of Sciences, Chernogolovka,
Russia  Andrei Ionov  Institute of Solid State Physics, Russian Academy of
Sciences, Chernogolovka, Russia  David D. O’Regan  School of Physics and
Centre for Research on Adaptive Nanostructures and Nanodevices (CRANN),
Trinity College Dublin, The University of Dublin, Dublin 2, Ireland  Igor V.
Shvets  School of Physics and Centre for Research on Adaptive Nanostructures
and Nanodevices (CRANN), Trinity College Dublin, The University of Dublin,
Dublin 2, Ireland  Kuanysh Zhussupbekov<EMAIL_ADDRESS>School of Physics and
Centre for Research on Adaptive Nanostructures and Nanodevices (CRANN),
Trinity College Dublin, The University of Dublin, Dublin 2, Ireland  School of
Chemistry, Trinity College Dublin, The University of Dublin, Dublin 2, Ireland

###### Abstract

The Niobium surface is almost always covered by a native oxide layer which
greatly influences the performance of superconducting devices. Here we
investigate the highly stable Niobium oxide overlayer of Nb(110), which is
characterised by its distinctive nanocrystal structure as observed by scanning
tunnelling microscopy (STM). Our ab-initio density functional theory (DFT)
calculations show that a subtle reconstruction in the surface Niobium atoms
gives rise to rows of 4-fold coordinated oxygen separated by regions of 3-fold
coordinated oxygen. The 4-fold oxygen rows are determined to be the source of
the nanocrystal pattern observed in STM, and the two chemical states of oxygen
observed in core-level X-ray photoelectron spectroscopy (XPS) are ascribed to
the 3-fold and 4-fold oxygens. Furthermore, we find excellent agreement
between the DFT calculated electronic structure with scanning tunnelling
spectroscopy and valence XPS measurements.

##  I I. Introduction

Superconducting electronics are a cornerstone of modern science and
technology, from superconducting radio-frequency cavities (SRFs) in particle
accelerators[1, 2, 3], to qubits in quantum computers [4, 5]. Niobium has
emerged as one of the most attractive materials for superconducting
electronics due to its high transition temperature and critical field [6], as
well as its ease of fabrication [7]. Niobium samples typically contain a large
amount of interstitial oxygen and quickly form an oxide layer when exposed to
atmosphere [8, 9]. While the superconducting gap of Niobium is sensitive only
to the presence of bulk interstitial oxygen [10, 11, 12, 13], device
properties such as coherence time in Josephson junctions [14, 15], qubit
relaxation time [16], and quality factor in SRF cavities [7, 17, 18, 19, 20,
21] are greatly affected by the precise nature of the surface oxide layer. For
SRF cavities, it has been shown that annealing the sample in ultra-high vacuum
conditions can compensate for the “Q-drop” [22], i.e. the sharp drop of in
quality factor (Q-factor) above 80 mT [23, 22]. The Q-drop is believed to be a
surface effect, with proposed mechanisms including local Joule heating of
inhomogeneities with high surface impedance [24], or the motion of magnetic
vortices pinned at the surface [25]. Understanding the precise structure of
the surface oxides is therefore vital for improving device performance.

The (110) surface of Niobium has been studied by a variety of experimental
techniques, including scanning tunnelling microscopy/spectroscopy (STM/STS)
[26, 27, 28, 29], low-energy electron diffraction (LEED) [26, 30], grazing
incidence X-ray diffraction (GIXRD) [8, 31], Auger electron spectroscopy (AES)
[28, 9], and X-ray photoelectron spectroscopy (XPS) [32, 33, 26]. When exposed
to atmosphere, a layered oxide structure of NbO, NbO2, and amorphous Nb2O5
[31] is formed. The NbO2 and Nb2O5 layers can be removed by annealing in
ultra-high vacuum (UHV), leaving only the NbO layer [8]. The remaining oxide
is very difficult to remove, requiring annealing with temperatures close to
the melting point of Niobium, in excess of 2400 oC [34, 11, 35, 36, 37]. This
structure will of course greatly influence the electronic structure of the
surface, as well as the structure of additional layers formed on top of the
surface [26]. Our STM experiments on this surface show a distinctive pattern
of finite rows of wide protrusions, consistent with previous reports [28, 27,
26]. These rows of protrusions are termed the “nanocrystals”, separated by
disordered regions dubbed “channels” (see Figure 1(a)).

Despite the technological and scientific relevance of the oxidised Nb(110)
surface, very few calculations have been performed to understand its
structure. While some previous works have performed calculations for the clean
Nb(110) surface [38, 34], the oxidised surface has received sparse attention.
Kilimis et al. [39] performed DFT calculations studying the effect of 2-fold
coordinated oxygen covering the surface, with a view towards understanding the
early stages of the oxidation process. Other works have computed the minimum
energy paths for diffusing oxygen atoms through the surface [40, 41]. However,
no previous reports have tackled the equilibrium structure or electronic
properties of the nanocrystals, which plays a vital role in device
performance. In this work, we investigate this surface structure by ab-initio
density functional theory (DFT) calculations, along with complementary STM,
STS, LEED, XPS and UPS experiments. We show that this surface structure can be
explained by a subtle oxygen induced reconstruction, with no need for Niobium
adatoms. In the supplemental material [42] (see also references [43, 44, 45]
therein) we show a similar analysis for models previously presented in the
literature from Razinkin et al. [27] and Arfouai et al. [46], and demonstrate
that these types of model cannot correctly explain the experimental data.

Figure 1: (a) STM image of the surface with $V$=2 mV, $I$=60 pA, showing the
nanocrystal and channel regions of the surface (with measured LEED pattern
inset). (b) Unrelaxed and (c) relaxed structures of a full monolayer of
oxygen, showing the emergence of a 4-fold site (top view). (d) Schematic of
the NW epitaxial relationship predicted from LEED experiments. (e) top view of
1st and 2nd Nb layers with oxygen removed to highlight the fcc-bcc epitaxial
relationship. (f) Side view of relaxed geometry.

##  II II. Methods

###  II.1 A. Computational Details

DFT calculations were carried out utilising the Quantum Espresso plane-wave
self-consistent field (PWscf) package [47, 48]. Throughout, the non-spin-
polarized PBE exchange-correlation functional [49] and projector augmented
wave (PAW) pseudopotentials [50] were used. We utilise symmetric Nb(110) slabs
with 9 atomic layers and vacuum padding of at least 10 Å, along with a
wavefunction cutoff energy of 45 Ry, k-point sampling of 16$\times$16$\times$1
in the Nb[-111], Nb[-11-1], and Nb[110] directions respectively, and Marzari-
Vanderbilt smearing [51] of 0.005 Ry giving a total energy convergence of $<$
1 meV/atom. The number of k-points was scaled when constructing the surface
reconstruction according to the unit cell dimensions. Crystal structures were
visualised in VESTA [52]. Simulated STM images and STS curves were obtained
within the Tersoff-Hamann approximation [53, 54], by integrating the local
density of states (LDOS) over a small 0.037 Å3 volume at a position of 4 Å
above the surface [34, 55, 56]. Changes in binding energy are calculated by
comparing the value of the electric potential at the atomic centres as in
Refs. [57, 58]. The work function was calculated by comparing the electric
potential in the vacuum region $\phi_{\mathrm{vac}}$ of the slab with the
Fermi level $E_{f}$ as in Ref. [59].

###  II.2 B. Experimental Details

All experiments were performed in-situ on the same (110) terminated niobium
single crystal. The crystal was annealed at
$850\text{\,}\mathrm{\SIUnitSymbolCelsius}$ under ultra-high-vacuum (UHV)
conditions. During annealing the sample temperature was measured from a K-type
thermocouple up to $600\text{\,}\mathrm{\SIUnitSymbolCelsius}$, with
temperatures above $600\text{\,}\mathrm{\SIUnitSymbolCelsius}$ estimated via
an infrared optical pyrometer ($\epsilon$ = 0.25). The crystal was transferred
between two UHV systems via a UHV suitcase with a base pressure of low
$10^{-10}$ mbar. All STM images demonstrated were obtained with a commercial
Createc slider-type STM in constant-current mode at 77 K. The STM tips
utilised were [001]-oriented single-crystalline W, which were
electrochemically etched in NaOH. The bias was applied to the sample with
respect to the tip. The UPS spectra were obtained with an excitation energy of
He I (21.2 eV). XPS measurements were performed on an Omicron MultiProbe XPS
system using monochromated Al Kα X-rays (XM 1000, 1486.7 eV) with an
instrumental resolution of 0.6 eV.

##  III III. Results

###  III.1 A. Crystal Structure

To arrive at this model, we first consider a single isolated oxygen atom on
top of the Niobium surface. Several absorption sites are examined, and our
calculations confer with that of Tafen et al. [41] i.e. the oxygen atom sits
in a 3-fold site on the surface. One might then expect that for a full
monolayer of oxygen on top of the surface, the oxygen would simply fill the
3-fold sites as in Figure 1(b). However, when a full monolayer of oxygen is
added we see a reconstruction take place, with the surface Niobium atoms
shifting to open up 4-fold sites on part of the surface. The relaxation of
this structure is shown in Figure 1(b-c), and is accompanied by an energy drop
of 0.11 eV/atom. The only choice that we make when constructing this model is
the size and shape of the supercell in which the structure is allowed to
relax. We simply choose a supercell which matches the periodicity observed in
the experimental STM images (see Figure 1(a) and Figure 2(a)) (i.e. 4 unit
cells along bcc[-11-1]). Initially, we consider an “infinite” nanocrystal as
shown in Figure 1(c), where the pattern extends infinitely along the bcc[-111]
direction and the disordered channels are not included. This is sufficient for
the purposes of calculating the local properties of the nanocrystals such as
STM images and STS curves. However, we do also perform calculations with
finite nanocrystals (full unit cell including channels), which are necessary
for calculating global electronic properties for comparison with area averaged
techniques such as XPS.

As shown in the top down crystal structure (Figure 1(c, e)), the oxygen
monolayer model displays a transition to the fcc (111) symmetry in the regions
where the oxygen is 3-fold coordinated at the surface. In our calculations the
epitaxial relationship closely resembles the Nishiyama-Wasserman (NW)
orientation ((fcc)[1-10]$||$(bcc)[001]) observed in most experiments [8, 26].
This is in agreement with the experimentally measured LEED pattern, shown in
the inset of Figure 1(a). Regarding the finite nature of the nanocrystals,
previous STM studies have determined the average length of a nanocrystal to be
between roughly 3.0 nm [27, 9] to 3.5 nm [28]. Since this model still contains
areas with fcc (111) symmetry, the previously proposed explanation for this
behaviour based on rigid lattice theory [28, 27] still holds. Along the
nanocrystals (fcc [110]/bcc [-111]) the mismatch is roughly $\simeq$4-5%
between the lattices ($a_{\mathrm{fcc}}=2.98\ \mathrm{\AA}$ and
$\frac{\sqrt{3}}{2}a_{\mathrm{bcc}}=2.86\ \mathrm{\AA}$), meaning after
roughly $\simeq$10-12 unit cells the two lattices will be out of phase. This
causes the finite rows with roughly 10-12 protrusions each, with length total
length of 3.0-3.5 nm.

###  III.2 B. Local Electronic Structure

Figure 2(a-b) shows an experimental STM image of the nanocrystal structure, as
compared to the simulated STM image from the reconstructed oxygen monolayer
model at the same bias (2 mV). In the simulated image we can see that the
distinctive rows of wide protrusions are present, with the area in between
these rows comparing favourably also. Since the “monolayer of oxygen” model
does not include any extra adatoms or major height variation on the surface,
it is interesting to consider the origins of the protrusions observed in STM.
By considering the overlayed surface atomic structure on the simulated STM
image in Figure 2(b) we can see that the protrusions arise from the
reconstructed 4-fold area of the surface. This can be understood looking at
the calculated superimposed atomic structure. In these reconstructed 4-fold
areas, the oxygen atoms sink deeper into the surface (see side view in Figure
1(f)), leaving the Niobium atoms in this area more exposed to the vacuum. In
the unreconstructed areas with 3-fold oxygen, the oxygen sits on top of the
Niobium, passivating the Niobium orbitals that protrude from the surface, and
reducing the LDOS at the Fermi level. Since almost all of the density near the
Fermi level comes from the Niobium 4d states (Figure 2(c)), it is reasonable
that the areas of the surface with the Niobium more exposed to the vacuum
would show a higher LDOS around the Fermi level. The distinctive “wide” shape
of the protrusions clearly arises from the 4-fold square nature of the
reconstructed area.

Figure 2: Comparison between experimental and simulated STM (a) Experimental
STM image at bias 2 mV with suggested atomic structure superimposed. (b)
Simulated STM image at a bias of 2 mV for the oxygen monolayer model with
superimposed atomic structure. (c) Relative contribution of surface atom
orbitals to the DOS at the Fermi level.

Turning to the STS, in Figure 3(a) we can see good agreement between the
experimentally measured STS and the LDOS as calculated in the vacuum above the
slab for bias in the region -1 V to +2 V. In this region we observe minimal
difference in the experimental STS on vs between the nanocrystal, and this is
reflected also in the DFT simulation. The theoretical STS captures the
relatively flat behaviour around the Fermi level from -0.5 to 0.5 eV, and the
sharp increase above 0.5 eV. We see no evidence of resonances arising from
surface states in the STS, such as was observed by Odobesko et al. [34] for
the clean Nb(110) surface. Our results for this oxygen saturated surface align
with their findings that this feature is suppressed in oxygen rich areas,
confirming that the resonance is unique to the clean Nb(110) surface. The
agreement between the theoretical and measured STS breaks down only at large
negative bias (below -1 V), where we see a resonance at -1.5 V in the theory,
which is most pronounced on the nanocrystal. This lack of agreement below -1 V
has been observed for the clean Nb(110) surface as well by Odobesko et al.
[34], and is not unexpected as tip states tend to dominate the tunnelling
matrix elements in this regime [34, 60, 61].

In order to better understand the nature of the electronic states at this
surface we plot the $m_{l}$ projected surface band structure (Figure 3(b)). In
the case of the nanocrystal structure, there are two equivalent N points
(along the nanocrystal (N1) vs across the nanocrystal (N2)). We see clearly
the absence of the z2 surface state observed by Odobesko et al. [34] for the
clean surface. Instead, the surface states below the Fermi level exist mostly
around -2.2 eV and have a xy orbital character. As expected, we do not see any
resonance at -2.2 eV coming from these states in the theoretical STS (Figure
3(a)) due to the orientation of the xy orbitals, where the majority of
electron density lies in the plane. We can understand the origin of the peak
at -1.5 eV in the theoretical STS by looking at the $m_{l}$ projected density
of states for the surface atoms on and between the nanocrystals (Figures 3(c)
and (d)) respectively. We see that around -1.5 eV, both contain a significant
z2 component, but on the nanocrystal, there is a significant peak of zy
character, which is missing for the atoms between the nanocrystal. For these
atoms between the nanocrystal, the in plane xy orbitals play a more
significant role. Therefore, we assign the resonance at -1.5 V observed in the
theoretical STS to these zy states. We note that this does not point to a zy
surface state, as can be seen from the band structure no such heavily
localised state exists. The states responsible for this peak are bulk like,
but with a high enough density to cause the peak in the calculated STS. The
combination of the bulk like nature of these states, and the tip states
dominating the tunnelling process in this regime, cause this feature to be
unseen in the experimental STS. Overall, we see strong agreement between the
DFT calculations and STM/STS experiments for this model.

Figure 3: (a) Simulated LDOS at a distance 4 Å above the surface, compared
with measured experimental STS spectrum. (b) ml projected band structure for
the surface nanocrystal atoms, (c-d) ml resolved DOS on and between the
nanocrystal  Figure 4: (a-b) Simulated total DOS over the whole slab with
orbital decomposition, compared against measured valence band XPS for infinite
and finite nanocrystals respectively. (c) Side and top views for the three
different channel structures considered here, with the channel area
highlighted by the blue rectangles and the nanocrystal area highlighted by the
green rectangles. (d) Total DOS calculated for the three finite nanocrystal
structures compared to infinite nanocrystal. (e) Experimental UPS spectrum
compared with calculated value for the secondary electron cutoff $h\nu-\phi$.

###  III.3 C. Global Electronic Structure

One of the most intriguing parts of this surface reconstruction is the
measured core level binding energies. Despite the Nb/O ratio being close to
1[9, 28], the Nb 3d level shows an anomalous binding energy of 203.55 eV, a
change in binding energy ($\Delta$B.E.) of 1.55 eV relative to the bulk [26,
33]. This does not match up with any of the known bulk Niobium oxides. We
calculate a change in binding energy for the surface Nb atoms of 1.43 eV
relative to the bulk, in good agreement with the experimentally determined
change. For the O 1s level, two different chemical states OI and OII are
observed with energies 530.3 eV and 531.8 eV, $\Delta$B.E. of 1.5 eV with
respect to one another [26, 33]. The latter binding energy (531.8 eV) matches
up exactly with the O 1s binding energy in bulk NbO, while the former (530.3
eV) does not match with any bulk Niobium oxide. Within this oxygen monolayer
model, the two chemical states are the 3-fold oxygen and the 4-fold oxygen in
Figure 1(c, f). For this change in binding energy, we calculate a value of
1.63 eV, again in good agreement with the experimental value. Because the
exact ratio between the two chemical states of oxygen varies significantly
between studies, we cannot compare this directly to the DFT calculations.
However, in all studies there is always an asymmetry between the two chemical
states, the OI chemical state is always more abundant than the OII chemical
state. This agrees with the oxygen monolayer model presented here, where the
3-fold oxygens correspond to OI , and the 4-fold oxygens correspond to OII .
Additionally, the OII oxygen is slightly deeper into the surface (see side
view in Figure 1(f)). This accounts for the angle resolved XPS of Razinkin et
al. [33], where they observe a higher OI /OII ratio at glancing angle compared
to normal incidence. Table 1 shows the comparison between the experimentally
fitted change in binding energy and the calculated values. Overall, there is
strong agreement between XPS experiments and the oxygen monolayer model.

Table 1:  Comparison of simulated and experimental core level binding energies and work function.  | Calculated | Experimental
---|---|---
Nb 3d $\Delta$B.E. [eV] | 1.43 [this work] | 1.55 [26, 33]
O 1s $\Delta$B.E. [eV] | 1.63 [this work] | 1.50 [26, 33]
Work Function $\phi$ [eV] | 5.2 [this work] | 5.0 [this work]
$h\nu-\phi$ [eV] | 16.0 [this work] | 16.2 [this work]

Looking now at the valence band XPS, we start by comparing the calculated DOS
for the infinite nanocrystal model Figure 4(a) to the experimentally measured
XPS. We can see broadly good agreement between the two curves, however, the
peaks in the vicinity of -5 eV give rise to a higher intensity than that
observed in the experiment. In order to improve agreement between the valence
band XPS and calculated DOS, we must account for the finite nature of the
crystal by including the channels separating the nanocrystals (Figure 4(b)).
The finite nanocrystal including the channels drastically improves agreement
between the simulated DOS and measured valence band XPS. The behaviour of the
DOS around -5 eV is now fully captured, and agreement near the Fermi level
remains strong. From the orbital resolved DOS in Figure 4(a-b) we see that the
main broad peak from 0 to -5 eV arises from mostly Nb 4d states, whereas the
shoulder around -5 to -7 eV comes mostly from the O 2p. The deep feature
around -18 eV is almost entirely arising from semicore O 2s states.

Overall, it is not surprising that to obtain the correct DOS, the finite
nature of the pattern must be included. While simulated STM images are highly
local, and depend mostly on the local charge density on a surface atom, the
total DOS will be highly sensitive to the total band structure across the
entire Brillouin zone. When moving to the finite model the band structure will
change dramatically, with the bcc[-111]/fcc[0-11] direction in the Brillouin
zone (along the nanocrystals) shrinking by as much as a factor $\simeq$10\.
Therefore, a significant change in parts of the DOS is to be expected. It is
also worth noting that since the channels separating the nanocrystals appear
disordered in STM, simulating it using periodic DFT poses a challenge. For the
purpose of ruling out the effect of the channel structure on the DOS we
evaluate the 3 possible extremes of the channel structure. Figure 4(c) shows
the three structures of finite nanocrystals considered, with different atomic
structures in the channels i.e., no oxygen/Niobium in the channel (i.e. a step
edge) (Type 1), partial oxygen but no Niobium (Type 2), and partial
oxygen/partial Niobium (Type 3). Figure 4(d) shows the calculated DOS for
these 3 structures compared to the infinite nanocrystal. We can see that while
there is a clear difference between the infinite nanocrystal calculation (no
channel) and the calculations including the channels, there is little
difference between the calculated DOS for the different types of channel
structures.

From our UPS experiment (Figure 4(e)), the work function ($\phi$) of this
surface can be obtained by comparing the incident photon energy to the
secondary electron cutoff ($E_{\mathrm{sec}}=h\nu-\phi$). Using this method we
obtain a work function of 5.0 eV (table 1). Comparing this result with the
work function of Nb metal (4.3 eV) and Nb2O5 (5.2 eV) [62], we see the
expected trend is upheld, with the work function increasing as more oxygen is
added. Within DFT we can estimate the work function by comparing the classical
electric potential at a point far from the surface slab with the Fermi level.
From this method we obtain a calculated work function of 5.2 eV, showing
decent agreement with the experimentally measured value.

##  IV IV. Conclusions

From the above discussion, it seems clear that the oxygen monolayer surface is
the best candidate for explaining the surface reconstruction. Due to the
unusual structure of this surface, it is worth discussing how correct it is to
call this surface structure a NbO structure. While the stoichiometry of the
surface measured from XPS suggests the ratio of oxygen and Niobium to be very
close to 1, the anomalous Nb 3d core level binding energy suggests that the
chemical environment of these atoms is not similar to that of bulk NbO. Our
model confirms this, with none of the Niobium atoms being square planer
coordinated (as they are in bulk NbO). Some papers have discussed this
anomalous binding energy in terms of a sub-oxide Nb2O [63, 64, 65]. However,
in these studies, the atoms assigned to that sub-oxide are buried underneath a
Nb2O5 layer, and as previously mentioned the stoichiometry contradicts this
sub-oxide interpretation. The crystallography also suggests that this layer is
not simply a NbO monolayer. This surface therefore appears to be a novel 2D
Niobium oxide layer, with no bulk counterpart.

Having determined the precise local structure of this oxide layer, there is
now a foundation in place upon which effect of the Niobium surface structure
on device performance can be understood. This surface structure will be
present on any Nb(110) facet, either buried underneath higher valence oxides
or on its own. In the case of SRF cavities specifically, this surface
structure may be the dominant surface oxide present in samples showing
improved performance after vacuum annealing [22]. Whatever the mechanism for
the improvement in device performance may be, it is likely that this surface
structure plays a significant role.

##  V Acknowledgements

This work was supported by Irish Research Council (IRC) Laureate Award
(IRCLA/2019/171), the Government of the Republic of Kazakhstan under the
Bolashak program, the Russian Academy of Sciences through the state task of
Institute of Solid State Physics RFBR Grant 19-29-03021 and Erasmus Plus
mobility grants (2017-1-IE02-KA107-000538 & 2018-1-IE02-KA107-000589). All
calculations were performed on the Boyle cluster maintained by the Trinity
Centre for High Performance Computing. This cluster was funded through grants
from the European Research Council and Science Foundation Ireland. K.Z. and
A.Z. would also like to acknowledge funding from IRC through GOIPD/2022/774
and GOIPD/2022/443 awards. This paper concerns work carried out from 2019 to
2021.

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Supporting Information
Unravelling the atomic and electronic structure of nanocrystals on
superconducting Nb(110): Impact of the oxygen monolayer

###  VI.1 DFT simulation of literature models

There have been two main models proposed in the literature, one from Razinkin
[27] and one from Arfouai [46]. Both of these models attribute the observed
periodic arrays of protrusions to rows of Niobium atoms on top of a fcc bulk
like NbO(111) plane with some arrangement of oxygen surrounding the structure.
They differ in how they arrange these additional oxygen atoms. In the Arfaoui
model a NbO(111) oxygen plane is stacked on top of the Nb(110) surface,
followed by a NbO(111) Niobium plane, followed by another NbO(111) oxygen
plane, and then finally the rows of Niobium atoms are placed on top in a row
like arrangement. Bulk NbO takes on an unusual “ordered defect” structure
[43], and these models include these ordered defects in their NbO layers. In
the Razinkin model a NbO(111) Niobium plane is stacked on top of the Nb(110)
surface, followed by a NbO(111) oxygen plane, followed by the rows of Niobium
atoms, this time with additional oxygen atoms installed inside the
nanocrystals. These additional oxygen atoms provide the second chemical state
of oxygen OII. Again they include the ordered vacancy structure in the
NbO(111) layers reminiscent of bulk NbO. The finite nature of the nanocrystals
is explained in terms of the mismatch between NbO(111) and Nb(110), as
explained previously.

Firstly, both of these models include the ordered vacancies present in bulk
NbO in their surface layer (Figure SI1(a)), as well as Niobium adatoms, in
their proposed surface layer. Razkinin et al. [27] do mention the possibility
of buckling of the nanocrystal in their model due to the vacancy, and our
calculations indicate that when these ordered vacancies are present, the
nanocrystal not only buckles but almost fully collapses, filling the vacancies
(Figure SI1(c-d)). This can be understood for the following reason: in bulk
NbO the ordered lattice of vacancies generate Nb6 octohedra (Figure SI1(b)).
The Nb d orbitals can then overlap through these octahedra, leading to a
release of delocalisation energy [44, 45]. If this delocalisation energy is
greater than the energy needed to create the ordered defects, this structure
will be the most energetically favourable. It turns out that due to the
dispersed nature of the Nb 4d orbitals, this does in fact occur, so bulk NbO
takes on the ordered vacancy structure as opposed to the rocksalt structure.
However, at the surface, there is only a monolayer of NbO, oriented in the
[111] direction, so there is no opportunity for Nb6 octahedra to form. Thus,
since there is no release of delocalisation energy, there is no reason to
think that the ordered vacancy structure would be stable for a monolayer of
NbO. This is corroborated by our DFT calculations. Therefore, we can say that
models which include ordered vacancies cannot account for this surface
reconstruction. Even if we modify these literature models to fill the
vacancies, (which does lead to stable adatom nanocrystals) the agreement with
experiment is still poor. Figures SI2 and SI3 show the structure and simulated
STM/STS from these models (without vacancies). We can see these simulated STM
images have poor agreement with the experimental images. The Niobium adatoms
present in these models do not produce the distinctively shaped/wide
protrusions observed in experiment. Hence, we can conclude that the observed
protrusions in STM are not due to Niobium adatoms.

Figure SI1: (a) Demonstrates a top down view of the Razinkin model, with
nanocrystal atoms highlighted by the green circles, and Niobium vacancies
highlighted by the gold circles. (b) Shows the bulk NbO crystal structure with
the Nb vacancy at the centre of the Nb6 octahedron highlighted by the yellow
sphere. (c-d) show a side view of our DFT structural relaxation of the
Razinkin model. Figure SI2:  Atomic structure and simulated STM/STS for
Razinkin like model with vacancies filled. Figure SI3: Structure and simulated
STM/STS for an Arfoaui like model with vacancies filled and second oxygen
layer removed.


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# Secrecy Analysis of CSI Ratio-Based Transmitter Selection with Unreliable
Backhaul

Burhan Wafai1, Ankit Dubey2, and Chinmoy Kundu3 {1burhan.wafai,
2ankit.dubey}<EMAIL_ADDRESS>12Department of EE,
Indian Institute of Technology Jammu, Jammu & Kashmir, India  3Wireless
Communications Laboratory, Tyndall National Institute, Dublin, Ireland

###### Abstract

This paper explores the secrecy performance of a multi-transmitter system with
unreliable backhaul links. To improve secrecy, we propose a novel transmitter
selection (TS) scheme that selects a transmitter with the maximum ratio of the
destination channel power gain to the eavesdropping channel power gain. The
backhaul reliability factor is incorporated with the distribution of the
channel power gain through the utilization of a mixture distribution. We
evaluate the non-zero secrecy rate (NZR) and the secrecy outage probability
(SOP) as well as their asymptotes in two scenarios of backhaul activity
knowledge, where it is available and where it is unavailable. The results
illustrate that because of the unreliable backhaul, the proposed destination-
to-eavesdropper channel power gain ratio-based TS scheme is constrained in
terms of secrecy performance. However, performance enhancements are observed
when the backhaul knowledge activity is utilized. Furthermore, the proposed
scheme outperforms all the sub-optimal TS schemes and achieves nearly optimal
performance without requiring noise power or the evaluation of the exact
secrecy rate measurement.

##  I Introduction

The Internet of Things (IoT) in the future wireless technologies will require
high connectivity and data rates to accommodate a substantial volume of
applications and devices [1]. Future wireless technologies will be supported
by heterogeneous small-cell networks to provide IoT services. In such
heterogeneous networks, the backhaul establishes a connection between the
access point and the small-cell base stations [2]. These backhaul links are
either wireless or wired. Notably, wireless backhaul is becoming widely used
rather than wired backhaul due to lower deployment costs [3].

The wireless nature of IoT networks exposes them to the risk of eavesdropping.
Physical layer security offers a solution to provide security to such networks
[4]. Transmitter selection (TS) is one of the techniques for improving secrecy
in multi-source wireless networks by increasing diversity gain of the
legitimate channel without employing multiple antennas [5]. Depending on
whether the channel state information (CSI) is available, TS schemes are
broadly classified as optimal and sub-optimal selection schemes. Optimal TS
requires global CSI to select the transmitter with maximum secrecy rate,
whereas the sub-optimal TS maximizes the legitimate channel rate, not
requiring the eavesdropper channel CSI [6, 7, 8, 9, 10, 11]. Furthermore, the
CSI of the backhaul significantly affects the system’s performance [11, 12,
13, 14]. The prior knowledge of the backhaul activity enables us to select the
transmitter with an active backhaul link, whereas a transmitter corresponding
to an inactive backhaul link may be selected because of the unavailability of
backhaul activity knowledge.

###  I-A Related Works

The security of the TS in a network with multiple transmitters has been
extensively explored in literature [6, 7, 10, 11, 8, 12, 9, 13, 14, 15].
Different TS schemes were investigated in [10] for a system with a destination
and an eavesdropper, with backhaul knowledge activity unavailable. Sub-
optimal, minimal eavesdropping, minimal interference, and optimal TS schemes
were studied for a cognitive radio (CR) network with unreliable backhaul. The
non-zero secrecy rate (NZR), secrecy outage probability (SOP), and ergodic
secrecy rate (ESR) were derived. A similar system for a CR network was
considered in [11], where the SOP of sub-optimal and optimal TS schemes were
evaluated, with backhaul knowledge activity available. However, the works in
[10] and [11] were primarily focused on the security of a CR network.

A sub-optimal and the optimal TS schemes were studied in [13] in a system
consisting of a destination and multiple eavesdroppers. The SOP and ESR were
derived under frequency selective fading for backhaul knowledge activity
unavailable and available cases. The ESR of the optimal TS scheme was obtained
for frequency flat fading channels with backhaul knowledge activity available
in [12] for a system with a destination and multiple eavesdroppers. Recently,
a system consisting of a destination and multiple eavesdroppers was considered
in [14] for frequency flat fading channels. A sub-optimal, minimum
eavesdropping, and optimal TS schemes were studied for backhaul knowledge
activity unavailable and available cases.

###  I-B Motivation and Novelty

Although various TS schemes were explored in [12, 13, 14], none have
specifically investigated a scheme that takes into account the maximum ratio
of legitimate-to-eavesdropper channel CSI. While the optimal TS scheme studied
in [10, 11, 12, 13, 14] involves the ratio of SNRs of the legitimate channel
to the eavesdropping channel, the proposed TS scheme differs from the optimal
TS scheme in the selection criteria. Unlike the optimal TS scheme, which aims
to maximize the achievable secrecy rate, the proposed scheme focuses on
maximizing the legitimate-to-eavesdropper channel power gains ratio to improve
security. Furthermore, the optimal TS scheme requires noise power of the
receiving nodes (destination and eavesdropper) for the TS. In contrast, the
proposed scheme selects the transmitter based on the CSIs of the legitimate
and eavesdropping channels without requiring noise power. In practical
applications, measuring noise power can be challenging and resource-intensive.
Therefore, a TS without the requirement of noise power simplifies the
selection process and reduces complexity. Furthermore, we will illustrate that
it also provides secrecy performance close to the performance of the optimal
TS scheme.

Motivated by the above discussion, a wireless backhaul serving a small-cell
network with a destination and an eavesdropper is considered where a novel
channel CSI ratio-based transmitter selection (RTS) scheme is proposed that
relies on the ratio of legitimate-to-eavesdropper channel power gains for the
best transmitter selection without requiring the receiver noise power
information. To the best of the authors’ knowledge, the RTS scheme has not
been explored yet in the context of the secrecy of a multi-transmitter system.
Despite the selection strategy, the system may still experience secrecy outage
due to channel conditions and backhaul unreliability. Therefore, we evaluate
the NZR and SOP for the RTS scheme incorporating the backhaul reliability
factor into the analysis while ensuring that the transmitter selection is
implemented. This makes the analysis challenging. The main contributions are:

  * •

With the global CSI available, we introduce a novel CSI RTS scheme in which a
transmitter with a maximum ratio of the legitimate link channel power gain to
the eavesdropping link channel power gain is selected.

  * •

The closed-form NZR and SOP expressions are obtained for the backhaul
knowledge activity available and unavailable cases of the RTS scheme. We also
evaluate the asymptotic NZR and SOP using the high SNR approximation to get
better insights into the system.

  * •

We demonstrate that the proposed scheme performs better than all the sub-
optimal schemes studied in [14]. Notably, it achieves performance comparable
to the optimal TS scheme without requiring noise power or the evaluation of
secrecy rate which simplifies the TS process in comparison to the optimal TS
scheme.

  * •

The results reveal that with the backhaul knowledge activity unavailable,
employing additional transmitters does not lead to an improvement in
performance. However, the resources can be judiciously used when the backhaul
knowledge activity is available.

##  II System model

Figure 1: A small-cell network with backhaul links.

A multi-transmitter system consisting of an access point (AP) serving $K$
transmitter nodes $T_{k}$ where $k\in\\{1,\ldots,K\\}$ through the unreliable
wireless backhaul links is considered, as depicted in Fig. 1. $K$ transmitters
communicate with a destination node $D$, and an eavesdropper $E$ intercepts
the transmission, with each node having a single antenna. We assume that the
eavesdropper is a part of the network, and thus, its CSI is available [16].
All the links are assumed to be independent; specifically, we also assume that
the channel gains of the links $T_{k}$-$D$, $h_{TD}^{(k)}$, for each
$k\in\\{1,\ldots,K\\}$ are independent and identically distributed. Similarly,
channel gains of the links $T_{k}$-$E$, $h_{TE}^{(k)}$, for each
$k\in\\{1,\ldots,K\\}$ are also assumed to be independent and identically
distributed. However, the channel gains $h_{TD}^{(k)}$ and $h_{TE}^{(k)}$ for
any $k$ are not necessarily identical. Each link exhibits frequency flat
Rayleigh fading and, therefore, the channel power gain $|h_{TX}^{(k)}|^{2}$
where $X\in\\{D,E\\}$ is exponentially distributed with parameter
$\lambda_{X}$. The CDF and PDF of the channel power gain $|h_{TX}^{(k)}|^{2}$
is

$\displaystyle F_{|h_{TX}^{(k)}|^{2}}(x)$
$\displaystyle=1-\exp(-\lambda_{X}x),$ (1) $\displaystyle
f_{|h_{TX}^{(k)}|^{2}}(x)$ $\displaystyle=\lambda_{X}\exp(-\lambda_{X}x),$ (2)

respectively. Each backhaul link serving a transmitter has a probability of
link failure. The reliability of the backhaul link is modeled by an
independent Bernoulli random variable with success probability $\Delta$ and
failure probability $1-\Delta$. The end-to-end link distribution of
AP-$T_{k}$-$D$ link where $k\in\\{1,\ldots,K\\}$ is expressed with the help of
a mixture distribution [10], which incorporates the backhaul reliability
factor $\Delta$. A mixture distribution is used to express the PDF of the link
SNR from AP to $X$ via $T_{k}$ is expressed as [12]

$\displaystyle f_{|\hat{h}_{TX}^{(k)}|^{2}}(x)=(1-\Delta)\delta(x)+\Delta
f_{|h_{TX}^{(k)}|^{2}}(x)$ (3)

where $\delta(x)$ is the Dirac delta function, $\hat{h}_{TX}^{(k)}$ represents
the channel that includes the backhaul reliability factor. The corresponding
CDF will be

$\displaystyle F_{|\hat{h}_{TX}^{(k)}|^{2}}(x)$
$\displaystyle=(1-\Delta)u(x)+\Delta\left(1-e^{-\lambda_{D}x}\right),$ (4)

where $u(x)$ is the unit step function [11]. Additive white Gaussian noise
(AWGN) is assumed to be added at each receiver node with zero mean and
variance $\sigma_{X}$ where $X\in\\{D,E\\}$. We study the RTS scheme in two
cases: backhaul knowledge activity available and backhaul knowledge activity
unavailable.

###  II-A Backhaul Activity Knowledge Available

With the backhaul knowledge activity known, the transmitter is chosen among
those for which the backhaul links are operational. We select the transmitter
($k^{*}$) for which the ratio of legitimate-to-eavesdropper channel power gain
is maximum. Assuming $\mathcal{A}^{(q)}$ is the subset of transmitters
consisting of $q$ active backhaul links, i.e.,
$\mathcal{A}^{(q)}\subseteq\\{1,\ldots,K\\}$ and
$\mathcal{A}^{(q)}\neq\emptyset$, the ratio of $T_{k^{*}}$-$D$ and
$T_{k^{*}}$-$E$ channel power gains for the selected transmitter $k^{*}$ is

$\displaystyle\hat{\Gamma}^{(k^{*})}=\max_{k\in\mathcal{A}^{(q)}}\left(\frac{|\hat{h}_{TD}^{(k)}|}{|\hat{h}_{TE}^{(k)}|}\right)^{2}.$
(5)

Since the destination and eavesdropper link have a common backhaul link, the
backhaul reliability factor has to be incorporated in either the $T_{k}$-$D$
or $T_{k}$-$E$ link. Here, we include it in the $T_{k}$-$D$ link. Using the
CDF definition, the distribution of $\hat{\Gamma}^{(k^{*})}$ is given by

$\displaystyle F_{\hat{\Gamma}^{(k^{*})}}(x)$
$\displaystyle=\mathbb{P}\left[\max_{k\in\mathcal{A}^{(q)}}\Big{\\{}\frac{|\hat{h}_{TD}^{(k)}|^{2}}{|\hat{h}_{TE}^{(k)}|^{2}}\Big{\\}}\leq
x\right]$
$\displaystyle=\left(\int_{0}^{\infty}F_{|\hat{h}_{TD}^{(k)}|^{2}}(xy)f_{|\hat{h}_{TE}^{(k)}|^{2}}(y)dy\right)^{K}$
$\displaystyle=(1-\Delta)^{K}u(x)$
$\displaystyle+\sum_{k=1}^{K}\binom{K}{k}(-1)^{k+1}\Delta^{k}\left(1-\frac{\lambda_{E}}{\lambda_{D}x+\lambda_{E}}\right)^{k},$
(6)

where the distributions of $f_{|\hat{h}_{TE}^{(k)}|^{2}}(\cdot)$ and
$F_{|\hat{h}_{TD}^{(k)}|^{2}}(\cdot)$ are used from (2) and (4), respectively.
The corresponding PDF is

$\displaystyle f_{\hat{\Gamma}^{(k^{*})}}(x)$
$\displaystyle=(1-\Delta)^{K}\delta(x)$
$\displaystyle+\sum_{k=1}^{K}\binom{K}{k}(-1)^{k+1}\Delta^{k}\left(\frac{\lambda_{E}}{\lambda_{D}x+\lambda_{E}}\right)^{k}.$
(7)

###  II-B Backhaul Activity Knowledge Unavailable

In this case, the TS is carried out without taking into account whether the
backhaul link is active or inactive. The transmitter $k^{*}$ which has the
channel power gain ratio $\Gamma^{(k^{*})}$ maximum is selected without the
backhaul knowledge activity. Thus,

$\displaystyle\Gamma^{(k^{*})}=\max_{k\in\\{1,\ldots,K\\}}\left(\frac{|h_{TD}^{(k)}|}{|h_{TE}^{(k)}|}\right)^{2}.$
(8)

It is important to highlight that the selected $k^{*}$ link may be inactive.
The CDF of $\Gamma^{(k^{*})}$ is expressed as

$\displaystyle F_{\Gamma^{(k^{*})}}(x)$
$\displaystyle=\mathbb{P}\left[\max_{k\in\\{1,\ldots,K\\}}\Big{\\{}\frac{|h_{TD}^{(k)}|^{2}}{|h_{TE}^{(k)}|^{2}}\Big{\\}}\leq
x\right]$
$\displaystyle=1-\sum_{k=1}^{K}\binom{K}{k}(-1)^{k+1}\left(\frac{\lambda_{E}}{\lambda_{D}x+\lambda_{E}}\right)^{k}.$
(9)

The corresponding PDF is written as

$\displaystyle f_{\Gamma^{(k^{*})}}(x)$
$\displaystyle=\sum_{k=1}^{K}\binom{K}{k}(-1)^{k+1}\frac{k\lambda_{E}^{k}\lambda_{D}}{\left(\lambda_{D}x+\lambda_{E}\right)^{k+1}}.$
(10)

We incorporate the backhaul reliability factor to get the end-to-end PDF of
the AP to $D$ link following (3) as

$\displaystyle f_{\hat{\Gamma}^{(k^{*})}}(x)$
$\displaystyle=(1-\Delta)\delta(x)+\Delta f_{\Gamma^{(k^{*})}}(x).$ (11)

Next, we obtain the NZR and SOP for available and unavailable cases of
backhaul knowledge activity using the distributions derived in this section.

##  III Non-zero Secrecy Rate

The NZR of the RTS scheme for both the backhaul knowledge activity available
and unavailable cases is evaluated in this section. In this regard, we first
define the secrecy rate corresponding to any transmitter $T_{k}$ as

$\displaystyle
C_{S}^{(k)}=\max\Bigg{\\{}\log_{2}\left(\frac{1+\frac{|\hat{h}_{TD}^{(k)}|^{2}}{\sigma_{D}}}{1+\frac{|\hat{h}_{TE}^{(k)}|^{2}}{\sigma_{E}}}\right),0\Bigg{\\}},$
(12)

where $\frac{|\hat{h}_{TX}^{(k)}|^{2}}{\sigma_{X}}$ represents the SNR of the
link $T_{k}$-$X$ where $X\in\\{D,E\\}.$ Consequently, the NZR of the RTS
scheme is the probability that the secrecy rate of the selected transmitter
$C_{S}^{(k^{*})}$ exceeds zero.

###  III-A Backhaul Activity Knowledge Available

The availability of backhaul knowledge activity enables us to select the
transmitter among the active links. Let us assume there exist $q$ backhaul
links that are active, i.e., the cardinality of the set $\mathcal{A}^{(q)}$ is
$|\mathcal{A}^{(q)}|=q$. Since independent and identically distributed
Bernoulli random variables govern the activity of the backhaul links, the NZR
is obtained by summing up all the possible combinations of $q$ active links as

$\displaystyle\mathcal{P}_{NZ}=1-\mathcal{P}_{Z}=1-\sum_{q=1}^{K}\binom{K}{q}\mathcal{P}_{Z}^{(q)},$
(13)

where $\mathcal{P}_{Z}$ is the probability that $C_{S}^{(k^{*})}$ is less than
or equal to zero. $\mathcal{P}_{Z}$ is the sum of all probabilities
$\mathcal{P}_{Z}^{(q)}$ that the secrecy rate $C_{S}^{(k^{*})}$ is less than
or equal to zero when there exist $q$ active backhaul links. The term
$\binom{K}{q}$ in the equation signifies the number of possible ways $q$
backhauls can be active. $\mathcal{P}_{Z}^{(q)}$ takes $\Delta$ into account
because of the consideration of mixture distribution in (3). It is evaluated
first by finding the probability that $C_{S}^{(k)}$ is less than or equal to
zero for each link $k\in\mathcal{A}^{(q)}$ provided that the link has the
maximum channel power gain ratio, i.e., the channel power gain ratio is
greater than the maximum of the rest of the corresponding channel power gain
ratios. Then, by following the law of total probability,
$\mathcal{P}_{Z}^{(q)}$ is obtained by summing all the above probabilities
corresponding to each link in $\mathcal{A}^{(q)}$ as

$\displaystyle\mathcal{P}_{Z}^{(q)}=\sum\limits_{k\in\mathcal{A}^{(q)}}\mathbb{P}\left[C_{S}^{(k)}\leq
0\Bigg{|}\frac{|\hat{h}_{TD}^{(k)}|^{2}}{|\hat{h}_{TE}^{(k)}|^{2}}>\bar{\Gamma}^{(k)}\right]$
(14)
$\displaystyle=\sum\limits_{k\in\mathcal{A}^{(q)}}\mathbb{P}\left[\frac{|\hat{h}_{TD}^{(k)}|^{2}}{\sigma_{D}}<\frac{|\hat{h}_{TE}^{(k)}|^{2}}{\sigma_{E}}\Bigg{|}|\hat{h}_{TD}^{(k)}|^{2}>|\hat{h}_{TE}^{(k)}|^{2}\bar{\Gamma}^{(k)}\right]$
$\displaystyle=\sum\limits_{k\in\mathcal{A}^{(q)}}\frac{\sigma_{D}}{\sigma_{E}}\int_{0}^{\infty}\int_{0}^{1}\int_{yz}^{y\sigma_{D}/\sigma_{E}}$
$\displaystyle
f_{|\hat{h}_{TD}^{(k)}|^{2}}(x\sigma_{D}/\sigma_{E})f_{\bar{\Gamma}^{(k)}}(z)f_{|\hat{h}_{TE}^{(k)}|^{2}}(y)dxdzdy,$
(15)

where $\bar{\Gamma}^{(k)}=\max_{\begin{subarray}{c}n\in\mathcal{A}^{(q)}\\\
n\neq
k\end{subarray}}\Big{\\{}\frac{|\hat{h}_{TD}^{(n)}|^{2}}{|\hat{h}_{TE}^{(n)}|^{2}}\Big{\\}}$.
The distribution of $\bar{\Gamma}^{(k)}$ is easily derived following (II-A).
Consequently, the PDF of $\bar{\Gamma}^{(k)}$ will be same as (II-A), with $K$
replaced by $K-1$. Using (2), (3), (II-A), (13), and (III-A) the NZR is
finally written as

$\displaystyle\mathcal{P}_{NZ}=1-\Bigg{[}(1-\Delta)^{K}+\frac{K\Delta(1-\Delta)^{K-1}\sigma_{D}{\lambda}_{D}}{\sigma_{E}{\lambda}_{E}+\sigma_{D}{\lambda}_{D}}+\sum_{q=1}^{K}\sum_{n=1}^{q-1}$
$\displaystyle\binom{K}{q}\binom{q-1}{n}\frac{(-1)^{n+1}nq\Delta^{n+1}\sigma_{E}{\lambda}_{E}}{(\sigma_{E}{\lambda}_{E}+\sigma_{D}{\lambda}_{D})^{n+1}}\Bigg{(}-\frac{1}{n}\Big{(}-\sigma_{E}^{n}{\lambda}_{E}^{n}$
$\displaystyle+(\sigma_{E}{\lambda}_{E}+\sigma_{D}{\lambda}_{D})^{n}\Big{)}+\left(\frac{(\sigma_{E}{\lambda}_{E}+\sigma_{D}{\lambda}_{D})^{n+1}}{\sigma_{E}{\lambda}_{E}}-\sigma_{E}^{n}{\lambda}_{E}^{n}\right)$
$\displaystyle\times\Bigg{(}\sum_{i=1}^{n}(-1)^{n-i}\frac{(n-i)!(i-1)!}{n!}+\frac{(-1)^{n}}{(n+1)}\Bigg{)}\Bigg{)}\Bigg{]}.$
(16)

###  III-B Backhaul Activity Knowledge Unavailable

The NZR for this case is evaluated without prior knowledge of backhaul link
activity. As each link has a failure probability $(1-\Delta)$, the NZR is
expressed as

$\displaystyle\mathcal{P}_{NZ}$ $\displaystyle=(1-\Delta)\times
0+\Delta\left(1-\mathcal{P}_{Z}\right),$ (17)

where $\mathcal{P}_{Z}$ is the probability that $C_{S}^{(k^{*})}$ is less than
or equal to zero as if the backhaul link is active for selected transmitter
$T_{k^{*}}$, i.e., assuming channel power gain as $|h_{TX}^{(k^{*})}|^{2}$ in
$C_{S}^{(k^{*})}$ without incorporating the backhaul in (12).
$\mathcal{P}_{Z}$ is evaluated following the similar logic in (III-A) as

$\displaystyle\mathcal{P}_{Z}=$
$\displaystyle\sum_{k=1}^{K}\mathbb{P}\left[C_{S}^{(k)}\leq
0\Bigg{|}\frac{|{h}_{TD}^{(k)}|^{2}}{|{h}_{TE}^{(k)}|^{2}}>\bar{\Gamma}^{(k)}\right]$
(18) $\displaystyle=$
$\displaystyle\frac{K\sigma_{D}}{\sigma_{E}}\int_{0}^{\infty}\int_{0}^{1}\int_{yz}^{y\sigma_{D}/\sigma_{E}}$
$\displaystyle
f_{|{h}_{TD}^{(k)}|^{2}}(x\sigma_{D})f_{\bar{\Gamma}^{(k)}}(z)f_{|{h}_{TE}^{(k)}|^{2}}(y\sigma_{E})dxdzdy,$
(19)

where $\bar{\Gamma}^{(k)}=\max_{\begin{subarray}{c}n\in\\{1,\ldots,K\\}\\\
n\neq
k\end{subarray}}\Big{\\{}\frac{|h_{TD}^{(n)}|^{2}}{|h_{TE}^{(n)}|^{2}}\Big{\\}}$.
In contrast to (14), the summation in (18) is over $K$ as the backhaul
activity knowledge is not available. Due to the independent and identical
distributions of $C_{S}^{(k)}$ for any $k$ in (18), the probabilities in (18)
for each $k$ is the same. Hence, $K$ is multiplied with the probability
corresponding to any $k$ in (III-B). Using (3), (2), (10), (17), and (III-B)
we get

$\displaystyle\mathcal{P}_{NZ}=\Delta\Bigg{[}1-K\sum_{n=1}^{K-1}\binom{K-1}{n}\frac{(-1)^{n+1}n\sigma_{E}{\lambda}_{E}}{(\sigma_{D}{\lambda}_{D}+\sigma_{E}{\lambda}_{E})^{n+1}}$
$\displaystyle\times\Bigg{(}\frac{\sigma_{E}^{n}{\lambda}_{E}^{n}-(\sigma_{E}{\lambda}_{E}+\sigma_{D}{\lambda}_{D})^{n}}{n}+\Big{(}\frac{(\sigma_{E}{\lambda}_{E}+\sigma_{D}{\lambda}_{D})^{n+1}}{\sigma_{E}{\lambda}_{E}}$
$\displaystyle-\sigma_{E}^{n}{\lambda}_{E}^{n}\Big{)}\left(\sum_{i=1}^{n}(-1)^{n-i}\frac{(n-i)!(i-1)!}{n!}+\frac{(-1)^{n}}{(n+1)}\right)\Bigg{)}\Bigg{]}.$
(20)

It is noticed from (III-B) that the NZR is proportional to $\Delta$. This
signifies that the secrecy is enhanced with $\Delta$. We also note that the
NZR increases with $K$ due to the summation of $K-1$ terms in the second term
of (III-B).

##  IV Secrecy Outage Probability

The SOP is the probability that the achievable secrecy rate of the system
falls below a threshold secrecy rate $R_{th}$ [17]. The SOP of the RTS scheme
is evaluated for both cases of backhaul knowledge activity.

###  IV-A Backhaul Activity Knowledge Available

The SOP is obtained using the same procedure considered in (13) and (14) for
the NZR. It is derived by evaluating the probability corresponding to each
link that the secrecy rate of that link falls below the threshold $R_{th}$
provided the channel power gain ratio of that link is maximum. Then, due to
the law of total probability, we sum over all such probabilities corresponding
to each link in set $\mathcal{A}^{(q)}$ and also sum over all possibilities of
$q$ active links. Thus, the SOP is expressed as

$\displaystyle\mathcal{{P}}_{out}=\sum_{q=1}^{K}\binom{K}{q}\sum\limits_{k\in\mathcal{A}^{(q)}}\mathbb{P}\left[C_{S}^{(k)}<{R_{th}}\bigg{|}\frac{|\hat{h}^{(k)}_{TD}|^{2}}{|\hat{h}^{(k)}_{TE}|^{2}}>\bar{\Gamma}^{(k)}\right]$
$\displaystyle=\sum_{q=1}^{K}\binom{K}{q}\sum\limits_{k\in\mathcal{A}^{(q)}}$
$\displaystyle\mathbb{P}\left[\frac{|\hat{h}^{(k)}_{TD}|^{2}}{\sigma_{D}}<\rho\frac{|\hat{h}^{(k)}_{TE}|^{2}}{\sigma_{E}}+\rho-1\Bigg{|}\frac{|\hat{h}^{(k)}_{TD}|^{2}}{|\hat{h}^{(k)}_{TE}|^{2}}>\bar{\Gamma}^{(k)}\right]$
$\displaystyle=\sum_{q=1}^{K}\binom{K}{q}{q}\int_{0}^{\infty}\int_{0}^{\chi(x)/{y}}\int_{yz}^{\chi(x)}$
$\displaystyle
f_{|\hat{h}_{TD}^{(k)}|^{2}}\left(\chi(x)\right)f_{\bar{\Gamma}^{(k)}}(z)f_{|\hat{h}_{TE}^{(k)}|^{2}}(y)dxdzdy,$
(21)

where $\bar{\Gamma}^{(k)}=\max_{\begin{subarray}{c}n\in\mathcal{A}^{(q)}\\\
n\neq
k\end{subarray}}\Big{\\{}\frac{|\hat{h}_{TD}^{(n)}|^{2}}{|\hat{h}_{TE}^{(n)}|^{2}}\Big{\\}}$,
$\rho=2^{R_{th}}$ and $\chi(x)=(\rho
x+\sigma_{E}(\rho-1))\frac{\sigma_{D}}{\sigma_{E}}$. Finally, on substituting
(2), (3), and (II-A) into (IV-A) we get the closed-form expression as

$\displaystyle\mathcal{P}_{out}=(1-\Delta)^{K}+\Delta(1-\Delta)^{K-1}K\Big{(}\frac{-\sigma_{E}\lambda_{E}e^{-\sigma_{D}\lambda_{D}(\rho-1)}}{\rho\sigma_{D}\lambda_{D}+\sigma_{E}\lambda_{E}}$
$\displaystyle+1\Big{)}+\sum_{q=2}^{K}\binom{K}{q}q\Bigg{[}\sum_{n=1}^{K-1}\binom{K-1}{n}(-1)^{n+1}\Delta^{n+1}n\Bigg{[}\sum_{r=1}^{n}$
$\displaystyle\frac{(-1)^{n-r}}{{n!}}(r-1)!(n-r)!+\frac{(-1)^{n}\gamma(n+1)}{(n+1)}-\frac{\sigma_{D}{\lambda}_{D}^{n+1}}{a^{n+1}n!}\sum_{r=1}^{n}$
$\displaystyle(r-1)!(-1)^{n-r}\sum_{i=0}^{n}\binom{n}{i}(-b)^{n-i}\Gamma(i-r+1,b)+\frac{(-1)^{n}}{a^{n+1}}$
$\displaystyle+\frac{(\sigma_{E}{\lambda}_{E})^{n+1}}{(n+1)!}\Big{(}(-b)^{n+1}\text{Ei}(-b)-e^{-b}\sum_{l=0}^{n}(n-l)!(-b)^{l}\Big{)}$
$\displaystyle+\frac{(\sigma_{E}{\lambda}_{E})^{n+1}}{na^{n+1}}\sum_{p=0}^{n}\binom{n}{p}\frac{\Gamma(p-n+1,b)}{a(-b)^{p-n}}-\frac{e^{-b}}{na\sigma_{E}{\lambda}_{E}}\Bigg{]},$
(22)

where $\gamma(\cdot)$ is the lower gamma integral, $\Gamma(\cdot)$ is the
upper gamma integral, $\text{Ei}(\cdot)$ is the exponential integral,
$a=\rho\sigma_{D}{\lambda}_{D}+\sigma_{D}{\lambda}_{D}$, and
$b=\sigma_{D}{\lambda}_{D}(\rho-1)$.

###  IV-B Backhaul Activity Knowledge Unavailable

The SOP is obtained in this section following the approach used for the NZR in
(17). Therefore, by following the similar steps as in (18) and (III-B), and
using (2), (3), (10), and (IV-B), the SOP is expressed as

$\displaystyle\mathcal{P}_{out}=(1-\Delta)+\Delta\mathbb{P}\left[C_{S}^{(k)}<{R_{th}}\bigg{|}\frac{|h^{(k)}_{TD}|^{2}}{|h^{(k)}_{TE}|^{2}}>\bar{\Gamma}^{(k)}\right]$
$\displaystyle=(1-\Delta)+\Delta
K\sum_{n=1}^{K-1}\binom{K-1}{n}(-1)^{n+1}n\Bigg{[}\sum_{r=1}^{n}\frac{(-1)^{n-r}}{n!}$
$\displaystyle\times(r-1)!(n-r)!+\frac{(-1)^{n}\gamma(n+1)}{(n+1)}-\frac{(\sigma_{E}{\lambda}_{E})^{n+1}}{a^{n+1}n!}\sum_{r=1}^{n}$
$\displaystyle(r-1)!(-1)^{n-r}\sum_{i=0}^{n}\binom{n}{i}(-b)^{n-i}\Gamma(i-r+1,b)+\frac{(-1)^{n}}{a^{n+1}}$
$\displaystyle\times\frac{(\sigma_{E}{\lambda}_{E})^{n+1}}{(n+1)!}\Big{(}(-b)^{n+1}\text{Ei}(-b)-e^{-b}\sum_{l=0}^{n}(n-l)!(-b)^{l}\Big{)}$
$\displaystyle+\frac{(\sigma_{E}{\lambda}_{E})^{n+1}}{na^{n+1}}\sum_{p=0}^{n}\binom{n}{p}\frac{\Gamma(p-n+1,b)}{a(-b)^{p-n}}-\frac{e^{-b}}{na\sigma_{E}{\lambda}_{E}}\Bigg{]}.$
(23)

It is difficult to get insights from the closed-form equations derived in
Section III and IV because of the complex nature of the equations in (III-A),
(III-B), (IV-A) and (IV-B). Therefore, we determine the asymptotic NZR and SOP
in the next section.

We note that the noise power information is needed for the performance
analysis (NZR and SOP); however, the system does not require this information
for the implementation of the proposed RTS scheme.

##  V Asymptotic Analysis

The approximation of high SNR is utilized in this section to find the
asymptotic NZR and SOP for both the backhaul knowledge activity cases. We use
the approximation of high SNR, i.e., $1/\lambda_{D}\rightarrow\infty$ or
$\lambda_{D}\rightarrow 0$ in the derived closed-form expressions to get
insights and the effect of $\Delta$ and $K$ on the system’s performance.

###  V-A Asymptotic NZR

####  V-A1 Backhaul Activity Knowledge Available

Using the high-SNR approximation in (III-A), we notice that all the terms
approach zero except the first term inside the square bracket. Hence, the
asymptotic NZR for the RTS scheme with backhaul knowledge activity available
is $1-(1-\Delta)^{K}$. The asymptotic NZR depends on $\Delta$ and $K$,
illustrating that the availability of backhaul knowledge activity enables the
use of resources (number of transmitters) judiciously, thereby enhancing the
secrecy performance.

####  V-A2 Backhaul Activity Knowledge Unavailable

Similarly, we find that all the terms approach zero as
$1/\lambda_{D}\rightarrow\infty$ in (III-B), except the first term, and thus
the asymptotic NZR will be equal to $\Delta$. This implies that the NZR
becomes independent of all parameters at high SNRs, depending only on
$\Delta$.

###  V-B Asymptotic SOP

####  V-B1 Backhaul Activity Knowledge Available

Using the approximation $1/\lambda_{D}\rightarrow\infty$ in (IV-A), all the
terms, except the first one, tend to zero. Consequently, the asymptotic SOP
when backhaul knowledge activity is available is $(1-\Delta)^{K}$. This
observation is similar to the NZR knowledge available case where the asymptote
depends on $\Delta$ and $K$.

####  V-B2 Backhaul Activity Knowledge Unavailable

By following a similar procedure of $1/\lambda_{D}\rightarrow\infty$ in
(IV-B), we notice that the asymptotic SOP is $1-\Delta$. Thus, when backhaul
knowledge activity is unavailable, the asymptotic SOP depends only on
$\Delta$. A similar observation was found in the NZR knowledge unavailable
case.

##  VI Numerical Results

This section presents and discusses the analytical results of the RTS scheme.
We validate the mathematical equations by Monte Carlo simulations, and it is
evident from the figures that the analytical results perfectly match the
simulation. The parameters are set as $R_{th}=1$ bits/s/Hz, $1/\lambda_{E}=8$
dB, $\sigma_{D}=1$ dB and $\sigma_{E}=10$ dB. The results are presented for
backhaul knowledge available and when it is available.

Figure 2: The NZR versus $1/\lambda_{D}$ varying $K$ and $\Delta$.

Fig. 2 presents the NZR versus the SNR $1/\lambda_{D}$(dB) by changing
$K=\\{3,5\\}$ and $\Delta=\\{0.9,0.2\\}$. We observe that with an increase in
$K$ and $\Delta$, the secrecy is enhanced. We also notice that the NZR
saturates to a constant value at high SNR. We notice the backhaul knowledge
activity has a significant impact on secrecy performance and the effect of
backhaul knowledge activity is more significant when the backhaul is less
reliable, i.e., $\Delta$ = $0.2$. This shows that with prior knowledge of
backhaul link activity, the unreliability of backhaul can be compensated.

Figure 3: SOP versus $1/\lambda_{D}$ varying $K$ and $\Delta$.

In Fig. 3, the SOP is plotted against the SNR $1/\lambda_{D}$(dB) for
different $K$ and $\Delta$. The same trend is found for the SOP performance in
this figure as the NZR. We notice that the backhaul knowledge activity
available case outperforms the cases in which it is unavailable. Furthermore,
the SOP reaches a constant value at high SNR, which decreases with $\Delta$.
The saturation levels of the knowledge available cases are lower than the
knowledge unavailable cases. This signifies that the transmitters can be well
utilized when backhaul knowledge activity is available.

Figure 4: SOP comparison of different transmitter selection schemes.

Fig. 4 shows a comparison of the RTS scheme and various TS schemes from [14]
when backhaul knowledge activity is available with $K=5$ and $\Delta=0.9$. The
comparison includes traditional transmitter selection (TTS), which maximizes
legitimate channel rate; minimal eavesdropping selection (MIN-ES), in which
the transmitter with the worst eavesdropping channel is selected; and the
optimal TS scheme. We observe that the RTS scheme outperforms the TTS and MIN-
ES schemes while closely following the optimal TS scheme from [14]. This
signifies that achieving optimal performance can be achieved without
considering noise power or secrecy rate. Thus simplifying the selection
process.

##  VII Conclusion

A destination-to-eavesdropper channel CSI ratio-based transmitter selection
scheme has been proposed to improve the secrecy of the system. The NZR and SOP
have been evaluated in closed form along with their asymptotes for the RTS
scheme for the backhaul knowledge activity available and unavailable cases.
The results show that the secrecy performance increases when the number of
transmitters increases. The secrecy performance also increases with the
backhaul reliability factor. In the high SNR regime, we observe that when the
backhaul knowledge activity is unavailable, both the NZR and the SOP saturate
to a constant value that depends only on the backhaul reliability factor. The
prior knowledge of backhaul activity enables more efficient utilization of
resources. We demonstrate that the RTS scheme outperforms all the sub-optimal
TS schemes and achieves nearly optimal performance without requiring noise
power or the evaluation of the exact secrecy rate measurement.

##  VIII Acknowlwdgement

This work was supported in part by the Science Foundation Ireland (SFI) under
Grant Number 22/IRDIFA/10425, Tata Consultancy Services (TCS) Foundation
through its TCS Research Scholar Program, and the Ministry of Electronics and
Information Technology (MeitY), Govt. of India, through its project on
capacity building for human resource development in unmanned aircraft systems
(drone and related technology) with Ref. No. L14011/29/2021-HRD.

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