# Outage Performance of Multi-UAV Relaying-based Imperfect Hardware Hybrid
Satellite-Terrestrial Networks

Pankaj K. Sharma and Deepika Gupta Pankaj K. Sharma is with the Department of
Electronics and Communication Engineering, National Institute of Technology
Rourkela, India. Email<EMAIL_ADDRESS>Gupta is with the
Department of Electronics and Communication Engineering, Dr S P M
International Institute of Information Technology, Naya Raipur, India. Email:
<EMAIL_ADDRESS>
###### Abstract

In this paper, we consider an imperfect hardware hybrid satellite-terrestrial
network (HSTN) where the satellite communication with a ground user equipment
(UE) is aided by the multiple amplify-and-forward (AF) three-dimensional
($3$D) mobile unmanned aerial vehicle (UAV) relays. Herein, we consider that
all transceiver nodes are corrupted by the radio frequency hardware
impairments (RFHI). Further, a stochastic mixed mobility (MM) model is
employed to characterize the instantaneous location of $3$D mobile UAV relays
in a cylindrical cell with UE lying at its center on ground plane. Taking into
account the aggregate RFHI model for satellite and UAV relay transceivers and
the random $3$D distances-based path loss for UAV relay-UE links, we
investigate the outage probability (OP) and corresponding asymptotic outage
behaviour of the system under an opportunistic relay selection scheme in a
unified form for shadowed-Rician satellite links’ channels and Nakagami-_m_ as
well as Rician terrestrial links’ channels. We corroborate theoretical
analysis by simulations.

###### Index Terms:

Hybrid satellite-terrestrial network, unmanned aerial vehicle (UAV), mobile
relaying, hardware impairments.

##  I Introduction

Hybrid satellite-terrestrial networks (HSTNs) taking into account the
terrestrial relaying to aid satellite communication have received great
importance for mitigating the deleterious masking effect [1]. Therefore, the
works [2]-[5] have recently analyzed the performance of static relay-based
HSTNs with perfect transceiver hardware. However, the transceivers (e.g., low-
complexity relays) with imperfect hardware subject to radio-frequency hardware
impairments (RFHI) [6] resulting from IQ imbalances, amplifier non-
linearities, phase noise, etc. Consequently, the performance of terrestrial
networks with RFHI and HSTNs was analyzed in works [7] and [8], respectively.
On another hand, mobile unmanned aerial vehicle (UAV) relaying has recently
been considered as a key candidate for future wireless communications [9]. Due
to low cost, flexible deployment, and line-of-sight (LOS) communications, the
small rotary-wing UAVs have qualified as potential candidates for mobile
relaying. Eventually, the mobile UAV relaying has been recently employed in
[10] for terrestrial networks and in [11], [12] for HSTNs. Note that along
with the satellite, the UAVs may subject to RFHI due to limited onboard
sophisticated signal processing equipment. However, the impact of RFHI on UAV
relaying-based HSTNs has not been investigated in literature.

In contrast to previous decode-and-forward relaying-based HSTN in [12], this
paper investigates the outage probability (OP) and corresponding asymptotic
outage behaviour of an imperfect hardware HSTN system model with low-
complexity amplify-and-forward (AF) mobile UAV relays. Herein, we resort to
the aggregate RFHI model proposed in [6] where the satellite transmitter and
UAV receiver distortions are lumped together at satellite node. Likewise, the
UAV transmitter and ground user equipment (UE) distortions are lumped together
at UAV node. We adopt the stochastic mixed mobility (MM) from [13] to station
UAV relays in a cylindrical space at a given snapshot of time for calculating
the path loss based on three-dimensional ($3$D) UAV-to-ground user distance
distribution. More importantly, we present our analysis in a unified form for
shadowed-Rician satellite links’ and Nakagami-_m_ as well as analytically
challenging Rician terrestrial links’ fading channels which was missing in
[12]. Although the unified OP expression is exact only for terrestrial
Nakagami-_m_ fading, it is an accurate finite-series solution for otherwise
cumbersome terrestrial Rician fading. We present useful insights through our
analysis on the outage performance of considered HSTN.

##  II System Description

We consider an HSTN where information transmission from a geostationary earth
orbit (GEO) satellite $S$ to a fixed-location ground user equipment (UE) $D$
is facilitated by one of the opportunistically selected UAV relays $U_{i}$,
among $M$ candidate relays $i=1,...,M$ deployed at $3$D locations in a
cylindrical cell of height $H$ and radius $R$. The direct link from $S$ to $D$
is assumed to be heavily masked. Further, we assume that the satellite
transmitter $S$, UAV relay transceivers $U_{i}$, and UE $D$ are built using
simple hardware equipment resulting in RFHI. As followed in [6], instead of
considering individual RFHI components at various transmitter/receiver nodes,
we consider the lumped distortions at $S$ for satellite transmitter and UAV
receiver distortions, and at $U_{i}$ the corresponding UAV transmitter and UE
receiver distortions. Further, the $3$D cylindrical cell lies within the
footprint of $S$ around $D$ which is placed at its centre in ground plane. We
station the UAV relays randomly in the $3$D deployment region by first running
the MM model [13] for sufficient time duration, and then, taking a snapshot at
specific observation time. The channels from $S$ to $U_{i}$ and $U_{i}$ to $D$
are denoted by $g_{su_{i}}$ and ${g}_{u_{i}d}$, respectively. The noise at
various nodes is additive white Gaussian noise (AWGN) with zero mean and
variance $\sigma^{2}$.

###  II-A Mixed Mobility Model for UAV

We consider the multi-parameter MM model [13] which describes the $3$D
mobility of UAVs in a cylinder of height $H$ and radius $R$. The UAV makes
vertical transitions based on random waypoint mobility (RWPM) model with
random dwell time at each waypoint. at time $t$, the pdf of instantaneous
altitude of UAV $h_{i}(t)$ is given by the weighted sum of a static pdf
$f^{st}_{h_{i}}(x|t)$ and a mobility pdf $f^{mo}_{h_{i}}(x|t)$ as
$f_{h_{i}}(x|t)=p_{\mathrm{s}}f^{st}_{h_{i}}(x|t)+(1-p_{\mathrm{s}})f^{mo}_{h_{i}}(x|t)$,
where $f^{st}_{h_{i}}(x|t)=\frac{1}{H}$ and
$f^{mo}_{h_{i}}(x|t)=-\frac{6x^{2}}{H^{3}}+\frac{6x}{H^{2}}$, for $0\leq x\leq
H,$ with weight
$p_{\mathrm{s}}=\frac{\mathbb{E}[T_{\mathrm{s}}]}{\mathbb{E}[T_{\mathrm{s}}]+\mathbb{E}[T_{\mathrm{m}}]}$
as stay probability at waypoints.

Meanwhile, in the dwell time, the UAV makes random walk (RW) in horizontal
plane by following $z_{i}(t+1)=z_{i}(t)+\mathrm{u}_{i}(t)$ with probability
$p_{\mathrm{s}}$, where $z_{i}(t)$ denotes the projection of UAV’s location on
ground plane and $\mathrm{u}_{i}(t)$ is the uniform distribution in ball
$B(z_{i}(t),R^{\prime})$ with $R^{\prime}$ as the maximum spatial mobility
range. Whereas, it follows $z_{i}(t+1)=z_{i}(t)$ with probability
$1-p_{\mathrm{s}}$. Consequently, the pdf of distance $Z_{i}(t)=\|z_{i}(t)\|$
is given by $f_{Z_{i}}(z|t)=\frac{2z}{R^{2}},0\leq z\leq R$. Various
parameters associated with the MM model are described below:
$v_{1,i}(t)\sim[v_{min},v_{max}]$: Uniformly random velocity of vertical
transition at waypoints; $T_{\mathrm{s}}\sim[\tau_{min},\tau_{max}]$ and
$\mathbb{E}[T_{\mathrm{s}}]$: Uniformly random and mean dwell time;
$T_{\mathrm{m}}$ and
$\mathbb{E}[T_{\mathrm{m}}]=\frac{\ln(v_{max}/v_{min})}{v_{max}-v_{min}}\,\frac{H}{3}$:
Random and mean vertical movement time; and
${v}_{2,i}(t)=\|z_{i}(t)-z_{i}(t-1)\|$ and
$\mathbb{E}[{v}_{2,i}(t)]=\frac{R^{\prime}}{1.5}$: Random and mean velocity of
horizontal transition at waypoints.

###  II-B Channel Models

####  II-B1 Satellite-UAV links

For $g_{su_{i}}$ following SR fading, the probability density function (pdf)
of $|g_{su_{i}}|^{2}$ is given by

$\displaystyle
f_{|g_{su_{i}}|^{2}}(x)=\alpha_{u}\sum_{\kappa=0}^{m_{su}-1}\zeta(\kappa)x^{\kappa}\textmd{e}^{-(\beta_{u}-\delta_{u})x},$
(1)

where
$\alpha_{u}=(2\flat_{su}m_{su}/(2\flat_{su}m_{su}+\Omega_{su}))^{m_{su}}/2\flat_{su}$,
$\beta_{u}=1/2\flat_{su}$, and
$\delta_{u}=\Omega_{su}/(2\flat_{su})(2\flat_{su}m_{su}+\Omega_{su})$,
$\Omega_{su}$ and $2\flat_{su}$ are the average powers, respectively, of the
LOS and multipath components, $m_{su}$ is the fading severity,
$\zeta(\kappa)=(-1)^{\kappa}(1-m_{su})_{\kappa}\delta_{u}^{\kappa}/(\kappa!)^{2}$,
and $(\cdot)_{\kappa}$ denotes the Pochhammer symbol [15, p. xliii].
Additionally, a free space loss scale factor for satellite links is given as
[8]
$\sqrt{\mathcal{L}_{su_{i}}(t)\vartheta_{s}\vartheta(\theta_{u_{i}})}=\sqrt{\frac{\vartheta_{s}\vartheta(\theta_{u_{i}})}{\mathcal{K_{B}}\mathcal{TW}}}\left(\frac{c}{4\pi
f_{\mathrm{c}}\mathrm{d}_{u_{i}}(t)}\right)$, where
$\mathcal{K_{B}}=1.38\times 10^{-23}$J/K is the Boltzman constant,
$\mathcal{T}$ is the receiver noise temperature, $\mathcal{W}$ is the carrier
bandwidth, $c$ is the speed of light, $f_{\mathrm{c}}$ is the carrier
frequency, and $\mathrm{d}_{u_{i}}(t)$ is the distance between $S$ and
$U_{i}$. Here, $\vartheta_{s}$ denotes the antenna gain at satellite,
$\vartheta(\theta_{u_{i}})$ gives the beam gain of satellite towards $U_{i}$
which can be expressed as
$\vartheta(\theta_{u_{i}})=\vartheta_{u_{i}}\left(\frac{\mathcal{J}_{1}(\rho_{u_{i}})}{2\rho_{u_{i}}}+36\frac{\mathcal{J}_{3}(\rho_{u_{i}})}{\rho^{3}_{u_{i}}}\right)$,
where $\theta_{u_{i}}$ is the angular separation of $U_{i}$ from the satellite
beam center, $\vartheta_{u_{i}}$ is the antenna gain at $U_{i}$,
$\mathcal{J}_{\varrho}(\cdot)$, $\varrho\in\\{1,3\\}$ is the Bessel function,
and
$\rho_{u_{i}}=2.07123\frac{\sin\theta_{u_{i}}}{\sin\theta_{{u_{i}}3\textmd{dB}}}$
with $\theta_{{u_{i}}3\text{dB}}$ as $3$dB beamwidth.

####  II-B2 UAV-Ground UE links

The Nakagami-_m_ fading for terrestrial $U_{i}$ to $D$ links yields the pdf of
$|g_{u_{i}d}|^{2}$ as

$\displaystyle f_{|g_{u_{i}d}|^{2}}(x)$
$\displaystyle=\left(\frac{m_{ud}}{\Omega_{ud}}\right)^{m_{ud}}\frac{x^{m_{ud}-1}}{\Gamma(m_{ud})}\,\textmd{e}^{-\frac{m_{ud}}{\Omega_{ud}}x},$
(2)

where $m_{ud}$ and $\Omega_{ud}$ represent the fading severity and average
channel power, respectively. Further, the Rician fading of $U_{i}$ to $D$
links results in the pdf of $|g_{u_{i}d}|^{2}$ as

$\displaystyle f_{|g_{u_{i}d}|^{2}}(x)$
$\displaystyle=\frac{(1+K_{ud})}{\Omega_{ud}}\textmd{e}^{-K_{ud}}\textmd{e}^{-\frac{(1+K_{ud})x}{\Omega_{ud}}}$
(3)
$\displaystyle\times\mathcal{I}_{0}\left(2\sqrt{{K_{ud}(1+K_{ud})x}/{\Omega_{ud}}}\right),$

where $K_{ud}$ is the Rician $K$-factor and $\mathcal{I}_{0}(\cdot)$ is the
zeroth-order modified Bessel function of first kind [15, eq. 9.6.10]. Next,
the instantaneous free-space path loss from $U_{i}$ to $D$ can be given as
$W^{-\alpha}_{id}(t)={\left({h^{2}_{i}(t)+Z^{2}_{i}(t)}\right)^{-\frac{\alpha}{2}}}$,
where $w_{id}$ is the distance from $U_{i}$ to $D$ with $\alpha$ as path loss
exponent. Due to space constraint, we consider the path loss model with unity
LOS probability in this letter.

###  II-C Propagation Model

In considered RFHI-affected HSTN, a satellite $S$ communicates with UE $D$ in
two consecutive time slots through a selected AF UAV relay $U_{i}$. In the
first slot at time $t$, $S$ sends the message $x_{s}(t)$ with power
$P_{s}=\mathbb{E}[|x^{2}_{s}(t)|]$ to $U_{i}$. The received message signal at
$U_{i}$ under lumped RFHI distortions at $S$ can be given as

$\displaystyle y_{u_{i}}(t)\\!$
$\displaystyle=\sqrt{\mathcal{L}_{su_{i}}\\!(t)\vartheta_{s}\vartheta(\theta_{u_{i}})}{g}_{su_{i}}(t)(x_{s}(t)+\rho_{{s_{i}}})+\nu_{u_{i}},$
(4)

where $\rho_{{s_{i}}}\sim\mathcal{CN}(0,\kappa^{2}_{{s_{i}}}P_{s})$ is the
aggregate distortion noise at $S$ with $\kappa_{{s_{i}}}$ as RFHI parameter,
$\vartheta_{s}$ is the satellite antenna gain, $\nu_{u_{i}}$ is the AWGN at
$U_{i}$.

In the second slot at time $t+1$, the $U_{i}$ amplifies the received signal
$y_{u_{i}}(t)$ with a gain factor $G$ and forward the amplified signal to UE
$D$. Thus, the received signal at $D$ can be expressed as

$\displaystyle y_{id}(t\\!+\\!1)$
$\displaystyle=W^{-\frac{\alpha}{2}}_{id}(t\\!+\\!1){g}_{u_{i}d}(t\\!+\\!1)(Gy_{u_{i}}(t)\\!+\\!\rho_{u_{i}})\\!+\\!\nu_{d},$
(5)

where $\rho_{u_{i}}\sim\mathcal{CN}(0,\kappa^{2}_{{u_{i}}}P_{u})$, is the
aggregate distortion noise at $U_{i}$ with $\kappa_{{u_{i}}}$ as RFHI
parameter, $\nu_{d}$ is the AWGN at $D$. Here, the gain factor $G$ can be
computed as
$G=\sqrt{\frac{P_{u}}{P_{s}\mathcal{L}_{su_{i}}\\!(t)\vartheta_{s}\vartheta(\theta_{u_{i}})|g_{s_{u_{i}}}|^{2}(1+\kappa^{2}_{s_{i}})+\sigma^{2}}}$
to calculate the end-to-end signal-to-noise plus distortion ratio (SNDR) using
(5) as

$\displaystyle\Lambda_{id}(t+1)=$ (6)
$\displaystyle\frac{\Lambda_{su_{i}}(t)\Lambda_{u_{i}d}(t+1)}{\lambda_{1}\Lambda_{su_{i}}(t)\Lambda_{u_{i}d}(t\\!+\\!1)\\!+\\!\lambda_{2}\Lambda_{su_{i}}(t)\\!+\\!\lambda_{3}\Lambda_{u_{i}d}(t\\!+\\!1)\\!+\\!1},$

where
$\Lambda_{su_{i}}(t)=\frac{P_{s}\mathcal{L}_{su_{i}}(t)\vartheta_{s}\vartheta(\theta_{u_{i}})|{g}_{su_{i}}(t)|^{2}}{\sigma^{2}}$,
$\Lambda_{u_{i}d}(t+1)=\frac{P_{u}W^{-\alpha}_{id}(t+1)|{g}_{u_{i}d}(t+1)|^{2}}{\sigma^{2}}$,
$\lambda_{1}=\kappa^{2}_{s_{i}}+\kappa^{2}_{u_{i}}+\kappa^{2}_{s_{i}}\kappa^{2}_{u_{i}}$,
$\lambda_{2}=1+\kappa^{2}_{s_{i}}$, and $\lambda_{3}=1+\kappa^{2}_{u_{i}}$.
For analytical tractability, as in [ttd], we consider RFHI levels
$\kappa_{s_{i}}=\kappa_{s}$ and $\kappa_{u_{i}}=\kappa_{u}$ $\forall i$.
Further, we observe that the distance $\mathrm{d}_{u_{i}}(t)$ is very large
(e.g., $35,786$ Km, for GEO satellite), it can be reasonably considered
$\mathrm{d}_{u_{i}}(t)\approx\mathrm{d}_{u}(t)$,
$\mathcal{L}_{su_{i}}(t)\approx\mathcal{L}_{su}(t)$,
$\theta_{u_{i}}\approx\theta_{u}$, $\rho_{u_{i}}\approx\rho_{u}$, and
$\vartheta_{u_{i}}\approx\vartheta_{u}$, $\forall i$ to carry-out the OP
analysis under independent and identically distributed (i.i.d.) channels.

Let an opportunistic UAV relay selection be applied to maximize the SNDR at
$D$ as $i^{\star}(t)=\displaystyle\arg\max_{i\in 1,...,M}\Lambda_{id}(t)$.

##  III Outage Probability Analysis

We next drop the time notation $t$ to analyze the OP in one snapshot under
causal $S$ to $D$ transmission (i.e., $t>1$).

We first present the important cumulative distribution function (cdf) and pdf
expressions to be used later in this paper.

The cdf $F_{\Lambda_{su_{i}}}(x)$ can be expressed, by applying a variable
transformation for $\Lambda_{su_{i}}=\eta_{u}|g_{su_{i}}|^{2}$ in (1), as

$\displaystyle F_{\Lambda_{su_{i}}}(x)$
$\displaystyle=1-\alpha_{u}\\!\\!\\!\\!\sum_{\kappa=0}^{m_{su}-1}\\!\\!\\!\\!\frac{\zeta(\kappa)}{(\eta_{s})^{\kappa+1}}\sum_{p=0}^{\kappa}\frac{\kappa!}{p!}{\Theta_{u}}^{-(\kappa+1-p)}$
(7) $\displaystyle\times x^{p}\textmd{e}^{-\Theta_{u}x},$

where $\Theta_{u}=\frac{\beta_{u}-\delta_{u}}{\eta_{s}}$ and $\eta_{s}=$
$\frac{P_{s}\mathcal{L}_{su}\vartheta_{s}\vartheta(\theta_{u})}{\sigma^{2}}$.

Furthermore, the pdf of random distance $W_{id}$ under MM model [13] is given
as

$\displaystyle f_{W_{id}}(w)$
$\displaystyle=p_{s}f^{st}_{W_{id}}(w)+(1-p_{s})f^{mo}_{W_{id}}(w),$ (8)

where the pdfs $f^{st}_{W_{id}}(w)$ and $f^{mo}_{W_{id}}(w)$ correspond to the
UAV’s horizontal and vertical motions, respectively, and are given as

$\displaystyle f^{st}_{W_{id}}(w)$
$\displaystyle=\left\\{\begin{array}[]{l}\frac{2w^{2}}{R^{2}H},\textmd{ for
}0\leq w<H,\\\ \frac{2w}{R^{2}},\textmd{ for }H\leq w<R,\\\
\frac{2w}{R^{2}}-\frac{2w\sqrt{w^{2}-R^{2}}}{R^{2}H},\\\ \qquad\textmd{ for
}R\leq w\leq\sqrt{R^{2}+H^{2}}.\end{array}\right.$ (13)
$\displaystyle\textmd{and }f^{mo}_{W_{id}}(w)$
$\displaystyle=\left\\{\begin{array}[]{l}\frac{6w^{3}}{R^{2}H^{2}}-\frac{4w^{4}}{R^{2}H^{3}},\textmd{
for }0\leq w<H,\\\ \frac{2w}{R^{2}},\textmd{ for }H\leq w<R,\\\
\frac{2w}{R^{2}}-\frac{6w(w^{2}-R^{2})}{R^{2}H^{2}}+\frac{4w(w^{2}-R^{2})^{\frac{3}{2}}}{R^{2}H^{3}},\\\
\qquad\textmd{ for }R\leq w\leq\sqrt{R^{2}+H^{2}}.\end{array}\right.$ (18)

In addition, letting $\eta_{u}=\frac{P_{u}}{\sigma^{2}}$, the pdf
$f_{\Lambda_{u_{i}d}}(x)$ can be derived using
$f_{\Lambda_{u_{i}d}}(x)=\frac{d}{dx}F_{\Lambda_{u_{i}d}}(x)$, where
$F_{\Lambda_{u_{i}d}}(x)=\textmd{Pr}\left[{\eta_{u}W^{-\alpha}_{{i}d}|g_{u_{i}d}|^{2}}<x\right]$
for Nakagami-_m_ and Rician fading can be calculated based on the pdfs given
in (2) and (3) as

$\displaystyle F_{\Lambda_{u_{i}d}}(x)$
$\displaystyle=\int_{0}^{\sqrt{R^{2}+H^{2}}}\frac{\Upsilon\left(m_{ud},\frac{m_{ud}xr^{\alpha}}{\Omega_{ud}\eta_{u}}\right)}{\Gamma(m_{ud})}f_{W_{id}}(r)dr$
(19) $\displaystyle\textmd{and }F_{\Lambda_{u_{i}d}}(x)=$ (20) $\displaystyle
1-\int_{0}^{\sqrt{R^{2}+H^{2}}}\mathcal{Q}_{1}\left(\sqrt{2K_{ud}},\sqrt{\frac{2(1+K_{ud})xr^{\alpha}}{\Omega_{ud}\eta_{u}}}\right)f_{W_{id}}(r)dr,$

respectively, where $\mathcal{Q}_{1}(\cdot,\cdot)$ is the first-order Marcum-Q
function [10]. Using (19), we get the pdf

$\displaystyle f_{\Lambda_{u_{i}d}}(x)$
$\displaystyle=\frac{1}{\Gamma(m_{ud})}\left(\frac{m_{ud}}{\Omega_{ud}\eta_{u}}\right)^{m_{ud}}x^{m_{ud}-1}$
(21)
$\displaystyle\times\int_{0}^{\sqrt{R^{2}+H^{2}}}r^{m_{ud}\alpha}\textmd{e}^{-\frac{m_{ud}xr^{\alpha}}{\Omega_{ud}\eta_{u}}}\\!\\!\\!f_{W_{id}}(r)dr,$

for Nakagami-_m_ fading. To get the tractable pdf for Rician fading, in (20),
we first apply the approximation
$\mathcal{Q}_{1}(a,b)\approx\sum_{r_{1}=0}^{R_{1}}\chi_{r_{1}}\Gamma\left(1+r_{1},\frac{b^{2}}{2}\right)$
[10], where
$\chi_{r_{1}}=\frac{\Gamma(r_{1}+R_{1})R_{1}^{1-2r_{1}}a^{2r_{1}}2^{-r_{1}}}{\Gamma(r_{1}+1)^{2}\Gamma(R_{1}-r_{1}+1)\textmd{e}^{\frac{a^{2}}{2}}}$
and the term $R_{1}$ can be truncated as $50\max(1,a,b)$. Then, taking the
derivative, we get the pdf

$\displaystyle f_{\Lambda_{u_{i}d}}(x)$
$\displaystyle\approx\sum_{r_{1}=0}^{R_{1}}\chi_{r_{1}}\left(\frac{1+K_{ud}}{\Omega_{ud}\eta_{u}}\right)^{r_{1}+1}x^{r_{1}}$
(22)
$\displaystyle\times\int_{0}^{\sqrt{R^{2}+H^{2}}}r^{(r_{1}+1)\alpha}\textmd{e}^{-\frac{(1+K_{ud})xr^{\alpha}}{\Omega_{ud}\eta_{u}}}\\!\\!\\!f_{W_{id}}(r)dr,$

where for calculating $R_{1}$, $r$ can be set as upper integral limit. After a
careful observation of pdf expressions in (21) and (22), we represent them in
a unified mathematical form as

$\displaystyle f_{\Lambda_{u_{i}d}}(x)$
$\displaystyle\approx\sum_{r_{1}=\lambda_{4}}^{\lambda_{5}}\mathcal{B}_{r_{1}}\mathcal{A}^{r_{1}+1}x^{r_{1}}\int_{0}^{\sqrt{R^{2}+H^{2}}}r^{(r_{1}+1)\alpha}$
(23)
$\displaystyle\times\textmd{e}^{-{\mathcal{A}xr^{\alpha}}}f_{W_{id}}(r)dr,$

where
$\\{\lambda_{4},\lambda_{5},\mathcal{A},\mathcal{B}_{r_{1}}\\}=\\{m_{ud}-1,m_{ud}-1,\frac{m_{ud}}{\Omega_{ud}\eta_{u}},\frac{1}{\Gamma(m_{ud})}\\}$
and $\\{0,R_{1},\frac{1+K_{ud}}{\Omega_{ud}\eta_{u}},\chi_{r_{1}}\\}$ for
Nakagami-_m_ and Rician fading, respectively. Here, the analysis remains exact
for Nakagami-_m_ fading and tight approximation holds for Rician fading only.

###  III-A Outage Probability

For a threshold $\gamma_{\textmd{th}}$ and i.i.d. links, we derive the OP for
considered HSTN in a unified form using (6), (7), and (23) as

$\displaystyle\mathcal{P}_{\textmd{out}}(\gamma_{\textmd{th}})=[\textmd{Pr}\left[\Lambda_{id}<\gamma_{\textmd{th}}\right]]^{M}$
(24)
$\displaystyle=\\!\\!\left[1\\!-\\!\\!\int_{0}^{\infty}\\!\\!\left[1\\!-\\!F_{\Lambda_{su_{i}}}\\!\\!\left(\\!{\lambda_{6}\\!\left(1\\!+\\!\frac{\lambda_{7}}{y}\right)}\\!\right)\right]\\!\\!f_{\Lambda_{u_{i}d}}\\!\\!\left(\frac{y\\!+\\!\lambda_{8}}{\lambda_{9}}\right)\\!\frac{dy}{\lambda_{9}}\right]^{M},$

where
$\\{\lambda_{6},\lambda_{7},\lambda_{8},\lambda_{9}\\}$$=$$\\{\frac{\lambda_{3}\gamma_{\textmd{th}}}{1-\lambda_{1}\gamma_{\textmd{th}}},\lambda_{2}\gamma_{\textmd{th}}+\frac{{1-\lambda_{1}\gamma_{\textmd{th}}}}{\lambda_{3}},\lambda_{2}\gamma_{\textmd{th}},1-\lambda_{1}\gamma_{\textmd{th}}\\}$.

###### Theorem 1

For $\gamma_{\textmd{th}}<\frac{1}{\lambda_{1}}$, the OP in (24) can be
derived as

$\displaystyle\mathcal{P}_{\textmd{out}}(\gamma_{\textmd{th}})=[1-2\alpha_{u}\sum_{\kappa=0}^{m_{su}-1}\frac{\zeta(\kappa)}{(\eta_{s})^{\kappa+1}}\sum_{p=0}^{\kappa}\frac{\kappa!}{p!}\sum_{q=0}^{p}\binom{p}{q}$
(25)
$\displaystyle\times\textmd{e}^{-\frac{\Theta_{u}\lambda_{3}\gamma_{\textmd{th}}}{1-\lambda_{1}\gamma_{\textmd{th}}}}\sum_{r_{1}=\lambda_{4}}^{\lambda_{5}}\mathcal{B}_{r_{1}}\sum_{n=0}^{r_{1}}\binom{r_{1}}{n}\left(\frac{\mathcal{A}}{1-\lambda_{1}\gamma_{\textmd{th}}}\right)^{r_{1}-\frac{n-q-1}{2}}$
$\displaystyle\times\Theta^{\frac{n-q+1}{2}-(\kappa+1-p)}_{u}\left(\frac{\lambda_{3}\gamma_{\textmd{th}}}{1-\lambda_{1}\gamma_{\textmd{th}}}\right)^{p+\frac{n-q+1}{2}}$
$\displaystyle\times\left(\lambda_{2}\gamma_{\textmd{th}}+\frac{1-\lambda_{1}\gamma_{\textmd{th}}}{\lambda_{3}}\right)^{\frac{n+q+1}{2}}(\lambda_{2}\gamma_{\textmd{th}})^{r_{1}-n}$
$\displaystyle\times\int_{0}^{\sqrt{R^{2}+H^{2}}}r^{\left(r_{1}-\frac{n-q-1}{2}\right)\alpha}\textmd{e}^{-{\mathcal{A}r^{\alpha}}\left(\frac{\lambda_{2}\gamma_{\textmd{th}}}{1-\lambda_{1}\gamma_{\textmd{th}}}\right)}$
$\displaystyle\times\mathcal{K}_{n-q+1}\left(2\sqrt{\frac{\mathcal{A}\Theta_{u}\lambda_{3}\gamma_{\textmd{th}}r^{\alpha}}{(1-\lambda_{1}\gamma_{\textmd{th}})^{2}}\left(\lambda_{2}\gamma_{\textmd{th}}+\frac{1-\lambda_{1}\gamma_{\textmd{th}}}{\lambda_{3}}\right)}\right)$
$\displaystyle\times\left.f_{W_{id}}(r)dr\right]^{M},$

where $\mathcal{K}_{v}(\cdot)$ is the $v$th-order Bessel function [15, eq.
8.44].

###### Proof:

For the condition $\gamma_{\textmd{th}}<\frac{1}{\lambda_{1}}$, utilizing (7)
and (21) along with the relevant parameters $\lambda_{j}$, $j\in\\{6,7,8,9\\}$
in (24) followed by a change in the order of integration and applying [15, eq.
3.471.9], we obtain the required OP. ∎

To reveal insights on system diversity order, we simplify the OP in Theorem 1
at high SNDR ($\eta_{s},\eta_{u}\rightarrow\infty$).

###### Corollary 1

For $\gamma_{\textmd{th}}<\frac{1}{\lambda_{1}}$, the unified asymptotic OP
expression in can be obtained as

$\displaystyle\mathcal{P}_{\textmd{out}}(\gamma_{\textmd{th}})$
$\displaystyle{\simeq}\left[\frac{\alpha_{u}}{\eta_{s}}\left(\\!\frac{\lambda_{2}\gamma_{\textmd{th}}}{1\\!-\\!\lambda_{1}\gamma_{\textmd{th}}}\\!\right)\\!+\\!\sum_{r_{1}=\lambda_{4}}^{\lambda_{5}}\frac{\mathcal{B}_{r_{1}}\mathcal{A}^{r_{1}+1}}{r_{1}+1}\right.$
(26)
$\displaystyle\times\left.\left(\\!\frac{\lambda_{3}\gamma_{\textmd{th}}}{1\\!-\\!\lambda_{1}\gamma_{\textmd{th}}}\\!\right)^{r_{1}+1}\\!\\!\\!\\!\int_{0}^{\sqrt{R^{2}+H^{2}}}r^{(r_{1}+1)\alpha}f_{W_{id}}(r)dr\right]^{M},$

where
$\\{\lambda_{4},\lambda_{5},\mathcal{A},\mathcal{B}_{r_{1}}\\}=\\{m_{ud}-1,m_{ud}-1,\frac{m_{ud}}{\Omega_{ud}\eta_{u}},\frac{1}{\Gamma(m_{ud})}\\}$
and $\\{0,0,\frac{1+K_{ud}}{\Omega_{ud}\eta_{u}},\chi_{r_{1}}\\}$ for
Nakagami-_m_ and Rician fading, respectively.

###### Proof:

We first apply the bound $\frac{xy}{x+y}<\min(x,y)$ (after neglecting $1$ from
denominator) in (6) to get
$\Lambda_{id}\leq\frac{1}{\lambda_{1}+{1}/{\min\left({\Lambda_{su_{i}}}/{\lambda_{2}},{\Lambda_{u_{i}d}}/{\lambda_{3}}\right)}}$.
Then, invoking this in
$\mathcal{P}_{\textmd{out}}(\gamma_{\textmd{th}})=[\textmd{Pr}\left[\Lambda_{id}<\gamma_{\textmd{th}}\right]]^{M}$,
for $\gamma_{\textmd{th}}<\frac{1}{\lambda_{1}}$, we get
$\mathcal{P}_{\textmd{out}}(\gamma_{\textmd{th}}){\simeq}\\!\left[F_{\Lambda_{su_{i}}}\\!\\!\left(\\!\frac{\lambda_{2}\gamma_{\textmd{th}}}{1\\!-\\!\lambda_{1}\gamma_{\textmd{th}}}\\!\right)\\!+\\!F_{\Lambda_{u_{i}d}}\\!\\!\left(\\!\frac{\lambda_{3}\gamma_{\textmd{th}}}{1\\!-\\!\lambda_{1}\gamma_{\textmd{th}}}\\!\right)\right]^{M},$
where the product of cdfs is neglected. Herein, for small $x$, we simplify the
cdf $F_{\Lambda_{su_{i}}}(x)\simeq\frac{\alpha_{u}x}{\eta_{s}}$ [4]. Further,
we have
$F_{\Lambda_{u_{i}d}}(x)\simeq\sum_{r_{1}=\lambda_{4}}^{\lambda_{5}}\frac{\mathcal{B}_{r_{1}}\mathcal{A}^{r_{1}+1}}{r_{1}+1}x^{r_{1}+1}\int_{0}^{\sqrt{R^{2}+H^{2}}}r^{(r_{1}+1)\alpha}f_{W_{id}}(r)dr$
using the pdf in (23), for small $x$. Furthermore, for Rician fading, the
simplification for small $x$ is achieved by retaining only the term with index
$r_{1}=0$ in (23) along with $R_{1}=0$ [4]. ∎

Remark: Note that due to the term $\lambda_{4}$ in (24),
$\mathcal{P}_{\textmd{out}}(\gamma_{\textmd{th}})=1$, for
$\gamma_{\textmd{th}}\geq\frac{1}{\lambda_{1}}$. It is a so called SNDR
ceiling effect that ceases the communication beyond a threshold
$\gamma_{\textmd{th}}=\frac{1}{\lambda_{1}}$ under RFHI. Upon substituting the
appropriate parameters in (26) and taking $\eta_{s}=\eta_{u}=\eta$, for
$\gamma_{\textmd{th}}<\frac{1}{\lambda_{1}}$, one can visualize that
$\mathcal{P}_{\textmd{out}}(\gamma_{\textmd{th}})\propto\frac{1}{\eta^{M}}$.
Hence, the achievable diversity order of considered HSTN is only $M$ for both
Nakagami-_m_ and Rician fadings of terrestrial links due to the bottleneck by
SR channel.

##  IV Numerical Results

Here, using [8], we set the satellite parameters as $\mathcal{T}=300$ K,
$\mathcal{W}=15$ MHz, $c=3\times 10^{8}$m/s, $\mathrm{d}_{u}=35,786$ Km,
$f_{\mathrm{c}}=2$ GHz, $\vartheta_{u}=4.8$ dB, $\vartheta_{s}=53.45$ dB,
$\theta_{u}=0.8^{\circ}$, $\theta_{u3\textmd{dB}}=0.3^{\circ}$, and
$(m_{su},\flat_{su},\Omega_{su})=(1,0.063,0.0007)$ for heavy shadowing. For
UAVs, we set the parameters $v_{1,i}\sim[0.1,30]$ m/s, $v_{2,i}\sim[0,40]$
m/s, $H=40$ m, $R=80$ m, $\Omega_{ud}=1$, and $\alpha=2$. Here,
$p_{\mathrm{s}}=0.5$ is set by adjusting the distribution of $T_{\mathrm{s}}$.
Furthermore, we set $\eta_{s}=\eta_{u}=\eta$ as SNR.

Fig. 1 plots the curves OP versus SNR $\eta$ for considered HSTN. We set
$\gamma_{\textmd{th}}=0$ dB and $\kappa_{s}=\kappa_{u}=0.3$ for RFHI and
$\kappa_{s}=\kappa_{u}=0$ for perfect hardware. Here, we can see our unified
OP and asymptotic OP analyses for both types of terrestrial fading scenarios
are in well agreement with the simulations. By observing the curves for $M=1$
with $m_{ud}=1$ and $3$ along with $K_{ud}=0$ and $2$, we conclude that the
system diversity order is not affected by the UAV-to-UE links’ fading
channels. However, when $M$ changes from $1$ to $3$, the slope of various OP
curves changes and confirms a diversity order of $M$. We further see that OP
of the system deteriorates in the presence of RFHI without affecting the
system diversity.

Figure 1: OP versus $\eta$ under terrestrial Nakagami-_m_ /Rician fading.

Fig. 2 plots the curves OP versus SNR for different SR fading scenarios. Here,
we set $M=1$ and parameters
$(m_{su},\flat_{su},\Omega_{su})=(1,0.063,0.0007)$, $(10,0.158,1.29)$, and
$(5,0.251,0.279)$ for heavy, light, and average fading scenarios, respectively
[14]. We observed a performance bottleneck effect under terrestrial Rician
links where the OP for all three fading scenarios is indistinguishable. This
is due to the dominance of second-hop SNR term as it varies inversely with
order unity in asymptotic OP under Rician fading. However, for Nakagami
fading, the OP of the system improves when fading changes from heavy to
average and then to light shadowing at high SNR. Here, the second-hop SNR term
in asymptotic OP varies inversely with order $m_{ud}$. Hence, the first-hop
SNR term dominates and impact of SR fading is reflected.

Figure 2: OP versus $\eta$ under various SR fading scenarios.

Fig. 3 plots the curves OP versus threshold $\gamma_{\textmd{th}}$ for
considered HSTN. We plot curves for two values of RFHI parameters, i.e.,
$\kappa_{s}=\kappa_{u}=0.1$ and $0.3$. Here, we see that for given terrestrial
channel fading and RFHI parameters, an SNDR ceiling occurs at certain value of
$\gamma_{\textmd{th}}$ forcing the overall system in outage. The SNDR ceiling
threshold is $16.9$ dB and $7.23$ dB for RFHI parameters $0.1$ and $0.3$,
respectively. Thus, a higher value of RFHI parameters induces SNDR ceiling at
relatively lower thresholds. Moreover, the deviation in OP of system with RFHI
from perfect hardware case is relatively severe near ceiling threshold if $M$
increases from $1$ to $2$.

Figure 3: OP versus $\gamma_{\textmd{th}}$ under terrestrial Nakagami-_m_
/Rician fading.

##  V Conclusion

We have analyzed the OP of a multiple AF UAV relay-assisted HSTN in unified
form over Nakagami-_m_ and Rician terrestrial fading, where each node is
corrupted by the RFHI. We examined that RFHI have following effects on the
performance of HSTNs: (a) the system diversity order remains unaffected by the
RFHI and type of terrestrial channel fading; (b) system is forced into outage
at certain threshold which depends only on RFHI parameters.

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# Is the problem of Cuprate high-$T_{c}$ superconductivity a solved problem?

Navinder Singh  Theoretical Physics Division, Physical Research Laboratory,
Ahmedabad. India. PIN: 380009<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The recent experimental verification of the charge-transfer superexchange
mechanism as the microscopic pairing mechanism of high-$T_{c}$ cuprate
superconductivity by Seamus Davis and collaborators[1] is a tour de force! The
correct model for cuprates is the three band Emery model in which oxygen
p-orbitals are explicitly taken into account. The doped holes go into these
oxygen p-orbitals where they undergo charge-transfer superexchange with
unpaired electrons in copper d orbitals. This charge transfer superexchange is
the key which leads to bound pairs and superconductivity. In the experimental
verification[1], the system chosen is $Bi_{2}Sr_{2}CaCu_{2}O_{8+x}$. What is
achieved is the direct functional dependence of the local electron pair
density ($n_{P}(r)$) on local charge transfer energy ($\epsilon_{pd}(r)$)
using state of the art single-electron and electron-pair (Josephson) scanning
tunneling microscopy. The quantitative functional dependence of $n_{P}(r)$ on
$\epsilon_{pd}(r)$ matches with that indicated and deduced by theory[2, 3, 4,
5, 6, 7, 8, 9, 10, 11]. The verdict of the experiment settles the debates on
the microscopic mechanisms of the cuprate superconductivity in the clear favor
of charge-transfer superexchange mechanism. We discuss this development in
brief, and present a simple minded approach to the essence of cuprate
superconductivity. We discuss what is settled now, and what is not settled
yet. A ”theoretical minimum” of the high-$T_{c}$ problem is also discussed.

  * 19th Oct 2022

Keywords: Unconventional superconductivity in Cuprates, Charge-transfer
superexchange mechanism, Pseudogap.

“…Thus when one has found A way through the maze of conflicting requirements,
that is certain to be THE way, no matter how many deep-seated prejudices it
may violate…” —P. W. Anderson.

##  1 Recent experimental verification of charge-transfer superexchange in
brief

There is a growing consensus in the community that the single band Hubbard
model cannot fully address the physics of the hole doped cuprates, even though
it has some success[10, 11, 12]. In addition, the direct experimental evidence
has shown that hole doping removes electrons from oxygen p-orbitals[13]. Thus
a three band model that incorporates oxygen p-orbitals explicitly is a must to
fully address the key to the pairing mechanism. Very early on this job was
completed by V. J. Emery[14]. And he rightly pointed out that key to the
mechanism is the exchange interaction (exchange interaction between un-paired
electrons on oxygen p-orbitals (created due to hole doping) and an un-paired
electrons in the central Cu d${}_{x^{2}-y^{2}}$-orbitals). He could predict
right order of magnitude of $T_{c}$ with his three band model, now called the
Emery model. P. W. Anderson also rightly guessed that the key to the pairing
mechanism is the exchange interaction (superexchange) not the conventional
phonon mediated interactions[15, 16]. But he insisted on the single-band
Hubbard model as a low energy model appropriate for cuprates[16]. A reduction
from the Emery model to a single band Hubbard model was also achieved[17]. But
now we understand that oxygen p orbitals must be taken into account explicitly
as charge-transfer energy (from p-band to d-band) is the key parameter that
controls pairing111Although the Zhang-Rice reduction from three-band to
single-band model incorporates the charge transfer energy ($\epsilon_{pd}$) in
$J=\frac{4t_{0}^{4}}{\epsilon_{pd}^{2}U}+\frac{4t_{0}^{4}}{2\epsilon_{pd}^{3}}$,
but the single band $t-t^{\prime}-t^{\prime\prime}-J$ model proved to be
insufficient with respect to other probes[12].. The experiment by Seamus Davis
and collaborators has unequivocally settled it!

Before we present a simple model that explains the key to the pairing
mechanism in cuprates superconductors in layman language, we sketch a brief
review of this crucial experiment[1].

Authors[1] achieved a coterminous visualization of the local charge transfer
energy $\epsilon_{pd}(r)$ and local electron pair density $n_{P}(r)$ by
measuring the dependence of these quantities on $\delta(r)$, which is the
distance of the apical oxygen atom from the planner copper atom (just below
it). They selected the canonical cuprate $Bi_{2}Sr_{2}CaCu_{2}O_{8+x}$ at
optimal doping $p=0.17$ in which there is a crystal supermodulation in
$\delta(r)$ with a period of 26Å. They measure the differential conductance
$g(r,V)$ as a function of the location $r$ and the tip-sample voltage $V$.
From which they derive the spatially resolved EDOS ($N(r,E)\propto
g(r,V=E/e)$) and from that the local charge transfer energy $\epsilon_{pd}(r)$
is derived as the minimum distance between upper and lower bands at constant
conductance[1]. Thus a $\delta(r)$ dependent $\epsilon_{pd}(r)$ is determined.

Then using the superconducting scanning tunneling microscope (STM) tips (of
the same material), Josephson critical current (for electron pair tunneling)
is also measured as a function of location $r$ (thus $\delta(r)$) from which
electron pair density $n_{P}(r)$ is determined as a function of $r$ (thus
$\delta(r)$).

They find that the charge transfer energy decreases linearly as $\delta$
increases and the pair density increases linearly as $\delta$ increases. By
eliminating the common variable $\delta$ they find that the electron pair
density decreases linearly as the charge transfer energy increases
($\frac{dn_{P}}{d\epsilon}\simeq-0.81\pm 0.17~{}eV^{-1}$). This value agrees
reasonable well with theory
($\frac{dn_{P}}{d\epsilon}\simeq-0.9~{}eV^{-1}$)[10]. This is intuitively
understandable as the pairing energy scale $J$ roughly varies at
$t^{4}/\epsilon_{pd}^{3}$, that is, increase in the charge transfer energy
reduces the pair forming superexchange energy $J$.

It is a clear proof that the charge transfer superexchange is the key which
leads to bound pairs. The experimental and theoretical agreement of these
quantities settles the debates on the microscopic mechanisms of the cuprate
superconductivity in a clear favor of the charge-transfer superexchange
mechanism.

In the next section we explain the charge transfer exchange with a very simple
model to illustrate the pairing mechanism.

##  2 The charge-transfer exchange mechanism in a simple language

To understand the essence of the mechanism behind cuprate superconductivity in
a simple language, consider figure (1). There, four states are depicted
$|1\rangle,~{}|2\rangle,~{}|3\rangle,~{}$ and $|4\rangle$. There are un-paired
electrons in copper $3d_{x^{2}-y^{2}}$ orbitals with opposite spins, and there
is an un-paired electron on oxygen $p_{\sigma}-$orbital (after hole doping, as
hole is created in oxygen $p_{\sigma}$ orbital, leaving an un-paired electron
there). In state $|1\rangle$ the spin in oxygen $p_{\sigma}$ orbital is
polarized in the $+z$ direction and in state $|2\rangle$ the spin is polarized
in the $-z$ direction. Now consider that the un-paired electron on the copper
ion on the left hand side tunnels into oxygen $p_{\sigma}$ orbital making it
doubly occupied (state $|3\rangle$). It costs energy, the charge transfer
energy and Coulomb repulsion between two electrons in the oxygen $p_{\sigma}$
orbital. We denote the total energy cost with $\epsilon_{pd}$. The tunneling
happens with an amplitude $-t_{pd}$. There is another possibility that the un-
paired electron in the oxygen $p_{\sigma}$ orbital tunnels into copper
$3d_{x^{2}-y^{2}}$ orbital, making it doubly occupied. But that costs a large
amount of energy $U$ and that configuration we do not consider (No double
occupancy in $3d_{x^{2}-y^{2}}$ orbital of copper!). In state $|4\rangle$ the
un-paired electron from the copper ion on right hand side tunnels into oxygen
$p_{\sigma}$ orbital making it doubly occupied.

Figure 1: Simplest exchange mechanism between an un-paired electron on copper
atom and an un-paired electron (after hole doping) on oxygen atom.

These states are not still or stationary in time. The electron tunnels back
and forth between copper $3d_{x^{2}-y^{2}}$ orbital and oxygen $p_{\sigma}$
orbitals. These ”flip-flops” happen on a time scale of femto-seconds
($\sim\frac{\hbar}{\epsilon_{pd}}$). This exchange leads to a lower energy
state. The solution is simple. Our Hilbert space is 4 dimensional with basis
states $|1\rangle,~{}|2\rangle,~{}|3\rangle,~{}$ and $|4\rangle$. We want to
find the eigen-energies. We will write the equations of motion in the site
basis and transform them into energy basis to see this. The system will be in
the superposition state :
$|\chi\rangle=C_{1}|1\rangle+C_{2}|2\rangle+C_{3}|3\rangle+C_{4}|4\rangle$,
and the coefficients follow the EOM:

$i\hbar\dot{C}_{i}=\sum_{j}H_{ij}C_{j}.$ (1)

We set the energy (in site basis) of the states $|1\rangle$ and $|2\rangle$ as
$\epsilon_{0}=0$ and that of $|3\rangle$ and $|4\rangle$ as
$\epsilon_{0}+\epsilon_{pd}=\epsilon_{pd}$. The non-zero matrix elements of
the Hamiltonian are $H_{13}=H_{31}=H_{24}=H_{42}=-t_{pd}$ and
$H_{33}=H_{44}=\epsilon_{pd}$, and rest of all are zero. It is easy to show
that we end up having coupled (but in pairs) linear differential equations:

$\displaystyle i\hbar\dot{C}_{1}$ $\displaystyle=$ $\displaystyle-t_{pd}C_{3}$
(2) $\displaystyle i\hbar\dot{C_{3}}$ $\displaystyle=$ $\displaystyle-
t_{pd}C_{1}+\epsilon_{pd}C_{3}$ (3) $\displaystyle i\hbar\dot{C}_{2}$
$\displaystyle=$ $\displaystyle-t_{pd}C_{4}$ (4) $\displaystyle
i\hbar\dot{C_{4}}$ $\displaystyle=$ $\displaystyle-
t_{pd}C_{2}+\epsilon_{pd}C_{4}$ (5)

The solution is easy to obtain with the following eigenvalues of energy:

$E_{\pm}=\frac{1}{2}(\epsilon_{pd}\pm\sqrt{\epsilon_{pd}^{2}+4t_{pd}^{2}})$
(6)

Thus, there exists a $-ve$ energy solution (a bound state) with energy
$E_{-}=\frac{1}{2}(\epsilon_{pd}-\sqrt{\epsilon_{pd}^{2}+4t_{pd}^{2}})$, which
is $~{}\sim-\frac{t_{pd}^{2}}{\epsilon_{pd}}$ when $t_{pd}\ll\epsilon_{pd}$.
If the tunneling is stopped ($t_{pd}=0$) we get back unbound (positive and
zero energy states) states, as we should.

Thus, it is the ”flip-flops” (as Feynman put it[18] in the context of ammonia
molecule) of an un-paired electron between the oxygen p sigma orbital and the
copper $3d_{x^{2}-y^{2}}$ orbital, which is the experimental, the theoretical,
and the empirical crux of the matter! Pauling called it resonance lowering of
energy[19]. Kramers-Anderson called it exchange lowering (if the ”flip-flop”
happens between copper spins via oxygen orbitals, it is called the
superexchange). V. J. Emery called it ”exchange of holes”[14]. They meant the
same mechanism. Their intuition has withstood the test of time! Once there are
many bound pairs they can condense into the superconducting state below some
critical temperature.

###  2.1 Single hole

Consider a single hole doped into p sigma orbital of an oxygen atom (in what
follows we discuss in terms of the hole language as is more appropriate for
hole doped cuprates). In a femto-second time scale ($\sim\hbar/\epsilon_{pd}$)
it starts to resonate with the neighboring copper spins. Since all the oxygen
sites around the given copper atom are equivalent, it starts to resonate via
all the four oxygen p sigma orbitals. It lowers its energy further by doing
this. But it does not stay there. It can still lower its energy by tunneling
to a neighboring plaquette of four oxygen atoms around one copper atom in a
time scale of $\hbar/t$ where $t$ is the tunneling matrix element between two
nearby plaquettes. In fact it will do a random motion in the entire $CuO_{2}$
lattice, each time, on its way, resonating with the un-paired copper spins
(figure 2). A certain fraction of them (at typically 5-6 percent hole doping)
are effective in destroying the entire AFM spin alignment of copper spins. It
is something to do with its quantum random motion within the entire copper
oxide lattice.

Figure 2: Quantum random motion of a single doped hole, and the resonances it
creates on its way. Picture depicts schematic snapshots at different times at
$t_{1},~{}t_{2},~{}t_{3},~{}t_{4}$, and at $t_{5}$.

###  2.2 Two holes

It is not difficult to imagine the possibility of further lowering of energy
of the resonating holes in view the gain in $t$ terms in
$E_{-}\sim-(\frac{t_{pd}^{2}}{\epsilon_{pd}}~{}+~{}extra~{}t~{}terms)$ when
the two holes are around one copper spin. A further resonating process will
set up: resonance of the hole on the copper with two nearby holes in oxygen p
sigma orbitals. Mathematics becomes more complicated (but the essence of the
mechanism remains the same). Thus two holes doped at two different locations
will do random motion through the copper oxide plane, and ultimately they will
”catch” each other and a new resonance process will set up which will further
lower its energy. Then this hybrid (two holes around one cooper spin) will do
random motion (as a single entity) in the entire lattice, because by doing
this, it gains $t$ terms and further lowers its energy. Mathematical
formulation of these physical pictures can be made but that will be of no
consequence as experimental injection of a few holes in a copper-oxide layer
and their experimental study is out of question. Also actual lattice is never
perfect. The ”two-hole-composite” may end up trapped around some lattice
impurity killing its own random motion.

###  2.3 N holes

The topic of N holes brings us to Anderson’s resonating valence bond picture.
A complete mathematical treatment has been impossible so far. But then the
problem is captured from another powerful intuitive argument: Superconducting
state can be obtained from the mean-filed BCS wave-function of all the
resonating holes provided double occupancy is restricted on any given copper
spin. This and other alternative approaches implementing the Anderson’s RVB
idea are:[15, 16, 20, 21, 22, 23, 24, 25, 26, 27, 28]. An excellent review is
presented in:[29].

V. J. Emery solved this problem using a simple but powerful argument. He first
showed that the exchange mechanism between un-paired electrons on oxygen atoms
via an un-paired electron on the central copper atom (refer to figure 1 in his
paper[14]) leads to an attractive interaction with energy $-v_{0}$ (expression
of $v_{0}>0$ is given in his paper[14]).

In the second step he uses the simple BCS mean field theory and the BCS gap
equation replacing phonon interaction by his exchange mechanism interaction.
The estimated value of $T_{c}$ agrees reasonably well with experimental value.
This also shows that a simple-minded mean-field approach is not far from
truth!

In what follows we present a perspective and some critical comments on the way
field developed in the last 36 years.

##  3 Anderson’s ”dogmas”:

Anderson’s ”dogmas” are some factual inferences obtained from a careful
analysis of some of the important experiments and logical reasoning[30]. The
purpose of the ”dogmas” is to get-rid-of the irrelevant information and to
simplify the theoretical problem. Below we collect these dogmas; one by one:

DOGMA NO. 1:

Look at the copper oxide planes only: All the relevant carriers of spin and
charge reside in the copper oxide planes, and are present in the anti-bonding
band $Cu~{}3d_{x^{2}-y^{2}}--O~{}2p_{\sigma}$ which is split into two Hubbard
bands due to strong on-site interaction $U$. All other bands are filled thus
dead. Batlogg’s and more recent observations of universal $\rho_{ab}$ (same
in-plane resistivity per copper-oxide plane) is the direct experimental
support to dogma no. 1[30, 31, 32]. This establishes the two dimensionality of
the cuprate problem[30].

DOGMA NO. 2:

Magnetism and superconductivity: Magnetism and superconductivity are closely
related. The very same electrons that order AFM at $x=0$ become charge
carriers at finite doping, and below a critical temperature condense into
copper pairs, and exhibit superconductivity.

In defense, Anderson argues that at zero doping cuprates are Mott-Hubbard
insulators and exhibit antiferromagnetism, and there are other compelling
reasons from optical, photoelectron spectroscopy, and NMR data[30].

DOGMA NO. 3:

Dominant interactions are repulsive: Dominant interactions in copper-oxide
planes are repulsive and their energy scales are large. The major role is
played by $J$ and $U$. Other interactions like electron-phonon and electron-
impurity are sub-dominant.

Basis of this ”dogma” is that cuprates at zero doping are Mott-Hubbard
insulators (not band insulators)[30].

DOGMA NO. 4:

Non-Fermi liquid: The electronic state (”normal” state) in copper-oxide planes
is not a Fermi liquid in the sense that quasiparticle weight goes to zero
($Z\rightarrow 0$). It is the direct consequence of the experimentally
inferred self-energy:

$\Sigma^{{}^{\prime\prime}}\propto\omega,~{}~{}~{}~{}\Sigma^{{}^{\prime}}\propto\omega\log(\omega),$
(7)

and thus $Z=(1-\frac{\partial\Sigma}{\partial\omega})^{-1}\rightarrow 0$.
Therefore, the quasiparticles cannot be defined in the Landau sense. However,
experiments show that Fermi surface exists, and obeys the Luttinger theorem.

Anderson advances serval arguments to defend this assertion[30]. Leading ones
are the anisotropy in c-axis and ab-plane transport, and several conclusions
from ARPES and optical measurements[30].

Anderson’s ”dogma” 5 and 6 deal with his interlayer hopping theory, which has
turn out to be wrong.

So we notice that the above mentioned ”dogmas” of Anderson has withstood the
test of time. Only change is in ’dogma no. 1’. The oxygen orbitals cannot be
integrated out. Hole doping creates holes in oxygen $p_{\sigma}$ orbitals, and
three band Emery model captures the essential physics.

Also we reproduced these ”dogmas” here because these are essential facts
derived from experiments, and any successful theory (be it for the
superconducting state or for the strange metal regime) must respect these
facts. The verdict of the experiment is in favor of the super-exchange based
theories and these are in harmony with these dogmas as far as the
superconducting state is concerned.

##  4 What is settled now?

  1. 1.

Pairing mechanism:

Proposition No. 1:

The Quantum Critical Point (QCP) which is the point (on T=0 line) where
pseudogap ends (when superconductivity is killed in high magnetic fields) is
the crux of matter in the sense that the fluctuations of the nearby
antiferromagnetic order are the cause of pairing and superconductivity and the
strange metal regime is dominantly affected by it (as the scattering is
strongest at the QCP, marked by several probes including a dead $T-$linear
resistivity).

Proposition No. 2:

It is Emery’s ”exchange of holes” mechanism or Anderson’s superexchange
mechanism, and its RVB formulation which can explain pairing. But the strange
metal regime is outside the scope of the current developments of this idea.

The question is which one of the above proposition is the correct one as far
as the mechanism of superconductivity is concerned. This is a very difficult
matter. Author understands that the framing the issue in this way might be
rather offensive! But, out of the two propositions, as far as the
superconducting state is concerned, the proposition 2 has clearly gained
weight with the experimental verification of the superexchange mechanism[1].
Thus, the author is in the defense of proposition 2. Exact connection of
exchange physics (like RVB model) with QCP remained to be worked out, and more
serious thought and discussions are required.

But as far as the pairing mechanism is concerned, proposition no. 2 must be
defended!

  2. 2.

Issue of phonons:

There is no isotope effect at the optimal doping and at this doping the DC
resistivity is dead $T-$linear from very low mili-Kelvin temperatures to very
high about thousand-Kelvin temperatures[33, 34]. There is no $T^{5}$ like
bending at low temperatures at $T\ll T_{D}$(figure 3) which is a signature of
phonon scattering. There are many other indications, and according to
Anderson’s dogma no. 3 phonons remain sub-dominant both in the superconducting
state and in the strange metal regime. Other reasons that phonons play a sub-
dominant role are discussed in[20, 30].

Figure 3: DC resistivity in a simple metal and in cuprates.

  3. 3.

The pairing is local in Cuprates! This is evident from the fact that cooper
pair size is roughly of the order of $10~{}\AA$ to $30~{}\AA$[35]. This is
from the indirect penetration depth measurements. Actual pair size may even be
smaller. This is many orders of magnitude smaller than that found in the BCS
superconductors (of the order of $1000~{}\AA$). Small size of the pairs does
not mean that the mean-field approaches are not useful in the case of
cuprates. V. J. Emery’s solution using mean-field BCS gives a reasonable value
of $T_{c}$!

  4. 4.

Anderson’s assertion that the idea of a pairing glue is a folklore has turned
out to be true! Both $U$ and $J$ leads to a very high energy (high frequency)
dynamics and the Eliashberg type theories are not on the right track[36].

##  5 What is not settled yet?

  1. 1.

Theoretical deduction of the $T-$linear resistivity in the strange metal
regime is a major problem that remains. More specifically, how to derive
$A(x)\propto T_{c}(x)$? Where $A(x)$ is the coefficient in the $T-$linear term
in the resistivity and $T_{c}(x)$ is the critical temperature at that
doping[33, 37].

  2. 2.

Theoretical deduction of the non-Drude scattering rate
$\frac{1}{\tau(\omega)}\propto\omega$ in a formalism which is in harmony with
Anderson’s superexchange physics is another open issue.

  3. 3.

The rise of $T_{c}$ with doping (in the underdoped regime), that is the rising
part of the ”dome”, is understandable from the phase fluctuations and low
superfuild density perspective (simply less number of cooper pairs at low
doping)[38]. But this author believes that the justice is not yet done with
the mechanism why $T_{c}$ decreases with doping in the overdoped regime (that
is fall of the ”dome”). In the RVB model it is thought to be due to the
decrease in the pairing scale due to the decrease in the renormalization
factor $g_{s}(x)$ on doping[20]. But as far as author understands they miss
out a very crucial observation that the carrier number reduces from $1+p$
holes per $Cu$ atom at $p>p^{*}$ to $p$ holes per $Cu$ atom at $p<p^{*}$[37].
The accurate quantitative theory that incorporates this very important fact is
still lacking.

  4. 4.

Some other open issues are discussed in[39].

##  6 Whether ”a theoretical minimum” of high-$T_{c}$ superconductivity
possible?

The question of a ”metric” or of a ”theoretical minimum” of high-$T_{c}$
problem is vexed one. Which experimental result is the key result is
subjective to some extent. Let us try to design a set of sample ”theoretical
minima” and then we will argue.

Sample ”theoretical minimum” no. 1:

History has taught a lesson: the full understanding of the BCS theory came
with the understanding of the normal state (the Fermi Liquid Theory). FLT was
in place when BCS arrived. Similarly, to have a complete understanding of the
high-$T_{c}$ problem we must first understand the strange metal regime out of
which (on cooling) the superconductivity emerges. For example, any successful
theory should explain the marginal Fermi liquid phenomenology[40]. Then the
theory should explain right order of magnitude of $T_{c}$ and the ”dome” shape
structure with experimentally determined other required parameters.

Sample ”theoretical minimum” no. 2:

The parent compounds are Mott-Hubbard insulators and the doping of which leads
to superconductivity. Thus one must solve the problem of ”doping the Mott
insulator” first. Get the right doing at which AFM is killed. Then one should
explain superconductivity systematically building upon this development. Then
the theory should explain right order of magnitude of $T_{c}$ and the ”dome”
shape structure with experimentally determined other required parameters etc.

Sample ”theoretical minimum” no. 3:

QCP is of central importance! One must first understand the anomalies at this
point ($c_{el}\propto-T\ln T$ etc). One must first understand why does the
carrier number reduce from $1+p$ holes at $p>p^{*}$ to $p$ holes at $p<p^{*}$.
Then one should construct the theory, and then the theory should explain right
order of magnitude of $T_{c}$ and the ”dome” shape structure with
experimentally determined other parameters etc.

etc…

Which is the right way to follow? Again, this is a difficult state of affairs!
The choice will be affected by one’s prejudice and one’s vantage point. But
the author would like to defend a ”minimal theoretical minimum”. According to
this scheme of ”minimal theoretical minimum”: (1) a direct experimental
verification (the ”smoking gun signature”) of the proposed pairing mechanism,
and (2) the quantitative reproduction of the value of $T_{c}$ using
experimentally determined other parameters, are sufficient conditions.

And it has been achieved. It is the incontrovertible verdict of the
experiment[1] and settles the pairing problem once and for all. Other issues
(related to the strange metal regime) will be resolved with time.

##  7 In conclusion

The jury is still out on whether quantum criticality motivated approaches have
any essential relevance to the key of cuprate superconductivity? In defense,
the author would like to argue that as far as the paring mechanism is
concerned we have now a direct experimental verification of the charge-
transfer superexchange mechanism. So this debate is settled now. It remains to
be seen whether in the physics of the strange metal regime, quantum
criticality motivated approaches are required or not?

Finally, in this manuscript a brief review of the pivotal experiment is given.
The key mechanism of exchange interactions is explained using a very simple
model and in a very simple language. We discussed what is settled now and what
is not settled yet. We reasoned about a minimal ”theoretical minimum” that has
been achieved.

## Acknowledgments

Author dedicates this manuscript to the memory of his professor, prof. N.
Kumar.

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