# Exploring the Non-Overlapping Visibility Regions in XL-MIMO Random Access
Protocol

José Carlos Marinello Filho, Glauber Brante, Richard Demo Souza, Taufik Abrão
This work was supported in part by the National Council for Scientific and
Technological Development (CNPq) of Brazil under Grant 310681/2019-7.J. C.
Marinello is with the Electrical Engineering Department, Federal University of
Technology PR, Cornélio Procópio, PR, Brazil<EMAIL_ADDRESS>Brante is with the Department of Electrotechnics, Federal University of
Technology PR, Curitiba, PR, Brazil<EMAIL_ADDRESS>R. D. Souza is with
the Electrical and Electronics Engineering Department, Federal University of
Santa Catarina (UFSC), Florianópolis, SC, Brazil<EMAIL_ADDRESS>Abrão
is with Electrical Engineering Department, State University of Londrina, PR,
Brazil<EMAIL_ADDRESS>
###### Abstract

The recent extra-large scale massive multiple-input multiple-output (XL-MIMO)
systems are seen as a promising technology for providing very high data rates
in increased user-density scenarios. Spatial non-stationarities and visibility
regions (VRs) appear across the XL-MIMO array, since its large dimension is of
the same order of the distances to the user-equipments (UEs). Due to the
increased density of UEs in typical applications of XL-MIMO systems and the
scarcity of pilots, the design of random access (RA) protocols and scheduling
algorithms become challenging. In this paper, we propose a joint RA and
scheduling protocol, namely non-overlapping VR XL-MIMO (NOVR-XL) RA protocol,
which takes advantage of the different VRs of the UEs for improving RA
performance, besides of seeking UEs with non-overlapping VRs to be scheduled
in the same payload data pilot resource. Our results reveal that the proposed
scheme achieves significant gains in terms of sum-rate compared with
traditional RA schemes, as well as reducing access latency and improving
connectivity performance as a whole.

###### Index Terms:

Random access protocol, grant-based protocols, massive MIMO, XL-MIMO, non-
stationarity, visibility region (VR), scheduling.

##  I Introduction

Unusual channel conditions are seen in the recent extra-large scale massive
multiple-input multiple-output (XL-MIMO) technology [1]. By distributing the
antenna elements in a wide space, like in the facades of buildings, stadiums,
or shopping malls, the distance between user equipments (UEs) and the base
station (BS) antennas might become of the same order of the array dimension,
giving rise to spatial non-stationarities, in which the received signal energy
of a given UE varies significantly accross the array [1, 2, 3, 4, 5, 6].
Besides, since different regions of the large-aperture array see different
sets of scatterers, obstacles, and UEs, the concept of visibility regions
(VRs) is introduced in this channel model [4]. Through the concept of VRs,
some antennas, sometimes grouped in subarrays (SAs), are visible by a given
UE, _i.e._ , receive significant signal energy, while other antennas are not
[1, 2, 3, 4, 5, 6].

Some envisioned services within the future sixth generation of wireless
networks (6G) will demand a highly efficient method to deal with massive UE
access attempts, as massive Machine Type Communication (mMTC) and crowded
Mobile Broadband (cMBB) [7]. In these scenarios, the number of available pilot
sequences is lower than the number of UEs. Therefore, implementing efficient
random access (RA) protocols becomes essential to allow coherent
communication. An important RA protocol for spatially stationary massive MIMO
systems is proposed in [8], namely the strongest user collision resolution
(SUCRe) protocol. This protocol aims to resolve RA pilot collisions by
selecting the UE with the strongest signal gain to repeat its pilot after the
collision, in a completely decentralized and uncoordinated fashion [8].

Extending [8], the authors in [9] propose an idle pilot access (SUCR-IPA)
protocol, which allows a fraction of the users that failed to access the BS to
re-select their pilots by means of an access class barring (ACB) factor,
alleviating pilot collision. Another extension can be found in [10], in which
a graph-based pilot access (SUCR-GBPA) protocol is proposed, allowing all
users that lost the contention resolution to randomly select a new pilot.
Besides, in [11], the SUCRe protocol is modified to include a non-orthogonal
multiple access (NOMA) approach. The proposed NOMA-RA protocol allows to
resolve collisions between users who try to access the medium using the same
pilot signal, distinguishing them in the power domain and achieving a better
sum-rate performance with lower average delay in comparison with the
traditional SUCRe scheme.

In the SUCRe protocol, the UE repeats its RA pilot in step 3 if it judges
itself as the strongest contending UE, otherwise it stays silent. This can be
seen as a hard decision, as described in [12]. As an improvement, the authors
in [12] derive the probability of each UE being the strongest contender, based
on information available at the UE side. Then, in the proposed Soft-SUCRe
protocol, the UE repeats or not its chosen RA pilot in step 3 according to
this probability, achieving significant performance improvements in
overcrowded scenarios, _i.e._ , when the average number of accessing UEs is
higher than the number of available RA pilots.

An important drawback of the original SUCRe protocol is its lack of fairness,
since the strongest user retransmission criterion improves the connectivity
performance of the UEs closer to the BS, while the farther UEs typically
present increased probabilities of failed access attempts. In [13], the
authors propose the ACB with power control (ACBPC) protocol, which is able to
provide a good RA performance for all UEs, independently of their distances to
the BS, improving the RA performance for a substantial fraction of UEs within
the cell in comparison with SUCRe.

The SUCRe protocol has been recently adapted to the XL-MIMO scenario in [14],
in the so called SUCRe-XL. By allowing different UEs to access the network
through the same RA pilot, provided that the respective UE channels result in
non-overlapping VRs across the XL array of antennas, this protocol achieves
significant improvements in terms of reducing access latency and the fraction
of failed access attempts in overcrowded scenarios. However, the SUCRe-XL RA
protocol does not perform any specific action in the sense of favoring the
communication of UEs with non-overlapping VRs channels by employing the same
pilot, while expecting this to occur only at random.

Contribution: This paper proposes a joint RA and scheduling protocol for
crowded XL-MIMO systems, named Non-Overlapping VR XL-MIMO (NOVR-XL) random
access protocol. Different than other works regarding RA protocols, which
terminate the analysis when the UE succeeds in the RA, we also look at the
post-RA stage, in which payload data pilots are scheduled to the UEs that
succeeded in the RA stage. Due to the randomly distributed obstacles,
scatterers, and UEs, different UEs might have different VRs configurations. In
the RA stage, more than one UE may contend for the same RA pilot, giving rise
to a pilot collision. Our protocol takes advantage of the random VR
configuration of the different UEs to admit them to the network. Once a new UE
is admitted to the network, the proposed protocol tries to allocate a payload
data pilot already in use by another UE, but with non-overlapping VRs. In this
way, the proposed procedures not only improve the RA system performance, but
also increase the sum-rate, since more UEs can be connected to the same
payload data pilot in an interference-free manner, thanks to the large
extension of the XL antenna array. Furthermore, another important advantage of
our proposed NOVR-XL protocol is that it is just a 2-step RA protocol,
different than SUCRe-based ones which require four steps, allowing thus an
appreciable access latency reduction.

Notation: The conjugate, transpose and conjugate-transpose of a matrix
$\mathbf{A}$ are represented by $\mathbf{A}^{*}$, $\mathbf{A}^{T}$ and
$\mathbf{A}^{H}$, respectively. The $i,j$-th element of a matrix $\bf A$ is
$[{\bf A}]_{ij}$. $\mathbf{I}_{M}$ is the $M\times M$ identity matrix,
$\left|\cdot\right|$ and $\left\|\cdot\right\|$ represent the cardinality of a
set and the Euclidean norm of a vector, respectively. In addition,
$\mathcal{U}\backslash\mathcal{A}$ denotes the set containing elements that
are in $\mathcal{U}$ but not in $\mathcal{A}$. $\mathcal{N}(\cdot,\cdot)$
denotes a Gaussian distribution, $\mathcal{CN}(\cdot,\cdot)$ is a circularly-
symmetric complex Gaussian distribution, and $\mathcal{B}(\cdot,\cdot)$ is a
binomial distribution. $\mathbb{C}$ denotes spaces of complex-valued numbers,
while $\Gamma(\cdot)$ is the Gamma function, and ${\rm Pr}(\cdot)$ denotes
probability. The real part operator is $\Re(\cdot)$. $\mathbb{E}[\cdot]$ and
${\rm var}(\cdot)$ denote statistical expectation and variance, respectively.

##  II System model

We consider a time-division-duplexing (TDD) system, in which a BS operates
equipped with an extra-large uniform linear array (ULA) with $M$ antenna-
elements. Similarly as in [2], [14], we consider a simplified bipartite graph
model, being the array divided into $B$ SAs, each with $M_{b}=M/B$ antennas.
Let we define $\mathcal{M}$ as the set of all the BS subarrays and
$\mathcal{V}_{k}\subset\mathcal{M}$ the subset of SAs visible to $k$-th UE.
For the sake of mathematical modeling, we assume that the subset
$\mathcal{V}_{k}$ is generated at random, where each SA follows a Bernoulli
distribution with success probability $P_{b}$. This probability reproduces the
effect of random obstacles and scatterers in the environment, which interact
with the signals transmitted by/to the UEs, giving rise to the VRs. Thus, each
SA is visible with probability $P_{b}$, allowing also a binary vector
representation $\textbf{v}_{k}\in\mathbb{Z}^{B\times 1}$ for the VR of every
UE, in which $v_{k,b}=1$ if the $b$-th SA is visible by UE $k$, and 0
otherwise. It is important to note that when an antenna is visible by some
user, this means that the user can transmit/receive signals to/from this
antenna with a non-zero channel gain, not necessarily implying in the
existence of a line of sight (LOS) component.

Let $\mathcal{U}$ be the set of UEs in the cell,
$\mathcal{A}\subset\mathcal{U}$ be the set of active UEs, with temporary
dedicated data pilots, and $\mathcal{K}=\mathcal{U}\backslash\mathcal{A}$ be
the set of the inactive UEs (iUEs), which do not have dedicated data pilots
and need to be assigned one if they want to become active. Hence,
$K=|\mathcal{K}|$ is the number of iUEs in the cell. We assume a crowded
scenario where $K$ iUEs ($K>M$) are uniformly distributed within a circular
cell with internal and external radius of $r_{i}$ and $r_{e}$, respectively,
as depicted in Fig. 1.

Figure 1: Illustration of the considered crowded scenario composed by an XL
ULA and uniformly distributed UEs. The visibility region of a UE due to the
presence of scatterers and obstacles is represented.

We define $\tau_{\textsc{ra}}$ as the number of available RA pilot sequences
$\boldsymbol{s}_{1},\ldots,\boldsymbol{s}_{\tau_{\textsc{ra}}}\in\mathbb{C}^{\tau_{\textsc{ra}}\times
1}$. Each RA pilot has length $\tau_{\textsc{ra}}$ and
$\lVert\boldsymbol{s}_{t}\rVert^{2}=\tau_{\textsc{ra}}$, since we are
considering mutually orthogonal pilot sequences. Similarly,
$\tau_{\textsc{pd}}$ is the number of available payload data pilot (PDP)
sequences
$\boldsymbol{p}_{1},\ldots,\boldsymbol{p}_{\tau_{\textsc{pd}}}\in\mathbb{C}^{\tau_{\textsc{pd}}\times
1}$, which also are orthogonal and have
$\lVert\boldsymbol{p}_{t}\rVert^{2}=\tau_{\textsc{pd}}$. We define the set of
PDP sequences as $\mathcal{P}$, with cardinality
$|\mathcal{P}|=\tau_{\textsc{pd}}$. In a practical system, some sequences in
the set $\mathcal{P}$ would be temporarily allocated to the active UEs, and
some other would be idle, being available to admit new UEs which succeed in
the RA stage. However, in our model, we adopt the simplifying condition that
all PDP sequences in $\mathcal{P}$ are temporarily allocated to the active
UEs, while every UE succeeding in the RA stage can be admitted to the
network.111In practice, it is like if the PDP length could be adjusted to
match the number of PDPs allocated in that moment, or if the number of UEs
disconnecting from the network matches the number of connecting UEs. This is
assumed to avoid the problem of finding a suitable PDP length
$\tau_{\textsc{pd}}=|\mathcal{P}|$, trading-off the PDP transmission overhead
and the availability of PDPs for suitable scheduling new UEs, which is beyond
the scope of this paper.

The time-frequency resources are divided into coherence blocks of $T$ channel
uses, dimensioned such that the channel responses between the BS and UEs are
constant and frequency flat within each block, while varying between blocks.
This time-frequency organization can be implemented with orthogonal frequency-
division multiplexing (OFDM). Similarly as in [8], the coherence blocks are
divided into two categories: payload data blocks and RA blocks. The blocks in
the first category are used for uplink (UL) and downlink (DL) payload data
transmission by the users in the subset $\mathcal{A}$, which have been already
allocated a temporary dedicated PDP sequence when they succeeded in an RA
attempt. The second category is used for RA from iUEs in the set $\mathcal{K}$
when they wish to be admitted to the payload data blocks, which requires the
temporary allocation of a PDP sequence.

The large-scale fading includes path loss and shadowing, following an urban
micro scenario model given by [15]:

$\beta^{(b)}_{k,m}=10^{-\kappa\log(d^{(b)}_{k,m})+\frac{g+\chi}{10}},$ (1)

where $d^{(b)}_{k,m}$ is the distance between the $k$-th UE and the $m$-th
antenna ($m=1,\ldots,M_{b}$) of SA $b$, $\chi\sim\mathcal{N}(0,\sigma^{2}_{\rm
sf})$ is the shadow fading, with standard deviation $\sigma_{\rm sf}=10$ dB;
$\kappa=3.8$ is the adopted path loss exponent, and $g=-34.53$ dB is the path
loss at the reference distance. Similarly as in [14], we adopt the
simplification
$\beta^{(b)}_{k}=\frac{1}{M_{b}}\sum_{m=1}^{M_{b}}\beta^{(b)}_{k,m}$, and
blocked SAs resulting in $\beta^{(b)}_{k}=0$. Hence, $\beta^{(b)}_{k}$
represents the average large-scale fading coefficient for the $k$-th user at
the $b$-th SA.

Let $\mathbf{h}^{(b)}_{k}\in\mathbb{C}^{M_{b}\times 1}$ be the Rayleigh fading
channel vector between UE $k\in\mathcal{K}$ and SA $b$, following
$\mathbf{h}^{(b)}_{k}\sim\mathcal{CN}(0,\beta^{(b)}_{k}\mathbf{I}_{M_{b}})$.
The above XL-MIMO channel model assumes a non-line-of-sight (NLOS) channel
between UE and BS, in the same way as [2, 4, 14, 16]. This assumption can be
justified in practice for scenarios where a large number of obstacles and
scatterers takes place between UEs and the BS.

Under this model, as shown in [16], the signal to interference plus noise
ratio (SINR) of an active UE $k\in\mathcal{A}$ employing zero-forcing (ZF)
combiner/precoder is described by:

$\overline{\gamma}_{k}^{(\textsc{ZF})}=\frac{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\rho}}}{\sigma^{2}}\left(M_{b}\,\sum_{b=1}^{B}\beta^{(b)}_{k}-\sum_{j\in\mathcal{A},j\neq
k}\frac{\sum_{b=1}^{B}\beta^{(b)}_{k}\beta^{(b)}_{j}}{\sum_{b=1}^{B}\beta^{(b)}_{j}}\right),$
(2)

in which
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\rho}}$
is the transmit power of the UEs assuming a uniform power allocation policy in
the data transmission stage, and $\sigma^{2}$ is the noise variance. Besides,
if this UE performs $\mu^{\textsc{ra}}_{k}$ RA attempts until establishing a
connection, and then remains a number of $\mu^{\textsc{pd}}_{k}$ channel
coherence time ($t_{c}$) intervals transmitting payload data occupying the
whole bandwidth w, its achieved spectral efficiency is given by:

$R_{k}=\left(\frac{\mu^{\textsc{pd}}_{k}{\textsc{w}}\,t_{c}}{\mu^{\textsc{ra}}_{k}\varphi_{\textsc{ra}}+\mu^{\textsc{pd}}_{k}{\textsc{w}}\,t_{c}}\right)\left(1-\frac{\tau_{\textsc{pd}}}{T}\right)\log_{2}\left(1+\overline{\gamma}_{k}^{(\textsc{ZF})}\right),$
(3)

in which $\varphi_{\textsc{ra}}$ is the number of channel uses spent in each
RA attempt, considering the employed RA protocol, and recalling that $T$ is
the number of channel uses in a coherence time. It is noteworthy that the
first pre-log factor in (3) accounts for the RA overhead, while the second one
accounts for the PDPs transmission overhead for channel estimation purposes.

Finally, the system sum-rate ($\rm SR$) is given by

${\rm SR}={\textsc{w}}\cdot\sum_{k\in\mathcal{A}}R_{k}.$ (4)

The sum-rate metric is very useful in order to compare different RA protocols,
since it takes into account the number of channel uses spent per attempt, the
number of RA attempts that the UEs are requiring to connect, and the number of
active UEs that the protocol can admit to the network. Furthermore, since
$\tau_{\textsc{pd}}$ always coincide in our model to the number of PDP
temporarily allocated to the active UEs, and
$\tau_{\textsc{pd}}\leq|\mathcal{A}|$ since the same PDP can be allocated to
different active UEs if their VRs are non-overlapping, the sum-rate metric
also quantifies the capability of the scheduling algorithm in finding UEs with
orthogonal VRs to allocate the same PDP.

##  III Proposed NOVR-XL Protocol

In this section, we describe the two steps of our proposed NOVR-XL protocol to
jointly perform RA and PDP scheduling in the XL-MIMO system. In the first
step, all iUEs make an RA attempt with probability $P_{a}\leq 1$. Each iUE
$k\in\mathcal{K}$ making an RA attempt uniformly selects an uplink RA pilot
sequence $\boldsymbol{s}_{r(k)}\in\mathbb{C}^{\tau_{\textsc{ra}}\times 1}$,
where $r(k)\in\\{1,2,\ldots,\tau_{\textsc{ra}}\\}$, transmitting it with a
non-zero power $\rho_{k}>0$, otherwise it stays silent by setting
$\rho_{k}=0$. Since transmission is uncoordinated, it is possible and utmost
usual in crowded scenarios that more than one iUE choose the same RA pilot
sequence. Therefore, let $\mathcal{S}_{t}=\\{k:r(k)=t,\rho_{k}>0\\}$ represent
the set of iUEs indices transmitting pilot $t$. As in [8], its cardinality
follows:

$|\mathcal{S}_{t}|\sim\mathcal{B}\left(K,\frac{P_{a}}{\tau_{\textsc{ra}}}\right).$
(5)

where $\mathcal{B}(\cdot,\cdot)$ is the binomial distribution.

Step 1: Random Pilot Sequence UL Transmission. At this step, the inactive UEs
that want to establish a connection transmit the signal
$\textbf{x}_{k}=[\boldsymbol{s}_{r(k)};\textbf{d}^{\rm
ul}_{k}]\in\mathbb{C}^{(\tau_{\textsc{ra}}+n_{u})\times 1}$, composed by its
chosen UL RA pilot sequence and the additional information vector
$\textbf{d}^{\rm ul}_{k}\in\mathbb{C}^{n_{u}\times 1}$. The additional
information vector $\textbf{d}^{\rm ul}_{k}$ contains the identity of the UE,
and its estimated VR binary vector
$\widehat{\textbf{v}}_{k}\in\mathbb{Z}^{B\times 1}$. Our method assumes that
$\widehat{\textbf{v}}_{k}$ can be obtained at the UE’s side, employing the
periodic transmission of a broadcast signal that activates each SA
sequentially, which can be seen as a slight modification of the broadcast
signal transmission in step 0, typically assumed by RA protocols, as in [8],
[14]. Thus, the $b$-th SA receives the signal

$\textbf{Y}^{(b)}=[\textbf{Y}_{p}^{(b)},\textbf{Y}_{d}^{(b)}]=\sum_{k\in\mathcal{K}}\sqrt{\rho_{k}}\mathbf{h}_{k}^{(b)}\textbf{x}_{k}^{T}+\textbf{N}^{(b)},$
(6)

where $\textbf{Y}^{(b)}\in\mathbb{C}^{M_{b}\times(\tau_{\textsc{ra}}+n_{u})}$,
$\textbf{Y}_{p}^{(b)}\in\mathbb{C}^{M_{b}\times\tau_{\textsc{ra}}}$,
$\textbf{Y}_{d}^{(b)}\in\mathbb{C}^{M_{b}\times n_{u}}$, and
$\textbf{N}^{(b)}\in\mathbb{C}^{M_{b}\times(\tau_{\textsc{ra}}+n_{u})}$ is the
receiver noise with entries drawn from $\mathcal{CN}(0,\sigma^{2})$. Besides,
we have that

$\textbf{Y}_{p}^{(b)}=\sum_{k\in\mathcal{K}}\sqrt{\rho_{k}}\mathbf{h}_{k}^{(b)}\boldsymbol{s}_{r(k)}^{T}+\textbf{N}_{p}^{(b)},\,\,{\rm
and}$ (7)
$\textbf{Y}_{d}^{(b)}=\sum_{k\in\mathcal{K}}\sqrt{\rho_{k}}\mathbf{h}_{k}^{(b)}(\textbf{d}^{\rm
ul}_{k})^{T}+\textbf{N}_{d}^{(b)},$ (8)

with $\textbf{N}^{(b)}=[\textbf{N}_{p}^{(b)},\textbf{N}_{d}^{(b)}]$. Hence,
each SA correlates (7) with each RA pilot. For the case of an arbitrary RA
pilot $\boldsymbol{s}_{t}$, it yields:

$\textbf{y}_{t}^{(b)}=\textbf{Y}_{p}^{(b)}\frac{\boldsymbol{s}_{t}^{*}}{\lVert\boldsymbol{s}_{t}\rVert}=\sum_{i\in\mathcal{S}_{t}}\sqrt{\rho_{i}\tau_{\textsc{ra}}}\mathbf{h}_{i}^{(b)}+\mathbf{n}^{(b)}_{t},\quad
b=1,\ldots,B,$ (9)

where
$\mathbf{n}^{(b)}_{t}=\textbf{N}_{p}^{(b)}\frac{\boldsymbol{s}_{t}^{*}}{\lVert\boldsymbol{s}_{t}\rVert}$
is the effective receiver noise, so that
$\mathbf{n}^{(b)}_{t}\sim\mathcal{CN}(0,\sigma^{2}\mathbf{I}_{M_{b}})$. As a
result, each SA tries to decode the message contained in (8) using the
estimates $\textbf{y}_{t}^{(b)}$, evaluating222Note that, at this moment, the
BS does not know which RA pilot each UE has chosen. So, the BS tries do decode
the message in (8) using the estimates obtained from every RA pilot. Only when
the decoding is successful the BS identifies which UE transmitted that
message.:

$\displaystyle\widehat{\textbf{d}^{\rm
ul}_{k}}^{T}=\frac{(\textbf{y}_{t}^{(b)})^{H}}{\sqrt{\tau_{\textsc{ra}}}}\textbf{Y}_{d}^{(b)}.$
(10)

In the next section, we analyze the attainable SINR of
$\widehat{\textbf{d}^{\rm ul}_{k}}$ in (10), while pointing out the conditions
in which its information can be suitably decoded. This is very likely to occur
when the SA $b$ is exclusively visible to the $k$-th UE among the UEs
contending by RA pilot $t=s_{r(k)}$, _i.e._ , in $\mathcal{S}_{t}$. This means
that $b\in\mathcal{V}_{k}$ and $b\not\in\mathcal{V}_{k^{\prime}}$, $\forall
k^{\prime}\in\mathcal{S}_{t},k^{\prime}\neq k$. In this case,
$\textbf{y}_{t}^{(b)}=\sqrt{\rho_{k}\tau_{\textsc{ra}}}\mathbf{h}_{k}^{(b)}+\mathbf{n}^{(b)}_{t}$,
representing a very good channel estimate of $\mathbf{h}_{k}^{(b)}$. However,
there is a possibility that the $b$-th SA may be visible to more than one UE
in $\mathcal{S}_{t}$, with the $k$-th UE presenting a strong predominant
channel gain w.r.t. its contenders, so that the channel estimate in (9),
although partially contaminated, could be of sufficient quality to result in
an appropriate decoding of the additional information from this UE. All these
possibilities are considered in the SINR analysis conducted in the next
section.

Assuming from now on an appropriate decoding of the information in (10), the
UE $k$ is admitted to the set $\mathcal{C}$, which contains the indices of UEs
that wished to become active and had their additional information in (10)
suitably decoded in some SA. Besides, the BS gets to know the estimated VR of
UEs in the set $\mathcal{C}$, constructing the binary matrix
$\textbf{V}\in\mathbb{Z}^{B\times|\mathcal{C}|}$, composed by the VR vectors
$\textbf{v}_{k}$ of the UE $k\in\mathcal{C}$. It also creates the binary
matrix $\textbf{E}\in\mathbb{Z}^{B\times|\mathcal{C}|}$, whose $i,j$-th
element is set when the BS has reliable channel estimates in SA $i$ for the
$j$-th UE in $\mathcal{C}$. This occurs when the BS successfully decoded the
message $\textbf{d}^{\rm ul}_{j}$ from the $i$-th SA. In such cases, the BS
stores the channel estimation vectors $\widehat{\bf h}^{(b)}_{k}=\widehat{\bf
y}^{(b)}_{r(k)}$, when $[\textbf{E}]_{b,k}=1$; otherwise, $\widehat{\bf
h}^{(b)}_{k}={\bf 0}_{M_{b}}$ and $[\textbf{E}]_{b,k}=0$.

Conventionally, in a spatially stationary channel configuration, it is
necessary a number of $|\mathcal{C}|$ payload data pilot (PDP) sequences be
available to admit the UEs in $\mathcal{C}$ into the active set $\mathcal{A}$.
Due to the high density of UEs in crowded XL-MIMO scenarios and scarcity of
orthogonal PDP sequences, such exclusive allocation of PDPs to active UEs
would require sequences of long lengths, substantially increasing the training
overhead and spectral efficiency penalty in (3). However, due to the obstacles
and scatterers in typical wireless environment, combined with the XL-MIMO
propagation features, the spatial non-stationarities and distinct VRs per UE
emerge; hence, it is reasonable and usual that some UEs in $\mathcal{C}$
present non-overlapping VRs with respect to certain active UEs in
$\mathcal{A}$. Moreover, since the BS already knows their VRs, this implies
that the same PDP sequence can be allocated to such UEs, since they can
communicate with the same data pilot in an interference-free manner, due to
the orthogonality in the VR-spatial domain.

We propose a novel procedure for the BS to allocate PDP sequences. First, the
BS needs to update matrix ${\bf F}\in\mathbb{Z}^{|\mathcal{P}|\times B}$,
which contains the combined VRs of UEs allocated to each PDP sequence. The
combined VRs related to each allocated PDP is simply the sum of the VRs of the
UEs using it. Since the system starts to operate, for each UE $k$ succeeding
in the RA stage, the BS first seeks for a line in $\bf F$ that is orthogonal
to ${\bf v}_{k}$. By denoting the $j$-th line of ${\bf F}$ by ${\bf f}_{j}$,
this operation is equivalent to finding ${\bf f}_{j}\,{\bf v}_{k}=0$. Once
this condition is valid for some line $j$, the $j$-th PDP in $\mathcal{P}$ can
be allocated to the UE $k$. In this case, the matrix $\bf F$ is updated,
evaluating ${\bf f}_{j}\leftarrow{\bf f}_{j}+{\bf v}^{T}_{k}$. On the other
hand, if the VR of the UE $k$ is not orthogonal to the combined VR of any PDP
already allocated, the BS has to allocate a new PDP sequence to this UE. In
this case, ${\bf F}{\leftarrow}\left[{\bf F}^{T},{\bf v}_{k}\right]^{T}$, and
$\mathcal{P}{\leftarrow}\mathcal{P}\cup\boldsymbol{p}_{t}$, being
$\boldsymbol{p}_{t}$ the new PDP allocated to the UE $k$. Furthermore,
whenever the UE $k$ which was allocated the PDP sequence $j$ terminates its
connection, matrix $\bf F$ is updated evaluating ${\bf f}_{j}\leftarrow{\bf
f}_{j}-{\bf v}^{T}_{k}$. If this line becomes null, it is removed from matrix
$\bf F$, as well as set $\mathcal{P}$ is updated as
$\mathcal{P}{\leftarrow}\mathcal{P}\backslash\boldsymbol{p}_{j}$. Our proposed
PDP scheduling procedure is presented in Algorithm 1.

Algorithm 1 NOVR-XL Scheduling Procedure

Input: $\mathcal{P}$, F, ${\bf v}_{k}$.

1: Initialize the index of the PDP to be allocated to UE $k$ as
$\iota{\leftarrow}-1$;

2: for $j=1,2,\ldots,|\mathcal{P}|$ do

3: if ${\bf f}_{j}\,{\bf v}_{k}==0$ then

4: Evaluate $\iota{\leftarrow}j$;

5: Evaluate ${\bf f}_{j}\leftarrow{\bf f}_{j}+{\bf v}^{T}_{k}$;

6: break;

7: end if

8: end for

9: if $\iota==-1$ then

10: Obtain a new PDP $\boldsymbol{p}_{t}$ to be allocated to the UE $k$;

11: Evaluate ${\bf F}{\leftarrow}\left[{\bf F}^{T},{\bf v}_{k}\right]^{T}$;

12: Evaluate $\mathcal{P}{\leftarrow}\mathcal{P}\cup\boldsymbol{p}_{t}$;

13: Evaluate $\iota{\leftarrow}t$;

14: end if

Output: $\mathcal{P}$, F, $\iota$.

Step 2: Precoded Random Access DL Response. The $b$-th SA generates a precoded
DL signal

$\mathbf{G}^{(b)}=\sqrt{\frac{q}{B\,\delta_{b}}}\sum_{k\in\mathcal{C}}[{\bf
E}]_{b,k}\frac{\widehat{\mathbf{h}}_{k}^{(b)*}}{\lVert\widehat{\mathbf{h}}_{k}^{(b)}\rVert}(\textbf{d}^{\rm
dl}_{k})^{T},$ (11)

where the DL precoder matrix $\mathbf{G}^{(b)}\in\mathbb{C}^{M_{b}\times
n_{d}}$, $q$ is the BS transmit power, $\delta_{b}=\sum_{k\in\mathcal{C}}[{\bf
E}]_{b,k}$ is the number of UEs in $\mathcal{C}$ having reliable channel
estimates in SA $b$, and $\textbf{d}^{\rm dl}_{k}\in\mathbb{C}^{n_{d}\times
1}$ is the DL RA data destined to the $k$-th UE, containing the index of the
PDP allocated to it, according to the output of Algorithm 1, and other
information, like timing advance, etc. Note that in (11) the DL data of a
given UE is precoded only in the SAs where the BS has reliable channel
estimates, since $[{\bf E}]_{b,k}=0$ in the other SAs. Besides, it is not
necessary to send DL pilots at this step of the proposed NOVR-XL protocol,
differently than SUCRe-XL, since the UEs will not estimate the sum of
contenders signal gains in this new context.

Thus, the UE ${k}\in\mathcal{C}$ receives, in the reciprocal channel, the
signal $\widehat{\textbf{d}^{\rm dl}_{k}}^{T}\in\mathbb{C}^{1\times n_{d}}$,

$\displaystyle\widehat{\textbf{d}^{\rm dl}_{k}}^{T}$
$\displaystyle=\sum_{b\in\mathcal{V}_{k}}\mathbf{h}_{k}^{(b)T}\mathbf{G}^{(b)}+\boldsymbol{\eta}_{k}^{T},$
(12)
$\displaystyle=\sum_{b\in\mathcal{V}_{k}}\sqrt{\frac{q}{B\,\delta_{b}}}[{\bf
E}]_{b,k}\mathbf{h}_{k}^{(b)T}\frac{\widehat{\mathbf{h}}_{k}^{(b)*}}{\lVert\widehat{\mathbf{h}}_{k}^{(b)}\rVert}(\textbf{d}^{\rm
dl}_{k})^{T}+\ldots$
$\displaystyle+\sum_{b\in\mathcal{V}_{k}}\sqrt{\frac{q}{B\,\delta_{b}}}\mathbf{h}_{k}^{(b)T}\sum_{j\in\mathcal{C},j\neq
k}[{\bf
E}]_{b,j}\frac{\widehat{\mathbf{h}}_{j}^{(b)*}}{\lVert\widehat{\mathbf{h}}_{j}^{(b)}\rVert}(\textbf{d}^{\rm
dl}_{j})^{T}+\boldsymbol{\eta}_{k}^{T},$

where $\boldsymbol{\eta}_{k}\sim\mathcal{CN}(0,\sigma^{2}\textbf{I}_{n_{d}})$
is noise. In the next section, the SINR of $\widehat{\textbf{d}^{\rm dl}_{k}}$
in (12) is evaluated, pointing out the conditions in which it can be suitably
decoded at the UE.

At this point, all UEs that wished to connect to the network try do decode the
received signal. If the decoding is successful, this UE will complete the
admission process to the active users set $\mathcal{A}$, being temporarily
allocated a PDP sequence from $\mathcal{P}$, while achieving a spectral
efficiency given by (3). If the decoding fails, this UE performs another RA
attempt after a random backoff time, as in [8]. If the UE does not succeed in
connecting to the network after a predefined number of RA attempts, it gives
up and declares a failed access attempt [8]. Fig. 2 summarizes the main steps
of the proposed NOVR-XL RA procedure, highlighting the two steps of the RA
protocol, and the main actions performed by the UE and BS.

Randomly select RA pilot Step 1: RA pilot and $\hat{\bf v}_{k}$ Tries to
decode received messages Create matrix $\bf V$Run Algorithm 1 Step 2: Index of
data pilot allocated Tries to decode received messages Decoding successful?
Max RA attempts? Random backoff time FSUEBSNoNoYesYes

Figure 2: Diagram of the proposed procedures for joint RA and PDP scheduling
in XL-MIMO systems. "F" denotes a failed access attempt, while "S" denotes a
successful access attempt.

###  III-A Variant of SUCRe-XL RA protocol: Re-allocating PDPs due to
Orthogonal VRs

Besides considering the SUCRe-XL RA protocol as proposed in [14], for
performance comparison in the next section, we also consider a variant of this
protocol denominated modified SUCRe-XL (mSUCRe-XL) RA protocol, with the aim
of leveraging non-overlapping VRs features. In this variant, the four steps of
the SUCRe-XL protocol are performed similarly as in [14], except that, in step
3, the re-transmitting UEs transmit together with their repeated RA pilot the
additional information vector containing ${\bf\widehat{v}}_{k}$, estimated
from the modified broadcast signal in step 0. If the collision is successfully
resolved, the BS can decode such information and gets to know the VR of this
new UE. Then, the BS is able to run our proposed scheduling procedure
described in Algorithm 1 in order to seek a PDP sequence allocated to active
UEs whose combined VRs are orthogonal to that of UE $k$. This variant serves
as a challenging opponent to the proposed NOVR-XL RA protocol.

In summary, the original SUCRe-XL protocol from [14] does not provide to the
BS an explicit information about the VRs of the UEs. Therefore, each UE
admitted to the network has to be temporarily allocated an exclusive PDP
sequence to it, such that $|\mathcal{P}|=|\mathcal{A}|$. On the other hand, in
the mSUCRe-XL protocol discussed herein, little overhead is added in the step
3 of the RA protocol, aiming to allow a more efficient scheduling of PDP
resources, such that $|\mathcal{P}|\leq|\mathcal{A}|$. Moreover, the sum-rate
metric in (4) is numerically evaluated in the next section, and the presented
results can answer if the re-allocation of PDPs to UEs with non-overlapping
VRs compensates the additional RA overhead required.

Finally, it is worth mentioning that SUCRe-XL and mSUCRe-XL RA protocols
require four steps since they rely on the _strongest user_ retransmission
criterion, _i.e._ , only one UE (with the strongest channel gain) retransmits
the RA pilot in step 3, resolving the collision. For this purpose, it is
required the DL pilot transmission in step 2, allowing the UEs to estimate the
sum of contenders’ signal gains, and then judging if it is the strongest. This
approach is not effective from the connectivity perspective, since its
objective is to connect only one UE per RA pilot; also, it is unfair, since
such UE is the one closer to the BS, while increases access latency, since
four steps are required. Contrarily, our proposed NOVR-XL protocol does not
rely on the strongest user retransmission criterion; hence, the RA attempt for
a UE will be successful if their RA message can be decoded from at least one
SA, which is likely to occur due to the different random VRs of the UEs in the
XL-MIMO system. As a result, more UEs can connect through the same RA pilot
resource, improving RA performance in just 2 steps, since no DL pilot
transmission is required, reducing access latency.

##  IV SINR in the NOVR-XL Protocol

This section develops analytical expressions for the uplink and downlink SINR
of the proposed NOVR-XL protocol. We start analyzing the probability of a
specific UE channel generates a visible region under an exclusive SA. Hence,
the expressions for the SINR in step 1 (UL) and step 2 (DL) of the proposed
algorithm are obtained analytically.

###  IV-A Probability of a UE having an exclusive SA

We first evaluate the probability of an SA $b$ being exclusively visible by UE
$k$, among the UEs contending for the same RA pilot. In this case,
$b\in\mathcal{V}_{k}$ and $b\not\in\mathcal{V}_{k^{\prime}}$, $\forall
k^{\prime}\in\mathcal{S}_{t},k^{\prime}\neq k$. This condition is very
desirable for our proposed NOVR-XL protocol, because it provides reliable
channel estimates in that SA, and, consequently, reliable communication links
for that UE in both directions. This makes very likely that the BS will
properly decode the information message sent by the UE in step 1, besides of
allowing the BS to transmit in step 2 information about the PDP being
allocated for it. However, it is important to note that even if no SA is
exclusively visible by UE $k$ among its contenders, there is still a chance
for its information message sent in step 1 being successfully decodable. This
is the case when the $k$-th UE presents a strong predominant channel gain in
certain SA $b$ w.r.t. its contenders, so that the channel estimate in (9) for
this SA, although partially contaminated, is of sufficient quality for an
appropriate decoding of the additional information from this UE. All these
possibilities are discussed in the SINR analysis conducted in Subsection IV-B.

From the system model in Section II, the probability of SA $b$ being visible
by UE $k$ is $P_{b}$; therefore, given a number of UEs $|\mathcal{S}_{t}|$
contending for the same RA pilot, the probability of the SA $b$ being
exclusively visible to it can be written as

${P^{\textsc{sa}}_{\rm
exc}}=P_{b}\cdot\left(1-P_{b}\right)^{|\mathcal{S}_{t}|-1}.$ (13)

Since we need that at least one SA is exclusively visible by the UE, this
probability $P_{\rm exc}$ can be evaluated as

$P_{\rm exc}=1-\left(1-P^{{\textsc{sa}}}_{\rm exc}\right)^{B}.$ (14)

Fig. 3 depicts the probability of the UE having at least one SA exclusively
visible by it. As one can see, such probability achieves appreciable values
for a not so large number of UEs contending for the same RA pilot. This
corroborates the idea of the proposed NOVR-XL protocol in taking advantage of
the reliable channel links in the RA stage for increasing the number of UEs
admitted to the network.

Figure 3: Probability of a UE having at least one SA exclusively visible by
it, for $P_{b}=0.5$. If $P_{b}$ increases, the curves move towards the bottom
right corner, while if $P_{b}$ decreases, the curves move towards the upper
left corner.

###  IV-B SINR of the UL Information in Step 1

We derive herein the SINR performance for the UL data
$\widehat{\textbf{d}^{\rm ul}_{k}}$ as defined in (10). First, we expand the
components of the $\ell$-th term ($\ell=1,\ldots,n_{u}$) of this estimation in
(IV-B). Then, following the analysis in [17], we compute the expected power of
each term in (IV-B) as the expectation of its squared magnitude. This result
is shown in Table I, in terms of the squared mean and variance of each
component. Then, taking only $\mathbb{E}[|\varpi_{1}|]^{2}$ as the desired
signal power, while relegating
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbb{V}}{\rm
ar}[|\varpi_{1}|]$ as an additional interference as in [17], and after some
manipulations, we obtain the UL SINR in (16). One can recognize the first term
in the denominator as the directed interference in the case of a contending UE
also having the $b$-th SA visible, which can be seen as a coherent RA
interference. The second term can thus be seen as a non-coherent RA
interference, and relates to the product of the variance of the channel
estimate with the variance of the received signal at the $b$-th SA in step 1.

$\displaystyle\widehat{d^{\rm ul}_{k\ell}}$ $\displaystyle=\frac{(\textbf{y}_{t}^{(b)})^{H}}{\sqrt{\tau_{\textsc{ra}}}}[\textbf{Y}_{d}^{(b)}]_{:\ell},$ $\displaystyle=\frac{1}{\sqrt{\tau_{\textsc{ra}}}}\left[\sum_{i\in\mathcal{S}_{t}}\sqrt{\rho_{i}\tau_{\textsc{ra}}}\mathbf{h}_{i}^{(b)}+\mathbf{n}^{(b)}_{t}\right]^{H}\left[\sum_{j\in\mathcal{K}}\sqrt{\rho_{j}}\mathbf{h}_{j}^{(b)}d^{\rm ul}_{j\ell}+[\textbf{N}_{d}^{(b)}]_{:\ell}\right],$ $\displaystyle=\underbrace{\rho_{k}||\mathbf{h}_{k}^{(b)}||^{2}d^{\rm ul}_{k\ell}}_{\varpi_{1}}+\underbrace{\sum_{i\in\mathcal{S}_{t},i\neq k}\rho_{i}||\mathbf{h}_{i}^{(b)}||^{2}d^{\rm ul}_{i\ell}}_{\varpi_{2}}+\underbrace{\sum_{i\in\mathcal{S}_{t}}\sum_{j\in\mathcal{K},j\neq i}\sqrt{\rho_{i}\rho_{j}}(\mathbf{h}_{i}^{(b)})^{H}\mathbf{h}_{j}^{(b)}d^{\rm ul}_{j\ell}}_{\varpi_{3}}+\underbrace{\frac{(\mathbf{n}^{(b)}_{t})^{H}}{\sqrt{\tau_{\textsc{ra}}}}\sum_{j\in\mathcal{S}_{t}}\sqrt{\rho_{j}}\mathbf{h}_{j}^{(b)}d^{\rm ul}_{j\ell}}_{\varpi_{4}}+$ $\displaystyle+\underbrace{\frac{(\textbf{y}_{t}^{(b)})^{H}}{\sqrt{\tau_{\textsc{ra}}}}\sum_{j\in\mathcal{K}\backslash\mathcal{S}_{t}}\sqrt{\rho_{j}}\mathbf{h}_{j}^{(b)}d^{\rm ul}_{j\ell}}_{\varpi_{5}}+\underbrace{\frac{(\textbf{y}_{t}^{(b)})^{H}}{\sqrt{\tau_{\textsc{ra}}}}[\textbf{N}_{d}^{(b)}]_{:\ell}}_{\varpi_{6}}.$ (15) TABLE I: Expected power of the terms in (IV-B). Term $\varpi_{i}$ | $\mathbb{E}[|\varpi_{i}|]^{2}$ | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbb{V}{\rm ar}}[|\varpi_{i}|]$
---|---|---
$\varpi_{1}$ | $M^{2}_{b}\,\rho^{2}_{k}\,(\beta_{k}^{(b)})^{2}$ | $M_{b}\,\rho_{k}\,\beta_{k}^{(b)}$
$\varpi_{2}$ | $M^{2}_{b}\sum_{\begin{subarray}{c}i\in\mathcal{S}_{t}\\\ i\neq k\end{subarray}}\rho^{2}_{i}\,(\beta_{i}^{(b)})^{2}$ | $M_{b}\sum_{\begin{subarray}{c}i\in\mathcal{S}_{t}\\\ i\neq k\end{subarray}}\rho_{i}\,\beta_{i}^{(b)}$
$\varpi_{3}$ | 0 | $M_{b}\sum_{i\in\mathcal{S}_{t}}\sum_{j\in\mathcal{K},j\neq i}\rho_{i}\rho_{j}\beta_{i}^{(b)}\beta_{j}^{(b)}$
$\varpi_{4}$ | 0 | $\frac{M_{b}\sigma^{2}}{\tau_{\textsc{ra}}}\sum_{j\in\mathcal{S}_{t}}\rho_{j}\beta_{j}^{(b)}$
$\varpi_{5}$ | 0 | $M_{b}\left[\sum_{i\in\mathcal{S}_{t}}\rho_{i}\beta_{i}^{(b)}+\frac{\sigma^{2}}{\tau_{\textsc{ra}}}\right]\sum_{j\in\mathcal{K}\backslash\mathcal{S}_{t}}\rho_{j}\beta_{j}^{(b)}$
$\varpi_{6}$ | 0 | $M_{b}\left[\sum_{i\in\mathcal{S}_{t}}\rho_{i}\beta_{i}^{(b)}+\frac{\sigma^{2}}{\tau_{\textsc{ra}}}\right]\sigma^{2}$
$\gamma^{{\rm
ul}}_{b,k}=\frac{M_{b}\,\rho^{2}_{k}\,(\beta_{k}^{(b)})^{2}}{M_{b}\sum_{\begin{subarray}{c}i\in\mathcal{S}_{t}\\\
i\neq
k\end{subarray}}\rho^{2}_{i}\,(\beta_{i}^{(b)})^{2}+\left[\sum_{i\in\mathcal{S}_{t}}\rho_{i}\beta_{i}^{(b)}+\frac{\sigma^{2}}{\tau_{\textsc{ra}}}\right]\left[\sum_{j\in\mathcal{K}}\rho_{j}\beta_{j}^{(b)}+\sigma^{2}\right]}.$
(16)

In the cases that SA $b$ is exclusively visible by UE $k$ among its contenders
in $\mathcal{S}_{t}$, as described in the previous subsection, the SINR
performance in (16) can be simplified to

$\gamma^{{\rm
ul}}_{b,k}=\frac{M_{b}\,\rho^{2}_{k}\,(\beta_{k}^{(b)})^{2}}{\left[\rho_{k}\beta_{k}^{(b)}+\frac{\sigma^{2}}{\tau_{\textsc{ra}}}\right]\left[\sum_{j\in\mathcal{K}}\rho_{j}\beta_{j}^{(b)}+\sigma^{2}\right]}.$
(17)

###  IV-C SINR of the DL Information in Step 2

In order to derive the SINR of the DL received signal at the UE in step 2, as
described in (12), we need first to define the subset $\mathcal{I}_{k}^{(b)}$.
This subset contains the indices of the UEs that result in a directed DL
interference from SA $b$ to the UE $k$, _i.e._ , if
$k^{\prime}\in\mathcal{I}^{(b)}_{k}$, the DL signal from SA $b$ is intended to
UE $k^{\prime}$, but it is also partially beamformed to UE $k$. Although rare,
this might occur if the following four conditions simultaneously hold: i) UEs
$k$ and $k^{\prime}$ chose the same RA pilot in step 1: $r(k)=r(k^{\prime})$;
ii) SA $b$ is visible by UE $k$: $b\in\mathcal{V}_{k}$; iii) SA $b$ is visible
by UE $k^{\prime}$: $b\in\mathcal{V}_{k}^{\prime}$; and iv) the BS acquired
reliable channel estimates for UE $k^{\prime}$ from SA $b$: $[{\bf
E}]_{b,k^{\prime}}=1$. This last condition refers to the cases in which SA $b$
is not exclusively visible by UE $k^{\prime}$ among the UEs in the subset
$\mathcal{S}_{r(k^{\prime})}$, but it is also visible by UE $k$. However,
since $\beta^{(b)}_{k^{\prime}}\gg\beta^{(b)}_{k}$, the UL information from UE
$k^{\prime}$ in step 1 could be successfully decoded from SA $b$. Furthermore,
note that condition iv implies in condition iii, since an SA can only obtain
reliable channel estimates for a UE if this UE is visible by that SA.

We also define the constants $\alpha_{t}^{(b)}$ assuming that the channel
hardening condition holds:

$\left(\alpha_{t}^{(b)}\right)^{2}=\frac{||{\bf
y}_{t}^{(b)}||^{2}}{{M_{b}}}\xrightarrow{M_{b}\rightarrow\infty}\sum_{i\in\mathcal{S}_{t}}\rho_{i}\tau_{\textsc{ra}}\beta_{i}^{(b)}+\sigma^{2}.$
(18)

Then, following a similar analysis conducted in Subsection IV-B, the SINR of
the DL signal received at the UE $k$ in step 2 can be obtained as in (19). One
can note that the first term in the denominator refers to the directed DL
interference due to the UEs in $\mathcal{I}_{k}^{(b)}$. Besides, the second
term refers to conventional DL interference due to the other UEs in
$\mathcal{C}$, and the third term refers to the noise power.

$\gamma^{{\rm dl}}_{k}=\frac{M_{b}\sum_{b\in\mathcal{V}_{k}}[{\bf
E}]_{b,k}\frac{q\,\rho_{k}\,\tau_{\textsc{ra}}}{B\,\delta_{b}}\frac{(\beta_{k}^{(b)})^{2}}{(\alpha_{r(k)}^{(b)})^{2}}}{M_{b}\sum_{b\in\mathcal{V}_{k}}\sum_{k^{\prime}\in\mathcal{I}^{(b)}_{k}}\frac{q\,\rho_{k}\,\tau_{\textsc{ra}}}{B\,\delta_{b}}\frac{(\beta_{k}^{(b)})^{2}}{(\alpha_{r(k^{\prime})}^{(b)})^{2}}+\sum_{b\in\mathcal{V}_{k}}\sum_{k^{\prime}\in\mathcal{C}}[{\bf
E}]_{b,k^{\prime}}\frac{q}{B\,\delta_{b}}\frac{\beta_{k}^{(b)}}{\alpha_{r(k^{\prime})}^{(b)}}+\sigma^{2}}.$
(19)

##  V Numerical results

We assume a circular cell as depicted in Fig. 1, with $r_{e}=200$m and
$r_{i}=20$m. Each UE wants to become active with probability $P_{a}=0.01$,
randomly choosing one RA pilot out of $\tau_{\textsc{ra}}=10$ pilots
available. The BS is a ULA located in the center of the cell where $M=400$
antennas are uniformly distributed along a $40$ m length array, and divided
into $B$ SAs, each visible by a UE with probability $P_{b}=0.5$. The employed
XL-MIMO scenario is very similar to that assumed in [14].

Simulations are generated in sequential RA blocks, which are represented by
$10^{4}$ Monte Carlo realizations. In each round, new iUEs realize an access
attempt following (5). UEs that failed at the first attempt retransmit their
RA pilots with probability $0.5$ in the next blocks. It is considered a failed
access attempt if a UE do not succeed in a total of $10$ RA attempts, starting
from its first transmission.

###  V-A Random Access Performance

We compare in Fig. 4 the average number of access attempts and the probability
of failed access attempts as a function of the number of iUEs, considering our
proposed NOVR-XL RA protocol, SUCRe-XL [14], and mSUCRe-XL as discussed in
Subsection III-A. Note that SUCRe-XL and mSUCRe-XL achieve the same RA
performance, differing only in terms of sum-rate. When obtaining these results
we assumed that there is always an available PDP to be allocated for a UE that
succeeded in the RA stage. Besides, for the proposed NOVR-XL RA protocol we
assumed that the UL RA additional information vector of UE $k$ can be
successfully decoded from SA $b$ if the UL SINR of $\widehat{\textbf{d}^{\rm
ul}_{k}}$ in (10), as described in (16), is greater than a threshold of $0$
dB. In this case, the BS assumes that reliable channel estimates are available
for this UE in this SA, evaluating $\widehat{\bf h}^{(b)}_{k}=\widehat{\bf
y}^{(b)}_{r(k)}$ and $[\textbf{E}]_{b,k}=1$. Furthermore, the UE $k$ succeeds
in its RA attempt if the received DL message from step 2,
$\widehat{\textbf{d}^{\rm dl}_{k^{\prime}}}$ in (12), can be successfully
decoded, as illustrated in Fig. 2, assuming the same decoding threshold as in
the uplink. We have chosen a threshold of $0$ dB for successfully decoding the
messages since this is a reasonable value for decoding uplink control
information (UCI) in the physical uplink control channel (PUCCH) for most
allowed transmit formats in 5G [18].

Figure 4: Average number of access attempts and probability of failed access
attempts as a function of the number of inactive UEs $K$, for $P_{b}=0.5$,
$P_{a}=0.01$, $\tau_{\textsc{ra}}=10$ and $M=400$.

In [14], authors assumed that the RA attempt is successful for SUCRe-XL if
only one UE retransmits its chosen pilot in the step 3 of that protocol, or if
the UEs retransmitting the same pilot have non-overlapping VRs. Herein, for a
fair comparison, we have assumed that the UL signal received at the BS in step
3 of SUCRe-XL protocol is successfully decoded in SA $b$ if its SINR is above
a threshold of $0$ dB. As a result, the channel estimates processed in such SA
for this UE are used for precoding its DL signal in step 4. Finally, the UE
has its RA attempt successful if the SINR of the DL signal received in step 4
is above a threshold of $0$ dB. In order to obtain the SINR of the UL signal
received at the SA $b$ in step 3 of SUCRe-XL protocol, and the SINR of the DL
signal received at the UE $k$ in step 4, we need to first define the set
$\mathcal{R}_{t}$, composed by the indices of the UEs retransmitting the RA
pilot $t$ in step 3 of that protocol. Besides, we define the set
$\mathcal{R}$, composed by the indices of all UEs retransmitting any RA pilot
in step 3, _i.e._ ,
$\mathcal{R}=\mathcal{R}_{1}\cup\mathcal{R}_{2}\cup\ldots\mathcal{R}_{\tau_{\textsc{ra}}}$.
Then, the SINR of the UL signal received at the SA $b$ in step 3 of SUCRe-XL
protocol can be obtained by (16), but replacing $\mathcal{S}_{t}$ by
$\mathcal{R}_{t}$, and $\mathcal{K}$ by $\mathcal{R}$. Finally, the subsets
$\mathcal{C}$ and $\mathcal{I}_{k}$ can be obtained for this protocol in the
same way as described for NOVR-XL, and (19) is readily applicable for
obtaining the SINR of the DL signal received at the UE $k$ in step 4.

As one can see from Fig. 4, the proposed NOVR-XL protocol achieves significant
improved RA performance in comparison with SUCRe-based protocols. For
instance, considering $K=2000$ iUEs, and $B=10$ SAs, the average number of
access attempts decreases from $\bar{\mu}_{\textsc{ra}}=8.064$ attempts for
SUCRe-XL to $\bar{\mu}_{\textsc{ra}}=6.363$ attempts for NOVR-XL. In the same
way, the probability of failed access attempts decreases from $0.7393$ to
$0.5505$, representing a significant performance improvement of the NOVR-XL
protocol in (over)crowded mMTC scenarios.

Remark 1: It is noteworthy that the result discussed above is in accordance
with the Markov’s inequality [19]. Defining the random variable
$\mu_{\textsc{ra}}^{k}$ as the number of RA attempts the UE $k$ performed, the
Markov bound can be readily applied as:

${\rm Pr}(\mu_{\textsc{ra}}^{k}\geq
10)\leq\frac{\mathbb{E}[\mu_{\textsc{ra}}^{k}]}{10}.$ (20)

In (20), one can recognize ${\rm Pr}(\mu_{\textsc{ra}}^{k}\geq 10)$ as the
probability of failed access attempt, and $\mathbb{E}[\mu_{\textsc{ra}}^{k}]$
as the average number of access attempts.

###  V-B Sum-Rate

In order to evaluate the sum-rate performance of the investigated protocols,
we need first to obtain appropriate values for $\varphi_{\textsc{ra}}$, _i.e._
, the number of channel uses spent per RA attempt of each RA protocol. In the
SUCRe-XL protocol [14], the four steps present a very similar structure to
that of medium access control in 5G, as described in [20]. Therefore, we can
use this document as reference. In SUCRe-XL, the UE transmits an RA pilot in
step 1, spending $\tau_{\textsc{ra}}$ channel uses. The BS responds in step 2
with the precoded RA response, composed by the DL RA pilot, the RA resource
network temporary identifier (RA-RNTI), and a timing advance (TA) information.
While the DL RA pilot transmission spends $\tau_{\textsc{ra}}$ channel uses,
the RA-RNTI and TA are of $16$ and $8$ bits length [20], respectively. For
simplicity, we assume in our analysis that the modulation and coding scheme
(MCS) employed is chosen resulting in a unitary MCS spectral efficiency333This
means that the coding rate multiplied by the number of bits per modulated
symbol is equal to $1$.. Therefore, the number of channel uses in step 2 is
${24+}\tau_{\textsc{ra}}$. The UE transmits in step 3 the repeated RA pilot
($\tau_{\textsc{ra}}$ channel uses) and the cell radio network temporary
identifier (C-RNTI), of $16$ bits [20], such that the number of channel uses
in this step is ${16+}\tau_{\textsc{ra}}$. Finally, in step 4, the BS responds
with the C-RNTI and the contention resolution information (CRI), of 48 bits
[20], such that the number of channel uses in this step is $64$. In summary,
for SUCRe-XL, $\varphi^{\rm SUCRe-XL}_{\textsc{ra}}=104+3\tau_{\textsc{ra}}$.

For the mSUCRe-XL RA protocol discussed in Section III-A, the signals
transmitted in the four steps of the RA stage are very similar to that of
SUCRe-XL, except that $\widehat{\bf v}_{k}$ is transmitted in step 3. Such
estimate is a binary vector of length $B$, and thus we assume its transmission
requires $B$ channel uses. Therefore, we have that $\varphi^{\rm mSUCRe-
XL}_{\textsc{ra}}=104+B+3\tau_{\textsc{ra}}$.

Finally, for the proposed NOVR-XL protocol, in the RA stage we assume the UE
transmits in step 1 its chosen RA pilot, an identifier information (ID) with
$16$ bits, and $\widehat{\bf v}_{k}$, spending $16+B+\tau_{\textsc{ra}}$
channel uses at this step. Then, the BS responds in step 2 with the identifier
information, followed by TA and CRI, spending at this step $72$ channel uses.
Therefore, we have that $\varphi^{\rm NOVR-
XL}_{\textsc{ra}}=88+B+\tau_{\textsc{ra}}$. Table II summarizes the number of
channel uses per RA attempt for each protocol under consideration.

TABLE II: Channel usage per attempt for the three RA protocols. Step | SUCRe-XL | mSUCRe-XL | NOVR-XL
---|---|---|---
Step 1 |  $\tau_{\textsc{ra}}$ (pilot) |  $\tau_{\textsc{ra}}$ (pilot) |  $\tau_{\textsc{ra}}$ (pilot) +
|  |  |  $B$ ($\widehat{{\bf v}}_{k}$) +
|  |  |  $16$ (ID)
Step 2 |  $\tau_{\textsc{ra}}$ (pilot) + |  $\tau_{\textsc{ra}}$ (pilot) + |  $48$ (CRI) +
|  $16$ (RA-RNTI) + |  $16$ (RA-RNTI) + |  $16$ (ID) +
|  $8$ (TA) |  $8$ (TA) |  $8$ (TA)
Step 3 |  $\tau_{\textsc{ra}}$ (pilot) + |  $\tau_{\textsc{ra}}$ (pilot) + |
|  $16$ (C-RNTI) |  $16$ (C-RNTI) + | $--$
|  |  $B$ ($\widehat{{\bf v}}_{k}$) |
Step 4 |  $48$ (CRI) + |  $48$ (CRI) + | $--$
|  $16$ (C-RNTI) |  $16$ (C-RNTI) |
Total | $104+3\tau_{\textsc{ra}}$ | $104+B+3\tau_{\textsc{ra}}$ | $88+B+\tau_{\textsc{ra}}$

Given the $\varphi_{\textsc{ra}}$ values for the three investigated RA
procedures, Fig. 5 depicts the sum-rate performance obtained with each scheme,
for $B=10$ SAs, $t_{c}=1$ ms, ${\textsc{w}}=20$ MHz, $T=200$ channel uses, and
$\mu_{\textsc{pd}}^{k}=10$ intervals, $\forall k\in\mathcal{A}$. As one can
see, the proposed NOVR-XL protocol achieves the best sum-rate performance,
with substantial improvements in comparison with mSUCRe-XL and SUCRe-XL. The
Figure also shows that mSUCRe-XL outperforms SUCRe-XL, indicating that the
application of our proposed scheduling procedure described in Algorithm 1
compensates the overhead required by the transmission of $\widehat{\bf
v}_{k}$. In terms of sum-rate, this occurs because more UEs can be scheduled
in the same PDP sequence, requiring a smaller set of PDP sequences, and little
training overhead since we assume $\tau_{\textsc{PD}}=|\mathcal{P}|$ in our
system model, as described in Section II.

Figure 5: Average sum-rate as a function of the number of inactive UEs $K$,
for $B=10$ SAs, $P_{b}=0.5$, $P_{a}=0.01$, $\tau_{\textsc{ra}}=10$ and $M=400$
antennas.

Fig. 6 analyses the effectiveness of each protocol in admitting UEs to the
network, as well as efficiently scheduling PDP resources to them. The Figure
depicts the average number of active UEs, the average number of allocated PDPs
($|\mathcal{P}|=\tau_{\textsc{PD}}$), and the average number of active UEs
scheduled in the same PDP resource. One can see that the proposed NOVR-XL
protocol is able to manage a substantially higher number of active UEs in the
network, due to the better connectivity performance in the RA stage, as
presented in Fig. 4. Furthermore, the Figure reveals that the proposed
scheduling procedure for non-stationary XL-MIMO systems in Algorithm 1 is very
effective in terms of simultaneously allocating the same PDP resource to
different UEs, provided that they present non-overlapping VRs. In the case of
NOVR-XL protocol, on average up to $1.274$ UEs can be scheduled in the same
PDP resource. The mSUCRe-XL RA protocol also achieves a good number of active
UEs connected in the same PDP sequence for low/moderate number of iUEs, but
its limited RA performance in (over)crowded scenarios reduces the number of
active UEs in such cases.

Figure 6: Average number of a) active UEs, b) allocated PDPs, c) active UEs
per allocated PDP, as a function of the number of inactive UEs $K$, and
assuming $B=10$, $P_{b}=0.5$, $P_{a}=0.01$, $\tau_{\textsc{ra}}=10$, and
$M=400$.

##  VI Final Remarks

XL-MIMO systems are a new topology which promises appreciable performance
improvements for 5G and 6G systems. However, many difficulties have to be
overcome in order to make this technology applicable in practice. This paper
addresses the problems of RA and resource scheduling, proposing a joint RA and
scheduling protocol which takes advantage of the random VRs of the UEs for
improving both performance metrics. Our numerical results show that the
connectivity performance is substantially improved, reducing the average
number of access attempts and the probability of failed access attempts. Then,
a detailed analysis of each RA protocol indicated the number of channel uses
spent per RA attempt. This has allowed a sum-rate performance analysis, able
to measure in a unified way the overhead of each protocol, its connectivity
performance, and its capability of managing a large number of active UEs,
requiring for this the smallest possible set of PDP sequences. Again, the
proposed NOVR-XL protocol presented very improved performance results, showing
that an increased number of active UEs can be supported at the same time,
hence, demanding a smaller set of PDPs, taking advantage of the non-
overlapping VRs of the UEs. Therefore, our proposed protocol is able to reduce
access latency in the same time it improves the XL-MIMO system sum-rate.

## References

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# Orbital-ordered ferromagnetic insulating state in tensile-strained SrCoO3
thin films

Sheng-Chieh Huang  Department of Physics, National Central University, Taoyuan
City 320317, Taiwan  Kanchan Sarkar  Universität Ulm, Institut für
Theoretische Chemie, Ulm 89081, Germany  Department of Applied Physics and
Applied Mathematics, Columbia University, New York, New York 10027, USA
Renata M. Wentzcovitch  Department of Applied Physics and Applied Mathematics,
Columbia University, New York, New York 10027, USA  Department of Earth and
Environmental Sciences, Columbia University, New York, New York 10027, USA
Lamont–Doherty Earth Observatory, Columbia University, Palisades, New York
10964, USA  Han Hsu<EMAIL_ADDRESS>Department of Physics, National Central
University, Taoyuan City 320317, Taiwan

###### Abstract

At ambient pressure, bulk SrCoO3 is a ferromagnetic (FM) metal in cubic
perovskite structure. By contrast, magnetic properties of epitaxial SrCoO3
thin films, especially at high tensile strain ($\varepsilon\gtrsim 3$%),
remain unclear: Previous calculations had predicted antiferromagnetic (AFM)
states more energetically favorable in this regime, but recent experiments
indicated a FM insulating state. In this work, using first-principles
calculations, we perform an extensive search for the structural, spin,
magnetic, and orbital states of SrCoO3 thin films. Our calculations indicate
that at $0<\varepsilon\lesssim 2.5$%, SrCoO3 favors a FM half-metallic state
with intermediate-spin ($t_{2g}^{5}e_{g}^{1}$-like) Co exhibiting
$d^{6}\underline{L}$ character. At $\varepsilon\gtrsim 2.5$%, a FM insulating
state with high-spin ($t_{2g}^{4}e_{g}^{2}$-like) Co dominates. This FM
insulating state is achieved via complicated orbital ordering, cooperative
Jahn–Teller distortion, and octahedral tilting about all three crystal axes.

Perovskite oxides exhibit various properties of scientific and technological
importance, including ferroelectricity, (anti)ferromagnetism, colossal
magnetoresistance, and multiferroics. These properties, arising from the
lattice, charge, spin, and orbital degrees of freedom in perovskite oxides,
can be further engineered via epitaxial strains, offering this class of
materials broad functionalities (see Refs. 1, 2, 3, 4 for reviews). Among
diverse perovskite oxides, cobaltites are a unique family due to their tunable
magnetic properties. For example, bulk LaCoO3 is a nonmagnetic insulator with
low-spin (LS, $t_{2g}^{6}e_{g}^{0}$) Co3+ at low temperature ($T\lesssim 30$
K). Starting $\sim$30 K, bulk LaCoO3 undergoes a thermally induced spin
transition, becoming a paramagnetic insulator at $\sim$90 K. The detailed
mechanism of this spin transition has been debated for decades but still
remains unclear, as reviewed in Refs. 5, 6, 7, 8. By contrast, tensile-
strained LaCoO3 thin film is a ferromagnetic (FM) insulator at $T\lesssim 85$
K [9, 10, 11, 12]. To explain this strain-induced FM insulating state, a
mixture of high-spin (HS, $t_{2g}^{4}e_{g}^{2}$) and LS Co3+ was proposed
[13], and this model is supported by various calculations and experiments [14,
15, 16, 17, 18].

In recent years, SrCoO3-δ ($0\leq\delta\leq 0.5$) thin films have also
attracted significant attention due to their potential applications in
catalysis and fuel cells and due to their composition-dependent chemical,
structural, and magnetic properties [19, 20, 21, 22, 23, 24, 25, 26, 27, 28,
30, 29, 31, 32]. At ambient condition, bulk SrCoO3 is a FM metal in cubic
perovskite structure, and brownmillerite SrCoO2.5 is an antiferromagnetic
(AFM) insulator in orthorhombic structure [33]. Theory has shown that bulk
SrCoO3 contains intermediate-spin (IS, $t_{2g}^{5}e_{g}^{1}$-like) Co
exhibiting $d^{6}\underline{L}$ character (nearly Co3+ accompanied by O $2p$
holes) [34, 35, 36, 37, 38, 39], but magnetic properties of tensile-strained
stoichiometric SrCoO3 thin films remain controversial. Previous first-
principles calculations had predicted AFM ordering energetically favorable at
tensile strain $\varepsilon\gtrsim 2$% [19, 20]. Experiments, however,
indicated that SrCoO3 thin films remain FM metallic at $\varepsilon\approx 2$%
(grown on SrTiO3 substrates) [26, 27, 28, 31, 32]. At $\varepsilon\approx 3$%
(grown on DyScO3 substrates), while AFM ordering had been reported [30], other
experiments indicated that the AFM ordering is induced by oxygen deficiency
[31]. Furthermore, a recent work with high-quality stoichiometric SrCoO3 thin
films reported a FM insulating state at $\varepsilon\approx 3$% [32], in
direct contradiction with theoretical predictions. In light of the HS+LS
mixture in LaCoO3 thin films, FM insulating state in SrCoO3 thin films may
also arise from spin transition and possible spin or orbital orderings. These
factors, however, have long been ignored. An extensive search (via
calculation) for the structural, magnetic, spin, and orbital states of SrCoO3
thin films may thus clarify the above-mentioned controversy and further
elucidate the magnetic properties and potential functionalities of SrCoO3 thin
films.

In this work, all calculations are performed using the Quantum ESPRESSO (QE)
codes [40]. Projected augmented wave (PAW) datasets are generated using the
EPAW code [41, 42], as detailed in Sec. S1 of Supplemental Material (SM) [43].
The generalized-gradient approximation + Hubbard $U$ (GGA+$U$) method is
adopted, with PBE-type GGA [44]. Epitaxial thin films are modeled with
strained bulks: Constraining the in-plane lattice vectors while relaxing the
out-of-plan lattice vector and atomic positions. To balance the efficiency and
accuracy of our extensive search, a reasonably chosen $U$-parameter is
essential. Using the HP code (implemented in QE) based on linear response
theory [45, 46, 47], we compute the self-consistent $U$ and lattice constant
($a_{0}^{\text{GGA}+U}$) of bulk SrCoO3. The obtained $U=6.5$ eV is adopted
throughout this work, and $a_{0}^{\text{GGA}+U}=3.849$ Å is in good agreement
with experiments ($a_{0}^{\text{exp}}=3.829$ Å [33]).

As mentioned above, bulk SrCoO3 contains IS $d^{6}\underline{L}$ Co with two
minority-spin electrons partially occupying the degenerate $t_{2g}$ orbitals
[Fig. 1(a)]. With tensile strain, HS ($t_{2g}^{4}e_{g}^{2}$-like)
$d^{6}\underline{L}$ Co should also be considered. In distorted crystal
fields, minority-spin electrons of IS and HS Co occupy two (e.g. $d_{xz}$ and
$d_{yz}$) and one (e.g. $d_{xy}$) of the $t_{2g}$ orbitals, respectively, as
schematically shown in Fig. 1(b). For transition-metal oxides, metal–insulator
transition often coincides with charge, spin, and/or orbital orderings; these
orderings should thus be considered for the possible FM insulating state in
SrCoO3 thin films. Nevertheless, interplay between the crystal structure, spin
state, and magnetic/spin/orbital orderings are highly complicated. To better
control these variables in our calculation, we begin with reduced degrees of
freedom: Optimizing the Co-O bond lengths while keeping the CoO6 octahedra
untilted (Glazer notation $a^{0}a^{0}c^{0}$). To obtain spin or orbital
orderings, cooperative Jahn–Teller (JT) distortions are manually imposed,
followed by structural optimization. Within $a^{0}a^{0}c^{0}$, both IS and HS
Co can be stabilized; various spin and orbital states are obtained, in FM and
AFM ($A$, $C$, and $G$ types) orderings, as described below.

In Figs. 1(c)–1(j), spin densities $s(\mathbb{r})\equiv
n_{\uparrow}(\mathbb{r})-n_{\downarrow}(\mathbb{r})$ of the obtained states
are plotted [$n_{\uparrow/\downarrow}(\mathbb{r})$: spin-up/down electron
density]. For IS Co, FM ordering is favored; a metallic (IS-m) state [Fig.
1(c)] and a more energetically favorable half-metallic (IS-hm) state [Fig.
1(d)] are obtained (see Fig. 2 for their energies). For both states, spin
densities at the Co sites exhibit $e_{g}+d_{xy}$ character, indicating two
spin-down electrons occupying the $d_{xz}$ and $d_{yz}$ orbitals. The main
difference between these two states is that IS-hm has more prominent local
magnetic moments at the O sites, resulting from O $2p$ holes. For HS Co,
without orbital ordering, FM ordering [Fig. 1(e)] is not favored; $A$-type AFM
($A$-AFM) [Fig. 1(f)] is (see also Fig. 2). For both magnetic orderings, spin
densities at the Co sites exhibit prominent dimples with $d_{xy}$ character,
indicating one minority-spin electron occupying the $d_{xy}$ orbital. This
type of HS Co is thus referred to as HS-$d_{xy}$ Co. (Likewise, the minority-
spin electron of HS-$d_{xz}$/$d_{yz}$ Co occupies the $d_{xz}$/$d_{yz}$
orbital). In addition to pure IS and HS states, one spin-ordered and three
orbital-ordered states are obtained, as shown in Figs. 1(g)–1(j),
respectively: A mixture of IS+HS-$d_{xy}$ Co in rock-salt ordering (IH-rs), a
mixture of HS-$d_{xz}$+HS-$d_{yz}$ Co in rock-salt (HS-rs) and columnar
orderings (HS-c), and a mixture of all three types of HS Co in a more
complicated ordering (HS-o3): HS-$d_{xy}+$HS-$d_{yz}$ checkerboards inter-
layered with HS-$d_{xy}+$HS-$d_{xz}$ checkerboards. To better visualize the
spin/orbital orderings of these states, their spin-down $3d$ electrons are
schematically plotted in Figs. 1(k)–1(n), respectively. For these four states,
FM ordering is favored (see Fig. 2 and Table SIV in SM [43]).

In Fig. 2, total energies of the obtained states at various strains (or
equivalently, in-plane pseudocubic lattice parameter $a_{pc}$) are plotted,
where symbols indicate spin/orbital states, line formats indicate magnetic
orderings, and line colors indicate structures. The equilibrium energy and
lattice constant of cubic perovskite SrCoO3 are used as the reference for
energy ($0$ eV) and epitaxial strain $\varepsilon\equiv
a_{pc}/a_{0}^{\text{GGA}+U}-1$, respectively. By analyzing the results of the
untilted ($a^{0}a^{0}c^{0}$) structures [Fig. 2(a)], effects of Co spin state
and magnetic/orbital orderings can be deduced. Overall, IS Co favors FM
ordering, and the IS-hm state (empty diamond, solid line) is the most
energetically favorable state, with energy minimum $E_{min}^{\text{IS-
hm}}=-350$ meV/f.u. at $a_{pc}=3.880$ Å. Energy of the IS-m state (filled
diamond, solid line) is higher than IS-hm at $0<\varepsilon\leq 4$% and is
even higher in $A$-AFM (filled diamond, dotted line), $C$-AFM (filled diamond,
dashed line), and $G$-AFM orderings (beyond the graph range). By contrast, for
HS Co without orbital ordering, $A$-AFM ordering (square, dotted line) is more
favorable ($E_{min}^{\text{HS (A-AFM)}}=-261$ meV/f.u. at $a_{pc}=3.920$ Å)
than FM (square, solid line). Nevertheless, HS $A$-AFM is still not
energetically competitive at $\varepsilon>1$%; the spin/orbital-ordered states
(IH-rs, HS-rs, HS-c, and HS-o3) are, as highlighted by the shaded area. All
these spin/orbital-ordered states favor FM ordering. When in AFM orderings,
their energies increase by $25$–$215$ meV/f.u. (see Table SIV in SM [43] for
$E_{min}$’s of all the obtained states in all magnetic orderings). For
clarity, we only plot the FM results in Fig. 2(a).

To search for the most energetically favorable structure for each spin/orbital
state, octahedral tilting must be considered. Given that SrCoO3 thin films are
often grown on substrates with equivalent in-plane crystal axes, we require
the same type of octahedral tilts about the in-plane crystal axes: both in-
phase (+), both out-of-phase (–), or both untilted (0). Combined with the
tilting about the out-of-plane axis, nine types of tilts (including
$a^{0}a^{0}c^{0}$) are considered, as tabulated in Table I, along with the
associated space groups. (Note: Based on group-theoretical analysis, some of
the tabulated space groups are associated with less restricted tilts,
including: $Immm$, $Pnnn$, and $C2/m$ associated with $a^{+}b^{+}c^{+}$
instead of $a^{+}a^{+}c^{+}$, $C2/c$ associated with $a^{-}a^{-}c^{-}$ instead
of $a^{-}a^{-}c^{0}$, and $P\bar{1}$ associated with $a^{-}b^{-}c^{-}$ instead
of $a^{-}a^{-}c^{-}$ [48, 49, 50]. To our best knowledge, no group-theoretical
analysis is available for the structures of HS-o3.) For each spin/orbital
state, octahedral tilts listed in Table I are manually imposed, followed by
structural optimization. Energies of the most favorable structures for FM and
AFM orderings are plotted in Figs. 2(b) and 2(c), respectively. Overall, even
with octahedral tilting, AFM orderings are still unfavorable, despite that HS
$A$-AFM becomes quite competitive in the $I4/mcm$ structure [cyan square, Fig.
2(c)]. Effects of octahedral tilting on magnetic ordering are basically
insignificant. In Fig. 2(b), some states in the untilted structures are
included for references (black symbols). With octahedral tilting, the IS-hm
($Imma$) and HS-o3 ($P2_{1}/c$) states, indicated by red diamond and green
plus, respectively, are the most energetically favorable state at
$\varepsilon\lesssim 2.5$% and $\varepsilon\gtrsim 2.5$%, respectively. Their
energies are lower than their untilted counterparts in $P4/mmm$ (black
diamond) and $P4_{2}/mnm$ symmetries (black plus) by $\sim$$20$ and $\sim$$50$
meV/f.u., respectively.

In Figs. 3 and 4, density of states (DOS), band structure, and spin density of
the IS-hm ($Imma$) and HS-o3 ($P2_{1}/c$) states at $a_{pc}=3.963$ Å
($\varepsilon=3$%) are plotted. Expectedly, their spin densities [Figs. 3(c)
and 4(d)] show great resemblance to their $a^{0}a^{0}c^{0}$ counterparts
[Figs. 1(d) and 1(j)]. For the IS-hm ($Imma$) state, $d_{xz}$ and $d_{yz}$
orbitals are equivalent, and their DOS curves coincide [Fig. 3(b)]. Their
occupation numbers in the spin-down channel are both $0.998$, and the local
magnetic moment at the Co site $\mu_{\text{Co}}=2.078\mu_{B}$. Within the
$Imma$ structure, there are two inequivalent O sites, O1 and O2 [Fig. 3(c)],
with local magnetic moments $\mu_{\text{O1}}=-0.303\mu_{B}$ and
$\mu_{\text{O2}}=-0.504\mu_{B}$. For the HS-o3 ($P2_{1}/c$) state, HS-$d_{yz}$
and HS-$d_{xz}$ Co are crystallographically equivalent; there are thus only
two Co sites: Co1 (HS-$d_{xy}$) and Co2 (HS-$d_{yz}$/HS-$d_{xz}$). Here we
plot the projected DOS onto the $3d$ orbitals of HS-$d_{xy}$ and HS-$d_{yz}$
Co in Figs. 4(b) and 4(c) (see Fig. S1 in SM [43] for HS-$d_{xz}$ Co).
Clearly, they each have one spin-down electron, occupying the $d_{xy}$ and
$d_{yz}$ orbital, respectively; their local magnetic moments
$\mu_{\text{Co1}}=2.328\mu_{B}$ and $\mu_{\text{Co2}}=2.281\mu_{B}$. In the
$P2_{1}/c$ structure, there are three O sites [Fig. 4(d)]. Among them, O1 and
O3 have nearly the same DOS curves [Fig. 4(a)] and significant local magnetic
moments ($\mu_{\text{O1}}=0.337\mu_{B}$ and $\mu_{\text{O3}}=0.336\mu_{B}$),
while O2 is nearly nonmagnetic ($\mu_{\text{O2}}=-0.032\mu_{B}$). Evident from
Figs. 4(e) and 4(f), an energy gap of $0.188$ eV is opened. Remarkably, only
in the $P2_{1}/c$ structure is the HS-o3 state insulating; all other
structures are metallic (see Table SV and Fig. S2 in SM [43]). Furthermore,
the HS-rs and HS-c states are also metallic in all structures (see Tables SVI
and SVII in SM [43]). It can thus be concluded that the FM insulating state in
tensile-strained SrCoO3 is achieved via complicated orbital ordering,
cooperative JT distortion, and octahedral tilting about all three axes.
Despite its monoclinic structure, the out-of-plane lattice vector of the HS-o3
($P2_{1}/c$) state is almost perpendicular to the (001) plane, consistent with
the high-resolution transmission electron microscopy (HRTEM) image obtained in
experiments [32] (see Fig. S3 in SM [43]).

Comparing with previous calculations for SrCoO3 thin films (e.g. Refs. 19,
20), the major difference of this work includes a more rigorously determined
$U$-parameter, investigations for spin/orbital orderings, and a systematic
structure search for each spin/orbital state. In Ref. 19, a fairly small
$U=2.5$ and $J=1.0$ eV were adopted; the ground state of SrCoO3 was found to
be FM cubic perovskite, and AFM states with cation displacements ($Amm2$ and
$P4mm$ structures) were found favorable at $\varepsilon\gtrsim 2$%. With a
larger $U$, SrCoO3 favors distorted structures than cubic by up to $>300$
meV/f.u., as shown in Ref. 20 ($U=6.0$ eV) and this work, suggesting that
cubic perovskite may be a high-temperature phase of SrCoO3. Also reported in
Ref. 20 is that for $U=6.0$ eV, structures with cation displacements are
highly unfavorable (by hundreds of meV/f.u.). We therefore do not consider
cation displacements in this work. Remarkably, the ground state reported in
Ref. 20 is the same as our HS-c state in the $P4/mbm$ (untilted) structure,
indicated as black triangles in Fig. 2(b) (see Fig. S4 in SM [43] for a
detailed complarison). Nevertheless, spin/orbital orderings were not addressed
in these previous works.

In summary, we perform an extensive search for the structural, spin, magnetic,
and orbital states of tensile-strained SrCoO3 thin films via GGA+$U$
calculations. Our results indicate that at tensile strain $\varepsilon\lesssim
2.5$%, SrCoO3 favors an IS half-metallic ($Imma$) state with
$d^{6}\underline{L}$ character. At $\varepsilon\approx 2.5$%, a simultaneous
structural, spin, orbital, and metal–insulator transition occurs; a FM
insulating ($P2_{1}/c$) state with HS Co in a complicated orbital ordering
emerges. At $\varepsilon\gtrsim 2.5$%, SrCoO3 favors this FM insulating state,
consistent with recent experiments. The energy gap of this state is opened via
complicated orbital ordering, cooperative JT distortion, and octahedral
tilting about all three crystal axes.

Acknowledgments This work was primarily supported by the National Science and
Technology Council (formerly Ministry of Science and Technology) of Taiwan
under Grants No. MOST 107-2119-M-009-009-MY3, 111-2124-M-A49-009,
107-2112-M-008-022-MY3, 110-2112-M-008-033, and 111-2112-M-008-032. R.M.W.
acknowledges support from DOE grant DE-SC0019759 and NSF grant EAR-1819126. We
thank National Center for High-performance Computing (NCHC) of Taiwan for the
computation resources.

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Table 1: Imposed octahedral tilts and associated space groups (and lattice systems) for all spin/orbital states. T: tetragonal; O: orthorhombic; M: monoclinic; Tri: triclinic. Note: Some space groups are associated with less restricted tilts (see text). Imposed tilts |  IS-m/hm or HS | IH-rs |  HS-rs |  HS-c |  HS-o3
---|---|---|---|---|---
$a^{0}a^{0}c^{0}$ |  $P4/mmm$ (T) |  $I4/mmm$ (T) |  $I4/mcm$ (T) |  $P4/mbm$ (T) |  $P4_{2}/mnm$ (T)
$a^{0}a^{0}c^{+}$ |  $P4/mbm$ (T) |  $P4/mnc$ (T) |  $P4_{2}/mbc$ (T) |  $Pbam$ (O) |  $Pnnm$ (O)
$a^{0}a^{0}c^{-}$ |  $I4/mcm$ (T) |  $I4/m$ (T) |  $Ibam$ (O) |  $P4_{2}/mbc$ (T) |  $P4_{2}/m$ (T)
$a^{+}a^{+}c^{0}$ |  $I4/mmm$ (T) |  $P4_{2}/nnm$ (T) |  $P4/nnc$ (T) |  $I4/m$ (T) |  $P4_{2}/n$ (T)
$a^{+}a^{+}c^{+}$ |  $Immm$ (O) |  $Pnnn$ (O) |  $Pnnn$ (O) |  $C2/m$ (M) |  $P2/c$ (M)
$a^{+}a^{+}c^{-}$ |  $P4_{2}/nmc$ (T) |  $P4_{2}/n$ (T) |  $Ccca$ (O) |  $P4_{2}/n$ (T) |  $P4_{2}/n$ (T)
$a^{-}a^{-}c^{0}$ |  $Imma$ (O) |  $C2/m$ (M) |  $C2/c$ (M) |  $Pnma$ (O) |  $P2_{1}/c$ (M)
$a^{-}a^{-}c^{+}$ |  $Pnma$ (O) |  $P2_{1}/c$ (M) |  $P2_{1}/c$ (M) |  $Pnma$ (O) |  $P2_{1}/c$ (M)
$a^{-}a^{-}c^{-}$ |  $C2/c$ (M) |  $P\bar{1}$ (Tri) |  $C2/c$ (M) |  $P2_{1}/c$ (M) |  $P\bar{1}$ (Tri)

Figure 1: (a,b) Schematic diagrams for the orbital occupations of Co3+ in
cubic and distorted crystal fields; (c–j) spin densities of the obtained
spin/orbital states in the untilted ($a^{0}a^{0}c^{0}$) structures (see text
for the nomenclatures), with isosurface values $\pm 0.02$ a.u.-3
(yellow/cyan); (k–n) schematic diagrams for the minority-spin $3d$ electrons
of the spin/orbital states in panels (g)–(j), respectively.

Figure 2: Total energies of the obtained spin/orbital states at various
strains (or equivalently, $a_{pc}$) in (a) the untilted and (b,c) tilted
structures; for the latter, FM and AFM states are plotted in panels (b) and
(c), respectively. The equilibrium energy and lattice constant of bulk SrCoO3
are used as the reference for energy and strain (see text).

Figure 3: Electronic structure of the IS-hm ($Imma$) state at $a_{pc}=3.963$ Å
($\varepsilon=3$%). (a) Total (shade) and projected DOS onto each atomic
species (lines); (b) projected DOS onto the Co $3d$ orbitals; (c) spin
density; (d,e) band structure of the spin-up/down channel.

Figure 4: Electronic structure of the HS-o3 ($P2_{1}/c$) state at
$a_{pc}=3.963$ Å ($\varepsilon=3$%). (a) Total (shade) and projected DOS onto
each atomic species (lines); (b,c) projected DOS onto the $3d$ orbitals of Co1
and Co2 (see text); (d) spin density; (e,f) band structure of the spin-up/down
channel.