# QoE-Centric Multi-User mmWave Scheduling:
A Beam Alignment and Buffer Predictive Approach

Babak Badnava, Sravan Reddy Chintareddy, Morteza Hashemi  Department of
Electrical Engineering and Computer Science, University of Kansas

###### Abstract

In this paper, we consider the multi-user scheduling problem in millimeter
wave (mmWave) video streaming networks, which comprises a streaming server and
several users, each requesting a video stream with a different resolution. The
main objective is to optimize the long-term average quality of experience
(QoE) for all users. We tackle this problem by considering the physical layer
characteristics of the mmWave network, including the beam alignment overhead
due to pencil-beams. To develop an efficient scheduling policy, we leverage
the contextual multi-armed bandit (MAB) models to propose a beam alignment
overhead and buffer predictive streaming solution, dubbed B2P-Stream. The
proposed B2P-Stream algorithm optimally balances the trade-off between the
overhead and users’ buffer levels, and improves the QoE by reducing the beam
alignment overhead for users of higher resolutions. We also provide a
theoretical guarantee for our proposed method and prove that it guarantees a
sub-liner regret bound. Finally, we examine our proposed framework through
extensive simulations. We provide a detailed comparison of the B2P-Stream
against a uniformly random and Round-robin (RR) policies and show that it
outperforms both of them in providing a better QoE and fairness. We also
analyze the scalability and robustness of the B2P-Stream algorithm with
different network configurations.

###### Index Terms:

Quality of Experience, mmWave Networking, Multi-user Streaming and Scheduling

##  I Introduction

3GPP broadband wireless standards such as LTE-Advanced and fifth generation
(5G) technologies and IEEE wireless standards such as 802.11ad and 802.11ay
have enabled high data transfer and data-intensive applications, and are
moving towards all-connected small-cell networks. The annual data traffic
generated is expected to reach one Zettabyte by $2022$, and is estimated that
more than $75\%$ of the world’s mobile data traffic will correspond to mobile
video streaming [1, 2]. This deluge of data traffic, especially demands for
high resolution video streaming on portable mobile devices, will pose
significant challenges for the wireless and cellular network providers to meet
the quality of experience (QoE) requirements. In contrast to the quality of
service (QoS) that is usually quantified in terms of achieved rate and
latency, video streaming QoE depends on several factors such as the resolution
of video frames, playback buffer level for each user, number of re-buffering
events, and frequency of resolution switches.

In terms of required infrastructure, millimeter wave (mmWave) networks are
capable of providing multi-Gbps data rates, which makes them suitable to meet
the ever-increasing demand for video streaming applications [3, 4]. However,
unlike omni-directional communications in sub-6 GHz, high data rates in mmWave
systems come at the price of large coordination overhead due to highly
directional communications needed to compensate for large channel losses [5,
6, 7]. Although there are extensive works on providing more efficient beam
alignment111In this paper, beam alignment, collectively, refers to initial
beam search, beam tracking, beam refinement, and beam switching.solutions [8,
9, 10, 11], this process still consumes resources that otherwise could have
been utilized for high-bit-rate data transfer.

Figure 1: System model depicting mmWave capable base station (gNB) and an
Access Point (AP) serving K of N users simultaneously.

In mmWave networks, it is true that small wavelengths allow for large antenna
arrays to be packed into small chip areas. However, due to power consumption
and hardware complexity, the number of RF chains remains limited at the mmWave
transceivers, which constrains the number of users that can be served
concurrently. A base station, equipped with multiple RF chains ($K$), can
serve up to $K$ users at the same time. As such, multi-user management plays a
central role to guarantee low-latency and high QoE for all users [10, 11]. The
key point, however, is that due to beam alignment overhead, switching from one
mobile user to another one incurs a _switching cost_ , as denoted by $\tau(.)$
in Figure 1.

Assuming that the QoE is a function of the playback buffer level and
resolution of the video frames stored in the buffer, the system needs to
balance between users’ buffer levels vs. beam alignment overhead to optimize
the QoE across all the users in the network. For instance, in the extreme
situation, the system could serve only a fixed subset of $K$ users, with the
objective of reducing the beam alignment overhead. While this minimizes the
risk of zero playback buffer for those $K$ users, the other under-served users
would exhaust their playback buffers, which leads to significant QoE
degradation. On the other hand, quickly switching between users results in
significant beam alignment overheads that is a relatively very slow process
vis-à-vis data transfer.

In this paper, we consider the interplay between beam alignment overhead
(i.e., switching cost) and multi-user scheduling in order to enhance the QoE
across all users. On one hand, the optimal scheduling should take the
switching cost into account, and on the other hand, switching cost is a
function of the scheduling algorithm that determines the beam quality. This is
in contrast to the classical scheduling problems, where the switching overhead
is traditionally assumed to be negligible compared to the service time [12].
Indeed, the switching cost and overhead in the mmWave networks becomes
critical due to abundant mmWave capacity that would be wasted during the beam
alignment phase.

Within this context, and given that beam alignment overhead is a function of
the previous schedules, we develop a multi-user scheduling algorithm that
works based on predicting the _beam alignment overhead and buffer_. We refer
to this algorithm as B2P-Stream that is built upon the contextual multi-armed
bandit (MAB) models to optimally balance the trade-offs between buffer levels
and beam alignment overhead. To maximize the average QoE for all users, the
streaming server estimates the beam alignment overhead as well as the playback
buffer level at each user, and selects $K$ users out of $N$ users at each time
slot. There are several studies, such as [13, 14, 10, 15], that have
considered the scheduling task under different scenarios. However, our work
aims to integrate the unique characteristics of the mmWave communication
(i.e., beam alignment overhead) into a QoE-centric multi-user scheduling
framework. In summary, the main contributions of this paper are as follows:

  * •

We model the beam alignment overhead of individual users based on the last
time we served that particular user, and we propose a dynamic model for the
users’ buffer level prediction.

  * •

Given the playback buffer level, we model the QoE for each user and formulate
an optimization problem to improve the long-term average QoE for all the
users.

  * •

We develop a MAB-based scheduling policy, called B2P-Stream, which provides a
sub-linear regret bound, to solve the defined optimization problem. This
algorithm incorporates estimated buffer level of each user and schedule users
with the help of a heuristic trend function on the beam alignment overhead.

The rest of this paper is organized as follows. In Section II, we provide a
detailed review of the previous related works. In Section III, we present the
system model and formulate a dynamic model of users’ playback buffer level.
Section IV presents the B2P-Stream algorithm followed by its regret bound
analysis. We continue by providing an experimental evaluation of our proposed
method in Section V. Finally, Section VI concludes the paper.

##  II Related works

In this section, we review two classes of most related works on user
scheduling and QoE optimization.

Related Works on User Scheduling: Jiang et al. [16] proposed a multi-task deep
learning scheme for user scheduling and beamforming. They modeled both
beamforming and user scheduling as two classification tasks and used a
parameter sharing technique to train a multi-task deep neural network. Wu et
al. [17] proposed a two-stage scheduling scheme, based on the inter-user
channel correlation and the channel energy, to maximize the sum rate
achievable and minimize the overall computational complexity. Although this
method maximizes the overall achievable rate, it may provide a poor
performance in terms of fairness, since those users with low orthogonality are
vulnerable to starvation (e.g., zero buffer size, in our proposed streaming
model). For uplink scheduling scenarios, Pérez-Adán et al. in [18] improved
their previous work in [19] by developing a distributed quantizer linear
coding to cluster the users into different groups such that the number of
groups would be as close as to the number of RF chains. Then, they proposed
different scheduling schemes based on the users’ group. The authors in [10,
11] develop a policy such that under user mobility, the selected users remain
the same unless an abrupt change in the beam direction happens. While this
work is closet to our model, it does not consider the QoE, users’ buffer
level, and impact of frame resolutions for multi-user scheduling in video
streaming applications.

Xu et al. [20] proposed a multi-agent reinforcement learning framework for
user scheduling and beam selection that minimizes the long-term average
network delay while guaranteeing QoS requirements. While they have considered
the beam alignment task and QoS satisfaction, there could be discontinuity
between the QoS and QoE in multi-user video streaming scenarios. Wang et al.
[21] proposed a scheduling and resource allocation framework in which they use
Lyapunov optimization to maximize a utility function. Any utility measure that
is a non-decreasing concave function of transmission rate, such as
proportional-fairness and long-term sum-rate, can be utilized for the
scheduling. Similar to the previous works, this method does not consider the
QoE requirements, and it is specific to analog beamforming with a single RF
chain.

Related Works on QoE Optimization: On a different note, there is a multitude
of prior works for QoE monitoring, approximation, and prediction. Adarsh et
al. [14] proposes a predictive model of QoE based on low-cost QoS measurements
like reference signal received power (RSRP) and throughput; They claim that
their model accurately predicts re-buffering events and resolution switches
more than $80\%$ of the time. The authors in [22] obtain the QoS parameters,
such as packet loss, jitter, and delay using a mmWave NS-3 simulator and
devise a non-linear regression method to predict and monitor the QoE of a
video streaming service. In [23], a specific type of recurrent neural networks
called LSTM is used to design a QoE predictive model based on features such as
playback status, time elapsed since last re-buffering event, and a short term
subjective video quality measure. A different work in this category is
provided by Nightingale et al. [24]. They propose a predictive model of the
QoE for an ultra-high-definition (UHD) live video streaming application based
on congestion indicators of a 5G network. However, all these predictive models
needs to be incorporated into a decision-making scheduler to prevent the re-
buffering, resolution switches, and other undesirable events that degenerate
the QoE.

There exist many other works in the literature that approached the QoE
requirement satisfaction in different ways. For example, Li et al. [25]
proposed an optimal cache placement algorithm to maintain a high QoE in an
adaptive streaming application; Tuysuz and Aydin [26] designed a mobility-
aware collaborative video streaming client application in which a group of
users with different characteristic can stream a high quality video, while
maintaining the required QoE; and many other works that are less relevant to
our work [27, 28, 29].

Compared with the previous works, our QoE optimization framework at the
application layer (i.e., video streaming) takes into account the physical
layer characteristics of the mmWave networks and transceiver hardware
limitations expressed in terms of the number of RF chains. Indeed, signal
directionality is the most distinguished feature for mmWave systems, and thus
our multi-user scheduler considers the beam alignment overhead and its impacts
on QoE. Overall, our developed solution is tailored for mmWave systems that
will be one of the key enablers in the Next-G era.

##  III System Model and Problem Formulation

We consider a mmWave network that consists of $N$ users and a single base
station (BS) or access point (AP), referred to as the streaming server. At
each time slot, the streaming server selects $K$ users out of $N$ users to
serve simultaneously. The $K$ beams generated by the server are used to stream
different video frames to each of the $K$ users. Different resolution of each
video is available at the server side, and each user may ask for a different
resolution based on the channel condition.

###  III-A Beam Alignment Model and Assumptions

As shown in Figure 1, each time slot is divided into two phases: beam
alignment and data transmission. In this paper, we normalize the duration of
each time slot to be equal to 1 unit of time. Thus, given that beam alignment
takes $\tau(.)$ units, $1-\tau(.)$ is the amount of time left for the data
transfer phase. In the extreme, this overhead can occupy up to $45\%$ of the
time slot duration in the cellular networks [30]. However, in a more general
sense, the function $\tau(.)$ can be expressed in terms of the time interval
between two consecutive schedules of a user, i.e., the beam alignment overhead
for a user at a specific time depends on how long ago that user was served by
the server. This model captures the “freshness” of the beam for the user. We
can model this characteristic using a _non-decreasing function_ of the last
time a user has been served. Thus, the beam alignment overhead of user $i$ at
time $t$ is denoted by $\tau^{i}(x_{t}^{i})$, where $x_{t}^{i}$ is the amount
of time that has been passed since the last schedule of user $i$ at time $t$.
For instance, Figure 2 demonstrates such a function for $\tau$. The value of
the function $\tau^{i}(.)$ increases as the $x_{t}^{i}$ increases. A full beam
search, which consumes $45\%$ of the time slot, is required if it has been
passed more than 100 time steps since the last time the user has been served,
which is an assumption that can be determined based on the system requirements
as suggested in [10].

Figure 2: Beam alignment overhead $\tau(.)$ as a function of the time interval
between two consecutive schedules.

###  III-B Playback Buffer Dynamics

Each user $i$ has a finite playback buffer of size $s^{i}$ bytes to store
video frames. Considering the resolution of the video $v_{res}^{i}$ that the
user is playing and its bit rate $v_{rate}^{i}$, the user has at most
$s^{i}/v_{rate}^{i}$ seconds of video to play. We denote $b_{t}^{i}$ as the
buffer level of user $i$ at time $t$ in seconds. When the proper beam has been
created, the server can start streaming to the user at a specific rate
$\mathit{R}$. As such, the amount of data transferred to the user at a time
$t$ is obtained as follows:

$\displaystyle d_{t}^{i}=(1-\tau_{t}^{i}(x_{t}^{i}))\mathit{R}/v_{rate}^{i},$
(1)

where $\tau_{t}^{i}(x_{t}^{i})$ is the beam alignment overhead, and
$d_{t}^{i}$ determines the amount of data, in seconds, that the server sends
to the user. Due to a blockage and other environmental issues, the user may
not receive all the data that has been sent by the server. For the sake of
exposition, we assume that the probability of successful reception is given by
$\mathbb{P}_{t}^{i}$. Therefore, the amount of received data is given by
$y_{t}^{i}=d_{t}^{i}\mathbb{P}_{t}^{i}$. The probability value
$\mathbb{P}_{t}^{i}$ depends on the several factors such as user mobility,
blockage, and propagation environment.

Now that we know how much data a user receives, the dynamics of the playback
buffer level is given as follows:

$\displaystyle b_{t+1}^{i}=\max\\{b_{t}^{i}-1,0\\}+u_{t}^{i}y_{t}^{i},$ (2)

where $u_{t}^{i}$ is a binary control variable that determines whether the
user $i$ is scheduled at time $t$ or not. Therefore, $u_{t}^{i}y_{t}^{i}$
determines the amount of seconds of the video that would be successfully
transmitted to the user, if it is scheduled. We consider that each time slot
is one second, and thus we subtract a second from the previous buffer level of
user $i$, and then add the amounts of seconds that the user would receive in
case of selection and reception.

###  III-C Quality of Experience

The QoE for each user depends on the playback buffer level and the resolution
of the video frames stored in the buffer. To characterize the QoE, we consider
three factors. (1) Any interruption in the streaming is undesirable, and it
happens whenever the playback buffer becomes empty. We call this event “zero-
hit”. (2) The QoE increases as the buffer level increases, but it has a
diminishing return modeled as a logarithmic function. (3) The resolution of
the video frames impacts the QoE. For two different users with the same amount
of data in their playback buffers, the QoE of the user who plays a higher
resolution is higher. We denote the QoE of user $i$ at time $t$ by
$q_{t}^{i}$, and putting together these factors, the overall QoE can be
expressed as:

$\displaystyle
q_{t}^{i}=(1-\mathbbm{1}_{\textbf{0}}(b_{i}^{t}))\lambda(v_{res}^{i})+\alpha\log(1+b_{t}^{i})-\gamma\mathbbm{1}_{\textbf{0}}(b_{i}^{t}).$
(3)

The first term is an offset, which only depends on the resolution of the video
the user is playing. The second term captures the diminishing return of the
playback buffer, and the third term accounts for the zero-hit events that
penalizes the QoE by a factor of $\gamma$.

###  III-D QoE-Centric Optimization Problem

The objective of the server is to maximize the long-term average QoE for all
users, given that switching to a new user (a user that was not scheduled in
the previous time slot) incurs a beam alignment overhead $\tau(.)$. The
decision variable is $\mathbf{u}~{}\in~{}\\{0,1\\}^{N}$, which is a binary
vector of size $N$. At each time step, only $K$ elements of $\mathbf{u}$ can
be active, and the rest of them are zero. Therefore, we can formulate the
following optimization problem:

$\begin{cases}\max\limits_{\mathbf{u}}\quad&\lim\limits_{T\to\infty}\frac{1}{T}\sum\limits_{t=1}^{T}\sum\limits_{i=1}^{N}q_{t}^{i}\\\
\textrm{s.t.}\quad&\sum_{i=1}^{N}u^{i}_{t}\leq K\,\,\forall_{t=1..T}\\\
&b_{t+1}^{i}=\max\\{b_{t}^{i}-1,0\\}+u_{t}^{i}y_{t}^{i}\,\,\forall_{i=1..N,t=1..T}\end{cases}$
(4)

The first constraint addresses the hardware limitations in terms of the number
of RF chains, and the second constraints captures the playback buffer
dynamics. The size of the decision space in Eq. 4 scales with the number of
users in the network. In the next section, we establish an efficient
scheduling framework based on contextual multi-armed bandits.

##  IV Beam and Buffer Predictive Streaming: B2P-Stream

In this section, we first provide a brief introduction on multi-armed bandit
models. Then, we develop a solution for the problem defined in Eq. 4 and
present the B2P-Stream algorithm. Finally, we derive an upper-bound on the
regret for the proposed algorithm.

###  IV-A Multi-Armed Bandit Models: An Overview

A MAB problem is an interactive game between a learner and an environment
[31]. The game repeats for a finite number of times. In each round of the
game, the learner chooses an action (i.e., plays an arm) $\mathbf{u}$, and
receives a reward $\mathbf{r}$, that is revealed by the environment. The
reward can come from a stochastic distribution or chosen by the environment
itself. The learner tries to find an optimal policy using the history of
played actions and received rewards. To this end, the Upper Confidence Bound
(UCB) method [32, 33] handles the _exploration and exploitation trade-off_ by
providing an upper bound for the estimation of the expected reward of each
arm. The upper bound decreases as the number of reward samples from one arm
increases, which means that we are more certain about the estimation of the
expected value. There are other classes of MAB algorithms that are specified
to different cases, such as the case that the learner can choose more than one
arm at a time [34, 35], called combinatorial bandit problem; or another case
where there are some contextual information available [32].

The performance of bandit models is usually measured in terms of _regret_ that
quantifies the gap with respect to the optimal solution. Let $\mu^{i}$ be the
expected value of the rewards achieved by playing arm $i$ (i.e.,
$\mu^{i}=\mathbb{E}(r^{i})$), and $\mu^{i^{*}}=\max\limits_{i}\mu^{i}$ be the
expected value of the reward of the optimal arm. In this case, the immediate
regret is defined as $\Delta_{i}=\mu^{i^{*}}-\mu^{i}$, and the accumulated
stochastic regret is defined over $T$ rounds of playing the game, which is
given by [31]:

$R(T)=T\mu^{i^{*}}-\mathbb{E}\left[\sum_{t=1}^{T}r_{t}\right]=\sum_{i}\Delta_{i}\mathbb{E}(n_{i}(T)),$

in which $n_{i}(T)$ is the number of times that arm $i$ has been played over
the time interval $T$.

###  IV-B B2P-Stream Policy

In order to solve the optimization problem defined in Eq. 4, we model this
problem as an instance of the _contextual_ multi-armed bandit formulation. We
designate $r_{t}^{i}$ as the measurement of the QoE at time $t$ for user $i$,
and $\tilde{\mu}_{t}^{i}$ denotes the average of these measurements. In
addition, the action set in this model is
$\mathbb{U}\subseteq\\{\mathbf{u}\in\\{0,1\\}^{N}:\lVert\mathbf{u}\rVert_{1}\leq
K\\}$, which tells us that we have a $N$ dimensional binary action vector that
has at most $K$ active elements.

The contextual bandit model stems from the fact that the scheduler can
estimate the playback buffer level at each user as follows:

$\displaystyle\tilde{b}_{t+1}^{i}=\max\\{\tilde{b}_{t}^{i}-1,0\\}+u_{t}^{i}d_{t}^{i}.$
(5)

Note that the state of each user changes over time according to Eq. 2, but the
scheduler can only estimate the buffer level since there are unknown
parameters such as the probability of successful frame reception by the user.
This estimated playback buffer level along with the knowledge on beam
alignment overhead function $\tau^{i}(.)$, which is a _non-decreasing
function_ as a function of the last time served, provide contextual
information for the server. For the sake of presentation, we combine these two
factors into a single _trend function_ $\mathit{f}(\tilde{b}^{i})$ that
captures the estimated QoE for a user $i$. The trend function is then added to
the average reward measurements received by the algorithm. In fact, MAB models
with trend functions are finding applications in different domains [33].

Algorithm 1 B2P-Stream

Inputs:
$N$: Number of connected users
$K$: Maximum number of allowed users at each timestamp
$T$: Total number of timestamps
$\mathit{R}$: The network data transfer rate
$\eta_{0}$: Initial learning rate
Algorithm:

1: for $t=0$ to $T$ do

2: $A_{t}=\emptyset$

3: for $k=1$ to $K$ do

4: Select arm $i_{k}$ = $\operatorname*{arg\,max}_{i\notin
A_{t}}(\tilde{\mu}_{t}^{i}+c_{t}^{i}+\mathit{f}^{i}(\tilde{b}_{t}))$

5: $A_{t}=A_{t}\cup\\{i_{k}\\}$

6: end for

7: $\mathbf{u_{t}}=one\\_hot(A_{t},N)$

8: Perform $\mathbf{u_{t}}$ and observe QoE vector $\mathbf{r_{t}}$

9: for $i=1$ to $N$ do

10: $d_{t}^{i}=(1-\tau_{t}^{i}(x_{t}^{i}))\mathit{R}$

11: $\tilde{b}_{t+1}^{i}=\max\\{\tilde{b}_{t}^{i}-1,0\\}+u_{t}^{i}d_{t}^{i}$

12:
$\tilde{\mu}_{t+1}^{i}=\tilde{\mu}_{t}^{i}+\eta_{t}(r_{t}^{i}-\tilde{\mu}_{t}^{i})$

13: if $u_{t}^{i}==1$ then

14: $x_{t+1}^{i}=0$

15: else

16: $x_{t+1}^{i}=x_{t}^{i}+1$

17: end if

18: end for

19: $\eta_{t+1}=\eta_{0}e^{-t/T}$

20: end for

The complete process is shown in Algorithm 1 in which first we select $K$
users that provide the maximum outcome, and add them to a set $A_{t}$ (lines 2
to 6). Then, in line 7, we create a $N$ dimensional binary vector using
$A_{t}$, and based on this vector, we create $K$ beams and stream to the
selected users, and measure the QoE $\mathbf{r_{t}}$. Then using Eq. 5, we
update the playback buffer level estimation in line 11, for all the users.
Next, we update the vector $\tilde{\mu}$ using the new measurements in line
12. Finally, from line 13 to 17, we either set the last time served to zero if
we scheduled the user in the current time stamp or increase it if we did not
schedule the user. Finally, the learning rate $\eta$ is decreasing
exponentially at each iteration.

###  IV-C Regret Analysis of B2P-Stream

In this section, we provide an analysis of our algorithm and find an upper
bound for the regret. In Theorem 1, we show that using a
$L_{\mathit{f}}$-Lipschitz trend function, we achieve a sub-linear regret
bound for the B2P-Stream algorithm.

###### Theorem 1

Given an $L_{\mathit{f}}$-Lipschitz trend function, $b_{max}$ as the maximum
playback buffer level, and an $\alpha>0$, the regret for B2P-Stream algorithm
is upper-bounded by

$\displaystyle R(T)\leq\sum_{i\neq
i^{*}}\frac{2\alpha\log(T)}{\Delta_{i}-L_{\mathit{f}}b_{max}}+\frac{2\alpha}{\alpha-1}(\Delta_{i}+L_{\mathit{f}}b_{max}).$

where $\Delta_{i}=\mu^{i^{*}}-\mu^{i}$.

###### Proof:

During the learning process, we either underestimate the value of all the sub-
optimal actions, event $G_{t}$, overestimated the value of the optimal action,
event $H_{t}$, or complement of these two events.

$\displaystyle G_{t})\quad$
$\displaystyle\tilde{\mu}_{n_{i}}^{i}+\mathit{f}(\tilde{b}^{i})\leq\mu^{i}+\mathit{f}(b^{i})+c^{i};$
$\displaystyle H_{t})\quad$
$\displaystyle\tilde{\mu}_{n_{i^{*}}}^{i^{*}}+\mathit{f}(\tilde{b}^{i^{*}})\geq\mu^{i^{*}}+\mathit{f}(b^{i^{*}})-c^{i^{*}},$

where $c^{i}=\sqrt{\frac{\alpha\log(t)}{2n_{i}}}$. Superscript $i$ corresponds
to sub-optimal actions and $i^{*}$ corresponds to the optimal action. We know
$G_{t}$ fails when,
$\tilde{\mu}_{n_{i}}^{i}+\mathit{f}(\tilde{b}^{i})>\mu^{i}+\mathit{f}(\tilde{b}^{i})+c^{i}.$
By using Hoeffding’s inequality, we have

$\displaystyle\mathbb{P}(G_{t}^{c})$
$\displaystyle=\mathbb{P}(\tilde{\mu}_{n_{i}}^{i}+\mathit{f}(\tilde{b}^{i})-(\mu^{i}+\mathit{f}(b^{i}))>c^{i})$
$\displaystyle\leq\exp(-2t(c^{i})^{2})=\exp(-2t\frac{\alpha\log(t)}{2n_{i}})$
$\displaystyle\leq\exp(-2t\frac{\alpha\log(t)}{2t})=\exp(-\alpha\log(t))=t^{-\alpha}.$

We achieve the same result for the event $H_{t}^{c}$. This result illustrates
that as the time passes, the probability of underestimating the optimal action
or overestimating all other actions decreases.

(a) Low Quality (480p) (b) Medium Quality (1080p) (c) High Quality (2160p)

Figure 3: The average and standard deviation of measured QoE with 200 users in
the network. For lower resolution users, the B2P-Stream converges faster to be
out of the critical regions at the cost of a lower QoE for those users. On the
other hand, for users with higher resolutions, the B2P-Stream provides a
faster convergence and better QoE.

Now, let us assume that both $G_{t}$ and $H_{t}$ hold. We bound the number of
sub-optimal arm pulls. In this case, the sub-optimal arm $i$ is pulled because
of insufficient sampling up to this point, which means:

$\displaystyle\tilde{\mu}_{n_{i}}^{i}+\mathit{f}(\tilde{b}^{i})+\sqrt{\frac{\alpha\log(t)}{2n_{i}}}\geq\tilde{\mu}_{n_{i^{*}}}^{i^{*}}+\mathit{f}(b^{i^{*}})+\sqrt{\frac{\alpha\log(t)}{2n_{i^{*}}}}.$
(6)

Since $G_{t}$ and $H_{t}$ are assumed to be true, by adding $c^{i}$ and
$c^{i^{*}}$ to both sides of $G_{t}$ and $H_{t}$, respectively, we have:

$\displaystyle\mu^{i}+\mathit{f}(b^{i})+2\sqrt{\frac{\alpha\log(t)}{2n_{i}}}$
$\displaystyle\geq\tilde{\mu}_{n_{i}}^{i}+\mathit{f}(\tilde{b}^{i})+\sqrt{\frac{\alpha\log(t)}{2n_{i}}}$
(7)
$\displaystyle\tilde{\mu}_{n_{i^{*}}}^{i^{*}}+\mathit{f}(\tilde{b}^{i^{*}})+\sqrt{\frac{\alpha\log(t)}{2n_{i^{*}}}}$
$\displaystyle\geq\mu^{i^{*}}+\mathit{f}(b^{i^{*}})$ (8)

Now, by chaining equations 6, 7, 8, we have:

$\displaystyle\mu^{i}+\mathit{f}(b^{i})+2\sqrt{\frac{\alpha\log(t)}{2n_{i}}}\geq\mu^{i^{*}}+\mathit{f}(b^{i^{*}})$
$\displaystyle\Rightarrow\sqrt{\frac{\alpha\log(t)}{2n_{i}}}\geq\frac{\mu^{i^{*}}+\mathit{f}(b^{i^{*}})-(\mu^{i}+\mathit{f}(b^{i}))}{2}$
$\displaystyle\Rightarrow\frac{\alpha\log(t)}{2n_{i}}\geq\frac{(\mu^{i^{*}}-\mu^{i}+\mathit{f}(b^{i^{*}})-\mathit{f}(b^{i}))^{2}}{4}.$

Let $\Delta_{i}=\mu^{i^{*}}-\mu^{i}$ and
$\delta_{i}=\mathit{f}(b^{i^{*}})-\mathit{f}(b^{i})$, we have:

$\displaystyle n_{i}$
$\displaystyle\leq\frac{2\alpha\log(t)}{(\mu^{i^{*}}-\mu^{i}+\mathit{f}(b^{i^{*}})-\mathit{f}(b^{i}))^{2}}=\frac{2\alpha\log(t)}{(\Delta_{i}+\delta_{i})^{2}}.$

Recall that arm $i$ can be pulled when it has been sampled insufficiently
(fewer than $\frac{2\alpha\log(t)}{(\Delta_{i}+\delta_{i})^{2}}$) or either
event $G_{t}$ or $H_{t}$ fails. Hence, the expected number of times that it
has been played is given by:

$\displaystyle\mathbb{E}[n_{i}]$
$\displaystyle=\sum_{t=1}^{T}\mathbb{E}[\mathbbm{1}(I_{t}=i)]\leq\frac{2\alpha\log(T)}{(\Delta_{i}+\delta_{i})^{2}}+\sum_{t=1}^{T}\mathbb{E}[\mathbbm{1}\\{G_{t}^{c}\cup
H_{t}^{c}\\}]$
$\displaystyle\leq\frac{2\alpha\log(T)}{(\Delta_{i}+\delta_{i})^{2}}+\sum_{t=1}^{T}\Big{(}\mathbb{E}[\mathbbm{1}\\{G_{t}^{c}\\}]+\mathbb{E}[\mathbbm{1}\\{H_{t}^{c}\\}]\Big{)}$
$\displaystyle\leq\frac{2\alpha\log(T)}{(\Delta_{i}+\delta_{i})^{2}}+\sum_{t=1}^{T}\Big{(}t^{-\alpha}+t^{-\alpha}\Big{)}\leq\frac{2\alpha\log(T)}{(\Delta_{i}+\delta_{i})^{2}}+\frac{2\alpha}{\alpha-1}.$

Thus, we can bound the regret by

$\displaystyle R(T)$ $\displaystyle=\sum_{i\neq
i^{*}}(\Delta_{i}+\delta_{i})\mathbb{E}[n_{i}]\leq\sum_{i\neq
i^{*}}\frac{2\alpha\log(T)}{\Delta_{i}+\delta_{i}}+\frac{2\alpha}{\alpha-1}(\Delta_{i}+\delta_{i})$
$\displaystyle\leq\sum_{i\neq
i^{*}}\frac{2\alpha\log(T)}{\Delta_{i}+\delta_{i}}+\frac{2\alpha}{\alpha-1}(\Delta_{i}+\delta_{i})$
$\displaystyle\leq\sum_{i\neq
i^{*}}\frac{2\alpha\log(T)}{\Delta_{i}-\lvert\delta_{i}\rvert}+\frac{2\alpha}{\alpha-1}(\Delta_{i}+\lvert\delta_{i}\rvert)$
$\displaystyle\leq\sum_{i\neq
i^{*}}\frac{2\alpha\log(T)}{\Delta_{i}-L_{\mathit{f}}b_{max}}+\frac{2\alpha}{\alpha-1}(\Delta_{i}+L_{\mathit{f}}b_{max}).$

Let $\mathit{f}$ be a $L_{\mathit{f}}$-Lipschitz function then we can upper
bound
$\lvert\delta_{i}\rvert=\lvert\mathit{f}(b^{i^{*}})-\mathit{f}(b^{i})\rvert\leq
L_{\mathit{f}}\lvert b^{i^{*}}-b^{i}\rvert\leq L_{\mathit{f}}b_{max}$, where
$b_{max}$ is the maximum level of buffer. ∎

(a) 4 RF chains

(b) 8 RF chains

(c) 16 RF chains

Figure 4: Fraction of simulation time that users experienced zero-hits per the
total number of connected users to the network for B2P-Stream, Uniform and RR
policy. The users scheduled with B2P-Stream experience far less number of
zero-hits.

##  V Numerical Results

In this section, we present the simulation results to demonstrate the efficacy
of our algorithm compared with two other baselines.

###  V-A Simulation Setting

We evaluate the B2P-Stream’s performance under different conditions to make
sure about the robustness of the method. We compare the B2P-Stream with two
different baselines, namely the Uniform and the RR scheduling algorithms. In
our simulations, all the users are initialized with zero buffer level. The
zero buffer level is considered a highly unsatisfying situation for all the
users. Also, the users experience a lower QoE as their playback buffer level
approaches zero. Thus, we define two critical situations to be able to compare
the performance of different algorithms. We call the first critical situation
_“critical region,”_ that corresponds to the case when a user has less than
fifteen seconds of data in the playback buffer. The other one is named
_“highly critical region,”_ which corresponds to the case when the user has
less than five seconds of data in the playback buffer. For instance, the dark
grey and slate grey, in Figure 3 and Figure 5, correspond to the critical and
highly critical region, respectively. Since we are initializing all the users
with zero buffer levels, it is desirable that the scheduling algorithm avoid
these two critical regions.

Resolution | Bit Rate (Mbps) | Portion of Users
---|---|---
2160p (4K) | 40 | 0.05
1440p (2K) | 16 | 0.1
1080p | 8 | 0.4
720p | 5 | 0.3
480p | 2.5 | 0.1
360p | 1 | 0.05
TABLE I: Video resolution bit rates and the portion of users with a specific
resolution.

Each experiment has been run 10 times and for 500 time steps. At the beginning
of each run, the video resolution requested by each user is selected randomly
from the list of resolutions reported in Table I. The video bit rates and
portion of users created for each video quality are also have been set
according to Table I. The portion of users with specific resolution is
inspired by [36].

We consider the mean and standard deviation of the observed performance. Since
we initialize all the users with a highly critical state, the beginning steps
in a simulation are more informative and can reveal more about the underlying
events.

###  V-B QoE Comparison

Figure 3 shows the average QoE for $200$ users connected to the network. The
BS is equipped with $4$ RF chains and users play videos with different
resolutions. Each of the columns corresponds to a different group of users
with different video resolutions: low resolution (480p), medium resolution
(1080p), and high resolution (4K), respectively. In this set of results, we
ignore the impact of resolution on the QoE (i.e., the first term in Eq. 3) to
provide a fair comparison across different resolutions. Also, the dark grey
and slate grey, in Figure 3, correspond to the critical and highly critical
region, respectively. The users scheduled by B2P-Stream exit the critical
regions much earlier than the users scheduled with Uniform or RR policy. Even
though only $5\%$ of the users in the simulation are playing a 4K video, both
RR and Uniform policies cannot provide a satisfying experience. The situation
is the same for users who are playing a 2K video.

Although the B2P-Stream provides a better zero-hit statistics compared to two
other baselines, there is some cost needs to be paid. This cost is lower QoE
for users of lower resolution. From the results, we note that the B2P-Stream
algorithm maintains a lower QoE for users of lower resolution to compensate
for the users of higher resolutions, as they need to be served more often
because of the higher bit rate requirements. This does not mean that users
with lower video quality would experience a significantly lower QoE because
the QoE has a diminishing effect and there is not much of QoE difference as
long as they are out of critical regions.

###  V-C Zero-hit Performance

Next, we compare the average fraction of simulation time that each user
experiences a zero-hit when the streaming server is equipped with $4$, $8$, or
$16$ RF chains. The number of RF chains determines the maximum number of users
that can be scheduled simultaneously at each time slot. Figure 4 compares the
performance of B2P-Stream with respect to other baselines, as the number of RF
chains and total number of users increase. From the results, B2P-Stream
achieves a smaller zero-hit compared with the RR and Uniform scheduler, as the
number of RF chains is equal to $4$, $8$, or $16$.

###  V-D Scalability Analysis

Figure 5: The average and standard deviation of the measured QoE achieved by
B2P-Stream, as the total number of connected users ($N$) is changing.

Figure 5 reveals more about the scalability and reliability of B2P-Stream.
This figure reports the average and standard deviation of the measured QoE,
but it is specific to the users who are streaming a 4K video, since they are
more prone to unsatisfying QoE. Figure 5 demonstrates how the QoE behaves for
these users as the total number of connected users increases. From the
results, we note that even with $N=650$ connected users and $K=4$ RF chains,
it takes less than $25$ time steps for B2P-Stream to push all the high quality
users out of the critical regions and provide an improved QoE performance.

###  V-E Beam Alignment Overhead Results

Figure 6 shows the average of beam alignment overhead of B2P-Stream compared
to other algorithms. A moving average with windows size of 50 has been applied
to these curves to make them smoother and more comparable. The RR algorithm
maintains a fixed beam alignment overhead due to its deterministic nature. On
the other hand, the Uniform policy maintains a lower beam alignment overhead
compared to B2P-Stream and RR, but as it is depicted in Figure 3, it fails to
satisfy the fairness criterion. Note that although the B2P-Stream has a
slightly larger beam alignment overhead compared to Uniform policy, it
cleverly exploits the time resources to provide a fair QoE for all the users.

Figure 6: Average beam alignment overhead when $4$ users are allowed to be
scheduled at a time. The shaded area shows the standard deviation.

###  V-F Intuitions Behind B2P-Stream

To balance the trade-off between beam alignment overhead and playback buffer
levels and optimize the QoE metric, we can intuitively distinguish two groups
of users. The first group corresponds to those users who were served recently,
thus the beam alignment overhead would be small for them (small
$\tau^{i}_{t}$), and we can stream more data to this group. The second group
are those users that their buffer levels are approaching zero, meaning that it
has been a long time since the last time we served them (large $x^{i}_{t}$).
Figure 7 demonstrates the empirical distribution of scheduled users with
respect to the last time those users were served, i.e., $x^{i}_{t}$. From the
results, we can identify these two groups that correspond to the two peaks in
Figure 7, respectively.

Figure 8 illustrates the time interval between two consecutive schedules of
different groups of users in the simulation. The B2P-Stream may allocate more
resources to the users with higher resolution, which means they would be
scheduled more frequently, due to the fact that the video bit rate for them is
higher, and they would need more data to play a video for a specific period of
time compared with users with lower video quality. We can again see that even
though only $5\%$ of the users play a 4K video, they would be scheduled almost
every $17$ time steps on average, because their QoE requirement are higher
than other users. On the other hand, users with a medium video quality of
1080p, which applies to 40% of the users, would be scheduled every $62$ time
steps.

Figure 7: Empirical distribution of the scheduled users with respect to the
last time they have been served. The first peak corresponds to the users with
a lower beam alignment cost, and the second peak corresponds to the users with
an exhausted buffer level. Figure 8: Time interval between consecutive
schedules of users of different video resolution. The interval decreases as
the resolution increases, means we choose users with higher resolution more
frequently.

##  VI Conclusion

In this paper, we considered the problem of multi-user mmWave scheduling ($K$
users out of $N$) who are streaming videos with different resolutions. The
overall objective is to optimize the QoE across all the users. Leveraging the
contextual MAB models, we developed a QoE-centric scheduling policy that
considers the physical layer characteristics of the mmWave networks. The
proposed B2P-Stream algorithm is able to optimally balance the trade-off
between the beam alignment overhead and the users’ playback buffer level. In
particular, B2P-Stream uses an estimated buffer level as an input for a _trend
function_ that biases the scheduling policy towards those users with exhausted
buffer levels. We provided theoretical analysis to prove that the B2P-Stream
guarantees a sub-linear regret bound, and through simulations, we showed that
the proposed algorithm outperforms both Uniform and RR policies. Overall,
mmWave networks are considered as one of the key enablers for data-intensive
applications such as high quality video streaming. As such, developing
efficient and reliable multi-user management algorithms that guarantee high
QoE for all the users, is of utmost importance to enable ubiquitous mmWave
technologies.

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# Structural and Magnetic Properties of Fe-Al alloys: an Ab initio Studies

Mikhail A. Zagrebin  Mariya V. Matyunina<EMAIL_ADDRESS>Alexey B.
Koshkin  Vladimir V. Sokolovskiy  Vasiliy D. Buchelnikov  Chelyabinsk State
University, 454001 Chelyabinsk, Russia

###### Abstract

In the framework of density functional theory, the structural and magnetic
properties of Fe100-xAlx alloys (${5\leq x\leq 25}$ at.%) with the different
structural order are investigated. Using the Korringa-Kohn-Rostoker Green’s
function method with a coherent potential approximation, the equilibrium
lattice parameters, ground-state energy, and shear moduli for D03, B2, and A2
structures are calculated. For all structures, the optimized lattice constant
increases while the shear modulus demonstrates a decreasing behavior with
increasing Al content. The tetragonal magnetostriction constants are estimated
by the torque method. The A2 and B2 structures provide a positive contribution
to the tetragonal magnetostriction. With the help of Monte Carlo simulations
of the Heisenberg model, the Curie temperatures are obtained in a wide
concentration range.

###### keywords:

Ab initio methods, Monte Carlo simulations, Curie temperature, Shear modulus,
Magnetostriction

††journal: Journal of Magnetism and Magnetic Materials

##  1 Introduction

Fe-Al alloys are attractive materials that exhibit a good oxidation and
sulfidation resistance, excellent resistance to abrasive wear and erosion,
high strength, relatively low density and high magnetic permeability [1, 2,
3]. Due to these unique properties, Fe-Al alloys are promising as the high-
temperature functional materials for magnetic and diffusion barrier
applications, and interconnections in microelectronics corrosion stability.
The functional properties of ordered alloys strongly depend on their phase
structure. According to the experimental phase diagram (see Fig. 1), three bcc
structures (A2, B2 and D03) with different degrees of order are formed for
iron-rich region depending on temperature and composition. This fact makes Fe-
Al attractive for study of ordering reactions since bcc structure is stable in
the wide concentration and temperature range.

Figure 1: Experimental phase diagram of Fe100-xAlx (${0\leq x\leq 50}$)
adapted from [6]. The dash-dotted lines mark the transitions from
ferromagnetic (FM) to paramagnetic (PM) $\alpha-$Fe and Fe3Al. Dotted line
between about 23 and 24 at.% Al indicates ($\alpha-$Fe + Fe3Al)/Fe3Al
boundary.

In the concentration range up to $\approx 19$ at.%Al there is the
$\alpha$-region with disordered bcc A2 ($\alpha$-Fe) phase. Whereas the
ordered phases based on B2 (FeAl) and D03 (Fe3Al) structures form with a
higher Al content [1, 3, 4, 5]. The D03 phase is stable below 825 K for
compounds with $18<x<37$ at.%Al while the B2 phase occurs in concentration
range between 23 and 54 at.% Al depending on temperature (see Fig. 1). In
general, there are two ”order-disorder” (B2$\rightarrow$A2 and
D0${}_{3}\rightarrow$B2) and one paramagnetic (PM)$\rightarrow$ferromagnetic
(FM) (A$2^{\mathrm{PM}}\rightarrow$A$2^{\mathrm{FM}}$) transformation for Fe-
Al compounds [3, 6]. Both transformations (B2$\rightarrow$A2 and
D0${}_{3}\rightarrow$B2) are the second order ones, B2$\rightarrow$A2 observes
for compounds with $23<x<45$ at.%Al whereas D0${}_{3}\rightarrow$B2 takes
place in the narrow concentration range and the maximal transition temperature
of 818 K is observed at 26.5 at.%.

It should be noted that the Fe-Al phase diagram includes some features related
to the areas of the coexistence of these three structures. One of them is the
small triangular area between 815 and 900 K for compositions with
$21.3<x<22.8$at.%Al where the FM disordered A2 phase coexists with the PM B2
phase [3, 6, 7]. The other feature is the two-phase (A2 and D03) region in the
concentration range from 18 to 25 at.% below 814 K. This region is
characterized by the first-order ”order-disorder” transition.

The tetragonal magnetostriction $\lambda_{001}$ of Fe-Al alloys has an unusual
dependence as for Fe-rich Fe-Ga alloys [8]. Room temperature magnetostriction
measurements of Fe-Al alloys indicated a five-fold rise in magnetostriction
with Al content up to 30% Al. Nevertheless, the $\lambda_{001}$ values for Fe-
Al are smaller as compared to the Fe-Ga system. This finding can be explained
by less magnitude of magnetoelastic constant $-b_{1}$ and the decreasing of
shear modulus $C^{\prime}$ with Al content, which is not so large also as for
Fe-Ga [8].

The aim of this paper is a complex study of structural and magnetic properties
of Fe100-xAlx ($5\leq x\leq 25$ at.%) alloys by the density functional theory
at zero temperature with the finite-temperature Monte Carlo (MC) simulations.

##  2 Methodology and calculation details

The ab initio calculations were performed using the spin-polarized
relativistic Korringa-Kohn-Rostoker code (SPR-KKR) [9] implementing the KKR
Green’s function multiple scattering theory. The exchange correlation energy
was carried out in the Perdew–Burke–Ernzerhof (PBE) [10] generalized gradient
approximation (GGA). The single-site coherent potential approximation (CPA)
successfully describing the properties of many compositionally disordered
alloy systems was used to create the off-stoichiometric compositions. An
impurity of each atom type, Fe or Al, is placed in an effective CPA medium and
then considers the alloy to be described by the weighted average of the two
different impurity solutions. As shown in [11], although the CPA deals with
disorder systems, it can describe long-range order systems too.

The electronic structure calculations of Fe100-xAlx ($5\leq x\leq 25$ at.%)
were performed using the spin-polarized scalar-relativistic mode for three
crystal cubic structures with different atomic order. The fully ordered D03
structure with space group $Fm\bar{3}m$ (No. 225) was created by the unit cell
consisting of four atoms. The Al and Fe atoms were located at the $4a\ (0;\
0;\ 0)$ Wyckoff position, the Fe atoms were occupied $4b\ (0.5;\ 0.5;\ 0.5)$
and $8c\ (0.25;\ 0.25;\ 0.25)$ Wyckoff positions. For stoichiometric Fe75Al25,
$4a$ site was fully occupied by Al. The partially ordered B2 phase with space
group $Pm\bar{3}m$ (No. 221) was created using two atoms per unit cell, where
Fe and Al atoms randomly were occupied the $1b\ (0.5;\ 0.5;\ 0.5)$ site while
$1a\ (0;\ 0;\ 0)$ site was occupied by Fe atoms. For disordered A2 phase
($Im\bar{3}m$ space group, No. 229) Fe and Ga atoms were distributed at the
$2a\ (0;\ 0;\ 0)$ Wyckoff position. The calculations of optimized lattice
constants were performed as single-point energy calculations at several
different volumes for each structure under study. To evaluate equilibrium
volume more precisely, the total energy curves as a function of volume were
fitted by the Birch–Murnaghan equation of state.

For the determination of tetragonal shear modulus $C^{\prime}$, the cubic
structures were deformed from their optimized geometries along the $z$ axis
assuming a volume-conserving mode
$\varepsilon_{x}=\varepsilon_{y}=-1/2\varepsilon_{z}$ in the range of
$\varepsilon=\pm 2\%$. $C^{\prime}$ was calculated from the
$\varepsilon_{z}$-dependent total energy $E_{\mathrm{tot.}}$ according to the
following equation [12],

$C^{\prime}=\frac{(C_{11}-C_{12})}{2}=\displaystyle\frac{1}{3V_{0}}\frac{d^{2}E_{\mathrm{tot.}}}{d\varepsilon^{2}}$
(1)

here $V_{0}$ is the equilibrium volume of the unit cell.

For A2, B2, D03 cubic structures, magnetostrictive coefficient $\lambda_{001}$
was determined through the tetragonal dependence of magnetocrystalline
anisotropy energy $E_{\mathrm{{MCA}}}$ as [12]

$\lambda_{001}=\frac{2}{3V_{0}C^{\prime}}\frac{dE_{\mathrm{{MCA}}}}{d\varepsilon}=-\frac{b_{1}}{3C^{\prime}},$
(2)

here, $-b_{1}$ is magnetoelastic constant. The $E_{\mathrm{{MCA}}}$ was
calculated using the torque method implemented in the SPR-KKR package.

The Heisenberg exchange parameters $J_{ij}$ were evaluated by way of KKR
multiple scattering formalism proposed by Liechtenstein et al. [13]. The spin-
polarized scalar-relativistic Dirac Hamiltonian mode and PBE approach for
exchange-correlation potential were considered to calculate the $J_{ij}$
parameters as a function of distance between interacting atoms. Using the
calculated $J_{ij}$ and Fe magnetic moments ($\mu^{\mathrm{Fe}}$) as input
data, the finite-temperature MC simulations of Heisenberg model without
magnetic field and anisotropy terms were done.

${\cal H}=-\sum_{ij}J_{ij}\mathbf{S}_{i}\mathbf{S}_{j}.$ (3)

Here, $\mathbf{S}_{i}=\left(S_{i}^{x},\ S_{i}^{y},\ S_{i}^{z}\right)$ is a
classical Heisenberg spin variable $\left|S_{i}\right|=1$. The $J_{ij}$ were
taken into account up to the 6 coordinational spheres and can be ferromagnetic
(FM, $J_{ij}>0$) or antiferromagnetic (AFM, $J_{ij}<0$). The modelling was
performed on a 3925 atoms lattice with periodic boundary conditions using the
Metropolis algorithm [14]. The MC step at each site corresponded to an
enumeration of randomly selected $N$ lattice sites, where $N$ is the number of
atoms in the lattice. For a given temperature, the number of MC steps at each
site was $5\times 10^{5}$. The magnetic order parameter $m$ and total
magnetization $M$ are defined by the following way,

$\begin{array}[]{ll}m^{\mathrm{Fe}}=\displaystyle\frac{1}{N^{\mathrm{Fe}}}\sum_{i}\sqrt{\left(S_{i}^{x}\right)^{2}+\left(S_{i}^{y}\right)^{2}+\left(S_{i}^{z}\right)^{2}},\\\\[2.84526pt]
M=3\mu^{\mathrm{Fe}}m^{\mathrm{Fe}},\end{array}$ (4)

Curie temperature ($T_{C}$) was estimated from temperature dependencies of
magnetization by plotting $M^{1/\beta\left(T\right)}$ function that decreases
almost linearly with increasing temperature. The intersection of
$M^{1/\beta}\left(T\right)$ curve with the $T$ axis indicates $T_{C}$. Here,
$\beta$ is the critical index, and it is equal to $0.3646$ for the three-
dimensional Heisenberg model [15].

##  3 Results and discussion

In the first step of our calculations, we have done the geometric optimization
of cubic crystal structures. The calculated ground-state energies summarized
in Table 1 show that the D03 structure has the lowest energy for all
considered compounds. The energy difference between B2 and D03 structure
($\Delta E_{0}^{\mathrm{B}2-\mathrm{D}0_{3}}$) is smaller than the respective
values for A2 and D03 ones ($\Delta E_{0}^{\mathrm{A}2-\mathrm{D}0_{3}}$). For
both structures, the $\Delta E_{0}^{\mathrm{B}2-\mathrm{D}0_{3}}$ and $\Delta
E_{0}^{\mathrm{A}2-\mathrm{D}0_{3}}$ are increasing functions of Al
concentration and they reach 65 meV/atom and 101 meV/atom for stoichiometric
Fe75Al25, respectively.

Table 1: The ground state energy $E_{0}$ (eV/atom) of Fe100-xAlx (${5\leq x\leq 25}$ at.%) with different cubic crystal structures, obtained via ab initio calculations at zero temperature. $x$, at.% | D03 | B2FM | B2NM | A2
---|---|---|---|---
5 | $-33219.647$ | $-33219.644$ | $-33219.179$ | $-33219.642$
10 | $-31818.911$ | $-31818.898$ | $-31818.488$ | $-31818.893$
15 | $-30418.175$ | $-30418.148$ | $-30417.800$ | $-30418.137$
18 | $-29577.735$ | $-29577.697$ | $-29577.390$ | $-29577.681$
20 | $-29017.442$ | $-29017.397$ | $-29017.117$ | $-29017.376$
21 | $-28737.296$ | $-28737.247$ | $-28736.981$ | $-28737.223$
24 | $-27896.859$ | $-27896.797$ | – | $-27896.764$
25 | $-27616.713$ | $-27616.648$ | $-27616.437$ | $-27616.612$

According to the experimental data, the B2 structure is paramagnetic [16].
However, ab initio calculations generally predict a FM ground state for B2
phase [17, 18] as in our case (see Table 1). There are two ways to explain
this finding. One is the presence of defects in an experimental sample that
have a strong influence on the magnetic state. The other is the fact that the
magnetic state depends strongly on the degree of chemical order [17]. As it
was shown by Mohn et al. [19] and Rhee et al. [18] the correct ab initio
prediction of magnetic state could be done by describing exchange and
correlation within the local-density approximation including Hubbard
parameters (LDA +$U$). In the present work, we performed electronic structure
calculations for the nonmagnetic state of the B2 phase (labeled B2NM) with an
account of exchange-correlation energy in GGA approximation. As can be seen
from Table 1 the obtained ground state energies for the nonmagnetic state of
B2 are higher than those for the disordered A2 structure. In further
discussion, all results for the B2 structure will be present in the FM state.

Fig. 2 (a) shows the equilibrium lattice parameter $a_{0}$ ($a_{0}/2$ for D03)
of phases corresponding to a minimum value of energy $E_{0}$ as a function of
Al concentration for D03, B2, and A2 structures of Fe100-xAlx alloys in
comparison with available experimental data taken from [4, 20].

Figure 2: (Color online) (a) Calculated lattice parameters $a_{0}$ ($a_{0}/2$
for D03) and (b) shear moduli as a function of the Al composition of
Fe100-xAlx (${5\leq x\leq 25}$ at.%) alloys with A2, B2 and D03 structures.
The experimental results at room temperature were taken from Leamy et al. [4],
Balagurov et al. [20], and Restorff et al. [21].

For all cubic structures calculated herein, the lattice parameter is found to
increase with increasing Al content. The differences between
$a_{0}^{\mathrm{A}2}-a_{0}^{\mathrm{D}0_{3}}$ and
$a_{0}^{\mathrm{D}0_{3}}-a_{0}^{\mathrm{B}2}$ increase too and numerically are
quite similar. At the 18 at.% of Al concentration, data for A2 and D03 are
equally distant from the dash experimental line. With further increasing of Al
content, the slope of the theoretical curve $a_{0}^{\mathrm{A}2}(x)$ becomes
higher than the experimental one. The $a_{0}^{\mathrm{D}0_{3}}(x)$ curve slope
is similar to the experiment that corresponds to neutron diffraction
experiments by Balagurov et al. [20]. According to [20], the D03 phase
corresponds to the main structural phase in the range from 21 to 34 at.%.

The concentration dependencies of calculated shear moduli of D03, B2, and A2
structures are shown in Fig. 2 (b). For comparison, experimental data taken
from works [4, 21] are plotted on the graph. The $C^{\prime}$ decreases
linearly with increasing Al content, and magnitudes of shear modulus for
considered structures are close to each other for Al content up to 10 at.%.
The anomaly in the $C^{\prime}$ behavior takes place for compounds with $x>15$
at.% Al. It could be related to the structural transformations which occur in
the alloys as indicated by the atomic volume changes (See Fig. 2 (a)). Thus a
difference in the $C^{\prime}$ behavior between D03(B2) and A2 has been
attributed to the transition from disordered to ordered phase [22].

Let us consider next the results of the magnetocrystalline anisotropy energy
calculations $E_{\mathrm{MCA}}$ for Fe100-xAlx (${5\leq x\leq 25}$ at.%) with
A2, D03 and B2 structures using the torque method. To demonstrate the tendency
of $E_{\mathrm{MCA}}(\varepsilon)$ behavior for all structures, in Fig. 3 we
show the results for compounds with $x=10,\ 20,\ 24$ at.%. For A2 and B2
structures, the strain dependencies of $E_{\mathrm{MCA}}(\varepsilon)$
demonstrate the linear behavior with a positive slope, which slightly
increases for the A2 phase and it is nearly constant for the B2 phase with
increasing Al content. In the case of D03 structure, an increase in Al content
20 at.% leads to linear decrease of $E_{\mathrm{MCA}}(\varepsilon)$ curve and
the slope of $E_{\mathrm{MCA}}(\varepsilon)$ is changed from positive to
negative for Fe80Al20.

Figure 3: (Color online) Calculated strain dependencies of magnetocrystalline
anisotropy energy $E_{\mathrm{MCA}}$ of Fe100-xAlx ($x=10,\ 20,\ 24$ at.%)
with D03, B2, and A2 structures.

The theoretical results of tetragonal magnetostriction $\lambda_{001}$
calculated by Eq. (2) are presented in Fig. 4. As can be seen, the A2 and B2
structures provide a positive contribution on tetragonal magnetostriction,
while D03 has a positive magnitude only up to 18 at.%.

Figure 4: (Color online) Calculated tetragonal magnetostriction constants of
Fe100-xAlx (${5\leq x\leq 25}$ at.%) with D03, B2, and A2 structures. The
experimental results were taken from Restorff et al. [21].

This tendency is dependent on $E_{\mathrm{MCA}}$ behavior. The higher values
of theoretical shear moduli $C^{\prime}$ could explain the numerically lesser
value of $\lambda_{001}$ for the A2 structure compared with the B2 phase. A
similar investigation of tetragonal magnetostriction using the torque method
was done for Fe-Ga alloys [23]. It has been shown that the main contribution
to tetragonal magnetostriction in the region of Ga concentrations $21\leq
x\leq 25$ at.% makes the B2 phase. One of the earliest models proposed to
explain the giant magnetostriction of Fe-Ga alloys was the question of Ga 2-nd
nearest-neighbor (B2-like) pairing [24]. In the D0${}_{3}-$like ordered phase,
this pairing is strongly suppressed.

The magnetic exchange coupling parameters $J_{ij}$ were calculated using the
equilibrium lattice parameters and the SPR-KKR package by employing CPA. We
considered the FM state of D03, B2 and A2 structures of Fe100-xAlx (${5\leq
x\leq 25}$). In general, the $J_{ij}$ shows damping oscillatory behavior for
all compositions and structures under-considered. The largest FM-interaction
observes between the first nearest neighbour of Fe-Fe atoms. Fig. 5(a)–(c)
show the exchange as a function of distance $d/a$ between atoms $i$ and $j$
for D03, B2, and A2 structures in composition with 24 at.% of Al content.
Calculated $J_{ij}$ between first neighbors of iron atoms as a function of Al
concentration is depicted in Fig. 5 (d).

Figure 5: (Color online) Calculated exchange coupling parameters $J_{ij}$ as a
function of distance ($d/a$) between atoms $i$ and $j$ for (a) D03, (b) B2,
and (c) A2 structures of Fe76Al24 alloy. (d) Calculated $J_{ij}$ between first
neighbors of iron atoms as a function of Al concentration of Fe100-xAlx
(${5\leq x\leq 25}$). The subscripts of Fe atoms correspond to Wyckoff
positions.

As we mentioned in Section 2, in the fully ordered D03 structure, the Fe atoms
are located on the $4a$, $4b$, and $8c$ Wyckoff positions. In the B2 phase, Fe
atoms occupy $1b$ and $1a$ Wyckoff positions, and in A2, there is one type of
iron on the $2a$ Wyckoff position. According to these locations, in fig. 5 all
Fe atoms are marked with the subscripts. For the D03 and B2 structures, the
intersublattice interactions Fe4b-Fe8c and Fe1b-Fe1a provide the largest
contribution to the exchange due to shorter distances compared to the
intrasublattice interactions. As can be seen in fig. 5 (d), for the D03 and A2
structures, the behavior of $J_{ij}\left(x\right)$ curves are similar to each
other and the largest values of $J_{ij}$ are obtained in composition Fe75Al25.
In the case of the B2 structure, the $J_{ij}$ is a decreasing function of Al
content.

The constants of magnetic exchange interactions and magnetic moments obtained
from ab initio calculations were used as input parameters to simulate the
temperature dependences of magnetization and to estimate the Curie
temperatures using Monte Carlo simulation. The calculated Curie temperatures
for D03, B2 and A2 structures are listed in Table 2.

Table 2: Curie temperatures $T_{C}$ (K) of Fe100-xAlx (${5\leq x\leq 25}$) with D03, B2, and A2 structures, obtained via MC calculations. For comparison experimental data from [6] are shown.  $x$, at.% | MC calculations ($T_{C}$) | Experiment ($T_{C}$)
---|---|---
D03 | B2 | A2 | D03 | A2
5 | 907 | 1235 | 1213 | $-$ | 1035
10 | 1159 | 1350 | 1258 | $-$ | 1018
15 | 1300 | 1360 | 1238 | $-$ | 991
18 | 1328 | 1369 | 1214 | $-$ | 967
20 | 1338 | 1349 | 1193 | $-$ | 937
21 | 1345 | 1334 | 1175 | $-$ | $-$
24 | 1250 | 1218 | 1131 | 781 | $-$
25 | 1074 | 1206 | 1127 | 758 | $-$

It follows from experimental results, for the A2 structure, Curie temperatures
continuously decrease with increasing Al concentration. In opposite to
experimental results, calculated Curie temperature dependence for A2 structure
has a maximum value at 10 at.%. Curie temperatures for B2 and D03 phases also
increase up to 18 (for B2) and 20 (for D03) at.% and then decrease. It should
be noted calculated Curie temperatures have overestimated values in comparison
to experimental data. This tendency could be explained by increasing magnetic
exchange interaction parameters $J_{ij}$ (See Figure 5).

##  4 Conclusion

The complex investigation of structural and magnetic properties of Fe100-xAlx
alloys ($5\leq x\leq 25$ at.%) was done by means of ab initio methods and
Monte Carlo simulations. Experimentally observed crystal structures D03, B2,
and A2 were considered. Conducted theoretical calculations of the ground state
energies showed that all considered phases are stable, while D03 is
energetically favourable for the whole Al concentration range. It was found
that the calculated equilibrium lattice constants increased with Al content.
In the range from 20 to 25 at.%, the theoretical data of $a_{0}$ for the D03
phase are closer to the experiment that corresponds to the existence of D03 as
the main structural phase. Shear moduli $C^{\prime}$ of A2, B2 and D03
structures are linearly decreased with adding Al atoms in compositions. The
significant difference between D03 (B2) and A2 values of $C^{\prime}$ in an
area higher than 15 at.% Al content could be related to the atomic volume
changes.

In A2 and B2 structures, the tetragonal magnetostriction constants are
positive, while in D03, this parameter is positive only up to 18 at.%. This
tendency depends on magnetocrystalline anisotropy energy behaviour as a
function of Al content. It was shown that the obtained values of
$\lambda_{001}$ for the B2 structure are closer to the experimental one. This
result could be explained by the proposed model of formation B2-like short-
range order that provide the giant magnetostriction in Fe-based alloys.

Using exchange interaction constants as input parameters, the Curie
temperatures were estimated with the help of Monte Carlo simulations. The
calculated Curie temperature dependence for the A2 structure has a maximum
value at 10 at.% while from experimental results, for the A2 structure, Curie
temperatures continuously decrease with increasing of Al concentration.

## Acknowledgment

This work was supported by the Ministry of Science and Higher Education of the
Russian Federation within the framework of the Russian State Assignment under
contract No. 075-00992-21-00. MVM gratefully acknowledges the Advanced science
research foundation of the Chelyabinsk State University.

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