# Exploring UAV Networking from the Terrain Information Completeness
Perspective: A Tutorial

Zhengying Lou, Ruibo Wang,
Baha Eddine Youcef Belmekki, ,
Mustafa A. Kishk, Member, IEEE, and Mohamed-Slim Alouini, Fellow, IEEE
Zhengying Lou, Ruibo Wang, Baha Eddine Youcef Belmekki and Mohamed-Slim
Alouini are with King Abdullah University of Science and Technology (KAUST),
CEMSE division, Thuwal 23955-6900, Saudi Arabia. Mustafa A. Kishk is with the
Department of Electronic Engineering, Maynooth University, Maynooth, W23 F2H6,
Ireland. (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>mustafa.kishk @mu.ie;
slim.alouini@kaust.edu.sa).

###### Abstract

Terrain information is a crucial factor affecting the performance of unmanned
aerial vehicle (UAV) networks. As a tutorial, this article provides a unique
perspective on the completeness of terrain information, summarizing and
enhancing the research on terrain-based UAV deployment. In the presence of
complete terrain information, two highly discussed topics are UAV-aided map
construction and dynamic trajectory design based on maps. We propose a case
study illustrating the mutually reinforcing relationship between them. When
terrain information is incomplete, and only terrain-related feature parameters
are available, we discuss how existing models map terrain features to blockage
probabilities. By introducing the application of this model with stochastic
geometry, a case study is proposed to analyze the accuracy of the model. When
no terrain information is available, UAVs gather terrain information during
the real-time networking process and determine the next position by collected
information. This real-time search method is currently limited to relay
communication. In the case study, we extend it to a multi-user scenario and
summarize three trade-offs of the method. Finally, we conduct a qualitative
analysis to assess the impact of three factors that have been overlooked in
terrain-based UAV deployment.

###### Index Terms:

UAV deployment, terrain information, blockage verification, network
performance.

##  I Introduction

###  I-A Motivation

In urban areas, factors such as high population density, towering buildings,
and complex terrain may pose challenges to the stability and coverage of
wireless communication networks [1, 2]. Moreover, in places with large
gatherings and high population mobility, such as sports stadiums and event
venues, ensuring the reliability and stability of network connections is also
an important issue [3, 4, 5]. It is foreseeable that future networks will
achieve more granular and seamless coverage [6]. Furthermore, with the real-
time changes in user demands, network services should be adaptive and self-
evolving [7]. The UAV network is one of the best solutions to meet these
advanced communication requirements [8, 9]. It can assist ground networks in
providing coverage for wireless network blind spots. Additionally, it is
expected to serve forthcoming aerial users and vehicles, including electric
vertical take-off and landing (eVTOL) aircraft, in urban air mobility (UAM)
[10]. At the same time, the mobility of UAVs enables the network to be
deployed in a real-time and flexible manner according to actual needs [11].

However, UAV communication also faces significant limitations. Since UAVs are
often used for temporary and dynamic coverage, they frequently cannot obtain a
stable and powerful power supply [12]. Therefore, UAVs are unable to sustain
high signal transmission power, and as a result, they often operate at lower
flight altitudes to provide better network services by approaching users
closely. Given their relatively low flight altitudes, UAVs are often deployed
in the gaps of obstacles’ shadowing, casting frequent blockage of line-of-
sight (LoS) UAV-user link [13]. Unfortunately, due to the limited transmission
power of UAVs, if the LoS is obstructed by blockages such as trees, the
received signal at the user is unstable and experiences significant
attenuation. The above reasons make terrain one of the most critical factors
affecting UAV deployment [14].

TABLE I: Summary of terrain-based UAV networking studies

Ref. | Completeness | Topology | Stage | Research objective | Application scenario
---|---|---|---|---|---
[15] | Complete | Single-hop | Hybrid | Optimize channel parameters learning | UAV-aided wireless network
[16] | Complete | One-to-many | Hybrid | Minimize the distances between UAVs and targets | Monitor moving targets
[17] | Complete | One-to-many | Hybrid | Optimize coverage and reduce overload | Cover irregular terrain surface
[18] | Complete | One-to-many | Hybrid | Improve the coverage and capacity | Establish links from a BS to users
[19] | Complete | One-to-many | Hybrid | Optimize the performance of localization | Localizing outdoor ground users
[20] | Incomplete | Single-hop | Offline | Predict the geometric LoS probability | Different urban environments
[21] | Incomplete | Single-hop | Offline | Evaluate the performance of UAV network | Finite interference nodes
[22] | Incomplete | Relay | Offline | Enhance system performance analysis | Hybrid cellular networks
[23] | Incomplete | Relay | Offline | Optimize the placement of relay UAVs | UAV swarm application
[24] | Incomplete | One-to-many | Offline | Maximize coverage and energy efficiency | Centralized town
[25] | Incomplete | One-to-many | Offline | Maximize wireless coverage on the ground | Rapid deployable relief networks
[26] | Incomplete | One-to-many | Offline | Maximize coverage and reduce consumption | Assist the terrestrial network
[27] | Incomplete | One-to-many | Offline | Optimize coverage and connectivity | Provide emergency communication
[28] | No information | Single-hop | Real-time | Maintain communication with base station | Complete flight mission
[29] | No information | Single-hop | Real-time | Ensure link quality on the shortest path | Anti-jamming communication
[30] | No information | Relay | Real-time | Boost capacity with linear search length | Provide coverage extension
[31] | No information | Relay | Real-time | Develop search strategies for UAV placement | Densely obstructed areas
[32] | No information | Relay | Real-time | Optimize the performance of relay links | Construct relay links to two users

###  I-B Challenges of Terrain Information Utilization

Except for a few scenarios in remote areas with expansive visibility, a large
amount of literature on UAV communication networks has taken the influence of
terrain into account. In summary, terrain-based UAV deployment is a complex
problem that primarily involves the following three challenges.

Firstly, due to the highly complex topology of buildings, the UAV deployment
should be fine-grained. Once the UAV or user has a few meters of movement,
they may disappear within each other’s LoS region [33]. Secondly, as UAVs
approach users by reducing their altitude, the distance to obstacles also
decreases accordingly [25]. In other words, UAVs often incur a higher
probability of being blocked in exchange for reduced path loss. Lastly, when
UAVs prioritize providing better service to a few selected users by altering
deployment, it can potentially result in a degradation of communication
performance for other users. In summary, relevant studies focus on scenarios
with different levels of terrain information completeness and address
different challenges.

There are several surveys and tutorials delving into a wide range of topics
related to UAV communication, including channel modeling [14], integration
with 5G and future networks [34, 35, 36], and enhancements through artificial
intelligence [37]. However, none of these studies comprehensively explore UAV
deployment issues from the perspective of terrain information completeness.
Due to the widespread interest in terrain-based UAV deployment, it is
necessary to provide a comprehensive tutorial of the diverse research
conducted on this topic.

###  I-C Contribution

From the unique perspective of terrain information completeness, this article
provides a comprehensive tutorial and bibliographic index for readers
interested in the field of terrain-based UAV deployment. The core contents and
contributions of the article are outlined as follows:

  * •

In Sec. II, we categorize the existing studies into three classes from the
perspective of prior terrain information completeness.

  * •

In Sec. III, we review two pivotal aspects of the UAV-aided terrain
construction method and the UAV-based trajectory design with complete terrain
information. Then, a case study, for the first time, integrates these two
approaches, drawing meaningful conclusions based on the interaction between
these methods.

  * •

In Sec. IV, we review the air-to-ground LoS probability model (A2GLPM),
summarize how existing studies utilize this model in UAV deployment with
incomplete terrain information. Then, a case study is proposed to validate the
accuracy of the model.

  * •

In Sec. V, we review the core concepts of real-time search algorithm, which
can be implemented without prior terrain information. Additionally, in the
case study, we have advanced the algorithm’s capability from serving a single
user with one UAV to serving multiple users.

  * •

In Sec. VI, we discuss several factors that can not be ignored in terrain-
based UAV deployment but have received limited attention.

##  II Classifications of Existing Literature

In this subsection, we classify the existing terrain-based UAV positioning
studies from different perspectives. Table I displays the categories to which
the existing studies belong under different classifications. The
classification based on terrain information completeness determined the main
framework of this paper. The correspondence between this classification and
the other classification approaches is also a key focus in this subsection.

###  II-A Terrain Information Completeness

As mentioned, terrain information completeness is classified into complete,
incomplete, and no terrain information.

####  II-A1 Complete Terrain Information

Complete terrain information means the available terrain information enables
the terrain construction to a certain degree of granularity [33]. This
information should at least include the positions, heights, and shapes of
obstacles such as buildings. In such situations, the most critical issue is
how to acquire and periodically update three-dimensional (3D) terrain maps at
a relatively lower cost.

####  II-A2 Incomplete Terrain Information

Due to the high cost of getting and updating complete 3D maps, a compromise
solution is to record the features of occlusions with a few parameters. For
instance, we can characterize the building density by the ratio of the
building footprint area to the total area of the region [38]. Undoubtedly,
since the available terrain information is compressed as characteristic
parameters, UAVs are unable to accurately verify occlusion, resulting in a
coarse-grained deployment. In this case, the most critical task is to map
these characteristic parameters onto the blockage probability of the UAV-user
link.

####  II-A3 No Terrain Information

No terrain information refers to the condition that prior information can not
provide direct assistance or does not have a direct impact on the UAV
deployment. Without relying on prior terrain information, UAVs have to collect
terrain information during networking to achieve terrain-based deployment. Due
to the limitations on power, UAVs have a restricted search trajectory length
during real-time networking. Finding the optimal deployment position that
avoids obstacles’ occlusion has become the most critical issue, achieved
through searching the shortest possible distance [28].

###  II-B Network Topology

In terms of network topology, there are three types of transmission modes:
single-hop transmission, relay transmission, and one-to-many transmission.

####  II-B1 Single-Hop

Single-hop communication is the fundamental communication structural unit,
thus it has been mentioned in all three cases of terrain information
completeness [15, 21, 28]. The most extensively discussed aspect of single-hop
communication is the issue of blockage detection. This involves determining
the blockage probability based on the relative positions of the UAV and the
user, as well as designing a moving trajectory when blocked to re-establish
the LoS link.

####  II-B2 Relay

In the no terrain information scenario, researchers have simplified the
complex study of one-to-many transmission by focusing on relay transmission
scenarios to optimize UAV deployment [30]. Due to the absence of a direct LoS
link between the transmitter and receiver, they need to perform relay
transmission through a UAV located in their common LoS area. In relay
transmission, the key issue is how to find the common LoS region and
simultaneously optimize the performance of both links.

####  II-B3 One-to-Many

One-to-many communication is suitable for scenarios with prior terrain
information. When complete terrain information is available, UAVs achieve
real-time deployment through fine-grained occlusion assessment [16, 17].
However, in the absence of complete terrain information, UAVs can only resort
to statistical models for coarse-grained deployment [26, 27].

###  II-C Stage

Based on whether UAVs are initiating network services, the acquisition and
utilization of terrain information by UAVs can be divided into two stages:
real-time and offline.

####  II-C1 Real-Time

In the no terrain information scenario, UAVs acquire terrain information and
deploy based on terrain after starting service [28]. Therefore, no terrain
information corresponds to the real-time stage.

####  II-C2 Offline

Some tools can take terrain information as inputs, mapping them to target
performance metrics, such as coverage probability and data rate [39, 40]. As
the entire process can be completed before networking, the deployment in the
incomplete terrain information scenario is done in the offline stage.

####  II-C3 Hybrid

As for UAV deployment with complete terrain, it is primarily divided into the
offline stage of map construction and the real-time stage involving UAV flight
trajectory design with the assistance of the map [16]. Therefore, it is a
deployment method with a hybrid blend of offline and real-time stages.

##  III Complete Terrain Information

In this section, we review two crucial aspects in the context of complete
terrain information: the UAV-aided terrain construction method and the
trajectory design method for UAVs in dynamic scenarios. Then, a case study is
proposed as an example of the interaction of two aspects of research content.

###  III-A UAV-Aided Terrain Construction

The UAV-aided terrain construction method is one of the key techniques for
obtaining complete terrain information [41]. This method creates a map that
records the signal strength at different locations and infers the position and
shape of obstacles by map. Its principle is verifying blockage based on the
difference in signal strength between LoS and non-line-of-sight (NLoS) links.

According to the air-to-ground channel model proposed in [42], the received
power $S_{Q}$ can be expressed as

$S_{Q}=\zeta\,G_{Q}\,\eta_{Q}\,d^{-\alpha_{Q}},$ (1)

where $d$ is the Euclidean distance between the user and the UAV. $G_{Q}$
denotes the small-scale fading of the channel, which follows Nakagami-$m$
fading with scale parameter $m_{Q}$ [43]. The meanings and the recommended
values of the remaining parameters are provided in Table II. $Q$ in (1) is
substituted by LoS if a LoS UAV-user link is established, otherwise $Q$ is
replaced with NLoS.

TABLE II: Parameters of the air-to-ground channel model [42] Meaning | Notation | LoS | NLoS
---|---|---|---
Average additional loss | $\eta_{Q}$ |  $-35$ dB |  $-48$ dB
Path-loss exponent | $\alpha_{Q}$ | $2$ | $2.3$
Scale parameter | $m_{Q}$ | 1 | $2$
Transmission power of UAV | $\zeta$ |  $30$ dBm
Figure 1:  Schematic diagram of UAV–aided terrain construction principles.

The left part of Fig. 1 shows the principle of UAV-aided terrain construction.
The link between the middle UAV and the ground user (indicated by the orange
line) is blocked by a building, resulting in the user’s received power being
$S_{\mathrm{NLoS}}$. When the UAV approaches the user and crosses over the
upper contour of the building shadow, the UAV-user link (the blue line)
becomes LoS, and the user’s received power is $S_{\mathrm{LoS}}$. Typically,
the received signal power exhibits continuous variations with the continuous
movement of the UAV. However, when crossing over the shadow contour of the
building, according to (1), the received signal power generally experiences a
drastic change exceeding $20$ dB. Therefore, the upper contour of the building
should be located on the blue line. Similarly, the UAV can detect the right
contour of the building shadow by shifting to the right.

Furthermore, there are two remarks worth mentioning. Firstly, to accurately
determine the building’s position, height, and shape, it often requires the
participation of multiple users (or ground signal collectors) in the
construction process. In fact, with a larger number of users, the construction
can process faster. The right of Fig. 1 provides an example of multi-user-
assisted construction, where orange and green squares represent blocked and
unblocked areas respectively. Secondly, the segmented propagation model is
proposed to describe different degrees of attenuation caused by different
types of obstacles, such as solid obstacles (e.g., buildings) and light
obstacles (e.g., vegetation blockage). With this model, a greater variety of
obstacles can be constructed with more precision and detail [44].

###  III-B Trajectory Design in Dynamic Scenario

The design of UAV trajectories is a critical topic in many applications, such
as surveying, searching, and monitoring, as it directly influences the
effectiveness and performance of UAV mission execution [16, 19]. So far, with
the robust computational power of computers, the trajectory planning of UAVs
in static scenarios can be simulated through stochastic algorithms in the
offline stage. Terrain-based UAV trajectory design typically operates within a
small urban area, and in static scenarios with complete terrain information,
it can even be accomplished through exhaustive search. Therefore, trajectory
design in dynamic scenarios holds greater significance and poses more
challenges.

In dynamic scenarios, UAVs need to adjust their deployment positions based on
the real-time locations of users. At this point, if the user’s movement
direction is entirely random, the UAV can not determine its deployment
position for the next moment. Therefore, authors in [16] proposed a scenario
in which the UAV monitors the parade crowd moving according to a specific
trajectory. Given that complete terrain information is available, UAVs attempt
to get closer to the crowd to achieve better monitoring effectiveness while
avoiding obstruction from buildings.

###  III-C Case Study: UAV Tracking During the Parade

####  III-C1 Research Objective

In existing studies, the topics of UAV-aided terrain construction and
trajectory design in dynamic scenario have been discussed separately. However,
the two should be closely connected. On the one hand, evaluating the
effectiveness of terrain construction in real-world application scenarios is
the most direct approach. On the other hand, assuming the availability of
complete terrain information for dynamic UAV tracking does not align with
practical scenarios in many cases. Therefore, the most important objective of
this case study is to explore the interaction between terrain construction and
dynamic UAV tracking. Given the known parade route, UAVs can primarily focus
on terrain construction in the vicinity of the route to reduce the amount of
terrain information collected. Conversely, the impact of errors in terrain
construction on UAV tracking can also be studied.

In this case study, we first utilize UAVs for terrain construction, followed
by real-time UAV tracking of parade crowds. The constructed terrain assists
the UAVs in avoiding blocking from buildings during tracking trajectory
design. The algorithm provided in [16] has been demonstrated to effectively
track the crowd while avoiding blockage from buildings. However, the scenario
is simplified, with users’ potential positions restricted to several discrete
points along the parade route and the shadow areas of buildings are
represented by regular polygon topologies. Therefore, we test the
effectiveness of the proposed algorithm in a more general scenario.

####  III-C2 UAV-Aided Terrain Construction

Figure 2: UAV-aided terrain construction and parade crowd tracking.

The left upper and lower parts of Fig. 2 show the 3D and planar city topology,
respectively. The buildings are distributed in a square area of $1$
km$\times$$1$ km. To avoid the effect of the building orientation, we refer to
the recommendations in [20] and model the building as a cylinder with a radius
of $8$ m. The center of the cylindrical building follows a homogeneous Poisson
point process (PPP) with a density of
$500~{}{\mathrm{buildings}}/{\mathrm{km}}^{2}$. The height of the building
obeys log-normal distribution with parameters $3$ and $0.4$, which are the
mean and standard deviation of the height’s logarithm, respectively111 Note
that all of the case studies are performed by MATLAB. The buildings, UAVs, and
users’ location generations and sampling procedures are implemented through
self-written scripts (with ”Editor” in MATLAB)..

We only deploy ground signal receivers near the parade route, and UAVs also
hover only in these areas, constructing terrain based on the algorithm
proposed in [41]. As shown in the upper right part of Fig. 2, the constructed
buildings have an acceptable deviation from the original terrain in the
vicinity of the route. However, several buildings far from the route have a
significant error in terms of altitude, even some buildings have not been
successfully constructed.

####  III-C3 Parade Crowd Tracking

As shown in the right lower part of Fig. 2, two sets of crowds are moving at
an average pace of $50$ m/slot along the predefined parade trajectory on the
road. The shaded part provides the blocked area when the UAV is at $30$ m
height. Each crowd is tracked by one UAV. The positions of each user in the
two crowds at different time slots are marked in the right lower part of Fig.
2. The users in the crowd are generated near the preset central position and
follow PPP at slot $1$. One crowd departs from the bottom right and exits on
the left, while another crowd departs from the bottom left and exits on the
top right. There is a segment of overlapping routes between two intersections
at slots $3$ and $4$, leading to a certain degree of mixing of the crowds.
Furthermore, the coverage is significantly influenced by terrain blockage at
slots $3$ and $4$. With the help of the constructed terrain, the flight path
of the UAV can more accurately avoid the blockage of the building and adapt to
the more complex movement trajectory during these challenging slots for UAV
tracking.

##  IV Incomplete Terrain Information

This section reviews the A2GLPM, where the LoS probability is the probability
of establishing the LoS link between the transmitter and the receiver. After
that, we summarize how existing studies deploy UAV networks and analyze
network performance based on this model. Considering that the accuracy of the
model has not been fully validated, we propose a case study to analyze the
accuracy of the model itself and the accuracy of network performance analysis
based on this model.

###  IV-A LoS Probability Model

The International Telecommunication Union (ITU) has proposed a standardized
model for urban areas based on three simple terrain feature parameters that
can describe the general geometric statistics of cities [45]:

  * •

$\kappa$ represents the ratio of the buildings’ footprint area to the total
area (dimensionless).

  * •

$\iota$ represents the average number of buildings per unit area
(buildings$/$km2).

  * •

$\omega$ is the scale parameter of Rayleigh distribution that describes the
height of the building (dimensionless).

Figure 3:  Schematic diagram of user distribution and UAV-user relative
position.

Based on these three parameters, the probability of the UAV and user
establishing an LoS link $P_{\rm{LoS}}(\theta)$ is given by the A2GLPM [25],

$\begin{split}P_{\rm{LoS}}(\theta)=\frac{1}{1+a\exp\left({-b\left(\theta-a\right)}\right)},\end{split}$
(2)

where $\theta=\arctan\left({{h}/{\sqrt{d^{2}-h^{2}}}}\right)$ is the elevation
angle, $d$ is the Euclidean distance between the UAV and user, $h$ is the
altitude of UAV. An example of the elevation angle is given in Fig. 3. $a$ and
$b$ can be presented by the terrain feature parameters:

$Q=\sum_{j=0}^{3}\sum_{i=0}^{3-j}C_{i,j}\left(\kappa\iota\right)^{i}\omega^{j},$
(3)

where $Q\in\\{a,b\\}$, $C_{i,j}$ are polynomial coefficients. The values of
$C_{i,j}$ can refer to Table I and Table II in [25].

As a result, the A2GLPM maps the terrain feature parameters to the blockage
probability, given the relative positions of the UAV and user.

###  IV-B Stochastic Geometry (SG) Analytical Framework

The authors in [42] integrated the A2GLPM into the SG analytical framework,
laying the foundation for subsequent extensive research in this field. From
the perspective of a typical UAV, users can be categorized into LoS users and
NLoS users. Specifically, all of the users are assumed to follow a homogeneous
PPP with density $\lambda$. Then, the PPP is divided into two sub-PPPs:

  * •

An LoS-PPP with density $\lambda P_{\mathrm{LoS}}(\theta)$ records the
positions of users that are not blocked (referred to as LoS users).
$P_{\mathrm{LoS}}(\theta)$ is defined in (2).

  * •

An NLoS-PPP with density $\lambda\left(1-P_{\mathrm{LoS}}(\theta)\right)$
records the positions of users that are blocked (referred to as NLoS users).

As shown in Fig. 3, when the altitude of the UAV is fixed, the density of LoS
users decreases with increasing distance to the UAV.

Then, we take the downlink coverage probability from the perspective of a
typical user as an example to illustrate how terrain information influences
the SG analyzing process. The typical user is served by the UAV that provides
the maximum average received power, while other UAVs are interfering UAVs. The
interference power is the cumulative received power at the user side,
attributed to interfering UAVs. The coverage probability is defined as the
probability that the received signal-to-interference plus noise ratio is
larger than a given coverage threshold [46]. From this, it can be observed
that coverage probability is closely linked to the distances between users and
LoS and NLoS UAVs, and to a large extent, it depends on the accuracy of the
A2GLPM.

With the SG analytical framework, coverage probability can be expressed as a
function of terrain feature parameters and network distribution parameters,
such as UAV deployment density and altitude. Therefore, the derived expression
for coverage probability can not only be utilized for performance estimation
but also serve as a tool for optimizing the deployment of UAV networks.
However, due to incomplete terrain information, the deployment is coarse-
grained. Compared to existing deployment algorithms in the complete terrain
scenarios, the SG analytical framework also possesses its unique advantage,
that is, the capability for interference analysis [42].

###  IV-C Case Study: Accuracy of the Model

####  IV-C1 Research Objective

The previous subsection emphasized the importance of model accuracy in
performance analysis. Therefore, this subsection has three objectives
regarding accuracy verification. First of all, to achieve a concise
expression, A2GLPM neglects the correlation of user positions. In other words,
the probability of two adjacent users being blocked remains mutually
independent in A2GLPM, while two adjacent users are likely to be blocked
simultaneously in reality. Therefore, we hope to investigate whether the
correlation between user positions can be neglected in the blockage
verification. Secondly, the A2GLPM is presented as a modified Sigmoid function
containing two degrees of freedom, namely, $a$ and $b$. An exploratory issue
is whether other commonly applied functions involving two degrees of freedom
can provide a more accurate description of the relationship between elevation
angle and LoS probability. Finally, we aim to find the difference between the
approximate coverage probability results obtained through A2GLPM for LoS
probability and the accurate coverage probability obtained through simulation
with precise blockage verification.

####  IV-C2 Model Accuracy Analysis

In this case study, UAVs and users are distributed in a square area of $1$
km$\times$$1$ km, forming two homogeneous PPPs. The density of users is fixed
as $1000~{}{\mathrm{users}}/{\mathrm{km}}^{2}$. The height of UAVs is $80$ m.
The distribution of buildings is the same as in Sec. III-III-C In Fig. 4, we
plot the sample data obtained by simulation with dots. For a fixed elevation
angle, the LoS probability is calculated by the proportion of the number of
samples of UAV-user links not blocked by cylindrical buildings in the total
number of samples.

Figure 4: LoS probability at different elevation and azimuth angles.

The relationship between elevation angle and LoS probability is fitted with
three different functions:

  * •

Modified Sigmoid function: $[a,b]=[2.8240,0.0628]$,

$P_{\mathrm{LoS}}^{\mathrm{Sig}}(\theta)=\frac{1}{1+{a\exp(-b(\theta-a))}}.$
(4)

  * •

Modified Tanh function: $[a,b]=[0.8794,2.0984]$,

$P_{\mathrm{LoS}}^{\mathrm{Tanh}}(\theta)=a\,\frac{\exp(2b\theta)-1}{\exp(2b\theta)+1}.$
(5)

  * •

Modified ReLu function: $[a,b]=[0.6775,0.2697]$,

$P_{\mathrm{LoS}}^{\mathrm{ReLu}}(\theta)=\max\\{0,a\theta+b\\}.$ (6)

The parameters $[a,b]$ are obtained from the minimum mean square error (MSE)
between the sample data and the corresponding value of the fitted curve. The
MSEs for Sigmoid, Tanh and ReLu functions are $\\{2.12,0.25,4.26\\}\times
10^{-3}$ respectively. The accuracy of the fitting by the three functions can
also be visually observed through Fig. 4. The above results indicate that the
improved Sigmoid function in [38] can accurately describe the relationship
between LoS probability and elevation angle. However, there might be better
alternatives, such as the improved Tanh function.

Next, the correlation between user positions is measured by the azimuth angle.
As drawn in Fig. 3 the azimuth angle takes the projection of the UAV on the
ground as the endpoint and the rays from the projections to two users (denoted
as user A and user B) as the edges. When calculating the LoS probability for
azimuth angle in Fig. 4, we verify whether the UAV-user B link is blocked,
given that a LoS link is established between UAV and user A. At a fixed
elevation angle, the LoS probability is independent of the azimuth angle as
long as the azimuth is greater than a threshold, which is around $10$ degrees
in this case, as shown in Fig. 4. The threshold of independence is related to
the footprint area of buildings. Furthermore, under a fixed elevation angle
$\theta_{0}$, the LoS probability for azimuth angle converges to
$P_{\mathrm{LoS}}(\theta_{0})$ as the azimuth angle increases.

####  IV-C3 Network Performance Accuracy Analysis

Now, we identify the variance between the approximate coverage probability
results obtained by the SG framework and the accurate coverage probability
obtained through simulation with accurate blockage verification. Unless
otherwise stated, channel parameters are set to their default values given in
Table II. The environmental noise power is $\sigma^{2}=-90$ dBm. The points
labeled as ”SG-based Analysis” in Fig. 5 are obtained by Theorem 2 in [42].
The A2GLPM follows the modified Sigmoid function with terrain related
parameters $[a,b]=[2.8240,0.0628]$. For lines labeled as ”Terrain-based
Analysis” in Fig. 5, complete terrain information is available for UAVs.
Blockage verification is no longer based on any statistical model but is
determined through simulation.

Figure 5: Coverage probability with different UAV densities.

Fig. 5 shows that the coverage probability obtained under the SG framework has
about a $9\%$ relative deviation compared to the accurate one. Given that the
SG-based approach relies on only two parameters to incorporate terrain
information, the relative deviation is acceptable, especially when the
coverage probability is high. Another interesting observation that can be made
from Fig. 5 is that the coverage probability estimated through the SG
framework is consistently slightly smaller than the accurate values. The
primary reason for this phenomenon is that the A2GLPM underestimates the LoS
probability for associated UAV or overestimates the LoS probability for
interfering UAVs.

Furthermore, this case study validates the feasibility of optimizing UAV
deployment density through SG-based analytical results. Fig. 5 shows the UAVs’
density that maximizes the coverage probability is around $12$ UAVs$/$km2,
which is independent of the coverage threshold. When the deployment density of
UAVs is low, users may be at a considerable distance from the serving UAV, and
there is a possibility of no UAV within the line of sight being available.
Conversely, if the deployment density is too high, interference becomes a
primary factor limiting coverage probability. It is evident that the
optimization of network-level parameters, such as UAV density, relies on the
unique interference analysis capability of the SG framework. The optimal value
is challenging to attain through the design of UAV tracking or search
algorithms mentioned earlier.

##  V No Terrain Information

This section centers on the exploration of the real-time search algorithm. A
detailed overview of the algorithm’s objectives, principles, and core concepts
is provided. Additionally, the specific execution process is illustrated
through heat maps. Considering that the existing methods only apply to the
relay scenario, we extended the algorithm to one-to-many scenario. Since the
real-time search does not require prior terrain information, it is limited by
many factors and requires trade-offs between practicability and performance.

###  V-A Real-time Search Method

In simple terms, the real-time search method is an online realization of UAV-
assisted terrain construction. Real-time search algorithms are typically
designed to address a class of max-min problems in the field of wireless
communication. The application scenario in [30] is taken as an example to
illustrate the optimization problem. In this scenario, the UAV serves as a
relay between a user and an aerial base station (ABS). The optimization
objective is to identify an optimal UAV deployment position that maximizes the
smaller of the two SNRs: the average SNR of the ABS-UAV link and the average
SNR of the UAV-user link. It can be mathematically represented as a special
case of the following optimization problem when $N=2$:

$\underset{x\in\mathcal{X}}{\arg\max}\
\min\left\\{\frac{\mathbb{E}_{G}[S_{Q,1}]}{\sigma^{2}},\frac{\mathbb{E}_{G}[S_{Q,2}]}{\sigma^{2}},\dots,\frac{\mathbb{E}_{G}[S_{Q,N}]}{\sigma^{2}}\right\\}$
(7)

$N$ represents the number of devices simultaneously served by the UAV. $x$
denotes the position of the UAV, and $\mathcal{X}$ is the set recording the
locations that are allowed for deployment. $S_{Q,1}$ and $S_{Q,2}$ are
receiver power of ABS-UAV link and UAV-user link, where
$Q=\\{\mathrm{LoS},\mathrm{NLoS}\\}$. Due to the fact that received power is
not only dependent on the distance between the UAV and the serviced device but
also on whether the link is blocked. Since the terrain is complex and
irregular, this problem is non-convex and can only be solved through
algorithmic search.

Although the real-time search algorithm can be considered as a specific type
of UAV-assisted terrain construction method, there are still distinct
differences between the two [47]. Firstly, the search trajectory of real-time
search algorithms should be linear, because the limited power supply of UAVs
makes it challenging to support search trajectories with quadratic or higher
complexity [48]. Linearity is typically in reference to the distance between
the two furthest serviced users. Due to strict limitations on the search
trajectory length, the search direction at each iteration needs to be
carefully designed based on terrain information gathered through previous
searches. In addition, real-time search involves the concept of granularity,
which can be understood as the step size of the UAV at each iteration. Due to
the need for the UAV to initiate communication with users each time it reaches
a new search position to assess whether the link is blocked, larger
granularity allows the UAV to search for potential optimal positions more
quickly. However, it also increases the probability of missing the optimal
position.

###  V-B Case Study: The Extension of Real-time Search

####  V-B1 Research Objective

In this section, we first illustrate the core idea of the algorithm through a
relatively simple example to provide an intuitive understanding. Additionally,
we will clarify that the algorithm’s search trajectory is linear, and how it
is designed to search for potential optimal positions. Secondly, since the
previous study focuses on a relatively simple relay scenario, and one UAV only
serves one user. Therefore, the case study aims to generalize the existing
method to a one-to-many scenario. Finally, we will compare the average
coverage performance under different levels of terrain completeness.

####  V-B2 Method Principle

Figure 6: An example of real-time search.

In this case study, the granularity of UAV search is $0.2$ m, and the coverage
threshold is set to $4$ dB. The optimization objective remains as specified in
(7). To facilitate the understanding of algorithmic concepts, we start from a
local area on the map and discuss a relatively simple scenario. The scenario
involves one UAV, two users, and three buildings, which is shown in the middle
part of Fig. 6. The real-time search algorithm is based on two core ideas:

  * •

$(c_{1})$: When the UAV is not blocked, it moves in the direction where there
is a steep increase in the average SNR.

  * •

$(c_{2})$: When the UAV is blocked, the UAV moves along a trajectory where the
average SNR does not change to avoid missing the next LoS region.

The objective of the max-min problem is to emphasize that UAVs can provide
equitable service quality for two users. Therefore, a significant portion of
the UAV’s search trajectory is likely to be along a plane where the optimal
deployment position is equidistant to both users, i.e., along the mid-
perpendicular plane. In the left part of Fig. 6, we plot a heat map to show
the smaller average SNR on the mid-perpendicular plane of two users. In this
heat map, the contour of buildings’ shadows is quite clear due to the intense
changes in SNR.

The UAV starts from above, initially descending rapidly based on the first
core idea $(c_{1})$ until it reaches the contour of the building’s shadow.
Then, part of the trajectory overlaps with the contour because the UAV
repeatedly enters the NLoS region of one user according to $(c_{1})$ and then
comes back to the common LoS area according to $(c_{2})$ in a bent route.
Finally, after reaching the suboptimal position, the UAV follows a left search
trajectory (purple curve) and a right search trajectory (blue curve) where the
average SNR remains constant on these trajectories according to $(c_{2})$. The
marked suboptimal position may not necessarily be optimal because the optimal
position may not lie within the mid-perpendicular plane222In fact, a
suboptimal position with an elevation angle to the users of less than 45
degrees is a sufficient but not necessary condition for it to be the optimal
relay position.. As shown in the right part of Fig. 6, the trajectory from the
perspective of the horizontal plane implies the searching trajectory is
linear.

####  V-B3 Generalized Algorithm

Next, we generalize the proposed algorithm into a multi-UAV serving multi-user
scenario. The buildings follow the same distribution as in Sec. III-III-C The
user distribution, UAV distribution, and channel model are the same as in Sec.
IV-IV-C. The density of UAV is set to optimal $12$ UAVs$/$km2 one obtained in
Sec. IV-IV-C.

The core idea of the generalization is simply described as follows. The UAV
focuses on the two most distant associated users to perform the proposed
algorithm. Starting from a random position, the UAV takes the shortest path to
reach the mid-perpendicular plane and then executes the proposed real-time
search algorithm. In this process, the two users farthest from the UAV may
undergo changes, thereby causing the algorithm to initiate a repetition of the
initial steps. The UAV stops when the maximum search length is reached, or the
algorithm finishes.

Figure 7: The influence of search length on the performance of real-time
search.

In Fig. 7, we compare the coverage performance in scenarios with complete,
incomplete, and no terrain information. The 3D exhaustive search is executed
in the scenario with complete terrain information to serve as an upper bound.
In the best-fitted SG-based deployment, UAVs follow a PPP at the optimal
altitude that maximizes the coverage probability given the density is $12$
UAVs$/$km2. This is considered as UAV deployment under incomplete terrain
information. In no terrain information, once the search trajectory length
reaches the preset upper limit ($x$-axis in Fig. 7), the search stops, and the
UAV is deployed to the position with the maximum coverage probability along
the trajectory. The $y$-axis in Fig. 7) represents the mean coverage
probability for all users in the area.

As shown in Fig. 7, with a smaller user density, the coverage performance of
the real-time search algorithm is better and closer to the upper bound
provided by exhaustive search. However, due to the limitations of a linear
search trajectory, the UAV can not achieve complete terrain information
through real-time searching. Therefore, regardless of the upper limit set for
the search trajectory, the coverage performance of real-time search can never
reach the upper bound provided by exhaustive search. The intersection point
between the curve corresponding to the real-time search algorithm and the
straight line corresponding to the SG-based deployment represents the moment
when the terrain information obtained by the UAV through real-time search is
equal to the terrain information contained by the terrain feature parameters.

###  V-C Trade-offs between Practicability and Performance

Without prior terrain information required, the real-time search method can
find the optimal position with a linear search distance. It is not difficult
to imagine that the method suffers several drawbacks, resulting in trade-offs
between practicability and performance.

####  V-C1 Optimality and Number of Served Users

The real-time search method can ensure optimality when one UAV serves one
user. When the UAV is serving two users, the optimality can only be satisfied
under certain conditions. In one-to-many network topology, when a UAV
approaches one user, the links with other users might be blocked, so achieving
optimality is challenging.

####  V-C2 Update Frequency and Power Consumption

Frequently updating real-time search will lead to a large amount of power
consumption, probably resulting in the UAV’s inability to maintain high-
quality communication. However, a low updating frequency may also cause a
degradation of communication quality as whether users enter the shadow of the
building is unknown to the UAV.

####  V-C3 Search Length and Prior Terrain Information

At each update, the determination of the starting search position partly
depends on the prior terrain information. More terrain information often leads
to a starting position closer to the potential optimal location, thus
shortening the search length. Utilizing the terrain information recorded in
the search process before updating can be an interesting research direction of
real-time search methods.

##  VI Further Discussions

This section explores three overlooked factors in terrain-based UAV deployment
that warrant attention.

###  VI-A Charging

As mentioned, power supply is an important factor affecting the hovering time
and communication quality of UAVs. When charged by wireless laser power beams
[49], UAVs should also pay attention to whether the power supply equipment’s
link is blocked by obstacles, especially in high-rise urban regions. The
establishment of aerial power stations with a higher altitude can improve
charging efficiency.

###  VI-B Backhaul

In the one-to-many scenario, the backhaul link between the UAV and the base
station from the core network bears the sum of data rates for all users [50].
For base stations with heights lower than buildings, UAVs can also ensure that
the backhaul link is not blocked by employing real-time search methods.
Additionally, UAVs should promptly reduce the distance to the base station
when the backhaul link experiences a significant communication load.

###  VI-C Electromagnetic field (EMF) exposure

When lowering EMF exposure is emphasized, UAVs might be deployed below the
optimal density to reduce EMF exposure from interference signals [51].
Compared to offline deployment relying on terrain information, real-time
search offers greater advantages. The optimal deployment positions identified
through real-time search are characterized by lower altitudes, resulting in
minimized interference to users.

##  VII Conclusion

The article classified and discussed terrain-based UAV network deployment
methods from the perspective of terrain information completeness. It
introduced the core algorithms and models involved, discussed their
corresponding application scenarios, and provided case studies as examples. In
the first case study, we demonstrated that UAVs could provide stable coverage
for regularly moving users using existing local terrain construction methods
with UAV sampling. In the second case study, a coarse-grained UAV deployment
was performed using a simple set of parameters that characterized the terrain
features. A stochastic geometry framework provided general analytical results
for the aforementioned coarse-grained UAV networks. In the last case study,
the UAV was able to avoid building blockage without prior terrain information
through real-time linear-trajectory search.

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# Quantum Hall effect originated from helical edge states in Cd3As2

Rui Chen  Shenzhen Institute for Quantum Science and Engineering and
Department of Physics, Southern University of Science and Technology
(SUSTech), Shenzhen 518055, China  School of Physics, Southeast University,
Nanjing 211189, China  C. M. Wang  Department of Physics, Shanghai Normal
University, Shanghai 200234, China  Shenzhen Institute for Quantum Science and
Engineering and Department of Physics, Southern University of Science and
Technology (SUSTech), Shenzhen 518055, China  Shenzhen Key Laboratory of
Quantum Science and Engineering, Shenzhen 518055, China  Tianyu Liu  Max-
Planck-Institut für Physik komplexer Systeme, 01187 Dresden, Germany  Shenzhen
Institute for Quantum Science and Engineering and Department of Physics,
Southern University of Science and Technology (SUSTech), Shenzhen 518055,
China  Hai-Zhou Lu  Corresponding author<EMAIL_ADDRESS>Shenzhen
Institute for Quantum Science and Engineering and Department of Physics,
Southern University of Science and Technology (SUSTech), Shenzhen 518055,
China  Shenzhen Key Laboratory of Quantum Science and Engineering, Shenzhen
518055, China  X. C. Xie  International Center for Quantum Materials, School
of Physics, Peking University, Beijing 100871, China  Beijing Academy of
Quantum Information Sciences, Beijing 100193,China  CAS Center for Excellence
in Topological Quantum Computation, University of Chinese Academy of Sciences,
Beijing 100190, China

###### Abstract

The recent experimental observations of the quantum Hall effect in 3D
topological semimetals have attracted great attention, but there are still
debates on its origin. We systematically study the dependence of the quantum
Hall effect in topological semimetals on the thickness, Fermi energy, and
growth direction, taking into account the contributions from the Fermi-arc
surface states, confinement-induced bulk subbands, and helical side-surface
edge states. In particular, we focus on the intensively studied Dirac
semimetal Cd3As2 and its slabs grown along experimentally accessible
directions, including [001], [110], and [112]. We reveal an ignored mechanism
from the Zeeman splitting of the helical edge states, which along with Fermi-
arc 3D quantum Hall effect, may give a non-monotonic dependence of the Hall
conductance plateaus on the magnetic field in the most experimentally studied
[112] direction slab. Our results will be insightful for exploring the quantum
Hall effects beyond two dimensions.

_Introduction_ \- Since the discovery of the quantum Hall effect in 2D
electron gases Klitzing  _et al._ (1980); Thouless  _et al._ (1982),
tremendous efforts have been devoted to generalizing the exotic phase of
matter to higher dimensions Halperin (1987); Zhang and Hu (2001); Zilberberg
_et al._ (2018), in the absence of magnetic fields Yu  _et al._ (2010); Chang
_et al._ (2013), or nonlinear-response regime Sodemann and Fu (2015); Low  _et
al._ (2015); Ma  _et al._ (2019); Kang  _et al._ (2019); Du  _et al._ (2018,
2019). Recently, quantized Hall conductance plateaus have been observed in the
topological Dirac semimetal Cd3As2, with sample thickness ranging from tens to
hundreds of nanometers Zhang  _et al._ (2017, 2019); Nishihaya  _et al._
(2019); Lin  _et al._ (2019); Schumann  _et al._ (2018); Galletti  _et al._
(2018); Uchida  _et al._ (2017); Nishihaya  _et al._ (2018); Kealhofer  _et
al._ (2020). One of the mechanisms is a 3D quantum Hall effect supported by
the Fermi-arc surface states in the topological semimetal Wang  _et al._
(2017); Lu (2018), which also can support a quantum oscillation Potter  _et
al._ (2014); Moll  _et al._ (2016); Zhang  _et al._ (2016). Nevertheless,
because the samples are grown along various directions, the nature of the
quantum Hall effect in 3D topological semimetals is still in debates (Tab. 1)
and has been attracting growing attention.

Table 1: The slab growth direction, thickness, and explanation in the recent experiments on the quantum Hall effect in the topological semimetal Cd3As2. Refs. |  Direction |  Thickness (nm) |  Explanation in Refs.
---|---|---|---
Moll  _et al._ (2016) |  [010] |  150-2000 |  Weyl orbit
Kealhofer  _et al._ (2020) |  [001] |  45-50 |  Topological insulator type surface states
Zhang  _et al._ (2017),Zhang  _et al._ (2019) |  [112] |  55-71,80-150 |  Weyl orbit
Nishihaya  _et al._ (2019),Lin  _et al._ (2019) |  [112] |  80,100 |  Mixed Fermi arcs
Schumann  _et al._ (2018),Galletti  _et al._ (2018) |  [112] |  20,38-43 |  Surface states
Uchida  _et al._ (2017),Nishihaya  _et al._ (2018) |  [112] |  12-23,35 |  Bulk subbands

In this Letter, we report a new mechanism of the quantum Hall effect in
topological semimetals. We numerically calculate the Hall conductance of the
Dirac-semimetal Cd3As2 slabs grown along three experimentally accessible and
widely investigated crystallographic directions. For the slab grown along the
[001] direction, the magnitude of the quantized Hall conductance increases
with the increasing magnetic field, as a result of the Zeeman splitting of the
helical edge states on the side surfaces [Fig. 1]. The mechanism was
previously ignored and is originated from the nontrivial topology of the
confinement induced bulk subbands characterized by the spin Chern number. In
contrast, for the slab grown along the [110] direction, the Hall plateaus
decrease with the increasing magnetic field, due to the Fermi-arc 3D quantum
Hall effect Wang  _et al._ (2017). As a result, the Hall conductance in the
slab grown along the [112] direction can be understood as a competition
between the helical edge states and Fermi-arc surface states, with Hall
plateaus decreasing in the weak-field region but growing in the strong-field
region.

_Model and method_ \- We start with an effective Hamiltonian for the Dirac
semimetal Cd3As2 Wang  _et al._ (2013), which reads

$\displaystyle H$ $\displaystyle=$
$\displaystyle\varepsilon_{0}(\mathbf{k})+\begin{bmatrix}M(\mathbf{k})&Ak_{+}&0&0\\\
Ak_{-}&-M(\mathbf{k})&0&0\\\ 0&0&M(\mathbf{k})&-Ak_{-}\\\
0&0&-Ak_{+}&-M(\mathbf{k})\\\ \end{bmatrix},$ (1)

where $k_{\pm}=k_{x}\pm ik_{y}$,
$\varepsilon_{0}(\mathbf{k})=C_{0}+C_{1}k_{z}^{2}+C_{2}(k_{x}^{2}+k_{y}^{2})$,
and $M(\mathbf{k})=M_{0}+M_{1}k_{z}^{2}+M_{2}(k_{x}^{2}+k_{y}^{2})$. The $x$,
$y$, and $z$ axes in the Hamiltonian are defined along the [100], [010], and
[001] crystallographic directions, respectively. The model hosts two pairs of
Weyl nodes at $\mathbf{k}=\left(0,0,\pm k_{\text{w}}\right)$ with the energy
$E_{\text{w}}=C_{0}-C_{1}M_{0}/M_{1}$ and
$k_{\text{w}}=\sqrt{\left|M_{0}/M_{1}\right|}$. We take the parameters for
Cd3As2 as $C_{0}=-0.0145$ eV, $C_{1}=10.59$ eVÅ2, $C_{2}=11.5$ eVÅ2,
$M_{0}=0.0205$ eV, $M_{1}=-18.77$ eVÅ2, $M_{2}=-13.5$ eVÅ2, $A=0.889$ eVÅ Cano
_et al._ (2017). The samples of Cd3As2 are usually cleaved or grown along the
[110] Schumann  _et al._ (2016), [001] Kealhofer  _et al._ (2020), or [112]
directions Nishihaya  _et al._ (2018, 2019). To obtain the dispersion of the
Dirac semimetal slab along an arbitrary growth direction, we rotate the $y$
axis to the $y^{\prime}$ axis through the rotation matrix

$\begin{pmatrix}k_{x}^{\prime}\\\ k_{y}^{\prime}\\\
k_{z}^{\prime}\end{pmatrix}=\begin{pmatrix}\cos\alpha&\sin\alpha&0\\\
-\cos\theta\sin\alpha&\cos\theta\cos\alpha&\sin\theta\\\
\sin\theta\sin\alpha&-\sin\theta\cos\alpha&\cos\theta\end{pmatrix}\begin{pmatrix}k_{x}\\\
k_{y}\\\ k_{z}\end{pmatrix}.$ (2)

The [110], [112], and [001] directions correspond to
$\left(\theta,\alpha\right)=\left(0,-\pi/4\right)$,
$\left(\arctan{\sqrt{2}},-\pi/4\right)$, and $\left(\pi/2,-\pi/4\right)$,
respectively. We include a magnetic field always normal to the cleave surface
$\mathbf{B}=\left(0,B,0\right)$. The Zeeman term has the form
$H_{\text{Zeeman}}=\frac{\mu_{B}}{2}({\bm{\sigma}}\cdot\mathbf{B})\otimes\frac{1}{2}[g_{s}(\sigma_{0}+\sigma_{z})+g_{p}(\sigma_{0}-\sigma_{z})]$,
where $\mu_{B}$ is the Bohr magneton, $g_{s}=18.6$ and $g_{p}=2$ are the $g$
factors Jeon  _et al._ (2014).

The Hall conductance for a slab of thickness $L$ can be found as
$\sigma_{\mathrm{H}}=\sigma L$, where the Hall conductivity can be found from
the Kubo formula Zhang  _et al._ (2014)

$\sigma=\frac{e^{2}\hbar}{iV_{\mathrm{eff}}}\sum_{\delta^{\prime}\neq\delta}\frac{\left\langle\Psi_{\delta}\left|v_{x}\right|\Psi_{\delta^{\prime}}\right\rangle\left\langle\Psi_{\delta^{\prime}}\left|v_{z}\right|\Psi_{\delta}\right\rangle\left[f\left(E_{\delta}\right)-f\left(E_{\delta^{\prime}}\right)\right]}{\left(E_{\delta}-E_{\delta^{\prime}}\right)\left(E_{\delta}-E_{\delta^{\prime}}+i\Gamma\right)},$
(3)

where $e$ is the elementary charge, $\hbar$ is the reduced Planck constant,
$V_{\text{eff }}$ is the volume of the slab,
$\left|\Psi_{\delta}\right\rangle$ is the eigenstate of energy $E_{\delta}$
for $H$ in the $y^{\prime}$-direction magnetic field and open boundaries at
$y^{\prime}=\pm L/2,v_{x}$ and $v_{z}$ are the velocity operators, $f(x)$ is
the Fermi distribution.

Figure 1: The [001]-direction slab of the Dirac semimetal. (a) The helical
edge states on the side surfaces in the
$\left(x^{\prime},y^{\prime},z^{\prime}\right)$ coordinates. The arrows denote
the directions of propagation, and the red and blue colors distinguish
opposite spin polarizations. (b) The confinement-induced energy gap
$\Delta_{E}$ (red solid line) and the spin Chern number $C_{s}$ (green dashed
line) as functions of the slab thickness $L$. (c) The Hall conductance
$\sigma_{\text{H}}$ (blue dots) and $\sigma_{\text{H}}^{\uparrow,\downarrow}$
(blue and green lines) as functions of the magnetic field $1/B$ for $L=100$
nm. The red line correspond to
$\sigma_{\text{H}}^{\uparrow}+\sigma_{\text{H}}^{\downarrow}$. (d)
$\sigma_{\text{H}}$ as a function of $L$ and $1/B$. The Fermi energy
$E_{\text{F}}$ is at the Weyl node $E_{\text{w}}$.

_The [001] slab_ \- For a Cd3As2 slab grown along the [001] direction, the
bulk spectrum is quantized into discrete gapped subbands (See Sec. SI of Sup )
because of the quantum confinement effect Xiao  _et al._ (2015); Pan  _et al._
(2015); Chen  _et al._ (2017). The spectrum opens a gap, which decays with
increasing $L$ (probably with an oscillation as well). The effective
Hamiltonian $\mathcal{H}_{nn}$ for each subband $n(=1,2,\ldots)$ is equivalent
to a quantum spin Hall insulator Qi and Zhang (2011); Hasan and Kane (2010);
Bernevig  _et al._ (2006) characterized by the spin Chern number Lu  _et al._
(2010); Sheng  _et al._ (2006); Li  _et al._ (2010)
$C_{s}^{n}=(C_{n}^{\uparrow}-C_{n}^{\downarrow})/2$, where
$C_{n}^{\uparrow,\downarrow}=\pm\frac{1}{2}\left[\text{sign}\left(M_{0}+M_{1}n^{2}\pi^{2}/L^{2}\right)-\text{sign}\left(M_{2}\right)\right]$
Lu  _et al._ (2010) are the valence-band Chern numbers of the spin-up and
spin-down blocks of the $n$-th subband. Each Chern number
$C_{n}^{\uparrow/\downarrow}$ represents a chiral edge state circulating
around the side surfaces [Fig. 1(a)]. The total spin Chern number
$C_{s}=\sum_{n}{C_{s}^{n}}$ is equal to the number of pairs of helical edge
states. As shown in Fig. 1(b), the oscillatory decay of the band gap with
increasing $L$ is always accompanied by the variation of the spin Chern number
$C_{s}$ at each dip. In the Dirac semimetal Na3Bi, a topological phase
transition to the quantum spin Hall state has been observed Collins  _et al._
(2018).

However, the spin Chern number is not measurable because the measurable Hall
conductance is associated with the total Chern number
$\sigma_{\text{H}}=\frac{e^{2}}{h}\sum_{n,s=\uparrow,\downarrow}{C_{n}^{s}}$,
which is zero in the absence of the magnetic field because of time-reversal
symmetry. A magnetic field can break time-reversal symmetry as well as the
balance between $C^{\uparrow}_{n}$ and $C^{\downarrow}_{n}$, leading to
measurable Hall conductance $\sigma_{\text{H}}$ whose magnitude increases with
increasing magnetic field, as shown in Figs. 1(c-d). We also plot the Hall
conductance $\sigma_{\text{H}}^{\uparrow,\downarrow}$ for the spin-up and
spin-down blocks of the Hamiltonian [Fig. 1(c)], which confirms that the non-
zero quantum Hall conductance is originated from the field-induced imbalance
between counter-propagating chiral edge states. Also, Fig. 1(d) shows that the
Hall conductance approaches zero for thinner slabs because of the mixing of
counter-propagating chiral edge states. This mechanism due to the splitting of
helical edge states was previously ignored and could benefit the further
experimental explorations.

Figure 2: The [110]-direction slab of the Dirac semimetal, which consists of a
Weyl semimetal and its time-reversal. (a) The real-space correspondence of
Fermi-arc surface states (red for top and blue for bottom) of the Weyl
semimetal described by the upper block of $H$ in Eq. (1), for a slab with open
boundary condition along the $y^{\prime}$ direction. $E_{\text{w}}$ marks the
Weyl-node energy. (b) The projection of the Fermi arc on the
$k_{x^{\prime}}$-$k_{z^{\prime}}$ plane. $S$ is the area of the Fermi loop.
(c) The area $S$ of the Fermi surface of the Fermi-arc surface states in the
$\left(k_{x^{\prime}},{y^{\prime}},k_{z^{\prime}}\right)$ plane as a function
of the slab thickness $L$. (d) The Hall conductance $\sigma_{\text{H}}$ as a
function of $L$ and magnetic field $1/B$. The Fermi energy $E_{\text{F}}$ is
at the Weyl node $E_{\text{w}}$.

_The [110] slab_ \- In contrast to the above [001] case, for a semimetal slab
grown along the [110] and equivalent [100] or [011] direction, there exists a
3D quantum Hall effect Wang  _et al._ (2017); Li  _et al._ (2020) arising from
the Fermi-arc surface states Potter  _et al._ (2014); Zhang  _et al._ (2016).
On each of the top and bottom surfaces of a Weyl semimetal, there are
topologically-protected surface states, which can be regarded as half of a 2D
electron gas. Their open Fermi surfaces are dubbed as the Fermi arcs. The 2D
Fermi-arc surface states on opposite surfaces can be connected through the
bulk states to form a complete 2D electron gas with a 3D spatial distribution
and closed Fermi surface [Figs. 2(a)-(c), red for top and blue for bottom
surfaces], to support a “3D” quantum Hall effect Wang  _et al._ (2017). A
Dirac semimetal can be regarded as two time-reversed Weyl semimetals to host
two copies of the Fermi-arc quantum Hall effect. In addition, in a Dirac
semimetal, the Fermi-arc surface states and their time-reversal partners on a
single surface can form a 2D electron gas Kargarian  _et al._ (2016), to
support a quantum Hall effect as well. For both cases, the Hall conductance
plateaus are supposed to decrease with increasing magnetic field, much like
those in conventional 2D electron gases Klitzing  _et al._ (1980), where the
magnetic field presses the occupation of electrons to lower Landau levels, as
shown in Figure 2(d) for different slab thickness $L$.

Moreover, Fig. 2(d) shows that the width of the quantized plateaus is stable
for thicker slabs ($L>50$ nm), while show obvious variations for ultrathin
slabs ($L<20$ nm) with decreasing thickness. This can be understood using
Figs. 2(b-c), where the area of the Fermi loop $S$ converges to a constant for
thick slabs but decreases exponentially with decreasing $L$, due to the
hybridization of the opposite surfaces in ultrathin slabs. According to the
Lifshitz-Onsager relation, $S$ determines the plateau width of the Hall
conductance, which explains the quantized pattern in Fig. 2(d). Moreover, in
Dirac semimetals $E_{\text{F}}$ shifts away from $E_{\text{w}}$ as the Zeeman
effect splits $E_{\text{w}}$ (See Sec. SIII of Sup ), leading to a systematic
shift of the Hall plateaus with increasing thickness [Fig. 2(d)].

Figure 3: The [112]-direction slab of the Dirac semimetal. (a) The area of the
Fermi surface $S$ (solid) and spin Chern number $C_{s}$ (dotted) as functions
of the thickness $L$. (b) The Hall conductance $\sigma_{\text{H}}$ as a
function of $L$ and magnetic field $1/B$. (c) Schematic of where the Fermi
energy $E_{\text{F}}$ and Weyl modes $E_{\text{w}}$ are in the bulk spectrum
of the Dirac semimetal. (d-f) $\sigma_{\text{H}}$ as a function of (d)
$\left(E_{\text{F}}-E_{\text{w}},1/B\right)$ and (e,f) $\left(L,1/B\right)$,
respectively. Here we take (d) $L=50$ nm, (e)
$E_{\text{F}}-E_{\text{w}}=5\text{ meV}$, and (f)
$E_{\text{F}}-E_{\text{w}}=-5\text{ meV}$.

_The [112] slab_ \- The slabs along the [112] direction inherits the
properties of both the confinement-induced helical edge states in the [001]
slab and Fermi-arc surface states in the [110] slab. Figure 3(b) shows the
Hall conductance of the slab along the [112] direction as a function of $1/B$
for different $L$. For weak magnetic fields [$1/B>0.4\text{ (1/T)}$], the
oscillation pattern of the quantized Hall conductance is similar to that of
the [110] slab, that is, the Hall plateaus decrease with increasing magnetic
field, which indicates the quantized conductance is mainly originated from the
Fermi-arc surface states. Compared to the [110] slab, the width of the
plateaus is larger because of the smaller area enclosed by the Fermi arc $S$
[Fig. 3(a)]. For strong magnetic fields [$1/B<0.4\text{ (1/T)}$], the Hall
conductance increases with increasing magnetic field, similar to that in the
[001] slab, indicating that it is mainly contributed by the imbalance of the
helical edge states.

Above, the Fermi energy is assumed to cross the Weyl nodes, i.e.,
$E_{\text{F}}=E_{\text{w}}$ [see Fig. 3(c) for the definitions of
$E_{\text{F}}$ and $E_{\text{w}}$]. Figure 3(d) shows $\sigma_{\text{H}}$ in
the $\left(E_{\text{F}}-E_{\text{w}},1/B\right)$ plane with $L=50$ nm. For the
Fermi energies far away from the Weyl nodes, the quantum Hall effect is
originated from the confinement-induced bulk subbands [Figs. 3(e)-(f)], and
the Hall conductance monotonically decreases with increasing magnetic field
for different thickness and Fermi energies, different from the non-monotonic
dependence when $E_{\text{F}}=E_{\text{w}}$ or small $E_{F}-E_{w}$.

Figure 4: The dependence on $\theta$, the angle between the line connecting
the Dirac nodes and $k_{x^{\prime}}$-$k_{z^{\prime}}$ plane. (a) Illustration
of $\theta$. (b) The area of the Fermi surface of the Fermi-arc surface states
$S$ (solid) and spin Chern number $C_{s}$ (dotted) as functions of $\theta$.
(c-d) The Hall conductance $\sigma_{\text{H}}$ as a function of
$\left(\theta,1/B\right)$ and $\left(E_{\text{F}}-E_{\text{w}},1/B\right)$,
respectively. The dashed line corresponds to the slab grown along the [112]
direction. Here we take (c) $E_{\text{F}}-E_{\text{w}}=0$ and (d) $B=5$ T.

_The angle dependence_ \- Figure 4(a) illustrates the dependence on $\theta$,
the angle between the line connecting the Dirac nodes and the
$k_{x^{\prime}}$-$k_{z^{\prime}}$ plane. For example, the Dirac nodes are
located on the $k_{x^{\prime}}$-$k_{z^{\prime}}$ plane when $\theta=0$ [Figs.
2(a-b)]. Figure 4(b) shows that $S$ decreases and $C_{s}$ increases with
increasing $\theta$, indicating the competition between the Fermi-arc surface
states and side-surface helical edge states. Figure 4(c) shows
$\sigma_{\text{H}}$ as a function of $\theta$ and $1/B$ for $L=100$ nm. For
$\theta=0$ ([110] direction) and $\theta=\pi/2$ ([001] direction), the
quantized conductance is only contributed by the Fermi-arc surface states and
imbalanced helical edge states, respectively. For other $\theta$, the
quantized Hall conductance originates from both the Fermi-arc surface states
and the helical edge states. Furthermore, $\sigma_{\text{H}}$ as a function of
$\left(E_{\text{F}}-E_{\text{w}},\theta\right)$ [Fig. 4(d)] shows that the
quantum Hall effect may be due to the confinement-induced bulk subbands,
giving another origin for the experimentally observed quantum Hall effect in
Cd3As2 Uchida  _et al._ (2017); Nishihaya  _et al._ (2018). In experiments, it
may be difficult to distinguish whether the thickness dependent conductance
plateaus are consequences of the bulk subbands or the Fermi-arc surface
states. The non-monotonic dependence of $\sigma_{\text{H}}$ on the magnetic
field may play another significant role to detect the side-surface helical
edge states or Fermi-arc 3D quantum Hall effect in a [112] Dirac semimetal
slab.

_Discussion_ \- Above, we have shown that the Hall plateaus can be attributed
to the Fermi-arc surface states, confinement-induced bulk subbands, and
helical side-surface edge states. It may be challenging to distinguish them in
a standard Hall-bar device. The helical side-surface edge states can be
identified through the nonlocal measurements Roth  _et al._ (2009).
Furthermore, in Sec. SIV of Sup , we propose two different local transport
devices, which may help revealing the mechanism of the observed quantum Hall
effects in experiments.

###### Acknowledgements.

We thank helpful discussions with Hua Jiang, Donghui Xu, and Bin Zhou. This
work was supported by the National Natural Science Foundation of China
(11534001, 11974249, and 11925402), the Strategic Priority Research Program of
Chinese Academy of Sciences (XDB28000000), Guangdong province (2016ZT06D348,
2020KCXTD001), the National Key R & D Program (2016YFA0301700), Shenzhen High-
level Special Fund (G02206304, G02206404), and the Science, Technology and
Innovation Commission of Shenzhen Municipality (ZDSYS20170303165926217,
JCYJ20170412152620376, KYTDPT20181011104202253). R.C. acknowledges support
from the project funded by the China Postdoctoral Science Foundation (Grant
No. 2019M661678) and the SUSTech Presidential Postdoctoral Fellowship. The
numerical calculations were supported by Center for Computational Science and
Engineering of Southern University of Science and Technology.

_Note Added._ \- Recently, we became aware of a complementary study, which
focuses on the quantum Hall states in topological semimetals confined in
different directions Nguyen  _et al._ (2021).

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