# Futuristic Intelligent Transportation System

Yilong Hui, Zhou Su, Tom H. Luan, , and Nan Cheng

###### Abstract

The emerging autonomous vehicles (AVs) will inevitably revolutionize the
transportation systems. This is because of a key feature of AVs; instead of
being managed by human drivers as the conventional vehicles, AVs are of the
complete capability to manage the driving by themselves. As a result, the
futuristic intelligent transportation system (FITS) can be a centrally managed
and optimized system with the fully coordinated driving of vehicles, which is
impossible by the current transportation systems controlled by humans. In this
article, we envision the operation of such FITS when AVs, advanced vehicular
networks (VANETs) and artificial intelligence (AI) are adopted. Specifically,
we first develop the autonomous vehicular networks (AVNs) based on the
advanced development of AVs and heterogeneous vehicular communication
technologies to achieve global data collection and real-time data sharing.
With this network architecture, we then integrate AVNs and AI based on the
intelligent digital twin (IDT) to design the FITS with the target of setting
up an accurate and efficient global traffic scheduling system. After that,
compared with the conventional schemes, a customized path planning case is
studied to evaluate the performance of the proposed FITS. Finally, we
highlight the emerging issues related to the FITS for future research.

##  I Introduction

With the rapid development of urbanization, the percentage of population
residing in cities is envisioned to rise to 70% in 2050 [1]. As a major means
of transportation in cities, commute by vehicles is an essential part in our
daily life. To this end, vehicular networks (VANETs) have attracted a lot of
attention with the target of developing intelligent transportation system
(ITS) and making our trip more convenient and enjoyable [2]. In ITS, vehicles
and roadside operating units are wirelessly connected using vehicular
communication technologies so as to provide drivers the real-time traffic
information and assist them to make driving decisions. Despite the efforts on
ITS, with vehicles controlled by human being, ITS is only an information
system with following issues:

Troublesome and dangerous work: Controlled by driver, the driving actions of a
vehicle is decided by the driver's personal experience and subjective
consciousness. In addition, the drivers in a boring driving environment may be
distracted by other things (e.g., phone ringing, scenery, etc.) along the way.
As reported by the Association for Safe International Road Travel [3], the
reason that causes accidents is mainly attributable to the inappropriate
operations of drivers. Apart from safety problem, as shown in Fig. 1, drivers
have to do a lot of labor intensive and time consuming works (e.g., park,
charge and repair their vehicles) in the ITS and thus fail to obtain the
comfortable travel experience.

Inefficient scheduling system: The ITS can provide information and recommend
driving decisions to drivers. However, not all the drivers drive according to
the recommended decisions. In addition, the road conditions provided by the
ITS, in general, are the events that have happened. This phenomenon mainly
stems from the fact that the traffic status is affected by the decisions of
all the drivers, where the topology dynamically changes with the high-speed
mobility of vehicles. Consequently, it is difficult to schedule and manage the
traffic in a global way.

Low quality of experience: With various travel intentions, different drivers
have different driving requirements. For example, some drivers may pay
attention to the driving time while the other drivers may care about the
driving costs. However, the driving decisions that can be recommended by the
ITS to each driver are limited, where the diverse driving requirements of
drivers can not be satisfied. Furthermore, the recommended decisions in the
ITS are not absolutely correct due to the lack of precise management and
global control. As a result, the drivers may suffer from a low quality of
experience (QoE) because the ITS fails to meet their diverse requirements.

Figure 1: The current ITS.

Fortunately, the rapid development of autonomous driving technology makes it
possible for revolutionizing the ITS. It is reported that all Tesla cars
shipped in 2019 are equipped with full autonomous driving capability. Embedded
with various sensors, computing devices and communication units, the
controllable autonomous vehicles (AVs) can understand the surrounding traffic
environment and make accurate driving decisions by themselves [4]. In this
way, drivers are released from getting into boring and dangerous works, where
the real-time data perception and precise vehicle control can significantly
improve the driving safety. Furthermore, by integrating VANETs with artificial
intelligence (AI), the AVs can be collaboratively managed with the target of
achieving the global scheduling system and satisfying the diverse driving
requirements. With these promising advantages, autonomous driving will
inevitably revolutionize the operation mode of the traditional transportation
system, namely, the ITS.

In this article, we represent the initial step toward the futuristic ITS
(FITS) by jointly considering the available vehicular communication
technologies, AI and the features of AVs. We first develop the autonomous
vehicular networks (AVNs) based on the rapidly development of AVs and the
advanced vehicular communication technologies to support globe data perception
and real-time data sharing. Then, we integrate the AVNs and AI to design the
FITS which is an accurate and efficient global traffic scheduling and
management system. After that, a case of customized path planning is studied
to evaluate the performance of the proposed framework. Finally, we discuss the
research issues to highlight the future research directions.

Figure 2: The principles of autonomous driving.

##  II Background of AVs and HetVNETs

Figure 3: The network architecture of the AVNs.

###  II-A AVs

In order to perceive the driving environment and make driving decisions like
humans, as shown in Fig. 2, an AV typically has the following components [5].

  * •

Sensing devices: AVs, as a type of robots, are equipped with various sensors
and devices such as lidar, radar sensors, video cameras, global position
system (GPS) and inertial measurement unit (IMU). With these sensors and
devices, each AV can understand the traffic environment and achieve a
controllable driving.

  * •

Storage device: An on-board unit (OBU) is embedded in each AV to cache
valuable information. On one hand, the private information of the AV can be
cached in the OBU, such as the interests of the owner. On the other hand, the
useful information, such as high precision maps, can be cached to facilitate
the autonomous driving.

  * •

Processing device: The central processing unit (CPU) is able to integrate the
data sensed from surrounding environment and the useful data cached in OBU to
make driving decisions.

  * •

Control device: The logic control unit (LCU) completes real-time control of
the onboard hardware devices to achieve safely autonomous driving.

With these components, the autonomous driving process of an AV can be divided
into the following steps.

  * •

Environment perception: The AV uses a group of sensors to sense the
surrounding static objects (e.g., road signs and number of lanes) and dynamic
objects (e.g., pedestrians, animals and other moving AVs), where the collected
data are used to support driving decisions.

  * •

Behavior decision: Behavior decision is the process that the CPU makes driving
decisions and generates the commands to control the AV's motion by jointly
analyzing the perceived traffic environment information and the AV's current
driving state.

  * •

Motion control: According to the driving decisions made by the CPU, the LCU
then transmits the corresponding control commands to engines, brakes, wheels
and so forth through wired signals with the target of realizing safe and
precise motion control.

###  II-B HetVNETs

With the development of space-air-ground integrated networks (SAGINs), a
variety of communication technologies can form heterogeneous vehicular
networks (HetVNETs) [6].

  * •

Satellite communication: With a globe communication coverage, satellite
communication is able to provide the network services for AVs which are out of
the coverage of other vehicular communication infrastructures [7].

  * •

DSRC: Based on IEEE 802.11p, the dedicated short range communication (DSRC) is
designed to provide network services to AVs [8]. To this end, a group of
roadside units (RSUs) can be deployed along roads so that each AV can connect
with its nearby RSU to obtain the requested information.

  * •

Cellular networks: The cellular networks can be regraded as the long term
evolution (LTE), 4G and 5G, which are advocated by the Third Generation
Partnership Project (3GPP), to provide AVs with vehicle-to-everything (V2X)
services [9]. Using the cellular base stations (CBS), AVs can obtain network
access services by connecting the cellular networks.

  * •

UAVs: Unmanned aerial vehicles (UAVs), which are equipped with dedicated
sensors and communication devices, can undertake various vehicular services to
facilitate the autonomous driving [10]. With flexible mobility, UAVs can be
treated as flying vehicular nodes to enhance the communication performance of
the RSUs or OBUs.

##  III AVNs

In this section, we design the AVNs by considering the features of AVs and the
HetVNETs.

###  III-A Network Architecture of AVNs

As shown in Fig. 3, the architecture of the network consists of three layers,
which are AV layer, edge layer and cloud layer.

AV layer: This layer consists of all the AVs. Equipped with advanced sensors
and OBU, each AV can sense the surrounding environment, cache valuable
information and execute computing tasks for autonomous driving. Additionally,
the perceived information can be shared among a group of AVs to make the
driving decisions more accurately. However, caused by the perception range of
sensors, the decisions made by AVs can only make driving decisions based on
limited information.

Edge layer: This layer is composed of edge infrastructures with caching,
communication and computing abilities, such as RSUs, CBSs and UAVs. The RSUs
and CBSs are fixed nodes while the UAVs in the contrast have high flexibility.
The devices deployed at the edge layer are closer to the AVs than the cloud
layer. Consequently, the edge layer can support local services with low
latency. On the other hand, the edge devices can pre-process the data that
need to be transmitted to the cloud layer, where the data traffic in the cloud
can be reduced.

Cloud layer: The cloud layer has more powerful caching and computing abilities
than the AV layer and edge layer. On one hand, the historical data related to
the traffic can be cached in this layer. For example, the information of an AV
user can be cached in cloud server so that the travel habit of the user can be
further estimated. On the other hand, the cloud layer has the global view of
the traffic. It can obtain the information generated by the dynamic traffic
and the AVs to make scheduling decisions and manage the traffic as a whole.

###  III-B Features of AVNs Enabled Transportation

In this subsection, we analyze the features of AVNs enabled transportation to
pave the way for developing the FITS.

Global data perception: With the advanced sensors and devices, the traffic
environment around an AV can be efficiently collected. Furthermore, with the
assist of AVNs, the data generated by the AVs together with the data generated
by the traffic in the road sections can be collected by the infrastructures
deployed at the edge layer to achieve the traffic data perception in a global
way.

Real time data transmission: With the support of the AVNs, the network
coverage of the city can be realized and the traffic data can be shared among
all the communication devices and infrastructures. On one hand, an AV can
connect to the nearby AVs to deliver warning information or share driving
information using V2V communication. On the other hand, the AVNs consist of
various communication technologies can communicate with AVs to obtain real-
time data sharing.

Multilayer data computing: In the AVNs, all the devices in the three layers
have the computing ability and can be used to complete different tasks. For
the AV layer, each AV can compute task by itself or execute task with other
AVs collaboratively [11]. Furthermore, the devices in the edge layer can
execute computing tasks locally. If an AV has a task that cannot be executed
by itself, the AV can offload the task to the nearby edge devices. Although
the AVs and edge devices can assist the transportation with computing
services, the resources that they have cannot satisfy the heavy computing
tasks generated by the traffic. Therefore, some tasks need to be completed by
the cloud server which can provide rich resources.

Distributed data caching: The data with various types and values are cached in
different devices in the AVNs. Specifically, the data about the AVs and their
owners can be cached in the cloud server to develop insurance and diagnostic
plans. The information related to geographic location can be distributed in
the devices at the edge layer. For example, the high precision maps of all the
roads in the city can be cached in the cloud server and a part of them can be
cached in the edge devices and AVs selectively. Besides, the data related to
the dynamic traffic are also distributed in different devices to facilitate
the real time traffic scheduling. In this way, the retrieve time of the cached
data can be reduced as some of them can be cached at the edge of the network.

##  IV FITS

Figure 4: The integration of AVNs and AI to support FITS.

According to the features of the AVNs enabled transportation, in this section,
as the architecture shown in Fig. 4, we briefly introduce the process of
combining AVNs and AI. With the AVNs, the data generated by the transportation
(i.e., physical network) can be efficiently collected, shared, computed and
cached. Then, based on the collected data, a virtual network can be developed
by constructing the intelligent digital twin (IDT) of the physical network
[12]. In the virtual network, the data can be processed, analyzed, and
analyzed to support various vehicular applications by using well-designed AI-
based algorithms. With the AI-based algorithms, the traffic management and
scheduling strategies and schemes can be obtained to facilitate the
transportation system in the physical network. Based on the combination of
AVNs and AI, we then design the FITS which includes the following systems, as
shown in Fig. 5.

Information collection system: The information collection system is able to
manage all the data and information related to the transportation, including
data generation, sensing, collection and utilization. These data can be
collected to achieve different goals. With the analysis of the data by using
AI-based algorithms, 1) the information of each AV and its owner cached in the
cloud server can be used to make scheduling plans. For example, machine
learning can be designed to detect the misbehavior of AVs [13]; 2) the
information of the components equipped on each AV, such as the lifetime of
various on-board sensors, can be used to ensure the safety of the AV and
develop insurance and diagnostic plans; 3) the environmental information
around each AV can be utilized to enable safe and efficient autonomous
driving; 4) the data generated by the traffic of each road segment can be
explored to forecast and schedule the traffic flow. Besides, compared with the
high simulation cost in the actual system, the valuable data can be used to
simulate and test new algorithms or train learning models. With the collected
information, the transportation system then has a global view of the traffic.
By using the global information to train AI algorithms, the cloud layer can
make full use of the available resources in the FITS, such as charging station
and parking lot.

AV control system: The AV control system refers to the full automatic control
process of an AV. In this process, each AV uses the AI-based algorithms to
make driving decisions based on the collected data from the surrounding
environment and the information received from the neighboring AVs or edge
devices. With these decisions, there is no need for human intervention in
driving and stopping operations as the motion of each AV can be controlled
accurately and efficiently. For example, deep learning can significantly
improve the AV's ability to detect and classify the surrounding objects, thus
facilitating the fusion of data obtained by different sensors. With the AV
control system, the accurate motion control of each AV can free drivers from
getting into boring driving and significantly reduce the probability of
accidents.

Intelligent driving system: Based on the information obtained from AVs and the
edge devices in the AVNs, the cloud server is in charge of the intelligent
driving system. Specifically, by using the AI-based algorithms to analyze the
collected data, the cloud layer can forecast the city traffic and dynamically
decides a driving speed which will be assigned to the edge devices in the
given road segment. Based on the assigned driving speed, the edge devices then
control the AVs that drive on this road segment. For the AVs that join in this
road segment, it keeps the driving speed as the same as others. On the
contrary, for the AV that intends to change its lane, it needs to adjust its
driving speed and follows the given speed in the new road segment. Obviously,
with the traffic control of each road segment, the probability of traffic jams
in each road segment can be reduced.

Intelligent traffic scheduling system: With the intelligent driving system,
the traffic in each road segment can be managed efficiently. We then introduce
the intelligent traffic scheduling system to schedule the traffic in a global
way. In the intelligent traffic scheduling system, the cloud layer can use AI-
based algorithms to predict the traffic flow of each road section and then
regulate the direction of traffic flow on each road to break the layout of
roads so as to make flexible use of road resources. For example, with the
efficient scheduling, traffic lights will no longer be needed to manage
traffic at intersections [14]. Besides, rewards or compensations can be
further used to dispatch traffic flow. Specifically, according to AI-based
data analysis in the cloud layer, the system can give different rewards or
compensations to different road sections. For example, the system can charge
high prices on congested road sections and give rewards on non-congested road
sections. In this way, some AVs will be attracted by rewards and drive to non-
congested road sections, even if the non-congested road sections need to drive
a longer distance. According to the precise management of traffic in different
road segments and intersections, the system can also develop real-time plans
to deal with emergencies. For example, to make sure the rescue AVs (e.g.,
police cars and ambulances) to drive with a high-speed and barrier-free
environment, the system will provide the best driving route to them and notify
the AVs in the road segments associated with the selected route in advance.
With the traffic scheduling, the transportation becomes smart, where the
probability of traffic jams can be reduced and the traffic efficiency can be
significantly improved.

Electronic trading system: In order to minimize human participation, it is
necessary to establish an electronic trading system to enable AVs to
charge/pay fees according to their behaviors in the FITS. With the electronic
trading system, each person, which intends to join in the FITS, has a virtual
currency account (VCA) to complete the transactions with various types. Each
AV owner needs to deposit enough money in advance in its VCA to make sure that
its AV can interact with other devices successfully. For the people who do not
have AVs, the VCA is also developed for each of them as they may travel by
using the public transportation. After completing the transaction, the related
fees will be automatically transferred among all the participants, thus
reducing the number of brakes on AVs and avoiding the wrong payment caused by
human error. As the resources in city transportation such as parking spaces
and charge stations are limited and dynamically changing, AI-based algorithms
can be used to dynamically price these resources. For the owner of an AV, when
the AV charges, occupies the parking space or executes other consumption
applications, the related fees will be deducted automatically from the owner's
VCA based on the ID of the AV. For the people who use public transport, AI-
based algorithms can be used to train their biological features so that they
can pay for services more convenient. In this way, people in the FITS do not
need cash or credit cards to pay for any transactions. Thus, it can greatly
enhance people's travel experience and improve the utilization rate of AVs.

Figure 5: The architecture of FITS.

Customized service system: In the fully controlled traffic environment,
passengers no longer need to focus on the driving process and thus pay more
attention to the driving requirements. Generally, different people have
different driving requirements (e.g., driving time and costs) even for the
same trip. Therefore, we also need to establish the customized service system
in the FITS to satisfy the needs of different users. For example, when an AV
intends to park with low cost while does not care about the driving distance,
the customized service system then provides the optimal choice to the AV for
parking using the AI-based algorithms. In the aspect of traffic dispatching,
the FITS can charge/compensate the cost to reduce/increase the travel time of
users. By designing AI-based algorithms to analyze the collected data, the
customized service system can set different charging/compensation schemes for
users with different driving requirements. Specifically, if a user plans to
reduce the travel time by selecting the shortest path and the fastest speed,
it has to pay the system for the driving request. Conversely, the
corresponding compensation will be paid the users who are willing to increase
their travel time. In addition, according to the requirements of users, AI-
based algorithms also can be explored to customize their driving routes.
Obviously, the customized service system can lubricate the city traffic
scheduling and enhance the travel experience of each user.

##  V Case Study

In this section, a case of customized path planning is studied to demonstrate
the efficiency of the proposed FITS. We first introduce the scenario setting,
followed by the result analysis.

###  V-A Scenario Setting

Consider a general scenario that an area has 100 road sections, where the
distance of each road section is randomly generated between 1km and 10km. In
addition, we set the maximum speed (i.e., $v_{\max}$) and the maximum density
(i.e., $k_{\max}$) of the AVs to be 110km/h and 80veh/km, respectively. In
this way, we can obtain the velocity that an AV drives on each road section by
$v=v_{\max}(1-\frac{k}{k_{\max}})$, where $k$ is the density of AVs in this
road section [15]. By dividing the distance by the speed, the cloud layer can
collect the driving time to determine the compensation price for each road
section, where the price changes from -1 to 1. Then the cloud layer delivers
the driving time and compensation price to the edge layer which broadcasts
these information to all the AVs. After receiving the driving time and
compensation price of each road section, each AV uses the Q-learning algorithm
to customize the optimal driving path to obtain maximum utility based on its
preference for compensation price and driving time, where the preference
varies from 0 to 1 to weight the above two factors.

###  V-B Simulation Result

With the aforementioned scenario, we compare the proposal with the following
conventional schemes: 1) Shortest driving time (SDT): The AV only considers
the driving time to obtain the optimal driving path; 2) Maximum driving
compensation (MDC): The AV only considers the compensation price to plan its
driving path; 3) Minimum number of sections (MNS): The AV selects the path
which has the least number of road sections. Fig. 6 is the utility of the AV
by changing the preference of the AV for the compensation price and driving
time. From this figure, we can see that the proposal can lead to the highest
utility to the AV compared with the other conventional schemes. The reasons
are as follows. For the SDT and MDC, they only consider one of the two factors
(i.e., the driving time and the compensation price) to determine the optimal
driving path, where the other factor is not considered to guarantee the
utility of the AV. In addition, the AV in the MNS scheme focuses on the number
of road sections, where both the driving time and the compensation price are
not considered to obtain high utility.

Figure 6: The simulation results.

##  VI Emerging Research Issues

###  VI-A High Performance Sensing

To perceive the complex and dynamic traffic information around an AV, it needs
to be equipped with sensing devices with high precision. At present, the
performance of the sensing devices, however, have constraints in different
aspects. For example, the lidar has high resolution and strong anti-
interference ability, while its price is very expensive. In addition, the GPS
must be used in a non-closed environment with the result that it can not be
used in many scenarios, such as tunnels. Therefore, more advanced sensors and
the efficient collaboration among sensors need to be further developed to
support the FITS.

###  VI-B Collaborative Networking, Computing and Caching

With the rapidly development of the sixth generation (6G) networks, the 6G
HetVNETs is no longer a simple communication network, but an information
network integrating communication, computing, and storage. On the one hand,
infrastructures or devices in different layers are different in transmission,
computing and caching capabilities. On the other hand, users in the FITS also
have diverse requirements for different services. Therefore, the heterogeneous
networking, computing and caching resources in the FITS should be well
organized and allocated with the target of satisfying various requirements of
services and providing AVs with the customized services.

###  VI-C AI-based Algorithms

AI-based algorithms can significantly improve the AV's ability to detect the
surrounding objects and facilitate the fusion of data obtained by different
sensors. In addition, by analyzing the massive data generated by
transportation through AI-based algorithms, the in-depth understanding of road
traffic demand and network state can be provided to decision-makers from
different network layers. However, different AI-based algorithms are suitable
for different applications and require different training time. Therefore, how
to design the optimal AI-based algorithm for all the vehicular applications
supported by the FITS should be further studied.

###  VI-D Network Security

The FITS will generate massive amounts of data to guide the efficient
operation of the entire system. In other words, the authenticity and
reliability of data are the key factors affecting the efficiency of the FITS.
Therefore, how to detect the authenticity and reliability of data in a
dynamically evolving FITS is a topic worthy of research. Furthermore, with the
designed electronic trading system, a large volume of transactions will occur
among network nodes. 1) The system needs to set different rewards for
different road sections to dispatch AVs. 2) Rewards should be paid to the
nodes in the system which share storage and computing resources. 3) The AVs
occupying public facilities need to pay for their requested services. However,
the FITS is an open system, some malicious or aggressive nodes may launch
attacks to seek personal benefits, which will bring unpredictable losses to
normal AVs. Therefore, a green and safe environment needs to be established to
solve the security problems in FITS.

##  VII Conclusion

The traffic in the city is getting worse with the rapidly increase of
vehicles. In order to break the operation mode of the traditional ITS, in this
article, we establish a framework of FITS to provide people with safe,
efficient and comfortable travel experience. Specifically, we have proposed
the network architecture of AVNs based on the rapidly development of AVs and
HetNETs. After that, by using the IDT to integrate AVNs and AI, we have
designed the FITS to set up an accurate and efficient global traffic
scheduling and management system. After that, a case is studied to evaluate
the performance of the proposed FITS. Finally, the emerging issues are
discussed to highlight the future research directions.

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##  VIII Biography

Yilong Hui received the Ph.D. degree in control theory and control engineering
from Shanghai University, Shanghai, China, in 2018. He is currently a lecturer
with the State Key Laboratory of Integrated Services Networks, and with the
School of Telecommunication Engineering, Xidian University, China. He has
published over 30 scientific articles in leading journals and international
conferences. His research interests include wireless communication, vehicular
networks, intelligent transportation systems and autonomous driving. He was
the recipient of the Best Paper Award of International Conference WiCon2016
and IEEE Cyber-SciTech2017.

Zhou Su (Corresponding author) is an Associate Editor of IEEE Internet of
Things Journal, IEEE Open Journal of Computer Society, and IET Communications.
He is the Chair of the Multimedia Services and Applications over Emerging
Networks Interest Group (MENIG) of the IEEE Comsoc Society, the Multimedia
Communications Technical Committee. His research interests include multimedia
communication, wireless communication and network traffic. He received the
best paper award of IEEE ICC2020, IEEE BigdataSE2019, IEEE ComSoc GCCTC2018,
IEEE CyberSciTech2017, etc.

Tom H. Luan is a professor with the School of Engineering, Xidian University,
Xi'an, China. He received the B.E. degree from the Xi'an Jiaotong University,
China, in 2004, the Master degree from the Hong Kong University of Science and
Technology, Hong Kong, in 2007, and the Ph.D. degree from the University of
Waterloo, Canada, in 2012, all in Electrical and Computer Engineering. Dr.
Luan's research mainly focuses on the content distribution and media streaming
in the vehicular ad hoc networks and peer-to-peer networking, protocol design
and performance evaluation of wireless cloud computing and fog computing. Dr.
Luan has authored/coauthored around 60 journal papers and 30 technical papers
in conference proceedings, and awarded one US patent. He served as a TPC
member for IEEE Globecom, ICC, PIMRC and the technical reviewer for multiple
IEEE Transactions including TMC, TPDS, TVT, TWC and ITS.

Nan Cheng received the Ph.D. degree from the Department of Electrical and
Computer Engineering, University of Waterloo in 2016, and B.E. degree and the
M.S. degree from the Department of Electronics and Information Engineering,
Tongji University, Shanghai, China, in 2009 and 2012, respectively. He is
currently a professor with State Key Lab of ISN, and with School of
Telecommunication Engineering, Xidian University, Shaanxi, China. He worked as
a Post-doctoral fellow with the Department of Electrical and Computer
Engineering, University of Toronto, from 2017 to 2018. His current research
focuses on space-air-ground integrated system, big data in vehicular networks,
and selfdriving system. His research interests also include performance
analysis, MAC, opportunistic communication, and application of AI for
vehicular networks.


# Effects of the geometry on the downstream conductance in quantum Hall-
superconductor hybrid systems

A. David  J. S. Meyer  M. Houzet  Univ. Grenoble Alpes, CEA, Grenoble INP,
IRIG, Pheliqs, 38000 Grenoble, France

###### Abstract

We consider a quantum Hall (QH) region in contact with a superconductor (SC),
i.e., a QH-SC junction. Due to successive Andreev reflections, the QH-SC
interface hosts hybridized electron and hole edge states called chiral Andreev
edge states (CAES). We theoretically study the transport properties of these
CAES by using a microscopic, tight-binding model. We find that the transport
properties strongly depend on the contact geometry and the value of the
filling factor. We notice that it is necessary to add local barriers at the
corners of the junction in order to reproduce such properties, when using
effective one-dimensional models.

##  I Introduction

Combining systems displaying a quantum Hall effect and superconductors is a
difficult task, as the magnetic field needed to realize the quantum Hall
effect tends to suppress superconductivity. If successful, it leads to
interesting phenomena as the superconductor may induce correlations in the
chiral edge states of the quantum Hall system. In particular, the formation of
so-called chiral Andreev edge states (CAES) has been predicted.
Semiclassically, these CAES result from skipping orbits of electrons and holes
involving Andreev reflections at the quantum Hall-superconductor (QH-SC)
interface [1, 2, 3, 4]. Quantum-mechanically, the edge states along that
interface are described as hybridized electron and hole states [5, 6, 7, 8,
9]. Their use for topologically protected quantum computing was also
considered [10, 11, 12].

A number of recents experiments have succeeded in creating QH-SC hybrid
systems, using either graphene [13, 14, 15, 16] or InAS two-dimensional
electron gas (2DEG) [17], and observing evidence for CAES in the so-called
downstream conductance. Namely the downstream conductance measures the
conversion of electrons into holes, involving the transfer of Cooper pairs
into the superconductor, along the interface. The larger the conversion
probability the smaller the downstream conductance and, in particular, it
becomes negative when the conversion probability exceeds one half. While the
experiments [13, 14, 15, 16, 17] did indeed measure negative downstream
conductances, questions remain about the magnitude and the parameter
dependence of the effect that do not match simple models: the observed signal
is much smaller than expected. Furthermore, it shows either an irregular
pattern [13, 14, 15] or remains roughly constant [17] when sweeping the field
or the gate voltage, while simple models predict a regular oscillation. This
stimulated further theoretical research. A suppression of the measured signal
may be explained by the absorption of quasiparticles in the superconductor,
for example by subgap states in nearby vortices [14, 18, 19, 20], whereas the
oscillations may be strongly affected by disorder [18, 19].

Here we explore a different aspect that has not been addressed before: the
role of the geometry. Namely the downstream conductance does not probe only
the properties of the QH-SC interface, but also the scattering properties at
the point where this interface meets the QH-vacuum interface. We find that
these scattering probabilities strongly depend on the geometry of the contact
region. In particular, a pronounced dependence of the angle between the QH-
vacuum interface and the QH-SC interface is observed. Interestingly this opens
the possibility of creating asymmetric structures, where these angles are
different on the two sides of the superconductor, that may display an enhanced
overall electron-hole conversion probability. This may even lead to a
situation where the downstream conductance becomes negative on average.

Note that to study the effect of geometry, a full twodimensional description
of the system is necessary – simple onedimensional models commonly used in the
literature are not sufficient. Some aspects may be captured by using a
generalized onedimensional model, though there is no obvious way to determine
its parameters.

The paper is organized as follows. In Sec. II, we present the system and the
downstream conductance formula based on edge state transport, whose parameters
have to be computed. To do so, we first use a twodimensional model in Sec.
III. In particular, we start by studying a continuous model in Sec. III.1 that
allows one to determine the properties of the edge states at an infinitely
long interface. We then use a tight-binding model in Sec. III.2 to obtain the
scattering probabilities at the points where two different interfaces, i.e.,
QH-vacuum and QH-SC, meet. With these two ingredients, we have all that is
needed to compute the downstream conductance. In Sec. IV, we address the
question whether the prior results may be obtained from an effective
onedimensional model. Further considerations on the role of additional non-
chiral edge states and the effects of temperature can be found in Sec. V,
before we conclude in Sec. VI. Some details were relegated to the appendices.

##  II System and conductance formula

The conductance along the edge of a system in the quantum Hall regime can be
attributed to the properties of its chiral edge states. We are interested in
the regime, where one spin-degenerate Landau level is occupied in the quantum
Hall region, i.e., there are two chiral edge states. Introducing particle-hole
space to be able to incorporate superconductivity, we can describe one spin
state as an electron state and the other spin state as a hole state. While the
chiral edge states along an edge with the vacuum are either pure electron or
hole states, the chiral Andreev edge states (CAES) along an edge with a
superconductor are a superposition of electron and hole components. In the
following, we will call them quasi-electron when their momentum at the Fermi
level is negative and quasi-hole when their momentum at the Fermi level is
positive. As we will see below, for the system under consideration, this
choice is in agreement with the pure electron and hole states obtained when
Andreev processes are suppressed.

Figure 1: QH-SC setup: the edge of the quantum Hall region is in contact with
a grounded superconductor over a finite length $L$. The geometry of the
corners at the beginning and end of that region can be characterized by two
angles each: $\theta_{QH,i}$ and $\theta_{SC,i}$. Both the QH-vacuum and QH-SC
interface host chiral edge states that can be probed by measuring the
differential downstream conductance $G_{d}=\partial I/\partial V$, where $V$
is the voltage applied to the upstream reservoir and $I$ is the current
flowing into the downstream reservoir. A typical process contributing to
$G_{d}$ is shown: an incoming electron $|e\rangle$ scatters at the first
corner, propagates along the QH-SC interface as a superposition of quasi-
electron $|qe\rangle$ and quasi-hole $|qh\rangle$ CAES, then scatters at the
second corner, and finally exits the superconductor in a superposition of
electron $|e\rangle$ and hole $|h\rangle$. The hole probability
$P_{h}=|p_{h}|^{2}$ of the outgoing state depends on the scattering processes
at the corners as well as the interference of the CAES propagation along the
QH-SC interface.

We want to study the situation where the edge of the quantum Hall system is in
contact with a superconductor over a region with finite length $L$ as shown in
Fig. 1. In that case, we can define a probability $P_{h}$ that an incoming
electron-like state is transformed into an outgoing hole-like state. It
depends on the properties of the CAES along the QH-SC interface as well as the
scattering amplitudes at the two corners, which begin and end that interface.
Assuming ballistic propagation along the interface with a given material, the
probability $P_{h}$ can be written as [8]

$\displaystyle P_{h}$ $\displaystyle=$
$\displaystyle\tau_{1}(1-\tau_{2})+\tau_{2}(1-\tau_{1})$ (1)
$\displaystyle+2\sqrt{\tau_{1}(1-\tau_{2})\tau_{2}(1-\tau_{1})}\cos(2k_{0}L+\phi_{12}),$

where $\tau_{1}$ is the probability that the electron is converted into a
quasi-hole at the beginning of the QH-SC interface whereas $\tau_{2}$ is the
probability that a quasi-electron is converted into a hole at the end of the
QH-SC interface. The second line describes the interference resulting from the
fact that the particle may propagate along the QH-SC interface either as a
quasi-electron with momentum $-k_{0}$ or as a quasi-hole with momentum
$+k_{0}$. The phase shift $\phi_{12}$ depends on the phases of the scattering
amplitudes at the two corners. At zero temperature, the differential
downstream conductance, $G_{d}(0)=\partial I/\partial V|_{V=0}$, where $V$ is
the voltage applied to the upstream reservoir and $I$ is the current flowing
into the downstream reservoir (see Fig. 1), is directly related to the
probability $P_{h}$ at the Fermi level, namely $G_{d}(0)=G_{0}(1-2P_{h})$,
where $G_{0}=2e^{2}/h$ is the conductance quantum. A negative downstream
conductance is a clear signature of the Andreev conversion taking place at the
QH-SC interface. Note that the average conductance is given as
$\bar{G}_{d}=G_{0}\prod_{i=1,2}(1-2\tau_{i})$. For $\tau_{1}=\tau_{2}$ it is
limited to positive values, whereas $\tau_{1}\neq\tau_{2}$ allows one to
realize $\bar{G}_{d}<0$. For completeness, let us mention that the maximal
downstream conductance is $G_{d}^{\rm
max}=G_{0}[1-2(\sqrt{\tau_{1}(1-\tau_{2})}-\sqrt{\tau_{2}(1-\tau_{1})}\,)^{2}]$
while the minimal downstream conductance is $G_{d}^{\rm
min}=G_{0}[1-2(\sqrt{\tau_{1}(1-\tau_{2})}+\sqrt{\tau_{2}(1-\tau_{1})}\,)^{2}]$.
In the symmetric case $\tau_{1}=\tau_{2}\equiv\tau$ this yields $G_{d}^{\rm
max}=G_{0}$ and $G_{d}^{\rm min}=G_{0}[1-8\tau(1-\tau)]$

Thus, to model the experimentally measured downstream conductance, we need to
determine $k_{0}$ as well as the probabilities $\tau_{i}$ associated with the
contact points between the QH region, the vacuum, and the superconductor. In
the following, we show that $k_{0}$ can be obtained semi-analytically from a
microscopic model of an infinite QH-SC interface. By contrast, there is no
simple model for the probabilities $\tau_{i}$. We study their dependence on
system parameters and in particular the geometry of the contact points using
tight-binding simulations. To conclude, we compare with an effective 1D model.

##  III Two-dimensional model

###  III.1 Continuum model of an infinite QH-SC interface

We will consider an interface along the $y$-axis such that the region $x<0$ is
in the quantum Hall regime whereas the region $x>0$ is a superconductor. The
microscopic Hamiltonian can be written in the form,

$H=\begin{pmatrix}H_{0}-\mu(x)&\Delta(x)\\\
\Delta^{*}(x)&-H_{0}^{*}+\mu(x)\end{pmatrix}\ $ (2)

with ${\bf r}=(x,y)$ and

$H_{0}=\frac{1}{2m}\left(-i{\bf{\nabla}}-e{\bf A}(x)\right)^{2}+V(x),$ (3)

using units where $\hbar=1$. Here,
$\mu(x)=\mu_{QH}\Theta(-x)+\mu_{SC}\Theta(x)$ accounts for the drop of the
chemical potential measured from the band bottom in the 2DEG and the
superconductor, $\mu_{QH}$ and $\mu_{SC}$, respectively,
$\Delta(x)=\Delta\Theta(x)$ is the superconducting order parameter with
amplitude $\Delta$ (that we will choose to be real in the following), the
potential $V(x)=V_{0}\delta(x)$ with strength $V_{0}$ models an interface
barrier, and $\Theta(x)$ is the Heaviside function. Note that we neglect self-
consistency of the order parameter. Furthermore, we assume that the magnetic
field in the superconductor is screened. Thus, choosing the Landau gauge that
preserves translational invariance along the interface, we set ${\bf
A}(x)=Bx\Theta(-x)\hat{u}_{y}$. The wave functions can then be written in the
form

$\Psi({\bf r})=\frac{e^{ik_{y}y}}{\sqrt{L_{y}}}\psi_{k_{y}}(x),$ (4)

where $L_{y}$ is the length of the system along the $y$-direction and
$\psi_{k_{y}}$ is the transverse wave function associated with longitudinal
wave vector $k_{y}$. Following [21], we can determine the CAES by writing the
wave functions $\psi_{k_{y}}^{QH}(x)$ in the half-space $x<0$ and
$\psi_{k_{y}}^{SC}(x)$ in the half-space $x>0$ and matching them at the
interface, so to obtain an eigenstate of Eq. (2) at energy $E$.

In the QH region, one obtains

$\psi_{k_{y}}^{QH}(x)=c^{QH}_{+}\begin{pmatrix}1\\\
0\end{pmatrix}\chi_{+}(x)+c^{QH}_{-}\begin{pmatrix}0\\\
1\end{pmatrix}\chi_{-}(x)$ (5)

with,

$\displaystyle\chi_{\pm}(x)$ $\displaystyle=N_{\pm}U\left(-\frac{\mu_{QH}\pm
E}{\omega_{c}},-\frac{\sqrt{2}}{l_{B}}(x\mp k_{y}l_{B}^{2})\right),$ (6)

where $U(a,z)$ are parabolic cylinder functions that vanish as
$z\rightarrow-\infty$ (see Ref. [22] for the formal definition), and $N_{\pm}$
are normalization coefficients such that
$\int_{-\infty}^{0}dx\;|\chi_{\pm}(x)|^{2}=1$. Here we introduced the
cyclotron frequency $\omega_{c}=eB/m$ and the magnetic length
$l_{B}=1/\sqrt{eB}$.

Restricting ourselves to the regime $|E|<\Delta$ , the wavefunctions in the SC
region take the form [23],

$\psi_{k_{y}}^{SC}(x)=c^{SC}_{+}\frac{1}{\sqrt{2}}\begin{pmatrix}\gamma\\\
1\end{pmatrix}\phi(x)+c^{SC}_{-}\frac{1}{\sqrt{2}}\begin{pmatrix}\gamma^{*}\\\
1\end{pmatrix}\phi^{*}(x)$ (7)

with $\phi(x)=\sqrt{2\,\text{Im}\,q}\;e^{iqx}$,

$\displaystyle q^{2}$
$\displaystyle=(k_{F}^{SC})^{2}-k_{y}^{2}+2im\Delta\sqrt{1-\epsilon^{2}}$ (8)

and $\gamma=\epsilon+i\sqrt{1-\epsilon^{2}},$ where $\epsilon=E/\Delta$, and
$k_{F}^{SC}=\sqrt{2m\mu_{SC}}$.

The matching procedure, $\psi_{k_{y}}^{QH}(0)=\psi_{k_{y}}^{SC}(0)$ and
$\partial_{x}\psi_{k_{y}}^{SC}(0)-\partial_{x}\psi_{k_{y}}^{QH}(0)=2mV_{0}\psi_{k_{y}}^{SC}(0)$,
yields the following secular equation for the energy $E(k_{y})$ [5],

$\displaystyle f(E,k_{y})\equiv\,$ $\displaystyle
GH\left(c^{2}+d^{2}\right)+G^{\prime}H^{\prime}+d(G^{\prime}H+GH^{\prime})$
$\displaystyle+c\frac{\epsilon}{\sqrt{1-\epsilon^{2}}}(G^{\prime}H-GH^{\prime})=0,$
(9)

with the shorthand notations $c=\mathrm{Re}\,q$, $d=\mathrm{Im}\,q+2mV_{0}$,
$G=\chi_{+}(0)$, $G^{\prime}=\chi_{+}^{\prime}(0)$, $H=\chi_{-}(0)$, and
$H^{\prime}=\chi_{-}^{\prime}(0)$, where the primes denote derivatives with
respect to $x$.

When the filling factor $\nu\equiv 2\mu_{QH}/\omega_{c}$ is in the range
$1<\nu<3$, the chemical potential lies between the first and second (spin-
degenerate) Landau levels of the QH region, and one obtains a single pair of
CAES. An example of the spectrum is shown in Fig. 2, where we considered an
ideal interface, i.e., $\mu_{QH}=\mu_{SC}$ and $V_{0}=0$. At low energies, we
see the two linearly dispersing CAES with energies $E_{\pm}(k_{y})=v_{\rm
CAES}(k_{y}\pm k_{0})$.

The Fermi momentum $k_{0}$ appearing in the interference term for the
downstream conductance, cf. (1), can be obtained by solving $f(0,\mp
k_{0})=0$, which in general has to be done numerically. We observe that, as
long as $\Delta\ll\mu_{QH},\mu_{SC}$, the momentum $k_{0}$ does not depend on
$\Delta$. In the limit of a large interface barrier $Z\equiv
2mV_{0}\big{/}k_{F}^{QH}\gg 1$ and $\nu\to 3$, we find $k_{0}l_{B}\ll 1$ and
an analytical solution is possible, namely
$k_{0}\approx(3-\nu)\sqrt{\pi}/4l_{B}$ [9]. In Fig. 3, we show the evolution
of $k_{0}$ as a function of the barrier strength $Z$ for various values of the
filling factor. Typically $k_{0}$ decreases with increasing $\nu$, except for
a small region of intermediate values of $Z$ and fillings $\nu$ close to 3.

The velocity of the low-energy states is given as $v_{\rm
CAES}=-\left.{\partial_{k_{y}}f(E,k_{y})}/{\partial_{E}f(E,k_{y})}\right|_{E=0,|k_{y}|=k_{0}}$.
One may also compute the electron and hole content of the states,
$\psi(x)=(\psi_{e}(x),\psi_{h}(x))$. We define

$f_{h}=\int_{-\infty}^{\infty}dx\;|\psi_{h}|^{2}=1-\int_{-\infty}^{\infty}dx\;|\psi_{e}|^{2}.$

In particular, the result for the states at the Fermi level reads

$\displaystyle\\!\\!\\!f_{h}^{+}=1-$
$\displaystyle\,c_{0}^{2}H_{0}^{2}\left(1\\!+\\!\frac{1}{4q_{0}"}\left(G_{0}^{2}+\frac{(g_{0}+q_{0}"G_{0})^{2}}{|q_{0}|^{2}}\right)\right)\times$
$\displaystyle\,\times\left[g_{0}^{2}+c_{0}^{2}H_{0}^{2}\left(1\\!+\\!\frac{1}{2q_{0}"}\left(G_{0}^{2}+\frac{g_{0}^{2}}{c_{0}^{2}}\right)\right)\right]^{-1},$
(10)

where $g_{0}=G_{0}^{\prime}+d_{0}G_{0}$ and $q_{0}"={\rm Im}\,q_{0}$.
Furthermore, the subscript 0 indicates that the previously introduced
quantities have to be taken at $E=0$ and $k_{y}=-k_{0}$. Particle-hole
symmetry implies $f_{h}^{-}=1-f_{h}^{+}$. In the limit $Z\to 0$, we recover
pure electron and hole states, $f_{h}^{+}=0$ and $f_{h}^{-}=1$. By contrast,
at $Z=0$ and in the limit $\Delta\to 0$, one finds an equal repartition
between electron and hole components, $f_{h}^{+}=f_{h}^{-}=1/2$. As an
illustrative example, we represent the content $f_{h}^{+}$ of the quasi-
electron CAES as a function of the barrier strength $Z$ for various values of
the filling factor $\nu$ in Fig. 4.

Figure 2: Energy spectrum of the states along the QH-SC interface obtained
from Eq. (9). The crossings of the CAES with the Fermi level are indicated by
red lines. Here the parameters are $\mu_{QH}=\mu_{SC}=10\Delta$, $\nu=2.4$,
and $V_{0}=0$.  Figure 3: Fermi momentum $k_{0}$ as a function of barrier
strength $Z$ for various values of the filling factor $\nu$ at
$\mu_{QH}=\mu_{SC}=10\Delta$. Except for a small region of intermediate $Z$
and $\nu$ close to 3, the momentum $k_{0}$ decreases with increasing $\nu$.
Figure 4: Hole content $f_{h}^{+}$ of the quasi-electron CAES versus the
barrier’s strength $Z$ for various values of the filling factor $\nu$ at
$\mu_{QH}=\mu_{SC}=10\Delta$. While at $Z=0$ the hole content is close to
$1/2$, it vanishes as $Z\gg 1$. Interestingly it is enhanced in an
intermediate region for $\nu>2$.

###  III.2 Tight-binding simulation and scattering probabilities

We now turn to the scattering probabilities at the corners where the QH-vacuum
interface and the QH-SC interface meet. In addition to the system parameters,
this corner can be characterized by two angles as shown in Fig. 1: the angle
$\theta_{QH,i}$ that the QH-vacuum interface forms with the continuation of
the QH-SC interface and the angle $\theta_{SC,i}$ that the SC-vacuum interface
forms with the continuation of the QH-SC interface. To ensure that there is no
overlap, the angles must satisfy $\theta_{QH,i}+\theta_{SC,i}>0$. To compute
the scattering probabilities $\tau_{i}$ as a function of these angles and
system parameters, we perform tight-binding simulations with a discretized
version of the Hamiltonian (3) on a square lattice using the Kwant software
[24]. Introducing the Nambu spinor $\Psi_{i}=(c_{i},c_{i}^{\dagger})^{T}$,
where $c_{i}^{\dagger}$ ($c_{i}$) is the operator that creates (annihilates)
an electron at the position $\mathbf{r}_{i}=(x_{i},y_{i})$, the second-
quantized tight-binding Hamiltonian reads,

$\displaystyle\mathcal{H}_{TB}=$
$\displaystyle\sum_{i}\psi_{i}^{\dagger}\left[\left(4t-\mu_{i}+V_{i}\right)\sigma_{z}+\Delta_{i}\sigma_{x}\right]\psi_{i}$
$\displaystyle+\sum_{\langle
i,j\rangle}\psi_{i}^{\dagger}\left[te^{i\phi_{ij}\sigma_{z}}\sigma_{z}\right]\psi_{j},$
(11)

where $\sigma_{x/z}$ are Pauli matrices in Nambu space, and $\langle
i,j\rangle$ denotes pairs of nearest neighbor sites. The barrier potential is
given as
$V_{i}=V_{0}\delta_{x_{i},0}\Theta\left(\frac{L}{2}\\!-\\!|y_{i}|\right)$,
where $\delta_{i,j}$ is the Kronecker delta. In the QH region,
$\mu_{i}=\mu_{QH}$ and $\Delta_{i}=0$, whereas in the SC region,
$\mu_{i}=\mu_{SC}$ and $\Delta_{i}=\Delta$. Using a Peierls substitution, the
hopping matrix element $t=1/(2ma^{2})$, where $a$ is the lattice spacing,
acquires a field-dependent phase [25]

$\displaystyle\phi_{ij}$ $\displaystyle=-\frac{\pi
B}{\phi_{0}}(x_{i}+x_{j})(y_{j}-y_{i})\theta\left(-\frac{x_{i}+x_{j}}{2}\right)$
(12)

with $\phi_{0}$ the flux quantum. This lattice model matches the continuum
model as long as the hopping energy is the largest energy scale,
$\Delta,\mu_{QH},\mu_{SC}\ll t$. We further make realistic assumptions
$\Delta\ll\mu_{QH}\leq\mu_{SC}$.

As the conversion probability from electron to quasi-hole at the first corner
is equal to the conversion probability from quasi-electron to hole at the
second corner when parameters are chosen the same [8],
$\tau_{1}(\theta_{QH},\theta_{SC})=\tau_{2}(\theta_{QH},\theta_{SC})\equiv\tau(\theta_{QH},\theta_{SC})$,
it is sufficient to simulate the first QH-SC corner. The python code is
available in Ref. [26]. When not specified, we set $t=1$ and $\mu_{SC}=t/20$.

Fig. 5 shows the dependence of $\tau$ on angles for
$\mu_{QH}=\mu_{SC}=10\Delta$, In Fig. 5a, $\theta_{QH}=90^{\circ}$ is fixed
while $\theta_{SC}$ varies. We see a weak dependence of $\tau$ on
$\theta_{SC}$ for angles up to $90^{\circ}$. This is not surprising as the
propagation of the chiral edge states does not involve the SC-vacuum
interface. The residual effect of $\theta_{SC}$ on the scattering probability
is due to the modified decay of the edge state wave function into the bulk in
the vicinity of the corner. We illustrate this in Fig. 6 by plotting the
probability density $|\psi_{e}(\mathbf{r})|^{2}-|\psi_{h}(\mathbf{r})|^{2}$ of
an incoming electron state: it can be seen that it is vanishingly small at
angles $<90^{\circ}$ within the SC region. By contrast, $\tau$ decreases as
$\theta_{QH}$ is increased. This is shown in Fig. 5b, where
$\theta_{SC}=90^{\circ}$ is fixed while $\theta_{QH}$ varies. The stronger
sensitivity of $\tau$ on $\theta_{QH}$ can be understood as stemming from the
fact that this angle directly determines the propagation direction of the edge
state and thus the projection of the momentum of the incoming state onto the
direction of the interface.

A more realistic interface is obtained when allowing for different values of
$\mu_{QH}$ and $\mu_{SC}$ as well as for an interface barrier $Z\neq 0$. As an
example, we show in Fig. 7 the evolution of $\tau$ as a function of
$\theta_{QH}$ with $\mu_{SC}=2\mu_{QH}$ and $Z=0.7$. The behavior is
qualitatively similar though the variation with angle is less pronounced. The
stronger variation with $\nu$ reflects the the stronger variation of
$f_{h}^{+}$ at intermediate values of $Z$ shown in Fig. 4.

(a)  (b)

Figure 5: Conversion probability $\tau$ for various values of the filling
factor $\nu$ as a function of (a) the SC angle $\theta_{SC}$ with
$\theta_{QH}=90^{\circ}$ and (b) the QH angle $\theta_{QH}$ with
$\theta_{SC}=90^{\circ}$. The parameters are $\mu_{QH}=\mu_{SC}=10\Delta$ and
$Z=0$. To minimize lattice effects, we only show commensurate angles. The
solid lines are a guide to the eye.  Figure 6: Probability density
$|\psi_{e}(\mathbf{r})|^{2}-|\psi_{h}(\mathbf{r})|^{2}$ of an incoming
electron state for $\theta_{SC}=45^{\circ}$ and $\theta_{QH}=90^{\circ}$. The
interference of CAES along the QH-SC interface (black line) can be clearly
seen. Note that the wave function does not have any weight in the vicinity of
the SC-vacuum boundary. The parameters are $\nu=2$,
$\mu_{QH}=\mu_{SC}=10\Delta$ and $Z=0$. Figure 7: Conversion probability
$\tau$ versus the angle $\theta_{QH}$ for a non-ideal interface at various
values of $\nu$. Here $\theta_{SC}=90^{\circ}$, $\mu_{SC}=2\mu_{QH}=20\Delta$,
and $Z=0.7$. As in Fig. 5, we only show commensurate angles, and the solid
lines are a guide to the eye.

##  IV One-dimensional model

Effective one-dimensional models are very useful to obtain a qualitative
understanding of the edge state physics. They have been extensively used in
the recent works [14, 17, 19, 20, 15] to describe the CAES. In this section,
we address the question how to incorporate the effects discussed in previous
sections into such an effective model.

The starting point is the one-dimensional Bogoliubov-de Gennes Hamiltonian,

$\displaystyle\\!\\!\\!\tilde{H}=\begin{pmatrix}-\frac{i}{2}\\{{v}(y),\partial_{y}\\}-{\mu}(y)\\!\\!\\!&\tilde{\Delta}(y)\\\
\tilde{\Delta}^{*}(y)&\\!\\!\\!-\frac{i}{2}\\{{v}(y),\partial_{y}\\}+{\mu}(y)\end{pmatrix},$
(13)

where $y$ denotes the coordinate along the QH edge, $\tilde{\Delta}(y)$ are
the induced superconducting correlations, $v(y)$ is the edge state velocity in
the absence of superconducting correlations, and $\mu(y)$ is an effective
chemical potential. Furthermore, $\\{.,.\\}$ is the anti-commutator.

Choosing all the parameters to be independent of $y$ allows one to extract the
zero-energy momentum $k_{0}$, the velocity $v_{\rm CAES}$ as well as the hole
content $f_{h}^{\pm}$ of the CAES as introduced in Sec. III.1. Diagonalizing
$\tilde{H}$, one finds
$E_{\pm}(k_{y})=vk_{y}\pm\sqrt{\mu^{2}+\tilde{\Delta}^{2}}$ and
$f_{h}^{\pm}=\left(1\pm\mu/\sqrt{\mu^{2}+\tilde{\Delta}^{2}}\right)/2$. To
match the results of Sec. III.1, we thus set $v=v_{\rm CAES}$,

$\displaystyle\mu$ $\displaystyle=$ $\displaystyle-v_{\rm
CAES}k_{0}(1-2f_{h}^{+}),$ (14) $\displaystyle\tilde{\Delta}$ $\displaystyle=$
$\displaystyle v_{\rm CAES}k_{0}\sqrt{f_{h}^{+}(1-f_{h}^{+})}.$ (15)

The simplest model often used to describe scattering at the corner consists of
choosing a step function for the induced correlations,
$\tilde{\Delta}(y)=\tilde{\Delta}\Theta(y)$. Matching of the wave functions at
the position of the step, $y=0$, directly yields the conversion probability of
an electron into a quasi-hole,

$\tau_{0}=f_{h}^{+}.$

This clearly is not sufficient to correctly describe the scattering – if only
because it doesn’t depend on the geometry of the contact point. Furthermore,
it can be shown that choosing a different velocity $v_{\rm vac}\neq v$ and/or
effective chemical potential $\tilde{\mu}_{\rm vac}\neq\tilde{\mu}$ for the
QH-vacuum interface at $y<0$ does not modify this result. To obtain a
conversion probability $\tau\neq\tau_{0}$, one needs to include a spatial
variation of the induced correlations $\tilde{\Delta}(y)$ in the vicinity of
$y=0$.

We thus consider a more general model with a barrier region,
$-L_{b}/2<y<L_{b}/2$, characterized by the parameters $v(y)=v_{b}$,
$\mu(y)=\mu_{b}$, and $\tilde{\Delta}(y)=\Delta_{b}e^{i\phi_{b}}$. Note that
the relative superconducting phase between the barrier and the bulk is allowed
as time-reversal symmetry is broken by the applied field. Solving the
Schrödinger equation in the three regions (QH-vacuum interface at
$y<-L_{b}/2$, barrier, and QH-SC interface at $y>L_{b}/2$), matching the
solutions at $y=\pm L_{b}/2$, and solving the resulting system, we obtain

$\displaystyle\tau$ $\displaystyle=$
$\displaystyle\left(\sqrt{\tau_{0}}\cos\beta_{b}+\sqrt{1-\tau_{0}}\sin\beta_{b}\right)^{2}$
$\displaystyle-4\sqrt{\tau_{0}(1-\tau_{0})}\sin\beta_{b}\cos\beta_{b}\cos^{2}\frac{\phi_{b}-\delta_{b}}{2}$

with $\alpha_{b}=\sqrt{\mu_{b}^{2}+\Delta_{b}^{2}}L_{b}/v_{b}$,
$\sin\beta_{b}=\sin\alpha_{b}\,\Delta_{b}/\sqrt{\mu_{b}^{2}+\Delta_{b}^{2}}$,
and $\tan\delta_{b}=\cot\alpha_{b}\sqrt{\mu_{b}^{2}+\Delta_{b}^{2}}/\mu_{b}$.
If $\Delta_{b}\neq 0$, $\tau\neq\tau_{0}$ is possible, and the model has
sufficient parameters to obtain an arbitrary value of $\tau$ for a given
$\tau_{0}$. Thus, in principal, this effective one-dimensional model can be
used to describe an arbitrary geometry. However, there is no straightforward
way to estimate parameters.

For illustration, we show in Fig. 8 the downstream conductance as a function
of the length of the QH-SC interface obtained from a full tight-binding
simulation of the structure shown in Fig. 1. Here the same parameters were
used as in Fig. 7 with $\nu=2.8$. It is compared with the result of an
effective 1D model, where we set $L_{b}=\xi/10$ with $\xi=v_{F}^{SC}/\Delta$
the BCS coherence length, $v_{b}=v_{CAES}$, and $\mu_{b}=\mu_{SC}$. We use a
numerical minimization procedure to find the values of $\Delta_{b}$ and
$\phi_{b}$ that give the scattering probabilities $\tau_{1},\tau_{2}$ obtained
from the tight-binding model. Fitting parameters for Fig. 8 were
$\Delta_{b1}=10.08\Delta$, $\phi_{b1}=3.382$ and $\Delta_{b2}=0.32\Delta$,
$\phi_{b2}=3.002$. (Note that the choice is not unique.) In addition, we
adjust the scattering phase $\phi_{12}$ appearing in Eq. (1) so that the
effective model matches the simulation at large $L$. A small mismatch between
the values of $k_{0}$ can be attributed to lattice effects. Furthermore,
deviations are visible at small lengths when the two corners cannot be treated
independently, as assumed in Eq. (1).

Figure 8: Conductance oscillations as a function of length $L$ of an
asymmetric junction, $\theta_{QH,1}=0$ and $\theta_{QH,2}=90^{\circ}$ whereas
$\theta_{SC,1}=\theta_{SC,2}=90^{\circ}$. We compare a full tight-binding
simulation (TB) with the results of an effective one-dimensional model, where
the parameters have been chosen as discussed in Sec. IV. Here $\nu=2.8$,
$\mu_{SC}=2\mu_{QH}=20\Delta$, and $Z=0.7$. The scattering phase $\phi_{12}$
in Eq. (1) is adjusted to match the results of the tight-binding simulation at
large $L$.

##  V Further considerations

In addition to the difficulty of determining parameters, effective 1D models
have other obvious limitations.

The effective 1D model only describes the topologically-protected chiral edge
states. As can be seen in Fig. 2, in a full 2D calculation, additional subgap
states may appear. While for the parameters chosen in Fig 2, these states are
close to the gap edge, they may cross the Fermi level in other parameter
regimes. An example is shown in Fig. 9. We studied their parameter dependence
and found zero-energy crossings only happen for $\nu\lesssim 3$ and close to
ideal interfaces. In experimentally relevant regimes, they are not expected to
play a role as discussed in App. A. Note that additional in-gap states may
appear as well when the interface is smooth. This question has been addressed
in Ref. [18].

Figure 9: Spectrum with additional non-chiral zero-energy edge states. Here we
set $\nu=2.8$, $\mu_{QH}=\mu_{SC}=20\Delta$ and $Z=0$. These additional states
appear for $\nu\lesssim 3$ and close to ideal interfaces.

The downstream conductance at finite temperature is more likely affected by
these non-chiral states. Furthermore, at finite $T$, the linear approximation
for the dispersion of the CAES may not be sufficient. Namely, as long as
$k_{B}T\ll\Delta$ and continuum contributions may be neglected, the downstream
conductance $G_{d}(T)$ takes the form

$\displaystyle G_{d}(T)\approx
G_{0}\int_{-\Delta}^{\Delta}dE\frac{1-2P_{h}(E)}{4k_{B}T\cosh^{2}\left(\frac{E}{2k_{B}T}\right)},$
(17)

where $P_{h}(E)$ is given by Eq. (1) by replacing $2k_{0}$ with $\delta
k(E)=k_{qe}(E)-k_{qh}(E)$, where $k_{qe/qh}(E)=E/v\pm k_{0}$, and using the
transmission probabilities $\tau_{i}$ at energy $E$. If $\delta k$ varies
significantly with energy on the scale $k_{B}T$, this leads to an averaging of
the oscillations of the downstream conductance. Numerically we find that the
effect is small in experimentally relevant parameter regimes, see App. B.

##  VI Conclusion

In this paper, we have studied the downstream conductance mediated by CAES in
QH-SC junctions. In particular, we found that the geometry plays an important
role. This limits the applicability of simple effective 1D models that are
often used to describe such systems. We showed that the most general effective
1D model containing a complex pairing potential localized in the region where
the QH-vacuum edge meets the QH-SC edge allows one to model an arbitrary
electron-hole conversion probability – however, there is no clear prescription
as to how parameters have to be chosen. We note that the geometry dependence
may be exploited to device asymmetric junctions, where the overall electron-
hole conversion probability is enhanced and the average downstream conductance
can become negative. This may be a way to obtain clearer signatures of the
Andreev conversion at the QH-SC interface. Our work concentrated on the clean
case. It will be interesting to explore how these features are modified by
disorder.

###### Acknowledgements.

We thank X. Waintal for help with Kwant. A.D. gratefully acknowledges
interesting discussions with A. Bondarev. Furthermore, we acknowledge support
from the ANR through Grant Nos. ANR-17-PIRE-0001 and ANR-21-CE30-0035.

##  Appendix A Additional non-chiral edge states

As discussed in the main text, see Fig. 9, additional non-chiral edge states
may cross the Fermi level in certain parameter regimes as $\nu$ approaches 3.
Using the continuous model of Sec. III.1, we may determine the value $\nu_{c}$
above which such states are present as a function of system parameters. To do
so we need to solve the secular equation, Eq. (9), at $E=0$ and determine the
value $\nu_{c}$ at which a second solution with $k_{y}>k_{0}$ appears. The
results are shown in Fig. 10 as a function of $\mu_{QH}/\Delta$ for an ideal
interface as well as a function of the interface barrier strength $Z$ and
mismatch $\mu_{SC}/\mu_{QH}$ at $\Delta=\mu_{QH}\times 10^{-6}$. We see that
at $\Delta/\mu_{QH}\ll 1$ additional zero-energy states appear for
$\nu>\nu_{c}\approx 2.63$ in the case of an ideal interface. An interface
barrier as well as potential mismatch push that critical value up. It reaches
3 at $\mu_{SC}/\mu_{QH}\gtrapprox 3.73$ or $Z\gtrapprox 0.65$. Beyond these
values, one never finds additional zero-energy states, which is likely the
case in experiments.

(a)  (b)  (c)

Figure 10:  Plots of $\nu_{c}$, indicating the appearance of additional non-
chiral edge states at the Fermi level, as a function of different parameters.
(a) Dependence of $\nu_{c}$ on $\mu_{QH}/\Delta$ for an ideal interface,
$\mu_{SC}=\mu_{QH}$ and $Z=0$. In the limit $\mu_{QH}/\Delta\to\infty$, the
critical value tends to $\nu_{c}\approx 2.63$. (b) Dependence of $\nu_{c}$ on
the mismatch $\mu_{SC}/\mu_{QH}$ at $\Delta=\mu_{QH}\times 10^{-6}$ and $Z=0$.
As $\nu_{c}$ reaches 3, the additional non-chiral subgap states disappear at
moderate values of the mismatch. (c) Dependence of $\nu_{c}$ on the barrier
strength $Z$ at $\Delta=\mu_{QH}\times 10^{-6}$ and $\mu_{SC}=\mu_{QH}$. As
$\nu_{c}$ reaches 3, the additional non-chiral subgap states disappear at
moderate values of the barrier strength.

##  Appendix B Downstream conductance at finite temperature

(a)  (b)  (c)  (d)

Figure 11:  Energy dependence of various parameters necessary to determine the
downstream conductance at finite temperature. Parameters are the same as in
Fig. 8. (a) Energy spectrum. (b) Variation of the relative momentum difference
$|\delta k(E)-2k_{0}|/2k_{0}$ of the pair of CAES. (c) & (d) Conversion
probability $\tau_{1}=\tau(\theta_{QH}=0,\theta_{SC}=90^{\circ})$ and (d)
$\tau_{2}=\tau(\theta_{QH}=90^{\circ},\theta_{SC}=90^{\circ})$. Variations in
Figs. (b-d) are seen to be small as long as $|E|\ll\Delta$.

As the downstream conductance at finite temperature involves an integral over
the hole conversion probabilities at different energies, it is important to
know the dependence of the parameters determining the hole conversion
probability ion energy. In particular, if the momentum mismatch $\delta k$ or
the phase $\phi_{12}$ strongly varies with energy, the oscillations of the
conductance should be averaged out upon increasing temperature.

The momentum mismatch $\delta k(E)$ can be obtained from the continuum model.
We find that, even beyond the regime where the edge state spectrum is linear,
the variation of $\delta k$ remains small. We illustrate our findings in Fig.
11. Here the same parameters as in Fig. 8 were used. The spectrum is shown in
Fig. 11a. Additional non-chiral edge states are visible at energies
$|E|\gtrsim\Delta/2$. The relative deviations of $\delta k(E)$ from $\delta
k(0)=2k_{0}$ are shown in Fig. 11b. For small enough energies, the deviations
are small, implying a nearly constant period of the oscillations. Figs. 11c
and 11d show the energy dependence of the conversion probabilities $\tau_{1}$
and $\tau_{2}$. Again the variation is weak is up to the energy where
additional subgap states appear. Note that this is consistent with what one
would obtain from our effective 1D model, where there is no energy dependence.
The scattering phase $\phi_{12}$ (not shown) remains approximatively constant
in this regime as well. These findings suggest that the zero-temperature
results obtained for the downstream conductance are robust as long as
$k_{B}T\ll\Delta$. This is confirmed by a full tight-binding simulation, shown
in Fig. 12. For $k_{B}T/\Delta=0.1$, the result is almost unaffected. By
contrast, at the larger temperature $k_{B}T/\Delta=0.5$, a clear suppression
of the amplitude of the oscillations is observed while the mean value
increases as energies close to $\Delta$ start to contribute, where variations
of $\delta k$ become non-negligible and $\tau_{i}\to 0$.

Figure 12: Downstream conductance at different temperatures. The zero-
temperature result is shown by blue dots. At $k_{B}T=\Delta/10$ (orange line),
there is almost no change. By contrast, a clear reduction of the amplitude of
the oscillations is observed at $k_{B}T=\Delta/2$ (green line). Parameters are
the same as in Fig. 8.

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