# Dynamic collaborative filtering Thompson Sampling for cross-domain
advertisements recommendation

Shion Ishikawa<EMAIL_ADDRESS>Rakuten Institute of
TechnologyJapan ,  Young-joo Chung<EMAIL_ADDRESS>Rakuten
Institute of TechnologyThe United States of America and  Yu Hirate
<EMAIL_ADDRESS>Rakuten Institute of TechnologyJapan

###### Abstract.

Recently online advertisers utilize Recommender systems (RSs) for display
advertising to improve users’ engagement. Contextual bandit model is a widely
used RS to exploit and explore users’ engagement and maximize the long-term
rewards such as clicks or conversions. However, the current models aim to
optimize a set of ads only in a specific domain and do not share information
with other models in multiple domains. In this paper, we propose dynamic
collaborative filtering Thompson Sampling (DCTS), the novel yet simple model
to transfer knowledge among multiple bandit models. DCTS exploits similarities
between users and between ads to estimate a prior distribution of Thompson
sampling. Such similarities are obtained based on contextual features of users
and ads. Similarities enable models in a domain that didn’t have much data to
converge more quickly by transferring knowledge. Moreover, DCTS incorporates
temporal dynamics of users to track the user’s recent change of preference. We
first show transferring knowledge and incorporating temporal dynamics improve
the performance of the baseline models on a synthetic dataset. Then we conduct
an empirical analysis on a real-world dataset and the result showed that DCTS
improves click-through rate by 9.7% than the state-of-the-art models. We also
analyze hyper-parameters that adjust temporal dynamics and similarities and
show the best parameter which maximizes CTR.

††conference: ; August 15, 2022; Washington, D.C.

##  1\. Introduction

Recently online advertisers utilize Recommender systems (RSs) for display
advertising to improve users’ engagement. Traditionally, collaborative
filtering (su2009survey, ) is one of the popular recommender systems
(anastasakos2009collaborative, ). However, it suffers from the cold start
problem (silva2019pure, ); Collaborative filtering cannot make a good
recommendation for new items and new users without historical interactions.
Content-based methods and hybrid models (DBLP:conf/recsys/Kula15,
),(guo2017deepfm, ) leverage contextual features (i.e., side information) of
items and users to overcome the cold-start problem.

Meanwhile, reinforcement learning (RL) -based methods have shown its
effectiveness on Recommender systems. By utilizing both exploration and
exploitation, it can maximize the long-term profits for RS. It is suited for
advertisement recommendation, where the ads are shown in the limited time
period. Among reinforcement learning-based methods, bandit models were
frequently applied to that problem (pandey2006handling, ), which dynamically
selects arms (corresponding to actions in RL) and maximizes total rewards.
Contextual bandit that utilizes side-information of users and ads as contexts
(e.g., user demography, creative characteristics) achieved great success in
various areas such as movie and book recommendation. (LinUCB
(li2010contextual, ) and LinTS (agrawal2013thompson, )). Ads, unlike movies
and products, are created when new sales campaigns begin and are removed from
displays when campaigns are over. Therefore, ratios of newly created ads are
higher compared to new movies or products. Traditional methods such as
collaborative filtering can’t perform well for ad-recommendation because of
the sparsity of training data and it is not suited for long-term profit
optimization. RL solved long-term profit problem and can deal with cold-start
cases relatively well, but it still suffers from this extreme cold-start
conditions. In this scenario, the agent tends to focus on exploration and
sacrifice short-term reward for long-term reward. The $\epsilon$-first policy
(tran2010epsilon, ), for instance, purely explores in cold-start conditions
via random recommendations, which goes overboard with the exploitation and
exploration balancing and results in even worse performance. Here previous
models were mainly created to optimize a specific widget on a web domain and
each model was independent of each other. Considering that the number of
active users in a domain (a business unit) varies in large companies with
multiple business units, it is natural to transfer model’s knowledge from
popular domain to unpopular/new domains to improve the recommendation
performance and convergence speed. Thus, in this paper we propose a model to
improve performance of existing RL-based recommend systems by utilizing
knowledge transfer.

Figure 1. Use-case of the dynamic collaborative filtering bandit.

We therefore propose multi-domain optimization to allow all model to
communicate and share knowledge with each other. Our method, dynamic
collaborative filtering Thompson Sampling (DCTS), can accelerate convergence
with only a small amount of data by utilizing information from models in
different domains. We here naturally expanded the concept of similarities
among ads (slivkins2011contextual, ) (li2016collaborative, ) to those in
different domains. In addition, we added the concept of temporal dynamics by
incorporating temporal dynamics to reflect user’s sudden preference change.
Recently collaborative filtering with dynamic terms (koren2009collaborative, )
and a RNN-based model (ma2020temporal, ) have improved existing performances.
We here added the concept of temporal dynamics by incorporating discounting
rewards (kocsis2006discounted, ).

In summary, our proposed model incorporates temporal dynamics in the multi-
domains and it enables our model to track the entire user’s journey among
multi-domains. We showed one use-case of multi-domains optimization in Fig. 1.
For example, a user was interested in earning reward points and clicked point-
related advertisements. Recently this user came to get an interest in eating
delicious food and clicked food-related ads on the e-commerce site (domain A).
Our model will recommend the related restaurant in the restaurant reservation
website (domain B), can recommend related hotels in the newly created website
for lodging reservations (domain C), and improve the cold start issue of
domain C. Our model also recommends food-related ads clicked by the user to
the similar user in domain A. These kinds of cross-domain and cross-user
recommendations will be conducted based on similarity function $S_{ad}$ and
$S_{user}$. To evaluate the performance of our model for this kind of use-
case, in this research, we first compare the performance of our model with the
one of existing bandit models on synthetic dataset, and show our model
leveraged knowledge in other domains and adapt the sudden change of customer’s
preference. Then, we empirically showed our model could also outperformed
existing bandits on the real-world dataset. Here we used our internal dataset
of online ads in our Rakuten Ichiba and Rakuten Travel , and observed
improvement of CTR.

Our contributions are as follows:

  1. (1)

We propose a dynamic collaborative filtering Thompson Sampling (DCTS), which
improve the estimation of prior distribution for Thompson Sampling by
utilizing similarity among ads and users, and discounting rewards. DCTS
leverages contextual features of ads and users for similarity computation. Our
model will leverage this similarity and transfer knowledge across
heterogeneous domains. It leads to track recent crosss-domain behavior of
users and gives a prior estimation of arms. As a result, it accelerates
convergence.

  2. (2)

We conducted empirical analyses on real-world dataset for advertisements in
multiple domains. We compared our method with the state-of-the-art algorithms
such as Transferrable LinUCB (labille2021transferable, ), hybrid LinUCB, and
Thompson Sampling. As a result, DCTS performed better than T-LinUCB by 9.7 %
and better than LinUCB and TS by around 37 % in terms of click-through rate
(CTR).

  3. (3)

We also evaluated the performance of hyper-parameters. Regarding the parameter
$\gamma$ which adjusts decays of rewards, we observed the best performance
when $\gamma$ = 0.25. It was better than when using responses in the latest
batch ($\gamma$ = 0) and no decay ($\gamma$ = 1). Regarding the parameter
$\gamma$ which adjusts the importance of global rewards, the best values when
$g$ =1 or $g$ =0.1. We assume this is because real-world dataset are sparse.
This suggests it’s still important to retain global rewards even when
personalized rewards are considered.

##  2\. Related Work

In this section, we discuss the existing bandit models for recommend systems.
First of all, Thompson sampling (chapelle2011empirical, ) was introduced to
select an arm that maximizes the expected reward in a randomly drawn belief.
It picked the best arm based on the sampled score from the posterior
distribution. Upper-Confidence-Bound methods deterministically take the action
according to an optimistic estimateof the reward they will yield. LinUCB
(li2010contextual, ) proposed the contextual bandit models to improve the
rewards.

To incorporate temporal dynamics with bandit models, UCB with discounting
reward was proposed (kocsis2006discounted, ). It utilized discount to reflect
recent change of user’s behaviour. Then, Rotting Bandit (levine2017rotting, )
was proposed. Rotting Bandit handles discount as a function of the number of
times each arm has been pulled. These previous studies showed that
incorporating dynamic change of user’s preference could increase the accuracy
for the Yahoo! new recommendation and KDD Cup 2012 Online Ads.

To add pre-estimation of rewards by utilizing similarities, contextual zooming
algorithms (slivkins2011contextual, ) and bandit with collaborative filtering
(li2016collaborative, ) were proposed. Bandit with collaborative filtering
utilized similarities between items in one domain, also dynamically built
clusters of users and items and leveraged them for the calculation of
similarities. Our research expanded the concept of bandits with similarities
to the dynamic optimizations across multiple heterogeneous domains. Also, we
utilized Locally-sensitive Hashing (charikar2002similarity, ) to calculate
similarities efficiently.

As for bandits with transfer learning, TCB (liu2018transferable, ) introduced
the translation matrix among domains to utilize similarities of domains.
T-LinUCB (liu2018transferable, ) utilized this similarities as prior
estimation of domains and alleviated cold-start problems. Here we describe the
difference between our model and TCB/T-LinUCB. First, our model is an
expansion of Thompson sampling whereas TCB/T-LinUCB were expansions of UCB;
Second, our model leverages similarity between users as well as domains;
Finally, only our model incorporates dynamic behaviour, captures recent change
of preference in the cross-domains. Also, B-kl-UCB (zhang2017transfer, ) is a
bandit considering transfer learning but this models assume the set of actions
are same across domain; on the other hand, our model can handle different
action set in the heterogeneous domains by leveraging similarities.

##  3\. Model

In this section, we propose the dynamic collaborative filtering Thompson
sampling (DCTS). First, we assume we have $N$ available sources. Each source
is corresponded to a widget where ads are displayed. In each source, we have a
set of ads. Let a set of ads in source $s$ be $\mathbf{A_{s}}$. Also let
$\mathbf{X}$ be a matrix of $m$ users which have $d_{u}$ features where
$\mathbf{X}\in\mathbb{R}^{m\times d_{u}}$, and let $\mathbf{Y^{s}}$ be a
matrix of $k_{s}$ ads which are in source $s$ and have $d_{a}$ features where
$\mathbf{Y^{s}}\in\mathbb{R}^{k_{s}\times d_{a}}$. Then, we denote $i_{t}^{s}$
as user $i$ in source $s$ at time step $t$. For each time step $t$ and source
$s$, we observe $x_{i_{t}^{s}}\in\mathbf{X}$ and
$y_{a^{s}_{t}}\in\mathbf{Y^{s}}$ as contexts, and user will see ad
$a^{s}_{t}\in\mathbf{A_{s}}$ and we observe reward $r_{i_{t}^{s}a_{t}^{s}t}$
where $r_{i_{t}^{s}a_{t}^{s}t}$ is an implicit reward which indicates whether
user $i_{t}^{s}$ clicked ad $a^{s}_{t}\in\mathbf{A_{s}}$ or not. We here
denote observation as
$O_{t}^{s}=(x_{i_{t}^{s}},y_{a^{s}_{t}},r_{i_{t}^{s}a_{t}^{s}t})$.

The objective of DCTS is to pick up an ad $a_{t}^{s}$ to display as an action
to maximize cumulative rewards
$\sum_{s}^{N}\sum_{t=0}^{T}r_{i_{t}^{s}a_{t}^{s}t}$ where T is a maximum time
step. This can be represented as minimizing total regret of user $i$ as
follows:

(1) $\begin{split}{\text{minimize
}}\mathbb{E}[\text{regret}_{i}(T)]=\sum_{s=0}^{N}\mathbb{E}[\underset{a_{t}^{s}\in
A_{s}}{\text{max}}\sum_{t=0}^{T}r^{*}_{i_{t}^{s}a_{t}^{s}t}-\sum_{t=0}^{T}r_{i_{t}^{s}a_{t}^{s}t}]\end{split}$

where $r^{*}$ indicates a reward from the best action for user $i_{t}^{s}$.

###  3.1. transferring knowledge via similarities of sources and users

DCTS learns a policy to pick up ads from observations of all ad sources
$\mathbf{O}=\\{O^{s}\\}_{s=0,...,N}$. DCTS utilizes connections between
sources and transfers knowledge among sources and users, which leads a policy
to be aware of more generalized user’s behaviors. DCTS transfers rewards among
users and among ads based on their similarities. Here we utilize cosine
similarity as a degree of transferring:

(2) $\begin{split}\mathcal{S}_{user}(x_{i},x_{j})=\frac{x_{i}\cdot
x_{j}}{|x_{i}||x_{j}|}\end{split}$

where $x_{i}$ and $x_{j}$ are contextual features of user $i$ and $j$. In the
same way, we define $\mathcal{S}_{ad}(y_{i},y_{j})$ by contextual features of
ads.

The number of users is huge in the real-world dataset and calculating
similarities among all pairs of users is not scalable. Therefore, we leveraged
Locally-sensitive Hashing (charikar2002similarity, ) to obtain similarities of
users efficiently.

###  3.2. Dynamic collaborative filtering Thompson sampling

Before describing an algorithm of DCTS, we first describe the original
Thompson sampling. Thompson sampling in the bandit problem is a method for
pulling an arm that maximizes the expected reward. This is equal to sampling
estimated scores from the posterior distributions of the arm in each round and
choosing the arm with the highest score. In Bernoulli Bandit case, the
likelihood function is formulated by Bernoulli distribution whose prior
distribution is Beta distribution as a natural conjugate prior. Beta
distribution can be written as

(3)
$\begin{split}\mathcal{B}(\theta_{k};\alpha_{k},\beta_{k})=\frac{\mathbf{\Gamma}(\alpha_{k}+\beta_{k})}{\mathbf{\Gamma}(\alpha_{k})\mathbf{\Gamma}(\beta_{k})}\theta_{k}^{\alpha_{k}-1}(1-\theta_{k})^{\beta_{k}-1}\end{split}$

where $\theta_{k}$ is a probability that action $a_{k}$ produces rewards, and
$\alpha_{k}$ and $\beta_{k}$ are parameters. As a non-informative prior,
previous studies assumed the special case where $\alpha_{k}=1$ and
$\beta_{k}=1$. _However, for the most practical cases, values of actions can
be estimated by utilizing historical data._ With leveraging user-user and ad-
ad similarity, DCTS offers the better estimation of prior distribution. First,
we formulate the pre-estimated parameters for each user $i$ and ad $k$ at time
step $t$ as follows:

(4) $\begin{split}\alpha^{0}_{ik}(t)=\sum_{l\neq
k}S_{ad}(y_{k},y_{l})s_{il}(t)+\sum_{j\neq i}S_{user}(x_{i},x_{j})s_{jk}(t)\\\
\beta^{0}_{ik}(t)=\sum_{l\neq k}S_{ad}(y_{k},y_{l})f_{il}(t)+\sum_{j\neq
i}S_{user}(x_{i},x_{j})f_{jk}(t)\end{split}$

where $s_{il}(t)$ is a discount-aware cumulative reward

(5)
$\begin{split}s_{il}(t)=\sum_{\tau=0}^{t}\gamma^{t-\tau}s_{ij\tau}\end{split}$

and $s_{ij\tau}$ is a binary variable, which will be $1$ if we observed reward
for user $i$ and ad $k$ at $\tau$ and $0$ otherwise, and $\gamma$ indicates a
discount ratio. In the same way, $f_{jk}(t)$ will be defined as a discount-
aware cumulative negative reward by $f_{ij\tau}$ and a discount ratio
$\gamma$. $f_{ij\tau}$ represent the number of failure of recommnedation; it
will be $1$ if user $i$ viewed ad $k$ at $\tau$ but didn’t click the ad. Here
we naturally assume the value of ad $k$ for user $i$ can be estimated by
similarity between them. Therefore, we transferred rewards based on
similarities of users and ads. We also assumed user’s preferences will change
with time, so we discount rewards and put a high value on rewards from user’s
recent behaviors.

Second, we formulate posterior distribution. For the basic Thompson sampling,
posterior distribution will be formulated by Beta distribution. With uniform
prior knowledge, the parameters were $\alpha_{k}=s_{k}+1$ and
$\beta_{k}=f_{k}+1$. In this research, with using prior knowledge in Eq. 4, we
formulate parameters of posterior Beta distribution as follows:

(6)
$\begin{split}\alpha_{ik}(t)=\lambda(s)\alpha^{0}_{ik}(t)+gs_{k}(t)+s_{ik}(t)+1\\\
\beta_{ik}(t)=\lambda(f)\beta^{0}_{ik}(t)+gf_{k}(t)+f_{ik}(t)+1\end{split}$

where $\lambda$ is a hyper-parameter which adjusts importance of prior
knowledge and $\lambda(s)=\frac{\lambda}{s_{ik}(t)+1}$,
$\lambda(f)=\frac{\lambda}{f_{ik}(t)+1}$. $g$ is a hyper-parameter that
adjusts the importance of global rewards. Like the original Thompson sampling,
we introduce $s_{k}(t)$ and $f_{k}(t)$ as a mean of rewards among users, since
the interactions of users and ads are sparse so that there are few valid
similar users will occur in real-world cases. 1 at the last term is a pseudo
count to avoid errors when historical reward were unavailable. We present
details of the policy DCTS in Algorithm 1.

Algorithm 1 Dynamic collaborative filtering Thompson sampling

0:

1: for $t=0,...,T$ do

2: Observe user $i_{t}^{s}$ and context $x_{i_{t}^{s}}$, action sets
$\bf{A_{s}}$ and their contexts $\bf{Y^{s}}$

3: for $k\in\bf{A_{s}}$ do

4: Calculate $\alpha^{0}_{i_{t}^{s}k}(t)$ and $\beta^{0}_{i_{t}^{s}k}(t)$
according to Eq. 4

5: Calculate $\alpha_{i_{t}^{s}k}(t)$ and $\beta_{i_{t}^{s}k}(t)$ according to
Eq. 6

6: Sample $\theta_{k}$ from the
$\mathcal{B}(\theta_{k};\alpha_{i_{t}^{s}k}(t)$, $\beta_{i_{t}^{s}k}(t))$

7: end for

8: Play action $k=\underset{k}{\mathop{\rm arg~{}max}\limits}$ $\theta_{k}$
and observe reward $r_{i_{t}^{s}kt}$

9: Add observation $O^{s}_{t}\leftarrow(x_{i_{t}^{s}},k,r_{i_{t}^{s}kt})$

10: end for

##  4\. Experiments

###  4.1. Synthetic Experiment

In this section, we evaluated the two advantages of DCTS with a simple
synthetic environment. The first advantage is a performance of transferring
knowledge among multi-domains. We prepared 10 ads and displayed 5 of the ads
on domain A, and the remaining 5 ads on domain B. In the first 500 steps,
there is only domain A and users were only allowed to visit domain A. In the
latter 500 steps, domain B was created and users were allowed to visit domain
B. We evaluated the performance when users first visited domain B. This kind
of scenario often happens in the real world when users newly registered domain
B, or when domain B or ads in domain B are newly created. In this synthetic
setting, we set $\lambda=1$, $g=1$, and $\gamma=1$. As for $\gamma$, we wanted
to investigate the performance of transferring, so we disabled discounting
rewards and set $\gamma$ as $1$. We also prepared the user’s response function
as an environment of reinforcement learning. This function receives contextual
attributes of users and ads and returns rewards with the probability, which is
obtained from the logistic regression with contextual attributes. Weights of
logistic regression were sampled from the standard normal distribution.

Figure 2. Synthetic simulation of transferring knowledge in 2 different pages

We drew a line plot of the average cumulative reward. The red vertical line
indicates the time step we changed a set of ads. The shaded band indicates 95%
confidence interval obtained by three times.

Figure 3. Synthetic simulation of changing user’s preference

We drew a line plot of the average cumulative reward across time steps. The
red vertical line indicates the time step when the change of user’s preference
occurred. The shaded band indicates 95% confidence interval obtained by three
times executions.

According to Fig. 2, at the very beginning TS and DCTS had the same
performance but over time DCTS outperformed than TS because DCTS could handle
a part of contextual preferences as personal rewards $s_{ik}$ and $f_{ik}$.
Also, as we assumed, the performance of DCTS right after the transition to the
new set of ads was much better than that of TS. This is because DCTS was able
to leverage knowledge of previous ad, while TS had to start exploration again.
We assume the average rewards of DCTS decreased until time step 700 because
the weight of prior knowledge $\lambda(s(t))$ decayed. In this case, higher
$\lambda$ seems to be better. Later in this paper, we will investigate the
effect of hyper-parameters.

To verify the second advantage of DCTS, namely its fast adaptation to user’s
preference change, we prepared 50 ads and made a user’s response function,
which is the same as the above one but its contextual weights will change at
the specified time. In this synthetic setting, we set $\lambda=1$ and $g=0$
and $\gamma=0.95$. We let the discount of rewards occur at every 10-time step
because in the real-world setting we update models in a batch and the time
step will be a certain duration of time. According to Fig. 3, for both TS and
DCTS rewards decreased right after the change of user’s preference. The
performance of DCTS recovered around time step 250 while TS recovered around
time step 300. This indicates that DCTS could learn the change of the user’s
preference more quickly than TS with the advantage of discounted rewards.

###  4.2. Offline experiment using real-world data.

In this section, we empirically compare DCTS with the TS, LinUCB with a hybrid
linear model (hLinUCB) (li2010contextual, ) and Transferable LinUCB (T-LinUCB)
(labille2021transferable, ) using real-world dataset. Details of baseline
models are as follows:

  * •

hLinUCB (li2010contextual, ): LinUCB is an expansion of UCB policy and
incorporates contextual features of arms. hybrid LinUCB incorporates contexts
of both arms and users.

  * •

T-LinUCB (labille2021transferable, ): This model is an expansion of LinUCB.
This calculates the prior evaluation of arms in the new domain with contextual
feature of arms in different domains.

  * •

Thompson Sampling (chapelle2011empirical, ): This model selects the action
that maximizes the expected reward with respect to a randomly drawn belief of
scores. In practice, this policy will sample scores from the posterior
distribution and pull the arm with maximum score.

Because there was no public dataset that includes ads among multi-domains with
contextual features of both users and ads, we leveraged our internal dataset
at Rakuten. Rakuten is an e-commerce company in Japan that serves over 70
services, such as online shopping, financial services, and e-books. Among
them, Rakuten Ichiba is an e-commerce platform and Rakuten Travel is an online
travel agency platform for lodging reservations. In this experiment, we used
data from Rakuten Ichiba for pre-training and let DCTS learn the rewards of
Rakuten Ichiba. We then evaluated performance for Rakuten Travel and compared
DCTS with other baselines.

Regarding evaluation methods, there are various methods for offline policy
evaluation of the bandit algorithm. Replay method (li2011unbiased, ) is often
leveraged but it requires random assignment of ads. Most datasets in the real
world, however, cannot afford random assignment because of the counterfactual
estimations. In other words, we can’t know the true responses if we showed
other ads to users. To overcome this issue, corrections such as Doubly Robust
estimation (jiang2016doubly, ) were proposed. However, here we propose another
simple way for the counterfactual estimation. In the real world, multiple ads
are often displayed at the same time in the widget like a carousel. In this
offline simulation, we regarded our model to pick only one of these ads to
users at each step. The advantage of this method is: in this situation not
only we know the user’s response of that ad, but also we know user’s responses
for other ads because in the real-world these ads were displayed together.
Moreover, for each impression of the carousel, we only have, at most, one
clicked ad because otherwise users will see the landing page after clicking an
ad. In this paper, we leveraged this technique and treated ads displayed at
the same time in the carousel as displayed in the one slot to obtain
counterfactual information.

Table 1. Statistics of Rakuten Ichiba and Rakuten Travel data. | impressions | responses |  | unique
---
users
ads
| Rakuten
---
Ichiba
191202 | 23394 | 33750 | 14
| Rakuten
---
Travel
16029 | 801 | 4552 | 5

Utilizing the technique above, we showed one ad at each time step $\tau$, and
if user $i$ clicked ad $k$ within 15 min, we regarded $s_{ik\tau}=1$ and
$f_{ik\tau}=0$. Otherwise $s_{ik\tau}=0$ and $f_{ik\tau}=1$. We discarded
subsequent impressions within 5 min after the first impression from the same
user to avoid the bias of repeated bot traffic. We summarized the Rakuten
Ichiba data and Rakuten Travel data we used in Table 1. Please note that the
sampled data does not represent the entire dataset. Datasets have a timestamp
and we served them in chronological order to the model.

Figure 4. t-SNE projection of users labeled with age groups. Users in the same
age group were clustered together.
---
Figure 5. t-SNE projection of users labeled with reward point status. Users
with low reward status were mostly placed in the upper part, while users with
high reward status were placed in the lower part.
Figure 6. t-SNE projection of ads labeled with ad types. Labels without
“travel_” in the prefix means the ads in “E-commerce service (Ichiba)”. Ads
which are specific for travel such as promotions for sightseeing spots and
hotels were clustered together. Also, point related ads for both Rakuten
Ichiba and Travel were clustered together.

DCTS also requires similarities among users and ads. To calculate them, we
utilized Customer DNAs in Rakuten. Customer DNAs serves thousands of
contextual attributes for each user. They consist of basic demographics such
as gender and age, and users behaviors such as the preference of purchases and
status of earned reward points. We leveraged 308 attributes among them and
calculated the cosine similarity of users by Eq. 2. We didn’t have contextual
attributes for ads so we regarded median values of contextual attributes of
users who clicked the ads as the context of ads. We visualized the users and
ads by t-SNE (van2008visualizing, ) to check that the above similarities were
reliable. As Fig. 6 and Fig. 6 showed, similarity of users could be largely
explained by ages and point reward status. In other words, users with similar
ages were placed closely. Also Fig. 6 showed similar type of ads were placed
closely with each other in the latent structure. This implies that user and ad
similarity indeed exist and can be utilized by DCTS.

In Fig. 7, we empirically verified that knowledge transfer among multi-
domains, users, and temporal dynamics could improve the bandit model. Total
CTRs of all policies were initially worse than random policy. Performance of
hLinUCB improved at the step around 1000. At the step around 2000, T-LinUCB
and DCTS outperformed hLinUCB and DCTS became the best, but DCTS had high
variance. We observed improvement of performance at step around 9000 for TS
and also observed that CTR of hLinUCB decreased after step around 11000. We
currently assume this was because, at step 11000, customer preferences
changed. As a result, DCTS performed better than T-LinUCB by 9.7 % and better
than LinUCB and TS by around 37 %. We set DCTS’s hyper-parameters as
$\lambda=10$ and $g=1$ and $\gamma=0.5$ and discounted values by multiplying
$\gamma$ every hour.

Figure 7. Relative CTR on Rakuten Travel data.

We compared dynamic collaborative filtering bandit (DCTS) with Transferable
LinUCB (T-LinUCB), hybrid LinUCB (hLinUCB), Thompson Sampling (TS) and random.
We initially pre-trained DCTS and T-LinUCB by Rakuten Ichiba data and then
utilized them for Rakuten Travel data.

We also evaluated the performance change with different hyper-parameters. Fig.
8 shows relative CTRs by various $\gamma$, and we normalized values by the
value of when $\gamma=0.5$. According to Fig. 8, we observed the best
performance when $\gamma=0.25$. It suggests this value is the most suitable
decay for Rakuten Travel dataset.

##  5\. Conclusion

In this paper, we proposed a dynamic collaborative filtering Thompson sampling
(DCTS) for a cross-domain and dynamic recommendation. DCTS leveraged the
concept of discounting reward and collaborative filtering. As a result, it
improved the prior distribution of Thompson sampling through leveraging
similarity in users and ads. DCTS can adjust exploration and exploitation by
choosing arm based on posterior probability and accelerate the convergence of
the model, which led to maximizing cumulative rewards. We evaluated DCTS on
the real-world data and showed that DCTS outperformed TS, hLinUCB, and random.
We also plan to build a model which incorporates the pattern of the user’s
behaviour as a more accurate recommendation in the user’s temporal dynamics.

Figure 8. Relative CTR on Travel data with various $\gamma$ parameters

## References

  * [1] Xiaoyuan Su and Taghi M Khoshgoftaar.  A survey of collaborative filtering techniques.  Advances in artificial intelligence, 2009, 2009.
  * [2] Tasos Anastasakos, Dustin Hillard, Sanjay Kshetramade, and Hema Raghavan.  A collaborative filtering approach to ad recommendation using the query-ad click graph.  In Proceedings of the 18th ACM conference on Information and knowledge management, pages 1927–1930, 2009.
  * [3] Nícollas Silva, Diego Carvalho, Adriano CM Pereira, Fernando Mourão, and Leonardo Rocha.  The pure cold-start problem: A deep study about how to conquer first-time users in recommendations domains.  Information Systems, 80:1–12, 2019.
  * [4] Maciej Kula.  Metadata embeddings for user and item cold-start recommendations.  In Toine Bogers and Marijn Koolen, editors, Proceedings of the 2nd Workshop on New Trends on Content-Based Recommender Systems co-located with 9th ACM Conference on Recommender Systems (RecSys 2015), Vienna, Austria, September 16-20, 2015., volume 1448 of CEUR Workshop Proceedings, pages 14–21. CEUR-WS.org, 2015.
  * [5] Huifeng Guo, Ruiming Tang, Yunming Ye, Zhenguo Li, and Xiuqiang He.  Deepfm: a factorization-machine based neural network for ctr prediction.  arXiv preprint arXiv:1703.04247, 2017.
  * [6] Sandeep Pandey and Christopher Olston.  Handling advertisements of unknown quality in search advertising.  In NIPS, volume 20, pages 1065–1072, 2006.
  * [7] Lihong Li, Wei Chu, John Langford, and Robert E Schapire.  A contextual-bandit approach to personalized news article recommendation.  In Proceedings of the 19th international conference on World wide web, pages 661–670, 2010.
  * [8] Shipra Agrawal and Navin Goyal.  Thompson sampling for contextual bandits with linear payoffs.  In International Conference on Machine Learning, pages 127–135. PMLR, 2013.
  * [9] Long Tran-Thanh, Archie Chapman, Enrique Munoz De Cote, Alex Rogers, and Nicholas R Jennings.  Epsilon–first policies for budget–limited multi-armed bandits.  In Twenty-Fourth AAAI Conference on Artificial Intelligence, 2010\.
  * [10] Aleksandrs Slivkins.  Contextual bandits with similarity information.  In Proceedings of the 24th annual Conference On Learning Theory, pages 679–702. JMLR Workshop and Conference Proceedings, 2011.
  * [11] Shuai Li, Alexandros Karatzoglou, and Claudio Gentile.  Collaborative filtering bandits.  In Proceedings of the 39th International ACM SIGIR conference on Research and Development in Information Retrieval, pages 539–548, 2016.
  * [12] Yehuda Koren.  Collaborative filtering with temporal dynamics.  In Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining, pages 447–456, 2009.
  * [13] Yifei Ma, Balakrishnan Narayanaswamy, Haibin Lin, and Hao Ding.  Temporal-contextual recommendation in real-time.  In Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pages 2291–2299, 2020.
  * [14] Levente Kocsis and Csaba Szepesvári.  Discounted ucb.  2006\.
  * [15] Kevin Labille, Wen Huang, and Xintao Wu.  Transferable contextual bandits with prior observations.  In Pacific-Asia Conference on Knowledge Discovery and Data Mining, pages 398–410. Springer, 2021.
  * [16] Olivier Chapelle and Lihong Li.  An empirical evaluation of thompson sampling.  Advances in neural information processing systems, 24:2249–2257, 2011.
  * [17] Nir Levine, Koby Crammer, and Shie Mannor.  Rotting bandits.  arXiv preprint arXiv:1702.07274, 2017.
  * [18] Moses S Charikar.  Similarity estimation techniques from rounding algorithms.  In Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, pages 380–388, 2002.
  * [19] Bo Liu, Ying Wei, Yu Zhang, Zhixian Yan, and Qiang Yang.  Transferable contextual bandit for cross-domain recommendation.  In Proceedings of the AAAI Conference on Artificial Intelligence, volume 32, 2018.
  * [20] Junzhe Zhang and Elias Bareinboim.  Transfer learning in multi-armed bandit: a causal approach.  In Proceedings of the 16th Conference on Autonomous Agents and MultiAgent Systems, pages 1778–1780, 2017.
  * [21] Lihong Li, Wei Chu, John Langford, and Xuanhui Wang.  Unbiased offline evaluation of contextual-bandit-based news article recommendation algorithms.  In Proceedings of the fourth ACM international conference on Web search and data mining, pages 297–306, 2011.
  * [22] Nan Jiang and Lihong Li.  Doubly robust off-policy value evaluation for reinforcement learning.  In International Conference on Machine Learning, pages 652–661. PMLR, 2016.
  * [23] Laurens Van der Maaten and Geoffrey Hinton.  Visualizing data using t-sne.  Journal of machine learning research, 9(11), 2008.


# Transverse Cooper-Pair Rectifier

Pei-Hao Fu<EMAIL_ADDRESS>Science, Mathematics and Technology,
Singapore University of Technology and Design, Singapore 487372, Singapore
Department of Physics, National University of Singapore, Singapore 117542
Jun-Feng Liu  School of Physics and Materials Science, Guangzhou University,
Guangzhou 510006, China  Ching Hua Lee<EMAIL_ADDRESS>Department of
Physics, National University of Singapore, Singapore 117542  Yee Sin Ang
<EMAIL_ADDRESS>Science, Mathematics and Technology, Singapore
University of Technology and Design, Singapore 487372, Singapore

###### Abstract

Non-reciprocal devices like diodes are key components in modern electronics
covering broad applications ranging from transistors to logic circuits thanks
to the output rectified signal in the direction parallel to the input. In this
work, we predict a transverse Cooper-pair rectifier in which a non-reciprocal
current is driven perpendicular to the driving field, when inversion, time
reversal, and mirror symmetries are broken simultaneously. The Blonder-
Tinkham-Klapwijk formalism is developed to describe the transverse current-
voltage relation in a normal-metal/superconductor tunneling junction with an
effective built-in supercurrent which leads to an asymmetric and anisotropic
Andreev reflection due to the lack of symmetry constraints. The asymmetry in
the Andreev reflection is induced when inversion and time reversal symmetry
are broken by the supercurrent component parallel to the junction while the
anisotropy occurs when the mirror symmetry with respect to the normal of the
junction interface is broken by the perpendicular supercurrent component to
the junction. Compared to the conventional longitudinal one, the transverse
rectifier supports fully polarized diode efficiency and colossal nonreciprocal
conductance rectification, completely decoupling the path of the input
excitation from the output rectified signal. This work provides a formalism
for realizing transverse non-reciprocity in superconducting junctions, which
is expected to be achieved after some modification to the current experimental
setups and may pave the way for future low-dissipation and fast
superconducting electronics.

Introduction.- As fundamental electronic components, diodes conduct electric
current primarily in one direction [Fig. 1], offering low (high) resistance in
the forward (reverse) bias [1, 2, 4, 3]. Typically made by semiconducting p-n
junctions, diodes play essential roles in such as rectification and voltage
regulation [1, 2]. In these devices, the rectified current direction is
parallel to the applied bias direction, potentially introducing variability in
the output rectification signal and subsequently complicating voltage
regulation and stability, particularly for the a.c. inputs [5].

Beyond these longitudinal ratification effects, the recently revealed Hall
rectifier [7, 6, 8, 9] supports output d.c. signals perpendicular to the input
a.c. ones, based on the intrinsic nonreciprocal nonlinear Hall response of
noncentrosymmetric quantum materials [8, 9, 10]. By decoupling the direction
of the output and input powers, the Hall rectifier is expected to be a
promising stable device for converting terahertz electromagnetic waves into
d.c. output with the high rectification efficiency [6].

Figure 1:  Schematic transport and $I$-$V$ relations of the conventional
electronic diode, $I=\exp(eV/k_{B}T)-1$ [1] and two types of Cooper-pair
rectifier [Eq. (8)] assisted by the built-in supercurrent flowing deflected
from the junction direction ($+x$) by an angle $\theta_{q}$. Currents with
negative bias are flipped to the positive side for comparison. The current
$I$s are exhibited in an arbitrary unit (a.u.). The doubling conductance
$G_{x+}=2G_{x-}$ indicates the direction-selective Cooper-pair (paired
electrons) transferring process. Fully polarized rectified current efficiency
[Eq. (10)] is expected in the transverse Cooper-pair rectifier.

In this work, we predict a transverse Cooper-pair rectifier [Fig. 1] which is
based on an alternative mechanism for Hall rectification from existing Hall
rectifiers [7, 6, 8]. Our proposed transverse diodes (i) are based on a normal
metal-superconductor (N-S) junction and, thus do not require the nonlinear
Hall effect from noncentrosymmetric quantum materials [7, 6, 8] as components
or quantum fluctuation of finite-momentum Cooper pairs [9] and (ii) support
dissipationless currents, dominated by Cooper pairs flowing perpendicular to
the bias direction, distinct from the existing longitudinal superconducting
diodes [11, 12]. As a superconducting tunneling junction, the transverse
Cooper-pair rectifier is a low-dispassion cryogenic electronics device in low
working bias ($\sim$meV) and temperature ($\sim$mK) conditions [13] and can be
controlled by an external electromagnet knot and determined by the
configuration of the junction. The proposed device enables highly efficient
($\eta_{y}=100\%$) direction-selective Cooper pair transport and significant
nonreciprocal conductance rectification [Fig. 1] and supports a colossal
nonreciprocal conductance rectification.

Symmetry constraints.- To characterize the transverse Cooper-pair rectifier,
we developed a universal current-voltage ($I$-$V$) relation from the symmetry
constraints (Table 1). Generally, the linearized $I$-$V$ relation of a two-
dimensional tunneling junction with a bias $V$ applied in the $x$-direction is
[14]

$I_{x,y}(eV)=G_{{}_{x,y}}(eV)V\text{,}$ (1)

where
$G_{{}_{x,y}}(eV)=G_{0}\int_{-\pi/2}^{\pi/2}d\theta_{k}\cos\theta_{k}\hat{k}_{F_{x,y}}T_{x,y}(eV,\theta_{k})$
is the longitudinal ($x$) and transverse ($y$) conductance in a unit $G_{0}$,
$\bm{\hat{k}}_{F}=\bm{k}_{F}/k_{F}=(\cos\theta_{k},\sin\theta_{k})$ denotes
the direction of the incident electrons around the Fermi surface, $\theta_{k}$
is the azimuth angle measured from $x$-axis, and $T_{x/y}(eV,\theta_{k})$ is
the direction-resolved transmission probability parallel/perpendicular to the
bias. The merely breaking of inversion symmetry $\mathcal{I}_{x}$ results in a
conventional semiconducting diode effect [1] with $I_{x}(eV)\neq I_{x}(-eV)$,
due to the asymmetry of the conductance $G_{x}(eV)\neq G_{x}(-eV)$ with
respect to zero bias. The further breaking of time-reversal symmetry
$\mathcal{T}$ give rise to the longitudinal superconducting diode [15, 16, 17]
embodying in both $I_{x}(eV)\neq I_{x}(-eV)$ and
$G_{{}_{x}}(eV)=2G_{{}_{x}}(-eV)$, where the factor $2$ comes from the paired
electrons.

To generate the transverse Cooper-pair rectification effect, the mirror
symmetry with respect to the bias direction $\mathcal{M}_{y}$ should be broken
simultaneously with $\mathcal{I}_{x}$ ad $\mathcal{T}$. The breaking of
$\mathcal{M}_{y}$ generates a transverse current, $I_{y}\neq 0$, because of
disparate current response from upward-going ($+\theta_{k}$) and downward-
going ($-\theta_{k}$) electrons, resulting in an anisotropic transmission
probability $T_{y}(eV,\theta_{k})\neq T_{y}(eV,-\theta_{k})$. This is a
universal argument for all tunneling Hall (transverse) currents in both non-
superconducting [18] and superconducting junction [20, 21, 19, 22].
Accompanied by the breaking $\mathcal{I}$ and $\mathcal{T}$, the anisotropic
and asymmetric transmission probability leads to the transverse Cooper-pair
rectification effect with $I_{y}(eV)\neq I_{y}(-eV)$.

The symmetry constraints above for rectifiers are universal and can be
specified in the N-S junction where the transmission probabilities

$T_{x(y)}(E,\theta_{k})=1-(+)R(E,\theta_{k})+R_{a}(E,\theta_{k})\text{,}$ (2)

contains the effect of (i) the retro-Andreev (electron-hole) reflection
$R_{a}(E,\theta_{k})$ always contributing to current by forming Cooper pairs
and (ii) the specular normal (electron-electron) reflection $R(E,\theta_{k})$
which reduces the longitudinal current because of the backscattering towards
the N lead, but enhances the transverse one near the N-S interface. The
Blonder-Tinkham-Klapwijk (BTK) formalism [23] is restored in the longitudinal
transport.

The required anisotropic and asymmetric transmission probability can be
achieved by an effective built-in unidirectional supercurrent [27, 24, 25, 26,
28], whose component is parallel to the N-S junction break $\mathcal{T}$ and
$\mathcal{I}_{x}$ and the one perpendicular to the biased direction breaks
$\mathcal{M}_{y}$. The effective built-in unidirectional supercurrent is the
key ingredient for the existing superconducting non-reciprocity in junction-
free non-centrosymmetric superconductors [29, 30, 31, 32, 33, 34, 35, 36, 37,
38, 39, 40, 41, 42, 44, 45, 46, 47, 43, 48, 49, 50, 51, 52, 53, 54, 55, 56,
57, 58, 59, 60] or Josephson junctions [61, 62, 63, 64, 66, 75, 76, 77, 67,
78, 79, 80, 81, 82, 65, 85, 86, 68, 70, 88, 69, 93, 87, 71, 89, 72, 94, 95,
96, 97, 73, 74, 98, 99, 100, 101, 102, 103, 104, 105, 90, 91, 106, 83, 107,
108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 84, 118, 119, 120, 121, 122,
123, 124, 125, 92, 126, 127, 128, 129] , whose origination is diverse
including finite-momentum Cooper pairs [130, 131, 132] induced by Meissner
effect [105], quantum interference effect [88, 87, 89, 65, 91, 90, 92], vortex
currents [83, 82, 84, 42, 43], the interplay between spin-orbit coupling and
in-plane Zeeman field [67, 68, 74, 66, 73, 72, 69, 65, 71, 70, 67], the non-
zero net velocity between paired electrons [127].

Table 1: The symmetry constrains for different rectifier, including inversion symmetry $\mathcal{I}_{x}$, time-reversal symmetry $\mathcal{T}$, and mirror symmetry $\mathcal{M}_{y}$. Symbols ✗ and ✓ denote symmetry breaking and preservation, respectively. Device | Conventional | Cooper-pair rectifier
---|---|---
| diode | longitudinal | transverse
$\mathcal{I}_{x}$ | ✗ | ✗ | ✗
$\mathcal{T}$ | ✓ | ✗ | ✗
$\mathcal{M}_{y}$ | ✓ | ✓ | ✗

BdG spectrum with built-in supercurrent.- In a microscopic picture, the
effective built-in unidirectional supercurrent modulates the Bogoliubov-de
Gennes (BdG) spectrum and subsequently leads to an anisotropic and asymmetric
transmission probability. The mechanism is exhibited in Fig. 2.

To elaborate on the relation between built-in unidirectional supercurrent and
the transmission probability, we first introduce this asymmetric and
anisotropic BdG spectrum in a conventional isotropic $s$-wave superconductor.
The concerning Hamiltonian with a gauge transformation in Nambu space is [27,
133]

$H_{\bm{q}}(\bm{k})=\left(\begin{array}[]{cc}h_{\text{e}}(\bm{k})&\Delta\\\
\Delta^{\dagger}&h_{\text{h}}(\bm{k})\end{array}\right)\text{,}$ (3)

where $\Delta$ is the isotropic conventional $s$-wave superconducting order
parameter, $\bm{q}=(q_{x},q_{y})=q(\cos\theta_{q},\sin\theta_{q})$ is the
Cooper-pair drifting momentum with a direction $\theta_{q}$,
$h_{\text{e}}(\bm{k})=\hbar^{2}/(2m)|\bm{q+k}|^{2}-E_{F}$ and
$h_{\text{h}}(\bm{k})=-h_{\text{e}}^{\ast}(-\bm{k})$ is the electron-like and
hole-like Hamiltonian with a Fermi level $E_{F}$, respectively. The spectrum
of the BdG quasiparticle with momentum
$\bm{k}=(k_{x},k_{y})=k(\cos\theta_{k},\sin\theta_{k})$ is

$E_{\pm}^{\bm{q}}(\bm{k})=-\hbar^{2}/m(\bm{k\cdot
q})\pm\sqrt{\epsilon_{q}^{2}+\Delta^{2}}\text{,}$ (4)

where $\epsilon_{q}=\hbar^{2}/(2m)(k^{2}+q^{2})-E_{F}$ is ”average” or ”center
of mass” energy [27].

As the built-in supercurrent is turned on ($\bm{q}\neq 0$), the BdG spectrum
becomes asymmetric [Fig. 2(d)] and anisotropic [Fig. 2(f)]. Both features are
due to the center shifting of the paraboloid dispersion of electron (hole)
from the origin [Fig. 2(c, e)] to $-\bm{q}$ ($+\bm{q}$) [Fig. 2(d, f)]. (i) An
anisotropy occurs because the original isotropic Fermi surfaces reduce to
mirror-symmetric with respect to the direction of the supercurrent. (ii)
Subsequently, the symmetry between the electron and hole dispersions with
respect to $E=0$ is destroyed, resulting in an indirect gapped asymmetric BdG
spectrum.

Figure 2: Mechanism of the Cooper-pair rectifier explained by the (c, d)
asymmetric and (e, f) anisotropic BdG spectrum and (g, h) AR spectroscopy. The
schematic (a) side and (b) top views of the biased N-S junction with a built-
in supercurrent with a drifting Cooper-pair momentum $\bm{q}$. The solid
(hollow) blue circles are electrons (holes) and the Cooper pairs composited by
two correlated electrons are circled by the green dashed lines. The blue
arrows represent the motions of the quasi-particles. (c, d) Schematic
dispersion and (e, f) Fermi surface of N and S affected by the built-in
supercurrent. The solid (dashed) lines denote the electron (hole) bands and
the green lines denote the BdG dispersion with multiple band edges
$\Delta_{\pm,\pm}$ [Eq. (5)]. (g) The AR spectroscopy [Eq. (7)] of a normally
incident electron ($\theta_{k}=0$) with different tunneling barriers. (h) The
energy and incident-direction resolved AR probability with $V=0.5E_{F}$.
Hereafter, we choose the Fermi energy $E_{F}=1$ as the energy unit and set
$\hbar^{2}/(2m)=1$ keeping the Fermi wave vector $k_{F}=1$. The
superconducting order parameter is $\Delta=0.01E_{F}$. The magnitude of the
Cooper-pair momentum is $q=0.5q_{c}$ and the direction is (g) $\theta_{q}=0$
and (h) $\theta_{q}=\pi/4$.

The asymmetric and asymmetric BdG spectrum is characterized by four
$\theta_{k}$-dependent band edges

$\Delta_{\pm,\pm}(\theta_{k})=\pm\Delta+D_{\pm}(\theta_{k})\text{,}$ (5)

as exhibited in Fig. 2(d) and (h), which is due to direction-selective Doppler
energy shift [134, 135, 136, 137, 127, 138]

$D_{\pm}(\theta_{k})=\frac{\hbar^{2}}{2m}[q^{2}\pm
2qk_{F}\cos(\theta_{q}\mp\theta_{k})]\text{,}$ (6)

enhancing/reducing the energy of states with a velocity parallel/anti-parallel
to the built-in supercurrent. Thereby, the center of the band gap gains a
$\theta_{k}$-dependent shift from $E=0$ to $D_{\pm}(\theta_{k})$ at $\pm
k_{Fx}=\pm k_{F}\cos\theta_{k}$ with $\theta_{k}=\arctan(k_{Fy}/k_{Fx})$ and
$\theta_{k}\in[-\pi/2,\pi/2]$, resulting in an asymmetric and asymmetric BdG
spectrum. A gapless BdG spectrum is expected when $q$ exceeds the critical
value, $q_{c}=k_{F}\Delta/(2E_{F})$. Effective built-in supercurrents in
recent experiments are inherited from the proximity effect [105, 139] and
thereby do not have to obey the self-consistency equation [24, 25, 26, 27,
28].

Asymmetric and anisotropic AR.- The BdG spectrum [Fig. 2(d, f)] implies that
the inherited AR probability is asymmetric and anisotropic and eventually
causes a non-reciprocal current in the N-S junction with a prioritized
transport direction provided by the built-in supercurrent. To elaborate on
this transport feature, a junction pointing in $+x$-direcion is studied, whose
Hamiltonian is
$H_{\text{NS}}(x)=[\hbar^{2}/(2m)(-\partial_{x}^{2}+k_{y}^{2})-E_{F}+U_{b}\delta(x)]\tau_{z}+[\Delta\tau_{x}+\hbar^{2}/(2m)(q_{0}^{2}\tau_{z}+2q_{y}k_{y}\tau_{0})]\Theta(x)-i\hbar^{2}/(2m)q_{x}\tau_{0}[\partial_{x}\Theta(x)+\Theta(x)\partial_{x}]$.
Here $\bm{\tau}$ ($\tau_{0}$) is Pauli (unit) matrices in particle-hole space,
and $\Theta(x)$ is the Heaviside step function. It is assumed that (i)
translational symmetry in $y$ direction is preserved, (ii) the effective mass
$m$ and the Fermi level $E_{F}$ are shared by both the N and S sides, and
(iii) the interface tunnel barrier $U_{b}$ at $x=0$ is the only boundary
condition involved. (iv) Andreev approximation [140, 23] is applied in a
supercurrent-modulated superconductor, assuming that $E_{F}\gg$ $\Delta$,
$\hbar^{2}q^{2}/(2m)$. Thereby, the Fermi wave number mismatch between the N
and S region is negligible due to $k_{F}\gg q$, and the effect of the built-in
supercurrent is revealed in the Dopper energy shift $D_{\pm}$. Approximation
(iv) is valid for incident angles $\theta_{k}\lesssim\pi/2$ but becomes
inaccurate in the vicinity of $\theta_{k}\sim\pi/2$ [141]. However,
quasiparticles with momentum nearly parallel to the interface do not
contribute significantly to both the longitudinal and transverse current,
thereby their effects in the $I$-$V$ relation and related quantities will thus
be negligible [141]. Our results are checked numerically [142] beyond the
assumption (ii-iv) and are qualitatively consistent with the analytical ones
[143].

The scattering properties of the N-S junction are analytically obtained with
the assumptions above. By solving the quantum tunneling problem with boundary
conditions, the AR probability for incident electrons with energy $E$ and
direction $\theta_{k}$ is [143]

$R_{a}(E,\theta_{k})=|4Z^{-1}\gamma_{-}^{\text{e}}\gamma_{+}^{\text{h}}k_{Fx}^{2}|^{2}\text{,}$
(7)

where
$Z=\gamma_{+}^{\text{e}}\gamma_{+}^{\text{h}}[u_{b}^{2}+(2k_{Fx}+q_{x})^{2}]-\gamma_{-}^{\text{e}}\gamma_{-}^{\text{h}}(u_{b}^{2}+q_{x}^{2})$,
$\gamma_{\tau=\pm
1}^{\text{e(h)}}=\sqrt{1+(-)\tau\Omega_{\tau}/\epsilon_{\tau}}$,
$\Omega_{\tau}=sign(E-D_{\tau})\sqrt{\epsilon_{\tau}^{2}-\Delta^{2}}$,
$u_{b}=2mU_{b}/\hbar^{2}$ and
$\epsilon_{\tau}=E-D_{\tau}-\hbar^{2}q^{2}/(2m)$. Probability conservation
relating to the scattering coefficients is preserved in both directions.

Corresponding to the BdG spectrum, the asymmetric and anisotropic AR
probability is characterized by the band edges as exhibited in Fig. 2(g) and
(h). For a transplant interface ($U_{b}=0$), AR of normal incident electron
($\theta_{k}=0$) keeps unit within the regime
$E\in[\Delta_{-,+}(0),\Delta_{+,+}(0)]$, where Cooper-pair transferring
process is dominant. While for energy beyond this regime, AR exponentially
decreases and single-electron transmission becomes dominant. The interfacial
barrier $U_{b}$ suppresses the unit AR differently for
$E\in[\Delta_{-,+}(0),\Delta_{+,-}(0)]$ and
$E\in[\Delta_{+,-}(0),\Delta_{+,+}(0)]$, the higher AR in the later regime
corresponds to the shifted superconducting gap due to supercurrent.

Therefore, in a microscopic picture, incident electrons experience a
direction-selective superconducting gap resulting in the anisotropic AR except
for the fully gaped regime, i.e.
$\max_{\theta_{k}}[\Delta_{-,+}(\theta_{k})]<E<\min_{\theta_{k}}[\Delta_{+,-}(\theta_{k})]$
Fig. 2(h). The anisotropy is the consequence of the prioritized supercurrent
orientation. Electrons with incident direction $\theta_{k}$ confront a
reducing Doppler energy shift depending on the deviation from the supercurrent
direction, i.e. $|\theta_{k}-\theta_{q}|$. As a result, electrons are mainly
reflected as holes transferring Cooper pairs for
$E\in[\Delta_{-,+}(\theta_{k}),\Delta_{+,+}(\theta_{k})]$, but transmits as an
electron-like quasiparticle for energy out of this regime.

Remarkably, apart from $\mathcal{I}$ and $\mathcal{T}$, $\mathcal{M}_{y}$ is
broken, when the supercurrent deviates from the direction of the junction
($q_{y}\not=0$). The broken $\mathcal{M}_{y}$ embodies in
$R_{a}(E,\theta_{k})\not=R_{a}(E,-\theta_{k})$ when $E$ is out of the regime
$[\max_{\theta_{k}}\Delta_{-,+}(\theta_{k}),\min_{\theta_{k}}\Delta_{+,-}(\theta_{k})]$
[Fig. 2(h)]. Schematic in Fig. 2(d), AR is the dominant process for electrons
with $\theta_{k}>0$ that nearly align with the supercurrent, while the single-
electron transmission process becomes dominant for electrons with
$\theta_{k}<0$ deviating from the supercurrent. The imbalance AR between
electrons with $\theta_{k}>0$ and $\theta_{k}<0$ is the origin of the
transverse current in the N-S junction [20, 21].

Figure 3: Non-reciprocal transport performance of transverse Cooper-pair
rectifier. Top panel: (a) $I_{y}$-$V$ relations [Eq. (8)] for various built-in
supercurrent directions ($\theta_{q}$) and (b) differential conductance
$G_{y}$, in the unit of $I_{0}=eWk_{F}v_{F}\Delta/(2\pi)$ and $G_{0}=eI_{0}$.
(c) The negative-bias currents are flipped to compare with the positive-bias
ones. Bottom panel: The performance of the Cooper-pair diode to the built-in
supercurrent direction ($\theta_{q}$) is revealed in (d) The current
difference [Eq. (11)] (e) current efficiency [Eq. (10)] and (f) conductance
efficiency [Eq. (9)]. Fully polarized transverse diode current
($\eta_{y}=100\%$) and a colossal nonreciprocal conductance rectification
[inserted in (d)] ($|\Gamma_{y}|\rightarrow\infty$) are expected. In (d), the
bias is $e|V|/\Delta=1$. The temperature is $k_{B}T=0.01\Delta$. Other
parameters are the same as those in Fig. 2.

Non-reciprocal $I$-$V$ relation and efficiency.- To investigate the currents
in a biased N-S junction, we develop the non-reciprocal $I$-$V$ relation by
following the quasiparticle current argument in the BTK formalism [23, 144].
By considering electron and hole currents and the quasiparticle probability
conservation, the $I_{\beta=x,y}$-$V$ relation is [143]

$\displaystyle I_{\beta}(eV)$ $\displaystyle=$
$\displaystyle\frac{1}{e}\int_{-\infty}^{+\infty}dEG_{\beta}(E)[f_{\text{N}}(E,eV)-f_{\text{S}}(E,eV)]\text{,}$
(8)

where $G_{0}=eI_{0}$, $I_{0}=eW\hbar k_{F}^{2}/(\pi m)$ and $W$ is the width
of the sample, and $f_{\text{S}}(E,eV)=f_{0}(E)=[1+\exp(E/k_{B}T)]^{-1}$ and
$f_{\text{N}}(E)=f_{0}(E-eV)$ are the Fermi-Dirac distribution between S and N
sides, respectively. The transmission probabilities are shown in Eq. (2), and
Eq. (1) is obtain by linearized Eq. (8) in small bias and low-temperature
condiction [143, 20, 21].

The transverse current $I_{y}$ occurs when $\mathcal{M}_{y}$ is broken by the
supercurrent component deviating from the direction of the junction
($q_{y}\not=0$) as shown in Fig. 3(a). The net contribution between
anisotropic AR with $\theta_{k}>0$ and $\theta_{k}<0$ generate a non-zero
$I_{y}$ as the bias exceeds the regime
$[\max|\Delta_{-,+}(\theta_{k})|,\min|\Delta_{+,-}(\theta_{k})|]$ [Fig. 2(g)].

A transverse Cooper-pair rectifier is expected when $\mathcal{T}$ and
$\mathcal{I}_{x}$ is further broken by the supercurrent component deviating
from the direction of the junction ($q_{x}\not=0$). This device is
characterized by the fact that the forward-bias current is dominated by Cooper
pairs transferring while single-electron transmission processes mainly
contribute to the reverse-bias current. This carrier alteration manifests in
the differential conductance with opposite biases in Fig. 3(b). For positive
bias ($eV\sim 0.7\Delta$), Cooper pairs dominate the current in all
directions, i.e. $G_{y,\theta_{k}>0}(eV)\approx G_{y,\theta_{k}<0}(eV)\approx
2G_{y,\theta_{k}<0}(eV\gg\Delta)$. While negative bias, Cooper-pair
transferring and single-electron transmission processes gain a direction-
selective contribution to the current, i.e. $G_{y,\theta_{k}<0}(-eV)\sim
2G_{y,\theta_{k}>0}(-eV)$. As a result, devices characterized by the
$I_{y}$-$V$ relation and the conductance shown in Fig. 3(a) and (b) can be
dubbed a transverse Cooper-pair rectifier as schematic in Fig. 1(c).

High rectification efficiency is expected due to this bias-direction-selective
Cooper pair transverse transport, which manifests in distinct two aspects. (i)
A divergent rectified conductance efficiency is expected when
$\theta_{q}=\pi/2$ [Fig. 3(d)], which is defined as

$\Gamma_{x/y}(e|V|)=\frac{G_{x/y,+}(e|V|)-G_{x/y,-}(e|V|)}{G_{x/y,+}(e|V|)+G_{x/y,-}(e|V|)}\text{.}$
(9)

The colossal conductance rectification indicates that identical transverse
currents are dominated by Cooper-pairs ($e|V|<\Delta$) or single electrons
($e|V|>\Delta$) regardless of the bias polarity [Fig. 3(c)]. (ii) When
$q_{x,y}\neq 0$ and $e|V|/\Delta\sim 1$, a fully rectified current efficiency

$\eta_{x/y}(e|V|)=\frac{\delta
I_{x/y}(e|V|)}{|I_{x/y,+}(e|V|)|+|I_{x/y,-}(e|V|)|}\text{,}$ (10)

is expected [Fig. 3(e)], where

$\delta I_{x/y}=sign(I_{x/y,+})(|I_{x/y,+}-|I_{x/y,-}|)\text{, }$ (11)

When the biases are around the superconducting gap, the transverse current is
negligible due to the compensation Cooper-pair current flowing perpendicular
to the junction under one bias and becomes finite stemming from the direction-
selective current of Cooper-pairs and single electrons, as schematic in Fig.
1. These two aspects endow the transverse Cooper-pair diode beyond its
longitudinal counterpart, whose diode efficiency is inherently constrained, as
it necessitates the transfer of carriers (either Cooper pairs or single
electrons) upon reversal of the bias.

The symmetry constrain (Table 1) is revealed in Eq. (11) and Fig. 3(f). The
supercurrent component parallel to the junction direction
($q_{x}=\cos\theta_{q}$) breaks both $\mathcal{I}_{x}$ and $\mathcal{T}$ and
subsequently causes the longitudinal Cooper pair effect embodying in $\delta
I_{x}\sim\cos\theta_{q}$. While the transverse current difference is $\delta
I_{y}\sim\sin 2\theta_{q}$ implying that the addition $\mathcal{M}_{y}$ broken
by the supercurrent component perpendicular to the junction direction
($q_{y}=\sin\theta_{q}$).

Conclusion and discussions.- Thanks to the recent experimental developments,
both the longitudinal and transverse Cooper-pair diode effects are expected to
be observed after some modifications to setup based on semiconducting [67, 68,
74, 66, 73, 72, 69, 65, 71, 70, 67] or topological materials like Bi2Te3-NbSe2
hybrid structure [139] One of the promising platforms is the InGaAs/InAs with
patterned epitaxial Al as proximate superconductor, where the effective built-
in supercurrent is contorted by an external in-plane magnetic field interplay
with spin-orbit coupling [145]. The asymmetric AR is observed in the
differential conductance [145]. The Cooper-pair diode effect should be
observed by detecting the $I$-$V$ relation and measuring the transverse
transport feature in the Hall-bar-like setup [146, 146, 147, 148]. The
mechanism for generating effective built-in supercurrent or Cooper-pair
momentum is also suitable for topological surface states with in-plane Zeeman
field and proximate superconducting [64, 139]. The Cooper-pair diode effects
are expected in NiTe2 [64] and Bi2Te3 [139]. We expect that the proposed
Cooper-pair diode effect will be observed soon by modifying the existing
experimental setup.

Results present in this work are in ballistic transport regimes since the
scattering coefficients are solved in the coherent transport picture with
assumptions (i-iv). Thus, the effects in more realistic situations with
finite-size effects in $y$-direction, impurities, disorders, dephasing, etc,
are open questions deserving further detailed studies. A four-terminal setup
[149] is necessary to detect the transverse non-reciprocity, based on multi-
terminal $I$-$V$ relation involving superconducting leads. Finally, the
transverse current in the Dirac system such as topological surface states may
be helpful to distinguish the retro- and specular AR [150] by modifying Eq.
(2) [21].

In conclusion, we predict Cooper-pair diodes by developing the BTK formalism
in characterizing the $I$-$V$ relation of the N-S junction with a built-in
supercurrent. Remarkably, a transverse Cooper-pair diode is expected when the
built-in supercurrent breaks $\mathcal{I}$, $\mathcal{T}$, and
$\mathcal{M}_{y}$ simultaneously. The transverse Cooper-pair diode enables
highly efficient direction-selective Cooper pair transport decoupling the path
of the input excitation from the output rectified signal. With fully polarized
diode efficiency and colossal nonreciprocal conductance rectification, the
proposed diode effects may be of potential application in non-reciprocal
superconducting electronics.

###### Acknowledgements.

P.-H. Fu thanks Weiping Xu and Alessandro Braggio for inspiring discussions.
P.-H. F. & Y. S. A. are supported by the Singapore Ministry of Education (MOE)
Academic Research Fund (AcRF) Tier 2 Grant (MOE-T2EP50221-0019). C.H. L. is
supported by Singapore’s NRF Quantum engineering grant NRF2021-QEP2-02-P09 and
Singapore’s MOE Tier-II grant Proposal ID: T2EP50222-0003. J.-F. L. is
supported by the National Natural Science Foundation of China (Grants No.
12174077), the Bureau of Education of Guangzhou Municipality (Grant No.
202255464), and the Natural Science Foundation of Guangdong Province (Grant
No. 2021A1515012363).

## References

  * [1] S. M. Sze, & M.-K. Lee, Semiconductor Devices: Physics and Technology 3rd edn (Wiley, 2012).
  * [2] T. Ideue, K. Hamamoto, S. Koshikawa, M. Ezawa, S. Shimizu, Y. Kaneko, Y. Tokura, N. Nagaosa, and Y. Iwasa, Bulk Rectification Effect in a Polar Semiconductor, Nat. Phys. 13, 578 (2017).
  * [3] Y. Tokura and N. Nagaosa, Nonreciprocal Responses from Non-Centrosymmetric Quantum Materials, Nat. Commun. 9, 3740 (2018).
  * [4] N. Nagaosa and Y. Yanase, Nonreciprocal Transport and Optical Phenomena in Quantum Materials, Annu. Rev. Condens. Matter Phys. 15, 63 (2024).
  * [5] D. Farrah, K. E. Smith, D. Ardila, C. M. Bradford, M. Dipirro, C. Ferkinhoff, J. Glenn, P. Goldsmith, D. Leisawitz, and T. Nikola et al., Far-infrared instrumentation and technology development for the next decade, J. Astron. Telesc. Instrum. Syst. 5(2), 020901 (2019).
  * [6] L. Min et al., Colossal Nonreciprocal Hall Effect and Broadband Frequency Mixing Due to a Room Temperature Nonlinear Hall Effect, arXiv: 2303.03738 (2023).
  * [7] H. Isobe, S.-Y. Xu, and L. Fu, High-Frequency Rectification via Chiral Bloch Electrons, Sci. Adv. 6, eaay2497 (2020).
  * [8] Y. Onishi and L. Fu, High-Efficiency Energy Harvesting Based on Nonlinear Hall Rectifier, arXiv: 2211.17219 (2022).
  * [9] A. Daido and Y. Yanase, Rectification and Nonlinear Hall Effect by Fluctuating Finite-Momentum Cooper Pairs, arXiv: 2302.10677 (2023).
  * [10] F. Qin, R. Chen, and C. H. Lee, Light-Enhanced Nonlinear Hall Effect, arXiv: 2401.18038 (2024).
  * [11] M. Nadeem, M. S. Fuhrer, and X. Wang, The Superconducting Diode Effect, Nat. Rev. Phys. 5, 558 (2023).
  * [12] R. Wakatsuki, Y. Saito, S. Hoshino, Y. M. Itahashi, T. Ideue, M. Ezawa, Y. Iwasa, and N. Nagaosa, Nonreciprocal Charge Transport in Noncentrosymmetric Superconductors, Sci. Adv. 3, e1602390 (2017).
  * [13] A. I. Braginski, Superconductor Electronics: Status and Outlook, J. Supercond. Nov. Magn. 32, 23 (2019).
  * [14] P. Jacquod, R. S. Whitney, J. Meair, and M. Büttiker, Onsager Relations in Coupled Electric, Thermoelectric, and Spin Transport: The Tenfold Way, Phys. Rev. B 86, 155118 (2012).
  * [15] B. Zinkl, K. Hamamoto, and M. Sigrist, Symmetry Conditions for the Superconducting Diode Effect in Chiral Superconductors, Phys. Rev. Res. 4, 033167 (2022).
  * [16] J. Hu, C. Wu, and X. Dai, Proposed Design of a Josephson Diode, Phys. Rev. Lett. 99, 067004 (2007).
  * [17] D. Wang, Q.-H. Wang, and C. Wu, Symmetry Constraints on Direct-Current Josephson Diodes, arXiv: 2209.12646 (2022).
  * [18] B. Scharf, A. Matos-Abiague, J. E. Han, E. M. Hankiewicz, and I. Žutić, Tunneling Planar Hall Effect in Topological Insulators: Spin Valves and Amplifiers, Phys. Rev. Lett. 117, 166806 (2016).
  * [19] F. Zhou and B. Spivak, Hall Effect in SN and SNS Junctions, Phys. Rev. Lett. 80, 3847 (1998).
  * [20] J. Ren and J.-X. Zhu, Asymmetric Andreev Reflection Induced Electrical and Thermal Hall-like Effects in Metal/Anisotropic Superconductor Junctions, Phys. Rev. B 89, 064512 (2014).
  * [21] M. Salehi, Anisotropic Angle-Dependent Andreev Reflection at the Ferromagnet/Superconductor Junction on the Surface of Topological Insulators, Phys. Scr. 98, 025822 (2023).
  * [22] A. V. Parafilo, V. M. Kovalev, and I. G. Savenko, Photoinduced Anomalous Supercurrent Hall Effect, Phys. Rev. B 108, L180509 (2023).
  * [23] G. E. Blonder, M. Tinkham, and T. M. Klapwijk, Transition from Metallic to Tunneling Regimes in Superconducting Microconstrictions: Excess Current, Charge Imbalance, and Supercurrent Conversion, Phys. Rev. B 25, 4515 (1982).
  * [24] J. Sáñchez Canizares and F. Sols, Self-Consistent Current-Voltage Characteristics of Normal-Superconductor Interfaces, J. Phys. Condens. Matter 7, L317 (1995).
  * [25] J. Sáñchez and F. Sols, Self-Consistent Scattering Description of Transport in Normal-Superconductor Structures, Phys. Rev. B 55, 531 (1997).
  * [26] R. Riedel and P. F. Bagwell, Excess Currents Larger than the Point Contact Limit in Normal-Metal/Superconducting Junctions, Superlattices Microstruct. 25, 683 (1999).
  * [27] P. F. Bagwell, Critical Current of a One-Dimensional Superconductor, Phys. Rev. B 49, 6841 (1994).
  * [28] V. Lukic and E. J. Nicol, Conductance Characteristics between a Normal Metal and a Two-Band Superconductor Carrying a Supercurrent, Phys. Rev. B 76, 144508 (2007).
  * [29] E. Zhang, X. Xu, Y.-C. Zou, L. Ai, X. Dong, C. Huang, P. Leng, S. Liu, Y. Zhang, Z. Jia, X. Peng, M. Zhao, Y. Yang, Z. Li, H. Guo, S. J. Haigh, N. Nagaosa, J. Shen, and F. Xiu, Nonreciprocal Superconducting NbSe2 Antenna, Nat. Commun. 11, 5634 (2020).
  * [30] F. Ando, Y. Miyasaka, T. Li, J. Ishizuka, T. Arakawa, Y. Shiota, T. Moriyama, Y. Yanase, and T. Ono, Observation of Superconducting Diode Effect, Nature 584, 373 (2020).
  * [31] T. Schumann, L. Galletti, H. Jeong, K. Ahadi, W. M. Strickland, S. Salmani-Rezaie, and S. Stemmer, Possible Signatures of Mixed-Parity Superconductivity in Doped Polar SrTiO3 Films, Phys. Rev. B 101, 100503(R) (2020).
  * [32] L. Bauriedl, C. Bäuml, L. Fuchs, C. Baumgartner, N. Paulik, J. M. Bauer, K.-Q. Lin, J. M. Lupton, T. Taniguchi, K. Watanabe, C. Strunk, and N. Paradiso, Supercurrent Diode Effect and Magnetochiral Anisotropy in Few-Layer NbSe2, Nat. Commun. 13, 4266 (2022).
  * [33] Y. M. Itahashi, T. Ideue, Y. Saito, S. Shimizu, T. Ouchi, T. Nojima, and Y. Iwasa, Nonreciprocal Transport in Gate-Induced Polar Superconductor SrTiO3, Sci. Adv. 6, eaay9120 (2020).
  * [34] J. Yun, S. Son, J. Shin, G. Park, K. Zhang, Y. J. Shin, J.-G. Park, and D. Kim, Magnetic Proximity-Induced Superconducting Diode Effect and Infinite Magnetoresistance in a van Der Waals Heterostructure, Phys. Rev. Res. 5, L022064 (2023).
  * [35] Y.-Y. Lyu, J. Jiang, Y.-L. Wang, Z.-L. Xiao, S. Dong, Q.-H. Chen, M. V. Milošević, H. Wang, R. Divan, J. E. Pearson, P. Wu, F. M. Peeters, and W.-K. Kwok, Superconducting Diode Effect via Conformal-Mapped Nanoholes, Nat. Commun. 12, 2703 (2021).
  * [36] K. Yasuda, H. Yasuda, T. Liang, R. Yoshimi, A. Tsukazaki, K. S. Takahashi, N. Nagaosa, M. Kawasaki, and Y. Tokura, Nonreciprocal Charge Transport at Topological Insulator/Superconductor Interface, Nat. Commun. 10, 2734 (2019).
  * [37] Y. Wu, Q. Wang, X. Zhou, J. Wang, P. Dong, J. He, Y. Ding, B. Teng, Y. Zhang, Y. Li, C. Zhao, H. Zhang, J. Liu, Y. Qi, K. Watanabe, T. Taniguchi, and J. Li, Nonreciprocal Charge Transport in Topological Kagome Superconductor CsV3Sb5, arXiv: 2210.10437 (2022).
  * [38] M. Masuko, M. Kawamura, R. Yoshimi, M. Hirayama, Y. Ikeda, R. Watanabe, J. J. He, D. Maryenko, A. Tsukazaki, K. S. Takahashi, M. Kawasaki, N. Nagaosa, and Y. Tokura, Nonreciprocal Charge Transport in Topological Superconductor Candidate Bi2Te3/PdTe2 Heterostructure, Npj Quantum Mater. 7, 104 (2022).
  * [39] Y. Hou, F. Nichele, H. Chi, A. Lodesani, Y. Wu, M. F. Ritter, D. Z. Haxell, M. Davydova, S. Ilić, O. Glezakou-Elbert, A. Varambally, F. S. Bergeret, A. Kamra, L. Fu, P. A. Lee, and J. S. Moodera, Ubiquitous Superconducting Diode Effect in Superconductor Thin Films, Phys. Rev. Lett. 131, 027001 (2023).
  * [40] R. Kealhofer, H. Jeong, A. Rashidi, L. Balents, and S. Stemmer, Anomalous Superconducting Diode Effect in a Polar Superconductor, Phys. Rev. B 107, L100504 (2023).
  * [41] X. Zhang et al., Superconducting Nanowire Diode, Educ. Theatr. J. 23, 343 (2023).
  * [42] A. Mehrnejat, M. C. Hatnean, M. C. Rosamond, N. Banerjee, G. Balakrishnan, S. E. Savel’ev, and F. K. Dejene, Flux-Pinning Mediated Superconducting Diode Effect in the NbSe2/CrGeTe3 Heterostructure, 2D Mater. 11, 021002 (2024).
  * [43] A. Sundaresh, J. I. Väyrynen, Y. Lyanda-Geller, and L. P. Rokhinson, Diamagnetic Mechanism of Critical Current Non-Reciprocity in Multilayered Superconductors, Nat. Commun. 14, 1628 (2023).
  * [44] Y. Zhang, J. Cai, P. Dong, J. He, Y. Ding, J. Wang, X. Zhou, K. Cao, Y. Wu, and J. Li, Intrinsic Supercurrent Diode Effect in NbSe2 Nanobridge, arXiv: 2402.16264 (2024).
  * [45] T. Wakamura, M. Hashisaka, S. Hoshino, M. Bard, S. Okazaki, T. Sasagawa, T. Taniguchi, K. Watanabe, K. Muraki, and N. Kumada, Gate-Tunable Giant Superconducting Nonreciprocal Transport in Few-Layer $T_{d}$-MoTe2, Phys. Rev. Res. 6, 013132 (2024).
  * [46] P. Dong et al., Nonreciprocal Charge Transport in the Titanium Sesquioxide Heterointerface Superconductor, arXiv: 2401.13072 (2024).
  * [47] D. Margineda, A. Crippa, E. Strambini, Y. Fukaya, M. T. Mercaldo, M. Cuoco, and F. Giazotto, Sign Reversal Diode Effect in Superconducting Dayem Nanobridges, Commun. Phys. 6, 343 (2023).
  * [48] J.-X. Lin, P. Siriviboon, H. D. Scammell, S. Liu, D. Rhodes, K. Watanabe, T. Taniguchi, J. Hone, M. S. Scheurer, and J. I. A. Li, Zero-Field Superconducting Diode Effect in Small-Twist-Angle Trilayer Graphene, Nat. Phys. 18, 1221 (2022).
  * [49] J. Jiang, M. V. Milošević, Y.-L. Wang, Z.-L. Xiao, F. M. Peeters, and Q.-H. Chen, Field-Free Superconducting Diode in a Magnetically Nanostructured Superconductor, Phys. Rev. Appl. 18, 034064 (2022).
  * [50] H. Narita, J. Ishizuka, R. Kawarazaki, D. Kan, Y. Shiota, T. Moriyama, Y. Shimakawa, A. V. Ognev, A. S. Samardak, Y. Yanase, and T. Ono, Field-Free Superconducting Diode Effect in Noncentrosymmetric Superconductor/Ferromagnet Multilayers, Nat. Nanotechnol. 17, 823 (2022).
  * [51] S. Hoshino, R. Wakatsuki, K. Hamamoto, and N. Nagaosa, Nonreciprocal Charge Transport in Two-Dimensional Noncentrosymmetric Superconductors, Phys. Rev. B 98, 054510 (2018).
  * [52] R. Wakatsuki and N. Nagaosa, Nonreciprocal Current in Noncentrosymmetric Rashba Superconductors, Phys. Rev. Lett. 121, 026601 (2018).
  * [53] B. Zhai, B. Li, Y. Wen, F. Wu, and J. He, Prediction of Ferroelectric Superconductors with Reversible Superconducting Diode Effect, Phys. Rev. B 106, L140505 (2022).
  * [54] J. J. He, Y. Tanaka, and N. Nagaosa, A Phenomenological Theory of Superconductor Diodes, New J. Phys. 24, 053014 (2022).
  * [55] H. D. Scammell, J. I. A. Li, and M. S. Scheurer, Theory of Zero-Field Superconducting Diode Effect in Twisted Trilayer Graphene, 2D Mater. 9, 025027 (2022).
  * [56] S. Ilić and F. S. Bergeret, Theory of the Supercurrent Diode Effect in Rashba Superconductors with Arbitrary Disorder, Phys. Rev. Lett. 128, 177001 (2022).
  * [57] A. Daido, Y. Ikeda, and Y. Yanase, Intrinsic Superconducting Diode Effect, Phys. Rev. Lett. 128, 037001 (2022).
  * [58] A. Daido and Y. Yanase, Superconducting Diode Effect and Nonreciprocal Transition Lines, Phys. Rev. B 106, 205206 (2022).
  * [59] H. F. Legg, D. Loss, and J. Klinovaja, Superconducting Diode Effect Due to Magnetochiral Anisotropy in Topological Insulators and Rashba Nanowires, Phys. Rev. B 106, 104501 (2022).
  * [60] H. Hayashi and K. Ando, Two-Dimensional Rashba Superconductivity in Ni/Bi Bilayers Evidenced by Nonreciprocal Transport, Appl. Phys. Rev. 11, 011401 (2024).
  * [61] E. Bocquillon, R. S. Deacon, J. Wiedenmann, P. Leubner, T. M. Klapwijk, C. Brüne, K. Ishibashi, H. Buhmann, and L. W. Molenkamp, Gapless Andreev Bound States in the Quantum Spin Hall Insulator HgTe, Nat. Nanotechnol. 12, 137 (2017).
  * [62] J. Diez-Merida, A. Diez-Carlon, S. Y. Yang, Y.-M. Xie, X.-J. Gao, K. Watanabe, T. Taniguchi, X. Lu, K. T. Law, and D. K. Efetov, Magnetic Josephson Junctions and Superconducting Diodes in Magic Angle Twisted Bilayer Graphene, arXiv: 2110.01067 (2021).
  * [63] E. Portolés, S. Iwakiri, G. Zheng, P. Rickhaus, T. Taniguchi, K. Watanabe, T. Ihn, K. Ensslin, and F. K. de Vries, A Tunable Monolithic SQUID in Twisted Bilayer Graphene, Nat. Nanotechnol. 17, 1159 (2022).
  * [64] B. Pal, A. Chakraborty, P. K. Sivakumar, M. Davydova, A. K. Gopi, A. K. Pandeya, J. A. Krieger, Y. Zhang, M. Date, S. Ju, N. Yuan, N. B. M. Schröter, L. Fu, and S. S. P. Parkin, Josephson Diode Effect from Cooper Pair Momentum in a Topological Semimetal, Nat. Phys. 18, 1228 (2022).
  * [65] C. Ciaccia, R. Haller, A. C. C. Drachmann, T. Lindemann, M. J. Manfra, C. Schrade, and C. Schönenberger, Gate-Tunable Josephson Diode in Proximitized InAs Supercurrent Interferometers, Phys. Rev. Res. 5, 033131 (2023).
  * [66] B. Turini, S. Salimian, M. Carrega, A. Iorio, E. Strambini, F. Giazotto, V. Zannier, L. Sorba, and S. Heun, Josephson Diode Effect in High Mobility InSb Nanoflags, Nano Lett. 22, 8502 (2022).
  * [67] A. Costa, C. Baumgartner, S. Reinhardt, J. Berger, S. Gronin, G. C. Gardner, T. Lindemann, M. J. Manfra, D. Kochan, J. Fabian, N. Paradiso, and C. Strunk, Sign Reversal of the Josephson Inductance Magnetochiral Anisotropy and $0$-$\pi$-like Transitions in Supercurrent Diodes Nat. Nanotechnol. 18, 1266 (2023).
  * [68] K.-R. Jeon, B. K. Hazra, J.-K. Kim, J.-C. Jeon, H. Han, H. L. Meyerheim, T. Kontos, A. Cottet, and S. S. P. Parkin, Chiral Antiferromagnetic Josephson Junctions as Spin-Triplet Supercurrent Spin Valves and d.c. SQUIDs, Nat. Nanotechnol. 18, 747 (2023).
  * [69] D. Margineda, A. Crippa, E. Strambini, Y. Fukaya, M. T. Mercaldo, M. Cuoco, and F. Giazotto, Sign Reversal Diode Effect in Superconducting Dayem Nanobridges, Commun. Phys. 6, 343 (2023).
  * [70] H. Su et al., Microwave-Assisted Unidirectional Superconductivity in Al-InAs Nanowire-Al Junctions under Magnetic Fields, arXiv: 2402.02137 (2024).
  * [71] S. Reinhardt et al., Link between Supercurrent Diode and Anomalous Josephson Effect Revealed by Gate-Controlled Interferometry, arXiv: 2308.01061 (2023).
  * [72] N. Lotfizadeh, B. Pekerten, P. Yu, W. Strickland, A. Matos-Abiague, and J. Shabani, Superconducting Diode Effect Sign Change in Epitaxial Al-InAs Josepshon Junctions, arXiv: 2303.01902 (2023).
  * [73] K.-R. Jeon, J.-K. Kim, J. Yoon, J.-C. Jeon, H. Han, A. Cottet, T. Kontos, and S. S. P. Parkin, Zero-Field Polarity-Reversible Josephson Supercurrent Diodes Enabled by a Proximity-Magnetized Pt Barrier, Nat. Mater. 21, 1008 (2022).
  * [74] G. P. Mazur, N. van Loo, D. van Driel, J.-Y. Wang, G. Badawy, S. Gazibegovic, E. P. A. M. Bakkers, and L. P. Kouwenhoven, The Gate-Tunable Josephson Diode, arXiv: 2211.14283 (2022).
  * [75] M. Gupta, G. V. Graziano, M. Pendharkar, J. T. Dong, C. P. Dempsey, C. Palmstrøm, and V. S. Pribiag, Gate-Tunable Superconducting Diode Effect in a Three-Terminal Josephson Device, Nat. Commun. 14, 3078 (2023).
  * [76] C. Baumgartner, L. Fuchs, A. Costa, S. Reinhardt, S. Gronin, G. C. Gardner, T. Lindemann, M. J. Manfra, P. E. Faria Junior, D. Kochan, J. Fabian, N. Paradiso, and C. Strunk, Supercurrent Rectification and Magnetochiral Effects in Symmetric Josephson Junctions, Nat. Nanotechnol. 17, 39 (2022).
  * [77] C. Baumgartner, L. Fuchs, A. Costa, J. Picó-Cortés, S. Reinhardt, S. Gronin, G. C. Gardner, T. Lindemann, M. J. Manfra, P. E. Faria Junior, D. Kochan, J. Fabian, N. Paradiso, and C. Strunk, Effect of Rashba and Dresselhaus Spin-Orbit Coupling on Supercurrent Rectification and Magnetochiral Anisotropy of Ballistic Josephson Junctions, J. Phys. Condens. Matter 34, 154005 (2022).
  * [78] M. S. Anwar, T. Nakamura, R. Ishiguro, S. Arif, J. W. A. Robinson, S. Yonezawa, M. Sigrist, and Y. Maeno, Spontaneous Superconducting Diode Effect in Non-Magnetic Nb/Ru/Sr2RuO4 Topological Junctions, Commun. Phys. 6, 290 (2023).
  * [79] S. Ghosh, V. Patil, A. Basu, Kuldeep, D. A. Jangade, R. Kulkarni, A. Thamizhavel, and M. M. Deshmukh, High-Temperature Superconducting Diode, arXiv: 2210.11256 (2022).
  * [80] J. Chiles, E. G. Arnault, C.-C. Chen, T. F. Q. Larson, L. Zhao, K. Watanabe, T. Taniguchi, F. Amet, and G. Finkelstein, Nonreciprocal Supercurrents in a Field-Free Graphene Josephson Triode, Nano Lett. 23, 5257 (2023).
  * [81] M. Trahms, L. Melischek, J. F. Steiner, B. Mahendru, I. Tamir, N. Bogdanoff, O. Peters, G. Reecht, C. B. Winkelmann, F. von Oppen, and K. J. Franke, Diode Effect in Josephson Junctions with a Single Magnetic Atom, Nature 615, 628 (2023).
  * [82] X. Song, S. Suresh-Babu, Y. Bai, D. Golubev, I. Burkova, A. Romanov, E. Ilin, J. N. Eckstein, and A. Bezryadin, Interference, Diffraction, and Diode Effects in Superconducting Array Based on Bismuth Antimony Telluride Topological Insulator, Commun. Phys. 6, 177 (2023).
  * [83] R. Haenel and O. Can, Superconducting Diode from Flux Biased Josephson Junction Arrays, arXiv: 2212.02657 (2022).
  * [84] Y. Fukaya, M. T. Mercaldo, D. Margineda, A. Crippa, E. Strambini, F. Giazotto, C. Ortix, and M. Cuoco, Design of Supercurrent Diode by Vortex Phase Texture, arXiv: 2403.04421 (2024).
  * [85] P. K. Sivakumar, M. T. Ahari, J. Kim, Y. Wu, A. Dixit, G. J. de Coster, A. K. Pandeya, M. J. Gilbert, and S. S. P. Parkin, Long-Range Phase Coherence and Tunable Second Order ${\phi}_{0}$-Josephson Effect in a Dirac Semimetal $1T-PtTe_{2}$, arXiv: 2403.19445 (2024).
  * [86] S. Matsuo, T. Imoto, T. Yokoyama, Y. Sato, T. Lindemann, S. Gronin, G. C. Gardner, M. J. Manfra, and S. Tarucha, Josephson Diode Effect Derived from Short-Range Coherent Coupling, Nat. Phys. 19, 1636 (2023).
  * [87] M. Valentini et al., Parity-Conserving Cooper-Pair Transport and Ideal Superconducting Diode in Planar Germanium, Nat. Commun. 15, 169 (2024).
  * [88] A. Leblanc et al., From Nonreciprocal to Charge-4e Supercurrents in Ge-Based Josephson Devices with Tunable Harmonic Content, arXiv: 2311.15371 (2023).
  * [89] A. Greco, Q. Pichard, and F. Giazotto, Josephson Diode Effect in Monolithic dc-SQUIDs Based on 3D Dayem Nanobridges, Appl. Phys. Lett. 123, 092601 (2023).
  * [90] R. S. Souto, M. Leijnse, and C. Schrade, Josephson Diode Effect in Supercurrent Interferometers, Phys. Rev. Lett. 129, 267702 (2022).
  * [91] Y. V. Fominov and D. S. Mikhailov, Asymmetric Higher-Harmonic SQUID as a Josephson Diode, Phys. Rev. B 106, 134514 (2022).
  * [92] J. J. Cuozzo, W. Pan, J. Shabani, and E. Rossi, Microwave-Tunable Diode Effect in Asymmetric SQUIDs with Topological Josephson Junctions, arXiv: 2303.16931 (2023).
  * [93] D. Margineda, A. Crippa, E. Strambini, Y. Fukaya, M. T. Mercaldo, C. Ortix, M. Cuoco, and F. Giazotto, Back-Action Supercurrent Diodes, arXiv: 2311.14503 (2023).
  * [94] P. Chen et al., Edelstein Effect Induced Superconducting Diode Effect in Inversion Symmetry Breaking MoTe 2 Josephson Junctions, Adv. Funct. Mater. 34, 2311229 (2024).
  * [95] T. Golod and V. M. Krasnov, Demonstration of a Superconducting Diode-with-Memory, Operational at Zero Magnetic Field with Switchable Nonreciprocity, Nat. Commun. 13, 3658 (2022).
  * [96] F. Zhang, M. T. Ahari, A. S. Rashid, G. J. de Coster, T. Taniguchi, K. Watanabe, M. J. Gilbert, N. Samarth, and M. Kayyalha, Reconfigurable Magnetic-Field-Free Superconducting Diode Effect in Multi-Terminal Josephson Junctions, Phys. Rev. Appl. 21, 034011 (2024).
  * [97] H. Wu, Y. Wang, Y. Xu, P. K. Sivakumar, C. Pasco, U. Filippozzi, S. S. P. Parkin, Y.-J. Zeng, T. McQueen, and M. N. Ali, The Field-Free Josephson Diode in a van Der Waals Heterostructure, Nature 604, 653 (2022).
  * [98] C.-Z. Chen, J. J. He, M. N. Ali, G.-H. Lee, K. C. Fong, and K. T. Law, Asymmetric Josephson Effect in Inversion Symmetry Breaking Topological Materials, Phys. Rev. B 98, 075430 (2018).
  * [99] A. A. Kopasov, A. G. Kutlin, and A. S. Mel’nikov, Geometry Controlled Superconducting Diode and Anomalous Josephson Effect Triggered by the Topological Phase Transition in Curved Proximitized Nanowires, Phys. Rev. B 103, 144520 (2021).
  * [100] K. Misaki and N. Nagaosa, Theory of the Nonreciprocal Josephson Effect, Phys. Rev. B 103, 245302 (2021).
  * [101] K. Halterman, M. Alidoust, R. Smith, and S. Starr, Supercurrent Diode Effect, Spin Torques, and Robust Zero-Energy Peak in Planar Half-Metallic Trilayers, Phys. Rev. B 105, 104508 (2022).
  * [102] Y. Wei, H.-L. Liu, J. Wang, and J.-F. Liu, Supercurrent Rectification Effect in Graphene-Based Josephson Junctions, Phys. Rev. B 106, 165419 (2022).
  * [103] Y.-M. Xie, D. K. Efetov, and K. T. Law, $\varphi_{0}$-Josephson Junction in Twisted Bilayer Graphene Induced by a Valley-Polarized State, Phys. Rev. Res. 5, 023029 (2023).
  * [104] Y. Zhang, Y. Gu, P. Li, J. Hu, and K. Jiang, General Theory of Josephson Diodes, Phys. Rev. X 12, 041013 (2022).
  * [105] M. Davydova, S. Prembabu, and L. Fu, Universal Josephson Diode Effect, Sci. Adv. 8, eabo0309 (2022).
  * [106] J.-X. Hu, Z.-T. Sun, Y.-M. Xie, and K. T. Law, Josephson Diode Effect Induced by Valley Polarization in Twisted Bilayer Graphene, Phys. Rev. Lett. 130, 266003 (2023).
  * [107] J. F. Steiner, L. Melischek, M. Trahms, K. J. Franke, and F. von Oppen, Diode Effects in Current-Biased Josephson Junctions, Phys. Rev. Lett. 130, 177002 (2023).
  * [108] T. H. Kokkeler, A. A. Golubov, and F. S. Bergeret, Field-Free Anomalous Junction and Superconducting Diode Effect in Spin-Split Superconductor/Topological Insulator Junctions, Phys. Rev. B 106, 214504 (2022).
  * [109] B. Lu, S. Ikegaya, P. Burset, Y. Tanaka, and N. Nagaosa, Tunable Josephson Diode Effect on the Surface of Topological Insulators, Phys. Rev. Lett. 131, 096001 (2023).
  * [110] Y. Tanaka, B. Lu, and N. Nagaosa, Theory of Giant Diode Effect in $d$-Wave Superconductor Junctions on the Surface of a Topological Insulator, Phys. Rev. B 106, 214524 (2022).
  * [111] H. F. Legg, K. Laubscher, D. Loss, and J. Klinovaja, Parity-Protected Superconducting Diode Effect in Topological Josephson Junctions, Phys. Rev. B 108, 214520 (2023).
  * [112] T. Karabassov, I. V. Bobkova, V. M. Silkin, B. G. Lvov, A. A. Golubov, and A. S. Vasenko, Phase Diagrams of the Diode Effect in Superconducting Heterostructures, Phys. Scr. 99, 015010 (2024).
  * [113] A. Maiani, K. Flensberg, M. Leijnse, C. Schrade, S. Vaitiekėnas, and R. S. Souto, Nonsinusoidal Current-Phase Relations in Semiconductor-Superconductor-Ferromagnetic Insulator Devices, Phys. Rev. B 107, 245415 (2023).
  * [114] A. Soori, Josephson Diode Effect in Junctions of Superconductors with Band Asymmetric Metals, arXiv: 2312.14084 (2023).
  * [115] T. Liu, M. Smith, A. V. Andreev, and B. Z. Spivak, Giant Nonreciprocity of Current-Voltage Characteristics of Noncentrosymmetric Supercondctor-Normal Metal-Superconductor Junctions, Phys. Rev. B 109, L020501 (2023).
  * [116] P. A. Volkov, É. Lantagne-Hurtubise, T. Tummuru, S. Plugge, J. H. Pixley, and M. Franz, Josephson Diode Effects in Twisted Nodal Superconductors, Phys. Rev. B 109, 094518 (2024).
  * [117] S. Fracassi, S. Traverso, N. T. Ziani, M. Carrega, S. Heun, and M. Sassetti, Anomalous Supercurrent and Diode Effect in Locally Perturbed Topological Josephson Junctions, arXiv: 2403.17894 (2024).
  * [118] J. Cayao, N. Nagaosa, and Y. Tanaka, Enhancing the Josephson Diode Effect with Majorana Bound States, Phys. Rev. B 109, L081405 (2023).
  * [119] J. Wang, Y. Jiang, J. J. Wang, and J.-F. Liu, Efficient Josephson Diode Effect on a Two-Dimensional Topological Insulator with Asymmetric Magnetization, Phys. Rev. B 109, 075412 (2024).
  * [120] Z. Liu, L. Huang, and J. Wang, Josephson Diode Effect in Topological Superconductor, arXiv: 2311.09009 (2023).
  * [121] Y. Wei, J.-J. Wang, and J. Wang, Josephson Diode Effect in a Line-Centered Honeycomb Lattice Based Superconductor Junction, Phys. Rev. B 108, 054521 (2023).
  * [122] A. Zazunov, J. Rech, T. Jonckheere, B. Grémaud, T. Martin, and R. Egger, Approaching Ideal Rectification in Superconducting Diodes through Multiple Andreev Reflections, arXiv: 2307.14698 (2023).
  * [123] A. Zazunov, J. Rech, T. Jonckheere, B. Grémaud, T. Martin, and R. Egger, Nonreciprocal Charge Transport and Subharmonic Structure in Voltage-Biased Josephson Diodes, Phys. Rev. B 109, 024504 (2024).
  * [124] Y. Zhang and Z. Wang, Kramers Fulde-Ferrell State and Superconducting Spin Diode Effect, Phys. Rev. B 107, 224510 (2023).
  * [125] Y. Mao, Q. Yan, Y.-C. Zhuang, and Q.-F. Sun, Universal Spin Superconducting Diode Effect from Spin-Orbit Coupling, arXiv: 2306.09113 (2023).
  * [126] A. Costa, J. Fabian, and D. Kochan, Microscopic Study of the Josephson Supercurrent Diode Effect in Josephson Junctions Based on Two-Dimensional Electron Gas, Phys. Rev. B 108, 054522 (2023).
  * [127] P.-H. Fu, Y. Xu, C. H. Lee, S. A. Yang, Y. S. Ang, J.-F. Liu, Field-Effect Josephson Diode via Asymmetric Spin-Momentum Locking States, arXiv: 2212.01980v3 (2022).
  * [128] P. Ding, C. H. Lee, X. Wu, and R. Thomale, Diagnosis of Pairing Symmetry by Vortex and Edge Spectra in Kagome Superconductors, Phys. Rev. B 105, 174518 (2022).
  * [129] P. Ding, T. Schwemmer, C. H. Lee, X. Wu, and R. Thomale, Hyperbolic Fringe Signal for Twin Impurity Quasiparticle Interference, Phys. Rev. Lett. 130, 256001 (2023).
  * [130] N. F. Q. Yuan and L. Fu, Zeeman-Induced Gapless Superconductivity with a Partial Fermi Surface, Phys. Rev. B 97, 115139 (2018).
  * [131] N. F. Q. Yuan and L. Fu, Topological Metals and Finite-Momentum Superconductors, Proc. Natl. Acad. Sci. 118 (3) e2019063118 (2021).
  * [132] S. Hart, H. Ren, M. Kosowsky, G. Ben-Shach, P. Leubner, C. Brüne, H. Buhmann, L. W. Molenkamp, B. I. Halperin, and A. Yacoby, Controlled Finite Momentum Pairing and Spatially Varying Order Parameter in Proximitized HgTe Quantum Wells, Nat. Phys. 13, 87 (2017).
  * [133] P. G. De Gennes, Superconductivity of Metals and Alloys (Benjamin, New York, 1966).
  * [134] G. Tkachov, P. Burset, B. Trauzettel, and E. M. Hankiewicz, Quantum interference of edge supercurrents in a two-dimensional topological insulator, Phys. Rev. B 92, 045408 (2015).
  * [135] B. Scharf, A. Braggio, E. Strambini, F. Giazotto, and E. M. Hankiewicz, Thermodynamics in Topological Josephson Junctions, Phys. Rev. Res. 3, 033062 (2021).
  * [136] F. Dolcini, M. Houzet, and J. S. Meyer, Topological Josephson $\phi_{0}$ junctions, Phys. Rev. B 92, 035428 (2015).
  * [137] P.-H. Fu, Y. Xu, X.-L. Yu, J.-F. Liu, and J. Wu, Electrically Modulated Josephson Junction of Light-Dressed Topological Insulators, Phys. Rev. B 105, 064503 (2022).
  * [138] F. Rohlfing, G. Tkachov, F. Otto, K. Richter, D. Weiss, G. Borghs, and C. Strunk, Doppler Shift in Andreev Reflection from a Moving Superconducting Condensate in Nb/InAs Josephson Junctions, Phys. Rev. B 80, 220507 (2009).
  * [139] Z. Zhu et al., Discovery of Segmented Fermi Surface Induced by Cooper Pair Momentum, Science. 374, 1381 (2021).
  * [140] A. F. Andreev, Thermo Conductivity of The Intermediate State of Superconductors, Zh. Eksp. Teor. Fiz. 46, 1823 (1964) [Sov. Phys.-JETP 19, 1228 (1964)].
  * [141] N. A. Mortensen, K. Flensberg, and A.-P. Jauho, Angle Dependence of Andreev Scattering at Semiconductor-Superconductor Interfaces, Phys. Rev. B 59, 10176 (1999).
  * [142] P.-H. Fu, J.-F. Liu, and J. Wu, Transport Properties of Majorana Drumhead Surface States in Topological Nodal-Ring Superconductors, Phys. Rev. B 102, 075430 (2020).
  * [143] Supplemental Material will be uploaded soon.
  * [144] S. Datta, P. F. Bagwell, and M. P. Anantram, Scattering Theory of Transport for Mesoscopic Superconductors, Phys. Low Dim. Struct. 3, 1 (1996).
  * [145] M. Kjaergaard et al., Quantized Conductance Doubling and Hard Gap in a Two-Dimensional Semiconductor–Superconductor Heterostructure, Nat. Commun. 7, 12841 (2016).
  * [146] E. Strambini et al., Superconducting Spintronic Tunnel Diode, Nat. Commun. 13, 2431 (2022).
  * [147] C. I. L. de Araujo, P. Virtanen, M. Spies, C. González-Orellana, S. Kerschbaumer, M. Ilyn, C. Rogero, T. T. Heikkilä, F. Giazotto, and E. Strambini, Superconducting Spintronic Heat Engine, arXiv: 2310.18132 (2023).
  * [148] Z. Geng et al., Superconductor-Ferromagnet Hybrids for Non-Reciprocal Electronics and Detectors, Supercond. Sci. Technol. 36, 123001 (2023).
  * [149] M. P. Anantram and S. Datta, Current Fluctuations in Mesoscopic Systems with Andreev Scattering, Phys. Rev. B 53, 16390 (1996).
  * [150] C. W. J. Beenakker, Specular Andreev Reflection in Graphene, Phys. Rev. Lett. 97, 067007 (2006).