# An Uncertainty Estimation Framework for Probabilistic Object Detection

Zongyao Lyu, Nolan B. Gutierrez, and William J. Beksi Z. Lyu, N.B. Gutierrez,
and W.J. Beksi are with the Department of Computer Science and Engineering,
University of Texas at Arlington, Arlington, TX, USA. Emails:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, we introduce a new technique that combines two popular methods
to estimate uncertainty in object detection. Quantifying uncertainty is
critical in real-world robotic applications. Traditional detection models can
be ambiguous even when they provide a high-probability output. Robot actions
based on high-confidence, yet unreliable predictions, may result in serious
repercussions. Our framework employs deep ensembles and Monte Carlo dropout
for approximating predictive uncertainty, and it improves upon the uncertainty
estimation quality of the baseline method. The proposed approach is evaluated
on publicly available synthetic image datasets captured from sequences of
video.

##  I Introduction

Object detection is a central task in computer vision and a necessary
capability for many robotic operations. Models for object detection have
achieved high mean average precision (mAP) on datasets such as ImageNet [1],
the PASCAL Visual Object Classes [2, 3], and the Microsoft Common Objects in
Context (COCO) [4]. However, these models can fail when evaluated in dynamic
scenarios which commonly contain objects from outside the training dataset.
The mAP metric encourages detectors to output many detections for each image,
yet it does not provide a consistent measure of confidence other than a higher
score indicating that the system is more accurate. Nonetheless, robotic vision
systems must be able to cope with diverse operating conditions such as
variations in illumination and occlusions in the surrounding environment.

The ACRV Robotic Vision Challenge on Probabilistic Object Detection (PrOD) [5]
introduces a variation on conventional object detection tasks, which requires
quantification of spatial and semantic uncertainty with the use of a new
evaluation measure called the probability-based detection quality (PDQ) [6].
Different from mAP, the PDQ jointly evaluates the quality of spatial and label
uncertainty, foreground and background separation, and the number of true
positive (correct), false positive (spurious), and false negative (missed)
detections. In addition, the datasets provided by the challenge are video
sequences comprising various scenes that have been generated from high-
fidelity simulations.

(a)

(b)

(c)

Figure 1: An example of a raw input image (a), object detections (b), and the
corresponding heatmaps associated with the bounding boxes (c). Our object
detection system represents the location of objects as probabilistic bounding
boxes. This allows a detector to express spatial uncertainty by inducing a
probability distribution over the pixels of the object.

In this work, we extend our initial groundwork [7] for addressing the issue of
uncertainty estimation in object detection (Fig. 1). This includes the
exploration of two popular uncertainty estimation techniques: deep ensembles
[8] and Monte Carlo (MC) dropout [9]. We apply these techniques jointly on two
detectors, Hybrid Task Cascade [10] and Grid R-CNN [11], to produce
probabilistic bounding boxes which allow a detector to express spatial
uncertainty. Furthermore, semantic (i.e., label) uncertainty is quantified by
a probability distribution over the known classes for each detection. In
summary, our contributions are the following.

  * •

We devise a novel aggregation of two prominent techniques to quantify
uncertainty in object detection models.

  * •

We create a new heuristic for improving false positive label quality.

  * •

We provide an easy-to-implement method for merging ensembles of detections.

  * •

We introduce high-speed tuning of the hyperparameters for increasing PDQ
scores.

  * •

We show that non-maximum suppression (NMS) can be used as a post-processing
step to keep all detections.

Our source code is available at [12].

The remainder of this paper is organized as follows. We describe related work
in Section II. Section III provides the background information for calculating
the PDQ score. Our framework for probabilistic object detection is presented
in Section IV. In Section V, we report our experimental results on the
validation data provided by the PrOD challenge. We conclude in Section VI with
a discussion of future work.

##  II Related Work

###  A Object Detection

The goal of object detection is to localize and recognize instances of
particular categories of objects within an image. Current state-of-the-art
object detection systems utilize the power of deep neural networks (DNNs) [13]
to output a softmax score indicating the class label and a bounding box
denoting the location of each object of a known class. DNN-based object
detectors can be categorized into one-stage detection models such as YOLO [14]
and its variations [15, 16, 17, 18], and two-stage detection models such as
R-CNN [19] and its variants [20, 21, 22, 23]. Both categories have advantages
over each other.

Two-stage detectors include a region proposal generator that produces a set of
proposals and extracts features from each proposal in the first stage,
followed by a second stage of classification and bounding-box regression.
Conversely, one-stage detectors conduct the prediction of object classes and
bounding boxes directly on the image feature maps without a region proposal
generation step. Two-stage detectors generally achieve higher accuracy, but
they have a relatively slow detection speed. On the other hand, one-stage
detectors are more time efficient and have greater applicability to real-time
detection, nevertheless they exhibit lower accuracy.

Although the performance of object detectors has been greatly improved with
the recent development of deep learning algorithms, current state-of-the-art
DNN-based detectors may give high classification probabilities to falsely
detected objects. This occurs due to the fact that they output deterministic
predictions of bounding boxes without information regarding how certain the
predicted results are. Thus, DNN detectors can be unreliable in practical
applications and relying on softmax scores and predicted boxes alone is not
sufficient.

###  B Bayesian Neural Networks for Uncertainty Estimation

To cope with these limitations, one may extend the classic neural network to
produce additional data which can be adapted to supplement the following
values: label and spatial uncertainty. This information can then be used to
augment the classification score and regressed bounding boxes, thus producing
more reliable predictions. Traditionally, Bayesian methods automatically infer
hyperparameters by marginalizing them out of the posterior distribution. They
can naturally express uncertainty in parameter estimates and propagate it to
the predicted results.

Bayesian neural networks (BNNs) [24, 25, 26] offer a natural way to integrate
Bayesian modeling with DNNs. BNNs provide a mathematically grounded foundation
to reason about uncertainty in DNNs by learning a prior distribution over the
weights of a neural network. Not only is each weight in a BNN a random
variable, but also the output of the network is a random variable thereby
providing a measure of uncertainty. However, compared to non-Bayesian neural
networks, BNNs are often computationally intractable which makes them hard to
apply in practice.

###  C Dropout as a Bayesian Approximation

Dropout [27, 28] was first proposed as a technique for training neural
networks by randomly omitting neuron units during training to prevent
overfitting. It helps prevent co-adaptations whereby a unit relies on several
other specific hidden units being present. Dropout also provides a way of
approximately combining many disparate neural network models efficiently.
Maeda et al. [29] provide an explanation of dropout from a Bayesian
standpoint. From this perspective, we can view dropout as a way to deal with
the model selection problem by Bayesian model averaging where each model is
weighted in accordance with the posterior distribution.

Gal et al. [30, 9] extend the idea of dropout as approximate Bayesian
inference in deep Gaussian processes. They show that dropout can be used at
test time to impose a Bernoulli distribution over the network’s weights, thus
requiring no additional model parameters. They refer to this technique as MC
dropout. Following this idea, Miller et al. [31, 32] estimate spatial and
semantic uncertainties of detections in open-set conditions as a rejection
criteria using MC dropout. Kraus et al. [33] incorporate the methods provided
by [30] into an object detector to estimate uncertainty, while improving the
detection performance of the baseline approach.

###  D Uncertainty Estimation using Deep Ensembles

Ensembles are one way of improving accuracy in object detection by
incorporating detection models. The basic idea is that detectors with
contrasting network structures have a different detection performance for
distinct objects in an image. As a result, the accuracy and robustness of the
detector predictions can be improved due to the application of ensemble
methods. For example, Lakshminarayanan et al. [8] propose a non-Bayesian
alternative to BNNs by training an ensemble of multiple networks
independently. When combined together, the networks behave differently for
given inputs hence expressing uncertainty.

###  E Probabilistic Object Detection

As opposed to assuming constant uncertainty, a detection system can assess its
own quality by estimating the ambiguity of its task performance. Kendall and
Gal [34] identify evaluating spatial and semantic uncertainties as paramount
importance, especially in applications where classifications need to be safely
clarified. Sirinukunwattana et al. [35] regress probability values for the
classification of nuclei in routine colon cancer histology images. Tanno et
al. [36] apply Bayesian image quality transfer to dMRI super-resolution by
applying sub-pixel convolutions to 3D tractography. In addition, the authors
provide uncertainty maps on a pixel-wise basis. A method for improving
detection accuracy by modeling bounding boxes with Gaussian parameters is put
forth by Choi et al. [37]. Wang et al. [38] propose optimizing PDQ scores by
focusing on training-time augmentations. A key contribution of their work was
the insight that a system could become more certain of its predictions by
feeding in gamma corrected images during training time. Ammirato and Berg [39]
examine how a system designed for mAP, specifically Mask-RCNN [22], performs
in an environment that evaluates the PDQ metric.

In contrast to the aforementioned works, we combine MC dropout and deep
ensembles, and apply it to the object detection problem. We show that the
uncertainty estimation performance can be improved over simply using deep
ensembles. In addition, whereas previous approaches merged observations by
averaging, our approach takes the merged box to be the box for which the
system is more confident. When dealing with ensembles, this approach makes
sense since object detectors are often distinctly confident with predictions
of classes and objects with certain characteristics such as appearance or
frequency in the dataset.

##  III Background

Consider a ground-truth object $G$ and a detection $D$. Associated with $G$ is
a segmentation mask $S$ and a label $c$, along with a bounding box $B$. The
segmentation mask $S$ corresponds to the true bounding box and determines
which pixels belong to $G$, while $B$ is the set of pixels within the bounding
box associated with the object. $D$ consists of a probability distribution
over a label vector $\bm{l}$, a segmentation mask $\hat{S}$, and a probability
that some pixel belongs to the segmentation mask $P(\bm{x}\in\hat{S})$.
$\hat{S}$ coincides to the pixels within the detection’s predicted bounding
box. The spatial quality $Q_{\hat{S}}(G,D)$ of this detection is defined by

$exp(\frac{1}{N}\sum\limits_{\bm{x}\in
S}log(P(\bm{x}\in\hat{S}))+\frac{1}{N}\sum\limits_{\bm{x}\in\\{S-B\\}}log(1-P(\bm{x}\in\hat{S}))),$
(1)

where $N$ is the number of pixels in the true bounding box $S$, and the two
terms in the exponent are known as the foreground and background losses,
respectively. $Q_{S}$ is one if the pixels in the ground-truth segmentation
are each given probability one, and all the remaining pixels in the set $S-B$
are assigned probability zero. Next, we can calculate the label quality
$Q_{\bm{l}}(G,D)$ by

$Q_{L}(G,D)=\bm{l}_{c},$ (2)

where $\bm{l}_{c}$ is the probability of the detection belonging to class $c$
as predicted by the object detector. The pairwise probabilistic detection
quality score (pPDQ) can then be calculated by

$\text{pPDQ}(G,D)=\sqrt{Q_{\hat{S}}\cdot Q_{L}}.$ (3)

Let $\bm{q}_{ij}$ represent the pPDQ of the $j$th detection in the $i$th
frame. Then the overall PDQ score can be found by

$\text{PDQ}(G,D)=\frac{1}{\sum\limits_{i=1}^{F}(\text{TP}_{i}+\text{FN}_{i}+\text{FP}_{i})}\sum\limits_{i=1}^{F}\sum\limits_{j=1}^{\text{TP}_{i}}\bm{q}_{ij},$
(4)

where $F$ constitutes the number of frames, and
$\text{TP}_{i},\text{FN}_{i},\text{FP}_{i}$ represent the number of true
positives, false negatives, and false positives in the $i$th frame,
respectively. The PDQ measure allows for uncertainty to be modeled in the
location of an object which is represented as a probabilistic bounding box. In
particular, the corners of the bounding box are 2D Gaussians and a covariance
matrix is given for each corner to express a spatial uncertainty over the
pixels. The label uncertainty is described as a full probability distribution
over the known classes for each detection.

##  IV Probabilistic Object Detection Framework

Figure 2: An overview of our deep ensemble architecture for probabilistic
object detection.

We detail the key components of our framework for probabilistic object
detection in this section. To begin, we describe the datasets and data
preprocessing steps. Next, we introduce the detection models and explain how
our system utilizes deep ensembles with MC dropout for uncertainty estimation.
This is followed by a discussion of the heuristics, merging methods, and post-
processing techniques we implemented to improve the PDQ scores. An overview of
our system is shown in Fig. 2.

###  A Datasets and Data Preprocessing

The video sequences provided by the PrOD challenge are divided into test,
test-dev, and validation sets. The validation set contains the released ground
truth and is used to evaluate our framework. It consists of around 21500
images with 56580 ground-truth objects in total. The 30 classes of the
validation dataset are a subset of 80 classes annotated from COCO. We use
models pretrained on COCO and filter out the 50 classes that are not evaluated
by the challenge.

###  B Detection Models

We employ two recent well-performing detection models for the task of first-
stage detection. The first model is Grid-RCNN [11], which applies a grid-
guided mechanism in place of the traditional box offset regression, for high-
quality localization. The second model is Hybrid Task Cascade (HTC) [10], an
effective cascade architecture for object detection and instance segmentation.

###  C Combining Deep Ensembles with Monte Carlo Dropout

For each detector, we retain dropout layers in the model during test time and
then make an ensemble of the two models. Our approach is similar to NMS [40].
We simplify and improve the method proposed by Miller et al. [31] in which
detections are grouped into observations by IoU. A key difference in our
approach is we take the merged bounding box from each group to be the box that
the model is most confident about (where confidence is measured by the highest
label score). To merge predictions, we group detections by IoU via NMS, and we
keep the most confident detections from each observation.

First, NMS partitions boxes $B$ into groups $G$ such that any two boxes in a
group $G_{i}$ has an IoU above a threshold $\lambda$ between 0 and 1. Each
group must be sorted by confidence scores. The list of boxes contains the top-
left and bottom-right corners, and the confidence of each box’s label. Next,
each of these observations represented as groups of bounding boxes are reduced
to the most confident bounding box and label. Finally, the covariance matrices
of every detection are computed. The procedure for merging the detections from
our deep ensemble is shown in Algorithm 1.

Algorithm 1 Merging of Deep Ensembles

1:I = input image

2:$D=$ $d_{1},d_{2},\ldots,d_{k}$ $\triangleright$ $d_{i}$: $i^{th}$ detector

3:$\lambda=[0,1]$ $\triangleright$ IoU between boxes in a cluster

4:$d_{i}(I)_{B},d_{i}(I)_{L}$ $\triangleright$ Detected bounding boxes and
labels

5:$B^{new},L^{new}$ $\triangleright$ Merged list of bounding boxes, labels

6:$B:=\underset{d\in D}{||}d(I)_{B},L:=\underset{d\in D}{||}d(I)_{L}$
$\triangleright$ $||:$ Concatenation

7:G $:=\\{\,g\mid\forall x,y\in g,\text{IoU}(B_{x},B_{y})\geq\lambda\,\\}$

8:for $i$ in $1,2,3,\ldots,|G|$ do

9: Sort $G_{i}$ by scores in $L$ in descending order

10: $B_{i}^{new}=B_{G_{i,0}}$

11: $L_{i}^{new}=L_{G_{i,0}}$

###  D Heuristics for Improving the PDQ Score

We can maximize the PDQ score by optimizing metrics that evaluate the quality
of object detections. As an example, we may increase the PDQ score by reducing
the number of objects in a detection that are incorrectly classified [39].
Alternatively, researchers have demonstrated that inflating label scores to
one increases the average label quality and thus the PDQ score. To
systematically find the optimal parameters, we save the boxes that are
obtained from each configuration of the model and uniformly search the
hyperparameter space. In the following paragraphs, we describe the details for
each heuristic employed to improve the PDQ score.

Threshold Filtering: Rejection of bounding boxes that have low-confidence has
the greatest impact on localization performance and the PDQ score. Succinctly,
the number of false negatives, false positives, and true positives is
affected, therefore producing a wide range of PDQ scores. While other work on
this topic has found an optimal confidence threshold of 0.5 [39], we’ve
discovered that higher thresholds significantly increase the number of false
negatives and thus reduce the PDQ score. In our search of the choice
hyperparameters, we’ve found that we can set the confidence threshold as low
as possible but only when running NMS once more with a low IoU threshold.

Bounding Box Reduction: The PDQ score can be substantially lowered by
confidently labeling pixels in the background. To lessen this issue, we center
crop the bounding box whereby the box’s width and height are reduced by a
specified proportion. The PDQ score increases by reducing the ratio of
background pixels to foreground pixels. We specify our reduction ratio to be
0.1 thereby gaining better results.

Improving False Positive Quality: False positive label quality is lower when
incorrectly classified objects assign low probabilities to correct labels. We
conceive a novel heuristic by noting that uniformly distributing probabilities
over the rest of the labels in the label vector $l$ can maximize this metric
over all testing data. Concretely,

$l_{i}=\begin{cases}S,&l_{i}\text{ corresponds to the detection label,}\\\
\frac{1-S}{30},&\text{otherwise,}\end{cases}$ (5)

where $S$ is the score of the bounding box’s label and 30 is the number of
classes.

Covariance Matrix Calculation: We experimented with traditional covariance
calculations and fixed covariances. Notwithstanding, we improve our results by
setting the diagonal of the covariance matrix, which represents the spatial
uncertainty, proportional to the bounding box’s width and height [39]. As an
example, if we use a proportion of $0.10$ for a 50 by 60 bounding box, then
the covariance matrix would be defined as $[[5,0],[0,6]]$ for both the top-
left and bottom-right corners.

Non-Maximal Suppression: Contemporary work has shown that the PDQ score can be
improved by removing low-confidence detections. We’ve verified that this
heuristic does improve the score, nevertheless this result can be outperformed
by adding an additional NMS post-processing step. However, allowing the
evaluation of low-confidence boxes may increase the number of false positives.
To avoid this situation, we allow larger higher-confident boxes to absorb
smaller lower-confident boxes which reduces the number of false positives.

##  V Experimental Results

This section presents our experimental results on the validation data provided
by the PrOD challenge. Our implementation is based on MMDetection [41], an
open-source object detection toolbox. The experiments were performed on a
Ubuntu 18.04 machine with an Intel Core i7-8700 CPU, 32GB of RAM, and an
NVIDIA Quadro P4000 GPU.

Both detectors used in the experiments are models pretrained on the COCO
dataset with the ResNeXt101 backbone. The results of using an ensemble model
composed of HTC and Grid R-CNN with various threshold parameters are shown in
Table I. In our experiments, lower threshold values provided higher scores.
Table II shows the effect of reducing the bounding box size of all detections
on the PDQ scores. Here we found that reduction of the bounding box size by
10% results in better performance. Table III shows how the PDQ score changes
when scaling the covariance to a different percentage of the box size. We
experimented with covariance matrices dependent on the widths and heights of
the bounding boxes. The effectiveness of this technique was demonstrated in
[39], and our final results applied this heuristic. After exploring various
values in our implementation, we found that setting the top-left and bottom-
right elements of the covariance matrix equal to a scale of 30% of the
bounding box size gave the best results. We report these findings with
different IoU threshold values in Table IV.

Table V shows the efficacy of the individual HTC and Grid R-CNN detectors, and
their ensembles. We observed that a single HTC detector outperforms an
ensemble of HTC and Grid R-CNN in our experiments. This indicates that a
careful selection of integrated detection models and parameter tuning are
necessary to bring about an effective ensemble model, thus further exploration
is required. We examined MC dropout as the uncertainty estimation technique on
the HTC and Grid R-CNN detectors individually, and also their ensembles. In
practice, this method is equivalent to performing several stochastic forward
passes through the network and then taking an average of the results. We chose
to sample three passes by adding a dropout rate of 0.3 to the second shared
fully-convolutional layer of HTC’s region of interest head. After testing a
large number of post-processing parameters, we found that models using dropout
consistently demonstrated better quality when compared to models without
dropout. The result of an ensemble model with dropout outperforming an
ensemble without dropout demonstrates the effectiveness of our idea for
integrating these two approaches. The results of the models with dropout are
displayed in Table V.

Finally, we compared our most confident box merging (MCBM) strategy with
averaging in Table VI. With all other parameters being the same, we present
evidence that even when boxes in an observation must have the same label, the
detection quality is best whenever we take the merged box to be the most
confident box in the observation. Our hypothesis is we should allow the
detection system that is most confident about its result have its prediction
take over the observation. As expected, the MCBM method achieved the same
results with and without the constraint that the boxes in an observation must
have the same label.

Threshold | PDQ Score | Avg. pPDQ | Avg. FP | Avg. SQ | Avg. LQ | TPs | FPs | FNs
---|---|---|---|---|---|---|---|---
0.018 | 22.569 | 0.409 | 0.835 | 0.464 | 0.498 | 58961 | 112689 | 29128
0.05 | 22.534 | 0.446 | 0.747 | 0.483 | 0.551 | 53036 | 66395 | 35053
0.1 | 22.468 | 0.481 | 0.648 | 0.498 | 0.602 | 48176 | 42652 | 39913
0.3 | 22.006 | 0.544 | 0.424 | 0.520 | 0.703 | 39940 | 18615 | 48149
0.5 | 20.919 | 0.591 | 0.270 | 0.537 | 0.780 | 33809 | 10277 | 54280
TABLE I: The results of an ensemble model using different threshold values on the validation dataset (Avg. SQ=average spatial quality, Avg. LQ=average label quality, TPs=true positives, FPs=false positives, FNs=false negatives). Box Ratio | PDQ Score | Avg. pPDQ | Avg. FP | Avg. SQ | Avg. LQ | TPs | FPs | FNs
---|---|---|---|---|---|---|---|---
0.05 | 22.084 | 0.399 | 0.835 | 0.445 | 0.497 | 59026 | 112799 | 29063
0.1 | 22.569 | 0.409 | 0.835 | 0.464 | 0.498 | 58961 | 112689 | 29128
0.2 | 19.861 | 0.360 | 0.835 | 0.383 | 0.499 | 58848 | 112414 | 29241
0.3 | 12.882 | 0.235 | 0.834 | 0.219 | 0.500 | 58509 | 112471 | 29580
TABLE II: The results of an ensemble model using different bounding box ratios on the validation dataset (Avg. SQ=average spatial quality, Avg. LQ=average label quality, TPs=true positives, FPs=false positives, FNs=false negatives). Covariance Scale | PDQ Score | Avg. pPDQ | Avg. FP | Avg. SQ | Avg. LQ | TPs | FPs | FNs
---|---|---|---|---|---|---|---|---
0.1 | 21.894 | 0.402 | 0.834 | 0.462 | 0.502 | 58275 | 113517 | 29814
0.2 | 22.536 | 0.411 | 0.835 | 0.473 | 0.500 | 58605 | 113146 | 29484
0.3 | 22.569 | 0.409 | 0.835 | 0.464 | 0.498 | 58961 | 112689 | 29128
0.4 | 22.458 | 0.403 | 0.835 | 0.450 | 0.495 | 59421 | 112020 | 28668
0.5 | 22.283 | 0.397 | 0.835 | 0.437 | 0.492 | 59737 | 111557 | 28352
TABLE III: The results of an ensemble model using different covariance scales on the validation dataset (Avg. SQ=average spatial quality, Avg. LQ=average label quality, TPs=true positives, FPs=false positives, FNs=false negatives). IoU Threshold | PDQ Score | Avg. pPDQ | Avg. FP | Avg. SQ | Avg. LQ | TPs | FPs | FNs
---|---|---|---|---|---|---|---|---
0.1 | 21.966 | 0.419 | 0.817 | 0.468 | 0.514 | 54194 | 84036 | 33895
0.2 | 22.395 | 0.411 | 0.827 | 0.465 | 0.504 | 57398 | 100192 | 30691
0.3 | 22.569 | 0.409 | 0.835 | 0.464 | 0.498 | 58961 | 112689 | 29128
0.4 | 22.530 | 0.405 | 0.844 | 0.465 | 0.486 | 60824 | 136205 | 27265
0.5 | 22.314 | 0.405 | 0.856 | 0.472 | 0.477 | 62338 | 175034 | 25751
TABLE IV: The results of an ensemble model using different IoU threshold values on the validation set (Avg. SQ=average spatial quality, Avg. LQ=average label quality, TPs=true positives, FPs=false positives, FNs=false negatives). Method | PDQ Score | Avg. pPDQ | Avg. FP | Avg. SQ | Avg. LQ | TPs | FPs | FNs
---|---|---|---|---|---|---|---|---
HTC | 22.769 | 0.415 | 0.833 | 0.472 | 0.496 | 56054 | 84565 | 32035
HTC + MC dropout | 22.822 | 0.414 | 0.824 | 0.467 | 0.504 | 58498 | 102610 | 29591
Grid R-CNN | 18.014 | 0.410 | 0.799 | 0.465 | 0.514 | 42383 | 42162 | 45706
Grid R-CNN + MC dropout | 18.222 | 0.395 | 0.789 | 0.445 | 0.514 | 48340 | 79423 | 39749
Ensemble | 22.569 | 0.409 | 0.835 | 0.464 | 0.498 | 58961 | 112689 | 29128
Ensemble + MC dropout | 22.599 | 0.417 | 0.818 | 0.469 | 0.515 | 59847 | 122280 | 28242
TABLE V: The results of a single and an ensemble model before and after adding dropout on the validation dataset (Avg. SQ=average spatial quality, Avg. LQ=average label quality, TPs=true positives, FPs=false positives, FNs=false negatives). Method | PDQ Score | Avg. pPDQ | Avg. FP | Avg. SQ | Avg. LQ | TPs | FPs | FNs
---|---|---|---|---|---|---|---|---
Ensemble (averaging) | 12.642 | 0.223 | 0.907 | 0.324 | 0.238 | 54622 | 90024 | 33467
Ensemble (same labels, averaging) | 14.286 | 0.246 | 0.887 | 0.323 | 0.294 | 57425 | 101779 | 30664
Ensemble (ours) | 22.569 | 0.409 | 0.835 | 0.464 | 0.498 | 58961 | 112689 | 29128
TABLE VI: A comparison of ensemble models using averaging and our merging
strategy on the validation dataset (Avg. SQ=average spatial quality, Avg.
LQ=average label quality, TPs=true positives, FPs=false positives, FNs=false
negatives).

##  VI Conclusion and Future Work

We presented an uncertainty estimation framework which makes use of the
recently introduced PDQ evaluation metric and achieves a top score on the PrOD
challenge validation dataset. By extensively exploring the hyperparameter
space, we showed the quality of object detections can be maintained by
introducing additional post-processing techniques. Likewise, we’ve shown that
post-processing methods designed to increase detection quality may influence
one another and the optimization of these techniques can be done offline. We
highlighted that the IoU parameter of NMS designed for mAP does not lead to
high-quality detections. In fact, we experimentally demonstrated that very low
IoU thresholds group more detections together and reduce the number of false
positives.

Our forthcoming work in probabilistic object detection will explore the
incorporation of manual heuristics into an end-to-end model. For example, it
is advantageous to add bias for smaller bounding boxes. Moreover, post-
processing steps must be adaptive, i.e., the hyperparameters for these steps
may be image dependent. This warrants further investigation into the adequacy
of applying computationally intensive methods that occasionally decrease the
quality of the performance. More importantly, when applying these methods one
should clearly select detectors based on their strengths and weaknesses while
being aware of any potential increase in computation.

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# Sign reversal diode effect in superconducting Dayem nanobridges

Daniel Margineda<EMAIL_ADDRESS>NEST Istituto Nanoscienze-CNR and
Scuola Normale Superiore, I-56127, Pisa, Italy  present address: CIC nanoGUNE
BRTA, E-20018 Donostia-San Sebastián, Spain.  Alessandro Crippa  NEST Istituto
Nanoscienze-CNR and Scuola Normale Superiore, I-56127, Pisa, Italy  Elia
Strambini  NEST Istituto Nanoscienze-CNR and Scuola Normale Superiore,
I-56127, Pisa, Italy  Yuri Fukaya  SPIN-CNR, I-84084 Fisciano (SA), Italy
Maria Teresa Mercaldo  Dipartimento di Fisica “E. R. Caianiello”, Università
di Salerno, I-84084 Fisciano (SA), Italy  Mario Cuoco  SPIN-CNR, I-84084
Fisciano (SA), Italy  Francesco Giazotto<EMAIL_ADDRESS>NEST
Istituto Nanoscienze-CNR and Scuola Normale Superiore, I-56127, Pisa, Italy

Superconducting Diode Effect, Superconducting electronics

Abstract
Supercurrent diodes are nonreciprocal electronic elements whose switching
current depends on their flow direction. Recently, a variety of composite
systems combining different materials and engineered asymmetric
superconducting devices have been proposed. Yet, ease of fabrication and
tunable sign of supercurrent rectification joined to large efficiency have not
been assessed in a single platform so far. We demonstrate that all-metallic
superconducting Dayem nanobridges naturally exhibit nonreciprocal
supercurrents under an external magnetic field, with a rectification
efficiency up to $\sim 27\%$. Our niobium nanostructures are tailored so that
the diode polarity can be tuned by varying the amplitude of an out-of-plane
magnetic field or the temperature in a regime without magnetic screening. We
show that sign reversal of the diode effect may arise from the high-harmonic
content of the current phase relation in combination with vortex phase
windings present in the bridge or an anomalous phase shift compatible with
anisotropic spin-orbit interactions.

Introduction
Non-reciprocal charge transport is an essential element in modern electronics
as a building block for multiple components such as rectifiers,
photodetectors, and logic circuits. For instance, pn-junctions and Schottky-
barrier devices are archetypal semiconductor-based examples of systems, known
as diodes, with direction-selective charge propagation. Their operation stems
from the spatial asymmetry of the heterojunction that provides inversion
symmetry breaking. Likewise, dissipationless rectification refers to the
asymmetric switching of the critical current $(I_{sw})$ required to turn a
superconductor into the normal state depending on the current bias polarity.
Breaking both inversion and time-reversal symmetry, which are preserved in
conventional superconductors, is the foundational aspect to enable the diode
effect, as recently observed in superconducting materials wak17 ; mer23 ;
lin22 ; paolucci2023gate  and heterostructures and20 ; pal22 ; baur22 ; bau22
; jeo22 ; pal22 ; sun23 ; lot23 ; tur22 . Recent experimental findings have
boosted a number of theoretical investigations in superconductors dai22 ;
ber22 ; yua22 ; he22  and Josephson junctions (JJs) mis21 ; dav22 ; zha22 . In
particular, several mechanisms have been proposed to account for the
supercurrent diode effect (SDE). On the one hand, those based on intrinsic
depairing currents focus on finite momentum pairing that arise from the
combination of spin-orbit coupling (SOC) and Zeeman field dai22 ; ber22 ;
yua22 ; he22 ; Scammell_2022 , or from Meissner currents dav22 . On the other
hand, other works underline the role of Abrikosov vortices, magnetic fluxes,
screening currents or self-field effects as key elements for setting out non-
reciprocal charge transport in superconductors wambaugh99 ; vodolazov05 ;
villegas05 ; devondel05 ; car09 ; cerbu_2013 , such as in systems with trapped
Abrikosov vortices gol21 ; lyu21 ; gol22  or in micron-sized Nb-based strips
sur22 ; cha23 ; hou22 ; sat23 ; vav13 .

Till now, most of the research efforts have aimed to the realization of a SDE
that maximizes the rectification efficiency, while the change of its polarity
has been reported in a few cases only lot23 ; pal22 ; cos22 ; sun23 ;
kawarazaki2022 ; Gupta2023 . The SDE sign reversal has been interpreted as a
consequence of finite momentum pairing pal22 ; cos22 ; kawarazaki2022 ; lot23
requiring in-plane magnetic fields or diamagnetic currents and Josephson
vortices gol22 ; sun23 , as well as ascribed to vortex ratchet and asymmetric
pinning effects Gillijns07 ; He19 ; Ideue20 ; Ji21 . All these outcomes
suggest the need for an effective mastering of the polarity change of the SDE
and its implementation in a simple and monolithic platform suitable for
nanoscale miniaturization, not accomplished yet.

Fig. 1: Nb Dayem nanobridge diodes and basic electrical characterization. a,
Samples and schematic setup to measure the voltage characteristics $V$ as a
function of a bias current $I$ with an applied out-of-plane magnetic field
$B_{z}$. In the upper part, scanning electron micrographs of two weak links
with different length $l$ and width $w$: one is a constriction of Nb strip
(with characteristic dimensions $l\sim 80$ nm and $w\sim 180$ nm), the other
is a quasi-1D wire ($l\sim 1$ $\mu$m, $w\sim 80$ nm) connecting the banks;
they are labeled as short and long, respectively. The electrodes used in the
experiment are false-colored in orange. b, Temperature dependence of the zero-
bias resistance for the two Dayem bridges. Thin film $(T_{TF})$ and weak link
$(T_{c})$ critical temperatures are marked by dashed lines for the short
device. c, Temperature $(T)$ dependence of the switching supercurrents for the
short (blue dots) and the long (black dots) bridges. Red dashed lines are the
fit to Bardeen equation, as described in the text. $IV$ characteristics of the
short, d, and long bridge, e. The curves are vertically offset for clarity,
and the superconducting region is highlighted in grey to visualize the
temperature-induced decay of the dissipationless current. Black arrows
indicate the direction of the bias current swept back and forth starting at
zero amplitude.

Here, we experimentally demonstrate a sign reversal tunable SDE in elemental
superconducting weak links made on niobium (Nb). Nano-sized constrictions of
Nb realize Dayem bridges whose switching currents for positive and negative
sweep direction, $I^{+}_{sw}$ and $I^{-}_{sw}$, respectively, differ in the
absolute value. This difference can be tuned both in amplitude and sign by an
out-of-plane magnetic field $(B_{z})$, without inverting the polarity of
$B_{z}$. Thermal effects can lead to two different energy scales for the
maximal amplitude and the sign reversal of the diode efficiency which is
inconsistent with physical scenarios related to Meissner currents or self-
field effects. We show that sign reversal of the non-reciprocal response may
arise from the phase shift due to the vortex phase winding or from spin-orbit
effects due to the material granularity, in either case jointly with a few-
harmonic content of the current-phase relation (CPR) of the weak link.

Fig. 2: Diode effect in a short Dayem bridge. a, Out-of-plane magnetic field
dependence of the switching current for positive $I_{sw}^{+}$(green dots) and
negative $|I_{sw}^{-}|$(orange dots) bias current recorded at 300 mK. b,
$\Delta I_{sw}$ obtained from a. The magnetic field values at which the
maximum $(B_{max})$ and sign reversal $(B_{R})$ of the rectification occur are
marked by red and black dashed lines, respectively. c, $IV$ curves with
positive $I_{sw}^{+}>|I_{sw}^{-}|$ and negative $I_{sw}^{+}<|I_{sw}^{-}|$
rectification recorded at the magnetic fields marked by bars in panel a. d,
Color plot of the rectification efficiency $\eta(B_{z},T)$ versus bath
temperature and magnetic field. e, Rectification parameters:
$\eta_{max}\equiv(\eta_{max}(B_{z}>0)+\eta_{max}(B_{z}<0))/2$ (green dots),
field-to-rectification efficiency transfer function $\Gamma$ (blue dots) (i),
$B_{max}$ (red dots) and $B_{R}$ (black dots) magnetic fields (ii) versus
normalized temperature. $T_{c}^{S}$ denotes the critical temperature of the
short bridge. $\eta_{max}$ is the rectification value of the low-field peak at
$B_{max}$. f, $\eta(B_{z})$ for selected values of bath temperature marked by
dashed lines in panel d. Curves are vertically offset for clarity.
$\eta(B_{z})$ function exhibits two extrema, below and above $B_{R}$. The
rectification peak $\eta_{max}$, identified by $B_{max}$, decreases in
magnitude and field for $T\geq$ 1.75 K = 0.4 $T_{c}^{S}$, until the minimum in
the rectification at $B>B_{R}$ becomes an absolute extremum at T = 3 K (orange
dots). For discussion, we keep $B_{max}$ and $\eta_{max}$ as the nomenclature
for the maximum rectification.

Results
Metallic diode architectures. We analyze two different geometries of Nb Dayem
bridges, i.e., weak links made of a constant-thickness and all-metallic
constriction between two superconducting banks lik79 . The schematics of the
electronic circuitry and false-color scanning electron micrographs of the
devices are shown in Fig. 1a. In the first type of samples, 25-nm-thick
micrometer-wide banks are connected via a link whose length $l$ is $\sim 80$
nm and width $w\sim 180$ nm. The second type consists of 55-nm-thick banks
connected via a quasi-one-dimensional wire with $l\simeq 1$ $\mu$m, and
$w\simeq 80$ nm. Hereafter, we shall refer to the first and second types of
bridges as short and long, respectively. Both device families are patterned
through a single electron-beam lithography step followed by sputter deposition
of the Nb thin film and lift-off. A 4-nm-thick Ti layer is pre-sputtered for
adhesion purposes.

The differential resistance $R=dV/dI$ versus temperature $T$ of two
representative bridges is shown in Fig. 1b. The first abrupt reduction of $R$
marks the critical temperature of the Nb films $T_{TF}\simeq 8.1(7.9)$ K for
the 55(25)-nm-thick sample, suggesting minor impact of the film thickness on
the superconducting state. The resistance drops to zero at the critical
temperature of the weak link $(T_{c})$, which strongly depends, along with its
normal-state resistance $R_{N}$, on the geometry lik79 . While the short
bridge exhibits $R_{N}\sim$ 40 $\Omega$, in the long one has $R_{N}\sim$ 270
$\Omega$.

Below $T_{c}$, dissipationless transport occurs in the bridges owing to Cooper
pairs supercurrent. The temperature dependence of the switching current
$I_{sw}$ of both devices is displayed in Fig 1c. From the fit to the Bardeen
equation $I_{sw}(T)=I_{sw}^{0}[1-(\frac{T}{T_{c}})^{2}]^{\frac{3}{2}}$, we
extract a zero-temperature switching current $I_{sw}^{0}\simeq 720$ $\mu$A and
a critical temperature $T_{c}^{S}\simeq 4.3$ K for the shor bridge. Similarly,
for the long weak link we obtain $I_{sw}^{0}\simeq 42\,\mu$A and
$T_{c}^{L}\simeq 2.1$ K. From these values, we determine a zero-temperature
BCS energy gap $\Delta_{0}$=1.764 $k_{B}T_{c}^{S(L)}\simeq$ 650(320) $\mu$eV
for the short(long) bridge, where $k_{B}$ is the Boltzmann constant. For the
long bridge, we deduce a superconducting coherence length $\xi_{0}=\sqrt{\hbar
l/(R_{N}wte^{2}N_{F}\Delta_{0})}\simeq$ 11 nm, where $t$ is the film
thickness, $N_{F}\simeq 5.33\times 10^{47}J^{-1}m^{-3}$ is the density of
states at the Fermi level of Nb jan88 , and $e$ is the electron charge.
Similarly, we can evaluate the London penetration depth
$\lambda_{L}=\sqrt{\hbar R_{N}wt/(\pi l\mu_{0}\Delta_{0})}\simeq$ 790 nm,
where $\mu_{0}$ is the vacuum magnetic permeability. Since
$w,t\ll\lambda_{L}$, the bridges can be uniformly penetrated by an external
magnetic field.

The current vs voltage $(IV)$ characteristics of the short and long bridges
are shown in Fig.1d and e, respectively, for selected values of bath
temperature. The devices show an abrupt transition to the normal state at the
switching current $I_{sw}$ and display the typical hysteresis of metallic
junctions which originates from Joule heating induced in the bridge when the
bias current is swept back from the resistive to the dissipationless state
sko76 .

Fig. 3: Diode effect in a long Dayem bridge. a, Magnetic field dependence of
the switching current for positive $I_{sw}^{+}$(green dots) and negative
$|I_{sw}^{-}|$(orange dots) bias current recorded at 50 mK. b, $\Delta I_{sw}$
obtained from panel a. The rectification efficiency increases linearly in
$B_{z}$ until $|B_{z}|\simeq 015$ T. Red and black dashed lines mark the
magnetic field values corresponding to sign reversal $(B_{R})$ and maximum
rectification $(B_{max})$, respectively. Inset: Blow-up of $\Delta I_{sw}$ for
large positive (black) and negative (brown) fields displaying several changes
of signs. c, $IV$ characteristics with negative $I_{sw}^{+}<|I_{sw}^{-}|$ and
positive $I_{sw}^{+}>|I_{sw}^{-}|$ rectification recorded at the magnetic
fields marked by bars in panel a. d, Color plot of the rectification
efficiency as a function of temperature and magnetic field, $\eta(B_{z},T)$.
Dashed lines are guides for the eye to highlight the different temperature
trends in $B_{max}$ and $B_{R}$. e, Rectification parameters:
$\eta_{max}\equiv(\eta_{max}(B_{z}>0)+\eta_{max}(B_{z}<0))/2$ (green dots),
field-to-rectification efficiency transfer function $\Gamma$ (blue dots) (i),
$B_{max}$ (red dots) and $B_{R}$ (black dots) magnetic fields (ii) versus
normalized temperature. $T_{c}^{L}$ denotes the critical temperature of the
long bridge. $\eta_{max}$ is the rectification value of the low-field peak at
$B_{max}$. f $\eta_{B_{z}}$ for selected bath temperatures marked by dashed
lines in panel d. Curves are vertically offset for clarity.

Short Dayem bridge diode performance. Let us now discuss how the short Dayem
bridge in Fig.1a can be used as a supercurrent diode. Non-reciprocal
dissipationless transport is revealed by comparing the switching currents
while sweeping the biasing current from zero to positive values ($I_{sw}^{+}$)
or from zero to negative values ($I_{sw}^{-}$) in the presence of an out-of-
plane magnetic field $B_{z}$. The switching currents at $T=0.3$ K are reported
in Fig. 2a. The magnetic field increasingly reduces the superconducting gap
and thereby both the switching currents. A linear decrease in $B_{z}$ of
$I_{sw}^{+}$ and $|I_{sw}^{-}|$ is observed up to $\sim 0.07$ T. At larger
fields, the dependence of the switching currents on $B_{z}$ is sublinear.
Notably, both $I_{sw}(B_{z})$ are not antisymmetric with respect to the
magnetic field ($I_{sw}(B_{z})\neq I_{sw}(-B_{z})$) while the symmetry
relation $I_{sw}^{+}(B_{z})\simeq-I_{sw}^{-}(-B_{z})$ is respected, within the
small experimental fluctuations, as theoretically expected. This symmetry
relation is further confirmed in the switching currents difference $\Delta
I_{sw}\equiv I_{sw}^{+}-|I_{sw}^{-}|$ displaying an odd-in-$B_{z}$
superconducting diode effect ($\Delta I_{sw}(B_{z})\simeq-\Delta
I_{sw}(-B_{z})$) as shown in Figure 2b. $\Delta I_{sw}$ is characterized by a
maximum at $B_{max}\simeq 0.05$ T and a sign inversion at $B_{R}\simeq 0.1$ T
where $I_{sw}^{+}$ and $|I_{sw}^{-}|$ have a crossing (see Fig. 2a). From now
on, $B_{max}$ indicates the position in field of the rectification peak. Two
$IV$ curves, recorded for magnetic fields lower and larger than $B_{R}$, are
plotted in Fig. 2c to emphasize the sign change in the rectification.

Nonreciprocal transport can be conveniently quantified by the rectification
efficiency defined as
$\eta=\frac{I_{sw}^{+}-|I_{sw}^{-}|}{I_{sw}^{+}+|I_{sw}^{-}|}$. Figure 2d
shows the evolution of $\eta$ versus $B_{z}$ and $T$. $\eta(B_{z})$ is
substantially unaffected by thermal effects up to $T\simeq 1.75$ K
$=0.41\,T_{c}^{S}$ where a maximum rectification $\eta_{max}\sim 27\%$ is
obtained. The evolution of $\eta_{max}$ in temperature is displayed in Fig. 2e
(i). In addition, we parametrize the diode sensitivity to the magnetic field
in the vicinity of the abrupt sign change as
$\Gamma=\eta_{max}/(|B_{max}-B_{R}|)$. A maximum value $\Gamma\sim 650$ T-1 is
achieved around $2.25$ K (see Fig. 2e (i)) At higher temperatures, the
quantities $\eta_{max}$, $\Gamma$, and the characteristic magnetic fields
$B_{max}$ and $B_{R}$ related to rectification (see Fig. 2e (ii)), all
decrease in a similar fashion. The full profile of the rectification
efficiency versus $B_{z}$ is better visualized in Fig. 2f where $\eta(B_{z})$
is plotted for a few selected values of temperature.

Long Dayem bridge diode performance. Next, we characterize the long nanobridge
shown in Fig. 1a. Figure 3a reports the decay of $I_{sw}^{+}$ and
$|I_{sw}^{-}|$ as a function of $B_{z}$. At first, we notice that the
switching currents are damped down to $\sim 60$% of their zero-field value at
$B_{z}\simeq 0.3$ T, whereas in the previous sample, the same damping is
achieved for lower fields ($B_{z}\simeq 0.07$ T see Fig. 2a). Figure 3b
displays $\Delta I_{sw}$ versus $B_{z}$. For low magnetic fields, $\Delta
I_{sw}(B_{z})$ exhibits a linear relation. While increasing $|B_{z}|$ further,
$\Delta I_{sw}$ bends and then inverts its trend: an abrupt jump realizes a
sign reversal at $|B_{R}|\sim 0.34$ T. Then, a relative peak at
$|B_{max}|\simeq 0.38$ T, marked by a red dashed line, represents the field at
which maximum rectification efficiency is achieved as before. Next to the sign
change, $\Delta I_{sw}$ leaves the clean trend and looks noisy. Such small
jumps are reproducible, thus ruling out a stochastic nature of the underlying
processes. Finally, $\Delta I_{sw}$ oscillates at higher magnetic fields, as
shown in the inset of Fig. 3b. Two $IV$ curves, for fields lower and larger
than $B_{R}$, are plotted in Fig. 3c to highlight that the rectification sign
changes from negative to positive as the field $B_{z}>0$ increases contrary to
the short bridge. This change in symmetry is attributed to the vortex
nucleation as discussed later.

The magnetic field and temperature dependence of the rectification efficiency
$\eta$ is presented as a color plot in Fig. 3d. The sign change and the
maximum rectification are affected by temperature in a different way as
compared to the short constriction. The linear increase of the rectification
at low fields smears out with temperature, reducing $B_{R}$ until it vanishes
at $T\simeq 1.1$ K $=0.5\,T_{c}^{L}$. Rectification parameters displayed in
Fig. 3e shows that $B_{max}$ is more robust in temperature than $B_{R}$: it is
still observable at $T\simeq 1.8$ K $=0.8\,T_{c}^{L}$. The sudden change of
sign is quantified by a maximum $\Gamma\sim 360$ T-1 at $0.15$ K. As before,
the profile of rectification efficiency as a function of $B_{z}$ is shown in
Fig. 3f for a few selected values of temperature. The difference in the
temperature trend between $B_{R}$ and $B_{max}$ (see Fig. 3e (ii)) suggests
two different energy scales responsible for the sign reversal and the maximum
rectification, as confirmed by measurements obtained in other devices with
same nominal dimensions. (see Supplementary Figure 1). Rectification on the
second sample exhibits similar $\eta(B_{z})$ lineshape with almost identical
$B_{max}$ and $\eta_{max}$ values and temperature dependence. In this sample,
low-field features fade more rapidly with temperature, which appears to be
sample-dependent.

Fig. 4: Modelling supercurrent across a nanobridge for sign-tunable diode
effect. a, Sketch of the theoretical framework for model I, with the arrows
indicating the phase winding associated with a vortex nucleated near the grain
boundary. Characteristic CPR (red line) of the weak link hosting a vortex
close to the grain boundary. b, Non-reciprocal rectification efficiency $\eta$
calculated for a few values of the second harmonic at a given position of the
vortex core with $(x_{v},y_{v})=(0.4\,w,0.4\,w)$ and $\gamma_{v}=1$. c, Non-
reciprocal rectification efficiency $\eta$ for different positions $(x_{v})$
of the vortex with $y_{v}=0.4w$ and $I_{2,0}=0.2$. Variation of the vortex
position leads to substantial modifications in $\eta$ at low magnetic fields.
d, Sketch of the nanobridge, with SS’S indicating the regions with different
amplitudes of the superconducting gap. Here we assume the presence of Rashba
and Dresselhaus spin-orbit couplings. Representative skewed and asymmetric CPR
(red line) originated by high-harmonic components (up to the third one) and an
anomalous phase offset $\varphi_{0}$. e, Non-reciprocal rectification
efficiency $\eta$ calculated for several values of the third harmonic
component $I_{3,0}$, assuming that $I_{1,0}=1$, $I_{2,0}=-0.3$ and
$\Gamma_{B}=0.2B/B^{*}$. $\eta$ changes sign after maximum rectification is
reached. f, Impact of the fourth harmonic in the rectification, with
$I_{1,0}=1$, $I_{2,0}=-0.3$ and $I_{3,0}=0.25$.

Modeling the sign reversal of the diode effect. Previous SDE reported in
conventional superconductor are normally attributed to Abrikosov vortices and
self-field effects vav13 ; gol22  or Meissner currents sur22 ; cha23 ; hou22 ;
sat23 . We have neglected those mechanisms since they are weak in amplitude.
Such conclusion can be directly deduced from the stark contrast of the profile
of the switching current vs B-field data points. In both scenarios
$I_{sw}(B_{z})$ reaches a maximum at nonzero magnetic fields. Moreover,
amplitude asymmetry does not change sign as a function of the magnetic field
without reversing its orientation before the critical current vanishes.
Instead, the switching current-field profile in our devices decays
monotonically at low fields with no signature of the characteristic skewness.
For this reason, we propose two physical scenarios compatible with our devices
that may explain our experimental findings. Both of them rely on non-
sinusoidal CPRs, typical of superconducting nanobridges lik79 , combined with
an inversion-symmetry breaker. In model I, this is represented by a
supercurrent vortex, while in model II by spin-orbit couplings. An out-of-
plane magnetic field parametrized by $B^{*}$ is considered, where $B^{*}$ is
given by the zero rectification field $\eta(B^{*})=0$ and by the vanishing of
high-harmonic amplitude.

In model I, the Dayem nanobridge is schematized as a one-dimensional chain of
weak links of width $w$ formed by the Nb grains. A supercurrent vortex can
nucleate in one of these weak links Aranson1994 ; Aranson_1996 , as sketched
in Fig. 4a, which induces a phase winding in the superconducting order
parameter. These vortices have a typical size of the order of $\xi$, so only a
few of them can be accommodated within the bridge. Notice that such vorticity
is not a screening of the $B$-field, since the small dimensions of the bridge
($w\ll\lambda_{L}$) allow full penetration of $B_{z}$. In this framework, the
CPR is affected by two phase shifts: the conventional vector potential
associated with $B_{z}$ and the phase winding of the vortex. It is indeed the
interplay of these two contributions that is responsible for a sign change in
the rectification parameter. Though on a different length scale, this physical
scenario is similar to that of Josephson phase vortices krasnov20 ; gol22 .
The rectification parameter $\eta$ is then evaluated by determining the
maximum and minimum values of the Josephson current with respect to the phase
bias, see Methods for details.

Figure 4b, reports the evolution of $\eta$ in $B$ for different amplitudes of
the second harmonic, $I_{2}$, of the CPR. The magnitude of $\eta$ scales with
$I_{2}$ showing multiple nodes, whose position in $B$ is independent of
$I_{2}$. The sign change also depends on the position of the vortex, as
displayed in Fig. 4c, where $\eta(B)$ is evaluated for a vortex nucleated at
different distances from the lateral edge of the bridge ($x_{\nu}$). This
behavior suggests a phase-shift competition dominated at low fields by the
vortex phase slip and at large fields by the vector potential.

Another scenario able to describe the sign reversal in the diode rectification
can be envisioned by combining the colored CPR of the nanobridge with an
anomalous phase shift ass19 ; str20 ; may20 ; mar23 ; ber22  induced by spin-
orbit interactions and magnetic fields. Nb is a heavy metal and thus possesses
an atomistic spin-orbit interaction of the order of $\sim 100$ meV mac69 . We
extend such concept to our weak links where we call for a spin-orbit coupling
of the order of a few meV, i.e., 2 orders of magnitude weaker than the atomic
limit gul22 . In particular, we can expect that mirror symmetry can be locally
or globally broken in polycrystalline films Park19 , thereby leading to spin-
orbit interaction of both Rashba and Dresselhaus types (see Methods). Figure
4d shows a sketch of the bridge modeled as an effective $SS^{\prime}S$
structure, where the $S$ and $S^{\prime}$ components have different amplitudes
of the superconducting gap and different spin-orbit couplings breaking
horizontal and vertical mirror symmetries. The anisotropic spin-orbit
interaction generates an anomalous phase shift in the CPR that varies with the
magnetic field, as explicitly shown in Supplementary Figure 2. The anomalous
phase is then introduced in the CPR via a phenomenological parameter
$\Gamma_{B}$ providing a first-order cosine component in the Fourier
expansion, i.e., $I=\sum_{n}I_{n}\sin(\varphi)+\Gamma_{B}\cos(\varphi)$ (see
Methods for details). The anomalous phase $\varphi_{0}$ is then related to the
amplitude of $\Gamma_{B}$. We assume a linear damping of the harmonics
$I_{n}=I_{n,0}(1-B/B^{*})$, which defines the scale parameter $B^{*}$.

From model II, the diode sign reversal takes place only in the presence of a
sizable third harmonic component. Figure 4e reports $\eta(B)$ for some values
of $I_{3,0}$. By increasing $I_{3,0}$, the sign inversion gets more
pronounced, whereas the maxima and minima of the rectification
($B_{max}^{s}\simeq 0.74B^{*}$, $B_{min}^{s}\simeq 0.89B^{*}$) are barely
affected by the weight of the harmonic. Moreover, by including more harmonics
in the CPR, the lineshape of $\eta(B)$ is modified. For instance, Figure 4f
shows that a fourth-order harmonic affects the magnetic field dependence by
substantially removing the sign change.

Discussion
The comparison between our experimental findings and the proposed models
reveals some important features supporting the proposed mechanisms. In
particular, for both bridges $\eta$ has an almost monotonic damping in
temperature that can be explained in both models by the reduction of high-
order harmonics. This is expected in long metallic weak links where the CPR
evolves from highly distorted to sinusoidal-like shapes at large temperatures
lik79 ; gol04 . Moreover, as shown in Figs. 2e and 3e, $B_{max}$ is
temperature resilient until $T\geq 0.5\,T_{c}$. This feature is fairly
captured by both models, as the maximum rectification looks almost independent
of the harmonic content (see Fig. 4b for model I and Fig. 4e for model II).
However, long bridges exhibit features that are mostly accounted by model I,
while short ones are more compatible with model II. For example, in long
(short) bridges the sign reversal is present below (above) $B_{max}$ as shown
in Fig. 3b and Fig. 2b to be compared with Fig. 4b and e, respectively.
Multiple sign reversal nodes appear in high fields only for long bridges, as
shown in Fig. 3b and well described by the interferometric mechanism of model
I. While the rectification lineshape given by model II presents only one
inversion node. Moreover, the quick damping of the rectification inversion
observed at low fields ($<$ 0.3 T) in Fig. 3f is captured by the vortex
dynamics described in Fig. 4c. The relative size of the vortex $\xi/w$ is
temperature-dependent and influences the vortex position in the bridge. Thus,
it is plausible to expect that variations of the vortex size mostly affect the
rectification lineshape at low fields while remaining substantially unchanged
for larger fields, as shown in Fig.4c.

Finally, it is interesting to note that by extending the proposed models to
in-plane magnetic fields, a sizable supercurrent rectification is anticipated
but without sign reversal. In particular, for model I, no phase shift is
expected from the spatial dependence of the vector potential, since the
orbital coupling between an in-plane field and the electron momentum becomes
negligible. Thus, the source of phase interference with the vortex winding is
eliminated. For model II, the anomalous phase and the harmonic content would
be differently affected by an in-plane Zeeman field compared to the out-of-
plane orientation not resulting in sign reversal.

Conclusions
In summary, we have demonstrated the implementation of supercurrent diodes in
Nb Dayem nanobridges. By breaking the time-reversal symmetry with an out-of-
plane magnetic field, we demonstrate that both the amplitude and sign of the
rectification can be tuned without inverting the polarity of the applied
field. The evolution of the critical current on $B$ field decays monotonically
without showing any skewness, which rules out self-field effects or Meissner
currents. We have developed two theoretical models to account for the sources
of time- and inversion-symmetry breaking, one based on a vortex phase winding,
and one that takes into account the spin-orbit interactions present in
polycrystalline heavy materials. Yet, a quantitative description of the
supercurrent diode effect in metallic nanoconstrictions should account for
both scenarios, which complement each other and can coexist. Furthermore, the
fabrication process is simple when compared to that of other platforms, a
compelling step towards scalability. Analogous nanobridges can be realized
from several elemental superconductors currently at the base of other
architectures, such as nanocryotrons mccaughan14 , rapid single-flux quanta
(RSFQ) likharev91  and memories ligato21 , which would ease a potential
integration.

Finally, the sharp sign reversal of the diode rectification allows us to
envisage applications of Dayem nanobridges as $B$-field threshold detectors.
When biased in the vicinity of the rectification node, small variations of an
environmental magnetic field would result in modifications of the sign of the
rectification parameter.

Methods

Sample fabrication. Nb strips and constrictions are patterned by e-beam
lithography on AR-P 679.04 (PMMA) resist. PMMA residuals are removed by
O2-plasma etching after developing. Nb thin films were deposited by sputtering
with a base pressure of 2 $\times$ 10-8 Torr in a 4 mTorr Ar (6N purity)
atmosphere and liftoff by acetone or AR-P 600.71 remover. A thin Ti layer was
previously sputtered to improve Nb adhesion and base pressure in the
deposition chamber, resulting in a nominal thickness Ti(4nm)/Nb(25nm) and
Ti(4nm)/Nb(55nm) for the so-called short and long nanobridges, respectively.

Transport measurements. Transport measurements were carried out in filtered
(two-stage RC and $\pi$ filters) cryogen-free 3He-4He dilution refrigerators
by a standard 4-point probe technique. DC current-voltage characteristics were
measured by sweeping a low-noise current bias positively and negatively, and
by measuring the voltage drop across the weak links with a room-temperature,
low-noise pre-amplifier for each current value every $\sim 20$ ms. Current
step size $\Delta I$ was adapted depending on the switching current keeping
the values lower than $0.002I_{sw}(B=0)\sim 1\mu$ A and 100 nA for short and
long devices, respectively. The switching currents are extracted from the
maximum of the derivative $dV/dI$ and averaged from 5-10 reiterations of the
$IV$ curves. Error bars account for the standard deviation and propagation of
errors in $I_{sw}$, $\Delta I_{sw}$, and $\eta$. For the characteristic
fields, the error is given by the magnetic field step $\Delta B_{z}<$ 10 mT.
Joule heating is minimized by automatically switching the current off once the
device turns into the normal state. A delay between sweeps was optimized to
keep the stability of the fridge temperature lower than 50 mK. Furthermore, no
changes in the switching currents (up to the accuracy given by the standard
deviation) were observed in different cooling cycles, by changing the order of
the sweeps or by adding an extra delay in the acquisition protocol concluding
that hysteretic behavior or local heating is negligible.

Theoretical models. The current phase relation for the model I is given by
$I=\sum_{n=1,2}\int_{0}^{w}I_{n}\sin\left(\phi(x)+n\varphi\right)dx$. The
critical current in both directions is evaluated by determining the maximum
and minimum values with respect to the phase bias $\varphi$. Here, the
spatial-dependent phase difference $\phi(x)$ is given by the magnetic field
and the phase winding contribution due to the vortex krasnov20 . We have
assumed that the supercurrent has a subdominant second harmonic contribution,
as expected in long, diffusive weak links lik79 ; gol04 . The spatial
dependence of the phase along the transverse direction $x$ is expressed as
$\phi(x)=\frac{2\pi d_{b}B_{z}x}{\Phi_{0}}+\phi_{v}(x)$ where $d_{b}$ is a
characteristic length of the weak link, related to the width of the junction,
and $\phi_{v}(x)=\gamma_{v}\arctan\left[\frac{(x-x_{v})}{y_{v}}\right]$ is the
spatially-inhomogeneous phase offset due to the vortex structure. $\gamma_{v}$
indicates the sign and amplitude of the winding and $(x_{v},y_{v})$ the
position of the vortex core with respect to the boundaries (see Fig. 4a). We
also assume that the second harmonic amplitude of the supercurrent is
vanishing at $B=B^{*}$.

Concerning model II, the current phase relation is given by
$I=\sum_{n}I_{n}\sin(\varphi)+\Gamma_{B}\cos(\varphi)$. The critical currents
are evaluated by determining the maximum and minimum values of the Josephson
current. For the examined model, we assume that the amplitudes of the
$n$-harmonic $I_{n}$ get suppressed by $B$ with a linear rate, i.e.
$I_{n}=I_{n,0}(1-\frac{B}{B^{*}})$ for $n\geq 1$. The linear trend is
compatible with the observed behavior of the overall supercurrent amplitude as
shown in Fig. 2a and 3a for the range of applied field where the rectification
is nonvanishing. Furthermore, the suppression of the amplitude of the
harmonics with the magnetic field can be related to the reduction of the
effective transmission across the grains due to depairing and magnetic
interference. The decrease in the transmission implies a reduction of the non-
harmonic amplitudes lik79 ; gol04 . In this model, we have performed a real
space simulation for the examined geometry. By solving the Bogoliubov-de
Gennes equations on a finite-size slab in the presence of an out-of-plane
magnetic field, we demonstrate that an anomalous phase shift can be obtained.
The simulation is performed for a system size $N_{x}\times N_{y}$ with
$N_{x}=150$ and $N_{y}=100$. The employed tight-binding model includes a
nearest neighbour hopping amplitude $t$, and the conventional spin-singlet
local pairing amplitude. We apply a phase bias across the weak link and
determine the free energy as shown in the Supplementary Figure 2a. The
resulting anomalous phase increases with the magnetic field and depends on the
strength of the Rashba and Dresselhaus interactions as shown in the
Supplementary Figure 2b. The linear Rashba term on a lattice for a two-
dimensional geometry is expressed as
$H_{R}=\alpha_{R}[\sin(k_{x})\sigma_{y}-\sin(k_{y})\sigma_{x}]$ while the
Dresselhaus term is given by
$H_{D}=\alpha_{D}[\sin(k_{x})\sigma_{x}-\sin(k_{y})\sigma_{y}]$ with
$\sigma_{i}(i=x,y)$ being the Pauli matrices associated with the spin angular
momentum. We notice that $H_{R}$ breaks the horizontal mirror symmetry while
$H_{D}$ breaks both the vertical and horizontal mirror symmetries. This
reduction of mirror symmetry is expected to be locally or globally broken in
granular films Park19 . Notably, the presence of the Dresselhaus term is
crucial to induce an anomalous phase shift in the supercurrent in the presence
of an out-of-plane magnetic field.

Concerning the size of the Rashba and Dresselhaus couplings, our analysis
shows a sizable, nonvanishing anomalous phase in the presence of an out-of-
plane magnetic field for values from 0.01 to 0.3, in units of the effective
electron hopping amplitude $t$. We notice that $t$ is an effective electronic
amplitude that describes the low energy properties of the superconductor which
can be taken in the order of 20 meV. For such value, Rashba/Dresselhaus
couplings able to generate the anomalous phase will range from 0.2 to 6 meV.
Although there is no experimental estimations of Rashba and Dresselhaus
coupling for Nb thin films, these values are plausible for heavy metals like
Nb, since Rashba spin splitting scales with the size of the atomic spin-orbit
coupling Sunko2017 ; Bihlmayer2022 . Nb has a multi-orbital electronic
structure Russmann2022 , in this case, the breaking of the inversion or mirror
symmetry primarily leads to an orbital Rashba coupling Park2011 ; Mercaldo2020
among orbitals with different mirror parity. Then, the spin Rashba interaction
arises through the atomic spin-orbit coupling. The large amplitude of the
atomic spin-orbit coupling in Nb can thus yield a sizable spin Rashba
interaction Sunko2017 ; Ishizaka2011 ; Yaji2010 ; Bihlmayer2022 .

Data availability
The data that support the findings of this study are available from the
corresponding author upon reasonable request.

Code availability
The code that support the findings of this study are available from the
corresponding author upon reasonable request.

References

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Acknowledgments
This work was funded by the EU’s Horizon 2020 Research and Innovation
Framework Program under Grant Agreement No. 964398 (SUPERGATE), No. 101057977
(SPECTRUM), and by the PNRR MUR project PE0000023-NQSTI.

Author contributions
D.M. fabricated the samples, conducted the experiments, and analyzed data with
inputs from A.C., E.S. and F.G. The theoretical models describing the
experiment was developed by Y.F., M.T.M., and M.C. The manuscript was written
by D.M., A.C., E.S., M.C., and F.G. with inputs from all the authors. E.S. and
F.G. conceived the experiment. F.G. supervised and coordinated the project.
All authors discussed the results and their implications equally at all
stages.

Competing interests
The authors declare no competing interests.

Additional Information
Supplementary Information The online version contains supplementary material.

Correspondence and requests for materials should be addressed to Daniel
Margineda or Francesco Giazotto.