# Deep neural network loses attention to adversarial images

Shashank Kotyan  Danilo Vasconcellos Vargas Kyushu University
<EMAIL_ADDRESS>
###### Abstract

Adversarial algorithms have shown to be effective against neural networks for
a variety of tasks. Some adversarial algorithms perturb all the pixels in the
image minimally for the image classification task in image classification. In
contrast, some algorithms perturb few pixels strongly. However, very little
information is available regarding why these adversarial samples so diverse
from each other exist. Recently, Vargas and Su (2020) showed that the
existence of these adversarial samples might be due to conflicting saliency
within the neural network. We test this hypothesis of conflicting saliency by
analysing the Saliency Maps (SM) and Gradient-weighted Class Activation Maps
(Grad-CAM) of original and few different types of adversarial samples. We also
analyse how different adversarial samples distort the attention of the neural
network compared to original samples. We show that in the case of Pixel
Attack, perturbed pixels either calls the network attention to themselves or
divert the attention from them. Simultaneously, the Projected Gradient Descent
Attack perturbs pixels so that intermediate layers inside the neural network
lose attention for the correct class. We also show that both attacks affect
the saliency map and activation maps differently. Thus, shedding light on why
some defences successful against some attacks remain vulnerable against other
attacks. We hope that this analysis will improve understanding of the
existence and the effect of adversarial samples and enable the community to
develop more robust neural networks. ††Copyright © 2021 for this paper by its
authors. Use permitted under Creative Commons License Attribution 4.0
International (CC BY 4.0).

##  1 Introduction

Figure 1: Illustration of the methodology to generate Saliency Maps with
respect to the predicted class for original and adversarial images.

Since adversarial samples were discovered some years ago in Szegedy (2014) for
neural networks, the variety of adversarial samples and corresponding
adversarial algorithms has grown in both number and types. From Universal
perturbations, Moosavi-Dezfooli et al. (2017) that can be added to almost any
image to generate an adversarial sample, to the addition of crafted patches
Brown et al. (2017) or in fact, even the addition of one-pixel Su et al.
(2019) was also shown to cause networks to be enough to misclassify. Some
approaches rely on detecting adversarial samples to mitigate the adverse
effects of adversarial algorithms, while some approaches rely on defensive
algorithms. However, most defences rely on obfuscating gradients Athalye et
al. (2018) which can be broken by black-box and stronger attacks.

It was shown in Vargas and Su (2020) that changes in pixels of an image
propagate and expand throughout the layers to either disappear or cause
significant changes in the classification. Additionally, it was also shown
that perturbation in nearby pixels of successful one-pixel attack has high
attack accuracy. This suggests that changes in a pixel may increase or
decrease the influence of a receptive field (small group of nearby pixels).
This is a direct relationship of the convolution, which is a linear operation.

In the adversarial setting, the analysis of the spatial distribution of
saliency proves helpful to interpret why changing some pixels Su et al. (2019)
in the network corresponds to misclassification. It was hypothesised in Vargas
and Su (2020) that the existence of adversarial samples is due to conflicting
saliency, which causes enough disturbance in the neural network forcing it to
misclassify. Hence, adversarial samples are not naively fooling neural
networks but diverting their attention towards another part of the image.

Contributions: In this article, we analyse how these spatial distribution
changes based on perturbed pixels. Figure 1 shows the illustration of the
methodology of generating saliency maps and the difference in saliency caused
by adversarial perturbations. We analyse the spatial distribution changes
caused by perturbing pixels by evaluating Saliency Maps and Gradient-weighted
Class Activation Maps. We use ResNet trained on CIFAR-10 for our evaluation.
We also assess the effect for adversarial samples generated by Pixel Attack
(an $L_{0}$ norm black box attack) Su et al. (2019), and Projected Gradient
Descent Attack (an $L_{\infty}$ norm white-box attack) Madry et al. (2018).

We evaluate the Saliency Maps and Gradient-weighted Class Activation Maps
concerning the predicted class, the true class and the misclassified class of
both original and adversarial images. Our experiments reveal that both Pixel
Attack and Projected Gradient Descent Attack distorts the saliency maps and
activation maps differently. Where the Pixel Attack calls the network’s
attention towards the perturbed pixels, effectively changing the region of
interest for the neural networks. The Projected Gradient Descent Attack
diffuses the image’s saliency and effectively destroys the class activation
maps for true class for the adversarial images.

We also investigate the hypothesis that adversarial samples exist due to a
conflicting image saliency raised in Vargas and Su (2020). Our experiments
demonstrate that pixels define both the strength as well as the type of
feature (i.e., changing pixels change the relationship between pixels of a
receptive field and, therefore, the feature). Therefore, it is possible to
either destroy a feature or decrease its intensity by modifying the pixels
slightly. It follows that the same should be valid for modifying pixels
slightly all over the image (in this case, however, it is natural that the
relative modification is more important).

##  2 Related Works

###  2.1 Adversarial Machine Learning

It was exhibited in Szegedy (2014) that neural networks behave oddly for
almost the same images. Afterwards, in Nguyen et al. (2015), the authors
demonstrated that neural networks show high confidence when presented with
textures and random noise. This led to discovering a series of vulnerabilities
in neural networks, which were then exploited by adversarial attacks.

Defensive distillation Papernot et al. (2016), a defence was proposed, in
which a smaller neural network squeezes the content learned by the original
one was proposed as a defence. However, it was shown not to be robust enough
in Carlini and Wagner (2017). Adversarial training was also proposed, in which
adversarial samples are used to augment the training dataset Goodfellow et al.
(2014), Madry et al. (2018). Augmentation of the dataset is done so that the
neural network should classify the adversarial samples, thus increasing their
robustness. Although adversarial training can increase the robustness
slightly, the resulting neural network is still vulnerable to attacks Tramèr
et al. (2018).

Regarding understanding the phenomenon, it is argued in Goodfellow et al.
(2014) that neural networks’ linearity is one of the main reasons. Another
investigation proposes the conflicting saliency added by adversarial samples
as the reason for misclassification Vargas and Su (2020). A geometric
perspective is analysed in Moosavi-Dezfooli et al. (2018), where it is shown
that adversarial samples lie in shared subspace, along which the decision
boundary of a classifier is positively curved. Further, in Fawzi et al.
(2018), a relationship between sensitivity to additive perturbations of the
inputs and the curvature of the decision boundary of deep networks is shown.
Another aspect of robustness is discussed in Madry et al. (2018), where
authors suggest that the capacity of the neural networks’ architecture is
relevant to the robustness.

###  2.2 Visualisation and understanding of neural networks

We can visualise the attention of the neural networks to understand which part
of the image neural network focuses. It is well known that some parts of the
image affect the output more than others; in fact, the most influential areas
can be seen by plotting the saliency maps of a given sample.

In Itti et al. (1998) saliency maps are defined as maps to represent
conspicuity at every location in the visual field by a scalar quantity. These
saliency maps also help guide the selection of attended locations based on the
spatial distribution of saliency. Therefore, saliency maps contribute to
finding the pixels of the image, which contribute more than the other pixels
based on the spatial distribution of saliency to categorise the image in
neural networks.

There are two variations of the backpropagation used in saliency maps in the
literature. One is Rectified/ Deconv Backpropagation Zeiler and Fergus (2014),
where only positive gradient information is propagated through the layers,
which correspond to the increase in output. An increase in output can be
interpreted as an increase in output probability for a class in a
classification problem. Another is Guided Backpropagation Springenberg et al.
(2014), where only positive gradient information is propagated through the
layers, which have positive activation for the layers.

##  3 Adversarial Machine Learning and Saliency Maps

Let us suppose that for the image classification problem,
$x\in\mathbb{R}^{m\times n\times c}$ be the image which is to be classified.
Here $m,n$ is the width and the height of the image, and $c$ is the number of
colour channels. A neural network comprises several neural layers composed of
a set of perceptrons (artificial neurons) linked together. Each of these
perceptrons maps a set of inputs to output values with an activation function.

Thus, function of the neural network (formed by a chain) can be defined as:

$\displaystyle g(x)=f^{(k)}(\ldots f^{(2)}(f^{(1)}(x)))$ (1)

where $f^{(i)}$ is the function of the $i^{\text{th}}$ layer of the network,
where $i=1,2,3,\ldots,k$ and $k$ is the last layer of the neural network. In
the image classification problem, $g(x)\in\mathbb{R}^{N}$ is the probabilities
(confidence) for all the available $N$ classes.

Also, in adversarial machine learning, one type of adversarial samples
$\hat{x}$ are defined as:

$\displaystyle\hat{x}=x+\epsilon_{x}$ (2)
$\displaystyle\\{\hat{x}\in\mathbb{R}^{m\times n\times
3}\mid{\operatorname{argmax}}[g(x)]\neq{\operatorname{argmax}}[g(\hat{x})]\\}$

in which $\epsilon_{x}$ is the perturbation added to the input. There exists,
a wide variety of norm-constraints imposed upon $\epsilon_{x}$ such as
$L_{0}$, $L_{1}$, $L_{2}$, and $L_{\infty}$ which allows for different
adversarial attacks. $L_{0}$ norm allows attacks to perturb a few pixels
strongly, $L_{\infty}$ norm allow all pixels to change slightly, and both
$L_{1}$ and $L_{2}$ allow for a mix of both strategies.

Making use of the definition of adversarial samples, adversarial machine
learning thus, can be formally defined as the following optimization problem
for untargeted attacks:

$\displaystyle\underset{\epsilon_{x}}{\text{minimize}}$ $\displaystyle
g(x+\epsilon_{x})_{C}$ subject to $\displaystyle\|\epsilon_{x}\|_{p}\leq th$
(3)

Similarly optimization problem for the targeted attacks can be defined as:

$\displaystyle\underset{\epsilon_{x}}{\text{maximize}}$ $\displaystyle
g(x+\epsilon_{x})_{T}$ subject to $\displaystyle\|\epsilon_{x}\|_{p}\leq th$
(4)

where $g()_{C}$ is the soft-label for the correct class, $g()_{T}$ is the
soft-label for the target class, $p\in\\{0,1,2,\infty\\}$ is the constraint
norm on $\epsilon_{x}$ and $th$ is the threshold value for the constraint
norm.

Saliency Maps (pixel-attribution maps or attribution maps or sensitivity maps)
Simonyan et al. (2013) is used to assess how output changes concerning a
change in input. It can be used to determine the region of interest in the
image for neural networks. Mathematically, the saliency map,
$SM\in\mathbb{R}^{m,n}$ can be defined as:

$\displaystyle SM=\frac{\partial\text{ Output}}{\partial\text{ Input}}$ (5)

Hence, we can evaluate saliency map of an image with respect to predicted
class of the image $C$ as:

$\displaystyle SM_{C}=max_{c}\frac{\partial g(x)_{C}}{\partial x}$ (6)

Similarly, we can evaluate the saliency map of the adversarial image
concerning the predicted class of adversarial image $\hat{C}$ as:

$\displaystyle\widehat{SM}_{\hat{C}}=max_{c}\frac{\partial
g(\hat{x})_{\hat{C}}}{\partial\hat{x}}=max_{c}\frac{\partial
g(x+\epsilon_{x})_{\hat{C}}}{\partial(x+\epsilon_{x})}$ (7)

Gradient-weighted Class Activation Map (Grad-CAM) Selvaraju et al. (2017) is
another type of saliency map evaluation that assesses the attention of a
convolution layer concerning a label. It can be used to determine the
attention region of a convolution layer in the image for neural networks.
Mathematically, Gradient-weighted Class Activation Map with respect to label
$l$ for $n^{\text{th}}$ layer $AM_{l}\in\mathbb{R}^{m,n}$ can be defined as:

$\displaystyle AM_{l}=f^{(n)}(x)\boldsymbol{\cdot}mean_{c}\frac{\partial
g(x)_{l}}{\partial f^{(n)}(x)}$ (8)

Based on this, we can evaluate activation maps of the image concerning
predicted class $C$ as:

$\displaystyle AM_{C}=f^{(n)}(x)\boldsymbol{\cdot}mean_{c}\frac{\partial
g(x)_{C}}{\partial f^{(n)}(x)}$ (9)

and we can evaluate activation maps of adversarial image with respect to
predicted class $\hat{C}$

$\displaystyle\widehat{AM}_{\hat{C}}=f^{(n)}(\hat{x})\boldsymbol{\cdot}mean_{c}\frac{\partial
g(\hat{x})_{C}}{\partial f^{(n)}(\hat{x})}$ (10)
$\displaystyle\quad\quad=f^{(n)}(x+\epsilon_{x})\boldsymbol{\cdot}mean_{c}\frac{\partial
g(x+\epsilon_{x})_{C}}{\partial f^{(n)}(x+\epsilon_{x})}$

This article assesses Saliency Maps $(SM)$ and Gradient-weighted Class
Activations Maps $(AM)$ concerning correctly predicted classes (or the true
class) $(C)$ and misclassified/incorrectly predicted class (or the adversarial
class) $(\hat{C})$ of both adversarial and original images. Analysing Saliency
Maps and Activations Maps for other unrelated classes are left as future work.
We also focus on the Activations Maps of the last convolution layer in the
neural network and leave the activation maps of other layers as future work.

##  4 Experimental results and analysis

Experimental settings: We use ResNet He et al. (2016) which is trained on
CIFAR-10 dataset Krizhevsky et al. (2009). For adversarial attacks, we
evaluate Pixel Attack ($L_{0}$ norm black-box attack) Su et al. (2019) and
Projected Gradient Descent Attack ($L_{\infty}$ norm white-box attack) Madry
et al. (2018) using the Adversarial Robustness Toolbox library Nicolae et al.
(2018). For evaluating saliency maps, we replace the traditional ReLU
backpropagation with Guided ReLU backpropagation Springenberg et al. (2014).

###  4.1 Saliency Maps

Pixel Attack
$x$ $SM_{C}$ $\hat{x}$ $\widehat{SM}_{\hat{C}}$

$x$ $SM_{C}$ $\hat{x}$ $\widehat{SM}_{\hat{C}}$

$x$ $SM_{C}$ $\hat{x}$ $\widehat{SM}_{\hat{C}}$

Projected Gradient Descent Attack
$x$ $SM_{C}$ $\hat{x}$ $\widehat{SM}_{\hat{C}}$

$x$ $SM_{C}$ $\hat{x}$ $\widehat{SM}_{\hat{C}}$

Figure 2: Overlayed Saliency Maps with respect to predicted class.

Pixel Attack
$x$ $SM_{C}$ $\hat{x}$ $\widehat{SM}_{C}$

$x$ $SM_{C}$ $\hat{x}$ $\widehat{SM}_{C}$

$x$ $SM_{C}$ $\hat{x}$ $\widehat{SM}_{C}$

Projected Gradient Descent Attack
$x$ $SM_{C}$ $\hat{x}$ $\widehat{SM}_{C}$

$x$ $SM_{C}$ $\hat{x}$ $\widehat{SM}_{C}$

Figure 3: Overlayed Saliency Maps with respect to true/correct class.

Pixel Attack
$x$ $SM_{\hat{C}}$ $\hat{x}$ $\widehat{SM}_{\hat{C}}$

$x$ $SM_{\hat{C}}$ $\hat{x}$ $\widehat{SM}_{\hat{C}}$

$x$ $SM_{\hat{C}}$ $\hat{x}$ $\widehat{SM}_{\hat{C}}$

Projected Gradient Descent Attack
$x$ $SM_{\hat{C}}$ $\hat{x}$ $\widehat{SM}_{\hat{C}}$

$x$ $SM_{\hat{C}}$ $\hat{x}$ $\widehat{SM}_{\hat{C}}$

Figure 4: Overlayed Saliency Maps with respect to adversarial/incorrect class.

We visualise the saliency maps of original sample $(SM)$ and saliency maps of
adversarial sample $(\widehat{SM})$ with respect to predicted class of the
model (Figure 2), true/correct class $(C)$ (Figure 3), and
adversarial/incorrect class $(\hat{C})$ (Figure 4) for different attacks. For
original sample predicted class is true/correct class $(C)$ where as for
adversarial sample predicted class adversarial/incorrect class $(\hat{C})$.
Intuitively, adversarial perturbations should distort the saliency of the
image as seen by the neural network to induce misclassification. From the
figures, it can be seen that both Pixel Attack and Projected Gradient Attack
distorts the saliency differently.

We observe that, in the case of PGD Attack, the saliency of the image is
diffused for the adversarial image created by the minimal change in pixels
(Figure 2). Moreover, we can visualise that PGD Attack effectively diffuses
the attention around the region of interest. Further, Figures 3 and 4 show
that this diffusion occurs for both true and adversarial class. This shows
that PGD Attack does not distort the region of interest determined by the
neural network but only manipulating the activations inside layers such that
the adversarial perturbation induces misclassification.

We also observe some peculiar characteristics of pixel perturbations generated
by Pixel Attack. Differently from the Projected Gradient Descent Attack, we
can see from the saliency maps (Figure 2) that the attention of the neural
network shifts to the perturbed region and not diffuses as in Projected
Gradient Descent Attack. This diversion of attention in the neural network is
the consequence of only introducing perturbed pixels. Figure 3 shows that the
saliency map with respect to the true class changes minimally as the saliency
in the adversarial image is in the same region as the original image. However,
Figure 4 shows that the saliency map changes with respect to the adversarial
class and this distorted saliency dominate the attention with respect to the
correct class. This shows that perturbed pixels created by Pixel Attack call
the neural network’s attention towards them and diverts the network’s
attention towards the adversarial class. Thus, Pixel Attack induces
misclassification by changing the region of interest for the neural network
through adversarial perturbations.

Further, this effect of diversion in attention due to perturbed pixels in
Pixel Attack is propagated through the network to intensify, which causes
misclassification in the neural network, as shown by Vargas and Su (2020).
This further builds on the hypothesis of conflicting saliency as adversarial
perturbations severely affect the saliency of the image determined by neural
networks. This also shows that distortion in saliency proves that adversarial
perturbations do not fool the neural network naively but diverts attention
towards the perturbations.

###  4.2 Gradient-weighted Class Activation Maps

Pixel Attack
$x$ $AM_{C}$ $\hat{x}$ $\widehat{AM}_{\hat{C}}$

$x$ $AM_{C}$ $\hat{x}$ $\widehat{AM}_{\hat{C}}$

$x$ $AM_{C}$ $\hat{x}$ $\widehat{AM}_{\hat{C}}$

Projected Gradient Descent Attack
$x$ $AM_{C}$ $\hat{x}$ $\widehat{AM}_{\hat{C}}$

$x$ $AM_{C}$ $\hat{x}$ $\widehat{AM}_{\hat{C}}$

Figure 5: Overlayed Gradient-weighted Class Activation Maps with respect to
predicted class.

Pixel Attack
$x$ $AM_{C}$ $\hat{x}$ $\widehat{AM}_{C}$

$x$ $AM_{C}$ $\hat{x}$ $\widehat{AM}_{C}$

$x$ $AM_{C}$ $\hat{x}$ $\widehat{AM}_{C}$

Projected Gradient Descent Attack
$x$ $AM_{C}$ $\hat{x}$ $\widehat{AM}_{C}$

$x$ $AM_{C}$ $\hat{x}$ $\widehat{AM}_{C}$

Figure 6: Overlayed Gradient-weighted Class Activation Maps with respect to
true/correct class.

Pixel Attack
$x$ $AM_{\hat{C}}$ $\hat{x}$ $\widehat{AM}_{\hat{C}}$

$x$ $AM_{\hat{C}}$ $\hat{x}$ $\widehat{AM}_{\hat{C}}$

$x$ $AM_{\hat{C}}$ $\hat{x}$ $\widehat{AM}_{\hat{C}}$

Projected Gradient Descent Attack
$x$ $AM_{\hat{C}}$ $\hat{x}$ $\widehat{AM}_{\hat{C}}$

$x$ $AM_{\hat{C}}$ $\hat{x}$ $\widehat{AM}_{\hat{C}}$

Figure 7: Overlayed Gradient-weighted Class Activation Maps with respect to
adversarial/incorrect class.

We visualise the gradient-weighted class activation maps of original sample
$(AM)$ and activation maps of adversarial sample $(\widehat{AM})$ with respect
to predicted class of the model (Figure 5), true/correct class $(C)$ (Figure
6), and adversarial/incorrect class $(\hat{C})$ (Figure 7) for different
attacks. For original sample predicted class is true/correct class $(C)$ where
as for adversarial sample predicted class adversarial/incorrect class
$(\hat{C})$.

Figure 5 shows that the Projected Gradient Descent, a white-box attack, keeps
the attention of the convolution layer in the same region using the gradients.
However, from Figure 6, we can see that attention for the correct class is
strongly distorted for adversarial image compared to the original. Whereas the
attention for the misclassified class for the adversarial image is distorted
to bring it close to the attention for the correct class for the original
image. This points out that white-box attacks find the perturbation keeping
the region of interest intact and thus only effectively manipulating the
activations inside the neural network (convolution layers) to call for
misclassification.

From the activation maps of the last convolution layer for Pixel Attack
(Figure 5), we observe that adversarial perturbations distort the attention of
the convolution layer. This distortion either calls the attention towards or
diverts the network’s attention from the perturbed pixels. Similar to the
characteristic shown by the Saliency Maps, Figures 6 and 7 show that this
distortion in activation maps is mainly focused on adversarial class’s
attention, and the true class’s attention is changed minimally.

An interesting observation of the activation maps is when the activation map
of a particular class retracts from the perturbed pixel, and the activation
map of another class gets intensely focused on the perturbed pixel. This sheds
light on the fact that some of the features learned are complementary to each
other, and when the network focuses on one feature, it loses the attention on
other complementary feature. Further, the influence of perturbed pixels can be
visualised through propagation maps Vargas and Su (2020). As the perturbed
pixels call attention towards another part of the image, another texture is
analysed and weighted more.

We believe each adversarial attack has its inherent characteristic strategy to
distort the saliency maps and activation maps of adversarial samples, as both
Projected Gradient Descent Attack and Pixel Attack changes the saliency maps
and activation maps differently. This sheds light on why some defences are
successful against some attacks while remaining vulnerable to other attacks.
This also calls for a deeper understanding of the effect of adversarial
samples and attacks in general on neural networks to propose effective
defences by understanding how each attack affects the attention of the
network.

##  5 Conclusions

This paper analysed Saliency Maps (SM) and Gradient-weighted Class Activation
Maps (Grad-CAM) for original images and adversarial images. We analyse the
distortion in the saliency of adversarial images compared to original images
to verify the hypothesis of conflicting saliency. We used adversarial images
created by Pixel Attack and Projected Gradient Descent Attack for analysing
the saliency. Both the attacks differed the way adversarial perturbations are
found (Pixel Attack being black-box and PGD being white box) and how
adversarial perturbations are added to the image (Pixel Attack being $L_{0}$
norm attack while PGD being $L_{\infty}$ attack).

Experimental results show that both black-box and white-box attacks,
irrespective of the norm constraint, distorts the neural networks’ saliency to
make adversarial images misclassify, and adversarial attacks do not naively
fool the neural networks. Moreover, results reveal that both Pixel Attack and
Projected Gradient Descent Attack distorts the saliency maps and activation
maps differently. While Pixel Attack, a black-box attack call the image’s
saliency to perturb pixels or divert their saliency from them, effectively
changing the region of interest for intermediate convolution layers of the
neural network to change their region of attention. The Projected Gradient
Descent Attack, a white box attack, on the other hand, diffuses the saliency
around the region of interest to induce misclassification. Further, this
diffusion of saliency causes intermediate convolution layers to lose attention
for the correct class around the region of interest while gaining attention
for the misclassified class on the intended region.

Thus, this paper analysed saliency for the adversarial images and shed light
on the conflicting saliency hypothesis raised in Vargas and Su (2020). It also
opens up the understanding of adversarial attacks by analysing the effect of
adversarial samples on neural networks. As we show, both the attacks evaluated
differ in their strategy to distort the attention of the neural network. We
believe this sheds light on why some adversarial defences mitigate some
adversarial attacks while remaining vulnerable to others. We hope this
analysis will help the community understand the existence of adversarial
samples and their effect on neural networks and help the community to develop
more robust neural networks and, at the same time, develop better adversarial
defences.

## Acknowledgments

This work was supported by JST, ACT-I Grant Number JP-50243 and JSPS KAKENHI
Grant Number JP20241216.

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##  Appendix A Extra Images of Saliency Maps For Pixel Attack

Figures 8 and 9 shows the Saliency Maps of original image and adversarial
image with respect to correct class and misclassified class for Pixel Attack.
First column shows the input image to the model. Middle column shows
activation maps concerning correctly predicted class $(C)$. Rightmost column
shows activation maps concerning misclassified class $(\hat{C})$.

Figure 8:  Overlayed Saliency Maps with respect to correctly predicted class
and the misclassified class of Original Image and Adversarial Image generated
by Pixel Attack.

Figure 9:  Overlayed Saliency Maps with respect to correctly predicted class
and the misclassified class of Original Image and Adversarial Image generated
by Pixel Attack.

##  Appendix B Extra Images of Gradient-weighted Class Activation Maps For
Pixel Attack

Figures 10 and 11 shows the Gradient-Weighted Class Activation Maps of
original image and adversarial image with respect to correct class and
misclassified class for Pixel Attack. First column shows the input image to
the model. Middle column shows activation maps concerning correctly predicted
class $(C)$. Rightmost column shows activation maps concerning misclassified
class $(\hat{C})$.

Figure 10:  Overlayed Gradient-weighted Class Activation Maps with respect to
correctly predicted class and the misclassified class of Original Image and
Adversarial Image generated by Pixel Attack.

Figure 11:  Overlayed Gradient-weighted Class Activation Maps with respect to
correctly predicted class and misclassified class of Original Image and
Adversarial Image generated by Pixel Attack.

##  Appendix C Extra Images of Saliency Maps For Projected Gradient Descent
Attack

Figures 12 and 13 shows the Saliency Maps of original image and adversarial
image with respect to correct class and misclassified class for Projected
Gradient Descent Attack. First column shows the input image to the model.
Middle column shows activation maps concerning correctly predicted class
$(C)$. Rightmost column shows activation maps concerning misclassified class
$(\hat{C})$.

Figure 12:  Overlayed Saliency Maps with respect to correctly predicted class
and the misclassified class of Original Image and Adversarial Image generated
by Projected Gradient Descent Attack.

Figure 13:  Overlayed Saliency Maps with respect to correctly predicted class
and the misclassified class of Original Image and Adversarial Image generated
by Projected Gradient Descent Attack.

##  Appendix D Extra Images of Gradient-weighted Class Activation Maps For
Projected Gradient Descent Attack

Figures 14 \- 15 shows the Gradient-Weighted Class Activation Maps of original
image and adversarial image with respect to correct class and misclassified
class for Projected Gradient Descent Attack. First column shows the input
image to the model. Middle column shows activation maps concerning correctly
predicted class $(C)$. Rightmost column shows activation maps concerning
misclassified class $(\hat{C})$.

Figure 14:  Overlayed Gradient-weighted Class Activation Maps with respect to
correctly predicted class and the misclassified class of Original Image and
Adversarial Image generated by Projected Gradient Descent Attack.

Figure 15:  Overlayed Gradient-weighted Class Activation Maps with respect to
correctly predicted class and the misclassified class of Original Image and
Adversarial Image generated by Projected Gradient Descent Attack.


# Physical properties and electronic structure of the two-gap superconductor
V2Ga5

P.-Y. Cheng  Department of Physics, National Cheng Kung University, Tainan
701, Taiwan  Center for Quantum Frontiers of Research & Technology (QFort),
National Cheng Kung University, Tainan 701, Taiwan  Taiwan Consortium of
Emergent Crystalline Materials, National Science and Technology Council,
Taipei 10622, Taiwan  Mohamed Oudah  Stewart Blusson Quantum Matter Institute,
University of British Columbia, Vancouver, BC, V6T 1Z4, Canada  T.-L. Hung
Institute of Physics, Academia Sinica, 128 Sec. 2, Academia Rd., Nankang,
Taipei 11529, Taiwan  C.-E. Hsu  Department of Physics, Tamkang University,
New Taipei 251301, Taiwan C.-C. Chang  Department of Physics, National Cheng
Kung University, Tainan 701, Taiwan  Center for Quantum Frontiers of Research
& Technology (QFort), National Cheng Kung University, Tainan 701, Taiwan
Taiwan Consortium of Emergent Crystalline Materials, National Science and
Technology Council, Taipei 10622, Taiwan  National Synchrotron Radiation
Research Center (NSRRC), Hsinchu 30076, Taiwan J.-Y. Haung  Department of
Physics, National Tsing Hua University, Hsinchu 30013, Taiwan T.-C. Liu
Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan
C.-M. Cheng  National Synchrotron Radiation Research Center (NSRRC), Hsinchu
30076, Taiwan Department of Physics, National Sun Yat-sen University,
Kaohsiung 80424, Taiwan Department of Electrophysics, National Yang Ming Chiao
Tung University, Hsinchu 300 Taiwan Taiwan Consortium of Emergent Crystalline
Materials, National Science and Technology Council, Taipei 10622, Taiwan M.-N.
Ou  Institute of Physics, Academia Sinica, 128 Sec. 2, Academia Rd., Nankang,
Taipei 11529, Taiwan  W.-T. Chen  Center for Condensed Matter Sciences,
National Taiwan University, Taipei 10617, Taiwan  Center of Atomic Initiative
for New Materials, National Taiwan University, Taipei 10617, Taiwan  Taiwan
Consortium of Emergent Crystalline Materials, National Science and Technology
Council, Taipei 10622, Taiwan  L. Z. Deng  Texas Center for Superconductivity
and Department of Physics, University of Houston, Houston, Texas 77204, United
States  C.-C. Lee  Department of Physics, Tamkang University, New Taipei
251301, Taiwan Y.-Y. Chen  Institute of Physics, Academia Sinica, 128 Sec. 2,
Academia Rd., Nankang, Taipei 11529, Taiwan  C.-N. Kuo  Department of Physics,
National Cheng Kung University, Tainan 701, Taiwan  Taiwan Consortium of
Emergent Crystalline Materials, National Science and Technology Council,
Taipei 10622, Taiwan  C.-S. Lue  Department of Physics, National Cheng Kung
University, Tainan 701, Taiwan  Taiwan Consortium of Emergent Crystalline
Materials, National Science and Technology Council, Taipei 10622, Taiwan
Janna Machts  TRIUMF Centre for Molecular and Material Science (TRIUMF-CMMS),
University of British Columbia, Vancouver, BC, V6T 2A3, Canada  School of
Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3FD, United
Kingdom  Kenji M. Kojima  Stewart Blusson Quantum Matter Institute, University
of British Columbia, Vancouver, BC, V6T 1Z4, Canada  TRIUMF Centre for
Molecular and Material Science (TRIUMF-CMMS), University of British Columbia,
Vancouver, BC, V6T 2A3, Canada  Alannah M. Hallas  Stewart Blusson Quantum
Matter Institute, University of British Columbia, Vancouver, BC, V6T 1Z4,
Canada  Department of Physics & Astronomy, University of British Columbia,
Vancouver, BC, V6T 1Z1, Canada  Canadian Institute for Advanced Research,
Toronto, ON, M5G1M1, Canada  C.-L. Huang<EMAIL_ADDRESS>Department of
Physics, National Cheng Kung University, Tainan 701, Taiwan  Center for
Quantum Frontiers of Research & Technology (QFort), National Cheng Kung
University, Tainan 701, Taiwan  Taiwan Consortium of Emergent Crystalline
Materials, National Science and Technology Council, Taipei 10622, Taiwan

###### Abstract

We present a thorough investigation of the physical properties and
superconductivity of the binary intermetallic V2Ga5. Electrical resistivity
and specific heat measurements show that V2Ga5 enters its superconducting
state below $T_{sc}=$ 3.5 K, with a critical field of $H_{\rm{c2},\perp
c}(H_{\rm{c2},||c})$ = 6.5(4.1) kOe. With $H\perp c$, the peak effect was
observed in resistivity measurements, indicating the ultrahigh quality of the
single crystal studied. The resistivity measurements under high pressure
reveal that the $T_{sc}$ is suppressed linearly with pressure and reaches
absolute zero around 20 GPa. Specific heat and muon spin relaxation
measurements both indicate that the two-gap $s$-wave model best describes the
superconductivity of V2Ga5. The spectra obtained from angle-resolved
photoemission spectroscopy measurements suggest that two superconducting gaps
open at the Fermi surface around the Z and $\Gamma$ points. These results are
verified by first-principles band structure calculations. We therefore
conclude that V2Ga5 is a phonon-mediated two-gap $s$-wave superconductor.

##  I Introduction

Intermetallic compounds composed of metals of $d-$ and $p-$shell electrons
have been one of the most studied series in condensed matter physics owing to
their abundance and their chemical and physical properties. In particular, due
to the low-melting points of metals with $p-$shell electrons, high-quality
single crystals can often be grown by the common flux method and hence
detailed property measurements can be conducted Canfield and Fisk (1992);
Kanatzidis  _et al._ (2005).

Figure 1: (a) Observed (circle), calculated (red line), and differences (blue
line) x-ray powder-diffraction patterns of V2Ga5 at room temperature. Bragg
reflections for V2Ga5 of the space group $P$4/$mbm$ are labeled as vertical
bars. The inset shows the crystal structure of one unit cell. Note that $a$
and $b$ are equivalent in tetragonal symmetry. (b) Laue x-ray pattern taken
along the tetragonal [001] direction. The inset shows a photo of the crystal.
(c) Field-cooled (FC) and zero-field-cooled (ZFC) magnetic susceptibility
measurements at $H=10$ Oe applied perpendicular to the $c$ axis. (d) Zero-
field temperature dependence of electrical resistivity with a fit by the
Bloch-Grueneissen model.

Among mentioned intermetallic compounds, the V-Ga binary system has gained
extra interest because some compounds in this family are superconductors. The
superconductor V3Ga is best known for its relatively high transition
temperature ($T_{sc}$ = 15 K), high critical currents, and fields Goodman
(1962); Junod  _et al._ (1971). In contrast, for another superconductor V2Ga5,
relevant research reports are quite scarce. The electrical resistivity,
magnetic susceptibility, and specific heat measurements conducted down to 1.8
K have been reported, which showed V2Ga5 enters the superconducting state
below $T_{sc}$ = 3.5 K and its critical field is about 4-5 kOe Cruceanu  _et
al._ (1974); Lobring  _et al._ (2002); Teruya  _et al._ (2015). Very recently,
Xu et al. report multigap superconductivity in V2Ga5 via specific heat and
thermal conductivity measurements Xu  _et al._ (2024). To verify this result
and acquire a deeper understanding of the superconductivity and the physical
properties of V2Ga5, it is essential to expand the specific heat measurements
to lower temperatures and conduct a comprehensive study using various
experimental techniques. We carried out electrical resistivity measurements
with magnetic fields applied parallel and perpendicular to the
crystallographic $c$ axis and constructed the critical field vs. $T_{sc}$
phase diagrams. Under high-pressure, $T_{sc}$ is suppressed linearly to
absolute zero around $P=20$ GPa. This result is in line with the phonon
mediated superconductivity in V2Ga5. A consistent conclusion about the two-gap
$s$-wave order parameter is drawn from two complimentary experiments: specific
heat as a bulk probe and muon spin relaxation as a local probe. This
conclusion was further validated by angle-resolved photoemission spectroscopy
experiments and first-principles calculations.

##  II Research methods

V2Ga5 single crystals were prepared by a flux method using excess Ga Lobring
_et al._ (2002); Teruya  _et al._ (2015). High-purity V powder and Ga pieces
were weighed according to the composition V6Ga94 and placed in an alumina
crucible, which was then sealed in an evacuated quartz tube. Higher vanadium
concentrations were also tested but yielded needle-shape samples, as been
reported in Ref. Cruceanu  _et al._ (1974); Lobring  _et al._ (2002); Teruya
_et al._ (2015). The quartz tube was heated to 1000 °C at a rate of 100 °C/h,
held at this temperature for 48 h and then cooled to 550 °C at a rate of 2.5
°C/h. Excess Ga flux was spun off using a centrifuge. The remaining Ga
droplets on the surface were removed by etching in diluted HCl. As shown in
the inset of Fig. 1(b), the obtained single crystals have rod-like shape along
the c-axis with typical dimensions $\sim 2\times 2\times 8$ mm3. Pulverized
V2Ga5 was characterized by x-ray powder diffraction with Cu-K$\alpha$
radiation (Bruker D2 Phaser diffractometor) at room temperature. The crystal
structure was analyzed using DIFFRAC.TOPAS program Coelho (2018) with the
Rietveld refinement method. Single crystal x-ray diffraction data were
collected with a Rigaku Synergy-S diffractometer with Mo-K radiation at 300
and 100 K. Images of representative reciprocal planes are shown in
Supplemental Material SM . The orientation of crystal was confirmed by a Laue
X-ray diffraction method. Reciprocal planes from single crystal x-ray
diffraction and Laue diffractograms validate the good quality of the single-
crystalline samples and an example is shown in Fig. 1(b).

Table 1: Crystal data and structural refinement for V2Ga5. $a$ and $c$ are
lattice parameters. $V$ is the unit-cell volume. $B_{iso}$ is the thermal
parameter. $\chi^{2}$ represents the goodness of fit. $R_{wp}$ and $R_{p}$ are
weighted and unweighted profile $R$ factors, respectively, obtained from the
refinement results. The gallium atom occupies Wyckoff positions Ga1 2$d$
($\nicefrac{{1}}{{2}}$, 0, 0) and Ga2 8$i$ ($x$, $y$, 0), while vanadium atom
occupies V 4$h$ ($x$, $x$+$\nicefrac{{1}}{{2}}$, $\nicefrac{{1}}{{2}}$)
position. Tetragonal $P$4/$mbm$ (No. 127)
---
$a$ = 8.9716(1) Å, $c$ = 2.69272(4) Å, $V$ = 216.737(7) Å3
Atom | $x$ | $y$ | $z$ | Occupancy | $B_{iso}$ | Site
Ga1 | 0.2942(1) | 0.5611(1) | 0 | 1.000 | 0.02(4) | 8$i$
Ga2 | 0.5 | 0 | 0 | 1.000 | 0.02 | 2$d$
V | 0.3212(2) | 0.8212(2) | 0.5 | 1.000 | 0.09(6) | 4$h$
$\chi^{2}$ = 1.60, $R_{wp}=11.36$ % and $R_{p}=8.73$ %

Magnetic susceptibility was measured using a Quantum Design (QD) magnetic
property measurement system. Ambient-pressure resistivity and specific heat
were measured using a QD Dynacool physical properties measurement system
(PPMS) equipped. For high-pressure resistivity measurements, we used a diamond
anvil cell (DAC) placed in a 3He cryostat. For high-pressure Hall effect
measurements, we used a DAC placed in QD Dynacool PPMS.

Muon spin relaxation ($\mu$SR) measurements were performed on the M15 beam
line at TRIUMF using the Pandora dilution refrigerator spectrometer. Several
dozen needle-like crystals of V2Ga5 were co-aligned by eye on a silver sample
plate using GE varnish. Measurements were performed over a temperature range
of 0.025 K to 6 K in both zero field and in an applied transverse field of 300
Oe. To achieve an accurate zero field, reference measurements were performed
on silicon, which through the formation of muonium atoms with a two orders of
magnitude larger gyromagnetic ratio than the muon, allows the magnetic field
to be zeroed to within 10-2 Oe Morris and Heffner (2003). In the zero field
measurements, the muons were implanted with their spins parallel to the
$ab$-plane of V2Ga5 while in the transverse field measurements, the muons are
implanted with their spins along the $c$-axis and the transverse field is
applied parallel to the $ab$-plane. The collected data were analyzed with
MUSRFIT Suter and Wojek (2012).

Angle-resolved Photoemission Spectroscopy (ARPES) experiment was conducted at
beamline BL21B1 of the Taiwan Light Source, affiliated with the National
Synchrotron Radiation Research Center. The whole experiment data were
collected by Scienta R4000 analyzer within ultra-high vacuum (UHV) condition,
utilizing an angular resolution calibrated to 0.5 degrees. Single crystal of
V2Ga5 was cleaved in the preparation chamber maintained under UHV conditions
to ensure the freshness of the crystal plane, with a pressure of approximately
8.9×$10^{-11}$ torr. The spectra were obtained at 89 K, using photon energies
between 52 eV and 86 eV, with the measurements achieving an overall energy
resolution exceeding 22 meV.

The first-principles calculations were performed using Quantum Espresso
Giannozzi  _et al._ (2017) code with PBE exchange-correlation functional. The
atomic positions were optimized at the experimental lattice constants. The
plane wave cutoff was set to 100 Ry and the k points were sampled on a
4$\times$4$\times$12 grid. The electronic structure was calculated with the
inclusion of spin-orbit coupling. For the electron-phonon properties, the EPW
Poncé  _et al._ (2016) software was used to interpolate the electron-phonon
coupling matrix elements from the coarse grid (4$\times$4$\times$12 k and
2$\times$2$\times$6 q grids) to the finer grid (20$\times$20$\times$60 k and
10$\times$10$\times$30 q grids). For the computational efficiency, the effect
of spin-orbit coupling has been neglected when calculating the electron-
phonon-related properties using the linear response framework. Note that the
band structure obtained here without adopting any Hubbard correction is
similar to the GGA+U result with U=1 eV, where the band topology at the Fermi
level is consistent with the result of U = 3.5 eV Xu  _et al._ (2024).

##  III Results and discussions

###  III.1 Crystal structure

The refined structural parameters are summarized in Table 1. It was shown that
the crystal is single phase and crystallized in tetragonal space group
$P$4/$mbm$ (No. 127), which is in consistent with reported crystal structure
Vucht  _et al._ (1964). The crystal structure can be viewed as connecting and
piling of slightly distorted vanadium-centered VGa10 pentagonal pillars, in
which consisting of eight longer V-Ga1 bonds of 2.72 Å and two shorter V-Ga2
bonds of 2.59 Å.

###  III.2 Electrical resistivity

As V2Ga5 enters the superconducting state below $T\sim 3.5$ K, a diamagnetic
signal is clearly seen in magnetization measurements, as shown in Fig. 1(c).
The demagnetization effect has been taken into account Cullity and Graham
(2008). Figure 1(d) shows the temperature dependence of electrical resistivity
$\rho$ for current along the tetragonal $c$ axis under zero magnetic field.
The residual resistivity $\rho_{0}\sim 1.0$ $\mu\Omega$ cm (the value at the
temperature right above the superconducting transition) and the residual
resistivity ratio RRR = $\rho_{300\rm{K}}/\rho_{4\rm{K}}=36$ are both
comparable with the previous result Teruya  _et al._ (2015). The normal state
$\rho$ could be described by the Bloch-Grueneisen model,

$\rho=\rho_{0}+A(\frac{T}{\theta_{R}})^{n}\int_{0}^{\frac{\theta_{R}}{T}}\frac{x^{n}}{(e^{x}-1)(1-e^{-x})}dx,$
(1)

where $A$ is a constant related electron-phonon interactions, $\theta_{R}$ is
the Debye temperature as obtained from resistivity measurements, and $n$ is an
integer that depends upon the nature of interaction. The excellent fit results
in $A=88.0$ ${\mu\Omega}\rm{cm}$ and $\theta_{R}=369$ K. An exponent $n=3$ can
best describe the data, implying the resistance in V2Ga5 is mainly due to
$s-d$ scattering, as is the case for transition metals Kittel (2004).

Figure 2: Field and pressure dependence of the superconducting transition in
resistivity of V2Ga5. (a) $\rho$ with $H$ applied perpendicular to the $c$
axis for V2Ga5. From right to left, $H=0,0.5,1,1.5,2,2.5,3,3.5,4,5,6,7,$ and 8
kOe. The downward red arrows point out the local resistivity maximum. (b)
$\rho$ with $H$ applied parallel to the $c$ axis. From right to left,
$H=0,0.5,1,1.5,2,2.5,3,3.5,4,5,$ and 6 kOe. Current $i=$ 0.1 mA is applied
along the $c$ axis. (c) Temperature dependence of resistivity at different
pressures $P$ for V2Ga5. The applied current is along the $c$ axis. An arrow
demonstrates how $T_{SC}$ was determined. (d) $T_{sc}$ vs. $P$. The inset
shows the relative pressure shift of $T_{sc}$ as function of the relative
volume change. The dashed and solid lines are linear fits through data points.

Next we turn to magnetic field effect on low-$T$ resistivity. Figure 2(a) and
(b) show the data measured with the magnetic field applied perpendicular to
and parallel to the $c$ axis, respectively. Two different samples were used.
The difference in normal state $\rho$ values between the two field directions
is mostly attributed to errors in measuring the dimensions of the samples. The
superconducting transition temperature $T_{sc}$ was determined as the
temperature where two linear fits above and below the resistivity drop
intersect, as shown by an arrow in Fig. 2(c). The good quality of our samples
is confirmed by a sharp transition into the superconducting state at zero
field. Upon application of a magnetic field, the transition broadens and moves
to lower temperatures as expected. The critical field of $H_{c2,\perp
c}(H_{c2,||c})$ = 6.5 (4.1) kOe is much smaller than the Pauli limited value
($H_{c2}=1.84T_{sc}=6.4$ T) and comparable to the value reported in the
literature Xu  _et al._ (2024). Interestingly, $\rho$ shows the peak effect,
which has been dubbed a rise in $\rho$ for a certain range of magnetic fields
below the onset of the superconducting transition Tomy  _et al._ (1997), with
$H\perp c$ axis, as shown in Fig. 2(a). This sudden increase in $\rho$ is
observable for $H$ between 1.5 and 3.5 kOe at the measured current density of
$\sim$1 A/cm2, and is reversible for increasing and decreasing the
temperatures. The peak effect has been observed in several single crystal
superconductors with two- or three-dimensional crystal structures Autler  _et
al._ (1962); Higgins and Bhattacharya (1996); Tomy  _et al._ (1997); Hedo  _et
al._ (1998); Chaudhary  _et al._ (2001); Jang  _et al._ (2008); Kaluarachchi
_et al._ (2016). The peak effect may be attributed to the absence of pinning
centers in the sample, indicating high sample quality. This allows the vortex
to flow around the sample, thereby forming a route for the current to
dissipate energy Willa  _et al._ (2018). We note that this is the first time
the peak effect is observed in V2Ga5. It is absent in the data reported by Xu
et al. where the sample has a higher residual resistivity Xu  _et al._ (2024).

Table 2: Superconducting parameters derived from current work and the literature. $\xi$ stands for the coherence length of superconducting Cooper pairs. parameters | Ref. Lobring  _et al._ (2002) | Ref. Teruya  _et al._ (2015) | Ref. Xu  _et al._ (2024) | present work
---|---|---|---|---
$T_{sc}$ | 3.59 | 3.3 | $3.6$ | 3.5
$H_{c2,\perp c}(0)$(kOe) | 3-4 | $-$ | 5.7 | 6.5
$H_{c2,||c}(0)$(kOe) | $-$ | $-$ | 4.9 | 4.1
$H_{c1,\perp c}(0)$(kOe) | $-$ | $-$ | $-$ | 0.036
$H_{c1,||c}(0)$(kOe) | $-$ | $-$ | $-$ | 0.072
$dH_{c2,\perp c}/dT|_{T=T_{sc}}$(kOe/K) | $-$ | $-$ | $-$ | $-1.85$
$dH_{c2,||c}/dT|_{T=T_{sc}}$(kOe/K) | $-$ | $-$ | $-$ | $-1.01$
$\xi_{ab}/\xi_{c}$ | $-$ | $-$ | $-$ | $1.83$
$\gamma_{n}$ (mJ mol K2) | $-$ | 16.7$\pm 0.2$ | 17.92 | 17.8$\pm 0.1$
$\beta$ (mJ mol K4) | $-$ | 0.448 | 0.024 | 0.20$\pm 0.01$
$\theta_{\rm D}(K)$ | $-$ | 312 | $-$ | 408
$C_{e}/\gamma_{n}T_{sc}$ | $-$ | 0.64 | 0.92 | 1.0
2$\Delta_{1}/k_{\rm B}T_{sc}$ | $-$ | $-$ | $-$ | 3.6
2$\Delta_{2}/k_{\rm B}T_{sc}$ | $-$ | $-$ | $-$ | 1.5

###  III.3 Pressure dependence of superconductivity

The pressure effect is a useful tool for varying the primary parameters that
determine superconductivity, such as the density of states at the Fermi
energy, the phonon frequency, and the coupling constant of electrons and
phonons, without introducing chemical disorder. Lorenz and Chu (2005). We
investigated the pressure effect on superconductivity of V2Ga5 as shown in
Fig. 2(c-d). Upon applying pressure, $T_{sc}$ decreases linearly at a rate of
$-0.18$ K/GPa up to $P=18.1$ GPa, from which we extrapolate $T_{sc}\rightarrow
0$ around $P=20$ GPa.

In order to convert the applied pressure to volume compression, the bulk
modulus $B\equiv V\times(dP/dV)$ needs to be known. Since there are no
available experimental data of $B_{\rm{V_{2}Ga_{5}}}$, we estimate it using a
Voigt-Reuss approximation method: from the bulk moduli of pure constituents
$B_{i}$ with relative concentration $f_{i}$ in the compound, we calculate
$B=(B_{Voigt}+B_{Reuss})/2$ with $B_{Voigt}=\sum f_{i}B_{i}$ and
1/$B_{Reuss}=\sum f_{i}/B_{i}$ yielding $B_{\rm{V_{2}Ga_{5}}}\sim 78.4$ GPa
Larson and Dreele (1999); Kittel (2004). For a range of intermetallic
compounds, this method yields $B$ values in agreement of with experimental
data Man and Huang (2011). We then used a Murnaghan equation of state:

$B=B_{0}+p\frac{dB}{dP}|_{p=0}$ (2)

and for cell volume one has:

$\frac{V}{V_{0}}=\left[1+\frac{\frac{dB}{dP}|_{p=0}}{B_{0}}p\right]^{-1/\frac{dB}{dP}|_{p=0}},$
(3)

where $B_{0}$ and $V_{0}$ are bulk modulus and cell volume at ambient
pressure, respectively. For $\frac{dB}{dP}|_{p=0}$, an empirical value of 5
was used, which was found for most compounds Grube . The obtained $T_{sc}$ as
a function of relative volume change is shown in the inset of Fig. 2(d) where
a linear relationship is observed. This behavior has been shown in a plethora
of low temperature superconductors, which indicates phonon hardening under
pressure, and hence suggest a phonon-mediated superconductivity for V2Ga5
Lorenz and Chu (2005).

###  III.4 Specific heat

Figure 3(a) shows the temperature dependence of the specific heat $C/T$ of
V2Ga5 under different magnetic fields $H=0$ to 8 kOe. In zero field, a jump in
$C(T)$ representing the superconducting transition is observed at around 3.5
K, consistent with the finding in the previous specific-heat studies Teruya
_et al._ (2015); Xu  _et al._ (2024). As the temperature decreases, $C/T$
approaches zero, indicating a complete superconducting volume of the sample.
Below $T\sim 0.1$ K, an upturn of $C/T$ due to a nuclear Schottky contribution
from the vanadium isotope is observed. This upturn occurs at higher
temperatures with an increase of the magnetic field as expected Rai  _et al._
(2018). The superconducting transition can be traced up to $H=5$ kOe and
cannot be observed at $H\geq 6$ kOe. Thus, the normal-state specific heat
$C_{n}(T)$ is extracted from the $H=8$ kOe data between $T=1$ to 4 K to avoid
the nuclear contribution at lower temperatures. $C_{n}(T)$ can be described as
$C_{n}(T)=\gamma_{n}T+\beta T^{3}$, where $\gamma_{n}T$ is the normal
electronic contribution and $\beta T^{3}$ represents the phonon contribution.
The fit (not shown) leads to $\gamma_{n}=17.82\pm 0.10$ mJ/mol K2 and
$\beta=0.200\pm 0.012$ mJ/mol K4. From $\beta$, we derive the corresponding
Debye temperature $\theta_{\rm D}=408$ K, which is close to the value
determined from our Bloch-Grueneissen fit to the resistivity, $\theta_{R}=369$
K (Fig. 1(d)). The obtained $\gamma_{n}T$ value is consistent with that in
Ref. Xu  _et al._ (2024), but our $\beta$ value is almost one order of
magnitude higher. This is most probably due to the fact that Xu et al.
introduced a higher order term ($C_{n}(T)=\gamma_{n}T+\beta T^{3}+\delta
T^{5}$) to describe the phonon contribution to the specific heat. Their fit
yields an unphysically low $\theta_{\rm D}$, and we did not find a $T^{5}$
term necessary to describe the data.

Figure 3: (a) Temperature dependence of specific heat plotted as $C/T$ vs $T$
at various magnetic fields applied perpendicular to the $c$ axis for V2Ga5.
(b) Temperature dependence of the superconducting-transition contribution to
the specific heat $C_{e}/T$. The solid, dashed, and dash-dot lines are fits
using a two-gap $s$-wave, a single-gap $s$-wave, and a line node $d$-wave
models, respectively. Inset shows that the entropy ($S$), calculated from the
data in the main panel, is conserved for the superconducting phase transition.

The zero-field superconducting electronic contribution to the specific heat
$C_{e}(T)$ is obtained with $C_{e}=C(H=0)-C_{n}$, as shown in Fig. 3(b). The
superconducting transition is a second-order phase transition which requires
the entropy conservation. The entropy conservation is evidenced in the inset
of Fig. 3(b). The dimensionless specific-heat jump $C_{e}/\gamma_{n}T_{sc}\sim
1.0$ at $T_{sc}$ is smaller than the weak coupling value 1.43 described in the
BCS theory Bardeen  _et al._ (1957), and is comparable to the value reported
in Ref. Xu  _et al._ (2024). To examine the superconducting order parameter of
V2Ga5, the data above the nuclear Schottky anomaly ($T>0.2$ K) are fitted to a
single-gap $s$-wave, a line-node with $d$-wave, and two-gap $s$-wave models
with the following equation

$\begin{split}C_{e}=2N(0)\beta
k\frac{1}{4\pi}\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta\textup{sin}\theta\\\
\times\int_{-\hbar\omega_{D}}^{\hbar\omega_{D}}-\frac{\partial f}{\partial
E}\left(E^{2}+\frac{1}{2}\beta\frac{d\Delta^{2}}{d\beta}\right)d\varepsilon,\end{split}$
(4)

where $N(0)$ is the density of states at the Fermi surface, $\beta=1/k_{\rm
B}T$, $E=(\varepsilon^{2}+\Delta^{2})^{1/2}$, $f=(1+e^{\beta E})^{-1}$,
$\Delta=\Delta_{0}$ the superconducting gap for an isotropic $s$-wave,
$\Delta=\Delta_{0}$cos$2\phi$ for line nodes with $d$-wave symmetry, and
$\Delta=\Delta_{1}\cdot x+\Delta_{2}\cdot(1-x)$ where $x$ is relative
weighting for two-gap $s$-wave. From the fitting result, it is obvious the
one-gap $s$-wave cannot describe the data. For the line-nodal $d-$wave model,
the fit agrees well with the data between $T_{sc}$ and 0.5 K, but becomes
unsatisfactory at lower temperatures. Finally, an excellent fit using a two-
gap $s$-wave model with 2$\Delta_{1}/k_{\rm B}T_{sc}=3.6$ and
2$\Delta_{2}/k_{\rm B}T_{sc}=1.5$, with a relative weighting of 55% and 45%,
respectively, across the entire fitting range is observed. The two
superconducting gap values obtained from this fit are 0.543 meV and 0.226 meV.
Therefore, we conclude V2Ga5 is a two-gap $s$-wave superconductor. This
conclusion can only be drawn if specific heat measurements are conducted below
0.5 K. Above 0.5 K, both the two-gap $s$-wave and $d$-wave models can describe
the data equally well Xu  _et al._ (2024).

Figure 4: Normalized specific heat coefficient $\gamma/\gamma_{n}$ vs.
normalized magnetic field $H/H_{c2}(0)$ for V2Ga5 and other two-gap
superconductors. $\gamma$ is obtained from a linear fit of $C/T$ vs. $T^{2}$
and its extrapolation to absolute zero (see inset and text for details). The
legends in the inset are identical to the ones of Fig. 3(a). Dashed and
dashed-dotted lines represent one-gap theoretical models with isotropic and
line nodal gap structures, respectively. The data of reference two-gap
superconductors are adopted from Refs. Bouquet  _et al._ (2001); Huang  _et
al._ (2006, 2007); Chen  _et al._ (2013, 2017); Shang  _et al._ (2020).

We further studied the vortex excitations in the mixed state by considering
the field-dependence of the electronic specific heat coefficient $\gamma(H)$.
Figure 4 shows $\gamma(H)$ of V2Ga5 where the data points were obtained from
the linear extrapolations in a $C/T$ vs. $T^{2}$ plot, as shown in the inset.
The fitting of the data was obtained from the relatively high-temperature
region to avoid the influence of nuclear contributions from the low-
temperature regime. Apparently, $\gamma(H)$ does not follow the linear
behavior for an isotropic $s$-wave model. It is however in line with the
power-law behavior $\gamma\sim H^{1/2}$ for nodal superconductor, as shown by
the dashed-dotted line. Such a power-law dependence of $\gamma(H)$ has also
been observed in other two-gap superconductors YNi2B2CHuang  _et al._ (2006),
NbSe2Huang  _et al._ (2007), and FeSeChen  _et al._ (2017). At first glance,
the nodal model describing $\gamma(H)$ seems incompatible with the two-gap
$s$-wave model that describes $C_{e}/T$ (see Fig. 3(b)). However, when the
superconducting gap is highly anisotropic, such a discrepancy could be
reconciled Huang  _et al._ (2006). In certain extreme cases among two-gap
superconductors, such as MgB2 Bouquet  _et al._ (2001) and LaNiC2 Chen  _et
al._ (2013), the smaller superconducting gap is closed at the magnetic field
strength much smaller than $H_{c2}$. This results a rapid increase of $\gamma$
at low fields and then an gradual saturation with increasing $H$, as shown in
Fig. 4.

We summarize the data from resistivity and specific heat measurements to
construct the $H-T_{sc}$ phase diagrams with the magnetic field applied
parallel and perpendicular to the $c$-axis, as shown in Fig. 5. The
$T_{sc,onset}$ value determined from $\rho(T)$ coincides with the one
determined from $C(T)$.

Figure 5: (a) $H-T_{sc}$ phase diagram with the magnetic field applied
perpendicular to the $c$ axis. A yellow oval represents the region of peak
effect. $T^{*}$ denotes the temperature of the resistivity maximum within the
superconducting state, i.e., the red arrows in Fig. 2(a). $T_{sc,onset}$
denotes the temperatures where $\rho$ starts to drop. $T_{sc,0}$ denotes the
temperature where $\rho$ becomes zero. $T_{sc}$ is the transition temperature
determined from specific heat measurements. (b) $H-T_{sc}$ phase diagram with
the magnetic field applied parallel to the $c$ axis. Same legends are applied
for (a) and (b). Figure 6: (a) Representative transverse field $\mu$SR spectra
for V2Ga5 above and below its $T_{sc}$ collected with a magnetic field applied
parallel to the $ab$-plane and muon spin parallel to the $c$-direction. The
muon precession in the applied field, which gives rise to the oscillatory
behavior in the decay asymmetry, shows an increasing relaxation upon entering
the vortex state. The fits to the data (solid lines) are given by Eq. 5 as
described in the text. These fits yield (b) the superconducting contribution
to the muon Gaussian relaxation rate $\sigma_{SC}$ as a function of
temperature. An improved fit is obtained by fitting the data with two $s$-wave
gaps (blue line) as compared to a single $s$-wave gap (black line). The two
$s$-wave gap fit is consistent with the results from specific heat, as shown
by the dashed red line. (c) Zero-field $\mu$SR spectra above and below
$T_{sc}$ collected with the muon spin parallel to the $ab$-plane show no
evidence for time-reversal symmetry breaking. The data are fitted by a Kubo-
Toyabe function (solid lines).

###  III.5 Muon spin relaxation

To further clarify the nature of the superconducting state in V2Ga5, we have
performed muon spin relaxation ($\mu$SR) measurements, as summarized in Fig.
6. Muons are highly sensitive to their local magnetic environment, making them
an excellent probe of the vortex state of type II superconductors as well as
spontaneous time-reversal symmetry breaking. We begin by investigating the
former, where considering the values of $H_{c1,\perp c}$ and $H_{c2,\perp c}$
extracted from our bulk characterization, the sample is expected to be in the
vortex state for an applied field of 300 Oe applied $\perp c$ used in our
transverse field $\mu\rm{SR}$ measurement. In the vortex state, we expect an
inhomogenous field distribution due to the presence of a flux line lattice
(FLL), which results in a decay of the precession signal as a function of
time. The decay in the superconducting state can be clearly seen if we compare
the transverse field spectra collected above and below $T_{sc}$, at 25 mK and
4.5 K as shown in Fig. 6(a).

We fit the transverse field asymmetry spectra with a two term sinusoidal
decaying function,

$\displaystyle G_{\mathrm{TF}}(t)$
$\displaystyle=A\left[f\exp\left(\frac{-\sigma^{2}t^{2}}{2}\right)\cos\left(\omega_{1}t+\phi\right)\right.$
(5) $\displaystyle\left.+(1-f)\exp(-\psi
t)\cos\left(\omega_{2}t+\phi\right)\right],$

where the first term captures the signal coming from fraction, $f$, of muons
stopping inside the sample, while the second term captures the background
signal from the fraction, $1-f$ of muons stopping inside the silver sample
holder. The precession frequencies of muons in the sample and the background
are given by $\omega_{1}$ and $\omega_{2}$, respectively, $A$ is the total
asymmetry, $\phi$ is the initial phase of the muons, and $\sigma$ and $\psi$
are depolarization rates of the sample and the background signals,
respectively. These fits are shown by the solid-lines in Fig. 6(a).

The superconducting relaxation rate $\left(\sigma_{sc}\right)$ reflects the
mean square inhomogeniety in the field experienced by the muons due to the FLL
Brandt (1988). This quantity was extracted from the fitted Gaussian relaxation
rate $\sigma$ by subtracting off a smaller, temperature-independent
contribution from randomly oriented nuclear dipole moments
$\left(\sigma_{N}\right)$, which give rise to a constant relaxation above
$T_{sc}$. The resulting temperature dependence of $\sigma_{sc}$ for V2Ga5 at
300 Oe is plotted in Fig. 6(b). The penetration depth can be calculated from
the relaxation rate Brandt (2003), which then allows a direct calculation of
the temperature dependence of the superconducting energy gap Carrington and
Manzano (2003). The full protocol for performing these calculations is
provided in the Supplemental Material.

In order to further investigate the superconducting gap structure for V2Ga5,
we have attempted fits of $\sigma_{SC}$ with both a single $s$-wave gap and
two $s$-wave gaps. We note a hump in $\sigma_{SC}$ between 2.5 and 3.0 K that
cannot be explained by either of the gap structures under consideration here.
We attribute this feature to the movement of vortices and it is likely also
related to the peak effect observed in resistivity measurements below
$T_{sc}$. In our fits, we therefore exclude this range of temperatures. The
resulting single $s$-wave gap and two $s$-wave gaps, as shown with solid black
and blue lines in Fig. 6(b) with the fitting parameters given in the
Supplemental Material. The overall quality of the fit is higher for the two
gap function, consistent with heat capacity, particularly in the temperature
range close to 1 K. For comparison, we show the calculated temperature
dependence of $\sigma_{SC}$ based on the two gaps extracted from specific heat
by a red dashed-line. Considering the level of noise in the data, we consider
the gap values extracted from heat capacity to be more reliable.

Finally, we performed zero field $\mu$SR measurements to look for any sign of
spontaneous magnetic fields and time-reversal symmetry breaking as V2Ga5
enters its superconducting state. The magnetic field at the sample position
was carefully zeroed following the protocol described in the methods. High
statistics zero field spectra were collected above and below $T_{sc}$ with
representative spectra shown in Fig. 6(c). In the absence of any static
electronic moments, the muon polarization decay is due to randomly oriented
nuclear magnetic moments, as described by the Gaussian Kubo-Toyabe function,

$G_{\mathrm{KT}}(t)=\frac{1}{3}+\frac{2}{3}\left(1-\sigma^{2}t^{2}\right)e^{-\frac{\sigma^{2}t^{2}}{2}},$
(6)

where $\sigma$ reflects the width of the field experienced by muons due to
nuclear dipoles. The total fitting function applied to our zero field data is

$A(t)=A[fG_{\mathrm{KT}}(t)e^{-\Lambda t}+(1+f)],$ (7)

where $A$ is the sample asymmetry, $f$ represents the asymmetry fraction
originating from muons that land in the sample, and $\Lambda$, represents any
additional relaxation inside the sample, such as that coming from spontaneous
fields inside the superconducting state. The zero field muon spin relaxation
rate for V2Ga5 is mainly attributed to a temperature-independent contribution
from nuclear dipole moments. No change in $\Lambda$ is observed at
temperatures well below $T_{sc}$, as can be evidenced from the comparison of
spectra collected at 25 mK and 4.5 K shown in Fig. 6(c). We conclude from our
data that no significant breaking of time-reversal symmetry occurs in the
superconducting state. Considering the lack of evidence for TRS breaking and
the gap being well fitted with two $s$-wave gaps, we conclude that the
superconductivity in V2Ga5 arises from a singlet pairing.

###  III.6 Angle-resolved photoemission spectroscopy and band-structure
calculations

Figure 7: (a) ARPES spectrum of V2Ga5 were acquired along the kz direction at
89 K using a photon energy of 78 eV. (b) Calculated band dispersion of V2Ga5
along kz direction at (kx, ky) = (0.133 Å-1, 0). The amount of the projected
orbital weight of V $d$ orbital and Ga $p$ orbital is represented by red and
blue curves, respectively. The color bar indicates the corresponding mixed
contribution. (c) Constant energy contours of V2Ga5 at distinct binding
energies with respect to the Fermi surface: 0 eV, 0.1 eV, and 0.2 eV, which
were measured at 89 K with 78 eV photon energy. (d) Quasiparticle density of
states for V2Ga5 in the superconducting state at different temperatures (3 K
and 7 K).

To elucidate the nature of superconductivity in V2Ga5, we conducted ARPES
experiments and performed the density functional theory (DFT) calculations to
unveil the band structure of V2Ga5. Figure 7(a) illustrates the band mapping
result of ARPES spectra along the $\Gamma$-Z direction, measured at 89 K
utilizing a photon energy of 78 eV. Two discernible bands across the Fermi
level were observed: one electron band below $-0.25$ eV of binding energy
around the Z point disperses toward the Fermi level, while another, with
weaker photoemission intensity, behaves as a parabolic band around the
$\Gamma$ point. To gain insight into the out-of-plane band dispersion, we
employed a photon energy-dependent ARPES experiment Damascelli (2004), using
energies ranging from 52 eV to 86 eV. No significant changes are observed in
bands around the Fermi level, suggesting only weakly dispersive behavior along
the kx direction, which is revealed in the Supplemental Material SM .
Additionally, constant energy mapping in Fig. 7(c) demonstrates a less
dispersive behavior for these bands around the Fermi level along the ky
direction, without the presence of nodes on the Fermi surface SM .

Figure 7(b) presents the calculated band structure of V2Ga5 along the
$\Gamma$-Z direction at (kx, ky) = (0.133 Å-1, 0), where the orbital
contribution for each energy eigenstate is analyzed and the projected weight
of V $d$ orbital and Ga $p$ orbital is represented by red and blue curves,
respectively. The measured bands in the low binding energy region are mainly
composed of the V 3d and Ga 4p orbitals, and the calculated band structure at
kx = 0.133 Å-1 aligns well with the ARPES spectra. As presented in the
Supplemental Material SM , these bands show less dispersion along the kx
direction, suggesting a quasi-one-dimensional behavior.

Through first-principles calculations, we have identified three bands that
cross the Fermi level, and two of them are responsible for the observed two-
gap superconductivity. Using the linear response framework (see Supplemental
Material SM ), the electron-phonon coupling strength ($\lambda$) has been
calculated. The result yields a theoretical electron-phonon coupling strength
of 0.75 and a superconducting critical temperature ($T_{sc}$) of 7.38 K based
on the McMillan-Allen-Dynes formula McMillan (1968); Allen and Dynes (1975):

$T_{sc}=\frac{\omega_{log}}{1.2}\exp\Big{[}\frac{-1.04(1+\lambda)}{\lambda-\mu_{c}^{*}(1+0.62\lambda)}\Big{]}.$
(8)

Here, the typical Coulomb pseudopotential value $\mu^{*}_{c}$ is set to $0.1$.
We have further solved the anisotropic Migdal-Eliashberg equations to
investigate the temperature dependence of the superconducting gap. In Fig.
7(d), the quasiparticle density of states in the superconducting state is
presented at 3 K and 7 K. The quasiparticle density of states clearly shows
the two-gap behavior, similar to that in the two-gap superconductor MgB2. The
calculated $T_{sc}$ and the nodeless Fermi surface are in agreement with our
experimental observations, and our theoretical result of the two-gap $s$-wave
behavior is also consistent with the recent experimental findings Xu  _et al._
(2024). However, we note that the theoretical two-gap values are overestimated
compared to the experimental ones. This discrepancy requires further
theoretical work to reconcile, such as first-principles calculations of the
Coulomb pseudopotential.

##  IV Summary

We present a detailed study of the physical properties and electronic
structure of the superconductor V2Ga5. From resistivity measurements with the
magnetic field applied parallel and perpendicular to the $c$ axis, the
$H_{c2}$ vs. $T$ phase diagrams are constructed, where small magnetic
anisotropy has been registered. Resistivity under high pressure shows that the
superconductivity is suppressed linearly with pressure, suggesting the
superconductivity is phonon-mediated. Low-temperature specific heat and
$\mu$SR experiments both reveal that the two-gap $s$-wave model best describes
the superconducting order parameter. These two gaps can be directly inferred
from ARPES measurements and are further supported by band structure
calculations. By combining various experiments with theoretical calculations,
our findings intricately detail the physical properties and superconductivity
of V2Ga5, paving the way for future studies.

###### Acknowledgements.

We thank R. Pierre and T. Klein for fruitful discussions. We thank Dr. M.-K.
Lee and C.-C. Yang at PPMS-16T and SQUID VSM Labs, Instrumentation Center,
National Cheng Kung University (NCKU) for technical support. We are grateful
to P.-Z. Hsu and L.-J. Chang for the help of Laue diffraction. CLH would like
to thank Academia Sinica for "Short-term Domestic Visiting Scholars" program
in the Nanomaterial and Low Temperature Physics Laboratory, IOP, Academia
Sinica. This work is supported by National Science and Technology Council in
Taiwan (grant number NSTC 109-2112-M-006-026-MY3, NSTC 110-2124-M-006-011, and
NSTC 112-2112-M-032-010) and the Higher Education Sprout Project, Ministry of
Education to the Headquarters of University Advancement at NCKU. WTC
acknowledges the National Science and Technology Council in Taiwan, for
funding 111-2112-M-002-044-MY3, 112-2124-M-002-012, Academia Sinica project
number AS-iMATE-111-12, and the Featured Areas Research Center Program within
the framework of the Higher Education Sprout Project by the Ministry of
Education of Taiwan 113L9008. LZD would like to thank U.S. Air Force Office of
Scientific Research Grants FA9550-15-1-0236 and FA9550-20-1-0068; the T. L. L.
Temple Foundation; the John J and Rebecca Moores Endowment; the State of Texas
through the Texas Center for Superconductivity at the University of Houston.
AMH acknowledges support from the Natural Sciences and Engineering Research
Council of Canada (NSERC), the CIFAR Azrieli Global Scholars program, and the
Sloan Research Fellowships program. Research at UBC was undertaken, in part,
thanks to funding from the Canada First Research Excellence Fund, Quantum
Materials and Future Technologies Program. CCL acknowledges support from the
National Center for Theoretical Sciences (NCTS) of Taiwan.

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